Resilience in reaction-diffusion systems
NASA Astrophysics Data System (ADS)
van Vuuren, J. H.
1999-10-01
Reaction-diffusion systems with zero-flux Neumann boundaries are widely used to model various kinds of interaction in, for example, the scientific fields of ecology, biology, chemistry, medicine and industry. The physical systems within these fields are often known to be (conditionally or unconditionally) resilient with respect to shocks, disturbances or catastrophies in the immediate environment. In order to be good mathematical models of such situations the reaction-diffusion systems must have the same resilient or asymptotic behaviour as that of the physical situation. Three fundamentally different kinds of reaction terms are usually distinguished according to the entry signs of the reaction Jacobian: mutualism, mixed (predator-prey) interaction and competition. The asymptotic stability (in the Poincare sense) of mutualistic systems has already been studied extensively, but the results cannot be generalized (globally) to the other two fundamental types, which are not order-preserving. A partial (local) generalization is, however given here for these two types, involving simple Jacobian inequalities and knowledge (often prompted by the underlying physical situation) of invariant sets in solution space. The return time of resilient systems and the approach rate of asymptotically stable solutions are also estimated. Key words: reaction-diffusion system; competition; resilience; asymptotic stability.
Parametric spatiotemporal oscillation in reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Ghosh, Shyamolina; Ray, Deb Shankar
2016-03-01
We consider a reaction-diffusion system in a homogeneous stable steady state. On perturbation by a time-dependent sinusoidal forcing of a suitable scaling parameter the system exhibits parametric spatiotemporal instability beyond a critical threshold frequency. We have formulated a general scheme to calculate the threshold condition for oscillation and the range of unstable spatial modes lying within a V-shaped region reminiscent of Arnold's tongue. Full numerical simulations show that depending on the specificity of nonlinearity of the models, the instability may result in time-periodic stationary patterns in the form of standing clusters or spatially localized breathing patterns with characteristic wavelengths. Our theoretical analysis of the parametric oscillation in reaction-diffusion system is corroborated by full numerical simulation of two well-known chemical dynamical models: chlorite-iodine-malonic acid and Briggs-Rauscher reactions.
Parametric spatiotemporal oscillation in reaction-diffusion systems.
Ghosh, Shyamolina; Ray, Deb Shankar
2016-03-01
We consider a reaction-diffusion system in a homogeneous stable steady state. On perturbation by a time-dependent sinusoidal forcing of a suitable scaling parameter the system exhibits parametric spatiotemporal instability beyond a critical threshold frequency. We have formulated a general scheme to calculate the threshold condition for oscillation and the range of unstable spatial modes lying within a V-shaped region reminiscent of Arnold's tongue. Full numerical simulations show that depending on the specificity of nonlinearity of the models, the instability may result in time-periodic stationary patterns in the form of standing clusters or spatially localized breathing patterns with characteristic wavelengths. Our theoretical analysis of the parametric oscillation in reaction-diffusion system is corroborated by full numerical simulation of two well-known chemical dynamical models: chlorite-iodine-malonic acid and Briggs-Rauscher reactions. PMID:27078346
Design and control of patterns in reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Vanag, Vladimir K.; Epstein, Irving R.
2008-06-01
We discuss the design of reaction-diffusion systems that display a variety of spatiotemporal patterns. We also consider how these patterns may be controlled by external perturbation, typically using photochemistry or temperature. Systems treated include the Belousov-Zhabotinsky (BZ) reaction, the chlorite-iodide-malonic acid and chlorine dioxide-malonic acid-iodine reactions, and the BZ-AOT system, i.e., the BZ reaction in a water-in-oil reverse microemulsion stabilized by the surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT).
Design and control of patterns in reaction-diffusion systems
Vanag, Vladimir K.; Epstein, Irving R.
2008-06-15
We discuss the design of reaction-diffusion systems that display a variety of spatiotemporal patterns. We also consider how these patterns may be controlled by external perturbation, typically using photochemistry or temperature. Systems treated include the Belousov-Zhabotinsky (BZ) reaction, the chlorite-iodide-malonic acid and chlorine dioxide-malonic acid-iodine reactions, and the BZ-AOT system, i.e., the BZ reaction in a water-in-oil reverse microemulsion stabilized by the surfactant sodium bis(2-ethylhexyl) sulfosuccinate (AOT)
Wave Phenomena in Reaction-Diffusion Systems
NASA Astrophysics Data System (ADS)
Steinbock, Oliver; Engel, Harald
2013-12-01
Pattern formation in excitable and oscillatory reaction-diffusion systems provides intriguing examples for the emergence of macroscopic order from molecular reaction events and Brownian motion. Here we review recent results on several aspects of excitation waves including anomalous dispersion, vortex pinning, and three-dimensional scroll waves. Anomalies in the speed-wavelength dependence of pulse trains include nonmonotonic behavior, bistability, and velocity gaps. We further report on the hysteresis effects during the pinning-depinning transition of twodimensional spiral waves. The pinning of three-dimensional scroll waves shows even richer dynamic complexity, partly due to the possibility of geometric and topological mismatches between the unexcitable, pinning heterogeneities and the one-dimensional rotation backbone of the vortex. As examples we present results on the pinning of scroll rings to spherical, C-shaped, and genus-2-type heterogeneities. We also review the main results of several experimental studies employing the Belousov-Zhabotinsky reaction and briefly discuss the biomedical relevance of this research especially in the context of cardiology.
NASA Astrophysics Data System (ADS)
Gaskins, Delora K.; Pruc, Emily E.; Epstein, Irving R.; Dolnik, Milos
2016-07-01
Turing patterns in the chlorine dioxide-iodine-malonic acid reaction were modified through additions of sodium halide salt solutions. The range of wavelengths obtained is several times larger than in the previously reported literature. Pattern wavelength was observed to significantly increase with sodium bromide or sodium chloride. A transition to a uniform state was found at high halide concentrations. The observed experimental results are qualitatively well reproduced in numerical simulations with the Lengyel-Epstein model with an additional chemically realistic kinetic term to account for the added halide and an adjustment of the activator diffusion rate to allow for interhalogen formation.
Gaskins, Delora K; Pruc, Emily E; Epstein, Irving R; Dolnik, Milos
2016-07-29
Turing patterns in the chlorine dioxide-iodine-malonic acid reaction were modified through additions of sodium halide salt solutions. The range of wavelengths obtained is several times larger than in the previously reported literature. Pattern wavelength was observed to significantly increase with sodium bromide or sodium chloride. A transition to a uniform state was found at high halide concentrations. The observed experimental results are qualitatively well reproduced in numerical simulations with the Lengyel-Epstein model with an additional chemically realistic kinetic term to account for the added halide and an adjustment of the activator diffusion rate to allow for interhalogen formation. PMID:27517779
Stochastic resonance with a mesoscopic reaction-diffusion system.
Mahara, Hitoshi; Yamaguchi, Tomohiko; Parmananda, P
2014-06-01
In a mesoscopic reaction-diffusion system with an Oregonator reaction model, we show that intrinsic noise can drive a resonant stable pattern in the presence of the initial subthreshold perturbations. Both spatially periodic and aperiodic stochastic resonances are demonstrated by employing the Gillespies stochastic simulation algorithm. The mechanisms for these phenomena are discussed. PMID:25019857
Spiral defect chaos in an advection-reaction-diffusion system
NASA Astrophysics Data System (ADS)
Affan, H.; Friedrich, R.
2014-06-01
This paper comprises numerical and theoretical studies of spatiotemporal patterns in advection-reaction-diffusion systems in which the chemical species interact with the hydrodynamic fluid. Due to the interplay between the two, we obtained the spiral defect chaos in the activator-inhibitor-type model. We formulated the generalized Swift-Hohenberg-type model for this system. Then the evolution of fractal boundaries due to the effect of the strong nonlinearity at the interface of the two chemical species is studied numerically. The purpose of the present paper is to point out that spiral defect chaos, observed in model equations of the extended Swift-Hohenberg equation for low Prandtl number convection, may actually be obtained also in certain advection-reaction-diffusion systems.
Hybrid stochastic simulations of intracellular reaction-diffusion systems
Kalantzis, Georgios
2009-01-01
With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction-diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a novel hybrid stochastic approach is introduced for simulating reaction-diffusion systems. We developed a dynamic partitioning strategy using fractional propensities. In that way processes with high frequency are simulated mostly with deterministic rate-based equations, and those with low frequency mostly with the exact stochastic algorithm of Gillespie. In this way we preserve the stochastic behavior of cellular pathways while being able to apply it to large populations of molecules. In this article we describe this hybrid algorithmic approach, and we demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors. PMID:19414282
A Discrete Model to Study Reaction-Diffusion-Mechanics Systems
Weise, Louis D.; Nash, Martyn P.; Panfilov, Alexander V.
2011-01-01
This article introduces a discrete reaction-diffusion-mechanics (dRDM) model to study the effects of deformation on reaction-diffusion (RD) processes. The dRDM framework employs a FitzHugh-Nagumo type RD model coupled to a mass-lattice model, that undergoes finite deformations. The dRDM model describes a material whose elastic properties are described by a generalized Hooke's law for finite deformations (Seth material). Numerically, the dRDM approach combines a finite difference approach for the RD equations with a Verlet integration scheme for the equations of the mass-lattice system. Using this framework results were reproduced on self-organized pacemaking activity that have been previously found with a continuous RD mechanics model. Mechanisms that determine the period of pacemakers and its dependency on the medium size are identified. Finally it is shown how the drift direction of pacemakers in RDM systems is related to the spatial distribution of deformation and curvature effects. PMID:21804911
Simulating mesoscopic reaction-diffusion systems using the Gillespie algorithm
Bernstein, David
2004-12-12
We examine an application of the Gillespie algorithm to simulating spatially inhomogeneous reaction-diffusion systems in mesoscopic volumes such as cells and microchambers. The method involves discretizing the chamber into elements and modeling the diffusion of chemical species by the movement of molecules between neighboring elements. These transitions are expressed in the form of a set of reactions which are added to the chemical system. The derivation of the rates of these diffusion reactions is by comparison with a finite volume discretization of the heat equation on an unevenly spaced grid. The diffusion coefficient of each species is allowed to be inhomogeneous in space, including discontinuities. The resulting system is solved by the Gillespie algorithm using the fast direct method. We show that in an appropriate limit the method reproduces exact solutions of the heat equation for a purely diffusive system and the nonlinear reaction-rate equation describing the cubic autocatalytic reaction.
Simulating mesoscopic reaction-diffusion systems using the Gillespie algorithm.
Bernstein, David
2005-04-01
We examine an application of the Gillespie algorithm to simulating spatially inhomogeneous reaction-diffusion systems in mesoscopic volumes such as cells and microchambers. The method involves discretizing the chamber into elements and modeling the diffusion of chemical species by the movement of molecules between neighboring elements. These transitions are expressed in the form of a set of reactions which are added to the chemical system. The derivation of the rates of these diffusion reactions is by comparison with a finite volume discretization of the heat equation on an unevenly spaced grid. The diffusion coefficient of each species is allowed to be inhomogeneous in space, including discontinuities. The resulting system is solved by the Gillespie algorithm using the fast direct method. We show that in an appropriate limit the method reproduces exact solutions of the heat equation for a purely diffusive system and the nonlinear reaction-rate equation describing the cubic autocatalytic reaction. PMID:15903653
Stochastic operator-splitting method for reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Choi, TaiJung; Maurya, Mano Ram; Tartakovsky, Daniel M.; Subramaniam, Shankar
2012-11-01
Many biochemical processes at the sub-cellular level involve a small number of molecules. The local numbers of these molecules vary in space and time, and exhibit random fluctuations that can only be captured with stochastic simulations. We present a novel stochastic operator-splitting algorithm to model such reaction-diffusion phenomena. The reaction and diffusion steps employ stochastic simulation algorithms and Brownian dynamics, respectively. Through theoretical analysis, we have developed an algorithm to identify if the system is reaction-controlled, diffusion-controlled or is in an intermediate regime. The time-step size is chosen accordingly at each step of the simulation. We have used three examples to demonstrate the accuracy and robustness of the proposed algorithm. The first example deals with diffusion of two chemical species undergoing an irreversible bimolecular reaction. It is used to validate our algorithm by comparing its results with the solution obtained from a corresponding deterministic partial differential equation at low and high number of molecules. In this example, we also compare the results from our method to those obtained using a Gillespie multi-particle (GMP) method. The second example, which models simplified RNA synthesis, is used to study the performance of our algorithm in reaction- and diffusion-controlled regimes and to investigate the effects of local inhomogeneity. The third example models reaction-diffusion of CheY molecules through the cytoplasm of Escherichia coli during chemotaxis. It is used to compare the algorithm's performance against the GMP method. Our analysis demonstrates that the proposed algorithm enables accurate simulation of the kinetics of complex and spatially heterogeneous systems. It is also computationally more efficient than commonly used alternatives, such as the GMP method.
Selecting spatio-temporal patterns by substrate injection in a reaction-diffusion system
NASA Astrophysics Data System (ADS)
Ghosh, Shyamolina; Ray, Deb Shankar
2015-07-01
We have explored the growth of patterns in chlorine dioxide-iodine-malonic acid reaction-diffusion system when the injection rate of the activator and inhibitor is varied over a range. The transition from spot to stripe and their mixture and finally the target wave which appears at the Hopf bifurcation boundary are observed. Our numerical simulations have been corroborated by theoretical analysis of amplitude equation for targets.
Reaction -Diffusion Systems in Intracellular Molecular Transport and Control
Soh, Siowling; Byrska, Marta; Kandere-Grzybowska, Kristiana
2013-01-01
Chemical reactions make cells work only if the participating chemicals are delivered to desired locations in a timely and precise fashion. While most research to date has focused on the so-called active-transport mechanisms, “passive” diffusion is often equally rapid and is always energetically less costly. Capitalizing on these advantages, cells have developed sophisticated reaction-diffusion (RD) systems that control a wide range of cellular functions – from chemotaxis and cell division, through signaling cascades and oscillations, to cell motility. Despite their apparent diversity, these systems share many common features and are “wired” according to “generic” motifs involving non-linear kinetics, autocatalysis, and feedback loops. Understanding the operation of these complex (bio)chemical systems requires the analysis of pertinent transport-kinetic equations or, at least on a qualitative level, of the characteristic times describing constituent sub-processes. Therefore, in reviewing the manifestations of cellular RD, we also attempt to familiarize the reader with the basic theory of these processes. PMID:20518023
Dichotomous-noise-induced pattern formation in a reaction-diffusion system
NASA Astrophysics Data System (ADS)
Das, Debojyoti; Ray, Deb Shankar
2013-06-01
We consider a generic reaction-diffusion system in which one of the parameters is subjected to dichotomous noise by controlling the flow of one of the reacting species in a continuous-flow-stirred-tank reactor (CSTR) -membrane reactor. The linear stability analysis in an extended phase space is carried out by invoking Furutzu-Novikov procedure for exponentially correlated multiplicative noise to derive the instability condition in the plane of the noise parameters (correlation time and strength of the noise). We demonstrate that depending on the correlation time an optimal strength of noise governs the self-organization. Our theoretical analysis is corroborated by numerical simulations on pattern formation in a chlorine-dioxide-iodine-malonic acid reaction-diffusion system.
Noisy transport in reaction-diffusion systems with quenched disorder
NASA Astrophysics Data System (ADS)
Missel, Andrew Royce
Reaction-diffusion (RD) models are useful tools for studying a wide variety of natural phenomena. The effects of quenched disorder in the reaction rates on RD models is not completely understood, especially in parameter regimes where internal noise or stochasticity is also important. In the first part of this dissertation, I will examine an RD model in which both quenched disorder and stochasticity are important. The model consists of particles (labeled A) which diffuse around space and reproduce (A → 2A), die (A → 0), and compete with one another (2A → A) with rates that depend on position. Specifically, birth is only allowed in localized patches called "oases," while death is allowed in the "desert" that comprises the rest of space. In the limit of low oasis density, transport through the system is achieved via rare "hopping" events when small concentrations of particles make it through the desert from one oasis to another. To correctly account for this hopping---and to accurately describe the nature of transport in the system---it is necessary to take stochasticity into account. In order to determine the nature of transport in this system analytically, I will use a variety of tools drawn from disparate sources, including the theory of hopping conduction in doped semiconductors and first passage percolation. These tools will allow for predictions to be made for a number of important transport-related features: the infection time, or time needed for the population to traverse the system; the velocity of the front moving through the system; and the dynamic roughening of the front. I will also present the results of simulations of the system that largely confirm the analytical predictions. Finally, in the second part of the dissertation I will study a pair of closely related RD models, one of which exhibits an active to absorbing state phase transition.
A priori [Formula: see text] estimates for solutions of a class of reaction-diffusion systems.
Du, Zengji; Peng, Rui
2016-05-01
In this short paper, we establish a priori [Formula: see text]-norm estimates for solutions of a class of reaction-diffusion systems which can be used to model the spread of infectious disease. The developed technique may find applications in other reaction-diffusion systems. PMID:26141826
Distributed order reaction-diffusion systems associated with Caputo derivatives
NASA Astrophysics Data System (ADS)
Saxena, R. K.; Mathai, A. M.; Haubold, H. J.
2014-08-01
This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. ["The fundamental solution of the space-time fractional diffusion equation," Fractional Calculus Appl. Anal. 4, 153-202 (2001); Mainardi et al. "Fox H-functions in fractional diffusion," J. Comput. Appl. Math. 178, 321-331 (2005)] for the fundamental solution of the space-time fractional equation, including Haubold et al. ["Solutions of reaction-diffusion equations in terms of the H-function," Bull. Astron. Soc. India 35, 681-689 (2007)] and Saxena et al. ["Fractional reaction-diffusion equations," Astrophys. Space Sci. 305, 289-296 (2006a)] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction diffusion, and time-fractional diffusion, see Schneider and Wyss ["Fractional diffusion and wave equations," J. Math. Phys. 30, 134-144 (1989)]. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-function in compact forms. The convergence conditions for the double series occurring in the solutions are investigated. It is interesting to observe that the double series comes out to be a special case of the Srivastava-Daoust hypergeometric function of two variables
A Simple Demonstration of Convective Effects on Reaction-Diffusion Systems: A Burning Cigarette.
ERIC Educational Resources Information Center
Pojman, John A.
1990-01-01
Described is a demonstration that provides an introduction to nonequilibrium reaction-diffusion systems and the coupling of hydrodynamics to chemical reactions. Experiments that demonstrate autocatalytic behavior that are effected by gravity and convection are included. (KR)
An integration factor method for stochastic and stiff reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Ta, Catherine; Wang, Dongyong; Nie, Qing
2015-08-01
Stochastic effects are often present in the biochemical systems involving reactions and diffusions. When the reactions are stiff, existing numerical methods for stochastic reaction diffusion equations require either very small time steps for any explicit schemes or solving large nonlinear systems at each time step for the implicit schemes. Here we present a class of semi-implicit integration factor methods that treat the diffusion term exactly and reaction implicitly for a system of stochastic reaction-diffusion equations. Our linear stability analysis shows the advantage of such methods for both small and large amplitudes of noise. Direct use of the method to solving several linear and nonlinear stochastic reaction-diffusion equations demonstrates good accuracy, efficiency, and stability properties. This new class of methods, which are easy to implement, will have broader applications in solving stochastic reaction-diffusion equations arising from models in biology and physical sciences.
NASA Astrophysics Data System (ADS)
Ersoy, Ozlem; Dag, Idris
2015-12-01
The solutions of the reaction-diffusion system are given by method of collocation based on the exponential B-splines. Thus the reaction-diffusion systemturns into an iterative banded algebraic matrix equation. Solution of the matrix equation is carried out byway of Thomas algorithm. The present methods test on both linear and nonlinear problems. The results are documented to compare with some earlier studies by use of L∞ and relative error norm for problems respectively.
Liu, Changchun; Sadik, Mohamed M; Mauk, Michael G; Edelstein, Paul H; Bushman, Frederic D; Gross, Robert; Bau, Haim H
2014-01-01
Real-time amplification and quantification of specific nucleic acid sequences plays a major role in medical and biotechnological applications. In the case of infectious diseases, such as HIV, quantification of the pathogen-load in patient specimens is critical to assess disease progression and effectiveness of drug therapy. Typically, nucleic acid quantification requires expensive instruments, such as real-time PCR machines, which are not appropriate for on-site use and for low-resource settings. This paper describes a simple, low-cost, reaction-diffusion based method for end-point quantification of target nucleic acids undergoing enzymatic amplification. The number of target molecules is inferred from the position of the reaction-diffusion front, analogous to reading temperature in a mercury thermometer. The method was tested for HIV viral load monitoring and performed on par with conventional benchtop methods. The proposed method is suitable for nucleic acid quantification at point of care, compatible with multiplexing and high-throughput processing, and can function instrument-free. PMID:25477046
The role of a reaction-diffusion system in the initiation of primary hair follicles.
Nagorcka, B N; Mooney, J R
1985-05-21
A mechanism based on a reaction-diffusion system is proposed for the initiation of hair follicles in the epidermis during fetal development. It is demonstrated that initiation of primary follicles in a series of waves, within the proposed mechanism, is a consequence of the size and shape dependent properties of the reaction-diffusion system without the need for the propagation of signals through the skin. The observed trio grouping of follicles and variation of primary follicle density per unit skin area during development are also correctly predicted. An explanation, based on the reaction-diffusion system and the variation of its characteristic spatial wavelength with time during development, is suggested for the termination of both primary and secondary follicle initiation as well as follicle neogenesis. The proposed initiation mechanism is basically the same as that used to explain various spatial patterns observed in hair fibre formation (Nagorcka & Mooney, 1982). PMID:4033155
Solitary travelling auto-waves in fractional reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Datsko, Bohdan; Gafiychuk, Vasyl; Podlubny, Igor
2015-06-01
In this article we study properties of solitary auto-waves in nonlinear fractional reaction-diffusion systems. As an example, the generalised FitzHugh-Nagumo model with time-fractional derivatives is considered. By a linear stability analysis and computer simulation it is shown that the order of the fractional derivative can substantially change the properties of solitary auto-waves and significantly enrich nonlinear system dynamics. The main properties of solitary travelling wave solutions, including the shape of the waves, the domain of their existence, as well as the parameters of their propagation in fractional reaction-diffusion systems, are investigated.
Trávnícková, Tereza; Kohout, Martin; Schreiber, Igor; Kubícek, Milan
2009-12-01
We analyze dynamics of stationary nonuniform patterns, traveling waves, and spatiotemporal chaos in a simple model of a tubular cross-flow reactor. The reactant is supplied continuously via convective flow and/or by diffusion through permeable walls of the reactor. First order exothermic reaction kinetics is assumed and the system is described by mass and energy balances forming coupled reaction-diffusion-convection equations. Dynamical regimes of the reaction-diffusion subsystem range from pulses and fronts to periodic waves and complex chaotic behavior. Two distinct types of chaotic patterns are identified and characterized by Lyapunov dimension. Next we examine the effects of convection on various types of the reaction-diffusion regimes. Remarkable zigzag fronts and steady state patterns are found despite the absence of differential flow. We employ continuation techniques to elucidate the existence and form of these patterns. PMID:20059221
NASA Astrophysics Data System (ADS)
Trávníčková, Tereza; Kohout, Martin; Schreiber, Igor; Kubíček, Milan
2009-12-01
We analyze dynamics of stationary nonuniform patterns, traveling waves, and spatiotemporal chaos in a simple model of a tubular cross-flow reactor. The reactant is supplied continuously via convective flow and/or by diffusion through permeable walls of the reactor. First order exothermic reaction kinetics is assumed and the system is described by mass and energy balances forming coupled reaction-diffusion-convection equations. Dynamical regimes of the reaction-diffusion subsystem range from pulses and fronts to periodic waves and complex chaotic behavior. Two distinct types of chaotic patterns are identified and characterized by Lyapunov dimension. Next we examine the effects of convection on various types of the reaction-diffusion regimes. Remarkable zigzag fronts and steady state patterns are found despite the absence of differential flow. We employ continuation techniques to elucidate the existence and form of these patterns.
Scaling of morphogenetic patterns in reaction-diffusion systems.
Rasolonjanahary, Manan'Iarivo; Vasiev, Bakhtier
2016-09-01
Development of multicellular organisms is commonly associated with the response of individual cells to concentrations of chemical substances called morphogens. Concentration fields of morphogens form a basis for biological patterning and ensure its properties including ability to scale with the size of the organism. While mechanisms underlying the formation of morphogen gradients are reasonably well understood, little is known about processes responsible for their scaling. Here, we perform a formal analysis of scaling for chemical patterns forming in continuous systems. We introduce a quantity representing the sensitivity of systems to changes in their size and use it to analyse scaling properties of patterns forming in a few different systems. Particularly, we consider how scaling properties of morphogen gradients forming in diffusion-decay systems depend on boundary conditions and how the scaling can be improved by passive modulation of morphogens or active transport in the system. We also analyse scaling of morphogenetic signal caused by two opposing gradients and consider scaling properties of patterns forming in activator-inhibitor systems. We conclude with a few possible mechanisms which allow scaling of morphogenetic patterns. PMID:27255960
Nonlinear stability in reaction-diffusion systems via optimal Lyapunov functions
NASA Astrophysics Data System (ADS)
Lombardo, S.; Mulone, G.; Trovato, M.
2008-06-01
We define optimal Lyapunov functions to study nonlinear stability of constant solutions to reaction-diffusion systems. A computable and finite radius of attraction for the initial data is obtained. Applications are given to the well-known Brusselator model and a three-species model for the spatial spread of rabies among foxes.
NASA Astrophysics Data System (ADS)
Bau, Haim; Liu, Changchun; Killawala, Chitvan; Sadik, Mohamed; Mauk, Michael
2014-11-01
Real-time amplification and quantification of specific nucleic acid sequences plays a major role in many medical and biotechnological applications. In the case of infectious diseases, quantification of the pathogen-load in patient specimens is critical to assessing disease progression, effectiveness of drug therapy, and emergence of drug-resistance. Typically, nucleic acid quantification requires sophisticated and expensive instruments, such as real-time PCR machines, which are not appropriate for on-site use and for low resource settings. We describe a simple, low-cost, reactiondiffusion based method for end-point quantification of target nucleic acids undergoing enzymatic amplification. The number of target molecules is inferred from the position of the reaction-diffusion front, analogous to reading temperature in a mercury thermometer. We model the process with the Fisher Kolmogoroff Petrovskii Piscounoff (FKPP) Equation and compare theoretical predictions with experimental observations. The proposed method is suitable for nucleic acid quantification at the point of care, compatible with multiplexing and high-throughput processing, and can function instrument-free. C.L. was supported by NIH/NIAID K25AI099160; M.S. was supported by the Pennsylvania Ben Franklin Technology Development Authority; C.K. and H.B. were funded, in part, by NIH/NIAID 1R41AI104418-01A1.
Bogdanov-Takens Bifurcation of a Class of Delayed Reaction-Diffusion System
NASA Astrophysics Data System (ADS)
Cao, Jianzhi; Wang, Peiguang; Yuan, Rong; Mei, Yingying
2015-06-01
In this paper, a class of reaction-diffusion system with Neumann boundary condition is considered. By analyzing the generalized eigenvector associated with zero eigenvalue, an equivalent condition for the determination of Bogdonov-Takens (B-T) singularity is obtained. Next, by using center manifold theorem and normal form method, we have a two-dimension ordinary differential system on its center manifold. Finally, two examples show that the given algorithm is effective.
Global Existence of Renormalized Solutions to Entropy-Dissipating Reaction-Diffusion Systems
NASA Astrophysics Data System (ADS)
Fischer, J.
2015-10-01
In the present work we introduce the notion of a renormalized solution for reaction-diffusion systems with entropy-dissipating reactions. We establish the global existence of renormalized solutions. In the case of integrable reaction terms our notion of a renormalized solution reduces to the usual notion of a weak solution. Our existence result in particular covers all reaction-diffusion systems involving a single reversible reaction with mass-action kinetics and (possibly species-dependent) Fick-law diffusion; more generally, it covers the case of systems of reversible reactions with mass-action kinetics which satisfy the detailed balance condition. For such equations the existence of any kind of solution in general was an open problem, thereby motivating the study of renormalized solutions.
Differential diffusivity of Nodal and Lefty underlies a reaction-diffusion patterning system
Müller, Patrick; Rogers, Katherine W.; Jordan, Ben M.; Lee, Joon S.; Robson, Drew; Ramanathan, Sharad; Schier, Alexander F.
2012-01-01
Biological systems involving short-range activators and long-range inhibitors can generate complex patterns. Reaction-diffusion models postulate that differences in signaling range are caused by differential diffusivity of inhibitor and activator. Other models suggest that differential clearance underlies different signaling ranges. To test these models, we measured the biophysical properties of the Nodal/Lefty activator/inhibitor system during zebrafish embryogenesis. Analysis of Nodal and Lefty gradients reveals that Nodals have a shorter range than Lefty proteins. Pulse-labelinganalysis indicates that Nodals and Leftys have similar clearance kinetics, whereas fluorescence recovery assays reveal that Leftys have a higher effective diffusion coefficient than Nodals. These results indicate that differential diffusivity is the major determinant of the differences in Nodal/Lefty range and provide biophysical support for reaction-diffusion models of activator/inhibitor-mediated patterning. PMID:22499809
Primal-mixed formulations for reaction-diffusion systems on deforming domains
NASA Astrophysics Data System (ADS)
Ruiz-Baier, Ricardo
2015-10-01
We propose a finite element formulation for a coupled elasticity-reaction-diffusion system written in a fully Lagrangian form and governing the spatio-temporal interaction of species inside an elastic, or hyper-elastic body. A primal weak formulation is the baseline model for the reaction-diffusion system written in the deformed domain, and a finite element method with piecewise linear approximations is employed for its spatial discretization. On the other hand, the strain is introduced as mixed variable in the equations of elastodynamics, which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction-diffusion system written in a deformed domain. The discrete mechanical problem yields a mixed finite element scheme based on row-wise Raviart-Thomas elements for stresses, Brezzi-Douglas-Marini elements for displacements, and piecewise constant pressure approximations. The application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed, and some illustrative examples are provided and extensively analyzed.
An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Padgett, Jill M. A.; Ilie, Silvana
2016-03-01
Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating the solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.
Nonlinear Waves in Reaction Diffusion Systems: The Effect of Transport Memory
HURD,ALAN J.; KENKRE,V.M.; MANNE,K.K.
1999-11-04
Motivated by the problem of determining stress distributions in granular materials, we study the effect of finite transport correlation times on the propagation of nonlinear wavefronts in reaction diffusion systems. We obtain new results such as the possibility of spatial oscillations in the wavefront shape for certain values of the system parameters and high enough wavefront speeds. We also generalize earlier known results concerning the minimum wavefront speed and shape-speed relationships stemming from the finiteness of the correlation times. Analytic investigations are made possible by a piece-wise linear representation of the nonlinearity.
Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK
2014-01-01
Background Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system’s set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This “code-based” approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution. Results As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations. Conclusions The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block
Square Turing patterns in reaction-diffusion systems with coupled layers
Li, Jing; Wang, Hongli E-mail: qi@pku.edu.cn; Ouyang, Qi E-mail: qi@pku.edu.cn
2014-06-15
Square Turing patterns are usually unstable in reaction-diffusion systems and are rarely observed in corresponding experiments and simulations. We report here an example of spontaneous formation of square Turing patterns with the Lengyel-Epstein model of two coupled layers. The squares are found to be a result of the resonance between two supercritical Turing modes with an appropriate ratio. Besides, the spatiotemporal resonance of Turing modes resembles to the mode-locking phenomenon. Analysis of the general amplitude equations for square patterns reveals that the fixed point corresponding to square Turing patterns is stationary when the parameters adopt appropriate values.
Formation of Somitogenesis-like Pattern in a Reaction-Diffusion System
NASA Astrophysics Data System (ADS)
Sakamoto, Fumitaka; Miyakawa, Kenji
2008-08-01
The Belousov-Zhabotinsky reaction system showing stationary patterns is realized on the basis of water-in-oil microemulsions with the surfactant sodium bis(2-ethylhexyl)sulfosuccinate. We experimentally demonstrate the formation of somitogenesis-like pattern in which chemical waves arising from spontaneous bulk oscillations are successively arrested and then stacked. The experimental results are qualitatively reproduced by numerical simulations using the two-variable oregonator model. These show that a somitogenesis can be accounted for by the genuine reaction-diffusion model.
Pattern formation in the iodate-sulfite-thiosulfate reaction-diffusion system.
Liu, Haimiao; Pojman, John A; Zhao, Yuemin; Pan, Changwei; Zheng, Juhua; Yuan, Ling; Horváth, Attila K; Gao, Qingyu
2012-01-01
Sodium polyacrylate-induced pH pattern formation and starch-induced iodine pattern formation were investigated in the iodate-sulfite-thiosulfate (IST) reaction in a one-side fed disc gel reactor (OSFR). As binding agents of the autocatalyst of hydrogen ions or iodide ions, different content of sodium polyacrylate or starch has induced various types of pattern formation. We observed pH pulses, striped patterns, mixed spots and stripes, and hexagonal spots upon increasing the content of sodium polyacrylate and observed iodine pulses, branched patterns, and labyrinthine patterns upon increasing the starch content in the system. Coexistence of a pH front and an iodine front was also studied in a batch IST reaction-diffusion system. Both pH and iodine front instabilities were observed in the presence of sodium polyacrylate, i.e., cellular fronts and transient Turing structures resulting from the decrease in diffusion coefficients of activators. The mechanism of multiple feedback may explain the different patterns in the IST reaction-diffusion system. PMID:22068976
Self-similar fast-reaction limits for reaction-diffusion systems on unbounded domains
NASA Astrophysics Data System (ADS)
Crooks, E. C. M.; Hilhorst, D.
2016-08-01
We present a unified approach to characterising fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit k → ∞ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line. For fixed k, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. Our results generalise earlier work of Hilhorst, van der Hout and Peletier to a much wider class of problems, and provide a quantitative description of the penetration of one substance into another in both the fast-reaction and long-time regimes.
Spatiotemporal patterns in reaction-diffusion system and in a vibrated granular bed
Swinney, H.L.; Lee, K.J.; McCormick, W.D.
1995-12-31
Experiments on a quasi-two-dimensional reaction-diffusion system reveal transitions from a uniform state to stationary hexagonal, striped, and rhombic spatial patterns. For other reactor conditions lamellae and self-replicating spot patterns are observed. These patterns form in continuously fed thin gel reactors that can be maintained indefinitely in well-defined nonequilibrium states. Reaction-diffusion models with two chemical species yield patterns similar to those observed in the experiments. Pattern formation is also being examined in vertically oscillated thin granular layers (typically 3-30 particle diameters deep). For small acceleration amplitudes, a granular layer is flat, but above a well-defined critical acceleration amplitude, spatial patterns spontaneously form. Disordered time-dependent granular patterns are observed as well as regular patterns of squares, stripes, and hexagons. A one-dimensional model consisting of a completely inelastic ball colliding with a sinusoidally oscillating platform provides a semi-quantitative description of most of the observed bifurcations between the different spatiotemporal regimes.
Mahara, Hitoshi; Okada, Koichi; Nomura, Atsushi; Miike, Hidetoshi; Sakurai, Tatsunari
2009-07-01
We found a rotating global structure induced by the dynamical force of local chemical activity in a thin solution layer of excitable Belousov-Zhabotinsky reaction coupled with diffusion. The surface flow and deformation associated with chemical spiral waves (wavelength about 1 mm) represents a global unidirectional structure and a global tilt in the entire Petri dish (100 mm in diameter), respectively. For these observations, we scanned the condition of hierarchal pattern selection. From this result, the bromomalonic acid has an important role to induce the rotating global structure. An interaction between a reaction-diffusion process and a surface-tension-driven effect leads to such hierarchal pattern with different scales. PMID:19658764
Transition to Spatio-Temporal Chaos with Increasing Length in the Reaction-Diffusion System
NASA Astrophysics Data System (ADS)
Trail, Collin; Tomlin, Brett; Olsen, Thomas; Wiener, Richard J.
2003-11-01
Calculations based up the Reaction-Diffusion model (H. Riecke and H.-G. Paap, Europhys. Lett. 14), 1235 (1991).have proven to be suggestive for a wide variety of pattern forming systems, including Taylor-Couette flow with hourglass geometry(Richard J. Wiener et al), Phys. Rev. E 55, 5489 (1997).. Seeking insight to guide experimental investigations, we extend these calculations. Previous calculations indicated that in smaller systems, only temporal chaos, located in a small region, would be observed, while in longer systems instabilities would form over a wide region. Our simulations explore this transition from purely temporal chaos to spatio-temporal chaos as the length of the system is increased.
Global existence and asymptotic stability of equilibria to reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Wang, Rong-Nian; Tang, Zhong Wei
2009-06-01
In this paper, we study weakly coupled reaction-diffusion systems in unbounded domains of {\\bb R}^2 or {\\bb R}^3 , where the reaction terms are sums of quasimonotone nondecreasing and nonincreasing functions. Such systems are more complicated than those in many previous publications and little is known about them. A comparison principle and global existence, and boundedness theorems for solutions to these systems are established. Sufficient conditions on the nonlinearities, ensuring the positively Ljapunov stability of the zero solution with respect to H2-perturbations, are also obtained. As samples of applications, these results are applied to an autocatalytic chemical model and a concrete problem, whose nonlinearities are nonquasimonotone. Our results are novel. In particular, we present a solution to an open problem posed by Escher and Yin (2005 J. Nonlinear Anal. Theory Methods Appl. 60 1065-84).
A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces
NASA Astrophysics Data System (ADS)
Glitzky, Annegret; Mielke, Alexander
2013-02-01
We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient-flow formulation for electro-reaction-diffusion systems with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.
Hybrid stochastic simulation of reaction-diffusion systems with slow and fast dynamics
Strehl, Robert; Ilie, Silvana
2015-12-21
In this paper, we present a novel hybrid method to simulate discrete stochastic reaction-diffusion models arising in biochemical signaling pathways. We study moderately stiff systems, for which we can partition each reaction or diffusion channel into either a slow or fast subset, based on its propensity. Numerical approaches missing this distinction are often limited with respect to computational run time or approximation quality. We design an approximate scheme that remedies these pitfalls by using a new blending strategy of the well-established inhomogeneous stochastic simulation algorithm and the tau-leaping simulation method. The advantages of our hybrid simulation algorithm are demonstrated on three benchmarking systems, with special focus on approximation accuracy and efficiency.
NASA Astrophysics Data System (ADS)
Fernandes, Ryan I.; Fairweather, Graeme
2012-08-01
An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only O(N) operations where N is the number of unknowns. Moreover, it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties.
Spatial propagation for a two component reaction-diffusion system arising in population dynamics
NASA Astrophysics Data System (ADS)
Ducrot, Arnaud
2016-06-01
In this work a two component epidemic reaction-diffusion system posed on the whole space RN is considered. Uniform boundedness of the solutions is proved using suitable local Lp-estimates. The spatial invasion of a localized introduced amount of infective is studied yielding to the derivation of the asymptotic speed of spread for the infection. This part is achieved using uniform persistence ideas. The state of the population after the epidemic is further investigated using different Lyapunov like arguments. The solution is shown to converge the endemic equilibrium point behind the front in the equi-diffusional case. For general diffusion coefficient unique ergodicity of the tail of invasion is obtained by constructing a suitable sub-harmonic map.
Signaling gradients in cascades of two-state reaction-diffusion systems.
Berezhkovskii, Alexander M; Coppey, Mathieu; Shvartsman, Stanislav Y
2009-01-27
Biological networks frequently use cascades, generally defined as chain-like arrangements of similar modules. Spatially lumped cascades can serve as noise filters, time-delay, or thresholding elements. The operation and functional capabilities of spatially distributed cascades are much less understood. Motivated by studies of pattern formation in the early Drosophila embryo, we analyze cascades of 2-state reaction-diffusion systems. At each stage within such as a cascade, a diffusible particle is reversibly bound by immobile traps and can be annihilated in both mobile and immobile states. When trapped, these particles drive the next stage by converting mobile particles of a different type from a passive to active form. The cascade initiated by injection of mobile particles into the first stage. We derive analytical expressions for the steady-state concentration profiles of mobile and immobile particles and analyze how the output of a cascade is controlled by properties of the constituent stages. PMID:19147842
A Robust and Efficient Method for Steady State Patterns in Reaction-Diffusion Systems
Lo, Wing-Cheong; Chen, Long; Wang, Ming; Nie, Qing
2012-01-01
An inhomogeneous steady state pattern of nonlinear reaction-diffusion equations with no-flux boundary conditions is usually computed by solving the corresponding time-dependent reaction-diffusion equations using temporal schemes. Nonlinear solvers (e.g., Newton’s method) take less CPU time in direct computation for the steady state; however, their convergence is sensitive to the initial guess, often leading to divergence or convergence to spatially homogeneous solution. Systematically numerical exploration of spatial patterns of reaction-diffusion equations under different parameter regimes requires that the numerical method be efficient and robust to initial condition or initial guess, with better likelihood of convergence to an inhomogeneous pattern. Here, a new approach that combines the advantages of temporal schemes in robustness and Newton’s method in fast convergence in solving steady states of reaction-diffusion equations is proposed. In particular, an adaptive implicit Euler with inexact solver (AIIE) method is found to be much more efficient than temporal schemes and more robust in convergence than typical nonlinear solvers (e.g., Newton’s method) in finding the inhomogeneous pattern. Application of this new approach to two reaction-diffusion equations in one, two, and three spatial dimensions, along with direct comparisons to several other existing methods, demonstrates that AIIE is a more desirable method for searching inhomogeneous spatial patterns of reaction-diffusion equations in a large parameter space. PMID:22773849
The dynamics of localized spot patterns for reaction-diffusion systems on the sphere
NASA Astrophysics Data System (ADS)
Trinh, Philippe H.; Ward, Michael J.
2016-03-01
In the singularly perturbed limit corresponding to a large diffusivity ratio between two components in a reaction-diffusion (RD) system, quasi-equilibrium spot patterns are often admitted, producing a solution that concentrates at a discrete set of points in the domain. In this paper, we derive and study the differential algebraic equation (DAE) that characterizes the slow dynamics for such spot patterns for the Brusselator RD model on the surface of a sphere. Asymptotic and numerical solutions are presented for the system governing the spot strengths, and we describe the complex bifurcation structure and demonstrate the occurrence of imperfection sensitivity due to higher order effects. Localized spot patterns can undergo a fast time instability and we derive the conditions for this phenomena, which depend on the spatial configuration of the spots and the parameters in the system. In the absence of these instabilities, our numerical solutions of the DAE system for N = 2 to N = 8 spots suggest a large basin of attraction to a small set of possible steady-state configurations. We discuss the connections between our results and the study of point vortices on the sphere, as well as the problem of determining a set of elliptic Fekete points, which correspond to globally minimizing the discrete logarithmic energy for N points on the sphere.
Bursting regimes in a reaction-diffusion system with action potential-dependent equilibrium.
Meier, Stephen R; Lancaster, Jarrett L; Starobin, Joseph M
2015-01-01
The equilibrium Nernst potential plays a critical role in neural cell dynamics. A common approximation used in studying electrical dynamics of excitable cells is that the ionic concentrations inside and outside the cell membranes act as charge reservoirs and remain effectively constant during excitation events. Research into brain electrical activity suggests that relaxing this assumption may provide a better understanding of normal and pathophysiological functioning of the brain. In this paper we explore time-dependent ionic concentrations by allowing the ion-specific Nernst potentials to vary with developing transmembrane potential. As a specific implementation, we incorporate the potential-dependent Nernst shift into a one-dimensional Morris-Lecar reaction-diffusion model. Our main findings result from a region in parameter space where self-sustaining oscillations occur without external forcing. Studying the system close to the bifurcation boundary, we explore the vulnerability of the system with respect to external stimulations which disrupt these oscillations and send the system to a stable equilibrium. We also present results for an extended, one-dimensional cable of excitable tissue tuned to this parameter regime and stimulated, giving rise to complex spatiotemporal pattern formation. Potential applications to the emergence of neuronal bursting in similar two-variable systems and to pathophysiological seizure-like activity are discussed. PMID:25823018
Bursting Regimes in a Reaction-Diffusion System with Action Potential-Dependent Equilibrium
Meier, Stephen R.; Lancaster, Jarrett L.; Starobin, Joseph M.
2015-01-01
The equilibrium Nernst potential plays a critical role in neural cell dynamics. A common approximation used in studying electrical dynamics of excitable cells is that the ionic concentrations inside and outside the cell membranes act as charge reservoirs and remain effectively constant during excitation events. Research into brain electrical activity suggests that relaxing this assumption may provide a better understanding of normal and pathophysiological functioning of the brain. In this paper we explore time-dependent ionic concentrations by allowing the ion-specific Nernst potentials to vary with developing transmembrane potential. As a specific implementation, we incorporate the potential-dependent Nernst shift into a one-dimensional Morris-Lecar reaction-diffusion model. Our main findings result from a region in parameter space where self-sustaining oscillations occur without external forcing. Studying the system close to the bifurcation boundary, we explore the vulnerability of the system with respect to external stimulations which disrupt these oscillations and send the system to a stable equilibrium. We also present results for an extended, one-dimensional cable of excitable tissue tuned to this parameter regime and stimulated, giving rise to complex spatiotemporal pattern formation. Potential applications to the emergence of neuronal bursting in similar two-variable systems and to pathophysiological seizure-like activity are discussed. PMID:25823018
Hopping Conduction and Bacteria: Transport Properties of Disordered Reaction-Diffusion Systems
NASA Astrophysics Data System (ADS)
Missel, Andrew; Dahmen, Karin
2008-03-01
Reaction-diffusion (RD) systems are used to model everything from the formation of animal coat patterns to the spread of genes in a population to the seasonal variation of plankton density in the ocean. In all of these problems, disorder plays a large role, but determining its effects on transport properties in RD systems has been a challenge. We present here both analytical and numerical studies of a particular disordered RD system consisting of particles which are allowed to diffuse and compete for resources (2A->A) with spatially homogeneous rates, reproduce (A->2A) in certain areas (``oases''), and die (A->0) everywhere else (the ``desert''). In the low oasis density regime, transport is mediated through rare ``hopping events'' in which a small number of particles diffuse through the desert from one oasis to another; the situation is mathematically analogous to hopping conduction in doped semiconductors, and this analogy, along with some ideas from first passage percolation theory, allows us to make some quantitative predictions about the transport properties of the system on a large scale.
Parallel Solutions for Voxel-Based Simulations of Reaction-Diffusion Systems
D'Agostino, Daniele; Pasquale, Giulia; Clematis, Andrea; Maj, Carlo; Mosca, Ettore; Milanesi, Luciano; Merelli, Ivan
2014-01-01
There is an increasing awareness of the pivotal role of noise in biochemical processes and of the effect of molecular crowding on the dynamics of biochemical systems. This necessity has given rise to a strong need for suitable and sophisticated algorithms for the simulation of biological phenomena taking into account both spatial effects and noise. However, the high computational effort characterizing simulation approaches, coupled with the necessity to simulate the models several times to achieve statistically relevant information on the model behaviours, makes such kind of algorithms very time-consuming for studying real systems. So far, different parallelization approaches have been deployed to reduce the computational time required to simulate the temporal dynamics of biochemical systems using stochastic algorithms. In this work we discuss these aspects for the spatial TAU-leaping in crowded compartments (STAUCC) simulator, a voxel-based method for the stochastic simulation of reaction-diffusion processes which relies on the Sτ-DPP algorithm. In particular we present how the characteristics of the algorithm can be exploited for an effective parallelization on the present heterogeneous HPC architectures. PMID:25045716
Parallel solutions for voxel-based simulations of reaction-diffusion systems.
D'Agostino, Daniele; Pasquale, Giulia; Clematis, Andrea; Maj, Carlo; Mosca, Ettore; Milanesi, Luciano; Merelli, Ivan
2014-01-01
There is an increasing awareness of the pivotal role of noise in biochemical processes and of the effect of molecular crowding on the dynamics of biochemical systems. This necessity has given rise to a strong need for suitable and sophisticated algorithms for the simulation of biological phenomena taking into account both spatial effects and noise. However, the high computational effort characterizing simulation approaches, coupled with the necessity to simulate the models several times to achieve statistically relevant information on the model behaviours, makes such kind of algorithms very time-consuming for studying real systems. So far, different parallelization approaches have been deployed to reduce the computational time required to simulate the temporal dynamics of biochemical systems using stochastic algorithms. In this work we discuss these aspects for the spatial TAU-leaping in crowded compartments (STAUCC) simulator, a voxel-based method for the stochastic simulation of reaction-diffusion processes which relies on the Sτ-DPP algorithm. In particular we present how the characteristics of the algorithm can be exploited for an effective parallelization on the present heterogeneous HPC architectures. PMID:25045716
Szalai, István; Cuiñas, Daniel; Takács, Nándor; Horváth, Judit; De Kepper, Patrick
2012-08-01
In his seminal 1952 paper, Alan Turing predicted that diffusion could spontaneously drive an initially uniform solution of reacting chemicals to develop stable spatially periodic concentration patterns. It took nearly 40 years before the first two unquestionable experimental demonstrations of such reaction-diffusion patterns could be made in isothermal single phase reaction systems. The number of these examples stagnated for nearly 20 years. We recently proposed a design method that made their number increase to six in less than 3 years. In this report, we formally justify our original semi-empirical method and support the approach with numerical simulations based on a simple but realistic kinetic model. To retain a number of basic properties of real spatial reactors but keep calculations to a minimal complexity, we introduce a new way to collapse the confined spatial direction of these reactors. Contrary to similar reduced descriptions, we take into account the effect of the geometric size in the confinement direction and the influence of the differences in the diffusion coefficient on exchange rates of species with their feed environment. We experimentally support the method by the observation of stationary patterns in red-ox reactions not based on oxihalogen chemistry. Emphasis is also brought on how one of these new systems can process different initial conditions and memorize them in the form of localized patterns of different geometries. PMID:23919126
Stability of periodic traveling waves in the Aliev-Panfilov reaction-diffusion system
NASA Astrophysics Data System (ADS)
Gani, M. Osman; Ogawa, Toshiyuki
2016-04-01
We study the two-component Aliev-Panfilov reaction-diffusion system of cardiac excitation. It is known that the model exhibits spiral wave instability in two-dimensional spatial domains. In order to describe the spiral wave instability, it is important to understand periodic traveling wave instability resulting from the model. We determine the existence and stability of periodic traveling waves in the model. In addition, we calculate the stability boundary between stable and unstable periodic traveling waves in a two-dimensional parameter plane. It is observed that the periodic traveling waves express instability by a stability change of Eckhaus type. As a result, a stable wave bifurcates to an oscillating periodic traveling wave. We describe these phenomena by calculating the essential spectra of the waves. Furthermore, we study the stability of the waves as a function of the gaps between two nullclines. In two dimensions, we determine the spiral wave instability based on the stability boundary of the periodic traveling waves.
A transition in the spatially integrated reaction rate of bimolecular reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Arshadi, Masoud; Rajaram, Harihar
2015-09-01
Numerical simulations of diffusion with bimolecular reaction demonstrate a transition in the spatially integrated reaction rate—increasing with time initially, and transitioning to a decrease with time. In previous work, this reaction-diffusion problem has been analyzed as a Stefan problem involving a distinct moving boundary (reaction front), leading to predictions that front motion scales as √t, and correspondingly the spatially integrated reaction rate decreases as the square root of time 1/√t. We present a general nondimensionalization of the problem and a perturbation analysis to show that there is an early time regime where the spatially integrated reaction rate scales as √t rather than 1/√t. The duration of this early time regime (where the spatially integrated reaction rate is kinetically rather than diffusion controlled) is shown to depend on the kinetic rate parameters, diffusion coefficients, and initial concentrations of the two species. Numerical simulation results confirm the theoretical estimates of the transition time. We present illustrative calculations in the context of in situ chemical oxidation for remediation of fractured rock systems where contaminants are largely dissolved in the rock matrix. We consider different contaminants of concern (COCs), including TCE, PCE, MTBE, and RDX. While the early time regime is very short lived for TCE, it can persist over months to years for MTBE and RDX, due to slow oxidation kinetics.
NASA Astrophysics Data System (ADS)
Li, Xin-Zheng; Bai, Zhan-Guo; Li, Yan; Zhao, Kun
2016-03-01
In this paper, various kinds of spontaneous dynamic patterns are investigated based on a two-layer nonlinearly coupled Brusselator model. It is found that, when the Hopf mode or supercritical Turing mode respectively plays major role in the short or long wavelength mode layer, the dynamic patterns appear under the action of nonlinearly coupling interactions in the reaction-diffusion system. The stripe pattern can change its symmetrical structure and form other graphics when influenced by small perturbations sourced from other modes. If two supercritical Turing modes are nonlinearly coupled together, the transition from Turing instability to Hopf instability may appear in the short wavelength mode layer, and the twinkling-eye square pattern, traveling and rotating pattern will be obtained in the two subsystems. If Turing mode and subharmonic Turing mode satisfy the three-mode resonance relation, twinkling-eye patterns are generated, and oscillating spots are arranged as square lattice in the two-dimensional space. When the subharmonic Turing mode satisfies the spatio-temporal phase matching condition, the traveling patterns, including the rhombus, hexagon and square patterns are obtained, which presents different moving velocities. It is found that the wave intensity plays an important role in pattern formation and pattern selection.
Brownian-motion based simulation of stochastic reaction-diffusion systems for affinity based sensors
NASA Astrophysics Data System (ADS)
Tulzer, Gerhard; Heitzinger, Clemens
2016-04-01
In this work, we develop a 2D algorithm for stochastic reaction-diffusion systems describing the binding and unbinding of target molecules at the surfaces of affinity-based sensors. In particular, we simulate the detection of DNA oligomers using silicon-nanowire field-effect biosensors. Since these devices are uniform along the nanowire, two dimensions are sufficient to capture the kinetic effects features. The model combines a stochastic ordinary differential equation for the binding and unbinding of target molecules as well as a diffusion equation for their transport in the liquid. A Brownian-motion based algorithm simulates the diffusion process, which is linked to a stochastic-simulation algorithm for association at and dissociation from the surface. The simulation data show that the shape of the cross section of the sensor yields areas with significantly different target-molecule coverage. Different initial conditions are investigated as well in order to aid rational sensor design. A comparison of the association/hybridization behavior for different receptor densities allows optimization of the functionalization setup depending on the target-molecule density.
Tulzer, Gerhard; Heitzinger, Clemens
2016-04-22
In this work, we develop a 2D algorithm for stochastic reaction-diffusion systems describing the binding and unbinding of target molecules at the surfaces of affinity-based sensors. In particular, we simulate the detection of DNA oligomers using silicon-nanowire field-effect biosensors. Since these devices are uniform along the nanowire, two dimensions are sufficient to capture the kinetic effects features. The model combines a stochastic ordinary differential equation for the binding and unbinding of target molecules as well as a diffusion equation for their transport in the liquid. A Brownian-motion based algorithm simulates the diffusion process, which is linked to a stochastic-simulation algorithm for association at and dissociation from the surface. The simulation data show that the shape of the cross section of the sensor yields areas with significantly different target-molecule coverage. Different initial conditions are investigated as well in order to aid rational sensor design. A comparison of the association/hybridization behavior for different receptor densities allows optimization of the functionalization setup depending on the target-molecule density. PMID:26939610
Interface proliferation and the growth of labyrinths in a reaction-diffusion system
NASA Astrophysics Data System (ADS)
Goldstein, Raymond E.; Muraki, David J.; Petrich, Dean M.
1996-04-01
In the bistable regime of the FitzHugh-Nagumo model of reaction-diffusion systems, spatially homogeneous patterns may be nonlinearly unstable to the formation of compact "localized states." The formation of space-filling patterns from instabilities of such structures is studied in the context of a nonlocal contour dynamics model for the evolution of boundaries between high and low concentrations of the activator. An earlier heuristic derivation [D. M. Petrich and R. E. Goldstein,
NASA Astrophysics Data System (ADS)
Liang, Xing; Jiang, Jifa
The asymptotic behavior of discrete type-K monotone dynamical systems and reaction-diffusion equations is investigated. The studying content includes the index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one. A nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits. Applications are made to time-periodic Lotka-Volterra systems with diffusion, and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given. For more general time-periodic type-K monotone reaction-diffusion systems with spatial homogeneity, a simple condition is given to guarantee the convergence of all positive solutions.
Lecca, Paola; Ihekwaba, Adaoha E C; Dematté, Lorenzo; Priami, Corrado
2010-01-01
Reaction-diffusion systems are mathematical models that describe how the concentrations of substances distributed in space change under the influence of local chemical reactions, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose solution predicts how diffusion causes the concentration field to change with time. This change is proportional to the diffusion coefficient. If the solute moves in a homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and solute. However, in nonhomogeneous and structured media the assumption of constant intracellular diffusion coefficient is not necessarily valid, and, consequently, the diffusion coefficient is a function of the local concentration of solvent and solutes. In this paper we propose a stochastic model of reaction-diffusion systems, in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces. We then describe the software tool Redi (REaction-DIffusion simulator) which we have developed in order to implement this model into a Gillespie-like stochastic simulation algorithm. Finally, we show the ability of our model implemented in the Redi tool to reproduce the observed gradient of the bicoid protein in the Drosophila Melanogaster embryo. With Redi, we were able to simulate with an accuracy of 1% the experimental spatio-temporal dynamics of the bicoid protein, as recorded in time-lapse experiments obtained by direct measurements of transgenic bicoidenhanced green fluorescent protein. PMID:21098882
Clustering and Optimal Arrangement of Enzymes in Reaction-Diffusion Systems
NASA Astrophysics Data System (ADS)
Buchner, Alexander; Tostevin, Filipe; Gerland, Ulrich
2013-05-01
Enzymes within biochemical pathways are often colocalized, yet the consequences of specific spatial enzyme arrangements remain poorly understood. We study the impact of enzyme arrangement on reaction efficiency within a reaction-diffusion model. The optimal arrangement transitions from a cluster to a distributed profile as a single parameter, which controls the probability of reaction versus diffusive loss of pathway intermediates, is varied. We introduce the concept of enzyme exposure to explain how this transition arises from the stochastic nature of molecular reactions and diffusion.
Cluster geometry and survival probability in systems driven by reaction diffusion dynamics
NASA Astrophysics Data System (ADS)
Windus, Alastair; Jensen, Henrik J.
2008-11-01
We consider a reaction-diffusion model incorporating the reactions A→phi, A→2A and 2A→3A. Depending on the relative rates for sexual and asexual reproduction of the quantity A, the model exhibits either a continuous or first-order absorbing phase transition to an extinct state. A tricritical point separates the two phase lines. While we comment on this critical behaviour, the main focus of the paper is on the geometry of the population clusters that form. We observe the different cluster structures that arise at criticality for the three different types of critical behaviour and show that there exists a linear relationship for the survival probability against initial cluster size at the tricritical point only.
NASA Astrophysics Data System (ADS)
Huang, Wenzhang
2016-02-01
In this paper we further extend a recently developed method to investigate the existence of traveling waves solutions and their minimum wave speed for non-monotone reaction-diffusion systems. Our approach consists of two steps. First we develop a geometrical shooting argument, with the aid of the theorem of homotopy invariance on the fundamental group, to obtain the positive semi-traveling wave solutions for a large class of reaction-diffusion systems, including the models of predator-prey interaction (for both predator-independent/dependent functional responses), the models of combustion, Belousov-Zhabotinskii reaction, SI-type of disease transmission, and the model of biological flow reactor in chemostat. Next, we apply the results obtained from the first step to some models, such as the Beddinton-DeAngelis model and the model of biolocal flow reactor, to show the convergence of these semi-traveling wave solutions to an interior equilibrium point by the construction of a Lyapunov-type function, or the convergence of semi-traveling waves to another boundary equilibrium point by the further analysis of the asymptotical behavior of semi-traveling wave solutions.
Reaction-diffusion waves in biology.
Volpert, V; Petrovskii, S
2009-12-01
The theory of reaction-diffusion waves begins in the 1930s with the works in population dynamics, combustion theory and chemical kinetics. At the present time, it is a well developed area of research which includes qualitative properties of travelling waves for the scalar reaction-diffusion equation and for system of equations, complex nonlinear dynamics, numerous applications in physics, chemistry, biology, medicine. This paper reviews biological applications of reaction-diffusion waves. PMID:20416847
NASA Astrophysics Data System (ADS)
Olsen, Thomas; Hou, Yu; Trail, Collin; Wiener, Richard
2004-11-01
The Reaction-Diffusion model (H. Riecke and H.-G. Paap, Europhys. Lett. 14), 1235 (1991).has been applied to a wide variety of pattern forming systems. It correctly predicted a period doubling cascade to chaos in Taylor-Couette flow with hourglass geometry(Richard J. Wiener et al), Phys. Rev. E 55, 5489 (1997).. We have conducted a series of such simulations, varying the length of the system. This has enabled us to study the transition from a purely temporal chaos of the formation of new pairs of Taylor Vortices at a single location, to a spatio-temporal chaos of formation across a range of locations. Application to anticipated experiments will be discussed.
NASA Astrophysics Data System (ADS)
Budroni, M. A.; De Wit, A.
2016-06-01
When two solutions containing separate reactants A and B of an oscillating reaction are put in contact in a gel, localized spatiotemporal patterns can develop around the contact zone thanks to the interplay of reaction and diffusion processes. Using the Brusselator model, we explore analytically the deployment in space and time of the bifurcation diagram of such an A +B → oscillator system. We provide a parametric classification of possible instabilities as a function of the ratio of the initial reactant concentrations and of the reaction intermediate species diffusion coefficients. Related one-dimensional reaction-diffusion dynamics are studied numerically. We find that the system can spatially localize waves and Turing patterns as well as induce more complex dynamics such as zigzag spatiotemporal waves when Hopf and Turing modes interact.
NASA Astrophysics Data System (ADS)
Halmstad, Andrew; Olsen, Thomas; Wiener, Richard
2006-11-01
Previously, we have observed a period-doubling cascade to chaos in Modified Taylor-Couette Flow with Hourglass Geometry. Such behavior had been predicted by The Reaction-Diffusion model simulations. The chaotic formation of Taylor-Vortex pair formation was restricted to a very narrow band about the waist of the hourglass. It was suggested that with increasing lengths of systems, the chaotic region would expand. We present a battery of simulations to determine the variation of the size of the chaotic region with length, seeking the transition to spatio- temporal chaos. Richard J. Wiener et al, Phys. Rev. E 55, 5489 (1997). H. Riecke and H.-G. Paap, Europhys. Lett. 14, 1235 (1991).
Mukhopadhyay, B; Bhattacharyya, R
2006-02-01
The paper is concerned with the effect of variable dispersal rates on Turing instability of a non-Lotka-Volterra reaction-diffusion system. In ecological applications, the dispersal rates of different species tends to oscillate in time. This oscillation is modeled by temporal variation in the diffusion coefficient with large as well as small periodicity. The case of large periodicity is analyzed using the theory of Floquet multipliers and that of the small periodicity by using Hill's equation. The effect of such variation on the resulting Turing space is studied. A comparative analysis of the Turing spaces with constant diffusivity and variable diffusivities is performed. Numerical simulations are carried out to support analytical findings. PMID:16794932
Pennington, Matthew W.; Lubensky, David K.
2011-01-01
We examine a spatially discrete reaction diffusion model based on the interactions that create a periodic pattern in the Drosophila eye imaginal disc. This model is known to be capable of generating a regular hexagonal pattern of gene expression behind a moving front, as observed in the fly system. In order to better understand the novel “switch and template” mechanism behind this pattern formation, we present here a detailed study of the model's behavior in one dimension, using a combination of analytic methods and numerical searches of parameter space. We find that patterns are created robustly provided that there is an appropriate separation of timescales and that self-activation is sufficiently strong, and we derive expressions in this limit for the front speed and the pattern wavelength. Moving fronts in pattern-forming systems near an initial linear instability generically select a unique pattern, but our model operates in a strongly nonlinear regime where the final pattern depends on the initial conditions as well as on parameter values. Our work highlights the important role that cellularization and cell-autonomous feedback can play in biological pattern formation. PMID:20862598
Two- and three-dimensional standing waves in a reaction-diffusion system
NASA Astrophysics Data System (ADS)
Bánsági, Tamás, Jr.; Vanag, Vladimir K.; Epstein, Irving R.
2012-10-01
We observe standing waves of chemical concentration in thin layers [quasi-two-dimensional (2D)] and capillaries [three-dimensional (3D)] containing the aqueous Belousov-Zhabotinsky reaction in a reverse microemulsion stabilized by the ionic surfactant sodium bis-2-ethylhexyl sulfosuccinate (AOT) and with cyclo-octane as the continuous phase. The 3D structures are oscillatory lamellae or square-packed cylinders at high and low volume fractions, respectively, of aqueous droplets. These patterns correspond to oscillatory labyrinthine stripes and square-packed spots in the 2D configuration. Computer simulations, as well as observations in E. coli, give qualitative agreement with the observed patterns and suggest that, in contrast to Turing patterns, the structures are sensitive to the size and shape of the system.
Two- and three-dimensional standing waves in a reaction-diffusion system.
Bánsági, Tamás; Vanag, Vladimir K; Epstein, Irving R
2012-10-01
We observe standing waves of chemical concentration in thin layers [quasi-two-dimensional (2D)] and capillaries [three-dimensional (3D)] containing the aqueous Belousov-Zhabotinsky reaction in a reverse microemulsion stabilized by the ionic surfactant sodium bis-2-ethylhexyl sulfosuccinate (AOT) and with cyclo-octane as the continuous phase. The 3D structures are oscillatory lamellae or square-packed cylinders at high and low volume fractions, respectively, of aqueous droplets. These patterns correspond to oscillatory labyrinthine stripes and square-packed spots in the 2D configuration. Computer simulations, as well as observations in E. coli, give qualitative agreement with the observed patterns and suggest that, in contrast to Turing patterns, the structures are sensitive to the size and shape of the system. PMID:23214640
NASA Astrophysics Data System (ADS)
Vanag, Vladimir K.
2004-09-01
Advances in nonequilibrium pattern formation in reaction-diffusion systems are reviewed. Special emphasis is placed on patterns found in the spatially extended Belousov-Zhabotinsky reaction dispersed in aerosol OT water-in-oil microemulsions (BZ-AOT system): Turing patterns, packet and standing waves, antispirals and segmented spirals, and accelerating waves and oscillons. All experimental results are explained theoretically and reproduced in computer simulations.
NASA Astrophysics Data System (ADS)
Iron, David; Rumsey, John; Ward, Michael J.; Wei, Juncheng
2014-10-01
The linear stability of steady-state periodic patterns of localized spots in for the two-component Gierer-Meinhardt (GM) and Schnakenberg reaction-diffusion models is analyzed in the semi-strong interaction limit corresponding to an asymptotically small diffusion coefficient of the activator concentration. In the limit , localized spots in the activator are centered at the lattice points of a Bravais lattice with constant area . To leading order in , the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies for some independent of the lattice and the Bloch wavevector . From a combination of the method of matched asymptotic expansions, Floquet-Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems, an explicit analytical formula for the continuous band of spectrum that lies within an neighborhood of the origin in the spectral plane is derived when , where is a detuning parameter. The periodic pattern is linearly stable when is chosen small enough so that this continuous band is in the stable left half-plane for all . Moreover, for both the Schnakenberg and GM models, our analysis identifies a model-dependent objective function, involving the regular part of the Bloch Green's function, that must be maximized in order to determine the specific periodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of . From a numerical computation, based on an Ewald-type algorithm, of the regular part of the Bloch Green's function that defines the objective function, it is shown within the class of oblique Bravais lattices that a regular hexagonal lattice arrangement of spots is optimal for maximizing the stability threshold in.
2012-01-01
Background Reaction-diffusion based models have been widely used in the literature for modeling the growth of solid tumors. Many of the current models treat both diffusion/consumption of nutrients and cell proliferation. The majority of these models use classical transport/mass conservation equations for describing the distribution of molecular species in tumor spheroids, and the Fick's law for describing the flux of uncharged molecules (i.e oxygen, glucose). Commonly, the equations for the cell movement and proliferation are first order differential equations describing the rate of change of the velocity of the cells with respect to the spatial coordinates as a function of the nutrient's gradient. Several modifications of these equations have been developed in the last decade to explicitly indicate that the tumor includes cells, interstitial fluids and extracellular matrix: these variants provided a model of tumor as a multiphase material with these as the different phases. Most of the current reaction-diffusion tumor models are deterministic and do not model the diffusion as a local state-dependent process in a non-homogeneous medium at the micro- and meso-scale of the intra- and inter-cellular processes, respectively. Furthermore, a stochastic reaction-diffusion model in which diffusive transport of the molecular species of nutrients and chemotherapy drugs as well as the interactions of the tumor cells with these species is a novel approach. The application of this approach to he scase of non-small cell lung cancer treated with gemcitabine is also novel. Methods We present a stochastic reaction-diffusion model of non-small cell lung cancer growth in the specification formalism of the tool Redi, we recently developed for simulating reaction-diffusion systems. We also describe how a spatial gradient of nutrients and oncological drugs affects the tumor progression. Our model is based on a generalization of the Fick's first diffusion law that allows to model
NASA Technical Reports Server (NTRS)
Hou, Jean W.; Sheen, Jeen S.
1987-01-01
The aim of this study is to find a reliable numerical algorithm to calculate thermal design sensitivities of a transient problem with discontinuous derivatives. The thermal system of interest is a transient heat conduction problem related to the curing process of a composite laminate. A logical function which can smoothly approximate the discontinuity is introduced to modify the system equation. Two commonly used methods, the adjoint variable method and the direct differentiation method, are then applied to find the design derivatives of the modified system. The comparisons of numerical results obtained by these two methods demonstrate that the direct differentiation method is a better choice to be used in calculating thermal design sensitivity.
NASA Astrophysics Data System (ADS)
Johnson, Margaret E.; Hummer, Gerhard
2014-07-01
We present a new algorithm for simulating reaction-diffusion equations at single-particle resolution. Our algorithm is designed to be both accurate and simple to implement, and to be applicable to large and heterogeneous systems, including those arising in systems biology applications. We combine the use of the exact Green's function for a pair of reacting particles with the approximate free-diffusion propagator for position updates to particles. Trajectory reweighting in our free-propagator reweighting (FPR) method recovers the exact association rates for a pair of interacting particles at all times. FPR simulations of many-body systems accurately reproduce the theoretically known dynamic behavior for a variety of different reaction types. FPR does not suffer from the loss of efficiency common to other path-reweighting schemes, first, because corrections apply only in the immediate vicinity of reacting particles and, second, because by construction the average weight factor equals one upon leaving this reaction zone. FPR applications include the modeling of pathways and networks of protein-driven processes where reaction rates can vary widely and thousands of proteins may participate in the formation of large assemblies. With a limited amount of bookkeeping necessary to ensure proper association rates for each reactant pair, FPR can account for changes to reaction rates or diffusion constants as a result of reaction events. Importantly, FPR can also be extended to physical descriptions of protein interactions with long-range forces, as we demonstrate here for Coulombic interactions.
Li, Bing-Wei; Cai, Mei-Chun; Zhang, Hong; Panfilov, Alexander V; Dierckx, Hans
2014-05-14
Chirality is one of the most fundamental properties of many physical, chemical, and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the Belousov-Zhabotinsky reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media. PMID:24832300
NASA Astrophysics Data System (ADS)
Li, Bing-Wei; Cai, Mei-Chun; Zhang, Hong; Panfilov, Alexander V.; Dierckx, Hans
2014-05-01
Chirality is one of the most fundamental properties of many physical, chemical, and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the Belousov-Zhabotinsky reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media.
NASA Astrophysics Data System (ADS)
Zhang, Chunxia; Zhang, Hong; Ouyang, Qi; Hu, Bambi; Gunaratne, Gemunu H.
2003-09-01
The transition from spiral waves to defect-mediated turbulence was studied in a spatial open reactor using Belousov-Zhabotinsky reaction. The experimental results show a new mechanism of the transition from spirals to spatiotemporal chaos, in which the gradient effects in the three-dimensional system are essential. The transition scenario consists of two stages: first, the effects of gradients in the third dimension cause a splitting of the spiral tip and a deletion of certain wave segments, generating new wave sources; second, the waves sent by the new wave sources undergo a backfire instability, and the back waves are laterally unstable. As a result, defects are automatically generated and fill all over the system. The result of numerical simulation using the FitzHugh-Nagumo model essentially agrees with the experimental observation.
Eliaš, Ján; Clairambault, Jean
2014-06-01
Spatio-temporal dynamics of a variety of proteins is, among other things, regulated by post-translational modifications of these proteins. Such modifications can thus influence stability and biochemical activities of the proteins, activity and stability of their upstream targets within specific signalling pathways. Commonly used mathematical tools for such protein-protein (and/or protein-mRNA) interactions in single cells, namely, Michaelis-Menten and Hill kinetics, yielding a system of ordinary differential equations, are extended here into (non-linear) partial differential equations by taking into account a more realistic spatial representation of the environment where these reactions occur. In the modelling framework under consideration, all interactions occur in a cell divided into two compartments, the nucleus and the cytoplasm, connected by the semipermeable nuclear membrane and bounded by the impermeable cell membrane. Passive transport mechanism, modelled by the so-called Kedem-Katchalsky boundary conditions, is used here to represent migration of species throughout the nuclear membrane. Nonlinear systems of partial differential equations are solved by the semi-implicit Rothe method. Examples of two spatial oscillators are shown. Namely, these are the circadian rhythm for concentration of the FRQ protein in Neurospora crassa and oscillatory dynamics observed in the activation and regulation of the p53 protein following DNA damage in mammalian cells. PMID:25210594
NASA Astrophysics Data System (ADS)
Zhang, Henggui; Garratt, Clifford J.; Kharche, Sanjay; Holden, Arun V.
2009-06-01
Human atrial tissue is an excitable system, in which myocytes are excitable elements, and cell-to-cell electrotonic interactions are via diffusive interactions of cell membrane potentials. We developed a family of excitable system models for human atrium at cellular, tissue and anatomical levels for both normal and chronic atrial fibrillation (AF) conditions. The effects of AF-induced remodelling of cell membrane ionic channels (reaction kinetics) and intercellular gap junctional coupling (diffusion) on atrial excitability, conduction of excitation waves and dynamics of re-entrant excitation waves are quantified. Both ionic channel and gap junctional coupling remodelling have rate dependent effects on atrial propagation. Membrane channel conductance remodelling allows the propagation of activity at higher rates than those sustained in normal tissue or in tissue with gap junctional remodelling alone. Membrane channel conductance remodelling is essential for the propagation of activity at rates higher than 300/min as seen in AF. Spatially heterogeneous gap junction coupling remodelling increased the risk of conduction block, an essential factor for the genesis of re-entry. In 2D and 3D anatomical models, the dynamical behaviours of re-entrant excitation waves are also altered by membrane channel modelling. This study provides insights to understand the pro-arrhythmic effects of AF-induced reaction and diffusion remodelling in atrial tissue.
Azimi, Mohammad; Jamali, Yousef; Mofrad, Mohammad R. K.
2011-01-01
Diffusion plays a key role in many biochemical reaction systems seen in nature. Scenarios where diffusion behavior is critical can be seen in the cell and subcellular compartments where molecular crowding limits the interaction between particles. We investigate the application of a computational method for modeling the diffusion of molecules and macromolecules in three-dimensional solutions using agent based modeling. This method allows for realistic modeling of a system of particles with different properties such as size, diffusion coefficients, and affinity as well as the environment properties such as viscosity and geometry. Simulations using these movement probabilities yield behavior that mimics natural diffusion. Using this modeling framework, we simulate the effects of molecular crowding on effective diffusion and have validated the results of our model using Langevin dynamics simulations and note that they are in good agreement with previous experimental data. Furthermore, we investigate an extension of this framework where single discrete cells can contain multiple particles of varying size in an effort to highlight errors that can arise from discretization that lead to the unnatural behavior of particles undergoing diffusion. Subsequently, we explore various algorithms that differ in how they handle the movement of multiple particles per cell and suggest an algorithm that properly accommodates multiple particles of various sizes per cell that can replicate the natural behavior of these particles diffusing. Finally, we use the present modeling framework to investigate the effect of structural geometry on the directionality of diffusion in the cell cytoskeleton with the observation that parallel orientation in the structural geometry of actin filaments of filopodia and the branched structure of lamellipodia can give directionality to diffusion at the filopodia-lamellipodia interface. PMID:21966493
A Lattice Boltzmann Model for Oscillating Reaction-Diffusion
NASA Astrophysics Data System (ADS)
Rodríguez-Romo, Suemi; Ibañez-Orozco, Oscar; Sosa-Herrera, Antonio
2016-07-01
A computational algorithm based on the lattice Boltzmann method (LBM) is proposed to model reaction-diffusion systems. In this paper, we focus on how nonlinear chemical oscillators like Belousov-Zhabotinsky (BZ) and the chlorite-iodide-malonic acid (CIMA) reactions can be modeled by LBM and provide with new insight into the nature and applications of oscillating reactions. We use Gaussian pulse initial concentrations of sulfuric acid in different places of a bidimensional reactor and nondiffusive boundary walls. We clearly show how these systems evolve to a chaotic attractor and produce specific pattern images that are portrayed in the reactions trajectory to the corresponding chaotic attractor and can be used in robotic control.
Physarum machines: encapsulating reaction-diffusion to compute spanning tree
NASA Astrophysics Data System (ADS)
Adamatzky, Andrew
2007-12-01
The Physarum machine is a biological computing device, which employs plasmodium of Physarum polycephalum as an unconventional computing substrate. A reaction-diffusion computer is a chemical computing device that computes by propagating diffusive or excitation wave fronts. Reaction-diffusion computers, despite being computationally universal machines, are unable to construct certain classes of proximity graphs without the assistance of an external computing device. I demonstrate that the problem can be solved if the reaction-diffusion system is enclosed in a membrane with few ‘growth points’, sites guiding the pattern propagation. Experimental approximation of spanning trees by P. polycephalum slime mold demonstrates the feasibility of the approach. Findings provided advance theory of reaction-diffusion computation by enriching it with ideas of slime mold computation.
Laser Spot Detection Based on Reaction Diffusion
Vázquez-Otero, Alejandro; Khikhlukha, Danila; Solano-Altamirano, J. M.; Dormido, Raquel; Duro, Natividad
2016-01-01
Center-location of a laser spot is a problem of interest when the laser is used for processing and performing measurements. Measurement quality depends on correctly determining the location of the laser spot. Hence, improving and proposing algorithms for the correct location of the spots are fundamental issues in laser-based measurements. In this paper we introduce a Reaction Diffusion (RD) system as the main computational framework for robustly finding laser spot centers. The method presented is compared with a conventional approach for locating laser spots, and the experimental results indicate that RD-based computation generates reliable and precise solutions. These results confirm the flexibility of the new computational paradigm based on RD systems for addressing problems that can be reduced to a set of geometric operations. PMID:26938537
Laser Spot Detection Based on Reaction Diffusion.
Vázquez-Otero, Alejandro; Khikhlukha, Danila; Solano-Altamirano, J M; Dormido, Raquel; Duro, Natividad
2016-01-01
Center-location of a laser spot is a problem of interest when the laser is used for processing and performing measurements. Measurement quality depends on correctly determining the location of the laser spot. Hence, improving and proposing algorithms for the correct location of the spots are fundamental issues in laser-based measurements. In this paper we introduce a Reaction Diffusion (RD) system as the main computational framework for robustly finding laser spot centers. The method presented is compared with a conventional approach for locating laser spots, and the experimental results indicate that RD-based computation generates reliable and precise solutions. These results confirm the flexibility of the new computational paradigm based on RD systems for addressing problems that can be reduced to a set of geometric operations. PMID:26938537
Fluorescence Correlation Spectroscopy and Nonlinear Stochastic Reaction-Diffusion
Del Razo, Mauricio; Pan, Wenxiao; Qian, Hong; Lin, Guang
2014-05-30
The currently existing theory of fluorescence correlation spectroscopy (FCS) is based on the linear fluctuation theory originally developed by Einstein, Onsager, Lax, and others as a phenomenological approach to equilibrium fluctuations in bulk solutions. For mesoscopic reaction-diffusion systems with nonlinear chemical reactions among a small number of molecules, a situation often encountered in single-cell biochemistry, it is expected that FCS time correlation functions of a reaction-diffusion system can deviate from the classic results of Elson and Magde [Biopolymers (1974) 13:1-27]. We first discuss this nonlinear effect for reaction systems without diffusion. For nonlinear stochastic reaction-diffusion systems there are no closed solutions; therefore, stochastic Monte-Carlo simulations are carried out. We show that the deviation is small for a simple bimolecular reaction; the most significant deviations occur when the number of molecules is small and of the same order. Extending Delbrück-Gillespie’s theory for stochastic nonlinear reactions with rapidly stirring to reaction-diffusion systems provides a mesoscopic model for chemical and biochemical reactions at nanometric and mesoscopic level such as a single biological cell.
NASA Astrophysics Data System (ADS)
Konkoli, Zoran
2004-01-01
Theoretical methods for dealing with diffusion-controlled reactions inevitably rely on some kind of approximation, and to find the one that works on a particular problem is not always easy. Here the approximation used by Bogolyubov to study a weakly nonideal Bose gas, referred to as the weakly nonideal Bose gas approximation (WBGA), is applied in the analysis of three reaction-diffusion models: (i) A+A→Ø, (ii) A+B→Ø, and (iii) A+A,B+B,A+B→Ø (the ABBA model). Two types of WBGA are considered, the simpler WBGA-I and the more complicated WBGA-II. All models are defined on the lattice to facilitate comparison with computer experiment (simulation). It is found that the WBGA describes the A+B reaction well, it reproduces the correct d/4 density decay exponent. However, it fails in the case of the A+A reaction and the ABBA model. (To cure the deficiency of WBGA in dealing with the A+A model, a hybrid of the WBGA and Kirkwood superposition approximations is suggested.) It is shown that the WBGA-I is identical to the dressed-tree calculation suggested by Lee [J. Phys. A 27, 2633 (1994)], and that the dressed-tree calculation does not lead to the d/2 density decay exponent when applied to the A+A reaction, as normally believed, but it predicts the d/4 decay exponent. Last, the usage of the small n0 approximation suggested by Mattis and Glasser [Rev. Mod. Phys. 70, 979 (1998)] is questioned if used beyond the A+B reaction-diffusion model.
Reaction-Diffusion Processes in Ultrathin Films of Photoresist
NASA Astrophysics Data System (ADS)
Perera, Ginusha; Stein, Gila
2011-03-01
Projection lithography is the primary technology used for patterning semiconductor devices. High-throughput manufacturing requires imaging materials (resists) that are highly sensitive to radiation, and this demand is satisfied through a process termed chemical amplification (CA). CA resists are comprised of a polymer resin (reactant) and photoacid generator (catalyst); a coupled reaction-diffusion mechanism drives image formation, where image resolution is limited by slow diffusion of the acid catalyst. There is evidence that thin film reaction rates deviate from the bulk behavior, and current models for image formation do not capture such effects. We demonstrate that X-Ray Diffraction can measure spatial extent-of-reaction in ultrathin films of a nanopatterned poly(4-hydroxystyrene-co-tertbutylacrylate) CA resist. The feedback acquired is used to construct predictive models for the coupled reaction-diffusion processes that incorporate the physics of confined polymers. Funded by NSF ECCS 0927147.
Transient spatiotemporal chaos in reaction-diffusion networks
NASA Astrophysics Data System (ADS)
Wackerbauer, Renate
2010-03-01
Complex transient dynamics is reported in various extended systems, including transient turbulence in shear flows, transient spatiotemporal chaos in reaction- diffusion models, and non-chaotic irregular transient dynamics in neural networks. The asymptotic stability is difficult to determine since the transient lifetime typically increases exponentially with the system size. Our studies show that transient spatiotemporal chaos is extensive in various reaction- diffusion systems; the Lyapunov dimension increases linearly with the network size. A master stability analysis provides insight into the asymptotic stability in the Baer- Eiswirth and the Gray-Scott systems. The asymptotic state is characterized by negative transverse Lyapunov exponents on the attractor of the invariant synchronization manifold. The average lifetime depends on the number of transverse directions that are unstable along a typical excitation cycle.
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
NASA Astrophysics Data System (ADS)
Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin
2013-02-01
In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized ( L 1) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.
Cohabitation reaction-diffusion model for virus focal infections
NASA Astrophysics Data System (ADS)
Amor, Daniel R.; Fort, Joaquim
2014-12-01
The propagation of virus infection fronts has been typically modeled using a set of classical (noncohabitation) reaction-diffusion equations for interacting species. However, for some single-species systems it has been recently shown that noncohabitation reaction-diffusion equations may lead to unrealistic descriptions. We argue that previous virus infection models also have this limitation, because they assume that a virion can simultaneously reproduce inside a cell and diffuse away from it. For this reason, we build a several-species cohabitation model that does not have this limitation. Furthermore, we perform a sensitivity analysis for the most relevant parameters of the model, and we compare the predicted infection speed with observed data for two different strains of the T7 virus.
Reaction diffusion equation with spatio-temporal delay
NASA Astrophysics Data System (ADS)
Zhao, Zhihong; Rong, Erhua
2014-07-01
We investigate reaction-diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction-diffusion equation with spatio-temporal delay. Applying this theory to Lotka-Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem's steady-state solution.
Reaction-diffusion processes at the nano- and microscales
NASA Astrophysics Data System (ADS)
Epstein, Irving R.; Xu, Bing
2016-04-01
The bottom-up fabrication of nano- and microscale structures from primary building blocks (molecules, colloidal particles) has made remarkable progress over the past two decades, but most research has focused on structural aspects, leaving our understanding of the dynamic and spatiotemporal aspects at a relatively primitive stage. In this Review, we draw inspiration from living cells to argue that it is now time to move beyond the generation of structures and explore dynamic processes at the nanoscale. We first introduce nanoscale self-assembly, self-organization and reaction-diffusion processes as essential features of cells. Then, we highlight recent progress towards designing and controlling these fundamental features of life in abiological systems. Specifically, we discuss examples of reaction-diffusion processes that lead to such outcomes as self-assembly, self-organization, unique nanostructures, chemical waves and dynamic order to illustrate their ubiquity within a unifying context of dynamic oscillations and energy dissipation. Finally, we suggest future directions for research on reaction-diffusion processes at the nano- and microscales that we find hold particular promise for a new understanding of science at the nanoscale and the development of new kinds of nanotechnologies for chemical transport, chemical communication and integration with living systems.
Programming reaction-diffusion: From theory to micro- and nanofabrication
NASA Astrophysics Data System (ADS)
Campbell, Christopher James
Nature often uses reaction-diffusion(RD) as a means of making structures and materials of unique properties or morphologies on scales from macro- (e.g., stripes in zebras, tigers, and seashells, and formations in trees, agates, and rocks) to microscopic (e.g., cellular growth, chemotaxis and biological waves). However, reaction-diffusion phenomena have not yet been applied in modern materials science and micro-/nanotechnology. In this context, RD systems are particularly promising for micropatterning of surfaces. Unlike conventional micropatterning techniques that modify the properties of the substrate only at the locations to which a modifying agent - be it a chemical or radiation - is delivered, RD can, in principle, evolve chemicals delivered onto a surface into structures of characteristic dimensions significantly smaller than those of the original pattern. In this Dissertation, I describe how reaction-diffusions are programmed and executed via a new micropatterning technique called Wet Stamping to (i) transform microscopic patterns of chemicals delivered onto thin films of dry gelatin into regular arrays of lines of submicrometer thicknesses, multicolor arrays on the micrometer scale, or three-dimensional microstructured surfaces; (ii) modify the properties of a surface by precisely delivering an oxidant to change hydrophilicity or deliver silanes or thiols to build a self-assembling monolayer; or (iii) cut into a metal, glass, or crystal surface by delivery of an etchant to form binary and curvilinear three-dimensional microstructures. This technique has allowed for a fundamental understanding and control of reaction-diffusion processes down to the nanoscale. In addition, this platform has allowed for the development of a range of applications on the micro- and nanoscale, including microlenses, microfluidic devices, and templates for studying cell motility and cancer metastasis.
Reaction rates for a generalized reaction-diffusion master equation
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules. PMID:26871190
Reaction rates for a generalized reaction-diffusion master equation
NASA Astrophysics Data System (ADS)
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
Synthesis of Programmable Reaction-Diffusion Fronts Using DNA Catalyzers
NASA Astrophysics Data System (ADS)
Zadorin, Anton S.; Rondelez, Yannick; Galas, Jean-Christophe; Estevez-Torres, André
2015-02-01
We introduce a DNA-based reaction-diffusion (RD) system in which reaction and diffusion terms can be precisely and independently controlled. The effective diffusion coefficient of an individual reaction component, as we demonstrate on a traveling wave, can be reduced up to 2.7-fold using a self-assembled hydrodynamic drag. The intrinsic programmability of this RD system allows us to engineer, for the first time, orthogonal autocatalysts that counterpropagate with minimal interaction. Our results are in excellent quantitative agreement with predictions of the Fisher-Kolmogorov-Petrovskii-Piscunov model. These advances open the way for the rational engineering of pattern formation in pure chemical RD systems.
Reaction-diffusion in the NEURON simulator
McDougal, Robert A.; Hines, Michael L.; Lytton, William W.
2013-01-01
In order to support research on the role of cell biological principles (genomics, proteomics, signaling cascades and reaction dynamics) on the dynamics of neuronal response in health and disease, NEURON's Reaction-Diffusion (rxd) module in Python provides specification and simulation for these dynamics, coupled with the electrophysiological dynamics of the cell membrane. Arithmetic operations on species and parameters are overloaded, allowing arbitrary reaction formulas to be specified using Python syntax. These expressions are then transparently compiled into bytecode that uses NumPy for fast vectorized calculations. At each time step, rxd combines NEURON's integrators with SciPy's sparse linear algebra library. PMID:24298253
Reaction-diffusion in the NEURON simulator.
McDougal, Robert A; Hines, Michael L; Lytton, William W
2013-01-01
In order to support research on the role of cell biological principles (genomics, proteomics, signaling cascades and reaction dynamics) on the dynamics of neuronal response in health and disease, NEURON's Reaction-Diffusion (rxd) module in Python provides specification and simulation for these dynamics, coupled with the electrophysiological dynamics of the cell membrane. Arithmetic operations on species and parameters are overloaded, allowing arbitrary reaction formulas to be specified using Python syntax. These expressions are then transparently compiled into bytecode that uses NumPy for fast vectorized calculations. At each time step, rxd combines NEURON's integrators with SciPy's sparse linear algebra library. PMID:24298253
Exact solutions for logistic reaction-diffusion equations in biology
NASA Astrophysics Data System (ADS)
Broadbridge, P.; Bradshaw-Hajek, B. H.
2016-08-01
Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.
Reactive radical facilitated reaction-diffusion modeling for holographic photopolymerization
Liu Jianhua; Pu Haihui; Gao Bin; Gao Hongyue; Yin Dejin; Dai Haitao
2010-02-08
A phenomenological concentration of reactive radical is proposed to take the role of curing light intensity in explicit proportion to the reaction rate for the conventional reaction-diffusion model. This revision rationally eliminates the theoretical defect of null reaction rate in modeling of the postcuring process, and facilitates the applicability of the model in the whole process of holographic photopolymerizations in photocurable monomer and nematic liquid crystal blend system. Excellent consistencies are obtained in both curing and postcuring processes between simulated and experimentally measured evolutions of the first order diffraction efficiency of the formed composite Bragg gratings.
Bosonic reaction-diffusion processes on scale-free networks
NASA Astrophysics Data System (ADS)
Baronchelli, Andrea; Catanzaro, Michele; Pastor-Satorras, Romualdo
2008-07-01
Reaction-diffusion processes can be adopted to model a large number of dynamics on complex networks, such as transport processes or epidemic outbreaks. In most cases, however, they have been studied from a fermionic perspective, in which each vertex can be occupied by at most one particle. While still useful, this approach suffers from some drawbacks, the most important probably being the difficulty to implement reactions involving more than two particles simultaneously. Here we develop a general framework for the study of bosonic reaction-diffusion processes on complex networks, in which there is no restriction on the number of interacting particles that a vertex can host. We describe these processes theoretically by means of continuous-time heterogeneous mean-field theory and divide them into two main classes: steady-state and monotonously decaying processes. We analyze specific examples of both behaviors within the class of one-species processes, comparing the results (whenever possible) with the corresponding fermionic counterparts. We find that the time evolution and critical properties of the particle density are independent of the fermionic or bosonic nature of the process, while differences exist in the functional form of the density of occupied vertices in a given degree class k . We implement a continuous-time Monte Carlo algorithm, well suited for general bosonic simulations, which allows us to confirm the analytical predictions formulated within mean-field theory. Our results, at both the theoretical and numerical levels, can be easily generalized to tackle more complex, multispecies, reaction-diffusion processes and open a promising path for a general study and classification of this kind of dynamical systems on complex networks.
Chemical computing with reaction-diffusion processes.
Gorecki, J; Gizynski, K; Guzowski, J; Gorecka, J N; Garstecki, P; Gruenert, G; Dittrich, P
2015-07-28
Chemical reactions are responsible for information processing in living organisms. It is believed that the basic features of biological computing activity are reflected by a reaction-diffusion medium. We illustrate the ideas of chemical information processing considering the Belousov-Zhabotinsky (BZ) reaction and its photosensitive variant. The computational universality of information processing is demonstrated. For different methods of information coding constructions of the simplest signal processing devices are described. The function performed by a particular device is determined by the geometrical structure of oscillatory (or of excitable) and non-excitable regions of the medium. In a living organism, the brain is created as a self-grown structure of interacting nonlinear elements and reaches its functionality as the result of learning. We discuss whether such a strategy can be adopted for generation of chemical information processing devices. Recent studies have shown that lipid-covered droplets containing solution of reagents of BZ reaction can be transported by a flowing oil. Therefore, structures of droplets can be spontaneously formed at specific non-equilibrium conditions, for example forced by flows in a microfluidic reactor. We describe how to introduce information to a droplet structure, track the information flow inside it and optimize medium evolution to achieve the maximum reliability. Applications of droplet structures for classification tasks are discussed. PMID:26078345
Cox process representation and inference for stochastic reaction-diffusion processes
NASA Astrophysics Data System (ADS)
Schnoerr, David; Grima, Ramon; Sanguinetti, Guido
2016-05-01
Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction-diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data. Here we use ideas from statistical physics and machine learning to provide a solution to the inverse problem of learning a stochastic reaction-diffusion process from data. Our solution relies on a non-trivial connection between stochastic reaction-diffusion processes and spatio-temporal Cox processes, a well-studied class of models from computational statistics. This connection leads to an efficient and flexible algorithm for parameter inference and model selection. Our approach shows excellent accuracy on numeric and real data examples from systems biology and epidemiology. Our work provides both insights into spatio-temporal stochastic systems, and a practical solution to a long-standing problem in computational modelling.
Guiding brine shrimp through mazes by solving reaction diffusion equations
NASA Astrophysics Data System (ADS)
Singal, Krishma; Fenton, Flavio
Excitable systems driven by reaction diffusion equations have been shown to not only find solutions to mazes but to also to find the shortest path between the beginning and the end of the maze. In this talk we describe how we can use the Fitzhugh-Nagumo model, a generic model for excitable media, to solve a maze by varying the basin of attraction of its two fixed points. We demonstrate how two dimensional mazes are solved numerically using a Java Applet and then accelerated to run in real time by using graphic processors (GPUs). An application of this work is shown by guiding phototactic brine shrimp through a maze solved by the algorithm. Once the path is obtained, an Arduino directs the shrimp through the maze using lights from LEDs placed at the floor of the Maze. This method running in real time could be eventually used for guiding robots and cars through traffic.
Field theory of propagating reaction-diffusion fronts
Escudero, C.
2004-10-01
The problem of velocity selection of reaction-diffusion fronts has been widely investigated. While the mean-field limit results are well known theoretically, there is a lack of analytic progress in those cases in which fluctuations are to be taken into account. Here, we construct an analytic theory connecting the first principles of the reaction-diffusion process to an effective equation of motion via field-theoretic arguments, and we arrive at results already confirmed by numerical simulations.
Generation of diversity in a reaction-diffusion-based controller.
Zahadat, Payam; Schmickl, Thomas
2014-01-01
A controller of biological or artificial organism (e.g., in bio-inspired cellular robots) consists of a number of processes that drive its dynamics. For a system of processes to perform as a successful controller, different properties can be mentioned. One of the desirable properties of such a system is the capability of generating sufficiently diverse patterns of outputs and behaviors. A system with such a capability is potentially adaptable to perform complicated tasks with proper parameterizations and may successfully reach the solution space of behaviors from the point of view of search and evolutionary algorithms. This article aims to take an early step toward exploring this capability at the levels of individuals and populations by introducing measures of diversity generation and by evaluating the influence of different types of processes on diversity generation. A reaction-diffusion-based controller called the artificial homeostatic hormone system (AHHS) is studied as a system consisting of different processes with various domains of functioning (e.g., internal or external to the control unit). Various combinations of these processes are investigated in terms of diversity generation at levels of both individuals and populations, and the effects of the processes are discussed representing different influences for the processes. A case study of evolving a multimodular AHHS controller with all the various process combinations is also investigated, representing the relevance of the diversity generation measures and practical scenarios. PMID:24730765
Reaction diffusion Voronoi diagrams: from sensors data to computing.
Vázquez-Otero, Alejandro; Faigl, Jan; Dormido, Raquel; Duro, Natividad
2015-01-01
In this paper, a new method to solve computational problems using reaction diffusion (RD) systems is presented. The novelty relies on the use of a model configuration that tailors its spatiotemporal dynamics to develop Voronoi diagrams (VD) as a part of the system's natural evolution. The proposed framework is deployed in a solution of related robotic problems, where the generalized VD are used to identify topological places in a grid map of the environment that is created from sensor measurements. The ability of the RD-based computation to integrate external information, like a grid map representing the environment in the model computational grid, permits a direct integration of sensor data into the model dynamics. The experimental results indicate that this method exhibits significantly less sensitivity to noisy data than the standard algorithms for determining VD in a grid. In addition, previous drawbacks of the computational algorithms based on RD models, like the generation of volatile solutions by means of excitable waves, are now overcome by final stable states. PMID:26035349
Adaptive mesh refinement for stochastic reaction-diffusion processes
Bayati, Basil; Chatelain, Philippe; Koumoutsakos, Petros
2011-01-01
We present an algorithm for adaptive mesh refinement applied to mesoscopic stochastic simulations of spatially evolving reaction-diffusion processes. The transition rates for the diffusion process are derived on adaptive, locally refined structured meshes. Convergence of the diffusion process is presented and the fluctuations of the stochastic process are verified. Furthermore, a refinement criterion is proposed for the evolution of the adaptive mesh. The method is validated in simulations of reaction-diffusion processes as described by the Fisher-Kolmogorov and Gray-Scott equations.
Modeling mammary gland morphogenesis as a reaction-diffusion process.
Grant, Mark R; Hunt, C Anthony; Xia, Lan; Fata, Jimmie E; Bissell, Mina J
2004-01-01
Mammary ducts are formed through a process of branching morphogenesis. We present results of experiments using a simulation model of this process, and discuss their implications for understanding mammary duct extension and bifurcation. The model is a cellular automaton approximation of a reaction-diffusion process in which matrix metalloproteinases represent the activator, inhibitors of matrix metalloproteinases represent the inhibitor, and growth factors serve as a substrate. We compare results from the simulation model with those from in-vivo experiments as part of an assessment of whether duct extension and bifurcation during morphogenesis may be a consequence of a reaction-diffusion mechanism mediated by MMPs and TIMPs. PMID:17271768
Transient spatiotemporal chaos is extensive in three reaction-diffusion networks
NASA Astrophysics Data System (ADS)
Stahlke, Dan; Wackerbauer, Renate
2009-11-01
Extensive (asymptotic) spatiotemporal chaos is comprised of statistically similar subsystems that interact only weakly. A systematic study of transient spatiotemporal chaos reveals extensive system behavior in all three reaction-diffusion networks for various boundary conditions. The Lyapunov dimension, the sum of positive Lyapunov exponents, and the logarithm of the transient lifetime grow linearly with the system size. The unstable manifold of the chaotic saddle has nearly the same dimension as the saddle itself, and the stable manifold is nearly space filling.
Analysis of some identification problems for the reaction-diffusion-convection equation
NASA Astrophysics Data System (ADS)
Alekseev, G. V.; Mashkov, D. V.; Yashenko, E. N.
2016-04-01
Identification problems for a linear stationary reaction-diffusion-convection model, considered in the bounded domain under Dirichlet boundary condition, are studied. Using an optimization method these problems are reduced to respective control problems. The reaction coefficient and the volume density of substance source play the role of controls in this control problem. The solvability of the direct and control problems is proved, the optimality system, which describes the necessary optimality conditions, is derived and the numerical algorithm is developed.
Cox process representation and inference for stochastic reaction-diffusion processes.
Schnoerr, David; Grima, Ramon; Sanguinetti, Guido
2016-01-01
Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction-diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data. Here we use ideas from statistical physics and machine learning to provide a solution to the inverse problem of learning a stochastic reaction-diffusion process from data. Our solution relies on a non-trivial connection between stochastic reaction-diffusion processes and spatio-temporal Cox processes, a well-studied class of models from computational statistics. This connection leads to an efficient and flexible algorithm for parameter inference and model selection. Our approach shows excellent accuracy on numeric and real data examples from systems biology and epidemiology. Our work provides both insights into spatio-temporal stochastic systems, and a practical solution to a long-standing problem in computational modelling. PMID:27222432
Kidney Tumor Growth Prediction by Coupling Reaction-Diffusion and Biomechanical Model
Chen, Xinjian; Summers, Ronald M.; Yao, Jianhua
2014-01-01
It is desirable to predict the tumor growth rate so that appropriate treatment can be planned in the early stage. Previously, we proposed a finite element method (FEM)-based 3D kidney tumor growth prediction system using longitudinal images. A reaction-diffusion model was applied as the tumor growth model. In this paper, we not only improve the tumor growth model by coupling the reaction-diffusion model with a biomechanical model, but also take the surrounding tissues into account. Different diffusion and biomechanical properties are applied for different tissue types. FEM is employed to simulate the coupled tumor growth model. Model parameters are estimated by optimizing an objective function of overlap accuracy using a hybrid optimization parallel search package (HOPSPACK). The proposed method was tested with kidney CT images of eight tumors from five patients with seven time points. The experimental results showed the performance of the proposed method improved greatly compared to our previous work. PMID:23047857
Untangling Knots Via Reaction-Diffusion Dynamics of Vortex Strings.
Maucher, Fabian; Sutcliffe, Paul
2016-04-29
We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolution preserves the topology of the knot and can untangle an unknot into a circle. Illustrative test case examples are presented, including the untangling of a hard unknot known as the culprit. Our approach to the unknotting problem has two novel features, in that it applies field theory rather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization. PMID:27176541
Untangling Knots Via Reaction-Diffusion Dynamics of Vortex Strings
NASA Astrophysics Data System (ADS)
Maucher, Fabian; Sutcliffe, Paul
2016-04-01
We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolution preserves the topology of the knot and can untangle an unknot into a circle. Illustrative test case examples are presented, including the untangling of a hard unknot known as the culprit. Our approach to the unknotting problem has two novel features, in that it applies field theory rather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization.
Field theory for a reaction-diffusion model of quasispecies dynamics.
Pastor-Satorras, R; Solé, R V
2001-11-01
RNA viruses are known to replicate with extremely high mutation rates. These rates are actually close to the so-called error threshold. This threshold is in fact a critical point beyond which genetic information is lost through a second-order phase transition, which has been dubbed as the "error catastrophe." Here we explore this phenomenon using a field theory approximation to the spatially extended Swetina-Schuster quasispecies model [J. Swetina and P. Schuster, Biophys. Chem. 16, 329 (1982)], a single-sharp-peak landscape. In analogy with standard absorbing-state phase transitions, we develop a reaction-diffusion model whose discrete rules mimic the Swetina-Schuster model. The field theory representation of the reaction-diffusion system is constructed. The proposed field theory belongs to the same universality class as a conserved reaction-diffusion model previously proposed [F. van Wijland et al., Physica A 251, 179 (1998)]. From the field theory, we obtain the full set of exponents that characterize the critical behavior at the error threshold. Our results present the error catastrophe from a different point of view and suggest that spatial degrees of freedom can modify several mean-field predictions previously considered, leading to the definition of characteristic exponents that could be experimentally measurable. PMID:11735970
Anomalous Impact in Reaction-Diffusion Financial Models
NASA Astrophysics Data System (ADS)
Mastromatteo, I.; Tóth, B.; Bouchaud, J.-P.
2014-12-01
We generalize the reaction-diffusion model A +B → /0 in order to study the impact of an excess of A (or B ) at the reaction front. We provide an exact solution of the model, which shows that the linear response breaks down: the average displacement of the reaction front grows as the square root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.
Reaction-diffusion modelling of bacterial colony patterns
NASA Astrophysics Data System (ADS)
Mimura, Masayasu; Sakaguchi, Hideo; Matsushita, Mitsugu
2000-07-01
It is well known from experiments that bacterial species Bacillus subtilis exhibit various colony patterns. These are essentially classified into five types in the morphological diagram, depending on the substrate softness and nutrient concentration. (A) diffusion-limited aggregation-like; (B) Eden-like; (C) concentric ring-like; (D) disk-like; and (E) dense branching morphology-like. There arises the naive question of whether the diversity of colony patterns observed in experiments is caused by different effects or governed by the same underlying principles. Our research has led us to propose reaction-diffusion models to describe the morphological diversity of colony patterns except for Eden-like ones.
A reaction-diffusion model of CO2 influx into an oocyte.
Somersalo, Erkki; Occhipinti, Rossana; Boron, Walter F; Calvetti, Daniela
2012-09-21
We have developed and implemented a novel mathematical model for simulating transients in surface pH (pH(S)) and intracellular pH (pH(i)) caused by the influx of carbon dioxide (CO(2)) into a Xenopus oocyte. These transients are important tools for studying gas channels. We assume that the oocyte is a sphere surrounded by a thin layer of unstirred fluid, the extracellular unconvected fluid (EUF), which is in turn surrounded by the well-stirred bulk extracellular fluid (BECF) that represents an infinite reservoir for all solutes. Here, we assume that the oocyte plasma membrane is permeable only to CO(2). In both the EUF and intracellular space, solute concentrations can change because of diffusion and reactions. The reactions are the slow equilibration of the CO(2) hydration-dehydration reactions and competing equilibria among carbonic acid (H(2)CO(3))/bicarbonate (HCO(3)(-)) and a multitude of non-CO(2)/HCO(3)(-) buffers. Mathematically, the model is described by a coupled system of reaction-diffusion equations that-assuming spherical radial symmetry-we solved using the method of lines with appropriate stiff solvers. In agreement with experimental data [Musa-Aziz et al. 2009, PNAS 106 5406-5411], the model predicts that exposing the cell to extracellular 1.5% CO(2)/10 mM HCO(3)(-) (pH 7.50) causes pH(i) to fall and pH(S) to rise rapidly to a peak and then decay. Moreover, the model provides insights into the competition between diffusion and reaction processes when we change the width of the EUF, membrane permeability to CO(2), native extra- and intracellular carbonic anhydrase-like activities, the non-CO(2)/HCO(3)(-) (intrinsic) intracellular buffering power, or mobility of intrinsic intracellular buffers. PMID:22728674
NASA Astrophysics Data System (ADS)
Dulos, E.; Hunding, A.; Boissonade, J.; de Kepper, P.
Since the seminal paper "The chemical basis of morphogenesis" by Alan Turing, the temporal and spatial self-organization phenomena produced in chemically reacting and diffusing systems are often thought as paradigms for biological development. The basic theoretical principles on which the development of stationary concentration patterns (Turing structures) rely on are briefly presented. We review different aspects of our contribution to the experimental observation of reaction-diffusion patterns in iodine-oxychlorine systems. The experimental techniques are emphasized. Phase diagrams gathering different standing and travelling patterns are presented, analyzed and modeled. A special attention is also given to some peculiar pattern growth dynamics (spot division, finger splitting).
Transient spatiotemporal chaos is extensive in three reaction-diffusion networks.
Stahlke, Dan; Wackerbauer, Renate
2009-11-01
Extensive (asymptotic) spatiotemporal chaos is comprised of statistically similar subsystems that interact only weakly. A systematic study of transient spatiotemporal chaos reveals extensive system behavior in all three reaction-diffusion networks for various boundary conditions. The Lyapunov dimension, the sum of positive Lyapunov exponents, and the logarithm of the transient lifetime grow linearly with the system size. The unstable manifold of the chaotic saddle has nearly the same dimension as the saddle itself, and the stable manifold is nearly space filling. PMID:20365064
Small-scale properties of a stochastic cubic-autocatalytic reaction-diffusion model
NASA Astrophysics Data System (ADS)
Gagnon, Jean-Sébastien; Hochberg, David; Pérez-Mercader, Juan
2015-10-01
We investigate the small-scale properties of a stochastic cubic-autocatalytic reaction-diffusion (CARD) model using renormalization techniques. We renormalize noise-induced ultraviolet divergences and obtain β functions for the decay rate and coupling at one loop. Assuming colored (power-law) noise, our results show that the behavior of both decay rate and coupling with scale depends crucially on the noise exponent. Interpreting the CARD model as a proxy for a (very simple) living system, our results suggest that power-law correlations in environmental fluctuations can both decrease or increase the growth of structures at smaller scales.
Breakdown of the reaction-diffusion master equation with nonelementary rates
NASA Astrophysics Data System (ADS)
Smith, Stephen; Grima, Ramon
2016-05-01
The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class, then the RDME converges to the CME of the same system, whereas if the RDME has propensities not in this class, then convergence is not guaranteed. These are revealed to be elementary and nonelementary propensities, respectively. We also show that independent of the type of propensity, the RDME converges to the CME in the simultaneous limit of fast diffusion and large volumes. We illustrate our results with some simple example systems and argue that the RDME cannot generally be an accurate description of systems with nonelementary rates.
Turing instability in reaction-diffusion models on complex networks
NASA Astrophysics Data System (ADS)
Ide, Yusuke; Izuhara, Hirofumi; Machida, Takuya
2016-09-01
In this paper, the Turing instability in reaction-diffusion models defined on complex networks is studied. Here, we focus on three types of models which generate complex networks, i.e. the Erdős-Rényi, the Watts-Strogatz, and the threshold network models. From analysis of the Laplacian matrices of graphs generated by these models, we numerically reveal that stable and unstable regions of a homogeneous steady state on the parameter space of two diffusion coefficients completely differ, depending on the network architecture. In addition, we theoretically discuss the stable and unstable regions in the cases of regular enhanced ring lattices which include regular circles, and networks generated by the threshold network model when the number of vertices is large enough.
Ford Versypt, Ashlee N.; Arendt, Paul D.; Pack, Daniel W.; Braatz, Richard D.
2015-01-01
A mathematical reaction-diffusion model is defined to describe the gradual decomposition of polymer microspheres composed of poly(D,L-lactic-co-glycolic acid) (PLGA) that are used for pharmaceutical drug delivery over extended periods of time. The partial differential equation (PDE) model treats simultaneous first-order generation due to chemical reaction and diffusion of reaction products in spherical geometry to capture the microsphere-size-dependent effects of autocatalysis on PLGA erosion that occurs when the microspheres are exposed to aqueous media such as biological fluids. The model is solved analytically for the concentration of the autocatalytic carboxylic acid end groups of the polymer chains that comprise the microspheres as a function of radial position and time. The analytical solution for the reaction and transport of the autocatalytic chemical species is useful for predicting the conditions under which drug release from PLGA microspheres transitions from diffusion-controlled to erosion-controlled release, for understanding the dynamic coupling between the PLGA degradation and erosion mechanisms, and for designing drug release particles. The model is the first to provide an analytical prediction for the dynamics and spatial heterogeneities of PLGA degradation and erosion within a spherical particle. The analytical solution is applicable to other spherical systems with simultaneous diffusive transport and first-order generation by reaction. PMID:26284787
Ford Versypt, Ashlee N; Arendt, Paul D; Pack, Daniel W; Braatz, Richard D
2015-01-01
A mathematical reaction-diffusion model is defined to describe the gradual decomposition of polymer microspheres composed of poly(D,L-lactic-co-glycolic acid) (PLGA) that are used for pharmaceutical drug delivery over extended periods of time. The partial differential equation (PDE) model treats simultaneous first-order generation due to chemical reaction and diffusion of reaction products in spherical geometry to capture the microsphere-size-dependent effects of autocatalysis on PLGA erosion that occurs when the microspheres are exposed to aqueous media such as biological fluids. The model is solved analytically for the concentration of the autocatalytic carboxylic acid end groups of the polymer chains that comprise the microspheres as a function of radial position and time. The analytical solution for the reaction and transport of the autocatalytic chemical species is useful for predicting the conditions under which drug release from PLGA microspheres transitions from diffusion-controlled to erosion-controlled release, for understanding the dynamic coupling between the PLGA degradation and erosion mechanisms, and for designing drug release particles. The model is the first to provide an analytical prediction for the dynamics and spatial heterogeneities of PLGA degradation and erosion within a spherical particle. The analytical solution is applicable to other spherical systems with simultaneous diffusive transport and first-order generation by reaction. PMID:26284787
A minimally-resolved immersed boundary model for reaction-diffusion problems.
Bhalla, Amneet Pal Singh; Griffith, Boyce E; Patankar, Neelesh A; Donev, Aleksandar
2013-12-01
We develop an immersed boundary approach to modeling reaction-diffusion processes in dispersions of reactive spherical particles, from the diffusion-limited to the reaction-limited setting. We represent each reactive particle with a minimally-resolved "blob" using many fewer degrees of freedom per particle than standard discretization approaches. More complicated or more highly resolved particle shapes can be built out of a collection of reactive blobs. We demonstrate numerically that the blob model can provide an accurate representation at low to moderate packing densities of the reactive particles, at a cost not much larger than solving a Poisson equation in the same domain. Unlike multipole expansion methods, our method does not require analytically computed Green's functions, but rather, computes regularized discrete Green's functions on the fly by using a standard grid-based discretization of the Poisson equation. This allows for great flexibility in implementing different boundary conditions, coupling to fluid flow or thermal transport, and the inclusion of other effects such as temporal evolution and even nonlinearities. We develop multigrid-based preconditioners for solving the linear systems that arise when using implicit temporal discretizations or studying steady states. In the diffusion-limited case the resulting linear system is a saddle-point problem, the efficient solution of which remains a challenge for suspensions of many particles. We validate our method by comparing to published results on reaction-diffusion in ordered and disordered suspensions of reactive spheres. PMID:24320369
A Local, Self-Organizing Reaction-Diffusion Model Can Explain Somite Patterning in Embryos.
Cotterell, James; Robert-Moreno, Alexandre; Sharpe, James
2015-10-28
During somitogenesis in embryos, a posteriorly moving differentiation front arrests the oscillations of "segmentation clock" genes, leaving behind a frozen, periodic pattern of expression stripes. Both mathematical theories and experimental observations have invoked a "clock and wavefront" model to explain this phenomenon, in which long-range molecular gradients control the movement of the front and therefore the placement of the stripes in the embryo. Here, we develop a fundamentally different model-a progressive oscillatory reaction-diffusion (PORD) system driven by short-range interactions. In this model, posterior movement of the front is a local, emergent phenomenon that, in contrast to the clock and wavefront model, is not controlled by global positional information. The PORD model explains important features of somitogenesis, such as size regulation, that previous reaction-diffusion models could not explain. Moreover, the PORD and clock and wavefront models make different predictions about the results of FGF-inhibition and tissue-cutting experiments, and we demonstrate that the results of these experiments favor the PORD model. PMID:27136055
A deterministic particle method for one-dimensional reaction-diffusion equations
NASA Technical Reports Server (NTRS)
Mascagni, Michael
1995-01-01
We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.
Reaction Diffusion Modeling of Calcium Dynamics with Realistic ER Geometry
Means, Shawn; Smith, Alexander J.; Shepherd, Jason; Shadid, John; Fowler, John; Wojcikiewicz, Richard J. H.; Mazel, Tomas; Smith, Gregory D.; Wilson, Bridget S.
2006-01-01
We describe a finite-element model of mast cell calcium dynamics that incorporates the endoplasmic reticulum's complex geometry. The model is built upon a three-dimensional reconstruction of the endoplasmic reticulum (ER) from an electron tomographic tilt series. Tetrahedral meshes provide volumetric representations of the ER lumen, ER membrane, cytoplasm, and plasma membrane. The reaction-diffusion model simultaneously tracks changes in cytoplasmic and ER intraluminal calcium concentrations and includes luminal and cytoplasmic protein buffers. Transport fluxes via PMCA, SERCA, ER leakage, and Type II IP3 receptors are also represented. Unique features of the model include stochastic behavior of IP3 receptor calcium channels and comparisons of channel open times when diffusely distributed or aggregated in clusters on the ER surface. Simulations show that IP3R channels in close proximity modulate activity of their neighbors through local Ca2+ feedback effects. Cytoplasmic calcium levels rise higher, and ER luminal calcium concentrations drop lower, after IP3-mediated release from receptors in the diffuse configuration. Simulation results also suggest that the buffering capacity of the ER, and not restricted diffusion, is the predominant factor influencing average luminal calcium concentrations. PMID:16617072
A Reaction-Diffusion Model of Cholinergic Retinal Waves
Lansdell, Benjamin; Ford, Kevin; Kutz, J. Nathan
2014-01-01
Prior to receiving visual stimuli, spontaneous, correlated activity in the retina, called retinal waves, drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their inter-wave interval and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In addition to simulation, we are thus able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how parameters such as the spontaneous activation rate and sAHP refractory period contribute to critical wave size variability. PMID:25474327
Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme
NASA Astrophysics Data System (ADS)
Peng, Rui; Yi, Feng-qi; Zhao, Xiao-qiang
Spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades. In this work, we are concerned with a reaction-diffusion model with the Degn-Harrison reaction scheme, which accounts for the qualitative feature of the respiratory process in a Klebsiella aerogenes bacterial culture. We study the global stability of the constant steady state, existence and nonexistence of nonconstant steady states as well as the Hopf and steady state bifurcations. In particular, our results show the existence of Turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous. These results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns.
Taylor-Couette Flow with Hourglass Geometry of Varying Lengths Simulated by Reaction-Diffusion
NASA Astrophysics Data System (ADS)
Zhao, Yunjie; Halmstad, Andrew; Olsen, Thomas; Wiener, Richard
2008-11-01
Previously, we have observed chaotic formation of Taylor-Vortex pairs in Modified Taylor- Couette Flow with Hourglass Geometry. In the experiment, the chaotic formation in a shorter system has been restricted to a narrow band about the waist of the hourglass. Such behavior has been modeled by The Reaction-Diffusion equation, which has been previously studied, by Riecke and Paap. Their calculation suggested that quadrupling length of the system would lead to spatial chaos in the vortex formation. We present a careful recreation of this result and consider an intermediate length. We demonstrate that doubling the length should be sufficient to observe spatially chaotic behavior. Richard J. Wiener et al, Phys. Rev. E 55, 5489 (1997). H. Riecke and H.-G. Paap, Europhys. Lett. 14, 1235 (1991).
Link between alginate reaction front propagation and general reaction diffusion theory.
Braschler, Thomas; Valero, Ana; Colella, Ludovica; Pataky, Kristopher; Brugger, Jürgen; Renaud, Philippe
2011-03-15
We provide a common theoretical framework reuniting specific models for the Ca(2+)-alginate system and general reaction diffusion theory along with experimental validation on a microfluidic chip. As a starting point, we use a set of nonlinear, partial differential equations that are traditionally solved numerically: the Mikkelsen-Elgsaeter model. Applying the traveling-wave hypothesis as a major simplification, we obtain an analytical solution. The solution indicates that the fundamental properties of the alginate reaction front are governed by a single dimensionless parameter λ. For small λ values, a large depletion zone accompanies the reaction front. For large λ values, the alginate reacts before having the time to diffuse significantly. We show that the λ parameter is of general importance beyond the alginate model system, as it can be used to classify known solutions for second-order reaction diffusion schemes, along with the novel solution presented here. For experimental validation, we develop a microchip model system, in which the alginate gel formation can be carried out in a highly controlled, essentially 1D environment. The use of a filter barrier enables us to rapidly renew the CaCl(2) solution, while maintaining flow speeds lower than 1 μm/s for the alginate compartment. This allows one to impose an exactly known bulk CaCl(2) concentration and diffusion resistance. This experimental model system, taken together with the theoretical development, enables the determination of the entire set of physicochemical parameters governing the alginate reaction front in a single experiment. PMID:21351747
Hybrid approaches for multiple-species stochastic reaction-diffusion models
NASA Astrophysics Data System (ADS)
Spill, Fabian; Guerrero, Pilar; Alarcon, Tomas; Maini, Philip K.; Byrne, Helen
2015-10-01
Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. In this way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.
An adaptive algorithm for simulation of stochastic reaction-diffusion processes
Ferm, Lars Hellander, Andreas Loetstedt, Per
2010-01-20
We propose an adaptive hybrid method suitable for stochastic simulation of diffusion dominated reaction-diffusion processes. For such systems, simulation of the diffusion requires the predominant part of the computing time. In order to reduce the computational work, the diffusion in parts of the domain is treated macroscopically, in other parts with the tau-leap method and in the remaining parts with Gillespie's stochastic simulation algorithm (SSA) as implemented in the next subvolume method (NSM). The chemical reactions are handled by SSA everywhere in the computational domain. A trajectory of the process is advanced in time by an operator splitting technique and the timesteps are chosen adaptively. The spatial adaptation is based on estimates of the errors in the tau-leap method and the macroscopic diffusion. The accuracy and efficiency of the method are demonstrated in examples from molecular biology where the domain is discretized by unstructured meshes.
Scalable implicit methods for reaction-diffusion equations in two and three space dimensions
Veronese, S.V.; Othmer, H.G.
1996-12-31
This paper describes the implementation of a solver for systems of semi-linear parabolic partial differential equations in two and three space dimensions. The solver is based on a parallel implementation of a non-linear Alternating Direction Implicit (ADI) scheme which uses a Cartesian grid in space and an implicit time-stepping algorithm. Various reordering strategies for the linearized equations are used to reduce the stride and improve the overall effectiveness of the parallel implementation. We have successfully used this solver for large-scale reaction-diffusion problems in computational biology and medicine in which the desired solution is a traveling wave that may contain rapid transitions. A number of examples that illustrate the efficiency and accuracy of the method are given here; the theoretical analysis will be presented.
Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model
NASA Astrophysics Data System (ADS)
Barbera, Elvira; Consolo, Giancarlo; Valenti, Giovanna
2013-11-01
A one-dimensional hyperbolic reaction-diffusion model of epidemics is developed to describe the dynamics of diseases spread occurring in an environment where three kinds of individuals mutually interact: the susceptibles, the infectives, and the removed. It is assumed that the disease is transmitted from the infected population to the susceptible one according to a nonlinear convex incidence rate. The model, based upon the framework of extended thermodynamics, removes the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models. Linear stability analyses are performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. Emphasis is given to the occurrence of Hopf and Turing bifurcations, which break the temporal and the spatial symmetry of the system, respectively. The existence of traveling wave solutions connecting two steady states is also discussed. The governing equations are also integrated numerically to validate the analytical results and to characterize the spatiotemporal evolution of diseases.
Symmetry breaking in a bulk-surface reaction-diffusion model for signalling networks
NASA Astrophysics Data System (ADS)
Rätz, Andreas; Röger, Matthias
2014-08-01
Signalling molecules play an important role for many cellular functions. We investigate here a general system of two membrane reaction-diffusion equations coupled to a diffusion equation inside the cell by a Robin-type boundary condition and a flux term in the membrane equations. A specific model of this form was recently proposed by the authors for the GTPase cycle in cells. We investigate here a putative role of diffusive instabilities in cell polarization. By a linearized stability analysis, we identify two different mechanisms. The first resembles a classical Turing instability for the membrane subsystem and requires (unrealistically) large differences in the lateral diffusion of activator and substrate. On the other hand, the second possibility is induced by the difference in cytosolic and lateral diffusion and appears much more realistic. We complement our theoretical analysis by numerical simulations that confirm the new stability mechanism and allow us to investigate the evolution beyond the regime where the linearization applies.
NASA Astrophysics Data System (ADS)
Ghosh, Atiyo; Leier, Andre; Marquez-Lago, Tatiana
2014-03-01
Spatial stochastic effects are prevalent in many biological systems spanning a variety of scales, from intracellular (e.g. gene expression) to ecological (plankton aggregation). The most common ways of simulating such systems involve drawing sample paths from either the Reaction Diffusion Master Equation (RDME) or the Smoluchowski Equation, using methods such as Gillespie's Simulation Algorithm, Green's Function Reaction Dynamics and Single Particle Tracking. The simulation times of such techniques scale with the number of simulated particles, leading to much computational expense when considering large systems. The Spatial Chemical Langevin Equation (SCLE) can be simulated with fixed time intervals, independent of the number of particles, and can thus provide significant computational savings. However, very little work has been done to investigate the behavior of the SCLE. In this talk we summarize our findings on comparing the SCLE to the well-studied RDME. We use both analytical and numerical procedures to show when one should expect the moments of the SCLE to be close to the RDME, and also when they should differ.
Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation
NASA Astrophysics Data System (ADS)
Strani, Marta
2015-12-01
We study the one-dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equation Metastable behaviour appears when the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches its steady state in a very long time interval. A rigorous analysis is used to study such behaviour by means of the construction of a one-parameter family {{≤ft\\{{{U}\\varepsilon}≤ft(x;ξ \\right)\\right\\}}ξ} of approximate stationary solutions and of a linearisation of the original system around an element of this family. We obtain a system consisting of an ODE for the parameter ξ, describing the position of the interface coupled with a PDE for the perturbation v and defined as the difference v:=u-{{U}\\varepsilon} . The key of our analysis are the spectral properties of the linearised operator around an element of the family ≤ft\\{{{U}\\varepsilon}\\right\\} : the presence of a first eigenvalue, small with respect to ε, leads to metastable behaviour when \\varepsilon \\ll 1 .
Local Perturbation Analysis: A Computational Tool for Biophysical Reaction-Diffusion Models
Holmes, William R.; Mata, May Anne; Edelstein-Keshet, Leah
2015-01-01
Diffusion and interaction of molecular regulators in cells is often modeled using reaction-diffusion partial differential equations. Analysis of such models and exploration of their parameter space is challenging, particularly for systems of high dimensionality. Here, we present a relatively simple and straightforward analysis, the local perturbation analysis, that reveals how parameter variations affect model behavior. This computational tool, which greatly aids exploration of the behavior of a model, exploits a structural feature common to many cellular regulatory systems: regulators are typically either bound to a membrane or freely diffusing in the interior of the cell. Using well-documented, readily available bifurcation software, the local perturbation analysis tracks the approximate early evolution of an arbitrarily large perturbation of a homogeneous steady state. In doing so, it provides a bifurcation diagram that concisely describes various regimes of the model’s behavior, reducing the need for exhaustive simulations to explore parameter space. We explain the method and provide detailed step-by-step guides to its use and application. PMID:25606671
tiReaction Diffusion Voronoi Diagrams: From Sensors Data to Computing
Vázquez-Otero, Alejandro; Faigl, Jan; Dormido, Raquel; Duro, Natividad
2015-01-01
In this paper, a new method to solve computational problems using reaction diffusion (RD) systems is presented. The novelty relies on the use of a model configuration that tailors its spatiotemporal dynamics to develop Voronoi diagrams (VD) as a part of the system's natural evolution. The proposed framework is deployed in a solution of related robotic problems, where the generalized VD are used to identify topological places in a grid map of the environment that is created from sensor measurements. The ability of the RD-based computation to integrate external information, like a grid map representing the environment in the model computational grid, permits a direct integration of sensor data into the model dynamics. The experimental results indicate that this method exhibits significantly less sensitivity to noisy data than the standard algorithms for determining VD in a grid. In addition, previous drawbacks of the computational algorithms based on RD models, like the generation of volatile solutions by means of excitable waves, are now overcome by final stable states. PMID:26035349
Roques, Lionel; Bonnefon, Olivier
2016-01-01
We propose and develop a general approach based on reaction-diffusion equations for modelling a species dynamics in a realistic two-dimensional (2D) landscape crossed by linear one-dimensional (1D) corridors, such as roads, hedgerows or rivers. Our approach is based on a hybrid "2D/1D model", i.e, a system of 2D and 1D reaction-diffusion equations with homogeneous coefficients, in which each equation describes the population dynamics in a given 2D or 1D element of the landscape. Using the example of the range expansion of the tiger mosquito Aedes albopictus in France and its main highways as 1D corridors, we show that the model can be fitted to realistic observation data. We develop a mechanistic-statistical approach, based on the coupling between a model of population dynamics and a probabilistic model of the observation process. This allows us to bridge the gap between the data (3 levels of infestation, at the scale of a French department) and the output of the model (population densities at each point of the landscape), and to estimate the model parameter values using a maximum-likelihood approach. Using classical model comparison criteria, we obtain a better fit and a better predictive power with the 2D/1D model than with a standard homogeneous reaction-diffusion model. This shows the potential importance of taking into account the effect of the corridors (highways in the present case) on species dynamics. With regard to the particular case of A. albopictus, the conclusion that highways played an important role in species range expansion in mainland France is consistent with recent findings from the literature. PMID:26986201
2016-01-01
We propose and develop a general approach based on reaction-diffusion equations for modelling a species dynamics in a realistic two-dimensional (2D) landscape crossed by linear one-dimensional (1D) corridors, such as roads, hedgerows or rivers. Our approach is based on a hybrid “2D/1D model”, i.e, a system of 2D and 1D reaction-diffusion equations with homogeneous coefficients, in which each equation describes the population dynamics in a given 2D or 1D element of the landscape. Using the example of the range expansion of the tiger mosquito Aedes albopictus in France and its main highways as 1D corridors, we show that the model can be fitted to realistic observation data. We develop a mechanistic-statistical approach, based on the coupling between a model of population dynamics and a probabilistic model of the observation process. This allows us to bridge the gap between the data (3 levels of infestation, at the scale of a French department) and the output of the model (population densities at each point of the landscape), and to estimate the model parameter values using a maximum-likelihood approach. Using classical model comparison criteria, we obtain a better fit and a better predictive power with the 2D/1D model than with a standard homogeneous reaction-diffusion model. This shows the potential importance of taking into account the effect of the corridors (highways in the present case) on species dynamics. With regard to the particular case of A. albopictus, the conclusion that highways played an important role in species range expansion in mainland France is consistent with recent findings from the literature. PMID:26986201
Reaction-diffusion waves in neuronal tissue and the window of cortical excitability
NASA Astrophysics Data System (ADS)
Dahlem, M. A.; Müller, S. C.
2004-07-01
Spreading depression (SD) is a dynamic wave phenomenon occurring in all gray matter regions of the central nervous systems (CNS). It is characterized by a sudden breakdown of neuronal activity and accompanied by a massive influx and efflux of ions across the membrane of neurons. The retina is a constituent of the CNS in which one can easily observe the dynamic behavior of the SD wave fronts, because SD changes the optical properties of the tissue. There is ample evidence that SD belongs to the self-organization processes due to the coupling of reaction with diffusion in excitable medium. It is assumed that the occurrence of SD is associated with some neurological symptoms of migraine with aura. A frequently reported aura symptom is a traveling visual blind region (scotoma) with a preceding figure of scintillating line segments. The characteristic form and development of the scotoma suggests that the underlying phenomenon is a wave propagating through the primary visual cortex, most likely the cortical spreading depression. In this article we discuss similarities between SD waves and the migraine aura on the basis of properties of reaction-diffusion waves known from other excitable media. In particular, the propagation velocities, the shape and the dynamics of the waves are compared with each other. We find that the assumption of the neuronal tissue to be in a state of only weak excitability explains some properties of the migraine aura, such as the confined appearance and its propagation with a stable velocity.
Discrete-continuous reaction-diffusion model with mobile point-like sources and sinks.
Kondrat, Svyatoslav; Zimmermann, Olav; Wiechert, Wolfgang; von Lieres, Eric
2016-01-01
In many applications in soft and biological physics, there are multiple time and length scales involved but often with a distinct separation between them. For instance, in enzyme kinetics, enzymes are relatively large, move slowly and their copy numbers are typically small, while the metabolites (being transformed by these enzymes) are often present in abundance, are small in size and diffuse fast. It seems thus natural to apply different techniques to different time and length levels and couple them. Here we explore this possibility by constructing a stochastic-deterministic discrete-continuous reaction-diffusion model with mobile sources and sinks. Such an approach allows in particular to separate different sources of stochasticity. We demonstrate its application by modelling enzyme-catalysed reactions with freely diffusing enzymes and a heterogeneous source of metabolites. Our calculations suggest that using a higher amount of less active enzymes, as compared to fewer more active enzymes, reduces the metabolite pool size and correspondingly the lag time, giving rise to a faster response to external stimuli. The methodology presented can be extended to more complex systems and offers exciting possibilities for studying problems where spatial heterogeneities, stochasticity or discreteness play a role. PMID:26830760
Bifurcation analysis of brown tide by reaction-diffusion equation using finite element method
Kawahara, Mutsuto; Ding, Yan
1997-03-01
In this paper, we analyze the bifurcation of a biodynamics system in a two-dimensional domain by virtue of reaction-diffusion equations. The discretization method in space is the finite element method. The computational algorithm for an eigenspectrum is described in detail. On the basis of an analysis of eigenspectra according to Helmholtz`s equation, the discrete spectra in regards to the physical variables are numerically obtained in two-dimensional space. In order to investigate this mathematical model in regards to its practical use, we analyzed the stability of two cases, i.e., hydranth regeneration in the marine hydroid Tubularia and a brown tide in a harbor in Japan. By evaluating the stability according to the linearized stability definition, the critical parameters for outbreaks of brown tide can be theoretically determined. In addition, results for the linear combination of eigenspectrum coincide with the distribution of the observed brown tide. Its periodic characteristic was also verified. 10 refs., 8 figs., 5 tabs.
NASA Astrophysics Data System (ADS)
Hellander, Andreas; Lawson, Michael J.; Drawert, Brian; Petzold, Linda
2014-06-01
The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps were adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the diffusive finite-state projection (DFSP) method, to incorporate temporal adaptivity.
Liang, Xiao; Wang, Linshan; Wang, Yangfan; Wang, Ruili
2016-09-01
In this paper, we focus on the long time behavior of the mild solution to delayed reaction-diffusion Hopfield neural networks (DRDHNNs) driven by infinite dimensional Wiener processes. We analyze the existence, uniqueness, and stability of this system under the local Lipschitz function by constructing an appropriate Lyapunov-Krasovskii function and utilizing the semigroup theory. Some easy-to-test criteria affecting the well-posedness and stability of the networks, such as infinite dimensional noise and diffusion effect, are obtained. The criteria can be used as theoretic guidance to stabilize DRDHNNs in practical applications when infinite dimensional noise is taken into consideration. Meanwhile, considering the fact that the standard Brownian motion is a special case of infinite dimensional Wiener process, we undertake an analysis of the local Lipschitz condition, which has a wider range than the global Lipschitz condition. Two samples are given to examine the availability of the results in this paper. Simulations are also given using the MATLAB. PMID:26259224
NASA Astrophysics Data System (ADS)
Hurdal, Monica K.; Striegel, Deborah A.
2011-11-01
Modeling and understanding cortical folding pattern formation is important for quantifying cortical development. We present a biomathematical model for cortical folding pattern formation in the human brain and apply this model to study diseases involving cortical pattern malformations associated with neural migration disorders. Polymicrogyria is a cortical malformation disease resulting in an excessive number of small gyri. Our mathematical model uses a Turing reaction-diffusion system to model cortical folding. The lateral ventricle (LV) and ventricular zone (VZ) of the brain are critical components in the formation of cortical patterning. In early cortical development the shape of the LV can be modeled with a prolate spheroid and the VZ with a prolate spheroid surface. We use our model to study how global cortex characteristics, such as size and shape of the LV, affect cortical pattern formation. We demonstrate increasing domain scale can increase the number of gyri and sulci formed. Changes in LV shape can account for sulcus directionality. By incorporating LV size and shape, our model is able to elucidate which parameters can lead to excessive cortical folding.
Dissipation and displacement of hotspots in reaction-diffusion models of crime.
Short, Martin B; Brantingham, P Jeffrey; Bertozzi, Andrea L; Tita, George E
2010-03-01
The mechanisms driving the nucleation, spread, and dissipation of crime hotspots are poorly understood. As a consequence, the ability of law enforcement agencies to use mapped crime patterns to design crime prevention strategies is severely hampered. We also lack robust expectations about how different policing interventions should impact crime. Here we present a mathematical framework based on reaction-diffusion partial differential equations for studying the dynamics of crime hotspots. The system of equations is based on empirical evidence for how offenders move and mix with potential victims or targets. Analysis shows that crime hotspots form when the enhanced risk of repeat crimes diffuses locally, but not so far as to bind distant crime together. Crime hotspots may form as either supercritical or subcritical bifurcations, the latter the result of large spikes in crime that override linearly stable, uniform crime distributions. Our mathematical methods show that subcritical crime hotspots may be permanently eradicated with police suppression, whereas supercritical hotspots are displaced following a characteristic spatial pattern. Our results thus provide a mechanistic explanation for recent failures to observe crime displacement in experimental field tests of hotspot policing. PMID:20176972
NASA Astrophysics Data System (ADS)
Gan, Qintao; Lv, Tianshi; Fu, Zhenhua
2016-04-01
In this paper, the synchronization problem for a class of generalized neural networks with time-varying delays and reaction-diffusion terms is investigated concerning Neumann boundary conditions in terms of p-norm. The proposed generalized neural networks model includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks as its special cases. By establishing a new inequality, some simple and useful conditions are obtained analytically to guarantee the global exponential synchronization of the addressed neural networks under the periodically intermittent control. According to the theoretical results, the influences of diffusion coefficients, diffusion space, and control rate on synchronization are analyzed. Finally, the feasibility and effectiveness of the proposed methods are shown by simulation examples, and by choosing different diffusion coefficients, diffusion spaces, and control rates, different controlled synchronization states can be obtained.
Gan, Qintao; Lv, Tianshi; Fu, Zhenhua
2016-04-01
In this paper, the synchronization problem for a class of generalized neural networks with time-varying delays and reaction-diffusion terms is investigated concerning Neumann boundary conditions in terms of p-norm. The proposed generalized neural networks model includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks as its special cases. By establishing a new inequality, some simple and useful conditions are obtained analytically to guarantee the global exponential synchronization of the addressed neural networks under the periodically intermittent control. According to the theoretical results, the influences of diffusion coefficients, diffusion space, and control rate on synchronization are analyzed. Finally, the feasibility and effectiveness of the proposed methods are shown by simulation examples, and by choosing different diffusion coefficients, diffusion spaces, and control rates, different controlled synchronization states can be obtained. PMID:27131492
NASA Astrophysics Data System (ADS)
Kuriakose, Jainy; Ghosh, Anandamohan; Ravi Kumar, V.; Kulkarni, B. D.
2004-03-01
Heterogeneous surface reactions exhibiting complex spatiotemporal dynamics and patterns can be studied as processes involving reaction-diffusion mechanisms. In many realistic situations, the surface has fractal characteristics. This situation is studied by isometric graphing and multidimensional scaling (IGMDS) of fractal surfaces for extracting geodesic distances (i.e., shortest scaled distances that obtain edges of neighboring surface nodes and their interconnections) and the results obtained used to model effects of surface diffusion with nonlinear reactions. Further analysis of evolved spatiotemporal patterns may be carried out by IGMDS because high-dimensional snapshot data can be efficiently projected to a transformed subspace with reduced dimensions. Validation of the IGMDS methodology is carried out by comparing results with reduction capabilities of conventional principal component analysis for simple situations of reaction and diffusion on surfaces. The usefulness of the IGMDS methodology is shown for analysis of complex patterns formed on both regular and fractal surfaces, and using generic nonlinear reaction-diffusion systems following FitzHugh Nagumo and cubic reaction kinetics. The studies of these systems with nonlinear kinetics and noise show that effects of surface disorder due to fractality can become very relevant. The relevance is shown by studying properties of dynamical invariants in IGMDS component space, viz., the Lyapunov exponents and the KS entropy for interesting situations of spiral formation and turbulent patterns.
Reaction-diffusion processes on interconnected scale-free networks
NASA Astrophysics Data System (ADS)
Garas, Antonios
2015-08-01
We study the two-particle annihilation reaction A +B →∅ on interconnected scale-free networks, using different interconnecting strategies. We explore how the mixing of particles and the process evolution are influenced by the number of interconnecting links, by their functional properties, and by the interconnectivity strategies in use. We show that the reaction rates on this system are faster than what was observed in other topologies, due to the better particle mixing that suppresses the segregation effect, in line with previous studies performed on single scale-free networks.
Flexible single molecule simulation of reaction-diffusion processes
Hellander, Stefan; Loetstedt, Per
2011-05-10
An algorithm is developed for simulation of the motion and reactions of single molecules at a microscopic level. The molecules diffuse in a solvent and react with each other or a polymer and molecules can dissociate. Such simulations are of interest e.g. in molecular biology. The algorithm is similar to the Green's function reaction dynamics (GFRD) algorithm by van Zon and ten Wolde where longer time steps can be taken by computing the probability density functions (PDFs) and then sample from the distribution functions. Our computation of the PDFs is much less complicated than GFRD and more flexible. The solution of the partial differential equation for the PDF is split into two steps to simplify the calculations. The sampling is without splitting error in two of the coordinate directions for a pair of molecules and a molecule-polymer interaction and is approximate in the third direction. The PDF is obtained either from an analytical solution or a numerical discretization. The errors due to the operator splitting, the partitioning of the system, and the numerical approximations are analyzed. The method is applied to three different systems involving up to four reactions. Comparisons with other mesoscopic and macroscopic models show excellent agreement.
A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation.
Wang, Kangle; Liu, Sanyang
2016-01-01
In this paper, a new Sumudu transform iterative method is established and successfully applied to find the approximate analytical solutions for time-fractional Cauchy reaction-diffusion equations. The approach is easy to implement and understand. The numerical results show that the proposed method is very simple and efficient. PMID:27386314
NASA Astrophysics Data System (ADS)
Wang, Zhi-Cheng; Bu, Zhen-Hui
2016-04-01
This paper is concerned with nonplanar traveling fronts in reaction-diffusion equations with combustion nonlinearity and degenerate Fisher-KPP nonlinearity. Our study contains two parts: in the first part we establish the existence of traveling fronts of pyramidal shape in R3, and in the second part we establish the existence and stability of V-shaped traveling fronts in R2.
Stability and bifurcations in a nonlocal delayed reaction-diffusion population model
NASA Astrophysics Data System (ADS)
Chen, Shanshan; Yu, Jianshe
2016-01-01
A nonlocal delayed reaction-diffusion equation with Dirichlet boundary condition is considered in this paper. It is shown that a positive spatially nonhomogeneous equilibrium bifurcates from the trivial equilibrium. The stability/instability of the bifurcated positive equilibrium and associated Hopf bifurcation are investigated, providing us with a complete picture of the dynamics.
Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect
NASA Astrophysics Data System (ADS)
Guo, Shangjiang
2015-08-01
In this paper, the existence, stability, and multiplicity of spatially nonhomogeneous steady-state solution and periodic solutions for a reaction-diffusion model with nonlocal delay effect and Dirichlet boundary condition are investigated by using Lyapunov-Schmidt reduction. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain.
Wavelet-Based Spatial Scaling of Coupled Reaction-Diffusion Fields
Mishra, Sudib; Muralidharan, Krishna; Deymier, Pierre; Frantziskonis, G.; Pannala, Sreekanth; Simunovic, Srdjan
2008-01-01
Multiscale schemes for transferring information from fine to coarse scales are typically based on homogenization techniques. Such schemes smooth the fine scale features of the underlying fields, often resulting in the inability to accurately retain the fine scale correlations. In addition, higher-order statistical moments (beyond mean) of the relevant field variables are not necessarily preserved. As a superior alternative to averaging homogenization methods, a wavelet-based scheme for the exchange of information between a reactive and diffusive field in the context of multiscale reaction-diffusion problems is proposed and analyzed. The scheme is shown to be efficient in passing information along scales, from fine to coarse, i.e., upscaling as well as from coarse to fine, i.e., downscaling. It incorporates fine scale statistics (higher-order moments beyond mean), mainly due to the capability of wavelets to represent fields hierarchically. Critical to the success of the scheme is the identification of dominant scales containing the majority of the useful information. The dominant scales in effect specify the coarsest resolution possible. The scheme is applied in detail to the analysis of a diffusive system with a chemically reacting boundary. Reactions are simulated using kinetic Monte Carlo (kMC) and diffusion is solved by finite differences (FDs). Spatial scale differences are present at the interface of the kMC sites and the diffusion grid. The computational efficiency of the scheme is compared to results obtained by averaging homogenization, and to results from a benchmark scheme that ensures spatial scale parity between kMC and FD.
NASA Astrophysics Data System (ADS)
Yochelis, Arik; Bar-On, Tomer; Gov, Nir S.
2016-04-01
Unconventional myosins belong to a class of molecular motors that walk processively inside cellular protrusions towards the tips, on top of actin filament. Surprisingly, in addition, they also form retrograde moving self-organized aggregates. The qualitative properties of these aggregates are recapitulated by a mass conserving reaction-diffusion-advection model and admit two distinct families of modes: traveling waves and pulse trains. Unlike the traveling waves that are generated by a linear instability, pulses are nonlinear structures that propagate on top of linearly stable uniform backgrounds. Asymptotic analysis of isolated pulses via a simplified reaction-diffusion-advection variant on large periodic domains, allows to draw qualitative trends for pulse properties, such as the amplitude, width, and propagation speed. The results agree well with numerical integrations and are related to available empirical observations.
Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity
NASA Astrophysics Data System (ADS)
Asvestas, M.; Sifalakis, A. G.; Papadopoulou, E. P.; Saridakis, Y. G.
2014-03-01
Motivated by proliferation-diffusion mathematical models for studying highly diffusive brain tumors, that also take into account the heterogeneity of the brain tissue, in the present work we consider a multi-domain linear reaction-diffusion equation with a discontinuous diffusion coefficient. For the solution of the problem at hand we implement Fokas transform method by directly following, and extending in this way, our recent work for a white-gray-white matter brain model pertaining to high grade gliomas. Fokas's novel approach for the solution of linear PDE problems, yields novel integral representations of the solution in the complex plane that, for appropriately chosen integration contours, decay exponentially fast and converge uniformly at the boundaries. Combining these method-inherent advantages with simple numerical quadrature rules, we produce an efficient method, with fast decaying error properties, for the solution of the discontinuous reaction-diffusion problem.
NASA Astrophysics Data System (ADS)
Gal, Ciprian G.; Warma, Mahamadi
2016-08-01
We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction-diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.
Delay-induced Turing-like waves for one-species reaction-diffusion model on a network
NASA Astrophysics Data System (ADS)
Petit, Julien; Carletti, Timoteo; Asllani, Malbor; Fanelli, Duccio
2015-09-01
A one-species time-delay reaction-diffusion system defined on a complex network is studied. Traveling waves are predicted to occur following a symmetry-breaking instability of a homogeneous stationary stable solution, subject to an external nonhomogeneous perturbation. These are generalized Turing-like waves that materialize in a single-species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time-delayed differential equations. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz network and with the inclusion of the delay.
Sato, Makoto; Yasugi, Tetsuo; Minami, Yoshiaki; Miura, Takashi; Nagayama, Masaharu
2016-08-30
Notch-mediated lateral inhibition regulates binary cell fate choice, resulting in salt and pepper patterns during various developmental processes. However, how Notch signaling behaves in combination with other signaling systems remains elusive. The wave of differentiation in the Drosophila visual center or "proneural wave" accompanies Notch activity that is propagated without the formation of a salt and pepper pattern, implying that Notch does not form a feedback loop of lateral inhibition during this process. However, mathematical modeling and genetic analysis clearly showed that Notch-mediated lateral inhibition is implemented within the proneural wave. Because partial reduction in EGF signaling causes the formation of the salt and pepper pattern, it is most likely that EGF diffusion cancels salt and pepper pattern formation in silico and in vivo. Moreover, the combination of Notch-mediated lateral inhibition and EGF-mediated reaction diffusion enables a function of Notch signaling that regulates propagation of the wave of differentiation. PMID:27535937
NASA Astrophysics Data System (ADS)
Nefedov, N. N.; Ni, Minkang
2015-12-01
A singularly perturbed boundary value problem for a second-order ordinary differential equation known in applications as a stationary reaction-diffusion equation is studied. A new class of problems is considered, namely, problems with nonlinearity having discontinuities localized in some domains, which leads to the formation of sharp transition layers in these domains. The existence of solutions with an internal transition layer is proved, and their asymptotic expansion is constructed.
Contribution to an effective design method for stationary reaction-diffusion patterns.
Szalai, István; Horváth, Judit; De Kepper, Patrick
2015-06-01
The British mathematician Alan Turing predicted, in his seminal 1952 publication, that stationary reaction-diffusion patterns could spontaneously develop in reacting chemical or biochemical solutions. The first two clear experimental demonstrations of such a phenomenon were not made before the early 1990s when the design of new chemical oscillatory reactions and appropriate open spatial chemical reactors had been invented. Yet, the number of pattern producing reactions had not grown until 2009 when we developed an operational design method, which takes into account the feeding conditions and other specificities of real open spatial reactors. Since then, on the basis of this method, five additional reactions were shown to produce stationary reaction-diffusion patterns. To gain a clearer view on where our methodical approach on the patterning capacity of a reaction stands, numerical studies in conditions that mimic true open spatial reactors were made. In these numerical experiments, we explored the patterning capacity of Rabai's model for pH driven Landolt type reactions as a function of experimentally attainable parameters that control the main time and length scales. Because of the straightforward reversible binding of protons to carboxylate carrying polymer chains, this class of reaction is at the base of the chemistry leading to most of the stationary reaction-diffusion patterns presently observed. We compare our model predictions with experimental observations and comment on agreements and differences. PMID:26117122
Contribution to an effective design method for stationary reaction-diffusion patterns
NASA Astrophysics Data System (ADS)
Szalai, István; Horváth, Judit; De Kepper, Patrick
2015-06-01
The British mathematician Alan Turing predicted, in his seminal 1952 publication, that stationary reaction-diffusion patterns could spontaneously develop in reacting chemical or biochemical solutions. The first two clear experimental demonstrations of such a phenomenon were not made before the early 1990s when the design of new chemical oscillatory reactions and appropriate open spatial chemical reactors had been invented. Yet, the number of pattern producing reactions had not grown until 2009 when we developed an operational design method, which takes into account the feeding conditions and other specificities of real open spatial reactors. Since then, on the basis of this method, five additional reactions were shown to produce stationary reaction-diffusion patterns. To gain a clearer view on where our methodical approach on the patterning capacity of a reaction stands, numerical studies in conditions that mimic true open spatial reactors were made. In these numerical experiments, we explored the patterning capacity of Rabai's model for pH driven Landolt type reactions as a function of experimentally attainable parameters that control the main time and length scales. Because of the straightforward reversible binding of protons to carboxylate carrying polymer chains, this class of reaction is at the base of the chemistry leading to most of the stationary reaction-diffusion patterns presently observed. We compare our model predictions with experimental observations and comment on agreements and differences.
Contribution to an effective design method for stationary reaction-diffusion patterns
Szalai, István; Horváth, Judit; De Kepper, Patrick
2015-06-15
The British mathematician Alan Turing predicted, in his seminal 1952 publication, that stationary reaction-diffusion patterns could spontaneously develop in reacting chemical or biochemical solutions. The first two clear experimental demonstrations of such a phenomenon were not made before the early 1990s when the design of new chemical oscillatory reactions and appropriate open spatial chemical reactors had been invented. Yet, the number of pattern producing reactions had not grown until 2009 when we developed an operational design method, which takes into account the feeding conditions and other specificities of real open spatial reactors. Since then, on the basis of this method, five additional reactions were shown to produce stationary reaction-diffusion patterns. To gain a clearer view on where our methodical approach on the patterning capacity of a reaction stands, numerical studies in conditions that mimic true open spatial reactors were made. In these numerical experiments, we explored the patterning capacity of Rabai's model for pH driven Landolt type reactions as a function of experimentally attainable parameters that control the main time and length scales. Because of the straightforward reversible binding of protons to carboxylate carrying polymer chains, this class of reaction is at the base of the chemistry leading to most of the stationary reaction-diffusion patterns presently observed. We compare our model predictions with experimental observations and comment on agreements and differences.
A Luenberger observer for reaction-diffusion models with front position data
NASA Astrophysics Data System (ADS)
Collin, Annabelle; Chapelle, Dominique; Moireau, Philippe
2015-11-01
We propose a Luenberger observer for reaction-diffusion models with propagating front features, and for data associated with the location of the front over time. Such models are considered in various application fields, such as electrophysiology, wild-land fire propagation and tumor growth modeling. Drawing our inspiration from image processing methods, we start by proposing an observer for the eikonal-curvature equation that can be derived from the reaction-diffusion model by an asymptotic expansion. We then carry over this observer to the underlying reaction-diffusion equation by an "inverse asymptotic analysis", and we show that the associated correction in the dynamics has a stabilizing effect for the linearized estimation error. We also discuss the extension to joint state-parameter estimation by using the earlier-proposed ROUKF strategy. We then illustrate and assess our proposed observer method with test problems pertaining to electrophysiology modeling, including with a realistic model of cardiac atria. Our numerical trials show that state estimation is directly very effective with the proposed Luenberger observer, while specific strategies are needed to accurately perform parameter estimation - as is usual with Kalman filtering used in a nonlinear setting - and we demonstrate two such successful strategies.
NASA Astrophysics Data System (ADS)
Musho, Matthew K.; Kozak, John J.
1984-10-01
A method is presented for calculating exactly the relative width (σ2)1/2/
Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period
NASA Astrophysics Data System (ADS)
Zhang, Liang; Wang, Zhi-Cheng; Zhao, Xiao-Qiang
2015-05-01
In this paper, we first propose a time-periodic reaction-diffusion epidemic model which incorporates simple demographic structure and the latent period of infectious disease. Then we introduce the basic reproduction number R0 for this model and prove that the sign of R0 - 1 determines the local stability of the disease-free periodic solution. By using the comparison arguments and persistence theory, we further show that the disease-free periodic solution is globally attractive if R0 < 1, while there is an endemic periodic solution and the disease is uniformly persistent if R0 > 1.
Reaction-diffusion analysis for one-step plasma etching and bonding of microfluidic devices
Rosso, Michel; Steijn, Volkert van; Smet, Louis C. P. M. de; Sudhoelter, Ernst J. R.; Kreutzer, Michiel T.; Kleijn, Chris R.
2011-04-25
A self-similar reaction front develops in reactive ion etching when the ions penetrate channels of shallow height h. This relates to the patterning of microchannels using a single-step etching and bonding, as described by Rhee et al. [Lab Chip 5, 102 (2005)]. Experimentally, we report that the front location scales as x{sub f{approx}}ht{sup 1/2} and the width is time-invariant and scales as {delta}{approx}h. Mean-field reaction-diffusion theory and Knudsen diffusion give a semiquantitative understanding of these observations and allow optimization of etching times in relation to bonding requirements.
Hopf bifurcations in a reaction-diffusion population model with delay effect
NASA Astrophysics Data System (ADS)
Su, Ying; Wei, Junjie; Shi, Junping
A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a "food-limited" population model with diffusion and delay effects as well as a weak Allee effect population model.
Conservation Laws of a Family of Reaction-Diffusion-Convection Equations
NASA Astrophysics Data System (ADS)
Bruzón, M. S.; Gandarias, M. L.; de la Rosa, R.
Ibragimov introduced the concept of nonlinear self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi-self-adjoint equations. Gandarias defined the concept of weak self-adjoint. In this paper, we found a class of nonlinear self-adjoint nonlinear reaction-diffusion-convection equations which are neither self-adjoint nor quasi-self-adjoint nor weak self-adjoint. From a general theorem on conservation laws proved by Ibragimov we obtain conservation laws for these equations.
Reaction-Diffusion Processes on Random and Scale-Free Networks
NASA Astrophysics Data System (ADS)
Banerjee, Subhasis; Mallick, Shrestha Basu; Bose, Indrani
We study the discrete Gierer-Meinhardt model of reaction-diffusion on three different types of networks: regular, random and scale-free. The model dynamics lead to the formation of stationary Turing patterns in the steady state in certain parameter regions. Some general features of the patterns are studied through numerical simulation. The results for the random and scale-free networks show a marked difference from those in the case of the regular network. The difference may be ascribed to the small world character of the first two types of networks.
A reaction-diffusion SIS epidemic model in an almost periodic environment
NASA Astrophysics Data System (ADS)
Wang, Bin-Guo; Li, Wan-Tong; Wang, Zhi-Cheng
2015-12-01
A susceptible-infected-susceptible almost periodic reaction-diffusion epidemic model is studied by means of establishing the theories and properties of the basic reproduction ratio {R0}. Particularly, the asymptotic behaviors of {R0} with respect to the diffusion rate {DI} of the infected individuals are obtained. Furthermore, the uniform persistence, extinction and global attractivity are presented in terms of {R0}. Our results indicate that the interaction of spatial heterogeneity and temporal almost periodicity tends to enhance the persistence of the disease.
Extension of Newman's method to electrochemical reaction-diffusion in a fuel cell catalyst layer
NASA Astrophysics Data System (ADS)
Duan, Tianping; Weidner, John W.; White, Ralph E.
A numerical technique is developed for solving coupled electrochemical reaction-diffusion equations. Through analyzing the nonlinearity of the problem, a trial and error iterating procedure is constructed. The coefficient matrix is arranged as a tridiagonal form with elements of block matrix and is decomposed to LU form. A compact forward and backward substitution algorithm based on the shift of inversing block matrix by Gauss-Jordan full pivoting is developed. A large number of node points is required to converge the calculation. Computation experiences show that the iteration converges very quickly. The effects of inner diffusion on the electrochemical reaction are analyzed by numerical solutions.
Wang, Chi-Jen
2013-01-01
In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.
Positional information and reaction-diffusion: two big ideas in developmental biology combine.
Green, Jeremy B A; Sharpe, James
2015-04-01
One of the most fundamental questions in biology is that of biological pattern: how do the structures and shapes of organisms arise? Undoubtedly, the two most influential ideas in this area are those of Alan Turing's 'reaction-diffusion' and Lewis Wolpert's 'positional information'. Much has been written about these two concepts but some confusion still remains, in particular about the relationship between them. Here, we address this relationship and propose a scheme of three distinct ways in which these two ideas work together to shape biological form. PMID:25804733
Self-assembly and plasticity of synaptic domains through a reaction-diffusion mechanism
NASA Astrophysics Data System (ADS)
Haselwandter, Christoph A.; Kardar, Mehran; Triller, Antoine; da Silveira, Rava Azeredo
2015-09-01
Signal transmission across chemical synapses relies crucially on neurotransmitter receptor molecules, concentrated in postsynaptic membrane domains along with scaffold and other postsynaptic molecules. The strength of the transmitted signal depends on the number of receptor molecules in postsynaptic domains, and activity-induced variation in the receptor number is one of the mechanisms of postsynaptic plasticity. Recent experiments have demonstrated that the reaction and diffusion properties of receptors and scaffolds at the membrane, alone, yield spontaneous formation of receptor-scaffold domains of the stable characteristic size observed in neurons. On the basis of these experiments we develop a model describing synaptic receptor domains in terms of the underlying reaction-diffusion processes. Our model predicts that the spontaneous formation of receptor-scaffold domains of the stable characteristic size observed in experiments depends on a few key reactions between receptors and scaffolds. Furthermore, our model suggests novel mechanisms for the alignment of pre- and postsynaptic domains and for short-term postsynaptic plasticity in receptor number. We predict that synaptic receptor domains localize in membrane regions with an increased receptor diffusion coefficient or a decreased scaffold diffusion coefficient. Similarly, we find that activity-dependent increases or decreases in receptor or scaffold diffusion yield a transient increase in the number of receptor molecules concentrated in postsynaptic domains. Thus, the proposed reaction-diffusion model puts forth a coherent set of biophysical mechanisms for the formation, stability, and plasticity of molecular domains on the postsynaptic membrane.
Turing-Hopf bifurcation in the reaction-diffusion equations and its applications
NASA Astrophysics Data System (ADS)
Song, Yongli; Zhang, Tonghua; Peng, Yahong
2016-04-01
In this paper, we consider the Turing-Hopf bifurcation arising from the reaction-diffusion equations. It is a degenerate case and where the characteristic equation has a pair of simple purely imaginary roots and a simple zero root. First, the normal form theory for partial differential equations (PDEs) with delays developed by Faria is adopted to this degenerate case so that it can be easily applied to Turing-Hopf bifurcation. Then, we present a rigorous procedure for calculating the normal form associated with the Turing-Hopf bifurcation of PDEs. We show that the reduced dynamics associated with Turing-Hopf bifurcation is exactly the dynamics of codimension-two ordinary differential equations (ODE), which implies the ODE techniques can be employed to classify the reduced dynamics by the unfolding parameters. Finally, we apply our theoretical results to an autocatalysis model governed by reaction-diffusion equations; for such model, the dynamics in the neighbourhood of this bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations; and then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found.
3D choroid neovascularization growth prediction based on reaction-diffusion model
NASA Astrophysics Data System (ADS)
Zhu, Shuxia; Chen, Xinjian; Shi, Fei; Xiang, Dehui; Zhu, Weifang; Chen, Haoyu
2016-03-01
Choroid neovascularization (CNV) is a kind of pathology from the choroid and CNV-related disease is one important cause of vision loss. It is desirable to predict the CNV growth rate so that appropriate treatment can be planned. In this paper, we seek to find a method to predict the growth of CNV based on 3D longitudinal Optical Coherence Tomography (OCT) images. A reaction-diffusion model is proposed for prediction. The method consists of four phases: pre-processing, meshing, CNV growth modeling and prediction. We not only apply the reaction-diffusion model to the disease region, but also take the surrounding tissues into consideration including outer retinal layer, inner retinal layer and choroid layer. The diffusion in these tissues is considered as isotropic. The finite-element-method (FEM) is used to solve the partial differential equations (PDE) in the diffusion model. The curve of CNV growth with treatment are fitted and then we can predict the CNV status in a future time point. The preliminary results demonstrated that our proposed method is accurate and the validity and feasibility of our model is obvious.
Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions
NASA Astrophysics Data System (ADS)
Bagchi, Partha; Das, Arpan; Sengupta, Srikumar; Srivastava, Ajit M.
2015-09-01
Reaction-diffusion equations with suitable boundary conditions have special propagating solutions which very closely resemble the moving interfaces in a first-order transition. We show that the dynamics of the chiral order parameter for the chiral symmetry breaking transition in heavy-ion collisions, with dissipative dynamics, is governed by one such equation; specifically, the Newell-Whitehead equation. Furthermore, required boundary conditions are automatically satisfied due to the geometry of the collision. The chiral transition is, therefore, completed by a propagating interface, exactly as for a first-order transition, even though the transition actually is a crossover for relativistic heavy-ion collisions. The same thing also happens when we consider the initial confinement-deconfinement transition with the Polyakov loop order parameter. The resulting equation, again with dissipative dynamics, can then be identified with the reaction-diffusion equation known as the FitzHugh-Nagumo equation which is used in population genetics. Observational constraints imply that the entire phase conversion cannot be achieved by such slow moving fronts, and some alternate faster dynamics needs also to be invoked; for example, involving fluctuations. We discuss the implications of these results for heavy-ion collisions. We also discuss possible extensions for the case of the early universe.
Hallock, Michael J.; Stone, John E.; Roberts, Elijah; Fry, Corey; Luthey-Schulten, Zaida
2014-01-01
Simulation of in vivo cellular processes with the reaction-diffusion master equation (RDME) is a computationally expensive task. Our previous software enabled simulation of inhomogeneous biochemical systems for small bacteria over long time scales using the MPD-RDME method on a single GPU. Simulations of larger eukaryotic systems exceed the on-board memory capacity of individual GPUs, and long time simulations of modest-sized cells such as yeast are impractical on a single GPU. We present a new multi-GPU parallel implementation of the MPD-RDME method based on a spatial decomposition approach that supports dynamic load balancing for workstations containing GPUs of varying performance and memory capacity. We take advantage of high-performance features of CUDA for peer-to-peer GPU memory transfers and evaluate the performance of our algorithms on state-of-the-art GPU devices. We present parallel e ciency and performance results for simulations using multiple GPUs as system size, particle counts, and number of reactions grow. We also demonstrate multi-GPU performance in simulations of the Min protein system in E. coli. Moreover, our multi-GPU decomposition and load balancing approach can be generalized to other lattice-based problems. PMID:24882911
NASA Astrophysics Data System (ADS)
Paez-Espejo, Miguel; Sy, Mouhamadou; Varret, François; Boukheddaden, Kamel
2014-01-01
We propose here a new theoretical treatment of the spatiotemporal properties in spin-crosser solids, based on the expansion of the free energy taking into account the spatial fluctuations of the high-spin (HS) fraction. This leads to an equation of motion on the HS fraction following a reaction diffusion equation (RDE), in which most of the parameters can be derived from the experiments. This equation involves the true temporal and spatial scales at variance from the previous stochastic microscopic models, which were based on a homogeneous treatment of the crystal's properties. We have illustrated this new treatment for a two-dimensional rectangularly shaped system with a square symmetry and we could reproduce quantitatively the process of nucleation, growth, and propagation of the HS fraction inside the thermal hysteresis loop, accompanying a first-order transition. The computed spatiotemporal evolution of the system allowed one to follow the propagation of a well-defined macroscopic HS:LS interface, which was found in excellent quantitative agreement with the experimental observations of optical microscopy on the switchable spin crossover crystal [{Fe(NCSe)(py)2}2(m-bpypz)]. The RDE treatment should generate predictive models for novel spatiotemporal effects in spin crossover solids and more generally for all kinds of switchable molecular solids.
A reaction-diffusion-based coding rate control mechanism for camera sensor networks.
Yamamoto, Hiroshi; Hyodo, Katsuya; Wakamiya, Naoki; Murata, Masayuki
2010-01-01
A wireless camera sensor network is useful for surveillance and monitoring for its visibility and easy deployment. However, it suffers from the limited capacity of wireless communication and a network is easily overflown with a considerable amount of video traffic. In this paper, we propose an autonomous video coding rate control mechanism where each camera sensor node can autonomously determine its coding rate in accordance with the location and velocity of target objects. For this purpose, we adopted a biological model, i.e., reaction-diffusion model, inspired by the similarity of biological spatial patterns and the spatial distribution of video coding rate. Through simulation and practical experiments, we verify the effectiveness of our proposal. PMID:22163620
NASA Astrophysics Data System (ADS)
Grytskyy, Dmytro; Diesmann, Markus; Helias, Moritz
2016-06-01
Self-organized structures in networks with spike-timing dependent synaptic plasticity (STDP) are likely to play a central role for information processing in the brain. In the present study we derive a reaction-diffusion-like formalism for plastic feed-forward networks of nonlinear rate-based model neurons with a correlation sensitive learning rule inspired by and being qualitatively similar to STDP. After obtaining equations that describe the change of the spatial shape of the signal from layer to layer, we derive a criterion for the nonlinearity necessary to obtain stable dynamics for arbitrary input. We classify the possible scenarios of signal evolution and find that close to the transition to the unstable regime metastable solutions appear. The form of these dissipative solitons is determined analytically and the evolution and interaction of several such coexistent objects is investigated.
The dilution wave in polymer crystallization is described by Fisher's reaction-diffusion equation
NASA Astrophysics Data System (ADS)
Higgs, Paul G.; Ungar, Goran
2001-04-01
Monodisperse long-chain alkanes such as C198H398 form lamellar crystals in both extended- and folded-chain forms. Folded-chain crystals are in a meta-stable equilibrium with polymer solution at a concentration CF. The crystal growth rate is virtually zero at this point, due to the self-poisoning phenomenon. If extended-chain crystallization is initiated from this state, a wave of crystallization proceeds through the solution, termed the dilution wave. The solution concentration falls as the wave passes, until a value CE is reached that is in equilibrium with the extended-chain crystal phase. We write down a reaction-diffusion equation to describe the dilution wave, and show that this is equivalent to Fisher's equation, which has previously been used to describe many other traveling wave phenomena. Numerical solutions of the equation are used to show examples of the wave shape.
The Hirota Method for Reaction-Diffusion Equations with Three Distinct Roots
Tanoglu, Gamze; Pashaev, Oktay
2004-10-04
The Hirota Method, with modified background is applied to construct exact analytical solutions of nonlinear reaction-diffusion equations of two types. The first equation has only nonlinear reaction part, while the second one has in addition the nonlinear transport term. For both cases, the reaction part has the form of the third order polynomial with three distinct roots. We found analytic one-soliton solutions and the relationships between three simple roots and the wave speed of the soliton. For the first case, if one of the roots is the mean value of other two roots, the soliton is static. We show that the restriction on three distinct roots to obtain moving soliton is removed in the second case by, adding nonlinear transport term to the first equation.
Hepburn, I; Chen, W; De Schutter, E
2016-08-01
Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-diffusion operator splitting in the literature and discuss their effects on accuracy. We introduce an operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification. PMID:27497550
NASA Astrophysics Data System (ADS)
Hepburn, I.; Chen, W.; De Schutter, E.
2016-08-01
Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-diffusion operator splitting in the literature and discuss their effects on accuracy. We introduce an operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification.
A Reaction-Diffusion-Based Coding Rate Control Mechanism for Camera Sensor Networks
Yamamoto, Hiroshi; Hyodo, Katsuya; Wakamiya, Naoki; Murata, Masayuki
2010-01-01
A wireless camera sensor network is useful for surveillance and monitoring for its visibility and easy deployment. However, it suffers from the limited capacity of wireless communication and a network is easily overflown with a considerable amount of video traffic. In this paper, we propose an autonomous video coding rate control mechanism where each camera sensor node can autonomously determine its coding rate in accordance with the location and velocity of target objects. For this purpose, we adopted a biological model, i.e., reaction-diffusion model, inspired by the similarity of biological spatial patterns and the spatial distribution of video coding rate. Through simulation and practical experiments, we verify the effectiveness of our proposal. PMID:22163620
Extremal equilibria for reaction-diffusion equations in bounded domains and applications
NASA Astrophysics Data System (ADS)
Rodríguez-Bernal, Aníbal; Vidal-López, Alejandro
We show the existence of two special equilibria, the extremal ones, for a wide class of reaction-diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation.
Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation
Singer, M A; Green, W H
2007-12-03
We introduce an adaptive POD method to reduce the computational cost of reacting flow simulations. The scheme is coupled with an operator-splitting algorithm to solve the reaction-diffusion equation. For the reaction sub-steps, locally valid basis vectors, obtained via POD and the method of snapshots, are used to project the minor species mass fractions onto a reduced dimensional space thereby decreasing the number of equations that govern combustion chemistry. The method is applied to a one-dimensional laminar premixed CH{sub 4}-air flame using GRImech 3.0; with errors less than 0:25%, a speed-up factor of 3:5 is observed. The speed-up results from fewer source term evaluations required to compute the Jacobian matrices.
NASA Astrophysics Data System (ADS)
Olsen, Thomas; Hou, Yu; Kowalski, Adam; Wiener, Richard
2006-05-01
The Reaction-Diffusion model predicted a period doubling cascade to chaos in a situation analagous Taylor- Couette flow with hourglass geometry. This cascade to chaos was discovered in the actual fluid flow experiments. We model Taylor-Couette flow in a cylindrical geometry with multiple waists of super-critical flow connected by regions of barely super-critical flow by corresponding Reaction-Diffusion models. We compare our results to the findings of an ongoing experimental program. H. Riecke and H.-G. Paap, Europhys. Lett. 14, 1235 (1991). Richard J. Wiener et al, Phys. Rev. E 55, 5489 (1997).
NASA Astrophysics Data System (ADS)
Bagchi, Partha; Das, Arpan; Sengupta, Srikumar; Srivastava, Ajit M.
2016-02-01
There are indications of formation of a thermalized medium in high multiplicity p p collisions at energies available at the CERN Large Hadron Collider. It is possible that such a medium may reach high enough energy density and temperature that a transient stage of quark-gluon plasma, where chiral symmetry is restored, may be achieved. Due to rapid three-dimensional expansion, the system will quickly cool, undergoing a spontaneous chiral symmetry breaking transition. We study the dynamics of the chiral field, after the symmetry breaking transition, for such an event using a reaction-diffusion equation approach which we have recently applied for studying QCD transitions in relativistic heavy-ion collisions. We show that the interior of such a rapidly expanding system is likely to lead to the formation of a single large domain of disoriented chiral condensate (DCC), which has been a subject of intensive search in earlier experiments. We argue that large multiplicity p p collisions naturally give rise to required boundary conditions for the existence of slowly propagating front solutions of the reaction-diffusion equation with resulting dynamics of the chiral field leading to the formation of a large DCC domain.
NASA Astrophysics Data System (ADS)
Pulikkottil, V. V.; Sujith, R. I.
2015-07-01
In this paper, instability mechanisms in a low Mach number reacting flow are investigated. Here, the emphasis is on the growth or decay of acoustic oscillations which arise from the acoustic-hydrodynamic interaction in a low Mach number reacting flow. Motivated by the studies in magnetohydrodynamics and atmospheric flows, we propose to investigate the acoustic-hydrodynamic coupling as a system of wave-mean flow interaction. For example, a comparison with the heat fluctuation modified hydrodynamics associated with magnetohydrodynamics is useful in understanding this coupling. The wavelike acoustic disturbance is introduced here as a compressibility correction to the mean flow. Accounting for the multiple scales introduced by the weak compressibility, we derive a set of equations governing wave-mean flow interaction in a reacting low Mach number flow. Sources such as volume expansion (which, in atmospheric flows arises due to the density variation with altitude) occur in reacting flows due to the heat release rate. This heat release rate, when coupled with the acoustic field, often leads to self-sustained thermo-acoustic oscillations. In the study of such oscillations, we discover a relation between the acoustic pressure and second order thermal fluctuations. Further, using this relation, we discover the nonlinear coupling mechanism that would lead to self-sustained oscillations in a reacting low Mach number flow. This mechanism, represented by a coupled convection reaction diffusion system, is presented here for the first time. In addition to the acoustic pressure and temperature fields, we also discover the role of acoustic velocity field in the acoustic-hydrodynamic interaction through a convective and lift-up mechanism.
NASA Astrophysics Data System (ADS)
Ding, Hongxia; Chen, Shangbin; Zeng, Shuai; Zeng, Shaoqun; Liu, Qian; Luo, Qingming
2008-12-01
Spreading depression (SD) shows as propagating suppression of electrical activity, which relates with migraine and focal cerebral ischaemia. The putative mechanism of SD is the reaction-diffusion hypothesis involving potassium ions. In part inspired by optical imaging of two SD waves collision, we aimed to show the merged and large wavefront but not annihilation during collision by experimental and computational study. This paper modified Reggia et al established bistable equation with recovery to compute and visualize SD. Firstly, the media tissue of SD was assumed as one-dimensional continuum. The Crank-Nicholson method was used to solve the modified equations with recovery term. Then, the computation results were extended to two-dimensional space by symmetry. One individual SD was visualized as a concentric wave initiating from the stimulation point. The mergence but not annihilation of two colliding waves of SD was demonstrated. In addition, the dynamics of SD depending on the parameters was studied and presented. The results allied SD with the emerging concepts of volume transmission. This work not only supplied a paradigm to compute and visualize SD but also became a tool to explore the mechanisms of SD.
Competing computational approaches to reaction-diffusion equations in clusters of cells
NASA Astrophysics Data System (ADS)
Stella, Sabrina; Chignola, Roberto; Milotti, Edoardo
2014-03-01
We have developed a numerical model that simulates the growth of small avascular solid tumors. At its core lies a set of partial differential equations that describe diffusion processes as well as transport and reaction mechanisms of a selected number of nutrients. Although the model relies on a restricted subset of molecular pathways, it compares well with experiments, and its emergent properties have recently led us to uncover a metabolic scaling law that stresses the common mechanisms that drive tumor growth. Now we plan to expand the biochemical model at the basis of the simulator to extend its reach. However, the introduction of additional molecular pathways requires an extensive revision of the reaction-diffusion part of the C++ code to make it more modular and to boost performance. To this end, we developed a novel computational abstract model where the individual molecular species represent the basic computational building blocks. Using a simple two-dimensional toy model to benchmark the new code, we find that the new implementation produces a more modular code without affecting performance. Preliminary results also show that a factor 2 speedup can be achieved with OpenMP multithreading, and other very preliminary results indicate that at least an order-of-magnitude speedup can be obtained using an NVidia Fermi GPU with CUDA code.
NASA Astrophysics Data System (ADS)
Cherniha, Roman; King, John R.; Kovalenko, Sergii
2016-07-01
Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on the form of diffusivity and source (sink) in the equations is derived. It is established that there exists a subclass in 1-D space admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable. A special power-law diffusivity with a fixed exponent, which leads to wider Lie invariance of the equations in question in 2-D space, is also derived. However, it is shown that the diffusion equation without a source term (which often arises in applications and is sometimes called the Perona-Malik equation) possesses no rich variety of Lie symmetries depending on the form of gradient-dependent diffusivity. The results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear 2-D equation with power-law diffusivity, also are included.
A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species.
Peng, Rui; Zhao, Xiao-Qiang
2016-02-01
In this article, we are concerned with a nonlocal reaction-diffusion-advection model which describes the evolution of a single phytoplankton species in a eutrophic vertical water column where the species relies solely on light for its metabolism. The new feature of our modeling equation lies in that the incident light intensity and the death rate are assumed to be time periodic with a common period. We first establish a threshold type result on the global dynamics of this model in terms of the basic reproduction number R0. Then we derive various characterizations of R0 with respect to the vertical turbulent diffusion rate, the sinking or buoyant rate and the water column depth, respectively, which in turn give rather precise conditions to determine whether the phytoplankton persist or become extinct. Our theoretical results not only extend the existing ones for the time-independent case, but also reveal new interesting effects of the modeling parameters and the time-periodic heterogeneous environment on persistence and extinction of the phytoplankton species, and thereby suggest important implications for phytoplankton growth control. PMID:26063527
Local traps as nanoscale reaction-diffusion probes: B clustering in c-Si
Pawlak, B. J.; Cowern, N. E. B.; Ahn, C.; Vandervorst, W.; Gwilliam, R.; Berkum, J. G. M. van
2014-12-01
A series of B implantation experiments into initially amorphized and not fully recrystallized Si, i.e., into an existing a/c-Si bi-layer material, have been conducted. We varied B dose, energy, and temperature during implantation process itself. Significant B migration has been observed within c-Si part near the a/c-interface and near the end-of-range region before any activation annealing. We propose a general concept of local trapping sites as experimental probes of nanoscale reaction-diffusion processes. Here, the a/c-Si interface acts as a trap, and the process itself is explored as the migration and clustering of mobile BI point defects in nearby c-Si during implantation at temperatures from 77 to 573 K. We find that at room temperature—even at B concentrations as high as 1.6 atomic %, the key B-B pairing step requires diffusion lengths of several nm owing to a small, ∼0.1 eV, pairing energy barrier. Thus, in nanostructures doped by ion implantation, the implant distribution can be strongly influenced by thermal migration to nearby impurities, defects, and interfaces.
A reaction-diffusion model of the Darien Gap Sterile Insect Release Method
NASA Astrophysics Data System (ADS)
Alford, John G.
2015-05-01
The Sterile Insect Release Method (SIRM) is used as a biological control for invasive insect species. SIRM involves introducing large quantities of sterilized male insects into a wild population of invading insects. A fertile/sterile mating produces offspring that are not viable and the wild insect population will eventually be eradicated. A U.S. government program maintains a permanent sterile fly barrier zone in the Darien Gap between Panama and Columbia to control the screwworm fly (Cochliomyia Hominivorax), an insect that feeds off of living tissue in mammals and has devastating effects on livestock. This barrier zone is maintained by regular releases of massive quantities of sterilized male screwworm flies from aircraft. We analyze a reaction-diffusion model of the Darien Gap barrier zone. Simulations of the model equations yield two types of spatially inhomogeneous steady-state solutions representing a sterile fly barrier that does not prevent invasion and a barrier that does prevent invasion. We investigate steady-state solutions using both phase plane methods and monotone iteration methods and describe how barrier width and the sterile fly release rate affects steady-state behavior.
Guterl, Jerome Smirnov, R. D.; Krasheninnikov, S. I.
2015-07-28
Desorption phase of thermal desorption spectroscopy (TDS) experiments performed on tungsten samples exposed to flux of hydrogen isotopes in fusion relevant conditions is analyzed using a reaction-diffusion model describing hydrogen retention in material bulk. Two regimes of hydrogen desorption are identified depending on whether hydrogen trapping rate is faster than hydrogen diffusion rate in material during TDS experiments. In both regimes, a majority of hydrogen released from material defects is immediately outgassed instead of diffusing deeply in material bulk when the evolution of hydrogen concentration in material is quasi-static, which is the case during TDS experiments performed with tungsten samples exposed to flux of hydrogen isotopes in fusion related conditions. In this context, analytical expressions of the hydrogen outgassing flux as a function of the material temperature are obtained with sufficient accuracy to describe main features of thermal desorption spectra (TDSP). These expressions are then used to highlight how characteristic temperatures of TDSP depend on hydrogen retention parameters, such as trap concentration or activation energy of detrapping processes. The use of Arrhenius plots to characterize retention processes is then revisited when hydrogen trapping takes place during TDS experiments. Retention processes are also characterized using the shape of desorption peaks in TDSP, and it is shown that diffusion of hydrogen in material during TDS experiment can induce long desorption tails visible aside desorption peaks at high temperature in TDSP. These desorption tails can be used to estimate activation energy of diffusion of hydrogen in material.
NASA Astrophysics Data System (ADS)
Cooper, Crystal Diane
A computer program was modified to model the dynamics of morphogen concentrations in a developing eye of a Xenopus laevis frog. The dynamics were modelled because it is believed that the behavior of the morphogen concentrations determine how the developing eye maps to the brain. The eye in the xenophus grows as a series of rings, and thus this is the model used. The basis for the simulation are experiments done by Sullivan et al. Following the experiment, aIl eye ring is 'split' in half, inverted, and then 'pasted' onto a donor half. The purpose of the program is to replicate and analyze the results that were found experimentally: a graft made on a north to south axis (dorsal to ventral) produces a change in vision along the east to west axis (anterior to posterior). Four modified Gierer-Meinhardt reaction- diffusion equations are used to simulate the operation. In the second part of the research, the program was further modified and a time series analysis was done on the results. It was found that the modified Gierer- Meinhardt equations demonstrated chaotic behavior under certain conditions. The dynamics included fixed points, limit cycles, transient chaos, intermittent chaos, and strange attractors. The creation and destruction of fractal torii was found.
Bifurcation analysis of a delay reaction-diffusion malware propagation model with feedback control
NASA Astrophysics Data System (ADS)
Zhu, Linhe; Zhao, Hongyong; Wang, Xiaoming
2015-05-01
With the rapid development of network information technology, information networks security has become a very critical issue in our work and daily life. This paper attempts to develop a delay reaction-diffusion model with a state feedback controller to describe the process of malware propagation in mobile wireless sensor networks (MWSNs). By analyzing the stability and Hopf bifurcation, we show that the state feedback method can successfully be used to control unstable steady states or periodic oscillations. Moreover, formulas for determining the properties of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Finally, we conduct extensive simulations on large-scale MWSNs to evaluate the proposed model. Numerical evidences show that the linear term of the controller is enough to delay the onset of the Hopf bifurcation and the properties of the bifurcation can be regulated to achieve some desirable behaviors by choosing the appropriate higher terms of the controller. Furthermore, we obtain that the spatial-temporal dynamic characteristics of malware propagation are closely related to the rate constant for nodes leaving the infective class for recovered class and the mobile behavior of nodes.
Quantitative modeling of the reaction/diffusion kinetics of two-chemistry photopolymers
NASA Astrophysics Data System (ADS)
Kowalski, Benjamin Andrew
Optically driven diffusion in photopolymers is an appealing material platform for a broad range of applications, in which the recorded refractive index patterns serve either as images (e.g. data storage, display holography) or as optical elements (e.g. custom GRIN components, integrated optical devices). A quantitative understanding of the reaction/diffusion kinetics is difficult to obtain directly, but is nevertheless necessary in order to fully exploit the wide array of design freedoms in these materials. A general strategy for characterizing these kinetics is proposed, in which key processes are decoupled and independently measured. This strategy enables prediction of a material's potential refractive index change, solely on the basis of its chemical components. The degree to which a material does not reach this potential reveals the fraction of monomer that has participated in unwanted reactions, reducing spatial resolution and dynamic range. This approach is demonstrated for a model material similar to commercial media, achieving quantitative predictions of index response over three orders of exposure dose (~1 to ~103 mJ cm-2) and three orders of feature size (0.35 to 500 microns). The resulting insights enable guided, rational design of new material formulations with demonstrated performance improvement.
Local traps as nanoscale reaction-diffusion probes: B clustering in c-Si
NASA Astrophysics Data System (ADS)
Pawlak, B. J.; Cowern, N. E. B.; Ahn, C.; Vandervorst, W.; Gwilliam, R.; van Berkum, J. G. M.
2014-12-01
A series of B implantation experiments into initially amorphized and not fully recrystallized Si, i.e., into an existing a/c-Si bi-layer material, have been conducted. We varied B dose, energy, and temperature during implantation process itself. Significant B migration has been observed within c-Si part near the a/c-interface and near the end-of-range region before any activation annealing. We propose a general concept of local trapping sites as experimental probes of nanoscale reaction-diffusion processes. Here, the a/c-Si interface acts as a trap, and the process itself is explored as the migration and clustering of mobile BI point defects in nearby c-Si during implantation at temperatures from 77 to 573 K. We find that at room temperature—even at B concentrations as high as 1.6 atomic %, the key B-B pairing step requires diffusion lengths of several nm owing to a small, ˜0.1 eV, pairing energy barrier. Thus, in nanostructures doped by ion implantation, the implant distribution can be strongly influenced by thermal migration to nearby impurities, defects, and interfaces.
Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations
NASA Astrophysics Data System (ADS)
Flegg, Mark B.; Hellander, Stefan; Erban, Radek
2015-05-01
In this paper, three multiscale methods for coupling of mesoscopic (compartment-based) and microscopic (molecular-based) stochastic reaction-diffusion simulations are investigated. Two of the three methods that will be discussed in detail have been previously reported in the literature; the two-regime method (TRM) and the compartment-placement method (CPM). The third method that is introduced and analysed in this paper is called the ghost cell method (GCM), since it works by constructing a "ghost cell" in which molecules can disappear and jump into the compartment-based simulation. Presented is a comparison of sources of error. The convergent properties of this error are studied as the time step Δt (for updating the molecular-based part of the model) approaches zero. It is found that the error behaviour depends on another fundamental computational parameter h, the compartment size in the mesoscopic part of the model. Two important limiting cases, which appear in applications, are considered: Δt → 0 and h is fixed; Δt → 0 and h → 0 such that √{ Δt } / h is fixed. The error for previously developed approaches (the TRM and CPM) converges to zero only in the limiting case (ii), but not in case (i). It is shown that the error of the GCM converges in the limiting case (i). Thus the GCM is superior to previous coupling techniques if the mesoscopic description is much coarser than the microscopic part of the model.
Wong, Ken C L; Summers, Ronald M; Kebebew, Electron; Yao, Jianhua
2015-10-01
The goal of tumor growth prediction is to model the tumor growth process, which can be achieved by physiological modeling and model personalization from clinical measurements. Although image-driven frameworks have been proposed with promising results, several issues such as infinitesimal strain assumptions, complicated personalization procedures, and the lack of functional information, may limit their prediction accuracy. In view of these issues, we propose a framework for pancreatic neuroendocrine tumor growth prediction, which comprises a FEM-based tumor growth model with coupled reaction-diffusion equation and nonlinear biomechanics. Physiological data fusion of structural and functional images is used to improve the subject-specificity of model personalization, and a derivative-free global optimization algorithm is adopted to facilitate the complicated model and accommodate flexible choices of objective functions. With this flexibility, we propose an objective function accounting for both the tumor volume difference and the root-mean-squared error of intracellular volume fractions. Experiments were performed on synthetic and clinical data to verify the parameter estimation capability and the prediction performance. Comparisons of using different biomechanical models and objective functions were also performed. From the experimental results of eight patient data sets, the average recall, precision, Dice coefficient, and relative volume difference between predicted and measured tumor volumes were 84.5 ± 6.9%, 85.8 ± 8.2%, 84.6 ± 1.7%, and 14.2 ± 8.4%, respectively. PMID:25962846
Cubic autocatalysis in a reaction-diffusion annulus: semi-analytical solutions
NASA Astrophysics Data System (ADS)
Alharthi, M. R.; Marchant, T. R.; Nelson, M. I.
2016-06-01
Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction-diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.
NASA Astrophysics Data System (ADS)
Lubuma, J. M.-S.; Mureithi, E.; Terefe, Y. A.
2011-11-01
The classical SIS epidemiological model is extended in two directions: (a) The number of adequate contacts per infective in unit time is assumed to be a function of the total population in such a way that this number grows less rapidly as the total population increases; (b) A diffusion term is added to the SIS model and this leads to a reaction diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the model (a), with the disease-free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, traveling wave solutions are found for the model (b). Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous models are presented. In particular, for the model (a), a nonstandard version of the Runge-Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided.
Evolutionary distributions and competition by way of reaction-diffusion and by way of convolution.
Cohen, Yosef; Galiano, Gonzalo
2013-12-01
Evolution by natural selection is the most ubiquitous and well understood process of evolution. We say distribution instead of the distribution of the density of populations of phenotypes across the values of their adaptive traits. A phenotype refers to an organism that exhibits a set of values of adaptive traits. An adaptive trait is a trait that a phenotype exhibits where the trait is subject to natural selection. Natural selection is a process by which populations of different phenotypes decline at different rates. An evolutionary distribution (ED) encapsulates the dynamics of evolution by natural selection. The main results are: (i) ED are derived by way of PDE of reaction-diffusion type and by way of integro-differential equations. The latter capture mutations through convolution of a kernel with the rate of growth of a population. The kernel controls the size and rate of mutations. (ii) The numerical solution of a logistic-like ED driven by competition corresponds to a bounded traveling wave solution of population models based on the logistic. (iii) Competition leads to increase in diversity of phenotypes on a single ED. Diversity refers to change in the number of local maxima (minima) within the bounds of values of adaptive traits. (iv) The principle of competitive exclusion in the context of evolution depends, smoothly, on the size and rate of mutations. (v) We identify the sensitivity—with respect to survival—of phenotypes to changes in values of adaptive traits to be an important parameter: increase in the value of this parameter results in decrease in evolutionary-based diversity. (vi) Stable ED corresponds to Evolutionary Stable Strategy; the latter refers to the outcome of a game of evolution. PMID:24122397
Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations
Flegga, Mark B.; Hellander, Stefan; Erban, Radek
2015-01-01
In this paper, three multiscale methods for coupling of mesoscopic (compartment-based) and microscopic (molecular-based) stochastic reaction-diffusion simulations are investigated. Two of the three methods that will be discussed in detail have been previously reported in the literature; the two-regime method (TRM) and the compartment-placement method (CPM). The third method that is introduced and analysed in this paper is called the ghost cell method (GCM), since it works by constructing a “ghost cell” in which molecules can disappear and jump into the compartment-based simulation. Presented is a comparison of sources of error. The convergent properties of this error are studied as the time step Δt (for updating the molecular-based part of the model) approaches zero. It is found that the error behaviour depends on another fundamental computational parameter h, the compartment size in the mesoscopic part of the model. Two important limiting cases, which appear in applications, are considered: (i) Δt → 0 and h is fixed; (ii) Δt → 0 and h → 0 such that √Δt/h is fixed. The error for previously developed approaches (the TRM and CPM) converges to zero only in the limiting case (ii), but not in case (i). It is shown that the error of the GCM converges in the limiting case (i). Thus the GCM is superior to previous coupling techniques if the mesoscopic description is much coarser than the microscopic part of the model. PMID:26568640
Manifolds and front propagation barriers in advection-reaction-diffusion systems
NASA Astrophysics Data System (ADS)
Solomon, Tom
2015-03-01
We present experiments on the propagation of reaction fronts in laminar, vortex-dominated flows. The fronts are produced by the excitable Belousov-Zhabotinsky chemical reaction. The flows studied are driven by magnetohydrodynamic forcing techniques and are composed of a single vortex, chains or arrays of vortices, or spatially-disordered flows. In all of these cases, one-way barriers appear that either inhibit front propagation or, in some cases, pin the reactions fronts. We analyze this behavior with a recent theory of burning invariant manifolds (BIMs) that are a generalization of the theory of invariant manifolds developed in the past to characterize chaotic mixing and transport of passive impurities. We demonstrate that the BIMs are responsible for the reaction barriers observed experimentally, and we discuss the applicability of this BIM formalism to a range of flows: time-independent, time-periodic and time-aperiodic. Supported by NSF Grants DMR-1004744, DMR-1361881 and PHY-1156964.
Reaction diffusion in the nickel-chromium-aluminum and cobalt-chromium-aluminum systems
NASA Technical Reports Server (NTRS)
Levine, S. R.
1977-01-01
The effects of MCrAl coating-substrate interdiffusion on oxidation life and the general mutliphase, multicomponent diffusion problem were examined. Semi-infinite diffusion couples that had sources representing coatings and sinks representing gas turbine alloys were annealed at 1,000, 1,095, 1,150, or 1,205 C for as long as 500 hours. The source and sink aluminum and chromium contents and the base metal (cobalt or nickel) determined the parabolic diffusion rate constants of the couples and predicted finite coating lives. The beta source strength concept provided a method (1) for correlating beta recession rate constants with composition; (2) for determining reliable average total, diffusion, and constitutional activation energies; and (3) for calculating interdiffusion coefficients.
Reaction-diffusion systems in natural sciences and new technology transfer
NASA Astrophysics Data System (ADS)
Keller, André A.
2012-12-01
Diffusion mechanisms in natural sciences and innovation management involve partial differential equations (PDEs). This is due to their spatio-temporal dimensions. Functional semi-discretized PDEs (with lattice spatial structures or time delays) may be even more adapted to real world problems. In the modeling process, PDEs can also formalize behaviors, such as the logistic growth of populations with migration, and the adopters’ dynamics of new products in innovation models. In biology, these events are related to variations in the environment, population densities and overcrowding, migration and spreading of humans, animals, plants and other cells and organisms. In chemical reactions, molecules of different species interact locally and diffuse. In the management of new technologies, the diffusion processes of innovations in the marketplace (e.g., the mobile phone) are a major subject. These innovation diffusion models refer mainly to epidemic models. This contribution introduces that modeling process by using PDEs and reviews the essential features of the dynamics and control in biological, chemical and new technology transfer. This paper is essentially user-oriented with basic nonlinear evolution equations, delay PDEs, several analytical and numerical methods for solving, different solutions, and with the use of mathematical packages, notebooks and codes. The computations are carried out by using the software Wolfram Mathematica®7, and C++ codes.
Analysis of reaction-diffusion systems for flame capturing in type IA supernova simulations
NASA Astrophysics Data System (ADS)
Zhyglo, Andriy V.
2009-06-01
We present a study of numerical behavior of a thickened flame used in Flame Capturing (FC, Khokhlov (1995)) for tracking thin physical flames in deflagration simulations. This technique, used extensively in astrophysics, utilizes artificial flame variable to evolve flame region, width of which is resolved in simulations, with physically motivated propagation speed. We develop a steady-state procedure for calibrating flame model used in FC, and test it against analytical results. Original flame model is properly calibrated with taking matter expansion into consideration and keeping artificial flame width at predetermined value regardless of expansion. We observe numerical noises generated by original realization of the technique. Alternative artificial burning rates are discussed, which produce acceptably quiet flames (relative dispersion in propagation speed within 0.1% at physically interesting ratios of fuel and ash densities). Two new quiet models are calibrated to yield required "flame" speed and width, and further studied in 2D and 3D setting. Landau-Darrieus type instabilities of the flames are observed. One model also shows significantly anisotropic propagation speed on the grid, both effects increasingly pronounced at larger matter expansion as a result of burning; these 2D/3D effects make that model unacceptable for use in type Ia supernova simulations at fuel densities below about 100 tons per cubic centimeter. Another model, first presented here, looks promising for use in flame capturing at fuel to ash density ratio of order 3 and below, the interval of most interest for astrophysical applications. No model was found to significantly inhibit LD instability development at larger expansions without increasing flame width. The model we propose, "Model B", yields flames completely localized within a region 6 cells wide at any expansion. We study Markstein effect (speed of the flame dependence on its curvature) for flame models described, through direct numerical simulations and through quasi- steady technique developed. By comparing results obtained by the 2 approaches we demonstrate that Markstein effect dominates instability effects on curved flame speed for Model B in 2D simulations for fuel to ash density of about 2.5 and below. We find critical wavelength of LD instability by direct simulations of perturbed nearly planar flames; these agree with analytical predictions with Markstein number values found in this work. We conclude that the behavior of model B is well understood, and optimal for FC applications among all flame models proposed to date.
Experimental studies of coherent structures in an advection-reaction-diffusion system.
Gowen, Savannah; Solomon, Tom
2015-08-01
We present experimental studies of reaction front propagation in a single vortex flow with an imposed external wind. The fronts are produced by the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. The flow is generated using an electromagnetic forcing technique: an almost-radial electrical current interacts with a magnetic field from a magnet below the fluid layer to produce the vortex. The magnet is mounted on crossed translation stages allowing for movement of the vortex through the flow. Reaction fronts triggered in or in front of the moving vortex form persistent structures that are seen experimentally for time-independent (constant motion), time-periodic, and time-aperiodic flows. These results are examined with the use of burning invariant manifolds that act as one-way barriers to front motion in the flows. We also explore the usefulness of finite-time Lyapunov exponent fields as an instrument for analyzing front propagation behavior in a fluid flow. PMID:26328574
NASA Astrophysics Data System (ADS)
Mena, Andres; Ferrero, Jose M.; Rodriguez Matas, Jose F.
2015-11-01
Solving the electric activity of the heart possess a big challenge, not only because of the structural complexities inherent to the heart tissue, but also because of the complex electric behaviour of the cardiac cells. The multi-scale nature of the electrophysiology problem makes difficult its numerical solution, requiring temporal and spatial resolutions of 0.1 ms and 0.2 mm respectively for accurate simulations, leading to models with millions degrees of freedom that need to be solved for thousand time steps. Solution of this problem requires the use of algorithms with higher level of parallelism in multi-core platforms. In this regard the newer programmable graphic processing units (GPU) has become a valid alternative due to their tremendous computational horsepower. This paper presents results obtained with a novel electrophysiology simulation software entirely developed in Compute Unified Device Architecture (CUDA). The software implements fully explicit and semi-implicit solvers for the monodomain model, using operator splitting. Performance is compared with classical multi-core MPI based solvers operating on dedicated high-performance computer clusters. Results obtained with the GPU based solver show enormous potential for this technology with accelerations over 50 × for three-dimensional problems.
Simulating Stochastic Reaction-Diffusion Systems on and within Moving Boundaries
Ghosh, Atiyo; Marquez-Lago, Tatiana T.
2015-01-01
Chemical reactions inside cells are generally considered to happen within fixed-size compartments. However, cells and their compartments are highly dynamic. Thus, such stringent geometrical assumptions may not reflect biophysical reality, and can highly bias conclusions from simulation studies. In this work, we present an intuitive algorithm for particle-based diffusion in and on moving boundaries, for both point particles and spherical particles. We first benchmark our proposed stochastic method against solutions of partial differential equations in appropriate scenarios, and further demonstrate that moving boundaries can give rise to super-diffusive motion as well as time-inhomogeneous reaction rates. Finally, we conduct a numerical experiment representing photobleaching of diffusing fluorescent proteins in dividing Saccharomyces cerevisiae cells to demonstrate that moving boundaries might cause important effects neglected in previously published studies of cell compartmentalization. PMID:26230406
Experimental studies of coherent structures in an advection-reaction-diffusion system
NASA Astrophysics Data System (ADS)
Gowen, Savannah; Solomon, Tom
2015-08-01
We present experimental studies of reaction front propagation in a single vortex flow with an imposed external wind. The fronts are produced by the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. The flow is generated using an electromagnetic forcing technique: an almost-radial electrical current interacts with a magnetic field from a magnet below the fluid layer to produce the vortex. The magnet is mounted on crossed translation stages allowing for movement of the vortex through the flow. Reaction fronts triggered in or in front of the moving vortex form persistent structures that are seen experimentally for time-independent (constant motion), time-periodic, and time-aperiodic flows. These results are examined with the use of burning invariant manifolds that act as one-way barriers to front motion in the flows. We also explore the usefulness of finite-time Lyapunov exponent fields as an instrument for analyzing front propagation behavior in a fluid flow.
Finite-scale singularity in the renormalization group flow of a reaction-diffusion system.
Gredat, Damien; Chaté, Hugues; Delamotte, Bertrand; Dornic, Ivan
2014-01-01
We study the nonequilibrium critical behavior of the pair contact process with diffusion (PCPD) by means of nonperturbative functional renormalization group techniques. We show that usual perturbation theory fails because the effective potential develops a nonanalyticity at a finite length scale: Perturbatively forbidden terms are dynamically generated and the flow can be continued once they are taken into account. Our results suggest that the critical behavior of PCPD can be either in the directed percolation or in a different (conjugated) universality class. PMID:24580152
Wu, Sheng; Li, Hong; Petzold, Linda R.
2015-01-01
The inhomogeneous stochastic simulation algorithm (ISSA) is a fundamental method for spatial stochastic simulation. However, when diffusion events occur more frequently than reaction events, simulating the diffusion events by ISSA is quite costly. To reduce this cost, we propose to use the time dependent propensity function in each step. In this way we can avoid simulating individual diffusion events, and use the time interval between two adjacent reaction events as the simulation stepsize. We demonstrate that the new algorithm can achieve orders of magnitude efficiency gains over widely-used exact algorithms, scales well with increasing grid resolution, and maintains a high level of accuracy. PMID:26609185
NASA Astrophysics Data System (ADS)
Xu, Zhaoquan; Xiao, Dongmei
2016-01-01
A class of reaction diffusion equation with spatio-temporal delays is systematically investigated. When the reaction function of this equation is nonlinear without monotonicity, it is shown that there exists a spreading speed c* > 0 for this equation such that c* is linearly determinate and coincides with the minimal wave speed of traveling waves, and that this equation admits a unique traveling wave (up to translation) with speed c >c* and no traveling wave with c
NASA Astrophysics Data System (ADS)
Hormuth, David A., II; Weis, Jared A.; Barnes, Stephanie L.; Miga, Michael I.; Rericha, Erin C.; Quaranta, Vito; Yankeelov, Thomas E.
2015-07-01
Reaction-diffusion models have been widely used to model glioma growth. However, it has not been shown how accurately this model can predict future tumor status using model parameters (i.e., tumor cell diffusion and proliferation) estimated from quantitative in vivo imaging data. To this end, we used in silico studies to develop the methods needed to accurately estimate tumor specific reaction-diffusion model parameters, and then tested the accuracy with which these parameters can predict future growth. The analogous study was then performed in a murine model of glioma growth. The parameter estimation approach was tested using an in silico tumor ‘grown’ for ten days as dictated by the reaction-diffusion equation. Parameters were estimated from early time points and used to predict subsequent growth. Prediction accuracy was assessed at global (total volume and Dice value) and local (concordance correlation coefficient, CCC) levels. Guided by the in silico study, rats (n = 9) with C6 gliomas, imaged with diffusion weighted magnetic resonance imaging, were used to evaluate the model’s accuracy for predicting in vivo tumor growth. The in silico study resulted in low global (tumor volume error <8.8%, Dice >0.92) and local (CCC values >0.80) level errors for predictions up to six days into the future. The in vivo study showed higher global (tumor volume error >11.7%, Dice <0.81) and higher local (CCC <0.33) level errors over the same time period. The in silico study shows that model parameters can be accurately estimated and used to accurately predict future tumor growth at both the global and local scale. However, the poor predictive accuracy in the experimental study suggests the reaction-diffusion equation is an incomplete description of in vivo C6 glioma biology and may require further modeling of intra-tumor interactions including segmentation of (for example) proliferative and necrotic regions.
MINIATURE ACID CONDENSATION SYSTEM: DESIGN AND OPERATION
An extractive source sampling system was designed and constructed. The sampling system measures gaseous sulfuric acid and sulfur dioxide in combustion emissions. The miniature acid condensation system (MACS) includes a high-temperature quartz probe and quartz-filter holder. Since...
Prabhu, Vivek M.; Kang, Shuhui; Sha, Jing; Bonnesen, Peter V; Satija, Sushil K.; Wu, Wen-li; Ober, Christoper K.
2012-01-01
Lithographic feature size requirements have approached a few radius of gyration of photoresist polymers used in thin-film patterning. Furthermore, the feature dimensions are commensurate with the photoacid diffusion length that defines the underlying latent image. Smaller imaging building blocks may enable reduced feature sizes; however, resolution limits are also dependent upon the spatial extent of the photoacid-catalyzed reaction diffusion front and subsequent dissolution mechanism. The reaction-diffusion front was characterized by neutron reflectivity for ccc stereoisomer-purified, deuterium-labeled tert-butoxycarbonyloxy calix[4]resorcinarene molecular resists. The spatial extent of the reaction front exceeds the size of the molecular resist with an effective diffusion constant of (0.13 0.06) nm2/s for reaction times longer than 60 s, with the maximum at shorter times. Comparison to a mean-field reaction-diffusion model shows that a photoacid trapping process provides bounds to the spatial and extent of reaction via a reaction-limited mechanism whereas the ratio of the reaction rate to trapping rate constants recovers the effective diffusion peak. Under the ideal step-exposure conditions, surface roughness was observed after either positive- or negative-tone development. However, negative-tone development follows a surface restructuring mechanism rather than etch-like dissolution in positive-tone development.
Beaghton, Andrea; Beaghton, Pantelis John; Burt, Austin
2016-04-01
Some genes or gene complexes are transmitted from parents to offspring at a greater-than-Mendelian rate, and can spread and persist in populations even if they cause some harm to the individuals carrying them. Such genes may be useful for controlling populations or species that are harmful. Driving-Y chromosomes may be particularly potent in this regard, as they produce a male-biased sex ratio that, if sufficiently extreme, can lead to population elimination. To better understand the potential of such genes to spread over a landscape, we have developed a series of reaction-diffusion models of a driving-Y chromosome in 1-D and radially-symmetric 2-D unbounded domains. The wild-type system at carrying capacity is found to be unstable to the introduction of driving-Y males for all models investigated. Numerical solutions exhibit travelling wave pulses and fronts, and analytical and semi-analytical solutions for the asymptotic wave speed under bounded initial conditions are derived. The driving-Y male invades the wild-type equilibrium state at the front of the wave and completely replaces the wild-type males, leaving behind, at the tail of the wave, a reduced- or zero-population state of females and driving-Y males only. In our simplest model of a population with one life stage and density-dependent mortality, wave speed depends on the strength of drive and the diffusion rate of Y-drive males, and is independent of the population dynamic consequences (suppression or elimination). Incorporating an immobile juvenile stage of fixed duration into the model reduces wave speed approximately in proportion to the relative time spent as a juvenile. If females mate just once in their life, storing sperm for subsequent reproduction, then wave speed depends on the movement of mated females as well as Y-drive males, and may be faster or slower than in the multiple-mating model, depending on the relative duration of juvenile and adult life stages. Numerical solutions are shown for
NASA Astrophysics Data System (ADS)
Benguria, R. D.; Depassier, M. C.
2016-04-01
We study the dynamics of the equation obtained by Schryer and Walker for the motion of domain walls. The reduced equation is a reaction diffusion equation for the angle between the applied field and the magnetization vector. If the hard-axis anisotropy Kd is much larger than the easy-axis anisotropy Ku, there is a range of applied fields where the dynamics does not select the Schryer-Walker solution. We give an analytic expression for the speed of the domain wall in this regime and the conditions for its existence.
21 CFR 862.1450 - Lactic acid test system.
Code of Federal Regulations, 2013 CFR
2013-04-01
... 21 Food and Drugs 8 2013-04-01 2013-04-01 false Lactic acid test system. 862.1450 Section 862.1450....1450 Lactic acid test system. (a) Identification. A lactic acid test system is a device intended to measure lactic acid in whole blood and plasma. Lactic acid measurements that evaluate the acid-base...
21 CFR 862.1450 - Lactic acid test system.
Code of Federal Regulations, 2010 CFR
2010-04-01
....1450 Lactic acid test system. (a) Identification. A lactic acid test system is a device intended to measure lactic acid in whole blood and plasma. Lactic acid measurements that evaluate the acid-base status... 21 Food and Drugs 8 2010-04-01 2010-04-01 false Lactic acid test system. 862.1450 Section...
21 CFR 862.1450 - Lactic acid test system.
Code of Federal Regulations, 2012 CFR
2012-04-01
... 21 Food and Drugs 8 2012-04-01 2012-04-01 false Lactic acid test system. 862.1450 Section 862.1450....1450 Lactic acid test system. (a) Identification. A lactic acid test system is a device intended to measure lactic acid in whole blood and plasma. Lactic acid measurements that evaluate the acid-base...
Simpson, Matthew J
2015-01-01
Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction—diffusion process on 0
Micro-electro-mechanical systems phosphoric acid fuel cell
Sopchak, David A.; Morse, Jeffrey D.; Upadhye, Ravindra S.; Kotovsky, Jack; Graff, Robert T.
2010-08-17
A phosphoric acid fuel cell system comprising a porous electrolyte support, a phosphoric acid electrolyte in the porous electrolyte support, a cathode electrode contacting the phosphoric acid electrolyte, and an anode electrode contacting the phosphoric acid electrolyte.
Micro-electro-mechanical systems phosphoric acid fuel cell
Sopchak, David A.; Morse, Jeffrey D.; Upadhye, Ravindra S.; Kotovsky, Jack; Graff, Robert T.
2010-12-21
A phosphoric acid fuel cell system comprising a porous electrolyte support, a phosphoric acid electrolyte in the porous electrolyte support, a cathode electrode contacting the phosphoric acid electrolyte, and an anode electrode contacting the phosphoric acid electrolyte.
Sulfuric acid thermoelectrochemical system and method
Ludwig, Frank A.
1989-01-01
A thermoelectrochemical system in which an electrical current is generated between a cathode immersed in a concentrated sulfuric acid solution and an anode immersed in an aqueous buffer solution of sodium bisulfate and sodium sulfate. Reactants consumed at the electrodes during the electrochemical reaction are thermochemically regenerated and recycled to the electrodes to provide continuous operation of the system.