Stochastic Reaction-diffusion Systems Research Articles

Hydrodynamic limits and propagation of chaos for interacting random walks in domains

A new non-conservative stochastic reaction–diffusion system in which two families of random walks in two adjacent domains interact near the interface is introduced and studied in this paper. Such a system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. We show that in the macroscopic limit, the particle densities converge to the solution of a coupled nonlinear heat equations. For this, we first prove that propagation of chaos holds by establishing the uniqueness of a new BBGKY hierarchy. A local central limit theorem for reflected diffusions in bounded Lipschitz domains is also established as a crucial tool.

Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion System

Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS), Stochastic Reaction-Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation.

The spatiotemporal master equation: Approximation of reaction-diffusion dynamics via Markov state modeling.

Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest. We consider the spatiotemporal chemical master equation (ST-CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. The space of motion is decomposed into metastable compartments, and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by the Gillespie method. We present the theory of Markov state models as a theoretical foundation of this intuitive approach. By means of Markov state modeling, both the number and shape of compartments and the transition rates between them can be determined. We consider the ST-CME for two reaction-diffusion systems and compare it to more detailed models. Moreover, a rigorous formal justification of the ST-CME by Galerkin projection methods is presented.

Brownian-motion based simulation of stochastic reaction–diffusion systems for affinity based sensors

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.

ROBUSTNESS OF RANDOM ATTRACTORS FOR A STOCHASTIC REACTION-DIFFUSION SYSTEM

Asymptotic pullback dynamics of a typical stochastic reactiondiffusion system, the reversible Schnackenberg equations, with multiplicative white noise is investigated. The robustness of random attractor with respect to the reverse reaction rate as it tends to zero is proved through the uniform pullback absorbing property and the uniform convergence of reversible to non-reversible cocycles. This result means that, even if the reverse reactions would be neglected, the dynamics of this class of stochastic reversible reaction-diffusion systems can still be captured by the random attractor of the non-reversible stochastic raction-diffusion system in a long run.

Existence and approximate controllability of stochastic semilinear reaction diffusion systems

In this paper, the approximate controllability of an abstract model of stochastic semilinear reaction diffusion equation in Hilbert spaces is considered. The existence and uniqueness of mild solutions of stochastic semilinear reaction diffusion equation is established by using contraction mapping principle and then a new set of sufficient conditions are formulated which guarantees the approximate controllability of main problem. Finally, the results are applied to stochastic Ornstein–Uhlenbeck equation.

Simulating Stochastic Reaction-Diffusion Systems on and within Moving Boundaries.

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.

Hybrid approaches for multiple-species stochastic reaction–diffusion models

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.

Stochastic Reaction-Diffusion Systems With Hölder Continuous Multiplicative Noise

We prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with a locally Lipschitz continuous reaction term of polynomial growth and Hölder continuous multiplicative noise. Under additional assumptions on the coefficients, we also prove positivity of the solutions.

Random Dynamics of Stochastic Reaction–Diffusion Systems with Additive Noise

For a typical autocatalytic stochastic reaction–diffusion system with additive noises, the multicomponent reversible Gray–Scott reaction–diffusion system on a two-dimensional bounded domain, the existence of a random attractor and its attracting regularity are proved through the sharp uniform estimates showing respectively the pullback absorbing, asymptotically compact, and flattening properties.

Keynote: In-Situ Visualization of Non-Linear Phenomena during Metastable Pitting Corrosion on Stainless Steel

Stainless steels and many other alloys are designed to be corrosion resistant. Nevertheless they can undergo localized pitting corrosion, which may rapidly lead to their failure, with the possibility of catastrophic events as a result. To study the initial steps of metastable pit formation in situ and in real time we adapted the ellipso-microscope for surface imaging (EMSI) [1] and a high resolution, contrast enhanced optical microscope to the electrolyte - stainless steel interface. Utilizing these methods we have been able to explain the sudden corrosion onset by an explosive autocatalytic growth in the number of metastable pits [2]. By applying those different but complementary imaging methods simultaneously the correlation between oxide film weakening and the nucleation of individual pits could be confirmed. The existence of front propagation as a component of the transition to pitting corrosion shows that characteristics of this process are consistent with the behavior of stochastic reaction-diffusion systems [3].Recently we implemented digital in-line holography as a third imaging tool simultaneously to gather three-dimensional information within the electrolyte, in close proximity to the exposed surface. This enhances the sensitivity for small pitting events and opens new avenues for investigating and understanding the underlying processes. For instance we are now able to follow the trajectories of particles ejected from a pit [4].Using this arsenal of in situ and real time imaging tools allows us to efficiently study methods to enhance the pitting corrosion resistance of stainless steels; initial results of simple treatments such as emerging the sample just in high purity deionized water at 90 °C for an hour, but nevertheless improving the pitting corrosion resistance in NaCl solutions dramatically, will be discussed [5].[1] H.H. Rotermund et al. Science 270, 608-610 (1995)[2] C. Punckt et al., Science 305, 1133-1136 (2004)[3] M. Dornhege et al., J. Electrochem. Soc. 154, C24-C27 (2007)[4] P.E. Klages et al., Corrosion Science 65, 128–135 (2012)[5] P.E. Klages et al., Electrochemistry Communications 15, 54–58 (2012)

Fluorescence correlation spectroscopy and nonlinear stochastic reaction-diffusion.

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. Our results show that current linear FCS theory could be adequate for measurements on biological systems that contain many other sources of uncertainties. At the same time, it provides a framework for future measurements of nonlinear, fluctuating chemical reactions with high-precision FCS. Extending Delbrück-Gillespie's theory for stochastic nonlinear reactions with rapid stirring to reaction-diffusion systems provides a mesoscopic model for chemical and biochemical reactions at nanometric and mesoscopic levels, such as a single biological cell.

Adaptive two-regime method: Application to front propagation

The Adaptive Two-Regime Method (ATRM) is developed for hybrid (multiscale) stochastic simulation of reaction-diffusion problems. It efficiently couples detailed Brownian dynamics simulations with coarser lattice-based models. The ATRM is a generalization of the previously developed Two-Regime Method [Flegg et al., J. R. Soc., Interface 9, 859 (2012)] to multiscale problems which require a dynamic selection of regions where detailed Brownian dynamics simulation is used. Typical applications include a front propagation or spatio-temporal oscillations. In this paper, the ATRM is used for an in-depth study of front propagation in a stochastic reaction-diffusion system which has its mean-field model given in terms of the Fisher equation [R. Fisher, Ann. Eugen. 7, 355 (1937)]. It exhibits a travelling reaction front which is sensitive to stochastic fluctuations at the leading edge of the wavefront. Previous studies into stochastic effects on the Fisher wave propagation speed have focused on lattice-based models, but there has been limited progress using off-lattice (Brownian dynamics) models, which suffer due to their high computational cost, particularly at the high molecular numbers that are necessary to approach the Fisher mean-field model. By modelling only the wavefront itself with the off-lattice model, it is shown that the ATRM leads to the same Fisher wave results as purely off-lattice models, but at a fraction of the computational cost. The error analysis of the ATRM is also presented for a morphogen gradient model.

Stability in Mean of Partial Variables for Coupled Stochastic Reaction-Diffusion Systems on Networks: A Graph Approach

This paper is devoted to investigating stability in mean of partial variables for coupled stochastic reaction-diffusion systems on networks (CSRDSNs). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations (SODE) and using Itô formula, we establish some novel stability principles for uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, and exponential stability in mean of partial variables for CSRDSNs. These stability principles have a close relation with the topology property of the network. We also provide a systematic method for constructing global Lyapunov function for these CSRDSNs by using graph theory. The new method can help to analyze the dynamics of complex networks. An example is presented to illustrate the effectiveness and efficiency of the obtained results.

Stability in mean of partial variables for stochastic reaction–diffusion systems with Markovian switching

This paper is denoted to investigating stability in mean of partial variables for stochastic reaction–diffusion equations with Markovian switching (SRDEMS). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations with Markovian switching (SODEMS) and using Itô formula, sufficient criteria on uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, exponential stability in mean of partial variables for SRDEMS are first derived. An example is presented to illustrate the effectiveness and efficiency of the obtained results.

Random attractors and robustness for stochastic reversible reaction-diffusion systems

For a typical stochastic reversible reaction-diffusion system with multiplicative white noise, the trimolecular autocatalytic Gray-Scott system on a three-dimensional bounded domain with random noise perturbation proportional to the state of the system, the existence of a random attractor and its robustness with respect to the reverse reaction rates are proved through sharp and uniform estimates showing the pullback uniform dissipation and the pullback asymptotic compactness.

Self-Similarity and Attraction in Stochastic Nonlinear Reaction-Diffusion Systems

Self-similarity solutions play an important role in many fields of science. We explore self-similarity in some stochastic partial differential equations (spdes). Important issues are not only the existence of stochastic self-similarity but also whether a self-similar solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of spdes in a form to which stochastic center manifold theory may be applied, we resolve these issues in this class. For definiteness, a first example of self-similarity in solutions of Burgers equation driven by some stochastic force is studied. Under suitable assumptions a stationary solution is constructed which yields the existence of a stochastically self-similar solution for the stochastic Burgers equation. Further, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems, stochastic center manifold theory provides a framework for constructing s...

Global stability analysis for stochastic coupled reaction–diffusion systems on networks

Coupled systems on networks (CSNs) can be used to model many real systems, such as food webs, ecosystems, metabolic pathways, the Internet, World Wide Web, social networks, and global economic markets. This paper is devoted to investigation of the stability problem for some stochastic coupled reaction–diffusion systems on networks (SCRDSNs). A systematic method for constructing global Lyapunov function for these SCRDSNs is provided by using graph theory. The stochastic stability, asymptotically stochastic stability and globally asymptotically stochastic stability of the systems are investigated. The derived results are less conservative than the results recently presented in Luo and Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493]. In fact, the system discussed in Q. Luo and Y. Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493] is a special case of ours. Moreover, our novel stability principles have a close relation to the topological property of the networks. Our new method which constructs a relation between the stability criteria of a CSN and some topology property of the network, can help analyzing the stability of the complex networks by using the Lyapunov functional method.

Nanosize pattern formation in overdamped stochastic reaction-diffusion systems with interacting adsorbate

We study overdamped stochastic model describing adsorption or desorption processes with nonequilibrium chemical reactions on the surface. It is shown that internal noise satisfying the fluctuation-dissipation relation at small intensities governs transitions between ordered thermodynamical dense and diluted phases. These phase transitions are characterized by an increase of fluctuations of the coverage filed and correlation radius of spatial modulation. At large noise intensity a transition towards disordered phase with chaotic spatial configuration is realized. We have shown that organized stationary patterns are of nanometer range. We define that both period of stationary structures and corresponding correlation radius depend on the noise intensity.

MesoRD 1.0: Stochastic reaction-diffusion simulations in the microscopic limit

MesoRD is a tool for simulating stochastic reaction-diffusion systems as modeled by the reaction diffusion master equation. The simulated systems are defined in the Systems Biology Markup Language with additions to define compartment geometries. MesoRD 1.0 supports scale-dependent reaction rate constants and reactions between reactants in neighbouring subvolumes. These new features make it possible to construct physically consistent models of diffusion-controlled reactions also at fine spatial discretization. MesoRD is written in C++ and licensed under the GNU general public license (GPL). MesoRD can be downloaded at http://mesord.sourceforge.net. The MesoRD homepage, http://mesord.sourceforge.net, contains detailed documentation and news about recently implemented features. johan.elf@icm.uu.se.