Stochastic Reaction-diffusion Model Research Articles

Modeling Stochastic Kinetics of Cathodic Cu Surface Corrosion and Ionic Interaction in Nanoscopic Water Pools

Cathodic Cu corrosion has been reported to occur in both the presence and absence of carbon dioxide in aqueous electrolytes and while the mechanism is not understood well it implies carbide-, hydride- and hydroxide-formation mediated pathways. Corrosion leads to roughening, which creates local pockets of electrolyte that are somewhat disconnected from the bulk. In this talk, we describe a combination of theoretical studies to identify reaction pathways, and kinetics modeling of them to obtain a working description of corrosion in water. Corrosion at metal surfaces is both influenced by and influences the formation of nanoscopic aqueous pools by trapping electrolyte solution in microscopic pockets. Such electrolyte nano-pools exhibit non-bulk behavior, and the lifetime of locally trapped ions can be significantly different than in the bulk. This can potentially have an effect on the chemistry in the local microenvironment. Stochastic reaction-diffusion models can be used to model kinetics of electrochemical processes at length and time scales relevant to laboratory experiments, allowing a direct comparison of transient observable quantities to time-resolved experimental data. Atomistic simulations employ methods of electronic structure theory (such as density functional theory (DFT)) and molecular dynamics (both classical and ab initio) to describe a detailed picture of the (electro-)chemistry at microscopic time and length scales. Scaling up these methods to macroscopic levels is not trivial but required for direct comparison to experiment. By using Gibbs free energies obtained from atomistic simulations it is possible to calculate rate constants for individual elementary steps in reaction schemes describing kinetics of relevant processes. We use Kinetiscope to implement stochastic kinetics models of two reaction schemes describing: 1. cathodic Cu surface corrosion through formation of copper hydroxides, through which we aim to describe essential steps in the cathodic corrosion mechanism of metal surfaces; 2. interactions between bicarbonate and hydronium ions, both present in the bicarbonate salt buffers used routinely in electrochemical reduction at copper surfaces, in a nanoscopic water droplet to understand time-dependent changes in the local concentrations of electrolytes trapped at or near metal surfaces. Figure 1

Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations.

Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely, restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka-Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka-Volterra model and its May-Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse graining in spatially extended stochastic reaction-diffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.

Self-organization of PIP3 waves is controlled by the topology and curvature of cell membranes

Phosphatidylinositol (3,4,5)-trisphosphate (PIP3) is a signaling lipid on the plasma membrane that plays a fundamental role in cell signaling with a strong impact on cell physiology and diseases. It is responsible for the protruding edge formation, cell polarization, macropinocytosis, and other membrane remodeling dynamics in cells. It has been shown that the membrane confinement and curvature affects the wave formation of PIP3 and F-actin. But, even in the absence of F-actin, a complex self-organization of the spatiotemporal PIP3 waves is observed. In recent findings, we have shown that these waves can be guided and pinned on strongly bended Dictyostelium membranes caused by molecular crowding and curvature-limited diffusion. Based on these experimental findings, we investigate the spatiotemporal PIP3 wave dynamics on realistic three-dimensional cell-like membranes to explore the effect of curvature-limited diffusion, as observed experimentally. We use an established stochastic reaction-diffusion model with enzymatic Michaelis-Menten-type reactions that mimics the dynamics of Dictyostelium cells. As these cells mimic the three-dimensional shape and size observed experimentally, we found that the PIP3 wave directionality can be explained by a Hopf-like and a reverse periodic-doubling bifurcation for uniform diffusion and curvature-limited diffusion properties. Finally, we compare the results with recent experimental findings and discuss the discrepancy between the biological and numerical results.

A dynamic partitioning mechanism polarizes membrane protein distribution

The plasma membrane is widely regarded as the hub of the numerous signal transduction activities. Yet, the fundamental biophysical mechanisms that spatiotemporally compartmentalize different classes of membrane proteins remain unclear. Using multimodal live-cell imaging, here we first show that several lipid-anchored membrane proteins are consistently depleted from the membrane regions where the Ras/PI3K/Akt/F-actin network is activated. The dynamic polarization of these proteins does not depend upon the F-actin-based cytoskeletal structures, recurring shuttling between membrane and cytosol, or directed vesicular trafficking. Photoconversion microscopy and single-molecule measurements demonstrate that these lipid-anchored molecules have substantially dissimilar diffusion profiles in different regions of the membrane which enable their selective segregation. When these diffusion coefficients are incorporated into an excitable network-based stochastic reaction-diffusion model, simulations reveal that the altered affinity mediated selective partitioning is sufficient to drive familiar propagating wave patterns. Furthermore, normally uniform integral and lipid-anchored membrane proteins partition successfully when membrane domain-specific peptides are optogenetically recruited to them. We propose “dynamic partitioning” as a new mechanism that can account for large-scale compartmentalization of a wide array of lipid-anchored and integral membrane proteins during various physiological processes where membrane polarizes.

Fluctuation analysis for particle-based stochastic reaction–diffusion models

Recent works have derived and proven the large-population mean-field limit for several classes of particle-based stochastic reaction–diffusion (PBSRD) models. These limits correspond to systems of partial integral–differential equations (PIDEs) that generalize standard mass-action reaction–diffusion PDE models. In this work we derive and prove the next order fluctuation corrections to such limits, which we show satisfy systems of stochastic PIDEs with Gaussian noise. Numerical examples are presented to illustrate how including the fluctuation corrections can enable the accurate estimation of higher order statistics of the underlying PBSRD model.

Numerical and renormalization group analysis of the phase diagram of a stochastic cubic autocatalytic reaction-diffusion system.

The renormalization group is a set of tools that can be used to incorporate the effect of fluctuations in a dynamical system as a rescaling of the system's parameters. Here, we apply the renormalization group to a pattern-forming stochastic cubic autocatalytic reaction-diffusion model and compare its predictions with numerical simulations. Our results demonstrate a good agreement within the range of validity of the theory and show that external noise can be used as a control parameter in such systems.

Stochastic effects in bacterial communication mediated by extracellular vesicles.

Quorum sensing (QS) allows bacterial cells to sense changes in local cell density and, hence, to regulate multicellular processes, including biofilm formation, regulation of virulence, and horizontal gene transfer. While, traditionally, QS was thought to involve the exchange of extracellular signal molecules free in solution, recent experiments have shown that for some bacterial systems a substantial fraction of signal molecules are packaged and delivered in extracellular vesicles. How the packaging of signal molecules in extracellular vesicles influences the ability of cells to communicate and coordinate multicellular behaviors remains largely unknown. We present here a stochastic reaction-diffusion model of QS that accounts for the exchange of both freely diffusing and vesicle-associated signal molecules. We find that the delivery of signal molecules via extracellular vesicles amplifies local fluctuations in the signal concentration, which can strongly affect the dynamics and spatial range of bacterial communication. For systems with multiple bacterial colonies, extracellular vesicles provide an alternate pathway for signal transport between colonies, and may be crucial for long-distance signal exchange in environments with strong degradation of free signal molecules.

Dynamics and optimal control of a stochastic Zika virus model with spatial diffusion.

Zika is an infectious disease with multiple transmission routes, which is related to severe congenital disabilities, especially microcephaly, and has attracted worldwide concern. This paper aims to study the dynamic behavior and optimal control of the disease. First, we establish a stochastic reaction-diffusion model (SRDM) for Zika virus, including human-mosquito transmission, human-human sexual transmission, and vertical transmission of mosquitoes, and prove the existence, uniqueness, and boundedness of the global positive solution of the model. Then, we discuss the sufficient conditions for disease extinction and the existence of a stationary distribution of positive solutions. After that, three controls, i.e. personal protection, treatment of infected persons, and insecticides for spraying mosquitoes, are incorporated into the model and an optimal control problem of Zika is formulated to minimize the number of infected people, mosquitoes, and control cost. Finally, some numerical simulations are provided to explain and supplement the theoretical results obtained.

Dynamics of a stochastic fractional nonlocal reaction-diffusion model driven by additive noise

In this paper, we are concerned with the long-time behavior of stochastic fractional nonlocal reaction-diffusion equations driven by additive noise. We use the techniques of random dynamical systems to transform the stochastic model into a random one. To deal with the new nonlocal term appeared in the transformed equation, we first use a generalization of Peano's theorem to prove the existence of local solutions, and then adopt the Galerkin method to prove existence and uniqueness of weak solutions. Next, the existence of pullback attractors for the equation and its associated Wong-Zakai approximation equation driven by colored noise are shown, respectively. Furthermore, we establish the upper semi-continuity of random attractors of the Wong-Zakai approximation equation as $ \delta\rightarrow0^{+} $.

Modeling convergent scale-by-scale skin color patterning in multiple species of lizards

Skin color patterning in vertebrates emerges at the macroscale from microscopic cell-cell interactions among chromatophores. Taking advantage of the convergent scale-by-scale skin color patterning dynamics in five divergent species of lizards, we quantify the respective efficiencies of stochastic (Lenz-Ising and cellular automata, sCA) and deterministic reaction-diffusion (RD) models to predict individual patterns and their statistical attributes. First, we show that all models capture the underlying microscopic system well enough to predict, with similar efficiencies, neighborhood statistics of adult patterns. Second, we show that RD robustly generates, in all species, a substantial gain in scale-by-scale predictability of individual adult patterns without the need to parametrize the system down to its many cellular and molecular variables. Third, using 3D numerical simulations and Lyapunov spectrum analyses, we quantitatively demonstrate that, given the non-linearity of the dynamical system, uncertainties in color measurements at the juvenile stage and in skin geometry variation explain most, if not all, of the residual unpredictability of adult individual scale-by-scale patterns. We suggest that the efficiency of RD is due to its intrinsic ability to exploit mesoscopic information such as continuous scale colors and the relations among growth, scales geometries, and the pattern length scale. Our results indicate that convergent evolution of CA patterning dynamics, leading to dissimilar macroscopic patterns in different species, is facilitated by their spontaneous emergence under a large range of RD parameters, as long as a Turing instability occurs in a skin domain with periodic thickness. VIDEO ABSTRACT.

STEPS 4.0: Fast and memory-efficient molecular simulations of neurons at the nanoscale

Recent advances in computational neuroscience have demonstrated the usefulness and importance of stochastic, spatial reaction-diffusion simulations. However, ever increasing model complexity renders traditional serial solvers, as well as naive parallel implementations, inadequate. This paper introduces a new generation of the STochastic Engine for Pathway Simulation (STEPS) project (http://steps.sourceforge.net/), denominated STEPS 4.0, and its core components which have been designed for improved scalability, performance, and memory efficiency. STEPS 4.0 aims to enable novel scientific studies of macroscopic systems such as whole cells while capturing their nanoscale details. This class of models is out of reach for serial solvers due to the vast quantity of computation in such detailed models, and also out of reach for naive parallel solvers due to the large memory footprint. Based on a distributed mesh solution, we introduce a new parallel stochastic reaction-diffusion solver and a deterministic membrane potential solver in STEPS 4.0. The distributed mesh, together with improved data layout and algorithm designs, significantly reduces the memory footprint of parallel simulations in STEPS 4.0. This enables massively parallel simulations on modern HPC clusters and overcomes the limitations of the previous parallel STEPS implementation. Current and future improvements to the solver are not sustainable without following proper software engineering principles. For this reason, we also give an overview of how the STEPS codebase and the development environment have been updated to follow modern software development practices. We benchmark performance improvement and memory footprint on three published models with different complexities, from a simple spatial stochastic reaction-diffusion model, to a more complex one that is coupled to a deterministic membrane potential solver to simulate the calcium burst activity of a Purkinje neuron. Simulation results of these models suggest that the new solution dramatically reduces the per-core memory consumption by more than a factor of 30, while maintaining similar or better performance and scalability.

Multiscale modeling of RQ-DCA storage of different pear cultivars using a hybrid physics-based stochastic approach

During respiratory quotient based dynamic controlled atmosphere (RQ-DCA) storage, the oxygen concentration in the room is reduced until the RQ measured in the air of the room (RQair) increases above a predefined threshold. This increase in RQ indicates an increased extent of fermentation within the fruit. Because of internal gas gradients, the oxygen concentration inside the fruit is lower than that of the storage atmosphere and, hence, the RQ value in the fruit (RQfruit,max) is larger than RQair. Moreover, due to biological variability and fluctuations in the storage conditions, both RQfruit,max and RQair vary. Currently, no information exists about how the RQ measured in the atmosphere correlates with the RQ value inside the fruit, and how representative this RQair is for all fruit inside the room. In this article, we investigated how variability affects the critical RQair threshold in RQ-DCA. Hereto we used a multiscale stochastic reaction-diffusion model to compute gas transport within pears of three different cultivars, ‘Cepuna’, ‘Conference’ and ‘Queen’s Forelle’. Biological variability in diffusion and respiration parameters and fluctuations in cool room conditions were taken into account by performing Monte-Carlo simulations. The results showed large variations in both RQfruit,max and RQair. The effect of cool room filling degree on both RQ values was negligible, while variations in fruit material properties (pear size, tissue gas diffusivity and respiration kinetics) showed to be of high importance. However, as RQair increased before fermentation started to dominate, the underestimation and variability of RQair did not inhibit a proper detection of hypoxia inside the fruit within the room. This finding indicates that RQair is a proper parameter to control DCA storage of various cultivars. Due to differences in fruit properties, the RQ-response varied between the cultivars. ‘Cepuna’ requires the highest O2 (0.61 kPa) to prevent fermentation, while ‘Conference’ and ‘Queen’s Forelle’ can be stored at lower O2 (0.33 and 0.22 kPa, respectively). The results obtained in this research can be used to optimize RQ-DCA storage of each cultivar.

Dendritic spine morphology regulates calcium-dependent synaptic weight change.

Dendritic spines act as biochemical computational units and must adapt their responses according to their activation history. Calcium influx acts as the first signaling step during postsynaptic activation and is a determinant of synaptic weight change. Dendritic spines also come in a variety of sizes and shapes. To probe the relationship between calcium dynamics and spine morphology, we used a stochastic reaction-diffusion model of calcium dynamics in idealized and realistic geometries. We show that despite the stochastic nature of the various calcium channels, receptors, and pumps, spine size and shape can modulate calcium dynamics and subsequently synaptic weight updates in a deterministic manner. Through a series of exhaustive simulations and analyses, we found that the calcium dynamics and synaptic weight change depend on the volume-to-surface area of the spine. The relationships between calcium dynamics and spine morphology identified in idealized geometries also hold in realistic geometries, suggesting that there are geometrically determined deterministic relationships that may modulate synaptic weight change.

Detailed balance for particle models of reversible reactions in bounded domains.

In particle-based stochastic reaction-diffusion models, reaction rates and placement kernels are used to decide the probability per time a reaction can occur between reactant particles and to decide where product particles should be placed. When choosing kernels to use in reversible reactions, a key constraint is to ensure that detailed balance of spatial reaction fluxes holds at all points at equilibrium. In this work, we formulate a general partial-integral differential equation model that encompasses several of the commonly used contact reactivity (e.g., Smoluchowski-Collins-Kimball) and volume reactivity (e.g., Doi) particle models. From these equations, we derive a detailed balance condition for the reversible A + B ⇆ C reaction. In bounded domains with no-flux boundary conditions, when choosing unbinding kernels consistent with several commonly used binding kernels, we show that preserving detailed balance of spatial reaction fluxes at all points requires spatially varying unbinding rate functions near the domain boundary. Brownian dynamics simulation algorithms can realize such varying rates through ignoring domain boundaries during unbinding and rejecting unbinding events that result in product particles being placed outside the domain.

Mean Field Limits of Particle-Based Stochastic Reaction-Diffusion Models

Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transport. In this work we derive coarse-grained deterministic partial integro-differential equation (PIDE) models that provide a mean field approximation to the volume reactivity PBSRD model, a model commonly used for studying cellular processes. We formulate a weak measure-valued stochastic process (MVSP) representation for the volume reactivity PBSRD model, demonstrating for a simplified but representative system that it is consistent with the commonly used Doi Fock space representation of the corresponding forward equation. We then prove the convergence of the general volume reactivity model MVSP to the mean field PIDEs in the large-population (i.e., thermodynamic) limit.

Finite-time stability and optimal control of a stochastic reaction-diffusion model for Alzheimer’s disease with impulse and time-varying delay

With increased longevity in a nowadays society, Alzheimer’s disease (AD), an incurable neurodegenerative disease, becomes a serious threat to human health. To address the challenges to understanding and predicting AD development and treatment, we focus on the development of a stochastic reaction-diffusion model with time-varying delay and impulsive perturbations, which is driven by Le´vy jump process to model the in vivo progression of AD. Moreover, based on an bounded impulsive interval method, certain sufficient conditions of finite-time stability are provided. Further, from a cost-benefit perspective, an optimal control problem of AD is formulated to minimize the pathogenic proteins and control cost. Several illustrative examples are presented to demonstrate and verify theoretical results of this study.

How Reaction-Diffusion PDEs Approximate the Large-Population Limit of Stochastic Particle Models

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 9 September 2020Accepted: 18 May 2021Published online: 13 December 2021Keywordsreaction-diffusion PDEs, particle-based stochastic reaction-diffusion models, mean-field models, Doi model, Volume Reactivity ModelAMS Subject Headings35K57, 82C22, 82C31, 35A35Publication DataISSN (print): 0036-1399ISSN (online): 1095-712XPublisher: Society for Industrial and Applied MathematicsCODEN: smjmap

Precision and dissipation of a stochastic Turing pattern.

Spontaneous pattern formation is a fundamental scientific problem that has received much attention since the seminal theoretical work of Turing on reaction-diffusion systems. In molecular biophysics, this phenomenon often takes place under the influence of large fluctuations. It is then natural to inquire about the precision of such pattern. In particular, spontaneous pattern formation is a nonequilibrium phenomenon, and the relation between the precision of a pattern and the thermodynamic cost associated with it remains largely unexplored. Here, we analyze this relation with a paradigmatic stochastic reaction-diffusion model, i.e., the Brusselator in one spatial dimension. We find that the precision of the pattern is maximized for an intermediate thermodynamic cost, i.e., increasing the thermodynamic cost beyond this value makes the pattern less precise. Even though fluctuations get less pronounced with an increase in thermodynamic cost, we argue that larger fluctuations can also have a positive effect on the precision of the pattern.

Stochastic sensitivity analysis of stationary patterns in spatially extended systems

In this paper, a spatially extended stochastic reaction–diffusion model is studied. Due to Turing instability, stable nonhomogeneous stationary patterns are generated in such models. A theoretical approach to estimating the mean‐square deviation of random solutions from the stable deterministic pattern–attractor is demonstrated. Stochastic sensitivity functions for stable stationary patterns are introduced. Theoretical evaluations are compared with statistically obtained data. Based on this approach, we investigate stochastic properties of different patterns in Brusselator. Variations in pattern sensitivity to noise and phenomenon of stochastic preference are discussed.

Stochastic Reaction-Diffusion Model of the Binding of Monoclonal Antibodies to CD4 Receptors on the Surface of T Cells.

A stochastic reaction–diffusion model was developed to describe the binding of labeled monoclonal antibodies (mAbs) to CD4 receptors on the surface of T cells. The mAbs diffused to, adsorbed on, and underwent monovalent and bivalent binding to CD4 receptors on the cell surface. The model predicted the time-dependent nature of all populations involved in the labeling process. At large time, the populations reached equilibrium values, giving the number of antibodies bound to the T cell (ABC) defined as the sum of monovalently and bivalently bound mAbs. The predicted coefficient of variation (CV%) of the (ABC) values translated directly to a corresponding CV% of the measured mean fluorescence intensity (MFI). The predicted CV% was about 0.2% from the intrinsic fluctuations of the stochastic reaction process, about 5% after inclusion of the known fluctuations in the number of available CD4 receptors, and about 11% when fluctuations in bivalent binding affinity were included. The fluorescence detection process is expected to contribute approximately 7%. The abovementioned contributions to CV% sum up to approximately 13%. Work is underway to reconcile the predicted values and the measured values of 17% to 22%.