Stochastic Reaction Research Articles

Averaging principle for stochastic quasi‐geostrophic flow equation with a fast oscillation

AbstractThis work is focused on the quasi‐geostrophic flow equation with a fast oscillation governed by a stochastic reaction–diffusion equation. It derives the well‐posedness of the slow–fast system, in which the fast component is ergodic and the slow component is tight. Applying the averaging principle, it is further proved that there exists a limit process, with respect to the singular perturbing parameter ε, where the fast component is averaged out. Moreover, the slow component of the slow–fast system converges to the solution of the averaged equation in some strong sense as ε tends to zero.

Virtual cell modeling and simulation software.

Virtual Cell (VCell, http://vcell.org) is free open-source software for modeling cell biological systems. Biology-based interface is designed for researchers with no programming/scripting experience: they enter reactions and reaction rules in an intuitive graphical way, and VCell automatically creates the math for you. Models and simulations can be accessed from anywhere using the VCell database; models can be shared among collaborators or made publicly available. Simulations can be run locally (on a user's computer) or using our server - a job can be sent to a server and results can be retrieved any time later. VCell provides a variety of simulation frameworks. The same reaction network can be simulated using deterministic (compartmental ODE) and/or stochastic reactions (SSA solvers) simulators. By adding a geometry for compartments and diffusion for species, the same reaction network can be simulated as a reaction-diffusion-advection PDE with support for 2D kinematics and/or using spatial stochastic (reaction-diffusion with Smoldyn, http://smoldyn.org). Geometries from 2D or 3D microscope images or from idealized analytical expressions can be used, and membrane flux, lateral membrane diffusion and electrophysiology could be incorporated. If elements of the model are defined by reaction rules (using a novel graphical user interface for defining, visualizing, and verifying rule-based models), the model can be simulated using BioNetGen (http://bionetgen.org) engine and/or network-free agent-based simulations. The VCell models can be exchanged with other tools using standard formats such as SBML and BNGL, as well as exported as MatLab, SED-ML, and COMBINE archive. The latest developments include better support of SBML features, the possibility to fully annotate the models, view models online, interact with ImageJ, and send simulations to runBiosimulations (https://run.biosimulations.org) service for online simulations and visualization with modified parameters. (Supported by NIGMS grant R24 GM137787)

4D dynamical whole-cell simulations of a growing minimal cell.

JCVI-syn3A is a genetically minimal bacterial cell with a single chromosome consisting of 493 genes that has retained few regulatory proteins or small RNAs. We recently developed a 3D fully dynamical whole-cell model that simulates the first 20 minutes of the cell cycle for this minimal cell before replication and growth. This model, based on experimental characterization of Syn3A and measurements from numerous biochemical experiments in related organisms, consisted of complete reaction networks for genetic information processing and metabolism totaling over 7200 chemical reactions. We now extend this 4D model to simulate the trajectory of a complete cell cycle while we dynamically update the shape of the membrane, chromosome configuration (including replication state), and polysome structures. To capture the broad range of length- and time-scales of the various dynamic processes, we periodically communicate cell state information among several coupled simulation techniques, including: reaction-diffusion and chemical master equation simulations for stochastic reactions, numerical integration of ODE's for metabolic reactions, and coarse-grained Brownian dynamics for the chromosome and membrane. Membrane shapes during division are inferred from experimental fluorescent images of dividing cells. The chromosome configurations are placed using physically realistic energy functions. From the model, we determine time-dependent concentrations, spatial distributions, and reaction fluxes that offer insight into the principles of life for this minimal cell, for example: prediction of doubling time matching the experimental value, realistic distributions of mRNA half-lives, and time-dependent balance of ATP costs and production for all reactions in the cell.

Extreme springing response statistics of a tethered platform by deconvolution

The research examines the motion response and hydrodynamic wave loads of a deep-water Tension Leg Platform (TLP), emphasising the impacts of the wave sum frequency on the restrained modes of heave, roll, and pitch. The stochastic TLP structural reaction in a random sea state was precisely computed using a Volterra series representation of the TLP corner vertical displacement, which was selected as a response process. The wave loading was evaluated using the second-order diffraction code WAMIT and applied to a linear damped mass-spring model representing the dynamic system. Then, platform displacement response at the design low probability level has been determined using a novel deconvolution approach. Since the Volterra series represented the analytical solution, the exact Volterra and the approximated predictions have been compared in this study. The latter provided an accurate way to validate the effectiveness and precision of the proposed novel deconvolution method. Compared to existing engineering techniques, the most attractive advantage of the proposed deconvolution method is that it does not rely on any pre-assumed asymptotic probability distribution class. The latter may be an attractive point for practical engineering design. Thus the primary objective of this work was to validate a novel deconvolution approach using exact quasi-analytical solutions. This work also highlights the limitations of mean up-crossing rate-based extrapolation methodologies for the situation of narrowband effects, including clustering, which are often included in the springing type of response.

Practical aspects of estimation of influence of groups of lobbyism are on priorities of strategy of development of rural areas and agricultural enterprises

In the article certainly priorities of the strategic planning of rural areas development and agricultural enterprises recognition weak stimulus of transformation of expectations on the real actions of groups of lobbyism in the conditions of risks which generate potentially negative risk decisions in accordance with standard rational expectations. There is disparity between subjective expectations and rational actions of groups of lobbyism in the conditions of institutional features of the partner organizations predefined prognoses in relation to strategy of rural areas development in previous periods and forming of subjective expectations about the prospects of deepening a particular branch aspects of European integration and subsequent institutional approchement of regional development policy for providing food security. Systematic uncoincidences between legalistically declared and really incarnate through the institutes of development strategy predetermine disparities of structure in a particular branch investments in an agro-food subcomplex for food security providing at national level. Inconsistency of priorities of agro-food subcomplex strategic development between entrepreneurs interested in permanent expansion of raw material export of plant-based products with the low degree of processing to the high marginal oversea markets and farms, oriented mainly to the internal consumers, predetermines disproportion of capital generuting for development of separate branches. Potential influence of groups of lobbyism at national level in transformation of development priorities of branches of agriculture recognition the limited influence of public opinion of habitants in rural areas at regional level predetermines strengthening of disproportion of development and standard of rural population living in different territorial areas, forming the important aspect of socio-economic reality. Disproportion and recurrence of dynamics of increase of the labour productivity increases for agricultural producers. Achievement of UN Sustainable Development Goals priorities becomes complicated in the conditions of stochastic network reactions of groups of lobbyism on calls in a national agro-food subcomplex. The network going near activity of groups of lobbyism predetermines strengthening of competition for bringing in of solvent agricultural producers to membership in institutional organization and their possibilities from development of analytical instruments. In a national agro-food subcomplex more widespread lobbyism is at parliamentary and ministerial level by comparison to private lobbyism of interests in concrete subject of entrepreneurial activity. Basic attention is spared development and realization at support of groups of lobbyism decisions from the questions of simplification access to land-tenure, enhancement output to the oversea markets with an export raw material agricultural produce and simplifications of terms of access foreign agricultural chemistry to the national market. At regional level influence of groups of lobbyism is related to the point forming of advantages for the profitability increase of separate subjects of entrepreneurial activity. A long-term income stimulates agricultural enterprises strategy priorities of development through the groups of lobbyism. Greater possibilities for lobbying of strategy priorities in a national agro-food subcomplex at large agricultural enterprises. Farms in the conditions of competition are limited in income generating and have less possibilities for lobbying of acceptable decisions in a national agro-food subcomplex. Potential possibilities of income generating in concrete industry of agriculture stimulate the subjects of entrepreneurial activity to increase activity of own participation on bringing in of different groups of lobbyism for advancement of the forecast administrative decisions from development of constituents of national agro-food subcomplex. The groups of lobbyism are oriented on simplification of procedures of regulator influence on development of branches of agriculture in Ukraine. In the period of post-war reсovery of national agro-food subcomplex influence of groups of lobbyism on making decision from the questions of industries development is transformed recognition necessity of food security providing at national and regional level.

Growth and depletion in linear stochastic reaction networks

This paper is about a class of stochastic reaction networks. Of interest are the dynamics of interconversion among a finite number of substances through reactions that consume some of the substances and produce others. The models we consider are continuous-time Markov jump processes, intended as idealizations of a broad class of biological networks. Reaction rates depend linearly on "enzymes," which are among the substances produced, and a reaction can occur only in the presence of sufficient upstream material. We present rigorous results for this class of stochastic dynamical systems, the mean-field behaviors of which are described by ordinary differential equations (ODEs). Under the assumption of exponential network growth, we identify certain ODE solutions as being potentially traceable and give conditions on network trajectories which, when rescaled, can with high probability be approximated by these ODE solutions. This leads to a complete characterization of the ω-limit sets of such network solutions (as points or random tori). Dimension reduction is noted depending on the number of enzymes. The second half of this paper is focused on depletion dynamics, i.e., dynamics subsequent to the "phase transition" that occurs when one of the substances becomes unavailable. The picture can be complex, for the depleted substance can be produced intermittently through other network reactions. Treating the model as a slow-fast system, we offer a mean-field description, a first step to understanding what we believe is one of the most natural bifurcations for reaction networks.

Stochastic suicide substrate reaction model

Nowadays, numeric paradigms have great significance in every field especially for solving the NDEs, PDEs, biochemical reactions etc. Here, we represent a realistic suicide substrate reaction specimen which can be represented by a system with four equations for the concentrations of the numerous molecules as functions of time. We present a technique to attain exact, approximate elucidations using the stochastic theory. This systematic technique provides more realistic numeric elucidations than other techniques which have been offered. Unfortunately, the numeric techniques like Euler–Maruyama, stochastic Euler, and stochastic RK4 fail for large step sizes. Stochastic nonstandard finite difference technique (SNSFD) has been fabricated for the suicide substrate reaction specimen and numeric tests are executed for diverse values of discretization parameter ‘h’. The consequences are matched with a well-known numeric scheme.

Noise in Biomolecular Systems: Modeling, Analysis, and Control Implications

While noise is generally associated with uncertainties and often has a negative connotation in engineering, living organisms have evolved to adapt to (and even exploit) such uncertainty to ensure the survival of a species or implement certain functions that would have been difficult or even impossible otherwise. In this article, we review the role and impact of noise in systems and synthetic biology, with a particular emphasis on its role in the genetic control of biological systems, an area we refer to as cybergenetics. The main modeling paradigm is that of stochastic reaction networks, whose applicability goes beyond biology, as these networks can represent any population dynamics system, including ecological, epidemiological, and opinion dynamics networks. We review different ways to mathematically represent these systems, and we notably argue that the concept of ergodicity presents a particularly suitable way to characterize their stability. We then discuss noise-induced properties and show that noise can be both an asset and a nuisance in this setting. Finally, we discuss recent results on (stochastic) cybergenetics and explore their relationships to noise. Along the way, we detail the different technical and biological constraints that need to be respected when designing synthetic biological circuits. Finally, we discuss the concepts, problems, and solutions exposed in the article; raise criticisms and concerns about current ideas and approaches; suggest current (open) problems with potential solutions; and provide some ideas for future research directions.

Operant conditioning of stochastic chemical reaction networks.

Adapting one's behavior to environmental conditions and past experience is a key trait of living systems. In the biological world, there is evidence for adaptive behaviors such as learning even in naturally occurring, non-neural, single-celled organisms. In the bioengineered world, advances in synthetic cell engineering and biorobotics have created the possibility of implementing lifelike systems engineered from the bottom up. This will require the development of programmable control circuitry for such biomimetic systems that is capable of realizing such non-trivial and adaptive behavior, including modification of subsequent behavior in response to environmental feedback. To this end, we report the design of novel stochastic chemical reaction networks capable of probabilistic decision-making in response to stimuli. We show that a simple chemical reaction network motif can be tuned to produce arbitrary decision probabilities when choosing between two or more responses to a stimulus signal. We further show that simple feedback mechanisms from the environment can modify these probabilities over time, enabling the system to adapt its behavior dynamically in response to positive or negative reinforcement based on its decisions. This system thus acts as a form of operant conditioning of the chemical circuit, in the sense that feedback provided based on decisions taken by the circuit form the basis of the learning process. Our work thus demonstrates that simple chemical systems can be used to implement lifelike behavior in engineered biomimetic systems.

Large deviations approach to a one-dimensional, time-periodic stochastic model of pattern formation

In this work we consider the problem of pattern formation modeled by a one dimensional stochastic reaction–diffusion equation with time periodic coefficients. In particular, we apply Large Deviations methods to obtain lower bounds on the probability that certain evenly spaced patterns will develop. Our estimates are optimized when the number of interfaces scales as (ϵT)−1, where ϵ is the length-scale and T is the time-scale. For large times T=ρ|lnϵ| our lower bound is of order exp(−ϵ2ρ), suggesting high likelihood for evenly spaced patterns whose number of interfaces is of order (ϵρ|lnϵ|)−1. Numerical simulations provide support to the idea that the more likely number of interfaces, even among unevenly spaced patterns, follows this law.

Bilateral constrained control for prosthesis walking on stochastically uneven terrain

The stochastic characteristic of uneven terrain is the leading cause of stochastic ground reaction forces, hindering gait coordination between the lower-limb prosthesis and healthy limb. Therefore, dedicated modeling and control methods for prosthesis walking on uneven terrain should be urgently provided. Noting that the stochastic feature originates from uneven ground contact, this paper first applies data-driven modeling to accurately represent the stochastic ground reaction forces. Then, an extended stochastic dynamic model for the amputee walking on uneven terrain is established based on this representation. Accordingly, a stochastic-gait-coordination (SGC) oriented control architecture is proposed and realized via the bilateral constrained adaptive neural controller (BC-ANC), where BC is designed to ensure disturbance boundness, and ANC is employed to handle system uncertainty. Rigorous stochastic stability analysis of the proposed architecture, i.e., BC-ANC, is also provided by deriving an asymmetric barrier Lyapunov function based on the backstepping technique. The study reveals that the controlled states in the stochastic dynamic system promise semi-globally uniformly ultimately bounded (SGUUB) in probability. Furthermore, prediction studies are deployed, showing that BC-ANC is more efficient than existing controllers.

Ergodicity of a nonlinear stochastic reaction–diffusion equation with memory

We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we show that the system in concern admits a unique weak solution. Also, any statistically steady state must possess regularity compatible with that of the solution. Moreover, if sufficiently many directions are stochastically forced, we employ the generalized coupling approach to prove that there exists a unique invariant probability measure and that the system is exponentially attractive. This extends ergodicity results previously established in Bonaccorsi et al., (2012).

Interval analysis of worst-case stationary moments for stochastic chemical reactions with uncertain parameters

The dynamics of cellular chemical reactions are variable due to stochastic noise from intrinsic and extrinsic sources. The intrinsic noise is the intracellular fluctuations of molecular copy numbers caused by the probabilistic encounter of molecules and is modeled by the chemical master equation. The extrinsic noise, on the other hand, represents the intercellular variation of the kinetic parameters due to the variation of global factors affecting gene expression. The objective of this paper is to propose a theoretical framework to analyze the combined effect of the intrinsic and the extrinsic noise modeled by the chemical master equation with uncertain parameters. More specifically, we formulate a semidefinite program to compute the intervals of the stationary solution of uncertain moment equations whose parameters are given only partially in the form of the statistics of their distributions. The semidefinite program is derived without approximating the governing equation in contrast with many existing approaches. Thus, we can obtain guaranteed intervals of the worst possible values of the moments for all parameter distributions satisfying the given statistics, which are prohibitively hard to estimate from sample-path simulations since sampling from all possible uncertain distributions is difficult. We demonstrate the proposed optimization approach using two examples of stochastic chemical reactions and show that the solution of the optimization problem gives informative upper and lower bounds of the statistics of the stationary copy number distributions.

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Kr\"uger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

AbstractThis paper provides a full classification of the dynamics for continuous-time Markov chains (CTMCs) on the nonnegative integers with polynomial transition rate functions and without arbitrary large backward jumps. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence versus transience, positive recurrence versus null recurrence, certain absorption, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and nonexistence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded forward jumps. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth–death processes. The approach is based on a mixture of Lyapunov–Foster-type results, the classical semimartingale approach, and estimates of stationary measures.

Approximating solutions of the Chemical Master equation using neural networks

SummaryThe Chemical Master Equation (CME) provides an accurate description of stochastic biochemical reaction networks in well-mixed conditions, but it cannot be solved analytically for most systems of practical interest. Although Monte Carlo methods provide a principled means to probe system dynamics, the large number of simulations typically required can render the estimation of molecule number distributions and other quantities infeasible. In this article, we aim to leverage the representational power of neural networks to approximate the solutions of the CME and propose a framework for the Neural Estimation of Stochastic Simulations for Inference and Exploration (Nessie). Our approach is based on training neural networks to learn the distributions predicted by the CME from relatively few stochastic simulations. We show on biologically relevant examples that simple neural networks with one hidden layer can capture highly complex distributions across parameter space, thereby accelerating computationally intensive tasks such as parameter exploration and inference.

Passivity-based boundary control for stochastic delay reaction–diffusion systems

This paper considers the passivity-based boundary control for stochastic delay reaction–diffusion systems (SDRDSs) with boundary input–output. Delay-dependent sufficient conditions are obtained in the sense of expectation for SDRDSs by the use of Lyapunov functional method, for the input strict passivity and the output strict passivity, respectively. Then delay-dependent sufficient conditions are presented to ensure the uncertainty SDRDSs to achieve the robust passivity. Moreover, by utilising the techniques of Green formula and Poincare's inequality, the difficulty is handled caused by the boundary input and boundary controller. Besides, the troubles brought by the existence of the stochastic term and uncertain parameters are resolved by stochastic inequality techniques. Finally, numerical simulations are provided to show the validity of our theoretical results.

A Simple Stochastic Reaction Model for Heterogeneous Polymerizations.

The stochastic reaction model (SRM) treats polymerization as a pure probability‐based issue, which is widely applied to simulate various polymerization processes. However, in many studies, active centers were assumed to react with the same probability, which cannot reflect the heterogeneous reaction microenvironment in heterogeneous polymerizations. Recently, we have proposed a simple SRM, in which the reaction probability of an active center is directly determined by the local reaction microenvironment. In this paper, we compared this simple SRM with other SRMs by examining living polymerizations with randomly dispersed and spatially localized initiators. The results confirmed that the reaction microenvironment plays an important role in heterogeneous polymerizations. This simple SRM provides a good choice to simulate various polymerizations.

A thermalized electrokinetics model including stochastic reactions suitable for multiscale simulations of reaction–advection–diffusion systems

We introduce a scheme to simulate the spatial and temporal evolution of the densities of charged species, taking into account diffusion, thermal fluctuations, coupling to a carrier fluid, and chemical reactions. To this end, the diffusive fluxes in the electrokinetic model by Capuani et al. (2004) are supplemented with thermal fluctuations. Chemical reactions are included via an additional source term in the mass balance equation. The diffusion–reaction model is then coupled to a solver for fluctuating hydrodynamics based on the lattice Boltzmann method. This combination is particularly useful for soft matter simulations, due to the ability to couple particles to the lattice-Boltzmann fluid. These could, e.g., be charged colloids or polymers, which then interact with an ion distribution. We describe one implementations based on the automatic code generation tools pystencils and lbmpy, and another one that is contained in the molecular dynamics package ESPResSo and that allows for an easy coupling of particles to the density fields. We validate our implementations by comparing to several known analytic results. Our method can be applied to coarse-grained catalysis problems as well as to many other multi-scale problems that require the coupling of explicit-particle simulations to flow fields, diffusion, and reaction problems in arbitrary geometries.

Stability of switched stochastic reaction–diffusion systems

As one of important hybrid dynamical systems, switched system has practical applications in some practical systems. Generally, switched system is made up of a family of continuous-time subsystems and discrete switching signals. Affected by environmental disturbance, stochastic switched system should be considered. This paper deals with a class of switched stochastic reaction-diffusion systems (SSRDSs, in short), where the discrete switching signal is a right continuous and piecewise function. The switching instants are stopping times or deterministic times. Criterions on stability in probability and exponential stability in mean square of the trivial solution for SSRDSs are derived by means of Lyapunov functional methods. We propose two examples to verify the obtained theoretical results.