Surfaces serve as highly efficient catalysts for a vast variety of chemical reactions. Typically, such surface reactions involve billions of molecules which diffuse and react over macroscopic areas. Therefore, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations, which are based on the mean-field approximation. However, in case that the surface is partitioned into a large number of disconnected microscopic domains, the number of reactants in each domain becomes small and it strongly fluctuates. This is, in fact, the situation in the interstellar medium, where some crucial reactions take place on the surfaces of microscopic dust grains. In this case rate equations fail and the simulation of surface reactions requires stochastic methods such as the master equation. However, in the case of complex reaction networks, the master equation becomes infeasible because the number of equations proliferates exponentially. To solve this problem, we introduce a stochastic method based on moment equations. In this method the number of equations is dramatically reduced to just one equation for each reactive species and one equation for each reaction. Moreover, the equations can be easily constructed using a diagrammatic approach. We demonstrate the method for a set of astrophysically relevant networks of increasing complexity. It is expected to be applicable in many other contexts in which problems that exhibit analogous structure appear, such as surface catalysis in nanoscale systems, aerosol chemistry in stratospheric clouds and genetic networks in cells.
A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily runcated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the anal…
The classical models for irreversible diffusion-influenced reactions can be derived by introducing absorbing boundary conditions to over-damped continuous Brownian motion (BM) theory. As there is a clear corresponding stochastic process, the mathematical description takes both Kolmogorov forward equation for the evolution of the probability distribution function and the stochastic sample trajectories. This dual description is a fundamental characteristic of stochastic processes and allows simple particle based simulations to accurately match the expected statistical behavior. However, in the t…
Classical transition state theory (TST) is the cornerstone of reaction rate theory. It postulates a partition of phase space into reactant and product regions, which are separated by a dividing surface that reactive trajectories must cross. In order not to overestimate the reaction rate, the dynamics must be free of recrossings of the dividing surface. This no-recrossing rule is difficult (and sometimes impossible) to enforce, however, when a chemical reaction takes place in a fluctuating environment such as a liquid. High-accuracy approximations to the rate are well known when the solvent for…
The microscopic structure and movement of reaction fronts in reaction diffusion systems far from equilibrium are investigated. We show that some three-site interaction models exhibit exact diffusive shock measures, i.e. domains of different densities connected by a sharp wall without correlations. In all cases fluctuating domains grow at the expense of ordered domains, the absence of growth is possible between ordered domains. It is shown that these models give rise to aspects not seen in nearest neighbor models, viz. double shocks and additional symmetries. A classification of the systems by …
We analyzed the stochastic behavior of systems controlled by autocatalytic reaction A+X -> X+X. Assuming the distribution of reacting particles in the system volume to be uniform, we introduced the notion of the point model of reaction kinetics, and derived a system of differential equations for probabilities of finding n=0,1,... autocatalytic particles at a given time moment. It has been found that the kinetic law of the mass action cannot be supported by stochastic model.
The position of a reaction front, propagating into a metastable state, fluctuates because of the shot noise of reactions and diffusion. A recent theory [B. Meerson, P.V. Sasorov, and Y. Kaplan, Phys. Rev. E 84, 011147 (2011)] gave a closed analytic expression for the front diffusion coefficient in the weak noise limit. Here we test this theory in stochastic simulations involving reacting and diffusing particles on a one-dimensional lattice. We also investigate a small noise-induced systematic shift of the front velocity compared to the prediction from the spatially continuous deterministic rea…
Many approaches to modelling reaction-diffusion systems with anomalous transport rely on deterministic equations and ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasi-patterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the fo…
We study a reaction model that presents stochastic resonance purely due to internal noise. This means that the only source of fluctuations comes from the discrete character of the reactants, and no more noises enter into the system. Our analysis reveals that the phenomenon is highly complex, and that is generated by the interplay of different stochasticity at the three fixed points of a bistable system.
The presence of fluctuations and non-linear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations. Stochastic dynamics can be formulated in terms of functional integrals. In this paper we apply the heat kernel method to study the short distance renormalizability of a stochastic (polynomial) reaction-diffusion equation with real additive noise. We calculate the one-loop {\emph{effective action}} and its ultraviolet scale dependent divergences. We show that for white noise a polynomial reaction-diffusion equation is one-loop {\emph{finite}} in…
For chemical reaction networks described by a master equation, we define energy and entropy on a stochastic trajectory and develop a consistent nonequilibrium thermodynamic description along a single stochastic trajectory of reaction events. A first-law like energy balance relates internal energy, applied (chemical) work and dissipated heat for every single reaction. Entropy production along a single trajectory involves a sum over changes in the entropy of the network itself and the entropy of the medium. The latter is given by the exchanged heat identified through the first law. Total entropy…
A stochastic model of autocatalytic chemical reactions is studied both numerically and analytically. The van Kampen perturbative scheme is implemented, beyond the second order approximation, so to capture the non Gaussianity traits as displayed by the simulations. The method is targeted to the characterization of the third moments of the distribution of fluctuations, originating from a system of four populations in mutual interaction. The theory predictions agree well with the simulations, pointing to the validity of the van Kampen expansion beyond the conventional Gaussian solution.
An idea for evaluating transition probabilities in chemical reaction systems is proposed, which is efficient for repeated calculations with various rate constants. The idea is based on duality relations; instead of direct time-evolutions of the original reaction system, the dual process is dealt with. Usually, if one changes rate constants of the original reaction system, the direct time-evolutions should be performed again, using the new rate constants. On the other hands, only one solution of an extended dual process can be re-used to calculate the transition probabilities for various rate c…
Understanding anomalous transport and reaction kinetics due to microscopic physical and chemical disorder is a long-standing goal in many fields including geophysics, biology, and engineering. We consider reaction-diffusion characterized by fluctuations in both transitions times and decay rates. We introduce and analyze a model framework that explicitly connects microscopic fluctuations with the mescoscopic description. For broad distributions of transport and reaction time scales we compute the particle density and derive the equations governing its evolution, finding power-law decay of the s…
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 beta 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 incr…
We analyzed the stochastic behavior of systems controlled by autocatalytic reaction A+X -> X+X, X+X -> A+X, X -> B provided that the distribution of reacting particles in the system volume is uniform, i.e. the point model of reaction kinetics introduced in arXiv:cond-mat/0404402 can be applied. Assuming the number of substrate particles A to be kept constant by a suitable reservoir, we derived the forward Kolmogorov equation for the probability of finding n=0,1,... autocatalytic particles X in the system at a given time moment. We have shown that the stochastic model results in an equation for…
How do single cell fate decisions induced by activation of key signaling proteins above threshold concentrations within a time interval are affected by stochastic fluctuations in biochemical reactions? We address this question using minimal models of stochastic chemical reactions commonly found in cell signaling and gene regulatory systems. Employing exact solutions and semi-analytical methods we calculate distributions of the maximum value ($N$) of activated species concentrations ($P_{max}(N)$) and the time ($t$) taken to reach the maximum value ($P_{max}(t)$) within a time window in the min…
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-diffusio…
In this paper, we address the modeling issues of cell movement and division with a special focus on the phenomenon of volume exclusion in a lattice-based, exact stochastic simulation framework. We propose a new exact method, called Reduced Rate Method -- RRM, that is substantially quicker than the previously used exclusion method, for large number of cells. In addition, we introduce three novel reaction types: the contact-inhibited, the contact-promoted, and the spontaneous reactions. To the best of our knowledge, these reaction types have not been taken into account in lattice-based stochasti…
We first derive the Hamilton-Jacobi theory underlying continuous-time Markov processes, and then use the construction to develop a variational algorithm for estimating escape (least improbable or first passage) paths for a generic stochastic chemical reaction network that exhibits multiple fixed points. The design of our algorithm is such that it is independent of the underlying dimensionality of the system, the discretization control parameters are updated towards the continuum limit, and there is an easy-to-calculate measure for the correctness of its solution. We consider several applicatio…
The application of the quasi-steady-state approximation to the Michaelis-Menten reaction embedded in large open chemical reaction networks is a popular model reduction technique in deterministic and stochastic simulations of biochemical reactions inside cells. It is frequently assumed that the predictions of the reduced master equations obtained using the stochastic quasi-steady-state approach are in very good agreement with the predictions of the full master equations, provided the conditions for the validity of the deterministic quasi-steady-state approximation are fulfilled. We here use the…
The quasi-steady-state approximation is widely used to develop simplified deterministic or stochastic models of enzyme catalyzed reactions. In deterministic models, the quasi-steady-state approximation can be mathematically justified from singular perturbation theory. For several closed enzymatic reactions, the homologous extension of the quasi-steady-state approximation to the stochastic regime, known as the stochastic quasi-steady-state approximation, has been shown to be accurate under the analogous conditions that permit the quasi-steady-state reduction of the deterministic counterpart. Ho…
