On the existence and the applications of\ud modified equations for stochastic differential\ud equations

In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive a SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed

Derivation of a dual porosity model for the uptake of nutrients by root hairs

Root hairs are thought to play an important role in mediating nutrient uptake by plants. We develop a mathematical model for the nutrient transport and uptake in the root hair zone of a single root in the soil. Nutrients are assumed to diffuse both in the soil fluid phase and within the soil particles. Nutrients can also be bound to the soil particle surfaces by reversible reactions. Using homogenization techniques we derive a macroscopic dual porosity model for nutrient diffusion and reaction in the soil which includes the effect of all root hair surfaces

Fast stochastic simulation of biochemical reaction systems by\ud alternative formulations of the Chemical Langevin Equation

The Chemical Langevin Equation (CLE), which is a stochastic differential equation (SDE) driven by a multidimensional Wiener process, acts as a bridge between the discrete Stochastic Simulation Algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions only m1+m2 Wiener processes are required in the formulation of the CLE, whereas the standard approach uses 2m1 + m2. We illustrate our findings by considering alternative formulations of the CLE for a\ud HERG ion channel model and the Goldbeter–Koshland switch. We show that there are considerable computational savings when using our insights

Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation

Higher-order numerical methods for stochastic simulation of\ud chemical reaction systems

In this paper, using the framework of extrapolation, we present an approach for obtaining higher-order -leap methods for the Monte Carlo simulation of stochastic chemical kinetics. Specifically, Richardson extrapolation is applied to the expectations of functionals obtained by a fixed-step -leap algorithm. We prove that this procedure gives rise to second-order approximations for the first two moments obtained by the chemical master equation for zeroth- and first-order chemical systems. Numerical simulations verify that this is also the case for higher-order chemical systems of biological importance. This approach, as in the case of ordinary and stochastic differential equations, can be repeated to obtain even higher-order approximations. We illustrate the results of a second extrapolation on two systems. The biggest barrier for observing higher-order convergence is the Monte Carlo error; we discuss different strategies for reducing it

Homogenization for advection-diffusion in a perforated domain

The volume of a Wiener sausage constructed from a diffusion process with periodic, mean-zero, divergence-free velocity field, in dimension 3 or more, is shown to have a non-random and positive asymptotic rate of growth. This is used to establish the existence of a homogenized limit for such a diffusion when subject to Dirichlet conditions on the boundaries of a sparse and independent array of obstacles. There is a constant effective long-time loss rate at the obstacles. The dependence of this rate on the form and intensity of the obstacles and on the velocity field is investigated. A Monte Carlo algorithm for the computation of the volume growth rate of the sausage is introduced and some numerical results are presented for the Taylor–Green velocity field

Analysis of Brownian dynamics simulations of reversible biomolecular reactions

A class of Brownian dynamics algorithms for stochastic reaction-diffusion models which include reversible bimolecular reactions is presented and analyzed. The method is a generalization of the λ-rho model for irreversible bimolecular reactions which was introduced in [11]. The formulae relating the experimentally measurable quantities (reaction rate constants and diffusion constants) with the algorithm parameters are derived. The probability of geminate recombination is also investigated

Analysis of Brownian Dynamics Simulations of Reversible Bimolecular Reactions

A class of Brownian dynamics algorithms for stochastic reaction-diffusion models which include reversible bimolecular reactions is presented and analyzed. The method is a generalization of the $\lambda$--\newrho model for irreversible bimolecular reactions which was introduced in [arXiv:0903.1298]. The formulae relating the experimentally measurable quantities (reaction rate constants and diffusion constants) with the algorithm parameters are derived. The probability of geminate recombination is also investigated.Comment: 16 pages, 13 figures, submitted to SIAM Appl Mat

A Constrained Approach to Multiscale Stochastic Simulation of\ud Chemically Reacting Systems

Stochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper we introduce a multiscale methodology suitable to address this problem. It is based on the Conditional Stochastic Simulation Algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the Constrained Multiscale Algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Stochastic Differential Equation (SDE) approximation, we can in turn approximate average switching times in stochastic chemical systems

Adsorption and desorption dynamics of citric acid anions in soil

The functional role of organic acid anions (e.g. citrate, oxalate, malonate, etc) in soil has been intensively investigated with special focus either on (i) microbial respiration and soil carbon dynamics, (ii) nutrient solubilization, or (iii) metal detoxification. Considering the potential impact of sorption processes on the functional significance of these effects, comparatively little is known about the adsorption and desorption dynamics of organic acid anions in soils. The aim of this study therefore was to experimentally characterize the adsorption and desorption dynamics of organic acid anions in different soils using citrate as a model carboxylate. Results showed that both adsorption and desorption processes were fast, reaching a steady state equilibrium solution concentration within approximately 1 hour. However, for a given total soil citrate concentration(ctot) the steady state value obtained was critically dependent on the starting conditions of the experiment (i.e. whether most of the citrate was initially present in solution (cl) or held on the solid phase (cs)). Specifically, desorption-led processes resulted in significantly lower equilibrium solution concentrations than adsorption led processes indicating time-dependent sorption hysteresis. As it is not possible to experimentally distinguish between different sorption pools in soil (i.e. fast, slow, irreversible adsorption/desorption), a new dynamic hysteresis model was developed that relies only on measured soil solution concentrations. The model satisfactorily explained experimental data and was able to predict dynamic adsorption and desorption behaviour. To demonstrate its use we applied the model to two relevant scenarios (exudation and microbial degradation), where the dynamic sorption behaviour of citrate occurs. Overall, this study highlights the complex nature of citrate sorption in soil and concludes that existing models need to incorporate both a temporal and sorption hysteresis component to realistically describe the role and fate of organic acids in soil processes