Several stochastic simulation algorithms (SSAs) have recently been proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A + B --> C or A + A --> C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site).

Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time-and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time we obtain a single second-order hyperbolic equation for the populations density, and using a parabolic scaling we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement.

Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker-Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker-Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker-Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the self-propelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker-Planck equation coefficient estimation approach, can be implemented in the selfpropelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.collective behavior | locusts | density-dependent switching | coarse-graining | swarming W hile recent years have seen an explosion in the number of simulation models of moving animal groups, there is little detailed comparison between these models and experimental data (1, 2). The models usually produce motion that "looks like" that of a swarm of locusts, a school of fish, or a flock of birds, but the similarities are difficult to quantify (3). Furthermore, the simulation models themselves are often difficult to understand from a mathematical viewpoint since, by their nature, they resist simple mean-field descriptions. These complications make it difficult to use models to predict, for example, the rate at which groups change direction of travel or how spatial patterns evolve through time (4, 5). We are left with a multitude of models, all of which seem to relate to the available experimental data, but none of which provide clear predictive power.One approach to the problem of linking experimental data to model behavior is the detailed study of the local interactions between animals. This approach has yielded better understanding of the rules ...

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.

A planar bistable liquid crystal device, reported in Tsakonas et al. [Appl. Phys. Lett. 90, 111913 (2007)], is modeled within the Landau-de Gennes theory for nematic liquid crystals. This planar device consists of an array of square micrometer-sized wells. We obtain six different classes of equilibrium profiles and these profiles are classified as diagonal or rotated solutions. In the strong anchoring case, we propose a Dirichlet boundary condition that mimics the experimentally imposed tangent boundary conditions. In the weak anchoring case, we present a suitable surface energy and study the multiplicity of solutions as a function of the anchoring strength. We find that diagonal solutions exist for all values of the anchoring strength W ≥ 0, while rotated solutions only exist for W ≥ W_{c}>0, where W_{c} is a critical anchoring strength that has been computed numerically. We propose a dynamic model for the switching mechanisms based on only dielectric effects. For sufficiently strong external electric fields, we numerically demonstrate diagonal-to-rotated and rotated-to-diagonal switching by allowing for variable anchoring strength across the domain boundary.

Nonlinear independent component analysis is combined with diffusion-map data analysis techniques to detect good observables in high-dimensional dynamic data. These detections are achieved by integrating local principal component analysis of simulation bursts by using eigenvectors of a Markov matrix describing anisotropic diffusion. The widely applicable procedure, a crucial step in model reduction approaches, is illustrated on stochastic chemical reaction network simulations.slow manifold | dimensionality reduction | chemical reactions E volution of dynamical systems often occurs on two or more time scales. A simple deterministic example is given by the coupled system of ordinary differential equations (ODEs)with the small parameter 0 < τ 1, where α(u, v) and β(u, v) are O(1). For any given initial condition (u 0 , v 0 ), already at t = O(τ) the system approaches a new value (u 0 , v), where v satisfies the asymptotic relation β(u 0 , v) = 0. Although the system is fully described by two coordinates, the relation β(u, v) = 0 defines a slow one-dimensional manifold which approximates the slow dynamics for t τ. In this example, it is clear that v is the fast variable whereas u is the slow one. Projecting onto the slow manifold here is rather easy: The fast foliation is simply "vertical", i.e. u = const. However, when we observe the system in terms of the variables x = x(u, v) and y = y (u, v) which are unknown nonlinear functions of u and v, then the "observables" x and y have both fast and slow dynamics. Projecting onto the slow manifold becomes nontrivial, because the transformation from (x, y) to (u, v) is unknown. Detecting the existence of an intrinsic slow manifold under these conditions and projecting onto it are important in any model reduction technique. Knowledge of a good parametrization of such a slow manifold is a crucial component of the equation-free framework for modeling and computation of complex/multiscale systems (1-3).Principal component analysis (PCA, also known as POD) (4-6) has traditionally been used for data and model reduction in contexts ranging from meteorology (7) and transitional flows (8) to protein folding (9, 10); in these contexts the PCA procedure is used to detect good global reduced coordinates that best capture the data variability. In recent years, diffusion maps (11-17) have been used in a similar spirit to detect low-dimensional, nonlinear manifolds underlying high-dimensional datasets.In this paper, we integrate ensembles of local PCA analyses in the diffusion-map framework to enable the detection of slow variables in high-dimensional data arising from dynamic model simulations. The proposed algorithm is built along the lines of the nonlinear independent component analysis method recently introduced in ref. 18. The approach takes into account the time dependence of the data, whereas in the diffusion-map approach the time labeling of the data points is not included. We demonstrate our algorithm for stochastic simulators arising in the context of chemical/biochemical react...

Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is an almost ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last 20 years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs.

Abstract. The collective behavior of bacterial populations provides an example of how cell-level decision making translates into population-level behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individual-level behavior into population-level models. Here we focus on the flagellated bacterium E. coli, for which a great deal is known about signal detection, transduction, and cell-level swimming behavior. We review the biological background on individual-and population-level processes and discuss the velocity-jump approach used for describing population-level behavior based on individual-level intracellular processes. In particular, we generalize the moment-based approach to macroscopic equations used earlier [R. Erban and H. G. Othmer, SIAM J. Appl. Math., 65 (2004), pp. 361-391] to higher dimensions and show how aspects of the signal transduction and response enter into the macroscopic equations. We also discuss computational issues surrounding the bacterial pattern formation problem and technical issues involved in the derivation of macroscopic equations.