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Heterogeneity and variability is ubiquitous in biology and physiology and one of the great modelling challenges is how we cope with and quantify this variability. There are a wide variety of approaches. We can attempt to ignore spatial effects and represent the heterogeneity through stochastic models that evolve only in time, or we can attempt to capture some key spatial components. Alternatively we can perform very detailed spatial simulations or we can attempt to use other approaches that mimic stochasticity in some way, such as by the use of delay models, or by using populations of deterministic models. The skill is knowing when a particular model is appropriate to the questions that are being addressed. In this review, we give a brief introduction to modelling and simulation in Computational Biology and discuss the various different sources of heterogeneity, pointing out useful modelling and analysis approaches. The starting point is how we deal with intrinsic noise; that is, the uncertainty of knowing when a chemical reaction takes place and which that reaction is. These discrete stochastic methods do not follow individual molecules over time; rather they track only total molecular numbers. This leads, in the first instance, to the Stochastic Simulation Algorithm that describes the time evolution of a discrete nonlinear Markov process. From there we consider approaches that are more efficient and effective but still preserve the discreteness of the simulation, so-called tau-leaping algorithms. We then move to approximations that are continuous in time based around the Chemical Langevin stochastic differential equation. In these contexts we will focus, later in this chapter, on a particular application, namely the behaviour of ion channels dynamics. In the second part of the review, we address the question of spatial heterogeneity. This involves consideration on the nature of diffusion in crowded spaces and, in particular, anomalous diffusion, a relevant topic (

In order to systematically understand the qualitative and quantitative behaviour of chemical reaction networks, scientists must derive and analyse associated mathematical models. However, biochemical systems are often very large, with reactions occurring at multiple time scales, as evidenced by signalling pathways and gene expression kinetics. Owing to the associated computational costs, it is then many times impractical, if not impossible, to solve or simulate these systems with an appropriate level of detail. By consequence, there is a growing interest in developing techniques for the simplification or reduction of complex biochemical systems. Here, we extend our recently presented methodology on exact reduction of linear chains of reactions with delay distributions in two ways. First, we report that it is now possible to deal with fully bi-directional monomolecular systems, including degradations, synthesis and …

One of the fundamental motivations underlying computational cell biology is to gain insight into the complicated dynamical processes taking place, for example, on the plasma membrane or in the cytosol of a cell. These processes are often so complicated that purely temporal mathematical models cannot adequately capture the complex chemical kinetics and transport processes of, for example, proteins or vesicles. On the other hand, spatial models such as Monte Carlo approaches can have very large computational overheads. This chapter gives an overview of the state of the art in the development of stochastic simulation techniques for the spatial modelling of dynamic processes in a living cell.

This chapter focuses on the interactions and roles between delays and intrinsic noise effects within cellular pathways and regulatory networks. We address these aspects by focusing on genetic regulatory networks that share a common network motif, namely the negative feedback loop, leading to oscillatory gene expression and protein levels. In this context, we discuss computational simulation algorithms for addressing the interplay of delays and noise within the signaling pathways based on biological data. We address implementational issues associated with efficiency and robustness. In a molecular biology setting we present two case studies of temporal models for the Hes1 gene (Monk, 2003; Hirata et al., 2002), known to act as a molecular clock, and the Her1/Her7 regulatory system controlling the periodic somite segmentation in vertebrate embryos (Giudicelli and Lewis, 2004; Horikawa et al., 2006).

Background In order to provide insights into the complex biochemical processes inside a cell, modelling approaches must find a balance between achieving an adequate representation of the physical phenomena and keeping the associated computational cost within reasonable limits. This issue is particularly stressed when spatial inhomogeneities have a significant effect on system's behaviour. In such cases, a spatially-resolved stochastic method can better portray the biological reality, but the corresponding computer simulations can in turn be prohibitively expensive. Results We present a method that incorporates spatial information by means of tailored, probability distributed time-delays. These distributions can be directly obtained by single in silico or a suitable set of in vitro experiments and are subsequently fed into a delay stochastic simulation algorithm (DSSA), achieving a good compromise between computational costs and a much more accurate representation of spatial processes such as molecular diffusion and translocation between cell compartments. Additionally, we present a novel alternative approach based on delay differential equations (DDE) that can be used in scenarios of high molecular concentrations and low noise propagation. Conclusions Our proposed methodologies accurately capture and incorporate certain spatial processes into temporal stochastic and deterministic simulations, increasing their accuracy at low computational costs. This is of particular importance given that time spans of cellular processes are generally larger (possibly by several orders of magnitude) than those achievable by current spatially-resolved stochastic simulators. Hence, our methodology allows users to explore cellular scenarios under the effects of diffusion and stochasticity in time spans that were, until now, simply unfeasible. Our methodologies are supported by theoretical considerations on the different modelling regimes, i.e. spatial vs. delay-temporal, as indicated b

Recent studies have shown that small genetic regulatory networks (GRNs) can be evolved in silico displaying certain dynamics in the underlying mathematical model. It is expected that evolutionary approaches can help to gain a better understanding of biological design principles and assist in the engineering of genetic networks. To take the stochastic nature of GRNs into account, our evolutionary approach models GRNs as biochemical reaction networks based on simple enzyme kinetics and simulates them by using Gillespie’s stochastic simulation algorithm (SSA). We have already demonstrated the relevance of considering intrinsic stochasticity by evolving GRNs that show oscillatory dynamics in the SSA but not in the ODE regime. Here, we present and discuss first results in the evolution of GRNs performing as stochastic switches.

We introduce a genetic programming (GP) approach for evolving genetic networks that demonstrate desired dynamics when simulated as a discrete stochastic process. Our representation of genetic networks is based on a biochemical reaction model including key elements such as transcription, translation and post-translational modifications. The stochastic, reaction-based GP system is similar but not identical with algorithmic chemistries. We evolved genetic networks with noisy oscillatory dynamics. The results show the practicality of evolving particular dynamics in gene regulatory networks when modelled with intrinsic noise.

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.