The genome encodes thousands of genes whose pro ducts enable cell survival and numerous cellular func tions. The amounts and the temporal pattern in which these products appear in the cell are crucial to the pro cesses of life. Gene regulatory networks govern the levels of these gene products. A gene regulatory net work is the collection of molecular species and their inter actions, which together control geneproduct abundance. Numerous cellular processes are affected by regulatory networks. Innovations in experimental methods have ena bled largescale studies of gene regulatory networks and can reveal the mechanisms that underlie them. Consequently, biologists must come to grips with extremely complex networks and must analyse and integrate great quantities of experimental data. Essential to this challenge are computational tools, which can answer various questions: what is the full range of behaviours that this system exhibits under different conditions? What changes are expected in the dynamics of the system if certain parts stop functioning? How robust is the system under extreme conditions? Various computational models have been developed for regulatory network analysis. These models can be roughly divided into three classes. The first class, logi cal models, describes regulatory networks qualitatively. They allow users to obtain a basic understanding of the different functionalities of a given network under dif ferent conditions. Their qualitative nature makes them flexible and easy to fit to biological phenomena, although they can only answer qualitative questions. To under stand and manipulate behaviours that depend on finer timing and exact molecular concentrations, a second class of models was developed -continuous models. For example, to simulate the effects of dietary restriction on yeast cells under different nutrient concentrations 1 , users must resort to the finer resolution of continuous models. A third class of models was introduced follow ing the observation that the functionality of regulatory networks is often affected by noise. As the majority of these models account for interactions between individual molecules, they are referred to here as singlemolecule level models. Singlemolecule level models explain the relationship between stochasticity and gene regulation. Predictive computational models of regulatory net works are expected to benefit several fields. In medi cine, mechanisms of diseases that are characterized by dysfunction of regulatory processes can be elucidated. Biotechnological projects can benefit from predictive models that will replace some tedious and costly lab experiments. And, computational analysis may con tribute to basic biological research, for example, by explaining developmental mechanisms or new aspects of the evolutionary process. Here we review the available methodologies for mod elling and analysing regulatory networks. These meth odologies have already proved to be a valuable research tool, both for the development of network models and for the analysis of their functionality. We discuss their relative advantages and limitations, and outline some open questions regarding regulatory networks, includ ing how structure, dynamics and functionality relate to each other, how organisms use regulatory networks to adapt to their environments, and the interplay between regulatory networks and other cellular processes, such as metabolism. Stochasticity The property of a system whose behaviour depends on probabilities. In a model with stochasticity, a single initial state can evolve into several different trajectories, each with an associated probability. Modelling and analysis of gene regulatory networks Guy Karlebach and Ron Shamir Abstract | Gene regulatory networks have an important role in every process of life, including cell differentiation, metabolism, the cell cycle and signal transduction. By understanding the dynamics of these networks we can shed light on the mechanisms of diseases that occur when these cellular processes are dysregulated. Accurate prediction of the behaviour of regulatory networks will also speed up biotechnological projects, as such predictions are quicker and cheaper than lab experiments. Computational methods, both for supporting the development of network models and for the analysis of their functionality, have already proved to be a valuable research tool.

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Formalized rules for protein-protein interactions have recently been introduced to represent the binding and enzymatic activities of proteins in cellular signaling. Rules encode an understanding of how a system works in terms of the biomolecules in the system and their possible states and interactions. A set of rules can be as easy to read as a diagrammatic interaction map, but unlike most such maps, rules have precise interpretations. Rules can be processed to automatically generate a mathematical or computational model for a system, which enables explanatory and predictive insights into the system’s behavior. Rules are independent units of a model specification that facilitate model revision. Instead of changing a large number of equations or lines of code, as may be required in the case of a conventional mathematical model, a protein interaction can be introduced or modified simply by adding or changing a single rule that represents the interaction of interest. Rules can be defined and visualized by using graphs, so no specialized training in mathematics or computer science is necessary to create models or to take advantage of the representational precision of rules. Rules can be encoded in a machine-readable format to enable electronic storage and exchange of models, as well as basic knowledge about protein-protein interactions. Here, we review the motivation for rule-based modeling; applications of the approach; and issues that arise in model specification, simulation, and testing. We also discuss rule visualization and exchange and the software available for rule-based modeling.

This paper presents a simulation algorithm for the stochastic π-calculus, designed for the efficient simulation of biological systems with large numbers of molecules. The cost of a simulation depends on the number of species, rather than the number of molecules, resulting in a significant gain in efficiency. The algorithm is proved correct with respect to the calculus, and then used as a basis for implementing the latest version of the SPiM stochastic simulator. The algorithm is also suitable for generating graphical animations of simulations, in order to visualise system dynamics.

Abstract. A stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS). pSHSs correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. We show that for pSHSs, the dynamics of the statistical moments of its continuous states, evolves according to infinite-dimensional linear ordinary differential equations (ODEs), which can be approximated by finite-dimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this types of approximation is provided. This procedure is used to construct approximate stochastic models for a variety of chemical reactions that have appeared in literature. These reactions include a simple bimolecular reaction, for which one can solve the master equation; a decaying-dimerizing reaction set which exhibits two distinct time scales; a reaction for which the chemical rate equations have a continuum of equilibrium points; and the bistable Schögl reaction. The accuracy of the approximate models is investigated by comparing with Monte Carlo simulations or the solution to the Master equation, when available. 1

A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. This chapter is devoted to the mathematical study of such stochastic models. We begin by developing much of the mathematical machinery we need to describe the stochastic models we are most interested in. We show how one can represent counting processes of the type we need in terms of Poisson processes. This random time-change representation gives a stochastic equation for continuous-time Markov chain models. We include a discussion on the relationship between this stochastic equation and the corresponding martingale problem and Kolmogorov forward (master) equation. Next, we exploit

Science is an iterative process of experiments and hypotheses. Experiments produce surprising results; hypotheses are created to explain the results; new experiments are designed to test the hypotheses, of which some agree, some fail without yielding useful information and some produce more surprises; and the cycle continues. As a field matures, knowledge grows and the hypotheses become more elaborate, eventually exceeding the limits of what a scientist can mentally grasp. This is where computational modeling becomes necessary, and where cell biology is today. Modeling serves the same purposes as scientific cartoons or calculations on the backs of envelopes, but is much more precise. A model can definitively show if an hypothesis can explain a set of data, make experimental predictions and help identify system aspects that are poorly understood. After many iterations of experiments and theory, models are often sufficiently supported by evidence that they represent the current understanding of a system, against which new results are compared. This primer focuses on simulations of biochemical reaction networks, which is a core component of most cell biological models. We leave aside the related arts of model building, model analysis such as sensitivity analysis, and model/ data comparison. We explore a range of simulation methods that vary in their level of physical approximation and abstraction. For concreteness, each is presented for the same generic elementary reversible chemical reaction:

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The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A dynamical system is a process whose state varies with time. A mechanistic approach to understanding such a system is to write down equations, based on scientific understanding of the system, which describe how it evolves with time. Further equations describe the relationship of the state of the system to available observations on it. Mechanistic time series analysis concerns drawing inferences from the available data about the hypothesized equations

Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner, and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described. 1