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Petri nets for systems and synthetic biology.
 Formal Methods for Computational Systems Biology, Lecture Notes in Computer Science,
, 2008
"... Abstract. We give a description of a Petri netbased framework for modelling and analysing biochemical pathways, which unifies the qualitative, stochastic and continuous paradigms. Each perspective adds its contribution to the understanding of the system, thus the three approaches do not compete, b ..."
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Cited by 80 (23 self)
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Abstract. We give a description of a Petri netbased framework for modelling and analysing biochemical pathways, which unifies the qualitative, stochastic and continuous paradigms. Each perspective adds its contribution to the understanding of the system, thus the three approaches do not compete, but complement each other. We illustrate our approach by applying it to an extended model of the three stage cascade, which forms the core of the ERK signal transduction pathway. Consequently our focus is on transient behaviour analysis. We demonstrate how qualitative descriptions are abstractions over stochastic or continuous descriptions, and show that the stochastic and continuous models approximate each other. Although our framework is based on Petri nets, it can be applied more widely to other formalisms which are used to model and analyse biochemical networks. Motivation Biochemical reaction systems have by their very nature three distinctive characteristics. (1) They are inherently bipartite, i.e. they consist of two types of game players, the species and their interactions. (2) They are inherently concurrent, i.e. several interactions can usually happen independently and in parallel. (3) They are inherently stochastic, i.e. the timing behaviour of the interactions is governed by stochastic laws. So it seems to be a natural choice to model and analyse them with a formal method, which shares exactly these distinctive characteristics: stochastic Petri nets. However, due to the computational efforts required to analyse stochastic models, two abstractions are more popular: qualitative models, abstracting away from any time dependencies, and continuous models, commonly used to approximate stochastic behaviour by a deterministic one. We describe an overall framework to unify these three paradigms, providing a family of related models with high analytical power. The advantages of using Petri nets as a kind of umbrella formalism are seen in the following: M. Bernardo, P. Degano, and G. Zavattaro (Eds.): SFM
Modeling and simulating chemical reactions
 SIAM Review
, 2007
"... 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 largescale limit of a sequence of finerscale probabilistic models. In studying this hierarchy of models, st ..."
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Cited by 34 (1 self)
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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 largescale limit of a sequence of finerscale 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
Stronger computational modelling of signalling pathways using both continuous and discretestate methods
 In To appear in Computational Methods in Systems Biology 2006, LNCS
, 2006
"... Abstract. Starting from a biochemical signalling pathway model expressed in a process algebra enriched with quantitative information we automatically derive both continuousspace and discretestate representations suitable for numerical evaluation. We compare results obtained using implicit numerica ..."
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Cited by 28 (15 self)
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Abstract. Starting from a biochemical signalling pathway model expressed in a process algebra enriched with quantitative information we automatically derive both continuousspace and discretestate representations suitable for numerical evaluation. We compare results obtained using implicit numerical differentiation formulae to those obtained using approximate stochastic simulation thereby exposing a flaw in the use of the differentiation procedure producing misleading results. 1
Studying genetic regulatory networks at the molecular level: delayed reaction stochastic models
, 2007
"... Abstract Current advances in molecular biology enable us to access the rapidly increasing body of genetic information. It is still challenging to model gene systems at the molecular level. Here, we propose two types of reaction kinetic models for constructing genetic networks. Time delays involved ..."
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Cited by 26 (5 self)
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Abstract Current advances in molecular biology enable us to access the rapidly increasing body of genetic information. It is still challenging to model gene systems at the molecular level. Here, we propose two types of reaction kinetic models for constructing genetic networks. Time delays involved in transcription and translation are explicitly considered to explore the effects of delays, which may be significant in genetic networks featured with feedback loops. One type of model is based on delayed effective reactions, each reaction modeling a biochemical process like transcription without involving intermediate reactions. The other is based on delayed virtual reactions, each reaction being converted from a mathematical function to model a biochemical function like gene inhibition. The latter stochastic models are derived from the corresponding meanfield models. The former ones are composed of single gene expression modules. We thus design a model of gene expression. This model is verified by our simulations using a delayed stochastic simulation algorithm, which accurately reproduces the stochastic kinetics in a recent experimental study. Various simplified versions of the model are given and evaluated. We then use the two methods to study the genetic toggle switch and the repressilator. We define the ''on'' and ''off'' states of genes and extract a binary code from the stochastic time series. The binary code can be described by the corresponding Boolean network models in certain conditions. We discuss these conditions, suggesting a method to connect Boolean models, meanfield models, and stochastic chemical models. r
Robustness: confronting lessons from physics and biology
 Biological Reviews of the Cambridge Philosophical Society
, 2008
"... The term robustness is encountered in very different scientific fields, from engineering and control theory to dynamical systems to biology. The main question addressed herein is whether the notion of robustness and its correlates (stability, resilience, selforganisation) developed in physics are r ..."
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Cited by 14 (0 self)
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The term robustness is encountered in very different scientific fields, from engineering and control theory to dynamical systems to biology. The main question addressed herein is whether the notion of robustness and its correlates (stability, resilience, selforganisation) developed in physics are relevant to biology, or whether specific extensions and novel frameworks are required to account for the robustness properties of living systems. To clarify this issue, the different meanings covered by this unique term are discussed; it is argued that they crucially depend on the kind of perturbations that a robust system should by definition withstand. Possible mechanisms underlying robust behaviours are examined, either encountered in all natural systems (symmetries, conservation laws, dynamic stability) or specific to biological systems (feedbacks and regulatory networks). Special attention is devoted to the (sometimes counterintuitive) interrelations between robustness and noise. A distinction between dynamic selection and natural selection in the establishment of a robust behaviour is underlined. It is finally argued that nested notions of robustness, relevant to different time scales and different levels of organisation, allow one to reconcile the seemingly contradictory requirements for robustness and adaptability in living systems.
Formalisms for Specifying Markovian Population Models
"... We compare several languages for specifying Markovian population models such as queuing networks and chemical reaction networks. These languages —matrix descriptions, stochastic Petri nets, stochastic process algebras, stoichiometric equations, and guarded command models — all describe continuoust ..."
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Cited by 13 (3 self)
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We compare several languages for specifying Markovian population models such as queuing networks and chemical reaction networks. These languages —matrix descriptions, stochastic Petri nets, stochastic process algebras, stoichiometric equations, and guarded command models — all describe continuoustime Markov chains, but they differ according to important properties, such as compositionality, expressiveness and succinctness, executability, ease of use, and the support they provide for checking the wellformedness of a model and for analyzing a model.
Chemical Master Equation and Langevin regimes for a gene transcription model
 in Computational Mathematics and Systems Biology
, 2007
"... Gene transcription models must take account of intrinsic stochasticity. The Chemical Master Equation framework is based on modelling assumptions that are highly appropriate for this context, and the Stochastic Simulation Algorithm (also known as Gillespie’s algorithm) allows for practical simulation ..."
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Cited by 11 (2 self)
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Gene transcription models must take account of intrinsic stochasticity. The Chemical Master Equation framework is based on modelling assumptions that are highly appropriate for this context, and the Stochastic Simulation Algorithm (also known as Gillespie’s algorithm) allows for practical simulations to be performed. However, for large networks and/or fast reactions, such computations can be prohibitatively expensive. The Chemical Langevin regime replaces the massive ordinary differential equation system with a small stochastic differential equation system that is more amenable to computation. Although the transition from Chemical Master Equation to Chemical Langevin Equation can be justified rigorously in the large system size limit, there is very little guidance available about how closely the two models match for a fixed system. Here, we consider a transcription model from the recent literature and show that it is possible to compare first and second moments in the two stochastic settings. To analyse the Chemical Master Equation we use some recent work of Gadgil, Lee and Othmer, and to analyse the Chemical Langevin Equation we use Ito’s Lemma. We find that there is a perfect match—both modelling regimes give the same means, variances and correlations for all components in the system. The model that we analyse involves ‘unimolecular reactions’, and we finish with some numerical simulations involving dimerization to show that the means and variances in the two regimes can also be close when more general ‘bimolecular reactions ’ are involved.
Approximation of event probabilities in noisy cellular processes
 In Proc. of CMSB
, 2009
"... Abstract. Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discretestate continuoustime stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain ..."
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Cited by 9 (4 self)
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Abstract. Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discretestate continuoustime stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required. 1
Process Noise: An Explanation for the Fluctuations in the Immune Response during Acute Viral Infection
"... ABSTRACT The parameters of the immune response dynamics are usually estimated by the use of deterministic ordinary differential equations that relate data trends to parameter values. Since the physical basis of the response is stochastic, we are investigating the intensity of the data fluctuations r ..."
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Cited by 7 (1 self)
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ABSTRACT The parameters of the immune response dynamics are usually estimated by the use of deterministic ordinary differential equations that relate data trends to parameter values. Since the physical basis of the response is stochastic, we are investigating the intensity of the data fluctuations resulting from the intrinsic response stochasticity, the socalled process noise. Dealing with the CD8 1 Tcell responses of virusinfected mice, we find that the process noise influence cannot be neglected and we propose a parameter estimation approach that includes the process noise stochastic fluctuations. We show that the variations in data can be explained completely by the process noise. This explanation is an alternative to the one resulting from standard modeling approaches which say that the difference among individual immune responses is the consequence of the difference in parameter values.
An alternative to Gillespie’s algorithm for simulating chemical reactions
 Computational Methods in Systems Biology (CMSB’05
, 2005
"... Abstract. We introduce a probabilistic algorithm for the simulation of chemical reactions, which can be used as an alternative to the wellestablished stochastic algorithm proposed by D.T. Gillespie in the ’70s. We show that the probabilistic evolution of systems derived by means of our algorithm can ..."
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Cited by 7 (4 self)
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Abstract. We introduce a probabilistic algorithm for the simulation of chemical reactions, which can be used as an alternative to the wellestablished stochastic algorithm proposed by D.T. Gillespie in the ’70s. We show that the probabilistic evolution of systems derived by means of our algorithm can be compared to the stochastic time evolution of chemical reactive systems described by Gillespie. Moreover, we use our algorithm in the definition of a formal model based on multiset rewriting, and we show some simulation results of enzymatic activity, which we compare with results of real experiments. 1