Petri nets for systems and synthetic biology

We give a description of a Petri net-based framework for modelling and analysing biochemical pathways, which uni¯es the qualita- tive, stochastic and continuous paradigms. Each perspective adds its con- tribution 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 quali- tative descriptions are abstractions over stochastic or continuous descrip- tions, 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

The interplay between discrete noise and nonlinear chemical kinetics in a signal amplification cascade

We used various analytical and numerical techniques to elucidate signal propagation in a small enzymatic cascade which is subjected to external and internal noise. The nonlinear character of catalytic reactions, which underlie protein signal transduction cascades, renders stochastic signaling dynamics in cytosol biochemical networks distinct from the usual description of stochastic dynamics in gene regulatory networks. For a simple 2-step enzymatic cascade which underlies many important protein signaling pathways, we demonstrated that the commonly used techniques such as the linear noise approximation and the Langevin equation become inadequate when the number of proteins becomes too low. Consequently, we developed a new analytical approximation, based on mixing the generating function and distribution function approaches, to the solution of the master equation that describes nonlinear chemical signaling kinetics for this important class of biochemical reactions. Our techniques work in a much wider range of protein number fluctuations than the methods used previously. We found that under certain conditions the burst-phase noise may be injected into the downstream signaling network dynamics, resulting possibly in unusually large macroscopic fluctuations. In addition to computing first and second moments, which is the goal of commonly used analytical techniques, our new approach provides the full time-dependent probability distributions of the colored non-Gaussian processes in a nonlinear signal transduction cascade.Comment: 16 pages, 9 figure

Approximating solutions of the chemical master equation using neural networks

The Chemical Master Equation (CME) provides an accurate description of stochastic biochemical reaction networks in well-mixed conditions, but it cannot be solved analytically for most systems of practical interest. Although Monte Carlo methods provide a principled means to probe system dynamics, the large number of simulations typically required can render the estimation of molecule number distributions and other quantities infeasible. In this article, we aim to leverage the representational power of neural networks to approximate the solutions of the CME and propose a framework for the Neural Estimation of Stochastic Simulations for Inference and Exploration (Nessie). Our approach is based on training neural networks to learn the distributions predicted by the CME from relatively few stochastic simulations. We show on biologically relevant examples that simple neural networks with one hidden layer can capture highly complex distributions across parameter space, thereby accelerating computationally intensive tasks such as parameter exploration and inference

Analysis of signalling pathways using the prism model checker

We describe a new modelling and analysis approach for signal transduction networks in the presence of incomplete data. We illustrate the approach with an example, the RKIP inhibited ERK pathway [1]. Our models are based on high level descriptions of continuous time Markov chains: reactions are modelled as synchronous processes and concentrations are modelled by discrete, abstract quantities. The main advantage of our approach is that using a (continuous time) stochastic logic and the PRISM model checker, we can perform quantitative analysis of queries such as if a concentration reaches a certain level, will it remain at that level thereafter? We also perform standard simulations and compare our results with a traditional ordinary differential equation model. An interesting result is that for the example pathway, only a small number of discrete data values is required to render the simulations practically indistinguishable

Complete integrability of information processing by biochemical reactions

Statistical mechanics provides an effective framework to investigate information processing in biochemical reactions. Within such framework far-reaching analogies are established among (anti-) cooperative collective behaviors in chemical kinetics, (anti-)ferromagnetic spin models in statistical mechanics and operational amplifiers/flip-flops in cybernetics. The underlying modeling -- based on spin systems -- has been proved to be accurate for a wide class of systems matching classical (e.g. Michaelis--Menten, Hill, Adair) scenarios in the infinite-size approximation. However, the current research in biochemical information processing has been focusing on systems involving a relatively small number of units, where this approximation is no longer valid. Here we show that the whole statistical mechanical description of reaction kinetics can be re-formulated via a mechanical analogy -- based on completely integrable hydrodynamic-type systems of PDEs -- which provides explicit finite-size solutions, matching recently investigated phenomena (e.g. noise-induced cooperativity, stochastic bi-stability, quorum sensing). The resulting picture, successfully tested against a broad spectrum of data, constitutes a neat rationale for a numerically effective and theoretically consistent description of collective behaviors in biochemical reactions.Comment: 24 pages, 10 figures; accepted for publication in Scientific Report

Investigating modularity in the analysis of process algebra models of biochemical systems

Compositionality is a key feature of process algebras which is often cited as one of their advantages as a modelling technique. It is certainly true that in biochemical systems, as in many other systems, model construction is made easier in a formalism which allows the problem to be tackled compositionally. In this paper we consider the extent to which the compositional structure which is inherent in process algebra models of biochemical systems can be exploited during model solution. In essence this means using the compositional structure to guide decomposed solution and analysis. Unfortunately the dynamic behaviour of biochemical systems exhibits strong interdependencies between the components of the model making decomposed solution a difficult task. Nevertheless we believe that if such decomposition based on process algebras could be established it would demonstrate substantial benefits for systems biology modelling. In this paper we present our preliminary investigations based on a case study of the pheromone pathway in yeast, modelling in the stochastic process algebra Bio-PEPA