Stochastic Control of Metabolic Pathways

📝 Abstract

We study the effect of extrinsic noise in metabolic networks. We introduce external random fluctuations at the kinetic level, and show how these lead to a stochastic generalization of standard Metabolic Control Analysis. While Summation and Connectivity Theorems hold true in presence of extrinsic noise, control coefficients incorporate its effect through an explicit dependency on the noise intensity. New elasticities and response coefficients are also defined. Accordingly, the concept of control by noise is introduced as a way of tuning the systemic behaviour of metabolisms. We argue that this framework holds for intrinsic noise too, when time-scale separation is present in the system.

💡 Analysis

We study the effect of extrinsic noise in metabolic networks. We introduce external random fluctuations at the kinetic level, and show how these lead to a stochastic generalization of standard Metabolic Control Analysis. While Summation and Connectivity Theorems hold true in presence of extrinsic noise, control coefficients incorporate its effect through an explicit dependency on the noise intensity. New elasticities and response coefficients are also defined. Accordingly, the concept of control by noise is introduced as a way of tuning the systemic behaviour of metabolisms. We argue that this framework holds for intrinsic noise too, when time-scale separation is present in the system.

📄 Content

arXiv:0906.0274v1 [q-bio.SC] 1 Jun 2009 Stochastic Control of Metabolic Pathways Andrea Rocco Department of Mathematical Sciences, University of Bath BA2 7AY Bath, United Kingdom (Dated: October 25, 2018) We study the effect of extrinsic noise in metabolic networks. We introduce external random fluctuations at the kinetic level, and show how these lead to a stochastic generalization of standard Metabolic Control Analysis. While Summation and Connectivity Theorems hold true in presence of extrinsic noise, control coefficients incorporate its effect through an explicit dependency on the noise intensity. New elasticities and response coefficients are also defined. Accordingly, the concept of control by noise is introduced as a way of tuning the systemic behaviour of metabolisms. We argue that this framework holds for intrinsic noise too, when time-scale separation is present in the system.

1. INTRODUCTION Stochastic fluctuations represent an important contribution to complex behaviours of biological systems. Stochas- ticity appears as a fundamental dynamical mechanism, which does not only generate phenotypic diversity [6], but also plays a major role at many different levels. Selection of alternative pathways in epigenetic switches [4], or synchroni- sation of multicellular systems [33], are just two of the many processes which not only are influenced by stochasticity, but seem to use it to perform in optimal way [32]. In molecular biology two classes of stochastic fluctuations are particularly relevant. The first class is related to the low copy number of chemical species. In particular, if N is the number of molecules in the system, fluctuations in N lead to an associated statistical noise with intensity of the order of N −1/2. While continuous deterministic descriptions in terms of average concentrations can effectively capture the relevant dynamics for N large, when N is small fluctuations may become huge, and noise cannot be neglected [9]. Gene regulation is a typical example, as it may be affected by large fluctuations due to the low copy number of transcription factors [5]. Fluctuations associated with the intrinsic discreteness of the collisional processes among single molecules are usually referred to as intrinsic (or internal) noise. On the other hand, the behaviour of a biochemical system depends also on a number of control parameters. Some of these relate directly to the macroscopic environment, such as for instance illumination conditions, or pH levels. Others have a more microscopic origin, but still exhert a control on the sytem which is independent of its intrinsic dynamics. In gene regulatory networks, for instance, factors acting globally on all genes, such as abundance of RNA polymerase, can change the global efficiency of transcription factors, and ultimately contribute to tuning gene expression levels [26]. Similarly, reaction constants, enzyme activities, or input signals critically control the functional behaviour of metabolic pathways. In thise sense, all these parameters are external to the system. Fluctuations of external parameters define a second type of noise, usually referred to as extrinsic, or external. The effect of extrinsic noise can be highly non-trivial and counterintuitive. Extrinsic noise in homogeneous chemical systems has been shown to provoke noise-induced transitions [22], and recently it has been identified as a mechanism for creating and sustaining spatio-temporal patterns in spatially extended systems [11]. In gene networks the interplay between intrinsic and extrinsic noise has been recently analysed in [25, 28, 29]. In this paper we plan to extend these findings to metabolic networks. We shall focus on the case when the copy number of molecules is large enough for intrinsic noise to be negligible, so that a continuous description in terms of average concentrations is feasible, but at the same time the system experiences extrinsic stochastic fluctuations. Deterministic rate equations become thus stochastic differential equations. Given the description in terms of rate equations, it is ideally desirable to define other, more “systemic” approaches, which explain how global properties of the pathway, such as fluxes and concentrations, depend on local variables, such as enzyme activities. To this aim, in the specific case of metabolic networks a useful strategy –the so-called Metabolic Control Analysis (MCA) [15]– has been developed, and has become nowadays a popular quantitative framework for investigating control and regulation of metabolisms. Mathematically MCA is a sensitivity analysis, and is based on a proper manipulation of the rate equations. It consists in perturbing the parameters of the system, and evaluating the corresponding change in steady state fluxes and concentrations. Perturbations need to be small so that a linear approximation can hold, even if in principle the analysis can be carried on including higher order terms. The effect of the perturbation is represented by a control

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