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Analytical distributions for stochastic gene expression.

ABSTRACT: Gene expression is significantly stochastic making modeling of genetic networks challenging. We present an approximation that allows the calculation of not only the mean and variance, but also the distribution of protein numbers. We assume that proteins decay substantially more slowly than their mRNA and confirm that many genes satisfy this relation by using high-throughput data from budding yeast. For a two-stage model of gene expression, with transcription and translation as first-order reactions, we calculate the protein distribution for all times greater than several mRNA lifetimes and thus qualitatively predict the distribution of times for protein levels to first cross an arbitrary threshold. If in addition the fluctuates between inactive and active states, we can find the steady-state protein distribution, which can be bimodal if fluctuations of the promoter are slow. We show that our assumptions imply that protein synthesis occurs in geometrically distributed bursts and allows mRNA to be eliminated from a master equation description. In general, we find that protein distributions are asymmetric and may be poorly characterized by their mean and variance. Through maximum likelihood methods, our expressions should therefore allow more quantitative comparisons with experimental data. More generally, we introduce a technique to derive a simpler, effective dynamics for a stochastic system by eliminating a fast variable.

SUBMITTER: Shahrezaei V

PROVIDER: S-EPMC2582303 | biostudies-literature | 2008 Nov

REPOSITORIES: biostudies-literature

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Analytical distributions for stochastic gene expression.

Shahrezaei Vahid V Swain Peter S PS

Proceedings of the National Academy of Sciences of the United States of America 20081106 45

Gene expression is significantly stochastic making modeling of genetic networks challenging. We present an approximation that allows the calculation of not only the mean and variance, but also the distribution of protein numbers. We assume that proteins decay substantially more slowly than their mRNA and confirm that many genes satisfy this relation by using high-throughput data from budding yeast. For a two-stage model of gene expression, with transcription and translation as first-order reacti ...[more]

PMID: 18988743

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Project description:The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments only provide limited information, especially when complex, non-linear dynamics are involved.

Dataset Information

Analytical distributions for stochastic gene expression.

Publications

Analytical distributions for stochastic gene expression.

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