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Uncoupled analysis of stochastic reaction networks in fluctuating environments.

Zechner C, Koeppl H - PLoS Comput. Biol. (2014)

Bottom Line: While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze.In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment's effect onto the network into a new model.Using several case studies, we demonstrate the significance of the approach.

View Article: PubMed Central - PubMed

Affiliation: Department of Information Technology and Electrical Engineering, ETH Zurich, Zurich, Switzerland.

ABSTRACT
The dynamics of stochastic reaction networks within cells are inevitably modulated by factors considered extrinsic to the network such as, for instance, the fluctuations in ribosome copy numbers for a gene regulatory network. While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze. In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment's effect onto the network into a new model. More technically, we show how such fluctuating extrinsic components (e.g., chemical species) can be marginalized in order to obtain this decoupled model. We derive its corresponding process- and master equations and show how stochastic simulations can be performed. Using several case studies, we demonstrate the significance of the approach.

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Analytical protein distributions through the slow noise approximation.(a) Two-stage gene expression model. Transcription and translation are modeled through mass-action kinetics with reaction rate constants -. Fluctuations on the mRNA are considered environmental and hence, integrated out in order to obtain a one-dimensional stochastic process describing only the protein. (b) Accuracy of the slow noise approximation. The SNA was compared to the QSS- and CE-approximations by means of the Kolmogorov distance between the respective approximate and exact distribution (SSA) as a function of the relative speed of the mRNA fluctuations defined by  and . The mRNA birth- and death-rates were varied between  and 1 while maintaining a constant ratio of . The remaining parameters were chosen as  and . QSS- and CE approximations break down for slow or fast environmental fluctuations respectively, whereas the SNA yields accurate distributions regardless of the mRNA's timescale. (c) Exemplary distributions obtained through the different approaches in three different regimes (slow, intermediate, fast).
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pcbi-1003942-g005: Analytical protein distributions through the slow noise approximation.(a) Two-stage gene expression model. Transcription and translation are modeled through mass-action kinetics with reaction rate constants -. Fluctuations on the mRNA are considered environmental and hence, integrated out in order to obtain a one-dimensional stochastic process describing only the protein. (b) Accuracy of the slow noise approximation. The SNA was compared to the QSS- and CE-approximations by means of the Kolmogorov distance between the respective approximate and exact distribution (SSA) as a function of the relative speed of the mRNA fluctuations defined by and . The mRNA birth- and death-rates were varied between and 1 while maintaining a constant ratio of . The remaining parameters were chosen as and . QSS- and CE approximations break down for slow or fast environmental fluctuations respectively, whereas the SNA yields accurate distributions regardless of the mRNA's timescale. (c) Exemplary distributions obtained through the different approaches in three different regimes (slow, intermediate, fast).

Mentions: Using the effective noise, we now aim to find a master equation, which describes the time-evolution of the marginal probability distribution . Since depends on rather than , it appears natural to formulate the master equation in and as well. For the example considered here, one can show that the probability distribution satisfies a GME of the form (22)that can be solved analytically using generating functions (see S.5.1 in S1 Text). From we compute the distribution of as(23)i.e., a negative binomial distribution. Eq. 23 provides a surprisingly simple approximate solution for the transient probability distribution of birth death processes in a fluctuating environment. In order to check its validity, we compared the analytical approximate distributions to the ones obtained through SSA for a gene expression model, where the environmental fluctuations are assumed to be due to the mRNA dynamics (see Fig. 5). More specifically, we computed the Kolmogorov distance between the resulting protein distributions as a function of the environmental timescale. Apart from the exact correspondence for the limiting time-scales, Fig. 5 indicates that the SNA provides a good approximation regardless of the environmental timescale.


Uncoupled analysis of stochastic reaction networks in fluctuating environments.

Zechner C, Koeppl H - PLoS Comput. Biol. (2014)

Analytical protein distributions through the slow noise approximation.(a) Two-stage gene expression model. Transcription and translation are modeled through mass-action kinetics with reaction rate constants -. Fluctuations on the mRNA are considered environmental and hence, integrated out in order to obtain a one-dimensional stochastic process describing only the protein. (b) Accuracy of the slow noise approximation. The SNA was compared to the QSS- and CE-approximations by means of the Kolmogorov distance between the respective approximate and exact distribution (SSA) as a function of the relative speed of the mRNA fluctuations defined by  and . The mRNA birth- and death-rates were varied between  and 1 while maintaining a constant ratio of . The remaining parameters were chosen as  and . QSS- and CE approximations break down for slow or fast environmental fluctuations respectively, whereas the SNA yields accurate distributions regardless of the mRNA's timescale. (c) Exemplary distributions obtained through the different approaches in three different regimes (slow, intermediate, fast).
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getmorefigures.php?uid=PMC4256010&req=5

pcbi-1003942-g005: Analytical protein distributions through the slow noise approximation.(a) Two-stage gene expression model. Transcription and translation are modeled through mass-action kinetics with reaction rate constants -. Fluctuations on the mRNA are considered environmental and hence, integrated out in order to obtain a one-dimensional stochastic process describing only the protein. (b) Accuracy of the slow noise approximation. The SNA was compared to the QSS- and CE-approximations by means of the Kolmogorov distance between the respective approximate and exact distribution (SSA) as a function of the relative speed of the mRNA fluctuations defined by and . The mRNA birth- and death-rates were varied between and 1 while maintaining a constant ratio of . The remaining parameters were chosen as and . QSS- and CE approximations break down for slow or fast environmental fluctuations respectively, whereas the SNA yields accurate distributions regardless of the mRNA's timescale. (c) Exemplary distributions obtained through the different approaches in three different regimes (slow, intermediate, fast).
Mentions: Using the effective noise, we now aim to find a master equation, which describes the time-evolution of the marginal probability distribution . Since depends on rather than , it appears natural to formulate the master equation in and as well. For the example considered here, one can show that the probability distribution satisfies a GME of the form (22)that can be solved analytically using generating functions (see S.5.1 in S1 Text). From we compute the distribution of as(23)i.e., a negative binomial distribution. Eq. 23 provides a surprisingly simple approximate solution for the transient probability distribution of birth death processes in a fluctuating environment. In order to check its validity, we compared the analytical approximate distributions to the ones obtained through SSA for a gene expression model, where the environmental fluctuations are assumed to be due to the mRNA dynamics (see Fig. 5). More specifically, we computed the Kolmogorov distance between the resulting protein distributions as a function of the environmental timescale. Apart from the exact correspondence for the limiting time-scales, Fig. 5 indicates that the SNA provides a good approximation regardless of the environmental timescale.

Bottom Line: While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze.In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment's effect onto the network into a new model.Using several case studies, we demonstrate the significance of the approach.

View Article: PubMed Central - PubMed

Affiliation: Department of Information Technology and Electrical Engineering, ETH Zurich, Zurich, Switzerland.

ABSTRACT
The dynamics of stochastic reaction networks within cells are inevitably modulated by factors considered extrinsic to the network such as, for instance, the fluctuations in ribosome copy numbers for a gene regulatory network. While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze. In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment's effect onto the network into a new model. More technically, we show how such fluctuating extrinsic components (e.g., chemical species) can be marginalized in order to obtain this decoupled model. We derive its corresponding process- and master equations and show how stochastic simulations can be performed. Using several case studies, we demonstrate the significance of the approach.

Show MeSH
Related in: MedlinePlus