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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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Propagation and suppression of environmental fluctuations.(a) Linear birth-death process in a fluctuation environment. The birth-rate is assumed to be linearly modulated by its environment  modeled as a CIR process (see Eq. 19). (b) Calculation of suppressed and effective noise. Individual components were computed analytically by solving an ordinary differential equation (see Eq. 22 in S1 Text) with , ,  and . For orientation, we also show the information gain between  and , computed using the marginal simulation algorithm (green); it can be understood as the gain in information about  through observing  and it exhibits a monotone relationship with the effective noise. (c) Relation between the effective noise and the speed of the environmental fluctuations. Noise contributions were computed by numerically solving Eq. 22 in S1 Text for , ,  and different values of . Note that the results are independent of the death rate . Also shown are realizations of  and  for the case for two different time-scales. If  is fast, the output  is able to suppress most of the variability in . In contrast, if  is slow, fluctuations are largely transferred to  (i.e., has a large effective noise) such that extrinsic noise in  is substantial.
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pcbi-1003942-g004: Propagation and suppression of environmental fluctuations.(a) Linear birth-death process in a fluctuation environment. The birth-rate is assumed to be linearly modulated by its environment modeled as a CIR process (see Eq. 19). (b) Calculation of suppressed and effective noise. Individual components were computed analytically by solving an ordinary differential equation (see Eq. 22 in S1 Text) with , , and . For orientation, we also show the information gain between and , computed using the marginal simulation algorithm (green); it can be understood as the gain in information about through observing and it exhibits a monotone relationship with the effective noise. (c) Relation between the effective noise and the speed of the environmental fluctuations. Noise contributions were computed by numerically solving Eq. 22 in S1 Text for , , and different values of . Note that the results are independent of the death rate . Also shown are realizations of and for the case for two different time-scales. If is fast, the output is able to suppress most of the variability in . In contrast, if is slow, fluctuations are largely transferred to (i.e., has a large effective noise) such that extrinsic noise in is substantial.

Mentions: The mean in Eq.18 is just the unconditional mean of , while the derivative of the expected variance shows an additional negative term, causing it to be smaller than the unconditional variance. Let us for instance consider the case where follows a Cox-Ingersoll-Ross process governed by the SDE (19)with , and as real process parameters and as a standard Wiener process. Note that in this case, Eq. 18 reduces to an autonomous ODE, which for large yields the relative effective noise at stationarity, i.e.,(20)where can be considered a normalized timescale of (see section S.4 in S1 Text). The computation of the effective noise and its dependency on the environmental timescale is illustrated in Fig. 4.


Uncoupled analysis of stochastic reaction networks in fluctuating environments.

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

Propagation and suppression of environmental fluctuations.(a) Linear birth-death process in a fluctuation environment. The birth-rate is assumed to be linearly modulated by its environment  modeled as a CIR process (see Eq. 19). (b) Calculation of suppressed and effective noise. Individual components were computed analytically by solving an ordinary differential equation (see Eq. 22 in S1 Text) with , ,  and . For orientation, we also show the information gain between  and , computed using the marginal simulation algorithm (green); it can be understood as the gain in information about  through observing  and it exhibits a monotone relationship with the effective noise. (c) Relation between the effective noise and the speed of the environmental fluctuations. Noise contributions were computed by numerically solving Eq. 22 in S1 Text for , ,  and different values of . Note that the results are independent of the death rate . Also shown are realizations of  and  for the case for two different time-scales. If  is fast, the output  is able to suppress most of the variability in . In contrast, if  is slow, fluctuations are largely transferred to  (i.e., has a large effective noise) such that extrinsic noise in  is substantial.
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pcbi-1003942-g004: Propagation and suppression of environmental fluctuations.(a) Linear birth-death process in a fluctuation environment. The birth-rate is assumed to be linearly modulated by its environment modeled as a CIR process (see Eq. 19). (b) Calculation of suppressed and effective noise. Individual components were computed analytically by solving an ordinary differential equation (see Eq. 22 in S1 Text) with , , and . For orientation, we also show the information gain between and , computed using the marginal simulation algorithm (green); it can be understood as the gain in information about through observing and it exhibits a monotone relationship with the effective noise. (c) Relation between the effective noise and the speed of the environmental fluctuations. Noise contributions were computed by numerically solving Eq. 22 in S1 Text for , , and different values of . Note that the results are independent of the death rate . Also shown are realizations of and for the case for two different time-scales. If is fast, the output is able to suppress most of the variability in . In contrast, if is slow, fluctuations are largely transferred to (i.e., has a large effective noise) such that extrinsic noise in is substantial.
Mentions: The mean in Eq.18 is just the unconditional mean of , while the derivative of the expected variance shows an additional negative term, causing it to be smaller than the unconditional variance. Let us for instance consider the case where follows a Cox-Ingersoll-Ross process governed by the SDE (19)with , and as real process parameters and as a standard Wiener process. Note that in this case, Eq. 18 reduces to an autonomous ODE, which for large yields the relative effective noise at stationarity, i.e.,(20)where can be considered a normalized timescale of (see section S.4 in S1 Text). The computation of the effective noise and its dependency on the environmental timescale is illustrated in Fig. 4.

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