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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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Uncoupled stochastic dynamics.The environmental process  modulates the dynamics of the process under study , e.g., through one of its hazard functions. Marginalization with respect to  yields the uncoupled dynamics of , whereas the original dependency on the environment  is replaced by its optimal estimator given the history of . Consequently, the marginal process  is self-exciting, i.e., it exerts a feedback on itself.
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pcbi-1003942-g001: Uncoupled stochastic dynamics.The environmental process modulates the dynamics of the process under study , e.g., through one of its hazard functions. Marginalization with respect to yields the uncoupled dynamics of , whereas the original dependency on the environment is replaced by its optimal estimator given the history of . Consequently, the marginal process is self-exciting, i.e., it exerts a feedback on itself.

Mentions: In case of the coupled processes and , we analogously marginalize the joint Markov chain with respect to the environmental process . While such a marginalization involves several difficulties, the idea remains the same: we try to construct an uncoupled process which directly admits the marginal path distribution . As a result, we obtain a jump process, which - in contrast to the conditional process - no longer depends on the environment . We remark that a straightforward marginalization of the joint master equation of and generally leads to intractable propensities [16], [5]. Based on the innovation theorem [17] we demonstrate in section S.1 in S1 Text that the hazard functions of the uncoupled process (later referred to as the marginal hazard functions) can be generally written as (2)where the expectation is taken with respect to the conditional distribution . The latter describes the conditional probability of the environmental process given the entire history (or filtration) of process until time . Using the expected value of that distribution, the feed-forward influence of on the hazard functions of can be replaced by a deterministic function of , which no longer depends on the actual state of . Instead, the uncoupled process becomes self-exciting, meaning that it exerts a feedback on itself. Hence, given that we can evaluate Eq.2, we have a means to simulate while bypassing the need to draw realizations of . This has for instance been exploited for the exact simulation of diffusion-driven Poisson processes [18]. Note that the uncoupled process is no longer Markovian, since the conditional expectation - and hence the hazard function - possibly depends on the full process history . A schematic illustration of that uncoupling is given in Fig. 1.


Uncoupled analysis of stochastic reaction networks in fluctuating environments.

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

Uncoupled stochastic dynamics.The environmental process  modulates the dynamics of the process under study , e.g., through one of its hazard functions. Marginalization with respect to  yields the uncoupled dynamics of , whereas the original dependency on the environment  is replaced by its optimal estimator given the history of . Consequently, the marginal process  is self-exciting, i.e., it exerts a feedback on itself.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4256010&req=5

pcbi-1003942-g001: Uncoupled stochastic dynamics.The environmental process modulates the dynamics of the process under study , e.g., through one of its hazard functions. Marginalization with respect to yields the uncoupled dynamics of , whereas the original dependency on the environment is replaced by its optimal estimator given the history of . Consequently, the marginal process is self-exciting, i.e., it exerts a feedback on itself.
Mentions: In case of the coupled processes and , we analogously marginalize the joint Markov chain with respect to the environmental process . While such a marginalization involves several difficulties, the idea remains the same: we try to construct an uncoupled process which directly admits the marginal path distribution . As a result, we obtain a jump process, which - in contrast to the conditional process - no longer depends on the environment . We remark that a straightforward marginalization of the joint master equation of and generally leads to intractable propensities [16], [5]. Based on the innovation theorem [17] we demonstrate in section S.1 in S1 Text that the hazard functions of the uncoupled process (later referred to as the marginal hazard functions) can be generally written as (2)where the expectation is taken with respect to the conditional distribution . The latter describes the conditional probability of the environmental process given the entire history (or filtration) of process until time . Using the expected value of that distribution, the feed-forward influence of on the hazard functions of can be replaced by a deterministic function of , which no longer depends on the actual state of . Instead, the uncoupled process becomes self-exciting, meaning that it exerts a feedback on itself. Hence, given that we can evaluate Eq.2, we have a means to simulate while bypassing the need to draw realizations of . This has for instance been exploited for the exact simulation of diffusion-driven Poisson processes [18]. Note that the uncoupled process is no longer Markovian, since the conditional expectation - and hence the hazard function - possibly depends on the full process history . A schematic illustration of that uncoupling is given in Fig. 1.

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