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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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Schematic illustration of the marginal simulation algorithm.The red line shows the computation of the marginal hazard function  of the telegraph model (i.e., using Eq.14). The blue dots indicate the corresponding transcription events. Also shown is a possible corresponding realization of the promoter state (gray bars indicate the time the promoter has been active). The figure illustrates that using the MSA algorithm, only the events associated with X need to be simulated (in this case the transcription of mRNA). In contrast, SSA requires explicit simulation of all environmental states, which – depending on the time-scale – may become computationally prohibitive.
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pcbi-1003942-g002: Schematic illustration of the marginal simulation algorithm.The red line shows the computation of the marginal hazard function of the telegraph model (i.e., using Eq.14). The blue dots indicate the corresponding transcription events. Also shown is a possible corresponding realization of the promoter state (gray bars indicate the time the promoter has been active). The figure illustrates that using the MSA algorithm, only the events associated with X need to be simulated (in this case the transcription of mRNA). In contrast, SSA requires explicit simulation of all environmental states, which – depending on the time-scale – may become computationally prohibitive.

Mentions: Upon normalization of the distribution we may express its expectation between two consecutive firing times and with as (14)with and . Furthermore, normalization of shows that when the next reaction at time fires, the conditional expectation instantaneously changes to one. This is consistent with the fact that for a transcription event to happen at , the promoter must be in its active state at least until . Fig. 2 illustrates the computation of the marginal hazard function used during MSA.


Uncoupled analysis of stochastic reaction networks in fluctuating environments.

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

Schematic illustration of the marginal simulation algorithm.The red line shows the computation of the marginal hazard function  of the telegraph model (i.e., using Eq.14). The blue dots indicate the corresponding transcription events. Also shown is a possible corresponding realization of the promoter state (gray bars indicate the time the promoter has been active). The figure illustrates that using the MSA algorithm, only the events associated with X need to be simulated (in this case the transcription of mRNA). In contrast, SSA requires explicit simulation of all environmental states, which – depending on the time-scale – may become computationally prohibitive.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003942-g002: Schematic illustration of the marginal simulation algorithm.The red line shows the computation of the marginal hazard function of the telegraph model (i.e., using Eq.14). The blue dots indicate the corresponding transcription events. Also shown is a possible corresponding realization of the promoter state (gray bars indicate the time the promoter has been active). The figure illustrates that using the MSA algorithm, only the events associated with X need to be simulated (in this case the transcription of mRNA). In contrast, SSA requires explicit simulation of all environmental states, which – depending on the time-scale – may become computationally prohibitive.
Mentions: Upon normalization of the distribution we may express its expectation between two consecutive firing times and with as (14)with and . Furthermore, normalization of shows that when the next reaction at time fires, the conditional expectation instantaneously changes to one. This is consistent with the fact that for a transcription event to happen at , the promoter must be in its active state at least until . Fig. 2 illustrates the computation of the marginal hazard function used during MSA.

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