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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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Evaluation of the marginal simulation algorithm.(a) Simple three-stage model. Species A, B and C are modeled as coupled linear birth-death processes. Filled arrows correspond to mass-action reactions, whereas empty arrows connecting species S and reaction  indicate that S linearly modulates the rate of that reaction, i.e., . The rate constants were chosen to be . The uncoupled marginal dynamics of C are obtained under a second-order zero-cumulants closure. (b, c) Evaluation of the marginal simulation algorithm. Simulations based on the QSS-approximation neglect a significant portion of variability as opposed to assuming a constant environment (CE) in which case the variability is overestimated. In contrast, the uncoupled dynamics correctly predict the fluctuations on the protein level, while yielding a reduction in computational effort when compared to standard SSA (i.e., a simulation time of approximately 20 min instead of 46 min). For each of the schemes, 7000 sample paths have been used to compute the respective histograms (and moments). (d) Nonlinear reaction network. A ten-dimensional reaction network consisting of species (A, B,…, J) modulates the production rate of a birth-death process X (rate constants ). (e) The uncoupled dynamics of X were computed under a second-order zero-cumulants closure. Different environmental time-scales were realized by multiplying the vector of rate constants by a constant factor . For each , 2000 samples paths were simulated to assess the performance of MSA and SSA, respectively. In all cases, the estimated Kolmogorov distance between the SSA and MSA distributions was below . (f) Bistable reaction network. The environmental part was modeled as a one-dimensional Schloegl system [33] modulating the production rate of a birth-death process X (rate constants ). (g) The uncoupled dynamics were computed by integrating Eq.5 on a truncated state space (i.e., feasible states ). The distribution over X was computed at  minutes using 50000 MSA and SSA samples (estimated Kolmogorov distance of ). The computation of 50 MSA and SSA samples took 13.79 and 255.05 seconds, respectively.
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pcbi-1003942-g003: Evaluation of the marginal simulation algorithm.(a) Simple three-stage model. Species A, B and C are modeled as coupled linear birth-death processes. Filled arrows correspond to mass-action reactions, whereas empty arrows connecting species S and reaction indicate that S linearly modulates the rate of that reaction, i.e., . The rate constants were chosen to be . The uncoupled marginal dynamics of C are obtained under a second-order zero-cumulants closure. (b, c) Evaluation of the marginal simulation algorithm. Simulations based on the QSS-approximation neglect a significant portion of variability as opposed to assuming a constant environment (CE) in which case the variability is overestimated. In contrast, the uncoupled dynamics correctly predict the fluctuations on the protein level, while yielding a reduction in computational effort when compared to standard SSA (i.e., a simulation time of approximately 20 min instead of 46 min). For each of the schemes, 7000 sample paths have been used to compute the respective histograms (and moments). (d) Nonlinear reaction network. A ten-dimensional reaction network consisting of species (A, B,…, J) modulates the production rate of a birth-death process X (rate constants ). (e) The uncoupled dynamics of X were computed under a second-order zero-cumulants closure. Different environmental time-scales were realized by multiplying the vector of rate constants by a constant factor . For each , 2000 samples paths were simulated to assess the performance of MSA and SSA, respectively. In all cases, the estimated Kolmogorov distance between the SSA and MSA distributions was below . (f) Bistable reaction network. The environmental part was modeled as a one-dimensional Schloegl system [33] modulating the production rate of a birth-death process X (rate constants ). (g) The uncoupled dynamics were computed by integrating Eq.5 on a truncated state space (i.e., feasible states ). The distribution over X was computed at minutes using 50000 MSA and SSA samples (estimated Kolmogorov distance of ). The computation of 50 MSA and SSA samples took 13.79 and 255.05 seconds, respectively.

Mentions: The impact of environmental fluctuations on a dynamical system of interest is as diverse as the timescale on which they operate. For instance, extrinsic noise in the context of gene expression might be slowly varying (e.g., correlates well with the cell-cycle [29], [30]), while fluctuations in transcription factor abundance might be significantly faster than the expression kinetics downstream. From a technical point of view, timescales range from constant environmental conditions that are random but fixed [31] to regimes where the fluctuations are very fast, such that quasi-steady-state (QSS) assumptions become applicable [16], [32]. A QSS-based approach for simulating a system in the presence of extrinsic noise corresponds to simulating the conditional CTMC , where is replaced by the mean of . Alternatively, one may try to replace a fluctuating environment through a random but fixed environment of same variance but this leads to an overestimation of the process variance in [5], as discussed in a later section. To investigate the two above simplifying assumptions and compare them to the exact solution obtained via SSA and MSA, we performed a simulation study on a linear three-stage birth-death model given in Fig. 3a. In this case only species C is considered of interest whose uncoupled dynamics are obtained by marginalizing A and B. The results from Fig. 3b and Fig. 3c show that MSA facilitates accurate and fast approximations also under intermediate environmental time-scales where QSS- and static environmental assumptions break down.


Uncoupled analysis of stochastic reaction networks in fluctuating environments.

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

Evaluation of the marginal simulation algorithm.(a) Simple three-stage model. Species A, B and C are modeled as coupled linear birth-death processes. Filled arrows correspond to mass-action reactions, whereas empty arrows connecting species S and reaction  indicate that S linearly modulates the rate of that reaction, i.e., . The rate constants were chosen to be . The uncoupled marginal dynamics of C are obtained under a second-order zero-cumulants closure. (b, c) Evaluation of the marginal simulation algorithm. Simulations based on the QSS-approximation neglect a significant portion of variability as opposed to assuming a constant environment (CE) in which case the variability is overestimated. In contrast, the uncoupled dynamics correctly predict the fluctuations on the protein level, while yielding a reduction in computational effort when compared to standard SSA (i.e., a simulation time of approximately 20 min instead of 46 min). For each of the schemes, 7000 sample paths have been used to compute the respective histograms (and moments). (d) Nonlinear reaction network. A ten-dimensional reaction network consisting of species (A, B,…, J) modulates the production rate of a birth-death process X (rate constants ). (e) The uncoupled dynamics of X were computed under a second-order zero-cumulants closure. Different environmental time-scales were realized by multiplying the vector of rate constants by a constant factor . For each , 2000 samples paths were simulated to assess the performance of MSA and SSA, respectively. In all cases, the estimated Kolmogorov distance between the SSA and MSA distributions was below . (f) Bistable reaction network. The environmental part was modeled as a one-dimensional Schloegl system [33] modulating the production rate of a birth-death process X (rate constants ). (g) The uncoupled dynamics were computed by integrating Eq.5 on a truncated state space (i.e., feasible states ). The distribution over X was computed at  minutes using 50000 MSA and SSA samples (estimated Kolmogorov distance of ). The computation of 50 MSA and SSA samples took 13.79 and 255.05 seconds, respectively.
© Copyright Policy
Related In: Results  -  Collection

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pcbi-1003942-g003: Evaluation of the marginal simulation algorithm.(a) Simple three-stage model. Species A, B and C are modeled as coupled linear birth-death processes. Filled arrows correspond to mass-action reactions, whereas empty arrows connecting species S and reaction indicate that S linearly modulates the rate of that reaction, i.e., . The rate constants were chosen to be . The uncoupled marginal dynamics of C are obtained under a second-order zero-cumulants closure. (b, c) Evaluation of the marginal simulation algorithm. Simulations based on the QSS-approximation neglect a significant portion of variability as opposed to assuming a constant environment (CE) in which case the variability is overestimated. In contrast, the uncoupled dynamics correctly predict the fluctuations on the protein level, while yielding a reduction in computational effort when compared to standard SSA (i.e., a simulation time of approximately 20 min instead of 46 min). For each of the schemes, 7000 sample paths have been used to compute the respective histograms (and moments). (d) Nonlinear reaction network. A ten-dimensional reaction network consisting of species (A, B,…, J) modulates the production rate of a birth-death process X (rate constants ). (e) The uncoupled dynamics of X were computed under a second-order zero-cumulants closure. Different environmental time-scales were realized by multiplying the vector of rate constants by a constant factor . For each , 2000 samples paths were simulated to assess the performance of MSA and SSA, respectively. In all cases, the estimated Kolmogorov distance between the SSA and MSA distributions was below . (f) Bistable reaction network. The environmental part was modeled as a one-dimensional Schloegl system [33] modulating the production rate of a birth-death process X (rate constants ). (g) The uncoupled dynamics were computed by integrating Eq.5 on a truncated state space (i.e., feasible states ). The distribution over X was computed at minutes using 50000 MSA and SSA samples (estimated Kolmogorov distance of ). The computation of 50 MSA and SSA samples took 13.79 and 255.05 seconds, respectively.
Mentions: The impact of environmental fluctuations on a dynamical system of interest is as diverse as the timescale on which they operate. For instance, extrinsic noise in the context of gene expression might be slowly varying (e.g., correlates well with the cell-cycle [29], [30]), while fluctuations in transcription factor abundance might be significantly faster than the expression kinetics downstream. From a technical point of view, timescales range from constant environmental conditions that are random but fixed [31] to regimes where the fluctuations are very fast, such that quasi-steady-state (QSS) assumptions become applicable [16], [32]. A QSS-based approach for simulating a system in the presence of extrinsic noise corresponds to simulating the conditional CTMC , where is replaced by the mean of . Alternatively, one may try to replace a fluctuating environment through a random but fixed environment of same variance but this leads to an overestimation of the process variance in [5], as discussed in a later section. To investigate the two above simplifying assumptions and compare them to the exact solution obtained via SSA and MSA, we performed a simulation study on a linear three-stage birth-death model given in Fig. 3a. In this case only species C is considered of interest whose uncoupled dynamics are obtained by marginalizing A and B. The results from Fig. 3b and Fig. 3c show that MSA facilitates accurate and fast approximations also under intermediate environmental time-scales where QSS- and static environmental assumptions break down.

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