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Tensor methods for parameter estimation and bifurcation analysis of stochastic reaction networks.

Liao S, Vejchodský T, Erban R - J R Soc Interface (2015)

Bottom Line: Another difficulty is to study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation).This approach enables simultaneous computation of the model properties for all parameter values within a parameter space.The TPA is illustrated by studying the parameter estimation, robustness, sensitivity and bifurcation structure in stochastic models of biochemical networks.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK.

ABSTRACT
Stochastic modelling of gene regulatory networks provides an indispensable tool for understanding how random events at the molecular level influence cellular functions. A common challenge of stochastic models is to calibrate a large number of model parameters against the experimental data. Another difficulty is to study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation). In this paper, tensor-structured parametric analysis (TPA) is developed to address these computational challenges. It is based on recently proposed low-parametric tensor-structured representations of classical matrices and vectors. This approach enables simultaneous computation of the model properties for all parameter values within a parameter space. The TPA is illustrated by studying the parameter estimation, robustness, sensitivity and bifurcation structure in stochastic models of biochemical networks. A Matlab implementation of the TPA is available at http://www.stobifan.org.

No MeSH data available.


Visualization of the bifurcation structure of the stochastic cell cycle model. (a–d) Marginal steady-state distributions of the phosphorylated cyclin (YP), the inactive MPF (pM) and the active MPF (M). (e–h) Marginal stationary distributions of the cyclin (Y), the phosphrylated cdc2 (CP) and the inactive MPF (pM). Each column corresponds to the same value of the bifurcation parameter k1, which from the left to the right are 0.24, 0.3, 0.35 and 0.4, respectively. All figures show log2 of the marginal steady-state distribution for better visualization.
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RSIF20150233F5: Visualization of the bifurcation structure of the stochastic cell cycle model. (a–d) Marginal steady-state distributions of the phosphorylated cyclin (YP), the inactive MPF (pM) and the active MPF (M). (e–h) Marginal stationary distributions of the cyclin (Y), the phosphrylated cdc2 (CP) and the inactive MPF (pM). Each column corresponds to the same value of the bifurcation parameter k1, which from the left to the right are 0.24, 0.3, 0.35 and 0.4, respectively. All figures show log2 of the marginal steady-state distribution for better visualization.

Mentions: In figure 5, we use the computed tensor-structured parametric probability distribution to visualize the stochastic bifurcation structure of the cell cycle model. As the bifurcation parameter k1 increases, the expected oscillation tube is formed and amplified in the marginalized YP-pM-M state space (figure 5a–d). In figure 5e–h, the marginal distribution in the Y-CP-pM subspace is plotted. We see that it changes from a unimodal (figure 5e) to a bimodal distribution (figure 5f). Cell cycle models have been studied in the deterministic context either as oscillatory [24] or bistable [42,43] systems. In figure 5, we see that the presented stochastic cell cycle model can appear to have both oscillations and bimodality, when different subsets of observables are considered.Figure 5.


Tensor methods for parameter estimation and bifurcation analysis of stochastic reaction networks.

Liao S, Vejchodský T, Erban R - J R Soc Interface (2015)

Visualization of the bifurcation structure of the stochastic cell cycle model. (a–d) Marginal steady-state distributions of the phosphorylated cyclin (YP), the inactive MPF (pM) and the active MPF (M). (e–h) Marginal stationary distributions of the cyclin (Y), the phosphrylated cdc2 (CP) and the inactive MPF (pM). Each column corresponds to the same value of the bifurcation parameter k1, which from the left to the right are 0.24, 0.3, 0.35 and 0.4, respectively. All figures show log2 of the marginal steady-state distribution for better visualization.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4528587&req=5

RSIF20150233F5: Visualization of the bifurcation structure of the stochastic cell cycle model. (a–d) Marginal steady-state distributions of the phosphorylated cyclin (YP), the inactive MPF (pM) and the active MPF (M). (e–h) Marginal stationary distributions of the cyclin (Y), the phosphrylated cdc2 (CP) and the inactive MPF (pM). Each column corresponds to the same value of the bifurcation parameter k1, which from the left to the right are 0.24, 0.3, 0.35 and 0.4, respectively. All figures show log2 of the marginal steady-state distribution for better visualization.
Mentions: In figure 5, we use the computed tensor-structured parametric probability distribution to visualize the stochastic bifurcation structure of the cell cycle model. As the bifurcation parameter k1 increases, the expected oscillation tube is formed and amplified in the marginalized YP-pM-M state space (figure 5a–d). In figure 5e–h, the marginal distribution in the Y-CP-pM subspace is plotted. We see that it changes from a unimodal (figure 5e) to a bimodal distribution (figure 5f). Cell cycle models have been studied in the deterministic context either as oscillatory [24] or bistable [42,43] systems. In figure 5, we see that the presented stochastic cell cycle model can appear to have both oscillations and bimodality, when different subsets of observables are considered.Figure 5.

Bottom Line: Another difficulty is to study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation).This approach enables simultaneous computation of the model properties for all parameter values within a parameter space.The TPA is illustrated by studying the parameter estimation, robustness, sensitivity and bifurcation structure in stochastic models of biochemical networks.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK.

ABSTRACT
Stochastic modelling of gene regulatory networks provides an indispensable tool for understanding how random events at the molecular level influence cellular functions. A common challenge of stochastic models is to calibrate a large number of model parameters against the experimental data. Another difficulty is to study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation). In this paper, tensor-structured parametric analysis (TPA) is developed to address these computational challenges. It is based on recently proposed low-parametric tensor-structured representations of classical matrices and vectors. This approach enables simultaneous computation of the model properties for all parameter values within a parameter space. The TPA is illustrated by studying the parameter estimation, robustness, sensitivity and bifurcation structure in stochastic models of biochemical networks. A Matlab implementation of the TPA is available at http://www.stobifan.org.

No MeSH data available.