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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: A common challenge of stochastic models is to calibrate a large number of model parameters against the experimental data.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.

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.


Circular representation [34] of estimated parameter combinations for the Schlögl model. Each spoke represents the corresponding parameter range listed in electronic supplementary material, table S3. The true parameter values are specified by the intersection points between the spokes and the dashed circle. Each triangle (or polygon in general) of a fixed colour corresponds to one admissible parameter set with ɛ = 0.25%. Each panel (a–d) shows the situation with one parameter fixed at its true value, namely (a) k1 is fixed; (b) k2 is fixed; (c) k3 is fixed; and (d) k4 is fixed.
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RSIF20150233F2: Circular representation [34] of estimated parameter combinations for the Schlögl model. Each spoke represents the corresponding parameter range listed in electronic supplementary material, table S3. The true parameter values are specified by the intersection points between the spokes and the dashed circle. Each triangle (or polygon in general) of a fixed colour corresponds to one admissible parameter set with ɛ = 0.25%. Each panel (a–d) shows the situation with one parameter fixed at its true value, namely (a) k1 is fixed; (b) k2 is fixed; (c) k3 is fixed; and (d) k4 is fixed.

Mentions: We illustrate the tensor-structured parameter estimation using the Schlögl chemical system [23], which is written for N = 1 molecular species and has M = 4 reaction rate constants ki, i = 1, 2, 3, 4. A detailed description of this system is provided in electronic supplementary material, appendix S2.1. We prescribe true parameter values as k1 = 2.5 × 10−4, k2 = 0.18, k3 = 2250 and k4 = 37.5, and use a long-time stochastic simulation to generate a time series as pseudo-experimental data (for a short segment, see figure 1a). These pseudo-experimental data are then used for estimating the first three empirical moments , i = 1, 2, 3, using (3.2). While the moments of the model output, , i = 1, 2, 3, are derived from the tensor-structured data p(x/k), computed using (2.2). Moment matching is sensitive to the choice of weights [33]. However, for the sake of simplicity, we choose the weights βi, i = 1, 2, 3, in a way that the contributions of the different orders of moments are of similar magnitude within the parameter space. Having the stationary distribution stored in the tensor format (2.2), we can then efficiently iterate steps (b1)–(b4) in table 3 to search for parameter values that produce adequate fit to the samples using the measure given in equation (3.1). We consider ɛ = 0.25% and visualize in figure 2 the admissible parameter values satisfying .Figure 1.


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

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

Circular representation [34] of estimated parameter combinations for the Schlögl model. Each spoke represents the corresponding parameter range listed in electronic supplementary material, table S3. The true parameter values are specified by the intersection points between the spokes and the dashed circle. Each triangle (or polygon in general) of a fixed colour corresponds to one admissible parameter set with ɛ = 0.25%. Each panel (a–d) shows the situation with one parameter fixed at its true value, namely (a) k1 is fixed; (b) k2 is fixed; (c) k3 is fixed; and (d) k4 is fixed.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSIF20150233F2: Circular representation [34] of estimated parameter combinations for the Schlögl model. Each spoke represents the corresponding parameter range listed in electronic supplementary material, table S3. The true parameter values are specified by the intersection points between the spokes and the dashed circle. Each triangle (or polygon in general) of a fixed colour corresponds to one admissible parameter set with ɛ = 0.25%. Each panel (a–d) shows the situation with one parameter fixed at its true value, namely (a) k1 is fixed; (b) k2 is fixed; (c) k3 is fixed; and (d) k4 is fixed.
Mentions: We illustrate the tensor-structured parameter estimation using the Schlögl chemical system [23], which is written for N = 1 molecular species and has M = 4 reaction rate constants ki, i = 1, 2, 3, 4. A detailed description of this system is provided in electronic supplementary material, appendix S2.1. We prescribe true parameter values as k1 = 2.5 × 10−4, k2 = 0.18, k3 = 2250 and k4 = 37.5, and use a long-time stochastic simulation to generate a time series as pseudo-experimental data (for a short segment, see figure 1a). These pseudo-experimental data are then used for estimating the first three empirical moments , i = 1, 2, 3, using (3.2). While the moments of the model output, , i = 1, 2, 3, are derived from the tensor-structured data p(x/k), computed using (2.2). Moment matching is sensitive to the choice of weights [33]. However, for the sake of simplicity, we choose the weights βi, i = 1, 2, 3, in a way that the contributions of the different orders of moments are of similar magnitude within the parameter space. Having the stationary distribution stored in the tensor format (2.2), we can then efficiently iterate steps (b1)–(b4) in table 3 to search for parameter values that produce adequate fit to the samples using the measure given in equation (3.1). We consider ɛ = 0.25% and visualize in figure 2 the admissible parameter values satisfying .Figure 1.

Bottom Line: A common challenge of stochastic models is to calibrate a large number of model parameters against the experimental data.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.

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.