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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.


The computation of the stationary distribution using the TPA for a 20-dimensional reaction chain. The CFPE is successively solved on seven grid levels with an increasing number of nodal points. The logarithm of the marginal stationary distribution in the X10–X15 plane computed on (a) the initial coarsest level; and (b) the finest grid level. (c) The convergence of the total error versus the computational time. The error is obtained by comparing the marginal distribution of the computed steady-state distribution with the exact solution of the corresponding CME. The vertical dashed lines correspond to the grid levels. The grid size details on each grid level are given in electronic supplementary material, table S8.
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RSIF20150233F7: The computation of the stationary distribution using the TPA for a 20-dimensional reaction chain. The CFPE is successively solved on seven grid levels with an increasing number of nodal points. The logarithm of the marginal stationary distribution in the X10–X15 plane computed on (a) the initial coarsest level; and (b) the finest grid level. (c) The convergence of the total error versus the computational time. The error is obtained by comparing the marginal distribution of the computed steady-state distribution with the exact solution of the corresponding CME. The vertical dashed lines correspond to the grid levels. The grid size details on each grid level are given in electronic supplementary material, table S8.

Mentions: To overcome technical (numerical) challenges, we have introduced two main approaches for successful computation of the steady-state distribution. First, we compute it using the CFPE approximation which provides additional flexibility in discretizing the state space . The CFPE admits larger grid sizes for numerical simulations than the unit grid size of the CME. In this way, the resulting discrete operator is better conditioned. We illustrate this using a 20-dimensional problem introduced in the last line of table 1 and in electronic supplementary material, appendix S2.4. To compute the stationary distribution, a multi-level approach is implemented, where the steady-state distribution is first approximated on a coarse grid, and then interpolated to a finer grid as the initial guess (see electronic supplementary material, appendix S1.3, for more details). The results are plotted in figure 7. Second, we introduce the adaptive inverse power iteration scheme tailored to current tensor solvers of linear systems, see electronic supplementary material, appendix S1.3, for technical details. As tensor linear solvers are less robust especially for ill-conditioned problems, it is necessary to carefully adapt the shift value during the inverse power iterations in order to balance the conditioning and sufficient speed of the convergence. We would like to emphasize the importance of these improvements, because the TPA is mainly limited by the efficiency of computing steady-state distributions, rather than by the problem dimension, N + K. Both the computational efficiency and the separation rank R are negatively correlated with the relaxation time of the reaction network. Reaction networks exhibiting bistable or oscillating behaviours usually have larger relaxation times. This explains some counterintuitive results in table 1, namely the smaller memory requirements and shorter computational times of the 20-dimensional reaction chain in comparison with the seven-dimensional cell cycle model. In particular, the TPA can be applied to systems with dimensionality N + K greater than 20, provided that they have small relaxation times.Figure 7.


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

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

The computation of the stationary distribution using the TPA for a 20-dimensional reaction chain. The CFPE is successively solved on seven grid levels with an increasing number of nodal points. The logarithm of the marginal stationary distribution in the X10–X15 plane computed on (a) the initial coarsest level; and (b) the finest grid level. (c) The convergence of the total error versus the computational time. The error is obtained by comparing the marginal distribution of the computed steady-state distribution with the exact solution of the corresponding CME. The vertical dashed lines correspond to the grid levels. The grid size details on each grid level are given in electronic supplementary material, table S8.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSIF20150233F7: The computation of the stationary distribution using the TPA for a 20-dimensional reaction chain. The CFPE is successively solved on seven grid levels with an increasing number of nodal points. The logarithm of the marginal stationary distribution in the X10–X15 plane computed on (a) the initial coarsest level; and (b) the finest grid level. (c) The convergence of the total error versus the computational time. The error is obtained by comparing the marginal distribution of the computed steady-state distribution with the exact solution of the corresponding CME. The vertical dashed lines correspond to the grid levels. The grid size details on each grid level are given in electronic supplementary material, table S8.
Mentions: To overcome technical (numerical) challenges, we have introduced two main approaches for successful computation of the steady-state distribution. First, we compute it using the CFPE approximation which provides additional flexibility in discretizing the state space . The CFPE admits larger grid sizes for numerical simulations than the unit grid size of the CME. In this way, the resulting discrete operator is better conditioned. We illustrate this using a 20-dimensional problem introduced in the last line of table 1 and in electronic supplementary material, appendix S2.4. To compute the stationary distribution, a multi-level approach is implemented, where the steady-state distribution is first approximated on a coarse grid, and then interpolated to a finer grid as the initial guess (see electronic supplementary material, appendix S1.3, for more details). The results are plotted in figure 7. Second, we introduce the adaptive inverse power iteration scheme tailored to current tensor solvers of linear systems, see electronic supplementary material, appendix S1.3, for technical details. As tensor linear solvers are less robust especially for ill-conditioned problems, it is necessary to carefully adapt the shift value during the inverse power iterations in order to balance the conditioning and sufficient speed of the convergence. We would like to emphasize the importance of these improvements, because the TPA is mainly limited by the efficiency of computing steady-state distributions, rather than by the problem dimension, N + K. Both the computational efficiency and the separation rank R are negatively correlated with the relaxation time of the reaction network. Reaction networks exhibiting bistable or oscillating behaviours usually have larger relaxation times. This explains some counterintuitive results in table 1, namely the smaller memory requirements and shorter computational times of the 20-dimensional reaction chain in comparison with the seven-dimensional cell cycle model. In particular, the TPA can be applied to systems with dimensionality N + K greater than 20, provided that they have small relaxation times.Figure 7.

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.