The stochastic dynamics of networks of biochemical reactions in living cells are typically modelled using chemical master equations (CMEs ). The stationary distributions of CMEs are seldom solvable analytically, and few methods exist that yield numerical estimates with computable error bounds. Here, we present two such methods based on mathematical programming techniques . First, we use semidefinite programming to obtain increasingly tighter upper and lower bounds on the moments of the stationary distribution for networks with rational propensities. Second, we employ linear programming to compute convergent upper and lower bounds on the stationary distributions themselves. The bounds obtained provide a computational test for the uniqueness of the stationary distribution . In the unique case, the bounds collectively form an approximation of the stationary distribution accompanied with a computable \ell^1-error bound . In the non-unique case, we explain how to adapt our approach so that it yields approximations of the ergodic distributions , also accompanied with computable error bounds . We illustrate our methodology through two biological examples : Schl\"ogl's model and a toggle switch model.
The stochastic dynamics of biochemical networks is usually modelled with the chemical master equation (CME ). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. To fill this gap, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy . First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves. The bounds obtained provide a computational test for the uniqueness of these distributions . In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error . In the non-unique case, our approach yields converging approximations of the ergodic distributions . We illustrate our methodology through two biochemical networks that exhibit bifurcations to multimodal behaviour : Schl\"ogl's model and a toggle switch model.
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networks of biochemical reactions in living cells are typically modelled using chemical master equations (CMEs
biochemical networks is usually modelled with the chemical master equation (CME
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few methods exist that yield numerical estimates with computable error bounds. Here, we present two such methods based on mathematical programming techniques
numerical methods typically produce estimates with uncontrolled errors. To fill this gap, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy
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obtain
compute
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distribution
distributions
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employ linear programming to compute
use these moment bounds to formulate linear programs that yield
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the stationary distribution
these distributions
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collectively
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accompanied
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\ell^1-error bound
bound on its error
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we explain how to adapt our approach so that it yields
our approach yields converging
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, also accompanied with computable error bounds
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biological examples
biochemical networks that exhibit bifurcations to multimodal behaviour