Solution of the chemical master equation by the separation of variables and alternating optimization methods

Last modified: 2014-06-09

#### Abstract

The chemical master equation (CME) serves as an accurate model of

stochastic fluctuations in chemical kinetics, which are ubiquitous

in systems with small amounts of reacting species, such as cells,

viruses and others.

However, the solution of the CME is the probability distribution function,

which is defined on all states of the system,

and may require enormously large storage, growing exponentially with

the number of species.

In this talk, we employ algebraic methods for separation of variables

to reduce the complexity,

namely, the so-called Matrix Product States (MPS)/Tensor Train format.

This format is an elegant generalization of the low-rank matrix

factorization to high dimensions,

and we demonstrate that the CME operator may be efficiently

represented and the solution sought directly in the MPS form, avoiding

exponential cost dependence on the number of species.

Besides, the spectral time discretization may be simply incorporated

into the scheme,

by considering time as an additional dimension.

The resulting large linear system is solved in the MPS format by the

recently developed

Alternating Minimal Energy (AMEn) algorithm.

The latter supports the alternating optimization technique (so-called

DMRG in quantum physics)

by the classical residual direction, similarly to the steepest descent method.

This scheme avoids traps of spurious optima and manifests rapid

convergence even for

non-symmetric systems, arising from the CME.

We present a couple of convincing examples, such as the cascade gene

regulatory network

and the phage-lambda model.

stochastic fluctuations in chemical kinetics, which are ubiquitous

in systems with small amounts of reacting species, such as cells,

viruses and others.

However, the solution of the CME is the probability distribution function,

which is defined on all states of the system,

and may require enormously large storage, growing exponentially with

the number of species.

In this talk, we employ algebraic methods for separation of variables

to reduce the complexity,

namely, the so-called Matrix Product States (MPS)/Tensor Train format.

This format is an elegant generalization of the low-rank matrix

factorization to high dimensions,

and we demonstrate that the CME operator may be efficiently

represented and the solution sought directly in the MPS form, avoiding

exponential cost dependence on the number of species.

Besides, the spectral time discretization may be simply incorporated

into the scheme,

by considering time as an additional dimension.

The resulting large linear system is solved in the MPS format by the

recently developed

Alternating Minimal Energy (AMEn) algorithm.

The latter supports the alternating optimization technique (so-called

DMRG in quantum physics)

by the classical residual direction, similarly to the steepest descent method.

This scheme avoids traps of spurious optima and manifests rapid

convergence even for

non-symmetric systems, arising from the CME.

We present a couple of convincing examples, such as the cascade gene

regulatory network

and the phage-lambda model.