Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

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

Last modified: 2014-06-09


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.