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

Tensor-structured treatment of the chemical master equation
Vladimir Kazeev

Last modified: 2014-06-09


The chemical master equation (CME) is a cornerstone of the stochastic analysis and simulation of models of biochemical reaction networks. Nevertheless, the direct numerical treatment of the CME has remained elusive due to the curse of dimensionality, i.e. the exponential growth of the storage cost and computational complexity with respect to the dimensionality of the problem.

We consider a novel approach based on the use of the quantized tensor train (QTT) decomposition for the low-parametric representation of tensors. As a measure of the storage cost and computational complexity, the size of the state space considered is replaced with the QTT ranks of the operator and solution. Those are intrinsic characteristics, which govern the number of parameters in the corresponding QTT representations and depend on the desired accuracy.

First, we discuss how the CME can be recast in QTT format efficiently and present bounds on the QTT ranks of the operator depending on the desired accuracy. We consider the classical mass-action and single-enzyme Michaelis–Menten kinetics, which correspond to two widely used classes of propensity functions. Arbitrary combinations of these two types of kinetics are also covered by our results.

Second, for evolution problems, we consider the hp-discontinuous Galerkin discretization in time to reduce the CME evolution problem to a sequence of QTT-structured system of linear equations, which are solved in the course of time marching. For solving the linear systems, we use an algorithm of the TT Toolbox inspired by an approach from quantum chemistry. We demonstrate the efficiency of the approach in three different examples from systems biology: independent birth-death process, an enzymatic futile cycle, and a stochastic switch model. The numerical results demonstrate dramatic speedups and storage savings over standard approaches with elementwise data representation.

Third, for the types of kinetics mentioned above, we consider systems of reacting species with a weakly-reversible reaction network of zero deficiency in the sense of Feinberg. We bound the QTT ranks of stationary distributions of such systems depending on the desired accuracy. We show that the complexity of the approximation scales linearly with respect to the number of species and logarithmically in the maximum copy numbers and desired accuracy. This rigorous theoretical result partly justifies the experimental observations of the efficiency of the QTT representation.

The talk is based on joint works with Christoph Schwab and Mustafa Khammash (ETH Zurich) and Michael Nip (UC Santa Barbara).