The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches.

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We consider two operations in the QTT format: composition of a multilevel Toeplitz matrix generated by a given multidimensional vector and convolution of two given multidimensional vectors. We show that low-rank QTT structure of the input is preserved in the output and propose efficient algorithms for these operations in the QTT format. For a d-dimensional 2n ×. .. × 2n-vector x given in a QTT representation with ranks bounded by p we show how a multilevel Toeplitz matrix generated by x can be obtained in the QTT format with ranks bounded by 2p in O dp 2 log n operations. We also describe how the convolution x y of x and a d-dimensional n ×. .. × n-vector y can be computed in the QTT format with ranks bounded by 2t in O dt 2 log n operations, provided that the matrix xy is given in a QTT representation with ranks bounded by t. We exploit approximate matrix-vector multiplication in the QTT format to accelerate the convolution algorithm dramatically. We demonstrate high performance of the convolution algorithm with numerical examples including computation of the Newton potential of a strong cusp on fine grids with up to 2 20 ×2 20 ×2 20 points in 3D.

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