It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA.

翻译：越来越认识到，考虑随机效应对研究生物细胞至关重要。然而，相应的数学形式——化学主方程（CME）——遭受维数灾难，因此直接求解对于大多数实际问题不可行。本文针对化学主方程提出一种动态低秩算法，通过将反应网络划分为区块来降低问题的维度。只有跨区块的反应会引入近似误差（其他所有反应均精确计算）。与常用的随机模拟算法（SSA，一种蒙特卡洛方法）相比，该方法的优势在于完全无噪声。这在需要解析概率分布尾部时尤为重要。我们证明，在某些情况下（例如λ噬菌体），本方法能显著降低内存消耗和运行时间，并提供比SSA更高的精度。