Fig. 3: Scalability of the RB-CME solver in a class of linear networks.
From: Advanced methods for gene network identification and noise decomposition from single-cell data
A A class of linear networks that consists of three types of reactions: the production, degradation, and conversion of Si into Si+1 (i = 1, …, n − 1). All the reactions follow mass-action kinetics, and their reaction constants are k1 = 2.4, k3n−1 = 1.6, k3i+1 = 0.9, k3i−1 = 0.6, and k3i = 1 (i = 1, …, n − 1). At the initial time, each species has a Poisson probability with mean 0.5, and all of them are independent. B The scalability of the Monte Carlo method (with 105 samples), RB-CME solver (with 104 samples), and finite state projection approach (with the truncated space \({\bigotimes }_{i=1}^{n}\{0,1,\ldots,9\}\)) when solving the CME at time 10. We used the filtered FSP as our chosen filtering approach in the RB-CME solver. In the third block, the error is evaluated by the L1 distance between the numerical solution and the exact solution of the CME. This panel tells that the computational time of the Monte Carlo method and that of the RB-CME solver both grow quadratically with the system dimension (n), whereas the computational time of the FSP method grows exponentially with n. Moreover, the error of the Monte Carlo method and that of the RB-CME solver both grow exponentially with n, but the latter grows much slower than the former (see the slopes of the linear-like curves in the log-domain). Notably, the RB-CME solver is as accurate as the FSP method when n = 2. It is because, in this case, no leader-level species exist, and, therefore, the RB-CME solver with the filtered FSP as the chosen filtering approach is equivalent to the FSP method. Source data are provided as a Source Data file.
