GPU-Based Steady-State Solution of the Chemical Master Equation

The Chemical Master Equation (CME) is a stochastic and discrete-state continuous-time model for macromolecular reaction networks inside the cell. Under this theoretical framework, the solution of a sparse linear system provides the steady-state probability landscape over the molecular microstates. The CME framework can in fact reveal important insights into basic principles on how biological networks function, having critical applications in stem cell study and cancer development. However, the exploratory nature of system biology research involves the solution of the same reaction network under different conditions. As a result, the application of the CME framework is feasible only if we are able to solve several large linear systems in a reasonable amount of time. In recent years, GPU has emerged as a cost-effective high performance architecture easily available to bioscientists around the world. In this paper, we propose an efficient GPUbased Jacobi iteration for steady-state probability calculation. We provide several optimization strategies based on the problem structure with the aim of outperforming the conventional multicore implementation while minimizing the GPU memory footprint. We combine an ELL+DIAG sparse matrix format with DFS ordering to leverage the diagonal density. Moreover, we devise an improved sliced ELL sparse matrix representation based on warp granularity and local rearrangement. Experimental results demonstrate an average double-precision performance of 14.212 GFLOPS in solving the CME (a speedup of 15.67x compared to the optimized Intel MKL library). Our implementation of the warp-grained sliced ELL format outperforms the state-of-the-art in terms of SpMV performance (a speedup of 1.24x over clSpMV). Moreover, it shows consistent improvements for a wider set of application domains and a good memory footprint reduction. The results achieved in this work provide the foundation for applying the CME framework to realistic biochemical systems. I- addition, our GPU-based steady-state computation can be generalized to operation on stochastic matrices (Markov models), achieving good performance with matrix structures similar to biological reaction networks.