A Numerical Study of the Stochastic Reaction-Diffusion Master Equation Using Tensors and Parallelism with Application to Biological Models.

Biological systems often exhibit both stochasticity in chemical reactions and spatial diffusion of molecules, making their modeling inherently complex. The Reaction-Diffusion Master Equation (RDME) provides a stochastic framework to describe such systems by partitioning space into compartments and considering well-mixed species within each. Compared to the Chemical Master Equation (CME), the RDME introduces a significantly larger state space due to the inclusion of jump processes between compartments. The Stochastic Simulation Algorithm (SSA) is commonly used for CME analysis, but is computationally intensive for large-scale systems, which grow worse in RDME problems. Given this complexity, efficient numerical methods are crucial for analyzing and predicting the dynamics of the systems. In this dissertation, we explore biological models using the RDME formulation and leverage tensor-based techniques to manage the large state space efficiently. Tensors, as multidimensional arrays, offer powerful mechanisms for processing and analyzing data in complex biological systems, making them well-suited for RDME applications. To address the computational challenges associated with simulating RDME trajectories over time with the SSA, we implement a parallelized approach using OpenMP with FORTRAN, distributing computations across multiple processors to enhance efficiency. Building on tensor-based methods and high-performance computing, our study aims to significantly accelerate RDME simulations while maintaining accuracy. Advanced numerical techniques enable more effective modeling of stochastic reaction-diffusion systems, providing deeper insights into the dynamics of biological processes.