Stochastic Modeling of Advection-Diffusion-Reaction Processes in Biological Systems

This dissertation deals with complex and multi-scale biological processes. In general, these phenomena can be described by ordinary or partial differential equations and treated with deterministic methods such as Runge-Kutta and alternating direction implicit algorithms. However, these approaches cannot predict the random effects caused by the low number of molecules involved and can result in severe stability and accuracy problem due to wide range of time or length scales depending upon the system being studied. In the first part of the dissertation, therefore, we develop the stochastic hybrid algorithm for complex reaction networks. Deterministic models of biochemical processes at the subcellular level might become inadequate when a cascade of chemical reactions is induced by a few molecules. Inherent randomness of such phenomena calls for the use of stochastic simulations. However, being computationally intensive, such simulations become infeasible for large and complex reaction networks. To improve their computational efficiency in handling these networks, we present a hybrid approach, in which slow reactions and fluxes are handled through exact stochastic simulation and their fast counterparts are treated partially deterministically through chemical Langevin equation. The classification of reactions as fast or slow is accompanied by the assumption that in the time-scale of fast reactions, slow reactions do not occur and hence do not affect the probability of the state. In the second and third part of the dissertation, we employ stochastic operator splitting algorithm for (chemotaxis-)diffusion-reaction processes. The reaction and diffusion steps employ stochastic simulation algorithm and Brownian dynamics, respectively. Through theoretical analysis, we develop an algorithm to identify if the system is reaction-controlled, diffusion-controlled or is in an intermediate regime. The time-step size is chosen accordingly at each step of the simulation. We apply our algorithm to several examples in order to demonstrate the accuracy, efficiency and robustness of the proposed algorithm comparing with the solutions obtained from deterministic partial differential equations and Gillespie multi-particle method. The third part deals with application of the stochastic-operator splitting approach to model the chemotaxis of leukocytes as part of the inflammation process during wound healing. We analyze both chemotaxis as well as the diffusion process as a drift phenomenon. We use two dimensionless numbers, Damkohler and Peclet number, in order to analyze the system. Damkohler number determines if the system is reaction-controlled or drift controlled and Peclet number identifies which phenomenon is dominant between diffusion and chemotaxis.