PhD Defence: Stochastic simulations for extended Becker-Döring models applied to self-assembly processes

Candidate: Pradeep Kunwar

Supervisor: Dr. Katrin Rohlf

Abstract
Understanding the aggregation and fragmentation of molecules in self-assembly processes has important applications in biological systems. Biological systems of interest include for example, red blood cell aggregation/attachment and break-up/detachment known as rouleaux formation, as well as HIV-1 capsid formation. Typically, these coagulation-fragmentation systems are modeled using a system of ordinary differential equations in the form of Becker-Döring (BD) equations, or fully Extended Becker-Döring models (EBDM) as in this Thesis. Cast in the form of stochastic master equations, theoretical estimates for biologically relevant quantities such as the mean first assembly time (MFAT) can be obtained. The MFAT is the time required for the largest-possible sized cluster (or molecule) to be formed in a finite system, from a given initial state. Typically initial populations would be small. Theoretical expressions for MFATs can be obtained from the stochastic master equations, with corresponding stochastic simulations leading to numerical MFAT estimates. In this Thesis, stochastic master equations are considered for chemically-reacting analog systems for extended Becker-Döring models (EBDM) for nite systems. We apply existing forward and backward master equation based methods to various EBDM models, that are then used to obtain theoretical expressions for MFATs. The Gillespie algorithm (or Stochastic Simulation Algorithm (SSA)) is used to obtain numerical simulation results for the stochastic systems, as well as the Reactive Multiparticle Collision Dynamics (RMPC). In the case of our novel RMPC simulations, a relevant analog chemical system for the approximate dynamics is proposed in this Thesis together with the chemical master equation and corresponding theoretical MFAT results are derived. Different rates (constant, additive and multiplicative) are considered, together with a time-rescaling mechanism that lowers the computational time required to obtain numerical RMPC-MFAT estimates and allows for simulation of MFATs that are smaller than the non-scaled RMPC time step of unity. The novel theoretical and numerical EBDM-RMPC results presented in this Thesis, provide a means to evaluate the applicability of an approximate particle - count - conserving numerical method to the self-assembly process. Despite the approximate nature of the RMPC implementation, the behaviour of the approximate RMPC system agrees with the expected physics of an exact implementation, making the approximate RMPC system an ideal simulation paradigm for large-scale applications in future extensions of this work. The approximation was assessed both in a large population (ODE) context, as well as for small population (MFAT) examples. In the case of our SSA results, we explored stochastic rates for analogous chemical rate law interpretations for the self assembly process, and EBDM stochastic rates. There are small differences between the two models. 

Development of stochastic simulation methods like RMPC, that are capable of incorporating velocity dependence in the cluster formation dynamics, are crucial in furthering our understanding of how self-assembly of molecules is coupled to flow. This Thesis provides the foundational aspects necessary for future development in that area.