Multiscale Simulation Of Reaction Dynamics In Chemical, Biological And Materials Systems

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In this dissertation, we introduce a novel accelerated-stochastic simulation method, known as the ‘partitioned-leaping algorithm’ (PLA), for efficiently simulating chemical reaction networks. The technique is multiscale in that it considers dynamics at scales ranging from the discrete-stochastic to the continuousdeterministic. It is particularly useful when considering nanoscale-sized systems that exhibit fluctuating dynamics and contain species with large disparities in populations. We present the theoretical foundations of the PLA, discuss various extensions and variants of the method and provide illustrative examples demonstrating its practical utility in chemistry, biology and materials science. In Chapter 1, we provide a general overview of the origins and consequences of stochastic “noise” in nanoscale-sized systems. We elucidate the implications of this phenomenon, which arises because of the discrete and probabilistic nature of molecular interactions, in both biological and materials settings and discuss mathematical approaches that have been applied previously to model such behaviors. The shortcomings of these methods provide the primary motivation for the work presented in this dissertation. In Chapter 2, we present the theoretical foundations of so-called “exact” stochastic simulation approaches. This material lays the foundation for all that is to follow. It can be seen as a review/tutorial of the subject at the level of advanced undergraduate and beginning graduate students. Our presentation closely follows the work of Gillespie ca. 1976. Though many equivalent formalisms have been presented in the literature, Gillespie’s has the advantage of being developed within the language of chemistry and, thus, being more accessible to chemical engineers than other approaches that are often cited, e.g., within the physics literature. In Chapter 3, we present the main contribution of this dissertation, the partitioned-leaping algorithm. Building upon the work of Gillespie ca. 2000 and concepts presented in Chapter 2, we develop an accelerated-stochastic simulation approach that efficiently describes stochastic effects in chemical reaction networks with very little loss in accuracy relative to exact methods. The method is simple, relatively easy to implement and is based on firm theoretical grounds. We also consider numerous variants of the method and discuss areas of possible future extension. In Chapter 4, we proceed to select applications of the PLA. We consider example systems inspired by chemistry, biology and materials science. We begin with various toy problems and then advance to simple, yet relevant, biochemical networks. In all cases, we compare the performance characteristics of the PLA, in terms of accuracy and efficiency, to exact approaches. We also identify conditions where the method does not perform particularly well, investigate the underlying reasons for this and discuss possible strategies for overcoming them. Finally, we conclude in Chapter 5 by summarizing the main results of this dissertation and laying out a vision for the future.

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