Stochastic Modeling Of Reversible Biochemical Reaction Diffusion Systems And High Resolution Shock Capturing Methods For Fluid Interfaces

My thesis contains two parts, both of which are motivated by biological problems. One is on stochastic reaction-diffusion for biochemical systems and the other on shock-capturing methods for fluid interfaces. In both parts, conservation laws are key to determine the dynamics and effective numerical methods. The first part is motivated by the need for quantitative mathematical models for cell-scale biological systems. Such a mathematical description must be inherently stochastic where the chancy reaction process is mediated by diffusion encounter. Diffusion-influenced reaction theory describes this coupling between diffusion and reaction. We apply this theory to theoretical and numerical kinetic Monte Carlo studies of the robustness of fluorescence correlation spectroscopy (FCS) theory, a widely used experimental method to determine chemical rate constants and diffusion coefficients of stochastic reaction-diffusion systems. We found that current FCS theory can produce significant errors at cell-scales. In addition, we developed a framework to understand diffusion-influenced reaction theory from a stochastic perspective. For irreversible bimolecular reactions, the theory is derived by introducing absorbing boundary conditions to overdamped Brownian motion theory. This provides a clear stochastic interpretation that describes the probability distribution dynamics and the stochastic sample trajectories. However, the stochastic interpretation is not clear for reversible reactions modeled with a back-reaction boundary condition. In order to address this, we developed a discrete stochastic model that conserves probability and recovers the classical equations in the continuous limit. In the case of reversible reactions, it recovers the back-reaction boundary condition and provides an accurate stochastic interpretation. We also explore extensions of this model and its relation to nonequilibrium stochastic processes as well as extensions into volume reactivity using coupled-diffusion processes. The second part was inspired by a collaboration with experimentalists at Seattle's Veterans Administration (VA) Hospital, who are studying the underlying biological mechanisms behind blast-induced traumatic brain injury (TBI). To better understand the effect of shock waves on the brain, we have investigated an in vitro model in which blood-brain barrier endothelial cells are grown in fluid-filled transwell vessels, placed inside a shock tube and exposed to shocks. As it is difficult to experimentally measure the forces inside the transwell, we developed a computational model of the experimental setup to measure them. First, we implemented a one-dimensional model using Euler equations coupled with a Tammann equation of state (EOS) to model the different materials and interfaces within the experimental setup. From this model, we learned that we can neglect very thin interfaces in our computations. Using this result, we implemented a three-dimensional wave propagation framework modeled with two-dimensional axisymmetric Euler equations and a Tammann EOS. In order to solve these equations, we used high-resolution conservative methods and implemented new Riemann solvers into the Clawpack software in a mixed Eulerian/Lagrangian frame of reference. We found that pressures can fall below vapor pressure due to the interaction of reflecting and diffracting shock waves, suggesting that cavitation bubbles could be a damage mechanism. We also show extensions of this model that allow the implementation of mapped grids and adaptive mesh refinement.

Stochastic kinetic methods are currently considered to be the most realistic and elegant means of representing and simulating the dynamics of biochemical and biological networks. Deterministic versus stochastic modelling in biochemistry and systems biology introduces and critically reviews the deterministic and stochastic foundations of biochemical kinetics, covering applied stochastic process theory for application in the field of modelling and simulation of biological processes at the molecular scale. Following an overview of deterministic chemical kinetics and the stochastic approach to biochemical kinetics, the book goes onto discuss the specifics of stochastic simulation algorithms, modelling in systems biology and the structure of biochemical models. Later chapters cover reaction-diffusion systems, and provide an analysis of the Kinfer and BlenX software systems. The final chapter looks at simulation of ecodynamics and food web dynamics. Introduces mathematical concepts and formalisms of deterministic and stochastic modelling through clear and simple examples Presents recently developed discrete stochastic formalisms for modelling biological systems and processes Describes and applies stochastic simulation algorithms to implement a stochastic formulation of biochemical and biological kinetics

This book provides an introduction to the analysis of stochastic dynamic models in biology and medicine. The main aim is to offer a coherent set of probabilistic techniques and mathematical tools which can be used for the simulation and analysis of various biological phenomena. These tools are illustrated on a number of examples. For each example, the biological background is described, and mathematical models are developed following a unified set of principles. These models are then analyzed and, finally, the biological implications of the mathematical results are interpreted. The biological topics covered include gene expression, biochemistry, cellular regulation, and cancer biology. The book will be accessible to graduate students who have a strong background in differential equations, the theory of nonlinear dynamical systems, Markovian stochastic processes, and both discrete and continuous state spaces, and who are familiar with the basic concepts of probability theory.

The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley. It presents the modern ideas in these fields in a way that is accessible to a wider audience than just mathematicians.The book is divided into four main parts: linear theory, reaction-diffusion equations, shock-wave theory, and the Conley index. For the second edition numerous typographical errors and other mistakes have been corrected and a new chapter on recent results has been added. The new chapter contains discussions of the stability of traveling waves, symmetry-breaking bifurcations, compensated compactness, viscous profiles for shock waves, and general notions for construction traveling-wave solutions for systems of nonlinear equations.

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 developed 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.

"The communication within a cell is taken care of by an intriguing system of 'messages in molecular bottles' known as intracellular signaling. This signaling system is coupled to other biochemical reaction systems that regulate specific intracellular processes. For a good understanding of the biochemical reaction networks involved in signaling and regulation, molecular biological techniques are essential. These techniques provide valuable information about the individual chemical reactions and interactions that constitute the reaction network. However, even with this knowledge, the complex interplay between those reactions cannot fully be understood. Therefore, there is an increasing demand for mathematical modeling and simulations to help to unravel the secrets of intracellular signaling and regulation. The most commonly used methods for mathematical modeling of biochemical reaction systems are deterministic. In such methods, differential equations are used to describe the average behavior of a reaction system. These methods are suitable for relatively large systems. However, of some types of molecules that are involved in intracellular signaling and regulation, only several hundred or a few thousand copies are present in a cell. In those cases, random events (intrinsic noise) may play an important role and so-called stochastic modeling methods may provide a better description of the behavior of the reaction network. Such stochastic methods take into account that from an initial state various other states can be visited subsequently. In this dissertation, we investigate biochemical reaction systems for which both deterministic and stochastic models are constructed. For those systems, we study how the behavior of the stochastic model depends on the total number of molecules. In addition, we relate this dependency to the behavior of the corresponding deterministic model." -- Provided by the publisher.

Biochemical reactions make up most of the activity in a cell. There is inherent stochasticity in the kinetic behavior of biochemical reactions which in turn governs the fate of various cellular processes. In this work, the precision of a method for dimensionality reduction for stochastic modeling of biochemical reactions is evaluated. Further, a method of stochastic simulation of reaction kinetics is implemented in case of a specific biochemical network involved in maintenance of long-term potentiation (LTP), the basic substrate for learning and memory formation. The dimensionality reduction method diverges significantly from a full stochastic model in prediction the variance of the fluctuations. The application of the stochastic simulation method to LTP modeling was used to find qualitative dependence of stochastic fluctuations on reaction volume and model parameters.

The most common theoretical approach to model the interactions in a biochemical process is through chemical reactions. Often for these reactions, the dynamics of the first M-order statistical moments of the species populations do not form a closed system of differential equations, in the sense that the time-derivatives of first M-order moments generally depend on moments of order higher than M. However, for analysis purposes, these dynamics are often made to be closed by approximating the needed derivatives of the first M-order moments by nonlinear functions of the same moments. These functions are called the moment closure functions. This paper presents a systematic procedure to construct these moment closure functions. This is done by first assuming that they exhibit a certain separable form, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. Using these results a stochastic model for gene expression is investigated. We show that in gene expression mechanisms, in which a protein inhibits its own transcription, the resulting negative feedback reduces stochastic variations in the protein populations.