Stochastic Reaction Diffusion Methods For Modeling Gene Expression And Spatially Distributed Chemical Kinetics

The aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations. Another main objective is to illustrate and compare diverse numerical simulation schemes and their computational effort. Unlike related works, this book presents a broad scope in its applications, from offering a detailed introduction to hybrid approaches for the case of multiple population scales to discussing the setting of time-scale separation resulting from widely varying firing rates of reaction channels. Additionally, it also addresses modeling approaches for non well-mixed reaction-diffusion dynamics, including deterministic and stochastic PDEs and spatiotemporal master equations. Finally, by translating and incorporating complex theory to a level accessible to non-mathematicians, this book effectively bridges the gap between mathematical research in computational biology and its practical use in biological, biochemical, and biomedical systems.

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.

This practical introduction to stochastic reaction-diffusion modelling is based on courses taught at the University of Oxford. The authors discuss the essence of mathematical methods which appear (under different names) in a number of interdisciplinary scientific fields bridging mathematics and computations with biology and chemistry. The book can be used both for self-study and as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics. New mathematical approaches are explained using simple examples of biological models, which range in size from simulations of small biomolecules to groups of animals. The book starts with stochastic modelling of chemical reactions, introducing stochastic simulation algorithms and mathematical methods for analysis of stochastic models. Different stochastic spatio-temporal models are then studied, including models of diffusion and stochastic reaction-diffusion modelling. The methods covered include molecular dynamics, Brownian dynamics, velocity jump processes and compartment-based (lattice-based) models.

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.

This comprehensive guide, by pioneers in the field, brings together, for the first time, everything a new researcher, graduate student or industry practitioner needs to get started in molecular communication. Written with accessibility in mind, it requires little background knowledge, and provides a detailed introduction to the relevant aspects of biology and information theory, as well as coverage of practical systems. The authors start by describing biological nanomachines, the basics of biological molecular communication and the microorganisms that use it. They then proceed to engineered molecular communication and the molecular communication paradigm, with mathematical models of various types of molecular communication and a description of the information and communication theory of molecular communication. Finally, the practical aspects of designing molecular communication systems are presented, including a review of the key applications. Ideal for engineers and biologists looking to get up to speed on the current practice in this growing field.

This book develops the theory of continuous and discrete stochastic processes within the context of cell biology. In the second edition the material has been significantly expanded, particularly within the context of nonequilibrium and self-organizing systems. Given the amount of additional material, the book has been divided into two volumes, with volume I mainly covering molecular processes and volume II focusing on cellular processes. A wide range of biological topics are covered in the new edition, including stochastic ion channels and excitable systems, molecular motors, stochastic gene networks, genetic switches and oscillators, epigenetics, normal and anomalous diffusion in complex cellular environments, stochastically-gated diffusion, active intracellular transport, signal transduction, cell sensing, bacterial chemotaxis, intracellular pattern formation, cell polarization, cell mechanics, biological polymers and membranes, nuclear structure and dynamics, biological condensates, molecular aggregation and nucleation, cellular length control, cell mitosis, cell motility, cell adhesion, cytoneme-based morphogenesis, bacterial growth, and quorum sensing. The book also provides a pedagogical introduction to the theory of stochastic and nonequilibrium processes – Fokker Planck equations, stochastic differential equations, stochastic calculus, master equations and jump Markov processes, birth-death processes, Poisson processes, first passage time problems, stochastic hybrid systems, queuing and renewal theory, narrow capture and escape, extreme statistics, search processes and stochastic resetting, exclusion processes, WKB methods, large deviation theory, path integrals, martingales and branching processes, numerical methods, linear response theory, phase separation, fluctuation-dissipation theorems, age-structured models, and statistical field theory. This text is primarily aimed at graduate students and researchers working in mathematical biology, statistical and biological physicists, and applied mathematicians interested in stochastic modeling. Applied probabilists should also find it of interest. It provides significant background material in applied mathematics and statistical physics, and introduces concepts in stochastic and nonequilibrium processes via motivating biological applications. The book is highly illustrated and contains a large number of examples and exercises that further develop the models and ideas in the body of the text. It is based on a course that the author has taught at the University of Utah for many years.

A unified Bayesian treatment of the state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models.