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This is a monograph on the emerging branch of mathematical
biophysics combining asymptotic analysis with numerical and
stochastic methods to analyze partial differential equations
arising in biological and physical sciences. In more detail, the
book presents the analytic methods and tools for approximating
solutions of mixed boundary value problems, with particular
emphasis on the narrow escape problem. Informed throughout by
real-world applications, the book includes topics such as the
Fokker-Planck equation, boundary layer analysis, WKB approximation,
applications of spectral theory, as well as recent results in
narrow escape theory. Numerical and stochastic aspects, including
mean first passage time and extreme statistics, are discussed in
detail and relevant applications are presented in parallel with the
theory. Including background on the classical asymptotic theory of
differential equations, this book is written for scientists of
various backgrounds interested in deriving solutions to real-world
problems from first principles.
This book focuses on the modeling and mathematical analysis of
stochastic dynamical systems along with their simulations. The
collected chapters will review fundamental and current topics and
approaches to dynamical systems in cellular biology. This text aims
to develop improved mathematical and computational methods with
which to study biological processes. At the scale of a single cell,
stochasticity becomes important due to low copy numbers of
biological molecules, such as mRNA and proteins that take part in
biochemical reactions driving cellular processes. When trying to
describe such biological processes, the traditional deterministic
models are often inadequate, precisely because of these low copy
numbers. This book presents stochastic models, which are necessary
to account for small particle numbers and extrinsic noise sources.
The complexity of these models depend upon whether the biochemical
reactions are diffusion-limited or reaction-limited. In the former
case, one needs to adopt the framework of stochastic
reaction-diffusion models, while in the latter, one can describe
the processes by adopting the framework of Markov jump processes
and stochastic differential equations. Stochastic Processes,
Multiscale Modeling, and Numerical Methods for Computational
Cellular Biology will appeal to graduate students and researchers
in the fields of applied mathematics, biophysics, and cellular
biology.
This book covers recent developments in the non-standard
asymptotics of the mathematical narrow escape problem in stochastic
theory, as well as applications of the narrow escape problem in
cell biology. The first part of the book concentrates on
mathematical methods, including advanced asymptotic methods in
partial equations, and is aimed primarily at applied mathematicians
and theoretical physicists who are interested in biological
applications. The second part of the book is intended for
computational biologists, theoretical chemists, biochemists,
biophysicists, and physiologists. It includes a summary of output
formulas from the mathematical portion of the book and concentrates
on their applications in modeling specific problems in theoretical
molecular and cellular biology. Critical biological processes, such
as synaptic plasticity and transmission, activation of genes by
transcription factors, or double-strained DNA break repair, are
controlled by diffusion in structures that have both large and
small spatial scales. These may be small binding sites inside or on
the surface of the cell, or narrow passages between subcellular
compartments. The great disparity in spatial scales is the key to
controlling cell function by structure. This volume reports recent
progress on resolving analytical and numerical difficulties in
extracting properties from experimental data, biophysical models,
and from Brownian dynamics simulations of diffusion in multi-scale
structures.
This book covers recent developments in the non-standard
asymptotics of the mathematical narrow escape problem in stochastic
theory, as well as applications of the narrow escape problem in
cell biology. The first part of the book concentrates on
mathematical methods, including advanced asymptotic methods in
partial equations, and is aimed primarily at applied mathematicians
and theoretical physicists who are interested in biological
applications. The second part of the book is intended for
computational biologists, theoretical chemists, biochemists,
biophysicists, and physiologists. It includes a summary of output
formulas from the mathematical portion of the book and concentrates
on their applications in modeling specific problems in theoretical
molecular and cellular biology. Critical biological processes, such
as synaptic plasticity and transmission, activation of genes by
transcription factors, or double-strained DNA break repair, are
controlled by diffusion in structures that have both large and
small spatial scales. These may be small binding sites inside or on
the surface of the cell, or narrow passages between subcellular
compartments. The great disparity in spatial scales is the key to
controlling cell function by structure. This volume reports recent
progress on resolving analytical and numerical difficulties in
extracting properties from experimental data, biophysical models,
and from Brownian dynamics simulations of diffusion in multi-scale
structures.
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