Stochastic Models in the Life Sciences and Their Methods of Analysis

The following sections are included:

• Dedication
• Contents
• Preface

The following sections are included:

• Terminology
• Intuitive Probability for Finite Sample Space
• The Binomial Distribution
• Conditional Probability

The following sections are included:

• Introduction
• A Simple Mouse Experiment
• Transition Matrix for a DMC
• Regular Markov Chains
• DNA Mutation
• Linear Difference Equations
• Appendix A — Proof of a Basic Existence Theorem for Regular MC

The following sections are included:

• Introduction
• Gambler’s Ruin
• Expected Transient Stops to an Absorbing State
• Birth and Death Processes
• A Simplified Infectious Disease Problem

The following sections are included:

• Mendelian Genetics and Difference Equation
• Hardy–Weinberg Stability Theorem
• Selective Breeding I
• Gene Frequencies
• Selective Breeding II
• Mutation
• A Nonlinear Infectious Disease Model
• Single Nonlinear Difference Equations

Up to now, the biological phenomena evolving with uncertainty encountered in this volume all share several distinctive features. Most conspicuous of these is the Markovian property of the processes whose evolution with time t depends only on its recent past. It is a feature that persists among the evolving processes treated in this volume since most biological organisms have relatively short term memory and models built on this assumption are more tractable mathematically and simpler for numerical simulations. Two other features common to the random processes in the previous chapters are:

• The sample space of the phenomena modeled is countable (and hence may be indexed by the integers) with elementary events {E₁, E₂, …}).
• The processes evolve with outcomes recorded only in stages indexed by n with a finite time interval between consecutive stages, ${\Delta }_n=t_{n+1}-t_n$ (not necessarily with same actual elapsed time as illustrated by the mouse experiment).

The following sections are included:

• Stochastic Models for the Development of C. Trachomatis
• A Birth and Death Process Model
• Uncertain Host Death
• Uniform Density on a Finite Interval
• Theory and Experimental Results

In this chapter, we begin with another application of the branching process to the problem of one-dimensional random walk along the x-axis. We then deduce from the relevant Kolmogorov ODE system the simplest diffusion equation in the limiting case of the drunkard’s small step size δx tending to zero. This limiting partial differential equation is a special case of the general linear PDE for diffusive phenomena including the phenomenon of heat conduction (see [72] for example). To the extent that heat and temperature distribution and other diffusive phenomena are important in the life sciences, we take the appearance of the heat equation as an opportunity to review some basic theory and solution techniques for linear PDE. These theory and techniques will also be needed for the solution of stochastic partial differential equations in later chapters.

The following sections are included:

• Random Variables
• Multivariate Random Variables
• Mean Square Convergence
• Chebyshev Inequality and Sample Size
• Characteristic Functions and Central Limit Theorem

The following sections are included:

• Introduction
• Functions of a Random Variable
• A Function of Two Random Variables
• Several Functions of Several Random Variables
• Applications to Stochastic ODE
• Liouville Equation for Random Initial Data
• ODE with Random Coefficients

The following sections are included:

• Random Variables with Continuous Indexing
• Moments and Characteristic Functions
• Stationary Stochastic Processes
• Random Walk and the Wiener Process
• Mean Square Continuity
• Mean Square Differentiation
• Mean Square Integration
• Additional Tools in Mean Square Calculus
• White Noise

The following sections are included:

• Existence and Uniqueness of a Mean Square Solution
• Integrate and Fire Models of Motoneuron
• The Scalar Linear Problem
• White Noise Excitation
• Correlated Forcing
• Linear Vector IVP with Random Forcing
• Time-Invariant Systems
• Storage Reduction for Separable PDE
• The Nonautonomous Case

The following sections are included:

• Nonlinear Stochastic ODE Models
• Existence and Uniqueness
• Kinetic Equation for a Stochastic Process
• Langevin’s Equation
• Diffusion Processes
• The Ito Type SDE
• Stochastic Stability

The following sections are included:

• Cable Model Neuron with Random Forcing
• The Method of Spatial Correlations
• Mean and Correlation
• Temporal White Noise Excitation
• Linear Stochastic PDE Systems
• Cable Model Neuron with O-U Input Current

Morphogens are molecular substances that bind to cell surface receptors and other molecules. The gradients of morphogen-receptor complex concentrations (or signaling gradients for brevity) are known to be responsible for the patterning of biological tissues during the developmental phase of the host biological organism. For a number of morphogen families, it is known experimentally that the signaling gradients are formed by morphogens transported from a localized production site and bound to surface receptors of cells near and away from the production site (see references cited in [28]). A theoretical basis for diffusion as a mechanism of morphogen transport was addressed in [28, 29] by analyzing appropriate PDE that model the known morphogen activities in the wing imaginal disc of Drosophila fruit flies. The results show that, when observations and data are correctly interpreted, diffusive transport offers a possible and likely mechanism for signaling gradient formation. We summarize in the sections below the simplest extracellular diffusive model of [29] and use it to investigate some stochastic effects on signaling gradients.

The following sections are included:

• Threshold for Action Potential in Nerve Axon
• The Moments of First Exit Time
• Input Variability and Neuronal Firing
• Stochastic HIV-1 Models
• First Exit Time with a Moving Threshold

The following sections are included:

• The Hodgkin–Huxley Model
• The Fitzhugh–Nagumo Model
• A Model for the Fast Variables

The following sections are included:

• Stochastic Simulations
• Simpler Approaches for Less Information
• Genetic Instability and Carcinogenesis

It has been said that different people learn in different ways. However, it is the author’s experience that most students learn better by an active learning process. The experience has led him to require graded weekly assignments as an important component of his courses and have the students’ performance contribute significantly to their final course grade. The official policy on late homework calls for a 10% reduction of the actual score for every business day late but no more than a 50% reduction for submissions before the final exam week. A typical set of these weekly assignments is included in this appendix.

The following sections are included:

• Bibliography
• Index