Stochastic Models in Biology

1st Edition - January 1, 1974
Authors: Narendra S. Goel, Nira Richter-Dyn
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 7 8 1 0 - 0

Stochastic Models in Biology describes the usefulness of the theory of stochastic process in studying biological phenomena. The book describes analysis of biological systems and… Read more

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Stochastic Models in Biology describes the usefulness of the theory of stochastic process in studying biological phenomena. The book describes analysis of biological systems and experiments though probabilistic models rather than deterministic methods. The text reviews the mathematical analyses for modeling different biological systems such as the random processes continuous in time and discrete in state space. The book also discusses population growth and extinction through Malthus' law and the work of MacArthur and Wilson. The text then explains the dynamics of a population of interacting species. The book also addresses population genetics under systematic evolutionary pressures known as deterministic equations and genetic changes in a finite population known as stochastic equations. The text then turns to stochastic modeling of biological systems at the molecular level, particularly the kinetics of biochemical reactions. The book also presents various useful equations such as the differential equation for generating functions for birth and death processes. The text can prove valuable for biochemists, cellular biologists, and researchers in the medical and chemical field who are tasked to perform data analysis.

Preface

1 Introduction

2 Random Processes Continuous in Time and Discrete in State Space

2.0 Notation and Formulation

2.1 Unrestricted Processes

2.2 Restricted Processes

References

3 Random Processes Continuous in Time and State Space

3.0 Diffusion Equations—Derivations and Solutions

3.1 Unrestricted Processes (Singular Processes)

3.2 Restricted Processes

References

4 Population Growth and Extinction

4.1 Extinction of a Colonizing Species

4.2 Population Growth in a Random Environment

References

5 Population Growth of Two-Species Systems

5.1 Epidemics

5.2 Bacteriophage Growth

References

6 Dynamics of a Population of Interacting Species

6.1 Species Diversity on Islands

6.2 Population Growth of Individual Species in an Interacting Population of Species

References

7 Population Genetics

7.1 Genetic Changes under Systematic Evolutionary Pressures - Deterministic Equations

7.2 Genetic Changes in a Finite Population—Stochastic Equations

7.3 Genetic Changes in a Finite Population with Random Mating and No Systematic Evolutionary Pressures

7.4 Genetic Changes in a Finite Population with Random Mating and Mutation or Migration

7.5 Genetic Changes in a Finite Population with Random Mating and Selection

7.6 Steady-State Distributions in a Finite Population with Random Mating, Mutation, Migration, and Selection

References

8 Firing of a Neuron

8.1 Discrete Models

8.2 Continuous Models

References

9 Chemical Kinetics

9.1 Conformational Changes of Biopolymers

9.2 Biosynthesis of Macromolecules

9.3 Enzyme Kinetics

References

10 Photosynthesis

References

11 Epilogue

References

Appendixes

A Calculation of < n > and var(n)

B Differential Equation for the Generating Function of Birth and Death Processes

C Calculation of Pn,m(t) for a Process with Constant Transition Probabilities and a Reflecting State

D Analysis of a Process Confined between an Absorbing and a Reflecting State

E Analysis of a Process Confined between Two Absorbing States

F Explicit Calculation of fUtm(s) for a Process with Linear Transition Probability Rates, Confined between Two Absorbing States

G Calculation of P(x\y,t) for the OU and Wiener Processes

H Derivation of Moments of First Passage Time for a Process in a Continuous State Space

Index