# Mathematical Concepts and Methods in Modern Biology

## 1st Edition

### Using Modern Discrete Models

**Editors:**Raina Robeva Terrell Hodge

**Hardcover ISBN:**9780124157804

**eBook ISBN:**9780124157934

**Imprint:**Academic Press

**Published Date:**15th January 2013

**Page Count:**372

## Description

- Contributors
- Preface
- Chapter 1. Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli
- 1.1 Introduction
- 1.2 E. Coli and the LAC Operon
- 1.3 Boolean Network Models of the LAC Operon
- 1.4 Determining the Fixed Points of Boolean Networks
- 1.5 Conclusions and Discussion
- Acknowledgments
- 1.6 Supplementary Materials
- References
- Chapter 2. Bistability in the Lactose Operon of Escherichia coli: A Comparison of Differential Equation and Boolean Network Models
- 2.1 Introduction
- 2.2 The Lactose Operon of Escherichia Coli
- 2.3 Modeling Biochemical Reactions with Differential Equations
- 2.4 The Yildirim-Mackey Differential Equation Models for the Lactose Operon
- 2.5 Boolean Modeling of Biochemical Interactions
- 2.6 Boolean Approximations of the Yildirim-Mackey Models
- 2.7 Conclusions and Discussion
- Acknowledgment
- 2.8 Supplementary Materials
- References
- Chapter 3. Inferring the Topology of Gene Regulatory Networks: An Algebraic Approach to Reverse Engineering
- 3.1 Introduction
- 3.2 Polynomial Dynamical Systems (PDSs)
- 3.3 Computational Algebra Preliminaries
- 3.4 Construction of the Model Space: A Reverse Engineering Algorithm
- 3.5 Model Selection
- 3.6 Discretization
- References
- Chapter 4. Global Dynamics Emerging from Local Interactions: Agent-Based Modeling for the Life Sciences
- 4.1 Introduction
- 4.2 Axon Guidance
- 4.3 An Agent-Based Model for Cholera and the Importance of Replication
- 4.4 Use and Description of ABM in Research: Tick-Borne Disease Agent-Based Models
- 4.5 Comments for Instructors
- Acknowledgments
- 4.6 Supplementary Materials
- References
- Chapter 5. Agent-Based Models and Optimal Control in Biology: A Discrete Approach
- 5.1 Introduction
- 5.2 A First Example
- 5.3 Netlogo: An Introduction
- 5.4 An Introduction to Agent-Based Models
- 5.5 Optimization and Optimal Control
- 5.6 Scaling and Aggregation
- 5.7 A Heuristic Approach
- 5.8 Mathematical Framework for Representing Agent-Based Models
- 5.9 Translating Agent-Based Models into Polynomial Dynamical Systems
- 5.10 Summary
- 5.11 Supplementary Materials
- References
- Chapter 6. Neuronal Networks: A Discrete Model
- 6.1 Introduction and Overview
- 6.2 Neuroscience in a Nutshell
- 6.3 The Discrete Model
- 6.4 Exploring the Model for Some Simple Connectivities
- 6.6.5 Exploring the Model for Some Random Connectivities
- 6.6 Another Interpretation of the Model: Disease Dynamics
- 6.7 More Neuroscience: Connection with ODE Models
- 6.8 Directions of Further Research
- 6.9 Supplementary Materials
- References
- Chapter 7. Predicting Population Growth: Modeling with Projection Matrices
- 7.1 Introduction
- 7.2 Life Cycles and Population Growth
- 7.3 Determining Stages in the Life Cycle
- 7.4 Determining the Number of Individuals in a Stage at Time
- 7.5 Constructing a Projection Matrix
- 7.6 Predicting How a Population Changes after One Year
- 7.7 The Stable Distribution of Individuals across Stages
- 7.8 Theory Supporting the Calculation of Stable Distributions
- 7.9 Determining Population Growth Rate and the Stable Distribution
- 7.10 Further Applications of the Projection Matrix
- References
- Chapter 8. Metabolic Pathways Analysis: A Linear Algebraic Approach
- 8.1 Introduction
- 8.2 Biochemical Reaction Networks, Metabolic Pathways, and the Stoichiometry Matrix
- 8.3 Extreme Paths and Model Improvements
- Acknowledgments
- 8.4 Supplementary Data
- References
- Chapter 9. Identifying CpG Islands: Sliding Window and Hidden Markov Model Approaches
- 9.1 Introduction
- 9.2 Quantitative Characteristics of the CpG Island Regions and Sliding Windows Algorithms
- 9.3 Definition and Basic Properties of Markov Chains and Hidden Markov Models
- 9.4 Three Canonical Problems for HMMs with Applications to CGI Identification
- 9.5 Conclusions and Discussion
- Acknowledgments
- 9.6 Supplementary Materials
- References
- Chapter 10. Phylogenetic Tree Reconstruction: Geometric Approaches
- 10.1 Introduction
- 10.2 Basics on Trees and Phylogenetic Trees
- 10.3 Tree Space
- 10.4 Neighbor-Joining and BME
- 10.5 Summary
- References
- Index

## Key Features

- Features self-contained chapters with real biological research examples using freely available computational tools
- Spans several mathematical techniques at basic to advanced levels
- Offers broad perspective on the uses of algebraic geometry/polynomial algebra in molecular systems biology

## Readership

Researchers, educators, and students engaged in Biological Research and Mathematics

## Table of Contents

- Contributors
- Preface
- Chapter 1. Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli
- 1.1 Introduction
- 1.2 E. Coli and the LAC Operon
- 1.3 Boolean Network Models of the LAC Operon
- 1.4 Determining the Fixed Points of Boolean Networks
- 1.5 Conclusions and Discussion
- Acknowledgments
- 1.6 Supplementary Materials
- References
- Chapter 2. Bistability in the Lactose Operon of Escherichia coli: A Comparison of Differential Equation and Boolean Network Models
- 2.1 Introduction
- 2.2 The Lactose Operon of Escherichia Coli
- 2.3 Modeling Biochemical Reactions with Differential Equations
- 2.4 The Yildirim-Mackey Differential Equation Models for the Lactose Operon
- 2.5 Boolean Modeling of Biochemical Interactions
- 2.6 Boolean Approximations of the Yildirim-Mackey Models
- 2.7 Conclusions and Discussion
- Acknowledgment
- 2.8 Supplementary Materials
- References
- Chapter 3. Inferring the Topology of Gene Regulatory Networks: An Algebraic Approach to Reverse Engineering
- 3.1 Introduction
- 3.2 Polynomial Dynamical Systems (PDSs)
- 3.3 Computational Algebra Preliminaries
- 3.4 Construction of the Model Space: A Reverse Engineering Algorithm
- 3.5 Model Selection
- 3.6 Discretization
- References
- Chapter 4. Global Dynamics Emerging from Local Interactions: Agent-Based Modeling for the Life Sciences
- 4.1 Introduction
- 4.2 Axon Guidance
- 4.3 An Agent-Based Model for Cholera and the Importance of Replication
- 4.4 Use and Description of ABM in Research: Tick-Borne Disease Agent-Based Models
- 4.5 Comments for Instructors
- Acknowledgments
- 4.6 Supplementary Materials
- References
- Chapter 5. Agent-Based Models and Optimal Control in Biology: A Discrete Approach
- 5.1 Introduction
- 5.2 A First Example
- 5.3 Netlogo: An Introduction
- 5.4 An Introduction to Agent-Based Models
- 5.5 Optimization and Optimal Control
- 5.6 Scaling and Aggregation
- 5.7 A Heuristic Approach
- 5.8 Mathematical Framework for Representing Agent-Based Models
- 5.9 Translating Agent-Based Models into Polynomial Dynamical Systems
- 5.10 Summary
- 5.11 Supplementary Materials
- References
- Chapter 6. Neuronal Networks: A Discrete Model
- 6.1 Introduction and Overview
- 6.2 Neuroscience in a Nutshell
- 6.3 The Discrete Model
- 6.4 Exploring the Model for Some Simple Connectivities
- 6.6.5 Exploring the Model for Some Random Connectivities
- 6.6 Another Interpretation of the Model: Disease Dynamics
- 6.7 More Neuroscience: Connection with ODE Models
- 6.8 Directions of Further Research
- 6.9 Supplementary Materials
- References
- Chapter 7. Predicting Population Growth: Modeling with Projection Matrices
- 7.1 Introduction
- 7.2 Life Cycles and Population Growth
- 7.3 Determining Stages in the Life Cycle
- 7.4 Determining the Number of Individuals in a Stage at Time
- 7.5 Constructing a Projection Matrix
- 7.6 Predicting How a Population Changes after One Year
- 7.7 The Stable Distribution of Individuals across Stages
- 7.8 Theory Supporting the Calculation of Stable Distributions
- 7.9 Determining Population Growth Rate and the Stable Distribution
- 7.10 Further Applications of the Projection Matrix
- References
- Chapter 8. Metabolic Pathways Analysis: A Linear Algebraic Approach
- 8.1 Introduction
- 8.2 Biochemical Reaction Networks, Metabolic Pathways, and the Stoichiometry Matrix
- 8.3 Extreme Paths and Model Improvements
- Acknowledgments
- 8.4 Supplementary Data
- References
- Chapter 9. Identifying CpG Islands: Sliding Window and Hidden Markov Model Approaches
- 9.1 Introduction
- 9.2 Quantitative Characteristics of the CpG Island Regions and Sliding Windows Algorithms
- 9.3 Definition and Basic Properties of Markov Chains and Hidden Markov Models
- 9.4 Three Canonical Problems for HMMs with Applications to CGI Identification
- 9.5 Conclusions and Discussion
- Acknowledgments
- 9.6 Supplementary Materials
- References
- Chapter 10. Phylogenetic Tree Reconstruction: Geometric Approaches
- 10.1 Introduction
- 10.2 Basics on Trees and Phylogenetic Trees
- 10.3 Tree Space
- 10.4 Neighbor-Joining and BME
- 10.5 Summary
- References
- Index

## Details

- No. of pages:
- 372

- Language:
- English

- Copyright:
- © Academic Press 2013

- Published:
- 15th January 2013

- Imprint:
- Academic Press

- eBook ISBN:
- 9780124157934

- Hardcover ISBN:
- 9780124157804

## About the Editor

### Raina Robeva

Raina Robeva was born in Sofia, Bulgaria. She has a PhD in Mathematics from the University of Virginia and has led multiple NSF-funded curriculum development projects at the interface of mathematics and biology. She is the lead author of the textbook An Invitation to Biomathematics (2008) and the lead editor of the volume Mathematical Concepts and Methods in Modern Biology: Using Modern Discrete Models (2013), both published by Academic Press. Robeva is the founding Chief Editor of the research journal Frontiers in Systems Biology. She is a professor of Mathematical Sciences at Sweet Briar College and lives in Charlottesville, Virginia.

### Affiliations and Expertise

Professor of Mathematical Sciences, Sweet Briar College, Sweet Briar, VA, USA

### Terrell Hodge

### Affiliations and Expertise

Western Michigan University, Kalamazoo, MI, USA

## Reviews

*"Contributors in biology, in mathematics, and in bioinformatics introduce undergraduate students and their instructors to more applications of discrete mathematics to biology than can be found in standard textbooks. The goal is not to be comprehensive, but to open the door to more advanced and specialized resources."--**Reference and Research Book News,* August 2013