Open Questions: Mathematics and Biology

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Introduction

Differential equations

Dynamical systems

Knot theory

Swarming behavior

Recommended references: Web sites

Recommended references: Magazine/journal articles

Recommended references: Books

Introduction

Recommended references: Web sites

Site indexes

Math Forum Internet Mathematics Library: Biology: Alphabetized list of links with extensive annotations.
Open Directory Project: Mathematical Biology: Categorized and annotated mathematical biology links. A version of this list is at Google, with entries sorted in "page rank" order.
Links to Mathematical Biology Sites: A substantial, categorized list of links, by Jason Matthiopoulos. Also has a good bibliography of textbooks in mathematical biology.
Mathematical Biology: Extensive categorized but unannotated list of links, at the Mathematical BBS.
Galaxy: Mathematical Biology: Categorized site directory. Entries usually include descriptive annotations. More here.
Society for Mathematical Biology: Other Sites of Interest: Unannotated links.

Sites with general resources

Mathematics in Biology: Site hosted at Brandeis University. Deals with all aspects of use of mathematics in the life sciences. There is a lot of substantial content, but much of it requires sophisticated mathematics.
Geometric models of biological phenomena: An outline (literally) and lecture notes related to combinatorial, statistical, and geometric issues in mathematical models of biological phenomena. Part of the Workshop Website Network of the American Institute of Matheamtics.
The Society for Mathematical Biology: Professional society which aims "to provide a forum for discussion of research in biology, mathematical-biology, and mathematics applied to or motivated by biology." Site includes useful external links.

Surveys, overviews, tutorials

Mathematical biology: Article from Wikipedia.
Mathematics and Biology: The Interface: This workshop report is an excellent summary of the current interfaces between mathematics and biology. It describes the impact of each field on the other. On the mathmematics side, nonlinear dynamics, topology, and knot theory are especially important. In biology, many areas are affected, but especially molecular biology and evolutionary biology.
Genome Multimedia Site: Interactive course on the analysis of the genetic sequence using information theory, probability, and statistics.

Special topics

Magical numbers in nature: October 12, 2001 interview from Nature Science Update with mathematician Ian Stewart on the use of mathematical ideas in biology. Topics include symmetry, dynamics, chaos theory, fractals, and pattern formation.
Professor Bonnie Berger: Home page of MIT professor specializing in computational biology. Site includes information on research, publications, and course notes.
Biomedical Models of Cellular and Physiological Systems and Disease: Research project led by Leah Keshet.
Mathematical Modelling in Pharmaceutical Development: Research Project led by Jack Tuszynski. Site contains project information and mini-tutorials.

Recommended references: Magazine/journal articles

Mathematical Challenges from Genomics and Molecular Biology
Richard M. Karp
Notices of the AMS, May 2002, pp. 544-553: A fundamental goal of biology is to understand the functioning of cells. Genomics and molecular biology have yielded a whole new kind of insight into this question. Among the biological questions most amenable to mathematical analysis are genome sequencing, gene finding, construction of evolutionary phylogenies, analysis of data from DNA microarrays, and understanding the process of gene expression. The most useful mathematical techniques are in the areas of probability, optimization, and computational theory.
[Article in PDF format]
The Unreasonable Effectiveness of Mathematics in Molecular Biology
Arthur M. Lesk
Mathematical Intelligencer, Spring 2000, pp. 28-37: The article presents an examination of the notion that mathematics can be as useful for (some) biological problems as it is in physics. One example is the study of protein folding and structure.
We Got Rhythm: Dynamical Systems of the Nervous System
Nancy Kopell
Notices of the AMS, January 2000, pp. 6-16: A mamalian nervous system is always active and exhibits a variety of periodic, rhythmical phenomena. Some of these "brain waves" are associated with mental processes and cognitive states, which are of great interest but poorly understood. The article presents mathematical ways to model such dynamical systems.
[Article in PDF format]
Using Mathematics to Understand HIV Immune Dynamics
Denise Kirschner
Notices of the AMS, February 1996, pp.191-202: Much effort has been expended in modeling the interaction of HIV and the immune system. A model is presented here involving the interaction of HIV with one type of immune system cell, using a system of three first order differential equations governing three dependent variables.
[Article in PDF format]
Getting Started in Mathematical Biology
Frank Hoppensteadt
Notices of the AMS, September 1995, pp. 969-975: The general nature of mathematical models in biology is discussed using examples from neurobiology. Advice is offered on how to do collaborative research in mathematical biology.
[Article in PDF format]
Mapping Heredity: Using Probabilistic Models and Algorithms to Map Genes and Genomes
Eric S. Lander
Notices of the AMS, July 1995, pp. 747-753; August 1995, pp. 854-858: Elements of genome mapping are presented, and appropriate statistical techniques are described.
[Article in PDF format] (Part one)
[Article in PDF format] (Part two)
Lifting the Curtain: Using Topology to Probe the Hidden Action of Enzymes
De Witt Sumners
Notices of the AMS, May 1995, pp. 528-537: Because DNA consists of two entwined and very long linear molecules, the processes of transcription and replication of DNA by enzymes must satisfy geometric and topological conditions. Understanding these constraints can help determine otherwise unobservable details of how various enzymes perform their tasks. Sophisticated results of knot theory play an important role in this analysis.
[Article in PDF format]

Recommended references: Books

Nicholas F. Britton – Essential Mathematical Biology
Springer-Verlag, 2003: The book provides a self-contained introduction to mathematical biology for students with a mathematical background (calculus and differential equations). Topics include population dynamics, infectious diseases, population genetics and evolution, molecular and cellular biology, and cancer modeling.
Iam Stewart -- Life's Other Secret: The New Mathematics of the Living World
John Wiley & Sons, 1998: The author, a professional mathematician presents ways in which some mathematical concepts are applicable to biology. Since almost no formulas or equations are used in the whole book, this limits what can be offered. The main concept that is applied heavily is symmetry. Topic covered include DNA and the genetic code, artificial life, the Fibonacci sequence, and neural networks.

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