# Mathematics applied to biology

Can anyone suggest reference material on mathematics applied to biology, in particular the study of the behavior of say simple unicellular organisms or cells? Ideally the level of complexity should be aimed at a competent undergrad. Thank you.

Added: Also, I am wondering if there are any areas of biology where more abstract branches of mathematics is/can potentially be used -- that would be very interesting. By abstract I mean not just crunching through scores of differential equations, but using say geometry, group theory and whatnot.

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what do you mean by "behavior" of organisms? their movement or metabolism or ... ? – begeistzwerst May 3 '12 at 20:50
@begeistzwerst: Anything related, really... – asker May 3 '12 at 21:11
@lhf: Thanks! I have a look at the contents and am impressed by the scope of the subject! – asker May 3 '12 at 21:16
A competent mathematics undergraduate, or a competent biology undergraduate? – Chris Taylor May 28 '12 at 9:02

The study of protein folding and protein structure uses a very wide variety of tools. First, there are statistical physics interpretations of the folding energy of a protein which heavily relies on theories from statistical mechanics and probability theory. Check out the introduction and few sections of this paper for a taste (the rest is pretty involved): http://arxiv.org/pdf/1002.5013v1.pdf

Second, there are techniques applied from differential geometry and group theory to study curvatures in protein folding and various structures that can arise, see here for an example: http://www.lmm.jussieu.fr/~neukirch/articles/GoHaNe_diff_geom_proteins_BioPhysRevLett_2008.pdf

Group theory as well has foreys in mathematical biology, see here: http://www.dur.ac.uk/mathematical.sciences/biomaths/events/iop08/

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Thanks for the references! I am especially excited about the geometry and groups applications. – asker May 3 '12 at 21:13

my encounters with "biomathematics" were restricted to deterministic (ODEs) and probabilistic models (markov processes) for describing epidemics or population genetics. we had a rather unappealing german book as learning material, which i can neither recommend nor find at the moment.

apart from that, game theory seems to be another major topic. karl sigmund wrote, among others, a much-praised book about this (i didn't read it), which can be found e.g. on this list. but game theory probably is not "abstract" enough for you?

unfortunately i have never read or heard anything about models for cell behaviour. but the Wikipedia entry on biomaths lists a large number of references. did you have a look there?

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A classic and excellent book is Murray's Mathematical Biology.

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Disclaimer: I hope it is OK to post "late" answers. The original question is quite vague, therefore, my answer will be quite general and lengthy.

General answer: There are so many books on Mathematical Biology now, that it is very hard to recommend just one book. The two volume book by James Murray, recommended in one of the answers, is not an exception, because it is primarily built on the works by Murray himself and his co-workers, and is not a textbook per se, however, it gives a very nice introduction to the mathematical models in biology, described by ODEs and PDEs. This is especially true for the population dynamics models, chemical kinetics and pattern formation.

Some people approach Mathematical Biology having originally biological background, and for them there are numerous textbooks, which provide basics of the required math. This site is devoted to mathematicians and people studying math, therefore I would like, in my answer, to point out mathematical books, mathematics of which was generated by biologically motivated problems.

Specifics:

1. Your area of expertise is Dynamical systems theory generally, ODEs in particular, including Stability theory. A recent book I would recommend is Dynamical Systems and Population Persistence by Hal Smith and Horst Thieme.

2. You are interested in Game theoretic models, including the analysis of ODEs. The point of entry, which also includes some classical works in mathematical genetics, is Evolutionary Games and Population Dynamics by Josef Hofbauer and Karl Sigmund. For people having necessary background a paper by the same authors is much shorter way to start research.

3. Your want to apply your knowledge of probability theory, especially of stochastic processes (SDEs and diffusion processes), to the real-world problems in mathematical genetics. This is an extremely mathematical area of research, good starting points are Probability Models for DNA Sequence Evolution and a now classic by Warren J. Ewens Mathematical Population Genetics: I. Theoretical Introduction.

4. Deterministic models of theoretical genetics are treated at length in a very mathematically involved book by Burger The Mathematical Theory of Selection, Recombination, and Mutation. The models include interesting integral equations of Quantitative Genetics with a lot of open mathematical problems.

5. (Might be close to the original question) There are a lot of mathematical models that study PDEs arising as a limit of some stochastic spatial processes for interacting particles, including populations of cells. This is a huge area of research, but one book, which may be interested to a mathematically mature person is Transport Equations in Biology by Benoît Perthame.

6. A huge number of models in Mathematical Biology (especially in Mathematical Epidemiology) deal with Random Graphs dynamics. A good book to start is Random Graphs Dynamics by Rick Durrett.

7. If you are an expert in optimization problems, Monte Carlo simulations, etc, then it would be interesting to look into Inferring Phylogenies by Joseph Felsenstein, or in the mathematical problems of the protein folding, which was also recommended in one of the answers. There is no unique mathematical book, which I consider a good introduction (one can always start with Physical Approaches to Biological Evolution by Mikhail V. Volkenstein and A. Beknazarov).

The list can be made much longer, but in my opinion, the mentioned books are a good start for a mathematically prepared person.

Edit:

1. And, of course, for pure mathematicians, it may be very interesting to read the ideas, review papers, and an introduction to biology by Misha Gromov.
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