[Math] Category Theory & Biology

applicationsbig-listbiologycategory-theoryintuition

Category theory is becoming more and more used in the following fields (besides others)

(1) Quantum physics (e.g. C. Isham and B. Coecke et al)

(2) General relativity (e.g. A.K. Guts et al)

(3) Linguistics and natural language processing (e.g. J. Lambek and S. Abramsky et al)

(4) Computer science (anyway)

Question. Does someone know if there are any connections so far between category theory and biology (genetics or ecology perhaps or any other discipline)?

Remark. I am aware that this question could also be posted on biology stack exchange. However my idea is that it could be better to ask it on the mathematics site, since it might rather be mathematicians that have some knowledge about such connections than biologists themselves.

Best Answer

You could have a look at the following two papers about genetics and category theory:

Category theory for genetics I: mutations and sequence alignments, https://arxiv.org/abs/1805.07002

Category theory for genetics II: genotype, phenotype and haplotype, https://arxiv.org/abs/1805.07004

These papers are revisions of the following one:

https://arxiv.org/abs/1708.05255

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[Math] Mathematics applied to biology

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:

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.
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.
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.
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.
(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.
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.
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:

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.

Category Theory – Real World Applications

The blog entry "Why Category Theory Matters" by Robert Seaton ends with a quite impressive reference list of applications of category theory to the sciences:

Category theory has been used to study grammar and human language.
In building a spreadsheet application.
As a descriptive tool in neuroscience.
In the analysis and design of cognitive neural network architectures.
In programming languages, especially Haskell and most famously monads, but also, for instance, a typed assembly language and work on the typed lambda calculus.
Generating program optimizations.
To model systems of interacting agents.
To generalize sorting algorithms.
To understand collaborative text editing. See also this blog.
To understand optimal play in sequential games like chess.
To formalize the notion of algorithm.
In the study of analogy.
As “a language for experimental design patterns” and “a new vocabulary in which to think and communicate.”
In definitions of emergence and discussions of biology.

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[Math] Mathematics applied to biology

Category Theory – Real World Applications

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