# Applications of group theory to mathematical biology (pharmacology)

Are there applications of group theory — broadly, say, representation theory, Lie algebras, $$q$$-groups, etc — to mathematical biology?

In particular, I am interested in applications to pharmacology — especially pharmacokinetics and pharmacodynamics. But I would be happy to hear about any applications to biology/pharmacology.

Some related questions:

Finite group theory is really basic in chemistry, it is commonly used by chemists. Derek Lowe, chemist and leading pharma blogger, and his commenters (many, perhaps most, of which pharma industry biochemists) regularly mentions simple symmetry concepts, c.f. 1, 2, 3, 4. E.g. it can be used to compute statistics, from enumeration problems on subgroups, conjugacy classes, etc, and to better understand the structure of a molecule where some bonds allow a finite number of rotations.

Chirality is $$\mathbb Z/2$$ symmetry, a transformation of order 2 of your molecule/object (for instance your left hand looks like the right when seen in a mirror, and when seen in 2 mirrors it looks like itself again, etc.). This is extremely important and common in biology, many molecules have dramatically different behaviors in living organisms depending on which of 2 forms they have, and overall billions of dollars have been spent trying to synthesize some form preferentially, 1, 2.

Crystallography uses finite and discrete (reflection) group theory quite heavily. This is important in biopharma, protein (or other) structure determination, it is a workhorse.

Finally (finite- or infinite-dimensional) dynamical systems are not as widely used but they do illuminate the deeper theory of chemical and biological networks, and symmetry has much to say in specific instances. There is also seeing a dynamical system as a semigroup (even just taking iterates of a transformation), or using ergodic theory consideration, with basic groups like $$\mathbb Z^n$$, or even interesting Lie groups if you find a system with much symmetry -a homogeneous space, though I do not have good examples in mind now. see here for something recent, and the works of Golubitsky and Stewart in general, for symmetry.

Maybe it is too far fetched. But I heard of the so called Conley index theory which deals with the question of existence/non-existence of equilibria in dynamical systems. It involves homology groups of the occurring manifolds. So I think of dynamical systems which one might find in some situation in biology/pharmacology etc. and apply the Conley index theory to it. Check also http://wwwb.math.rwth-aachen.de/~barakat/MTNS2010/Conley.pdf

My research group is using group theory to model the evolution of bacterial genome. Here are a couple of relevant papers:

"An algebraic view of bacterial genome evolution" AR Francis. Published in Journal of mathematical biology

"Group-theoretic models of the inversion process in bacterial genomes" A Egri-Nagy, V Gebhardt, MM Tanaka, AR Francis Published in Journal of mathematical biology

In addition, this is a kind of survey paper (not written by us).

Rietman, Edward A., Robert L. Karp, and Jack A. Tuszynski. "Review and application of group theory to molecular systems biology." Theor Biol Med Modell 8.21 (2011).

The only thing that comes to my mind right now is the notion of symmetry. I found a rather old paper that seems to be concerned with this:

http://www.sciencedirect.com/science/article/pii/0022519377903319

Biological similarity and group theory by Jean-Robert Derome

• If someone having access can send me this paper I would be indebted. al.mysurname at gmail.com Apr 23, 2012 at 12:27

1. There are a few old papers by Robert Rosen where he (seemingly) applies free semigroups to DNA-protein coding problem and argues about biological significance of the notion of freeness. These papers seem to be forgotten by now.

• The DNA-protein coding problem, Bull. Math. Biophysics 21 (1959), N1, 71-95 DOI:10.1007/BF02476459
• Some further comments on the DNA-protein coding problem, Bull. Math. Biophysics 21 (1959), N3, 289-297 - DOI:10.1007/BF02477917
• Some further comments on the DNA-protein coding problem: A correction and a note, Bull. Math. Biophysics 22 (1960), N2, 199-205 DOI:10.1007/BF02478006
• An hypothesis of Freese and the DNA-protein coding problem, Bull. Math. Biophys. 23 (1961), 305--318 DOI:10.1007/BF02476743

2. There were some efforts to describe evolution and symmetry breaking of genetic code in terms of symmetries of some algebraic structures, including Lie groups and quantum groups, see Bashford, J.D. and Jarvis, P.D., The genetic code as a periodic table: algebraic aspects, arXiv:physics/0001066, and references therein.

3. (Elementary) representation theory of symmetric group was used in some classical genetics: see, for example, Bennett, J.H., A general class of enumerations arising in genetics, Biometrics 23 (1967), No.3, 517-537 [JSTOR link]. There are probably more current treatements in the same direction, of which I am unaware of.

See references here: http://www.dur.ac.uk/mathematical.sciences/biomaths/events/iop08/

It appears that some people in the computational anatomy community might be doing things with group actions: https://en.wikipedia.org/wiki/Group_actions_in_computational_anatomy