Applications of group theory to math. biology (pharmacology) ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:26:24Z http://mathoverflow.net/feeds/question/94907 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94907/applications-of-group-theory-to-math-biology-pharmacology Applications of group theory to math. biology (pharmacology) ? Alexander Chervov 2012-04-23T06:06:43Z 2012-08-14T07:17:14Z <p>Are there applications of group theory (take it broadly: representation theory, Lie algs., q-groups, whatever ... ) to math. biology ?</p> <p>I am in particular interested about applications to pharmacology (in particular <a href="http://en.wikipedia.org/wiki/Pharmacokinetics" rel="nofollow">pharmacokinetics</a>, <a href="http://en.wikipedia.org/wiki/Pharmacodynamics" rel="nofollow">pharmacodynamics</a> ) ? But would be happy to hear about any applications to biology/pharmacology.</p> <p>PS</p> <p>Some related question:</p> <p><a href="http://mathoverflow.net/questions/94840/any-applications-integrable-systems-pde-ode-q-to-math-biology-pharmakin" rel="nofollow">http://mathoverflow.net/questions/94840/any-applications-integrable-systems-pde-ode-q-to-math-biology-pharmakin</a></p> <p>And not so related, but still:</p> <p><a href="http://mathoverflow.net/questions/87575/mathematics-and-cancer-research" rel="nofollow">http://mathoverflow.net/questions/87575/mathematics-and-cancer-research</a></p> http://mathoverflow.net/questions/94907/applications-of-group-theory-to-math-biology-pharmacology/94923#94923 Answer by Felix Goldberg for Applications of group theory to math. biology (pharmacology) ? Felix Goldberg 2012-04-23T10:19:08Z 2012-04-23T10:19:08Z <p>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:</p> <p><a href="http://www.sciencedirect.com/science/article/pii/0022519377903319" rel="nofollow">http://www.sciencedirect.com/science/article/pii/0022519377903319</a></p> <p><em>Biological similarity and group theory</em> by Jean-Robert Derome</p> http://mathoverflow.net/questions/94907/applications-of-group-theory-to-math-biology-pharmacology/94963#94963 Answer by Mark Sapir for Applications of group theory to math. biology (pharmacology) ? Mark Sapir 2012-04-23T17:33:48Z 2012-04-23T17:33:48Z <p>See references here: <a href="http://www.dur.ac.uk/mathematical.sciences/biomaths/events/iop08/" rel="nofollow">http://www.dur.ac.uk/mathematical.sciences/biomaths/events/iop08/</a></p> http://mathoverflow.net/questions/94907/applications-of-group-theory-to-math-biology-pharmacology/94969#94969 Answer by Natalie for Applications of group theory to math. biology (pharmacology) ? Natalie 2012-04-23T18:46:24Z 2012-04-23T18:46:24Z <p>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 <a href="http://wwwb.math.rwth-aachen.de/~barakat/MTNS2010/Conley.pdf" rel="nofollow">http://wwwb.math.rwth-aachen.de/~barakat/MTNS2010/Conley.pdf</a></p> http://mathoverflow.net/questions/94907/applications-of-group-theory-to-math-biology-pharmacology/95006#95006 Answer by plm for Applications of group theory to math. biology (pharmacology) ? plm 2012-04-24T09:23:46Z 2012-04-24T13:09:32Z <p>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. <a href="http://pipeline.corante.com/archives/2008/10/20/fearful_symmetry.php" rel="nofollow">1</a>, <a href="http://pipeline.corante.com/archives/2011/10/05/a_quasicrystal_nobel_prize.php" rel="nofollow">2</a>, <a href="http://pipeline.corante.com/archives/2010/12/15/chiral_what_chiral_how.php" rel="nofollow">3</a>, <a href="http://pipeline.corante.com/archives/2010/12/15/what_a_paper_doesnt_have_in_it.php" rel="nofollow">4</a>. 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.</p> <p><a href="http://en.wikipedia.org/wiki/Chirality_%28chemistry%29" rel="nofollow">Chirality</a> 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, <a href="http://www.neuvitro.com/pll-coated-coverslips.htm" rel="nofollow">1</a>, <a href="http://www.neuvitro.com/pdl-coated-coverslips.htm" rel="nofollow">2</a>.</p> <p>Crystallography uses finite and discrete (reflection) group theory quite heavily. This is important in biopharma, protein (or other) structure determination, it is a workhorse.</p> <p>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 <a href="http://www.ams.org/journals/bull/2006-43-03/S0273-0979-06-01108-6/" rel="nofollow">here</a> for something recent, and the works of Golubitsky and Stewart in general, for symmetry.</p> http://mathoverflow.net/questions/94907/applications-of-group-theory-to-math-biology-pharmacology/104671#104671 Answer by Pasha Zusmanovich for Applications of group theory to math. biology (pharmacology) ? Pasha Zusmanovich 2012-08-14T07:17:14Z 2012-08-14T07:17:14Z <p>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.</p> <ul> <li>The DNA-protein coding problem, Bull. Math. Biophysics 21 (1959), N1, 71-95 DOI:10.1007/BF02476459</li> <li>Some further comments on the DNA-protein coding problem, Bull. Math. Biophysics 21 (1959), N3, 289-297 - DOI:10.1007/BF02477917</li> <li>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</li> <li>An hypothesis of Freese and the DNA-protein coding problem, Bull. Math. Biophys. 23 (1961), 305--318 DOI:10.1007/BF02476743</li> </ul> <p>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.</p> 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 [<a href="http://links.jstor.org/sici?sici=0006-341X(196709)23%3A3%3C517%3AAGCOEA%3E2.0.CO%3B2-0" rel="nofollow">JSTOR link</a>]. There are probably more current treatements in the same direction, of which I am unaware of.