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:
Best Answer
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.