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I've heard that there's a relatively new field of science called Mathematical Biology. It will certainly apply well known and less known mathematical techniques to the understanding of some biological phenomena such as strings of DNA knotting together, proteins bending, cell membranes, population dynamics,...

1) Which interesting advances from the point of view of a mathematician are there that have been inspired by Biology and related areas?

For example, I heard of "Membrane computation", but I don't know if it's genuinely inspired by biology in some nontrivial way or if it just bears that name by a loose analogy...

2) Which fields of Mathematics are currently used in the discipline called Mathematical Biology?

If I were asked about Economy and Mathematical Finance I would loosely quote probability theory, stochastic differential equations, Ito integrals, ... So, what about Biology?

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You may want to take a look at the page of Martin Nowak, at Harvard. –  Andres Caicedo Sep 1 '11 at 21:42
    
(btw, should I make it community wiki?) –  Qfwfq Sep 1 '11 at 21:50
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Community Wiki would probably be a good idea. –  Willie Wong Sep 1 '11 at 22:03
    
CW-fied . –  Qfwfq Sep 1 '11 at 22:24
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14 Answers 14

I guess you might find this article of Bernd Sturmfels interesting: CAN BIOLOGY LEAD TO NEW THEOREMS?

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As regards interesting mathematics arising in biology:

  • A mathematically fascinating class of integro-PDEs arise in the study of age-structured population models. The independent variables are age $a$ and time $t$ ; the systems are first-order PDE; and the boundary conditions on the curve age=0 are given in terms of integrals of the dependent variables, $u$. That is, $u(0,t)= \int_{a=0}^T \phi(u(s,t)) ds$ where $\phi$ may be a nonlinear function. Such models arise frequently in physiology. It's my impression that this is a field with many interesting open mathematical questions to be asked.

  • PDEs arising in pattern-forming systems in biology exhibit interesting mathematical behavior; questions about long-time regularity of such PDE are mathematically interesting. One may wish, for example, to characterize finite-time blow-up, or development of geometric singularities on interfaces.

  • Dynamical systems with delays (functional differential equations) for the form $\frac{dy}{dt} = A(y(t-\tau),t)$ arise naturally in biology. This is a field which is not as mathematically developed as the theory of ODE.

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Quite a number of delay differential equations (DDEs) turn up in modeling things as various as the transmission of electrical signals of neurons and the transmission and propagation of victims in an epidemic. –  J. M. Sep 3 '11 at 14:25
    
Indeed, J.M. DDE occur in many applications. The mathematical analysis of DDE, however, is not as complete (yet) as that of ODE systems. –  Nilima Nigam Sep 3 '11 at 15:16
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Mathematical biology is a huge area which is not so young.

Statistics is a major research tool in biology (as in most other areas of natural and social science) so biology questions rely and have led to substantial progress in mathematical statistics and related probability theory.

Like in most natural sciences differential equations of various types (and some unique types of euations) arose in biology. It makes sense to mention in particular the pioneering work of Alan Turing in his paper entitled The Chemical Basis of Morphogenesis. Nilima Nigam's answer mentioned several related connections and this is just the tip of the iceberg.

Biology have led to vast questions regarding algorithms, computation, and optimization. The fields of computational biology and bioinformatics are closely related.

Mathematical biology have led to Evolutionary game theory which had strong impact on mathematical game theory.

Sometimes a single question in biology is related to a large number of mathematical disciplines. For example, Amit Singer and Yoel Shkolnisky are involved in a long term project aiming to determine the 3-dimensional shape of a certain molecule based on noisy 2-dimensional pictures from an electronic microscope. Their research is related to fascinating questions in harmonic analysis (and wavelets), graph theory, representation theory, semidefinite programmings, probability theory, and even the notion of unique games fron theoretical computer science enter.

A last example: John Bush and David Hu had remarlable mathematical models explaining how insects walk on waters. Bush's homepage is a good source for various issues in mathematical biology.

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Indeed, the Hardy-Weinberg principle (yes, Hardy is G.H. Hardy) in genetics goes back to 1908: en.wikipedia.org/wiki/Hardy%E2%80%93Weinberg_principle#History –  j.c. Sep 2 '11 at 18:32
    
Two surveys by J. Calvin Giddings, relating to mathematics in concrete (mainly biological) phenomena, are Transport, space, entropy, diffusion and flow: elements underlying separation by electrophoresis, chromatography, field-flow fractionation and related methods (1987) and Unified Separation Science (1987). Sample: "Separation is the art and science of maximizing the ratio of separative transport to dissipative transport." The word "quantum" is absent, yet what is quantum computing but the separative transport of qubits via unitary versus dissipative dynamics? There's work to do here. –  John Sidles Sep 2 '11 at 21:13
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For a broad overview, an entry point seems to be: this Wikipedia page

To partially address your second question, here is a very short list:

  1. Stochastic process; stochastic differential equations etc.
  2. Graph theory and combinatorics
  3. PDEs
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A very thorough introduction to some now classical topics can be found in James D. Murray's now two-volume book published by Springer. Expect lots of ODE's and PDE's in that one.

As far as more exotic math is concerned, a complete overview would be difficult: it seems people throw everything they have and see what works. I've seen some interesting talks involving combinatorics, others involving algebraic geometry.

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If you haven't seen it yet, this article by M. Gromov has, at the very least, a lot of references.

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You might also look at this article by Avner Friedman from the Notices of the AMS which gives a survey of the field.

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I wound up with two copies of a book, Math & Bio 2010 edited by Lynn Arthur Steen. It was mainly aimed at improving undergraduate interdisciplinary education. But it also has a huge bibliography on research topics, several pages of researchers with websites. It says ISBN 0-88385-818-5. Alright, table of contents HERE

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Also try to partially answer your second question here, I would say:

1: Differential Equations including Ordinary Differential Equations, Difference Equations, Delay Differential Equations, Integral-Differential or Integral-Difference Equations, 2: Dynamical Systems; 3: Game Theory; 4: Numerical Analysis; 5: Programming techniques: MatLab, Maple, Python, etc.

In the workshop I recently attended, I heard talks about using Algebraic Geometry and Lie Theory in this field as well.

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Differential geometry. For example, see this answer. This also relates to your first question- study of writhe was heavily influenced by mathematical biology, with many central papers having been written by biologists.

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Knot theory has been used to understand the mechanism by which enzymes like gyrase relieve tension on DNA molecules. See here.

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Bernd Sturmfels wrote a book called Algebraic Statistics for Computational Biology.

A variety (no pun intended) of mathematical topics occur. Phylogenetic trees are studied by means of "tropical" algebraic geometry. It's been long enough since I've looked at it that I'm not going to trust myself to give more examples yet. Here's the amazon.com page: http://www.amazon.com/Algebraic-Statistics-Computational-Biology-Pachter/dp/0521857007

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In the 1930's Etherington developed several nonassociative algebras that express much of (what was known at the time about) genetics and inheritance. Here's a survey from the early 1990's by Mary Lynn Reed.

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