Interesting mathematical topics arising from Biology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:15:45Z http://mathoverflow.net/feeds/question/74295 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology Interesting mathematical topics arising from Biology Qfwfq 2011-09-01T21:34:42Z 2011-09-03T01:48:43Z <p>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,...</p> <blockquote> <p>1) Which interesting advances <em>from the point of view of a mathematician</em> are there that have been inspired by Biology and related areas?</p> </blockquote> <p>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...</p> <blockquote> <p>2) Which fields of Mathematics are currently used in the discipline called Mathematical Biology?</p> </blockquote> <p>If I were asked about Economy and Mathematical Finance I would loosely quote probability theory, stochastic differential equations, Ito integrals, ... So, what about Biology?</p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74299#74299 Answer by Willie Wong for Interesting mathematical topics arising from Biology Willie Wong 2011-09-01T22:05:14Z 2011-09-01T22:05:14Z <p>If you haven't seen it yet, <a href="http://www.ams.org/journals/bull/2011-48-02/S0273-0979-2010-01319-7/home.html" rel="nofollow">this article by M. Gromov</a> has, at the very least, a lot of references. </p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74300#74300 Answer by S. Sra for Interesting mathematical topics arising from Biology S. Sra 2011-09-01T22:07:33Z 2011-09-01T22:07:33Z <p>For a broad overview, an entry point seems to be: <a href="http://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology" rel="nofollow">this Wikipedia page</a></p> <p>To partially address your second question, here is a very short list:</p> <ol> <li>Stochastic process; stochastic differential equations etc.</li> <li>Graph theory and combinatorics</li> <li>PDEs </li> </ol> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74301#74301 Answer by Michael A Warren for Interesting mathematical topics arising from Biology Michael A Warren 2011-09-01T22:08:52Z 2011-09-01T22:08:52Z <p>You might also look at this <a href="http://www.ams.org/notices/201007/rtx100700851p.pdf" rel="nofollow">article</a> by Avner Friedman from the Notices of the AMS which gives a survey of the field.</p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74304#74304 Answer by Isabel Li for Interesting mathematical topics arising from Biology Isabel Li 2011-09-01T22:31:45Z 2011-09-01T22:31:45Z <p>Also try to partially answer your second question here, I would say:</p> <p>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. </p> <p>In the workshop I recently attended, I heard talks about using Algebraic Geometry and Lie Theory in this field as well. </p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74305#74305 Answer by Hans Stricker for Interesting mathematical topics arising from Biology Hans Stricker 2011-09-01T22:37:10Z 2011-09-01T22:37:10Z <p>I guess you might find this article of Bernd Sturmfels interesting: <a href="http://math.berkeley.edu/~bernd/ClayBiology.pdf" rel="nofollow">CAN BIOLOGY LEAD TO NEW THEOREMS?</a></p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74309#74309 Answer by Thierry Zell for Interesting mathematical topics arising from Biology Thierry Zell 2011-09-01T23:09:09Z 2011-09-01T23:09:09Z <p>A very thorough introduction to some now classical topics can be found in <a href="http://en.wikipedia.org/wiki/James_D._Murray" rel="nofollow">James D. Murray's</a> now two-volume book published by Springer. Expect lots of ODE's and PDE's in that one. </p> <p>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.</p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74318#74318 Answer by Daniel Moskovich for Interesting mathematical topics arising from Biology Daniel Moskovich 2011-09-02T00:32:08Z 2011-09-02T00:32:08Z <p>Differential geometry. For example, see <a href="http://mathoverflow.net/questions/74252/things-that-should-be-positive-integers-really/74314#74314" rel="nofollow">this answer</a>. 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.</p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74323#74323 Answer by Nilima Nigam for Interesting mathematical topics arising from Biology Nilima Nigam 2011-09-02T01:33:39Z 2011-09-02T01:33:39Z <p>As regards interesting mathematics arising in biology: </p> <ul> <li><p>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.</p></li> <li><p>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.</p></li> <li><p>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.</p></li> </ul> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74327#74327 Answer by Will Jagy for Interesting mathematical topics arising from Biology Will Jagy 2011-09-02T02:11:35Z 2011-09-02T02:11:35Z <p>I wound up with two copies of a book, <em>Math &amp; Bio 2010</em> 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 <a href="http://www.maa.org/mtc/mathbiotoc.html" rel="nofollow">HERE</a></p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74357#74357 Answer by Steven Landsburg for Interesting mathematical topics arising from Biology Steven Landsburg 2011-09-02T13:08:51Z 2011-09-02T13:08:51Z <p>Knot theory has been used to understand the mechanism by which enzymes like <a href="http://en.wikipedia.org/wiki/Gyrase" rel="nofollow">gyrase</a> relieve tension on DNA molecules. See <a href="http://www.sciencemag.org/content/206/4422/1081.abstract" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74374#74374 Answer by Gil Kalai for Interesting mathematical topics arising from Biology Gil Kalai 2011-09-02T16:24:38Z 2011-09-02T16:24:38Z <p>Mathematical biology is a huge area which is not so young.</p> <p>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.</p> <p>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 <a href="http://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf" rel="nofollow">The Chemical Basis of Morphogenesis</a>. Nilima Nigam's <a href="http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74323#74323" rel="nofollow">answer</a> mentioned several related connections and this is just the tip of the iceberg. </p> <p>Biology have led to vast questions regarding algorithms, computation, and optimization. The fields of <a href="http://en.wikipedia.org/wiki/Computational_biology" rel="nofollow">computational biology</a> and <a href="http://en.wikipedia.org/wiki/Bioinformatics" rel="nofollow">bioinformatics</a> are closely related.</p> <p>Mathematical biology have led to <a href="http://en.wikipedia.org/wiki/Evolutionary_game_theory" rel="nofollow">Evolutionary game theory</a> which had strong impact on mathematical game theory.</p> <p>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.</p> <p>A last example: John Bush and David Hu had remarlable mathematical models explaining how insects walk on waters. <a href="http://math.mit.edu/~bush/" rel="nofollow">Bush's homepage</a> is a good source for various issues in mathematical biology. </p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74384#74384 Answer by Michael Hardy for Interesting mathematical topics arising from Biology Michael Hardy 2011-09-02T18:18:27Z 2011-09-02T18:18:27Z <p>Bernd Sturmfels wrote a book called <em>Algebraic Statistics for Computational Biology</em>.</p> <p>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: <a href="http://www.amazon.com/Algebraic-Statistics-Computational-Biology-Pachter/dp/0521857007" rel="nofollow">http://www.amazon.com/Algebraic-Statistics-Computational-Biology-Pachter/dp/0521857007</a></p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74385#74385 Answer by Michael Hardy for Interesting mathematical topics arising from Biology Michael Hardy 2011-09-02T18:23:16Z 2011-09-02T18:23:16Z <p>Warren Ewens wrote this book: <a href="http://www.amazon.com/Mathematical-Population-Genetics-Introduction-Interdisciplinary/dp/1441918981" rel="nofollow">http://www.amazon.com/Mathematical-Population-Genetics-Introduction-Interdisciplinary/dp/1441918981</a></p> <p>He is the eponym of the Ewens sampling formula: <a href="http://en.wikipedia.org/wiki/Ewens%27s_sampling_formula" rel="nofollow">http://en.wikipedia.org/wiki/Ewens%27s_sampling_formula</a></p> http://mathoverflow.net/questions/74295/interesting-mathematical-topics-arising-from-biology/74412#74412 Answer by Scott McKuen for Interesting mathematical topics arising from Biology Scott McKuen 2011-09-03T01:48:43Z 2011-09-03T01:48:43Z <p>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 <a href="http://www.ams.org/journals/bull/1997-34-02/S0273-0979-97-00712-X/" rel="nofollow">Mary Lynn Reed</a>.</p>