# Mathematical formula for biological phenomena?

Math is strongly intervened with Physics and Chemistry: it's used for an assortment of calculations and experiments. However, I find that Biology (at least elementary Biology) is severely lacking in mathematical models. Do mathematical models of biological phenomena (i.e. cell reproduction, anatomical systems etc.) exist? Are they simply too complicated and inapplicable to be taught in general classes, or have we just not been able to attribute mathematical models to these phenomena?

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Yes, mathematical models of biology abound. There's even an entire branch of math (Mathematical Biology) that deals with creating and studying models of biological phenomenon. At least at my university, the first courses on this are taught at the senior undergraduate level. This is not "mathematical logic", though. It's mathematical modeling. –  Arturo Magidin May 23 '12 at 2:12
Models do exist, and the study of such models is a very hot field right now called mathematical biology. But explicit calculations are not generally feasible, as the systems in question are ** unbelievably** complicated. Take the nervous system for example. The human nervous system has approximately 100 billion neurons. In contrast, the C. elegans has 302 neurons, and only recently have computers and graph-theoretic algorithms become powerful enough to analyze the system. –  Alex Becker May 23 '12 at 2:16
You might look at "Mathematical Models in Biology" by my colleague Leah Edelstein-Keshet ec-securehost.com/SIAM/CL46.html –  Robert Israel May 23 '12 at 2:22
Also Martin Novak's text on evolutionary dynamics might be of interest: amazon.com/Evolutionary-Dynamics-Exploring-Equations-Life/dp/… –  student May 23 '12 at 4:01

Mathematical biology is a very active field. As a starting point, you might look at the Wikipedia article on Mathematical and theoretical biology. The Society for Mathematical Biology publishes the Bulletin of Mathematical Biology. There are also a Journal of Mathematical Biology and the open access Journal of Mathematical Neuroscience, both published by Springer. In the August 1010 Notices of the AMS there’s a seven-page essay on What Is Mathematical Biology and How Useful Is It? by Avner Friedman.

It’s true that the subject has only relatively recently percolated into undergraduate curricula, though I remember teaching some very elementary modeling of epidemics back in the 70s. For one thing, modern computing has made parts of it considerably more accessible than they used to be. But it’s getting there. Links on this Math Archives page show that there are courses in aspects of the subject at the undergraduate as well as the graduate level. Indeed, the Society for Mathematical Biology lists several schools offering undergraduate majors in some sort of mathematical biology. The list isn’t complete, either: the University of Houston also offers such a major, as does the University of Pittsburgh, and McGill offers a joint major in biology and mathematics. A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, by Sarah P. Otto and Troy Day, is expressly designed to make the techniques of mathematical modeling available to students and biologists who don’t already have more mathematical background than first-year calculus.

And of course biostatistics has become an indespensable part of biology and medicine and is increasingly showing up in undergraduate statistics programs.

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Here is one of my favorites, the Lotka-Volterra equations used to model predator/prey relations: http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation.

Not to mention the essential use of statistics in biology and medicine to tease out correlations only observable in large data sets. In fact some of the most important statistical tests used today were developed by the biologist and mathematician Ronald Fisher: http://en.wikipedia.org/wiki/Ronald_Fisher

However the comparative lack of mathematical methods in biology compared to other scientific fields is something worth thinking about. One possible answer is that full blown biological systems (with perhaps hundreds, thousands, or more interacting elements) are just too complicated to accurately model using mathematics. This I think is quickly becoming outdated I think, especially with the rise of computational tools and large data sets in biology. A pessimist about progress in biology might say that biology is not just well developed enough. Chemistry and physics too had long historical phases where very little was done using quantitative methods. Pushing back against this, people like Peter Godfrey-Smith have argued that perhaps this idea drawn from the history of physics and chemistry of "mature" sciences needs to reexamined. For biology has been quite successful, practical and interesting even without heavy use of mathematics. (This is more of an aside to your original question but interesting nonetheless.)

For a cutting edge mathematical model, there are quite serious attempts to model complicated biological systems mathematically. One intriguing example is the Blue Brain project in Switzerland http://bluebrain.epfl.ch/. Their first major goal was to model a rat neocortical column using one virtual neuron for every real neuron (a real column has something like 10,000 neurons and $10^8$ synapses). Needless to say there is lots of mathematics and computation involved in this project!

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It seems fitting that an answer emphasizing the important role of statistics is given by "student". –  KCd May 23 '12 at 3:09
Oh yes, why fitting? –  student May 23 '12 at 3:24
@student: Ever heard of Student's $t$ distribution? –  Arturo Magidin May 23 '12 at 3:50
hahah, oh yes I have –  student May 23 '12 at 3:51