# Allostery

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Cosider the folowing definiton:

Allostery is the process by which biological macromolecules (mostly proteins) transmit the effect of binding at one site to another, often distal, functional site, allowing for regulation of activity.

Many allosteric effects can be explained by the concerted MWC model put forth by Monod, Wyman, and Changeux, or by the sequential model described by Koshland, Nemethy, and Filmer.  The concerted model of allostery, also referred to as the symmetry model or MWC model, postulates that enzyme subunits are connected in such a way that a conformational change in one subunit is necessarily conferred to all other subunits. Thus, all subunits must exist in the same conformation. The model further holds that, in the absence of any ligand (substrate or otherwise), the equilibrium favors one of the conformational states, T (tensed) or R (relaxed). The equilibrium can be shifted to the R or T state through the binding of one ligand (the allosteric effector or ligand) to a site that is different from the active site (the allosteric site). [Wikipedia]

In this post, I want to draw attention towards application of mathematics in understanding biological process, allostery. Consider the following equation which relates the difference between $n$, the number of binding sites, and $n'$, the Hill coefficient, to the ratio of the ligand binding function, $\overline{Y}$, for oligomers with $n-1$ and $n$ ligand binding sites $\displaystyle{\boxed{n-n' = (n-1) \frac{\overline{Y}_{n-1}}{\overline{Y}_n}}}$

This is known as Crick-Wyman Equation  in enzymology, where $\displaystyle{\overline{Y}_n = \frac{\alpha(1+\alpha)^{n-1}+ Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}}$ and $\displaystyle{n' =\frac{d( \ln(\overline{Y}_n) - \ln(1-\overline{Y}_n))}{d\ln\alpha}}$; $L$ is allosteric constant and $\alpha$ is the concentration of ligand under some normalizaton conditions.

It’s not uncommon to find simple differential equations in biochemistry (like Michaelis-Menten kinetics), but the above equation stated above is not a kinetics equation but rather a mathematical model for a biological phenomina. Comparable to the Hardy-Weinberg Equation discussed earlier.

# Geometry of Virus

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This semester I am taking a course about protein structures. Here is a quick intro to proteins:

Though I have taken some other biology courses in past years, I found this course very much relatable to mathematics. Proteins are made up of “amino acids”. Though, chemistry allows large number of possible structures for amino acids (considering steric hindrance etc.), nature uses only 20 unique amino acids to make billions of different proteins. In my opinion, these 20 amino acids are “axioms” of protein building just like the 5 axioms of euclidean geometry.

Using just 20 amino acids we can get a large variety of protein structures, just like creating any kind of shape in euclidean space using just 5 axioms. Even more fascinating is the existence of “Quasisymmetry in Icosahedral Viruses”. An awesome article explaining this is available here. Note that, the term “triangulation number” stated in that article was not borrowed from mathematics. It’s a term used to study symmetries in icosahedral viruses and refers to “the square of the distance between 2 adjacent 5-fold vertices.” 200 Icosahedral Viruses from the PDB (source: http://pdb101.rcsb.org/learn/resource/200-icosahedral-viruses-poster)

Moreover, the structures which don’t conform to classic quasisymmetry are similar to Escher print and Penrose tiling, as visible in following picture: If you are interested in doing a fun activity, you may refer to: http://pdb101.rcsb.org/learn/resource/quasisymmetry-in-icosahedral-viruses-activity-page

# Hyperbolic Plane Example

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Few months ago I gave a lecture on Non-euclidean geometry and it was a bit difficult for me to give audience an example of hyperbolic surface from their day-to-day life. While reading Donal O’ Shea’s book on Poincaré Conjecture I came across following interesting example on pp. 97 : Negatively curved cloth will drape a woman’s side (© Donal O’ Shea, 2007)

Estrogen causes fat to be stored in the buttocks, thighs, and hips in women. Thus females generally have relatively narrow waists and large buttocks, and this along with wide hips make for a wider hip section and a lower waist-hip ratio compared to men. The saddle-shaped area on a woman’s side above her hip has negative curvature.

One can imagine cloth (it is flexible but does not stretch, hence an isometry) that would drape it perfectly. Here the region inside a circle of given radius contains more material than the same circle on the plane, and to make the cloth the tailor might start with a flat piece of fabric, make a cut as if he/she were going to make a dart, but instead of stitching the cut edges together, insert an extra piece of fabric or a gusset. Negatively curved cloth would have lots of folds if one tried to lay it flat in  dresser.

If one tries to extend a cloth with constant positive curvature (like a cap), in all directions, it would close up, making a sphere. On the other hand, if one imagines extending a cloth with constant negative curvature in all directions, one gets a surface called hyperbolic plane.

# Mathematical Relations

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In this post I will share my perception of relation of mathematics with other academic disciplines. All this is based on my very limited knowledge of various disciplines. Shape doesn’t signify anything.

Mathematics deals with study of properties of numbers (or the symbols representing them) and geometric objects (not in classical sense, it can mean manifolds also). In my opinion, there is no partition of mathematics into “applied” or “pure”, but intersections with other subjects. The term applied Mathematics doesn’t make any sense to me. Mathematics is somehow applicable in various places. For me, mathematics is what people call “pure” mathematics (what about “impure” Mathematics??).  Also now I agree with the vastly established belief that art and mathematics are similar, since both involve abstract ideas motivated but physical situations (at some point). Truth Lies Deception and Coverups – Democracy Under Fire (Source: http://goo.gl/yUHi93)

All experimental sciences (physics, chemistry, biology, economics) are based on statistics. Since statistics is a young discipline (only a couple of centuries old) many times we get wrong interpretation of results. As far as real life is concerned, study of statistics gives us a powerful tool for predicting future and Probability Theory acts as the connecting link between statistics and mathematics. Understanding of statistics affects us on daily basis since (effective) government policies are framed keeping statistical analysis in mind. Unfortunately, most of universities don’t have separate department for statistics. P vs NP Problem in Relationships (http://ctp200.com/comic/6; CC BY-NC 4.0)

Study of algorithms is one of the most important aspect of computer science (I am not talking about software industry…). What surprises me is that Euclid’s division algorithm is  one of the most efficient division algorithm even for computers! The neglected subject of Logic, which is supposed to be foundations of mathematics, flourishes in computer science. P vs NP is another “millennium open problem“. Convincing (http://xkcd.com/833/ ; CC BY-NC 2.5)

For me, Economics like Statistics is full of imperfections due to real life complications (so many dependencies to account for). Game Theory appears to be the connecting link between mathematics and economics. We all know that the needs of physicists are responsible for development of calculus and study of differential equations. On the other hand, theoretical physics (quantum mechanics, string theory) depends heavily on the developments in algebra.

# G.H. Hardy contributes to Biology !!!

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Last month I appeared for my Biology mid-sem exam. I discovered Hardy-Weinberg Equation, which states that:

The Hardy-Weinberg equation is a mathematical equation that can be used to calculate the genetic variation of a population at equilibrium. The Hardy-Weinberg equation is expressed as: $p^2 +2pq+q^2=1$

You can understand this equation here:

Now the interesting point for me here was that Hardy preferred his work to be considered pure mathematics, perhaps because of his detestation of war and the military uses to which mathematics had been applied. He made several statements similar to that in his autobiography (A Mathematician’s Apology):

“I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”