### A Mathematical Model for Protein Folding

Yi Fang^{1}, Warren Kaplan^{2}

^{1}yi@maths.anu.edu.au, CBiS, ANU; ^{2}w.kaplan@garvan.org.au, Garvan Institute

The native structures of globular proteins are: compactly packed, with a hydrophobic
core, and a small surface area. We have established a mathematical model that pursues these three features. The protein model is presented as a space-filled model. Each atom is represented as a sphere with its respective van der Waals radius. Chemical bonds are represented as the intersections of spheres that can adopt sterically permissible conformations .

For each conformation P, we define a simply connected surface, S_P, and call it the aqueous surface of the conformation P. For any closed surface enclosing the conformation Pin its interior, we can calculate the following three quantities: the area A(S_P), the enclosed volume V(S_P), and the area that is exposed to the hydrophobic part of the protein, H(S_P). Then we feed the three quantities into a positive and increasing function E of three variables. among all simply connected surfaces that enclosing the conformation P in their interior. For all simply connected surfaces that enclosing the conformation P in their interior - the aqueous surface S_P then is defined as the minimizing simply connected surfaces of the E function

Thus via this aqueous surface S_P we have defined a value for each conformation P, E(A(S_P), V(S_P), H(S_P)). We can then define an energy function F(P) = E(A(S_P), V(S_P), H(S_P)), for any conformation P. Of all the conformations, the conformation Q that minimizes the energy function F will
achieve simultaneous and coherent minimization of the three afore mentioned geometric
parameters. We hypothesize that this minimizing conformation Q to be the native structure of the globular protein.

It can be proven that the aqueous surface S_P of a conformation P has piecewise constant surface tension, or mean curvature. It therefore fits the equilibrium features of dividing surface between different chemicals.

Collaborations are welcome, preprints are available. Please contact
yi@maths.anu.edu.au