Minimal surface as a model of beta-sheets

The purpose of this article is to present arguments based on experimental data that the beta-sheet structures in proteins are the result of the tendency to minimize surface areas. Thus, we propose the model that all beta-sheet structures are almost minimal surfaces, namely, their mean curvatures are nearly zero. To support this model, we chose 1740 disjoint beta-sheets with less than 10 strands from the all beta-protein class in a nonredundant 40% Structural Classification of Proteins (SCOP) database and applied the least-squares method to fit the minimal surface catenoid (and in some rare cases, the plane) to the beta-sheet structures. The fitting errors were extremely small: The error of 1729 beta-sheets with catenoid minimal surface is 0.90 +/- 0.55 A and the error of the remaining 11 flat sheets with the plane is 0.64 +/- 0.46 A. The fact that the commonly used models for some beta-sheet surfaces (i.e., the hyperboloid and strophoid) have very small mean curvatures (< 0.05) supports our model. Moreover, we showed that this model also includes the isotropically stressed configuration model proposed by Salemme, in which the intrastrand tendency of the individual chains to twist or coil is in equilibrium with the tendency of the interstrand hydrogen bonding to resist twisting of the sheet as a whole. As an application we used our model to quantify the two principal independent modes in the flexibility of beta-sheets, that is, the bending parameter of beta-sheets and the inclined angle of beta-strands in a sheet.