Are protein folds atypical?

Abstract

Protein structures are a very special class among all possible structures. It has been suggested that a "designability principle" plays a crucial role in nature's selection of protein sequences and structures. Here, we provide a theoretical base for such a selection principle, using a simple model of protein folding based on hydrophobic interactions. A structure is reduced to a string of 0s and 1s, which represent the surface and core sites, respectively, as the backbone is traced. Each structure is therefore associated with one point in a high dimensional space. Sequences are represented by strings of their hydrophobicities and thus can be mapped into the same space. A sequence that lies closer to a particular structure in this space than to any other structures will have that structure as its ground state. Atypical structures, namely those far away from other structures in the high dimensional space, have more sequences that fold into them and are thermodynamically more stable. We argue that the most common folds of proteins are the most atypical in the space of possible structures.

Figure 1

Structures are represented by strings s of 0s and 1s, according to whether a site is on the surface or in the core, respectively. Shown are two examples of compact 6 × 6 lattice structures. (a) A typical structure. Dotted lines indicate local changes that can be performed to transform it to other compact structures. Note that the change at the lower right corner does not change the string pattern, so this structure is a degenerate one. (b) The most designable structure.

Figure 2

Schematic plot of the sequence and the structure spaces and the Voronoi construction. The Voronoi polytope is the shaded region.

Figure 3

Histogram of the designability obtained by random sampling by using 19,492,200 sequences.

Figure 4

Correlation functions for the structures.

Figure 5

Number of structures vs. the Hamming distance for three structures with: low (circles), intermediate (triangles), and high (squares) designability.