Geometry and symmetry presculpt the free-energy

landscape of proteins

Protein folding (1-5) is complex because of the sheer size ofprotein molecules, the twenty types of constituent amino acids with distinct side chains, and the essential role played by theenvironment. Nevertheless, proteins fold into a limited number (6,7) of evolutionarily conserved structures (8, 9). It is a familiar, yet remarkable, consequence of symmetry and geometry that ordinary matter crystallizes in a limited number of distinct forms. Indeed, crystalline structures transcend the specifics of the various entities housed in them. Here, we ask the analogous question (10): is the menu of protein folds also determined by geometry and symmetry?

We show that a simple model that encapsulates a few general attributes common to all polypeptide chains, such as steric constraints (11-13), hydrogen bonding (14 -16), and hydrophobicity (17), gives rise to the emergent free-energy landscape of globular proteins. The relatively few minima in the resulting landscape correspond to putative marginally compact native-state structures of proteins, which are assemblies of helices, hairpins, and planar sheets. A superior fit (18, 19) of a given protein or sequence of amino acids to one of these predeter-mined folds dictates the choice of the topology of its native-state structure. Instead of each sequence shaping its own free energy landscape, we find that the overarching principles of geometry and symmetry determine the menu of possible folds that the sequence can choose from.

Following Bernal (20), the protein problem can be divided into two distinct steps: first, analogous to the elucidation of crystal structures, one must identify the essential features that account for the common characteristics of all proteins; second, one must understand what makes one protein different from another. Guided by recent work (21, 22) that has shown that a faithful description of a chain molecule is a tube and using information from known protein native-state structures, our focus, in this paper, is on the first step: we demonstrate that the native-state folds of proteins emerge from considerations of symmetry and geometry within the context of a simple model.

We model a protein as a chain of identical amino acids, represented by their C atoms, lying along the axis of a self-avoiding flexible tube. The preferential parallel placement of nearby tube segments approximately mimics the effects of the anisotropic interaction of hydrogen bonds whereas the space needed for the clash-free packing of side chains is approximately captured by the non-zero tube thickness (21, 22). Here, we carefully incorporate these key geometrical features by means of an extensive statistical analysis of experimentally determined native-state structures in the Protein Data Bank (PDB).

A tube description places constraints on the radii of circles drawn through both local and nonlocal triplets of C positions of a protein native structure (22, 23). Furthermore, when one deals with a chain molecule, the tube picture underscores the crucial importance of knowing the context that an amino acid is in within the chain. The standard coarse-grained approach con-siders the locations of interacting amino acid pairs. Here, instead, we incorporate the strongly directional hydrogen bond-ing between a pair of amino acids, through an analysis of the PDB to determine the constraints on the mutual orientation of the local coordinate systems defined from a knowledge of thelocations of the C atoms (see Methods and Fig. 1). Thegeometrical constraints associated with the tube and hydrogen bonds that we consider here are representative of the typical aspecific behavior of the interacting amino acids.

There are two other ingredients in the model: a local bending penalty, which is related to the steric hindrance of the amino acid side chains, and a pair-wise interaction of the standard type mediated by the water (17). Even though these two properties clearly depend on the specific amino acids involved in the interaction, here, we choose to study the phase diagram of a homo-peptide chain by varying its overall hydrophobicity and local bending penalty, while keeping them constant along the chain. This is the simplest and most general way to assess their relevance in shaping the free-energy landscape.

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