Coarse grained models, bond orientational order, and the structural characterization of residue networks

Abstract

As a second goal, we show why contact number is the dominant factor that controls protein fluctuations. We show that the variations of temperature factors of residues can be tied to the BOO parameters established around each residue. BOO parameters constructed from bond density expansions with degree l=2 spherical harmonics are shown to successfully explain the variations in mean squared displacements (MSD) computed from the anisotropic network model (ANM). To elucidate the origin of residue flexibility, we reformulate the BOO parameter with Legendre polynomials and show that it reduces to the sum of reciprocal coordination number and a geometric sum that characterizes the spatial distribution of contacts. Using this new formulation of BOO parameter, we show that within the commonly used cut-off range of 9-16 Å, a special distribution of contact angles is established that largely nullifies the geometric term in the BOO parameter. Therefore, the contact number stands out as the prominent variable that controls the protein fluctuations.

Proteins are polypeptide chains that fold into compact 3-dimensional shapes, which determine their molecular function. Due their unique chemistry, coarse graining with hard spheres at the amino acid level works well for proteins. Following this framework, the first goal of the current study is to portray protein structure as a sphere packing problem and decipher the universal characteristics of globular proteins at the residue level. One of the most direct ways of spotting structural features that are shared among proteins is to find proper analogues from simpler condensed matter systems. In this respect, sphere packing arguments provide the most direct route for structural comparisons as they successfully characterize a wide array of materials such as close packed crystals, dense liquids, and structural glasses.

In the current study, we show that a set of simple but precisely defined rules is sufficient for robust generation of protein-like structures on a simple cubic (SC) lattice when proteins are coarse grained at the amino acid level. In the proposed formulation, cubic clusters of hard spheres are carved out from an infinite SC crystal. Then, vacancies up to 60% by volume are randomly introduced into these SC crystals. Finally, these initial structures are relaxed by either moving particles from their perfect lattice positions randomly or with a combination of reverse Monte Carlo and simulated annealing (RMC-SA) procedure. The RMC-SA procedure targets the average radial distribution function (RDF) of 210 globular proteins. All resulting structures created by either method are shown to mimic average residue contact number and bond orientational order (BOO) parameter distribution of real protein chains. Our results indicate that residue networks closely resemble defect laden SC crystalline clusters. Therefore, we conclude that the maximally dense packing of rigid spheres is not necessarily the prevalent organizational principle for globular proteins at the residue level.