Small-world communication of residues and significance for protein dynamics

Abstract

It is not merely the position of residues that is critically important for a protein's function and stability, but also their interactions. We illustrate, by using a network construction on a set of 595 nonhomologous proteins, that regular packing is preserved in short-range interactions, but short average path lengths are achieved through some long-range contacts. Thus, lying between the two extremes of regularity and randomness, residues in folded proteins are distributed according to a "small-world" topology. Using this topology, we show that the core residues have the same local packing arrangements irrespective of protein size. Furthermore, we find that the average shortest path lengths are highly correlated with residue fluctuations, providing a link between the spatial arrangement of the residues and protein dynamics.

FIGURE 1

Network construction from a protein. Here the structure of human interleukin 1-β converting enzyme (PDB code: 1ice) is shown on the left. The network constructed from the C^β coordinates of the residues (C^α for Gly) at 7 Å cutoff is shown on the right.

FIGURE 2

Residue contact distribution at r_c = 7 Å, computed as an average over all the residues in a set of 54 proteins. The familiar form of the contact distribution is captured (see, for example, Fig. 4 in Miyazawa and Jernigan, 1996). The contact distributions of core and surface residues are also displayed. Gaussian distribution of coordination numbers is valid for both the hydrophobic core and the molten surface.

FIGURE 3

The depth dependence of the characteristic path length (open symbols) and the clustering coefficient (filled symbols) for proteins of fixed sizes (N = 150: squares, 24 proteins; N = 210: triangles, 15 proteins; N = 310: circles, 15 proteins). The characteristic path length consistently decreases for residues at greater depths; moreover, its value depends on system size. On the other hand, at depths >4 Å, the clustering coefficient attains a fixed value of ∼0.35 irrespective of system size and the location of the residue. Even for the surface residues, the clustering coefficient is independent of system size, although its value is location dependent and somewhat higher than 0.35.

FIGURE 4

A good correlation between the shortest path lengths (○) and residue fluctuations (•) is observed. Four examples, one of each from α, β, α + β, and α/β class of proteins, are displayed.

FIGURE 5

Shortest paths to relieve the impact upon binding of CI2. Ile-56 that is in the binding pocket and that makes many contacts with the substrate, subtilisin novo, is shown with its accessible surface in yellow. Thr-55 and Val-57 (shown in red) are bonded to this residue, and they have their side chains pointing toward the very stable residues of the inhibitor, Arg-67 and Phe-69 (shown in orange). These are in turn connected to the most stable residue of CI2, Leu-68 (shown in yellow). To avoid unfolding, the impact is finally communicated to the stabilizing residues Ala-35 and Ile-76 (shown in purple), which propagate it to the rest of the protein.

FIGURE 6

In a SWN, characteristic path length, L, is on the same order of magnitude as its randomized counterpart, whereas clustering density, C, is at least one order of magnitude larger. The variation of the ratios L/L_random (right ordinate) and C/C_random (left ordinate) in the residue networks with the cutoff distance, r_c, used in forming the networks is shown. Note that asr_c → ∞ both L and C approach 1, because every node will be connected to every other node at this limit. (Inset) Radial distribution function of the residue networks. All data are averages over 595 nonhomologous proteins.

FIGURE 7

In a SWN, the characteristic path length, L, should show a logarithmic dependence on the system size, N. Thus, the relation L α log(N) should hold up to a cutoff value of ∼8.5 Å. An example case for r_c = 7 Å is shown. Also shown is the logarithmic dependence of L on N for the randomized networks, the slope of which is the inverse logarithm of the connectivity. That the relationship L = log N/log K should hold for Poisson and Gaussian distributed random networks is a well-known result (Newman et al., 2001).