Reassessing random-coil statistics in unfolded proteins

Abstract

The Gaussian-distributed random coil has been the dominant model for denatured proteins since the 1950s, and it has long been interpreted to mean that proteins are featureless, statistical coils in 6 M guanidinium chloride. Here, we demonstrate that random-coil statistics are not a unique signature of featureless polymers. The random-coil model does predict the experimentally determined coil dimensions of denatured proteins successfully. Yet, other equally convincing experiments have shown that denatured proteins are biased toward specific conformations, in apparent conflict with the random-coil model. We seek to resolve this paradox by introducing a contrived counterexample in which largely native protein ensembles nevertheless exhibit random-coil characteristics. Specifically, proteins of known structure were used to generate disordered conformers by varying backbone torsion angles at random for approximately 8% of the residues; the remaining approximately 92% of the residues remained fixed in their native conformation. Ensembles of these disordered structures were generated for 33 proteins by using a torsion-angle Monte Carlo algorithm with hard-sphere sterics; bulk statistics were then calculated for each ensemble. Despite this extreme degree of imposed internal structure, these ensembles have end-to-end distances and mean radii of gyration that agree well with random-coil expectations in all but two cases.

Fig. 1.

Flexibility profile for lysozyme (PDB ID code 1HEL). Secondary structure is indicated by bars beneath the plot, which are color-coded as follows: red, α-helices; blue, β-strands; and green, turns. Secondary structure determinations are based on backbone torsions, as described in ref. .

Fig. 2.

End-to-end distance histogram for lysozyme using 5,000 chains generated from the rigid-segment model. Chains were grouped into 10-Å bins based on the distance from the N-terminal nitrogen to the C-terminal oxygen. For comparison, a Gaussian curve having the same mean and SD as the actual distribution is also shown (dashed line).

Fig. 3.

Representative lysozyme structures from rigid-segment simulations. The entire chain was held fixed in its x-ray-determined conformation, except for 11 flexible hinge residues (shown as yellow space-filling spheres). Ribbon diagram depicts elements of secondary structure, defined here from the Protein Data Bank header records and generated by using molscript (49) and raster3d (50). Termini are color-coded as follows: blue, N termini; red, C termini.

Fig. 4.

Coil dimensions for 33 proteins using the rigid-segment model. (A) Radius of gyration (〈R_G〉) versus chain length in residues for 33 ensembles from rigid-segment simulations. The curve is well fitbyEq. 2, with R₀ = 1.98 ± 0.37 Å and ν = 0.602 ± 0.035. (B) Mean-squared end-to-end distance (〈L²〉) versus chain length in residues for the same 33 ensembles. The best-fit value of L₀, the slope of the line, is 81.8 ± 3.4 Ų. These fitted parameters are in close agreement with accepted random-coil values.

Fig. 5.

Comparison between our values of R_G from the rigid-segment model and corresponding values of R_G from random-coil expectations by using Eq. 1. All data points fall near the diagonal line. To aid in visualization, a shaded region marks the ±15% boundary, ranging between y = 1.15x and y = 0.85x.

Fig. 6.

Kratky plots of rigid-segment simulations. (A) Calculated Kratky plot for 1,296 structures chosen at random from the lysozyme ensemble. (B) Calculated Kratky plot for the same structures after removal of side-chain atoms beyond Cβ. The maximum in A suggests a native protein, whereas B resembles a denatured protein, suggesting the fact that the hump in A is caused by sidechain rigidity and not by lack of backbone flexibility.