Context and force field dependence of the loss of protein backbone entropy upon folding using realistic denatured and native state ensembles

Abstract

The loss of conformational entropy is the largest unfavorable quantity affecting a protein's stability. We calculate the reduction in the number of backbone conformations upon folding using the distribution of backbone dihedral angles (ϕ,ψ) obtained from an experimentally validated denatured state model, along with all-atom simulations for both the denatured and native states. The average loss of entropy per residue is TΔS(BB)(U-N) = 0.7, 0.9, or 1.1 kcal·mol(-1) at T = 298 K, depending on the force field used, with a 0.6 kcal·mol(-1) dispersion across the sequence. The average equates to a decrease of a factor of 3-7 in the number of conformations available per residue (f = Ω(Denatured)/Ω(Native)) or to a total of f(tot) = 3(n)-7(n) for an n residue protein. Our value is smaller than most previous estimates where f = 7-20, that is, our computed TΔS(BB)(U-N) is smaller by 10-100 kcal mol(-1) for n = 100. The differences emerge from our use of realistic native and denatured state ensembles as well as from the inclusion of accurate local sequence preferences, neighbor effects, and correlated motions (vibrations), in contrast to some previous studies that invoke gross assumptions about the entropy in either or both states. We find that the loss of entropy primarily depends on the local environment and less on properties of the native state, with the exception of α-helical residues in some force fields.

Figure 1. Loss of backbone entropy upon folding

(Upper Panels) The backbone entropies corrected for nearest neighbor correlations for the folded and denatured states, along with the differences between the two states, for residues 3–74 calculated using both the OPLS/AA-L (Left) and G-S A94 FFs (Middle), as well as the CHARMM FF in explicit solvent (Right). The entropy calculations for the native and DSE implicitly depend on the pixel resolution used to construct the probability distributions. We eliminate this dependence by computing the entropy for multiple bin widths and fitting the difference in entropy as a function of the ratio of pixel sizes (see Suppl. Methods, Suppl. Fig. 1). (Lower Panels) The change in backbone entropy during folding is presented with the residues colored according to native secondary structure elements. While the loss of entropy varies across the sequence, no strong dependence on sequence appears, except for the unstructured carboxy-terminal, proline, and pre-proline residues that incur smaller changes in entropy during folding.

Figure 2. Loss of backbone entropy for secondary structure elements

Calculated changes in backbone entropy are averaged over various secondary structure types. Glycines and helical residues on average yield a slightly larger loss in entropy than coil and sheet residues. Proline residues exhibit little change in entropy between states. Pre-proline residues likewise have a reduced change in entropy. Individual values are shown for each secondary structure type along with a box-whisker plot covering the interquartile range (IQ = Q2-Q1) and the upper inner (Q2 + 1.5·IQ) and lower inner (Q1 − 1.5·IQ) fence values, respectively.

Figure 3. Ramachandran plots of alanine and glycine residues

Free energy landscapes in Ramachandran space are displayed for Ala-28 and Ala-46 and for Gly-35 and Gly-53 in both the denatured and native state ensembles. Data are taken from simulations using the OPLS/AA-L FF. The probability distributions are calculated using a pixel size of 10°×10° and are converted to free energy distributions using –RT lnP. The color scale ranges from red (ground state) to blue (6 kcal·mol⁻¹). Dihedral angles with free energies larger than 6 kcal·mol⁻¹ are represented in black.

Figure 4. Nearest neighbor contributions to the backbone entropy

The contributions of conformational correlations between nearest neighbors to the backbone entropy are displayed for both the folded and denatured states and for both the OPLS/AA-L and G-S A94 FFs. The contributions are larger in magnitude in the native state ensemble than in the DSE (TΔSⁿⁿ = −0.3±0.1 and −0.2±0.1 kcal·mol⁻¹, respectively). The turn regions between the β1-β2 hairpin and α-helix and the β4-β5 hairpin yield the greatest contributions in the native state, but pronounced contributions occur along other regions of the protein as well. The largest contributions in the denatured state are associated with glycine residues and their nearest neighbors. The OPLS/AA-L FF yields a slightly larger contribution to glycines and pre-glycine residues (TΔSⁿⁿ = −0.22 ± 0.03 kcal·mol⁻¹), whereas the average for all other residues is TΔSⁿⁿ = −0.17 ± 0.05 kcal·mol⁻¹. However, the contributions in the denatured state are larger for the G-S A94 FF, i.e., the contributions for glycine residues exceed those for pre- and post-glycine residues and all other residues (TΔSⁿⁿ = −0.63 ± 0.08, −0.49 ± 0.04, −0.46 ± 0.13, −0.33 ± 0.08 kcal·mol⁻¹, respectively).