Protein unfolding: rigidity lost

Abstract

We relate the unfolding of a protein to its loss of structural stability or rigidity. Rigidity and flexibility are well defined concepts in mathematics and physics, with a body of theorems and algorithms that have been applied successfully to materials, allowing the constraints in a network to be related to its deformability. Here we simulate the weakening or dilution of the noncovalent bonds during protein unfolding, and identify the emergence of flexible regions as unfolding proceeds. The transition state is determined from the inflection point in the change in the number of independent bond-rotational degrees of freedom (floppy modes) of the protein as its mean atomic coordination decreases. The first derivative of the fraction of floppy modes as a function of mean coordination is similar to the fraction-folded curve for a protein as a function of denaturant concentration or temperature. The second derivative, a specific heat-like quantity, shows a peak around a mean coordination of <r> = 2.41 for the 26 diverse proteins we have studied. As the protein denatures, it loses rigidity at the transition state, proceeds to a state where just the initial folding core remains stable, then becomes entirely denatured or flexible. This universal behavior for proteins of diverse architecture, including monomers and oligomers, is analogous to the rigid to floppy phase transition in network glasses. This approach provides a unifying view of the phase transitions of proteins and glasses, and identifies the mean coordination as the relevant structural variable, or reaction coordinate, along the unfolding pathway.

Figure 1

The fractional number of floppy modes. f = F/3N in a glass as a function of the mean coordination, 〈r〉. The Maxwell approximation (Eq. 3) is shown as a thick black dashed line. (Left) The results for random glass networks (purple line, where the transition is second-order; orange line, for a network with an absence of small rings where the transition is first-order; ref. 21). (Right) Similar results for a representative set of 26 structurally and functionally diverse proteins (blue lines, monomers; red lines, dimers; green lines, tetramers; see Methods for details). The gray shaded region indicates the range in which protein folding/unfolding takes place.

Figure 2

Change in the fraction of floppy modes as a function of mean coordination for the set of 26 representative proteins shown in Fig. 1. Gray shading shows the transition region where folding takes place. The curves for the two kinds of glass networks from Fig. 1 Left (thick purple and orange lines) are shown superimposed on the protein curves. The notations at top indicate Denatured, Folding Core, Transition, and Native states of the proteins. For comparison with results for a typical thermal denaturation experiment, the Inset sketches the decrease in fraction of folded protein as temperature increases (adapted from figure 7.11 in ref. 42).

Figure 3

Modeling hydrophobic contacts within proteins. Pairs of carbon and/or sulfur atoms, shown in A, are considered to make hydrophobic contacts if their van der Waals surfaces, represented by sphere radii r_a and r_b (without correction for attached hydrogen atoms) are within R = 0.25 Å. This allows atoms to either be in contact or lightly separated as shown in B, but without enough space between for water to intervene. Because first represents the protein as interatomic constraints, multijointed tethers with pseudoatoms at the joints shown in C are used to flexibly join atoms to form hydrophobic interactions. The flexible tethers allow two atoms forming a hydrophobic interaction to slip relative to one another, while remaining in the same vicinity.

Figure 4

Rigid cluster decompositions of barnase (PDB ID code 1a2p). Each image corresponds to a different value of 〈r〉 along the unfolding pathway shown in Figs. 2 and 5. Calculations were carried out for the entire protein structure, but only the backbone is shown for clarity, with main chain to main chain hydrogen bonds drawn as thinner black lines. Each bond is colored according to the rigid cluster to which it belongs. Bonds split into two colors indicate that the bond remains rotatable, and small regions of alternating color indicate a sequence of flexible bonds. Note how the largest rigid cluster, shown in dark blue, shrinks as the protein goes from the Native state at 〈r〉 = 2.45, through the Transition state at 〈r〉 = 2.41, to the Folding Core at 〈r〉 = 2.39, which just precedes the onset of complete flexibility.

Figure 5

The second derivative of the fraction of floppy modes as a function of mean coordination for the set of 26 proteins from Figs. 1 and 2. The Inset shows a sketch of the specific heat for a protein with the Denatured state, Folding Core, Transition state, and Native state indicated. The x axis of the Inset has the temperature increasing to the left.