A critical assessment of the topomer search model of protein folding using a continuum explicit-chain model with extensive conformational sampling

Abstract

Recently, a series of closely related theoretical constructs termed the "topomer search model" (TSM) has been proposed for the folding mechanism of small, single-domain proteins. A basic assumption of the proposed scenarios is that the rate-limiting step in folding is an essentially unbiased, diffusive search for a conformational state called the native topomer defined by an overall native-like topological pattern. Successes in correlating TSM-predicted folding rates with that of real proteins have been interpreted as experimental support for the model. To better delineate the physics entailed, key TSM concepts are examined here using extensive Langevin dynamics simulations of continuum C(alpha) chain models. The theoretical native topomers of four experimentally well-studied two-state proteins are characterized. Consistent with the TSM perspective, we found that the sizes of the native topomers increase with experimental folding rate. However, a careful determination of the corresponding probabilities that the native topomers are populated during a random search fails to reproduce the previously predicted folding rates. Instead, our results indicate that an unbiased TSM search for the native topomer amounts to a Levinthal-like process that would take an impossibly long average time to complete. Furthermore, intraprotein contacts in all four native topomers considered exhibit no apparent correlation with the experimental phi-values determined from the folding kinetics of these proteins. Thus, the present findings suggest that certain basic, generic yet essential energetic features in protein folding are not accounted for by TSM scenarios to date.

Figure 1.

Simple schematic illustration of the energy landscape in the TSM, for a slow-folding (left) and a fast-folding (right) protein. The horizontal dimensions represent protein conformational variation or conformational entropy; the vertical dimension provides the free energy of every given protein conformation—with appropriate averaging of solvent degrees of freedom (Dill and Chan 1997).

Figure 2.

Illustrations of the native topomers for the proteins 1aps, 1ci2, 1urn, and 1lmb. Shown are the top 25 conformations (red) selected by a simple clustering procedure on the full l_c=12, r_c=8 Å topomer ensembles. Each of the 25 conformations is optimally superimposed on the corresponding native structure (dark trace). In the clustering procedure, the highest ranking conformation is the one with the largest number of conformational neighbors, where two conformations are considered to be neighbors if their rmsd is less than a certain cutoff (1.3–4.0 Å). The next highest ranking conformation is the conformation with most neighbors in the reduced ensemble, where the highest-ranking conformation and its neighbors have been excluded, and so on.

Figure 3.

Probability distributions P(rmsd), where rmsd is from the corresponding native pdb structure, for 1aps, 2ci2, 1urn, and 1lmb, as obtained using l_c=12 and r_c=8 Å in Equation 1, and sampling at T=1.0.

Figure 4.

Probability distributions P(Q) for 1aps, 2ci2, 1urn, and 1lmb, as obtained using l_c=12 and r_c=8 Å in Equation 1, and sampling at T=1.0.

Figure 5.

Comparison between TSM-predicted (+) and experimental (○) φ-values. The lines between data points serve merely as a guide for the eye. The secondary structure of the proteins along the sequences is indicated by large green and small black rectangular shapes for α-helical and β-sheet regions, respectively.

Figure 6.

Probabilities of native contacts being formed (ψ^ij_calc-values) in the l_c=12, r_c=8 Å topomer ensembles. The color scale goes from =0 (green) to =1 (red). The triangular regions above and below the main diagonal provide identical information.

Figure 7.

Free-energy profiles for the four proteins are given by the negative logarithm of the conformational distribution P(Q) in Q, obtained in this study by simulations of native-centric models (Kaya and Chan 2003a) at T ≈ T_f. Results from two native contact sets are shown: (1) contact pairs are defined by native C_α–C_α distance rⁿ_ij<8 Å (dashed curves); and (2) contact pairs are defined by the shortest spatial separation between non-hydrogen atoms in the two amino acid residues as described by Chavez et al. (2004) and in the text (solid curves). For comparison with the hypothetical TSM picture, the range of variation of 〈Q〉 of the corresponding native topomers across different (l_c, r_c) criteria (Table 3) is indicated by the shaded areas.

Figure 8.

Comparison between the TSM quantity Λ_Da^ΛD (a=0.86) (Makarov and Plaxco 2003) of the four proteins and the corresponding folding rate k_f ^nc computed in this study from direct kinetic simulations of the native-centric Langevin dynamics models at each model’s T_f. Results for k_f ^nc denoted by × and ⊡ are, respectively, for the native contact sets (1) and (2) specified in the caption for Fig. 7 ▶.

Figure 9.

Marginal distributions P_λi (E_bias) for eight linearly separated λ_i values (from λ₁=0 to λ₈=1.6), as obtained from a simulation of the energy function E_λ=E₀+λE_bias for 2ci2 and l_c=12 and r_c=8 Å in which λ is a dynamical parameter (see text).