Cooperativity and modularity in protein folding

Abstract

A simple statistical mechanical model proposed by Wako and Saitô has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Muñoz and Eaton and has since been referred to as the Wako-Saitô-Muñoz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related models, Saitô emphasized the importance of the hierarchical pathway in protein folding; folding starts with the creation of contiguous segments having a native-like configuration and proceeds as growth and coalescence of these segments. The Φ-values calculated for barnase with the WSME model suggested that segments contributing to the folding nucleus are similar to the structural modules defined by the pattern of native atomic contacts. The WSME model was extended to explain folding of multi-domain proteins having a complex topology, which opened the way to comprehensively understanding the folding process of multi-domain proteins. The WSME model was also extended to describe allosteric transitions, indicating that the allosteric structural movement does not occur as a deterministic sequential change between two conformations but as a stochastic diffusive motion over the dynamically changing energy landscape. Statistical mechanical viewpoint on folding, as highlighted by the WSME model, has been renovated in the context of modern methods and ideas, and will continue to provide insights on equilibrium and dynamical features of proteins.

Keywords: WSME model; energy landscape; statistical mechanics.

Figure 1

The native interaction in the WSME model. Residues in the native-like configuration are shown with white circles, and residues in non-native configurations are shown with filled circles. A) The native interaction (a blue dashed line) between the residues within a contiguous native-like segment is taken into account in the WSME model. B) The interaction becomes ineffective when an intervening residue is in the non-native configuration. C) If the linker chain connecting two native-like segments is long enough, a number of residues with random configurations can compensate each other to allow two segments to reach the positions where native interactions are effective. This type of interaction, however, is not taken into account in the WSME model. D) Interactions as in C can be suitably calculated with the WSME Hamiltonian if we consider that the N- and C-termini are connected by a virtual link, as explained in the section “The WSME Model for Multi-domain Proteins”.

Figure 2

The hierarchical process of protein folding. Folding starts with the creation of contiguous segments with a native-like configuration. After nucleation, folding proceeds as those segments grow and coalesce into larger regions to reach native conformation.

Figure 3

Application of the WSME model to the B domain of Staphylococcal protein A (BdpA). A) Native conformation of BdpA (Protein Data Bank (PDB) code: 1bdd). B) Two-dimensional free-energy landscape, F(n₁, n₂), calculated with the WSME model, where n₁ is the folding order parameter of the N-terminal half, and n₂ is the one of the C-terminal half. A contour is drawn every 0.5k_BT. F(n₁, n₂) has two basins: the unfolded state basin (n₁≈0.3, n₂≈0.3) and the basin of the native state (n₁≈1.0, n₂≈1.0). Two transition states, TS1 and TS2, are shown; there are two dominant pathways of folding, which proceed through TS1 and TS2. C) Comparison of the calculated and observed Φ-values. The calculated values are shown with a line and the observed values [37] are green squares shown with error bars. Bars on the bottom represent the positions of α helices. Modified from Figures 1, 3, and 5 of [15].

Figure 4

Application of the WSME model to barnase. A) Native conformation of barnase from Bacillus amyloliquefaciens (PDB code: 1a2p). B) Two-dimensional free-energy landscape, F(n₁, n₂), calculated with the WSME model, where n₁ is the order parameter of folding of the N-terminal half, and n₂ is the one of the C-terminal half. Contour is drawn in every 2k_BT. F(n₁, n₂) has four basins; basin of unfolded state (n₁≈0.2, n₂≈0.2), basin of native state (n₁≈1.0, n₂≈1.0), and two basins of intermediate states, I₁ (n₁≈0.2, n₂≈0.8) and I₂ (n₁≈0.9, n₂≈0.2). Saddles around the basin I₁ are much lower in free energy than those around I₂ are; therefore, a pathway through I₁ is a dominant pathway, and I₁ is a dominant intermediate. I₂ could be detected as an off-pathway intermediate. Along the dominant pathway, there are two transition states, TS1 and TS2. Modified from Figure 14 of [20] with permission.

Figure 5

Calculated and observed Φ-values at the two transition states, TS1 and TS2, of barnase. Lines shaded with gray correspond to the calculated Φ-values with the WSME model. Dots are the experimentally observed values [39,40]. Red arrows are boundaries of modules defined by the pattern of atomic contacts in the native conformation [44,45]. Bars shown on the bottom represent secondary structure elements, helices (blue) and strands (yellow). Modified from Figure. 15 of [20] with permission.

Figure 6

Examples of multi-domain proteins with non-trivial topology. A) Dihydrofolate reductase (DHFR) (PDB code: 1rx1) has two domains, DLD and ABD. B) Adenylate kinase (AdK) (PDB code: 4ake) has three domains, CORE, NMP, and LID. Topological connectivity of the chain is illustrated at the bottom.

Figure 7

Free-energy landscape and kinetics of DHFR folding calculated by the eWSME model. A) Free-energy landscape of DHFR folding represented in the two-dimensional space of M_DLD and M_ABD. The landscape has basins corresponding to the unfolded state U, the native state N, and the intermediates, I_A, I_B, and I_α. B–D) Evolution of the population of 200 molecules simulated with the Monte Carlo calculation at B) 3.3×10⁵ t₀, C) 1.6×10⁶ t₀, and D) 3.0×10⁶ t₀, where t₀ is a unit of time in simulation. Reproduced from [23].

Figure 8

Free-energy landscape and folding kinetics of the circular permutant of DHFR calculated by the eWSME model. A) Difference in the free-energy landscape between the wild type and the circular permutant of DHFR. B–D) Evolution of the population of 200 molecules simulated with the Monte Carlo calculation at B) 3.3×10⁵ t₀, C) 1.6×10⁶ t₀, and D) 3.0×10⁶ t₀, where t₀ is a unit of time in simulation. Reproduced from [23].

Figure 9

Allosteric transition of NtrC. Upon phosphorylation of Asp54, the NtrC structure switches from a state around the inactive (I) conformation (PDB code: 1dc7) to another state around the active (A) conformation (PDB code: 1dc8). Asp54 is shown with blue colored spheres. “3445 face” (the region comprises helices and strands, α3, β4, α4, and β5) is colored red. Reproduced from [31].

Figure 10

Free-energy landscape F_α(x, n) of allosteric transition of NtrC calculated with the aWSME model. x is the order parameter of allosteric transition and n is the order parameter of folding transition. (x, n)=(0, 1) is the I conformation, (1, 1) is the A conformation, and (0, 0) is the completely disordered state. A) F_dephos(x, n) in the dephosphorylated state and B) F_phos(x, n) in the phosphorylated state. C) and D) are closeups of A) and B), respectively, at n≈1. Contour is drawn for every 2k_BT. Reproduced from [31].

Figure 11

Pre-existing structural fluctuation of NtrC. (Top) The parameter ξ^A showing the extent of the A-like structure development in the dephosphoryated state. ξ^A calculated with the aWSME model under the constraint of each fixed χ and n=1 is plotted in gray scale. Even in conformations near the I conformation with small x, the A-like structure appears as a fluctuation around the 3445 face. (Bottom) R_ex observed in the relaxation measurement of NMR in the dephosphorylated state [61] are shown with red dots. R_ex is larger than a threshold for the blue dots [61]. Reproduced from [31].

Figure 12

An example of a non-sequential structure alignment. A) Structure of Q8ZRJ2 (PDB code: 2es9), B) structure of the eukaryotic clamp loader (PDB code: 1sxj), and C) the superimposition of Q8ZRJ2 and the eukaryotic clamp loader obtained by the non-sequential alignment program MICAN [70]. In A–C, the structurally equivalent regions are drawn with the same color. It can be clearly seen that all helices are well superimposed if both the chain direction and the connectivity are ignored. D and E are two-dimensional diagrams of protein topology of Q8ZRJ2 (A) and eukaryotic clamp loader (B), respectively. F) Correspondence relation of helices obtained by MICAN. Reproduced from [73] with permission.