Structural origin of slow diffusion in protein folding

Abstract

Experimental, theoretical, and computational studies of small proteins suggest that interresidue contacts not present in the folded structure play little or no role in the self-assembly mechanism. Non-native contacts can, however, influence folding kinetics by introducing additional local minima that slow diffusion over the global free-energy barrier between folded and unfolded states. Here, we combine single-molecule fluorescence with all-atom molecular dynamics simulations to discover the structural origin for the slow diffusion that markedly decreases the folding rate for a designed α-helical protein. Our experimental determination of transition path times and our analysis of the simulations point to non-native salt bridges between helices as the source, which provides a quantitative glimpse of how specific intramolecular interactions influence protein folding rates by altering dynamics and not activation free energies.

Fig. 1.. One-dimensional free energy surface description of kinetics and dynamics of protein folding.

Protein folding dynamics are most often described by diffusion on a free energy surface obtained from a one-dimensional (1D) projection of a multidimensional free energy landscape. The folding transition path of a single molecule in an equilibrium trajectory is one that crosses q₀ on the reaction coordinate and reaches q₁ on the other side of the free energy barrier without recrossing q₀. The transition path appears as a jump in the experimental property being monitored in a single molecule trajectory. The folding rate coefficient, k_f, is the inverse of the average waiting (residence) time spent exploring the configurations of the unfolded well from which numerous unsuccessful attempts are made at crossing the free energy barrier.

Fig. 2.. α₃D folding transition path at neutral pH from MD simulation (13).

Exemplar structures from the unfolded state, the folded state, and the transition path are shown. The side chains of charged residues are represented in sticks and colored in red if they are involved in a non-native salt bridge, or in blue if the salt bridge is native. All other side chains are omitted for the sake of clarity. The transition path is defined as the region of the free energy surface (black) enclosed between the two dashed lines that separates the unfolded from the folded basin. The average number of native (blue) and non-native (red) salt bridges is also reported as a function of an optimized reaction coordinate. Salt bridges are determined using the stable-state picture (24) with cutoff distances of 4.5 and 8.0 Å between the Cγ of Asp or the Cδ of Glu and the Nζ of Lys or the Cζ of Arg. Salt bridges are considered native if they are within the longest cutoff distance in the experimentally determined structure . Note that due to the large number of highly flexible charged side chains present, ~2 non-native salt bridges can be transiently formed, on average, even in the folded state.

Fig. 3.. pH and viscosity dependence of α₃D folding kinetics close to the chemical denaturant midpoint.

(A) FRET efficiency histograms as a function of pH. The concentrations of urea are 5 M at pH 7.5 and 6 M at all other pHs (Table S2). The FRET efficiency histograms were constructed from 1 ms bins in the trajectories with the mean photon count rate > 50 ms⁻¹. The near identity of the measured histograms (wide bars) and the histograms constructed from re-colored photon trajectories (red narrow bars), using the parameters obtained from the maximum likelihood method with the two-state model, indicate that this model is adequate (10, 26). (B) pH dependence of relaxation rate, k. (C) FRET efficiency histograms at different relative solvent viscosities. (D) Viscosity dependence of the inverse of the relaxation rate at pH 7.6 (Table S1) and (E) at pH 3.6 under conditions where the stability is only slightly altered (Table S3). The data was fitted to the power-law function A(η/η₀)^α (orange) or a linear equation A(σ + η/ η₀) (green) (27), where A is an amplitude and σ is the internal viscosity, which reflects the internal friction. The blue dashed lines show the dependence expected when the folding times are proportional to the first power of the solvent viscosity.

Fig. 4.. Analysis of single-molecule photon trajectories.

The difference of the log likelihood, Δln L = ln L(t_TP) – ln L(0), is plotted as a function of the lifetime of the virtual intermediate state (= t_TP), for folding and unfolding transitions. L(0) is the likelihood for a model with instantaneous folding and unfolding transitions. Therefore, Δln L quantifies how much better or worse a model with a finite transition path time describes the data than a model with an instantaneous transition (Fig. S4). The transition path time is determined from the maximum of the likelihood function above the upper 95% confidence limit (Δln L = +3, red dashed line), which is 13.3 (± 1.8) μs at pH 5.3. The error is the standard deviation obtained from the curvature at the maximum of the likelihood function. The upper bound of transition path time is determined by the time when the likelihood curve crosses the 95% confidence limit (Δln L = −3, red dashed line) (30).

Fig. 5.. Trajectories, free energy surfaces, and diffusion coefficients from MD simulations of reversible folding of α₃D.

(A) Time series of the Cα RMSD from the experimental native structure for a 360-μs simulation of α₃D performed at 370 K and neutral pH. (B) One-dimensional projection of the folding free energy surface along an optimized reaction coordinate (30). The cutoffs used for determining rates and transition path times in the analysis to define the folded and unfolded states are indicated by two vertical lines. The different simulations are distinguished using the color scheme described in panel C. (C) Diffusion coefficients along the optimized reaction coordinate for neutral-pH (black) and low-pH (red) simulations performed at 370 K. Results are also reported for simulations performed with weak (orange) or strong (blue) salt bridges and for a 370-K, neutral-pH simulation performed at low viscosity (black dashed line). (D) Time series of the Cα RMSD for a 65-μs simulation of α₃D performed at 370 K and low pH. (E) One-dimensional projection of the folding free energy surface for low-pH simulations performed at 350 K and either normal viscosity (black solid line) or low viscosity (black dashed line). (F) Diffusion coefficients (350 K) along the optimized reaction coordinate. The different simulations are distinguished using the same scheme employed in panel E.

Fig. 6.. Comparison of the effect of pH and viscosity on folding and transition path times from experiments and simulations and the average number and persistence time of non-native interactions during the transition path.

(A) The ratio of the folding and transition path times in simulation (yellow bars) were calculated from the values in Table S6. The experimental values (light blue bars) were calculated as follows. 1: The ratio of the relaxation times at pH 3.6 (low) and pH 5.3 (neutral) at 295 K; 2: The ratio of the folding time of 6 μs at pH 2 and 346 K(21) and the folding time of 10 μs at pH 7.5 obtained from the GdmCl dependence (23); 3: The transition path time at pH 3.6 (low) is an upper bound; 4: The transition path time at pH 3.6 was assumed to be 1 μs at 295 K, and it was extrapolated to 370 K using the viscosity dependence of t_TP ∝ η^0.7 obtained from the viscosity dependence of the kinetics (Fig. 3E). The transition path time at pH 7.5 and 370 K was calculated by extrapolation using the super-Arrhenius temperature dependence (D ∝ exp[-ε²/ (kT)²]) with the roughness (ε²) of 2.3 kT (T = 295 K) and t_TP = 12.3 μs at 295 K (8). The experimental viscosity dependence was calculated using the power law dependence (η^α) with the powers (α) of 0.24 (t_f, neutral pH, Fig. 3D), 0.30 (t_TP, neutral pH) (8), and 0.70 (t_f, low pH, Fig. 3E). See Table S6 for absolute times and additional details of the simulations. (B) Lowering pH significantly reduces the average number and persistence time of non-native salt bridges between positively charged side chains (Arg, Lys) and negatively charged side chains (Glu, Asp) and the non-native hydrophobic interactions during the transition path.

Fig. 7.. Viscosity dependence of persistence times for a number of ion pairs.

The persistence time is reported on a logarithmic plot as a function of viscosity relative to TIP3P for the Na…Cl ion pair (black), the guanidinum…acetate ion pair (red), the butyl-guanidinum…butyric acid ion pair (green, representing the side chains of Arg and Glu), a butyl-guanidinum…butyric acid ion pair where the terminal carbon atoms have been restrained at a distance of 10 Å (blue), and the Arg…Glu/Asp salt bridges in the α₃D transition path (purple). A power-law fit with exponents ranging between 0.51 and 0.56 is reported in solid lines, while the exponent of 1.0 predicted by hydrodynamic Kramers’ theory is shown as a dashed line.