A statistical mechanical model for beta-hairpin kinetics

Abstract

Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here, we describe a simple statistical mechanical model for the formation of a beta-hairpin, the minimal structural element of the antiparallel beta-pleated sheet. The model accurately describes the thermodynamic and kinetic behavior of a 16-residue, beta-hairpin-forming peptide, successfully explaining its two-state behavior and apparent negative activation energy for folding. The model classifies structures according to their backbone conformation, defined by 15 pairs of dihedral angles, and is further simplified by considering only the 120 structures with contiguous stretches of native pairs of backbone dihedral angles. This single sequence approximation is tested by comparison with a more complete model that includes the 2(15) possible conformations and 15 x 2(15) possible kinetic transitions. Finally, we use the model to predict the equilibrium unfolding curves and kinetics for several variants of the beta-hairpin peptide.

Figure 1

Chemical, structural, and schematic representations of the β-hairpin. The sequence corresponds to the C-terminal fragment containing residues 41–56 of protein G B1 (28). Dashed lines indicate hydrogen bonds or hydrophobic interactions.

Figure 2

Choice of dihedral angle pairs for motion in elementary kinetic steps.

Figure 3

Comparison of thermal unfolding curve for the β-hairpin predicted by standard single sequence (121 species) and complete (32,768 species) models. The fractional population of molecules containing the intact hydrophobic cluster is plotted vs. temperature. The points are derived from a two-state analysis of the fluorescence equilibrium curves. The dashed curve is the fit to the data using the standard single sequence (121 state model) partition function (Eqs. 1 and 2). The continuous curve is predicted by the 2¹⁵-state partition function using the parameters from the fit with the standard single sequence model (ΔS_conf = −3.09 cal mol⁻¹ K⁻¹, ΔH_hb = −0.86 kcal mol⁻¹, ΔG_sc = −2.19 kcal mol⁻¹).

Figure 4

Comparison of thermal unfolding curves and kinetics for modified single sequence and complete models. (a) Fractional population of the hydrophobic cluster as a function of temperature. Derived from a two-state analysis of fluorescence equilibrium curves (large dots). Fit to the data with the modified single sequence model (Eqs. 3 and 4), producing the parameters ΔS_conf = −2.74 cal mol⁻¹ K⁻¹, ΔH_hb = −0.96 kcal mol⁻¹, ΔG_sc = −1.94 kcal mol⁻¹ (dashed line). Calculated with the complete model using these parameters (continuous line). Fraction of native hydrogen bonds calculated using the model with modified single sequence approximation (dotted line). (b) Simulations of progress curves for the complete model (continuous line) and the model using the modified single sequence approximation (dotted line). The fractional population of the hydrophobic cluster vs. time is plotted following a temperature jump from 283 to 298 K. The dashed lines are single exponential fits to the simulated progress curves at times >10 ns, the resolution of the T-jump instrument. The fits of the modified single sequence model to the kinetic data were performed using the lsoda routine (36), which incorporates algorithms for solving both stiff and nonstiff systems of equations. The resulting parameters were k₀ = 8.0 × 10⁸ s⁻¹ and E₀ = 0 (equilibrium parameters same as in a. The equilibrium and kinetic parameters are slightly different from those reported by Muñoz et al. (27) for two reasons. One is that in the previous work the viscosity dependence was not included in the preexponential factor, and the second is that in the present work the kinetic and equilibrium data were fit simultaneously, whereas in the previous analysis (27), the equilibrium data were fit independently. (c) Arrhenius plot of relaxation times following 15 degree temperature jumps. The points are the experimental relaxation rates, whereas the dashed curve through the points is obtained from the fit to the data using the modified single sequence model. The continuous curve is obtained from single exponential fits to the kinetic progress curves generated by the complete 2¹⁵-state model using the kinetic parameters from the modified single sequence model.

Figure 5

Prediction of equilibrium and kinetic properties of β-hairpins with additional interactions. Red, original hairpin; blue, hairpin with interaction in the β-turn (residues D47–K50); and green, hairpin with interaction between end residues (residues R41–E56). (a) Free energy profiles (not including kinetic barriers). (b) Population of the hydrophobic cluster (continuous lines) and fraction of hydrogen bonds (dotted lines) for the three hairpins. (c) Arrhenius plot of the kinetics of the three hairpins. Relaxation rates (continuous lines), folding rates (dotted lines), and unfolding rates (dashed-dotted lines). The folding and unfolding rates have been calculated from a two-state fit to the relaxation rates and equilibrium constants generated with the modified single sequence model.

Figure 6

Prediction of equilibrium and kinetic properties of β-hairpins with repositioned hydrophobic cluster. Red, original hairpin; Blue, hairpin with hydrophobic cluster moved one residue closer to the β-turn; and green, hairpin with hydrophobic cluster moved one residue closer to the ends. (a) Free energy profiles. (b) Population of the hydrophobic cluster (continuous lines) and fraction of hydrogen bonds (dotted lines) for the three hairpins. (c) Arrhenius plot of the kinetics of the three hairpins. Relaxation rates (solid lines), folding rates (dotted lines), and unfolding rates (dashed-dotted lines). The folding and unfolding rates have been calculated as in Fig. 5.