Equilibrium and Nonequilibrium Effects in the Collapse of a Model Polypeptide

We present results of molecular simulations of a model polypeptide whose hydrophobic collapse proceeds as a cascade of downhill transitions between distinct intermediate states. Different intermediates are stabilized by means of appropriate harmonic constraints, allowing explicit calculation of the equilibrium free energy landscape. Nonequilibrium collapse trajectories are simulated independently and compared to diffusion on the calculated free energy surface. We find that collapse generally adheres to this surface, but quantitative agreement is complicated by nonequilibrium effects and by dependence of the diffusion coefficient on position on the surface.

Figure 1

(a) Polymer conformations corresponding to different states. H monomers are gray (dark), $P$ monomers are green (light). Numbers in brackets indicate monomers comprising compact globules. (b) Free energies $F$ and $F^{*}$ for the impeding force $f_0=0$ (solid line) and $f_0=0.02\epsilon /\sigma$ (dashed line), $0.04\epsilon /\sigma$ (dash-dotted line). $F$, $F^{*}$ are given in units of $k_{B}T$ and $R_G$ in units of $\sigma$. The dash-dotted line is shifted up by $40k_{B}T$.

Figure 2

(a) Free energy surface for polymer collapse from state 2. Two clearly visible trenches belong to two distinct pathways connecting states 2 and 0. (b) Contour plot of the same free energy surface together with three typical collapse trajectories from state 2. $R_{\mathrm{c}.\mathrm{m}.}$ and $R_{G1}$ are introduced in the main text and shown in units of $\sigma$.

Figure 3

(a) Collapse times $t_{\mathrm{eq}}$ (squares) and $t_{\mathrm{neq}}$ (circles) from different initial positions in the $1\to 0$ trench. The solid curve is the fit of $t_{\mathrm{eq}}$ with $D_1=6.4\times {10}^{-5}$, describing lateral vibrations of collapse trajectories in the trench, and $D_2$, shown as dashed curve in (b). Triangles (right axis) are $t_{\mathrm{neq}}/t_{\mathrm{eq}}$. In (b), the solid curve is the planar cut $R_{G1}=2\sigma$ of the 2D free energy surface. The units are $k_{B}T$ for $F$, $\sigma$ for $R_{\mathrm{c}.\mathrm{m}.}$, and $k_{B}T{\gamma }^{-1}$ for $D_1$, $D_2$.