Geometry of the energy landscape and folding transition in a simple model of a protein

A geometric analysis of the global properties of the energy landscape of a minimalistic model of a polypeptide is presented, which is based on the relation between dynamical trajectories and geodesics of a suitable manifold, whose metric is completely determined by the potential energy. We consider different sequences, some with a definite proteinlike behavior, a unique native state and a folding transition, and others undergoing a hydrophobic collapse with no tendency to a unique native state. The global geometry of the energy landscape appears to contain relevant information on the behavior of the various sequences: in particular, the fluctuations of the curvature of the energy landscape, measured by means of numerical simulations, clearly mark the folding transition and allow the proteinlike sequences to be distinguished from the others.

Figure 1

(Color online) $V_{\text{dihedral}}$ vs dihedral angle ${\psi }_i$: the red (lower) curve corresponds to the case in which at least two out of the four beads defining the angle ${\psi }_i$ are neutral (N), the blue (upper) curve to the other cases; the central minimum corresponds to a planar configuration of the four monomers defining ${\psi }_i$.

Figure 2

(Color online) $⟨N_n⟩$ vs temperature $T$ for the two good folders $S_{\text{g}}^{22}$ (a), $S_{\text{g}}^{46}$ (b). The curves are a guide to the eye.

Figure 3

(Color online) Specific heat $c_V$ vs temperature $T$ for the homopolymer $S_{\text{h}}^{22}$ (a) and for the good folder $S_{\text{g}}^{22}$ (b). The curves are a guide to the eye.

Figure 4

(Color online) As in Fig. 3 for the homopolymer $S_{\text{h}}^{46}$ (a) and the good folder $S_{\text{g}}^{46}$ (b).

Figure 5

(Color online) Relative curvature fluctuation $\sigma$ vs temperature $T$ for the homopolymer $S_{\text{h}}^{22}$ (a) and for the good folder $S_{\text{g}}^{22}$ (b). The curves are a guide to the eye.

Figure 6

(Color online) As in Fig. 5 for the homopolymer $S_{\text{h}}^{46}$ and the proteinlike sequence $S_{\text{g}}^{46}$.

Figure 7

(Color online) $\sigma \left(T\right)$ for the sequence $S_{\text{b}}^{22}$ (a) and for the sequence $S_{\text{i}}^{22}$ (b). The curves are a guide to the eye.

Figure 8

$K_R\left(t\right)$ for the homopolymer $S_{\text{h}}^{22}$ at $T\approx T_{\theta }$ (a) and for the proteinlike sequence $S_{\text{g}}^{22}$ at $T\approx T_f$ (b).

Figure 9

$K_R\left(t\right)$ for the proteinlike sequence $S_{\text{g}}^{22}$ at $T\simeq 0.68$, i.e., at the temperature of the peak in the relative curvature fluctuations $\sigma \left(T\right)$.

Figure 10

(Color online) Normalized distributions of $K_R$ for the homopolymer $S_{\text{h}}^{22}$ at $T\approx T_{\theta }$ (broad histogram) and for the proteinlike sequence $S_{\text{g}}^{22}$ at $T\simeq 0.68$ (peaked histogram). The values on the vertical axis must be multiplied by ${10}^{-7}$. The skewness of the distributions is 0.6 for the homopolymer case and 1.4 for the proteinlike case.

Figure 11

(Color online) Average curvature (per degree of freedom) $⟨K_R⟩/N$ for the proteinlike sequence $S_{\text{g}}^{22}$ as a function of $T$.

Figure 12

(Color online) Synopsis of the instantaneous values of the curvature $K_R$ (upper curve) and of the number of native contacts $N_n$ (lower curve) for the proteinlike sequence $S_{\text{g}}^{22}$ in a simulation run at $T\simeq 0.68$.