Stochastic dynamics of model proteins on a directed graph

A method for reconstructing the potential energy landscape of simple polypeptidic chains is described. We show how to obtain a faithful representation of the energy landscape in terms of a suitable directed graph. Topological and dynamical indicators of the graph are shown to yield an effective estimate of the time scales associated with both folding and equilibration processes. This conclusion is drawn by comparing molecular dynamics simulations at constant temperature with the dynamics on the graph, defined as a temperature-dependent Markov process. The main advantage of the graph representation is that its dynamics can be naturally renormalized by collecting nodes into “hubs” while redefining their connectivity. We show that the dynamical properties at large time scales are preserved by the renormalization procedure. Moreover, we obtain clear indications that the heteropolymers exhibit common topological properties, at variance with the homopolymer, whose peculiar graph structure stems from its spatial homogeneity. In order to distinguish between “fast” and “slow” folders, one has to look at the kinetic properties of the corresponding directed graphs. In particular, we find that the average time needed to the fast folder for reaching its native configuration is two orders of magnitude smaller than its equilibration time while for the bad folder these time scales are comparable.

Figure 1

Folding probability $P_f$ estimated from Eq. (11) as a function of temperature $T$. The full dotted-dashed and dashed lines correspond to S0, S2, and S1, respectively.

Figure 2

(a) Number of nodes $N$ and (b) average connectivity $\overline{\sigma }$ of the renormalized graph versus temperature $T$ for S0 (empty circles), S1 (crosses), and S2 (filled squares). In (a) the dashed lines are exponential fits of the data, due to the adopted log-linear representation. The full lines in (b) are drawn to guide the eyes.

Figure 3

Sequence S0: the fraction of nodes with connectivity $\sigma$, $\nu \left(\sigma \right)$, in log-log scale at $T=0$ (full line), $T=0.04$ (dashed line), and $T=0.08$ (dotted-dashed line).

Figure 4

Sequence S1: the fraction of nodes with connectivity $\sigma$, $\nu \left(\sigma \right)$, in log-log scale at $T=0$ (full line), $T=0.04$ (dashed line), and $T=0.08$ (dotted-dashed line).

Figure 5

Sequence S2: the fraction of nodes with connectivity $\sigma$, $\nu \left(\sigma \right)$, in log-log scale at $T=0$ (full line), $T=0.04$ (dashed line), and $T=0.08$ (dotted-dashed line).

Figure 6

Log-log plot of the spectrum of eigenvalues of the topological Laplacian matrix $W_0$ of the zero-temperature directed graph of S0 (empty circles), S1 (crosses), and S2 (filled squares). The dashed lines refer to power law with exponents of 3.3.

Figure 7

Log-log plot of the spectrum of eigenvalues of the topological Laplacian matrix $W_0$ of S1 for five different temperatures: $T=0.00$ (○), $0.02(+)$, 0.04 (◻), and 0.06 (△), $0.08(\times )$. The data obtained for different temperatures are shifted horizontally by an arbitrary constant factor in order to obtain a better view. Notice that the power-law fit passes from its maximum value of 3.4 at $T=0$ to a minimum value of 2.5 above $T=0.02$.

Figure 8

The eigenvalues $r_k$, for $k=2,3,4$ versus temperature $T$ in log-reciprocal scale for (a) S1 and (b) S2. The dashed lines are Arrhenius fits to the data.

Figure 9

Components of $w^{\left(1\right)}$ and absolute value of the components of $w^{\left(2\right)}$, $w^{\left(3\right)}$, and $w^{\left(4\right)}$ for the slow-folder S2 at $T=0.06$.

Figure 10

Components of $w^{\left(1\right)}$ and absolute value of the components of $w^{\left(2\right)}$, $w^{\left(3\right)}$, and $w^{\left(4\right)}$ for the fast-folder S1 at $T=0.04$.

Figure 11

Absolute value of the components of the normalized eigenvectors $\overline{w^{\left(2\right)}}$, $\overline{w^{\left(3\right)}}$, and $\overline{w^{\left(4\right)}}$ for the fast-folder S1 at $T=0.04$.