Graph theoretical analysis of the energy landscape of model polymers

In systems characterized by a rough potential-energy landscape, local energetic minima and saddles define a network of metastable states whose topology strongly influences the dynamics. Changes in temperature, causing the merging and splitting of metastable states, have nontrivial effects on such networks and must be taken into account. We do this by means of a recently proposed renormalization procedure. This method is applied to analyze the topology of the network of metastable states for different polypeptidic sequences in a minimalistic polymer model. A smaller spectral dimension emerges as a hallmark of stability of the global energy minimum and highlights a nonobvious link between dynamic and thermodynamic properties.

Figure 1

(Color online) Sketch of temperature dependence of the partitioning in metastable states in a three-well potential model. At low temperature the saddle separating the minima A and B is sufficiently high to provide a separation of time scales. When thermal agitation increase, the explored regions around the minima of the energy landscape (shaded areas) become wider.

Figure 2

Coalescence of two nodes. If one of the energy barriers separating 1 from 2, $W_{12}$ or $W_{21}$, is smaller than the reference temperature, the two nodes coalesce together forming a new node $A$ that is connected to node 3 by a single energy barrier, which is $\text{min}\left(W_{13},W_{23}\right)$ to go from $A$ to 3, and $\text{min}\left(W_{31},W_{32}\right)$ to come back from 3 to $A$.

Figure 3

Specific heat (circles) and derivative of the gyration radius (squares) as a function of temperature for the hydrophobic homopolymer of length 12. To ease comparison both quantities are normalized to their maximum value.

Figure 4

Specific heat (circles) and derivative of the gyration radius (squares) as a function of temperature for the unstable heteropolymer. To ease comparison both quantities are normalized to their maximum value.

Figure 5

Specific heat (circles) and derivative of the gyration radius (squares) as a function of temperature for the stable heteropolymer. To ease comparison both quantities are normalized to maximum of their absolute value.

Figure 6

Histogram of the rescaled potential $\overline{V}$ of the minima of the potential energy for the homopolymers of lengths $N=10,11,12$. The rescaling factor is $N^{-2}$. The circles radius is proportional to $N$.

Figure 7

Number of minima $N_{\text{min}}$ (circles) and saddles $N_{\text{sad}}$ (squares) for increasing chain length $N$. In the inset the ratio $N_{\text{sad}}/N_{\text{min}}$ is reported together with a power law fit (exponent 1.07).

Figure 8

Histogram of the rescaled entropy $\overline{S}=S/N^{1.15}$ of the minima of the potential energy for the homopolymers of lengths $N=10,11,12$. The symbol size is proportional to $N$.

Figure 9

Histogram of energy barriers $W$ between connected minima for homopolymers with $N=10,11,12$. The symbol size increases with $N$. Inset: distribution of energy barriers for the unstable folder (squares), stable folder (circles), and homopolymers with $N=12$ (crosses).

Figure 10

Histograms of the multiplicities $m_i$ of each node of the renormalized graph for the $N=12$ homopolymer at temperatures $T=0.7,1.4,2.1$. Increasing temperatures are represented by lines of increasing thickness.

Figure 11

Histograms of the number $n_i$ of contacts of each node for the $N=12$ and $N=10$ (inset) homopolymers at temperatures $T=0.0$, 0.7, 1.4, 2.1 and $T=0.0,2.1$, respectively. Increasing temperatures are represented by lines of increasing thickness.

Figure 12

Rank-to-eigenvalue plot for the Laplacian matrix of the hydrophobic homopolymer of length 9 at increasing temperature: $T=0.0,0.7,1.4,2.1$. The circles radius is proportional to temperature. Dashed lines represent least-square power-law fits to the data.

Figure 13

Rank-to-eigenvalue plot for the Laplacian matrix of the hydrophobic homopolymer of length 12 for different random rewirings.

Figure 14

Rank-to-eigenvalue plot for the Laplacian matrix of the hydrophobic homopolymer of length 12 at increasing temperature: $T=0.0,0.7,1.4,2.1$. Larger symbols correspond to higher temperature. Rank data were multiplied for an arbitrary factor to ease reading. Dashed lines represent least-square power-law fits to the scale invariant portion of the curves.

Figure 15

Rank-to-eigenvalue plot for the Laplacian matrix of unstable folder (squares), stable folder (circles) and homopolymer with $N=12$ at $T=0$. Continuous lines represent least-square power-law fits to the data.