A toy model of a protein prototype reveals nontrivial ultrametricity of the energy landscape

Abstract

A model for studying the ultrametricity of the energy landscape in a disordered heteropolymer is presented. It is treated as a simplified model of a protein molecule in which amino acid residues are modeled as point masses. Pairwise interactions include universal repulsion, the Lennard-Jones potential, the Coulomb potential with screening, and the elastic potential for bonds between adjacent residues. An analogy with spin glass models is used, allowing the application of replica theory methods. Unlike the standard approach to disordered systems, averaging over disorder is not performed. The overlap between replicas is defined as the Pearson correlation coefficient between the vectors of average pairwise energies, which corresponds to a comparison of thermodynamic averages in the spirit of spin glass theory. The results of a computational experiment conducted using the developed algorithm on a graphics processing unit (GPU) are presented. The simulations were performed using a 128-residue-long sequence, with 50 independent disorder realizations and 50 replicas for each sequence at a temperature of T = 1.0. It was found that for 90.0% of the sequences, the distance matrix between replicas contains more than half of the ultrametric triangles, and nontrivial ultrametricity predominates in 97.8% of them, indicating a hierarchical organization of the energy landscape. A repeated computational experiment for selected sequences confirms the reliability of the observations: 95.5% of them again demonstrated ultrametricity, of which 97.7% showed a predominance of the nontrivial type of ultrametricity. The obtained results confirm Frauenfelder's hypothesis of protein ultrametricity and pave the way for a systematic study of ultrametric properties in more realistic protein models.

Figures (3)

• Figure 1: Histogram of the distribution of fractions of ultrametric triangles over the ensemble of 50 random sequences (first stage). Columns outlined with a solid line show the distribution for the total fraction of ultrametric triangles $f_{\text{ultra}}$. Columns outlined with a dashed line with gray fill show the distribution for the fraction of nontrivial ultrametric triangles $f_{\text{nontriv}}$. The peak of the $f_{\text{nontriv}}$ distribution occurs in the region $0.45-0.50$, indicating the predominance of nontrivial hierarchical organization.
• Figure 2: Correlation between the Edwards--Anderson parameter $q_{\text{EA}}$ and the fraction of nontrivial ultrametric triangles $f_{\text{nontriv}}$ for 50 sequences (first stage). Filled black circles denote sequences for which the total fraction of ultrametric triangles exceeds 0.5 (ultrametric), empty squares denote the remaining (non-ultrametric) ones. A non-monotonic structure is observed: maximum values of $f_{\text{nontriv}}$ are achieved at $q_{\text{EA}}\approx0.25-0.30$, indicating the existence of an optimal level of frustration for the formation of a hierarchical landscape. At higher values of $q_{\text{EA}}$ (> 0.35), ultrametricity is typically not detected.
• Figure 3: Dendrogram of hierarchical clustering of 50 replicas for the sequence with the maximum fraction of nontrivial ultrametric triangles at the second stage (sequence with $f_{\text{nontriv}}=51.02\%$, $f_{\text{ultra}}=76.37\%$). The distance matrix is defined as $D=1-q$, where $q$ is the overlap between replicas. The average linkage method was used in constructing the dendrogram. An asymmetric structure is clearly visible: in the left part of the dendrogram, the merging levels of clusters ($D\sim0.7-0.9$) are significantly higher than in the right part ($D\sim0.3-0.5$). This corresponds to a hierarchy of macrostates: large clusters separated by high barriers (left part) contain internal subclusters corresponding to closer substates (right part). Such a structure is a characteristic feature of ultrametric organization of state space.