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

TL;DR

This study introduces a simplified protein model to investigate the ultrametric structure of its energy landscape, revealing hierarchical organization consistent with spin glass theories and supporting the hypothesis of protein ultrametricity.

Contribution

It presents a novel computational approach combining a simplified heteropolymer model with GPU-accelerated simulations to demonstrate ultrametricity in protein-like energy landscapes.

Findings

90% of sequences exhibit ultrametric triangles in their energy landscape

97.8% of sequences show nontrivial ultrametricity

Repeated experiments confirm the robustness of ultrametricity observations

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.