Graph’s Topology and Free Energy of a Spin Model on the Graph

In this Letter we investigate a direct relationship between a graph’s topology and the free energy of a spin system on the graph. We develop a method of separating topological and energetic contributions to the free energy, and find that considering the topology is sufficient to qualitatively compare the free energies of different graph systems at high temperature, even when the energetics are not fully known. This method was applied to the metal lattice system with defects, and we found that it partially explains why point defects are more stable than high-dimensional defects. Given the energetics, we can even quantitatively compare free energies of different graph structures via a closed form of linear graph contributions. The closed form is applied to predict the sequence-space free energy of lattice proteins, which is a key factor determining the designability of a protein structure.

Figure 1

(a),(b),(c) Examples of three-link multigraphs. (a) is a parent graph of (c) [i.e., (c) is obtained by a node contraction operation on (a)], and (b) is also a parent graph of (c), but there is no such relationship between (a) and (b).

Figure 2

(a),(b) Two different types of defects, drawn by vesta 3 [18]. Broken bonds are marked in red. (a) Diamond structure with a grain boundary indicated by a red plane. (b) Same structure with a vacancy defect. (c) Difference between Monte Carlo (MC) free energies of two Si-Ge alloy lattice systems with different types of defects, as a function of inverse temperature. Here, the $y$ axis shows $\mathrm{\Delta }F=F(\text{vacancy defect})-F(\text{grain boundary})$ for the entire system.

Figure 3

(a),(b) Examples of subgraphs (black) contributing to the term [$g$] where $g$ is a three-link path graph [Fig. 1] on the given graph architecture (gray). Panel (b) shows a child graph of a three-link path graph [Fig. 1].

Figure 4

Scatter plots of sequence-space free-energy (SSFE) distributions for 10 000 lattice protein structures, comparing exact and predicted SSFEs at $\beta J=0.1$. Equations (16) (red) and (17) (blue) were used to calculate predicted SSFEs. (Upper left) A cartoon of a prototypical lattice protein; (lower right) zoom of the boxed region.