Universal and Accessible Entropy Estimation Using a Compression Algorithm

Entropy and free-energy estimation are key in thermodynamic characterization of simulated systems ranging from spin models through polymers, colloids, protein structure, and drug design. Current techniques suffer from being model specific, requiring abundant computation resources and simulation at conditions far from the studied realization. Here, we present a universal scheme to calculate entropy using lossless-compression algorithms and validate it on simulated systems of increasing complexity. Our results show accurate entropy values compared to benchmark calculations while being computationally effective. In molecular-dynamics simulations of protein folding, we exhibit unmatched detection capability of the folded states by measuring previously undetectable entropy fluctuations along the simulation timeline. Such entropy evaluation opens a new window onto the dynamics of complex systems and allows efficient free-energy calculations.

Figure 1

Schematic asymptotic entropy calculation. Simulations of physical systems are preprocessed and encoded into data files [22]. Entropy is directly calculated from the size of the compressed ($C_d$) and calibration ($C_0$, $C_1$) data, as well as the entropy range ($S_{\min }$, $S_{\max }$).

Figure 2

Validation to asymptotic entropy calculation to Monte Carlo simulation data on benchmark model systems, by comparing analytical entropy calculation (lines) and compression algorithm method (symbols). (a) Discrete energy levels; Ising models on (b) square lattice, and (c) triangular lattice, either with antiferromagnetic ($J<0$, triangles) or ferromagnetic ($J>0$, circles); (d) Ideal chains held at varying end-to-end distance ($R$). (e) and (f) Entropy derivatives of the Ising model simulations [(b) and (c), respectively].

Figure 3

(a) Residuals from analytical entropy for various 1D reductions of the Ising model on a square lattice: row by row (squares), spiral (triangles), Hilbert (circles), Hilbert with 1 site/byte (pentagons), Hilbert with 3 sites/byte (diamonds). (b) and (c) Asymptotic entropy convergence for the Ising model on a square lattice. (b) Varying sample size ($N$) and fixed sampling interval ($140L^2$). (c) Varying sampling interval with fixed sample size ($10^4$), exhibiting exponential convergence (solid line). (d) Effect of coarse-graining level on $S_A$ for the ideal chain. Lines are a guide for the eye.

Figure 4

(a) Representative scan through the Villin headpiece simulation timeline, with mean-subtracted $S_A$ (blue bars), overlaying regions identified as folded using transition-based assignment (gray area). (b) Enthalpy–entropy population diagram at $T=360\mathrm{K}$. (c) Protein structure collected from randomly sampled low $S_A$ states simulated at 360 K (red) overlaid with the protein fragment (2F4K) crystal structure (blue) [7, 35], generated with pymol [36]. (d) Distributions of sliding-window $S_A$, at three simulation temperatures.