Equilibrium properties of realistic random heteropolymers and their relevance for globular and naturally unfolded proteins

Random heteropolymers do not display the typical equilibrium properties of globular proteins, but are the starting point to understand the physics of proteins and, in particular, to describe their non-native states. So far, they have been studied only with mean-field models in the thermodynamic limit, or with computer simulations of very small chains on lattice. After describing a self-adjusting parallel-tempering technique to sample efficiently the low-energy states of frustrated systems without the need of tuning the system-dependent parameters of the algorithm, we apply it to random heteropolymers moving in continuous space. We show that if the mean interaction between monomers is negative, the usual description through the random energy model is nearly correct, provided that it is extended to account for non-compact conformations. If the mean interaction is positive, such a simple description breaks out and the system behaves in a way more similar to Ising spin glasses. The former case is a model for the denatured state of glob- ular proteins, the latter of naturally-unfolded proteins, whose equilibrium properties thus result qualitatively different.

💡 Research Summary

The paper investigates the equilibrium properties of random heteropolymers in continuous three‑dimensional space, aiming to bridge the gap between idealized mean‑field models and realistic protein physics. Traditional studies have relied on the Random Energy Model (REM) applied to lattice polymers or on mean‑field replica calculations in the thermodynamic limit, both of which neglect the conformational richness of real proteins. To overcome these limitations, the authors first develop a self‑adjusting adaptive simulated tempering (AST) algorithm that automatically determines an optimal set of temperatures and weight factors without any system‑specific tuning. The algorithm starts with a high‑temperature Monte Carlo run, extracts the density of states via a multiple‑histogram method, and iteratively inserts new temperatures so that the Metropolis transition rates match a preset value. It also optimizes the temperature ladder by maximizing the product of forward and backward transition probabilities, ensuring efficient diffusion across the temperature space. This approach enables reliable sampling down to temperatures as low as 0.1 (in reduced units) for chains of length N = 20–60, with a typical runtime of about ten hours on a single CPU.

The polymer model consists of an inextensible chain of beads spaced 3.8 Å apart. Each bead carries one of 20 “amino‑acid” types, and pairwise interactions are defined by a 20 × 20 Gaussian random matrix Bαβ with mean μ and unit variance. Interactions are implemented as a spherical‑well potential of width R = 5.5 Å and hard‑core radius R₀ = 4 Å. By varying μ (−1, −0.5, 0, +1) the authors generate four families of interaction matrices, each sampled over 20 independent realizations, and study chains of length N = 20, 25, 30, and 60.

Key observables include the structural overlap q(α,β), which measures the fraction of common contacts between two conformations, and the number of contacts n_c. The distribution p(q) and the Binder cumulant B(T) are used to characterize replica symmetry breaking (RSB) patterns.

Two distinct thermodynamic regimes emerge:

1. Negative mean interaction (μ < 0).

   • p(q) displays two well‑separated peaks at q ≈ 1 (identical structures) and q ≈ 0 (completely different structures). As temperature decreases, the high‑q peak shrinks while the low‑q peak grows, a hallmark of one‑step RSB.
   • The Binder parameter shows a modest bump reminiscent of the 3‑spin glass transition.
   • The ground‑state energy per monomer ε_c = E_c/N is essentially independent of chain length in the studied range, but its standard deviation remains a few percent of the mean, indicating significant sequence‑to‑sequence variability.
   • ε_c correlates strongly with the contact density z = 2 n_c/N, revealing that fluctuations in compactness drive energy variability.
   • The entropy S(E) deviates from the parabolic form predicted by the REM because the number of contacts is not fixed; instead S(E) is asymmetric and reflects a mixture of compact and coil‑like states.
   • This behavior aligns with the traditional picture of a protein’s denatured state: a broad ensemble of partially compact conformations that can be described by an extended REM that accounts for variable compactness.

2. Positive mean interaction (μ > 0).

   • p(q) exhibits a broad high‑q peak that never reaches q = 1 and a pronounced peak at q = 0, indicating a spectrum of partially overlapping structures. The high‑q peak shifts to lower q values as N increases, suggesting that no single dominant native‑like basin exists.
   • The Binder cumulant decreases monotonically with temperature, closely resembling the Sherrington‑Kirkpatrick model’s full RSB behavior.
   • Energy fluctuations are less correlated with contact density, and low‑energy states are not necessarily maximally compact.
   • This regime mirrors the physics of intrinsically disordered proteins (IDPs), which lack a unique folded structure and instead sample a rugged landscape with many nearly degenerate minima.

The authors also discuss evolutionary implications. Because ε_c varies appreciably with the random interaction matrix, natural selection must design sequences whose native energy E_N is sufficiently lower than the typical ε_c to ensure robustness against mutations and environmental changes. Simply increasing chain length to gain stability would also increase ε_c variability, making it an inefficient evolutionary strategy.

In summary, the study demonstrates that random heteropolymers with negative average interactions conform reasonably to an extended REM, providing a realistic model for the denatured ensemble of globular proteins. Conversely, polymers with positive average interactions behave like Ising spin glasses with full replica symmetry breaking, offering a plausible statistical‑mechanical framework for intrinsically disordered proteins. The adaptive simulated tempering method introduced here enables thorough exploration of these frustrated systems and could be applied to more detailed protein models in future work.