Equilibrium characterization of the folded phase of the IMP random heteropolymer: genericity of few stable states and low-temperature equilibration

The IMP model of Iori, Marinari and Parisi (1991-1993) is a random heteropolymer — a bead chain in continuous three-dimensional space whose Lennard-Jones-type pair interactions carry quenched Gaussian random components. At strong disorder it exhibits a 'folded phase' with a non-trivial distribution $P(\delta)$ of the roto-translation-minimized distance between chain conformations, extremely long correlation times, and apparent dynamics among only a few stable states, observed at chain length $N\approx30$ on the APE-100 supercomputer. The source paper left the equilibrium characterization of this phase incomplete. The problem is to complete it: (1) compute the disorder-averaged distance distribution $\overline{P(\delta)}$ over a much larger statistical sample than the original 128 disorder realizations and decide whether the few-stable-states phenomenology is generic to random heteropolymers or a feature of some disorder realizations; (2) establish whether the non-trivial $P(\delta)$ is a genuine equilibrium property deeper in the folded phase (below $T\approx1.5$, where correlation times of order $3\times10^7$ Monte Carlo sweeps defeated the original runs). Adjacent analytic and computational work on random heteropolymers indicates glassy many-state structure, and tempering-based sampling methods are now available; however, the specific equilibrium computations for the IMP model itself have not been reported, so the open problem is model-specific.

Background

Protein folding poses a statistical-mechanics puzzle: a linear amino-acid chain reproducibly reaches one or a few native three-dimensional structures out of an astronomically large conformational space. As a reference point, physicists from the late 1980s studied random heteropolymers — polymer chains whose monomer-monomer interactions contain quenched disorder, i.e. random interaction parameters drawn once per chain and then frozen, in the same sense as the random couplings of a spin glass (a disordered magnetic system with a rugged free-energy landscape and many nearly degenerate metastable states). Mean-field analyses of such chains (Bryngelson and Wolynes 1987; Garel and Orland 1988; Sfatos, Gutin and Shakhnovich 1993) predicted a frustrated, glassy low-temperature landscape with many metastable states, often approximated by the random energy model (REM), the idealized limit in which the total energies of distinct conformations behave as independent random variables. Real proteins fold to few states, so the open question was whether generic quenched disorder alone can produce a phase with only a few stable conformations.

The IMP model of Iori, Marinari and Parisi ('Random self-interacting chains: a mechanism for protein folding', 1991; 'Statistical mechanics of heteropolymer folding', 1992) is a minimal continuous-space realization. A chain of $N$ sites in three dimensions carries pair energies

$$E_{i,j}=\delta_{i,j+1}\,r_{i,j}^{2}+\frac{R}{r_{i,j}^{12}}-\frac{A}{r_{i,j}^{6}}+\frac{\eta_{i,j}}{r_{i,j}^{6}},$$

with total Hamiltonian $H=\sum_{i<j}E_{i,j}$; here $r_{i,j}$ is the Euclidean distance between sites $i$ and $j$, the Kronecker-delta harmonic term binds nearest neighbors along the chain, $R/r^{12}-A/r^{6}$ is a Lennard-Jones potential (the standard short-range repulsion-plus-attraction pair potential of simple fluids), and the quenched variables $\eta_{i,j}=\eta_{j,i}$ are Gaussian with zero mean and variance $\epsilon$, encoding random sequence-specific interactions; $\epsilon$ measures the disorder strength.

A homopolymer ($\epsilon=0$) shows the standard coil-globule transition: on cooling, the extended random-walk coil collapses into a compact globule that has no definite shape. To distinguish mere compactness from shape freezing, the authors introduced shape-sensitive distances between two conformations $\alpha,\beta$ of the same chain: the roto-translation-minimized positional distance $\delta^{2}_{(\alpha,\beta)}=(1/N)\sum_{i}|x^{(\alpha)}_{i}-x^{(\beta)}_{i}|^{2}$ (minimized over rigid rotations and translations) and an energy-based analogue $\Delta^{2}$ built from pair-energy differences. In Monte Carlo simulations (one sweep = one attempted update per chain degree of freedom), the distribution $P(\delta)$ of the positional distance is trivial — concentrated at the values typical of unrelated or identical configurations — in the coil and globule phases. At sufficiently strong disorder ($\epsilon\gtrsim\epsilon_c$, where $\epsilon_c$ denotes the critical disorder strength above which the folded phase sets in), the model instead enters a new folded phase: $P(\delta)$ becomes non-trivial, relaxation after perturbation is a stretched exponential, correlation times grow enormous, and the chain repeatedly returns to one or a few stable shapes (at least two distinct recurrent states were observed), in contrast to the proliferation of unrelated minima in spin glasses.

In the APE-100 study (Iori, Marinari and Parisi 1993, the source of this problem), simulations run on the APE-100 — a massively parallel array-processor supercomputer developed at the Italian INFN for lattice field-theory calculations — assigned one disorder realization to each processor and evolved two independent copies per realization. The disorder average $\overline{P(\delta)}$ — the average of $P(\delta)$ over independent realizations of the random $\eta_{i,j}$ — was estimated in two ways: a history estimator (distances between configurations of the same copy at different times) and an equal-time estimator (distances between the two independent copies at the same time); agreement between the two is a standard necessary test of thermalization. At $N=30$ with 128 disorder realizations: at temperatures $T=2.5$ and $T=2.0$ the system equilibrated but $\overline{P(\delta)}$ was trivial; at $T=1.0$ equilibrium was not reached; at $T=1.5$ a non-trivial equilibrium $\overline{P(\delta)}$ was obtained, but with an estimated correlation time of order $3\times10^7$ sweeps the runs barely exceeded it, and no error bars over realizations were reported. The equilibrium characterization this study left incomplete, which defines this problem, is: whether the few-stable-states observation is generic or realization-specific (requiring $\overline{P(\delta)}$ on a much larger statistical sample), and whether the non-trivial $P(\delta)$ deeper in the folded phase is an equilibrium property rather than a long-lived transient. The companion study Pliszka and Marinari (EPL 22, 167, 1993) documented the dramatic critical slowing down, noted that large-scale simulations would be needed to get quantitative precise results, and observed that the new tempering approach 'has chances to alleviate the situation'. Tempering (simulated tempering, Marinari and Parisi 1992) is a Monte Carlo scheme in which the temperature itself is treated as a stochastic variable that moves through a preset set of values with suitably chosen weights, letting the system heat up temporarily to escape kinetic traps; its replica-parallel variant is parallel tempering (Hansmann 1997). In the modern theory of random heteropolymers, the low-temperature state structure is described by replica symmetry breaking (RSB) — the pattern of ergodicity breaking in disordered systems, where one-step RSB means a discrete set of widely separated thermodynamic states and full RSB a continuous hierarchy of states nested at all scales.

Problem Statement

For the IMP random heteropolymer — the continuous-space chain with Hamiltonian $H=\sum_{i<j}E_{i,j}$, $E_{i,j}=\delta_{i,j+1}r_{i,j}^{2}+R/r_{i,j}^{12}-A/r_{i,j}^{6}+\eta_{i,j}/r_{i,j}^{6}$, with quenched Gaussian $\eta_{i,j}$ of zero mean and variance $\epsilon$, in the folded-phase regime identified by Iori, Marinari and Parisi (chain length $N\approx30$ and larger, disorder strength $\epsilon\gtrsim\epsilon_c$, the critical strength above which the folded phase appears; the companion studies used $A=3.8$, $R=2.0$, $\epsilon=6$) — complete the equilibrium characterization of the folded phase by determining:

1. (Genericity of few stable states) whether the few-stable-states phenomenology — a non-trivial disorder-averaged distance distribution $\overline{P(\delta)}$ with recurrent stable states — is a generic property of the random heteropolymer or only a feature of some disorder realizations. Here $\delta$ is the roto-translation-minimized mean-square positional distance between two conformations of the same chain, $P(\delta)$ its distribution within one disorder realization, and the overbar the average over realizations of the quenched $\eta_{i,j}$. This requires a reliable computation of $\overline{P(\delta)}$ at the folded-phase temperature $T=1.5$ of the source's non-trivial equilibrium result, over a statistical sample much larger than the 128 disorder realizations of the 1993 study, with the history-based and equal-time estimators of $P(\delta)$ compared — distances within a single copy at different times versus distances between the two independent copies at the same time — and with sample-to-sample fluctuations quantified. The phenomenology counts as generic when it is present across essentially all realizations, and not only for a fraction of them.

2. (Equilibrium versus transient) whether the non-trivial $P(\delta)$ is a genuine equilibrium property deeper in the folded phase — at temperatures below $T\approx1.5$, where the estimated correlation time of order $3\times10^7$ Monte Carlo sweeps prevented equilibration in the original runs.

These two determinations jointly answer one question: whether the folded phase of the IMP model is a genuine thermodynamic phase with few equilibrium states — which would distinguish random heteropolymers from spin-glass-like many-state systems and support disorder as a mechanism for protein-like folding — or a finite-sample and kinetic artifact of glassy dynamics.

References

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