Entropic droplets and activated events near the glass transition of a random heteropolymer

The barriers between metastable states near the glass transition of a random heteropolymer are studied using replicas by describing inhomogeneous states. The instanton solution for a replica space free energy functional is found numerically to estimate the size of activation barriers and of the critical nuclei themselves between the dynamic and the static glass transition temperatures. Typeset using REVTEX 1 Quantifying escape from configurational traps on a rugged free energy landscape is important for understanding the dynamics of spin glasses [1], structural glasses [2], and folded proteins [3], and for protein folding [4]. A first step towards understanding barriers is to appreciate the organization of the stable minima [5]. This has only been carried out completely for infinite range spin glasses. That organizational structure inspires many of the dynamical theories [4,6–9]. Barriers in mean field spin glasses scale with the system size, but the finite range of interactions allows escape from traps through localized reconfigurations with finite barriers in the thermodynamic limit. The Vogel-Fulcher law for viscosity of structural glasses has been explained through such a mechanism [10]. Various mean field theories of structural glasses resemble those for spin glasses lacking reflection symmetry. In strict mean field theory these models undergo a dynamical transition at a temperature TA, where a macroscopic number of frozen free energy minima appear, and a static transition at TK, the Kauzmann temperature, where the configurational entropy of the minima disappears. Kirkpatrick and Wolynes (KW) pointed out that that individual free energy minima between TA and TK will be inherently unstable for short range interaction models because of entropic droplets. The extensive configurational entropy provides a driving force for a localized region in a local minimum to reconfigure and escape the trap. Their analysis gave a modified Vogel-Fulcher law while a later scaling picture incorporating entropic droplets gave precisely the usual form used empirically [11]. Parisi has presented a novel instanton argument in replica space yielding the original KW form [12]. Here, we use replica instanton calculations to quantify reconfiguration barriers for the random heteropolymer. Reconfiguration barriers determine the configurational diffusion coefficients that enter the theory of protein folding times [4,13]. At low T , trap escape is rate limiting. Whether the escape barrier is extensive can be tested experimentally in a crude fashion since proteins are mesoscopic with finite chain length N . The size scaling is presently controversial [13–15]. The barriers computed in the strict mean field limit of uniform transitions are of order ∼ 0.1NkBTK at the static transition TK using parameters fit to the lattice model thermodynamics [9]. This seems to be consistent with recent simulation results at low tem2 perature [14]. Our calculations suggest entropic droplets are often large in the random heteropolymer so that the mean field arguments are a good starting point for mesoscopic systems of the size of the smaller proteins. One caveat is that polymers have additional entanglement constraints not present in structural or spin glasses. These are neglected here. We focus on the temperature range near the glass-like transition of a finite-size heteropolymer in a poor solvent so that it is collapsed. The droplet we deal with is a small globular region of the polymer that may take on many configurations, local in space but not necessarily in sequence, buried in a remaining frozen glassy portion. We utilize replica formalism and derive a Landau-like free energy functional in terms of the Debye-Waller factor for a residue which plays the role of an Edwards-Anderson order parameter which is taken as spatially varying. We introduce the standard bead contact Hamiltonian for a random heteropolymer which includes finite range random interactions between monomers;