Frozen phases with re-entrant transition for random heteropolymers with composition specific and annealed cross-links

Abstract

A field theoretic formalism of random heteropolymers (RHPs) with composition specific cross-links that are annealed, is developed in this work. This system can serve as a simple meaningful model of proteins with disulfide bonds. The agent that cross-links the macromolecules, as in real proteins, is assumed to maintain an equilibrium cross-linking density within the heteropolymer globule; this situation is reminiscent to in vivo reagent induced cross-linking, and protrudes the experimental desired control on formation and dissolution of cross-links. The starting point of our analysis, i.e. the random heteropolymer Hamiltonian, captures both the chain connectivity and the essentially quenched nature of amino-acids distribution reminiscent of linear uncross-linked polypeptides. The conformational statistics of the RHP is determined within the replica approach. For experimentally realizable values of the RHP interaction parameters, we predict the appearance of two frozen phases of RHPs wherein a small number of chain conformations of order O(1) are being sampled. At the interim between the two phases our calculation predicts the appearance of a re-entrant transition wherein the number of chain conformations sampled is of order O(N). The occurrence of the re-entrant transition is due to a tight competition between the intersegment interactions (measured by χF), and the propensity of composition specific and annealed cross-links to form (determined by the chemical potential of the cross-linking agent). We suggest how to traverse the re-entrant transition line, by chemically manipulating the competition of dissimilar interactions of the disparate segments, and the specific propensity of the peptides to cross-link, and also, we provide an explanation for the observation that in nature cross-linking between alike rather than dissimilar segments is observed.