Solvable lattice gas models of random hetero-polymers at finite density: I. Statics

Abstract

We introduce ∞-dimensional lattice gas versions of three common models of random heteropolymers, in which both the polymer density and the density of the polymer-solvent mixture are finite. These solvable models give valuable insight into the problems related to the (quenched) average over the randomness in statistical mechanical models of proteins, without having to deal with the hard geometrical constraints occurring in finite-dimensional models. Our exact solution, which is specific to the ∞-dimensional case, is compared to the results obtained by a saddle-point analysis and by the grand ensemble approach, both of which can also be applied to models of finite dimension. We find, somewhat surprisingly, that the saddle-point analysis can lead to qualitatively incorrect results.PACS. 61.41.+e Polymers, elastomers, and plastics -75.10.Nr Spin-glass and other random models

Solvable lattice gas models of random heteropolymers at finite density: II. Dynamics and transitions to compact states

“…In the latter approach the limit n c → ∞ poses no problems (although the method would not apply to models with bond disorder, in contrast to that of [1]). Working out our equations for models which just a single monomer type shows that for n c → ∞ a hydrophilic model will not undergo phase transitions at any temperature (as expected), but that the hydrophobic system will be in a collapsed state, at any finite temperature.…”

“…In statics such models have been solved using replica theory, see [1,[3][4][5][6]8,9], and it is to be anticipated that the method of path integrals [13] would have to be used for solving the dynamics, as in e.g. [7].…”

“…arrangement with n − ∈ {0, 2}. Hence the observed evolution of the system towards a state where only c (0,0) , c (0,2) and c (1,0) are nonzero.…”

“…The movements of the latter are controlled only by energetic considerations, by the maximum number of monomers allowed to occupy any given site, and by trivial restrictions on the number of moves allowed to occur simultaneously. In a recent paper [1], which was devoted to the equilibrium solution of a particular infinite dimensional model, this property was shown to enable an exact solution, in spite of the presence of quenched disorder. (viz.…”

“…The present paper is concerned with the further analysis and exploration of the model introduced in [1], to which we refer for a more detailed and extensive discussion of the relevant literature and of the motivation underlying the study of random heteropolymers in general (e.g. their role as precursors of models for proteins), and lattice gas models of random heteropolymers in particular (see also e.g.…”