Swollen-Collapsed Transition in Random Hetero-Polymers

A lattice model of a hetero-polymer with random hydrophilic-hydrophobic charges interacting with the solvent is introduced, whose continnuum counterpart has been proposed by T. Garel, L. Leibler and H. Orland {J. Phys. II France 4, 2139 (1994)]. The transfer matrix technique is used to study various constrained annealed systems which approximate at various degrees of accuracy the original quenched model. For highly hydrophobic chains an ordinary $\theta$-point transition is found from a high temperature swollen phase to a low temperature compact phase. Depending on the type of constrained averages, at very low temperatures a swollen phase or a coexistence between compact and swollen phases are found. The results are carefully compared with the corresponding ones obtained in the continuum limit, and various improvements in the original calculations are discussed.