Typeset Using Revt E X

The Langevin dynamics of a random heteropolymer and its dynamic glass transition are studied using elementary mode coupling theory. Contrary to recent reports using a similar framework, a discontinuous ergodic-nonergodic phase transition is predicted for all Rouse modes at a finite temperature TA. For sufficiently long chains, TA is almost independent of chain length and is in good agreement with the value previously estimated by a static replica theory. Typeset using REVTEX 1 The self-organization of evolved biopolymers such as foldable proteins ultimately depends upon the chain dynamics of heteropolymers. The modern statistical mechanical view using energy landscape ideas focuses on the analogy of folding with phase transitions in finite size systems and exploits to a large extent our understanding of the quasi-thermodynamic and static features of both regular systems and frustrated disordered systems such as spin glasses [1]. The connection with detailed analytical theories of the time dependence of fluctuational motions of a randomly interacting chain has only recently received attention [2–5]. For completely random heteropolymers (RHP) the quasi-thermodynamic analyses [6,7] using replica techniques give results parallel to those for other systems with random first order phase transitions such as the mean field Potts spin glasses [8]. An important feature of theories of random first order transitions is the presence of two transitions; one is static, while the other is dynamic and generally occurs at a higher temperature. The dynamical transition signals a crossover to a motional mechanism involving activated motions [8,9] (We denote this dynamical transition temperature TA). For RHP’s two different kinds of approximate quasi-static theories based on replicas [6,7] and on the generalized random energy model [10] that takes into account correlation in the energy landscape yield these two transitions. Replica based techniques also yield estimates for the free energy barrier heights of activated motions between the two characteristic temperatures [6,7]. Purely dynamical theories based on mode coupling ideas generally yield a transition in harmony with the quasi-static analysis of the dynamical transition which is in some sense a spinodal. While mode coupling theory (MCT) is perturbatively well defined for spin systems with long range random interactions [11,9], there are various versions developed in the theory of fluids [12] and polymers [13] that are forced to make uncontrolled approximations. On the basis of two such calculations, the validity of the emerging picture of the dynamics and the energy landscape of RHP’s have been questioned [4]. Roan and Shakhnovich derive a mode coupling theory of the RHP and explicitly solve it for a polymer in a good solvent [4]. It is no surprise that an uncollapsed chain has no dynamical transition, but the authors further claim that this is actually a structural fea2 ture of the RHP mode coupling equations independent of the state of collapse. Thirumalai et al. [5] derive another set of mode coupling equations for a somewhat different model that exhibits no static replica symmetry breaking [14] and conclude there is a dynamical transition with a transition temperature that depends on the length scale of the motional mode considered [5]. A numerical treatment of self-consistent equations for heteropolymer collapse, on the other hand, does show evidence for a dynamic freezing transition [3]. The inconsistency of these results with each other and in the first two cases with replica calculations is disturbing. The technical intricacy of these mode coupling calculations is a barrier to understanding their inconsistencies. We have therefore derived mode coupling equations using ‘elementary’ methods like those used decades ago in the theory of critical phenomena [12]. The resulting equations differ in some respects from those of the earlier workers but clearly yield transitions in harmony with the quasi-static results. These equations also lead to an understanding of how the dynamical freezing depends on chain length and state of collapse as well as how the dynamics varies for different modes of chain motion. We consider a standard Hamiltonian for N interacting beads