INTRODUCTION

It has been known for some time that for many proteins the information necessary to specify the native structure is contained within the amino acid sequence. There has been a tremendous amount of research aimed at deciphering this code and determining the final structure from the sequence. Solving this problem is of paramount importance; however, simply knowing how to map sequences to structures would leave many interesting questions unanswered. How do proteins fold to their native structure and, more specifically, how do they manage to fold so quickly? What are the key factors that determine whether or not a given sequence will fold and what the folding time will be? One may argue that it might be necessary to solve these problems before it will be possible to solve the folding problem (i.e., predicting structure from sequence).

A great deal of work (both experimental and theoretical) has been done on the kinetics of protein folding. One extremely useful theoretical technique is to study simple heteropolymer models. The idea is to reduce the complex system of proteins in solution to its bare essentials, leaving only the key features. The advantage of studying these simpler models is that an in-depth analysis (sometimes even an exhaustive one) can be performed, yielding detailed answers and information. This information should, in turn, provide insights into real proteins.

One class of model that is often used in theoretical polymer work is the lattice model, where the monomers are constrained to lie on lattice sites. Excluded volume is included by allowing only one monomer per site. To study dynamics, the Monte Carlo algorithm with a variety of move sets is used. Some of the earliest work using lattice models on proteins was done by Go and others[1, 2] using two- and three-dimensional lattices to examine the folding process. However, the interaction potential they used was somewhat unusual. The native state was explicitly built into the potential. The energy of any given conformation was determined by counting the number of native contacts, i.e., contacts found in the native structure. An attractive contribution to the energy was added for each native contact formed. This potential is somewhat unphysical, depending on an a priori knowledge of the native structure. Although much of this early work on lattice models was on simple cubic lattices, Skolnick and others[3, 4, 5, 6, 7] have used more complex lattices which are able to more faithfully represent the structure of actual proteins. Using these lattices they are able to model real protein structures (e.g., secondary structure) and study the dynamics of folding and the formation of these structures. However, with increasing complexity it becomes more difficult to study these models in great detail.

Rather than trying to model real proteins exactly, some have opted for simpler models which permit a more thorough analysis. Chan and Dill[8, 9, 10, 11] have used a two-dimensional simple cubic lattice model with two monomer types (a polar monomer, P, and a hydrophobic one, H). The potential used models the hydrophobic interaction and is equal to times the number of hydrophobic contacts (HH). They studied short chains, which allowed them to do exhaustive enumeration to measure a variety of properties (both static and dynamic). For dynamics they used both Monte Carlo[9] and transfer matrix methods.[10, 11] By using short polymers, they were able to construct the full transfer matrix (this matrix determines the probability of one state transforming to another) and use it to solve exactly for the dynamics of the system. Although their model is simpler than an actual protein, it has yielded a wealth of interesting information and provided valuable insight into proteins and heteropolymers. Their models show a two-phase process similar to that found in proteins. There is a rapid collapse to compact states, followed by slower reconfiguring of the chains to the native structure. Fiebig and Dill [12] show that simple searching strategies, such as the formation of opportunistic hydrophobic contacts, can lead to the globally optimal conformation (native state), suggesting a possible mechanism for folding. Shakhnovich and others[13, 14] have studied the folding of random heteropolymers (the interaction between monomers is picked from a random distribution) on the three-dimensional simple cube lattice. They examined 27 monomer polymers using Monte Carlo dynamics and also found a two-stage collapse process in folding. They found that by examining an overlap function, which measures how low-energy conformations differ, they could distinguish the difference between foldable and not foldable sequences. From examination of many different sequences, they conclude that the existence of a pronounced energy gap between the native state and the remaining conformations distinguishes good folding sequences.[15] To examine how the specific form of the interaction affects the dynamics of folding, Camacho and Thirumalai[16] looked at two-dimensional lattice systems. They studied the kinetics of three different types of interaction potentials. They found two transition temperatures: A collapse temperature at which the chain forms a compact structure and a folding temperature at which the native structure is formed. They found three stages in the transition from open coil to native structure.

In this work we will continue using the three-dimensional simple cubic lattice model. The polymers will be 27 monomers long and consist of two monomer types. Monte Carlo dynamics will be used to study the collapse and folding kinetics. The chains are too long for exhaustive enumeration of all conformations but are short enough to permit exhaustive enumeration of all maximally compact configurations. This information will be used to determine the minimum energy structure (native state) which will allow us to measure the folding time from extended conformations. We will examine several different sequences and measure collapse and folding time as a function of temperature and sequence. One question to be addressed is which kinetic quantities are sequence dependent and which are sequence independent ( self-averaging). In addition, we will examine how the glass transition affects the ability of a sequence to fold. A major goal is to define, as precisely as possible, various physically important quantities. Of particular importance will be the determination of the important time scales. One problem with Monte Carlo dynamic simulations is the relation between Monte Carlo steps and physical time. There is no simple connection; in fact, the precise relation may depend on the move set.[10, 11] To circumvent this problem, we will relate Monte Carlo steps to physical time by looking for the natural time scales in the problem, such as the collapse and the folding time. Using these time scales, we will then be able to define the glass transition temperature ( ) of this model. In the past others have speculated that the relation between the folding temperature ( ) and the glass temperature ( ) would play an important role in protein folding. Bryngelson and Wolynes[17, 18] have proposed that in order for a chain to fold, the folding transition must occur before the glass transition of the system, and the optimal folding temperature would lie between and . Specifically, Wolynes and others state that to optimize folding potentials for structure prediction, one should maximize the ratio of the folding temperature to the glass temperature ( ).[19, 20] To calculate the glass transition, they used a random energy modellike assumption; i.e., for each given value of the degree of folding, the energies of the different conformation are independent random variables. In our work we will give a direct kinetic definition of the glass temperature that does not rely on this assumption, and show explicitly that the relative values of and will determine the folding properties of a given sequence.