We have studied the folding mechanism of lattice model 36-mer proteins. Using a simulated annealing procedure in sequence space, we have designed sequences to have sufficiently low energy in a given target conformation, which plays the role of the native structure in our study. The sequence design algorithm generated sequences for which the native structures is a pronounced global energy minimum. Then, designed sequences were subjected to lattice Monte Carlo simulations of folding. In each run, starting from a random coil conformation, the chain reached its native structure, which is indicative that the model proteins solve the Levinthal paradox. The folding mechanism involved nucleation growth. Formation of a specific nucleus, which is a particular pattern of contacts, is shown to be a necessary and sufficient condition for subsequent rapid folding to the native state. The nucleus represents a transition state of folding to the molten globule conformation. The search for the nucleus is a rate-limiting step of folding and corresponds to overcoming the major free energy barrier. We also observed a folding pathway that is the approach to the native state after nucleus formation; this stage takes about 1% of the simulation time. The nucleus is a spatially localized substructure of the native state having 8 out of 40 native contacts. However, monomers belonging to the nucleus are scattered along the sequence, so that several nucleus contacts are long-range while other are short-range. A folding nucleus was also found in a longer chain 80-mer, where it also constituted 20% of the native structure. The possible mechanism of folding of designed proteins, as well as the experimental implications of this study is discussed.

The statistical mechanics of protein folding implies that the best-folding proteins are those that have the native conformation as a pronounced energy minimum. We show that this can be obtained by proper selection of protein sequences and suggest a simple practical way to find these sequences. The statistical mechanics of these proteins with opimized native structure is discussed. These concepts are tested with a simple lattice model of a protein with full enumeration ofcompact conformations. Selected sequences are shown to have a native state that is very stable and kinetically accessible.How and why proteins fold to their native structure is an intriguing unsolved problem in molecular biophysics. From the theoretical side there are two different approaches to this problem (1-10).The first approach-initiated by Taketomi and Go (1) and then, with several significant modifications, continued by several groups (2-5)-was to define some special model with certain biases to the known native state and to investigate its folding properties from both thermodynamic (1, 2, 4) and kinetic (3,5,11) perspectives. The basic feature of these models is "ultraspecificity," which means the introduction of some special force fields biasing the polypeptide chain to the native state. Investigation of such models provided many interesting insights and allowed analysis of the sufficient conditions for folding.Another approach is the investigation of the necessary conditions for folding using simple, completely unbiased protein models. Such an unbiased model is a random heteropolymer. Statistical mechanics of random heteropolymers has been considered (refs. 7, 8, and 12; Studies of heteropolymers have revealed that their energy spectrum (i.e., set of conformations and their energies) consists of two parts: the "continuous" part, to which the majority of random conformations belong, and the discrete part, representing a few conformations with best-fit contacts. In the continuous part, energy levels are highly populated (so that an exponentially large number of conformations belongs to each such level). More important, this part is selfaveraging; i.e., its features do not depend on specific realization of a sequence but rather on gross properties of the sequence ensemble (such as composition). However, the bottom part of the spectrum is very sequence-sensitive, so that different sequences deliver significantly different energies to their native conformations. Then, as temperatureThe publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.decreases the system undergoes a freezing folding transition passing from the continuous to the discrete part of the spectrum (a detailed discussion of the analytical theory of heteropolymers is in refs. 7 and 8; a less mathematical review is in ref. 13).Monte Carlo simulations of the kinetics of folding of heteropolymers in a model with fully enum...

Exhaustive enumeration of all compact self-avoiding conformations of a chain of 27 monomers on the 3*3*3 fragment of a simple cubic lattice is given. Total number of conformations unrelated by symmetry is 103 346. This number is relatively small which makes it possible to make a numerically exact calculation of all thermodynamic functions this chain. Heteropolymers with random sequence of links were considered, and the freezing transition at finite temperature was observed. This transition is analogous to folding transition in proteins where unique structure is formed. The numeric results demonstrate the equivalence between random 3-dimensional heteropolymers and the random energy model found previously in analytical investigations. The possible application of these results to some problems of combinational optimization is discussed.

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Folding of protein-like heteropolymers into unique 3D structures is investigated using Monte Carlo simulations on a cubic lattice. We found that folding time of chains of length N scales as N λ at temperature of fastest folding. For chains with random sequences of monomers λ ≈ 6, and for chains with sequences designed to provide a pronounced minimum of energy to their ground state conformation λ ≈ 4. Folding at low temperatures exhibits an Arrheniuslike behavior with the energy barrier E b ≈ φ|E n |, where E n is the energy of the native conformation. φ ≈ 0.18 both for random and designed sequences.