The number of all possible conformations of a polypeptide chain is too large to be sampled exhaustively. Nevertheless, protein sequences do fold into unique native states in seconds (the Levinthal paradox). To determine how the Levinthal paradox is resolved, we use a lattice Monte Carlo model in which the global minimum (native state) is known. The necessary and sufficient condition for folding in this model is that the native state be a pronounced global minimum on the potential surface. This guarantees thermodynamic stability of the native state at a temperature where the chain does not get trapped in local minima. Folding starts by a rapid collapse from a random-coil state to a random semi-compact globule. It then proceeds by a slow, rate-determining search through the semi-compact states to find a transition state from which the chain folds rapidly to the native state. The elements of the folding mechanism that lead to the resolution of the Levinthal paradox are the reduced number of conformations that need to be searched in the semi-compact globule (approximately 10(10) versus approximately 10(16) for the random coil) and the existence of many (approximately 10(3)) transition states. The results have evolutionary implications and suggest principles for the folding of real proteins.

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...

The entropic component of the free energy of assembly for multiparticle hydrogen-bonded aggregates is analyzed using a model based on balls connected by rigid rods or flexible strings. The entropy of assembly, ΔS, is partitioned into translational, rotational, vibrational, and conformational components. While previously reported theoretical treatments of rotational and vibrational entropies for assembly are adequate, treatments of translational entropy in solution and of conformational entropyoften the two largest components of ΔSare not. This paper provides improved estimates and illustrates the methods used to obtain them. First, a model is described for translational entropy of molecules in solution (ΔS trans(sol)); this model provides physically intuitive corrections for values of ΔS trans(sol) that are based on the Sackur−Tetrode equation. This model is combined with one for rotational entropy to estimate the difference in entropy of assembly between a 4-particle aggregate and a 6-particle one. Second, an approximate analysis of a model based on balls connected by rods or strings gives an approximate estimate of the maximum contribution of conformational entropy to the difference in free energy of assembly of flexible and of rigid molecular assemblies. This analysis, although approximate, is easily applied by all types of chemists and biochemists; it serves as a guide to the design of stable molecular aggregates, and the qualitative arguments apply generally to any form of self-assembly.

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Classical population genetics a priori assigns fitness to alleles without considering molecular or functional properties of proteins that these alleles encode. Here we study population dynamics in a model where fitness can be inferred from physical properties of proteins under a physiological assumption that loss of stability of any protein encoded by an essential gene confers a lethal phenotype. Accumulation of mutations in organisms containing ⌫ genes can then be represented as diffusion within the ⌫-dimensional hypercube with adsorbing boundaries determined, in each dimension, by loss of a protein's stability and, at higher stability, by lack of protein sequences. Solving the diffusion equation whose parameters are derived from the data on point mutations in proteins, we determine a universal distribution of protein stabilities, in agreement with existing data. The theory provides a fundamental relation between mutation rate, maximal genome size, and thermodynamic response of proteins to point mutations. It establishes a universal speed limit on rate of molecular evolution by predicting that populations go extinct (via lethal mutagenesis) when mutation rate exceeds approximately six mutations per essential part of genome per replication for mesophilic organisms and one to two mutations per genome per replication for thermophilic ones. Several RNA viruses function close to the evolutionary speed limit, whereas error correction mechanisms used by DNA viruses and nonmutant strains of bacteria featuring various genome lengths and mutation rates have brought these organisms universally Ϸ1,000-fold below the natural speed limit. biological evolution ͉ genome sizes ͉ lethal mutagenesis ͉ mutation load ͉ population genetics

The algorithm along with the model of interresidue interactions can serve as a tool for studying the thermodynamics and kinetics of protein models.