SynopsisA three-dimensional lattice model of protein designed to assimilate lysozyme is introduced. An attractive interaction is assumed to work between preassigned specific pairs of units, when they occupy the nearest-neighbor lattice points. The behavior of this lattice lysozyme is studied by a Monte Carlo simulation method. Because of the specific interunit interactions, "native state" of the lattice lysozyme is stable at low temperatures. Conformational fluctuations in the native state are observed to occur a t both termini and loop regions of the main chain existing on the surface. The process of unfolding and the denatured states of this model are discussed. Complete refolding from a denatured state was not observed. However, by starting from partially folded structures, the native conformation could be attained. From these observations it is concluded that, in the process of folding of proteins as simplified in a lattice model, nucleation is a rate-limiting factor. The artificial character of this model and possible improvement are discussed.

A lattice model of proteins is introduced. “A protein molecule” is a chain of non‐intersecting units of a given length on the two‐dimensional square lattice. The copolymeric character of protein molecules is incorporated into the model in the form of specificities of inter‐unit interactions. This model proved most effective for studying the statistical mechanical characteristics of protein folding, unfolding and fluctuations. The specificities of inter‐unit interactions are shown to be the primary factors responsible for the all‐or‐none type transition from native to denatured states of globular proteins. The model has been studied by the Monte Carlo method of Metropolis et al., which is now shown applied to approximately simulating a kinetic process. In the strong limit of the specificity of the inter‐unit interaction the native conformation was reached in this method by starting from an extended conformation. The possible generalization and application of this method for finding the native conformation of proteins from their amino acid sequence are discussed.

ABSTRACr A lattice model of protein is studied by a Monte Carlo simulation method. The native conformation of the lattice protein molecule is stabilized by specific long-range and short-range interactions. By comparing results of simulation for different relative weights of the long-and short-range interactions, it is concluded that the specific long-range interactions are essential for highly cooperative stabilization of the native conformation and that the short-range interactions accelerate the folding and unfolding transitions.The importance of both the short-range and long-range interactions in protein folding has long been recognized. The importance of the short-range interactions was inferred first by the fair success of predicting secondary structures in the native structure of proteins from their amino acid sequences (1). The importance of the long-range interactions can be deduced from various facts, among which we will cite the following two. (i) Large protein fragments do not conserve the same conformations they possess in the native structure of an intact (uncleaved) protein when isolated from their complementing fragments (2), or the probability of assuming such a conformation is very low (3). This means that, for maintaining the native structure of a protein, an indispensable role is played by interfragment interactions, most of which are long range. (ii) Denaturational transitions in globular proteins take place in a more-or-less all-or-none manner. Even when the existence of intermediate states is discussed, the transition is certainly not of such a diffuse type as observed in the helix-coil transition of homopolypeptides. If the long-range interactions could be neglected and only the short-range interactions were assumed to be operative, then a protein could be regarded essentially as a one-dimensional system. Transitions in any one-dimensional systems are inevitably of a diffuse type (4). Therefore, the more-or-less all-ornone character of denaturational transitions in globular proteins means that the long-range interactions play an essential role in the transitions (5).Because both the short-range and long-range interactions have been shown to be important, it is necessary to understand the respective roles of these two types of interactions in protein folding. A powerful method is to study this problem in terms of a simplified theoretical model introduced by focusing only on this point. For this purpose we incorporate short-range as well as long-range interactions into the lattice model of protein that we previously studied (6). We will describe in the present paper results obtained in the two-dimensional square lattice. A "two-dimensional protein" is admittedly a very idealized model. However, the results obtained in this paper regarding the respective roles of the short-range and long-range interactions are expected to hold in real three-dimensional proteins. LATTICE PROTEINWe first consider the two-dimensional square lattice in a computer. A "protein molecule" is a self-avoiding chain pol...

SynopsisF'rotein-folding and -unfolding transitions were studied by the method of computer simulation. The protein was modeled as a two-dimensional lattice polymer. Various energy terms were assumed to be operative between units composing the polymer. But hydrophobic interactions were neglected explicitly. Both thermodynamic and kinetic quantities were obtained from the simulation, and from their temperature dependence in the transition zone characteristics of the conformational transition of proteins were discussed. Two amino acid substituted models, differing in the location of substitution, were studied and compared with the original in order to clarify the effect of substitution on conformational transition of proteins. The following conclusions were reached in this study: (1) The relaxation time of the slow mode, which reflects the overall folding and unfolding processes, shows a peak near the transition temperature, while that of the fast mode is almost independent of temperature. The peak of the slow mode occurs at a slightly lower temperature than the transition temperature. (2) The dependence of the logarithm of the rate constants on the inverse of temperature (Arrhenius plot) is linear. Therefore, the plot of the free energy of activation vs temperature is linear. (3) The values of kinetic parameters obtained suggest that in the activated state the intramolecular interactions are half broken, while the state is close to the native state on the entropy axis. (4) The amino acid substitution, which is modeled as having slightly unfavorable short-range interactions, causes the substituted ones to be slightly unstable. Moreover, it causes the folding transition to slow. From the analysis of the way slowing down is observed in the two substituted models, we conclude that a structure, designed to model a /3-sheet, is formed before it gets assembled with other structures, which are designed to model a-helices. The process of assembly occurs nearly at the activated state of the folding and unfolding transition. (5) It is suggested from this study that the maximum of folding rate constant in the Arrhenius plot that has been observed experimentally in real proteins is likely due to hydrophobic interactions.

The theoretical model of proteins on the two‐dimensional square lattice, introduced previously, is extended to include the hydrophobic interactions. Two proteins, whose native conformations have different folded patterns, are studied. Units in the protein chains are classified into polar units and nonpolar units. If there is a vacant lattice point next to a nonpolar unit, it is interpreted as being occupied by solvent water and the entropy of the system is assumed to decrease by a certain amount. Besides these hydrophobic free energies, the specific longrange interactions studied in previous papers are assumed to be operative in a protein chain. Equilibrium properties of the folding and unfolding transitions of the two proteins are found to be similar, even though one of them was predicted, based on the one globule model of the transitions, to unfold through a significant intermediate state (or at least to show a tendency toward such a behavior), when the hydrophobic interactions are strongly weighted. The failure of this prediction led to the development of a more refined model of transitions; a non‐interacting local structure model. The hydrophobic interactions assumed here have a character of non‐specific long‐range interactions. Because of this character the hydrophobic interactions have the effect of decelerating the folding kinetics. The deceleration effect is less pronounced in one of the two proteins, whose native conformation is stabilized by many pairs of medium‐range interactions. It is therefore inferred that the medium‐range interactions have the power to cope with the decelerating effect of the non‐specific hydrophobic interactions.