Statistical mechanical treatment of protein conformation. II. A three-state model for specific-sequence copolymers of amino acids.

A one-dimensional three-state Ising model [involving alpha-helical (alpha), extended (epsilon), and coil (or other) (c) states] for specific-sequence copolymers of amino acids ahs been formulated in order to treat the conformational states of proteins. This model involves four parameters (wh,iota, vh, iota, v episilon, iota, and uc, iota), and requires a 4 X 4 matrix for generating statistical weights. Some problems in applying this model to a specific-sequence copolymer of amino acids are discussed. A nearest-neighbor approximation for treating this three-state model is also formulated; it requires a 3 X 3 matrix, in which the same four parameters appear, but (as with the 4 X 4 matrix treatment) only three parameters (wh, uh, and v epsilon) are required if relative statistical weights are used. The relationship between the present three-state model (3 X 3 matrix treatment) and models of the helix--coil transition is discussed. Then, the three-state model (3 X 3 matrix treatment) is incorporated into an earlier (Tanaka--Scheraga) model of the helix-coil transition, in which asymmetric nucleation of helical sequences is taken into account. A method for calculating molecular averages and conformational-sequence probabilities, P(iota/eta/(rho)), i.e., the probability of finding a sequence of eta residues in a specific conformational state (rho), starting at the iotath position of the chain, is described. Two alternative methods for calculating P(iota/eta/(rho)), that can be applied to a model involving any number of states, are proposed and presented; one is the direct matrix-multiplication method, and the other uses a first-order a priori probability and a conditional probability. In this paper, these calculations are performed with the nearest-neighbor model, and without the feature of asymmetric nucleation. Finally, it is indicated how the three-state model and the methods for computing P(iota/eta/(rho)) can be applied to predict protein conformation.