Synopsis
Methods for calculating the rate of cooperative transitions on a linear lattice, for which the helix-coil transition of polypeptides is an example, are reported. The problem is to determine the kinetic characteristics of the transition given the rate constants for a set of elementary steps: in this case, the transformations of individual segments between the helix and coil states. The most straightforward method is to store the state of a long lattice (in which helix and coil segments are represented by 1's and 0's) in a computer and to use random-number techniques to generate its behavior as a function of time. This is, in principle, a solution to the problem, but it requires very large amounts of computer time. We have devised a matrix iteration procedure which allows much faster computation while reproducing the results of the random-number method accurately. In this procedure the computer operates repeatedly with a transition probability matrix on a vector which represents the time-dependent state of a finite group of units. The choice of a finite group neglects kinetic correlations between the state changes of units inside and outside the group, but comparison with the randomnumber method indicates that these correlations are not important. Thus it is possible to generate the kinetic behavior of the model under essentially any conditions, for either relaxation or large perturbations. Examination of these calculated curves suggests a simple and quite generally applicable solution to the inverse problem-that of evaluating the rate constants from kinetic curves. The initial slope is well defined in almost every case, and since an analytic equation can be written relating this to the rate constants, these can be obtained directly from the initial rate. This latter is therefore the most convenient single measure of transition rate.