Brownian Motion of the End-to-end Distance in Oligopeptide Molecules: Numerical Solution of the Diffusion Equations as Coupled First Order Linear Differential Equations

The Brownian motion of one end of an oligopeptide molecule relative to its other end was previously studied by the measurement of non-radiative energy transfer between chromophores attached to the molecular ends, and was found to conform to a model which describes this motion as a diffusional process in a force field (Haas et al., 1978a). The theoretical treatment of this diffusional problem is performed here by an approach which is different from the one used previously. In this approach advantage is taken of the fact that in the case under consideration the rate of change of the concentration of molecules with a given end-to-end distance is linearly dependent on the instantaneous concentration at this and neighboring end-to-end distances. One thus obtains a set of coupled first-order linear differential equations, which can be solved by standard techniques involving the diagonalization of the matrix of the rate constants in the above set of coupled equations. The concentration distribution at any instant is subsequently obtained as a linear combination of the eigenvectors weighted according to their respective exponential decay with time with rate constants which are related to their respective eigenvalues. Some of the advantages offered by this approach are as follows: one does not have to start the procedure from the beginning if new initial conditions are desired, and the concentration distribution at any given instant is obtained directly without the need of a stepwise build-up of the solution with time. The latter point is especially useful if one is interested in the asymptotic behavior of the changes in concentration at long times, since this behavior can be readily expressed as an exponential decay of the longest-lived eigenvector (or the sum of a few exponentially decaying eigenvectors which have the longest decay times). The above approach is used to treat the energy-transfer experiments performed previously by Haas et al. (1978b), as well as to simulate the dynamics of ring-opening of cyclic oligopeptides and ring closure of linear peptides.