Matrix method for fluctuations and noise in kinetic systems.

In a series of papers we were concerned with the question of how to calculate the concentration noise power spectra of an ensemble of multi-state linear kinetic systems when the rate constants of the systems are assumed to be known. We have used a standard eigenvalue-eigenfunction method to solve the differential equations which govern the regression of the means and derived the noise power spectrum as a function of the eigenvalues and eigenfunctions of the relaxation matrix of the system. In this paper, we have obtained an equation which relates the noise spectrum matrix of the fluctuations directly to the relaxation matrix of the means. As a result, the noise power spectrum can be calculated through matrix operations without the necessity of an eigenvalue-eigenfunction calculation. The present formalism is particularly useful in the evaluation of kinetic rate constants when the noise spectrum data of concentration fluctuations are given. Possible applications to biochemical systems are briefly discussed.