Dynamic
biological processes such as
enzyme catalysis,
molecular motor translocation, and protein and
nucleic acid conformational dynamics are inherently
stochastic processes. However, when such processes are studied on a nonsynchronized ensemble, the inherent fluctuations are lost, and only the
average rate of the process can be measured. With the recent development of methods of
single-molecule manipulation and detection, it is now possible to follow the progress of an individual molecule, measuring not just the
average rate but the fluctuations in this rate as well. These fluctuations can provide a great deal of detail about the underlying kinetic cycle that governs the dynamical behavior of the system. However, extracting this information from experiments requires the ability to calculate the general properties of arbitrarily complex theoretical kinetic schemes. We present here a general technique that determines the exact analytical
solution for the
mean velocity and for measures of the fluctuations. We adopt a formalism based on the
master equation and show how the probability
density for the position of a
molecular motor at a given time can be solved exactly in Fourier-Laplace space. With this analytic
solution, we can then calculate the
mean velocity and fluctuation-related parameters, such as the
randomness parameter (a dimensionless ratio of the
diffusion constant and the velocity) and the dwell time distributions, which fully characterize the fluctuations of the system, both commonly used kinetic parameters in
single-molecule measurements. Furthermore, we show that this formalism allows calculation of these parameters for a much wider class of general kinetic models than demonstrated with previous methods.
