  A method is presented that, when used in conjunction with single molecule
experimental techniques, allows for the extraction of rates and mechanical
properties of a biomolecule undergoing transitions between mechanically
distinct states. This analysis enables the exploration of systems where the
lifetimes of survival are of order of the intrinsic time constant of the
experimental apparatus; permitting the study of kinetic events whose transition
rates are an order of magnitude (or two) larger than those that can be studied
with traditional averaging methods. Using current experimental techniques, this
allows for the study of biomolecules whose lifetime of survival in a given
state are as low as milliseconds down to microseconds. The relevant observable
is the auto-correlation function of the experimental probe that is attached to
the biomolecule of interest. General solutions are expressed in terms of a
series. Closed form solutions are found for two physically opposing limits:
where transitions between the mechanically distinct states of the biomolecule
occur on either much faster or much slower time scales than those governing the
motion of the experimental probe. Motivated by the derivation of these opposing
bounds, two series solutions for the general case are then presented. We
present an error analysis for truncating each series at arbitrary order and
obtain a range of parameters for which this method could be applicable to the
study of the two state biomolecular problem. Finally, both series (up to third
order) are expressed for the two state problem when the system obeys Markov
statistics. These solutions should be amiable to the analysis of experimental
data, expanding temporal analysis of data from single molecule experiments.
