Expanding the Temporal Analysis in Single-Molecule Switching Experiments   Through the Auto-Correlation Function: Mathematical Framework
  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. 
Darren E. Segall