Generic Schemes for Single-Molecule Kinetics. 1: Self-Consistent Pathway Solutions for Renewal Processes

In this paper, we discuss a strategy for reducing a complex single molecule kinetic process to a set of generic structures (motifs) that are building blocks for a general kinetic scheme. In general, these motifs have complex kinetics (i.e., waiting time distribution functions) which are composed of fundamental kinetic steps. (1) First, we treat four different experimental single molecule measurements within both the usual kinetic framework (i.e., using the rate matrix) and the waiting time distribution function framework. The two frameworks are then shown to be equivalent and can be formulated on the basis of the first passage time distribution function of monitored single molecule events. (2) Second, to calculate this basic quantity, we decompose a complex kinetic scheme with the help of two kinetic motifs, sequential and branching, and derive self-consistent equations by convoluting waiting time distributions and first passage time distribution(s) along the reaction pathway(s). (3) As examples, two experimental systems, a chain reaction model with a special case of enzymatic reactions and a general kinetic model for fluorescence emission, are analyzed on the basis of a genetic scheme composed of a monitored link, controlled link, and unknown link, each representing a possible subscheme associated with a complex waiting time distribution function. As a result, single molecule measurements of the generic scheme retain the same functional form when a kinetic link is altered within a subscheme, and different measurements can be classified and analyzed within the same framework. (4) Finally, to explore the physical reasons for nonexponential waiting time distribution, we use the example of blinking phenomena to discuss several scenarios of dynamic and static disorder and their implications for observed memory effects. The self-consistent pathway formalism is presented in this paper for renewal processes and will be generalized to nonrenewal processes with memory effects in a future publication.