Statistical kinetics of macromolecular dynamics

Fluctuations in biochemical processes can provide insights into the underlying kinetics beyond what can be gleaned from studies of average rates alone. Historically, analysis of fluctuating transmembrane currents supplied information about ion channel conductance states and lifetimes before single-channel recording techniques emerged. More recently, fluctuation analysis has helped to define mechanochemical pathways and coupling ratios for the motor protein kinesin as well as to probe the contributions of static and dynamic disorder to the kinetics of single enzymes. As growing numbers of assays are developed for enzymatic or folding behaviors of single macromolecules, the range of applications for fluctuation analysis increases. To evaluate specific biochemical models against experimental data, one needs to predict analytically the distribution of times required for completion of each reaction pathway. Unfortunately, using traditional methods, such calculations can be challenging for pathways of even modest complexity. Here, we derive an exact expression for the distribution of completion times for an arbitrary pathway with a finite number of states, using a recursive method to solve algebraically for the appropriate moment-generating function. To facilitate comparisons with experiments on processive motor proteins, we develop a theoretical formalism for the randomness parameter, a dimensionless measure of the variance in motor output. We derive the randomness for motors that take steps of variable sizes or that move on heterogeneous substrates, and then discuss possible applications to enzymes such as RNA polymerase, which transcribes varying DNA sequences, and to myosin V and cytoplasmic dynein, which may advance by variable increments.