General methods for analysis of sequential n-step'' kinetic mechanisms:: Application to single turnover kinetics of helicase-catalyzed DNA unwinding

Helicase-catalyzed DNA unwinding is often studied using all or none assays that detect only the final product of fully unwound DNA. Even using these assays, quantitative analysis of DNA unwinding time courses for DNA duplexes of different lengths, L, using n-step sequential mechanisms, can reveal information about the number of intermediates in the unwinding reaction and the kinetic step size, m, defined as the average number of basepairs unwound between two successive rate limiting steps in the unwinding cycle. Simultaneous nonlinear least-squares analysis using n-step sequential mechanisms has previously been limited by an inability to float the number of unwinding steps, n, and m, in the fitting algorithm. Here we discuss the behavior of single turnover DNA unwinding time courses and describe novel methods for nonlinear least-squares analysis that overcome these problems. Analytic expressions for the time courses, f(ss)(t), when obtainable, can be written using gamma and incomplete gamma functions. When analytic expressions are not obtainable, the numerical solution of the inverse Laplace transform can be used to obtain f(ss)(t). Both methods allow n and m to be continuous fitting parameters. These approaches are generally applicable to enzymes that translocate along a lattice or require repetition of a series of steps before product formation.