Progress curve analysis for enzyme and microbial kinetic reactions using explicit solutions based on the Lambert W function

We present a simple method for estimating kinetic parameters from progress curve analysis of biologically catalyzed reactions that reduce to forms analogous to the Michaelis–Menten equation. Specifically, the Lambert W function is used to obtain explicit, closed-form solutions to differential rate expressions that describe the dynamics of substrate depletion. The explicit nature of the new solutions greatly simplifies nonlinear estimation of the kinetic parameters since numerical techniques such as the Runge–Kutta and Newton–Raphson methods used to solve the differential and integral forms of the kinetic equations, respectively, are replaced with a simple algebraic expression. The applicability of this approach for estimating V max and K m in the Michaelis–Menten equation was verified using a combination of simulated and experimental progress curve data. For simulated data, final estimates of V max and K m were close to the actual values of 1 μM/h and 1 μM, respectively, while the standard errors for these parameter estimates were proportional to the error level in the simulated data sets. The method was also applied to hydrogen depletion experiments by mixed cultures of bacteria in activated sludge resulting in V max and K m estimates of 6.531 μM/h and 2.136 μM, respectively. The algebraic nature of this solution, coupled with its relatively high accuracy, makes it an attractive candidate for kinetic parameter estimation from progress curve data.