On the time course of the reversible Michaelis-Menten reaction

The methods of Padé approximants and Euler transformation are used to construct approximate solutions for the time course of the reversible Michaelis-Menten reaction. The solutions are found to describe the concentrations of the various species quite accurately throughout and beyond the transient phase. To illustrate the results, the ratio of the reverse bi-molecular rate constant to the forward bi-molecular rate constant, k2/k1, is varied from 0.1 to 5, and the initial enzyme-to-substrate concentration ratio is changed from 0.01 to 5. Only when k-2/k1 is less than one, the concentration of the intermediate complex, y(t), undergoes a maximum (steady state); for all other values of this ratio, y(t) increases monotonically with time t, to the equilibrium value, i.e. no maximum is attained. The present methods are particularly useful when the total enzyme concentration is comparable to, or greater than the initial substrate concentration, a situation commonly found under in vivo conditions.