Statistical properties of fluctuating enzymes with dynamic cooperativity using a first passage time distribution formalism.

We study the temporal fluctuations in catalytic rates for single enzyme reactions undergoing slow transitions between two active states. We use a first passage time distribution formalism to obtain the closed-form analytical expressions of the mean reaction time and the randomness parameter for reaction schemes where conformational fluctuations are present between two free enzyme conformers. Our studies confirm that the sole presence of free enzyme fluctuations yields a non Michaelis-Menten equation and can lead to dynamic cooperativity. The randomness parameter, which is a measure of the dynamic disorder in the system, converges to unity at a high substrate concentration. If slow fluctuations are present between the enzyme-substrate conformers (off-pathway mechanism), dynamic disorder is present at a high substrate concentration. Our results confirm that the dynamic disorder at a high substrate concentration is determined only by the slow fluctuations between the enzyme-substrate conformers and the randomness parameter is greater than unity. Slow conformational fluctuations between free enzymes are responsible for the emergence of dynamic cooperativity in single enzymes. Our theoretical findings are well supported by comparison with experimental data on the single enzyme beta-galactosidase.