Single-Molecule Michaelis−Menten Equations

Abstract

This paper summarizes our present theoretical understanding of single-molecule kinetics associated with the Michaelis-Menten mechanism of enzymatic reactions. Single-molecule enzymatic turnover experiments typically measure the probability density f(t) of the stochastic waiting time t for individual turnovers. While f(t) can be reconciled with ensemble kinetics, it contains more information than the ensemble data; in particular, it provides crucial information on dynamic disorder, the apparent fluctuation of the catalytic rates due to the interconversion among the enzyme's conformers with different catalytic rate constants. In the presence of dynamic disorder, f(t) exhibits a highly stretched multiexponential decay at high substrate concentrations and a monoexponential decay at low substrate concentrations. We derive a single-molecule Michaelis-Menten equation for the reciprocal of the first moment of f(t), 1/<t>, which shows a hyperbolic dependence on the substrate concentration [S], similar to the ensemble enzymatic velocity. We prove that this single-molecule Michaelis-Menten equation holds under many conditions, in particular when the intercoversion rates among different enzyme conformers are slower than the catalytic rate. However, unlike the conventional interpretation, the apparent catalytic rate constant and the apparent Michaelis constant in this single-molecule Michaelis-Menten equation are complicated functions of the catalytic rate constants of individual conformers. We also suggest that the randomness parameter r, defined as <(t - <t>)2> / t2, can serve as an indicator for dynamic disorder in the catalytic step of the enzymatic reaction, as it becomes larger than unity at high substrate concentrations in the presence of dynamic disorder.

Single Enzyme Studies Reveal the Existence of Discrete Functional States for Monomeric Enzymes and How They Are “Selected” upon Allosteric Regulation

“…Inspired by the conformational selection hypothesis, we employed a previously reported two-state statistical model 32 to describe stochastic single-substrate turnovers; see the SI for detailed statistical analysis. A catalytic cycle begins when a substrate binds reversibly with the enzyme to form the enzyme−substrate complex.…”

“…As a consistency check, we examined C s (t), which for the two activity states model should decay monoexponentially ( Figure 2C) with an exponent that equals the sum of the two interconversion rates. 9,32 Indeed, the exponent of the intensity autocorrelation decay K (I) = 0.06 ± 0.001 ms −1 equals the sum of interconversion rates k 12 + k 21 = 0.065 ± 0.03 ms −1 (see SI Figure S15 and Table S2). An alternative method employed to quantify activity fluctuations is the activity autocorrelation K(t).…”

Dynamic Simulations of Single-Molecule Enzyme Networks

“…For instance, 3 discuss the influence of dynamic disorder on Michaelis-Menten kinetics. Dynamic disorder assumes there are multiple channels for the same enzymatic reaction that work at slightly different speeds.…”

Fluctuating Enzymes: Lessons from Single-Molecule Studies

“…[8][9][10][11][12] The absence of ensemble averaging yields a wealth of dynamic information, which has profoundly changed the way enzymatic dynamics is studied. [13][14][15][16][17][18][19][20][21][22][23][24][25][26] New knowledge derived from the single-molecule approach continues to emerge.…”

“…This phenomenon, referred to broadly as dynamic disorder, 27,28 has been inferred previously from ensemble measurements 29 and quantitatively verified by recent single-molecule experiments. 14,[18][19][20][21][22][23][24][25] Dynamic disorder is not described within the framework of conventional transition state theory 30 and Kramers' theory 31,32 of condensed phase chemical reactions.…”