Cooperativity and specificity in enzyme kinetics: a single-molecule time-based perspective

Abstract

An alternative theoretical approach to enzyme kinetics that is particularly applicable to single-molecule enzymology is presented. The theory, originated by Van Slyke and Cullen in 1914, develops enzyme kinetics from a "time perspective" rather than the traditional "rate perspective" and emphasizes the nonequilibrium steady-state nature of enzymatic reactions and the significance of small copy numbers of enzyme molecules in living cells. Sigmoidal cooperative substrate binding to slowly fluctuating, monomeric enzymes is shown to arise from association pathways with very small probability but extremely long passage time, which would be disregarded in the traditional rate perspective: A single enzyme stochastically takes alternative pathways in serial order rather than different pathways in parallel. The theory unifies dynamic cooperativity and Hopfield-Ninio's kinetic proofreading mechanism for specificity amplification.

FIGURE 1

The simple, classic enzyme kinetic scheme in panel A corresponds to a cyclic reaction of a single enzyme in panel B. It is easy to obtain the steady-state probability for E and ES, namely, p_E = (k_–1 + k₂)/(k₁[S] + k_–1 + k₂ + k_–2[P]) and p_ES = (k₁[S] + k_–2[P])/(k₁[S] + k_–1 + k₂ + k_–2[P]), and the steady-state cycle flux = k₁[S]p_E – k_–1p_ES = k₂p_ES – k_–2[P]p_E. If [P] = 0, we have Eq. 1.

FIGURE 2

Waiting time probability distributions for product arrivals as functions of the number of enzyme molecules, m. For an irreversible enzyme reaction with a single enzyme (m = 1), k₁[S] = 0.667, k_–1 = 0.083, and k₂ = 0.75, the expected waiting time distribution is shown by the open squares (simulation) and dashed line: (e^−t/2 – e^−t) (Eq. 5 with λ₁ = 1, λ₂ = 0.5). It has a mean waiting time 〈T〉 = 3 (Eq. 4). The solid squares and open circles are the waiting time distributions, from simulations, for three (m = 3) and ten (m = 10) enzyme molecules, respectively. For large m, the waiting time distribution becomes exponential with a mean time 〈T〉/m (30,41); the solid line represents 3.33e^−(10/3)t.

FIGURE 3

(A) The simplest kinetic model with two unbound enzyme states that exhibits dynamic cooperativity. One can simplify the notation by taking (B) Assuming k₄ ≪ k₃ in panel A, and under the condition k₁[S] ≫ β ≫ k₂[S] ≫ α, the steady-state probability of ES, p_ES, is proportional to [S]². (C) The steady-state velocity for the enzymatic reaction in panel B, with k₁ = 100, k₂ = 0.01, α = 0, β = 1, and k₃ = 100, shows a sigmoidal shape. Note that

FIGURE 4

Schematic plots illustrating the dynamical difference between slow and rapid fluctuating states E. In the former, the enzyme substrate associations occur in serial order (A), while in the latter, they are in parallel. The overall association is determined by the average time in panel A but average rate in panel B.