[Paper Review] Waiting Cycle Times and Generalized Haldane Equality in the Steady-state Cycle Kinetics of Single Enzymes

Hao Ge|ArXiv.org|Apr 15, 2009

Advanced Thermodynamics and Statistical Mechanics31 references3 citations

TL;DR

This paper introduces waiting cycle times $T$, $T_+$, and $T_-$ for single-enzyme turnover in a reversible three-step Michaelis-Menten mechanism, showing their means are reciprocals of the steady-state cycle fluxes $J^{ss}$, $J^{ss}_+$, and $J^{ss}_-$. The key contribution is the generalized Haldane equality: the distribution of $T_+$ conditioned on $T_+ < T_-$ equals that of $T_-$ conditioned on $T_- < T_+$, implying $\langle T_+|T_+<T_-\rangle = \langle T_-|T_-<T_+\rangle = \langle T\rangle$, with extensions to n-step cycles and experimental validation.

ABSTRACT

Enzyme kinetics are cyclic. A more realistic reversible three-step mechanism of the Michaelis-Menten kinetics is investigated in detail, and three kinds of waiting cycle times $T$, $T_{+}$, $T_{-}$ are defined. It is shown that the mean waiting cycle times $$, $$, and $$ are the reciprocal of the steady-state cycle flux $J^{ss}$, the forward steady-state cycle flux $J^{ss}_{+}$ and the backward steady-state cycle flux $J^{ss}_{-}$ respectively. We also show that the distribution of $T_{+}$ conditioned on $T_{+}$) and the one of $T_{-}$ conditioned on $T_{-}$) are both just the same as $$. In addition, the forward and backward stepping probabilities $p^{+},p^{-}$ are also defined and discussed, especially their relationship with the cycle fluxes and waiting cycle times. Furthermore, we extend the same results to the $n$-step cycle, and finally, experimental and theoretically based evidences are also included.

Motivation & Objective

• To model single-enzyme turnover as a stochastic cycle process under nonequilibrium steady states (NESS), moving beyond deterministic Michaelis-Menten kinetics.
• To define three waiting cycle times—$T$, $T_+$, and $T_-$—corresponding to full, forward, and backward cycles in a reversible three-step mechanism.
• To establish the generalized Haldane equality, showing symmetry in the first-passage time distributions of forward and backward cycles.
• To connect mean waiting times to cycle fluxes via reciprocal relationships, enabling direct experimental inference from single-molecule trajectories.
• To extend the framework to general $n$-step cycles and validate results with theoretical and experimental evidence.

Proposed method

• Define three waiting cycle times: $T$ for full cycles, $T_+$ for forward cycles, and $T_-$ for backward cycles in a reversible three-state Markov model.
• Apply the strong Markov property to compute mean waiting times and conditional distributions from single-molecule trajectories.
• Use ergodic theory and renewal theory to relate time-averaged cycle counts to steady-state fluxes $J^{ss}$, $J^{ss}_+$, and $J^{ss}_-$.
• Derive the generalized Haldane equality by proving that the conditional distribution of $T_+$ given $T_+ < T_-$ is identical to that of $T_-$ given $T_- < T_+$, using detailed balance and first-passage time analysis.
• Introduce forward and backward stepping probabilities $p^+$, $p^-$ and relate them to fluxes and waiting times via stochastic process theory.
• Generalize the results to $n$-step cyclic Markov chains, maintaining the reciprocal and symmetry relationships.

Experimental results

Research questions

• RQ1 How do the mean waiting cycle times $\langle T\rangle$, $\langle T_+\rangle$, and $\langle T_-\rangle$ relate to the steady-state cycle fluxes $J^{ss}$, $J^{ss}_+$, and $J^{ss}_-$ in a reversible three-step enzyme mechanism?
• RQ2 Is there a symmetry between the first-passage time distributions of forward and backward cycles under nonequilibrium steady states, and if so, what is its form?
• RQ3 Can the generalized Haldane equality be extended from the three-step model to general $n$-step cyclic systems?
• RQ4 How can experimental single-molecule trajectory data be used to extract cycle fluxes, waiting times, and stepping probabilities without ensemble averaging?
• RQ5 What is the relationship between the conditional mean waiting times $\langle T_+|T_+<T_-\rangle$ and $\langle T_-|T_-<T_+\rangle$ and the overall mean cycle time $\langle T\rangle$?

Key findings

• The mean waiting cycle time $\langle T\rangle$ is the reciprocal of the steady-state cycle flux $J^{ss}$, and similarly $\langle T_+\rangle = 1/J^{ss}_+$, $\langle T_-\rangle = 1/J^{ss}_-$.
• The distribution of $T_+$ conditioned on $T_+ < T_-$ is identical to the distribution of $T_-$ conditioned on $T_- < T_+$, which defines the generalized Haldane equality.
• The conditional mean waiting times satisfy $\langle T_+|T_+<T_-\rangle = \langle T_-|T_-<T_+\rangle = \langle T\rangle$, showing symmetry in first-passage behavior.
• Forward and backward stepping probabilities $p^+$, $p^-$ are related to fluxes and waiting times, with $p^+ = J^{ss}_+ / J^{ss}$, $p^- = J^{ss}_- / J^{ss}$.
• The results are extended to $n$-step cyclic Markov chains, preserving the reciprocal and symmetry relationships.
• Experimental and theoretical evidence supports the generalized Haldane equality, with measurable implications for single-molecule data analysis.