Novel Chemical Kinetics for a Single Enzyme Reaction: Relationship between Substrate Concentration and the Second Moment of Enzyme Reaction Time

Abstract

We report a robust quadratic relation between the inverse substrate concentration and the second moment, <t(2)>, of the catalytic turnover time distribution for enzyme reactions. The results hold irrespective of the mechanism and dynamics of the enzyme reaction and suggest a novel single molecule experimental analysis that provides information about reaction processes of the enzyme-substrate complex and ergodicity of the enzyme reaction system, which is beyond the reach of the conventional analysis for the mean reaction time, <t>. It turns out that <t(2)> - 2<t>(2) is linear in inverse substrate concentration for an ergodic homogeneous enzyme system given that the enzyme substrate encounter is a simple rate process, and its value at the high substrate concentration limit provides direct information about if any non-Poisson reaction process of the enzyme-substrate complex. For a nonergodic heterogeneous reaction system, the corresponding quantity becomes a quadratic function of the inverse substrate concentration. This leads us to suggest an ergodicity measure for single enzyme reaction systems. We obtain a simple analytic expression of the randomness parameter for the single catalytic turnover time, which could provide a quantitative explanation about the previously reported randomness data of the beta-galactosidase enzyme. In obtaining the results, we introduce novel chemical kinetics applicable to a non-Poisson reaction network with arbitrary connectivity, as a generalization of the conventional chemical kinetics.

Extracting signal from noise: kinetic mechanisms from a Michaelis–Menten‐like expression for enzymatic fluctuations

“…For all linear kinetic models, (Eqn 21), we have proven that this statement is true [16]. Moreover, Jung et al [35] have independently extended this proof to a much broader family of kinetic models. Using the concept of an extended waiting time distribution [34,54,57] (as noted above), they proved that an expression equivalent to Eqn (23) but with N L = 1 applies for any kinetic model of the form:…”

“…Similar to k cat and K M , the values adopted by these new parameters are a function of the specific kinetic rates in a given kinetic model. Equations (23,24) generalize or reparameterize expressions that have been derived previously by ourselves [18], as well as by Jung et al [35], and, more recently, by Chaudhury et al [36]. We prefer Eqns (23,24) over these alternative expressions because the parameters that Eqns (23,24) introduce are simpler to interpret than the equivalent parameters of these other expressions.…”

Nonclassical Chemical Kinetics for Description of Chemical Fluctuation in a Dynamically Heterogeneous Biological System

“…9,14 It is now known that the average enzymatic turnover time obeys the MM relation for a variety of different models of enzyme reactions involving reaction rate fluctuation. [15][16][17] Recently, a generalized MM relation is established for the mean turnover time of a general multistate enzyme reaction model, which reduces to the MM relation whenever the detailed balance condition between different states is satisfied. 18 However, the conventional chemical kinetics is not so satisfactory in description of statistical fluctuations contained in biopolymer reaction trajectories.…”

“…19 Recently, a general quantitative description of fluctuations in single enzymatic turnover times was also achieved, 17,18,20 for which a generalization of chemical kinetics was made in description of non-Poisson reaction processes of enzyme-substrate (ES) complex.…”

“…16,17 In this approach, the turnover time distribution ψ(t) of the single enzyme Michaelis-Menten (MM) reaction is represented in terms of the reaction time distributions, φ 1 0 (t), φ −1 (t), and φ 2 (t) for the three elementary reaction processes, , , and . The reaction time distribution (RTD) for each of the elementary reactions represents the probability density of the time elapsed for a completion of the elementary reaction process.…”