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Enzyme Kinetics at the Molecular Level∗

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Enzyme Kinetics at the Molecular Level∗ GENERAL ARTICLE Enzyme Kinetics at the Molecular Level∗ Arti Dua The celebrated Michaelis–Menten (MM) expression provides a fundamental relation between the rate of enzyme cataly- sis and substrate concentration. The validity of this classical expression is, however, restricted to macroscopic amounts of enzymes and substrates and, thus, to processes with negligi- ble fluctuations. Recent experiments have measured fluctu- ations in the catalytic rate to reveal that the MM equation, though valid for bulk amounts, is not obeyed at the molecu- Arti Dua is an Associate lar level. In this review, we show how new statistical measures Professor in the Indian of fluctuations in the catalytic rate identify a regime in which Institute of Technology Madras. She is a theoretical the MM equation is always violated. This regime, character- chemist, working in the area ized by temporal correlations between enzymatic turnovers, of stochastic reaction is absent for a single enzyme and is unobservably short in the dynamics and statistical classical limit. mechanics of complex fluids. 1. Classical Enzyme Kinetics Enzymes are biological catalysts that accelerate chemical reac- tions manyfold, without getting consumed in the catalytic pro- cess. Several biological processes involving the conversion of substrates to products, thus, rely crucially on the catalytic activity of enzymes. Specific enzymes control and regulate a wide range of life-sustaining processes that vary from digestion, metabolism, absorption, and blood clotting to reproduction. While specificity Keywords depends on the detailed chemical structure of enzyme proteins, Biological catalysts, enzymes, rate kinetics, stochastic enzyme the rate at which the enzymes carry out the catalytic conversion kinetics, Michaelis–Menten depends less on their chemical structure but more on the physical mechanism, enzymatic velocity, parameters, including the amounts of enzymes, substrates, tem- molecular noise. perature, pH, and so on [1]. ∗DOI: https://doi.org/10.1007/s12045-019-0781-9 RESONANCE | March 2019 297 GENERAL ARTICLE In 1903, Victor Henri, in In 1903, Victor Henri, in his doctoral thesis, studied the rate of hy- his doctoral thesis, drolysis of sucrose into glucose and fructose by the enzyme inver- studied the rate of tase and laid the foundation for the understanding of enzymatic hydrolysis of sucrose into glucose and fructose mechanisms from reaction rates [2]. In 1913, Leonor Michaelis by the enzyme invertase and Maud Menten, building on the work of Victor Henri and and laid the foundation many others, introduced the initial rate method for kinetic anal- for the understanding of ysis. Using data for the initial rate of hydrolysis of sucrose, a enzymatic mechanisms from reaction rates. In simple reaction mechanism for enzyme-catalyzed reactions, the 1913, Leonor Michaelis Michaelis–Menten (MM) mechanism, was proposed [3–5]. and Maud Menten, E S building on the work of According to MM mechanism, enzyme binds with substrate Victor Henri and many to form an enzyme-substrate complex ES, which either dissoci- others, introduced the ates irreversibly to form product P, regenerating the free enzyme initial rate method for E or dissociates reversibly to release the substrate: kinetic analysis. k1 k E + S ES →2 P + E . (1) k−1 For thermodynamically large numbers of enzymes and substrates, deterministic mass action kinetics provides the rate equations for the MM mechanism in terms of the temporal variation of the concentrations of E, ES and P. The mean rate of product for- mation d[P]/dt, then, quantifies the catalytic activity of enzymes through its dependence on the rate parameters for substrate bind- ing k1[S ], substrate release k−1 and product formation k2.As- suming a time scale separation in which the intermediate com- plex reaches a steady-state faster than the reactants and products, d[ES]/dt ≈ 0, the mean rate of product formation yields: Vmax[S ] Vss = , (2) [S ] + KM the celebrated Michaelis–Menten (MM) equation, the steady-state ‘enzymatic velocity’, which has widespread applicability in bio- k− +k chemical catalysis [6]. Here, KM = 1 2 is the Michaelis con- k1 stant and Vmax = k2[E0] is the maximum velocity at saturat- ing substrate concentration, with [E0] being the initial enzyme concentration. The MM equation, in its double reciprocal form, V−1 = V−1 + KM S −1 V−1 ss max Vmax [ ] , yields a linear variation of ss with 298 RESONANCE | March 2019 GENERAL ARTICLE [S ]−1 [7]. Kinetic data for the variation of the initial rate of prod- uct formation with [S ], when fitted to this linear form, provides a simple way to estimate the rate parameters of several biochemical reactions. The hyperbolic dependence of the catalytic rate on sub- strate concentration, which the classical MM expression predicts, has had an enormous influence on the progress of biochemistry. 2. Enzyme Kinetics at the Molecular Level Enzymatic reactions at the molecular level, however, do not pro- Fluctuations of both ceed deterministically. Fluctuations of both quantum mechanical quantum mechanical and and thermal origin, termed as ‘molecular noise’, are inherent to thermal origin, termed as ‘molecular noise’, are reactions catalyzed by individual enzyme molecules. These im- inherent to reactions part stochasticity to each step in the MM mechanism, such that catalyzed by individual neither the lifetime of a given enzymatic state and nor the state enzyme molecules. to which it transits can be known with certainty. The effect of fluctuations, and thus the uncertainty, diminish progressively with increasing number of enzymes and vanish for their macroscopic amounts. It is in the latter limit that the MM kinetics acquires its deterministic character, and the classical description of enzymatic velocity as the ‘mean rate of product formation’, is sufficient to characterize the catalytic activity of enzymes [1]. Biochemical catalysis, under physiological conditions, involves enzyme concentrations that are not thermodynamically large, but vary from nanomolar to micromolar. The substrates are typically between ten and ten thousand times more than the number of en- zymes. At these low concentrations, termed as mesoscopic, the presence of molecular noise is inherent to enzymatic reactions [9]. Once this fact is realized, a series of questions arise. Is clas- sical enzymatic velocity, the MM equation, a valid description of the catalytic activity at physiologically relevant concentrations of enzymes? How do fluctuations in the rates of substrate bind- ing, substrate release and product formation influence the rate of enzyme catalysis? How are uncertainties in the product forma- tion times measured and characterized? What new information do these uncertainties carry about the catalytic mechanism that is RESONANCE | March 2019 299 GENERAL ARTICLE Figure 1. Panel (a) shows (a) a schematic of the MM S P S P S P P mechanism for a single en- E ES E ES E ES ES τ τ τ τp zyme forming products, one 1 2 3 First Product Second Product Third Product p-th Product by one, in discrete turnover T1 T2 events. Here, T p with Tp p = 1, 2,... is the p-th (b) (c) S SSS S S S S turnover time and τp = T p − E ES E ES E ES E ES E E ES E ES E ES E P P P P P P T p−1 is the waiting time be- 3rd product (d) 3rd product (e) tween two consecutive prod- T3 T3 τ4 (b) 2nd product tE 2nd product uct bursts. Panels T tES T2 2 τ τ3 (c) 1st product 3 1st product and show two stochastic p p n T n T 1 τ MM networks, each form- 1 τ2 2 ing three products in suc- τ1 τ1 substrate substrate cession, but following dif- release binding ferent pathways due to in- t t herent stochasticity of the MM reaction at the molecu- lar level. Panels (d) and (e) lost in going over to the deterministic limit? show two stochastic trajec- To address some of these questions, it is important to understand tories, corresponding to (b) how single-enzyme kinetic data is collected and analyzed. and (c), for the number of products np formed in time t. The lifetimes of E and 2.1 Single-Molecule Kinetic Data ES, tE and tES , are indi- cated by red and blue colors, In the last two decades, advances in experimental techniques have respectively. Stochasticity finally made it possible to measure, with precision, the effect of in three consecutive waiting molecular noise in enzyme-catalyzed reactions, involving a sin- and turnover times is indi- gle enzyme and numerous substrates [10, 11]. Such techniques cated by solid and dashed have made it possible to monitor, in real time, the catalytic con- lines. version of non-fluorescent substrates to fluorescent products, one substrate at a time, and yield a time series of the turnover times T1, T2,..., for the first, second,...product bursts. The turnover time series yields an equivalent series of the waiting times τ1,τ2,...between two consecutive product bursts τp = Tp − Tp−1, with turnover number, p = 1, 2,.... A schematic of the MM mechanism for a single enzyme form- ing products in succession, one product per enzyme turnover, is 300 RESONANCE | March 2019 GENERAL ARTICLE shown in panel (a)ofFigure 1. The waiting and turnover times for p = 1, 2,...product formation are indicated as τp and Tp,re- spectively. Each waiting time is the sum of the lifetimes of E and ES states, tE and tES , τp = tE, j + tES, j, where the subscript index j = 1, 2,...denotes the number of times a given state E or ES is visited. Due to the presence of molecular noise, the lifetimes of E and Enzyme kinetics at the ES states, and hence τp and Tp, are random variables. Enzyme molecular level, thus, kinetics at the molecular level, thus, requires a stochastic de- requires a stochastic E description that includes scription that includes intrinsic fluctuations in the lifetimes of intrinsic fluctuations in and ES states, the waiting times between two consecutive prod- the lifetimes of E and uct turnovers, and the turnover times for p-th product formation.
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