View source for Enzyme kinetics

{{short description|Study of biochemical reaction rates catalysed by an enzyme}}
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{{Use dmy dates|date=August 2018}}
[[File:EcDHFR raytraced.png|thumb|right|upright=1.2|[[Dihydrofolate reductase]] from ''[[Escherichia coli|E. coli]]'' with its two substrates [[Dihydrofolic acid|dihydrofolate]] (right) and [[Nicotinamide adenine dinucleotide phosphate|NADPH]] (left), bound in the active site. The protein is shown as a [[ribbon diagram]], with alpha helices in red, beta sheathes in yellow and loops in blue. ({{Pdb|7DFR}})]]
'''Enzyme kinetics''' is the study of the rates of [[enzyme catalysis|enzyme-catalysed]] [[chemical reaction]]s. In enzyme kinetics, the [[reaction rate]] is measured and the effects of varying the conditions of the reaction are investigated. Studying an enzyme's [[chemical kinetics|kinetics]] in this way can reveal the catalytic mechanism of this enzyme, its role in [[metabolism]], how its activity is controlled, and how a [[drug]] or a modifier ([[Enzyme inhibitor|inhibitor]] or [[Enzyme activator|activator]]) might affect the rate.

An enzyme (E) is a [[protein]] [[molecule]] that serves as a biological catalyst to facilitate and accelerate a chemical reaction in the body. It does this through binding of another molecule, its [[Substrate (biochemistry)|substrate]] (S), which the enzyme acts upon to form the desired product. The substrate binds to the [[active site]] of the enzyme to produce an enzyme-substrate complex ES, and is transformed into an enzyme-product complex EP and from there to product P, via a [[transition state]] ES*. The series of steps is known as the [[enzyme catalysis|mechanism]]:

: E + S ⇄ ES ⇄ ES* ⇄ EP ⇄ E + P

This example assumes the simplest case of a reaction with one substrate and one product. Such cases exist: for example, a [[mutase]] such as [[phosphoglucomutase]] catalyses the transfer of a phosphate group from one position to another, and [[isomerase]] is a more general term for an enzyme that catalyses any one-substrate one-product reaction, such as [[Triosephosphateisomerase|triosephosphate isomerase]]. However, such enzymes are not very common, and are heavily outnumbered by enzymes that catalyse two-substrate two-product reactions: these include, for example, the NAD-dependent [[dehydrogenase]]s such as [[alcohol dehydrogenase]], which catalyses the oxidation of ethanol by NAD<sup>+</sup>. Reactions with three or four substrates or products are less common, but they exist. There is no necessity for the number of products to be equal to the number of substrates; for example, [[glyceraldehyde 3-phosphate dehydrogenase]] has three substrates and two products.

When enzymes bind multiple substrates, such as [[dihydrofolate reductase]] (shown right), enzyme kinetics can also show the sequence in which these substrates bind and the sequence in which products are released.  An example of enzymes that bind a single substrate and release multiple products are [[protease]]s, which cleave one protein substrate into two polypeptide products. Others join two substrates together, such as [[DNA polymerase]] linking a [[nucleotide]] to [[DNA]]. Although these mechanisms are often a complex series of steps, there is typically one ''rate-determining step'' that determines the overall kinetics.  This [[rate-determining step]] may be a chemical reaction or a [[conformational isomerism|conformational]] change of the enzyme or substrates, such as those involved in the release of product(s) from the enzyme.

Knowledge of the [[protein structure|enzyme's structure]] is helpful in interpreting kinetic data.  For example, the structure can suggest how substrates and products bind during catalysis; what changes occur during the reaction; and even the role of particular [[amino acid]] residues in the mechanism.  Some enzymes change shape significantly during the mechanism; in such cases, it is helpful to determine the enzyme structure with and without bound substrate analogues that do not undergo the enzymatic reaction.

Not all biological catalysts are protein enzymes: [[RNA]]-based catalysts such as [[ribozymes]] and [[ribosomes]] are essential to many cellular functions, such as [[splicing (genetics)|RNA splicing]] and [[translation (biology)|translation]]. The main difference between ribozymes and enzymes is that RNA catalysts are composed of nucleotides, whereas enzymes are composed of amino acids.  Ribozymes also perform a more limited set of reactions, although their [[reaction mechanism]]s and kinetics can be analysed and classified by the same methods.

==General principles==
[[File:KinEnzymo(en).svg|thumb|300px|right|As larger amounts of [[Substrate (biochemistry)|substrate]] are added to a reaction, the available enzyme [[binding site]]s become filled to the limit of [[Michaelis-Menten kinetics|<math>V_\max</math>]].  Beyond this limit the enzyme is saturated with substrate and the reaction rate ceases to increase.]]

The reaction catalysed by an enzyme uses exactly the same reactants and produces exactly the same products as the uncatalysed reaction. Like other [[catalysts]], enzymes do not alter the position of [[chemical equilibrium|equilibrium]] between substrates and products.<ref>{{cite book| vauthors = Wrighton MS, Ebbing DD |title=General chemistry |publisher=Houghton Mifflin |location=Boston |year=1993 |isbn=978-0-395-63696-1 |edition=4th}}</ref> However, unlike uncatalysed chemical reactions, enzyme-catalysed reactions display saturation kinetics. For a given enzyme concentration and for relatively low substrate concentrations, the reaction rate increases linearly with substrate concentration; the enzyme molecules are largely free to catalyse the reaction, and increasing substrate concentration means an increasing rate at which the enzyme and substrate molecules encounter one another.  However, at relatively high substrate concentrations, the reaction rate [[asymptote|asymptotically]] approaches the theoretical maximum; the enzyme active sites are almost all occupied by substrates resulting in saturation, and the reaction rate is determined by the intrinsic turnover rate of the enzyme.<ref name="Fromm H.J. 2012">Fromm H.J., Hargrove M.S. (2012) Enzyme Kinetics. In: Essentials of Biochemistry. Springer, Berlin, Heidelberg</ref> The substrate concentration midway between these two limiting cases is denoted by ''K''<sub>M</sub>. Thus, ''K''<sub>M</sub> is the substrate concentration at which the reaction velocity is half of the maximum velocity.<ref name="Fromm H.J. 2012"/>

The two important properties of enzyme kinetics are how easily the enzyme can be saturated with a  substrate, and the maximum rate it can achieve. Knowing these properties suggests what an enzyme might do in the cell and can show how the enzyme will respond to changes in these conditions.

==Enzyme assays==
{{Main|Enzyme assay}}
[[File:Enzyme progress curve.svg|thumb|250px|left|Progress curve for an enzyme reaction. The slope in the initial rate period is the '''initial rate of reaction''' ''v''.  The [[Michaelis–Menten kinetics|Michaelis–Menten equation]] describes how this slope varies with the concentration of substrate.]]
[[Enzyme assay]]s are laboratory procedures that measure the rate of enzyme reactions. Since enzymes are not consumed by the reactions they catalyse, enzyme assays usually follow changes in the concentration of either substrates or products to measure the rate of reaction. There are many methods of measurement. [[Ultraviolet-visible spectroscopy|Spectrophotometric]] assays observe the change in the [[absorbance]] of light between products and reactants; radiometric assays involve the incorporation or release of [[radioactivity]] to measure the amount of product made over time. Spectrophotometric assays are most convenient since they allow the rate of the reaction to be measured continuously. Although radiometric assays require the removal and counting of samples (i.e., they are discontinuous assays) they are usually extremely sensitive and can measure very low levels of enzyme activity.<ref>{{cite book| vauthors = Danson M, Eisenthal R |title=Enzyme assays: a practical approach |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2002 |isbn=978-0-19-963820-8 }}</ref>  An analogous approach is to use [[mass spectrometry]] to monitor the incorporation or release of [[stable isotope]]s as the substrate is converted into product. Occasionally, an assay fails and approaches are essential to resurrect a failed assay. 

The most sensitive enzyme assays use [[laser]]s focused through a [[microscope]] to observe changes in single enzyme molecules as they catalyse their reactions. These measurements either use changes in the [[fluorescence]] of [[Cofactor (biochemistry)|cofactors]] during an enzyme's reaction mechanism, or of [[fluorescent dyes]] added onto specific sites of the [[protein]] to report movements that occur during catalysis.<ref>{{cite journal | vauthors = Xie XS, Lu HP | title = Single-molecule enzymology | journal = The Journal of Biological Chemistry | volume = 274 | issue = 23 | pages = 15967–15970 | date = June 1999 | pmid = 10347141 | doi = 10.1074/jbc.274.23.15967 | doi-access = free }}</ref> These studies provide a new view of the kinetics and dynamics of single enzymes, as opposed to traditional enzyme kinetics, which observes the average behaviour of populations of millions of enzyme molecules.<ref>{{cite journal | vauthors = Lu HP | title = Single-molecule spectroscopy studies of conformational change dynamics in enzymatic reactions | journal = Current Pharmaceutical Biotechnology | volume = 5 | issue = 3 | pages = 261–269 | date = June 2004 | pmid = 15180547 | doi = 10.2174/1389201043376887 }}</ref><ref>{{cite journal | vauthors = Schnell JR, Dyson HJ, Wright PE | title = Structure, dynamics, and catalytic function of dihydrofolate reductase | journal = Annual Review of Biophysics and Biomolecular Structure | volume = 33 | pages = 119–140 | year = 2004 | pmid = 15139807 | doi = 10.1146/annurev.biophys.33.110502.133613 | author-link2 = Jane Dyson }}</ref>

An example progress curve for an enzyme assay is shown above. The enzyme produces product at an initial rate that is approximately linear for a short period after the start of the reaction.  As the reaction proceeds and substrate is consumed, the rate continuously slows (so long as the substrate is not still at saturating levels). To measure the initial (and maximal) rate, enzyme assays are typically carried out while the reaction has progressed only a few percent towards total completion.  The length of the initial rate period depends on the assay conditions and can range from milliseconds to hours.  However, equipment for rapidly mixing liquids allows fast kinetic measurements at initial rates of less than one second.<ref>{{cite book|author=Gibson QH |title=Rapid mixing: Stopped flow |volume=16 |pages=187–228 |year=1969 |doi=10.1016/S0076-6879(69)16009-7 |isbn=978-0-12-181873-9|series=Methods in Enzymology |chapter=&#91;6&#93; Rapid mixing: Stopped flow }}</ref> These very rapid assays are essential for measuring pre-steady-state kinetics, which are discussed below.

Most enzyme kinetics studies concentrate on this initial, approximately linear part of enzyme reactions. However, it is also possible to measure the complete reaction curve and fit this data to a non-linear [[rate equation]]. This way of measuring enzyme reactions is called progress-curve analysis.<ref>{{cite book|author=Duggleby RG |title=Analysis of enzyme progress curves by non-linear regression |volume=249 |pages=61–90 |year=1995 |doi=10.1016/0076-6879(95)49031-0 |pmid=7791628 |isbn=978-0-12-182150-0|series=Methods in Enzymology |chapter=&#91;3&#93; Analysis of enzyme progress curves by nonlinear regression }}</ref> This approach is useful as an alternative to [[rapid kinetics]] when the initial rate is too fast to measure accurately.

The [[Standards for Reporting Enzymology Data]] Guidelines provide minimum information required to comprehensively report kinetic and equilibrium data from investigations of enzyme activities including corresponding experimental conditions. The guidelines have been developed to report functional enzyme data with rigor and robustness.

==Single-substrate reactions==
Enzymes with single-substrate mechanisms include [[isomerase]]s such as [[triosephosphateisomerase]] or [[bisphosphoglycerate mutase]], intramolecular [[lyase]]s such as [[adenylate cyclase]] and the [[hammerhead ribozyme]], an RNA lyase.<ref>{{cite journal | vauthors = Murray JB, Dunham CM, Scott WG | title = A pH-dependent conformational change, rather than the chemical step, appears to be rate-limiting in the hammerhead ribozyme cleavage reaction | journal = Journal of Molecular Biology | volume = 315 | issue = 2 | pages = 121–130 | date = January 2002 | pmid = 11779233 | doi = 10.1006/jmbi.2001.5145 | s2cid = 18102624 | url = https://escholarship.org/uc/item/28h464wj }}</ref> However, some enzymes that only have a single substrate do not fall into this category of mechanisms. [[Catalase]] is an example of this, as the enzyme reacts with a first molecule of [[hydrogen peroxide]] substrate, becomes oxidised and is then reduced by a second molecule of substrate. Although a single substrate is involved, the existence of a modified enzyme intermediate means that the mechanism of catalase is actually a ping–pong mechanism, a type of mechanism that is discussed in the ''Multi-substrate reactions'' section below.

===Michaelis–Menten kinetics===
{{multiple image
 | direction= vertical
 | width    = 325
 | footer   =

 | image1   = Enzyme mechanism 2.svg
 | alt1     = Schematic reaction diagrams for uncatalzyed (Substrate to Product) and catalyzed (Enzyme + Substrate to Enzyme/Substrate complex to Enzyme + Product)
 | caption1 = A chemical reaction mechanism with or without [[enzyme catalysis]]. The enzyme (E) binds [[substrate (chemistry)|substrate]] (S) to produce [[product (chemistry)|product]] (P).

 | image2   = Michaelis Menten curve 2.svg
 | alt2     = A two dimensional plot of substrate concentration (x axis) vs. reaction rate (y axis).  The shape of the curve is hyperbolic.  The rate of the reaction is zero at zero concentration of substrate and the rate asymptotically reaches a maximum at high substrate concentration.
 | caption2 = [[Michaelis–Menten kinetics|Saturation curve]] for an enzyme reaction showing the relation between the substrate concentration and reaction rate.
 }}

{{Main|Michaelis–Menten kinetics}}

As enzyme-catalysed reactions are saturable, their rate of catalysis does not show a linear response to increasing substrate. If the initial rate of the reaction is measured over a range of substrate concentrations (denoted as [S]), the initial reaction rate (<math>v_0</math>) increases as [S] increases, as shown on the right. However, as [S] gets higher, the enzyme becomes saturated with substrate and the initial rate reaches ''V''<sub>max</sub>, the enzyme's maximum rate.

The [[Michaelis–Menten kinetics|Michaelis–Menten kinetic model of a single-substrate reaction]] is shown on the right. There is an initial [[chemical kinetics|bimolecular reaction]] between the enzyme E and substrate S to form the enzyme–substrate complex ES. The rate of enzymatic reaction increases with the increase of the substrate concentration up to a certain level called V<sub>max</sub>; at V<sub>max</sub>, increase in substrate concentration does not cause any increase in reaction rate as there is no more enzyme (E) available for reacting with substrate (S). Here, the rate of reaction becomes dependent on the ES complex and the reaction becomes a [[chemical kinetics|unimolecular reaction]] with an order of zero. Though the enzymatic mechanism for the [[chemical kinetics|unimolecular reaction]] <small><chem> ES ->[k_{cat}] E + P </chem></small> can be quite complex, there is typically one rate-determining enzymatic step that allows this reaction to be modelled as a single catalytic step with an apparent unimolecular rate constant [[turnover number|''k''<sub>cat</sub>]].
If the reaction path proceeds over one or several intermediates, ''k''<sub>cat</sub> will be a function of several elementary rate constants, whereas in the simplest case of a single elementary reaction (e.g. no intermediates) it will be identical to the elementary unimolecular rate constant ''k''<sub>2</sub>. The apparent unimolecular rate constant ''k''<sub>cat</sub> is also called [[turnover number]], and denotes the maximum number of enzymatic reactions catalysed per second.

The [[Michaelis–Menten kinetics|Michaelis–Menten equation]]<ref>Michaelis L. and Menten M.L. ''Kinetik der Invertinwirkung'' Biochem. Z. 1913; 49:333–369 [http://web.lemoyne.edu/~giunta/menten.html English translation] Accessed 6 April 2007</ref> describes how the (initial) reaction rate ''v''<sub>0</sub> depends on the position of the substrate-binding [[chemical equilibrium|equilibrium]] and the rate constant ''k''<sub>2</sub>.

:<math chem> v_0 = \frac{V_{\max}[\ce{S}]}{K_M + [\ce{S}]} </math>&nbsp;&nbsp;&nbsp;&nbsp;''(Michaelis–Menten equation)''
with the constants
:<math chem> \begin{align} 
K_M \ &\stackrel{\mathrm{def}}{=}\  \frac{k_{2} + k_{-1}}{k_{1}} \approx K_D\\
V_\max \ &\stackrel{\mathrm{def}}{=}\   k_{cat}\ce{[E]}_{tot}
\end{align} </math>

This Michaelis–Menten equation is the basis for most single-substrate enzyme kinetics. Two  crucial assumptions underlie this equation (apart from the general assumption about the mechanism only involving no intermediate or product inhibition, and there is no [[Allosteric regulation|allostericity]] or [[Cooperative binding|cooperativity]]). The first assumption is  the so-called 
[[steady state (chemistry)|quasi-steady-state assumption]] (or pseudo-steady-state hypothesis), namely that the concentration of the substrate-bound enzyme (and hence also the unbound enzyme) changes much more slowly than those of the product and substrate and thus the change over time of the complex can be set to zero
<math chem> d\ce{[ES]}/{dt}  \; \overset{!} = \;0 </math>. The second assumption is that the total enzyme concentration does not change over time, thus <math chem> \ce{[E]}_\text{tot} = \ce{[E]} + \ce{[ES]}  \; \overset{!} = \; \text{const} </math>.

The Michaelis constant ''K''<sub>M</sub> is experimentally defined as the concentration at which the rate of the enzyme reaction is half ''V''<sub>max</sub>, which can be verified by substituting [S] = ''K''<sub>M</sub> into the Michaelis–Menten equation and can also be seen graphically. If the rate-determining enzymatic step is slow compared to substrate dissociation (<math>k_2 \ll k_{-1} </math>), the Michaelis constant ''K''<sub>M</sub> is roughly the [[dissociation constant]] ''K''<sub>D</sub> of the ES complex.

If <chem>[S]</chem> is small compared to <math>K_M</math> then the term <math chem>[\ce S] / (K_M + [\ce S]) \approx [\ce S] / K_M </math> and also very little ES complex is formed, thus <chem>[E]_{\rm tot} \approx [E]</chem>.  Therefore, the rate of product formation is
:<math chem>v_0 \approx  \frac{k_{cat}}{K_M} \ce{[E] [S]} \qquad \qquad \text{if } [\ce S] \ll K_M</math>
Thus the product formation rate depends on the enzyme concentration as well as on the substrate concentration, the equation resembles a bimolecular reaction with a corresponding pseudo-second order rate constant <math>k_2 / K_M</math>. This constant is a measure of [[catalytic efficiency]]. The most efficient enzymes reach a <math>k_2 / K_M</math> in the range of {{math|10<sup>8</sup> – 10<sup>10</sup>&nbsp;[[Molar concentration|M]]<sup>−1</sup>&nbsp;[[second|s]]<sup>−1</sup>}}. These enzymes are so efficient they effectively catalyse a reaction each time they encounter a substrate molecule and have thus reached an upper theoretical limit for efficiency ([[Diffusion limited enzyme|diffusion limit]]); and are sometimes referred to as [[Diffusion limited enzyme|kinetically perfect enzymes]].<ref>{{cite journal | vauthors = Stroppolo ME, Falconi M, Caccuri AM, Desideri A | title = Superefficient enzymes | journal = Cellular and Molecular Life Sciences | volume = 58 | issue = 10 | pages = 1451–1460 | date = September 2001 | pmid = 11693526 | doi = 10.1007/PL00000788 | s2cid = 24874575 | pmc = 11337273 }}</ref> But most enzymes are far from perfect: the average values of <math>k_{2}/K_{\rm M}</math> and <math>k_{2}</math> are about <math> 10^5 {\rm s}^{-1}{\rm M}^{-1}</math> and <math>10 {\rm s}^{-1}</math>, respectively.<ref name="Bar-Even_2011">{{cite journal | vauthors = Bar-Even A, Noor E, Savir Y, Liebermeister W, Davidi D, Tawfik DS, Milo R | title = The moderately efficient enzyme: evolutionary and physicochemical trends shaping enzyme parameters | journal = Biochemistry | volume = 50 | issue = 21 | pages = 4402–4410 | date = May 2011 | pmid = 21506553 | doi = 10.1021/bi2002289 }}</ref>

===Direct use of the Michaelis–Menten equation for time course kinetic analysis===
{{Further|Rate equation}}{{Further|Reaction Progress Kinetic Analysis}}

The observed velocities predicted by the Michaelis–Menten equation can be used to directly model the [[Reaction progress kinetic analysis|time course disappearance of substrate]] and the production of product through  incorporation of the Michaelis–Menten equation into the equation for first order chemical kinetics.  This can only be achieved however if one recognises the problem associated with the use of [[Euler's number]] in the description of first order chemical kinetics. i.e. ''e''<sup>−''k''</sup> is a split constant that introduces a systematic error into calculations and can be rewritten as a single constant which represents the remaining substrate after each time period.<ref>{{cite journal | vauthors = Walsh R, Martin E, Darvesh S | title = A method to describe enzyme-catalyzed reactions by combining steady state and time course enzyme kinetic parameters | journal = Biochimica et Biophysica Acta (BBA) - General Subjects | volume = 1800 | issue = 1 | pages = 1–5 | date = January 2010 | pmid = 19840832 | doi = 10.1016/j.bbagen.2009.10.007 }}</ref>

:<math>[S]=[S]_0(1-k)^{t}\,</math>

:<math>[S]=[S]_0(1-v/[S]_0)^{t}\,</math>

:<math>[S]=[S]_0(1-(V_{\max} [S]_0 / (K_M + [S]_0)/[S]_0))^{t}\,</math>

In 1983 Stuart Beal (and also independently [[Santiago Schnell]] and Claudio Mendoza in 1997) derived a closed form solution for the time course kinetics analysis of the Michaelis-Menten mechanism.<ref>{{cite journal | vauthors = Beal SL | title = Computation of the explicit solution to the Michaelis-Menten equation | journal = Journal of Pharmacokinetics and Biopharmaceutics | volume = 11 | issue = 6 | pages = 641–657 | date = December 1983 | pmid = 6689584 | doi = 10.1007/BF01059062 | s2cid = 32571415 }}</ref><ref>{{cite journal | vauthors = Schnell S, Mendoza C | title = Closed Form Solution for Time-dependent Enzyme Kinetics | journal = Journal of Theoretical Biology | volume = 187 | issue = 2 | pages = 207–212 | year = 1997 | doi = 10.1006/jtbi.1997.0425 | bibcode = 1997JThBi.187..207S }}</ref> The solution, {{failed verification span|text=known as the Schnell-Mendoza equation|date=November 2024}}, has the form:

:<math>\frac{[S]}{K_M} = W \left[ F(t) \right]\,  </math>

where W[ ] is the [[Lambert W function|Lambert-W function]].<ref>{{cite journal | vauthors = Goudar CT, Sonnad JR, Duggleby RG | title = Parameter estimation using a direct solution of the integrated Michaelis-Menten equation | journal = Biochimica et Biophysica Acta (BBA) - Protein Structure and Molecular Enzymology | volume = 1429 | issue = 2 | pages = 377–383 | date = January 1999 | pmid = 9989222 | doi = 10.1016/s0167-4838(98)00247-7 | url = http://rdbiotech.drivehq.com/Biblio/Pubs/Pub103.pdf | url-status = dead | archive-url = https://www.webcitation.org/6cuXxfqIj?url=http://rdbiotech.drivehq.com/Biblio/Pubs/Pub103.pdf | archive-date = 9 November 2015 }}</ref><ref>{{cite journal | vauthors = Goudar CT, Harris SK, McInerney MJ, Suflita JM | title = Progress curve analysis for enzyme and microbial kinetic reactions using explicit solutions based on the Lambert W function | journal = Journal of Microbiological Methods | volume = 59 | issue = 3 | pages = 317–326 | date = December 2004 | pmid = 15488275 | doi = 10.1016/j.mimet.2004.06.013 }}</ref> and where F(t) is

:<math>F(t) = \frac{[S]_0}{K_M} \exp\!\left(\frac{[S]_0}{K_M} - \frac{V_\max}{K_M}\,t \right) \,  </math>

This equation is encompassed by the equation below, obtained by Berberan-Santos,<ref>{{cite journal | vauthors = Berberan-Santos MN  |date=2010|title=A General Treatment of Henri Michaelis Menten Enzyme Kinetics: Exact Series Solution and Approximate Analytical Solutions|url=http://match.pmf.kg.ac.rs/electronic_versions/Match63/n2/match63n2_283-318.pdf|journal=MATCH Communications in Mathematical and in Computer Chemistry|volume=63|pages=283 }}</ref> which is also valid when the initial substrate concentration is close to that of enzyme,

:<math>\frac{[S]}{K_M} = W \left[ F(t) \right]- \frac{V_\max}{k_{cat} K_M}\ \frac{W \left[ F(t) \right]}{1+W \left[ F(t) \right]}\,  </math>

where W[ ] is again the [[Lambert W function|Lambert-W function]].

===Linear plots of the Michaelis–Menten equation===
{{See also|Lineweaver–Burk plot|Eadie–Hofstee diagram|Hanes–Woolf plot}}

[[File:Lineweaver-Burke plot.svg|thumb|350px|right|Lineweaver–Burk or double-reciprocal plot of kinetic data, showing the significance of the axis intercepts and gradient.]]

The plot of ''v'' versus [S] above is not linear; although initially linear at low [S], it bends over to saturate at high [S].  Before the modern era of [[nonlinear regression|nonlinear curve-fitting]] on computers, this nonlinearity could make it difficult to estimate ''K''<sub>M</sub> and ''V''<sub>max</sub> accurately.  Therefore, several researchers developed linearisations of the Michaelis–Menten equation, such as the [[Lineweaver–Burk plot]], the [[Eadie–Hofstee diagram]] and the [[Hanes–Woolf plot]].  All of these linear representations can be useful for visualising data, but none should be used to determine kinetic parameters, as computer software is readily available that allows for more accurate determination by [[nonlinear regression]] methods.<ref>{{cite journal | vauthors = Jones ME | title = Analysis of algebraic weighted least-squares estimators for enzyme parameters | journal = The Biochemical Journal | volume = 288 ( Pt 2) | issue = Pt 2 | pages = 533–538 | date = December 1992 | pmid = 1463456 | pmc = 1132043 | doi = 10.1042/bj2880533 }}</ref>

The [[Lineweaver–Burk plot]] or double reciprocal plot is a common way of illustrating kinetic data. This is produced by taking the [[Multiplicative inverse|reciprocal]] of both sides of the Michaelis–Menten equation. As shown on the right, this is a linear form of the Michaelis–Menten equation and produces a straight line with the equation ''y'' = m''x'' + c with a ''y''-intercept equivalent to 1/''V''<sub>max</sub> and an ''x''-intercept of the graph representing −1/''K''<sub>M</sub>.

:<math>\frac{1}{v} = \frac{K_{M}}{V_{\max} [\mbox{S}]} + \frac{1}{V_\max}</math>

Naturally, no experimental values can be taken at negative 1/[S]; the lower limiting value 1/[S] = 0 (the ''y''-intercept) corresponds to an infinite substrate concentration, where ''1/v=1/V<sub>max</sub>'' as shown at the right; thus, the ''x''-intercept is an [[extrapolation]] of the experimental data taken at positive concentrations.  More generally, the Lineweaver–Burk plot skews the importance of measurements taken at low substrate concentrations and, thus, can yield inaccurate estimates of ''V''<sub>max</sub> and ''K''<sub>M</sub>.<ref name=Tseng>{{cite journal | vauthors = Tseng SJ, Hsu JP | title = A comparison of the parameter estimating procedures for the Michaelis-Menten model | journal = Journal of Theoretical Biology | volume = 145 | issue = 4 | pages = 457–464 | date = August 1990 | pmid = 2246896 | doi = 10.1016/S0022-5193(05)80481-3 | bibcode = 1990JThBi.145..457T }}</ref>  A more accurate linear plotting method is the [[Eadie–Hofstee diagram|Eadie–Hofstee plot]]. In this case, ''v'' is plotted against ''v''/[S].  In the third common linear representation, the [[Hanes–Woolf plot]], [S]/''v'' is plotted against [S].
In general, data normalisation can help diminish the amount of experimental work and can increase the reliability of the output, and is suitable for both graphical and numerical analysis.<ref>{{cite journal | vauthors = Bravo IG, Busto F, De Arriaga D, Ferrero MA, Rodríguez-Aparicio LB, Martínez-Blanco H, Reglero A | title = A normalized plot as a novel and time-saving tool in complex enzyme kinetic analysis | journal = The Biochemical Journal | volume = 358 | issue = Pt 3 | pages = 573–583 | date = September 2001 | pmid = 11577687 | pmc = 1222113 | doi = 10.1042/bj3580573 }}</ref>

===Practical significance of kinetic constants===
The study of enzyme kinetics is important for two basic reasons. Firstly, it helps explain how enzymes work, and secondly, it helps predict how enzymes behave in living organisms. The kinetic constants defined above, ''K''<sub>M</sub> and ''V''<sub>max</sub>, are critical to attempts to understand how enzymes work together to control [[metabolism]].

Making these predictions is not trivial, even for simple systems. For example, [[oxaloacetate]] is formed by [[malate dehydrogenase]] within the [[mitochondrion]]. Oxaloacetate can then be consumed by [[citrate synthase]], [[phosphoenolpyruvate carboxykinase]] or [[aspartate aminotransferase]], feeding into the [[citric acid cycle]], [[gluconeogenesis]] or [[aspartic acid]] biosynthesis, respectively. Being able to predict how much oxaloacetate goes into which pathway requires knowledge of the concentration of oxaloacetate as well as the concentration and kinetics of each of these enzymes. This aim of predicting the behaviour of metabolic pathways reaches its most complex expression in the synthesis of huge amounts of kinetic and [[gene expression]] data into mathematical models of entire organisms. Alternatively, one useful simplification of the metabolic modelling problem is to ignore the underlying enzyme kinetics and only rely on information about the reaction network's stoichiometry, a technique called [[flux balance analysis]].<ref>{{cite journal | vauthors = Almaas E, Kovács B, Vicsek T, Oltvai ZN, Barabási AL | title = Global organization of metabolic fluxes in the bacterium Escherichia coli | journal = Nature | volume = 427 | issue = 6977 | pages = 839–843 | date = February 2004 | pmid = 14985762 | doi = 10.1038/nature02289 | arxiv = q-bio/0403001 | s2cid = 715721 | bibcode = 2004Natur.427..839A }}</ref><ref>{{cite journal | vauthors = Reed JL, Vo TD, Schilling CH, Palsson BO | title = An expanded genome-scale model of Escherichia coli K-12 (iJR904 GSM/GPR) | journal = Genome Biology | volume = 4 | issue = 9 | pages = R54 | year = 2003 | pmid = 12952533 | pmc = 193654 | doi = 10.1186/gb-2003-4-9-r54 | doi-access = free }}</ref>

===Michaelis–Menten kinetics with intermediate===
One could also consider the less simple case
: <chem> 
 {E} + S 
   <=>[k_{1}][k_{-1}]
 ES
   ->[k_2]
 EI
   ->[k_3]
 {E} + P
</chem>
where a complex with the enzyme and an intermediate exists and the intermediate is converted into product in a second step. In this case we have a very similar equation<ref>for a complete derivation, see [[commons:File:Enzyme Kinetics.pdf|here]]</ref>
: <math chem> 
v_0 = k_{cat}\frac\ce{[S] [E]_0}{K_M^{\prime}+ \ce{[S]}}
</math>
but the constants are different
:<math> \begin{align} 
K_M^{\prime} \ &\stackrel{\mathrm{def}}{=}\  \frac{k_3}{k_2 + k_3} K_M = \frac{k_3}{k_2 + k_3} \cdot \frac{k_{2} + k_{-1}}{k_{1}}\\
k_{cat} \ &\stackrel{\mathrm{def}}{=}\  \dfrac{k_3 k_2}{k_2 + k_3}
\end{align} </math>
We see that for the limiting case <math>k_3 \gg k_2</math>, thus when the last step from <chem>EI -> E + P</chem> is much faster than the previous step, we get again the original equation. Mathematically we have then <math> K_M^{\prime} \approx K_M </math> and <math>k_{cat} \approx k_2</math>.

==Multi-substrate reactions==
Multi-substrate reactions follow complex rate equations that describe how the substrates bind and in what sequence. The analysis of these reactions is much simpler if the concentration of substrate A is kept constant and substrate B varied. Under these conditions, the enzyme behaves just like a single-substrate enzyme and a plot of ''v'' by [S] gives apparent ''K''<sub>M</sub> and ''V''<sub>max</sub> constants for substrate B. If a set of these measurements is performed at different fixed concentrations of A, these data can be used to work out what the mechanism of the reaction is. For an enzyme that takes two substrates A and B and turns them into two products P and Q, there are two types of mechanism: ternary complex and substituted-enzyme mechanisms.

===Ternary-complex mechanisms===
[[File:Random order ternary mechanism.svg|thumb|310px|right|Random-order ternary-complex mechanism for an enzyme reaction. The reaction path is shown as a line and enzyme intermediates containing substrates A and B or products P and Q are written below the line.]]
Source:<ref>In older papers these have also been called single-displacement mechanisms, but this term is now little used.</ref>

In these enzymes, both substrates bind to the enzyme at the same time to produce an EAB ternary complex. The order of binding can either be random (in a random mechanism) or substrates have to bind in a particular sequence (in an ordered mechanism). When a set of ''v'' by [S] curves (fixed A, varying B) from an enzyme with a ternary-complex mechanism are plotted in a [[Lineweaver–Burk plot]], the set of lines produced will intersect.

Enzymes with ternary-complex mechanisms include [[Glutathione S-transferase|glutathione ''S''-transferase]],<ref>{{cite journal | vauthors = Dirr H, Reinemer P, Huber R | title = X-ray crystal structures of cytosolic glutathione S-transferases. Implications for protein architecture, substrate recognition and catalytic function | journal = European Journal of Biochemistry | volume = 220 | issue = 3 | pages = 645–661 | date = March 1994 | pmid = 8143720 | doi = 10.1111/j.1432-1033.1994.tb18666.x | doi-access =  }}</ref> [[dihydrofolate reductase]]<ref>{{cite journal | vauthors = Stone SR, Morrison JF | title = Dihydrofolate reductase from Escherichia coli: the kinetic mechanism with NADPH and reduced acetylpyridine adenine dinucleotide phosphate as substrates | journal = Biochemistry | volume = 27 | issue = 15 | pages = 5493–5499 | date = July 1988 | pmid = 3052577 | doi = 10.1021/bi00415a016 }}</ref> and [[DNA polymerase]].<ref>{{cite book |author=Fisher PA |title=Enzymologic mechanism of replicative DNA polymerases in higher eukaryotes |volume=47 |pages=[https://archive.org/details/progressinnuclei0000unse/page/371 371–97] |year=1994 |pmid=8016325 |doi=10.1016/S0079-6603(08)60257-3 |isbn=978-0-12-540047-3 |series=Progress in Nucleic Acid Research and Molecular Biology |url=https://archive.org/details/progressinnuclei0000unse/page/371 }}</ref> The following links show short animations of the ternary-complex mechanisms of the enzymes dihydrofolate reductase{{Ref_label|B|β|none}} and DNA polymerase{{Ref_label|C|γ|none}}.

===Substituted-enzyme ("ping–pong") mechanisms===
Source:<ref>In older papers these have also been called double-displacement mechanisms, but this term is now little used.</ref>

{{Image frame|content=<chem>\overset{}{E ->[\ce{A \atop \downarrow}] EA <=> E^\ast P ->[\ce{P \atop \uparrow}] E^\ast ->[\ce{B \atop \downarrow}] E^\ast B <=> EQ ->[\ce{Q \atop \uparrow}] E}</chem>|width=410|align=right|caption=Ping–pong mechanism for an enzyme reaction. Intermediates contain substrates A and B or products P and Q.}}
As shown on the right, enzymes with a substituted-enzyme mechanism can exist in two states, E and a chemically modified form of the enzyme E*; this modified enzyme is known as an [[Reactive intermediate|intermediate]]. In such mechanisms, substrate A binds, changes the enzyme to E* by, for example, transferring a chemical group to the active site, and is then released. Only after the first substrate is released can substrate B bind and react with the modified enzyme, regenerating the unmodified E form. When a set of ''v'' by [S] curves (fixed A, varying B) from an enzyme with a substituted-enzyme mechanism are plotted in a Lineweaver–Burk plot, a set of parallel lines will be produced.  This is called a [[Secondary plot (kinetics)|secondary plot]].

Enzymes with substituted-enzyme mechanisms include some [[oxidoreductases]] such as [[peroxidase|thioredoxin peroxidase]],<ref>{{cite journal | vauthors = Akerman SE, Müller S | title = 2-Cys peroxiredoxin PfTrx-Px1 is involved in the antioxidant defence of Plasmodium falciparum | journal = Molecular and Biochemical Parasitology | volume = 130 | issue = 2 | pages = 75–81 | date = August 2003 | pmid = 12946843 | doi = 10.1016/S0166-6851(03)00161-0 }}</ref> [[transferases]] such as acylneuraminate cytidylyltransferase<ref>{{cite journal | vauthors = Bravo IG, Barrallo S, Ferrero MA, Rodríguez-Aparicio LB, Martínez-Blanco H, Reglero A | title = Kinetic properties of the acylneuraminate cytidylyltransferase from Pasteurella haemolytica A2 | journal = The Biochemical Journal | volume = 358 | issue = Pt 3 | pages = 585–598 | date = September 2001 | pmid = 11577688 | pmc = 1222114 | doi = 10.1042/bj3580585 }}</ref> and [[serine protease]]s such as [[trypsin]] and [[chymotrypsin]].<ref>{{cite journal | vauthors = Kraut J | title = Serine proteases: structure and mechanism of catalysis | journal = Annual Review of Biochemistry | volume = 46 | pages = 331–358 | year = 1977 | pmid = 332063 | doi = 10.1146/annurev.bi.46.070177.001555 }}</ref>  Serine proteases are a very common and diverse family of enzymes, including [[digestion|digestive]] enzymes (trypsin, chymotrypsin, and elastase), several enzymes of the [[Coagulation|blood clotting cascade]] and many others. In these serine proteases, the E* intermediate is an acyl-enzyme species formed by the attack of an active site [[serine]] residue on a [[peptide bond]] in a protein substrate. A short animation showing the mechanism of chymotrypsin is linked here.{{Ref_label|D|δ|none}}

===Memory effects===

Both of these two types of mechanism can display [[enzyme memory]], with very different causes and consequences in the two cases. In ternary complex mechanisms these are possible if the mechanism includes slow processes and the binding steps are not at quasi-equilibrium, because the intermediates may be swept away very fast. This can generate [[cooperativity]], even in monomeric enzymes.<ref>{{cite journal | doi= 10.1016/s0022-5193(87)80248-5 | journal = J. Theor. Biol.  | title = Co-operativity in monomeric enzymes |first2 = María Luz | last2 = Cárdenas |first1 = Athel|last1 = Cornish-Bowden | year =1987  | volume = 124 | number = 1 |  pages = 1–23| pmid = 3309473 | bibcode = 1987JThBi.124....1C }}</ref> In a substituted-enzyme mechanism slow steps are not needed to generate memory effects. Instead, for an enzyme with several alternative substrates the kinetic properties of the second half reaction may vary with different substrates in the first half reaction, even though the same substituted enzyme seems to be transformed.<ref name= jarabak>{{cite journal | doi= 10.1021/bi00713a007|title = Enzymic memory. Consequence of conformational mobility | first1 = Rebecca | last1 = Jarabak  | first2 = John | last2 = Westley | journal = Biochemistry | year=1974 | volume = 13 |number = 16 |pages = 3237–3239|pmid = 4842741 }}</ref>

== Reversible catalysis and the Haldane equation ==
{{main article|Reversible Michaelis-Menten kinetics}}
External factors may limit the ability of an enzyme to catalyse a reaction in both directions (whereas the nature of a catalyst in itself means that it cannot catalyse just one direction, according to the principle of [[microscopic reversibility]]). We consider the case of an enzyme that catalyses the reaction in both directions:

<chem> 
 {E} + {S} 
   <=>[k_{1}][k_{-1}]
 ES
   <=>[k_{2}][k_{-2}]
 {E} + {P}
</chem>

The steady-state, initial rate of the reaction is  <math> v_0 = \frac{d\, [{\rm P}]}{dt} = \frac{(k_1 k_2\, [{\rm S}]- k_{-1}k_{-2}[{\rm P}])[{\rm E}]_0}{k_{-1}+k_2+k_1\,
  [{\rm S}]+k_{-2}\, [{\rm P}]} </math>

<math>v_0</math> is positive if the reaction proceed in the forward direction (<math>S \rightarrow P</math>) and negative otherwise.

[[:en:Equilibrium constant|Equilibrium]] requires that <math>v=0</math>, which occurs when <math>
\frac{[{\rm P}]_{\rm eq}}{[{\rm S}]_{\rm eq}}=\frac{k_1 k_2}{k_{-1} k_{-2}}=K_{\rm eq}
</math>. This shows that [[:en:Equilibrium constant|thermodynamics]] forces a relation between the values of the 4 rate constants.

The values of the forward and backward ''maximal'' rates, obtained for <math>[{\rm S}] \rightarrow \infty</math>,  <math>[{\rm P}] =0</math>, and  <math>[{\rm S}] =0 </math>,  <math>[{\rm P}] \rightarrow \infty</math>, respectively, are <math>V_{\rm max}^f = k_{2} {\rm [E]}_{tot}</math> and <math>V_{\rm max}^b = -k_{-1} {\rm [E]}_{tot}</math>, respectively. Their ratio is not equal to the equilibrium constant, which implies that [[:en:Equilibrium constant|thermodynamics]] does not constrain the ratio of the maximal rates. This explains that enzymes can be much "better catalysts" (''in terms of maximal rates'') in one particular direction of the reaction.<ref>{{cite book|author-link1=Athel Cornish-Bowden|last=Cornish-Bowden |first=Athel | name-list-style = vanc |date=2012|title=Fundamentals of Enzyme Kinetics |publisher=Wiley-Blackwell |quote=Some enzymes are much more effective catalysts for one direction than the other. As a striking example, the limiting rates of the forward reaction catalyzed by methionine adenosyltransferase is about 10<sup>5</sup> greater than that for the reverse direction, even though the equilibrium constant is close to unity (page 59).}}</ref>

On can also derive the two Michaelis constants <math>K_M^S=(k_{-1}+k_2)/k_1</math>and  <math>K_M^P=(k_{-1}+k_2)/k_{-2}</math>. The Haldane equation is the relation <math>K_{\rm eq}=\frac{[{\rm P}]_{\rm eq}}{[{\rm S}]_{\rm eq}}=\frac{V_{\rm max}^f / K_M^S}{V_{\rm max}^b / K_M^P}</math>.

Therefore, [[:en:Equilibrium constant|thermodynamics]] constrains the ratio between the forward and backward <math>V_{\rm max}/K_M</math> values, not the ratio of <math>V_{\rm max}</math>values.

==Non-Michaelis–Menten kinetics==
{{Main|Allosteric regulation}}
[[File:Allosteric v by S curve.svg|thumb|300px|left|Saturation curve for an enzyme reaction showing sigmoid kinetics.]]
Many different enzyme systems follow non Michaelis-Menten behavior. A select few examples include kinetics of self-catalytic enzymes, cooperative and allosteric enzymes, interfacial and intracellular enzymes, processive enzymes and so forth. Some enzymes produce a [[sigmoid function|sigmoid]] ''v'' by [S] plot, which often indicates [[cooperative binding]] of substrate to the active site. This means that the binding of one substrate molecule affects the binding of subsequent substrate molecules. This behavior is most common in [[protein structure|multimeric]] enzymes with several interacting active sites.<ref>{{cite journal | vauthors = Ricard J, Cornish-Bowden A | title = Co-operative and allosteric enzymes: 20 years on | journal = European Journal of Biochemistry | volume = 166 | issue = 2 | pages = 255–272 | date = July 1987 | pmid = 3301336 | doi = 10.1111/j.1432-1033.1987.tb13510.x | doi-access = free }}</ref> Here, the mechanism of cooperation is similar to that of [[hemoglobin]], with binding of substrate to one active site altering the affinity of the other active sites for substrate molecules. Positive cooperativity occurs when binding of the first substrate molecule ''increases'' the affinity of the other active sites for substrate. Negative cooperativity occurs when binding of the first substrate ''decreases'' the affinity of the enzyme for other substrate molecules.

Allosteric enzymes include mammalian tyrosyl tRNA-synthetase, which shows negative cooperativity,<ref>{{cite journal | vauthors = Ward WH, Fersht AR | title = Tyrosyl-tRNA synthetase acts as an asymmetric dimer in charging tRNA. A rationale for half-of-the-sites activity | journal = Biochemistry | volume = 27 | issue = 15 | pages = 5525–5530 | date = July 1988 | pmid = 3179266 | doi = 10.1021/bi00415a021 }}</ref> and bacterial [[aspartate transcarbamoylase]]<ref>{{cite journal | vauthors = Helmstaedt K, Krappmann S, Braus GH | title = Allosteric regulation of catalytic activity: Escherichia coli aspartate transcarbamoylase versus yeast chorismate mutase | journal = Microbiology and Molecular Biology Reviews | volume = 65 | issue = 3 | pages = 404–21, table of contents | date = September 2001 | pmid = 11528003 | pmc = 99034 | doi = 10.1128/MMBR.65.3.404-421.2001 }}</ref> and [[phosphofructokinase]],<ref>{{cite journal | vauthors = Schirmer T, Evans PR | title = Structural basis of the allosteric behaviour of phosphofructokinase | journal = Nature | volume = 343 | issue = 6254 | pages = 140–145 | date = January 1990 | pmid = 2136935 | doi = 10.1038/343140a0 | s2cid = 4272821 | bibcode = 1990Natur.343..140S }}</ref> which show positive cooperativity.

Cooperativity is surprisingly common and can help regulate the responses of enzymes to changes in the concentrations of their substrates. Positive cooperativity makes enzymes much more sensitive to [S] and their activities can show large changes over a narrow range of substrate concentration. Conversely, negative cooperativity makes enzymes insensitive to small changes in [S].

The [[Hill equation (biochemistry)|Hill equation]]<ref>{{cite journal | vauthors = Hill AV | title = The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. | journal = J. Physiol. | date = 1910 | volume = 40 | pages = iv–vii }}</ref> is often used to describe the degree of cooperativity quantitatively in non-Michaelis–Menten kinetics.  The derived Hill coefficient ''n'' measures how much the binding of substrate to one active site affects the binding of substrate to the other active sites. A Hill coefficient of <1 indicates negative cooperativity and a coefficient of >1 indicates positive [[cooperativity]].

==Pre-steady-state kinetics==
[[File:Burst phase.svg|thumb|300px|right|Pre-steady state progress curve, showing the burst phase of an enzyme reaction.]]
In the first moment after an enzyme is mixed with substrate, no product has been formed and no [[reactive intermediate|intermediate]]s exist. The study of the next few milliseconds of the reaction is called pre-steady-state kinetics. Pre-steady-state kinetics is therefore concerned with the formation and consumption of enzyme–substrate intermediates (such as ES or E*) until their [[steady state (chemistry)|steady-state concentrations]] are reached.

This approach was first applied to the hydrolysis reaction catalysed by [[chymotrypsin]].<ref>{{cite journal | vauthors = Hartley BS, Kilby BA | title = The reaction of p-nitrophenyl esters with chymotrypsin and insulin | journal = The Biochemical Journal | volume = 56 | issue = 2 | pages = 288–297 | date = February 1954 | pmid = 13140189 | pmc = 1269615 | doi = 10.1042/bj0560288 }}</ref> Often, the detection of an intermediate is a vital piece of evidence in investigations of what mechanism an enzyme follows. For example, in the ping–pong mechanisms that are shown above, rapid kinetic measurements can follow the release of product P and measure the formation of the modified enzyme intermediate E*.<ref name=Fersht>{{cite book|author=Fersht, Alan |title=Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding |publisher=W.H. Freeman |location=San Francisco |year=1999 |isbn=978-0-7167-3268-6 }}</ref> In the case of chymotrypsin, this intermediate is formed by an attack on the substrate by the [[nucleophile|nucleophilic]] serine in the active site and the formation of the acyl-enzyme intermediate.

In the figure to the right, the enzyme produces E* rapidly in the first few seconds of the reaction. The rate then slows as steady state is reached. This rapid burst phase of the reaction measures a single turnover of the enzyme. Consequently, the amount of product released in this burst, shown as the intercept on the ''y''-axis of the graph, also gives the amount of functional enzyme which is present in the assay.<ref>{{cite journal | vauthors = Bender ML, Begué-Cantón ML, Blakeley RL, Brubacher LJ, Feder J, Gunter CR, Kézdy FJ, Killheffer JV, Marshall TH, Miller CG, Roeske RW, Stoops JK | display-authors = 6 | title = The determination of the concentration of hydrolytic enzyme solutions: alpha-chymotrypsin, trypsin, papain, elastase, subtilisin, and acetylcholinesterase | journal = Journal of the American Chemical Society | volume = 88 | issue = 24 | pages = 5890–5913 | date = December 1966 | pmid = 5980876 | doi = 10.1021/ja00976a034 | bibcode = 1966JAChS..88.5890B }}</ref>

==Chemical mechanism==
An important goal of measuring enzyme kinetics is to determine the chemical mechanism of an enzyme reaction, i.e., the sequence of chemical steps that transform substrate into product. The kinetic approaches discussed above will show at what rates [[Reactive intermediate|intermediates]] are formed and inter-converted, but they cannot identify exactly what these intermediates are.

Kinetic measurements taken under various solution conditions or on slightly modified enzymes or substrates often shed light on this chemical mechanism, as they reveal the rate-determining step or intermediates in the reaction. For example, the breaking of a [[covalent bond]] to a [[hydrogen]] [[atom]] is a common rate-determining step. Which of the possible hydrogen transfers is rate determining can be shown by measuring the kinetic effects of substituting each hydrogen by [[deuterium]], its stable [[isotope]]. The rate will change when the critical hydrogen is replaced, due to a primary [[kinetic isotope effect]], which occurs because bonds to deuterium are harder to break than bonds to hydrogen.<ref>{{cite journal | vauthors = Cleland WW | title = The use of isotope effects to determine enzyme mechanisms | journal = Archives of Biochemistry and Biophysics | volume = 433 | issue = 1 | pages = 2–12 | date = January 2005 | pmid = 15581561 | doi = 10.1016/j.abb.2004.08.027 }}</ref> It is also possible to measure similar effects with other isotope substitutions, such as <sup>13</sup>C/<sup>12</sup>C and <sup>18</sup>O/<sup>16</sup>O, but these effects are more subtle.<ref>{{cite journal | vauthors = Northrop DB | title = The expression of isotope effects on enzyme-catalyzed reactions | journal = Annual Review of Biochemistry | volume = 50 | pages = 103–131 | year = 1981 | pmid = 7023356 | doi = 10.1146/annurev.bi.50.070181.000535 }}</ref>

Isotopes can also be used to reveal the fate of various parts of the substrate molecules in the final products. For example, it is sometimes difficult to discern the origin of an [[oxygen]] atom in the final product; since it may have come from water or from part of the substrate. This may be determined by systematically substituting oxygen's stable isotope <sup>18</sup>O into the various molecules that participate in the reaction and checking for the isotope in the product.<ref>{{cite journal | vauthors = Baillie TA, Rettenmeier AW | title = Drug biotransformation: mechanistic studies with stable isotopes | journal = Journal of Clinical Pharmacology | volume = 26 | issue = 6 | pages = 448–451 | year = 1986 | pmid = 3734135 | doi = 10.1002/j.1552-4604.1986.tb03556.x | s2cid = 39193680 }}</ref> The chemical mechanism can also be elucidated by examining the kinetics and isotope effects under different pH conditions,<ref>{{cite journal | vauthors = Cleland WW | title = Use of isotope effects to elucidate enzyme mechanisms | journal = CRC Critical Reviews in Biochemistry | volume = 13 | issue = 4 | pages = 385–428 | year = 1982 | pmid = 6759038 | doi = 10.3109/10409238209108715 }}</ref> by altering the metal ions or other bound [[Cofactor (biochemistry)|cofactor]]s,<ref>{{cite journal |author1-link=David W. Christianson| vauthors = Christianson DW, Cox JD | title = Catalysis by metal-activated hydroxide in zinc and manganese metalloenzymes | journal = Annual Review of Biochemistry | volume = 68 | pages = 33–57 | year = 1999 | pmid = 10872443 | doi = 10.1146/annurev.biochem.68.1.33 }}</ref>  by [[site-directed mutagenesis]] of conserved amino acid residues, or by studying the behaviour of the enzyme in the presence of analogues of the substrate(s).<ref>{{cite journal | vauthors = Kraut DA, Carroll KS, Herschlag D | title = Challenges in enzyme mechanism and energetics | journal = Annual Review of Biochemistry | volume = 72 | pages = 517–571 | year = 2003 | pmid = 12704087 | doi = 10.1146/annurev.biochem.72.121801.161617 }}</ref>

==Enzyme inhibition and activation==
{{Main|Enzyme inhibitor}}
[[File:Reversible inhibition.svg|thumb|300px|right|Kinetic scheme for reversible enzyme inhibitors.]]

Enzyme inhibitors are molecules that reduce or abolish enzyme activity, while enzyme activators are molecules that increase the catalytic rate of enzymes. These interactions can be either ''reversible'' (i.e., removal of the inhibitor restores enzyme activity) or ''irreversible'' (i.e., the inhibitor permanently inactivates the enzyme).

===Reversible inhibitors===
Traditionally reversible enzyme inhibitors have been classified as [[Competitive inhibition|competitive]], [[Uncompetitive inhibition|uncompetitive]], or [[Non-competitive inhibition|non-competitive]], according to their effects on ''K''<sub>M</sub> and ''V''<sub>max</sub>. These different effects result from the inhibitor binding to the enzyme E, to the enzyme–substrate complex ES, or to both, respectively.  The division of these classes arises from a problem in their derivation and results in the need to use two different binding constants for one binding event.  The binding of an inhibitor and its effect on the enzymatic activity are two distinctly different things, another problem the traditional equations fail to acknowledge. In noncompetitive inhibition the binding of the inhibitor results in 100% inhibition of the enzyme only, and fails to consider the possibility of anything in between.<ref>{{cite journal | vauthors = Walsh R, Martin E, Darvesh S | title = Limitations of conventional inhibitor classifications | journal = Integrative Biology | volume = 3 | issue = 12 | pages = 1197–1201 | date = December 2011 | pmid = 22038120 | doi = 10.1039/c1ib00053e }}</ref>  In noncompetitive inhibition, the inhibitor will bind to an enzyme at its allosteric site; therefore, the binding affinity, or inverse of ''K''<sub>M</sub>, of the substrate with the enzyme will remain the same. On the other hand, the V<sub>max</sub> will decrease relative to an uninhibited enzyme. On a Lineweaver-Burk plot, the presence of a noncompetitive inhibitor is illustrated by a change in the y-intercept, defined as 1/V<sub>max</sub>. The x-intercept, defined as −1/''K''<sub>M</sub>, will remain the same. In competitive inhibition, the inhibitor will bind to an enzyme at the active site, competing with the substrate. As a result, the ''K''<sub>M</sub> will increase and the V<sub>max</sub> will remain the same.<ref>{{cite journal | vauthors = Cleland WW | title = The kinetics of enzyme-catalyzed reactions with two or more substrates or products. III. Prediction of initial velocity and inhibition patterns by inspection | journal = Biochimica et Biophysica Acta | volume = 67 | pages = 188–196 | date = February 1963 | pmid = 14021669 | doi = 10.1016/0006-3002(63)91816-x }}</ref> The common form of the inhibitory term also obscures the relationship between the inhibitor binding to the enzyme and its relationship to any other binding term be it the Michaelis–Menten equation or a dose response curve associated with ligand receptor binding. To demonstrate the relationship the following rearrangement can be made:

:<math>\cfrac{V_\max}{1 + \cfrac{[I]}{K_i}} 
=
\cfrac{V_\max}{\cfrac{[I]+K_i}{K_i}} </math>

Adding zero to the bottom ([I]-[I])

:<math>\cfrac{V_\max}{\cfrac{[I]+K_i}{[I]+K_i-[I]}} </math>

Dividing by [I]+K<sub>i</sub>

:<math>\cfrac{V_\max}{\cfrac{1}{1 - \cfrac{[I]}{[I]+K_i}}}
=
V_\max - V_\max \cfrac{[I]}{[I]+K_i} </math>

This notation demonstrates that similar to the Michaelis–Menten equation, where the rate of reaction depends on the percent of the enzyme population interacting with substrate, the effect of the inhibitor is a result of the percent of the enzyme population interacting with inhibitor.  The only problem with this equation in its present form is that it assumes absolute inhibition of the enzyme with inhibitor binding, when in fact there can be a wide range of effects anywhere from 100% inhibition of substrate turn over to just >0%.  To account for this the equation can be easily modified to allow for different degrees of inhibition by including a delta ''V''<sub>max</sub> term.

:<math>V_\max - \Delta V_\max \cfrac{[I]}{[I]+K_i} </math>

or

:<math>V_{\max1} -  (V_{\max1} - V_{\max2} ) \cfrac{[I]}{[I]+K_i} </math>

This term can then define the residual enzymatic activity present when the inhibitor is interacting with individual enzymes in the population.  However the inclusion of this term has the added value of allowing for the possibility of activation if the secondary ''V''<sub>max</sub> term turns out to be higher than the initial term. To account for the possibly of activation as well the notation can then be rewritten replacing the inhibitor "I" with a modifier term denoted here as "X".

:<math>V_{\max1} -  (V_{\max1} - V_{\max2} ) \cfrac{[X]}{[X]+K_x} </math>

While this terminology results in a simplified way of dealing with kinetic effects relating to the maximum velocity of the Michaelis–Menten equation, it highlights potential problems with the term used to describe effects relating to the ''K''<sub>M</sub>.  The ''K''<sub>M</sub> relating to the affinity of the enzyme for the substrate should in most cases relate to potential changes in the binding site of the enzyme which would directly result from enzyme inhibitor interactions.  As such a term similar to the one proposed above to modulate ''V''<sub>max</sub> should be appropriate in most situations:<ref>{{cite journal | vauthors = Walsh R, Martin E, Darvesh S | title = A versatile equation to describe reversible enzyme inhibition and activation kinetics: modeling beta-galactosidase and butyrylcholinesterase | journal = Biochimica et Biophysica Acta (BBA) - General Subjects | volume = 1770 | issue = 5 | pages = 733–746 | date = May 2007 | pmid = 17307293 | doi = 10.1016/j.bbagen.2007.01.001 }}</ref>

:<math>K_{m1} -  (K_{m1} - K_{m2} ) \cfrac{[X]}{[X]+K_x} </math>

===Irreversible inhibitors===
Enzyme inhibitors can also irreversibly inactivate enzymes, usually by covalently modifying active site residues. These reactions, which may be called suicide substrates, follow [[exponential decay]] functions and are usually saturable. Below saturation, they follow [[reaction rate|first order]] kinetics with respect to inhibitor. Irreversible inhibition could be classified into two distinct types. Affinity labelling is a type of irreversible inhibition where a functional group that is highly reactive modifies a catalytically critical residue on the protein of interest to bring about inhibition.  Mechanism-based inhibition, on the other hand, involves binding of the inhibitor followed by enzyme mediated alterations that transform the latter into a reactive group that irreversibly modifies the enzyme.

=== Philosophical discourse on reversibility and irreversibility of inhibition ===
Having discussed reversible inhibition and irreversible inhibition in the above two headings, it would have to be pointed out that the concept of reversibility (or irreversibility) is a purely theoretical construct exclusively dependent on the time-frame of the assay, i.e., a reversible assay involving association and dissociation of the inhibitor molecule in the minute timescales would seem irreversible if an assay assess the outcome in the seconds and vice versa. There is a continuum of inhibitor behaviors spanning reversibility and irreversibility at a given non-arbitrary assay time frame.  There are inhibitors that show slow-onset behavior and most of these inhibitors, invariably, also show tight-binding to the protein target of interest.

==Mechanisms of catalysis==
{{Main|Enzyme catalysis}}
[[File:Activation2 updated.svg|thumb|300px|The energy variation as a function of [[reaction coordinate]] shows the stabilisation of the transition state by an enzyme.]]

The favoured model for the enzyme–substrate interaction is the induced fit model.<ref>{{cite journal | vauthors = Koshland DE | title = Application of a Theory of Enzyme Specificity to Protein Synthesis | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 44 | issue = 2 | pages = 98–104 | date = February 1958 | pmid = 16590179 | pmc = 335371 | doi = 10.1073/pnas.44.2.98 | doi-access = free | bibcode = 1958PNAS...44...98K }}</ref> This model proposes that the initial interaction between enzyme and substrate is relatively weak, but that these weak interactions rapidly induce [[conformational change]]s in the enzyme that strengthen binding. These [[Tertiary structure|conformational]] changes also bring catalytic residues in the active site close to the chemical bonds in the substrate that will be altered in the reaction.<ref>{{cite journal | vauthors = Hammes GG | title = Multiple conformational changes in enzyme catalysis | journal = Biochemistry | volume = 41 | issue = 26 | pages = 8221–8228 | date = July 2002 | pmid = 12081470 | doi = 10.1021/bi0260839 }}</ref> Conformational changes can be measured using [[circular dichroism]] or [[dual polarisation interferometry]]. After binding takes place, one or more mechanisms of catalysis lower the energy of the reaction's [[transition state]] by providing an alternative chemical pathway for the reaction. Mechanisms of catalysis include catalysis by bond strain; by proximity and orientation; by active-site proton donors or acceptors; covalent catalysis and [[quantum tunnelling]].<ref name=Fersht/><ref>{{cite journal | vauthors = Sutcliffe MJ, Scrutton NS | title = A new conceptual framework for enzyme catalysis. Hydrogen tunnelling coupled to enzyme dynamics in flavoprotein and quinoprotein enzymes | journal = European Journal of Biochemistry | volume = 269 | issue = 13 | pages = 3096–3102 | date = July 2002 | pmid = 12084049 | doi = 10.1046/j.1432-1033.2002.03020.x | doi-access =  }}</ref>

Enzyme kinetics cannot prove which modes of catalysis are used by an enzyme. However, some kinetic data can suggest possibilities to be examined by other techniques. For example, a ping–pong mechanism with burst-phase pre-steady-state kinetics would suggest covalent catalysis might be important in this enzyme's mechanism. Alternatively, the observation of a strong pH effect on ''V''<sub>max</sub> but not ''K''<sub>M</sub> might indicate that a residue in the active site needs to be in a particular [[Ionization|ionisation]] state for catalysis to occur.

==History==

In 1902 [[Victor Henri]] proposed a quantitative theory of enzyme kinetics,<ref>{{cite journal | vauthors = Henri V | year = 1902 | title = Theorie generale de l'action de quelques diastases|journal=Compt. Rend. Acad. Sci. Paris|volume=135|pages=916–9}}</ref> but at the time the experimental significance of the [[pH|hydrogen ion concentration]] was not yet recognized. After [[S. P. L. Sørensen|Peter Lauritz Sørensen]] had defined the logarithmic pH-scale and introduced the concept of [[buffer solution|buffering]] in 1909<ref>{{cite journal | vauthors = Sørensen PL | year = 1909 | title = Enzymstudien {II}. Über die Messung und Bedeutung der Wasserstoffionenkonzentration bei enzymatischen Prozessen | journal = Biochem. Z. | volume = 21 | pages = 131–304 | language = de | trans-title = Enzyme studies III: About the measurement and significance of the hydrogen ion concentration in enzymatic processes  }}</ref> the German chemist [[Leonor Michaelis]] and Dr. [[Maud Leonora Menten]] (a postdoctoral researcher in Michaelis's lab at the time) repeated Henri's experiments and confirmed his equation, which is now generally referred to as [[Michaelis-Menten kinetics]] (sometimes also ''Henri-Michaelis-Menten kinetics'').<ref>{{cite journal | vauthors = Michaelis L, Menten M | year = 1913 | title = Die Kinetik der Invertinwirkung | journal = Biochem. Z. | volume = 49 | pages = 333–369 | language = de | trans-title = The Kinetics of Invertase Action }}; {{cite journal | vauthors = Michaelis L, Menten ML, Johnson KA, Goody RS | title = The original Michaelis constant: translation of the 1913 Michaelis-Menten paper | journal = Biochemistry | volume = 50 | issue = 39 | pages = 8264–8269 | date = October 2011 | pmid = 21888353 | pmc = 3381512 | doi = 10.1021/bi201284u }}</ref> Their work was further developed by [[George Edward Briggs|G. E. Briggs]] and [[J. B. S. Haldane]], who derived kinetic equations that are still widely considered today a starting point in modeling enzymatic activity.<ref>{{cite journal | vauthors = Briggs GE, Haldane JB | title = A Note on the Kinetics of Enzyme Action | journal = The Biochemical Journal | volume = 19 | issue = 2 | pages = 338–339 | year = 1925 | pmid = 16743508 | pmc = 1259181 | doi = 10.1042/bj0190338 }}</ref>

The major contribution of the Henri-Michaelis-Menten approach was to think of enzyme reactions in two stages. In the first, the substrate binds reversibly to the enzyme, forming the enzyme-substrate complex. This is sometimes called the Michaelis complex. The enzyme then catalyzes the chemical step in the reaction and releases the product. The kinetics of many enzymes is adequately described by the simple Michaelis-Menten model, but all enzymes have [[protein dynamics|internal motions]] that are not accounted for in the model and can have significant contributions to the overall reaction kinetics. This can be modeled by introducing several Michaelis-Menten pathways that are connected with fluctuating rates,<ref name=OF_2005>{{cite journal | vauthors = Flomenbom O, Velonia K, Loos D, Masuo S, Cotlet M, Engelborghs Y, Hofkens J, Rowan AE, Nolte RJ, Van der Auweraer M, de Schryver FC, Klafter J | display-authors = 6 | title = Stretched exponential decay and correlations in the catalytic activity of fluctuating single lipase molecules | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 102 | issue = 7 | pages = 2368–2372 | date = February 2005 | pmid = 15695587 | pmc = 548972 | doi = 10.1073/pnas.0409039102 | doi-access = free | bibcode = 2005PNAS..102.2368F }}</ref><ref name=English_2006>{{cite journal | vauthors = English BP, Min W, van Oijen AM, Lee KT, Luo G, Sun H, Cherayil BJ, Kou SC, Xie XS | display-authors = 6 | title = Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited | journal = Nature Chemical Biology | volume = 2 | issue = 2 | pages = 87–94 | date = February 2006 | pmid = 16415859 | doi = 10.1038/nchembio759 | s2cid = 2201882 }}</ref><ref name=Lu_1998>{{cite journal | vauthors = Lu HP, Xun L, Xie XS | title = Single-molecule enzymatic dynamics | journal = Science | volume = 282 | issue = 5395 | pages = 1877–1882 | date = December 1998 | pmid = 9836635 | doi = 10.1126/science.282.5395.1877 | bibcode = 1998Sci...282.1877P }}</ref> which is a mathematical extension of the basic Michaelis Menten mechanism.<ref name=XX_2006>{{cite journal | vauthors = Xue X, Liu F, Ou-Yang ZC | title = Single molecule Michaelis-Menten equation beyond quasistatic disorder | journal = Physical Review E| volume = 74 | issue = 3 Pt 1 | pages = 030902 | date = September 2006 | pmid = 17025584 | doi = 10.1103/PhysRevE.74.030902 | arxiv = cond-mat/0604364 | s2cid = 41674948 | bibcode = 2006PhRvE..74c0902X }}</ref>

==Software==

'''ENZO''' (Enzyme Kinetics) is a graphical interface tool for building kinetic models of enzyme catalyzed reactions. ENZO automatically generates the corresponding differential equations from a stipulated enzyme reaction scheme. These differential equations are processed by a numerical solver and a regression algorithm which fits the coefficients of differential equations to experimentally observed time course curves. ENZO allows rapid evaluation of rival reaction schemes and can be used for routine tests in enzyme kinetics.<ref>{{cite journal | vauthors = Bevc S, Konc J, Stojan J, Hodošček M, Penca M, Praprotnik M, Janežič D | title = ENZO: a web tool for derivation and evaluation of kinetic models of enzyme catalyzed reactions | journal = PLOS ONE | volume = 6 | issue = 7 | pages = e22265 | year = 2011 | pmid = 21818304 | pmc = 3139599 | doi = 10.1371/journal.pone.0022265 | doi-access = free | bibcode = 2011PLoSO...622265B }} 
[http://enzo.cmm.ki.si/ ENZO server]</ref>

== See also ==
*[[Protein dynamics]]
*[[Diffusion limited enzyme]]
*[[Langmuir adsorption model]]

==Footnotes==
:'''α.''' {{Note_label|A|α|none}}[https://web.archive.org/web/20120616044431/http://cti.itc.virginia.edu/~cmg/Demo/scriptFrame.html Link: Interactive Michaelis–Menten kinetics tutorial (Java required)]

:'''β.''' {{Note_label|B|β|none}}[https://web.archive.org/web/20060820062258/http://chem-faculty.ucsd.edu/kraut/dhfr.html Link: dihydrofolate reductase mechanism (Gif)]

:'''γ.''' {{Note_label|C|γ|none}}[https://web.archive.org/web/20060514070741/http://chem-faculty.ucsd.edu/kraut/dNTP.html Link: DNA polymerase mechanism (Gif)]

:'''δ.''' {{Note_label|D|δ|none}}[https://web.archive.org/web/20070319235224/http://courses.cm.utexas.edu/jrobertus/ch339k/overheads-2/06_21_chymotrypsin.html Link: Chymotrypsin mechanism (Flash required)]

== References ==
{{Reflist|35em}}

== Further reading ==
'''Introductory'''
* {{cite book| vauthors = Cornish-Bowden A |title=Fundamentals of enzyme kinetics |publisher=Wiley-Blackwell |location=Weinheim |year=2012 |isbn = 978-3-527-33074-4 |edition=4th}}
* {{cite book| vauthors = Stevens L, Price NC |title=Fundamentals of enzymology: the cell and molecular biology of catalytic proteins | url = https://archive.org/details/fundamentalsofen0000pric_h0c7 | url-access = registration |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=1999 |isbn=978-0-19-850229-6 }}
* {{cite book| vauthors = Bugg T |title=Introduction to Enzyme and Coenzyme Chemistry |publisher=Blackwell Publishers |location=Cambridge, MA |year=2004 |isbn=978-1-4051-1452-3 }}
* {{cite book| vauthors = Segel IH |title=Enzyme kinetics: behavior and analysis of rapid equilibrium and steady state enzyme systems |publisher=Wiley |location=New York |year=1993 |isbn=978-0-471-30309-1}}

'''Advanced'''
* {{cite book| vauthors = Fersht A |title=Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding |publisher=W.H. Freeman |location=San Francisco |year=1999 |isbn=978-0-7167-3268-6 }}
* {{cite journal| vauthors = Schnell S, Maini PK |title=A century of enzyme kinetics: Reliability of the K<sub>M</sub> and v<sub>max</sub> estimates |journal=Comments on Theoretical Biology |volume=8 |pages=169–87 |year=2004 |doi=10.1080/08948550302453 |issue=2–3 |citeseerx=10.1.1.493.7178 }}
* {{cite book| vauthors = Walsh C |title=Enzymatic reaction mechanisms |publisher=W. H. Freeman |location=San Francisco |year=1979 |isbn=978-0-7167-0070-8 }}
* {{cite book| vauthors = Cleland WW, Cook P |title=Enzyme kinetics and mechanism |publisher=Garland Science |location=New York |year=2007 |isbn=978-0-8153-4140-6 }}

== External links ==
* [http://www.kscience.co.uk/animations/model.swf Animation of an enzyme assay] — Shows effects of manipulating assay conditions
* [http://www.ebi.ac.uk/thornton-srv/databases/MACiE/ MACiE] — A database of enzyme reaction mechanisms
* [https://web.archive.org/web/20061006064937/http://us.expasy.org/enzyme/ ENZYME] — Expasy enzyme nomenclature database
* [http://enzo.cmm.ki.si/ ENZO] — Web application for easy construction and quick testing of kinetic models of enzyme catalyzed reactions.
* [http://ezcatdb.cbrc.jp/EzCatDB/ ExCatDB] — A database of enzyme catalytic mechanisms
* [http://www.brenda-enzymes.info/ BRENDA] — Comprehensive enzyme database, giving substrates, inhibitors and reaction diagrams
* [http://sabio.h-its.org/ SABIO-RK] — A database of reaction kinetics
* [https://web.archive.org/web/20060820062258/http://chem-faculty.ucsd.edu/kraut/dhfr.html Joseph Kraut's Research Group, University of California San Diego] — Animations of several enzyme reaction mechanisms
* [https://web.archive.org/web/20060712051732/http://www.chem.qmul.ac.uk/iubmb/kinetics/ Symbolism and Terminology in Enzyme Kinetics] — A comprehensive explanation of concepts and terminology in enzyme kinetics
* [https://web.archive.org/web/20040612065857/http://orion1.paisley.ac.uk/kinetics/contents.html An introduction to enzyme kinetics] — An accessible set of on-line tutorials on enzyme kinetics
* [http://www.wiley.com/college/pratt/0471393878/student/animations/enzyme_kinetics/index.html Enzyme kinetics animated tutorial] — An animated tutorial with audio

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