Michaelis–Menten kinetics

A chemical reaction mechanism with or without enzyme catalysis. The enzyme (E) binds substrate (S) to produce product (P).

Saturation curve for an enzyme reaction showing the relation between the substrate concentration and reaction rate.

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 (${\displaystyle v_{0}}$) 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_max, the enzyme's maximum rate.

The Michaelis–Menten kinetic model of a single-substrate reaction is shown on the right. There is an initial 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_max; at V_max, 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 unimolecular reaction with an order of zero. Though the enzymatic mechanism for the unimolecular reaction ${\displaystyle {\ce {ES ->[k_{cat}] E + P}}}$ 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 k_cat. If the reaction path proceeds over one or several intermediates, k_cat 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 k₂. The apparent unimolecular rate constant k_cat is also called turnover number, and denotes the maximum number of enzymatic reactions catalysed per second.

The Michaelis–Menten equation^[10] describes how the (initial) reaction rate v₀ depends on the position of the substrate-binding equilibrium and the rate constant k₂.

${\displaystyle v_{0}={\frac {V_{\max }[{\ce {S}}]}{K_{M}+[{\ce {S}}]}}}$    (Michaelis–Menten equation)

with the constants

${\displaystyle {\begin{aligned}K_{M}\ &{\stackrel {\mathrm {def} }{=}}\ {\frac {k_{2}+k_{-1}}{k_{1}}}\approx K_{D}\\V_{\max }\ &{\stackrel {\mathrm {def} }{=}}\ k_{cat}{\ce {[E]}}_{tot}\end{aligned}}}$

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 allostericity or cooperativity). The first assumption is the so-called 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 ${\displaystyle d{\ce {[ES]}}/{dt}\;{\overset {!}{=}}\;0}$. The second assumption is that the total enzyme concentration does not change over time, thus ${\displaystyle {\ce {[E]}}_{\text{tot}}={\ce {[E]}}+{\ce {[ES]}}\;{\overset {!}{=}}\;{\text{const}}}$. A complete derivation can be found here.

The Michaelis constant K_M is experimentally defined as the concentration at which the rate of the enzyme reaction is half V_max, which can be verified by substituting [S] = K_M into the Michaelis–Menten equation and can also be seen graphically. If the rate-determining enzymatic step is slow compared to substrate dissociation (${\displaystyle k_{2}\ll k_{-1}}$), the Michaelis constant K_M is roughly the dissociation constant K_D of the ES complex.

If ${\displaystyle {\ce {[S]}}}$ is small compared to ${\displaystyle K_{M}}$ then the term ${\displaystyle [{\ce {S}}]/(K_{M}+[{\ce {S}}])\approx [{\ce {S}}]/K_{M}}$ and also very little ES complex is formed, thus ${\displaystyle {\ce {[E]_{\rm {tot}}\approx [E]}}}$. Therefore, the rate of product formation is

${\displaystyle v_{0}\approx {\frac {k_{cat}}{K_{M}}}{\ce {[E][S]}}\qquad \qquad {\text{if }}[{\ce {S}}]\ll K_{M}}$

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 ${\displaystyle k_{2}/K_{M}}$. This constant is a measure of catalytic efficiency. The most efficient enzymes reach a ${\displaystyle k_{2}/K_{M}}$ in the range of 10⁸ – 10¹⁰ M⁻¹ s⁻¹. 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 limit); and are sometimes referred to as kinetically perfect enzymes.^[11] But most enzymes are far from perfect: the average values of ${\displaystyle k_{2}/K_{\rm {M}}}$ and ${\displaystyle k_{2}}$ are about ${\displaystyle 10^{5}{\rm {s}}^{-1}{\rm {M}}^{-1}}$ and ${\displaystyle 10{\rm {s}}^{-1}}$, respectively.^[12]

Direct use of the Michaelis–Menten equation for time course kinetic analysis

The observed velocities predicted by the Michaelis–Menten equation can be used to directly model the 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^−k 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.^[13]

${\displaystyle [S]=[S]_{0}(1-k)^{t}\,}$

${\displaystyle [S]=[S]_{0}(1-v/[S]_{0})^{t}\,}$

${\displaystyle [S]=[S]_{0}(1-(V_{\max }[S]_{0}/(K_{M}+[S]_{0})/[S]_{0}))^{t}\,}$

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.^[14][15] The solution, known as the Schnell-Mendoza equation, has the form:

${\displaystyle {\frac {[S]}{K_{M}}}=W\left[F(t)\right]\,}$

where W[ ] is the Lambert-W function.^[16][17] and where F(t) is

${\displaystyle F(t)={\frac {[S]_{0}}{K_{M}}}\exp \!\left({\frac {[S]_{0}}{K_{M}}}-{\frac {V_{\max }}{K_{M}}}\,t\right)\,}$

This equation is encompassed by the equation below, obtained by Berberan-Santos,^[18] which is also valid when the initial substrate concentration is close to that of enzyme,

${\displaystyle {\frac {[S]}{K_{M}}}=W\left[F(t)\right]-{\frac {V_{\max }}{k_{cat}K_{M}}}\ {\frac {W\left[F(t)\right]}{1+W\left[F(t)\right]}}\,}$

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

Linear plots of the Michaelis–Menten equation

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 curve-fitting on computers, this nonlinearity could make it difficult to estimate K_M and V_max 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.^[19]

The Lineweaver–Burk plot or double reciprocal plot is a common way of illustrating kinetic data. This is produced by taking the 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 = mx + c with a y-intercept equivalent to 1/V_max and an x-intercept of the graph representing −1/K_M.

${\displaystyle {\frac {1}{v}}={\frac {K_{M}}{V_{\max }[{\mbox{S}}]}}+{\frac {1}{V_{\max }}}}$

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_max 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_max and K_M.^[20] A more accurate linear plotting method is the 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.^[21]

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_M and V_max, 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.^[22][23]

Michaelis–Menten kinetics with intermediate

One could also consider the less simple case

${\displaystyle {\ce {{E}+S<=>[k_{1}][k_{-1}]ES->[k_{2}]EI->[k_{3}]{E}+P}}}$

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^[24]

${\displaystyle v_{0}=k_{cat}{\frac {{\ce {[S] [E]_0}}}{K_{M}^{\prime }+{\ce {[S]}}}}}$

but the constants are different

${\displaystyle {\begin{aligned}K_{M}^{\prime }\ &{\stackrel {\mathrm {def} }{=}}\ {\frac {k_{3}}{k_{2}+k_{3}}}K_{M}={\frac {k_{3}}{k_{2}+k_{3}}}\cdot {\frac {k_{2}+k_{-1}}{k_{1}}}\\k_{cat}\ &{\stackrel {\mathrm {def} }{=}}\ {\dfrac {k_{3}k_{2}}{k_{2}+k_{3}}}\end{aligned}}}$

We see that for the limiting case ${\displaystyle k_{3}\gg k_{2}}$, thus when the last step from ${\displaystyle {\ce {EI -> E + P}}}$ is much faster than the previous step, we get again the original equation. Mathematically we have then ${\displaystyle K_{M}^{\prime }\approx K_{M}}$ and ${\displaystyle k_{cat}\approx k_{2}}$.

Ternary-complex mechanisms

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.

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,^[25] dihydrofolate reductase^[26] and DNA polymerase.^[27] The following links show short animations of the ternary-complex mechanisms of the enzymes dihydrofolate reductase^[β] and DNA polymerase^[γ].

Ping–pong mechanisms

${\displaystyle {\ce {\overset {}{E->[{\ce {A \atop \downarrow }}]EA<=>E^{\ast }P->[{\ce {P \atop \uparrow }}]E^{\ast }->[{\ce {B \atop \downarrow }}]E^{\ast }B<=>EQ->[{\ce {Q \atop \uparrow }}]E}}}}$

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 ping-pong mechanism can exist in two states, E and a chemically modified form of the enzyme E*; this modified enzyme is known as an 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 ping–pong mechanism are plotted in a Lineweaver–Burk plot, a set of parallel lines will be produced. This is called a secondary plot.

Enzymes with ping–pong mechanisms include some oxidoreductases such as thioredoxin peroxidase,^[28] transferases such as acylneuraminate cytidylyltransferase^[29] and serine proteases such as trypsin and chymotrypsin.^[30] Serine proteases are a very common and diverse family of enzymes, including digestive enzymes (trypsin, chymotrypsin, and elastase), several enzymes of the 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.^[δ]

Reversible inhibitors

Traditionally reversible enzyme inhibitors have been classified as competitive, uncompetitive, or non-competitive, according to their effects on K_M and V_max. 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.^[46] In noncompetitive inhibition, the inhibitor will bind to an enzyme at its allosteric site; therefore, the binding affinity, or inverse of K_M, of the substrate with the enzyme will remain the same. On the other hand, the V_max 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_max. The x-intercept, defined as −1/K_M, 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_M will increase and the V_max will remain the same.^[47] 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:

${\displaystyle {\cfrac {V_{\max }}{1+{\cfrac {[I]}{K_{i}}}}}={\cfrac {V_{\max }}{\cfrac {[I]+K_{i}}{K_{i}}}}}$

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

${\displaystyle {\cfrac {V_{\max }}{\cfrac {[I]+K_{i}}{[I]+K_{i}-[I]}}}}$

Dividing by [I]+K_i

${\displaystyle {\cfrac {V_{\max }}{\cfrac {1}{1-{\cfrac {[I]}{[I]+K_{i}}}}}}=V_{\max }-V_{\max }{\cfrac {[I]}{[I]+K_{i}}}}$

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_max term.

${\displaystyle V_{\max }-\Delta V_{\max }{\cfrac {[I]}{[I]+K_{i}}}}$

or

${\displaystyle V_{\max 1}-(V_{\max 1}-V_{\max 2}){\cfrac {[I]}{[I]+K_{i}}}}$

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_max 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".

${\displaystyle V_{\max 1}-(V_{\max 1}-V_{\max 2}){\cfrac {[X]}{[X]+K_{x}}}}$

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_M. The K_M 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_max should be appropriate in most situations:^[48]

${\displaystyle K_{m1}-(K_{m1}-K_{m2}){\cfrac {[X]}{[X]+K_{x}}}}$

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 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.