Michaelis–Menten kinetics

The specificity constant ${\displaystyle k_{\text{cat}}/K_{\mathrm {m} }}$ (also known as the catalytic efficiency) is a measure of how efficiently an enzyme converts a substrate into product. Although it is the ratio of ${\displaystyle k_{\text{cat}}}$ and ${\displaystyle K_{\mathrm {m} }}$ it is a parameter in its own right, more fundamental than ${\displaystyle K_{\mathrm {m} }}$. Diffusion limited enzymes, such as fumarase, work at the theoretical upper limit of 10⁸ – 10¹⁰ M⁻¹s⁻¹, limited by diffusion of substrate into the active site.[21]

If we symbolize the specificity constant for a particular substrate A as ${\displaystyle k_{\mathrm {A} }=k_{\text{cat}}/K_{\mathrm {m} }}$ the Michaelis-Menten equation can be written in terms of ${\displaystyle k_{\mathrm {A} }}$ and ${\displaystyle K_{\mathrm {m} }}$ as follows:

${\displaystyle v={\dfrac {k_{\mathrm {A} }e_{0}a}{1+{\dfrac {a}{K_{\mathrm {m} }}}}}}$

The reaction changes from approximately first-order in substrate concentration at low concentrations to approximately sero order at high concentrations.

At small values of the substrate concentration this approximates to a first-order dependence of the rate on the substrate concentration:

${\displaystyle v\approx k_{\mathrm {A} }e_{0}a{\text{ when }}a\rightarrow 0}$

Conversely it approaches a zero-order dependence on ${\displaystyle a}$ when the substrate concentration is high:

${\displaystyle v\rightarrow k_{\mathrm {cat} }e_{0}{\text{ when }}a\rightarrow \infty }$

The capacity of an enzyme to distinguish between two competing substrates that both follow Michaelis-Menten kinetics depends only on the specificity constant, and not on either ${\displaystyle k_{\text{cat}}}$ or ${\displaystyle K_{\mathrm {m} }}$ alone. Putting ${\displaystyle k_{\mathrm {A} }}$ for substrate ${\displaystyle \mathrm {A} }$ and ${\displaystyle k_{\mathrm {A'} }}$ for a competing substrate ${\displaystyle \mathrm {A'} }$, then the two rates when both are present simultaneously are as follows:

${\displaystyle v_{\mathrm {A} }={\frac {k_{\mathrm {A} }e_{0}a}{1+{\dfrac {a}{K_{\mathrm {m} }^{\mathrm {A} }}}+{\dfrac {a'}{K_{\mathrm {m} }^{\mathrm {A'} }}}}},v_{\mathrm {A'} }={\frac {k_{\mathrm {A'} }e_{0}a'}{1+{\dfrac {a}{K_{\mathrm {m} }^{\mathrm {A} }}}+{\dfrac {a'}{K_{\mathrm {m} }^{\mathrm {A'} }}}}}}$

Although both denominators contain the Michaelis constants they are the same, and thus cancel when one equation is divided by the other:

${\displaystyle {\frac {v_{\mathrm {A} }}{v_{\mathrm {A'} }}}={\frac {k_{\mathrm {A} }\cdot a}{k_{\mathrm {A'} }\cdot a'}}}$

and so the ratio of rates depends only on the concentrations of the two substrates and their specificity constants.

As the equation originated with Henri, not with Michaelis and Menten, it is more accurate to call it the Henri–Michaelis–Menten equation,[22] though it was Michaelis and Menten who realized that analysing reactions in terms of initial rates would be simpler, and as a result more productive, than analysing the time course of reaction, as Henri had attempted. Although Henri derived the equation he made no attempt to apply it.

In their analysis, Michaelis and Menten (and also Henri) assumed that the substrate is in instantaneous chemical equilibrium with the complex, which implies[9][24]

${\displaystyle k_{+1}ea=k_{-1}x}$

in which e is the concentration of free enzyme (not the total concentration) and x is the concentration of enzyme-substrate complex EA.

Conservation of enzyme requires that[24]

${\displaystyle e=e_{0}-x}$

where ${\displaystyle e_{0}}$ is now the total enzyme concentration. After combining the two expressions some straightforward algebra leads to the following expression for the concentration of the enzyme-substrate complex:

${\displaystyle x={\frac {e_{0}a}{K_{\mathrm {diss} }+a}}}$

where ${\displaystyle K_{\mathrm {diss} }=k_{-1}/k_{+1}}$ is the dissociation constant of the enzyme-substrate complex. Hence the rate equation is the Michaelis–Menten equation,[24]

${\displaystyle v={\frac {k_{+2}e_{0}a}{K_{\mathrm {diss} }+a}}}$

where ${\displaystyle k_{+2}}$ corresponds to the catalytic constant ${\displaystyle k_{\mathrm {cat} }}$ and the limiting rate is ${\displaystyle V_{\mathrm {max} }=k_{+2}e_{0}=k_{\mathrm {cat} }e_{0}}$. Likewise with the assumption of equilibrium the Michaelis constant ${\displaystyle K_{\mathrm {m} }=K_{\mathrm {diss} }}$.

When studying urease at about the same time as Michaelis and Menten were studying invertase, Donald Van Slyke and G. E. Cullen[25] made essentially the opposite assumption, treating the first step not as an equilibrium but as an irreversible second-order reaction with rate constant ${\displaystyle k_{+1}}$. As their approach is never used today it is sufficient to give their final rate equation:

${\displaystyle v={\frac {k_{\mathrm {+2} }e_{0}a}{k_{+2}/k_{+1}+a}}}$

and to note that it is functionally indistinguishable from the Henri–Michaelis–Menten equation. One cannot tell from inspection of the kinetic behaviour whether ${\displaystyle K_{\mathrm {m} }}$ is equal to ${\displaystyle k_{+2}/k_{+1}}$ or to ${\displaystyle k_{-1}/k_{+1}}$ or to something else.

G. E. Briggs and J. B. S. Haldane undertook an analysis that harmonized the approaches of Michaelis and Menten and of Van Slyke and Cullen,[26][27] and is taken as the basic approach to enzyme kinetics today. They assumed that the concentration of the intermediate complex does not change on the time scale over which product formation is measured.[28] This assumption means that ${\displaystyle k_{+1}ea=k_{-1}x+k_{\mathrm {cat} }x=(k_{-1}+k_{\mathrm {cat} })x}$. The resulting rate equation is as follows:

${\displaystyle v={\frac {k_{\mathrm {cat} }e_{0}a}{K_{\mathrm {m} }+a}}}$

where

${\displaystyle k_{\mathrm {cat} }=k_{+2}{\text{ and }}K_{\mathrm {m} }={\frac {k_{-1}+k_{\mathrm {cat} }}{k_{+1}}}}$

This is the generalized definition of the Michaelis constant.[29]

All of the derivations given treat the initial binding step in terms of the law of mass action, which assumes free diffusion through the solution. However, in the environment of a living cell where there is a high concentration of proteins, the cytoplasm often behaves more like a viscous gel than a free-flowing liquid, limiting molecular movements by diffusion and altering reaction rates.[30] Note, however that although this gel-like structure severely restricts large molecules like proteins its effect on small molecules, like many of the metabolites that participate in central metabolism, is very much smaller.[31] In practice, therefore, treating the movement of substrates in terms of diffusion is not like to produce major errors. Nonetheless, Schnell and Turner consider that is more appropriate to model the cytoplasm as a fractal, in order to capture its limited-mobility kinetics.[32]

Determining the parameters of the Michaelis-Menten equation typically involves running a series of enzyme assays at varying substrate concentrations ${\displaystyle a}$, and measuring the initial reaction rates ${\displaystyle v}$, i.e. the reaction rates are measured after a time period short enough for it to be assumed that the enzyme-substrate complex has formed, but that the substrate concentration remains almost constant, and so the equilibrium or quasi-steady-state approximation remain valid.[33] By plotting reaction rate against concentration, and using nonlinear regression of the Michaelis–Menten equation with correct weighting based on known error distribution properties of the rates, the parameters may be obtained.

Before computing facilities to perform nonlinear regression became available, graphical methods involving linearisation of the equation were used. A number of these were proposed, including the Eadie–Hofstee plot of ${\displaystyle v}$ against ${\displaystyle v/a}$,[34][35] the Hanes plot of ${\displaystyle a/v}$ against ${\displaystyle a}$,[36] and the Lineweaver–Burk plot (also known as the double-reciprocal plot) of ${\displaystyle 1/v}$ against ${\displaystyle 1/a}$.[37] Of these,[38] the Hanes plot is the most accurate when ${\displaystyle v}$ is subject to errors with uniform standard deviation.[39] From the point of view of visualizaing the data the Eadie–Hofstee plot has an important property: the entire possible range of ${\displaystyle v}$ values from ${\displaystyle 0}$ to ${\displaystyle V}$ occupies a finite range of ordinate scale, making it impossible to choose axes that conceal a poor experimental design.

However, while useful for visualization, all three linear plots distort the error structure of the data and provide less precise estimates of ${\displaystyle v}$ and ${\displaystyle K_{\mathrm {m} }}$ than correctly weighted non-linear regression. Assuming an error ${\displaystyle \varepsilon (v)}$ on ${\displaystyle v}$, an inverse representation leads to an error of ${\displaystyle \varepsilon (v)/v^{2}}$ on ${\displaystyle 1/v}$ (Propagation of uncertainty), implying that linear regression of the double-reciprocal plot should include weights of ${\displaystyle v^{4}}$. This was well understood by Lineweaver and Burk,[37] who had consulted the eminent statistician W. Edwards Deming before analysing their data.[40] Unlike nearly all workers since, Burk made an experimental study of the error distribution, finding it consistent with a uniform standard error in ${\displaystyle v}$, before deciding on the appropriate weights.[41] This aspect of the work of Lineweaver and Burk received virtually no attention at the time, and was subsequently forgotten.

Many authors, for example Greco and Hakala,[42] have claimed that non-linear regression is always superior to regression of the linear forms of the Michaelis-Menten equation. However, that is correct only if the appropriate weighting scheme is used, preferably on the basis of experimental investigation, something that is almost never done. As noted above, Burk[41] carried out the appropriate investigation, and found that the error structure of his data was consistent with a uniform standard deviation in ${\displaystyle v}$. More recent studies found that a uniform coefficient of variation (standard deviation expressed as a percentage) was closer to the truth with the techniques in use in the 1970s.[43][44] However, this truth may be more complicated than any dependence on ${\displaystyle v}$ alone can represent.[45]

Santiago Schnell and Claudio Mendoza suggested a closed form solution for the time course kinetics analysis of the Michaelis–Menten kinetics based on the solution of the Lambert W function.[46] Namely,

${\displaystyle {\frac {a}{K_{\mathrm {m} }}}=W(F(t))}$

where W is the Lambert W function and

${\displaystyle F(t)={\frac {a}{K_{\mathrm {m} }}}\exp \!\left({\frac {a_{0}}{K_{\mathrm {m} }}}-{\frac {Vt}{K_{\mathrm {m} }}}\right)}$

The above equation, known nowadays as the Schnell-Mendoza equation,[47] has been used to estimate ${\displaystyle V}$ and ${\displaystyle K_{\mathrm {m} }}$ from time course data.[48][49]