Michaelis–Menten kinetics

thumb|upright=1.5|Curve of the Michaelis–Menten equation labelled in accordance with IUBMB recommendations

In biochemistry, Michaelis–Menten kinetics, named after Leonor Michaelis and Maud Menten, is the simplest case of enzyme kinetics, applied to enzyme-catalysed reactions involving the transformation of one substrate into one product. In 1913, Michaelis and Menten expanded on Victor Henri's fundamental equation of enzyme kinetics, which was established in 1902. It takes the form of a differential equation describing the reaction rate <math>v</math> (rate of formation of product P, with concentration <math chem>p</math>) as a function of <math chem>a</math>, the concentration of the substrate A (using the symbols recommended by the IUBMB). The formula below is given by the Michaelis–Menten equation:

:<math chem> v = \frac{\mathrm{d} p}{\mathrm{d} t} = \frac{V a}{K_\mathrm{m} + a} </math> .

<math>V</math>, which is often written as <math>V_\max</math>, represents the limiting rate approached by the system at saturating substrate concentration for a given enzyme concentration. The Michaelis constant <math>K_\mathrm{m}</math> has units of concentration, and for a given reaction is equal to the concentration of substrate at which the reaction rate is half of <math>V</math>. but this terminology is historically misleading, as Michaelis and Menten did not use such a plot. Instead, they plotted <math>v</math> against <math>\log a</math>, which has some advantages over the usual ways of plotting Michaelis–Menten data. If <math>v</math> is the dependent variable, then it does not distort any experimental errors in <math>v</math>. Michaelis and Menten did not attempt to estimate <math>V</math> directly from the limit approached at high <math>\log a</math>, something difficult to do accurately with data obtained with modern techniques, and almost impossible with their data. Instead they took advantage of the fact that the curve is almost straight in the middle range and has a maximum slope of <math>0.576V</math> i.e. <math>0.25\ln 10 \cdot V</math>. With an accurate value of <math>V</math> it was easy to determine <math>\log K_\mathrm{m}</math> from the point on the curve corresponding to <math>0.5V</math>.

This plot is virtually never used today for estimating <math>V</math> and <math>K_\mathrm{m}</math>, but it remains valuable to compare the properties of several enzymes across a broad range of substrate concentrations - such as isoenzymes. For example, the four mammalian isoenzymes of hexokinase are half-saturated by glucose at concentrations ranging from about 0.02 mM for hexokinase A (brain hexokinase) to about 50 mM for hexokinase D ("glucokinase", liver hexokinase), spanning a 2500-fold range. A conventional (linear) plot would compromise on readability for the high-affinity isoenzyme graphs, but a semi-logarithmic plot allows to read off the kinetic parameters for all isoenzymes.

Model

A decade before Michaelis and Menten, Victor Henri found that enzyme reactions could be explained by assuming a binding interaction between the enzyme and the substrate. This may be represented schematically as

:<chem>E{} + A <=>[\mathit{k_\mathrm{+1][\mathit{k_\mathrm{-1] EA ->[k_\ce{cat}] E{} + P</chem>

where <math>k_\mathrm{+1}</math> (forward rate constant), <math>k_\mathrm{-1}</math> (reverse rate constant), and <math>k_\mathrm{cat}</math> (catalytic rate constant) denote the rate constants, However at higher <math>a</math>, with <math>a \gg K_\mathrm{m}</math>, the reaction approaches independence of <math>a</math> (zero-order kinetics in <math>a</math>), Its value depends both on the identity of the enzyme and that of the substrate, as well as conditions such as temperature and pH.

The model is used in a variety of biochemical situations other than enzyme-substrate interaction, including antigen–antibody binding, DNA–DNA hybridization, and protein–protein interaction. and also, for example, to limiting nutrients and phytoplankton growth in the global ocean.

Specificity

The specificity constant <math>k_\text{cat}/K_\mathrm{m}</math> (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 <math>k_\text{cat}</math> and <math>K_\mathrm{m}</math> it is a parameter in its own right, more fundamental than <math>K_\mathrm{m}</math>. Diffusion limited enzymes, such as fumarase, work at the theoretical upper limit of , limited by diffusion of substrate into the active site. 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 addition, Michaelis and Menten understood the need for buffers to control the pH, but Henri did not.

Applications

Parameter values vary widely between enzymes. Some examples are as follows: made essentially the opposite assumption, treating the first step not as an equilibrium but as an irreversible second-order reaction with rate constant <math>k_{+1}</math>. As their approach is never used today it is sufficient to give their final rate equation:

:<math>v = \frac{k_\mathrm{+2}e_0 a}{k_{+2}/k_{+1} + a}</math>

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

Steady-state approximation

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, 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. This assumption means that <math>k_{+1} e a = k_{-1}x + k_\mathrm{cat} x = (k_{-1} + k_\mathrm{cat})x</math>. The resulting rate equation is as follows:

:<math>v = \frac{k_\mathrm{cat}e_0 a}{K_\mathrm{m} + a}</math>

where

:<math>k_\mathrm{cat} = k_{+2} \text { and }

K_\mathrm{m} = \frac{k_{-1} + k_\mathrm{cat{k_{+1</math>

This is the generalized definition of the Michaelis constant. In practice, therefore, treating the movement of substrates in terms of diffusion is not likely to produce major errors. Nonetheless, Schnell and Turner consider it more appropriate to model the cytoplasm as a fractal, in order to capture its limited-mobility kinetics. the Hanes plot of <math>a/v</math> against <math>a</math>, and the Lineweaver–Burk plot (also known as the double-reciprocal plot) of <math>1/v</math> against <math>1/a</math>. Of these, the Hanes plot is the most accurate when <math>v</math> is subject to errors with uniform standard deviation. From the point of view of visualizaing the data the Eadie–Hofstee plot has an important property: the entire possible range of <math>v</math> values from <math>0</math> to <math>V</math> 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 <math>v</math> and <math>K_\mathrm{m}</math> than correctly weighted non-linear regression. Assuming an error <math>\varepsilon (v)</math> on <math>v</math>, an inverse representation leads to an error of <math>\varepsilon (v)/v^2</math> on <math>1/v</math> (Propagation of uncertainty), implying that linear regression of the double-reciprocal plot should include weights of <math>v^4</math>. This was well understood by Lineweaver and Burk, Unlike nearly all workers since, Burk made an experimental study of the error distribution, finding it consistent with a uniform standard error in <math>v</math>, before deciding on the appropriate weights. This aspect of the work of Lineweaver and Burk received virtually no attention at the time, and was subsequently forgotten.

The direct linear plot is a graphical method in which the observations are represented by straight lines in parameter space, with axes <math>K_\mathrm{m}</math> and <math>V</math>: each line is drawn with an intercept of <math>-a</math> on the <math>K_\mathrm{m}</math> axis and <math>v</math> on the <math>V</math> axis. The point of intersection of the lines for different observations yields the values of <math>K_\mathrm{m}</math> and <math>V</math>.

Weighting

Many authors, for example Greco and Hakala, However, this truth may be more complicated than any dependence on <math>v</math> alone can represent.

Uniform standard deviation of <math>1/v</math>. If the rates are considered to have a uniform standard deviation the appropriate weight for every <math>v</math> value for non-linear regression is 1. If the double-reciprocal plot is used each value of <math>1/v</math> should have a weight of <math>v^4</math>, whereas if the Hanes plot is used each value of <math>a/v</math> should have a weight of <math>v^4/a^2</math>.

Uniform coefficient variation of <math>1/v</math>. If the rates are considered to have a uniform coefficient variation the appropriate weight for every <math>v</math> value for non-linear regression is <math>v^2</math>. If the double-reciprocal plot is used each value of <math>1/v</math> should have a weight of <math>v^2</math>, whereas if the Hanes plot is used each value of <math>a/v</math> should have a weight of <math>v^2/a^2</math>.

Ideally the <math>v</math> in each of these cases should be the true value, but that is always unknown. However, after a preliminary estimation one can use the calculated values <math>\hat v</math> for refining the estimation. In practice the error structure of enzyme kinetic data is very rarely investigated experimentally, therefore almost never known, but simply assumed. It is, however, possible to form an impression of the error structure from internal evidence in the data. This is tedious to do by hand, but can readily be done in the computer.

Closed form equation

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.

Namely,

:<math>\frac{a}{K_\mathrm{m = W(F(t))</math>

where W is the Lambert W function and

:<math>F(t) = \frac{a_0}{K_\mathrm{m \exp\!\left(\frac{a_0}{K_\mathrm{m - \frac{Vt}{K_\mathrm{m \right)</math>

The above equation, known nowadays as the Schnell-Mendoza equation, has been used to estimate <math>V</math> and <math>K_\mathrm{m}</math> from time course data.

Reactions with more than one substrate

Only a small minority of enzyme-catalysed reactions have just one substrate, and even if the number is increased by treating two-substrate reactions in which one substrate is water as one-substrate reactions the number is still small. One might accordingly suppose that the Michaelis–Menten equation, normally written with just one substrate, is of limited usefulness. This supposition is misleading, however. One of the common equations for a two-substrate reaction can be written as follows to express <math>v</math> in terms of two substrate concentrations <math>a</math> and <math>b</math>:

: <math>v = \frac{Vab}{K_\mathrm{iA}K_\mathrm{mB} + K_\mathrm{mB}a + K_\mathrm{mA}b + ab}</math>

the other symbols represent kinetic constants. Suppose now that <math>a</math> is varied with <math>b</math> held constant. Then it is convenient to reorganize the equation as follows:

: <math>v = \frac{Vb \cdot a}{K_\mathrm{iA}K_\mathrm{mB}+ K_\mathrm{mA}b +(K_\mathrm{mB} + b)a}

= \dfrac{\dfrac{Vb }{K_\mathrm{mB}+b}\cdot a}{\dfrac{K_\mathrm{iA}K_\mathrm{mB}+ K_\mathrm{mA}b}{K_\mathrm{mB}+b} +a}

</math>

This has exactly the form of the Michaelis–Menten equation

:<math> v = \frac{V^\mathrm{app} a}{K^\mathrm{app}_\mathrm{m} + a} </math>

with apparent values <math>V^\mathrm{app}</math> and <math>K^\mathrm{app}_\mathrm{m}</math> defined as follows:

: <math>V^\mathrm{app} = \dfrac{Vb}{K_\mathrm{mB}+b}</math>

: <math>K^\mathrm{app}_\mathrm{m} = \dfrac{K_\mathrm{iA}K_\mathrm{mB}+ K_\mathrm{mA}b}{K_\mathrm{mB}+b}

</math>

Linear inhibition

The linear (simple) types of inhibition can be classified in terms of the general equation for mixed inhibition at an inhibitor concentration <math>i</math>:

: <math>v = \dfrac{Va}{K_\mathrm{m}\left(1 + \dfrac {i}{K_\mathrm{ic \right)

+ a\left(1 + \dfrac {i}{K_\mathrm{iu \right)}

</math>

in which <math>K_\mathrm{ic}</math> is the competitive inhibition constant and <math>K_\mathrm{iu}</math> is the uncompetitive inhibition constant. This equation includes the other types of inhibition as special cases:

If <math>K_\mathrm{iu} \rightarrow \infty</math> the second parenthesis in the denominator approaches <math>1</math> and the resulting behaviour is competitive inhibition.
If <math>K_\mathrm{ic} \rightarrow \infty</math> the first parenthesis in the denominator approaches <math>1</math> and the resulting behaviour is uncompetitive inhibition.
If both <math>K_\mathrm{ic}</math> and <math>K_\mathrm{iu}</math> are finite the behaviour is mixed inhibition.
If <math>K_\mathrm{ic} = K_\mathrm{iu}</math> the resulting special case is pure non-competitive inhibition.

Pure non-competitive inhibition is very rare, being mainly confined to effects of protons and some metal ions. Cleland recognized this, and he redefined noncompetitive to mean mixed. Some authors have followed him in this respect, but not all, so when reading any publication one needs to check what definition the authors are using.

In all cases the kinetic equations have the form of the Michaelis–Menten equation with apparent constants, as can be seen by writing the equation above as follows:

: <math>v = \dfrac{\dfrac{V}{1 + i/K_\mathrm{iu \cdot a}

{\dfrac{K_\mathrm{m}(1 + i/K_\mathrm{ic})}

{1 + i/K_\mathrm{iu +a}

= \frac{V^\mathrm{app} a}{K^\mathrm{app}_\mathrm{m} + a}

</math>

with apparent values <math>V^\mathrm{app}</math> and <math>K^\mathrm{app}_\mathrm{m}</math> defined as follows:

: <math>V^\mathrm{app} = \dfrac{V}{1 + i/K_\mathrm{iu</math>

: <math>K^\mathrm{app}_\mathrm{m} =

\dfrac{K_\mathrm{m}(1 + i/K_\mathrm{ic})}{1 + i/K_\mathrm{iu

</math>

Footnotes

References

External links

Online <math>K_\mathrm{M}</math> <math>V_\max</math> Vmax calculator (ic50.tk/kmvmax.html) based on the C programming language and the non-linear least-squares Levenberg–Marquardt algorithm of gnuplot
Alternative online <math>K_\mathrm{M}</math> <math>V_\max</math> calculator (ic50.org/kmvmax.html) based on Python, NumPy, Matplotlib and the non-linear least-squares Levenberg–Marquardt algorithm of SciPy

Michaelis–Menten kinetics

Model

Specificity

Applications

Steady-state approximation

Weighting

Closed form equation

Reactions with more than one substrate

Linear inhibition

See also

Footnotes

References

External links

Further reading