Michaelis–Menten kinetics

The plot of v against a has often been called a "Michaelis–Menten plot", even recently, but this terminology is historically misleading, as Michaelis and Menten did not use such a plot. Instead, they plotted v against \log a, which has some advantages over the usual ways of plotting Michaelis–Menten data. If v is the dependent variable, then it does not distort any experimental errors in v. Michaelis and Menten did not attempt to estimate V directly from the limit approached at high \log a, 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 0.576V i.e. 0.25\ln 10 \cdot V. With an accurate value of V it was easy to determine \log K_\mathrm{m} from the point on the curve corresponding to 0.5V. This plot is virtually never used today for estimating V and K_\mathrm{m}, 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 :E{} + A [\mathit{k_\mathrm{+1}}][\mathit{k_\mathrm{-1}}] EA ->[k_\ce{cat}] E{} + P where k_\mathrm{+1} (forward rate constant), k_\mathrm{-1} (reverse rate constant), and k_\mathrm{cat} (catalytic rate constant) denote the rate constants, However at higher a, with a \gg K_\mathrm{m}, the reaction approaches independence of a (zero-order kinetics in a), 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 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 k_\text{cat} and K_\mathrm{m} it is a parameter in its own right, more fundamental than K_\mathrm{m}. 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: ==Derivation==

[E][S]), gives a system of four non-linear ordinary differential equations that define the rate of change of reactants with time t made essentially the opposite assumption, treating the first step not as an equilibrium but as an irreversible second-order reaction with rate constant k_{+1}. As their approach is never used today it is sufficient to give their final rate equation: :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 K_\mathrm{m} is equal to k_{+2}/k_{+1} or to k_{-1}/k_{+1} 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 k_{+1} e a = k_{-1}x + k_\mathrm{cat} x = (k_{-1} + k_\mathrm{cat})x. The resulting rate equation is as follows: :v = \frac{k_\mathrm{cat}e_0 a}{K_\mathrm{m} + a} where :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. 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. ==Estimation of Michaelis–Menten parameters==

Graphical methods Determining the parameters of the Michaelis–Menten equation typically involves running a series of enzyme assays at varying substrate concentrations a, and measuring the initial reaction rates 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. the Hanes plot of a/v against a, and the Lineweaver–Burk plot (also known as the double-reciprocal plot) of 1/v against 1/a. Of these, the Hanes plot is the most accurate when v 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 v values from 0 to 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 v and K_\mathrm{m} than correctly weighted non-linear regression. Assuming an error \varepsilon (v) on v, an inverse representation leads to an error of \varepsilon (v)/v^2 on 1/v (Propagation of uncertainty), implying that linear regression of the double-reciprocal plot should include weights of v^4. 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 v, 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 K_\mathrm{m} and V: each line is drawn with an intercept of -a on the K_\mathrm{m} axis and v on the V axis. The point of intersection of the lines for different observations yields the values of K_\mathrm{m} and V. Weighting Many authors, for example Greco and Hakala, However, this truth may be more complicated than any dependence on v alone can represent. Uniform standard deviation of 1/v. If the rates are considered to have a uniform standard deviation the appropriate weight for every v value for non-linear regression is 1. If the double-reciprocal plot is used each value of 1/v should have a weight of v^4, whereas if the Hanes plot is used each value of a/v should have a weight of v^4/a^2. Uniform coefficient variation of 1/v. If the rates are considered to have a uniform coefficient variation the appropriate weight for every v value for non-linear regression is v^2. If the double-reciprocal plot is used each value of 1/v should have a weight of v^2, whereas if the Hanes plot is used each value of a/v should have a weight of v^2/a^2. Ideally the v 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 \hat v 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, :\frac{a}{K_\mathrm{m}} = W(F(t)) where W is the Lambert W function and :F(t) = \frac{a_0}{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, has been used to estimate V and K_\mathrm{m} 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 v in terms of two substrate concentrations a and b: : v = \frac{Vab}{K_\mathrm{iA}K_\mathrm{mB} + K_\mathrm{mB}a + K_\mathrm{mA}b + ab} the other symbols represent kinetic constants. Suppose now that a is varied with b held constant. Then it is convenient to reorganize the equation as follows: : 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} This has exactly the form of the Michaelis–Menten equation : v = \frac{V^\mathrm{app} a}{K^\mathrm{app}_\mathrm{m} + a} with apparent values V^\mathrm{app} and K^\mathrm{app}_\mathrm{m} defined as follows: : V^\mathrm{app} = \dfrac{Vb}{K_\mathrm{mB}+b} : K^\mathrm{app}_\mathrm{m} = \dfrac{K_\mathrm{iA}K_\mathrm{mB}+ K_\mathrm{mA}b}{K_\mathrm{mB}+b} == Linear inhibition ==

The linear (simple) types of inhibition can be classified in terms of the general equation for mixed inhibition at an inhibitor concentration i: : v = \dfrac{Va}{K_\mathrm{m}\left(1 + \dfrac {i}{K_\mathrm{ic}} \right) + a\left(1 + \dfrac {i}{K_\mathrm{iu}} \right)} in which K_\mathrm{ic} is the competitive inhibition constant and K_\mathrm{iu} is the uncompetitive inhibition constant. This equation includes the other types of inhibition as special cases: • If K_\mathrm{iu} \rightarrow \infty the second parenthesis in the denominator approaches 1 and the resulting behaviour is competitive inhibition. • If K_\mathrm{ic} \rightarrow \infty the first parenthesis in the denominator approaches 1 and the resulting behaviour is uncompetitive inhibition. • If both K_\mathrm{ic} and K_\mathrm{iu} are finite the behaviour is mixed inhibition. • If K_\mathrm{ic} = K_\mathrm{iu} 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: : 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} with apparent values V^\mathrm{app} and K^\mathrm{app}_\mathrm{m} defined as follows: : V^\mathrm{app} = \dfrac{V}{1 + i/K_\mathrm{iu}} : K^\mathrm{app}_\mathrm{m} = \dfrac{K_\mathrm{m}(1 + i/K_\mathrm{ic})}{1 + i/K_\mathrm{iu}} ==See also==