Michaelis–Menten kinetics

Michaelis–Menten kinetics (occasionally also referred to as Michaelis–Menten–Henri kinetics) approximately describes the kinetics of many enzymes. It is named after Leonor Michaelis and Maud Menten. This kinetic model is relevant to situations where the concentration of enzyme is much lower than the concentration of substrate (i.e. where enzyme concentration is the limiting factor), and when the enzyme is not allosteric.

History

The modern relationship between substrate and enzyme concentration was proposed in 1903 by Victor Henri. A microscopic interpretation was thereafter proposed in 1913 by Leonor Michaelis and Maud Menten, following earlier work by Archibald Vivian Hill. It postulated that enzyme (catalyst) and substrate (reactant) are in fast equilibrium with their complex, which then dissociates to yield product and free enzyme.

The current derivation, based on the quasi steady state approximation (that the concentrations of the intermediate complexes remain constant) was proposed by Briggs and Haldane.

Determination of constants

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

To determine the maximum rate of an enzyme mediated reaction, a series of experiments is carried out where the substrate concentration ([S]) is increased until a constant initial rate of product formation is achieved. ('Initial' here is taken to mean that the reaction rate is measuremed after a relatively short time period, during which complex builds up but the substrate concentration remains approximately constant.) This is the maximum velocity (V_max) of the enzyme under the conditions of the experiment. In this state, enzyme active sites are saturated with substrate.

Reaction rate/velocity V

The reaction rate V is the number of reactions per second catalyzed per mole of the enzyme. The reaction rate increases with increasing substrate concentration [S], asymptotically approaching the maximum rate V_max. There is therefore no clearly-defined substrate concentration at which the enzyme can be said to be saturated with substrate. A more appropriate measure to characterise an enzyme is the substrate concentration at which the reaction rate reaches half of its maximum value (V_max/2). This concentration can be shown to be equal to the Michaelis constant (K_M).

Michaelis constant K_m

For enzymatic reactions which exhibit simple Michaelis–Menten kinetics and in which product formation is the rate-limiting step (i.e., when k₂ << k₋₁) K_m ≈ k₋₁/k₁ = K_d, where K_d is the dissociation constant (affinity for substrate) of the enzyme-substrate (ES) complex. However, often k₂ >> k₋₁, or k₂ and k₋₁ are comparable, in which case nothing can be said about the enzyme affinity from the Michaelis constant alone.

The Michaelis constant can be defined as:

$K_{m}={\frac {k_{-1}+k_{2}}{k_{1}}}.$

Equation

The most convenient derivation of the Michaelis–Menten equation, described by Briggs and Haldane, is obtained as follows:

The enzymatic reaction is assumed to be irreversible, and the product does not bind to the enzyme.

$E+S{\overset {k_{1}}{\underset {k_{-1}}{\begin{smallmatrix}\displaystyle \longrightarrow \\\displaystyle \longleftarrow \end{smallmatrix}}}}ES{\overset {k_{2}}{\longrightarrow }}E+P$

The rate of production of the product, d[P]/dt is referred to as the reaction rate, V in enzyme kinetics. It is dependent on the conversion rate constant, k₂ (often referred to as the catalytic constant, k_cat) and [ES], the concentration of enzyme that is bound to substrate. As [ES] is usually not measurable, it must be expressed in terms of the known parameters of the system, namely the concentration of enzyme and substrate originally added.

A key assumption in this derivation is the quasi-steady state approximation, namely that the concentration of the substrate-bound enzyme (and hence also the unbound enzyme) change much more slowly than those of the product and substrate. Starting from the quasi steady state approximation, this allows us to express the relationship between the substrate concentration and the bound and unbound enzyme concentrations in terms of the various rate constants:

${\frac {d[ES]}{dt}}=0=k_{1}[E][S]-[ES](k_{-1}+k_{2})(1).$

The total concentration of enzyme ([E₀]) is the sum of the free enzyme in solution [E] and that which is bound to the substrate ([ES]); mathematically, $[E_{0}]=[E]+[ES]$. This equation allows us to derive the free enzyme concentration,

$[E]=[E_{0}]-[ES]\qquad (2).$

Substituting into equation (1), we obtain

$0=k_{1}\left([E_{0}]-[ES]\right)[S]-[ES]\left(k_{-1}+k_{2}\right),$

which can be rearranged into

$0=k_{1}[E_{0}][S]-[ES](k_{1}[S]+k_{-1}+k_{2})\qquad (3).$

To simplify the equation, we define the Michaelis constant via:

$K_{m}={\frac {k_{-1}+k_{2}}{k_{1}}}.$

With this definition at hand, (3) may be written as

$0=[E_{0}][S]-[ES]([S]+K_{m})\qquad (4).$

Solving, now, this equation for [ES], we obtain

$[ES]=[E_{0}]{\frac {[S]}{[S]+K_{m}}},$

or, equivalently,

$[ES]=[E_{0}]{\frac {1}{1+{\frac {K_{m}}{[S]}}}}\qquad (5).$

The production rate is:

$V={\frac {d[P]}{dt}}=k_{2}[ES]\qquad (6).$

Substituting (5) in (6) and multiplying the numerator and denominator by $[S]$ gives:

${\frac {d[P]}{dt}}=k_{2}[E_{0}]{\frac {[S]}{K_{m}+[S]}}=V_{\max }{\frac {[S]}{K_{m}+[S]}}$

Because the concentration of substrate changes as the reaction takes place, the initial reaction rate (V₀) is used to simplify analysis, taking the initial concentration of substrate as $[S]$.

Lineweaver–Burk plot

This equation may be represented by a Lineweaver–Burk plot or a Hanes–Woolf plot.

If [S] is large compared to K_m, [S]/(K_m + [S]) approaches 1. Therefore, the rate of product formation is equal to k₂[E₀] in this case.

When [S] equals K_m, [S]/(K_m + [S]) equals 0.5. In this case, the rate of product formation is half of the maximum rate (1/2 V_max). By plotting V₀ against [S], one can easily determine V_max and K_m. The most common way of generating this data is with a series of experiments at constant E₀ and different substrate concentration [S].

The Michaelis–Menten equation describes the rates of irreversible reactions. A steady state solution for a chemical equilibrium modeled with Michaelis–Menten kinetics can be obtained with the Goldbeter–Koshland equation.

Limitations

The first source of limitations for the Michaelis-Menten kinetics has to do with the fact that it is an inexact model - in other words, an approximation - of the kinetics derived by the law of mass action. In particular, the Michaelis-Menten kinetics is based on the quasi-steady state assumption,

${\frac {d[ES]}{dt}}=0,$

which is only approximately true: the rate of change of the complex [ES] is not zero but, rather, small. The quality of the Michaelis-Menten approximation depends on the timescale separation present in the dynamics on the phase space, which controls the magnitude of the rate of change of [ES]. In particular, it has been shown that this timescale separation is measured by a small, positive parameter,

$\varepsilon ={\frac {E_{0}}{S_{0}+K_{m}}},$

where S₀ is the initial (that is, at time zero) concentration of the substrate and S₀, K_m have been defined above. The smaller $\varepsilon$ is, the larger the timescale separation present in the system and the more accurate the Michaelis-Menten approximation.

The second source of limitations Michaelis–Menten kinetics relies, like other classical biochemical kinetic theories, on the law of mass action which, in turn, is derived from the assumptions of free (Fickian) diffusion and thermodynamically-driven random collision. However, many biochemical or cellular processes deviate significantly from such conditions. For example, the cytoplasm inside a cell behaves more like a gel than a freely flowable or watery liquid, due to the very high concentration of protein (up to ~400 mg/mL) and other “solutes”, which can severely limit molecular movements (diffusion or collision) (see e.g. Olsen ). This causes macromolecular crowding, which can alter reaction rates and dissociation constants.

For heterogeneous enzymatic reactions, such as those of membrane enzymes, molecular mobility of the enzyme or substrates can also be severely restricted, due to the immobilization or phase-separation of the reactants. For some homogeneous enzymatic reactions, the mobility of the enzyme or substrate may also be limited, such as the case of DNA polymerase where the enzyme moves along a chained substrate, rather than having a three-dimensional freedom. The limitation on molecular mobility (as well as other “non-ideal” conditions) demands modifications on the conventional mass-action laws, and Michaelis–Menten kinetics, to better reflect certain real world situations. Although it has been shown that the law of mass action can be valid in heterogeneous environments (see, R. Grima and S. Schnell ). In general physics and chemistry, limited mobility-derived kinetics have been successfully described by the fractal-like kinetics. R. Kopelman , M.A. Savageau , and S. Schnell pioneered the “fractal enzymology”, which has been further developed by other researchers.

Further reading

• Biochemistry/Catalysis at Wikibooks

External links

• Enzyme assay development and analysis
• LMMpro, a Michaelis–Menten optimization software,Website