michaelis manten kinetics

8/12/2019 Michaelis Manten Kinetics

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MichaelisMenten kinetics From Wikipedia, the free encyclopedia

An example curve with parameters V max = 3.4 and K m = 0.4.

In biochemistry , Michaelis

Menten kinetics is one of best-known models of enzyme kinetics . It is namedafter German biochemist Leonor Michaelis and Canadian physician Maud Menten . The model takes the

form of an equation describing the rate of enzymatic reactions , by relating reaction rate to ,

the concentration of a substrate S . Its formula is given by

.

Here, represents the maximum rate achieved by the system, at maximum (saturating) substrate

concentrations. The Michaelis constant is the substrate concentration at which the reaction rate

is half of . Biochemical reactions involving a single substrate are often assumed to follow

Michaelis Menten kinetics, without regard to the model's underlying assumptions.

Contents

[hide ]

1 Model

2 Applications

3 Derivation

o 3.1 Equilibrium approximation

o 3.2 Quasi-steady-state approximation

o 3.3 Assumptions and limitations

4 Determination of constants

5 See also

6 References

7 Further reading

Model [edit ]
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Change in concentrations over time for enzyme E, substrate S, complex ES and product P

In 1903, French physical chemist Victor Henri found that enzyme reactions were initiated by a bond

(more generally, a binding interaction) between the enzyme and the substrate . [1] His work was taken

up by German biochemist Leonor Michaelis and Canadian physician Maud Menten , who investigated

the kinetics of an enzymatic reaction mechanism, invertase , that catalyzes

the hydrolysis of sucrose into glucose and fructose .[2] In 1913, they proposed a mathematical model of

the reaction .[3] It involves an enzyme E binding to a substrate S to form a complex ES, which in turn is

converted into a product P and the enzyme. This may be represented schematically as

where , , and denote the rate constants ,[4] and the double arrows between S and ES

represent the fact that enzyme-substrate binding is a reversible process.

Under certain assumptions such as the enzyme concentration being much less than the

substrate concentration the rate of product formation is given by

The reaction rate increases with increasing substrate concentration

, asymptotically approaching its maximum rate , attained when all enzyme is bound to

substrate. It also follows that , where is the enzyme

concentration. , the turnover number , is the maximum number of substrate molecules

converted to product per enzyme molecule per second.

The Michaelis constant is the substrate concentration at which the reaction rate is at

half-maximum, and is an inverse measure of the substrate's affinity for the enzyme as a

small indicates high affinity, meaning that the rate will approach more

quickly .[5] The value of is dependent on both the enzyme and 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
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interaction .[5][6] It can be used to characterise a generic biochemical reaction, in the same way

that the Langmuir equation can be used to model generic adsorption of biomolecular

species .[6] When an empirical equation of this form is applied to microbial growth, it is

sometimes called a Monod equation .

Applications [edit ]

Parameter values vary wildly between enzymes :[7]

Enzyme (M) (1/s) (1/M.s)

Chymotrypsin 1.5 10 2 0.14 9.3

Pepsin 3.0 104

0.50 1.7 10 3

Tyrosyl-tRNA synthetase 9.0 10 4 7.6 8.4 10 3

Ribonuclease 7.9 10 3 7.9 10 2 1.0 10 5

Carbonic anhydrase 2.6 10 2 4.0 10 5 1.5 10 7

Fumarase 5.0 10 6 8.0 10 2 1.6 10 8

The constant is a measure of how efficiently an enzyme converts a substrate

into product. It has a theoretical upper limit of 10 8 10 10 /M.s; enzymes working close to this,

such as fumarase, are termed superefficient .[8]

Michaelis Menten kinetics have also been applied to a variety of spheres outside of

biochemical reactions ,[4]

including alveolar clearance of dusts ,[9]

the richness ofspecies pools ,[10] clearance o f blood alcohol ,[11] the photosynthesis-irradiance relationship, and

bacterial phage infection .[12]

Derivation [edit ]

Applying the law of mass action , which states that the rate of a reaction is proportional to the

product of the concentrations of the reactants (i.e.[E][S]), gives a system of four non-

linear ordinary differential equations that define the rate of change of reactants with time :[13]
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In this mechanism, the enzyme E is a catalyst , which only facilitates the reaction, so that

its total concentration, free plus combined, is a constant. This

conservation law can also be observed by adding the first and third equations

above .[13][14]

Equilibrium approximation [edit ]

In their original analysis, Michaelis and Menten assumed that the substrate is in

instantaneous chemical equilibrium with the complex, and

thus .[3][14] Combining this relationship with the enzyme

conservation law, the concentration of complex i s [14]

where is the dissociation constant for the enzyme-substrate

complex. Hence the velocity of the reaction the rate at which P is formed is [14]

where is the maximum reaction velocity.

Quasi-steady-state approximation [edit ]

An alternative analysis of the system was undertaken by British botanist G. E.

Briggs and British geneticist J. B. S. Haldane in 1925 .[15] They assumed that the

concentration of the intermediate complex does not change on the time-scale of

product formation known as the quasi -steady-state assumption or pseudo-

steady-state-hypothesis. Mathematically, this assumption

means . Combining this relationship

with the enzyme conservation law, the concentration of complex i s [14]

where
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is known as the Michaelis constant, where , , and are,

respectively, the constants for substrate unbinding, conversion toproduct, and binding to the enzyme. Hence the velocity of the

reaction i s [14]

Assumptions and limitations [edit ]

The first step in the derivation applies the law of mass action ,

which is reliant on free diffusion . However, in the environment of a

living cell where there is a high concentration of proteins, the

cytoplasm often behaves more like a gel than a liquid, limiting

molecular movements and altering reaction rates .[16] Whilst the

law of mass action can be valid in heterogeneous

environments ,[17] it is more appropriate to model the cytoplasm as

a fractal , in order to capture its limited-mobility kinetics .[18]

The resulting reaction rates predicted by the two approaches are

similar, with the only difference being that the equilibrium

approximation defines the constant as , whilst the quasi-

steady-state approximation uses . However, each approach

is founded upon a different assumption. The Michaelis Menten

equilibrium analysis is valid if the substrate reaches equilibrium on

a much faster time-scale than the product is formed or, more

precisely, that [14]

By contrast, the Briggs Haldane quasi-steady-state analysis

is valid if [13][19]

Thus it holds if the enzyme concentration is much less

than the substrate concentration. Even if this is not

satisfied, the approximation is valid if is large.

In both the Michaelis Menten and Briggs Haldane

analyses, the quality of the approximation improves
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as decreases. However, in model building, Michaelis

Menten kinetics are often invoked without regard to the

underlying assumptions .[14]

It is also important to remember that, while irreversibilityis a necessary simplification in order to yield a tractable

analytic solution, in the general case product formation is

not in fact irreversible. The enzyme reaction is more

correctly described as

In general, the assumption of irreversibility is a good

one in situations where one of the below is true:

1. The concentration of substrate(s) is very much

larger than the concentration of products:

This is true under standard in vitro assay

conditions, and is true for many in

vivo biological reactions, particularly where the

product is continually removed by a subsequent

reaction.

2. The energy released in the reaction is very

large, that is

In situations where neither of these two

conditions hold (that is, the reaction is low

energy and a substantial pool of product(s)

exists), the Michaelis

Menten equationbreaks down, and more complex modelling

approaches explicitly taking the forward

and reverse reactions into account must be

taken to understand the enzyme biology.

Determination ofconstants [edit ]

The typical method for determining the

constants and involves
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running a series of enzyme assays at

varying substrate concentrations , and

measuring the initial reaction rate .

'Initial' here is taken to mean that the

reaction rate is measured after a relatively

short time period, during which it is

assumed that the enzyme-substrate

complex has formed, but that the substrate

concentration held approximately constant,

and so the equilibrium or quasi-steady-

state approximation remain valid .[19] By

plotting reaction rate against concentration,

and usin gnonlinear regression of the

Michaelis Menten equation, the

parameters may be obtained .[20]

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 diagram , Hanes Woolf

plot and Lineweaver Burk plot ; of these,

the Hanes Woolf plot is the most

accurate .[20] However, while useful for

visualization, all three methods distort the

error structure of the data and are inferior

to nonlinear regression .[21] Nonetheless,

their use can still be found in modern

literature .[22]

In 1997 Santiago Schnell and Claudio

Mendoza derived a closed form solution for

the time course kinetics analysis of the

Michaelis Menten kinetics .[23] The solution,

known as the Schnell-Mendoza equation,

has the form:
http://en.wikipedia.org/wiki/Enzyme_assayhttp://en.wikipedia.org/wiki/Enzyme_assayhttp://en.wikipedia.org/wiki/Enzyme_assayhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-segel89-19http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-segel89-19http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-segel89-19http://en.wikipedia.org/wiki/Nonlinear_regressionhttp://en.wikipedia.org/wiki/Nonlinear_regressionhttp://en.wikipedia.org/wiki/Nonlinear_regressionhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Eadie%E2%80%93Hofstee_diagramhttp://en.wikipedia.org/wiki/Eadie%E2%80%93Hofstee_diagramhttp://en.wikipedia.org/wiki/Eadie%E2%80%93Hofstee_diagramhttp://en.wikipedia.org/wiki/Eadie%E2%80%93Hofstee_diagramhttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Lineweaver%E2%80%93Burk_plothttp://en.wikipedia.org/wiki/Lineweaver%E2%80%93Burk_plothttp://en.wikipedia.org/wiki/Lineweaver%E2%80%93Burk_plothttp://en.wikipedia.org/wiki/Lineweaver%E2%80%93Burk_plothttp://en.wikipedia.org/wiki/Lineweaver%E2%80%93Burk_plothttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-greco79-21http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-greco79-21http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-greco79-21http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-hayakawa06-22http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-hayakawa06-22http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-hayakawa06-22http://en.wikipedia.org/wiki/Santiago_Schnellhttp://en.wikipedia.org/wiki/Santiago_Schnellhttp://en.wikipedia.org/wiki/Santiago_Schnellhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-23http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-23http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-23http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-23http://en.wikipedia.org/wiki/Santiago_Schnellhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-hayakawa06-22http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-greco79-21http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Lineweaver%E2%80%93Burk_plothttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Hanes%E2%80%93Woolf_plothttp://en.wikipedia.org/wiki/Eadie%E2%80%93Hofstee_diagramhttp://en.wikipedia.org/wiki/Eadie%E2%80%93Hofstee_diagramhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-leskovac03-20http://en.wikipedia.org/wiki/Nonlinear_regressionhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-segel89-19http://en.wikipedia.org/wiki/Enzyme_assay


8/8

where W[] is the Lambert-W

function and where F(t) is

The Schnell-Mendoza equation

has been used to

estimate and from

time course data .[24][25]

The equation below, obtained by

Berberan-Santos in 2010

(MATCH Commun. Math.

Comput. Chem. 63, 283),

encompasses the Schnell-

Mendoza equation, and is still

valid when the initial substrate

concentration is close to that of

enzyme,

where W[] is again

the Lambert-W function .
http://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-24http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-24http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-24http://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-24http://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics#cite_note-24http://en.wikipedia.org/wiki/Lambert_W_functionhttp://en.wikipedia.org/wiki/Lambert_W_function

michaelis manten kinetics

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