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Michaelis–Menten kinetics

From Wikipedia, the free encyclopedia
Model of enzyme kinetics
Curve of the Michaelis–Menten equation labelled in accordance with IUBMB recommendations

Inbiochemistry,Michaelis–Menten kinetics, named afterLeonor Michaelis andMaud Menten, is the simplest case ofenzyme kinetics, applied to enzyme-catalysed reactions involving the transformation of one substrate into one product. In 1913, Michaelis and Menten expanded onVictor Henri's fundamental equation of enzyme kinetics, which was established in 1902.[1][2] It takes the form of adifferential equation describing thereaction ratev{\displaystyle v} (rate of formation ofproduct P, with concentrationp{\displaystyle p}) as a function ofa{\displaystyle a}, theconcentration of thesubstrate A (using the symbols recommended by theIUBMB).[3][4][5][6] The formula below is given by theMichaelis–Menten equation:

v=dpdt=VaKm+a{\displaystyle v={\frac {\mathrm {d} p}{\mathrm {d} t}}={\frac {Va}{K_{\mathrm {m} }+a}}}

V{\displaystyle V}, which is often written asVmax{\displaystyle V_{\max }},[7] represents thelimiting rate approached by the system at saturating substrate concentration for a given enzyme concentration. TheMichaelis constantKm{\displaystyle K_{\mathrm {m} }} has units of concentration, and for a given reaction is equal to the concentration of substrate at which the reaction rate is half ofV{\displaystyle V}.[8] Biochemical reactions involving a single substrate are often assumed to follow Michaelis–Menten kinetics, without regard to the model's underlying assumptions. Only a small proportion of enzyme-catalysed reactions have just one substrate, but the equation still often applies if only one substrate concentration is varied.

"Michaelis–Menten plot"

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Semi-logarithmic plot of Michaelis–Menten data

The plot ofv{\displaystyle v} againsta{\displaystyle a} has often been called a "Michaelis–Menten plot", even recently,[9][10][11] but this terminology is historically misleading, asMichaelis andMenten did not use such a plot. Instead, they plottedv{\displaystyle v} againstlog⁡a{\displaystyle \log a}, which has some advantages over the usual ways of plotting Michaelis–Menten data. Ifv{\displaystyle v} is the dependent variable, then it does not distort any experimental errors inv{\displaystyle v}.Michaelis andMenten did not attempt to estimateV{\displaystyle V} directly from the limit approached at highlog⁡a{\displaystyle \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 of0.576V{\displaystyle 0.576V} i.e.0.25ln⁡10⋅V{\displaystyle 0.25\ln 10\cdot V}. With an accurate value ofV{\displaystyle V} it was easy to determinelog⁡Km{\displaystyle \log K_{\mathrm {m} }} from the point on the curve corresponding to0.5V{\displaystyle 0.5V}.

This plot is virtually never used today for estimatingV{\displaystyle V} andKm{\displaystyle K_{\mathrm {m} }}, but it remains valuable to compare the properties of several enzymes across a broad range of substrate concentrations - such asisoenzymes. For example, the four mammalian isoenzymes ofhexokinase 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.[12]

Model

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A decade beforeMichaelis andMenten,Victor Henri found that enzyme reactions could be explained by assuming a binding interaction between the enzyme and the substrate.[13] His work was taken up by Michaelis and Menten, who investigated thekinetics ofinvertase, an enzyme that catalyzes thehydrolysis ofsucrose intoglucose andfructose.[14] In 1913, they proposed a mathematical model of the reaction.[15] It involves anenzyme E binding to a substrate A to form acomplex EA that releases aproduct P regenerating the original form of the enzyme.[8] This may be represented schematically as

E+A⇌k−1k+1EA→kcatE+P{\displaystyle {\ce {E{}+A<=>[{\mathit {k_{\mathrm {+1} }}}][{\mathit {k_{\mathrm {-1} }}}]EA->[k_{\ce {cat}}]E{}+P}}}

wherek+1{\displaystyle k_{\mathrm {+1} }} (forward rate constant),k−1{\displaystyle k_{\mathrm {-1} }} (reverse rate constant), andkcat{\displaystyle k_{\mathrm {cat} }} (catalytic rate constant) denote therate constants,[16] the double arrows between A (substrate) and EA (enzyme-substrate complex) represent the fact that enzyme-substrate binding is areversible process, and the single forward arrow represents the formation of P (product).

Under certainassumptions – such as the enzyme concentration being much less than the substrate concentration – the rate of product formation is given by

v=dpdt=VmaxaKm+a=kcate0aKm+a{\displaystyle v={\frac {\mathrm {d} p}{\mathrm {d} t}}={\frac {V_{\max }a}{K_{\mathrm {m} }+a}}={\frac {k_{\mathrm {cat} }e_{0}a}{K_{\mathrm {m} }+a}}}

in whiche0{\displaystyle e_{0}} is the initial enzyme concentration. Thereaction order depends on the relative size of the two terms in the denominator. At low substrate concentrationa≪Km{\displaystyle a\ll K_{\mathrm {m} }}, so that the ratev=kcate0aKm{\displaystyle v={\frac {k_{\mathrm {cat} }e_{0}a}{K_{\mathrm {m} }}}} varies linearly with substrate concentrationa{\displaystyle a} (first-order kinetics ina{\displaystyle a}).[17] However at highera{\displaystyle a}, witha≫Km{\displaystyle a\gg K_{\mathrm {m} }}, the reaction approaches independence ofa{\displaystyle a} (zero-order kinetics ina{\displaystyle a}),[17]asymptotically approaching the limiting rateVmax=kcate0{\displaystyle V_{\mathrm {max} }=k_{\mathrm {cat} }e_{0}}. This rate, which is never attained, refers to the hypothetical case in which all enzyme molecules are bound to substrate.kcat{\displaystyle k_{\mathrm {cat} }}, known as theturnover number orcatalytic constant, normally expressed in s–1, is the limiting number of substrate molecules converted to product per enzyme molecule per unit of time. Further addition of substrate would not increase the rate, and the enzyme is said to be saturated.

The Michaelis constantKm{\displaystyle K_{\mathrm {m} }} is not affected by the concentration or purity of an enzyme.[18] 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, includingantigen–antibody binding,DNA–DNA hybridization, andprotein–protein interaction.[19][20] It can be used to characterize a generic biochemical reaction, in the same way that theLangmuir equation can be used to model genericadsorption of biomolecular species.[20] When an empirical equation of this form is applied to microbial growth, it is sometimes called aMonod equation.

Michaelis–Menten kinetics have also been applied to a variety of topics outside of biochemical reactions,[16] includingalveolar clearance of dusts,[21] therichness of species pools,[22] clearance ofblood alcohol,[23] thephotosynthesis-irradiance relationship, and bacterialphage infection.[24]

The equation can also be used to describe the relationship betweenion channelconductivity andligand concentration,[25] and also, for example, to limiting nutrients and phytoplankton growth in the global ocean.[26]

Specificity

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Thespecificity constantkcat/Km{\displaystyle k_{\text{cat}}/K_{\mathrm {m} }} (also known as thecatalytic efficiency) is a measure of how efficiently an enzyme converts a substrate into product. Although it is the ratio ofkcat{\displaystyle k_{\text{cat}}} andKm{\displaystyle K_{\mathrm {m} }} it is a parameter in its own right, more fundamental thanKm{\displaystyle K_{\mathrm {m} }}.Diffusion limited enzymes, such asfumarase, work at the theoretical upper limit of108 – 1010 M−1s−1, limited by diffusion of substrate into theactive site.[27]

If we symbolize the specificity constant for a particular substrate A askA=kcat/Km{\displaystyle k_{\mathrm {A} }=k_{\text{cat}}/K_{\mathrm {m} }} the Michaelis–Menten equation can be written in terms ofkA{\displaystyle k_{\mathrm {A} }} andKm{\displaystyle K_{\mathrm {m} }} as follows:

v=kAe0a1+aKm{\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 zeroth 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:

v≈kAe0a when a→0{\displaystyle v\approx k_{\mathrm {A} }e_{0}a{\text{ when }}a\rightarrow 0}

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

v→kcate0 when a→∞{\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 eitherkcat{\displaystyle k_{\text{cat}}} orKm{\displaystyle K_{\mathrm {m} }} alone. PuttingkA{\displaystyle k_{\mathrm {A} }} for substrateA{\displaystyle \mathrm {A} } andkA′{\displaystyle k_{\mathrm {A'} }} for a competing substrateA′{\displaystyle \mathrm {A'} }, then the two rates when both are present simultaneously are as follows:

vA=kAe0a1+aKmA+a′KmA′,vA′=kA′e0a′1+aKmA+a′KmA′{\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:

vAvA′=kA⋅akA′⋅a′{\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.

Nomenclature

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As the equation originated withHenri, not withMichaelis andMenten, it is more accurate to call it the Henri–Michaelis–Menten equation,[28] 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

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Parameter values vary widely between enzymes. Some examples are as follows:[29]

EnzymeKm{\displaystyle K_{\mathrm {m} }} (M)kcat{\displaystyle k_{\text{cat}}} (s−1)kcat/Km{\displaystyle k_{\text{cat}}/K_{\mathrm {m} }} (M−1s−1)
Chymotrypsin1.5 × 10−20.149.3
Pepsin3.0 × 10−40.501.7 × 103
tRNA synthetase9.0 × 10−47.68.4 × 103
Ribonuclease7.9 × 10−37.9 × 1021.0 × 105
Carbonic anhydrase2.6 × 10−24.0 × 1051.5 × 107
Fumarase5.0 × 10−68.0 × 1021.6 × 108

Derivation

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Equilibrium approximation

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In their analysis, Michaelis and Menten (and also Henri) assumed that the substrate is in instantaneouschemical equilibrium with the complex, which implies[15][30]

k+1ea=k−1x{\displaystyle k_{+1}ea=k_{-1}x}

in whiche is the concentration offree enzyme (not the total concentration) andx is the concentration of enzyme-substrate complex EA.

Conservation of enzyme requires that[30]

e=e0−x{\displaystyle e=e_{0}-x}

wheree0{\displaystyle e_{0}} is now thetotal enzyme concentration. After combining the two expressions some straightforward algebra leads to the following expression for the concentration of the enzyme-substrate complex:

x=e0aKdiss+a{\displaystyle x={\frac {e_{0}a}{K_{\mathrm {diss} }+a}}}

whereKdiss=k−1/k+1{\displaystyle K_{\mathrm {diss} }=k_{-1}/k_{+1}} is thedissociation constant of the enzyme-substrate complex. Hence the rate equation is the Michaelis–Menten equation,[30]

v=k+2e0aKdiss+a{\displaystyle v={\frac {k_{+2}e_{0}a}{K_{\mathrm {diss} }+a}}}

wherek+2{\displaystyle k_{+2}} corresponds to the catalytic constantkcat{\displaystyle k_{\mathrm {cat} }} and the limiting rate isVmax=k+2e0=kcate0{\displaystyle V_{\mathrm {max} }=k_{+2}e_{0}=k_{\mathrm {cat} }e_{0}}. Likewise with the assumption of equilibrium the Michaelis constantKm=Kdiss{\displaystyle K_{\mathrm {m} }=K_{\mathrm {diss} }}.

Irreversible first step

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When studyingurease at about the same time as Michaelis and Menten were studying invertase,Donald Van Slyke and G. E. Cullen[31] made essentially the opposite assumption, treating the first step not as an equilibrium but as an irreversible second-order reaction with rate constantk+1{\displaystyle k_{+1}}. As their approach is never used today it is sufficient to give their final rate equation:

v=k+2e0ak+2/k+1+a{\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 whetherKm{\displaystyle K_{\mathrm {m} }} is equal tok+2/k+1{\displaystyle k_{+2}/k_{+1}} or tok−1/k+1{\displaystyle k_{-1}/k_{+1}} or to something else.

Steady-state approximation

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G. E. Briggs andJ. B. S. Haldane undertook an analysis that harmonized the approaches of Michaelis and Menten and of Van Slyke and Cullen,[32][33] 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.[34] This assumption means thatk+1ea=k−1x+kcatx=(k−1+kcat)x{\displaystyle k_{+1}ea=k_{-1}x+k_{\mathrm {cat} }x=(k_{-1}+k_{\mathrm {cat} })x}. The resulting rate equation is as follows:

v=kcate0aKm+a{\displaystyle v={\frac {k_{\mathrm {cat} }e_{0}a}{K_{\mathrm {m} }+a}}}

where

kcat=k+2 and Km=k−1+kcatk+1{\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.[35]

Assumptions and limitations

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All of the derivations given treat the initial binding step in terms of thelaw of mass action, which assumes freediffusion through the solution. However, in the environment of a living cell where there is a high concentration ofproteins, thecytoplasm often behaves more like a viscousgel than a free-flowing liquid, limiting molecular movements bydiffusion and altering reaction rates.[36] 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.[37] 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 afractal, in order to capture its limited-mobility kinetics.[38]

Estimation of Michaelis–Menten parameters

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Graphical methods

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Determining the parameters of the Michaelis–Menten equation typically involves running a series ofenzyme assays at varying substrate concentrationsa{\displaystyle a}, and measuring the initial reaction ratesv{\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.[39] By plotting reaction rate against concentration, and usingnonlinear 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 theEadie–Hofstee plot ofv{\displaystyle v} againstv/a{\displaystyle v/a},[40][41] theHanes plot ofa/v{\displaystyle a/v} againsta{\displaystyle a},[42] and theLineweaver–Burk plot (also known as thedouble-reciprocal plot) of1/v{\displaystyle 1/v} against1/a{\displaystyle 1/a}.[43] Of these,[44] the Hanes plot is the most accurate whenv{\displaystyle v} is subject to errors with uniform standard deviation.[45] From the point of view of visualizaing the data the Eadie–Hofstee plot has an important property: the entire possible range ofv{\displaystyle v} values from0{\displaystyle 0} toV{\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 ofv{\displaystyle v} andKm{\displaystyle K_{\mathrm {m} }} than correctly weighted non-linear regression. Assuming an errorε(v){\displaystyle \varepsilon (v)} onv{\displaystyle v}, an inverse representation leads to an error ofε(v)/v2{\displaystyle \varepsilon (v)/v^{2}} on1/v{\displaystyle 1/v} (Propagation of uncertainty), implying that linear regression of the double-reciprocal plot should include weights ofv4{\displaystyle v^{4}}. This was well understood by Lineweaver and Burk,[43] who had consulted the eminent statisticianW. Edwards Deming before analysing their data.[46] Unlike nearly all workers since, Burk made an experimental study of the error distribution, finding it consistent with a uniform standard error inv{\displaystyle v}, before deciding on the appropriate weights.[47] This aspect of the work of Lineweaver and Burk received virtually no attention at the time, and was subsequently forgotten.

Thedirect linear plot is a graphical method in which the observations are represented by straight lines in parameter space, with axesKm{\displaystyle K_{\mathrm {m} }} andV{\displaystyle V}: each line is drawn with an intercept of−a{\displaystyle -a} on theKm{\displaystyle K_{\mathrm {m} }} axis andv{\displaystyle v} on theV{\displaystyle V} axis. The point of intersection of the lines for different observations yields the values ofKm{\displaystyle K_{\mathrm {m} }} andV{\displaystyle V}.[48]

Weighting

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Many authors, for example Greco and Hakala,[49] 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[47] carried out the appropriate investigation, and found that the error structure of his data was consistent with a uniform standard deviation inv{\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.[50][51] However, this truth may be more complicated than any dependence onv{\displaystyle v} alone can represent.[52]

Uniform standard deviation of1/v{\displaystyle 1/v}. If the rates are considered to have a uniform standard deviation the appropriate weight for everyv{\displaystyle v} value for non-linear regression is 1. If the double-reciprocal plot is used each value of1/v{\displaystyle 1/v} should have a weight ofv4{\displaystyle v^{4}}, whereas if the Hanes plot is used each value ofa/v{\displaystyle a/v} should have a weight ofv4/a2{\displaystyle v^{4}/a^{2}}.

Uniform coefficient variation of1/v{\displaystyle 1/v}. If the rates are considered to have a uniform coefficient variation the appropriate weight for everyv{\displaystyle v} value for non-linear regression isv2{\displaystyle v^{2}}. If the double-reciprocal plot is used each value of1/v{\displaystyle 1/v} should have a weight ofv2{\displaystyle v^{2}}, whereas if the Hanes plot is used each value ofa/v{\displaystyle a/v} should have a weight ofv2/a2{\displaystyle v^{2}/a^{2}}.

Ideally thev{\displaystyle 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 valuesv^{\displaystyle {\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.[53] This is tedious to do by hand, but can readily be done in the computer.

Closed form equation

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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 theLambert W function.[54]Namely,

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

whereW is the Lambert W function and

F(t)=a0Kmexp(a0Km−VtKm){\displaystyle 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,[55] has been used to estimateV{\displaystyle V} andKm{\displaystyle K_{\mathrm {m} }} from time course data.[56][57]

Reactions with more than one substrate

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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 expressv{\displaystyle v} in terms of two substrate concentrationsa{\displaystyle a} andb{\displaystyle b}:

v=VabKiAKmB+KmBa+KmAb+ab{\displaystyle 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 thata{\displaystyle a} is varied withb{\displaystyle b} held constant. Then it is convenient to reorganize the equation as follows:

v=Vb⋅aKiAKmB+KmAb+(KmB+b)a=VbKmB+b⋅aKiAKmB+KmAbKmB+b+a{\displaystyle 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=VappaKmapp+a{\displaystyle v={\frac {V^{\mathrm {app} }a}{K_{\mathrm {m} }^{\mathrm {app} }+a}}}

withapparent valuesVapp{\displaystyle V^{\mathrm {app} }} andKmapp{\displaystyle K_{\mathrm {m} }^{\mathrm {app} }} defined as follows:

Vapp=VbKmB+b{\displaystyle V^{\mathrm {app} }={\dfrac {Vb}{K_{\mathrm {mB} }+b}}}
Kmapp=KiAKmB+KmAbKmB+b{\displaystyle K_{\mathrm {m} }^{\mathrm {app} }={\dfrac {K_{\mathrm {iA} }K_{\mathrm {mB} }+K_{\mathrm {mA} }b}{K_{\mathrm {mB} }+b}}}

Linear inhibition

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The linear (simple) types of inhibition can be classified in terms of the general equation formixed inhibition at an inhibitor concentrationi{\displaystyle i}:

v=VaKm(1+iKic)+a(1+iKiu){\displaystyle v={\dfrac {Va}{K_{\mathrm {m} }\left(1+{\dfrac {i}{K_{\mathrm {ic} }}}\right)+a\left(1+{\dfrac {i}{K_{\mathrm {iu} }}}\right)}}}

in whichKic{\displaystyle K_{\mathrm {ic} }} is thecompetitive inhibition constant andKiu{\displaystyle K_{\mathrm {iu} }} is theuncompetitive inhibition constant. This equation includes the other types of inhibition as special cases:

Pure non-competitive inhibition is very rare, being mainly confined to effects of protons and some metal ions. Cleland recognized this, and he redefinednoncompetitive to meanmixed.[59] 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=V1+i/Kiu⋅aKm(1+i/Kic)1+i/Kiu+a=VappaKmapp+a{\displaystyle 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 {m} }^{\mathrm {app} }+a}}}

with apparent valuesVapp{\displaystyle V^{\mathrm {app} }} andKmapp{\displaystyle K_{\mathrm {m} }^{\mathrm {app} }} defined as follows:

Vapp=V1+i/Kiu{\displaystyle V^{\mathrm {app} }={\dfrac {V}{1+i/K_{\mathrm {iu} }}}}
Kmapp=Km(1+i/Kic)1+i/Kiu{\displaystyle K_{\mathrm {m} }^{\mathrm {app} }={\dfrac {K_{\mathrm {m} }(1+i/K_{\mathrm {ic} })}{1+i/K_{\mathrm {iu} }}}}

See also

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References

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  1. ^Heckmann, Christian M.; Paradisi, Francesca (2020-12-16)."Looking Back: A Short History of the Discovery of Enzymes and How They Became Powerful Chemical Tools".ChemCatChem.12 (24):6082–6102.doi:10.1002/cctc.202001107.ISSN 1867-3880.PMC 7756376.PMID 33381242.
  2. ^"Lois générales de l'Action des Diastases".Nature.68 (1758):221–221. 1903-07-09.doi:10.1038/068221d0.ISSN 0028-0836.
  3. ^"Symbolism and terminology in enzyme kinetics. Recommendations 1981".Eur. J. Biochem.128 (2–3):281–291. 1982.doi:10.1111/j.1432-1033.1982.tb06963.x.
  4. ^"Symbolism and terminology in enzyme kinetics. Recommendations 1981".Arch. Biochem. Biophys.234 (2):732–740. 1983.doi:10.1016/0003-9861(83)90262-X.
  5. ^"Symbolism and terminology in enzyme kinetics. Recommendations 1981".Biochem. J.213 (3):561–571. 1982.doi:10.1042/bj2130561.PMC 1152169.PMID 6615450.
  6. ^Cornish-Bowden, A. (2014)."Current IUBMB recommendations on enzyme nomenclature and kinetics".Perspectives in Science.1 (1–6):74–87.Bibcode:2014PerSc...1...74C.doi:10.1016/j.pisc.2014.02.006.
  7. ^The subscript max and term "maximum rate" (or "maximum velocity") often used are not strictly appropriate because this isnot a maximum in the mathematical sense.
  8. ^abCornish-Bowden, Athel (2012).Fundamentals of Enzyme Kinetics (4th ed.). Wiley-Blackwell, Weinheim. pp. 25–75.ISBN 978-3-527-33074-4.
  9. ^Busch, T.; Petersen, M. (2021)."Identification and biochemical characterisation of tyrosine aminotransferase fromAnthoceros agrestis unveils the conceivable entry point into rosmarinic acid biosynthesis in hornworts".Planta.253 (5): 98.Bibcode:2021Plant.253...98B.doi:10.1007/s00425-021-03623-2.PMC 8041713.PMID 33844079.S2CID 233212717.
  10. ^M. A. Chrisman; M. J. Goldcamp; A. N. Rhodes; J. Riffle (2023). "Exploring Michaelis–Menten kinetics and the inhibition of catalysis in a synthetic mimic of catechol oxidase: an experiment for the inorganic chemistry or biochemistry laboratory".J. Chem. Educ.100 (2):893–899.Bibcode:2023JChEd.100..893C.doi:10.1021/acs.jchemed.9b01146.S2CID 255736240.
  11. ^Huang, Y. Y.; Condict, L.; Richardson, S. J.; Brennan, C. S.; Kasapis, S. (2023). "Exploring the inhibitory mechanism of p-coumaric acid on α-amylasevia multi-spectroscopic analysis, enzymatic inhibition assay and molecular docking".Food Hydrocolloids.139 108524: 19)08524.doi:10.1016/j.foodhyd.2023.108524.S2CID 256355620.
  12. ^Cárdenas, M. L.; Cornish-Bowden, A.; Ureta, T. (1998)."Evolution and regulatory role of the hexokinases".Biochim. Biophys. Acta.1401 (3):242–264.doi:10.1016/S0167-4889(97)00150-X.PMID 9540816.
  13. ^Henri, Victor (1903).Lois Générales de l'Action des Diastases. Paris: Hermann.
  14. ^"Victor Henri". Whonamedit?. Retrieved24 May 2011.
  15. ^abMichaelis, L.; Menten, M.L. (1913). "Die Kinetik der Invertinwirkung".Biochem Z.49:333–369.(recent translation, and anolder partial translation)
  16. ^abChen, W.W.; Neipel, M.; Sorger, P.K. (2010)."Classic and contemporary approaches to modeling biochemical reactions".Genes Dev.24 (17):1861–1875.doi:10.1101/gad.1945410.PMC 2932968.PMID 20810646.
  17. ^abLaidler K.J. and Meiser J.H.Physical Chemistry (Benjamin/Cummings 1982) p.430ISBN 0-8053-5682-7
  18. ^Ninfa, Alexander; Ballou, David P. (1998).Fundamental laboratory approaches for biochemistry and biotechnology. Bethesda, Md.: Fitzgerald Science Press.ISBN 978-1-891786-00-6.OCLC 38325074.
  19. ^Lehninger, A.L.; Nelson, D.L.; Cox, M.M. (2005).Lehninger principles of biochemistry. New York: W.H. Freeman.ISBN 978-0-7167-4339-2.
  20. ^abChakraborty, S. (23 Dec 2009).Microfluidics and Microfabrication (1 ed.). Springer.ISBN 978-1-4419-1542-9.
  21. ^Yu, R.C.; Rappaport, S.M. (1997)."A lung retention model based on Michaelis–Menten-like kinetics".Environ Health Perspect.105 (5):496–503.Bibcode:1997EnvHP.105..496Y.doi:10.1289/ehp.97105496.PMC 1469867.PMID 9222134.
  22. ^Keating, K.A.; Quinn, J.F. (1998). "Estimating species richness: the Michaelis–Menten model revisited".Oikos.81 (2):411–416.Bibcode:1998Oikos..81..411K.doi:10.2307/3547060.JSTOR 3547060.
  23. ^Jones, A.W. (2010). "Evidence-based survey of the elimination rates of ethanol from blood with applications in forensic casework".Forensic Sci Int.200 (1–3):1–20.doi:10.1016/j.forsciint.2010.02.021.PMID 20304569.
  24. ^Abedon, S.T. (2009). "Kinetics of phage-mediated biocontrol of bacteria".Foodborne Pathog Dis.6 (7):807–15.doi:10.1089/fpd.2008.0242.PMID 19459758.
  25. ^Ding, Shinghua;Sachs, Frederick (1999)."Single Channel Properties of P2X2 Purinoceptors".The Journal of General Physiology.113 (5):695–720.doi:10.1085/jgp.113.5.695.PMC 2222910.PMID 10228183.
  26. ^Dugdale, RCJ (1967)."Nutrient limitation in the sea: Dynamics, identification, and significance".Limnology and Oceanography.12 (4):685–695.Bibcode:1967LimOc..12..685D.doi:10.4319/lo.1967.12.4.0685.
  27. ^Stroppolo, M.E.; Falconi, M.; Caccuri, A.M.; Desideri, A. (Sep 2001)."Superefficient enzymes".Cell Mol Life Sci.58 (10):1451–60.doi:10.1007/PL00000788.PMC 11337273.PMID 11693526.S2CID 24874575.
  28. ^Deichmann, U.; Schuster, S.; Mazat, J.-P.; Cornish-Bowden, A. (2013)."Commemorating the 1913 Michaelis–Menten paperDie Kinetik der Invertinwirkung: three perspectives".FEBS J.281 (2):435–463.doi:10.1111/febs.12598.PMID 24180270.S2CID 5183178.
  29. ^Mathews, C.K.; van Holde, K.E.; Ahern, K.G. (10 Dec 1999).Biochemistry (3 ed.). Prentice Hall.ISBN 978-0-8053-3066-3.
  30. ^abcKeener, J.; Sneyd, J. (2008).Mathematical Physiology: I: Cellular Physiology (2 ed.). Springer.ISBN 978-0-387-75846-6.
  31. ^Van Slyke, D. D.; Cullen, G. E. (1914)."The mode of action of urease and of enzymes in general".J. Biol. Chem.19 (2):141–180.doi:10.1016/S0021-9258(18)88300-4.
  32. ^Briggs, G.E.; Haldane, J.B.S. (1925)."A note on the kinetics of enzyme action".Biochem J.19 (2):338–339.doi:10.1042/bj0190338.PMC 1259181.PMID 16743508.
  33. ^Laidler, Keith J. (1978).Physical Chemistry with Biological Applications. Benjamin/Cummings. pp. 428–430.ISBN 0-8053-5680-0.
  34. ^In advanced work this is known as the quasi-steady-state assumption or pseudo-steady-state-hypothesis, but in elementary treatments thesteady-state assumption is sufficient.
  35. ^Murray, J.D. (2002).Mathematical Biology: I. An Introduction (3 ed.). Springer.ISBN 978-0-387-95223-9.
  36. ^Zhou, H.X.; Rivas, G.; Minton, A.P. (2008)."Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences".Annu Rev Biophys.37 (1):375–97.doi:10.1146/annurev.biophys.37.032807.125817.PMC 2826134.PMID 18573087.
  37. ^Mastro, A. M.; Babich, M. A.; Taylor, W. D.; Keith, A. D. (1984)."Diffusion of a small molecule in the cytoplasm of mammalian cells".Proc. Natl. Acad. Sci. USA.81 (11):3414–3418.Bibcode:1984PNAS...81.3414M.doi:10.1073/pnas.81.11.3414.PMC 345518.PMID 6328515.
  38. ^Schnell, S.; Turner, T.E. (2004). "Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws".Prog Biophys Mol Biol.85 (2–3):235–60.CiteSeerX 10.1.1.117.1997.doi:10.1016/j.pbiomolbio.2004.01.012.PMID 15142746.
  39. ^Segel, L.A.; Slemrod, M. (1989)."The quasi-steady-state assumption: A case study in perturbation".SIAM Review.31 (3):446–477.Bibcode:1989SIAMR..31..446S.doi:10.1137/1031091.
  40. ^Eadie, G. S. (1942)."The inhibition of cholinesterase by physostigmine and prostigmine".J. Biol. Chem.146 (1):85–93.Bibcode:1942JBiCh.146...85E.doi:10.1016/S0021-9258(18)72452-6.
  41. ^Hofstee, B. H. J. (1953)."Specificity of esterases".J. Biol. Chem.199 (1):357–364.doi:10.1016/S0021-9258(18)44843-0.
  42. ^Hanes, C.S. (1932)."Studies on plant amylases. I. The effect of starch concentration upon the velocity of hydrolysis by the amylase of germinated barley".Biochem. J.26 (2):1406–1421.doi:10.1042/bj0261406.PMC 1261052.PMID 16744959.
  43. ^abLineweaver, H.; Burk, D. (1934)."The Determination of Enzyme Dissociation Constants".Journal of the American Chemical Society.56 (3):658–666.Bibcode:1934JAChS..56..658L.doi:10.1021/ja01318a036.
  44. ^The name of Barnet Woolf is often coupled with that of Hanes, but not with the other two. However, Haldane and Stern attributed all three to Woolf in their bookAllgemeine Chemie der Enzyme in 1932, about the same time as Hanes and clearly earlier than the others.
  45. ^This is not necessarily the case!
  46. ^Lineweaver H, Burk D, Deming WE (1934). "The dissociation constant of nitrogen-nitrogenase inAzobacter".J. Amer. Chem. Soc.56 (1):225–230.Bibcode:1934JAChS..56..225L.doi:10.1021/ja01316a071.
  47. ^abBurk, D. "Nitrogenase".Ergebnisse der Enzymforschung.3:23–56.
  48. ^Eisenthal, R.; Cornish-Bowden, A. (1974)."The direct linear plot: a new graphical procedure for estimating enzyme kinetic parameters".Biochem. J.139 (3):715–720.doi:10.1042/bj1390715.PMC 1166335.PMID 4854723.
  49. ^Greco, W.R.; Hakala, M.T. (1979)."Evaluation of methods for estimating the dissociation constant of tight binding enzyme inhibitors".J Biol Chem.254 (23):12104–12109.doi:10.1016/S0021-9258(19)86435-9.PMID 500698.
  50. ^Storer, A. C.; Darlison, M. G.; Cornish-Bowden, A. (1975)."The nature of experimental error in enzyme kinetic measurements".Biochem. J.151 (2):361–367.doi:10.1042/bj1510361.PMC 1172366.PMID 1218083.
  51. ^Askelöf, P; Korsfeldt, M; Mannervik, B (1975)."Error structure of enzyme kinetic experiments: Implications for weighting in regression-analysis of experimental-data".Eur. J. Biochem.69 (1):61–67.doi:10.1111/j.1432-1033.1976.tb10858.x.PMID 991863.
  52. ^Mannervik, B.; Jakobson, I.; Warholm, M. (1986)."Error structure as a function of substrate and inhibitor concentration in enzyme kinetic experiments".Biochem. J.235 (3):797–804.doi:10.1042/bj2350797.PMC 1146758.PMID 3753447.
  53. ^Cornish-Bowden, A; Endrenyi, L. (1986)."Robust regression of enzyme kinetic data".Biochem. J.234 (1):21–29.doi:10.1042/bj2340021.PMC 1146522.PMID 3707541.
  54. ^Schnell, S.; Mendoza, C. (1997). "A closed form solution for time-dependent enzyme kinetics".Journal of Theoretical Biology.187 (2):207–212.Bibcode:1997JThBi.187..207S.doi:10.1006/jtbi.1997.0425.
  55. ^Olp, M.D.; Kalous, K.S.; Smith, B.C. (2020)."ICEKAT: an interactive online tool for calculating initial rates from continuous enzyme kinetic traces".BMC Bioinformatics.21 (1): 186.doi:10.1186/s12859-020-3513-y.PMC 7222511.PMID 32410570.S2CID 218624836.
  56. ^Goudar, C. T.; Sonnad, J. R.; Duggleby, R. G. (1999). "Parameter estimation using a direct solution of the integrated Michaelis–Menten equation".Biochimica et Biophysica Acta (BBA) - Protein Structure and Molecular Enzymology.1429 (2):377–383.doi:10.1016/s0167-4838(98)00247-7.PMID 9989222.
  57. ^Goudar, C. T.; Harris, S. K.; McInerney, M. J.; Suflita, J. M. (2004). "Progress curve analysis for enzyme and microbial kinetic reactions using explicit solutions based on the LambertW function".Journal of Microbiological Methods.59 (3):317–326.doi:10.1016/j.mimet.2004.06.013.PMID 15488275.
  58. ^According to the IUBMB Recommendations inhibition is classifiedoperationally, i.e. in terms of what is observed, not in terms of its interpretation.
  59. ^Cleland, W. W. (1963). "The kinetics of enzyme-catalyzed reactions with two or more substrates or products: II. Inhibition: Nomenclature and theory".Biochim. Biophys. Acta.67 (2):173–187.doi:10.1016/0926-6569(63)90226-8.PMID 14021668.

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