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| [[Image:Michaelis-Menten saturation curve of an enzyme reaction LARGE.svg|thumb|An example curve with parameters ''V''<sub>max</sub> = 3.4 and ''K''<sub>m</sub> = 0.4.]]
| | For over a decade, the Prostate Cancer Education Council (PCEC) has provided free and low-cost screenings to men throughout the 3rd week of September, called Prostate Cancer Awareness Week (PCAW).<br><br>The average petite proportion for petite size outfits females caters for those under 5 foot plus 4 inches. They have somewhat less wide shoulders and one to 1 / 2 inch less broad from the misses' size for the bust, waist and hips measurements. Petite clothes for women have the sleeves plus pant length usually half to an inch less whilst pant's inseam plus skirt length 2 to 3 inches shorter.<br><br>But, the National Institutes of Health does not really recommend a weight loss waist to height ratio goal for individuals with abdominal weight but that have BMIs inside the normal and obese range -- unless these individuals have two or more risk factors for cardiovascular condition, or just the want to lose several weight.<br><br>Waking up this morning I was thus hungry however, I decided to stick to the diet. After my grapefruit breakfast, I am going on another bike ride. Well my belly hurt so much it was hard to eat my lunch. I am thus hungry. I went over to my sisters home plus they you having pizza for dinner. I was tempted to eat it, nevertheless I didnt. Then Im getting ready to consume dinner. Im thus happy for dinner considering I feel that this is the best dinner from the 4 days. I got a hamburger patty and stewed tomatoes. I then dipped the patty into the stewed tomatoes, it tasted like ketchup on the hamburger. Tonights dinner had filled me up, which is superior considering I dont like to go to bed with stomach cramps again. Im going to bed now and it feels advantageous to be full.<br><br>To learn more, a research team searched medical literature for studies that looked at stroke risk plus body mass index with a minimal of 4 years of follow up.<br><br>According to the National Heart, Lung, plus [http://safedietplansforwomen.com/waist-to-height-ratio waist to height ratio] Blood Institute ladies should have a waist circumference (WC) of 35 inches or less to be fit. Men must have a WC of 40 inches or less.<br><br>The fat can be tied to any number of health conditions or diseases. Underweight is really as harmful because obese. The doctor will sit down with you and discuss the right weight for the height, age, body size, exercise, healthcare condition, etc. I'll bet we didn't learn there was more to weight than a "standard chart." Those just consider 2 things- height and pounds.<br><br>The expecting mother should equally maintain a healthy lifestyle, eat a diet that is wealthy in fresh fruits and greens, keep perfectly hydrated plus have at least 30 minutes of exercise each day. |
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| In [[biochemistry]], '''Michaelis–Menten kinetics''' is one of best-known models of [[enzyme kinetics]]. It is named after German biochemist [[Leonor Michaelis]] and Canadian physician [[Maud Menten]]. The model takes the form of an equation describing the rate of [[enzymatic reaction]]s, by relating [[reaction rate]] <math>v</math> to <math>[S]</math>, the [[concentration]] of a [[Enzyme substrate (biology)|substrate]] ''S''. Its formula is given by
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| :<math> v = \frac{d [P]}{d t} = \frac{ V_\max {[S]}}{K_m + [S]} </math>.
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| Here, <math>V_\max</math> represents the maximum rate achieved by the system, at maximum (saturating) substrate concentrations. The Michaelis constant <math>K_m</math> is the substrate concentration at which the reaction rate is half of <math>V_\max</math>. Biochemical reactions involving a single substrate are often assumed to follow Michaelis–Menten kinetics, without regard to the model's underlying assumptions.
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| ==Model==
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| [[Image:Michaelis Menten S P E ES.svg|thumb|Change in concentrations over time for enzyme E, substrate S, complex ES and product P]]
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| In 1903, French physical chemist [[Victor Henri]] found that enzyme reactions were initiated by a bond between the enzyme and the substrate.<ref name="henri03" /> His work was taken up by German biochemist [[Leonor Michaelis]] and Canadian physician [[Maud Menten]], who investigated the [[Chemical kinetics|kinetics]] of an enzymatic reaction mechanism, [[invertase]], that catalyzes the [[hydrolysis]] of [[sucrose]] into [[glucose]] and [[fructose]].<ref name="whonamedit" /> In 1913, they proposed a mathematical model of the reaction.<ref name="michaelis13" /> It involves an [[enzyme]] E binding to a [[enzyme substrate (biology)|substrate]] S to form a complex ES, which in turn is converted into a [[product (biology)|product]] P and the enzyme. This may be represented schematically as
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| :<math>
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| E + S \, \overset{k_f}{\underset{k_r} \rightleftharpoons} \, ES \, \overset{k_\mathrm{cat}} {\longrightarrow} \, E + P
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| </math>
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| where <math>k_f</math>, <math>k_r</math>, and <math>k_\mathrm{cat}</math> denote the [[reaction rate constant|rate constant]]s,<ref name="chen10" /> and the double arrows between S and ES represent the fact that enzyme-substrate binding is a [[reversible reaction|reversible]] process.
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| Under certain [[#Assumptions and limitations|assumptions]] – such as the enzyme concentration being much less than the substrate concentration – the rate of product formation is given by
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| :<math>v = \frac{d [P]}{d t} = V_\max \frac{[S]}{K_m + [S]} = k_\mathrm{cat} [E]_0 \frac{[S]}{K_m + [S]}</math>
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| The [[reaction rate]] increases with increasing substrate concentration <math>[S]</math>, [[asymptote|asymptotically]] approaching its maximum rate <math>V_\max</math>, attained when all enzyme is bound to substrate. It also follows that <math>V_\max = k_\mathrm{cat} [E]_0</math>, where <math>[E]_0</math> is the enzyme concentration. <math>k_\mathrm{cat}</math>, the [[turnover number]], is the maximum number of substrate molecules converted to product per enzyme molecule per second.
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| The Michaelis constant <math>K_m</math> is the substrate concentration at which the reaction rate is at half-maximum, and is an inverse measure of the substrate's [[affinity (pharmacology)|affinity]] for the enzyme—as a small <math>K_m</math> indicates high affinity, meaning that the rate will approach <math>V_\max</math> more quickly.<ref name="nelson00" /> The value of <math>K_m</math> is dependent on both the enzyme and the substrate, as well as conditions such as temperature and pH.
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| The model is used in a variety of biochemical situations other than enzyme-substrate interaction, including [[immune complex|antigen-antibody binding]], [[DNA-DNA hybridization]], and [[protein-protein interaction]].<ref name="nelson00" /><ref name="chakraborty09" /> 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.<ref name="chakraborty09" /> When an empirical equation of this form is applied to microbial growth, it is sometimes called a [[Monod equation]].
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| ==Applications==
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| Parameter values vary wildly between enzymes:<ref name="mathews99" />
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| {| class="wikitable sortable" style="text-align: center"
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| |-
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| ! Enzyme !! <math>K_m</math> (M) !! <math>k_\text{cat}</math> (1/s) !! <math>k_\text{cat}/K_m</math> (1/M.s)
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| |-
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| | [[Chymotrypsin]] || 1.5 × 10<sup>−2</sup> || 0.14 || 9.3
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| |-
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| | [[Pepsin]] || 3.0 × 10<sup>−4</sup> || 0.50 || 1.7 × 10<sup>3</sup>
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| |-
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| | Tyrosyl-tRNA synthetase || 9.0 × 10<sup>−4</sup> || 7.6 || 8.4 × 10<sup>3</sup>
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| |-
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| | [[Ribonuclease]] || 7.9 × 10<sup>−3</sup> || 7.9 × 10<sup>2</sup> || 1.0 × 10<sup>5</sup>
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| |-
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| | [[Carbonic anhydrase]] || 2.6 × 10<sup>−2</sup> || 4.0 × 10<sup>5</sup> || 1.5 × 10<sup>7</sup>
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| |-
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| | [[Fumarase]] || 5.0 × 10<sup>−6</sup> || 8.0 × 10<sup>2</sup> || 1.6 × 10<sup>8</sup>
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| |}
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| The constant <math>k_\text{cat}/K_m</math> is a measure of how efficiently an enzyme converts a substrate into product. It has a theoretical upper limit of {{nowrap|10<sup>8</sup> – 10<sup>10</sup> /M.s}}; enzymes working close to this, such as fumarase, are termed superefficient.<ref name="stroppolo01" />
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| Michaelis–Menten kinetics have also been applied to a variety of spheres outside of biochemical reactions,<ref name="chen10" /> including [[pulmonary alveolus|alveolar]] clearance of dusts,<ref name="yu97" /> the [[species richness|richness of species]] pools,<ref name="keating98" /> clearance of [[blood alcohol content|blood alcohol]],<ref name="jones10" /> the [[PI curve|photosynthesis-irradiance]] relationship, and bacterial [[bacteriophage|phage]] infection.<ref name="abedon09" />
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| ==Derivation==
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| 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 equation]]s that define the rate of change of reactants with time <math>t</math>:<ref name="murray02" />
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| :<math>\begin{align}
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| \frac{d[E]}{dt} &= - k_f [E][S] + k_r [ES] + k_{cat} [ES] \\
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| \frac{d[S]}{dt} &= - k_f [E][S] + k_r [ES] \\
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| \frac{d[ES]}{dt} &= k_f [E][S] - k_r [ES] - k_{cat} [ES] \\
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| \frac{d[P]}{dt} &= k_{cat} [ES].
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| \end{align}</math>
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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, <math>[E] + [ES] = [E]_0</math> is a constant. This conservation law can also be observed by adding the first and third equations above.<ref name="murray02" /><ref name="keener08" />
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| ===Equilibrium approximation===
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| In their original analysis, Michaelis and Menten assumed that the substrate is in instantaneous [[chemical equilibrium]] with the complex, and thus <math>k_f [E] [S] = k_r [ES]</math>.<ref name="michaelis13" /><ref name="keener08" /> Combining this relationship with the enzyme conservation law, the concentration of complex is<ref name="keener08" />
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| :<math>[ES] = \frac{[E]_0 [S]}{K_d + [S]}</math>
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| where <math>K_d = k_r / k_f</math> is the [[dissociation constant]] for the enzyme-substrate complex. Hence the velocity <math>v</math> of the reaction – the rate at which P is formed – is<ref name="keener08" />
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| :<math>v = \frac{d [P]}{d t} = \frac{V_\max {[S]}}{K_d + [S]}</math>
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| where <math>V_\max = k_\mathrm{cat} [E]_0</math> is the maximum reaction velocity.
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| ===Quasi-steady-state approximation===
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| An alternative analysis of the system was undertaken by British botanist [[George Edward Briggs|G. E. Briggs]] and British geneticist [[J. B. S. Haldane]] in 1925.<ref name="briggs25" /> 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 (chemistry)|steady-state]] assumption or pseudo-steady-state-hypothesis. Mathematically, this assumption means <math>k_f [E] [S] = k_r [ES] + k_\mathrm{cat} [ES] </math>. Combining this relationship with the enzyme conservation law, the concentration of complex is<ref name="keener08" />
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| :<math>[ES] = \frac{[E]_0 [S]}{K_m + [S]}</math>
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| where
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| :<math>K_m = \frac{k_r + k_\mathrm{cat}}{k_f}</math>
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| is known as the Michaelis constant, where <math>k_r</math>, <math>k_\mathrm{cat}</math>, and <math>{k_f}</math> are, respectively, the constants for substrate unbinding, conversion to product, and binding to the enzyme.
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| Hence the velocity <math>v</math> of the reaction is<ref name="keener08" />
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| :<math>v = \frac{d [P]}{d t} = \frac{V_\max {[S]}}{K_m + [S]}</math>
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| ===Assumptions and limitations===
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| 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.<ref name="zhou08" /> Whilst the law of mass action can be valid in heterogeneous environments,<ref name="grima07" /> it is more appropriate to model the cytoplasm as a [[fractal]], in order to capture its limited-mobility kinetics.<ref name="schnell04" />
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| The resulting reaction rates predicted by the two approaches are similar, with the only difference being that the equilibrium approximation defines the constant as <math>K_d</math>, whilst the quasi-steady-state approximation uses <math>K_m</math>. 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 <ref name="keener08" />
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| :<math>\epsilon_d = \frac{k_\mathrm{cat}}{k_r} \ll 1</math>
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| By contrast, the Briggs–Haldane quasi-steady-state analysis is valid if <ref name="murray02" /><ref name="segel89" />
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| :<math>\epsilon_m = \frac{[E]_0}{[S]_0 + K_m} \ll 1</math>
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| 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 <math>K_m</math> is large.
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| In both the Michaelis–Menten and Briggs–Haldane analyses, the quality of the approximation improves as <math>\epsilon\,\!</math> decreases. However, in model building, Michaelis–Menten kinetics are often invoked without regard to the underlying assumptions.<ref name="keener08" />
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| It is also important to remember that, while irreversibility is 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
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| :<math>
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| E + S \, \overset{k_{f_1}} {\underset{k_{r_1}} {\rightleftharpoons}} \, ES \, \overset{k_{f_2}} {\underset{k_{r_2}} {\rightleftharpoons}} \, E + P
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| </math>
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| In general, the assumption of irreversibility is a good one in situations where one of the below is true:<br>
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| 1. The concentration of substrate(s) is very much larger than the concentration of products:
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| :<math> [S] \gg [P] </math>
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| 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.<br>
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| 2. The energy released in the reaction is very large, that is
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| :<math>\Delta{G} \ll 0</math>
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| 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 equation breaks down, and more complex modelling approaches explicitly taking the forward and reverse reactions into account must be taken to understand the enzyme biology.
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| ==Determination of constants==
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| The typical method for determining the constants <math>V_\max</math> and <math>K_m</math> involves running a series of [[enzyme assay]]s at varying substrate concentrations <math>[S]</math>, and measuring the initial reaction rate <math>v_0</math>. '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.<ref name="segel89" /> By plotting reaction rate against concentration, and using [[nonlinear regression]] of the Michaelis–Menten equation, the parameters may be obtained.<ref name="leskovac03" />
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| 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.<ref name="leskovac03" /> However, while useful for visualization, all three methods distort the error structure of the data and are inferior to nonlinear regression.<ref name="greco79" /> Nonetheless, their use can still be found in modern literature.<ref name="hayakawa06" />
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| In 1997 [[Santiago Schnell]] and Claudio Mendoza derived a closed form solution for the time course kinetics analysis of the Michaelis–Menten kinetics.<ref>Schnell S, Mendoza C. A closed form solution for time-dependent enzyme kinetics. Journal of theoretical Biology, 187 (1997): 207-212 [http://dx.doi.org/10.1006/jtbi.1997.0425 DOI]</ref> The solution, known as the Schnell-Mendoza equation, has the form:
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| :<math>\frac{[S]}{K_M} = W \left[ F(t) \right]\, </math>
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| where W[] is the [[Lambert W function|Lambert-W function]] and where F(t) is
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| :<math>F(t) = \frac{[S]_0}{K_M} \exp\!\left(\frac{[S]_0}{K_M} - \frac{V_\max}{K_M}\,t \right) \, </math>
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| The Schnell-Mendoza equation has been used to estimate <math>V_\max</math> and <math>K_m</math> from time course data.<ref>C.T. Goudar, J.R. Sonnad, R.G. Duggleby (1999). Parameter estimation using a direct solution of the integrated Michaelis-Menten equation. Biochimica et Biophysica Acta - Protein Structure and Molecular Enzymology 1429, 377-383.</ref><ref>C.T. Goudar, S.K. Harris, M.J. McInerney, J.M Suflita (2004). Progress curve analysis for enzyme and microbial kinetic reactions using explicit solutions based on the Lambert W function. Journal of Microbiological Methods 59, 317-326</ref>
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| 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,
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| :<math>\frac{[S]}{K_M} = W \left[ F(t) \right]- \frac{V_\max}{k_{cat} K_M}\ \frac{W \left[ F(t) \right]}{1+W \left[ F(t) \right]}\, </math>
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| where W[] is again the [[Lambert W function|Lambert-W function]].
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| ==See also==
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| * [[Enzyme kinetics]]
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| * [[Reaction progress kinetic analysis]]
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| * [[Steady state (chemistry)]]
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| ==References==
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| {{reflist|2|refs=
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| <ref name="michaelis13">
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| {{citation
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| | last1 = Michaelis
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| | first1 = L.
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| | last2 = Menten
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| | first2 = M.L.
| |
| | year = 1913
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| | title = Die Kinetik der Invertinwirkung
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| | journal = Biochem Z
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| | volume = 49
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| | pages = 333–369
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| | doi =
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| }}
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| ([http://pubs.acs.org/doi/suppl/10.1021/bi201284u recent translation], and an [http://web.lemoyne.edu/~giunta/menten.html older partial translation])
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| </ref>
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| <ref name="murray02">
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| | |
| ==Further reading==
| |
| *{{Wikibooks-inline|Biochemistry/Catalysis}}
| |
| | |
| {{Enzymes}}
| |
| | |
| {{DEFAULTSORT:Michaelis-Menten Kinetics}}
| |
| [[Category:Enzyme kinetics]]
| |
| [[Category:Chemical kinetics]]
| |
| [[Category:Ordinary differential equations]]
| |
| [[Category:Catalysis]]
| |
| | |
| [[nl:Michaelis-Menten-vergelijking]]
| |
| [[ru:Уравнение Михаэлиса — Ментен]]
| |
| [[sk:Michaelisova konštanta]]
| |
| [[sv:Michaelis-Menten kinetik]]
| |
| [[zh:米氏方程]]
| |