The Michaelis-Menten equation

Leonor Michaelis (1875-1949) and Maud Menten (1879-1960)

It was a peculiar circumstance that brought together the legendary research duo of German biochemist Leonor Michaelis and Canadian physician Maud Menten. The latter, having gained a degree in medicine from the University of Toronto in 1911, was looking forward to pursuing her research ambitions. But it was not to be in her home country because in those days women were not allowed to do research in Canada.

And so it was that in 1912, Menten boarded a transatlantic steamer bound for Germany and the more emancipated atmosphere of Michaelis’ laboratory in Berlin. It is a testament to Menten’s courage and conviction that she undertook the journey within months of the sinking of the Titanic.

Michaelis and Menten published their classic paper, Die Kinetik der Invertinwirkung [The Kinetics of Invertase Activity] in 1913. In it, they formulated a rate equation based on a kinetic model where the substrate binds to the enzyme to form a complex, which in turn is converted into product(s) whose release regenerates free enzyme, allowing the process to be repeated.

Over a century later, the equation they developed still forms the basis for most single-substrate enzyme kinetics – a remarkable achievement by any standards. But neither Michaelis nor Menten were to receive the recognition they rightly deserved.

The outbreak of World War I cut short their further collaboration, with hostilities between Germany and Canada forcing Menten to return to North America in 1914. In that same year, Michaelis published a paper questioning the validity of the work of Emil Abderhalden, who held a powerful position in the German scientific establishment. Despite the fact that Abderhalden’s findings were lacking in scientific rigor and quite possibly fraudulent, Michaelis’ reputation was so tarnished by this episode that it effectively ended his career as an academic in Germany, and in 1922 he moved to the University of Nagoya in Japan.

For both Menten and Michaelis, the future that seemed so bright in 1913 faded into obscurity. This perhaps explains why so many bioscience students, although familiar with the equation to which the names of Michaelis and Menten are attached, know little or nothing about who Michaelis and Menten were.

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Why they studied invertase

The hospital in Berlin where Michaelis and Menten conducted their legendary research

Michaelis and Menten’s classic study didn’t come out of a vacuum, far from it. Enzyme kinetics had already been developed to some extent by AJ Brown in England and Victor Henri in France. Brown had theorized in 1902 that an enzyme-substrate complex was formed during the reaction sequence, and Henri used this idea to develop a reaction rate equation which he then tested using experimental results from the hydrolytic action of the enzyme invertase on sucrose. These data, published in 1903, did not however furnish a satisfactory proof-of-concept.

It took 10 years for the international bioscience community to recognize the importance of Victor Henri’s mathematical modelling and the shortcomings of his experimental technique. In particular, Leonor Michaelis in Berlin recognized that two important sources of error were present in Henri’s experimental data. Firstly, he had not taken into account variations in the hydrogen ion concentration (pH) in his experiments, and secondly he had neglected the mutarotation of the sugars (the reaction course was followed using a polarimeter).

Michaelis realized that by correcting these sources of error, he could conduct an improved test of the mathematically modelled relationship between the rate of enzyme-catalyzed reaction and the substrate concentration. His laboratory would be the first to do this work, he decided, and it was just at this moment that Maud Menten sailed onto the scene.

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Formulating the Michaelis-Menten equation

The purpose of Michaelis and Menten’s experimental work was to test the validity of the assumption put forward by Victor Henri that a very labile enzyme-substrate complex was formed during the reaction sequence. But they did not use Henri’s rate equation to test it. Instead they devised one of their own, which they arrived at in just two steps.

The key to the brevity of their formulation was the bold assertion that substrate and enzyme entered into rapid equilibrium, which enabled them to apply the law of mass action and write as their initial equation

I have used the same symbols as they did, [S] representing “the concentration of free sucrose, or since only a vanishingly small fraction of it is bound by the enzyme, the total concentration of sucrose”; Φ is the total enzyme concentration, φ is the concentration of the complexed enzyme, [Φ– φ] is accordingly the concentration of the free enzyme, and k is the dissociation constant.

Step 1 is a rearrangement of the above to

Step 2 involves the postulate that if the test assumption is correct, the initial reaction velocity v must be proportional to the prevailing concentration of the complexed enzyme φ. Therefore v = C.φ where C is the proportionality constant. Substituting the above expression for φ yields

This is the Michaelis-Menten equation as they themselves wrote it.

Two changes have subsequently been made to realize its modern manifestation. The first change is the substitution of C.Φ with its equivalent V_max. This is essentially a like-for-like replacement. The second change is a different matter altogether, and forms the subject of the next section.

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Reinventing the Michaelis-Menten equation

GE Briggs (1893-1985) and JBS Haldane (1892-1964)

In Michaelis and Menten’s original equation the term k is the dissociation constant for the reaction

expressed as

Adopting the modern representation of the kinetics involved

Michaelis and Menten’s k was the ratio of two rate constants: k _-1/k₁. But in the version of the equation we know today

the ‘Michaelis constant’ K_m is differently defined. So who made this change, and why?

Stand up Messrs. Briggs and Haldane, a pair of bright minds at the Botanical and Biochemical Laboratories, Cambridge, who in 1925 re-examined the enzyme-substrate model shown above and applied the steady-state approximation d[ES]/dt = 0 to the study of initial reaction velocities. By making this assumption, they were able to derive a general kinetics of which Michaelis-Menten kinetics is a special case.

Briggs-Haldane kinetics accommodated a further special case, known as Van Slyke-Cullen kinetics after the Dutch-American biochemist Donald Van Slyke and Glenn Cullen who, like Maud Menten, was Canadian. In 1914, a year after Michaelis and Menten’s paper, Van Slyke and Cullen published a paper on the kinetics of the enzyme urease. In it, they derived a kinetic equation which produced a result identical to the Michaelis-Menten equation.

This was an unusual finding because whereas Michaelis and Menten assumed that the reaction E + S ↔ ES was always practically in equilibrium, i.e. that k₂ is negligible in comparison with k _-1, Van Slyke and Cullen assumed the opposite – that k _-1 is negligible in comparison with k₂. So how could contrary assumptions lead to the same conclusion? Briggs and Haldane had the answer.

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Steady-state kinetics

Unlike either Michaelis and Menten or Van Slyke and Cullen, Briggs and Haldane made no assumptions about the relative values of k₂ and k _-1. Instead they assumed that in the initial reaction scenario [ES] is constant and thus d[ES]/dt = 0. Applying this constraint to the initial reaction rate equation

gives

Note that the denominator is a general constant, formed from the three rate constants. This term remains intact through the substitution pathway leading to the reaction velocity equation

By applying steady-state argumentation, Briggs and Haldane reinvented the constant k in the Michaelis-Menten equation, enabling them to explain why Van Slyke and Cullen had obtained the same result despite contrary reasoning.

The kinetics that Briggs and Haldane had formulated constituted a general case, within which there were two special cases. One was Michaelis Menten kinetics for which k _-1 >> k₂; the other was Van Slyke and Cullen kinetics for which k₂ >> k _-1. Since these terms were added in computing the general constant, the overall result in either case was the same.

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Graphing the Michaelis-Menten equation

Hans Lineweaver (1907-2009) and Dean Burk (1904-1988)

In 1934, the American physical chemist Hans Lineweaver was attending evening classes at George Washington University in Washington D.C., where the chemical engineer Paul H. Emmett was lecturing on contact catalysts. Emmett showed how kinetic data could be plotted in linear form to test hypotheses for catalytic mechanisms that involved adsorption of gases on catalytic surfaces. Lineweaver noted the similarity of the classical Langmuir adsorption isotherm

to the Michaelis-Menten equation (now in its Briggs-Haldane manifestation)

and realized that similar linear test plots could be made of enzyme kinetic data to test hypotheses and to evaluate characteristic constants. Lineweaver applied the simplest of the three possible linear transformations that can be made of the Michaelis-Menten equation

and showed it to Dean Burk, whom he was assisting in a laboratory at the US Department of Agriculture. Burk suggested publication, and was instrumental in getting their jointly-written paper accepted by the Journal of the American Chemical Society.

The famous double-reciprocal plot presented by Lineweaver and Burk in their 1934 paper would go on to be the most cited in the history of J Am Chem Soc, with more than 11,000 citations (as of 2011).

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An unanswered question

A remarkable feature of the kinetic model tested by Michaelis and Menten and re-examined by Briggs and Haldane is that it is unimolecular as regards the substrate sucrose. One molecule of sucrose combines reversibly with the enzyme to form a complex, which then releases one molecule of glucose and fructose, regenerating the free enzyme.

But this is an incomplete description of the catalyzed reaction. Sucrose is not simply cleaved into two hexoses. It is hydrolyzed – a molecule of water is required to convert a molecule of sucrose into glucose and fructose.

This raises the question of whether the model, and its associated kinetics, is quite complete. The fact that invertase specifically cleaves the O-C(fructose) bond suggests a particular orientation of the water molecule in the transition state. Does this amount to binding, and should this be reflected in the model?

If one considers the reverse situation in which a molecule of glucose and fructose combine with the enzyme, the transition state would necessarily include a water molecule, which would be released along with a sucrose molecule. This is not the reverse of a unimolecular process.

From a phase perspective, water is regarded as the solvent for the sucrose-enzyme system, and so does not possess concentration in the terms demanded by the rate equation. But water is indisputably a part of the mechanism by which sucrose is cleaved, which implies that any kinetic model of sucrose inversion will not be complete without it.

Leonor Michaelis in exile, Japan 1926.

Maud Menten in the 1930s

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Mouse-over links to works referred to in this post

L. Michaelis and Miss Maud L. Menten The Kinetics of Invertase action (1913)
[English translation of the original paper in German]

Donald D. Van Slyke and Glenn E. Cullen The mode of action of urease and of enzymes in general (1914)

G.E. Briggs and J.B.S. Haldane A note on the kinetics of enzyme action (1925)

Hans Lineweaver and Dean Burk The determination of enzyme dissociation constants (1934)
[pay to view]

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