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biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...
, Michaelis–Menten kinetics is one of the best-known models of enzyme kinetics. It is named after German biochemist
Leonor Michaelis Leonor Michaelis (16 January 1875 – 8 October 1949) was a German biochemist, physical chemist, and physician, known for his work with Maud Menten on enzyme kinetics in 1913, as well as for work on enzyme inhibition, pH and quinones. Ear ...
and Canadian physician Maud Menten. The model takes the form of an equation describing the rate of
enzymatic reaction Enzyme catalysis is the increase in the rate of a process by a biological molecule, an "enzyme". Most enzymes are proteins, and most such processes are chemical reactions. Within the enzyme, generally catalysis occurs at a localized site, called ...
s, by relating
reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per uni ...
v (rate of formation of product, ce P/math>) to ce S/math>, the
concentration In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', '' molar concentration'', ''number concentration'', ...
of a substrate ''S''. Its formula is given by : v = \frac = V_\max \frac This equation is called the Michaelis–Menten equation. Here, V_\max represents the maximum rate achieved by the system, happening at saturating substrate concentration for a given enzyme concentration. When the value of the Michaelis constant K_\mathrm is numerically equal to the substrate concentration, then the reaction rate is half of V_\max. Biochemical reactions involving a single substrate are often assumed to follow Michaelis–Menten kinetics, without regard to the model's underlying assumptions.


Model

In 1901, 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. His work was taken up by German biochemist
Leonor Michaelis Leonor Michaelis (16 January 1875 – 8 October 1949) was a German biochemist, physical chemist, and physician, known for his work with Maud Menten on enzyme kinetics in 1913, as well as for work on enzyme inhibition, pH and quinones. Ear ...
and Canadian physician Maud Menten, who investigated the
kinetics Kinetics ( grc, κίνησις, , kinesis, ''movement'' or ''to move'') may refer to: Science and medicine * Kinetics (physics), the study of motion and its causes ** Rigid body kinetics, the study of the motion of rigid bodies * Chemical kin ...
of an enzymatic reaction mechanism, invertase, that catalyzes the
hydrolysis Hydrolysis (; ) is any chemical reaction in which a molecule of water breaks one or more chemical bonds. The term is used broadly for substitution, elimination, and solvation reactions in which water is the nucleophile. Biological hydrolysi ...
of
sucrose Sucrose, a disaccharide, is a sugar composed of glucose and fructose subunits. It is produced naturally in plants and is the main constituent of white sugar. It has the molecular formula . For human consumption, sucrose is extracted and refine ...
into
glucose Glucose is a simple sugar with the molecular formula . Glucose is overall the most abundant monosaccharide, a subcategory of carbohydrates. Glucose is mainly made by plants and most algae during photosynthesis from water and carbon dioxide, u ...
and
fructose Fructose, or fruit sugar, is a ketonic simple sugar found in many plants, where it is often bonded to glucose to form the disaccharide sucrose. It is one of the three dietary monosaccharides, along with glucose and galactose, that are absorb ...
. In 1913, they proposed a mathematical model of the reaction. It involves an
enzyme Enzymes () are proteins that act as biological catalysts by accelerating chemical reactions. The molecules upon which enzymes may act are called substrates, and the enzyme converts the substrates into different molecules known as products ...
, E, binding to a substrate, S, to form a complex, ES, which in turn releases a product, P, regenerating the original enzyme. This may be represented schematically as :E + S <=> mathit\mathit] ES -> _\ceE + P where k_f (forward rate constant), k_r (reverse rate constant), and k_\mathrm (catalytic rate constant) denote the
rate constant In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction. For a reaction between reactants A and B to form product C the reaction rate is often found to have the ...
s, the double arrows between S (substrate) and ES (enzyme-substrate complex) represent the fact that enzyme-substrate binding is a reversible process, and the single forward arrow represents the formation of P (product). Under certain assumptions – such as the enzyme concentration being much less than the substrate concentration – the rate of product formation is given by :v = \frac = V_\max \frac = k_\mathrm ce E0 \frac . The
reaction order In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reac ...
depends on the relative size of the two terms in the denominator. At low substrate concentration ce S\ll K_M, so that the
reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per uni ...
v = k_\mathrm ce E0 \frac varies linearly with substrate concentration /chem> ( first-order kinetics). Laidler K.J. and Meiser J.H. ''Physical Chemistry'' (Benjamin/Cummings 1982) p.430 However at higher /chem> with ce S\gg K_M, the reaction becomes independent of /chem> (zero-order kinetics) and
asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
approaches its maximum rate V_\max = k_\ce ce E0, where 0 is the initial enzyme concentration. This rate is attained when all enzyme is bound to substrate. k_\mathrm, the
turnover number Turnover number has two different meanings: In enzymology, turnover number (also termed ''k''cat) is defined as the maximum number of chemical conversions of substrate molecules per second that a single active site will execute for a given enzyme ...
, is the maximum number of substrate molecules converted to product per enzyme molecule per second. Further addition of substrate does not increase the rate which is said to be saturated. The value of the Michaelis constant K_\mathrm is numerically equal to the /chem> at which the reaction rate is at half-maximum, and is a measure of the substrate's affinity for the enzyme—a small K_\mathrm indicates high affinity, meaning that the rate will approach V_\max with lower /chem> than those reactions with a larger K_\mathrm. The constant is not affected by the concentration or purity of an enzyme. The value of K_\mathrm is dependent on both the identity of 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, including antigen–antibody binding, DNA–DNA hybridization, and protein–protein interaction. 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 Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which a ...
of biomolecular species. When an empirical equation of this form is applied to microbial growth, it is sometimes called a Monod equation.


Applications

Parameter values vary widely between enzymes: The constant k_\text/K_\mathrm ( catalytic efficiency) is a measure of how efficiently an enzyme converts a substrate into product.
Diffusion limited enzyme A diffusion-limited enzyme catalyses a reaction so efficiently that the rate limiting step is that of substrate diffusion into the active site, or product diffusion out. This is also known as kinetic perfection or catalytic perfection. Since th ...
s, such as fumarase, work at the theoretical upper limit of , limited by diffusion of substrate into the
active site In biology and biochemistry, the active site is the region of an enzyme where substrate molecules bind and undergo a chemical reaction. The active site consists of amino acid residues that form temporary bonds with the substrate ( binding site) ...
. Michaelis–Menten kinetics have also been applied to a variety of spheres outside of biochemical reactions, including alveolar clearance of dusts, the richness of species pools, clearance of
blood alcohol Blood alcohol content (BAC), also called blood alcohol concentration or blood alcohol level, is a measurement of alcohol intoxication used for legal or medical purposes; it is expressed as mass of alcohol per volume or mass of blood. For example ...
, the photosynthesis-irradiance relationship, and bacterial phage infection. The equation can also be used to describe the relationship between
ion channel Ion channels are pore-forming membrane proteins that allow ions to pass through the channel pore. Their functions include establishing a resting membrane potential, shaping action potentials and other electrical signals by gating the flow of ...
conductivity and
ligand In coordination chemistry, a ligand is an ion or molecule (functional group) that binds to a central metal atom to form a coordination complex. The bonding with the metal generally involves formal donation of one or more of the ligand's elect ...
concentration.


Biological Oceanography

Michaelis Menten can also be applied to limiting nutrients and phytoplankton growth in the global ocean. The relationship is : V = \frac where V is uptake of the nutrient by phytoplankton, S is the concentration of the limiting nutrient, V_\max is the maximum uptake rate by phytoplankton, and K is the half saturation constant. The relationship can also be understood for growth rate V_\max and K also depend on phytoplankton physiology and genetics. Oligotrophic regions—regions that are nutrient depleted—often have phytoplankton populations characterized by low V_\max and low K, which help them to efficiently take up the nutrient that is at low concentrations. These phytoplankton tend to be small, with large area to volume ratios. Phytoplankton in eutrophic regions—nutrient-replete regions—are often characterized by high V_\max and K, which make them less efficient at taking up a given nutrient, but allows them to take up more of that nutrient. These phytoplankton tend to be large (e.g. diatoms) with a smaller area to volume ratio. Their larger V_\max enables them to become larger. This relationship is generally applied to nitrate concentrations, as that is the limiting nutrient in much of the ocean, but other nutrients—for example, iron and phosphate—might also be limiting.


Derivation

Applying the
law of mass action In chemistry, the law of mass action is the proposition that the rate of the chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dy ...
, which states that the rate of a reaction is proportional to the product of the concentrations of the reactants (i.e. /math>), gives a system of four non-linear
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s that define the rate of change of reactants with time t :\begin \frac &= - k_f ce Ece S+ k_r ce+ k_\mathrm ce\\ pt\frac &= - k_f ce Ece S+ k_r ce\\ pt\frac &= k_f ce Ece S- k_r ce- k_\mathrm ce\\ pt\frac &= k_\mathrm ce \end In this mechanism, the enzyme E is a
catalyst Catalysis () is the process of increasing the rate of a chemical reaction by adding a substance known as a catalyst (). Catalysts are not consumed in the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recyc ...
, which only facilitates the reaction, so that its total concentration, free plus combined, + S= 0 is a constant (i.e. 0 = ). This conservation law can also be observed by adding the first and third equations above.


Equilibrium approximation

In their original analysis, Michaelis and Menten assumed that the substrate is in instantaneous
chemical equilibrium In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the ...
with the complex, which implies :k_f ce E ce S= k_r ce From the enzyme conservation law, we obtain : ce E= ce E0 - ce Combining the two expressions above, gives us :\begin k_f( ce E0 - ce ce S&= k_r ce\\ ptk_f ce E0 ce- k_f cece S&= k_r ce\\ ptk_r ce+ k_f cece S&= k_f ce E0 ce\\ pt cek_r + k_f ce &= k_f ce0 ce\\ pt ce&= \frac \\ pt ce&= \frac \\ pt\end Upon simplification, we get : ce= \frac where K_d = k_r / k_f is the
dissociation constant In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex ...
for the enzyme-substrate complex. Hence the velocity v of the reaction – the rate at which P is formed – is :v = \frac = V_\max\frac where V_\max = k_\mathrm ce E0 is the maximum reaction velocity.


Quasi-steady-state approximation

An alternative analysis of the system was undertaken by British botanist G. E. Briggs and British geneticist
J. B. S. Haldane John Burdon Sanderson Haldane (; 5 November 18921 December 1964), nicknamed "Jack" or "JBS", was a British-Indian scientist who worked in physiology, genetics, evolutionary biology, and mathematics. With innovative use of statistics in biolo ...
in 1925. 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 In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties '' ...
assumption or pseudo-steady-state-hypothesis. Mathematically, this assumption means k_f ce E ce S= k_r ce+ k_\mathrm ce= (k_r + k_\mathrm) ce/math>. This is mathematically the same as the previous equation, with k_r replaced by k_r + k_\mathrm. Hence, following the same steps as above, the velocity v of the reaction is :v = V_\max \frac where :K_\mathrm = \frac is known as the Michaelis constant.


Assumptions and limitations

The first step in the derivation applies the
law of mass action In chemistry, the law of mass action is the proposition that the rate of the chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dy ...
, which is reliant on free
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
. However, in the environment of a living cell where there is a high concentration of
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, res ...
s, the
cytoplasm In cell biology, the cytoplasm is all of the material within a eukaryotic cell, enclosed by the cell membrane, except for the cell nucleus. The material inside the nucleus and contained within the nuclear membrane is termed the nucleoplasm. ...
often behaves more like a viscous
gel A gel is a semi-solid that can have properties ranging from soft and weak to hard and tough. Gels are defined as a substantially dilute cross-linked system, which exhibits no flow when in the steady-state, although the liquid phase may still di ...
than a free-flowing liquid, limiting molecular movements by
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
and altering reaction rates. Although the law of mass action can be valid in heterogeneous environments, it is more appropriate to model the cytoplasm as a
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
, in order to capture its limited-mobility kinetics. The resulting reaction rates predicted by the two approaches are similar, with the only difference being that the equilibrium approximation defines the constant as K_d, whilst the quasi-steady-state approximation uses K_\mathrm. 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 :\varepsilon_d = \frac \ll 1. By contrast, the Briggs–Haldane quasi-steady-state analysis is valid if :\varepsilon_m = \frac\ce \ll 1. Thus it holds if the enzyme concentration is much less than the substrate concentration or K_\mathrm or both. In both the Michaelis–Menten and Briggs–Haldane analyses, the quality of the approximation improves as \varepsilon\,\! decreases. However, in model building, Michaelis–Menten kinetics are often invoked without regard to the underlying assumptions. Importantly, 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 : E + S <=> mathit\mathit] ES <=> mathit\mathit] E + P. 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: :: \gg This is true under standard ''
in vitro ''In vitro'' (meaning in glass, or ''in the glass'') studies are performed with microorganisms, cells, or biological molecules outside their normal biological context. Colloquially called " test-tube experiments", these studies in biology a ...
'' assay conditions, and is true for many ''
in vivo Studies that are ''in vivo'' (Latin for "within the living"; often not italicized in English) are those in which the effects of various biological entities are tested on whole, living organisms or cells, usually animals, including humans, and p ...
'' 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 ::\Delta \ll 0. 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.


Determination of constants

The typical method for determining the constants V_\max and K_\mathrm involves running a series of enzyme assays at varying substrate concentrations /math>, and measuring the initial reaction rate v_0. '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. By plotting reaction rate against concentration, and using
nonlinear regression In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fi ...
of the Michaelis–Menten equation, 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 the Eadie–Hofstee diagram,
Hanes–Woolf plot In biochemistry, a Hanes–Woolf plot, Hanes plot, or plot of a/v against a, is a graphical representation of enzyme kinetics in which the ratio of the initial substrate concentration a to the reaction velocity v is plotted against a. It is ba ...
and Lineweaver–Burk plot; of these, the Hanes–Woolf plot is the most accurate. However, while useful for visualization, all three methods distort the error structure of the data and are inferior to nonlinear regression. Assuming a similar error dv_0 on v_0, an inverse representation leads to an error of dv_0/v_0^2 on 1/v_0 ( Propagation of uncertainty). Without proper estimation of dv_0 values, linearisation should be avoided. In addition, regression analysis using
Least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...
assumes that errors are normally distributed, which is not valid after a transformation of v_0 values. Nonetheless, their use can still be found in modern literature. In 1997
Santiago Schnell Santiago Schnell FRSC is a Venezuelan theoretical and mathematical biologist. He is the William K. Warren Foundation Dean of the College of Science at the University of Notre Dame, as well as a professor in the Department of Biological Sciences, an ...
and Claudio Mendoza suggested a closed form solution for the time course kinetics analysis of the Michaelis–Menten kinetics based on the solution of the Lambert W function. Namely, :\frac = W(F(t))\, where ''W'' is the Lambert W function and :F(t) = \frac \exp\!\left(\frac - \frac\,t \right) \,. The above equation, known nowadays as the Schnell-Mendoza equation, has been used to estimate V_\max and K_\mathrm from time course data.


Role of substrate unbinding

The Michaelis-Menten equation has been used to predict the rate of product formation in enzymatic reactions for more than a century. Specifically, it states that the rate of an enzymatic reaction will increase as substrate concentration increases, and that increased unbinding of enzyme-substrate complexes will decrease the reaction rate. While the first prediction is well established, the second is more elusive. Mathematical analysis of the effect of enzyme-substrate unbinding on enzymatic reactions at the single-molecule level has shown that unbinding of an enzyme from a substrate can reduce the rate of product formation under some conditions, but may also have the opposite effect. As substrate concentrations increase, a tipping point can be reached where an increase in the unbinding rate results in an increase, rather than a decrease, of the reaction rate. The results indicate that enzymatic reactions can behave in ways that violate the classical Michaelis-Menten equation, and that the role of unbinding in enzymatic catalysis still remains to be determined experimentally.


See also

* Eadie–Hofstee diagram * Enzyme kinetics *
Functional response A functional response in ecology is the intake rate of a consumer as a function of food density (the amount of food available in a given ecotope). It is associated with the numerical response, which is the reproduction rate of a consumer as a fu ...
* Gompertz function * Hill equation (biochemistry) * Hill contribution to Langmuir equation *
Langmuir adsorption model The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes. This model even explains the effect of pressu ...
(equation with the same mathematical form) * Lineweaver–Burk plot * Monod equation (equation with the same mathematical form) * Reaction progress kinetic analysis * Steady state (chemistry) * Victor Henri, who first wrote the general equation form in 1901 *
Von Bertalanffy function The von Bertalanffy growth function (VBGF), or von Bertalanffy curve, is a type of growth curve for a time series and is named after Ludwig von Bertalanffy. It is a special case of the generalised logistic function. The growth curve is used to ...


References


External links


Online K_\mathrm V_\max Vmax calculator
(ic50.tk/kmvmax.html) based on the
C programming language ''The C Programming Language'' (sometimes termed ''K&R'', after its authors' initials) is a computer programming book written by Brian Kernighan and Dennis Ritchie, the latter of whom originally designed and implemented the language, as well a ...
and the non-linear least-squares Levenberg–Marquardt algorithm of
gnuplot gnuplot is a command-line and GUI program that can generate two- and three-dimensional plots of functions, data, and data fits. The program runs on all major computers and operating systems (Linux, Unix, Microsoft Windows, macOS, FreeDOS, ...

Alternative online K_\mathrm V_\max calculator
(ic50.org/kmvmax.html) based on Python, NumPy,
Matplotlib Matplotlib is a plotting library for the Python programming language and its numerical mathematics extension NumPy. It provides an object-oriented API for embedding plots into applications using general-purpose GUI toolkits like Tkinter, wxPy ...
and the non-linear least-squares Levenberg–Marquardt algorithm of
SciPy SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing. SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, ...


Further reading

* {{DEFAULTSORT:Michaelis-Menten Kinetics Enzyme kinetics Chemical kinetics Ordinary differential equations Catalysis