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Enzyme kinetics is the study of the rates of enzyme-catalysed
chemical reaction A chemical reaction is a process that leads to the chemistry, chemical transformation of one set of chemical substances to another. When chemical reactions occur, the atoms are rearranged and the reaction is accompanied by an Gibbs free energy, ...
s. In enzyme kinetics, 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 u ...
is measured and the effects of varying the conditions of the reaction are investigated. Studying an enzyme's
kinetics Kinetics (, ''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 kinetics, the study of chemical ...
in this way can reveal the catalytic mechanism of this enzyme, its role in
metabolism Metabolism (, from ''metabolē'', "change") is the set of life-sustaining chemical reactions in organisms. The three main functions of metabolism are: the conversion of the energy in food to energy available to run cellular processes; the co ...
, how its activity is controlled, and how a
drug A drug is any chemical substance other than a nutrient or an essential dietary ingredient, which, when administered to a living organism, produces a biological effect. Consumption of drugs can be via insufflation (medicine), inhalation, drug i ...
or a modifier (
inhibitor Inhibitor or inhibition may refer to: Biology * Enzyme inhibitor, a substance that binds to an enzyme and decreases the enzyme's activity * Reuptake inhibitor, a substance that increases neurotransmission by blocking the reuptake of a neurotransmi ...
or activator) might affect the rate. An enzyme (E) is a
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residue (biochemistry), residues. Proteins perform a vast array of functions within organisms, including Enzyme catalysis, catalysing metab ...
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
that serves as a biological catalyst to facilitate and accelerate a chemical reaction in the body. It does this through binding of another molecule, its
substrate Substrate may refer to: Physical layers *Substrate (biology), the natural environment in which an organism lives, or the surface or medium on which an organism grows or is attached ** Substrate (aquatic environment), the earthy material that exi ...
(S), which the enzyme acts upon to form the desired product. The substrate binds to 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, the ''binding s ...
of the enzyme to produce an enzyme-substrate complex ES, and is transformed into an enzyme-product complex EP and from there to product P, via a
transition state In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked w ...
ES*. The series of steps is known as the
mechanism Mechanism may refer to: *Mechanism (economics), a set of rules for a game designed to achieve a certain outcome **Mechanism design, the study of such mechanisms *Mechanism (engineering), rigid bodies connected by joints in order to accomplish a ...
: : E + S ⇄ ES ⇄ ES* ⇄ EP ⇄ E + P This example assumes the simplest case of a reaction with one substrate and one product. Such cases exist: for example, a
mutase A mutase is an enzyme of the isomerase class that catalyzes the movement of a functional group from one position to another within the same molecule. In other words, mutases catalyze intramolecular group transfers. Examples of mutases include '' b ...
such as
phosphoglucomutase Phosphoglucomutase () is an enzyme that transfers a phosphate group on an α-D-glucose monomer from the 1 to the 6 position in the forward direction or the 6 to the 1 position in the reverse direction. More precisely, it facilitates the interconv ...
catalyses the transfer of a phosphate group from one position to another, and
isomerase In biochemistry, isomerases are a general class of enzymes that convert a molecule from one isomer to another. Isomerases facilitate intramolecular rearrangements in which chemical bond, bonds are Bond cleavage, broken and formed. The general form ...
is a more general term for an enzyme that catalyses any one-substrate one-product reaction, such as
triosephosphate isomerase Triose-phosphate isomerase (TPI or TIM) is an enzyme () that catalyzes the reversible interconversion of the triose phosphate isomers dihydroxyacetone phosphate and D-glyceraldehyde 3-phosphate. TPI plays an important role in glycolysis and i ...
. However, such enzymes are not very common, and are heavily outnumbered by enzymes that catalyse two-substrate two-product reactions: these include, for example, the NAD-dependent
dehydrogenase A dehydrogenase is an enzyme belonging to the group of oxidoreductases that oxidizes a substrate by reducing an electron acceptor, usually NAD+/NADP+ or a flavin coenzyme such as FAD or FMN. Like all catalysts, they catalyze reverse as well as ...
s such as
alcohol dehydrogenase Alcohol dehydrogenases (ADH) () are a group of dehydrogenase enzymes that occur in many organisms and facilitate the interconversion between alcohols and aldehydes or ketones with the reduction of nicotinamide adenine dinucleotide (NAD+) to N ...
, which catalyses the oxidation of ethanol by NAD+. Reactions with three or four substrates or products are less common, but they exist. There is no necessity for the number of products to be equal to the number of substrates; for example,
glyceraldehyde 3-phosphate dehydrogenase Glyceraldehyde 3-phosphate dehydrogenase (abbreviated GAPDH) () is an enzyme of about 37kDa that catalyzes the sixth step of glycolysis and thus serves to break down glucose for energy and carbon molecules. In addition to this long establis ...
has three substrates and two products. When enzymes bind multiple substrates, such as
dihydrofolate reductase Dihydrofolate reductase, or DHFR, is an enzyme that reduces dihydrofolic acid to tetrahydrofolic acid, using NADPH as an electron donor, which can be converted to the kinds of tetrahydrofolate cofactors used in one-carbon transfer chemistry. ...
(shown right), enzyme kinetics can also show the sequence in which these substrates bind and the sequence in which products are released. An example of enzymes that bind a single substrate and release multiple products are
protease A protease (also called a peptidase, proteinase, or proteolytic enzyme) is an enzyme that catalysis, catalyzes proteolysis, breaking down proteins into smaller polypeptides or single amino acids, and spurring the formation of new protein products ...
s, which cleave one protein substrate into two polypeptide products. Others join two substrates together, such as
DNA polymerase A DNA polymerase is a member of a family of enzymes that catalyze the synthesis of DNA molecules from nucleoside triphosphates, the molecular precursors of DNA. These enzymes are essential for DNA replication and usually work in groups to create t ...
linking a
nucleotide Nucleotides are Organic compound, organic molecules composed of a nitrogenous base, a pentose sugar and a phosphate. They serve as monomeric units of the nucleic acid polymers – deoxyribonucleic acid (DNA) and ribonucleic acid (RNA), both o ...
to
DNA Deoxyribonucleic acid (; DNA) is a polymer composed of two polynucleotide chains that coil around each other to form a double helix. The polymer carries genetic instructions for the development, functioning, growth and reproduction of al ...
. Although these mechanisms are often a complex series of steps, there is typically one ''rate-determining step'' that determines the overall kinetics. This
rate-determining step In chemical kinetics, the overall rate of a reaction is often approximately determined by the slowest step, known as the rate-determining step (RDS or RD-step or r/d step) or rate-limiting step. For a given reaction mechanism, the prediction of the ...
may be a chemical reaction or a conformational change of the enzyme or substrates, such as those involved in the release of product(s) from the enzyme. Knowledge of the enzyme's structure is helpful in interpreting kinetic data. For example, the structure can suggest how substrates and products bind during catalysis; what changes occur during the reaction; and even the role of particular
amino acid Amino acids are organic compounds that contain both amino and carboxylic acid functional groups. Although over 500 amino acids exist in nature, by far the most important are the 22 α-amino acids incorporated into proteins. Only these 22 a ...
residues in the mechanism. Some enzymes change shape significantly during the mechanism; in such cases, it is helpful to determine the enzyme structure with and without bound substrate analogues that do not undergo the enzymatic reaction. Not all biological catalysts are protein enzymes:
RNA Ribonucleic acid (RNA) is a polymeric molecule that is essential for most biological functions, either by performing the function itself (non-coding RNA) or by forming a template for the production of proteins (messenger RNA). RNA and deoxyrib ...
-based catalysts such as
ribozymes Ribozymes (ribonucleic acid enzymes) are RNA molecules that have the ability to catalyze specific biochemical reactions, including RNA splicing in gene expression, similar to the action of protein enzymes. The 1982 discovery of ribozymes demons ...
and
ribosomes Ribosomes () are macromolecular machines, found within all cells, that perform biological protein synthesis (messenger RNA translation). Ribosomes link amino acids together in the order specified by the codons of messenger RNA molecules to fo ...
are essential to many cellular functions, such as
RNA splicing RNA splicing is a process in molecular biology where a newly-made precursor messenger RNA (pre-mRNA) transcription (biology), transcript is transformed into a mature messenger RNA (Messenger RNA, mRNA). It works by removing all the introns (non-cod ...
and
translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
. The main difference between ribozymes and enzymes is that RNA catalysts are composed of nucleotides, whereas enzymes are composed of amino acids. Ribozymes also perform a more limited set of reactions, although their
reaction mechanism In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical reaction occurs. A chemical mechanism is a theoretical conjecture that tries to describe in detail what takes place at each stage ...
s and kinetics can be analysed and classified by the same methods.


General principles

The reaction catalysed by an enzyme uses exactly the same reactants and produces exactly the same products as the uncatalysed reaction. Like other
catalysts Catalysis () is the increase in reaction rate, rate of a chemical reaction due to an added substance known as a catalyst (). Catalysts are not consumed by the reaction and remain unchanged after it. If the reaction is rapid and the catalyst ...
, enzymes do not alter the position of
equilibrium Equilibrium may refer to: Film and television * ''Equilibrium'' (film), a 2002 science fiction film * '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film * "Equilibrium" (''seaQuest 2032'') * ''Equilibr ...
between substrates and products. However, unlike uncatalysed chemical reactions, enzyme-catalysed reactions display saturation kinetics. For a given enzyme concentration and for relatively low substrate concentrations, the reaction rate increases linearly with substrate concentration; the enzyme molecules are largely free to catalyse the reaction, and increasing substrate concentration means an increasing rate at which the enzyme and substrate molecules encounter one another. However, at relatively high substrate concentrations, the reaction rate
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 the theoretical maximum; the enzyme active sites are almost all occupied by substrates resulting in saturation, and the reaction rate is determined by the intrinsic turnover rate of the enzyme.Fromm H.J., Hargrove M.S. (2012) Enzyme Kinetics. In: Essentials of Biochemistry. Springer, Berlin, Heidelberg The substrate concentration midway between these two limiting cases is denoted by ''K''M. Thus, ''K''M is the substrate concentration at which the reaction velocity is half of the maximum velocity. The two important properties of enzyme kinetics are how easily the enzyme can be saturated with a substrate, and the maximum rate it can achieve. Knowing these properties suggests what an enzyme might do in the cell and can show how the enzyme will respond to changes in these conditions.


Enzyme assays

Enzyme assay Enzyme assays are laboratory methods for measuring enzymatic activity. They are vital for the study of enzyme kinetics and enzyme inhibition. Enzyme units The quantity or concentration of an enzyme can be expressed in molar amounts, as with a ...
s are laboratory procedures that measure the rate of enzyme reactions. Since enzymes are not consumed by the reactions they catalyse, enzyme assays usually follow changes in the concentration of either substrates or products to measure the rate of reaction. There are many methods of measurement.
Spectrophotometric Spectrophotometry is a branch of electromagnetic spectroscopy concerned with the quantitative measurement of the reflection or transmission properties of a material as a function of wavelength. Spectrophotometry uses photometers, known as spe ...
assays observe the change in the
absorbance Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". Alternatively, for samples which scatter light, absorbance may be defined as "the negative log ...
of light between products and reactants; radiometric assays involve the incorporation or release of
radioactivity Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is conside ...
to measure the amount of product made over time. Spectrophotometric assays are most convenient since they allow the rate of the reaction to be measured continuously. Although radiometric assays require the removal and counting of samples (i.e., they are discontinuous assays) they are usually extremely sensitive and can measure very low levels of enzyme activity. An analogous approach is to use
mass spectrometry Mass spectrometry (MS) is an analytical technique that is used to measure the mass-to-charge ratio of ions. The results are presented as a ''mass spectrum'', a plot of intensity as a function of the mass-to-charge ratio. Mass spectrometry is used ...
to monitor the incorporation or release of
stable isotope Stable nuclides are Isotope, isotopes of a chemical element whose Nucleon, nucleons are in a configuration that does not permit them the surplus energy required to produce a radioactive emission. The Atomic nucleus, nuclei of such isotopes are no ...
s as the substrate is converted into product. Occasionally, an assay fails and approaches are essential to resurrect a failed assay. The most sensitive enzyme assays use
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
s focused through a
microscope A microscope () is a laboratory equipment, laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic ...
to observe changes in single enzyme molecules as they catalyse their reactions. These measurements either use changes in the
fluorescence Fluorescence is one of two kinds of photoluminescence, the emission of light by a substance that has absorbed light or other electromagnetic radiation. When exposed to ultraviolet radiation, many substances will glow (fluoresce) with colore ...
of cofactors during an enzyme's reaction mechanism, or of
fluorescent dyes A fluorophore (or fluorochrome, similarly to a chromophore) is a fluorescent chemical compound that can re-emit light upon light excitation. Fluorophores typically contain several combined aromatic groups, or planar or cyclic molecules with sev ...
added onto specific sites of the
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residue (biochemistry), residues. Proteins perform a vast array of functions within organisms, including Enzyme catalysis, catalysing metab ...
to report movements that occur during catalysis. These studies provide a new view of the kinetics and dynamics of single enzymes, as opposed to traditional enzyme kinetics, which observes the average behaviour of populations of millions of enzyme molecules. An example progress curve for an enzyme assay is shown above. The enzyme produces product at an initial rate that is approximately linear for a short period after the start of the reaction. As the reaction proceeds and substrate is consumed, the rate continuously slows (so long as the substrate is not still at saturating levels). To measure the initial (and maximal) rate, enzyme assays are typically carried out while the reaction has progressed only a few percent towards total completion. The length of the initial rate period depends on the assay conditions and can range from milliseconds to hours. However, equipment for rapidly mixing liquids allows fast kinetic measurements at initial rates of less than one second. These very rapid assays are essential for measuring pre-steady-state kinetics, which are discussed below. Most enzyme kinetics studies concentrate on this initial, approximately linear part of enzyme reactions. However, it is also possible to measure the complete reaction curve and fit this data to a non-linear
rate equation In chemistry, the rate equation (also known as the rate law or empirical differential rate equation) is an Empirical relationship, empirical Differential equation, differential Expression (mathematics), mathematical expression for the reaction rat ...
. This way of measuring enzyme reactions is called progress-curve analysis. This approach is useful as an alternative to rapid kinetics when the initial rate is too fast to measure accurately. The
Standards for Reporting Enzymology Data Standards for Reporting Enzymology Data (STRENDA) is an initiative as part of the Minimum Information Standards which specifically focuses on the development of guidelines for reporting (describing metadata) enzymology experiments. The initiative i ...
Guidelines provide minimum information required to comprehensively report kinetic and equilibrium data from investigations of enzyme activities including corresponding experimental conditions. The guidelines have been developed to report functional enzyme data with rigor and robustness.


Single-substrate reactions

Enzymes with single-substrate mechanisms include
isomerase In biochemistry, isomerases are a general class of enzymes that convert a molecule from one isomer to another. Isomerases facilitate intramolecular rearrangements in which chemical bond, bonds are Bond cleavage, broken and formed. The general form ...
s such as
triosephosphateisomerase Triose-phosphate isomerase (TPI or TIM) is an enzyme () that catalyzes the reversible interconversion of the triose phosphate isomers dihydroxyacetone phosphate and D-glyceraldehyde 3-phosphate. TPI plays an important role in glycolysis and i ...
or
bisphosphoglycerate mutase Bisphosphoglycerate mutase (, BPGM) is an enzyme expressed in erythrocytes and placental cells. It is responsible for the catalytic synthesis of 2,3-Bisphosphoglycerate (2,3-BPG) from 1,3-bisphosphoglycerate. BPGM has both a mutase and a ph ...
, intramolecular
lyase In biochemistry, a lyase is an enzyme that catalyzes the breaking (an elimination reaction) of various chemical bonds by means other than hydrolysis (a substitution reaction) and oxidation Redox ( , , reduction–oxidation or oxidatio ...
s such as
adenylate cyclase Adenylate cyclase (EC 4.6.1.1, also commonly known as adenyl cyclase and adenylyl cyclase, abbreviated AC) is an enzyme with systematic name ATP diphosphate-lyase (cyclizing; 3′,5′-cyclic-AMP-forming). It catalyzes the following reaction: :A ...
and the
hammerhead ribozyme The hammerhead ribozyme is an RNA Sequence motif, motif that catalyzes reversible cleavage and Ligation (molecular biology), ligation reactions at a specific site within an RNA molecule. It is one of several catalytic RNAs (ribozymes) known to occ ...
, an RNA lyase. However, some enzymes that only have a single substrate do not fall into this category of mechanisms.
Catalase Catalase is a common enzyme found in nearly all living organisms exposed to oxygen (such as bacteria, plants, and animals) which catalyzes the decomposition of hydrogen peroxide to water and oxygen. It is a very important enzyme in protecting ...
is an example of this, as the enzyme reacts with a first molecule of
hydrogen peroxide Hydrogen peroxide is a chemical compound with the formula . In its pure form, it is a very pale blue liquid that is slightly more viscosity, viscous than Properties of water, water. It is used as an oxidizer, bleaching agent, and antiseptic, usua ...
substrate, becomes oxidised and is then reduced by a second molecule of substrate. Although a single substrate is involved, the existence of a modified enzyme intermediate means that the mechanism of catalase is actually a ping–pong mechanism, a type of mechanism that is discussed in the ''Multi-substrate reactions'' section below.


Michaelis–Menten kinetics

As enzyme-catalysed reactions are saturable, their rate of catalysis does not show a linear response to increasing substrate. If the initial rate of the reaction is measured over a range of substrate concentrations (denoted as , the initial reaction rate (v_0) increases as increases, as shown on the right. However, as gets higher, the enzyme becomes saturated with substrate and the initial rate reaches ''V''max, the enzyme's maximum rate. The Michaelis–Menten kinetic model of a single-substrate reaction is shown on the right. There is an initial bimolecular reaction between the enzyme E and substrate S to form the enzyme–substrate complex ES. The rate of enzymatic reaction increases with the increase of the substrate concentration up to a certain level called Vmax; at Vmax, increase in substrate concentration does not cause any increase in reaction rate as there is no more enzyme (E) available for reacting with substrate (S). Here, the rate of reaction becomes dependent on the ES complex and the reaction becomes a
unimolecular reaction In chemistry, molecularity is the number of molecules that come together to react in an elementary (single-step) reactionAtkins, P.; de Paula, J. Physical Chemistry. Oxford University Press, 2014 and is equal to the sum of stoichiometric coeffici ...
with an order of zero. Though the enzymatic mechanism for the
unimolecular reaction In chemistry, molecularity is the number of molecules that come together to react in an elementary (single-step) reactionAtkins, P.; de Paula, J. Physical Chemistry. Oxford University Press, 2014 and is equal to the sum of stoichiometric coeffici ...
ES -> _E + P can be quite complex, there is typically one rate-determining enzymatic step that allows this reaction to be modelled as a single catalytic step with an apparent unimolecular rate constant ''k''cat. If the reaction path proceeds over one or several intermediates, ''k''cat will be a function of several elementary rate constants, whereas in the simplest case of a single elementary reaction (e.g. no intermediates) it will be identical to the elementary unimolecular rate constant ''k''2. The apparent unimolecular rate constant ''k''cat is also called
turnover number In chemistry, the term "turnover number" has two distinct meanings. In enzymology, the turnover number () is defined as the limiting number of chemical conversions of substrate molecules per second that a single active site will execute for a g ...
, and denotes the maximum number of enzymatic reactions catalysed per second. The Michaelis–Menten equation describes how the (initial) reaction rate ''v''0 depends on the position of the substrate-binding
equilibrium Equilibrium may refer to: Film and television * ''Equilibrium'' (film), a 2002 science fiction film * '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film * "Equilibrium" (''seaQuest 2032'') * ''Equilibr ...
and the rate constant ''k''2. : v_0 = \frac     ''(Michaelis–Menten equation)'' with the constants : \begin K_M \ &\stackrel\ \frac \approx K_D\\ V_\max \ &\stackrel\ k_\ce_ \end This Michaelis–Menten equation is the basis for most single-substrate enzyme kinetics. Two crucial assumptions underlie this equation (apart from the general assumption about the mechanism only involving no intermediate or product inhibition, and there is no allostericity or
cooperativity Cooperativity is a phenomenon displayed by systems involving identical or near-identical elements, which act dependently of each other, relative to a hypothetical standard non-interacting system in which the individual elements are acting indepen ...
). The first assumption is the so-called quasi-steady-state assumption (or pseudo-steady-state hypothesis), namely that the concentration of the substrate-bound enzyme (and hence also the unbound enzyme) changes much more slowly than those of the product and substrate and thus the change over time of the complex can be set to zero d\ce/ \; \overset = \;0 . The second assumption is that the total enzyme concentration does not change over time, thus \ce_\text = \ce + \ce \; \overset = \; \text . The Michaelis constant ''K''M is experimentally defined as the concentration at which the rate of the enzyme reaction is half ''V''max, which can be verified by substituting = ''K''M into the Michaelis–Menten equation and can also be seen graphically. If the rate-determining enzymatic step is slow compared to substrate dissociation (k_2 \ll k_ ), the Michaelis constant ''K''M is roughly 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 ...
''K''D of the ES complex. If /chem> is small compared to K_M then the term ce S/ (K_M + ce S \approx ce S/ K_M and also very little ES complex is formed, thus \approx /chem>. Therefore, the rate of product formation is :v_0 \approx \frac \ce \qquad \qquad \text ce S\ll K_M Thus the product formation rate depends on the enzyme concentration as well as on the substrate concentration, the equation resembles a bimolecular reaction with a corresponding pseudo-second order rate constant k_2 / K_M. This constant is a measure of catalytic efficiency. The most efficient enzymes reach a k_2 / K_M in the range of . These enzymes are so efficient they effectively catalyse a reaction each time they encounter a substrate molecule and have thus reached an upper theoretical limit for efficiency (
diffusion limit Diffusion-controlled (or diffusion-limited) reactions are reactions in which the reaction rate is equal to the rate of transport of the reactants through the reaction medium (usually a solution). The process of chemical reaction can be considere ...
); and are sometimes referred to as kinetically perfect enzymes. But most enzymes are far from perfect: the average values of k_/K_ and k_ are about 10^5 ^^ and 10 ^, respectively.


Direct use of the Michaelis–Menten equation for time course kinetic analysis

The observed velocities predicted by the Michaelis–Menten equation can be used to directly model the time course disappearance of substrate and the production of product through incorporation of the Michaelis–Menten equation into the equation for first order chemical kinetics. This can only be achieved however if one recognises the problem associated with the use of
Euler's number The number is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can ...
in the description of first order chemical kinetics. i.e. ''e''−''k'' is a split constant that introduces a systematic error into calculations and can be rewritten as a single constant which represents the remaining substrate after each time period. : 0(1-k)^\, : 0(1-v/ 0)^\, : 0(1-(V_ 0 / (K_M + 0)/ 0))^\, In 1983 Stuart Beal (and also independently
Santiago Schnell Santiago Schnell FRSB FRSC is a Venezuelan-born scientist and academic administrator. He is the Provost of Dartmouth College, where he also holds appointments as Professor of Mathematics in the School of Arts and Sciences, Adjunct Professor of B ...
and Claudio Mendoza in 1997) derived a closed form solution for the time course kinetics analysis of the Michaelis-Menten mechanism. The solution, , has the form: :\frac = W \left F(t) \right, where W is the Lambert-W function. and where F(t) is :F(t) = \frac \exp\!\left(\frac - \frac\,t \right) \, This equation is encompassed by the equation below, obtained by Berberan-Santos, which is also valid when the initial substrate concentration is close to that of enzyme, :\frac = W \left F(t) \right \frac\ \frac\, where W is again the Lambert-W function.


Linear plots of the Michaelis–Menten equation

The plot of ''v'' versus above is not linear; although initially linear at low it bends over to saturate at high Before the modern era of nonlinear curve-fitting on computers, this nonlinearity could make it difficult to estimate ''K''M and ''V''max accurately. Therefore, several researchers developed linearisations of the Michaelis–Menten equation, such as the
Lineweaver–Burk plot In biochemistry, the Lineweaver–Burk plot (or double reciprocal plot) is a graphical representation of the Michaelis–Menten equation of enzyme kinetics, described by Hans Lineweaver and Dean Burk in 1934. The double reciprocal plot distort ...
, the Eadie–Hofstee diagram and the Hanes–Woolf plot. All of these linear representations can be useful for visualising data, but none should be used to determine kinetic parameters, as computer software is readily available that allows for more accurate determination by
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 ...
methods. The
Lineweaver–Burk plot In biochemistry, the Lineweaver–Burk plot (or double reciprocal plot) is a graphical representation of the Michaelis–Menten equation of enzyme kinetics, described by Hans Lineweaver and Dean Burk in 1934. The double reciprocal plot distort ...
or double reciprocal plot is a common way of illustrating kinetic data. This is produced by taking the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of both sides of the Michaelis–Menten equation. As shown on the right, this is a linear form of the Michaelis–Menten equation and produces a straight line with the equation ''y'' = m''x'' + c with a ''y''-intercept equivalent to 1/''V''max and an ''x''-intercept of the graph representing −1/''K''M. :\frac = \frac + \frac Naturally, no experimental values can be taken at negative 1/ the lower limiting value 1/ = 0 (the ''y''-intercept) corresponds to an infinite substrate concentration, where ''1/v=1/Vmax'' as shown at the right; thus, the ''x''-intercept is an
extrapolation In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
of the experimental data taken at positive concentrations. More generally, the Lineweaver–Burk plot skews the importance of measurements taken at low substrate concentrations and, thus, can yield inaccurate estimates of ''V''max and ''K''M. A more accurate linear plotting method is the Eadie–Hofstee plot. In this case, ''v'' is plotted against ''v''/ In the third common linear representation, the Hanes–Woolf plot, ''v'' is plotted against In general, data normalisation can help diminish the amount of experimental work and can increase the reliability of the output, and is suitable for both graphical and numerical analysis.


Practical significance of kinetic constants

The study of enzyme kinetics is important for two basic reasons. Firstly, it helps explain how enzymes work, and secondly, it helps predict how enzymes behave in living organisms. The kinetic constants defined above, ''K''M and ''V''max, are critical to attempts to understand how enzymes work together to control
metabolism Metabolism (, from ''metabolē'', "change") is the set of life-sustaining chemical reactions in organisms. The three main functions of metabolism are: the conversion of the energy in food to energy available to run cellular processes; the co ...
. Making these predictions is not trivial, even for simple systems. For example,
oxaloacetate Oxaloacetic acid (also known as oxalacetic acid or OAA) is a crystalline organic compound with the chemical formula HO2CC(O)CH2CO2H. Oxaloacetic acid, in the form of its conjugate base oxaloacetate, is a metabolic intermediate in many processes ...
is formed by
malate dehydrogenase Malate dehydrogenase () (MDH) is an enzyme that reversibly catalyzes the oxidation of malate to oxaloacetate using the reduction of NAD+ to NADH. This reaction is part of many metabolic pathways, including the citric acid cycle. Other malate ...
within the
mitochondrion A mitochondrion () is an organelle found in the cell (biology), cells of most eukaryotes, such as animals, plants and fungi. Mitochondria have a double lipid bilayer, membrane structure and use aerobic respiration to generate adenosine tri ...
. Oxaloacetate can then be consumed by
citrate synthase Citrate synthase ( E.C. 2.3.3.1 (previously 4.1.3.7)) is an enzyme that exists in nearly all living cells. It functions as a pace-making enzyme in the first step of the citric acid cycle (or Krebs cycle). Citrate synthase is located within euka ...
,
phosphoenolpyruvate carboxykinase Phosphoenolpyruvate carboxykinase (, PEPCK) is an enzyme in the lyase family used in the metabolic pathway of gluconeogenesis. It converts oxaloacetate into phosphoenolpyruvate and carbon dioxide. It is found in two forms, cytosolic and mitoc ...
or
aspartate aminotransferase Aspartate transaminase (AST) or aspartate aminotransferase, also known as AspAT/ASAT/AAT or (serum) glutamic oxaloacetic transaminase (GOT, SGOT), is a pyridoxal phosphate (PLP)-dependent transaminase enzyme () that was first described by Arthur ...
, feeding into the
citric acid cycle The citric acid cycle—also known as the Krebs cycle, Szent–Györgyi–Krebs cycle, or TCA cycle (tricarboxylic acid cycle)—is a series of chemical reaction, biochemical reactions that release the energy stored in nutrients through acetyl-Co ...
,
gluconeogenesis Gluconeogenesis (GNG) is a metabolic pathway that results in the biosynthesis of glucose from certain non-carbohydrate carbon substrates. It is a ubiquitous process, present in plants, animals, fungi, bacteria, and other microorganisms. In verte ...
or
aspartic acid Aspartic acid (symbol Asp or D; the ionic form is known as aspartate), is an α-amino acid that is used in the biosynthesis of proteins. The L-isomer of aspartic acid is one of the 22 proteinogenic amino acids, i.e., the building blocks of protei ...
biosynthesis, respectively. Being able to predict how much oxaloacetate goes into which pathway requires knowledge of the concentration of oxaloacetate as well as the concentration and kinetics of each of these enzymes. This aim of predicting the behaviour of metabolic pathways reaches its most complex expression in the synthesis of huge amounts of kinetic and
gene expression Gene expression is the process (including its Regulation of gene expression, regulation) by which information from a gene is used in the synthesis of a functional gene product that enables it to produce end products, proteins or non-coding RNA, ...
data into mathematical models of entire organisms. Alternatively, one useful simplification of the metabolic modelling problem is to ignore the underlying enzyme kinetics and only rely on information about the reaction network's stoichiometry, a technique called
flux balance analysis In biochemistry, flux balance analysis (FBA) is a mathematical method for simulating the metabolism of cells or entire unicellular organisms, such as ''Escherichia coli, E. coli'' or yeast, using genome-scale reconstructions of metabolic networks ...
.


Michaelis–Menten kinetics with intermediate

One could also consider the less simple case : + S <=> _k_] ES -> _2 EI -> _3 + P where a complex with the enzyme and an intermediate exists and the intermediate is converted into product in a second step. In this case we have a very similar equation : v_0 = k_\frac\ce but the constants are different : \begin K_M^ \ &\stackrel\ \frac K_M = \frac \cdot \frac\\ k_ \ &\stackrel\ \dfrac \end We see that for the limiting case k_3 \gg k_2, thus when the last step from EI -> E + P is much faster than the previous step, we get again the original equation. Mathematically we have then K_M^ \approx K_M and k_ \approx k_2.


Multi-substrate reactions

Multi-substrate reactions follow complex rate equations that describe how the substrates bind and in what sequence. The analysis of these reactions is much simpler if the concentration of substrate A is kept constant and substrate B varied. Under these conditions, the enzyme behaves just like a single-substrate enzyme and a plot of ''v'' by gives apparent ''K''M and ''V''max constants for substrate B. If a set of these measurements is performed at different fixed concentrations of A, these data can be used to work out what the mechanism of the reaction is. For an enzyme that takes two substrates A and B and turns them into two products P and Q, there are two types of mechanism: ternary complex and substituted-enzyme mechanisms.


Ternary-complex mechanisms

Source: In these enzymes, both substrates bind to the enzyme at the same time to produce an EAB ternary complex. The order of binding can either be random (in a random mechanism) or substrates have to bind in a particular sequence (in an ordered mechanism). When a set of ''v'' by curves (fixed A, varying B) from an enzyme with a ternary-complex mechanism are plotted in a
Lineweaver–Burk plot In biochemistry, the Lineweaver–Burk plot (or double reciprocal plot) is a graphical representation of the Michaelis–Menten equation of enzyme kinetics, described by Hans Lineweaver and Dean Burk in 1934. The double reciprocal plot distort ...
, the set of lines produced will intersect. Enzymes with ternary-complex mechanisms include glutathione ''S''-transferase,
dihydrofolate reductase Dihydrofolate reductase, or DHFR, is an enzyme that reduces dihydrofolic acid to tetrahydrofolic acid, using NADPH as an electron donor, which can be converted to the kinds of tetrahydrofolate cofactors used in one-carbon transfer chemistry. ...
and
DNA polymerase A DNA polymerase is a member of a family of enzymes that catalyze the synthesis of DNA molecules from nucleoside triphosphates, the molecular precursors of DNA. These enzymes are essential for DNA replication and usually work in groups to create t ...
. The following links show short animations of the ternary-complex mechanisms of the enzymes dihydrofolate reductase and DNA polymerase.


Substituted-enzyme ("ping–pong") mechanisms

Source: As shown on the right, enzymes with a substituted-enzyme mechanism can exist in two states, E and a chemically modified form of the enzyme E*; this modified enzyme is known as an intermediate. In such mechanisms, substrate A binds, changes the enzyme to E* by, for example, transferring a chemical group to the active site, and is then released. Only after the first substrate is released can substrate B bind and react with the modified enzyme, regenerating the unmodified E form. When a set of ''v'' by curves (fixed A, varying B) from an enzyme with a substituted-enzyme mechanism are plotted in a Lineweaver–Burk plot, a set of parallel lines will be produced. This is called a secondary plot. Enzymes with substituted-enzyme mechanisms include some
oxidoreductases In biochemistry, an oxidoreductase is an enzyme that catalyzes the transfer of electrons from one molecule, the reductant, also called the electron donor, to another, the oxidant, also called the electron acceptor. This group of enzymes usually ut ...
such as thioredoxin peroxidase,
transferases In biochemistry, a transferase is any one of a class of enzymes that catalyse the transfer of specific functional groups (e.g. a methyl group, methyl or glycosyl group) from one molecule (called the donor) to another (called the acceptor). They ...
such as acylneuraminate cytidylyltransferase and
serine protease Serine proteases (or serine endopeptidases) are enzymes that cleave peptide bonds in proteins. Serine serves as the nucleophilic amino acid at the (enzyme's) active site. They are found ubiquitously in both eukaryotes and prokaryotes. Serin ...
s such as
trypsin Trypsin is an enzyme in the first section of the small intestine that starts the digestion of protein molecules by cutting long chains of amino acids into smaller pieces. It is a serine protease from the PA clan superfamily, found in the dig ...
and
chymotrypsin Chymotrypsin (, chymotrypsins A and B, alpha-chymar ophth, avazyme, chymar, chymotest, enzeon, quimar, quimotrase, alpha-chymar, alpha-chymotrypsin A, alpha-chymotrypsin) is a digestive enzyme component of pancreatic juice acting in the duodenu ...
. Serine proteases are a very common and diverse family of enzymes, including digestive enzymes (trypsin, chymotrypsin, and elastase), several enzymes of the blood clotting cascade and many others. In these serine proteases, the E* intermediate is an acyl-enzyme species formed by the attack of an active site
serine Serine (symbol Ser or S) is an α-amino acid that is used in the biosynthesis of proteins. It contains an α- amino group (which is in the protonated − form under biological conditions), a carboxyl group (which is in the deprotonated − ...
residue on a
peptide bond In organic chemistry, a peptide bond is an amide type of covalent chemical bond linking two consecutive alpha-amino acids from C1 (carbon number one) of one alpha-amino acid and N2 (nitrogen number two) of another, along a peptide or protein cha ...
in a protein substrate. A short animation showing the mechanism of chymotrypsin is linked here.


Memory effects

Both of these two types of mechanism can display
enzyme memory Enzyme memory is a concept in enzyme kinetics based on the idea that the kinetic properties of an enzyme may vary according to conditions in its previous catalytic cycle. It can occur both in ternary-complex mechanisms and in substituted-enzyme En ...
, with very different causes and consequences in the two cases. In ternary complex mechanisms these are possible if the mechanism includes slow processes and the binding steps are not at quasi-equilibrium, because the intermediates may be swept away very fast. This can generate
cooperativity Cooperativity is a phenomenon displayed by systems involving identical or near-identical elements, which act dependently of each other, relative to a hypothetical standard non-interacting system in which the individual elements are acting indepen ...
, even in monomeric enzymes. In a substituted-enzyme mechanism slow steps are not needed to generate memory effects. Instead, for an enzyme with several alternative substrates the kinetic properties of the second half reaction may vary with different substrates in the first half reaction, even though the same substituted enzyme seems to be transformed.


Reversible catalysis and the Haldane equation

External factors may limit the ability of an enzyme to catalyse a reaction in both directions (whereas the nature of a catalyst in itself means that it cannot catalyse just one direction, according to the principle of
microscopic reversibility The principle of microscopic reversibility in physics and chemistry is twofold: * First, it states that the microscopic detailed dynamics of particles and fields is time-reversible because the microscopic equations of motion are symmetric with respe ...
). We consider the case of an enzyme that catalyses the reaction in both directions: + <=> _k_] ES <=> _k_] + The steady-state, initial rate of the reaction is v_0 = \frac = \frac v_0 is positive if the reaction proceed in the forward direction (S \rightarrow P) and negative otherwise. :en:Equilibrium constant, Equilibrium requires that v=0, which occurs when \frac=\frac=K_ . This shows that
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
forces a relation between the values of the 4 rate constants. The values of the forward and backward ''maximal'' rates, obtained for [] \rightarrow \infty, [] =0, and [] =0 , [] \rightarrow \infty, respectively, are V_^f = k_ _ and V_^b = -k_ _, respectively. Their ratio is not equal to the equilibrium constant, which implies that
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
does not constrain the ratio of the maximal rates. This explains that enzymes can be much "better catalysts" (''in terms of maximal rates'') in one particular direction of the reaction. On can also derive the two Michaelis constants K_M^S=(k_+k_2)/k_1and K_M^P=(k_+k_2)/k_. The Haldane equation is the relation K_=\frac=\frac. Therefore,
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
constrains the ratio between the forward and backward V_/K_M values, not the ratio of V_values.


Non-Michaelis–Menten kinetics

Many different enzyme systems follow non Michaelis-Menten behavior. A select few examples include kinetics of self-catalytic enzymes, cooperative and allosteric enzymes, interfacial and intracellular enzymes, processive enzymes and so forth. Some enzymes produce a
sigmoid Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include: * Sigmoid function, a mathematical function * Sigmoid colon, part of the l ...
''v'' by plot, which often indicates
cooperative binding Cooperative binding occurs in molecular binding systems containing more than one type, or species, of molecule and in which one of the partners is not mono-valent and can bind more than one molecule of the other species. In general, molecular bindi ...
of substrate to the active site. This means that the binding of one substrate molecule affects the binding of subsequent substrate molecules. This behavior is most common in multimeric enzymes with several interacting active sites. Here, the mechanism of cooperation is similar to that of
hemoglobin Hemoglobin (haemoglobin, Hb or Hgb) is a protein containing iron that facilitates the transportation of oxygen in red blood cells. Almost all vertebrates contain hemoglobin, with the sole exception of the fish family Channichthyidae. Hemoglobin ...
, with binding of substrate to one active site altering the affinity of the other active sites for substrate molecules. Positive cooperativity occurs when binding of the first substrate molecule ''increases'' the affinity of the other active sites for substrate. Negative cooperativity occurs when binding of the first substrate ''decreases'' the affinity of the enzyme for other substrate molecules. Allosteric enzymes include mammalian tyrosyl tRNA-synthetase, which shows negative cooperativity, and bacterial
aspartate transcarbamoylase Aspartate carbamoyltransferase (also known as aspartate transcarbamoylase or ATCase) catalyzes the first step in the pyrimidine biosynthetic pathway (). In ''E. coli'', the enzyme is a multi- subunit protein complex composed of 12 subunits (30 ...
and
phosphofructokinase Phosphofructokinase (PFK) is a kinase enzyme that phosphorylates fructose 6-phosphate in glycolysis. Function The enzyme-catalysed transfer of a phosphoryl group from ATP is an important reaction in a wide variety of biological processe ...
, which show positive cooperativity. Cooperativity is surprisingly common and can help regulate the responses of enzymes to changes in the concentrations of their substrates. Positive cooperativity makes enzymes much more sensitive to and their activities can show large changes over a narrow range of substrate concentration. Conversely, negative cooperativity makes enzymes insensitive to small changes in The Hill equation is often used to describe the degree of cooperativity quantitatively in non-Michaelis–Menten kinetics. The derived Hill coefficient ''n'' measures how much the binding of substrate to one active site affects the binding of substrate to the other active sites. A Hill coefficient of <1 indicates negative cooperativity and a coefficient of >1 indicates positive
cooperativity Cooperativity is a phenomenon displayed by systems involving identical or near-identical elements, which act dependently of each other, relative to a hypothetical standard non-interacting system in which the individual elements are acting indepen ...
.


Pre-steady-state kinetics

In the first moment after an enzyme is mixed with substrate, no product has been formed and no intermediates exist. The study of the next few milliseconds of the reaction is called pre-steady-state kinetics. Pre-steady-state kinetics is therefore concerned with the formation and consumption of enzyme–substrate intermediates (such as ES or E*) until their steady-state concentrations are reached. This approach was first applied to the hydrolysis reaction catalysed by
chymotrypsin Chymotrypsin (, chymotrypsins A and B, alpha-chymar ophth, avazyme, chymar, chymotest, enzeon, quimar, quimotrase, alpha-chymar, alpha-chymotrypsin A, alpha-chymotrypsin) is a digestive enzyme component of pancreatic juice acting in the duodenu ...
. Often, the detection of an intermediate is a vital piece of evidence in investigations of what mechanism an enzyme follows. For example, in the ping–pong mechanisms that are shown above, rapid kinetic measurements can follow the release of product P and measure the formation of the modified enzyme intermediate E*. In the case of chymotrypsin, this intermediate is formed by an attack on the substrate by the
nucleophilic In chemistry, a nucleophile is a chemical species that forms bonds by donating an electron pair. All molecules and ions with a free pair of electrons or at least one pi bond can act as nucleophiles. Because nucleophiles donate electrons, they a ...
serine in the active site and the formation of the acyl-enzyme intermediate. In the figure to the right, the enzyme produces E* rapidly in the first few seconds of the reaction. The rate then slows as steady state is reached. This rapid burst phase of the reaction measures a single turnover of the enzyme. Consequently, the amount of product released in this burst, shown as the intercept on the ''y''-axis of the graph, also gives the amount of functional enzyme which is present in the assay.


Chemical mechanism

An important goal of measuring enzyme kinetics is to determine the chemical mechanism of an enzyme reaction, i.e., the sequence of chemical steps that transform substrate into product. The kinetic approaches discussed above will show at what rates intermediates are formed and inter-converted, but they cannot identify exactly what these intermediates are. Kinetic measurements taken under various solution conditions or on slightly modified enzymes or substrates often shed light on this chemical mechanism, as they reveal the rate-determining step or intermediates in the reaction. For example, the breaking of a
covalent bond A covalent bond is a chemical bond that involves the sharing of electrons to form electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs. The stable balance of attractive and repulsive forces between atom ...
to a
hydrogen Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
is a common rate-determining step. Which of the possible hydrogen transfers is rate determining can be shown by measuring the kinetic effects of substituting each hydrogen by
deuterium Deuterium (hydrogen-2, symbol H or D, also known as heavy hydrogen) is one of two stable isotopes of hydrogen; the other is protium, or hydrogen-1, H. The deuterium nucleus (deuteron) contains one proton and one neutron, whereas the far more c ...
, its stable
isotope Isotopes are distinct nuclear species (or ''nuclides'') of the same chemical element. They have the same atomic number (number of protons in their Atomic nucleus, nuclei) and position in the periodic table (and hence belong to the same chemica ...
. The rate will change when the critical hydrogen is replaced, due to a primary
kinetic isotope effect In physical organic chemistry, a kinetic isotope effect (KIE) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes. Formally, it is the ratio of rate constants for t ...
, which occurs because bonds to deuterium are harder to break than bonds to hydrogen. It is also possible to measure similar effects with other isotope substitutions, such as 13C/12C and 18O/16O, but these effects are more subtle. Isotopes can also be used to reveal the fate of various parts of the substrate molecules in the final products. For example, it is sometimes difficult to discern the origin of an
oxygen Oxygen is a chemical element; it has chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen group (periodic table), group in the periodic table, a highly reactivity (chemistry), reactive nonmetal (chemistry), non ...
atom in the final product; since it may have come from water or from part of the substrate. This may be determined by systematically substituting oxygen's stable isotope 18O into the various molecules that participate in the reaction and checking for the isotope in the product. The chemical mechanism can also be elucidated by examining the kinetics and isotope effects under different pH conditions, by altering the metal ions or other bound cofactors, by
site-directed mutagenesis Site-directed mutagenesis is a molecular biology method that is used to make specific and intentional mutating changes to the DNA sequence of a gene and any gene products. Also called site-specific mutagenesis or oligonucleotide-directed mutagenes ...
of conserved amino acid residues, or by studying the behaviour of the enzyme in the presence of analogues of the substrate(s).


Enzyme inhibition and activation

Enzyme inhibitors are molecules that reduce or abolish enzyme activity, while enzyme activators are molecules that increase the catalytic rate of enzymes. These interactions can be either ''reversible'' (i.e., removal of the inhibitor restores enzyme activity) or ''irreversible'' (i.e., the inhibitor permanently inactivates the enzyme).


Reversible inhibitors

Traditionally reversible enzyme inhibitors have been classified as
competitive Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, indi ...
, uncompetitive, or
non-competitive Non-competitive inhibition is a type of enzyme inhibition where the inhibitor reduces the activity of the enzyme and binds equally well to the enzyme regardless of whether it has already bound the substrate. This is unlike competitive inhibition, ...
, according to their effects on ''K''M and ''V''max. These different effects result from the inhibitor binding to the enzyme E, to the enzyme–substrate complex ES, or to both, respectively. The division of these classes arises from a problem in their derivation and results in the need to use two different binding constants for one binding event. The binding of an inhibitor and its effect on the enzymatic activity are two distinctly different things, another problem the traditional equations fail to acknowledge. In noncompetitive inhibition the binding of the inhibitor results in 100% inhibition of the enzyme only, and fails to consider the possibility of anything in between. In noncompetitive inhibition, the inhibitor will bind to an enzyme at its allosteric site; therefore, the binding affinity, or inverse of ''K''M, of the substrate with the enzyme will remain the same. On the other hand, the Vmax will decrease relative to an uninhibited enzyme. On a Lineweaver-Burk plot, the presence of a noncompetitive inhibitor is illustrated by a change in the y-intercept, defined as 1/Vmax. The x-intercept, defined as −1/''K''M, will remain the same. In competitive inhibition, the inhibitor will bind to an enzyme at the active site, competing with the substrate. As a result, the ''K''M will increase and the Vmax will remain the same. The common form of the inhibitory term also obscures the relationship between the inhibitor binding to the enzyme and its relationship to any other binding term be it the Michaelis–Menten equation or a dose response curve associated with ligand receptor binding. To demonstrate the relationship the following rearrangement can be made: :\cfrac = \cfrac Adding zero to the bottom ( :\cfrac Dividing by Ki :\cfrac = V_\max - V_\max \cfrac This notation demonstrates that similar to the Michaelis–Menten equation, where the rate of reaction depends on the percent of the enzyme population interacting with substrate, the effect of the inhibitor is a result of the percent of the enzyme population interacting with inhibitor. The only problem with this equation in its present form is that it assumes absolute inhibition of the enzyme with inhibitor binding, when in fact there can be a wide range of effects anywhere from 100% inhibition of substrate turn over to just >0%. To account for this the equation can be easily modified to allow for different degrees of inhibition by including a delta ''V''max term. :V_\max - \Delta V_\max \cfrac or :V_ - (V_ - V_ ) \cfrac This term can then define the residual enzymatic activity present when the inhibitor is interacting with individual enzymes in the population. However the inclusion of this term has the added value of allowing for the possibility of activation if the secondary ''V''max term turns out to be higher than the initial term. To account for the possibly of activation as well the notation can then be rewritten replacing the inhibitor "I" with a modifier term denoted here as "X". :V_ - (V_ - V_ ) \cfrac While this terminology results in a simplified way of dealing with kinetic effects relating to the maximum velocity of the Michaelis–Menten equation, it highlights potential problems with the term used to describe effects relating to the ''K''M. The ''K''M relating to the affinity of the enzyme for the substrate should in most cases relate to potential changes in the binding site of the enzyme which would directly result from enzyme inhibitor interactions. As such a term similar to the one proposed above to modulate ''V''max should be appropriate in most situations: :K_ - (K_ - K_ ) \cfrac


Irreversible inhibitors

Enzyme inhibitors can also irreversibly inactivate enzymes, usually by covalently modifying active site residues. These reactions, which may be called suicide substrates, follow
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda Lambda (; uppe ...
functions and are usually saturable. Below saturation, they follow
first order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
kinetics with respect to inhibitor. Irreversible inhibition could be classified into two distinct types. Affinity labelling is a type of irreversible inhibition where a functional group that is highly reactive modifies a catalytically critical residue on the protein of interest to bring about inhibition. Mechanism-based inhibition, on the other hand, involves binding of the inhibitor followed by enzyme mediated alterations that transform the latter into a reactive group that irreversibly modifies the enzyme.


Philosophical discourse on reversibility and irreversibility of inhibition

Having discussed reversible inhibition and irreversible inhibition in the above two headings, it would have to be pointed out that the concept of reversibility (or irreversibility) is a purely theoretical construct exclusively dependent on the time-frame of the assay, i.e., a reversible assay involving association and dissociation of the inhibitor molecule in the minute timescales would seem irreversible if an assay assess the outcome in the seconds and vice versa. There is a continuum of inhibitor behaviors spanning reversibility and irreversibility at a given non-arbitrary assay time frame. There are inhibitors that show slow-onset behavior and most of these inhibitors, invariably, also show tight-binding to the protein target of interest.


Mechanisms of catalysis

The favoured model for the enzyme–substrate interaction is the induced fit model. This model proposes that the initial interaction between enzyme and substrate is relatively weak, but that these weak interactions rapidly induce
conformational change In biochemistry, a conformational change is a change in the shape of a macromolecule, often induced by environmental factors. A macromolecule is usually flexible and dynamic. Its shape can change in response to changes in its environment or othe ...
s in the enzyme that strengthen binding. These conformational changes also bring catalytic residues in the active site close to the chemical bonds in the substrate that will be altered in the reaction. Conformational changes can be measured using
circular dichroism Circular dichroism (CD) is dichroism involving circular polarization, circularly polarized light, i.e., the differential Absorption (electromagnetic radiation), absorption of left- and right-handed light. Left-hand circular (LHC) and right-hand ci ...
or
dual polarisation interferometry Dual-polarization interferometry (DPI) is an analytical technique that probes molecular layers adsorbed to the surface of a waveguide using the evanescent wave of a laser beam. It is used to measure the conformational change in proteins, or o ...
. After binding takes place, one or more mechanisms of catalysis lower the energy of the reaction's
transition state In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked w ...
by providing an alternative chemical pathway for the reaction. Mechanisms of catalysis include catalysis by bond strain; by proximity and orientation; by active-site proton donors or acceptors; covalent catalysis and
quantum tunnelling In physics, quantum tunnelling, barrier penetration, or simply tunnelling is a quantum mechanical phenomenon in which an object such as an electron or atom passes through a potential energy barrier that, according to classical mechanics, shoul ...
. Enzyme kinetics cannot prove which modes of catalysis are used by an enzyme. However, some kinetic data can suggest possibilities to be examined by other techniques. For example, a ping–pong mechanism with burst-phase pre-steady-state kinetics would suggest covalent catalysis might be important in this enzyme's mechanism. Alternatively, the observation of a strong pH effect on ''V''max but not ''K''M might indicate that a residue in the active site needs to be in a particular
ionisation Ionization or ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule i ...
state for catalysis to occur.


History

In 1902
Victor Henri Victor Henri (6 June 1872 – 21 June 1940) was a French-Russian Physical chemistry, physical chemist and physiologist. He was born in Marseilles as a son of Russian parents. He is known mainly as an early pioneer in enzyme kinetics. He publishe ...
proposed a quantitative theory of enzyme kinetics, but at the time the experimental significance of the hydrogen ion concentration was not yet recognized. After Peter Lauritz Sørensen had defined the logarithmic pH-scale and introduced the concept of buffering in 1909 the German chemist
Leonor Michaelis Leonor Michaelis (16 January 1875 – 8 October 1949) was a German biochemist, physical chemist, and physician. He is known for his work with Maud Menten on enzyme kinetics in 1913, as well as for work on enzyme inhibition, pH and quinones. ...
and Dr. Maud Leonora Menten (a postdoctoral researcher in Michaelis's lab at the time) repeated Henri's experiments and confirmed his equation, which is now generally referred to as Michaelis-Menten kinetics (sometimes also ''Henri-Michaelis-Menten kinetics''). Their work was further developed by G. E. Briggs and
J. B. S. Haldane John Burdon Sanderson Haldane (; 5 November 18921 December 1964), nicknamed "Jack" or "JBS", was a British-born scientist who later moved to India and acquired Indian citizenship. He worked in the fields of physiology, genetics, evolutionary ...
, who derived kinetic equations that are still widely considered today a starting point in modeling enzymatic activity. The major contribution of the Henri-Michaelis-Menten approach was to think of enzyme reactions in two stages. In the first, the substrate binds reversibly to the enzyme, forming the enzyme-substrate complex. This is sometimes called the Michaelis complex. The enzyme then catalyzes the chemical step in the reaction and releases the product. The kinetics of many enzymes is adequately described by the simple Michaelis-Menten model, but all enzymes have internal motions that are not accounted for in the model and can have significant contributions to the overall reaction kinetics. This can be modeled by introducing several Michaelis-Menten pathways that are connected with fluctuating rates, which is a mathematical extension of the basic Michaelis Menten mechanism.


Software

ENZO (Enzyme Kinetics) is a graphical interface tool for building kinetic models of enzyme catalyzed reactions. ENZO automatically generates the corresponding differential equations from a stipulated enzyme reaction scheme. These differential equations are processed by a numerical solver and a regression algorithm which fits the coefficients of differential equations to experimentally observed time course curves. ENZO allows rapid evaluation of rival reaction schemes and can be used for routine tests in enzyme kinetics.
ENZO server
/ref>


See also

*
Protein dynamics In molecular biology, proteins are generally thought to adopt unique structures determined by their amino acid sequences. However, proteins are not strictly static objects, but rather populate ensembles of (sometimes similar) conformations. Tran ...
*
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 ...
*
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 ...


Footnotes

:α.''
Link: Interactive Michaelis–Menten kinetics tutorial (Java required)
:β.''

:γ.''

:δ.''


References


Further reading

Introductory * * * * Advanced * * * *


External links


Animation of an enzyme assay
— Shows effects of manipulating assay conditions
MACiE
— A database of enzyme reaction mechanisms
ENZYME
— Expasy enzyme nomenclature database
ENZO
— Web application for easy construction and quick testing of kinetic models of enzyme catalyzed reactions.
ExCatDB
— A database of enzyme catalytic mechanisms
BRENDA
— Comprehensive enzyme database, giving substrates, inhibitors and reaction diagrams
SABIO-RK
— A database of reaction kinetics

— Animations of several enzyme reaction mechanisms
Symbolism and Terminology in Enzyme Kinetics
— A comprehensive explanation of concepts and terminology in enzyme kinetics

— An accessible set of on-line tutorials on enzyme kinetics

— An animated tutorial with audio {{DEFAULTSORT:Enzyme Kinetics Catalysis