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Page 1: Enzyme Kinetics

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Enzyme kinetics

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ContentsArticles

Enzyme kinetics 1Rate equation 16Michaelis–Menten kinetics 24Lineweaver–Burk plot 28

ReferencesArticle Sources and Contributors 30Image Sources, Licenses and Contributors 31

Article LicensesLicense 32

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Enzyme kinetics 1

Enzyme kinetics

Dihydrofolate reductase from E. coli with its two substrates, dihydrofolate(right) and NADPH (left), bound in the active site. The protein is shown as aribbon diagram, with alpha helices in red, beta sheets in yellow and loops in

blue. Generated from 7DFR [1].

Enzyme kinetics is the study of the chemicalreactions that are catalysed by enzymes. Inenzyme kinetics, the reaction rate is measured andthe effects of varying the conditions of thereaction investigated. Studying an enzyme'skinetics in this way can reveal the catalyticmechanism of this enzyme, its role inmetabolism, how its activity is controlled, andhow a drug or an agonist might inhibit theenzyme.

Enzymes are usually protein molecules thatmanipulate other molecules — the enzymes'substrates. These target molecules bind to anenzyme's active site and are transformed intoproducts through a series of steps known as theenzymatic mechanism. These mechanisms can bedivided into single-substrate andmultiple-substrate mechanisms. Kinetic studieson enzymes that only bind one substrate, such astriosephosphate isomerase, aim to measure theaffinity with which the enzyme binds thissubstrate and the turnover rate.

When enzymes bind multiple substrates, such asdihydrofolate reductase (shown right), enzymekinetics can also show the sequence in whichthese substrates bind and the sequence in whichproducts are released. An example of enzymes that bind a single substrate and release multiple products areproteases, which cleave one protein substrate into two polypeptide products. Others join two substrates together,such as DNA polymerase linking a nucleotide to DNA. Although these mechanisms are often a complex series ofsteps, there is typically one rate-determining step that determines the overall kinetics. This rate-determining stepmay be a chemical reaction or a conformational change of the enzyme or substrates, such as those involved in therelease of product(s) from the enzyme.

Knowledge of the enzyme's structure is helpful in interpreting kinetic data. For example, the structure can suggesthow substrates and products bind during catalysis; what changes occur during the reaction; and even the role ofparticular amino acid 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 notundergo the enzymatic reaction.

Not all biological catalysts are protein enzymes; RNA-based catalysts such as ribozymes and ribosomes are essentialto many cellular functions, such as RNA splicing and translation. The main difference between ribozymes andenzymes 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 mechanisms and kinetics can beanalysed and classified by the same methods.

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General principles

As larger amounts of substrate are added to a reaction, the available enzymebinding sites become filled to the limit of . Beyond this limit the enzyme is

saturated with substrate and the reaction rate ceases to increase.

The reaction catalysed by an enzyme usesexactly the same reactants and producesexactly the same products as the uncatalysedreaction. Like other catalysts, enzymes donot alter the position of equilibrium betweensubstrates and products.[2] However, unlikeuncatalysed chemical reactions,enzyme-catalysed reactions displaysaturation kinetics. For a given enzymeconcentration and for relatively lowsubstrate concentrations, the reaction rateincreases linearly with substrateconcentration; the enzyme molecules arelargely free to catalyse the reaction, andincreasing substrate concentration means an increasing rate at which the enzyme and substrate molecules encounterone another. However, at relatively high substrate concentrations, the reaction rate asymptotically approaches thetheoretical maximum; the enzyme active sites are almost all occupied and the reaction rate is determined by theintrinsic turnover rate of the enzyme. The substrate concentration midway between these two limiting cases isdenoted by KM.

The two most important kinetic properties of an enzyme are how quickly the enzyme becomes saturated with aparticular substrate, and the maximum rate it can achieve. Knowing these properties suggests what an enzyme mightdo in the cell and can show how the enzyme will respond to changes in these conditions.

Enzyme assays

Progress curve for an enzyme reaction. The slope in the initial rateperiod is the initial rate of reaction v. The Michaelis–Menten

equation describes how this slope varies with the concentration ofsubstrate.

Enzyme assays are laboratory procedures that measurethe rate of enzyme reactions. Because enzymes are notconsumed by the reactions they catalyse, enzymeassays usually follow changes in the concentration ofeither substrates or products to measure the rate ofreaction. There are many methods of measurement.Spectrophotometric assays observe change in theabsorbance of light between products and reactants;radiometric assays involve the incorporation or releaseof radioactivity to measure the amount of product madeover time. Spectrophotometric assays are mostconvenient since they allow the rate of the reaction tobe measured continuously. Although radiometric assaysrequire the removal and counting of samples (i.e., theyare discontinuous assays) they are usually extremelysensitive and can measure very low levels of enzymeactivity.[3] An analogous approach is to use mass

spectrometry to monitor the incorporation or release of stable isotopes as substrate is converted into product.

The most sensitive enzyme assays use lasers focused through a microscope to observe changes in single enzyme molecules as they catalyse their reactions. These measurements either use changes in the fluorescence of cofactors

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during an enzyme's reaction mechanism, or of fluorescent dyes added onto specific sites of the protein to reportmovements that occur during catalysis.[4] These studies are providing a new view of the kinetics and dynamics ofsingle enzymes, as opposed to traditional enzyme kinetics, which observes the average behaviour of populations ofmillions of enzyme molecules.[5] [6]

An example progress curve for an enzyme assay is shown above. The enzyme produces product at an initial rate thatis approximately linear for a short period after the start of the reaction. As the reaction proceeds and substrate isconsumed, the rate continuously slows (so long as 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 percenttowards total completion. The length of the initial rate period depends on the assay conditions and can range frommilliseconds to hours. However, equipment for rapidly mixing liquids allows fast kinetic measurements on initialrates of less than one second.[7] These very rapid assays are essential for measuring pre-steady-state kinetics, whichare discussed below.Most enzyme kinetics studies concentrate on this initial, approximately linear part of enzyme reactions. However, itis also possible to measure the complete reaction curve and fit this data to a non-linear rate equation. This way ofmeasuring enzyme reactions is called progress-curve analysis.[8] This approach is useful as an alternative to rapidkinetics when the initial rate is too fast to measure accurately.

Single-substrate reactionsEnzymes with single-substrate mechanisms include isomerases such as triosephosphateisomerase orbisphosphoglycerate mutase, intramolecular lyases such as adenylate cyclase and the hammerhead ribozyme, a RNAlyase.[9] However, some enzymes that only have a single substrate do not fall into this category of mechanisms.Catalase is an example of this, as the enzyme reacts with a first molecule of hydrogen peroxide substrate, becomesoxidised and is then reduced by a second molecule of substrate. Although a single substrate is involved, the existenceof a modified enzyme intermediate means that the mechanism of catalase is actually a ping–pong mechanism, a typeof mechanism that is discussed in the Multi-substrate reactions section below.

Michaelis–Menten kinetics

Saturation curve for an enzyme showing the relation between the concentration ofsubstrate and rate.

As enzyme-catalysed reactions aresaturable, their rate of catalysis does notshow a linear response to increasingsubstrate. If the initial rate of the reaction ismeasured over a range of substrateconcentrations (denoted as [S]), the reactionrate (v) increases as [S] increases, as shownon the right. However, as [S] gets higher,the enzyme becomes saturated withsubstrate and the rate reaches Vmax, theenzyme's maximum rate.

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Single-substrate mechanism for an enzyme reaction. k1, k-1 and k2 are the rateconstants for the individual steps.

The Michaelis–Menten kinetic model of a single-substrate reaction is shown on the right. There is an initialbimolecular reaction between the enzyme E and substrate S to form the enzyme–substrate complex ES. Although theenzymatic mechanism for the unimolecular reaction can be quite complex, there is typically onerate-determining enzymatic step that allows this reaction to be modelled as a single catalytic step with an apparentunimolecular rate constant kcat. If the reaction path proceeds over one or several intermediates, kcat will be a functionof several elementary rate constants, whereas in the simplest case of a single elementary reaction (e.g. nointermediates) it will be identical to the elementary unimolecular rate constant k2. The apparent unimolecular rateconstant kcat is also called turnover number and denotes the maximum number of enzymatic reactions catalysed persecond.The Michaelis–Menten equation[10] describes how the (initial) reaction rate v0 depends on the position of thesubstrate-binding equilibrium and the rate constant k2.

    (Michaelis–Menten equation)

with the constants

This Michaelis–Menten equation is the basis for most single-substrate enzyme kinetics. Two crucial assumptionsunderlie this equation (apart from the general assumption about the mechanism only involving no intermediate orproduct inhibition, and there is no allostericity or cooperativity). The first assumption is the so calledquasi-steady-state assumption (or pseudo-steady-state hypothesis), namely that the concentration of thesubstrate-bound enzyme (and hence also the unbound enzyme) changes much more slowly than those of the productand substrate and thus the change over time of the complex can be set to zero . The second

assumption is that the total enzyme concentration does not change over time, thus. A complete derivation can be found here.

The Michaelis constant KM is experimentally defined as the concentration at which the rate of the enzyme reaction ishalf Vmax, which can be verified by substituting [S] = Km into the Michaelis–Menten equation and can also be seengraphically. If the rate-determining enzymatic step is slow compared to substrate dissociation ( ), theMichaelis constant KM is roughly the dissociation constant KD of the ES complex.If is small compared to then the term and also very little ES complex isformed, thus . Therefore, the rate of product formation is

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 . This

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constant is a measure of catalytic efficiency. The most efficient enzymes reach a in the range of 108 - 1010 M−1 s−1.These enzymes are so efficient they effectively catalyse a reaction each time they encounter a substrate molecule andhave thus reached an upper theoretical limit for efficiency (diffusion limit); these enzymes have often been termedperfect enzymes.[11]

Direct use of the Michaelis–Menten equation for time course kinetic analysisFurther information: Rate equationThe observed velocities predicted by the Michaelis–Menten equation can be used to directly model the time coursedisappearance of substrate and the production of product through incorporation of the Michaelis–Menten equationinto the equation for first order chemical kinetics. This can only be achieved however if one recognises the problemassociated with the use of Euler's number in the description of first order chemical kinetics. i.e. e-k is a split constantthat introduces a systematic error into calculations and can be rewritten as a single constant which represents theremaining substrate after each time period.[12]

In 1997, Santiago Schnell and Claudio Mendoza derived a closed form solution for the time course kinetics analysisof the Michaelis-Menten mechanism.[13] The solution has the form:

where W[] is the Lambert-W function.[14] [15]

Linear plots of the Michaelis–Menten equation

Lineweaver–Burk or double-reciprocal plot of kinetic data, showing the significance ofthe axis intercepts and gradient.

Using an interactiveMichaelis–Menten kinetics tutorial atthe University of Virginia,[α] theeffects on the behaviour of an enzymeof varying kinetic constants can beexplored.

The plot of v versus [S] above is notlinear; although initially linear at low[S], it bends over to saturate at high[S]. Before the modern era of nonlinearcurve-fitting on computers, thisnonlinearity could make it difficult toestimate KM and Vmax accurately.Therefore, several researchersdeveloped linearisations of theMichaelis–Menten equation, such asthe Lineweaver–Burk plot, the Eadie–Hofstee diagram and the Hanes–Woolf plot. All of these linear representationscan be useful for visualising data, but none should be used to determine kinetic parameters, as computer software isreadily available that allows for more accurate determination by nonlinear regression methods.[16]

The Lineweaver–Burk plot or double reciprocal plot is a common way of illustrating kinetic data. This is produced by taking the reciprocal 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 = mx + c with a y-intercept

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equivalent to 1/Vmax and an x-intercept of the graph representing -1/KM.

Naturally, no experimental values can be taken at negative 1/[S]; the lower limiting value 1/[S] = 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 anextrapolation of the experimental data taken at positive concentrations. More generally, the Lineweaver–Burk plotskews the importance of measurements taken at low substrate concentrations and, thus, can yield inaccurateestimates of Vmax and KM.[17] A more accurate linear plotting method is the Eadie-Hofstee plot. In this case, v isplotted against v/[S]. In the third common linear representation, the Hanes-Woolf plot, [S]/v is plotted against [S]. Ingeneral, data normalisation can help diminish the amount of experimental work and can increase the reliability of theoutput, and is suitable for both graphical and numerical analysis.[18]

Practical significance of kinetic constantsThe study of enzyme kinetics is important for two basic reasons. Firstly, it helps explain how enzymes work, andsecondly, it helps predict how enzymes behave in living organisms. The kinetic constants defined above, KM andVmax, are critical to attempts to understand how enzymes work together to control metabolism.Making these predictions is not trivial, even for simple systems. For example, oxaloacetate is formed by malatedehydrogenase within the mitochondrion. Oxaloacetate can then be consumed by citrate synthase,phosphoenolpyruvate carboxykinase or aspartate aminotransferase, feeding into the citric acid cycle,gluconeogenesis or aspartic acid biosynthesis, respectively. Being able to predict how much oxaloacetate goes intowhich pathway requires knowledge of the concentration of oxaloacetate as well as the concentration and kinetics ofeach of these enzymes. This aim of predicting the behaviour of metabolic pathways reaches its most complexexpression in the synthesis of huge amounts of kinetic and gene expression data into mathematical models of entireorganisms. Alternatively, one useful simplification of the metabolic modelling problem is to ignore the underlyingenzyme kinetics and only rely on information about the reaction network's stoichiometry, a technique called fluxbalance analysis.[19] [20]

Michaelis–Menten kinetics with intermediateOne could also consider the less simple case

where a complex with the enzyme and an intermediate exists and the intermediate is converted into product in asecond step. In this case we have a very similar equation[21]

but the constants are different

We see that for the limiting case , thus when the last step from EI to E + P is much faster than theprevious step, we get again the original equation. Mathematically we have then and .

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Multi-substrate reactionsMulti-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 Bvaried. Under these conditions, the enzyme behaves just like a single-substrate enzyme and a plot of v by [S] givesapparent KM and Vmax constants for substrate B. If a set of these measurements is performed at different fixedconcentrations of A, these data can be used to work out what the mechanism of the reaction is. For an enzyme thattakes two substrates A and B and turns them into two products P and Q, there are two types of mechanism: ternarycomplex and ping–pong.

Ternary-complex mechanisms

Random-order ternary-complex mechanism for an enzyme reaction. The reaction pathis shown as a line and enzyme intermediates containing substrates A and B or

products P and Q are written below the line.

In these enzymes, both substrates bind tothe enzyme at the same time to produce anEAB ternary complex. The order ofbinding can either be random (in arandom mechanism) or substrates have tobind in a particular sequence (in anordered mechanism). When a set of v by[S] curves (fixed A, varying B) from anenzyme with a ternary-complexmechanism are plotted in aLineweaver–Burk plot, the set of linesproduced will intersect.

Enzymes with ternary-complexmechanisms include glutathione S-transferase,[22] dihydrofolate reductase[23] and DNA polymerase.[24] Thefollowing links show short animations of the ternary-complex mechanisms of the enzymes dihydrofolate reductase[β]

and DNA polymerase[γ].

Ping–pong mechanisms

Ping–pong mechanism for an enzyme reaction. Intermediates contain substrates Aand B or products P and Q.

As shown on the right, enzymes with aping-pong mechanism can exist in twostates, E and a chemically modified formof the enzyme E*; this modified enzymeis known as an intermediate. In suchmechanisms, substrate A binds, changesthe enzyme to E* by, for example, transferring a chemical group to the active site, and is then released. Only afterthe first substrate is released can substrate B bind and react with the modified enzyme, regenerating the unmodifiedE form. When a set of v by [S] curves (fixed A, varying B) from an enzyme with a ping–pong mechanism are plottedin a Lineweaver–Burk plot, a set of parallel lines will be produced. This is called a secondary plot.

Enzymes with ping–pong mechanisms include some oxidoreductases such as thioredoxin peroxidase,[25] transferasessuch as acylneuraminate cytydilyltransferase[26] and serine proteases such as trypsin and chymotrypsin.[27] Serineproteases 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 residue on a peptide bond in aprotein substrate. A short animation showing the mechanism of chymotrypsin is linked here.[δ]

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Non-Michaelis–Menten kinetics

Saturation curve for an enzyme reaction showing sigmoid kinetics.

Some enzymes produce a sigmoid v by [S]plot, which often indicates cooperativebinding of substrate to the active site. Thismeans that the binding of one substratemolecule affects the binding of subsequentsubstrate molecules. This behavior is mostcommon in multimeric enzymes withseveral interacting active sites.[28] Here, themechanism of cooperation is similar to thatof hemoglobin, with binding of substrate toone active site altering the affinity of theother active sites for substrate molecules.Positive cooperativity occurs when bindingof the first substrate molecule increases theaffinity of the other active sites for substrate.Negative cooperativity occurs when bindingof the first substrate decreases the affinity of

the enzyme for other substrate molecules.

Allosteric enzymes include mammalian tyrosyl tRNA-synthetase, which shows negative cooperativity,[29] andbacterial aspartate transcarbamoylase[30] and phosphofructokinase,[31] which show positive cooperativity.Cooperativity is surprisingly common and can help regulate the responses of enzymes to changes in theconcentrations of their substrates. Positive cooperativity makes enzymes much more sensitive to [S] and theiractivities can show large changes over a narrow range of substrate concentration. Conversely, negative cooperativitymakes enzymes insensitive to small changes in [S].The Hill equation (biochemistry)[32] is often used to describe the degree of cooperativity quantitatively innon-Michaelis–Menten kinetics. The derived Hill coefficient n measures how much the binding of substrate to oneactive site affects the binding of substrate to the other active sites. A Hill coefficient of <1 indicates negativecooperativity and a coefficient of >1 indicates positive cooperativity.

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Pre-steady-state kinetics

Pre-steady state progress curve, showing the burst phase of an enzyme reaction.

In the first moment after an enzyme ismixed with substrate, no product has beenformed and no intermediates exist. Thestudy of the next few milliseconds of thereaction is called Pre-steady-state kineticsalso referred to as Burst kinetics.Pre-steady-state kinetics is thereforeconcerned with the formation andconsumption of enzyme–substrateintermediates (such as ES or E*) until theirsteady-state concentrations are reached.

This approach was first applied to thehydrolysis reaction catalysed bychymotrypsin.[33] Often, the detection of anintermediate is a vital piece of evidence ininvestigations of what mechanism an enzyme follows. For example, in the ping–pong mechanisms that are shownabove, rapid kinetic measurements can follow the release of product P and measure the formation of the modifiedenzyme intermediate E*.[34] In the case of chymotrypsin, this intermediate is formed by an attack on the substrate bythe nucleophilic 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 slowsas 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, alsogives the amount of functional enzyme which is present in the assay.[35]

Chemical mechanismAn 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 willshow at what rates intermediates are formed and inter-converted, but they cannot identify exactly what theseintermediates are.Kinetic measurements taken under various solution conditions or on slightly modified enzymes or substrates oftenshed light on this chemical mechanism, as they reveal the rate-determining step or intermediates in the reaction. Forexample, the breaking of a covalent bond to a hydrogen atom is a common rate-determining step. Which of thepossible hydrogen transfers is rate determining can be shown by measuring the kinetic effects of substituting eachhydrogen by deuterium, its stable isotope. The rate will change when the critical hydrogen is replaced, due to aprimary kinetic isotope effect, which occurs because bonds to deuterium are harder to break than bonds tohydrogen.[36] It is also possible to measure similar effects with other isotope substitutions, such as 13C/12C and18O/16O, but these effects are more subtle.[37]

Isotopes can also be used to reveal the fate of various parts of the substrate molecules in the final products. Forexample, it is sometimes difficult to discern the origin of an oxygen atom in the final product; since it may havecome from water or from part of the substrate. This may be determined by systematically substituting oxygen's stableisotope 18O into the various molecules that participate in the reaction and checking for the isotope in the product.[38]

The chemical mechanism can also be elucidated by examining the kinetics and isotope effects under different pHconditions,[39] by altering the metal ions or other bound cofactors,[40] by site-directed mutagenesis of conservedamino acid residues, or by studying the behaviour of the enzyme in the presence of analogues of the substrate(s).[41]

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Enzyme inhibition and activation

Kinetic scheme for reversible enzyme inhibitors.

Enzyme inhibitors are molecules that reduceor abolish enzyme activity, while enzymeactivators are molecules that increase thecatalytic rate of enzymes. These interactionscan be either reversible (i.e., removal of theinhibitor restores enzyme activity) orirreversible (i.e., the inhibitor permanentlyinactivates the enzyme).

Reversible inhibitors

Traditionally reversible enzyme inhibitorshave been classified as competitive, uncompetitive, or non-competitive, according to their effects on Km and Vmax.These different effects result from the inhibitor binding to the enzyme E, to the enzyme–substrate complex ES, or toboth, respectively. The particular type of an inhibitor can be discerned by studying the enzyme kinetics as a functionof the inhibitor concentration. The three types of inhibition produce Lineweaver–Burke and Eadie–Hofstee plots[17]

that vary in distinctive ways with inhibitor concentration. For a straightforward qualitative explanation, see[42]

Non-linear regression fits of the enzyme kinetics data to rate equations[43] can yield accurate estimates ofdissociation constants and yield important information relating to the mechanism of action.

Adding zero to the bottom ([I]-[I])

Dividing by [I]+Ki

This notation demonstrates that similar to the Michaelis–Menten equation,where the rate of reaction depends on thepercent of the enzyme population interacting with substratefraction of the enzyme population bound by substrate

fraction of the enzyme population bound by inhibitor

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the effect of the inhibitor is a result of the percent of the enzyme population interacting with inhibitor. The onlyproblem with this equation in its present form is that it assumes absolute inhibition of the enzyme with inhibitorbinding, when in fact there can be a wide range of affects anywhere from 100% inhibition of substrate turn over tojust >0%. To account for this the equation can be easily modified to allow for different degrees of inhibition byincluding a delta Vmax term.

or

This term can then define the residual enzymatic activity present when the inhibitor is interacting with individualenzymes in the population. However the inclusion of this term has the added value of allowing for the possibility ofactivation if the secondary Vmax term turns out to be higher than the initial term. To account for the possibly ofactivation as well the notation can then be rewritten replacing the inhibitor "I" with a modifier term denoted here as"X".

While this terminology results in a simplified way of dealing with kinetic effects relating to the maximum velocity ofthe Michaelis–Menten equation, it highlights potential problems with the term used to describe effects relating to theKm. The Km relating to the affinity of the enzyme for the substrate should in most cases relate to potential changes inthe binding site of the enzyme which would directly result from enzyme inhibitor interactions. As such a term similarto the one proposed above to modulate Vmax should be appropriate in most situations.:[44]

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Irreversible inhibitorsEnzyme inhibitors can also irreversibly inactivate enzymes, usually by covalently modifying active site residues.These reactions, which may be called suicide substrates, follow exponential decay functions and are usuallysaturable. Below saturation, they follow first order kinetics with respect to inhibitor.

Mechanisms of catalysis

The energy variation as a function of reaction coordinate shows the stabilisation ofthe transition state by an enzyme.

The favoured model for theenzyme–substrate interaction is the inducedfit model.[45] This model proposes that theinitial interaction between enzyme andsubstrate is relatively weak, but that theseweak interactions rapidly induceconformational changes in the enzyme thatstrengthen binding. These conformationalchanges also bring catalytic residues in theactive site close to the chemical bonds in thesubstrate that will be altered in thereaction.[46] Conformational changes can bemeasured using circular dichroism or dualpolarisation interferometry. After bindingtakes place, one or more mechanisms ofcatalysis lower the energy of the reaction'stransition state by providing an alternativechemical pathway for the reaction.Mechanisms of catalysis include catalysis by bond strain; by proximity and orientation; by active-site proton donorsor acceptors; covalent catalysis and quantum tunnelling.[34] [47]

Enzyme kinetics cannot prove which modes of catalysis are used by an enzyme. However, some kinetic data cansuggest possibilities to be examined by other techniques. For example, a ping–pong mechanism with burst-phasepre-steady-state kinetics would suggest covalent catalysis might be important in this enzyme's mechanism.Alternatively, the observation of a strong pH effect on Vmax but not Km might indicate that a residue in the active siteneeds to be in a particular ionisation state for catalysis to occur.

Footnotesα.  Link: Interactive Michaelis–Menten kinetics tutorial (Java required) [48]

β.  Link: dihydrofolate reductase mechanism (Gif) [49]

γ.  Link: DNA polymerase mechanism (Gif) [50]

δ.  Link: Chymotrypsin mechanism (Flash required) [51]

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[13] Schnell S, Mendoza C. A closed form solution for time-dependent enzyme kinetics. Journal of theoretical Biology, 187 (1997): 207-212[14] C.T. Goudar, J.R. Sonnad, R.G. Duggleby (1999). Parameter estimation using a direct solution of the integrated Michaelis-Menten equation.

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analysis" (http:/ / www. biochemj. org/ bj/ 358/ 0573/ bj3580573. htm). Biochem. J. 358 (Pt 3): 573–83. PMC 1222113. PMID 11577687. .[19] Almaas E, Kovács B, Vicsek T, Oltvai ZN, Barabási AL (February 2004). "Global organization of metabolic fluxes in the bacterium

Escherichia coli". Nature 427 (6977): 839–43. doi:10.1038/nature02289. PMID 14985762.[20] Reed JL, Vo TD, Schilling CH, Palsson BO (2003). "An expanded genome-scale model of Escherichia coli K-12 (iJR904 GSM/GPR)".

Genome Biol. 4 (9): R54. doi:10.1186/gb-2003-4-9-r54. PMC 193654. PMID 12952533.[21] for a complete derivation, see here[22] Dirr H, Reinemer P, Huber R (March 1994). "X-ray crystal structures of cytosolic glutathione S-transferases. Implications for protein

architecture, substrate recognition and catalytic function". Eur. J. Biochem. 220 (3): 645–61. doi:10.1111/j.1432-1033.1994.tb18666.x.PMID 8143720.

[23] Stone SR, Morrison JF (July 1988). "Dihydrofolate reductase from Escherichia coli: the kinetic mechanism with NADPH and reducedacetylpyridine adenine dinucleotide phosphate as substrates". Biochemistry 27 (15): 5493–9. doi:10.1021/bi00415a016. PMID 3052577.

[24] Fisher PA (1994). "Enzymologic mechanism of replicative DNA polymerases in higher eukaryotes". Prog. Nucleic Acid Res. Mol. Biol..Progress in Nucleic Acid Research and Molecular Biology 47: 371–97. doi:10.1016/S0079-6603(08)60257-3. ISBN 9780125400473.PMID 8016325.

[25] Akerman SE, Müller S (August 2003). "2-Cys peroxiredoxin PfTrx-Px1 is involved in the antioxidant defence of Plasmodium falciparum"(http:/ / www. sciencedirect. com/ science?_ob=ArticleURL& _udi=B6T29-49321S6-2& _coverDate=08/ 31/ 2003& _alid=466052639&_rdoc=1& _fmt=& _orig=search& _qd=1& _cdi=4913& _sort=d& view=c& _acct=C000050221& _version=1& _urlVersion=0&_userid=10& md5=965e18a4c2ad0af711ba05d1a46dc855). Mol. Biochem. Parasitol. 130 (2): 75–81. doi:10.1016/S0166-6851(03)00161-0.PMID 12946843. .

[26] Bravo IG, Barrallo S, Ferrero MA, Rodríguez-Aparicio LB, Martínez-Blanco H, Reglero A (September 2001). "Kinetic properties of theacylneuraminate cytidylyltransferase from Pasteurella haemolytica A2" (http:/ / www. biochemj. org/ bj/ 358/ 0585/ bj3580585. htm).Biochem. J. 358 (Pt 3): 585–98. PMC 1222114. PMID 11577688. .

[27] Kraut J (1977). "Serine proteases: structure and mechanism of catalysis" (http:/ / arjournals. annualreviews. org/ doi/ abs/ 10. 1146/ annurev.bi. 46. 070177. 001555?url_ver=Z39. 88-2003& rfr_id=ori:rid:crossref. org& rfr_dat=cr_pub=ncbi. nlm. nih. gov). Annu. Rev. Biochem. 46:331–58. doi:10.1146/annurev.bi.46.070177.001555. PMID 332063. .

[28] Ricard J, Cornish-Bowden A (July 1987). "Co-operative and allosteric enzymes: 20 years on". Eur. J. Biochem. 166 (2): 255–72.doi:10.1111/j.1432-1033.1987.tb13510.x. PMID 3301336.

[29] Ward WH, Fersht AR (July 1988). "Tyrosyl-tRNA synthetase acts as an asymmetric dimer in charging tRNA. A rationale forhalf-of-the-sites activity". Biochemistry 27 (15): 5525–30. doi:10.1021/bi00415a021. PMID 3179266.

[30] Helmstaedt K, Krappmann S, Braus GH (September 2001). "Allosteric Regulation of Catalytic Activity: Escherichia coli AspartateTranscarbamoylase versus Yeast Chorismate Mutase" (http:/ / mmbr. asm. org/ cgi/ content/ full/ 65/ 3/ 404). Microbiol. Mol. Biol. Rev. 65(3): 404–21, table of contents. doi:10.1128/MMBR.65.3.404-421.2001. PMC 99034. PMID 11528003. .

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[31] Schirmer T, Evans PR (January 1990). "Structural basis of the allosteric behaviour of phosphofructokinase". Nature 343 (6254): 140–5.doi:10.1038/343140a0. PMID 2136935.

[32] Hill, A. V. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J. Physiol. (Lond.), 1910 40,iv-vii.

[33] Hartley BS, Kilby BA (February 1954). "The reaction of p-nitrophenyl esters with chymotrypsin and insulin". Biochem. J. 56 (2): 288–97.PMC 1269615. PMID 13140189.

[34] Fersht, Alan (1999). Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding. San Francisco: W.H.Freeman. ISBN 0-7167-3268-8.

[35] Bender ML, Begué-Cantón ML, Blakeley RL, et al. (December 1966). "The determination of the concentration of hydrolytic enzymesolutions: alpha-chymotrypsin, trypsin, papain, elastase, subtilisin, and acetylcholinesterase". J. Am. Chem. Soc. 88 (24): 5890–913.doi:10.1021/ja00976a034. PMID 5980876.

[36] Cleland WW (January 2005). "The use of isotope effects to determine enzyme mechanisms" (http:/ / www. sciencedirect. com/science?_ob=ArticleURL& _udi=B6WB5-4DD8DXJ-8& _coverDate=01/ 01/ 2005& _alid=466049795& _rdoc=1& _fmt=& _orig=search&_qd=1& _cdi=6701& _sort=d& view=c& _acct=C000050221& _version=1& _urlVersion=0& _userid=10&md5=cf322c4a1c7db6b9f89551a8469a1a2d). Arch. Biochem. Biophys. 433 (1): 2–12. doi:10.1016/j.abb.2004.08.027. PMID 15581561. .

[37] Northrop D (1981). "The expression of isotope effects on enzyme-catalyzed reactions". Annu. Rev. Biochem. 50: 103–31.doi:10.1146/annurev.bi.50.070181.000535. PMID 7023356.

[38] Baillie T, Rettenmeier A (1986). "Drug biotransformation: mechanistic studies with stable isotopes". Journal of clinical pharmacology 26(6): 448–51. PMID 3734135.

[39] Cleland WW (1982). "Use of isotope effects to elucidate enzyme mechanisms". CRC Crit. Rev. Biochem. 13 (4): 385–428.doi:10.3109/10409238209108715. PMID 6759038.

[40] Christianson DW, Cox JD (1999). "Catalysis by metal-activated hydroxide in zinc and manganese metalloenzymes". Annu. Rev. Biochem.68: 33–57. doi:10.1146/annurev.biochem.68.1.33. PMID 10872443.

[41] Kraut D, Carroll K, Herschlag D (2003). "Challenges in enzyme mechanism and energetics". Annu. Rev. Biochem. 72: 517–71.doi:10.1146/annurev.biochem.72.121801.161617. PMID 12704087.

[42] Waldrop GL (January 2009). "A qualitative approach to enzyme inhibition". Biochemistry and Molecular Biology Education 37 (1): 11–15.doi:10.1002/bmb.20243. PMID 21567682.

[43] Leatherbarrow RJ (December 1990). "Using linear and non-linear regression to fit biochemical data". Trends Biochem. Sci. 15 (12): 455–8.doi:10.1016/0968-0004(90)90295-M. PMID 2077683.

[44] Walsh R, Martin E, Darvesh S. A versatile equation to describe reversible enzyme inhibition and activation kinetics: modelingbeta-galactosidase and butyrylcholinesterase. Biochim Biophys Acta. 2007 1770:733-46.

[45] Koshland DE (February 1958). "Application of a Theory of Enzyme Specificity to Protein Synthesis". Proc. Natl. Acad. Sci. U.S.A. 44 (2):98–104. doi:10.1073/pnas.44.2.98. PMC 335371. PMID 16590179.

[46] Hammes G (2002). "Multiple conformational changes in enzyme catalysis". Biochemistry 41 (26): 8221–8. doi:10.1021/bi0260839.PMID 12081470.

[47] Sutcliffe M, Scrutton N (2002). "A new conceptual framework for enzyme catalysis. Hydrogen tunnelling coupled to enzyme dynamics inflavoprotein and quinoprotein enzymes" (http:/ / content. febsjournal. org/ cgi/ content/ full/ 269/ 13/ 3096). Eur. J. Biochem. 269 (13):3096–102. doi:10.1046/j.1432-1033.2002.03020.x. PMID 12084049. .

[48] http:/ / cti. itc. virginia. edu/ ~cmg/ Demo/ scriptFrame. html[49] http:/ / chem-faculty. ucsd. edu/ kraut/ dhfr. html[50] http:/ / chem-faculty. ucsd. edu/ kraut/ dNTP. html[51] http:/ / courses. cm. utexas. edu/ jrobertus/ ch339k/ overheads-2/ 06_21_chymotrypsin. html

Further readingIntroductory

• Cornish-Bowden, Athel (2004). Fundamentals of enzyme kinetics (3rd ed.). London: Portland Press.ISBN 1-85578-158-1.

• Stevens, Lewis; Price, Nicholas C. (1999). Fundamentals of enzymology: the cell and molecular biology ofcatalytic proteins. Oxford [Oxfordshire]: Oxford University Press. ISBN 0-19-850229-X.

• Bugg, Tim (2004). Introduction to Enzyme and Coenzyme Chemistry. Cambridge, MA: Blackwell Publishers.ISBN 1-4051-1452-5.

Advanced

• Segel, Irwin H. (1993). Enzyme kinetics: behavior and analysis of rapid equilibrium and steady state enzymesystems (New ed.). New York: Wiley. ISBN 0-471-30309-7.

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Enzyme kinetics 15

• Fersht, Alan (1999). Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding.San Francisco: W.H. Freeman. ISBN 0-7167-3268-8.

• Santiago Schnell, Philip K. Maini (2004). "A century of enzyme kinetics: Reliability of the KM and vmaxestimates" (http:/ / www. informatics. indiana. edu/ schnell/ papers/ ctb8_169. pdf). Comments on TheoreticalBiology 8 (2–3): 169–87. doi:10.1080/08948550302453.

• Walsh, Christopher (1979). Enzymatic reaction mechanisms. San Francisco: W. H. Freeman.ISBN 0-7167-0070-0.

• Cleland, William Wallace; Cook, Paul (2007). Enzyme kinetics and mechanism. New York: Garland Science.ISBN 0-8153-4140-7.

External links• Animation of an enzyme assay (http:/ / www. kscience. co. uk/ animations/ model. swf) — Shows effects of

manipulating assay conditions• MACiE (http:/ / www. ebi. ac. uk/ thornton-srv/ databases/ MACiE/ ) — A database of enzyme reaction

mechanisms• ENZYME (http:/ / us. expasy. org/ enzyme/ ) — Expasy enzyme nomenclature database• ENZO (http:/ / enzo. cmm. ki. si) — Web application for easy construction and quick testing of kinetic models of

enzyme catalyzed reactions.• ExCatDB (http:/ / mbs. cbrc. jp/ EzCatDB/ ) — A database of enzyme catalytic mechanisms• BRENDA (http:/ / www. brenda-enzymes. info/ ) — Comprehensive enzyme database, giving substrates,

inhibitors and reaction diagrams• SABIO-RK (http:/ / sabio. villa-bosch. de/ SABIORK/ ) — A database of reaction kinetics• Joseph Kraut's Research Group, University of California San Diego (http:/ / chem-faculty. ucsd. edu/ kraut/ dhfr.

html) — Animations of several enzyme reaction mechanisms• Symbolism and Terminology in Enzyme Kinetics (http:/ / www. chem. qmul. ac. uk/ iubmb/ kinetics/ ) — A

comprehensive explanation of concepts and terminology in enzyme kinetics• An introduction to enzyme kinetics (http:/ / orion1. paisley. ac. uk/ kinetics/ contents. html) — An accessible set

of on-line tutorials on enzyme kinetics• Enzyme kinetics animated tutorial (http:/ / www. wiley. com/ college/ pratt/ 0471393878/ student/ animations/

enzyme_kinetics/ index. html) — An animated tutorial with audio

Page 18: Enzyme Kinetics

Rate equation 16

Rate equationThe rate law or rate equation for a chemical reaction is an equation that links the reaction rate with concentrationsor pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders).[1] Todetermine the rate equation for a particular system one combines the reaction rate with a mass balance for thesystem.[2] For a generic reaction aA + bB → C with no intermediate steps in its reaction mechanism (that is, anelementary reaction), the rate is given by

where [A] and [B] express the concentration of the species A and B, respectively (usually in moles per liter(molarity, M)); x and y are not the respective stoichiometric coefficients of the balanced equation; they must bedetermined experimentally. k is the rate coefficient or rate constant of the reaction. The value of this coefficient kdepends on conditions such as temperature, ionic strength, surface area of the adsorbent or light irradiation. Forelementary reactions, the rate equation can be derived from first principles using collision theory. Again, x and y areNOT always derived from the balanced equation.The rate equation of a reaction with a multi-step mechanism cannot, in general, be deduced from the stoichiometriccoefficients of the overall reaction; it must be determined experimentally. The equation may involve fractionalexponential coefficients, or it may depend on the concentration of an intermediate species.The rate equation is a differential equation, and it can be integrated to obtain an integrated rate equation that linksconcentrations of reactants or products with time.If the concentration of one of the reactants remains constant (because it is a catalyst or it is in great excess withrespect to the other reactants), its concentration can be grouped with the rate constant, obtaining a pseudo constant:If B is the reactant whose concentration is constant, then . The second-order rate equationhas been reduced to a pseudo-first-order rate equation. This makes the treatment to obtain an integrated rateequation much easier.

Zeroth-order reactionsA Zeroth-order reaction has a rate that is independent of the concentration of the reactant(s). Increasing theconcentration of the reacting species will not speed up the rate of the reaction. Zeroth-order reactions are typicallyfound when a material that is required for the reaction to proceed, such as a surface or a catalyst, is saturated by thereactants. The rate law for a zeroth-order reaction is

where r is the reaction rate and k is the reaction rate coefficient with units of concentration/time. If, and only if, thiszeroth-order reaction 1) occurs in a closed system, 2) there is no net build-up of intermediates, and 3) there are noother reactions occurring, it can be shown by solving a mass balance equation for the system that:

If this differential equation is integrated it gives an equation often called the integrated zero-order rate law.

where represents the concentration of the chemical of interest at a particular time, and represents theinitial concentration.A reaction is zeroth order if concentration data are plotted versus time and the result is a straight line. The slope ofthis resulting line is the negative of the zero order rate constant k.The half-life of a reaction describes the time needed for half of the reactant to be depleted (same as the half-lifeinvolved in nuclear decay, which is a first-order reaction). For a zero-order reaction the half-life is given by

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Rate equation 17

Example of a zeroth-order reaction

• Reversed Haber process: It should be noted that the order of a reaction cannot be deduced from the chemical equation of the reaction.

First-order reactionsSee also Order of reaction.

A first-order reaction depends on the concentration of only one reactant (a unimolecular reaction). Other reactantscan be present, but each will be zero-order. The rate law for an elementary reaction that is first order with respect to areactant A is

k is the first order rate constant, which has units of 1/s.The integrated first-order rate law is

A plot of vs. time t gives a straight line with a slope of .

The half-life of a first-order reaction is independent of the starting concentration and is given by .

Examples of reactions that are first-order with respect to the reactant:

••

Further Properties of First-Order Reaction KineticsThe integrated first-order rate law

is usually written in the form of the exponential decay equation

A different (but equivalent) way of considering first order kinetics is as follows: The exponential decay equation canbe rewritten as:

where corresponds to a specific time period and is an integer corresponding to the number of time periods.At the end of each time period, the fraction of the reactant population remaining relative to the amount present at thestart of the time period, , will be:

Such that after time periods, the fraction of the original reactant population will be:

where: corresponds to the fraction of the reactant population that will break down in each time period. This equation indicates that the fraction of the total amount of reactant population that will break down in each time

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Rate equation 18

period is independent of the initial amount present. When the chosen time period corresponds to , the fraction of the

population that will break down in each time period will be exactly ½ the amount present at the start of the timeperiod (i.e. the time period corresponds to the half-life of the first-order reaction).The average rate of the reaction for the nth time period is given by:

Therefore, the amount remaining at the end of each time period will be related to the average rate of that time periodand the reactant population at the start of the time period by:

Since the fraction of the reactant population that will break down in each time period can be expressed as:

The amount of reactant that will break down in each time period can be related to the average rate over that timeperiod by:

Such that the amount that remains at the end of each time period will be related to the amount present at the start ofthe time period according to:

This equation is a recursion allowing for the calculation of the amount present after any number of time periods,without need of the rate constant, provided that the average rate for each time period is known. [3]

Second-order reactionsA second-order reaction depends on the concentrations of one second-order reactant, or two first-order reactants.For a second order reaction, its reaction rate is given by:

or or

In several popular kinetics books, the definition of the rate law for second-order reactions is .

Conflating the 2 inside the constant for the first, derivative, form will only make it required in the second, integratedform (presented below). The option of keeping the 2 out of the constant in the derivative form is considered morecorrect, as it is almost always used in peer-reviewed literature, tables of rate constants, and simulation software.[4]

The integrated second-order rate laws are respectively

or

[A]0 and [B]0 must be different to obtain that integrated equation.

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Rate equation 19

The half-life equation for a second-order reaction dependent on one second-order reactant is . For a

second-order reaction half-lives progressively double.Another way to present the above rate laws is to take the log of both sides: Examples of a Second-order reaction

Pseudo-first-orderMeasuring a second-order reaction rate with reactants A and B can be problematic: The concentrations of the tworeactants must be followed simultaneously, which is more difficult; or measure one of them and calculate the otheras a difference, which is less precise. A common solution for that problem is the pseudo-first-order approximation

If either [A] or [B] remains constant as the reaction proceeds, then the reaction can be consideredpseudo-first-order because, in fact, it depends on the concentration of only one reactant. If, for example, [B]remains constant, then:

where (k' or kobs with units s−1) and an expression is obtained identical to the first order expressionabove.One way to obtain a pseudo-first-order reaction is to use a large excess of one of the reactants ([B]>>[A] would workfor the previous example) so that, as the reaction progresses, only a small amount of the reactant is consumed, and itsconcentration can be considered to stay constant. By collecting for many reactions with different (but excess)concentrations of [B], a plot of versus [B] gives (the regular second order rate constant) as the slope.Example: The hydrolysis of esters by dilute mineral acids follows pseudo-first-order kinetics where theconcentration of water is present in large excess.

CH3COOCH3 + H2O → CH3COOH + CH3OH

Summary for reaction orders 0, 1, 2, and nElementary reaction steps with order 3 (called ternary reactions) are rare and unlikely to occur. However, overallreactions composed of several elementary steps can, of course, be of any (including non-integer) order.

Zero-Order First-Order Second-Order nth-Order

Rate Law [4]

Integrated Rate Law[4] [Except first

order]Units of RateConstant (k)

Linear Plot todetermine k [Except first order]

Half-life[4] [Except first order]

Where M stands for concentration in molarity (mol · L−1), t for time, and k for the reaction rate constant. Thehalf-life of a first-order reaction is often expressed as t1/2 = 0.693/k (as ln2 = 0.693).

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Rate equation 20

Equilibrium reactions or opposed reactionsA pair of forward and reverse reactions may define an equilibrium process. For example, A and B react into X and Yand vice versa (s, t, u, and v are the stoichiometric coefficients):

The reaction rate expression for the above reactions (assuming each one is elementary) can be expressed as:

where: k1 is the rate coefficient for the reaction that consumes A and B; k2 is the rate coefficient for the backwardsreaction, which consumes X and Y and produces A and B.The constants k1 and k2 are related to the equilibrium coefficient for the reaction (K) by the following relationship(set r=0 in balance):

Concentration of A (A0 = 0.25 mole/l) and B versus time reaching equilibrium kf =2 min-1 and kr = 1 min-1

Simple Example

In a simple equilibrium between twospecies:

Where the reactions starts with an initial concentration of A, , with an initial concentration of 0 for B at timet=0.Then the constant K at equilibrium is expressed as:

Where and are the concentrations of A and B at equilibrium, respectively.

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Rate equation 21

The concentration of A at time t, , is related to the concentration of B at time t, , by the equilibriumreaction equation:

Note that the term is not present because, in this simple example, the initial concentration of B is 0.This applies even when time t is at infinity; i.e., equilibrium has been reached:

then it follows, by the definition of K, that

and, therefore,

These equations allow us to uncouple the system of differential equations, and allow us to solve for the concentrationof A alone.The reaction equation, given previously as:

The derivative is negative because this is the rate of the reaction going from A to B, and therefore the concentrationof A is decreasing. To simplify annotation, let x be , the concentration of A at time t. Let be theconcentration of A at equilibrium. Then:

Since:

The reaction rate becomes:

which results in:

A plot of the negative natural logarithm of the concentration of A in time minus the concentration at equilibriumversus time t gives a straight line with slope kf + kb. By measurement of Ae and Be the values of K and the tworeaction rate constants will be known.[5]

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Rate equation 22

Generalization of Simple ExampleIf the concentration at the time t = 0 is different from above, the simplifications above are invalid, and a system ofdifferential equations must be solved. However, this system can also be solved exactly to yield the followinggeneralized expressions:

When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformationalanalysis of molecules, other methods are required for the determination of rate constants for instance by completelineshape analysis in NMR spectroscopy.

Consecutive reactionsIf the rate constants for the following reaction are and ; , then the rate equation is:

For reactant A:

For reactant B:

For product C:

With the individual concentrations scaled by the total population of reactants to become probabilities, linear systemsof differential equations such as these can be formulated as a master equation. The differential equations can besolved analytically and the integrated rate equations are

The steady state approximation leads to very similar results in an easier way.

Parallel or competitive reactionsWhen a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said totake place.• Two first order reactions:

and , with constants and and rate equations ,

and

The integrated rate equations are then ; and

.

One important relationship in this case is

Page 25: Enzyme Kinetics

Rate equation 23

• One first order and one second order reaction:[6]

This can be the case when studying a bimolecular reaction and a simultaneous hydrolysis (which can be treated aspseudo order one) takes place: the hydrolysis complicates the study of the reaction kinetics, because some reactant isbeing "spent" in a parallel reaction. For example A reacts with R to give our product C, but meanwhile the hydrolysisreaction takes away an amount of A to give B, a byproduct: and . The rate

equations are: and . Where is the pseudo first order

constant.

The integrated rate equation for the main product [C] is , which is

equivalent to . Concentration of B is related to that of C through

The integrated equations were analytically obtained but during the process it was assumed thattherefeore, previous equation for [C] can only be used for low concentrations of [C]

compared to [A]0

References[1] IUPAC Gold Book definition of rate law (http:/ / goldbook. iupac. org/ R05141. html). See also: According to IUPAC Compendium of

Chemical Terminology.[2] Kenneth A. Connors Chemical Kinetics, the study of reaction rates in solution, 1991, VCH Publishers. This book contains most of the rate

equations in this article and their derivation.[3] Walsh R, Martin E, Darvesh S. A method to describe enzyme-catalyzed reactions by combining steady state and time course enzyme kinetic

parameters... Biochim Biophys Acta. 2010 Jan;1800:1-5[4] NDRL Radiation Chemistry Data Center (http:/ / www. rcdc. nd. edu/ compilations/ Ali/ Ali. htm). See also: Christos Capellos and Bennon

H. Bielski "Kinetic systems: mathematical description of chemical kinetics in solution" 1972, Wiley-Interscience (New York) (http:/ / www.getcited. org/ puba/ 101600761).

[5] For a worked out example see: Determination of the Rotational Barrier for Kinetically Stable Conformational Isomers via NMR and 2D TLCAn Introductory Organic Chemistry Experiment Gregory T. Rushton, William G. Burns, Judi M. Lavin, Yong S. Chong, Perry Pellechia, andKen D. Shimizu J. Chem. Educ. 2007, 84, 1499. Abstract (http:/ / jchemed. chem. wisc. edu/ Journal/ Issues/ 2007/ Sep/ abs1499. html)

[6] José A. Manso et al."A Kinetic Approach to the Alkylating Potential of Carcinogenic Lactones" Chem. Res. Toxicol. 2005, 18, (7) 1161-1166

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MichaelisMenten kinetics 24

Michaelis–Menten kinetics

An example curve with parameters Vmax=3.4 andKm=0.4.

In biochemistry, Michaelis–Menten kinetics is one of the simplestand best-known models of enzyme kinetics. It is named after Germanbiochemist Leonor Michaelis and Canadian physician Maud Menten.The model takes the form of an equation describing the rate ofenzymatic reactions, by relating reaction rate to , theconcentration of a substrate S. Its formula is given by

Here, represents the maximum rate achieved by the system, at maximum (saturating) substrate concentrations.The Michaelis constant is the substrate concentration at which the reaction rate is half of . Biochemicalreactions involving a single substrate are often assumed to follow Michaelis–Menten kinetics, without regard to themodel's underlying assumptions.

Model

Change in concentrations over time for enzymeE, substrate S, complex ES and product P

In 1903, French physical chemist Victor Henri found that enzymereactions were initiated by a bond between the enzyme and thesubstrate.[1] His work was taken up by German biochemist LeonorMichaelis and Canadian physician Maud Menten who investigated thekinetics of an enzymatic reaction mechanism, invertase, that catalyzesthe hydrolysis of sucrose into glucose and fructose.[2] In 1913, theyproposed a mathematical model of the reaction.[3] It involves anenzyme E binding to a substrate S to form a complex ES, which in turnis converted into a product P and the enzyme. This may be representedschematically as

where , and denote the rate constants,[4] and the double arrows between S and ES represent the fact thatenzyme-substrate binding is a reversible process.Under certain assumptions – such as the enzyme concentration being much less than the substrate concentration –the rate of product formation is given by

The reaction rate increases with increasing substrate concentration ], asymptotically approaching its maximum rate , attained when all enzyme is bound to substrate. It also follows that , where is the enzyme concentration. , the turnover number, is maximum number of substrate molecules converted to

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MichaelisMenten kinetics 25

product per enzyme molecule per second.The Michaelis constant is the substrate concentration at which the reaction rate is at half-maximum, and is ameasure of the substrate's affinity for the enzyme. A small indicates high affinity, meaning that the rate willapproach more quickly.[5]

The model is used in a variety of biochemical situations other than enzyme-substrate interaction, includingantigen-antibody binding, DNA-DNA hybridization and protein-protein interaction.[5] [6] It can be used tocharacterise a generic biochemical reaction, in the same way that the Langmuir equation can be used to modelgeneric adsorption of biomolecular species.[6]

ApplicationsParameter values vary wildly between enzymes:[7]

Enzyme (M) (1/s) (1/M.s)

Chymotrypsin 1.5 × 10-2 0.14 9.3

Pepsin 3.0 × 10-4 0.50 1.7 × 103

Tyrosyl-tRNA synthetase 9.0 × 10-4 7.6 8.4 × 103

Ribonuclease 7.9 × 10-3 7.9 × 102 1.0 × 105

Carbonic anhydrase 2.6 × 10-2 4.0 × 105 1.5 × 107

Fumarase 5.0 × 10-6 8.0 × 102 1.6 × 108

The constant is a measure of how efficiently an enzyme converts a substrate into product. It has atheoretical upper limit of 108 – 1010 /M.s; enzymes working close to this, such as fumarase, are termedsuperefficient.[8]

Michaelis-Menten kinetics have also been applied to a variety of spheres outside of biochemical reactions,[4]

including alveolar clearance of dusts,[9] the richness of species pools,[10] clearance of blood alcohol,[11] thephotosynthesis-irradiance relationship and bacterial phage infection.[12]

DerivationApplying the law of mass action, which states that the rate of a reaction is proportional to the product of theconcentrations of the reactants, gives a system of four non-linear ordinary differential equations that define the rateof change of reactants with time :[13]

In this mechanism, the enzyme E is a catalyst, which only facilitates the reaction, so its total concentration, free pluscombined, is a constant. This conservation law can also be obtained by adding the second andthird equations above.[13] [14]

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Equilibrium approximationIn their original analysis, Michaelis and Menten assumed that the substrate is in instantaneous chemical equilibriumwith the complex, and thus .[3] [14] Combining this relationship with the enzymeconservation law, the concentration of complex is[14]

where is the dissociation constant for the enzyme-substrate complex. Hence the velocity of thereaction – the rate at which P is formed – is[14]

where is the maximum reaction velocity.

Quasi-steady-state approximationAn alternative analysis of the system was undertaken by British botanist G. E. Briggs and British geneticist J. B. S.Haldane in 1925.[15] They assumed that the concentration of the intermediate complex does not change on thetime-scale of product formation – known as the quasi-steady-state assumption or pseudo-steady-state-hypothesis.Mathematically, this assumption means . Combining this relationship with theenzyme conservation law, the concentration of complex is[14]

where

is known as the Michaelis constant. Hence the velocity of the reaction is[14]

Assumptions and limitationsThe first step in the derivation applies the law of mass action, which is reliant on free diffusion. However, in theenvironment of a living cell where there is a high concentration of protein, the cytoplasm often behaves more like agel than a liquid, limiting molecular movements and altering reaction rates.[16] Whilst the law of mass action can bevalid in heterogeneous environments,[17] it is more appropriate to model the cytoplasm as a fractal, in order tocapture its limited-mobility kinetics.[18]

The resulting reaction rates predicted by the two approaches are similar, with the only difference being that theequilibrium approximation defines the constant as , whilst the quasi-steady-state approximation uses .However, each approach is founded upon a different assumption. The Michaelis-Menten equilibrium analysis isvalid if the substrate reaches equilibrium on a much faster time-scale than the product is formed or, more precisely,that [14]

By contrast, the Briggs-Haldane quasi-steady-state analysis is valid if [13] [19]

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MichaelisMenten kinetics 27

Thus it holds if the enzyme concentration is much less than the substrate concentration. Even if this is not satisfied,the approximation is valid if is large.In both the Michaelis-Menten and Briggs-Haldane analyses, the quality of the approximation improves as decreases. However, in model building, Michaelis-Menten kinetics are often invoked without regard to theunderlying assumptions.[14]

Determination of constantsThe typical method for determining the constants and involves running a series of enzyme assays atvarying substrate concentrations , and measuring the initial reaction rate . 'Initial' here is taken to mean thatthe reaction rate is measured after a relatively short time period, during which it is assumed that theenzyme-substrate complex has formed, but that the substrate concentration held approximately constant, and so theequilibrium or quasi-steady-state approximation remain valid.[19] By plotting reaction rate against concentration, andusing nonlinear regression of the Michaelis-Menten equation, the parameters may be obtained.[20]

Before computing facilities to perform nonlinear regression became available, graphical methods involvinglinearisation of the equation were used. A number of these were proposed, including the Eadie–Hofstee diagram,Hanes–Woolf plot and Lineweaver–Burk plot; of these, the Hanes-Woolf plot is the most accurate.[20] However,while useful for visualization, all three methods distort the error structure of the data and are inferior to nonlinearregression.[21] Nonetheless, their use can still be found in modern literature.[22]

References[1] Henri, Victor (1903). Lois Générales de l’Action des Diastases. Paris: Hermann. Google books (US only) (http:/ / books. google. com/

books?id=Wcs9AAAAYAAJ)[2] "Victor Henri" (http:/ / www. whonamedit. com/ doctor. cfm/ 2881. html). Whonamedit?. . Retrieved 24 May 2011.[3] Menten, L.; Michaelis, M.I. (1913), "Die Kinetik der Invertinwirkung", Biochem Z 49: 333–369 ( recent translation (http:/ / pubs. acs. org/

doi/ suppl/ 10. 1021/ bi201284u), and an older partial translation (http:/ / web. lemoyne. edu/ ~giunta/ menten. html))[4] Chen, W.W.; Neipel, M.; Sorger, P.K. (2010). "Classic and contemporary approaches to modeling biochemical reactions". Genes Dev 24

(17): 1861–1875. doi:10.1101/gad.1945410. PMC 2932968. PMID 20810646.[5] Lehninger, A.L.; Nelson, D.L.; Cox, M.M. (2005). Lehninger principles of biochemistry. New York: W.H. Freeman.

ISBN 978-0-7167-4339-2.[6] Chakraborty, S. (23 Dec 2009). Microfluidics and Microfabrication (1 ed.). Springer. ISBN 978-1441915429.[7] Mathews, C.K.; van Holde, K.E.; Ahern, K.G. (10 Dec 1999). Biochemistry (http:/ / www. pearsonhighered. com/ mathews/ ) (3 ed.). Prentice

Hall. ISBN 978-0805330663. .[8] Stroppolo, M.E.; Falconi, M.; Caccuri, A.M.; Desideri, A. (Sep 2001). "Superefficient enzymes". Cell Mol Life Sci 58 (10): 1451–60.

doi:10.1007/PL00000788. PMID 11693526.[9] Yu, R.C.; Rappaport, S.M. (1997). "A lung retention model based on Michaelis-Menten-like kinetics". Environ Health Perspect 105 (5):

496–503. doi:10.1289/ehp.97105496. PMC 1469867. PMID 9222134.[10] Keating, K.A.; Quinn, J.F. (1998). "Estimating species richness: the Michaelis-Menten model revisited". Oikos 81 (2): 411–416.

doi:10.2307/3547060. JSTOR 3547060.[11] Jones, A.W. (2010). "Evidence-based survey of the elimination rates of ethanol from blood with applications in forensic casework".

Forensic Sci Int 200 (1–3): 1–20. doi:10.1016/j.forsciint.2010.02.021. PMID 20304569.[12] Abedon, S.T. (2009). "Kinetics of phage-mediated biocontrol of bacteria". Foodborne Pathog Dis 6 (7): 807–15.

doi:10.1089/fpd.2008.0242. PMID 19459758.[13] Murray, J.D. (2002). Mathematical Biology: I. An Introduction (3 ed.). Springer. ISBN 978-0387952239.[14] Keener, J.; Sneyd, J. (2008). Mathematical Physiology: I: Cellular Physiology (2 ed.). Springer. ISBN 978-0387758466.[15] Briggs, G.E.; Haldane, J.B.S. (1925). "A note on the kinematics of enzyme action". Biochem J 19 (2): 338–339. PMC 1259181.

PMID 16743508.[16] Zhou, H.X.; Rivas, G.; Minton, A.P. (2008). "Macromolecular crowding and confinement: biochemical, biophysical, and potential

physiological consequences". Annu Rev Biophys 37: 375–97. doi:10.1146/annurev.biophys.37.032807.125817. PMC 2826134.PMID 18573087.

[17] Grima, R.; Schnell, S. (Oct 2006). "A systematic investigation of the rate laws valid in intracellular environments". Biophys Chem 124 (1):1–10. doi:10.1016/j.bpc.2006.04.019. PMID 16781049.

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MichaelisMenten kinetics 28

[18] Schnell, S.; Turner, T.E. (2004). "Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws".Prog Biophys Mol Biol 85 (2–3): 235–60. doi:10.1016/j.pbiomolbio.2004.01.012. PMID 15142746.

[19] Segel, L.A.; Slemrod, M. (1989). "The quasi-steady-state assumption: A case study in perturbation". Thermochim Acta 31 (3): 446–477.doi:10.1137/1031091.

[20] Leskovac, V. (2003). Comprehensive enzyme kinetics. New York: Kluwer Academic/Plenum Pub.. ISBN 978-0-306-46712-7.[21] Greco, W.R.; Hakala, M.T. (1979). "Evaluation of methods for estimating the dissociation constant of tight binding enzyme inhibitors,". J

Biol Chem 254 (23): 12104–12109. PMID 500698.[22] Hayakawa, K.; Guo, L.; Terentyeva, E.A.; Li, X.K.; Kimura, H.; Hirano, M.; Yoshikawa, K.; Nagamine, T. et al. (2006). "Determination of

specific activities and kinetic constants of biotinidase and lipoamidase in LEW rat and Lactobacillus casei (Shirota)". J Chromatogr B AnalytTechnol Biomed Life Sci 844 (2): 240–50. doi:10.1016/j.jchromb.2006.07.006. PMID 16876490.

Lineweaver–Burk plotIn biochemistry, theLineweaver–Burk plot (or doublereciprocal plot) is a graphicalrepresentation of the Lineweaver–Burkequation of enzyme kinetics, describedby Hans Lineweaver and Dean Burk in1934.[1]

Derivation

The plot provides a useful graphicalmethod for analysis of theMichaelis–Menten equation:

Taking the reciprocal gives

where V is the reaction velocity (the reaction rate), Km is the Michaelis–Menten constant, Vmax is the maximumreaction velocity, and [S] is the substrate concentration.

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LineweaverBurk plot 29

Use

Lineweaver–Burk plots of different typesof reversible enzyme inhibitors. The arrow

shows the effect of increasingconcentrations of inhibitor.

The Lineweaver–Burk plot was widely used to determine important terms inenzyme kinetics, such as Km and Vmax, before the wide availability ofpowerful computers and non-linear regression software. The y-intercept ofsuch a graph is equivalent to the inverse of Vmax; the x-intercept of thegraph represents −1/Km. It also gives a quick, visual impression of thedifferent forms of enzyme inhibition.

The double reciprocal plot distorts the error structure of the data, and it istherefore unreliable for the determination of enzyme kinetic parameters.Although it is still used for representation of kinetic data,[2] non-linearregression or alternative linear forms of the Michaelis–Menten equationsuch as the Hanes-Woolf plot or Eadie–Hofstee plot are generally used forthe calculation of parameters.[3]

When used for determining the type of enzyme inhibition, theLineweaver–Burk plot can distinguish competitive, non-competitive anduncompetitive inhibitors. Competitive inhibitors have the same y-interceptas uninhibited enzyme (since Vmax is unaffected by competitive inhibitorsthe inverse of Vmax also doesn't change) but there are different slopes andx-intercepts between the two data sets. Non-competitive inhibition producesplots with the same x-intercept as uninhibited enzyme (Km is unaffected) butdifferent slopes and y-intercepts. Uncompetitive inhibition causes differentintercepts on both the y- and x-axes but the same slope.

Problems with the method

The Lineweaver–Burk plot is classically used in older texts, but is prone toerror, as the y-axis takes the reciprocal of the rate of reaction – in turnincreasing any small errors in measurement. Also, most points on the plot are found far to the right of the y-axis (dueto limiting solubility not allowing for large values of [S] and hence no small values for 1/[S]), calling for a largeextrapolation back to obtain x- and y-intercepts.

References[1] Lineweaver, H and Burk, D. (1934). "The Determination of Enzyme Dissociation Constants". Journal of the American Chemical Society 56

(3): 658–666. doi:10.1021/ja01318a036.[2] Hayakawa, K.; Guo, L.; Terentyeva, E.A.; Li, X.K.; Kimura, H.; Hirano, M.; Yoshikawa, K.; Nagamine, T. et al. (2006). "Determination of

specific activities and kinetic constants of biotinidase and lipoamidase in LEW rat and Lactobacillus casei (Shirota)". J Chromatogr B AnalytTechnol Biomed Life Sci 844 (2): 240–50. doi:10.1016/j.jchromb.2006.07.006. PMID 16876490.

[3] Greco, W. R. and Hakala, M. T., (1979). "Evaluation of methods for estimating the dissociation constant of tight binding enzyme inhibitors,"(http:/ / www. jbc. org/ cgi/ reprint/ 254/ 23/ 12104. pdf) (PDF). J. Biol. Chem. 254 (23): 12104–12109. PMID 500698. .

External links• NIH guide (http:/ / www. ncgc. nih. gov/ guidance/ section4. html#inhibition-constant), enzyme assay

development and analysis

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Article Sources and ContributorsEnzyme kinetics  Source: http://en.wikipedia.org/w/index.php?oldid=462305976  Contributors: 08cflin, A2a2a2, Adenosine, Adrian J. Hunter, Alan Liefting, Alansohn, Alexandria,AndrewWTaylor, Angmar09, Anylai, Arcadian, BanjoCam, Bbatsell, BecR, Blanchardb, Bobo192, BokicaK, Brighterorange, Brim, Britzingen, Cacycle, CarinaT, Carstensen, Chaser, Chymo72,Clemwang, ClockworkSoul, Cmprince, DRHagen, Dana boomer, Dasdasdase, Davo22, Diberri, Dmol, Domitori, Dr Zak, ESkog, Economo, Eskeptic, Espresso Addict, Evenap, Flyguy649,Forluvoft, Fourthhour, Gene Nygaard, Giftlite, Graft, Gökhan, Hannes Röst, Hede2000, I need a game, Igodard, Itub, Jacobandrew2012, Java13690, Jeff Dahl, JoNo67, Joshwa-hayswa,Karam.Anthony.K, Kimchi.sg, Kmg90, KnightRider, Knights who say ni, Koavf, Kozuch, Kyoko, Laurinavicius, Legalvoice123, Lightmouse, Luna Santin, Manysplintered, Mastercampbell,Mauegd, Michael A. Wilkins, Michael Hardy, Mion, Moonsword, MrOllie, Nigholith, NuclearWarfare, Oda Mari, OpenToppedBus, Outriggr, Poccil, Pro bug catcher, Qef, R'n'B, RDBrown,Rasikraj, Raul654, Rich Farmbrough, Richard001, Rjwilmsi, RobertG, RockMFR, Rschen7754, Ruute, Samsara, SandyGeorgia, Saravask, SebastianHelm, Shauun, Shyamal, Slashme, SnowFire,Sodin, Some standardized rigour, Sonjaaa, Spidrak, Spoladore, Stefano Garibaldi, Stillnotelf, Storm Rider, Suruena, Sxenko, Tbhotch, The Parsnip!, The Thing That Should Not Be, TheTweaker,TheoThompson, TimVickers, ToNToNi, Todd40324, Tom Doniphon, Tommy2010, Ukexpat, V8rik, Wabaman44, WillowW, Wknight94, Wnt, WolfmanSF, Xcomradex, Xiahou, Xijkix, Yaseral-Nabriss, 168 anonymous edits

Rate equation  Source: http://en.wikipedia.org/w/index.php?oldid=459042841  Contributors: A.Ou, Aceleo, Anetode, Arthena, Bob, Bomac, Canuckian89, Choij, Chris the speller,Connor.mcnulty, D. F. Schmidt, D6, Davitz, Discospinster, DoubleBlue, DougsTech, Dr.enh, Dr.sapangupta, Drphilharmonic, Ectomaniac, Eskeptic, Excirial, Fintelia, Fireswordfight,Futurechemist1, Gashead 12, Glanhawr, Goldenmug8, Halls12, Hamsterlopithecus, HansHermans, Hugo-cs, IchimaruSan, Ideal gas equation, Itub, Ixfd64, J.nuttall, JForget, Jackelfive, JohnBaez, John of Reading, JoshuaTree, Jrockley, Julia W, Knights who say ni, KnowledgeOfSelf, Lukov880, MagnInd, Mandarax, Markozeta, Maury Markowitz, Mbc362, Melaen, Michael Hardy,Michał Sobkowski, Misantropo, Mn-imhotep, MorganGreen, NHRHS2010, NHSavage, NewEnglandYankee, Nihiltres, Opestovsky, Pdch, QuantumEngineer, Raptur, Rifleman 82, Ronhjones,Ryulong, Saippuakauppias, Saittam, Shalom Yechiel, Sreekanthsreebhavan, Stan J Klimas, StradivariusTV, Strait, Svick, The Ronin, Thisismikesother, Tony1, V8rik, Vettaikannan,Voyagerfan5761, Weihao.chiu, Werson, WikHead, William Avery, ZPOT, 196 anonymous edits

Michaelis–Menten kinetics  Source: http://en.wikipedia.org/w/index.php?oldid=465037978  Contributors: 84user, Aa77zz, Agor153, Aliekens, Andkennard, Anrade, Arcadian, Athenray, Avihu,Axl, Bensaccount, Boleslaw, Cacycle, Calvero JP, Charles Matthews, Chris the speller, Christopherlin, Clayt85, Cschulthess, Cubic Hour, CuriousOliver, Cutler, CzarB, David Shear, Delta G,Denisarona, Deskana, Deviator13, Eumolpo, Expresser, Eykanal, Gegnome, Gene Nygaard, Genie05, Giftlite, Hannes Röst, Huhnra, Iamunknown, Ian Glenn, Ike9898, Isilanes, Itub, JaGa, Jao,Jrf, Kaarel, KnightRider, Lenov, Lexor, Lifeonahilltop, Lord Kelvin, Lostella, Lvzon, Maphyche, Martina Steiner, Materialscientist, Matro, Maurreen, Michael Hardy, Mion, NawlinWiki,Northfox, NotoriousOCG, Popnose, Pro bug catcher, Richmc, Rjwilmsi, SHL-at-Sv, Sasata, Shell Kinney, SjoerdC, Slashme, Smenden, Squidonius, Stepa, Stevestoker, Stillnotelf, Thorwald,TimVickers, Tiphaine800, Twisted86, U+003F, Ultraviolet scissor flame, V8rik, Vramasub, Vuo, Waveguide2, Wenlanzsw, Wfgiuliano, Wik, Wsloand, Xris0, 170 anonymous edits

Lineweaver–Burk plot  Source: http://en.wikipedia.org/w/index.php?oldid=462768499  Contributors: Arcadian, Banus, Barlovento, Bensaccount, Cacycle, CarinaT, CiaranAnthony,Clicketyclack, Cuahl, Devon Fyson, Diberri, Docfaust, Flewis, Giftlite, Hooperbloob, Huhnra, Isopropyl, Itub, J.delanoy, Jujutacular, KSchutte, KnightRider, Meconium, Michael Hardy,Mr0t1633, Nemesis of Reason, Newty23125, Nightstallion, Obradovic Goran, Pro bug catcher, Rjwilmsi, Slashme, Stillnotelf, Tagishsimon, The Anome, TimVickers, U+003F, Vermiculus,WillowW, 37 anonymous edits

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Image Sources, Licenses and Contributors 31

Image Sources, Licenses and ContributorsImage:EcDHFR raytraced.png  Source: http://en.wikipedia.org/w/index.php?title=File:EcDHFR_raytraced.png  License: Public Domain  Contributors: TimVickersImage:KinEnzymo(en).svg  Source: http://en.wikipedia.org/w/index.php?title=File:KinEnzymo(en).svg  License: Public Domain  Contributors: TimVickers, YassineMrabet, МихајлоАнђелковићImage:Enzyme progress curve.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Enzyme_progress_curve.svg  License: Copyrighted free use  Contributors: en:User:Poccil, Based onpublic domain JPG by TimVickers.Image:Michaelis-Menten saturation curve of an enzyme reaction.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Michaelis-Menten_saturation_curve_of_an_enzyme_reaction.svg License: Public Domain  Contributors: fullofstarsImage:Mechanism plus rates.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Mechanism_plus_rates.svg  License: Public Domain  Contributors: Atropos235, TimVickersImage:Lineweaver-Burke plot.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Lineweaver-Burke_plot.svg  License: GNU Free Documentation License  Contributors: Originaluploader was Pro bug catcher at en.wikipediaImage:Random order ternary mechanism.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Random_order_ternary_mechanism.svg  License: Public Domain  Contributors:Fvasconcellos, TimVickersImage:Ping pong.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Ping_pong.svg  License: Public Domain  Contributors: TimVickersImage:Allosteric v by S curve.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Allosteric_v_by_S_curve.svg  License: Copyrighted free use  Contributors: by TimVickers.Image:Burst phase.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Burst_phase.svg  License: Public Domain  Contributors: Original bitmap version by TimVickers, SVG versiontraced over it in Inkscape by Qef.Image:Reversible inhibition.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Reversible_inhibition.svg  License: Public Domain  Contributors: User Poccil on en.wikipediaImage:Activation2 updated.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Activation2_updated.svg  License: GNU Free Documentation License  Contributors: Originally uploadedby Jerry Crimson Mann, vectorized by Tutmosis, corrected by FvasconcellosImage:ChemicalEquilibrium.svg  Source: http://en.wikipedia.org/w/index.php?title=File:ChemicalEquilibrium.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:FinteliaImage:Michaelis-Menten saturation curve of an enzyme reaction LARGE.svg  Source:http://en.wikipedia.org/w/index.php?title=File:Michaelis-Menten_saturation_curve_of_an_enzyme_reaction_LARGE.svg  License: Creative Commons Zero  Contributors: U+003FImage:Michaelis Menten S P E ES.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Michaelis_Menten_S_P_E_ES.svg  License: Creative Commons Zero  Contributors: U+003FFile:Inhibition diagrams.png  Source: http://en.wikipedia.org/w/index.php?title=File:Inhibition_diagrams.png  License: Public Domain  Contributors: Brandon, Devon Fyson, Samsara,TimVickers

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