Enzyme kinetics

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ContentsArticlesEnzyme kinetics Rate equation MichaelisMenten kinetics LineweaverBurk plot 1 16 24 28

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Enzyme kinetics

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Enzyme kineticsEnzyme kinetics is the study of the chemical reactions that are catalysed by enzymes. In enzyme kinetics, the reaction rate is measured and the effects of varying the conditions of the reaction investigated. Studying an enzyme's kinetics in this way can reveal the catalytic mechanism of this enzyme, its role in metabolism, how its activity is controlled, and how a drug or an agonist might inhibit the enzyme. Enzymes are usually protein molecules that manipulate other molecules the enzymes' substrates. These target molecules bind to an enzyme's active site and are transformed into products through a series of steps known as the enzymatic mechanism. These mechanisms can be divided into single-substrate and multiple-substrate mechanisms. Kinetic studies on enzymes that only bind one substrate, such as triosephosphate isomerase, aim to measure the affinity with which the enzyme binds this substrate and the turnover rate. When enzymes bind multiple substrates, such as (right) and NADPH (left), bound in the active site. The protein is shown as a dihydrofolate reductase (shown right), enzyme ribbon diagram, with alpha helices in red, beta sheets in yellow and loops in [1] blue. Generated from 7DFR . 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 proteases, 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 of steps, there is typically one rate-determining step that determines the overall kinetics. This rate-determining step 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 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-based catalysts such as ribozymes and ribosomes are essential to many cellular functions, such as RNA splicing and translation. 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 mechanisms and kinetics can be analysed and classified by the same methods.Dihydrofolate reductase from E. coli with its two substrates, dihydrofolate

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General principlesThe reaction catalysed by an enzyme uses exactly the same reactants and produces exactly the same products as the uncatalysed reaction. Like other catalysts, enzymes do not alter the position of equilibrium between substrates and products.[2] 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 As larger amounts of substrate are added to a reaction, the available enzyme increases linearly with substrate binding sites become filled to the limit of . Beyond this limit the enzyme is concentration; the enzyme molecules are saturated with substrate and the reaction rate ceases to increase. 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 approaches the theoretical maximum; the enzyme active sites are almost all occupied and the reaction rate is determined by the intrinsic turnover rate of the enzyme. The substrate concentration midway between these two limiting cases is denoted by KM. The two most important kinetic properties of an enzyme are how quickly the enzyme becomes saturated with a particular 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 assaysEnzyme assays are laboratory procedures that measure the rate of enzyme reactions. Because 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 assays observe change in the absorbance of light between products and reactants; radiometric assays involve the incorporation or release of radioactivity 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 Progress curve for an enzyme reaction. The slope in the initial rate period is the initial rate of reaction v. The MichaelisMenten require the removal and counting of samples (i.e., they equation describes how this slope varies with the concentration of are discontinuous assays) they are usually extremely substrate. sensitive and can measure very low levels of enzyme activity.[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

Enzyme kinetics during an enzyme's reaction mechanism, or of fluorescent dyes added onto specific sites of the protein to report movements that occur during catalysis.[4] These studies are providing 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.[5] [6] 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 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 on initial rates of less than one second.[7] 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. This way of measuring enzyme reactions is called progress-curve analysis.[8] This approach is useful as an alternative to rapid kinetics when the initial rate is too fast to measure accurately.

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Single-substrate reactionsEnzymes with single-substrate mechanisms include isomerases such as triosephosphateisomerase or bisphosphoglycerate mutase, intramolecular lyases such as adenylate cyclase and the hammerhead ribozyme, a RNA lyase.[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, 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 pingpong mechanism, a type of mechanism that is discussed in the Multi-substrate reactions section below.

MichaelisMenten kineticsAs 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 [S]), the reaction rate (v) increases as [S] increases, as shown on the right. However, as [S] gets higher, the enzyme becomes saturated with substrate and the rate reaches Vmax, the enzyme's maximum rate.

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

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

The MichaelisMenten 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 enzymesubstrate complex ES. Although the enzymatic mechanism for the unimolecular reaction 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 kcat. If the reaction path proceeds over one or several intermediates, kcat 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 k2. The apparent unimolecular rate constant kcat is also called turnover number and denotes the maximum number of enzymatic reactions catalysed per second. The MichaelisMenten equation[10] describes how the (initial) reaction rate v0 depends on the position of the substrate-binding equilibrium and the rate constant k2. (MichaelisMenten equation) with the constants

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

The Michaelis constant KM is experimentally defined as the concentration at which the rate of the enzyme reaction is half Vmax, which can be verified by substituting [S] = Km into the MichaelisMenten equation and can also be seen graphically. If the rate-determining enzymatic step is slow compared to substrate dissociation ( ), the Michaelis 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 is formed, 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

Enzyme kinetics constant is a measure of catalytic efficiency. The most efficient enzymes reach a

5 in the range of 108 - 1010M1s1.

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); these enzymes have often been termed perfect enzymes.[11]

Direct use of the MichaelisMenten equation for time course kinetic analysisFurther information: Rate equation The observed velocities predicted by the MichaelisMenten equation can be used to directly model the time course disappearance of substrate and the production of product through incorporation of the MichaelisMenten 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 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.[12]

In 1997, Santiago Schnell and Claudio Mendoza derived a closed form solution for the time course kinetics analysis of the Michaelis-Menten mechanism.[13] The solution has the form: where W[] is the Lambert-W function.[14] [15]

Linear plots of the MichaelisMenten equationUsing an interactive MichaelisMenten kinetics tutorial at the University of Virginia,[] the effects on the behaviour of an enzyme of varying kinetic constants can be explored. The plot of v versus [S] above is not linear; although initially linear at low [S], it bends over to saturate at high [S]. Before the modern era of nonlinear curve-fitting on computers, this nonlinearity could make it difficult to estimate KM and Vmax accurately. LineweaverBurk or double-reciprocal plot of kinetic data, showing the significance of Therefore, several researchers the axis intercepts and gradient. developed linearisations of the MichaelisMenten equation, such as the LineweaverBurk plot, the EadieHofstee diagram and the HanesWoolf 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 methods.[16] The LineweaverBurk 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 MichaelisMenten equation. As shown on the right, this is a linear form of the MichaelisMenten equation and produces a straight line with the equation y = mx + c with a y-intercept

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

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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 an extrapolation of the experimental data taken at positive concentrations. More generally, the LineweaverBurk plot skews the importance of measurements taken at low substrate concentrations and, thus, can yield inaccurate estimates of Vmax and KM.[17] A more accurate linear plotting method is the Eadie-Hofstee plot. In this case, v is plotted against v/[S]. In the third common linear representation, the Hanes-Woolf plot, [S]/v is plotted against [S]. 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.[18]

Practical significance of kinetic constantsThe 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, KM and Vmax, 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 malate dehydrogenase 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 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 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.[19] [20]

MichaelisMenten 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 a second 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 the and .

previous step, we get again the original equation. Mathematically we have then

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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 B varied. Under these conditions, the enzyme behaves just like a single-substrate enzyme and a plot of v by [S] gives apparent KM and Vmax 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 pingpong.

Ternary-complex mechanismsIn 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 [S] curves (fixed A, varying B) from an enzyme with a ternary-complex mechanism are plotted in a LineweaverBurk plot, the set of lines produced will intersect.

Random-order ternary-complex mechanism for an enzyme reaction. The reaction path is shown as a line and enzyme intermediates containing substrates A and B or products P and Q are written below the line.

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

Pingpong mechanismsAs shown on the right, enzymes with a ping-pong mechanism can exist in two states, E and a chemically modified form of the enzyme E*; this modified enzyme Pingpong mechanism for an enzyme reaction. Intermediates contain substrates A is known as an intermediate. In such and B or products P and Q. 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 [S] curves (fixed A, varying B) from an enzyme with a pingpong mechanism are plotted in a LineweaverBurk plot, a set of parallel lines will be produced. This is called a secondary plot. Enzymes with pingpong mechanisms include some oxidoreductases such as thioredoxin peroxidase,[25] transferases such as acylneuraminate cytydilyltransferase[26] and serine proteases such as trypsin and chymotrypsin.[27] 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 residue on a peptide bond in a protein substrate. A short animation showing the mechanism of chymotrypsin is linked here.[]

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Non-MichaelisMenten kineticsSome enzymes produce a sigmoid v by [S] plot, which often indicates cooperative binding 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.[28] Here, the mechanism of cooperation is similar to that of 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

Saturation curve for an enzyme reaction showing sigmoid kinetics.

the enzyme for other substrate molecules. Allosteric enzymes include mammalian tyrosyl tRNA-synthetase, which shows negative cooperativity,[29] and bacterial 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 the concentrations of their substrates. Positive cooperativity makes enzymes much more sensitive to [S] and their activities can show large changes over a narrow range of substrate concentration. Conversely, negative cooperativity makes enzymes insensitive to small changes in [S]. The Hill equation (biochemistry)[32] is often used to describe the degree of cooperativity quantitatively in non-MichaelisMenten 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 positive cooperativity.

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Pre-steady-state kineticsIn 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 also referred to as Burst kinetics. Pre-steady-state kinetics is therefore concerned with the formation and consumption of enzymesubstrate intermediates (such as ES or E*) until their steady-state concentrations are reached. This approach was first applied to the hydrolysis reaction catalysed by Pre-steady state progress curve, showing the burst phase of an enzyme reaction. [33] chymotrypsin. Often, the detection of an intermediate is a vital piece of evidence in investigations of what mechanism an enzyme follows. For example, in the pingpong 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*.[34] In the case of chymotrypsin, this intermediate is formed by an attack on the substrate by the 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 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.[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 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 to a hydrogen atom 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, its stable isotope. The rate will change when the critical hydrogen is replaced, due to a primary kinetic isotope effect, which occurs because bonds to deuterium are harder to break than bonds to hydrogen.[36] It is also possible to measure similar effects with other isotope substitutions, such as 13C/12C and 18 16 O/ O, 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. For example, it is sometimes difficult to discern the origin of an oxygen 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.[38] The chemical mechanism can also be elucidated by examining the kinetics and isotope effects under different pH conditions,[39] by altering the metal ions or other bound cofactors,[40] by site-directed mutagenesis of conserved amino 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 activationEnzyme 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

Kinetic scheme for reversible enzyme inhibitors.

Traditionally reversible enzyme inhibitors have 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 enzymesubstrate complex ES, or to both, respectively. The particular type of an inhibitor can be discerned by studying the enzyme kinetics as a function of the inhibitor concentration. The three types of inhibition produce LineweaverBurke and EadieHofstee 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 of dissociation 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 MichaelisMenten equation,where the rate of reaction depends on the percent of the enzyme population interacting with substrate fraction of the enzyme population bound by substrate

fraction of the enzyme population bound by inhibitor

Enzyme kinetics

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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 affects 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 Vmax term.

or

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 Vmax 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".

While this terminology results in a simplified way of dealing with kinetic effects relating to the maximum velocity of the MichaelisMenten equation, it highlights potential problems with the term used to describe effects relating to the Km. The Km 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 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 usually saturable. Below saturation, they follow first order kinetics with respect to inhibitor.

Mechanisms of catalysisThe favoured model for the enzymesubstrate interaction is the induced fit model.[45] This model proposes that the initial interaction between enzyme and substrate is relatively weak, but that these weak interactions rapidly induce conformational changes 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.[46] Conformational changes can be measured using circular dichroism or dual polarisation interferometry. After binding takes place, one or more mechanisms of The energy variation as a function of reaction coordinate shows the stabilisation of catalysis lower the energy of the reaction's the transition state by an enzyme. transition state 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.[34] [47] 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 pingpong 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 Vmax but not Km might indicate that a residue in the active site needs to be in a particular ionisation state for catalysis to occur.

Footnotes. Link: Interactive MichaelisMenten kinetics tutorial (Java required) [48] . Link: dihydrofolate reductase mechanism (Gif) [49] . Link: DNA polymerase mechanism (Gif) [50] . Link: Chymotrypsin mechanism (Flash required) [51]

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Enzyme kinetics[5] Lu H (2004). "Single-molecule spectroscopy studies of conformational change dynamics in enzymatic reactions". Current pharmaceutical biotechnology 5 (3): 2619. doi:10.2174/1389201043376887. PMID15180547. [6] Schnell J, Dyson H, Wright P (2004). "Structure, dynamics, and catalytic function of dihydrofolate reductase". Annual review of biophysics and biomolecular structure 33: 11940. doi:10.1146/annurev.biophys.33.110502.133613. PMID15139807. [7] Gibson QH (1969). "Rapid mixing: Stopped flow". Methods Enzymol. Methods in Enzymology 16: 187228. doi:10.1016/S0076-6879(69)16009-7. ISBN9780121818739. [8] Duggleby RG (1995). "Analysis of enzyme progress curves by non-linear regression". Methods Enzymol. Methods in Enzymology 249: 6190. doi:10.1016/0076-6879(95)49031-0. ISBN9780121821500. PMID7791628. [9] Murray JB, Dunham CM, Scott WG (January 2002). 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"A normalized plot as a novel and time-saving tool in complex enzyme kinetic analysis" (http:/ / www. biochemj. org/ bj/ 358/ 0573/ bj3580573. htm). Biochem. J. 358 (Pt 3): 57383. PMC1222113. PMID11577687. . [19] Almaas E, Kovcs B, Vicsek T, Oltvai ZN, Barabsi AL (February 2004). "Global organization of metabolic fluxes in the bacterium Escherichia coli". Nature 427 (6977): 83943. doi:10.1038/nature02289. PMID14985762. [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. PMC193654. PMID12952533. [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): 64561. doi:10.1111/j.1432-1033.1994.tb18666.x. PMID8143720. [23] Stone SR, Morrison JF (July 1988). "Dihydrofolate reductase from Escherichia coli: the kinetic mechanism with NADPH and reduced acetylpyridine adenine dinucleotide phosphate as substrates". Biochemistry 27 (15): 54939. doi:10.1021/bi00415a016. PMID3052577. [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: 37197. doi:10.1016/S0079-6603(08)60257-3. ISBN9780125400473. PMID8016325. [25] Akerman SE, Mller 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): 7581. doi:10.1016/S0166-6851(03)00161-0. PMID12946843. . [26] Bravo IG, Barrallo S, Ferrero MA, Rodrguez-Aparicio LB, Martnez-Blanco H, Reglero A (September 2001). "Kinetic properties of the acylneuraminate cytidylyltransferase from Pasteurella haemolytica A2" (http:/ / www. biochemj. org/ bj/ 358/ 0585/ bj3580585. htm). Biochem. J. 358 (Pt 3): 58598. PMC1222114. PMID11577688. . [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: 33158. doi:10.1146/annurev.bi.46.070177.001555. PMID332063. . [28] Ricard J, Cornish-Bowden A (July 1987). "Co-operative and allosteric enzymes: 20 years on". Eur. J. Biochem. 166 (2): 25572. doi:10.1111/j.1432-1033.1987.tb13510.x. PMID3301336. [29] Ward WH, Fersht AR (July 1988). "Tyrosyl-tRNA synthetase acts as an asymmetric dimer in charging tRNA. A rationale for half-of-the-sites activity". Biochemistry 27 (15): 552530. doi:10.1021/bi00415a021. PMID3179266. [30] Helmstaedt K, Krappmann S, Braus GH (September 2001). "Allosteric Regulation of Catalytic Activity: Escherichia coli Aspartate Transcarbamoylase versus Yeast Chorismate Mutase" (http:/ / mmbr. asm. org/ cgi/ content/ full/ 65/ 3/ 404). Microbiol. Mol. Biol. Rev. 65 (3): 40421, table of contents. doi:10.1128/MMBR.65.3.404-421.2001. PMC99034. PMID11528003. .

13

Enzyme kinetics[31] Schirmer T, Evans PR (January 1990). "Structural basis of the allosteric behaviour of phosphofructokinase". Nature 343 (6254): 1405. doi:10.1038/343140a0. PMID2136935. [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): 28897. PMC1269615. PMID13140189. [34] Fersht, Alan (1999). Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding. San Francisco: W.H. Freeman. ISBN0-7167-3268-8. [35] Bender ML, Begu-Cantn ML, Blakeley RL, et al. (December 1966). "The determination of the concentration of hydrolytic enzyme solutions: alpha-chymotrypsin, trypsin, papain, elastase, subtilisin, and acetylcholinesterase". J. Am. Chem. Soc. 88 (24): 5890913. doi:10.1021/ja00976a034. PMID5980876. [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): 212. doi:10.1016/j.abb.2004.08.027. PMID15581561. . [37] Northrop D (1981). "The expression of isotope effects on enzyme-catalyzed reactions". Annu. Rev. Biochem. 50: 10331. doi:10.1146/annurev.bi.50.070181.000535. PMID7023356. [38] Baillie T, Rettenmeier A (1986). "Drug biotransformation: mechanistic studies with stable isotopes". Journal of clinical pharmacology 26 (6): 44851. PMID3734135. [39] Cleland WW (1982). "Use of isotope effects to elucidate enzyme mechanisms". CRC Crit. Rev. Biochem. 13 (4): 385428. doi:10.3109/10409238209108715. PMID6759038. [40] Christianson DW, Cox JD (1999). "Catalysis by metal-activated hydroxide in zinc and manganese metalloenzymes". Annu. Rev. Biochem. 68: 3357. doi:10.1146/annurev.biochem.68.1.33. PMID10872443. [41] Kraut D, Carroll K, Herschlag D (2003). "Challenges in enzyme mechanism and energetics". Annu. Rev. Biochem. 72: 51771. doi:10.1146/annurev.biochem.72.121801.161617. PMID12704087. [42] Waldrop GL (January 2009). "A qualitative approach to enzyme inhibition". Biochemistry and Molecular Biology Education 37 (1): 1115. doi:10.1002/bmb.20243. PMID21567682. [43] Leatherbarrow RJ (December 1990). "Using linear and non-linear regression to fit biochemical data". Trends Biochem. Sci. 15 (12): 4558. doi:10.1016/0968-0004(90)90295-M. PMID2077683. [44] Walsh R, Martin E, Darvesh S. A versatile equation to describe reversible enzyme inhibition and activation kinetics: modeling beta-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): 98104. doi:10.1073/pnas.44.2.98. PMC335371. PMID16590179. [46] Hammes G (2002). "Multiple conformational changes in enzyme catalysis". Biochemistry 41 (26): 82218. doi:10.1021/bi0260839. PMID12081470. [47] Sutcliffe M, Scrutton N (2002). "A new conceptual framework for enzyme catalysis. Hydrogen tunnelling coupled to enzyme dynamics in flavoprotein and quinoprotein enzymes" (http:/ / content. febsjournal. org/ cgi/ content/ full/ 269/ 13/ 3096). Eur. J. Biochem. 269 (13): 3096102. doi:10.1046/j.1432-1033.2002.03020.x. PMID12084049. . [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

14

Further readingIntroductory Cornish-Bowden, Athel (2004). Fundamentals of enzyme kinetics (3rd ed.). London: Portland Press. ISBN1-85578-158-1. Stevens, Lewis; Price, Nicholas C. (1999). Fundamentals of enzymology: the cell and molecular biology of catalytic proteins. Oxford [Oxfordshire]: Oxford University Press. ISBN0-19-850229-X. Bugg, Tim (2004). Introduction to Enzyme and Coenzyme Chemistry. Cambridge, MA: Blackwell Publishers. ISBN1-4051-1452-5. Advanced Segel, Irwin H. (1993). Enzyme kinetics: behavior and analysis of rapid equilibrium and steady state enzyme systems (New ed.). New York: Wiley. ISBN0-471-30309-7.

Enzyme kinetics Fersht, Alan (1999). Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding. San Francisco: W.H. Freeman. ISBN0-7167-3268-8. Santiago Schnell, Philip K. Maini (2004). "A century of enzyme kinetics: Reliability of the KM and vmax estimates" (http://www.informatics.indiana.edu/schnell/papers/ctb8_169.pdf). Comments on Theoretical Biology 8 (23): 16987. doi:10.1080/08948550302453. Walsh, Christopher (1979). Enzymatic reaction mechanisms. San Francisco: W. H. Freeman. ISBN0-7167-0070-0. Cleland, William Wallace; Cook, Paul (2007). Enzyme kinetics and mechanism. New York: Garland Science. ISBN0-8153-4140-7.

15

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

Rate equation

16

Rate equationThe rate law or rate equation for a chemical reaction is an equation that links the reaction rate with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction orders).[1] To determine the rate equation for a particular system one combines the reaction rate with a mass balance for the system.[2] For a generic reaction aA + bB C with no intermediate steps in its reaction mechanism (that is, an elementary 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 be determined experimentally. k is the rate coefficient or rate constant of the reaction. The value of this coefficient k depends on conditions such as temperature, ionic strength, surface area of the adsorbent or light irradiation. For elementary reactions, the rate equation can be derived from first principles using collision theory. Again, x and y are NOT always derived from the balanced equation. The rate equation of a reaction with a multi-step mechanism cannot, in general, be deduced from the stoichiometric coefficients of the overall reaction; it must be determined experimentally. The equation may involve fractional exponential 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 links concentrations 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 with respect 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 equation has been reduced to a pseudo-first-order rate equation. This makes the treatment to obtain an integrated rate equation much easier.

Zeroth-order reactionsA Zeroth-order reaction has a rate that is independent of the concentration of the reactant(s). Increasing the concentration of the reacting species will not speed up the rate of the reaction. Zeroth-order reactions are typically found when a material that is required for the reaction to proceed, such as a surface or a catalyst, is saturated by the reactants. 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, this zeroth-order reaction 1) occurs in a closed system, 2) there is no net build-up of intermediates, and 3) there are no other 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 the

initial concentration. A reaction is zeroth order if concentration data are plotted versus time and the result is a straight line. The slope of this 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-life involved in nuclear decay, which is a first-order reaction). For a zero-order reaction the half-life is given by

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 reactants can be present, but each will be zero-order. The rate law for an elementary reaction that is first order with respect to a reactant 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 can be rewritten as:

where

corresponds to a specific time period and , will be:

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 the start of the time period,

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

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 the amount present at the start of the time period (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 period and 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 time period by:

Such that the amount that remains at the end of each time period will be related to the amount present at the start of the 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, integrated form (presented below). The option of keeping the 2 out of the constant in the derivative form is considered more correct, 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.

Rate equation

19

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

. For a

Pseudo-first-orderMeasuring a second-order reaction rate with reactants A and B can be problematic: The concentrations of the two reactants must be followed simultaneously, which is more difficult; or measure one of them and calculate the other as 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 considered pseudo-first-order because, in fact, it depends on the concentration of only one reactant. If, for example, [B] remains constant, then: (k' or kobs with units s1) and an expression is obtained identical to the first order expression

where

above. One way to obtain a pseudo-first-order reaction is to use a large excess of one of the reactants ([B]>>[A] would work for the previous example) so that, as the reaction progresses, only a small amount of the reactant is consumed, and its concentration 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 the concentration 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, overall reactions composed of several elementary steps can, of course, be of any (including non-integer) order.Zero-Order Rate Law First-Order Second-Order [4] nth-Order

Integrated Rate Law

[4] order]

[Except first

Units of Rate Constant (k) Linear Plot to determine k Half-life [4] [Except first order]

[Except first order]

Where M stands for concentration in molarity (mol L1), t for time, and k for the reaction rate constant. The half-life of a first-order reaction is often expressed as t1/2 = 0.693/k (as ln2 = 0.693).

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 Y and 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 backwards reaction, 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):

Simple ExampleIn a simple equilibrium between two species:

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

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

, with an initial concentration of 0 for B at time

Where

and

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

Rate equation The concentration of A at time t, reaction equation: Note that the term is not present because, in this simple example, the initial concentration of B is 0. , is related to the concentration of B at time t, , by the equilibrium

21

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 concentration of 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 concentration of A is decreasing. To simplify annotation, let x be , the concentration of A at time t. Let be the concentration 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 equilibrium versus time t gives a straight line with slope kf + kb. By measurement of Ae and Be the values of K and the two reaction rate constants will be known.[5]

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 of differential equations must be solved. However, this system can also be solved exactly to yield the following generalized expressions:

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

Consecutive reactionsIf the rate constants for the following reaction are For reactant A: For reactant B: For product C: With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation. The differential equations can be solved analytically and the integrated rate equations are and ; , then the rate equation is:

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 to take place. Two first order reactions: and and The integrated rate equations are then . One important relationship in this case is ; and , with constants and and rate equations ,

Rate equation 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 as pseudo order one) takes place: the hydrolysis complicates the study of the reaction kinetics, because some reactant is being "spent" in a parallel reaction. For example A reacts with R to give our product C, but meanwhile the hydrolysis reaction takes away an amount of A to give B, a byproduct: and . The rate equations are: constant. The integrated rate equation for the main product [C] is equivalent to , which is and . Where is the pseudo first order

23

. Concentration of B is related to that of C through

The integrated equations were analytically obtained but during the process it was assumed that therefeore, 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 TLC An Introductory Organic Chemistry Experiment Gregory T. Rushton, William G. Burns, Judi M. Lavin, Yong S. Chong, Perry Pellechia, and Ken D. Shimizu J. Chem. Educ. 2007, 84, 1499. Abstract (http:/ / jchemed. chem. wisc. edu/ Journal/ Issues/ 2007/ Sep/ abs1499. html) [6] Jos A. Manso et al."A Kinetic Approach to the Alkylating Potential of Carcinogenic Lactones" Chem. Res. Toxicol. 2005, 18, (7) 1161-1166

MichaelisMenten kinetics

24

MichaelisMenten kineticsIn biochemistry, MichaelisMenten kinetics is one of the simplest and best-known models of enzyme kinetics. It is named after German biochemist Leonor Michaelis and Canadian physician Maud Menten. The model takes the form of an equation describing the rate of enzymatic reactions, by relating reaction rate to , the concentration of a substrate S. Its formula is given by

An example curve with parameters Vmax=3.4 and Km=0.4.

Here,

represents the maximum rate achieved by the system, at maximum (saturating) substrate concentrations. is the substrate concentration at which the reaction rate is half of . Biochemical

The Michaelis constant

reactions involving a single substrate are often assumed to follow MichaelisMenten kinetics, without regard to the model's underlying assumptions.

ModelIn 1903, French physical chemist Victor Henri found that enzyme reactions were initiated by a bond between the enzyme and the substrate.[1] His work was taken up by German biochemist Leonor Michaelis and Canadian physician Maud Menten who investigated the kinetics of an enzymatic reaction mechanism, invertase, that catalyzes the hydrolysis of sucrose into glucose and fructose.[2] In 1913, they proposed a mathematical model of the reaction.[3] It involves an enzyme E binding to a substrate S to form a complex ES, which in turn is converted into a product P and the enzyme. This may be represented schematically as

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

where

,

and

denote the rate constants,[4] and the double arrows between S and ES represent the fact that

enzyme-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 rate the enzyme concentration.

], asymptotically approaching its maximum , where is

, attained when all enzyme is bound to substrate. It also follows that

, the turnover number, is maximum number of substrate molecules converted to

MichaelisMenten kinetics product per enzyme molecule per second. The Michaelis constant approach is the substrate concentration at which the reaction rate is at half-maximum, and is a indicates high affinity, meaning that the rate will[5]

25

measure of the substrate's affinity for the enzyme. A small more quickly.

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

ApplicationsParameter values vary wildly between enzymes:[7]Enzyme Chymotrypsin Pepsin (M) 1.5 10-2 3.0 10-4 (1/s) 0.14 0.50 7.6 (1/M.s) 9.3 1.7 103 8.4 103 1.0 105 1.5 107 1.6 108

Tyrosyl-tRNA synthetase 9.0 10-4 Ribonuclease Carbonic anhydrase Fumarase

7.9 10-3 7.9 102 2.6 10-2 4.0 105 5.0 10-6 8.0 102

The constant

is a measure of how efficiently an enzyme converts a substrate into product. It has a

theoretical upper limit of 108 1010 /M.s; enzymes working close to this, such as fumarase, are termed superefficient.[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] the photosynthesis-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 the concentrations of the reactants, gives a system of four non-linear ordinary differential equations that define the rate of 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 plus combined, is a constant. This conservation law can also be obtained by adding the second and third equations above.[13] [14]

MichaelisMenten kinetics

26

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

where

is the dissociation constant for the enzyme-substrate complex. Hence the velocity[14]

of the

reaction the rate at which P is formed is

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 the time-scale of product formation known as the quasi-steady-state assumption or pseudo-steady-state-hypothesis. Mathematically, this assumption means . Combining this relationship with the enzyme 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 the environment of a living cell where there is a high concentration of protein, the cytoplasm often behaves more like a gel than a liquid, limiting molecular movements and altering reaction rates.[16] Whilst the law of mass action can be valid in heterogeneous environments,[17] it is more appropriate to model the cytoplasm as a fractal, in order to capture its limited-mobility kinetics.[18] The resulting reaction rates predicted by the two approaches are similar, with the only difference being that the equilibrium 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 is valid 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]

MichaelisMenten kinetics 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 the underlying assumptions.[14]

27

Determination of constantsThe typical method for determining the constants varying substrate concentrations and involves running a series of enzyme assays at . 'Initial' here is taken to mean that , and measuring the initial reaction rate

the reaction rate is measured after a relatively short time period, during which it is assumed that the enzyme-substrate complex has formed, but that the substrate concentration held approximately constant, and so the equilibrium or quasi-steady-state approximation remain valid.[19] By plotting reaction rate against concentration, and using nonlinear regression of the Michaelis-Menten equation, the parameters may be obtained.[20] Before computing facilities to perform nonlinear regression became available, graphical methods involving linearisation of the equation were used. A number of these were proposed, including the EadieHofstee diagram, HanesWoolf plot and LineweaverBurk 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 nonlinear regression.[21] Nonetheless, their use can still be found in modern literature.[22]

References[1] Henri, Victor (1903). Lois Gnrales de lAction 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: 333369 ( 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): 18611875. doi:10.1101/gad.1945410. PMC2932968. PMID20810646. [5] Lehninger, A.L.; Nelson, D.L.; Cox, M.M. (2005). Lehninger principles of biochemistry. New York: W.H. Freeman. ISBN978-0-7167-4339-2. [6] Chakraborty, S. (23 Dec 2009). Microfluidics and Microfabrication (1 ed.). Springer. ISBN978-1441915429. [7] Mathews, C.K.; van Holde, K.E.; Ahern, K.G. (10 Dec 1999). Biochemistry (http:/ / www. pearsonhighered. com/ mathews/ ) (3 ed.). Prentice Hall. ISBN978-0805330663. . [8] Stroppolo, M.E.; Falconi, M.; Caccuri, A.M.; Desideri, A. (Sep 2001). "Superefficient enzymes". Cell Mol Life Sci 58 (10): 145160. doi:10.1007/PL00000788. PMID11693526. [9] Yu, R.C.; Rappaport, S.M. (1997). "A lung retention model based on Michaelis-Menten-like kinetics". Environ Health Perspect 105 (5): 496503. doi:10.1289/ehp.97105496. PMC1469867. PMID9222134. [10] Keating, K.A.; Quinn, J.F. (1998). "Estimating species richness: the Michaelis-Menten model revisited". Oikos 81 (2): 411416. doi:10.2307/3547060. JSTOR3547060. [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 (13): 120. doi:10.1016/j.forsciint.2010.02.021. PMID20304569. [12] Abedon, S.T. (2009). "Kinetics of phage-mediated biocontrol of bacteria". Foodborne Pathog Dis 6 (7): 80715. doi:10.1089/fpd.2008.0242. PMID19459758. [13] Murray, J.D. (2002). Mathematical Biology: I. An Introduction (3 ed.). Springer. ISBN978-0387952239. [14] Keener, J.; Sneyd, J. (2008). Mathematical Physiology: I: Cellular Physiology (2 ed.). Springer. ISBN978-0387758466. [15] Briggs, G.E.; Haldane, J.B.S. (1925). "A note on the kinematics of enzyme action". Biochem J 19 (2): 338339. PMC1259181. PMID16743508. [16] Zhou, H.X.; Rivas, G.; Minton, A.P. (2008). "Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences". Annu Rev Biophys 37: 37597. doi:10.1146/annurev.biophys.37.032807.125817. PMC2826134. PMID18573087. [17] Grima, R.; Schnell, S. (Oct 2006). "A systematic investigation of the rate laws valid in intracellular environments". Biophys Chem 124 (1): 110. doi:10.1016/j.bpc.2006.04.019. PMID16781049.

MichaelisMenten kinetics[18] Schnell, S.; Turner, T.E. (2004). "Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws". Prog Biophys Mol Biol 85 (23): 23560. doi:10.1016/j.pbiomolbio.2004.01.012. PMID15142746. [19] Segel, L.A.; Slemrod, M. (1989). "The quasi-steady-state assumption: A case study in perturbation". Thermochim Acta 31 (3): 446477. doi:10.1137/1031091. [20] Leskovac, V. (2003). Comprehensive enzyme kinetics. New York: Kluwer Academic/Plenum Pub.. ISBN978-0-306-46712-7. [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): 1210412109. PMID500698. [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 Analyt Technol Biomed Life Sci 844 (2): 24050. doi:10.1016/j.jchromb.2006.07.006. PMID16876490.

28

LineweaverBurk plotIn biochemistry, the LineweaverBurk plot (or double reciprocal plot) is a graphical representation of the LineweaverBurk equation of enzyme kinetics, described by Hans Lineweaver and Dean Burk in 1934.[1]

DerivationThe plot provides a useful graphical method for analysis of the MichaelisMenten equation:

Taking the reciprocal gives

where V is the reaction velocity (the reaction rate), Km is the MichaelisMenten constant, Vmax is the maximum reaction velocity, and [S] is the substrate concentration.

LineweaverBurk plot

29

UseThe LineweaverBurk plot was widely used to determine important terms in enzyme kinetics, such as Km and Vmax, before the wide availability of powerful computers and non-linear regression software. The y-intercept of such a graph is equivalent to the inverse of Vmax; the x-intercept of the graph represents 1/Km. It also gives a quick, visual impression of the different forms of enzyme inhibition. The double reciprocal plot distorts the error structure of the data, and it is therefore unreliable for the determination of enzyme kinetic parameters. Although it is still used for representation of kinetic data,[2] non-linear regression or alternative linear forms of the MichaelisMenten equation such as the Hanes-Woolf plot or EadieHofstee plot are generally used for the calculation of parameters.[3] When used for determining the type of enzyme inhibition, the LineweaverBurk plot can distinguish competitive, non-competitive and uncompetitive inhibitors. Competitive inhibitors have the same y-intercept as uninhibited enzyme (since Vmax is unaffected by competitive inhibitors the inverse of Vmax also doesn't change) but there are different slopes and x-intercepts between the two data sets. Non-competitive inhibition produces plots with the same x-intercept as uninhibited enzyme (Km is unaffected) but different slopes and y-intercepts. Uncompetitive inhibition causes different intercepts on both the y- and x-axes but the same slope.

Problems with the method

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

LineweaverBurk plots of different types of reversible enzyme inhibitors. The arrow shows the effect of increasing concentrations of inhibitor.

References[1] Lineweaver, H and Burk, D. (1934). "The Determination of Enzyme Dissociation Constants". Journal of the American Chemical Society 56 (3): 658666. 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 Analyt Technol Biomed Life Sci 844 (2): 24050. doi:10.1016/j.jchromb.2006.07.006. PMID16876490. [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): 1210412109. PMID500698. .

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, Gkhan, Hannes Rst, 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, Yaser al-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, John Baez, John of Reading, JoshuaTree, Jrockley, Julia W, Knights who say ni, KnowledgeOfSelf, Lukov880, MagnInd, Mandarax, Markozeta, Maury Markowitz, Mbc362, Melaen, Michael Hardy, Micha 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 MichaelisMenten 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 Rst, 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 LineweaverBurk 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 ContributorsImage:EcDHFR raytraced.png Source: http://en.wikipedia.org/w/index.php?title=File:EcDHFR_raytraced.png License: Public Domain Contributors: TimVickers Image:KinEnzymo(en).svg Source: http://en.wikipedia.org/w/index.php?title=File:KinEnzymo(en).svg License: Public Domain Contributors: TimVickers, YassineMrabet, Image: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 on public 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: fullofstars Image:Mechanism plus rates.svg Source: http://en.wikipedia.org/w/index.php?title=File:Mechanism_plus_rates.svg License: Public Domain Contributors: Atropos235, TimVickers Image:Lineweaver-Burke plot.svg Source: http://en.wikipedia.org/w/index.php?title=File:Lineweaver-Burke_plot.svg License: GNU Free Documentation License Contributors: Original uploader was Pro bug catcher at en.wikipedia Image: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, TimVickers Image:Ping pong.svg Source: http://en.wikipedia.org/w/index.php?title=File:Ping_pong.svg License: Public Domain Contributors: TimVickers Image: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 version traced 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.wikipedia Image:Activation2 updated.svg Source: http://en.wikipedia.org/w/index.php?title=File:Activation2_updated.svg License: GNU Free Documentation License Contributors: Originally uploaded by Jerry Crimson Mann, vectorized by Tutmosis, corrected by Fvasconcellos Image:ChemicalEquilibrium.svg Source: http://en.wikipedia.org/w/index.php?title=File:ChemicalEquilibrium.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Fintelia Image: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+003F Image: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+003F File: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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