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Enzyme kinetics

2007 Schools Wikipedia Selection. Related subjects: General Chemistry

   Dihydrofolate reductase from E. coli with its two substrates,
   dihydrofolate (right) and NADPH (left), bound in the active site. The
   protein is shown as a ribbon diagram, with alpha helices in red, beta
   sheets in yellow and loops in blue. Generated from 7DFR.
   Enlarge
   Dihydrofolate reductase from E. coli with its two substrates,
   dihydrofolate (right) and NADPH (left), bound in the active site. The
   protein is shown as a ribbon diagram, with alpha helices in red, beta
   sheets in yellow and loops in blue. Generated from 7DFR.

   Enzyme kinetics is the study of the rates of chemical reactions that
   are catalysed by enzymes. The study of an enzyme's kinetics provides
   insights into the catalytic mechanism of this enzyme, its role in
   metabolism, how its activity is controlled in the cell and how drugs
   and poisons can inhibit its activity.

   Enzymes are 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. Some enzymes bind multiple substrates and/or
   release multiple products, such as a protease cleaving one protein
   substrate into two polypeptide products. Others join 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 visualizing the
   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 analogs that do not undergo the enzymatic reaction.

   Enzyme 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
   dihydrofolate reductase (shown right), enzyme kinetics can also show
   the sequence in which these substrates bind and the sequence in which
   products are released.

   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 the RNA catalysts perform a more
   limited set of reactions, although their reaction mechanisms and
   kinetics can be analysed and classified by the same methods.

General principles

   The rate of reaction will increase as substrate concentration
   increases, eventually becoming saturated at very high concentrations of
   substrate.
   Enlarge
   The rate of reaction will increase as substrate concentration
   increases, eventually becoming saturated at very high concentrations of
   substrate.

   The 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. However, unlike normal chemical
   reactions, enzymes are saturable. This means as more substrate is
   added, the reaction rate will increase, because more active sites
   become occupied. This can continue until all the enzyme becomes
   saturated with substrate and the rate reaches a maximum.

   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 environment of 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 rate
   period represents the initial rate of reaction v. Equations such as the
   Michaelis-Menten equation describe how this slope varies with the
   substrate and enzyme concentrations.
   Enlarge
   Progress curve for an enzyme reaction. The slope in the initial rate
   period represents the initial rate of reaction v. Equations such as the
   Michaelis-Menten equation describe how this slope varies with the
   substrate and enzyme concentrations.

   Enzyme 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 require the removal and counting of samples (i.e., they are
   discontinuous assays) they are usually extremely sensitive and can
   measure very low levels of enzyme activity. An analogous approach is to
   use mass spectrometry 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 during an enzyme's reaction mechanism, or
   fluorescent dyes added onto specific sites of the protein that report
   movements that occur during catalysis. These studies are providing a
   new view of the kinetics and dynamics of single molecules, as opposed
   to traditional enzyme kinetics, which observes the average behaviour of
   populations of millions of enzyme molecules.

   On the left is shown a typical progress curve for an enzyme assay. The
   enzyme produces product at a linear initial rate at the start of the
   reaction. Later in this progress curve, the rate slows down as
   substrate is used up or products accumulate. The length of the initial
   rate period depends on the assay conditions and can range from
   milliseconds to hours. Enzyme assays are usually set up to produce an
   initial rate lasting over a minute, to make measurements easier.
   However, equipment for rapidly mixing liquids allows fast kinetic
   measurements on initial rates of less than one second. These very rapid
   assays are essential for measuring pre-steady-state kinetics, which are
   discussed below.

   Most enzyme kinetics studies concentrate on this initial, 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. This approach is useful as an alternative to
   rapid kinetics when the initial rate is too fast to measure accurately.

Single-substrate reactions

   Enzymes 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. 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 ping–pong mechanism, a type of 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 of substrate and rate.
   Enlarge
   Saturation curve for an enzyme showing the relation between the
   concentration of substrate and rate.

   As enzyme-catalysed reactions are saturable, their rate of catalysis
   does not show a linear response to increasing substrate. If the initial
   rate of the reaction is measured over a range of substrate
   concentrations (denoted as [S]), the reaction rate (v) increases as [S]
   increases, as shown on the left. However, as [S] gets higher, the
   enzyme becomes saturated with substrate and the rate reaches V[max],
   the enzyme's maximum rate.
   Single-substrate mechanism for an enzyme reaction. k1, k-1 and k2 are
   the rate constants for the individual steps.
   Enlarge
   Single-substrate mechanism for an enzyme reaction. k[1], k[-1] and k[2]
   are the rate constants for the individual steps.

   The Michaelis-Menten kinetic model of a single-substrate reaction is
   shown on the right. There is an initial bimolecular reaction between
   the enzyme E and substrate S to form the enzyme–substrate complex ES.
   Although the enzymatic mechanism for the unimolecular reaction ES
   \rightarrow E + P reaction can be quite complex, there is typically one
   rate-determining enzymatic step that allows the mechanism to be modeled
   as a single kinetic step of rate constant k[2].

          \begin{matrix}v = k_2 [\mbox{ES}]\end{matrix}     (Equation 1).

   k[2] is also called k[cat] or the turnover number, the maximum number
   of enzymatic reactions catalyzed per second.

   At low concentrations of substrate [S], the enzyme exists in an
   equilibrium between both the free form E and the enzyme–substrate
   complex ES; increasing [S] likewises increases [ES] at the expense of
   [E], shifting the binding equilibrium to the right. Since the rate of
   the reaction depends on the concentration [ES], the rate is sensitive
   to small changes in [S]. However, at very high [S], the enzyme is
   entirely saturated with substrate, and exists only in the ES form.
   Under these conditions, the rate (v≈k[2][E][tot]=V[max]) is insensitive
   to small changes in [S]; here, [E][tot] is the total enzyme
   concentration

          [\mbox{E}]_{tot} \ \stackrel{\mathrm{def}}{=}\ [\mbox{E}] +
          [\mbox{ES}]

   which is approximately equal to the concentration [ES] under saturating
   conditions.

   The Michaelis–Menten equation describes how the reaction rate v depends
   on the position of the substrate-binding equilibrium and the rate
   constant k[2]. Michaelis and Menten showed when k[2] is much less than
   k[-1] (the equilibrium approximation) they could derive the following
   equation:

          v = \frac{V_{max}[\mbox{S}]}{K_m + [\mbox{S}]}     (Equation 2)

   This Michaelis-Menten equation is the basis for most single-substrate
   enzyme kinetics.

   The Michaelis constant K[m] is defined as the concentration at which
   the rate of the enzyme reaction is half V[max]. This may be verified by
   substituting [S] = K[m] into the Michaelis-Menten equation. If the
   rate-determining enzymatic step is slow compared to substrate
   dissociation (k[2] << k[-1]), the Michaelis constant K[m] is roughly
   the dissociation constant of the ES complex, although this situation is
   relatively rare.

   The more normal situation where k[2] > k[-1] is sometimes called
   Briggs- Haldane kinetics. The Michaelis–Menten equation still holds
   under these more general conditions, as may be derived from the
   steady-state approximation. During the initial-rate period, the
   reaction rate v is roughly constant, indicating that [ES] is similarly
   constant (cf. equation 1):

          \frac{d}{dt}[\mbox{ES}] = k_{1} [\mbox{E}][\mbox{S}] -
          k_{2}[\mbox{ES}] - k_{-1}[\mbox{ES}] \approx 0

   Therefore, the concentration [ES] is given by the formula

          [\mbox{ES}] \approx \frac{[\mbox{E}]_{tot}
          [\mbox{S}]}{[\mbox{S}] + K_{m}}

   where the Michaelis constant K[m] is defined

          K_{m} \ \stackrel{\mathrm{def}}{=}\ \frac{k_{2} + k_{-1}}{k_{1}}
          \approx \frac{[\mbox{E}][\mbox{S}]}{[\mbox{ES}]}

   ([E] is the concentration of free enzyme). Taken together, the general
   formula for the reaction rate v is again the Michaelis-Menten equation:

          v = k_{2} [\mathrm{ES}] = \frac{k_{2} [\mbox{E}]_{tot}
          [\mbox{S}]}{[\mbox{S}] + K_{m}} = \frac{V_{max}
          [\mbox{S}]}{[\mbox{S}] + K_{m}}

   The specificity constant k[cat] / K[m] is a measure of how efficiently
   an enzyme converts a substrate into product. Using the definition of
   the Michaelis constant K[m], the Michaelis-Menten equation may be
   written in the form

          v = k_{2} [\mathrm{ES}] = \frac{k_{2}}{K_{m}} [\mbox{E}]
          [\mbox{S}]

   where [E] is the concentration of free enzyme. Thus, the specificity
   constant is an effective bimolecular rate constant for free enzyme to
   react with free substrate to form product. The specificity constant is
   limited by the frequency with which the substrate and enzyme encounter
   each other in solution, roughly 10^10 M^-1 s^-1 at 25°C. Remarkably,
   this maximum does not depend on the size of either the substrate or the
   enzyme. The ratio of the specificity constants for two substrates is a
   quantitative comparison of how efficient the enzyme is in converting
   those substrates. The slope of the Michaelis-Menten equation at low
   substrate concentration [S] (when [S] << K[m]) also yields the
   specificity constant.

Linear plots of the Michaelis-Menten equation

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

   Using an interactive Michaelis–Menten 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 K[m] and V[max] accurately. Therefore,
   several researchers developed linearizations of the Michaelis-Menten
   equation, such as the Lineweaver-Burk plot and the Eadie-Hofstee
   diagram.

   The Lineweaver-Burk plot or double reciprocal plot is 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
   equivalent to 1/V[max] and an x-intercept of the graph representing
   -1/K[m].

          \frac{1}{v} = \frac{K_{m}}{V_{max} [\mbox{S}]} +
          \frac{1}{V_{max}}

   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/V[max] as shown at the
   right; thus, the x-intercept is an extrapolation of the experimental
   data taken at positive concentrations. More generally, the
   Lineweaver-Burk plot skews the importance of measurements taken at low
   substrate concentrations and, thus, can yield inaccurate estimates of
   V[max] and K[m]. A more accurate linear plotting method is the
   Eadie-Hofstee plot although, in modern research, all such
   linearizations have been superseded by more reliable nonlinear
   regression methods.

Practical significance of kinetic constants

   The study of enzyme kinetics is important for two basic reasons.
   Firstly, it helps explain how enzymes work, and secondly, it helps
   predict how enzymes behave in living organisms. The kinetic constants
   defined above, K[m] and V[max], are critical to attempts to understand
   how enzymes work together to control metabolism.

   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. Although this goal is far in the future for any eukaryote,
   attempts are now being made to achieve this in bacteria, with models of
   Escherichia coli metabolism now being produced and tested.

Multi-substrate reactions

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

Ternary-complex mechanisms

   In these enzymes, both substrates bind to the enzyme at the same time
   to produce an EAB ternary complex. The order of binding can either be
   random (in a random mechanism) or substrates have to bind in a
   particular sequence (in an ordered mechanism). When a set of v by [S]
   curves (fixed A, varying B) from an enzyme with a ternary-complex
   mechanism are plotted in a Lineweaver–Burk plot, the set of lines
   produced will intersect.

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

Ping–pong mechanisms

   Ping–pong mechanism for an enzyme reaction. Enzyme intermediates
   contain substrates A and B or products P and Q.
   Enlarge
   Ping–pong mechanism for an enzyme reaction. Enzyme intermediates
   contain substrates A and B or products P and Q.

   As 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 is known as an intermediate. In such mechanisms,
   substrate A binds, changes the enzyme to E* by, for example,
   transferring a chemical group to the active site, and is then released.
   Only after the first substrate is released can substrate B bind and
   react with the modified enzyme, regenerating the unmodified E form.
   When a set of v by [S] curves (fixed A, varying B) from an enzyme with
   a ping–pong mechanism are plotted in a Lineweaver–Burk plot, a set of
   parallel lines will be produced.

   Enzymes with ping–pong mechanisms include some oxidoreductases such as
   thioredoxin peroxidase and serine proteases such as trypsin and
   chymotrypsin. 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.

Non-Michaelis–Menten kinetics

   Saturation curve for an enzyme reaction showing sigmoid kinetics.
   Enlarge
   Saturation curve for an enzyme reaction showing sigmoid kinetics.

   Some 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 behaviour is most common in multimeric
   enzymes with several interacting active sites. Here, the mechanism of
   cooperation is similar to that of haemoglobin, with binding of
   substrate to one active site altering the affinity of the other active
   sites for substrate molecules. Positive cooperativity occurs when
   binding of the first substrate molecule increases the affinity of the
   other active sites for substrate. Negative cooperativity occurs when
   binding of the first substrate decreases the affinity of the enzyme for
   other substrate molecules.

   Allosteric enzymes include mammalian tyrosyl tRNA-synthetase, which
   shows negative cooperativity, and bacterial aspartate transcarbamoylase
   and phosphofructokinase, 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 is ofetn used to characterize the degree of
   cooperativity quantitatively for non-Michaelis–Menten enzymes. The
   derived Hill coefficient n measures how much the binding of substrate
   to one active site affects the binding of substrate to the other active
   sites. A Hill coefficient of <1 indicates negative cooperativity and a
   coefficient of >1 indicates positive cooperativity.

Pre-steady-state kinetics

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

   In the first moment after an enzyme is mixed with substrate, no product
   has been formed and no intermediates exist. The study of the next few
   milliseconds of the reaction is called pre-steady-state kinetics.
   Pre-steady-state kinetics is therefore concerned with the formation and
   consumption of enzyme–substrate intermediates (such as ES or E*) until
   their steady-state concentrations are reached.

   This approach was first applied to the hydrolysis reaction catalysed by
   chymotrypsin. Often, the detection of an intermediate is a vital piece
   of evidence in investigations of what mechanism an enzyme follows. For
   example, in the ping–pong mechanisms that are shown above, rapid
   kinetic measurements can follow the release of product P and measure
   the formation of the modified enzyme intermediate E*. In the case of
   chymotrypsin, this intermediate is formed by the attack of 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.

Chemical mechanism

   An important goal of measuring enzyme kinetics is to determine the
   chemical mechanism of an enzyme reaction, i.e., the sequence of
   chemical steps that transform substrate into product. The kinetic
   approaches discussed above will show at what rates intermediates are
   formed and inter-converted, but they cannot identify exactly what these
   intermediates are.

   Kinetic measurements taken under various solution conditions or on
   slightly modified enzymes or substrates often shed light on this
   chemical mechanism, as they reveal the rate-determining step or
   intermediates in the reaction. For example, the breaking of a covalent
   bond 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 then bonds to
   hydrogen. It is also possible to measure similar effects with other
   isotope substitutions, such as ^13C/^12C and ^18O/^16O, but these
   effects are more subtle.

   Isotopes can also be used to reveal the fate of various parts of the
   substrate molecules in the final products. For example, it is sometimes
   difficult to discern the origin of an oxygen atom in the final product;
   since it may have come from water or from part of the substrate. This
   may be determined by systematically substituting oxygen's stable
   isotope ^18O into the various molecules that participate in the
   reaction and checking for the isotope in the product. The chemical
   mechanism can also be elucidated by examining the kinetics and isotope
   effects under different pH conditions, by altering the metal ions or
   other bound cofactors, by site-directed mutagenesis of conserved amino
   acid residues, or by studying the behaviour of the enzyme in the
   presence of analogues of the substrate(s).

Enzyme inhibition

   Kinetic scheme for reversible enzyme inhibitors.
   Enlarge
   Kinetic scheme for reversible enzyme inhibitors.

   Enzyme inhibitors are molecules that reduce or abolish enzyme activity.
   These are either reversible (i.e., removal of the inhibitor restores
   enzyme activity) or irreversible (i.e., the inhibitor permanently
   inactivates the enzyme).

Reversible inhibitors

   Reversible enzyme inhibitors can be classified as competitive,
   uncompetitive, non-competitive or mixed, according to their effects on
   K[m] and V[max]. These different effects result from the inhibitor
   binding to the enzyme E, to the enzyme–substrate complex ES, or to
   both, as shown in the figure to the right and the table below. The
   particular type of an inhibitor can be discerned by studying the enzyme
   kinetics as a function of the inhibitor concentration. The four types
   of inhibition produce Lineweaver–Burke and Eadie–Hofstee plots that
   vary in distinctive ways with inhibitor concentration. For brevity, two
   symbols are used:

          \alpha = 1 + \frac{[\mbox{I}]}{K_{i}}       and
          \alpha^{\prime} = 1 + \frac{[\mbox{I}]}{K_{i}^{\prime}}

   where K[i] and K'[i] are the dissociation constants for binding to the
   enzyme and to the enzyme–substrate complex, respectively. In the
   presence of the reversible inhibitor, the enzyme's apparent K[m] and
   V[max] become (α/α')K[m] and (1/α')V[max], respectively, as shown below
   for common cases.

              Type of inhibition K[m] apparent V[max] apparent
      K[i] only ( \alpha^{\prime}=1 ) competitive K_m \alpha~ ~V_{max}~
     K[i]' only ( \alpha=1~ ) uncompetitive \frac{K_m}{\alpha^{\prime}}
                       \frac{V_{max}}{\alpha^{\prime}}
       K[i] = K[i]' ( \alpha = \alpha^{\prime} ) non-competitive ~K_m~
                       \frac{V_{max}}{\alpha^{\prime}}
            K[i] ≠ K[i]' ( \alpha \neq \alpha^{\prime} ) mixed
      \frac{K_m\alpha}{\alpha^{\prime}} \frac{V_{max}}{\alpha^{\prime}}

   Non-linear regression fits of the enzyme kinetics data to the rate
   equations above can yield accurate estimates of the dissociation
   constants K[i] and K'[i].

Irreversible inhibitors

   Enzyme inhibitors can also irreversibly inactivate enzymes, usually by
   covalently modifying active site residues. These reactions follow
   exponential decay functions and are usually saturable. 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 of the transition state by an enzyme.
   Enlarge
   The energy variation as a function of reaction coordinate shows the
   stabilisation of the transition state by an enzyme.

   The favoured model for the enzyme–substrate interaction is the induced
   fit model. This model proposes that the initial interaction between
   enzyme and substrate is relatively weak, but that these weak
   interactions rapidly induce conformational 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. After binding takes
   place, one or more mechanisms of catalysis lower the energy of the
   reaction's 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.

   Enzyme kinetics cannot prove which modes of catalysis are used by an
   enzyme. However, some kinetic data can suggest possibilities to be
   examined by other techniques. For example, a ping–pong mechanism with
   burst-phase pre-steady-state kinetics would suggest covalent catalysis
   might be important in this enzyme's mechanism. Alternatively, the
   observation of a strong pH effect on V[max] but not K[m] might indicate
   that a residue in the active site needs to be in a particular
   ionisation state for catalysis to occur.

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