Enzyme kinetics

But most enzymes are far from perfect: the average values of ${\displaystyle k_{2}/K_{\rm {M}}}$ and ${\displaystyle k_{2}}$ are about ${\displaystyle 10^{5}{\rm {s}}^{-1}{\rm {M}}^{-1}}$ and ${\displaystyle 10{\rm {s}}^{-1}}$, respectively.

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

The observed velocities predicted by the Michaelis–Menten equation can be used to directly model the

In 1983 Stuart Beal (and also independently Santiago Schnell and Claudio Mendoza in 1997) derived a closed form solution for the time course kinetics analysis of the Michaelis-Menten mechanism.^[14][15] The solution, known as the Schnell-Mendoza equation, has the form:

${\displaystyle {\frac {[S]}{K_{M}}}=W\left[F(t)\right]\,}$

where W[ ] is the Lambert-W function.^[16][17] and where F(t) is

${\displaystyle F(t)={\frac {[S]_{0}}{K_{M}}}\exp \!\left({\frac {[S]_{0}}{K_{M}}}-{\frac {V_{\max }}{K_{M}}}\,t\right)\,}$

This equation is encompassed by the equation below, obtained by Berberan-Santos,^[18] which is also valid when the initial substrate concentration is close to that of enzyme,

${\displaystyle {\frac {[S]}{K_{M}}}=W\left[F(t)\right]-{\frac {V_{\max }}{k_{cat}K_{M}}}\ {\frac {W\left[F(t)\right]}{1+W\left[F(t)\right]}}\,}$

where W[ ] is again the Lambert-W function.

Linear plots of the Michaelis–Menten equation

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 linearisations of the Michaelis–Menten equation, such as the Lineweaver–Burk plot, the Eadie–Hofstee diagram and the Hanes–Woolf plot. All of these linear representations can be useful for visualising data, but none should be used to determine kinetic parameters, as computer software is readily available that allows for more accurate determination by nonlinear regression methods.^[19]

The Lineweaver–Burk plot or double reciprocal plot is a common way of illustrating kinetic data. This is produced by taking the reciprocal of both sides of the Michaelis–Menten equation. As shown on the right, this is a linear form of the Michaelis–Menten equation and produces a straight line with the equation y = mx + c with a y-intercept equivalent to 1/V_max and an x-intercept of the graph representing −1/K_M.

${\displaystyle {\frac {1}{v}}={\frac {K_{M}}{V_{\max }[{\mbox{S}}]}}+{\frac {1}{V_{\max }}}}$

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.^[20] 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.^[21]

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,

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[24]

${\displaystyle v_{0}=k_{cat}{\frac {{\ce {[S] [E]_0}}}{K_{M}^{\prime }+{\ce {[S]}}}}}$

but the constants are different

${\displaystyle {\begin{aligned}K_{M}^{\prime }\ &{\stackrel {\mathrm {def} }{=}}\ {\frac {k_{3}}{k_{2}+k_{3}}}K_{M}={\frac {k_{3}}{k_{2}+k_{3}}}\cdot {\frac {k_{2}+k_{-1}}{k_{1}}}\\k_{cat}\ &{\stackrel {\mathrm {def} }{=}}\ {\dfrac {k_{3}k_{2}}{k_{2}+k_{3}}}\end{aligned}}}$

We see that for the limiting case ${\displaystyle k_{3}\gg k_{2}}$, thus when the last step from ${\displaystyle {\ce {EI -> E + P}}}$ is much faster than the previous step, we get again the original equation. Mathematically we have then ${\displaystyle K_{M}^{\prime }\approx K_{M}}$ and ${\displaystyle k_{cat}\approx k_{2}}$.

Multi-substrate reactions

Multi-substrate reactions follow complex rate equations that describe how the substrates bind and in what sequence. The analysis of these reactions is much simpler if the concentration of substrate A is kept constant and substrate B varied. Under these conditions, the enzyme behaves just like a single-substrate enzyme and a plot of v by [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.

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,^[25] dihydrofolate reductase^[26] and DNA polymerase.^[27] The following links show short animations of the ternary-complex mechanisms of the enzymes dihydrofolate reductase^[β] and DNA polymerase^[γ].

Ping–pong mechanisms

${\displaystyle {\ce {\overset {}{E->[{\ce {A \atop \downarrow }}]EA<=>E^{\ast }P->[{\ce {P \atop \uparrow }}]E^{\ast }->[{\ce {B \atop \downarrow }}]E^{\ast }B<=>EQ->[{\ce {Q \atop \uparrow }}]E}}}}$

Ping–pong mechanism for an enzyme reaction. 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. This is called a secondary plot.

Enzymes with ping–pong mechanisms include some

External factors may limit the ability of an enzyme to catalyse a reaction in both directions (whereas the nature of a catalyst in itself means that it cannot catalyse just one direction, according to the principle of microscopic reversibility). We consider the case of an enzyme that catalyses the reaction in both directions:

${\displaystyle {\ce {{E}+{S}<=>[k_{1}][k_{-1}]ES<=>[k_{2}][k_{-2}]{E}+{P}}}}$

The steady-state, initial rate of the reaction is ${\displaystyle v_{0}={\frac {d\,[{\rm {P}}]}{dt}}={\frac {(k_{1}k_{2}\,[{\rm {S}}]-k_{-1}k_{-2}[{\rm {P}}])[{\rm {E}}]_{0}}{k_{-1}+k_{2}+k_{1}\,[{\rm {S}}]+k_{-2}\,[{\rm {P}}]}}}$

${\displaystyle v_{0}}$ is positive if the reaction proceed in the forward direction (${\displaystyle S\rightarrow P}$) and negative otherwise.

Equilibrium requires that ${\displaystyle v=0}$, which occurs when ${\displaystyle {\frac {[{\rm {P}}]_{\rm {eq}}}{[{\rm {S}}]_{\rm {eq}}}}={\frac {k_{1}k_{2}}{k_{-1}k_{-2}}}=K_{\rm {eq}}}$. This shows that thermodynamics forces a relation between the values of the 4 rate constants.

The values of the forward and backward maximal rates, obtained for ${\displaystyle [{\rm {S}}]\rightarrow \infty }$, ${\displaystyle [{\rm {P}}]=0}$, and ${\displaystyle [{\rm {S}}]=0}$, ${\displaystyle [{\rm {P}}]\rightarrow \infty }$, respectively, are ${\displaystyle V_{\rm {max}}^{f}=k_{2}{\rm {[E]}}_{tot}}$ and ${\displaystyle V_{\rm {max}}^{b}=-k_{-1}{\rm {[E]}}_{tot}}$, respectively. Their ratio is not equal to the equilibrium constant, which implies that thermodynamics does not constrain the ratio of the maximal rates. This explains that enzymes can be much "better catalysts" (in terms of maximal rates) in one particular direction of the reaction.^[31]

On can also derive the two Michaelis constants ${\displaystyle K_{M}^{S}=(k_{-1}+k_{2})/k_{1}}$and ${\displaystyle K_{M}^{P}=(k_{-1}+k_{2})/k_{-2}}$. The Haldane equation is the relation ${\displaystyle K_{\rm {eq}}={\frac {[{\rm {P}}]_{\rm {eq}}}{[{\rm {S}}]_{\rm {eq}}}}={\frac {V_{\rm {max}}^{f}/K_{M}^{S}}{V_{\rm {max}}^{b}/K_{M}^{P}}}}$.

Therefore, thermodynamics constrains the ratio between the forward and backward ${\displaystyle V_{\rm {max}}/K_{M}}$ values, not the ratio of ${\displaystyle V_{\rm {max}}}$values.

Non-Michaelis–Menten kinetics

Many different enzyme systems follow non Michaelis-Menten behavior. A select few examples include kinetics of self-catalytic enzymes, cooperative and allosteric enzymes, interfacial and intracellular enzymes, processive enzymes and so forth. Some enzymes produce a sigmoid 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.^[32] 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 the enzyme for other substrate molecules.

Allosteric enzymes include mammalian tyrosyl tRNA-synthetase, which shows negative cooperativity,

The Hill equation^[36] is often used to describe the degree of cooperativity quantitatively in non-Michaelis–Menten kinetics. The derived Hill coefficient n measures how much the binding of substrate to one active site affects the binding of substrate to the other active sites. A Hill coefficient of <1 indicates negative cooperativity and a coefficient of >1 indicates positive cooperativity.

Pre-steady-state kinetics

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

This approach was first applied to the hydrolysis reaction catalysed by chymotrypsin.^[37] 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*.^[38] 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.^[39]

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 than bonds to hydrogen.^[40] 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.^[41]

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.^[42] The chemical mechanism can also be elucidated by examining the kinetics and isotope effects under different pH conditions,^[43] by altering the metal ions or other bound cofactors,^[44] 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).^[45]

Enzyme inhibition and activation

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

Reversible inhibitors

Traditionally reversible enzyme inhibitors have been classified as competitive, uncompetitive, or non-competitive, according to their effects on K_M and V_max. These different effects result from the inhibitor binding to the enzyme E, to the enzyme–substrate complex ES, or to both, respectively. The division of these classes arises from a problem in their derivation and results in the need to use two different binding constants for one binding event. The binding of an inhibitor and its effect on the enzymatic activity are two distinctly different things, another problem the traditional equations fail to acknowledge. In noncompetitive inhibition the binding of the inhibitor results in 100% inhibition of the enzyme only, and fails to consider the possibility of anything in between.^[46] In noncompetitive inhibition, the inhibitor will bind to an enzyme at its allosteric site; therefore, the binding affinity, or inverse of K_M, of the substrate with the enzyme will remain the same. On the other hand, the V_max will decrease relative to an uninhibited enzyme. On a Lineweaver-Burk plot, the presence of a noncompetitive inhibitor is illustrated by a change in the y-intercept, defined as 1/V_max. The x-intercept, defined as −1/K_M, will remain the same. In competitive inhibition, the inhibitor will bind to an enzyme at the active site, competing with the substrate. As a result, the K_M will increase and the V_max will remain the same.^[47] The common form of the inhibitory term also obscures the relationship between the inhibitor binding to the enzyme and its relationship to any other binding term be it the Michaelis–Menten equation or a dose response curve associated with ligand receptor binding. To demonstrate the relationship the following rearrangement can be made:

${\displaystyle {\cfrac {V_{\max }}{1+{\cfrac {[I]}{K_{i}}}}}={\cfrac {V_{\max }}{\cfrac {[I]+K_{i}}{K_{i}}}}}$

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

${\displaystyle {\cfrac {V_{\max }}{\cfrac {[I]+K_{i}}{[I]+K_{i}-[I]}}}}$

Dividing by [I]+K_i

${\displaystyle {\cfrac {V_{\max }}{\cfrac {1}{1-{\cfrac {[I]}{[I]+K_{i}}}}}}=V_{\max }-V_{\max }{\cfrac {[I]}{[I]+K_{i}}}}$

This notation demonstrates that similar to the Michaelis–Menten equation, where the rate of reaction depends on the percent of the enzyme population interacting with substrate, the effect of the inhibitor is a result of the percent of the enzyme population interacting with inhibitor. The only problem with this equation in its present form is that it assumes absolute inhibition of the enzyme with inhibitor binding, when in fact there can be a wide range of effects anywhere from 100% inhibition of substrate turn over to just >0%. To account for this the equation can be easily modified to allow for different degrees of inhibition by including a delta V_max term.

${\displaystyle V_{\max }-\Delta V_{\max }{\cfrac {[I]}{[I]+K_{i}}}}$

or

${\displaystyle V_{\max 1}-(V_{\max 1}-V_{\max 2}){\cfrac {[I]}{[I]+K_{i}}}}$

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

${\displaystyle V_{\max 1}-(V_{\max 1}-V_{\max 2}){\cfrac {[X]}{[X]+K_{x}}}}$

While this terminology results in a simplified way of dealing with kinetic effects relating to the maximum velocity of the Michaelis–Menten equation, it highlights potential problems with the term used to describe effects relating to the K_M. The K_M relating to the affinity of the enzyme for the substrate should in most cases relate to potential changes in the binding site of the enzyme which would directly result from enzyme inhibitor interactions. As such a term similar to the one proposed above to modulate V_max should be appropriate in most situations:^[48]

${\displaystyle K_{m1}-(K_{m1}-K_{m2}){\cfrac {[X]}{[X]+K_{x}}}}$

Irreversible inhibitors

Enzyme inhibitors can also irreversibly inactivate enzymes, usually by covalently modifying active site residues. These reactions, which may be called suicide substrates, follow exponential decay functions and are usually saturable. Below saturation, they follow first order kinetics with respect to inhibitor. Irreversible inhibition could be classified into two distinct types. Affinity labelling is a type of irreversible inhibition where a functional group that is highly reactive modifies a catalytically critical residue on the protein of interest to bring about inhibition. Mechanism-based inhibition, on the other hand, involves binding of the inhibitor followed by enzyme mediated alterations that transform the latter into a reactive group that irreversibly modifies the enzyme.

Philosophical discourse on reversibility and irreversibility of inhibition

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

Mechanisms of catalysis

The favoured model for the enzyme–substrate interaction is the induced fit model.

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.

History

In 1902

The major contribution of the Henri-Michaelis-Menten approach was to think of enzyme reactions in two stages. In the first, the substrate binds reversibly to the enzyme, forming the enzyme-substrate complex. This is sometimes called the Michaelis complex. The enzyme then catalyzes the chemical step in the reaction and releases the product. The kinetics of many enzymes is adequately described by the simple Michaelis-Menten model, but all enzymes have internal motions that are not accounted for in the model and can have significant contributions to the overall reaction kinetics. This can be modeled by introducing several Michaelis-Menten pathways that are connected with fluctuating rates,^[56][57][58] which is a mathematical extension of the basic Michaelis Menten mechanism.^[59]

Software

ENZO (Enzyme Kinetics) is a graphical interface tool for building kinetic models of enzyme catalyzed reactions. ENZO automatically generates the corresponding differential equations from a stipulated enzyme reaction scheme. These differential equations are processed by a numerical solver and a regression algorithm which fits the coefficients of differential equations to experimentally observed time course curves. ENZO allows rapid evaluation of rival reaction schemes and can be used for routine tests in enzyme kinetics.^[60]

See also

• Protein dynamics
• Diffusion limited enzyme
• Langmuir adsorption model

Footnotes

α. ^{^} Link: Interactive Michaelis–Menten kinetics tutorial (Java required)

β. ^{^} Link: dihydrofolate reductase mechanism (Gif)

γ. ^{^} Link: DNA polymerase mechanism (Gif)

δ. ^{^} Link: Chymotrypsin mechanism (Flash required)

References