Enzyme kinetics

Enzyme kinetics is the study of the rates of chemical reactions catalyzed by enzymes. The study of enzyme kinetics provides information on how enzymes work, how their activity is controlled in the body and how drugs and poisons inhibit their reactions.

Enzymes are molecular machines that manipulate other molecules. The information provided by an enzyme's structure is akin to a complete blueprint of a machine. However, the information provided by kinetic studies is similar to a movie of this machine in action. In order to completely understand how enzymes work, we need to know both their structures and their mechanisms.

Kinetic studies on enzymes that only bind one substrate, such as catalase, aim to measure the affinity with which the enzyme binds this substrate and how fast it can turn it into a product. When enzymes bind multiple substrates, enzyme kinetics can also show the order in which these substrates bind and the order in which products are released.

Enzyme assays

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Figure 1: Progress curve for an enzyme reaction

Enzyme assays are laboratory procedures that measure the rate of enzyme reactions.

Assays commonly use either changes in the absorbance of light between products and reactants (spectrophotometric assays), or the incorporation or release of radioactivity (radiometric assays) to measure the amount of product made over time. Spectrophometric assays are most convenient since they let you measure the rate of the reaction continuously. However, 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 amounts of enzyme activity.

Figure 1 shows 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 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.

Kinetics of single-substrate reactions

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Figure 2: Saturation curve for an enzyme reaction showing the relation between the substrate concentration (S) and rate (v).

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Figure 3: Single substrate mechanism for an enzyme reaction. k₁, k_-1 and k₂ are the rate constants for the individual steps.

Enzyme-catalysed reactions are saturable. As shown in figure 2, if the initial rate of the reaction is measured over a range of substrate concentrations (or [S]), as [S] increases the reaction rate (v) also increases. However, as [S] gets higher, the enzyme becomes saturated with substrate and the rate reaches V_max the enzyme's maximum rate.

A model for the mechanism of a single-substrate reactions is shown in figure 3. At low concentrations of [S] the enzyme exists in both the free form E and as the enzyme-substrate complex ES. The rate of the reaction will depend on the concentration of the enzyme-substrate complex ES and the rate of the chemical step k₃. Therefore the rate is sensitive to small changes in [S]. Or mathematically: ${\begin{matrix}v=k_{2}[ES]\end{matrix}}$ (Equation 1)

At very high [S] the position of this binding equilibrium shifts and the enzyme becomes saturated with substrate, existing only in the ES form. The rate under these conditions (V_max) is insensitive to small changes in [S]. (i.e. ${\begin{matrix}v=k_{2}\end{matrix}}$ (Equation 2))

k₂ is usually called the turnover number or k_cat. This number is the maximum number of substrate molecules one active site can handle per second.

At all concentrations of substrate below saturation, the rate of the reaction depends on both the position of the substrate-binding equilibrium and the rate of the chemical step. Substituting the equation describing this equilibrium into the rate equation 1 above allows us to derive the Michaelis-Menten equation. (a full derivation is given below)

$v={\frac {V_{max}[S]}{K_{m}+[S]}}$ (Equation 3)

To experiment with the Michaelis-Menten equation, try this interactive Michaelis-Menten Kinetics tutorial

The Michaelis constant K_m is defined as the concentration at which the rate of the enzyme reaction is half V_max. You can see this when you substitute v = V_max /2, into the Michaelis-Menten equation, the expression reduces to K_m = [S]

The Michaelis constant K_m is, in a few cases, equal to the disassociation constant of the ES complex. The value of K_m may therefore give you a measure of how tightly the enzyme binds a substrate. However this only applies when the chemical step is rate-limiting and k₂ is much lower than k_-1.

The specificity constant is a measure of how well an enzyme can use a substrate. Since V_max = k_cat.[ES] (see equation [2] above) you can substitute this into the Michaelis-menten equation to get

$v=k_{cat}{\frac {[E]_{total}[S]}{K_{m}+[S]}}$ (Equation 4)

Since [E]_total = [ES] + [E] and also [ES] = [E][S]/K_m the expression simplifies to

$v={\frac {k_{cat}}{K_{m}}}[E][S]$ (Equation 5)

Therefore, for any concentration of enzyme and substrate, the rate of reaction is dependent on the specificity constant for that substrate. This lets you compare how an enzyme would use two competing substrates. Equation 5 is also useful for measuring exactly how badly an enzyme uses very poor substrates where little saturation is seen. Here, if v is plotted as a function of [E] you get a straight line of equation y = mx with the slope equal to k_cat/K_m.[S]

Kinetics of multi-substrate reactions

Multi-substrate reactions follow complex rate equations that describe how the substrates bind and in what order. The analysis of these reactions is much simpler however if you keep the concentration of substrate A constant and vary substrate B. Under these conditions, the enzyme behaves just like a single-substrate enzyme and you can measure a K_mapp and V_maxapp for substrate B.

If you then perform a set of these measurements at different fixed concentrations of A, you can use these data to work out what the mechanism of the reaction is. There are two basic types.

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Figure 4: 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 at the same time to produce an EAB ternary complex. The order of binding can either be random (as shown in figure 4) or they can bind in a particular order.

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 two links below are to short animations showing enzymes with ternary complex mechanisms in action.

Dihydrofolate reductase.

DNA polymerase.

Figure 5: Ping-pong mechanism for an enzyme reaction. Enzyme intermediates contain substrates A and B or products P and Q.

Ping-pong mechanisms

As shown in figure 5, these enzymes can exist in two structures, E and a chemically-modified form of the enzyme E*. In these mechanisms, substrate A binds, changes the enzyme to E* by for example transferring a chemical group to the active site, and then is then released. Only after the fisrt 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 oxidoreductases such as thioredoxin reductase or serine proteases such as trypsin or chymotrypsin. Serine proteases, include both digestive enzymes (trypsin, chymotrypsin, and elastase) and several enzymes of the blood clotting cascade. These proteases contain an active site serine whose hydroxyl group forms a covalent bond with a carbonyl carbon of a peptide bond, thereby causing hydrolysis of the polypeptide. The link below is to a short animation showing the mechanism of chymotrypsin.

Chymotrypsin

Non-Michaelis-Menten kinetics

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Figure 6: Saturation curve for an enzyme reaction showing sigmoid kinetics.

Sometimes an enzyme will produce a sigmoid v by [S] plot. This often indicates cooperative binding of substrate to the active site.

This behavior is most common in multimeric enzymes with several interacting active sites. Here, the mechanism of co-operation 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 is when binding of the first substrate molecule increases the affinity of the other active sites for substrate. Negative cooperativity is when binding of the first substrate reduces 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].

It can be useful to apply the Hill equation to data with these non-Michaelis-Menten characteristics and calculate the hill coefficient. This is not a kinetic constant but 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 in the reaction is called pre-steady-state kinetics.

Pre-steady-state kinetics is concerned with the formation and interconversion of enzyme-substrate intermediates. Often, the detection of an intermediate is essential in proving what mechanism an enzyme follows. For example, in the ping-pong mechanism shown above in figure 5, these rapid kinetic measurements can follow the release of P and measure the formation of the modified enzyme intermediate E*.

Mechanisms of catalysis

Figure 7: Stabilisation ot the transition state by an enzyme.

The favored model for the enzyme-substrate interaction is known as 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 lowers the energy of the reaction's transition state, by providing an alternative chemical pathway for the reaction. There are four possible mechanisms of "over the barrier" catalysis as well as a "through the barrier" mechanism:

Catalysis by bond strain

Here, the induced structural rearrangements that take place with the binding of substrate and enzyme produce strained substrate bonds, which are closer to the conformation of the transition state. This lowers the energy difference between the substrate and transition state.

Catalysis by proximity and orientation

Here, enzyme-substrate interactions align reactive chemical groups and hold them close together.

Catalysis Involving Proton donors or acceptors

If the transition state is charged, residues in the active site accept or donate a proton to stabilize the intermediate.

Covalent Catalysis

Here, the substrate is reacts with residues in the active site and forms a covalent intermediate between the enzyme and the substrate. This requires a ping-pong mechanism and is discussed in more detail above.

Quantum Tunneling

These traditional "over the barrier" mechanisms have been challenged in some cases by models and observations of "through the barrier" mechanisms (quantum tunneling). Some enzymes operate with kinetics which are faster than diffusion rates, which is impossible according to traditional models. In "through the barrier" models, a proton or an electron can tunnel through activation barriers. . Quantum tunneling for protons has been observed in tryptamine oxidation by aromatic amine dehydrogenase.

Derivation of Michaelis-Menten Equation

This derivation of "Michaelis-Menten" was actually described by Briggs and Haldane. It is obtained as follows:

The enzymatic reaction is supposed to be irreversible, and the product does not rebind the enzyme.

$E+S{\begin{matrix}k_{1}\\\longrightarrow \\\longleftarrow \\k_{-1}\end{matrix}}ES{\begin{matrix}k_{2}\\\longrightarrow \\\ \end{matrix}}E+P$

Because we follow the steady state approximation:

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

$[ES]={\frac {k_{1}[E][S]}{k_{-1}+k_{2}}}$

Let's define:

$K_{m}={\frac {k_{-1}+k_{2}}{k_{1}}}$

Therefore:

$[ES]={\frac {[E][S]}{K_{m}}}$ (1)

The rate (or velocity) of the reaction is:

${\frac {d[P]}{dt}}=k_{2}[ES]$ (2)

The total concentration of enzyme is:

$[E_{0}]=[E]+[ES]$

Hence:

$[E]=[E_{0}]-[ES]$ (3)

Substituting (3) into (1) gives:

$[ES]={\frac {([E_{0}]-[ES])[S]}{K_{m}}}$

Rearranging gives:

$[ES]{\frac {K_{m}}{[S]}}=[E_{0}]-[ES]$

$[ES](1+{\frac {K_{m}}{[S]}})=[E_{0}]$

$[ES]=[E_{0}]{\frac {1}{1+{\frac {K_{m}}{[S]}}}}$ (4)

Substituting (4) in (2) and multiplying numerator and denominator by $[S]$:

${\frac {d[P]}{dt}}=k_{2}[E_{0}]{\frac {[S]}{K_{m}+[S]}}=V_{max}{\frac {[S]}{K_{m}+[S]}}$

See also

• List of enzymes
• Enzyme
• Enzyme inhibition
• Enzyme assay

• MACiE: a database of enzyme reaction mechanisms
• Symbolism and Terminology in Enzyme Kinetics, A comprehensive explanation of concepts and terminology in enzyme kinetics.
• An introduction to enzyme kinetics An accessible set of on-line tutorials on enzyme kinetics.

Further reading

• Athel Cornish-Bowden, Fundamentals of Enzyme Kinetics. (3rd edition), Portland Press 2004, ISBN 1855781581.
• Irwin H. Segel, Enzyme Kinetics : Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems. Wiley-Interscience; New Ed edition 1993, ISBN 0471303097.
• Alan Fersht, Structure and Mechanism in Protein Science : A Guide to Enzyme Catalysis and Protein Folding. W. H. Freeman, 1998. ISBN 0716732688
• Chris Walsh, Enzymatic Reaction Mechanisms. W. H. Freeman and Company. 1979. ISBN 0716700700
• Nicholas Price, Lewis Stevens, Fundamentals of Enzymology, Oxford University Press, 1999. ISBN 019850229X
• Tim Bugg, An Introduction to Enzyme and Coenzyme Chemistry Blackwell Publishing, 2004 ISBN 1405114525