Enzyme kinetics

File:EcDHFR.png

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 enzyme kinetics provides information on how enzymes work, how their activity is controlled in the cell and how drugs and poisons can inhibit their reactions.

Enzymes are molecules that manipulate other molecules—the enzymes' substrates. These target molecules bind to an enzyme's active site and then are transformed into products. A complete understanding of an enzyme requires knowledge of both its structure and its mechanism. Studying an enzyme's structure is akin to producing its complete blueprint. The operating mechanism of the enzyme is understood by the study of its kinetics, which provides information similar to a movie of the enzyme in action. These complementary analyses can be combined to give a complete picture of how substrates bind in the enzyme's active site and what steps are involved in the 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 how fast it can turn it into a product. 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.

However, not all biological catalysts are enzymes, and 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, but their reaction mechanisms and kinetics can be analysed and classified using the same methods as used for enzymes.

General principles

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 can achieve. Knowing these properties suggests what an enzyme may do in the environment of the cell and can show how the enzyme will respond to changes in these conditions.

Enzyme assays

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.^[1]

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.^[2] 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.^[3] 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.^[4] This approach is very 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.^[5] 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

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 (or [S]), as [S] increases the reaction rate (v) also 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.

A model for the mechanism of a single-substrate reactions is shown on the right. 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 k₂. 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].

${\begin{matrix}v=k_{2}\end{matrix}}$ (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 (this equilibrium assumption only applies when k₂ is much less than k_-1).^[6]

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

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. (Link: Java required).

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 shows 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].

If k₂ is similar or more than k_-1, the Michaelis–Menten equation can still be applied under the steady-state approximation. Here, the concentration of ES is regarded as constant during the initial-rate period. This approach is called Briggs–Haldane kinetics and gives equation 3, but K_m in this case is the product of a set of rate constants and does not equal the disassociation constant of ES.^[7]

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 are 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. 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 behavior of metabolic pathways reaches its most complex expression in the production of 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 such as Escherichia coli.^[8]^[9]

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 K_mapparent and V_maxapparent for substrate B. If a set of these measurements is then 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 substrates A and B and turns them into 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 or they can bind in a particular order; these two possibilities are therefore known as random or ordered mechanisms. 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,^[10] dihydrofolate reductase^[11] and DNA polymerase.^[12] These two links show short animations showing the mechanisms of dihydrofolate reductase (Gif) and DNA polymerase (Gif): two enzymes with ternary complex mechanisms.

Ping–pong mechanisms

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

As shown on the right, these enzymes can exist in two structures, E and a chemically-modified form of the enzyme E*. This modified enzyme is known as an intermediate. 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 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^[13] or serine proteases such as trypsin or chymotrypsin.^[14] Serine proteases, include both digestive enzymes (trypsin, chymotrypsin, and elastase) and several enzymes of the blood clotting cascade. 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. This link is to a short animation showing the mechanism of chymotrypsin (Flash required).

Non-Michaelis-Menten 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.^[15] 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,^[16] and bacterial aspartate transcarbamoylase,^[17] and phosphofructokinase^[18] 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^[19] 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

File:Burst phase.png

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.

This approach was first applied to the hydrolysis reaction catalysed by chymotrypsin^[20] 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*.^[21] In the case of chymotrypsin, the burst phase corresponds to 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 burst phase measures a single turnover of the enzyme. Consequently, the size of this burst, shown as the intercept on the y-axis of the graph, gives the amount of functional enzyme which is present in the assay.^[22]

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.^[23] 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 may also be elucidated by examining the kinetics and isotope effects under different pH conditions,^[24] by altering the metal ions or other bound cofactors,^[25] 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

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 left 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,^[26] that vary in distinctive ways with inhibitor concentration. For brevity, two symbols are used:

\alpha =1+{\frac {[I]}{K_{i}}}

and

\alpha ^{\prime }=1+{\frac {[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^[27] 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. The kinetics of these reactions follow exponential decay functions and are usually saturable, but they follow first order kinetics with respect to inhibitor below saturation concentrations.

Mechanisms of catalysis

Stabilisation of the transition state by an enzyme.

The favored model for the enzyme–substrate interaction is the induced fit model.^[28] 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.

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 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.^[29]^[30]

References

^ Eisenthal R. Danson M.J. (Eds), Enzyme Assays: A Practical Approach. Oxford University Press (2002) ISBN 0-19-963820-9
^ Xie XS, Lu HP. Single-molecule enzymology. J Biol Chem. 1999 Jun 4;274(23):15967-70. PMID 10347141
^ Gibson Q.H. Rapid mixing: Stopped flow Methods in Enzymology, (1969) 16:187-228
^ Duggleby, R.G. Analysis of enzyme progress curves by non-linear regression. Methods in Enzymology, (1995) 249:61-90.
^ Hammann C, Lilley DM. Folding and activity of the hammerhead ribozyme. Chembiochem. 2002 Aug 2;3(8):690-700. PMID 11779233
^ Michaelis L. and Menten M.L. Kinetik der Invertinwirkung Biochem. Z. 1913; 49:333–369
^ Briggs GE, Haldane JB. A Note on the Kinetics of Enzyme Action. Biochem J. 1925;19(2):338-9. PMID 16743508
^ Almaas E, Kovacs B, Vicsek T, Oltvai ZN, Barabasi AL. Global organization of metabolic fluxes in the bacterium Escherichia coli. Nature. 2004 Feb 26;427(6977):839-43. PMID 14985762
^ Reed JL, Vo TD, Schilling CH, Palsson BO. An expanded genome-scale model of Escherichia coli K-12 (iJR904 GSM/GPR). Genome Biol. 2003;4(9):R54. PMID 12952533
^ Dirr H, Reinemer P, Huber R. X-ray crystal structures of cytosolic glutathione S-transferases. Implications for protein architecture, substrate recognition and catalytic function. Eur J Biochem. 1994 Mar 15;220(3):645-61. PMID 8143720
^ Stone SR, Morrison JF. Dihydrofolate reductase from Escherichia coli: the kinetic mechanism with NADPH and reduced acetylpyridine adenine dinucleotide phosphate as substrates. Biochemistry. 1988 Jul 26;27(15):5493-9. PMID 3052577
^ Fisher PA. Enzymologic mechanism of replicative DNA polymerases in higher eukaryotes. Prog Nucleic Acid Res Mol Biol. 1994;47:371-97. PMID 8016325
^ Akerman SE, Muller S. 2-Cys peroxiredoxin PfTrx-Px1 is involved in the antioxidant defence of Plasmodium falciparum. Mol Biochem Parasitol. 2003 Aug 31;130(2):75-81. PMID 12946843
^ Kraut J. Serine proteases: structure and mechanism of catalysis. Annu Rev Biochem. 1977;46:331-58. PMID 332063
^ Ricard J, Cornish-Bowden A. Co-operative and allosteric enzymes: 20 years on. Eur J Biochem. 1987 Jul 15;166(2):255-72. PMID 3301336
^ Ward WH, Fersht AR., Tyrosyl-tRNA synthetase acts as an asymmetric dimer in charging tRNA. A rationale for half-of-the-sites activity. Biochemistry. 1988 Jul 26;27(15):5525-30. PMID 3179266
^ Helmstaedt K, Krappmann S, Braus GH., Allosteric regulation of catalytic activity: Escherichia coli aspartate transcarbamoylase versus yeast chorismate mutase. Microbiol. Mol. Biol. Rev. 2001 Sep;65(3):404-21 PMID 11528003
^ Schirmer T, Evans PR., Structural basis of the allosteric behaviour of phosphofructokinase. Nature. 1990 Jan 11;343(6254):140-5. PMID 2136935
^ 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.
^ Hartley B.S. and Kilby B.A. The reaction of p-nitrophenyl esters with chymotrypsin and insulin. Biochem J. 1954 Feb;56(2):288-97. PMID 13140189
^ Alan Fersht, Structure and Mechanism in Protein Science : A Guide to Enzyme Catalysis and Protein Folding. W. H. Freeman, 1998. ISBN 0-7167-3268-8
^ Bender ML, Begue-Canton ML, Blakeley RL, Brubacher LJ, Feder J, Gunter CR, Kezdy FJ, Killheffer JV Jr, Marshall TH, Miller CG, Roeske RW, Stoops JK. The Determination of the Concentration of Hydrolytic Enzyme Solutions : a-Chymotrypsin, Trypsin, Papain, Elastase, Subtilisin, and Acetylcholinesterase. J Am Chem Soc. 1966 Dec 20;88(24):5890-913. PMID 5980876
^ Cleland WW. The use of isotope effects to determine enzyme mechanisms. Arch Biochem Biophys. 2005 Jan 1;433(1):2-12. PMID 15581561
^ Cleland WW. Use of isotope effects to elucidate enzyme mechanisms. CRC Crit Rev Biochem. 1982;13(4):385-428. PMID 6759038
^ Christianson DW, Cox JD. Catalysis by metal-activated hydroxide in zinc and manganese metalloenzymes. Annu Rev Biochem. 1999;68:33-57. PMID 10872443
^ Tseng SJ, Hsu JP. A comparison of the parameter estimating procedures for the Michaelis–Menten model. J Theor Biol. 1990 Aug 23;145(4):457–64. PMID 2246896
^ Leatherbarrow RJ. Using linear and non-linear regression to fit biochemical data. Trends Biochem Sci. 1990 Dec;15(12):455–8. PMID 2077683
^ Koshland DE, Application of a Theory of Enzyme Specificity to Protein Synthesis. Proc. Natl. Acad. Sci. U.S.A. 1958 Feb;44(2):98-104. PMID 16590179
^ Garcia-Viloca M, Gao J, Karplus M, Truhlar DG. How enzymes work: analysis by modern rate theory and computer simulations. Science. 2004 Jan 9;303(5655):186-95. PMID 14716003
^ Olsson MH, Siegbahn PE, Warshel A. Simulations of the large kinetic isotope effect and the temperature dependence of the hydrogen atom transfer in lipoxygenase. J Am Chem Soc. 2004 Mar 10;126(9):2820-8. PMID 14995199

External links

MACiE A database of enzyme reaction mechanisms.
ENZYME Expasy enzyme nomenclature database.
ExCatDB A database of enzyme catalytic mechanisms.
BRENDA Comprehensive enzyme database, giving substrates, inhibitors and reaction diagrams.
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.

[1] Eisenthal R. Danson M.J. (Eds), Enzyme Assays: A Practical Approach. Oxford University Press (2002) ISBN 0-19-963820-9

[2] Xie XS, Lu HP. Single-molecule enzymology. J Biol Chem. 1999 Jun 4;274(23):15967-70. PMID 10347141

[3] Gibson Q.H. Rapid mixing: Stopped flow Methods in Enzymology, (1969) 16:187-228

[4] Duggleby, R.G. Analysis of enzyme progress curves by non-linear regression. Methods in Enzymology, (1995) 249:61-90.

[5] Hammann C, Lilley DM. Folding and activity of the hammerhead ribozyme. Chembiochem. 2002 Aug 2;3(8):690-700. PMID 11779233

[6] Michaelis L. and Menten M.L. Kinetik der Invertinwirkung Biochem. Z. 1913; 49:333–369

[7] Briggs GE, Haldane JB. A Note on the Kinetics of Enzyme Action. Biochem J. 1925;19(2):338-9. PMID 16743508

[8] Almaas E, Kovacs B, Vicsek T, Oltvai ZN, Barabasi AL. Global organization of metabolic fluxes in the bacterium Escherichia coli. Nature. 2004 Feb 26;427(6977):839-43. PMID 14985762

[9] Reed JL, Vo TD, Schilling CH, Palsson BO. An expanded genome-scale model of Escherichia coli K-12 (iJR904 GSM/GPR). Genome Biol. 2003;4(9):R54. PMID 12952533

[10] Dirr H, Reinemer P, Huber R. X-ray crystal structures of cytosolic glutathione S-transferases. Implications for protein architecture, substrate recognition and catalytic function. Eur J Biochem. 1994 Mar 15;220(3):645-61. PMID 8143720

[11] Stone SR, Morrison JF. Dihydrofolate reductase from Escherichia coli: the kinetic mechanism with NADPH and reduced acetylpyridine adenine dinucleotide phosphate as substrates. Biochemistry. 1988 Jul 26;27(15):5493-9. PMID 3052577

[12] Fisher PA. Enzymologic mechanism of replicative DNA polymerases in higher eukaryotes. Prog Nucleic Acid Res Mol Biol. 1994;47:371-97. PMID 8016325

[13] Akerman SE, Muller S. 2-Cys peroxiredoxin PfTrx-Px1 is involved in the antioxidant defence of Plasmodium falciparum. Mol Biochem Parasitol. 2003 Aug 31;130(2):75-81. PMID 12946843

[14] Kraut J. Serine proteases: structure and mechanism of catalysis. Annu Rev Biochem. 1977;46:331-58. PMID 332063

[15] Ricard J, Cornish-Bowden A. Co-operative and allosteric enzymes: 20 years on. Eur J Biochem. 1987 Jul 15;166(2):255-72. PMID 3301336

[16] Ward WH, Fersht AR., Tyrosyl-tRNA synthetase acts as an asymmetric dimer in charging tRNA. A rationale for half-of-the-sites activity. Biochemistry. 1988 Jul 26;27(15):5525-30. PMID 3179266

[17] Helmstaedt K, Krappmann S, Braus GH., Allosteric regulation of catalytic activity: Escherichia coli aspartate transcarbamoylase versus yeast chorismate mutase. Microbiol. Mol. Biol. Rev. 2001 Sep;65(3):404-21 PMID 11528003

[18] Schirmer T, Evans PR., Structural basis of the allosteric behaviour of phosphofructokinase. Nature. 1990 Jan 11;343(6254):140-5. PMID 2136935

[19] 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.

[20] Hartley B.S. and Kilby B.A. The reaction of p-nitrophenyl esters with chymotrypsin and insulin. Biochem J. 1954 Feb;56(2):288-97. PMID 13140189

[21] Alan Fersht, Structure and Mechanism in Protein Science : A Guide to Enzyme Catalysis and Protein Folding. W. H. Freeman, 1998. ISBN 0-7167-3268-8

[22] Bender ML, Begue-Canton ML, Blakeley RL, Brubacher LJ, Feder J, Gunter CR, Kezdy FJ, Killheffer JV Jr, Marshall TH, Miller CG, Roeske RW, Stoops JK. The Determination of the Concentration of Hydrolytic Enzyme Solutions : a-Chymotrypsin, Trypsin, Papain, Elastase, Subtilisin, and Acetylcholinesterase. J Am Chem Soc. 1966 Dec 20;88(24):5890-913. PMID 5980876

[23] Cleland WW. The use of isotope effects to determine enzyme mechanisms. Arch Biochem Biophys. 2005 Jan 1;433(1):2-12. PMID 15581561

[24] Cleland WW. Use of isotope effects to elucidate enzyme mechanisms. CRC Crit Rev Biochem. 1982;13(4):385-428. PMID 6759038

[25] Christianson DW, Cox JD. Catalysis by metal-activated hydroxide in zinc and manganese metalloenzymes. Annu Rev Biochem. 1999;68:33-57. PMID 10872443

[26] Tseng SJ, Hsu JP. A comparison of the parameter estimating procedures for the Michaelis–Menten model. J Theor Biol. 1990 Aug 23;145(4):457–64. PMID 2246896

[27] Leatherbarrow RJ. Using linear and non-linear regression to fit biochemical data. Trends Biochem Sci. 1990 Dec;15(12):455–8. PMID 2077683

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