Enzymes (Second Edition)

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might say that the minimum number of binding sites on this enzyme is 2 and that the sites display a more moderate level of cooperativity. However, there is no compelling evidence from this experiment that the enzyme has only two binding sites. It could have three or four or more binding sites with weaker intersite cooperativity. This is why the value of 2 in this example is said to represent the minimum number of possible binding sites.

As we saw in Chapter 5, the Hill equation can be linearized by taking the logarithm of both sides and rearranging to yield:

This equation can be used to construct linearized plots from which the values of h and K can be determined graphically. An example of a linearized Hill plot was given in Chapter 5 (Figure 5.16). Despite the form of Equation 12.8, the experimental graphs usually deviate from linearity in the low substrate region, where species with fewer than h substrate molecules bound can contribute to the overall velocity. Typically, the data conform well to a linear function between values of [S] yielding 10—90% saturation (i.e., V ). The slope of the best fit line between these limits is commonly taken as the average value of h .

The degree of sigmoidicity of the direct velocity plot is a measure of the strength of cooperativity between sites in an oligomeric enzyme. This is best measured by taking the ratio of substrate concentrations required to reach two velocities representing different fractions of V . Most commonly this is done using the substrate concentrations for which v 0.9V , known as [S] , and

for which v 0.1V , known as [S] . The ratio [S] /[S] , the cooperativity index, is an inverse measure of cooperatiave interactions; in other words,

the larger the difference in substrate concentration required to span the range

of v 0.1V to v 0.9V , the larger the value of [S] /[S] and the weaker the degree of cooperativity between sites. The value of the cooperativity

index is related to the Hill coefficient h, and K as follows:

Thus the Hill coefficient and the cooperativity index for an oligomeric enzyme can be related to each other, and together they provide a measure of the degree of cooperativity between binding sites on the enzyme and the minimum number of these interacting sites.

While the Hill coefficient is a convenient and commonly used index of cooperativity, it is not a direct measure of the change in free energy of binding ( G) that must exist in cooperative systems. A thermodynamic treatment of cooperativity for a two-site system presented by Forsen and Linse (1995) discusses the changes in binding affinities in terms of changes in binding free energies. This alternative treatment offers a straightforward means of describing the phenomenon of cooperativity in more familiar thermodynamic terms. It is well worth reading.

Another useful method for diagnosing the presence of cooperativity in enzyme kinetics is to plot the velocity curves in semilog form (velocity as a function of log[S]), as presented in Chapter 8 for dose—response plots of enzyme inhibitors. Such plots always yield a sigmoidal curve, regardless of whether cooperativity is involved. The steepness of the curve, however, is related to the degree of positive or negative cooperativity. When the enzyme displays positive cooperativity, the curves reach saturation with a much steeper slope than in the absence of cooperativity. Likewise, when negative cooperativity is in place, the saturation curve displays a much shallower slope (Neet, 1980). The data in these semilog plots is still well described by Equation 12.7, as illustrated in Figure 12.8 for examples of positive cooperativity. The steepness of the curves in these plots is directly related to the value of h that appears in Equation 12.7.

These plots are useful because the presence of cooperativity is very readily apparent in these plots. The effect of positive or negative cooperativity on the steepness of the curves is much more clearly observed in the semilog plot as opposed to the linear plot, especially in the case of small degrees of cooperativity. The steepness of the curves in such semilog plots is also diagnostic of cooperative effects in ligands other than substrate. Thus, for example, the IC equation introduced in Chapter 8 (Equation 8.20) can be modified to include a term to account for cooperative effects in inhibitor binding to enzymes as well.

382 COOPERATIVITY IN ENZYME CATALYSIS

Figure 12.8 Velocity as a function of substrate concentration plotted in semilog fashion: data for a noncooperative enzyme; squares and triangles, data for enzymes displaying positive cooperativity. Each solid line through the data represents the best fit of an individual data set to Equation 12.7.

12.4 SIGMOIDAL KINETICS FOR NONALLOSTERIC ENZYMES

The appearance of sigmoidal kinetics in enzyme velocity curves for allosteric enzymes is a reflection of the cooperativity of the substrate binding events that precede the catalytic steps at the enzyme active sites. The same cooperativity should be realized in direct studies of ligand binding by the enzyme, which can be performed by equilibrium dialysis, certain spectroscopic methods, and so on (Chapter 4). If true allostery is involved, the cooperativity of ligand binding should be measurable in the enzyme velocity curves and in the separate binding experiments as well. In some cases, however, the direct ligand binding experiments fail to display the same cooperativity observed in the velocity measurements. One must assume that such ligand binding events are not cooperative, which means that some other explanation must be sought to account for the sigmoidal velocity curve.

One way of observing sigmoidal kinetics in the absence of true cooperativity entails an enzyme preparation containing a mixture of enzyme isoforms that have different K values for the substrate (Palmer, 1985). In such cases the velocity curve will be the superposition of the individual curves for the varied isoforms. If two or more isoforms differ significantly in K for the substrate, a nonhyperbolic curve, resembling the sigmoidal behavior of cooperative enzyme, may result.

Also, it has been noted that a two-substrate enzyme that follows a random ordered mechanism can display sigmoidal kinetics without true cooperativity.

This occurs when one of the two ordered reactions proceeds faster than the competing ordered reaction: when, for example, formation of E · AX then E · AX · B and subsequent product release is faster than formation of E · B then E · B · AX and product release. In the case of two ordered reactions of unequal speed, the affinity of the free enzyme for substrate B is less than the affinity of the E · AX complex for B. If [E ] and [B] are held constant while [AX] is varied at low concentrations of AX the enzyme will react mainly with substrate B first, and thus will proceed through the slower of the two pathways to product. As the concentration of AX increases, there will be a greater probability of the enzyme first binding this substrate and proceeding via the faster pathway. The observed result of this pathway ‘‘switching’’ with increasing substrate concentration is a sigmoidal plot of velocity as a function of [AX].

Finally, sigmoidal kinetics can be observed even for a monomeric single binding site enzyme if substrate binding induces a catalytically required conformational transiton of the enzyme. If the isomerization step after substrate binding is rate limiting, the relative populations of the two isomers, E and E , can influence the overall reaction velocity. If the equilibrium between E and E is perturbed by substrate, the relative populations of these two forms of the enzyme will vary with increasing substrate concentration. Again, the end result is the appearance of a sigmoidal curve when velocity is plotted as a function of substrate concentration.

12.5 SUMMARY

In this chapter we presented the concept of cooperative interactions between distal binding sites on oligomeric enzymes, which communicate through conformational transitions of the polypeptide chain. These allosteric enzymes display deviations from the normal Henri—Michaelis—Menten behavior that is seen with single substrate binding enzymes, as introduced in Chapter 5. Examples of allosteric proteins and enzymes were described that provide some structural rationale for allosteric interactions in specific cases, and two theoretical models of cooperativity were described. The classic signature of cooperativity in enzyme kinetics is a sigmoidal shape to the curve of velocity versus [S]. The appearance of such sigmoidicity in the enzyme kinetics is not sufficient, however, to permit us to conclude that the substrate binding sites interact cooperatively. Direct measurements of ligand binding must be used to confirm the cooperativity of ligand binding. We saw that in some cases sigmoidal enzyme kinetics exist in the absence of true cooperativity — when, for example, multisubstrate enzymes proceed by different rates depending on the order of substrate addition, and when rate-limiting enzyme isomerization occurs after substrate binding.

The understanding of allostery and cooperativity in structural terms is an active area of research today. This fascinating subject was reviewed by one of the leading experts in the field of allostery, Max Perutz, who spent most of his

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career studying the structural determinants of cooperativity in hemoglobin. The text by Perutz (1990) is highly recommended for those interested in delving deeper into this subject.

REFERENCES AND FURTHER READING

Abelson, P. H. (1954) J. Biol. Chem. 206, 335.

Forsen, S., and Linse, S. (1995) Trends Biochem. Sci. 20, 495.

Koshland, D. E., Nemethy, G., and Filmer, D. (1966) Biochemistry, 5, 365. Monod, J., Wyman, J., and Changeux, J. P. (1965) J. Mol. Biol. 12, 88. Neet, K. E. (1980) Methods Enzymol. 64, 139.

Palmer, T. (1985) Understanding Enzymes, Wiley, New York, pp. 257—274.

Perutz, M. (1990) Mechanisms of Cooperativity and Allosteric Regulation in Proteins, Cambridge University Press, New York.

Segel, I. H. (1975) Enzyme Kinetics, Wiley, New York, pp. 346—464.

388 USEFUL COMPUTER SOFTWARE AND WEB SITES FOR ENZYME STUDIES

EZ-FIT. A practical curve-fitting program for the analysis of enzyme kinetic data. Reference: F. W. Perrella, Anal. Biochem. 174, 437 (1988).

Graphfit. A commercial package for data management and graphic display of enzyme kinetic data and other scientific data graphing. This program has extensive preprogrammed routines for enzyme kinetic analysis and allows global fitting of data of the form y f (x, z), which is very useful for analysis of inhibitor modality, and so on. Available from Erithacus Software Limited, P.O. Box 35, Staines, Middlesex, TW18 2TG, United Kingdom [http://www.erithacus.com]. Also distributed by Sigma Chemical Company.

Graphpad Prism. A general graphic package written for scientific applications. Contains specific equations and routines for enzyme kinetics and equilibrium ligand binding applications. The user’s guide and related Web site are quite informative. Available from Intuitive Software for Science, 10855 Sorrento Valley Road, Suite 203, San Diego, CA 92121. Telephone: 858-457-3909. E-mail: support graphpad.com.

Kaleidagraph. A commercial software package for general scientific graphing. Available from Synergy Software, 2457 Perkiomen Ave., Reading, PA 19606. Telephone: 610-779-0522. [http://www.synergy.com].

K·cat. A commercial package for data management and graphic displays of enzyme kinetic and receptor ligand binding data. Available from BioMetallics, Inc., P.O. Box 2251, Princeton, NJ 08543. Telephone: 800-999- 1961.

Kinlsq. A program for fitting kinetic data with numerically integrated rate equations. Provides data analysis routines for tight binding inhibitors as well as classical inhibitors. Reference: W. G. Gutheil, C. A. Kettner, and W. W. Bachovchin, Anal. Biochem. 223, 13 (1994).

Kinsim. A very useful program that allows the researcher to enter a chemical mechanism for a reaction in symbolic terms and have the computer translate this into a set of differential equations that can be solved to predict the concentrations of products and reactants as functions of time, based on the kinetic scheme and the values of the rate constants used. Reference: B. A. Barshop et al., Anal. Biochem. 130, 134 (1983). This program has been greatly expanded and is available free from KinTek Corporation under the name KinTekSim. For instructions on downloading this freeware see the KinTeck Corporation Web site [http://www.kintek-corp.com].

MPA. A program for analyzing enzyme rate data obtained from a microplate reader. Provides a convenient means of transforming and analyzing data directly from 96-well formated data arrays. Reference: S. P. J. Brooks, BioTechniques, 17, 1154 (1994).

Origin. A very robust commercial graphic program that allows fitting of data to equations of the form y f (x, z) and three-dimensional displays

of the resulting fits. Available from Microcal Software, Inc., One Roundhouse Plaza, Northampton, MA 01060. Telephone: 800-969-7720. [http:// www.microcal.com].

Ultrafit. Similar to Enzfitter software, but designed for Apple Macintosh computers. Available from Biosoft, P.O. Box 10938, Ferguson, MO. Telephone: 314-524-8029. E-mail: ab47 cityscape.co.uk.

WEB SITES

Another useful source of programs and information about enzymes and protein biochemistry in general is the Internet. A recent search of the term ‘‘Enzyme’’ yielded over 600 Internet addresses with useful information and analysis packages that the reader can assess. It is well worth the reader’s time to do a little ‘‘surfing’’ on this exciting medium. Rather than attempting a comprehensive listing of these many useful sites, I provide four particularly good starting points for exploring the Internet. Each of these sites contains a wealth of useful information to the biochemist and further provide links to additional Web sites that are of interest.

Data/Information for Enzymologists and Kineticists

http://www.med.umich.edu/biochem/enzresources/realenzymes.html

This site is part of an online contents page for an enzymology course taught at the University of Michigan. It contains a collection of links to useful Web sites of interest to enzymologists.

The Enzyme Data Bank

http://192.239.77.6/Dan/proteins/ec-enzymes.html

This site provides information on EC numbers, recommended and alternative names, catalytic activities, cofactor utilization, disease association, and other useful facts about enzymes. It is an excellent starting point for gaining information on a particular enzyme of interest.

ExPASy Molecular Biology Server

http://www.expasy.ch

This is the Web site for the molecular biology server of the Swiss Institute of Bioinformatics. It contains a number of useful databases and protein analysis tools. It also provides direct links to other molecular biology and biochemistry Web sites.