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Enzyme Kinetics, Inhibition and Activation 9
Figure 1.5A. Graphs v
i
= f ([S]) and Lineweaver
and Burk. Absolute non-competitive inhibition
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10 Enzymes and Drugs
Figure 1.5B. Eadie Hofstee and Dixon representations.
Absolute non-competitive inhibition
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Enzyme Kinetics, Inhibition and Activation 11
Figure 1.5 shows the v
i
= f ([S]) graph and the Lineweaver and Burk, Eadie
Hofstee and Dixon representations for an absolute non-competitive inhibitor (which
binds with the same affinity to the free enzyme and to the enzyme–substrate
complex with K
i
= K
i
’).
1.3.3. Reversible uncompetitive inhibitors
These inhibitors (I) only bind to the enzyme that has bound the substrate (ES
complex). The result is an ESI complex. The inhibitor does not compete with the
substrate, and its action requires prior binding of the substrate to the enzyme. A
conformational change reveals the inhibitor’s binding site. The reaction is as
follows:
Figure 1.6 shows the v
i
= f ([S]) graph and the Lineweaver and Burk, Eadie
Hofstee and Dixon representations for an uncompetitive inhibitor. The equations of
the curves are shown below. The particular characteristics of the lines obtained for
the relative graphs (parallelism, competition (converging lines), slopes of the lines
according to each case) give precise access to the kinetic parameters (V
max
, K
i
and
K
M
).
– Equation of the straight lines obtained for the Lineweaver and Burk
representation:
[
]
[
]
[]
[] []
Mi'
iMmaxmaxi'
max
KS(1I/K)
1/ v (K / V ).1/ S 1/ (V / (1 I /K ))
VS
++
==++
y slope x y-intercept
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12 Enzymes and Drugs
– Equation of the straight lines obtained for the Eadie Hofstee representation:
[
]
{
[
]
}
[
]
imax i' M i'i
vV/(1I/K)K(1I/K).v/S=+−+
y y-intercept slope x
– Equation of the lines obtained for the Dixon representation:
[
]
[
]
[]
{}
[] [ ]
Mi'
i i ' max M max
max
KS(1I/K)
1/ v 1/(K .V ) . I (1 K / S ) / V
VS
++
==++
y slope x y-intercept
Figure 1.6A. Graphs v
i
= f ([S]) and Lineweaver
and Burk. Uncompetitive inhibition
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Enzyme Kinetics, Inhibition and Activation 13
Figure 1.6B. Eadie Hofstee and Dixon representations.
Uncompetitive inhibition (continued)
1.4. Allostery: inhibitors and activators (Monod et al. 1965; Koshland
1970; Kamoun et al. 1992; Fenton 2008; Liu and Nussinov 2016)
Allosteric effectors are biological in nature. They are inhibitors, often
corresponding to products at the end of metabolic chains. They act by feedback on
one of the first enzymes in the chain that produced them. They may also be small
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14 Enzymes and Drugs
molecules that activate or inhibit (NAD
+
; NADH; ATP; ADP; AMP), providing
information on the cell’s energy state.
Allosteric enzymes are made up of several sub-units (monomers), linked by
numerous weak bonds. In addition to an active site, each sub-unit has one or two
additional sites to which allosteric effectors are attached via weak bonds. These
effectors are known as regulators.
1.4.1. Allosteric enzyme kinetics
In the case of these enzymes, Michaelis-Menten relationships are not found.
1.4.1.1. K-type allosteric system
Many of these catalysts produce sigmoid curves (representation v
i
= f ([S]) –
(Figure 1.7)). This is a feature of the cooperative action between the various
monomers. In a K-type allosteric system, these enzymes are characterized by the
fact that the first substrate molecules have difficulty binding to the enzyme. The
higher the concentration, the greater the binding and transformation of the substrate:
this is cooperativity or the positive homotropic effect of the substrate on its own
binding.
Figure 1.7. Graph v
i
= f ([S]) for a K-system allosteric enzyme (at least 10 different
substrate concentrations are tested). A sigmoid curve is obtained for this allosteric
kinetics. The K
1/2
designates the substrate concentration at which half the maxi mum
velocity is reached
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Enzyme Kinetics, Inhibition and Activation 15
The sigmoid equation is:
[
]
[]
h
max
i
h
h
1/ 2
V.S
v
KS
=
+
with h being the Hill coefficient related to the degree of inflection of the curve.
When v
i
is studied according to substrate concentration in the presence of an
excess allosteric activator, an equilateral hyperbola is obtained. Its V
max
does not
change and the K
1/2
decreases compared with the curve obtained without an effector.
This aspect is linked to the fact that the activating molecules will promote substrate
binding to the enzyme and/or its transformation. Here again, this is a cooperative
feature or positive heterotropic effect of the activator on substrate binding and/or its
transformation.
When v
i
is studied according to substrate concentration in the presence of an
excess allosteric inhibitor, the curve obtained tends towards an equilateral
hyperbola. Its V
max
does not vary and the K
1/2
increases compared with the curve
obtained without an effector. This aspect is linked to the fact that inhibitor molecules
will hinder substrate binding to the enzyme and/or its transformation. Here again,
this is a feature of cooperativity or a negative heterotropic effect of the inhibitor on
substrate binding and/or transformation.
1.4.1.2. V-type allosteric system
For other catalysts, the curves (representation v
i
= f ([S])) have the appearance of
an equilateral hyperbola. There is no indication of how the substrate is fixed at this
stage and therefore of any cooperativity.
When v
i
is studied according to substrate concentration in the presence of an
excess allosteric activator, an equilateral hyperbola is obtained. Its V
max
is increased
and K
1/2
is unchanged compared with the curve without an effector. There is nothing
to characterize the mode of binding of the activator and its effect on the substrate at
this stage.
A study of v
i
according to substrate concentration in the presence of an excess
allosteric inhibitor results in an equilateral hyperbola with decreased V
max
and no
change in K
1/2
, compared with the curve without an effector. There is nothing to
characterize the mode of inhibitor binding and its effect on the substrate at this
stage. It could be an absolute non-competitive inhibitor in a Michaelian system.
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16 Enzymes and Drugs
The curves (representation v
i
= f ([S])) for an enzyme operating in a V allosteric
system without an effector, on the one hand, with a saturating concentration of
activator, and with a saturating concentration of inhibitor, on the other hand, are
shown in Figure 1.8.
Figure 1.8. Graph v
i
= f ([S]) for a V-type allosteric enzyme system:
without effector; with saturating activator; with saturating inhibitor
To characterize the cooperative effects generated by allosteric activators and
inhibitors in a V system, further experiments are required. v
i
must be represented as
a function of effector concentration by setting a substrate concentration close to K
1/2
.
Sigmoids are obtained under these conditions and cooperativity traits can thus be
observed. These experiments can also be carried out for effectors in a K system.
1.4.2. Mechanism of allosteric enzyme regulation
Two main models have been proposed to explain the cooperative interactions
between sub-units of an allosteric enzyme. These are the symmetrical model of
Monod, Wyman and Changeux, and the sequential model of Koshland.
1.4.2.1. Symmetrical model
In the Monod model, the enzyme exists in two forms. One form, R, has a high
affinity for the substrate; the other, T, has a low affinity for the substrate. Both forms
transform the substrate (with R >> T).
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Enzyme Kinetics, Inhibition and Activation 17
In the case of a K system, in the absence of substrate and any effector, the
enzyme is predominantly in its T-form. The addition of substrate will produce a
sigmoidal curve, progressively shifting the T/R equilibrium towards the R-form.
Initially, in the presence of an activator (saturating concentration), the enzyme’s T/R
equilibrium will shift towards the R-form. The inhibitor binds preferentially to the
T-form.
In a V system, there are initially as many T-forms as R-forms, and the substrate
binds with as much affinity to the R-form as to the T-form, but is preferentially
catalyzed by the R-form.
1.4.2.2. Sequential model
In the case of Koshland’s concerted model, a tetramer, taken as an example, has
four sub-units which can independently be in R or T-form. Transformation of one
sub-unit to an R-form will facilitate that of subsequent sub-units. Here again, the
substrate is better transformed by R (R >> T).
In short, depending on the allosteric case studied, the most appropriate model
must be chosen.
The different modes of inhibition discussed are of vital importance in the
development of new drugs. Their application to molecules used in therapeutics will
be described in Chapters 2–4 of this book, with the respective advantages and
disadvantages of the different modes of inhibition.
1.5. References
Fenton, A.W. (2008). Allostery: An illustrated definition for the ‘second secret of life’.
Trends Biochem Sci, 33(9), 420–425.
Hopkins, A.L. and Groom, C.R. (2002). The druggable genome. Nat Rev Drug Discov, 1(9),
727–730.
Kamoun, P., Leroux, J.-P., Demaugre, F. (1992). Aide mémoire de biochimie. Flammarion,
Paris.
Kenneth, A.J. and Roger, S.G. (2011). The original Michaelis constant: Translation of the
1913 Michaelis-Menten paper. Biochemistry, 50(39), 8264–8269.
Koshland, D.E. (1970). The molecular basis for enzyme regulation. In The Enzymes,
Boyer, P.D. (ed.). Academic Press, New York, 341–396.
Liu, J. and Nussinov, R. (2016). Allostery: An overview of its history, concepts, methods, and
applications. PLoS Comput Biol, 12(6).
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18 Enzymes and Drugs
Michaelis, L. and Menten, M.L. (1913). Die kinetik der invertinwirkung. Biochemische
Zeitschrift, 191349, 333–369.
Monod, J., Wyman, J., Changeux, J.P. (1965). On the nature of allosteric transitions: A
plausible model. J. Mol. Biol, 88–118.
Rask-Andersen, M., Almén, M.S., Schiöth, H.B. (2011). Trends in the exploitation of novel
drug targets. Nat Rev Drug Discov., 10(8), 579–590.
Stein, R.L., Melandri, F., Dick, L. (1996). Kinetic characterization of the chymotryptic
activity of the 20S proteasome. Biochemistry, 35(13), 3899–3908.
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