Enzyme Kinetics and
Inhibition
Andy Howard
Introductory Biochemistry, Spring 2008
12 February 2008
12 Feb 2008
What’s the relationship
between concentration and
reaction rate?

In the presence of an enzyme,
there is a definable relationship
between substrate concentration
and reaction rate.
Biochemistry: Enzyme
Kinetics & inhibition
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Plans for Today
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Enzyme kinetics
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Michaelis-Menten
kinetics: overview
Kinetic Constants
Kinetic Mechanisms
Induced Fit
Measurements and
calculational tools
Multisubstrate
reactions
Biochemistry: Enzyme
Kinetics & inhibition
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Enzyme Inhibition
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Why study it?
The concept
Types of inhibitors
Kinetics of
inhibition
Pharmaceutical
inhibitors
What makes an
inhibitor a useful
drug?
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More complex cases
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More complicated than this if >1 reactant
involved or if a catalyst whose
concentration influences the production of
species B is present.
If >1 reactant required for making B, then
usually the reaction will be linear in the
concentration of the scarcest reactant and
nearly independent of the concentration of
the more plentiful reactants.
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Kinetics & inhibition
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Bimolecular reaction
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If in the reaction
A+DB
the initial concentrations of [A] and [D]
are comparable, then the reaction rate
will be linear in both [A] and [D]:
d[B]/dt = v = k[A][D] = k[A]1[D]1
I.e. the reaction is first-order in both A
and D, and it’s second-order overall
Biochemistry: Enzyme
Kinetics & inhibition
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Forward and backward

Rate of reverse reaction may
not be the same as the rate at
which the forward reaction
occurs. If the forward reaction
rate of reaction 1 is designated
as k1, the backward rate
typically designated as k-1.
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Kinetics & inhibition
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Multi-step reactions

In complex reactions, we may need to keep
track of rates in the forward and reverse
directions of multiple reactions. Thus in the
conversion A  B  C
we can write rate constants
k1, k-1, k2, and k-2
as the rate constants associated with converting
A to B, converting B to A, converting B to C, and
converting C to B.
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Kinetics & inhibition
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[ES]
Michaelis-Menten
kinetics
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t
A very common situation is one in which for
some portion of the time in which a reaction is
being monitored, the concentration of the
enzyme-substrate complex is nearly constant.
Thus in the general reaction
E + S  ES  E + P
where E is the enzyme, S is the substrate, ES is
the enzyme-substrate complex (or "enzymeintermediate complex"), and P is the product
We find that [ES] is nearly constant for a
considerable stretch of time.
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Kinetics & inhibition
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Michaelis-Menten rates
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Rate at which new ES molecules are being
produced in the first forward reaction is
equal to the rate at which ES molecules are
being converted to (E and P) and (E and S).
Rate of formation of ES from left =
vf = k1([E]tot - [ES])[S]
because the enzyme that is already
substrate-bound is unavailable!
Biochemistry: Enzyme
Kinetics & inhibition
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Equating the rates
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Rate of disappearance of ES on right and
left is
vd = k-1[ES] + k2[ES] = (k-1+ k2)[ES]
This rate of disappearance should be
equal to the rate of appearance
Under these conditions vf = vd.
Biochemistry: Enzyme
Kinetics & inhibition
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Derivation, continued
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Thus since vf = vd.
k1([E]tot - [ES])[S] = (k-1+ k2)[ES]
Km  (k-1+ k2)/k1 =
([E]tot - [ES])[S] / [ES]
[ES] = [E]tot [S] / (Km + [S])
But the rate-limiting reaction is the
formation of product: v0 = k2[ES]
Thus v0 = k2[E]tot [S] / (Km + [S])
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Kinetics & inhibition
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Maximum velocity
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What conditions would produce the
maximum velocity?
Answer: very high substrate
concentration ([S] >> [E]tot),
for which all the enzyme would be
bound up with substrate. Thus
under those conditions we get
Vmax = v0 = k2[ES] = k2[E]tot
Biochemistry: Enzyme
Kinetics & inhibition
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Using Vmax in
M-M kinetics
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Thus since
Vmax = k2[E]tot,
v0 = Vmax [S] / (Km+[S])
That’s the famous Michaelis-Menten
equation
Biochemistry: Enzyme
Kinetics & inhibition
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Graphical interpretation
0.01
Michaelis-Menten kinetics
0.009
Initial velocity v0, Ms-1
0.008
0.007
Vmax = 0.01 Ms -1
0.006
Km = 0.03M
0.005
[E]tot = 10 -7M
0.004
kcat = 10 5 s-1
0.003
0.002
0.001
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Substrate conc, M
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Kinetics & inhibition
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0.7
Physical meaning of Km
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As we can see from the plot, the
velocity is half-maximal when [S] = Km
Trivially derivable: if [S] = Km, then
v0 = Vmax[S] / ([S]+[S]) = Vmax /2
We can turn that around and say that
the Km is defined as the concentration
resulting in half-maximal velocity
Km is a property associated with
binding of S to E, not a property of
turnover
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Kinetics & inhibition
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Physical meaning of Vmax
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Vmax is the initial reaction velocity when
the substrate concentration is high
compared to Km:
v0 = Vmax when [S] >> Km.
This result follows directly from
Michaelis-Menten equation:
if [S] >> Km, then [S] + Km ~ [S],
so [S]/(Km+[S]) = 1, and v0 = Vmax.
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Kinetics & inhibition
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kcat
A quantity we often want is the
maximum velocity independent of
how much enzyme we originally
dumped in
 That would be kcat = Vmax / [E]tot
 Oh wait: that’s just the rate of our
rate-limiting step, i.e. kcat = k2
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Kinetics & inhibition
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What do Vmax and kcat tell us?
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These are properties associated with the
enzyme’s ability to turn over substrate,
not its ability to bind substrate.
So if we influence an enzyme’s substrate
affinity without altering its ability to
convert substrate to product, we won’t
change Vmax or kcat.
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Kinetics & inhibition
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Physical meaning of kcat
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Describes turnover of substrate to
product:
Number of product molecules produced
per sec per molecule of enzyme
More complex reactions may not have
kcat = k2, but we can often approximate
them that way anyway
Some enzymes very efficient:
kcat > 106 s-1
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Kinetics & inhibition
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Specificity constant, kcat/Km
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kcat/Km measures affinity of enzyme for a
specific substrate: we call it the specificity
constant for the enzyme for that
particular substrate
Useful in comparing primary substrate to
other substrates (e.g. ethanol vs.
propanol in alcohol dehydrogenase)
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Kinetics & inhibition
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Kinetic Mechanisms
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If a reaction involves >1 reactant or >1
product, there may be variations in
kinetics that occur as a result of the
order in which substrates are bound or
products are released.
Examine figure 5.7 in Horton, which
depicts bisubstrate reactions of various
sorts. As you can see, the possibilities
enumerated include sequential, random,
and ping-pong mechanisms.
Biochemistry: Enzyme
Kinetics & inhibition
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Historical thought
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Biochemists, 1935 - 1970 examined effect on
reaction rates of changing [reactants] and
[enzymes], and deducing the mechanistic
realities from kinetic data.
In recent years other tools have become
available for deriving the same information,
including static and dynamic structural studies
that provide us with slide-shows or even movies
of reaction sequences.
But things like fig. 5.7 still help!
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Kinetics & inhibition
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Sequential, ordered
reactions
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W.W.Cleland
Substrates, products must bind in
specific order for reaction to complete
A
B
P
Q
_____________________________
E
EA (EAB) (EPQ) EQ E
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Kinetics & inhibition
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Sequential, random reactions
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Substrates can come in in either order, and
products can be released in either order
A
B
P Q
EA
EQ
__
E
(EAB)(EPQ)
E
EB
EP
B
A
Q P
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Kinetics & inhibition
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Ping-pong mechanism
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First substrate enters, is altered, is
released, with change in enzyme
Then second substrate reacts with
altered enzyme, is altered, is released
Enzyme restored to original state
A
P
B
Q
E EA FA F
Biochemistry: Enzyme
Kinetics & inhibition
FB FQ E
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Historical thought



Biochemists, 1935 - 1970 examined effect on
reaction rates of changing [reactants] and
[enzymes], and deducing the mechanistic realities
from kinetic data.
In recent years other tools have become available
for deriving the same information, including static
and dynamic structural studies that provide us with
slide-shows or even movies of reaction
sequences.
But things like fig. 5.7 still help!
Biochemistry: Enzyme
Kinetics & inhibition
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Induced fit
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Conformations of enzymes don't change
enormously when they bind substrates,
but they do change to some extent. An
instance where the changes are fairly
substantial is the binding of substrates to
kinases.
Danger:enzyme will catalyze the
unproductive hydrolysis of ATP in the
same site where the kinase reaction
might occur
Biochemistry: Enzyme
Kinetics & inhibition
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Daniel
Koshland
12 Feb 2008
Kinase reactions
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unwanted reaction
ATP + H-O-H ⇒ ADP + Pi
will compete with the desired reaction
ATP + R-O-H ⇒ ADP + R-O-P
Kinases minimize the likelihood of this
unproductive activity by changing
conformation upon binding substrate so that
hydrolysis of ATP cannot occur until the
binding happens.
Illustrates the importance of the order in which
things happen in enzyme function
Biochemistry: Enzyme
Kinetics & inhibition
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Measurements and
calculations
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The standard Michaelis-Menten
formulation is v0=f([S]), but it’s not linear
in [S]. We seek linearizations of the
equation so that we can find Km and kcat,
and so that we can understand how
various changes affect the reaction.
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Kinetics & inhibition
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Lineweaver-Burk
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Dean
Burk
Simple linearization of Michaelis-Menten:
v0 = Vmax[S]/(Km+[S]). Take reciprocals:
1/v0 = (Km +[S])/(Vmax[S])
= (Km /(Vmax[S])) + [S]/(Vmax[S]))
Hans
Lineweaver
= (Km/Vmax)*1/[S] + 1/Vmax
Thus a plot of 1/[S] as the independent variable
vs. 1/v0 as the dependent variable will be linear
with Y-intercept = 1/Vmax and slope Km/Vmax
Biochemistry: Enzyme
Kinetics & inhibition
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How to use this
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Y-intercept is useful directly:
computeVmax = 1/(Y-intercept)
We can get Km/Vmax from slope and then
use our knowledge of Vmax to get Km; or
X intercept = -1/ Km
… that gets it for us directly!
Biochemistry: Enzyme
Kinetics & inhibition
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Demonstration that the
X-intercept is at -1/Km
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X-intercept means Y = 0
In Lineweaver-Burk plot,
0 = (Km/Vmax)*1/[S] + 1/Vmax
For nonzero 1/Vmax we divide through:
0 = Km /[S] + 1, -1 = Km/[S], [S] = -Km.
But the axis is for 1/[S], so the intercept is
at 1/[S] = -1/ Km.
Biochemistry: Enzyme
Kinetics & inhibition
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Graphical form of L-B
1/v0, sLmol-1
1/Vmax,
sLmol-1
Slope=Km/Vmax
1/[S], M-1
-1/Km, Lmol-1
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Kinetics & inhibition
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Are those values to the left of
1/[S] = 0 physical?
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No. It doesn’t make sense to talk about
negative substrate concentrations or
infinite substrate concentrations.
But if we can curve-fit, we can still use
these extrapolations to derive the kinetic
parameters.
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Kinetics & inhibition
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Advantages and
disadvantages of L-B plots
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Easy conceptual reading of Km and Vmax (but
remember to take the reciprocals!)
Suboptimal error analysis
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[S] and v0 values have errors
Error propagation can lead to significant uncertainty in
Km (and Vmax)
Other linearizations available
(see homework)
Better ways of getting Km and Vmax available
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Kinetics & inhibition
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Don’t fall into the trap!
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When you’re calculating Km and Vmax from
Lineweaver-Burk plots, remember that you
need the reciprocal of the values at the
intercepts
If the X-intercept is -5000 M-1, then
Km = -1/(X-intercept) =(-)(-1/5000 M-1) =
2*10-4M
Sanity check: typically 10-5M < Km < 10-3M
Typically kcat ~ 10 to 107 s-1 (table 5.1),
so for typical [E]=10-7M,
Vmax = [E]kcat = 10-6 Ms-1 to 1 Ms-1
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Kinetics & inhibition
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iClicker quiz
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1. The kinase reaction just described
probably operates according to a
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(a) sequential, random mechanism
(b) sequential, ordered mechanism
( c) ping-pong mechanism
(d) none of the above.
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Kinetics & inhibition
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iClicker quiz
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2. If we alter the kinetics of a reaction by
increasing Km but leaving Vmax alone, how will
the L-B plot change?
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(a) X intercept will move toward the origin and the Y
intercept will remain as it was
(b) X intercept will move away from origin and Y
intercept will remain as it was
( c) Y intercept will move away from origin and X
intercept will remain as it was
(d) Y intercept will move toward origin and X intercept
will remin as it was
(e) None of the above
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Kinetics & inhibition
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L-B plots
for ordered
sequential
reactions
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http://www-biol.paisley.ac.uk/
kinetics/Chapter_4/chapter4_3.html
Plot 1/v0 vs. 1/[A] for various [B] values;
flatter slopes correspond to larger [B]
Lines intersect @ a point
in between X intercept and Y intercept
Biochemistry: Enzyme
Kinetics & inhibition
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L-B plots
for pingpong
reactions
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Again we plot 1/v vs 1/[A] for various [B]
Parallel lines (same kcat/Km);
lower lines correspond to larger [B]
http://www-biol.paisley.ac.uk/kinetics/
Chapter_4/chapter4_3_2.html
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Kinetics & inhibition
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Why study inhibition?
• Let’s look at how enzymes get inhibited.
• At least two reasons to do this:
• We can use inhibition as a probe for
understanding the kinetics and properties of
enzymes in their uninhibited state;
• Many—perhaps most—drugs are inhibitors of
specific enzymes.
• We'll see these two reasons for
understanding inhibition as we work our
way through this topic.
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Kinetics & inhibition
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The concept of inhibition
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An enzyme is a biological catalyst, i.e. a
substance that alters the rate of a reaction
without itself becoming permanently
altered by its participation in the reaction.
The ability of an enzyme (particularly a
proteinaceous enzyme) to catalyze a
reaction can be altered by binding small
molecules to it
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sometimes at its active site
sometimes at a site distant from the active site.
Biochemistry: Enzyme
Kinetics & inhibition
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Inhibitors and Accelerators
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Usually these alterations involve a
reduction in the enzyme's ability to
accelerate the reaction; less commonly,
they give rise to an increase in the
enzyme's ability to accelerate a reaction.
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Kinetics & inhibition
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Why more inhibitors
than accelerators?
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Natural selection: if there were small molecules
that can facilitate the enzyme's propensity to
speed up a reaction, nature probably would
have found a way to incorporate those
facilitators into the enzyme over the billions of
years that the enzyme has been available.
Most enzymes are already fairly close to
optimal in their properties; we can readily mess
them up with effectors, but it's more of a
challenge to find ways to make enzymes better
at their jobs.
Biochemistry: Enzyme
Kinetics & inhibition
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Types of inhibitors
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Irreversible
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Inhibitor binds without possibility of release
Usually covalent
Each inhibition event effectively removes a
molecule of enzyme from availability
Reversible
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Usually noncovalent (ionic or van der Waals)
Several kinds
Classifications somewhat superseded by
detailed structure-based knowledge of
mechanisms, but not entirely
Biochemistry: Enzyme
Kinetics & inhibition
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Types of reversible inhibition
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Competitive
 Inhibitor binds at active site
 Prevents binding of substrate
Noncompetitive
 Inhibitor binds distant from active site
 Interferes with turnover
Uncompetitive (rare?)
 Inhibitor binds to ES complex
 Removes ES, interferes with turnover
Mixed
(usually Competitive + Noncompetitive)
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Kinetics & inhibition
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How to tell them apart
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Reversible vs irreversible
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dialyze an enzyme-inhibitor complex
against a buffer free of inhibitor
if turnover or binding still suffers, it’s
irreversible
Competitive vs. other reversible:
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Structural studies if feasible
Kinetics
Biochemistry: Enzyme
Kinetics & inhibition
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Competitive inhibition
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S
Put in a lot of substrate:
ability of the inhibitor to get
in the way of the binding is hindered:
out-competed by sheer #s of substrate
molecules.
How many substrate molecules it will take to
overwhelm the inhibitor depends on how
strongly the enzyme is attracted to the substrate
as compared to the attraction of the enzyme for
the inhibitor. However, once the substrate is
bound, the inhibitor does not influence how
quickly the enzyme turns the substrate over.
Biochemistry: Enzyme
Kinetics & inhibition
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Ic
Kinetics of competition
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Competitive inhibitor hinders binding of
substrate but not reaction velocity:
Affects the Km of the enzyme, not Vmax.
Which way does it affect it?
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

Km = amount of substrate that needs to be
present to run the reaction velocity up to half
its saturation velocity.
Competitive inhibitor requires us to shove
more substrate into the reaction in order to
achieve that half-maximal velocity.
So: competitive inhibitor increases Km,
Biochemistry: Enzyme
Kinetics & inhibition
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L-B: competitive inhibitor
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Km goes up so -1/ Km moves toward origin
Vmax unchanged so Y intercept unchanged
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Kinetics & inhibition
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Competitive inhibitor:
Quantitation of Ki
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Define inhibition constant Ki to be the
concentration of inhibitor that
increases Km by a factor of two.
Km,obs = Km(1+[Ic]/Ki)
So [Ic] that moves Km halfway to the
origin is Ki.
If Ki = 100 nM and [Ic] = 1 µM, then
we’ll increase Km,obs elevenfold!
Biochemistry: Enzyme
Kinetics & inhibition
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Noncompetitive I
inhibition


S
Noncompetitive inhibitor has no
influence on how available the
binding site for substrate is, so it
does not affect Km at all
However, it has a profound
inhibitory influence on the speed
of the reaction, i.e. turnover. So
it reduces Vmax and has no
influence on Km.
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Kinetics & inhibition
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L-B for non-competitives
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Decrease in Vmax  1/Vmax is larger
X-intercept unaffected
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Kinetics & inhibition
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Ki for noncompetitives
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Ki defined as concentration of inhibitor
that cuts Vmax in half
Vmax,obs = Vmax/(1 + [In]/Ki)
In previous figure the “high”
concentration of inhibitor is Ki
Biochemistry: Enzyme
Kinetics & inhibition
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Uncompetitive inhibition
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Inhibitor binds only if ES has already formed
It creates a ternary ESI complex
This removes ES, so by LeChatlier’s Principle it
actually drives the original reaction (E + S 
ES) to the right; so it decreases Km
But it interferes with turnover so Vmax goes
down
If Km and Vmax decrease at the same rate, then
it’s classical uncompetitive inhibition.
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Kinetics & inhibition
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L-B for uncompetitives
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Km moves toward origin
Vmax moves away from the origin
Slope ( Km/Vmax) is unchanged
Biochemistry: Enzyme
Kinetics & inhibition
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Ki for
uncompetitives
Defined as inhibitor concentration
that cuts Vmax or Km in half
 Easiest to read from Vmax value
 Iu labeled “high” is Ki in this plot

Biochemistry: Enzyme
Kinetics & inhibition
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iClicker quiz!

3. Treatment of enzyme E with
compound Y doubles Km and leaves
Vmax unchanged. Compound Y is:
 (a) an accelerator of the reaction
 (b) a competitive inhibitor
 (c) a non-competitive inhibitor
 (d) an uncompetitive inhibitor
Biochemistry: Enzyme
Kinetics & inhibition
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iClicker quiz, question 4

4. Treatment of enzyme E with
compound X doubles Vmax and leaves Km
unchanged. Compound X is:
 (a) an accelerator of the reaction
 (b) a competitive inhibitor
 (c) a non-competitive inhibitor
 (d) an uncompetitive inhibitor
Biochemistry: Enzyme
Kinetics & inhibition
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Most pharmaceuticals are
enzyme inhibitors

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
Inhibitors of enzymes that are necessary
for functioning of pathogens
Others are inhibitors of some protein
whose inappropriate expression in a
human causes a disease.
Others are targeted at enzymes that are
produced more energetically by tumors
than they are by normal tissues.
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Kinetics & inhibition
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Characteristics of
Pharmaceutical Inhibitors

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Usually competitive, i.e. they raise Km
without affecting Vmax
Some are mixed, i.e. Km up, Vmax down
Iterative design work will decrease Ki
from millimolar down to nanomolar
Sometimes design work is purely blind
HTS; other times, it’s structure-based
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Kinetics & inhibition
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Amprenavir


Competitive inhibitor of HIV
protease,
Ki = 0.6 nM for HIV-1
No longer sold: mutual
interference with rifabutin,
which is an antibiotic used
against a common HIV
secondary bacterial infection,
Mycobacterium avium
Biochemistry: Enzyme
Kinetics & inhibition
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When is a good inhibitor a
good drug?



It needs to be bioavailable and nontoxic
Beautiful 20nM inhibitor is often neither
Modest sacrifices of Ki in improving
bioavailability and non-toxicity are okay if
Ki is low enough when you start
sacrificing
Biochemistry: Enzyme
Kinetics & inhibition
P. 63 of 65
12 Feb 2008
How do we lessen toxicity
and improve bioavailability?



Increase solubility…
that often increases Ki because the van
der Waals interactions diminish
Solubility makes it easier to get the
compound to travel through the
bloodstream
Toxicity is often associated with fat
storage, which is more likely with
insoluble compounds
Biochemistry: Enzyme
Kinetics & inhibition
P. 64 of 65
12 Feb 2008
Drug-design timeline
2 years of research, 8 years of trials
-8
Research
0
2
Biochemistry: Enzyme
100
Cost/yr, 106 $
-3
Preliminary toxicity testing

10
Clinical Trials
Time, Yrs
Kinetics & inhibition
10
P. 65 of 65
12 Feb 2008