Bmm480 Enzymology lecture-3

• Enzyme kinetics: the rate of the enzymatic reaction and how it changes in
response to changes in experimental parameters.
• This is the oldest approach to understanding enzyme mechanisms and
remains the most important.

What is the key factor that effects the rate of enzymatic
reactions ??
• ES : enzyme-substrate complex
• EP: enzyme-product complex
• Keq: equilibrium constant

Substrate Concentration Affects the Rate
of Enzyme-Catalyzed Reactions
A key factor affecting the rate of a reaction
catalyzed by an enzyme is the concentration of
substrate, [S].
However, studying the effects of substrate
concentration is complicated by the fact that
[S] changes during the course of an in vitro
reaction as substrate is converted
to product.
One simplifying approach in kinetics
experiments is to measure the initial rate (or
initial velocity), designated V0, when [S] is
much greater than the concentration of
enzyme, [E].

In a typical reaction, the enzyme may be
present in nanomolar quantities, whereas [S]
may be five or six orders of magnitude higher !!
If only the beginning of the reaction is
monitored (often the first 60 seconds or less),
changes in [S] can be limited to a few percent,
and [S] can be regarded as constant.
V0 can then be explored as a function of [S],
which is adjusted by the investigator.
The effect on V0 of varying [S] when the
enzyme concentration is held
constant is shown in Figure 6–11.

At relatively low concentrations of
substrate, V0 increases almost linearly with
an increase in [S].
At higher substrate concentrations, V0
increases by smaller and smaller amounts
in response to increases in [S].
Finally, a point is reached beyond which
increases in V0 are vanishingly small as [S]
increases.
This plateau-like V0 region is close to the
maximum velocity, Vmax.

The combination of an enzyme with its substrate molecule to form an ES
complex is a necessary step in enzymatic catalysis.
Michaelis-Menten Theory
They postulated that the enzyme first combines reversibly with its substrate
to form an enzyme-substrate complex in a relatively fast reversible step:
The ES complex then breaks down in a slower second step to yield the free
enzyme and the reaction product P:
Equation 6-7
Equation 6-8

• Because the slower second reaction (Eqn 6–8) must limit the rate of the
overall reaction, the overall rate must be proportional to the
concentration of the species that reacts in the second step, that is, ES.
Equation 6-8
• At any given instant in an enzyme-catalyzed reaction, the enzyme exists in two
forms, the free or uncombined form E and the combined form ES.
• At low [S], most of the enzyme is in the uncombined form E.
• Here, the rate is proportional to [S] because the equilibrium of Equation 6–7 is
pushed toward formation of more ES as [S] increases.
Equation 6-7

• The maximum initial rate of the catalyzed reaction (Vmax) is
observed when virtually all the enzyme is present as the ES complex
and [E] is vanishingly small.
• Under these conditions, the enzyme is “saturated” with its
substrate, so that further increases in [S] have no effect on rate.
• This condition exists when [S] is sufficiently high that essentially all
the free enzyme has been converted to the ES form.
• After the ES complex breaks down to yield the product P, the
enzyme is free to catalyze reaction of another molecule of
substrate.

• The saturation effect is a distinguishing
characteristic of enzymatic catalysts and is
responsible for the plateau observed in Figure
6–11.
• The pattern seen in Figure 6–11 is sometimes
referred to as saturation kinetics.
• V0 is limited to the early part of the reaction,
and analysis of these initial rates is referred
to as steady-state kinetics.

The Relationship between Substrate Concentration
and Reaction Rate Can Be Expressed Quantitatively
• The curve expressing the relationship between [S] and
V0 (Fig. 6–11) has the same general shape for most
enzymes (it approaches a rectangular hyperbola), which
can be expressed algebraically by the Michaelis-Menten
equation.

• The important terms are [S], V0, Vmax, and a
constant called the Michaelis constant, Km.
• Km has units of concentration.
• All these terms are readily measured
experimentally.
Michaelis-Menten equation is the rate equation for a one-substrate enzyme-
catalyzed reaction.

• An important numerical relationship emerges from the Michaelis-Menten
equation in the special case when V0 is exactly one-half Vmax .
On dividing by Vmax, we obtain
Solving for Km, we get Km + [S] = 2[S], or
Km is equivalent to the substrate
concentration at which V0 is one-
half Vmax.

Transformations of the Michaelis-Menten
Equation: The Double-Reciprocal Plot
A double-reciprocal or Lineweaver-
Burk plot.

• This form of the Michaelis-Menten equation is
called the Lineweaver-Burk equation.
• For enzymes obeying the Michaelis-Menten
relationship, a plot of 1/V0 versus 1/[S] (the
“double reciprocal” of the V0 versus [S] plot we
have been using to this point) yields a straight line.
• This line has a slope of Km/Vmax
an intercept of 1/Vmax on the 1/V0 axis
an intercept of 1/Km on the 1/[S] axis.
Transformations of the Michaelis-Menten
Equation: The Double-Reciprocal Plot
Burk plot.
Compare this equation with the standard equation for a
straight line:
where m is the slope and b is the y intercept.

• Lineweaver-Burk plot has the great
advantage of allowing a more accurate
determination of Vmax, which can only
be approximated from a simple plot of
V0 versus [S]
• The double-reciprocal plot of enzyme
reaction rates is very useful in
distinguishing between certain types of
enzymatic reaction mechanisms and
in analyzing enzyme inhibition
Burk plot.

• Using linear transformations of the primary data for determining the
values of the kinetic constants has some limitations.
• Despite the errors associated with this method, the Lineweaver—Burk
double reciprocal plot has become the most popular means of graphically
representing enzyme kinetic data.
• Other transformations of the Michaelis-Menten equation have been
derived, each with some particular advantage in analyzing enzyme kinetic
data.
• Eadie—Hofstee
• Hanes—Wolff
• Eisenthal—Cornish- Bowden direct plots.
• the use of these transformation methods is no longer necessary because
most researchers have access to computer-based nonlinear curve-fitting
methods

Eadie‒Hofstee Plots
• If we multiply both sides of this equation by Km + [S], we obtain:
This is the Michaelis-Menten equation
• If we now divide both sides by [S] and rearrange, we obtain:
vKm + vS = Vmax S
vS = VmaxS – VKm
Divide by S

• If we plot v as a function of v/[S],
this equation would predict a
straight-line relationship with
slope of -Km and y intercept of
Vmax.
• Such a plot is referred to as an
Eadie—Hofstee plot. Eadie—Hofstee plot of enzyme kinetic data.

Hanes‒Wolff Plots
• If one multiplies both sides of the Lineweaver—Burk Equation by
[S], one obtains:
This treatment also leads to linear plots when [S]/v is plotted as a function
of [S].
Lineweaver—Burk Equation

• In Hanes—Wolff plot
– the slope is 1/Vmax
– the y intercept is Km/Vmax
– and the x intercept is -Km
Hanes—Wolff plot of enzyme kinetic data.

Eisenthal‒Cornish-Bowden Direct Plots
• Pairs of v, [S] data are plotted as follows:
• values of v along the y axis and the
negative values of [S] along the x axis.
• For each pair, one then draws a straight
line connecting the points on the two
axes and extrapolates these lines past
their point of intersection.
• When a horizontal line is drawn from the
point of intersection of these line to the y
axis, the value at which this horizontal
line crosses the y axis is equal to Vmax.
• Similarly, when a vertical line is dropped
from the point of intersection to the x
axis, the value at which this vertical line
crosses the x axis defines Km.
Eisenthal—Cornish-Bowden direct plot of
enzyme kinetic data.

• This is Eisenthal—Cornish-
Bowden direct plot
• considered to give the best
estimates of Km and Vmax of
any of the linear transformation
methods.
Eisenthal—Cornish-Bowden direct plot of
enzyme kinetic data.

• In real experimental data, small errors in the measured values of v are amplified by the
mathematical transformation of taking the reciprocal. !!
• The greatest percent error is likely to be associated with velocity values at low substrate
concentration.
• Unfortunately, in the reciprocal plot, the lowest values of [S] correspond to the highest
values of 1/[S], and because of the details of linear regression, these data points are
weighted more heavily in the analysis.
Hence the
experimental
error is amplified
and unevenly
weighted in this
analysis, resulting
in poor estimates
of the kinetic
constants even
when the
experimental
error is relatively
small.

• Nevertheless, the Lineweaver—Burk plots are still commonly used by many
researchers.
• the use of these transformation methods is no longer necessary because
most researchers have access to computer-based nonlinear curve-fitting
methods, and the direct fitting of untransformed data by these methods is
highly recommended.

Example
a) Draw the V vs. S graph
b) Draw the Lineveawer-Burk Plot
c) Determine the Km and Vmax from
both graphs and compare
substrate
concentration (mM)
Enzyme activity
(U/ml)
0 0
1 92,3
2 129,3
3,6 152,8
5 163,7
10 189,2
15 208,6
20 226,1
25 226,2
Arylesterase enzyme activity was measured at
different substrate concentrations
Hydrolysis of phenylacetate by arylesterase
(PON1) into phenol and acetate.

Kinetic Parameters Are Used to Compare
Enzyme Activities
• Enzymes that exhibit a hyperbolic dependence of V0 on [S] are said to
follow Michaelis- Menten kinetics.
• Km = [S] when V0 = 1⁄2Vmax holds for all enzymes that follow
Michaelis-Menten kinetics.
• The most important exceptions to Michaelis-Menten kinetics are the
regulatory enzymes.
Many enzymes that follow Michaelis-Menten kinetics have
quite different reaction mechanisms, and enzymes that
catalyze reactions with six or eight identifiable steps often
exhibit the same steady-state kinetic behavior.
Even though the equation Km = [S] when V0 = 1⁄2Vmax
holds true for many enzymes,
both the magnitude and the real meaning of Vmax and
Km can differ from one enzyme to the next.

• The parameters Vmax and Km can be obtained experimentally for
any given enzyme,
• but by themselves they provide little information about the
number, rates, or chemical nature of discrete steps in the reaction.

Interpreting Vmax and Km
• The Km can vary greatly from enzyme to enzyme, and even for different
substrates of the same enzyme (Table 6–6).
• The term is sometimes used (often inappropriately) as an indicator of the
affinity of an enzyme for its substrate.
Km is the substrate
concentration that results in half-
maximal velocity for the
enzymatic reaction.
An equivalent way of stating this is
that the Km represents the
substrate concentration at which
half of the enzyme active sites in
the sample are filled (i.e.,
saturated) by substrate molecules
in the steady state.

The significance of Km
• The actual meaning of Km depends on specific aspects of the reaction
mechanism such as the number and relative rates of the individual steps.
• For reactions with two steps,
• When k2 is rate-limiting, k2 << k1 and Km reduces to k-1/k1, which is
defined as the dissociation constant, Kd, of the ES complex.
Where these conditions hold, Km does represent a measure of the affinity
of the enzyme for its substrate in the ES complex.
k1, k-1 and k2 are rate constants.

http://www.ucl.ac.uk
/~ucbcdab/enzass/im
ages/subs5.png
If k-1 is small, k1 is large, than Km will be
small.
What does this mean?

• However, this scenario does not apply for most enzymes.
• Sometimes k2 >> k-1, and then Km = k2/k1.
• In other cases, k2 and k1 are comparable and Km
remains a more complex function of all three rate
constants ( )

• The quantity Vmax also varies greatly from one enzyme to the next.
• If an enzyme reacts by the two-step Michaelis-Menten mechanism,
Vmax = k2[Et], where k2 is rate-limiting.
• However, the number of reaction steps and the identity of the rate-
limiting step(s) can vary from enzyme to enzyme.
• For example, consider the quite common situation where product
release, EP E + P, is rate-limiting.
• Early in the reaction (when [P] is low), the overall reaction can be
described by the scheme
The significance of Vmax

• In this case, most of the enzyme is in the EP form at saturation, and
Vmax = k3[Et].
• It is useful to define a more general rate constant, kcat, to describe the limiting
rate of any enzyme-catalyzed reaction at saturation.
• If the reaction has several steps and one is clearly rate limiting, kcat is equivalent
to the rate constant for that limiting step.
• For the simple reaction seen on the right kcat = k2.
• For the reaction seen on above, kcat = k3.
• When several steps are partially rate-limiting, kcat can become a complex function
of several of the rate constants that define each individual reaction step.
• In the Michaelis-Menten equation, kcat = Vmax/[Et], and the Equation becomes:
K cat (Turnover number)

• The constant kcat is a first-order rate constant and hence has units
of reciprocal time.
• It is also called the turnover number.
• It is equivalent to the number of substrate molecules converted to
product in a given unit of time on a single enzyme molecule when
the enzyme is saturated with substrate.
K cat (Turnover number)

Comparing Catalytic Mechanisms and Efficiencies
• The kinetic parameters kcat and Km are generally useful for the study and
comparison of different enzymes, whether their reaction mechanisms are
simple or complex.
• Each enzyme has values of kcat and Km that reflect
– the cellular environment,
– the concentration of substrate normally encountered in vivo by the enzyme, and
– the chemistry of the reaction being catalyzed.

• The parameters kcat and Km also allow us to evaluate the kinetic
efficiency of enzymes, but either parameter alone is insufficient for this
task.
• Two enzymes catalyzing different reactions may have the same kcat
(turnover number), yet the rates of the uncatalyzed reactions may be
different and thus the rate enhancements brought about by the enzymes
may differ greatly.
• Experimentally, the Km for an enzyme tends to be similar to the cellular
concentration of its substrate.
• An enzyme that acts on a substrate present at a very low concentration in
the cell usually has a lower Km than an enzyme that acts on a substrate
that is more abundant.

• The best way to compare the catalytic efficiencies of different enzymes or the
turnover of different substrates by the same enzyme is to compare the ratio kcat/Km
for the two reactions.
• This parameter is sometimes called the specificity constant.
• There is an upper limit to kcat/Km, imposed by the rate at which E and S can diffuse
together in an aqueous solution.
• This diffusion controlled limit is 108 to 109 M-1s-1, and many enzymes have a kcat/Km
near this range !
• Such enzymes are said to have achieved catalytic perfection !!!
• Note that different values of kcat and Km can produce the maximum ratio.

Enzymes that Catalyze Reactions
with Two or More Substrates
• In most enzymatic reactions, two (and sometimes more) different
substrate molecules bind to the enzyme and participate in the reaction.
• For example, in the reaction catalyzed by hexokinase, ATP and glucose are
the substrate molecules, and ADP and glucose 6-phosphate are the
products:
The rates of such bisubstrate reactions can also
be analyzed
by the Michaelis-Menten approach.
Hexokinase has
a characteristic Km for each of its substrates

Enzymatic reactions with two substrates
• Enzymatic reactions with two substrates usually involve transfer of an atom or a
functional group from one substrate to the other.
• These reactions proceed by one of several different pathways.
A) In some cases, both substrates are bound to the enzyme concurrently at some
point in the course of the reaction, forming a noncovalent ternary complex (Fig)
• the substrates bind in a random sequence or in a specific order.
(a) The enzyme and both substrates
come together to form a ternary complex.
In ordered binding, substrate 1 must bind
before substrate 2 can bind productively.
In random binding, the substrates can
bind in either order.

B) In other cases, the first substrate is converted to product and dissociates
before the second substrate binds, so no ternary complex is formed.
• An example of this is the Ping-Pong, or double-displacement, mechanism (Fig).
(b) An enzyme-substrate complex
forms, a product leaves the complex,
the altered enzyme forms a second
complex with another substrate
molecule, and the second product
leaves, regenerating the enzyme.
Substrate 1 may transfer a functional
group to the enzyme (to form the
covalently modified E), which is
subsequently transferred to substrate
2. This is called a Ping-Pong or
double-displacement mechanism.

• Steady-state kinetics can often help distinguish among these possibilities.

DEVIATIONS FROM HYPERBOLIC
KINETICS
• Deviations from the hyperbolic dependence of velocity on substrate
concentration are seen occasional.
• Such anomalies occur for several reasons.
1) Some physical methods of measuring velocity, such as optical spectroscopies,
can lead to experimental artifacts that have the appearance of deviations from
the expected behavior
2) Nonhyperbolic behavior can also be
caused by the presence of certain types
of inhibitor as well.
In the most often encountered case,
substrate inhibition, a
second molecule of substrate can bind to
the ES complex to form an inactive
ternary complex, SES.

3) Another cause of nonhyperbolic kinetics is the presence of more than one
enzyme acting on the same substrate.
• Many enzyme studies are performed with only partially purified enzymes,
and many clinical diagnostic tests that rely on measuring enzyme activities
are performed on crude samples (of blood, tissue homogenates, etc.).
When the substrate for the reaction is
unique to the enzyme of interest, these
crude samples can be used with good
results.
If, however, the sample contains more
than one enzyme that can act on the
substrate, deviations from the expected
kinetic results occur.
DEVIATIONS FROM HYPERBOLIC KINETICS

4) Enzymes displaying cooperativity of substrate binding also deviate from
hyperbolic kinetics.
• It was assumed until here that the active sites of the enzyme molecules
behave independently of one another.
• Sometimes proteins occur as multimeric assemblies of subunits. Some
enzymes occur as homomultimers, each subunit containing a separate active
site.
allosteric enzymes

• It is possible that the binding of a substrate molecule at one of these active
sites could influence the affinity of the other active sites in the multisubunit
assembly.
• This effect is known as cooperativity.
• It is said to be positive when the binding of a substrate molecule to one active
site increases the affinity for substrate of the other active sites.
• On the other hand, when the binding of substrate to one active site lowers the
affinity of the other active sites for the substrate, the effect is called negative
cooperativity.
• The number of potential substrate binding sites on the enzyme and the degree
of cooperativity among them can be quantified by the Hill coefficient, h.
The influence of
cooperativity on the measured
values of velocity We will see how
cooperativity affects the
Michaelis—Menten and
Lineweaver—Burk plots
of an enzyme reaction
K’ is related to Km but also contains
terms related to the effect of
substrate occupancy at one site on the
substrate affinity of other sites
allosteric enzymes

SUMMARY
• Most enzymes have certain kinetic properties in common. When substrate
is added to an enzyme, the reaction rapidly achieves a steady state in
which the rate at which the ES complex forms balances the rate at which it
reacts.
• As [S] increases, the steady-state activity of a fixed concentration of
enzyme increases in a hyperbolic fashion to approach a characteristic
maximum rate, Vmax, at which essentially all the enzyme has formed a
complex with substrate.
• The substrate concentration that results in a reaction rate equal to one-
half Vmax is the Michaelis constant Km, which is characteristic for each
enzyme acting on a given substrate.
• The Michaelis-Menten equation
relates initial velocity to [S] and Vmax through the constant Km.
• Michaelis-Menten kinetics is also called steady-state kinetics.

• Km and Vmax have different meanings for different enzymes. The
limiting rate of an enzyme-catalyzed reaction at saturation is
described by the constant kcat, the turnover number.
• The ratio kcat/Km provides a good measure of catalytic efficiency.
• The Michaelis-Menten equation is also applicable to bisubstrate
reactions, which occur by ternary-complex or Ping-Pong (double-
displacement) pathways.

Bmm480 Enzymology lecture-3

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