This document discusses enzyme kinetics and the Michaelis-Menten model of enzyme kinetics. It defines key terms like reaction rate, elementary reactions, rate laws, and transition state theory. It then introduces the Michaelis-Menten equation, defines terms like Km, Vmax, and kcat. It discusses steady state kinetics and how the Michaelis-Menten equation was derived. It explains the meaning and uses of Km and Vmax and concludes by discussing the Lineweaver-Burk double reciprocal plot.

1. ENZYME KINETICS

2. Chemical Kinetics
Chemical kinetics: the study of reaction rate, a quantity
conditions affecting it,
the molecular events during a chemical reaction (mechanism), and
presence of other components (catalysis).
Factors affecting reaction rate:
Concentrations of reactants
Catalyst
Temperature
Surface area of solid reactants or catalyst
2

3. Elementary vs. overall reactions
Reactions are the result of molecular collisions; & almost invariably
depend on the collision of no more than 2 molecular species at a time.
Overall reactions, such as:
2 KAlSi3O8 + 2H + + 9H 2O
Al2Si2O5 (OH)4 + 4H 4SiO4 + 2K +
do not reveal the sequential, and possibly parallel, sets of molecular
interactions, i.e. elementary reactions, that are actually involved.

4. Example of fast reactions (only 1 elementary step):
Ag + + Cl - AgCl(s)
CO2,aq + OH - -
HCO3
Example of a two step reaction:
O3 O2 + O

O + O3 2 O2
_____________
Overall reaction: 2 O3 3 O2
Determining a rate law requires knowledge of the rate-limiting
elementary reaction (usually only one). Allows accounting for the
stoichiometry and the reaction order. If this is not possible (eg. for an
overall reaction), rate laws are determined experimentally.

5. Reaction Rate Defined
Reaction rate: changes in a
concentration of a product or a
[]
reactant per unit time.
[ ] concentration
Reaction rate = ——
t change [
]
t
t

6. Expressing reaction rates
For a chemical reaction, there are many ways to express the
reaction rate. The relationships among expressions depend on
the equation.
Note the expression and reasons for their relations for the
reaction
2 NO + O2 (g) = 2 NO2 (g)
[O2] 1 [NO] 1 [NO2]
Reaction rate = – ——— = – — ———— = — ———
t 2 t 2 t

9. Second Order Reaction

10. Half time
• the time for half of the reactant initially
present to decompose, its half-time or half-
life, t1/2,is a constant and hence independent
of the initial concentration of the reactant.
• For first order reaction :

12. Variation of Reaction rates and Order
2nd order, rate = k [A]2
rate
First order, rate = k [A]
k = rate, 0th
order
[A]
[A] = ___?

13. Transition State Theory
Applies statistical mechanics to individual elementary reactions.
Meaningless for overall reactions.
Focuses on the activated complex, the molecular configuration
present at the top of the energy barrier (actually a saddle point)
between reactants and products in an elementary reaction.
Assumes this complex is a true chemical species and assumes that the
initial reactants are always at equilibrium with the complex.
Predicts that the rate is proportional to the number of activated
complexes and to their rate of decomposition.
Applies near equilibrium.

14. TST is somewhat similar to the idea that the rate is proportional
to the deviation from equilibrium (or the degree of
supersaturation or undersaturation).
n G Q
Rate kdiss 1 exp G RT ln
RT K eq
kdiss G0
K eq exp
k precip RT
Activated Complex: a hypothetical species believed to
exist in an intermediate state (transition state) that lies
between the reactants & the products. It is a temporary
state where bonds are in the process of breaking &
forming; it is a very unstable species with a high
potential energy

15. • in the hypothetical reaction,
A2 + B2  2AB
A + B  A…B  A B
: : A B
A B A…B
reactants activated products
complex

16. • Let us consider the reaction of a hydrogen
atom with diatomic hydrogen (H2) to yield a
new H2 molecule and a different hydrogen
atom

17. Enzyme Kinetics Equation

18. Michaelis-Menten Equation

19. Initial Velocity (vo) and [S]
• The concentration of substrate [S] present will greatly
influence the rate of product formation, termed the
velocity (v) of a reaction. Studying the effects of [S] on
the velocity of a reaction is complicated by the
reversibility of enzyme reactions, e.g. conversion of
product back to substrate. To overcome this problem,
the use of initial velocity (vo) measurements are used.
At the start of a reaction, [S] is in large excess of [P],
thus the initial velocity of the reaction will be
dependent on substrate concentration

20. Michaelis-Menten Curve

21. Initial Velocity (vo) and [S] (cont)
• When initial velocity is plotted against [S], a
hyperbolic curve results, where Vmax
represents the maximum reaction velocity. At
this point in the reaction, if [S] >> E, all
available enzyme is "saturated" with bound
substrate, meaning only the ES complex is
present.

22. Michaelis-Menten Curve

23. Substrate Saturation of an Enzyme
A. Low [S] B. 50% [S] or Km C. High, saturating [S]

24. Steady State Assumption
• The M-M equation was derived in part by making
several assumptions. An important one was: the
concentration of substrate must be much greater than
the enzyme concentration. In the situation where [S]
>> [E] and at initial velocity rates, it is assumed that the
changes in the concentration of the intermediate ES
complex are very small over time (vo). This condition is
termed a steady-state rate, and is referred to as
steady-state kinetics. Therefore, it follows that the
rate of ES formation will be equal to the rate ES
breakdown.

25. Michaelis-Menten Equation
Derivation
• Rate of ES formation = k1([ET] - [ES])[S]
(where [ET] is total concentration of
enzyme E and k-2 is considered neglible)
• Rate of ES breakdown to product = k-
1[ES] + k2[ES]

26. Michaelis-Menten Equation
Derivation (cont)
• Thus for the steady state assumption:
• k1([ET] - [ES])[S] = k-1[ES] + k2[ES]
• This equation is the basis for the final Michaelis-
Menten following algebraic rearrangement and
substitution of Km and Vmax terms.

27. Meaning of Km
• An important relationship that can be derived from the
Michaelis-Menten equation is the following: If vo is set
equal to 1/2 Vmax, then the relation Vmax /2 = Vmax[S]/Km
+ [S] can be simplied to Km + [S] = 2[S], or Km = [S].
This means that at one half of the maximal
velocity, the substrate concentration at this
velocity will be equal to the Km. This relationship
has been shown experimentally to be valid for many
enzymes much more complex in regards to the number of
substrates and catalytic steps than the simple single
substrate model used to derive it.

28. Meaning of Km (cont)
• The significance of Km will change based on the different
rate constants and which step is the slowest (also called
the rate-limiting step). In the simplest assumption, the
rate of ES breakdown to product (k2) is the rate-
determining step of the reaction, so k-1 >> k2 and Km = k-
1/k1. This relation is also called a dissociation constant
for the ES complex and can be used as a relative
measure of the affinity of a substrate for an enzyme
(identical to Kd). However if k2 >> k-1 or k2 and k-1 are
similar, then Km remains more complex and cannot be
used as a measure of substrate affinity.

29. Uses of Km
• Experimentally, Km is a useful parameter for
characterizing the number and/or types of substrates that
a particular enzyme will utilize (an example will be
discussed). It is also useful for comparing similar
enzymes from different tissues or different organisms.
Also, it is the Km of the rate-limiting enzyme in many of
the biochemical metabolic pathways that determines the
amount of product and overall regulation of a given
pathway. Clinically, Km comparisons are useful for
evaluating the effects mutations have on protein function
for some inherited genetic diseases.

30. Meaning of Vmax
• The values of Vmax will vary widely for different enzymes and
can be used as an indicator of an enzymes catalytic efficiency.
It does not find much clinical use.
• There are some enzymes that have been shown to have the
following reaction sequence:
• In this situation, the formation of product is dependent on the
breakdown of an enzyme-product complex, and is thus the rate-
limiting step defined by k3.

31. Important Conclusions of Michaels -
Menten Kinetics
• when [S]= KM, the equation reduces to
• when [S] >> KM, the equation reduces to
• when [S] << KM, the equation reduces to

32. Derivation of kcat
• A more general term has been defined, termed
kcat, to describe enzymes in which there are
multiple catalytic steps and possible multiple
rate-limiting steps. The Michaelis-Menten
equation can be substituted with kcat

33. Definition and Use of kcat
• The constant, kcat (units of sec-1), is also called
the turnover number because under saturating
substrate conditions, it represents the number
of substrate molecules converted to product in
a given unit of time on a single enzyme
molecule. In practice, kcat values (not Vmax) are
most often used for comparing the catalytic
efficiencies of related enzyme classes or
among different mutant forms of an enzyme.

34. Lineweaver-Burk (double reciprocal
plot)
• If the reciprocal (1/X) of the Michaelis-Menten
equation is done, after algebraic simplification the
following equation results:
• This relation is written in the format of the equation
for a straight line, y = mx + b, where y = 1/vo, m
(slope) = Km/Vmax, x = 1/[S] and the y-intercept, b
= 1/Vmax. When this relation is plotted,the result is
a straight line graph

35. Lineweaver-Burk (double reciprocal
plot) (cont)

36. Uses of double reciprocal plot
• The x intercept value is equal to -1/Km.
The biggest advantage to using the double
reciprocal plot is a more accurate
determination of Vmax, and hence Km. It is
also useful in characterizing the effects of
enzyme inhibitors and distinguishing
between different enzyme mechanisms.