Wednesday 7th March, 2012 at 12:13 Noon Control
Theory for Biologists, Draft 0.81 www.sys-bio.org
3
Basic Enzyme Kinetics
3.1 Enzyme Kinetics
The vast majority of chemical transformations inside cells are catalyzed by
enzymes. Enzymes accelerate the rate of chemical reactions (both forward
and backward) without being consumed in the process and tend to be very
selective, with a particular enzyme accelerating only a specific reaction.
The model for enzyme action, first suggested by Brown and Henri but later
established more thoroughly Michaelis and Menten, suggests the binding
of free enzyme to the reactant forming a enzyme-reactant complex. This
complex undergoes a transformation, releasing product and free enzyme.
The free enzyme is then available for another round of binding to new
reactant. Traditionally, the reactant molecule that binds to the enzyme is
termed the substrate, S , and the mechanism is often written as:
k1
k2
E C S )* ES ! E C P
k
(3.1)
1
This mechanism illustrates the binding of substrate and release of product,
P . E is the free enzyme and ES the enzyme substrate complex. Note that
55
56
CHAPTER 3. BASIC ENZYME KINETICS
in this model substrate binding is reversible but product release is not. A
more realistic mechanism will always have some degree of reversibility in
product formation which leads to the following more general model:
k1
k2
E C S )* ES )* E C P
k
1
k
2
It is possible to model enzymes using the explicit mechanisms shown
above, however the rate constants for the binding and unbinding reactions
are either often unknown or difficult to determine. Instead, assumptions
are made about the dynamics of the mechanism which reduces the number
of constants required to characterize the enzyme. This leads to a discussion of aggregate rates laws, the most celebrated being Michaelis-Menten
kinetics.
3.1.1
Michaelis-Menten Kinetics
In practice we rarely build models using explicit elementary reactions unless it is absolutely necessary in order to capture a particular type of dynamical behavior. Quite apart from the huge increase in complexity, the
rate constants for the elementary reaction are in any case usually not known.
Instead we will often use approximations, sometimes called aggregate rate
laws. If we consider first the fully reversible mechanism for enzyme action:
k1
k2
E C S )* ES ! E C P
k
1
Two different assumptions have been employed to reduce this scheme to
a simpler formulation, the first termed rapid equilibrium was made in the
original derivation by Michaelis and Menten. They assumed that the first
step, that is binding of substrate to enzyme, was in equilibrium. The second approach was introduced by Briggs and Haldane, called the steady
state assumption (Fig. 3.1) and in enzyme kinetics is the most commonly
used approach. Rather than assume equilibration, Briggs and Haldane assumed that the enzyme substrate complex rapidly reached steady state.
This was less restrictive that the rapid equilibrium assumption. Enzyme
rate laws are often are derived using the steady state assumption, however
3.1. ENZYME KINETICS
57
Concentration
10
8
P
6
S
4
E
2
ES
0
0
0:2
0:4
0:6
0:8
Time
1
1:2
1:4
Figure 3.1: Progress curves for a simple irreversible enzyme catalyzed
reaction (3.1). Initial substrate concentration is set at 10 units. The
enzyme concentration is set to an initial concentration of 1 unit (E and
ES curves are scaled by two in order to make the changes in E and ES
easier to visualize). In the central portion of the plot one can observe
the relatively steady concentrations of ES and E (dES=dt 0). At the
same time, the rate of change of S and P are constant over this period.
k1 D 20; k
20
1
10
* ES ! E C P
D 1; k2 D 10, that is: E C S )
1
because the mathematics can become complicated, many complex mechanisms, such as cooperativity and gene expression are still derived using
the rapid equilibrium assumption. For this reason the rapid equilibrium
derivation will be briefly described here.
Rapid Equilibrium Assumption If we let Ks be the dissociation constant
for binding:
E:S
Ks D
ES
and noting that the total concentration of enzyme, E t , is the sum of free
enzyme, E and enzyme substrate complex, ES: E t D E C ES, it is easy
to show that the equilibrium concentration of ES is given by:
ES D
Et : S
Ks C S
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CHAPTER 3. BASIC ENZYME KINETICS
Since the rate of reaction is determined by the rate of release of product,
we can write down the rate of reaction as v D k2 ES. Combining this with
the previous relation for ES, yields our result:
vD
E t : k2 S
Ks C S
Steady State Assumption Instead of assuming rapid equilibrium, let us
follow the treatment of Briggs and Haldane by assuming that the enzyme
substrate complex rapidly reaches steady steady. Fig 3.1 shows progress
curves illustrating the changes in concentrations for the different enzymatic species. Note that the concentration of the enzyme substrate complex rapidly approaches a steady state and remains in this state until the
substrate level reaches a low level. The rate of change of the enzyme substrate complex (??) can be written down using the laws of mass-action:
d ES
D k1 E : S
dt
k
1
ES
k2 ES
The concentration of enzyme substrate complex is assumed to rapidly
reach steady-state (Fig. ??) so that the above equation can be set to zero:
0 D k1 E : S
k
1
ES
k2 ES
We also note that the total concentration of enzyme, E t , is the sum of free
enzyme, E and enzyme substrate complex, ES:
E t D E C ES
From these relationships, the steady-state concentration of enzyme substrate complex can be derived:
ES D
Et : S
.k 1 C k2 /=k1 C S
By assuming that the rate of reaction is given by v D k2 ES , we obtain:
vD
.k
E t k2 S
C
k2 /=k1 C S
1
(3.2)
3.1. ENZYME KINETICS
59
The Vmax can be expressed as the total enzyme concentration times the
rate constant for the product formation, E t k2 . We can also combine the
constants .k 1 C k2 /=k1 into a single new constant called the Michaelis
constant, or Km.
Reaction Rate
Vmax 1
0:8
0:6
0:4
0:2
0
0
5
10
Km
15
20
25
30
Substrate Concentration
Figure 3.2: Relationship between the rate of reaction for a simple
Michaelis-Menten rate law. The reaction rate reaches a limiting value
(saturates) called the Vmax. Km is set to 4.0 and Vmax to 1.0. Note
that the value of the Km is the substrate concentration that gives half
the maximal rate.
vD
Vmax S
Km C S
(3.3)
If we set the reaction velocity to half the Vmax, one can easily show that
the Km is the substrate concentration that gives half the maximal rate (Figure. 3.2).
Reversible Michaelis-Menten Rate law
The derivation of the irreversible Michaelis-Menten is an instructive exercise, however it is not a particularly realistic model to use in models be-
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CHAPTER 3. BASIC ENZYME KINETICS
cause there is no explicit product inhibition term. Instead, it is much better
to consider the reversible Michaelis-Menten rate law. The derivation of the
reversible form is very similar to the derivation of the irreversible rate law.
The main difference is that the steady-state rate is given by an expression
that incorporates both the forward and reverse rates for the product:
v D k2 ES
k
2
E:P
The expression that describes the steady-state concentration of the enzyme
substrate complex also has an additional term from the product binding
(k 2 EP ). Taking these into consideration leads to the general reversible
rate expression (See Appendix B for a full derivation):
vD
Vf S=KS Vr P =KP
1 C S=KS C P =KP
At equilibrium the rate of the reversible reaction is zero. When positive
the reaction is going in the forward direction and in the reverse direction
when negative. At equilibrium the equation reduces to
0 D Vf Seq =KS
Vr Peq =KP
where Seq and Peq represent the equilibrium concentrations for substrate
and product. Rearrangement yields
Keq D
Vf KP
Peq
D
Seq
Vr KS
This expression is known as the Haldane relationship and shows that the
four kinetic constants are not independent. The relationship can be used to
eliminate one of the kinetic constants and substitute the equilibrium constants in its place. This is useful because equilibrium constants tend to be
better known that kinetic constants. Incorporating the Haldane relationship
yields the equation
vD
Vf =KS .S P =Keq /
1 C S=KS C P =KP
Separating out the terms makes it easier to see that the equation has a
thermodynamic term .S P =Keq / and a kinetic term as shown in the
3.1. ENZYME KINETICS
61
following expression:
v D .S
P =Keq /
Vf =KS
1 C S=KS C P =KP
The fact that the equilibrium constant appears are a constant factor in the
expression suggests that enzymes do not change the equilibrium ratio, but
simply accelerate the approach to equilibrium.
Haldane Equilibrium Relations
At equilibrium the net rate of reaction is zero. When the net rate is positive
the reaction is going in the forward direction and in the reverse direction
when negative. At equilibrium the reversible Michaelis equation reduces
to
0 D Vf Seq =KS Vr Peq =KP
where Seq and Peq represent the equilibrium concentrations for substrate
and product. Rearrangement yields
Keq D
Vf KP
Peq
D
Seq
Vr KS
This expression is known as the Haldane relationship and shows that the
four kinetic constants are not independent and is directly related to the law
of detailed balance that was introduced in section 2.2. The relationship can
be used to eliminate one of the kinetic constants and substitute the equilibrium constants in its place. This is useful because equilibrium constants
tend to be better known that kinetic constants. Incorporating the Haldane
relationship yields the equation
vD
Vf =KS .S P =Keq /
1 C S=KS C P =KP
Separating out the terms makes it easier to see that the equation has a
thermodynamic term .S P =Keq / and a kinetic term as shown in the
following expression:
v D .S
P =Keq /
Vf =KS
1 C S=KS C P =KP
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CHAPTER 3. BASIC ENZYME KINETICS
The fact that the equilibrium constant appears are a constant factor in the
expression suggests that enzymes do not change the equilibrium ratio, but
simply accelerate the approach to equilibrium.
Product Inhibition
Sometimes reactions appear irreversible, that is there is no discernable
back rate, and yet the forward reaction is influenced by the accumulation
of product. This effect is caused by the product competing with substrate
for binding to the active site and is often called product inhibition. An important industrial example of this is the conversion of lactose to galactose
by the enzyme ˇ galactosidase where galactose will compete with lactose
and thereby slow the forward rate (Gekas and Lopex-Leiva, 1985). The reversible Michaelis-Menten rate law need not be used in these situations,
instead a modified form of the irreversible rate law can be employed. The
rate law below shows a simple modification to the irreversible rate law that
accommodates product inhibition:
vD
Vm S
S C Km 1 C P =Kp
Further discussion on this is given in more detail in section ?? when discussing competitive inhibition.
The steady-state approximation that allows us to derive convenient aggregate rate laws comes with a price. The approximation assumes that the
amount of substrate sequestered by the enzyme is negligible compared
to the free substrate. in vivo this assumption may not necessarily hold
where enzyme concentrations can be comparable to substrate concentrations. Models that employ the Michaelis-Menten laws compared to explicit mass-action models can exhibit changes in their behavior. In particular the presence of high levels of enzyme substrate complex compared
to free substrate can add buffering effects to the dynamics causing time
delays in the evolution of the system. Fortunately the steady-state behavior will be largely unaffected except in some cases where the dynamic
stability might change, for example leading to the onset of oscillatory behavior. Ideally one should check whether in a particular model the use
3.2. COOPERATIVE KINETICS
63
of Michaelis-Menten kinetics or any aggregate rate law has an effect on
the model dynamics by comparing the model to one built using explicit
mass-action rate laws.
3.1.2
Aggregate Rate Laws
The steady-state approximation that allows us to derive convenient aggregate rate laws comes with a price. The approximation assumes that the
amount of substrate sequestered by the enzyme is negligible compared
to the free substrate. in vivo this assumption may not necessarily hold
where enzyme concentrations can be comparable to substrate concentrations. Models that employ the Michaelis-Menten laws compared to explicit mass-action models can exhibit changes in their behavior. In particular the presence of high levels of enzyme substrate complex compared
to free substrate can add buffering effects to the dynamics causing time
delays in the evolution of the system. Fortunately the steady-state behavior will be largely unaffected except in some cases where the dynamic
stability might change, for example leading to the onset of oscillatory behavior. Ideally one should check whether in a particular model the use
of Michaelis-Menten kinetics or any aggregate rate law has an effect on
the model dynamics by comparing the model to one built using explicit
mass-action rate laws.
3.2 Cooperative Kinetics
Many proteins are known to be oligomeric, that is they are composed of
more than one identical protein subunit. For example, phosphofructokinase (E.C 2.7.1.11) from Escherichia coli is made of up four identical
subunits. Each subunit has at least three binding sites corresponding to
sites for ATP, Fructose-6-Phosphate (F6P) and one site for ADP and PEP.
Both the F6P and ADP/PEP sites are on subunit boundaries, this means
that their binding can change the binding affinities on the other subunits.
In general subunits in an oligomer will have one or more ligand binding
sites, which can, when occupied, affect the binding affinities in the other
subunits. The ability of a ligand to affect the binding affinity of sites on the
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CHAPTER 3. BASIC ENZYME KINETICS
other subunits is termed cooperative binding. If ligand binding increases
the affinity of subsequent ligand binding, then it is termed positive cooperativity, otherwise it is called negative cooperativity.
One of the characteristics of positive cooperativity on a reaction rate is to
generate a sigmoid curve. Such a curve is illustrated in the figure below, a
corresponding Michaelian curve is shown for comparison.
Reaction Rate
1
0:8
0:6
0:4
0:2
0
0
0:5
1
1:5
2
2:5
3
Substrate Concentration
Figure 3.3: Plot comparing positive cooperativity to a hyperbolic response.
Hill Equation
The Hill equation was originally derived empirically to describe the sigmoid character found in the binding of oxygen to hemoglobin. Only later
was a mechanism proposed that might explain the relationship. The model
however was simplistic, and even unrealistic, but it provided a baseline
from which to compare other models.
Consider an oligomer with n subunits and a binding site on each subunit
for a ligand, S. If we make the assumption that when the first ligand
binds, the binding affinity for the remaining n 1 sites change such that
all the remaining ligands also bind simultaneously, then we can represent
3.2. COOPERATIVE KINETICS
65
this situation as follows:
E CnS
! ES
Assuming the rapid equilibrium assumption we can write:
KD
ES
E : Sn
where K is the association constant for ligand binding. Using the conservation relation E t D E C ES, the relative saturation can be shown to be
given by:
ES
Sn
Sn
D
D
Et
1=K C S n
Kd C S n
This is the Hill equation where Kd is the dissociation constant. Often the
Hill equation is represented in the following way in the literature:
vD
Vmax S n
Kd C S n
where Kd is the dissociation constant and h the Hill coefficient. Sometimes the equation is also expressed in terms of the half-maximal activity
constant, KH . To do this we set the left-hand side to 0.5 and find the
relationship between S and Kd . If we do this then we find:
SD
p
n
Kd
p
n
p
That is Kd is the half-maximal activity value, or KH D n Kd , that is
n
KH
D Kd . We can therefore write the Hill equation in an alternative form
as:
vD
Vmax S n
Vmax .S=KH /n
D
n
n
KH
C Sn
1 C KSH
In the literature both forms are presented but they all have the same behavior. The equation in terms of the half-maximal activity has advantages
because half-maximal activity can be measured directly from experiments.
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CHAPTER 3. BASIC ENZYME KINETICS
If ligand binding acted in the way suggested in the derivation of the Hill
equation, n would represent the number of binding sites, an integer. However, fitting the Hill equation to real data rarely gives integer estimates to
n suggesting that the model is not a faithful representation of any real system. The utility of the Hill equation however lies in its ability to represent
sigmoid behavior for simple cooperative systems such as transcription factor binding and as a result it has found wide spread use in modeling circles.
However it is severely limited in other aspects, it is not possible to easily
add regulator molecules to the equation or model multi-reactant systems
and significantly it models an irreversible reaction.
1
nD8
nD4
Reaction Rate
0:8
nD2
0:6
0:4
0:2
0
0
0:5
1
1:5
2
2:5
3
Substrate Concentration
Figure 3.4: Plot showing the response of the rate and elasticity for the
Hill model, with n set to the indicated values and KH D 1.
3.2.1
Reversible Hill Equation
In the enzyme kinetics literature much attention is paid to the molecular
mechanisms that generate cooperativity. However for modeling purposes
simple rate models such as the the Hill equation can be sufficient. However
the main problem with the Hill equation is that it describes an irreversible
3.2. COOPERATIVE KINETICS
67
reaction. In recent years, Hofmeyr and Cornish-Bowden published a description of the reversible Hill equation with modifiers. The general form
of the reversible Hill equation without modifiers is given by:
Vf
vD
S
Ks
S
P
C
Keq
Ks
Kp
h
S
P
1C
C
Ks
Kp
h
1
1
Figure 3.5 illustrates the sigmoid behavior with respect to the substrate
concentration. The K constants in the equation are the half saturation constants.
is the mass-action ratio and Keq the equilibrium constant for
the reaction. What is significant about this formulation is that the thermodynamic terms are separated from the saturation terms, a structure also
found in all the variants. The equation also reduces to familiar forms when
certain restrictions are applied. For example if h D 1 the equation reduced to the non-cooperative reversible Michaelis rate law and of course
if reversibility is removed as well the equation reduces to the simple irreversible Michaelis-Menten rate law. The equation can also revert to the
product inhibited but irreversible rate law by setting the Keq to infinity.
The reversible Hill equation is therefore quite flexible and can be used in
my situations.
When modifiers are included an additional term appears in the denominator. In the equation below the modifier is indicated by the symbol M . The
˛ term can be used to determine whether the modifier is an activator or an
inhibitor. If ˛ < 1 then the modifier acts as an inhibitor otherwise it acts
as an activator.
S
S
P h
Vf
1
C
Ks
Keq
Ks
Kp
vD
h
1 C KMm
S
P h
C
C
h
Ks
Ks
1 C ˛ KMm
1
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CHAPTER 3. BASIC ENZYME KINETICS
˛<1
inhibitor
˛>1
activator
1
Reaction Rate
0:8
0:6
Ks = 1
4.0
2.4
0:4
0:2
0
0
2
4
6
8
Substrate Concentration
Figure 3.5: Plot showing the response of the reaction rate for a reversible Hill model with respect to the substrate as a function of the
substrate Michaelian constant. In this a the next figure, the parameters
were set as follows: V m D 1; D 2; Keq D 10:95; Kp D 0:5; n D
4:85; Ke D 2:75; ˛ D 10 5 , P = 0, M = 0
The reversible Hill equation also shows one additional property. Under
a certain set of parameter values, the product concentration can act as a
positive regulator (Figure ??). The possibility of positive activation can
lead to some interesting behavior which we will return to in a later chapter.
Hanekom ?? derived (along with many other variants) a generalized uniuni reversible Hill equation that incorporated multiple modulators:
3.2. COOPERATIVE KINETICS
69
0:5
Reaction Rate
0:4
0:3
0:2
Ke = 1.2
2.0
4.2
0:1
0
0
2
4
6
8
Inhibitor Concentration
Figure 3.6: Plot showing the response of the reaction rate for a reversible Hill model with respect to the inhibitor concentrations as a
function of the inhibitor Michaelis constant. Ks D 2; S D 1, all other
parameter were identical to the previous figure.
Vf ˛ 1 C Keq .˛ C /h 1
vD
Qnm
1Ch
i
C .˛ C /h
h
i D1
1Ci i
To simplify the notation in the above equation, ˛ D S=Ks , D P =Kp
and D M=Km . is the modifier factor that determines whether the
modifier is an activator (> 1) or an inhibitor. Kx are the Michaelian constants, S the substrate, P the product and M the modifier. This equation
assumes that each modifier binds independently of the other, that is the
binding of one modifier does not affect the binding of any other.
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CHAPTER 3. BASIC ENZYME KINETICS
3.3 Multiple Substrate Enzymes
It is probably fair to say that most enzyme catalyzed reactions involve two
substrates. For example, all oxidoreductases involve two substrates, one an
oxidant and the other a reductant. Even apparently single substrate reactions may actually involve water as a second substrate which we choose to
ignore because we assume that the concentration of water hardly changes
during the reaction.
The world of two substrate kinetics is however far more complex than single substrate kinetics. There are more possible variations in the rate laws
particularly when we consider how the substrates bind and products leave
the active site. The commonest reaction mechanisms include compulsoryorder, when one substrate must bind before the other, random-order where
substrates can bind in any order and double-displacement where one substrate binds, modifies the enzyme then leaves to allow the other substrate
to bind. These different mechanisms can generate subtlety different rate
laws. The question however is whether such subtlety is significant when
modeling pathways? For those interested in catalytic mechanisms, the difference in rate laws allow one to distinguish between the mechanisms and
is thus an important consideration. For modeling, the need to be so precise
is not so clear. Given the imprecision in kinetic data and the robustness
of pathways to parameter variation, such subtleties may not in fact be important. As a result some authors suggest the use of generalized rate laws
for modeling two substrate/product enzyme reactions. A number of these
generalizations exist in the literature although they are all closely related
to each other. For example, a generalized irreversible two substrate rate
laws was introduced by Alberty in 1953:
vD
Vm AB
KB A C MA B C AB C KiA KB
where KA and KB are Michaelian constants and KiA is a dissociation constant.
A useful reversible rate law is given by:
3.3. MULTIPLE SUBSTRATE ENZYMES
Reaction Scheme
Rate Law
A$B
Vf ˛ Vr ˇ
1C˛Cˇ
ACB $C
Vf ˛ ˇ Vr 1C˛CˇC˛ ˇC
ACB $C CD
Vf ˛ ˇ Vr ı
1C˛CˇC˛ ˇC CıC ı
71
Table 3.1: Generalized rate equations where Vf and Vr represent the
forward and reverse Vmax values and the greek symbols such as ˛,
represent the species concentrations divided by the Michaelisn constant,
for example: ˛ D A=KA .
Vm A B
P Q
1
Keq A B KA KB
vD
A
Q
B
P
1C
C
1C
C
KA
KQ
KB
KP
which uses the Haldane relationships to eliminate parameters in favor of
introducing the equilibria constant, Keq .
Leibermeister and Klipp describe what they called ‘convenience kinetics’
which is a further generalization that includes a range of different stoichiometric reaction schemes some of which are given in the table below.
Of more interest is the reversible Hill equation described in the last section. The reversible Hill equations can be generalized to accommodate
many different possibilities, including multi-substrate, multi-modulators
and irreversibility.
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CHAPTER 3. BASIC ENZYME KINETICS
3.4 Gene Regulatory Rate Laws
Rate laws associated with gene regulation will only be covered briefly here.
The companion book, Enzyme Kinetics for Systems Biology has a much
more extensive discussion with an entire chapter devoted to rate laws used
for modeling gene expression and regulation.
3.4.1
Structure of a Microbial Genetic Unit
In this chapter we address exclusively prokaryotic gene regulation because
it is much simpler than eukaryotic systems. However, many of the basic
principles still apply to both groups of organism.
The fundamental functional unit of the bacterial genome is the operon
which consists of a control sequence followed by one or more coding regions. The control sequence has a promoter together with zero or more
operator sites (Figure 3.7). The promoter is the specific sequence of DNA
recognized by RNA polymerase which in turn is responsible for transcribing the DNA coding sequence into messenger RNA (mRNA). The binding
of proteins called transcription factors (TF) to the operator sites are responsible for influencing the binding of RNA polymerase and thus can
modulate mRNA production.
Operators
...
Operators
...
Promoter
One or More Coding Sequences
Figure 3.7: Generic Bacterial Operon comprising of one or more coding
sequences, one promoter site for RNA polymerase binding, and zero or
more operator sites that may be upstream or downstream of the promoter. Operator sites that act as repressors are often found to overlap
with the promoter site.
Two other components are not shown in Figure 3.7, these include the ribosome binding site (RBS) and the terminator. The RBS is often a six to
seven base nucleotide base sequence located about eight nucleotides upstream from the coding sequence start codon and is used by the ribosome
3.4. GENE REGULATORY RATE LAWS
73
as a recognition site. The other component, the terminator, is used to stop
mRNA transcription at the end of the coding sequence.
Binding of transcription factors results in the activation or inhibition of
gene transcription. Multiple transcription factors may also interact to control the expression of a single operon. These interactions can emulate simple logical functions (such as AND, OR, etc.) or more elaborate computations. Gene regulatory networks range from a single controlled gene to
hundreds of genes interlinked with transcription factors forming a complex, decision making network.
Different classes of transcription factors also exist. For example, the binding of some transcription factors is modulated by small molecules, a well
known example being the binding of allolactose (a disaccharide very similar to lactose) to the lac repressor or cAMP to the catabolite activator
protein (CAP), also known as the cAMP receptor protein (CRP). Alternatively, a transcription factor may be expressed by one gene and either
directly modulate a second gene (which could be itself) or via other transcription factors. Additionally, some transcription factors only become
active when phosphorylated or unphosphorylated by protein kinases and
phosphatases (Figure 3.9).
The size of gene regulatory networks vary from organism to organism. The
genome of E. coli for example encodes for approximately 171 transcription
factors [19]. These proteins directly control all levels of gene expression.
The EcoCyc [19] database reports at least 48 small molecules and ions that
also influence transcription factors.
The most extensive gene regulatory network database is RegulonDB [16,
12] and another associated network database EcoCyc [19]. RegulonDB is
a database on the gene regulatory network of E. coli. More detail on the
structure of regulatory networks can be found in the work of Alon [35] and
Seshasayee [34].
3.4.2
Gene Regulation
Gene expression rates are controlled by transcription factors, RNA polymerase, and proteins called factors. factors are transcriptional initia-
74
CHAPTER 3. BASIC ENZYME KINETICS
tion proteins that influence the binding of RNA polymerase to the promoter
and can be thought of as global signals that are synthesized in response to
specific environmental conditions. Of more interest here is the role of
transcription factors. These proteins either enhance or reduce the ability of
RNA polymerase to bind to the promoter region and commence transcription. Transcription factors operate by recognizing and binding to specific
DNA sequences on the operator sites. When transcription factors bind to
operator sites they either block or help RNA polymerase bind to the promoter.
At the molecular level, it is assumed that a given transcription factor will
bind and unbind at a rapid rate. To quantify how transcription factors influence gene expression it is important to consider the state of an operator
site. For a single transcription factor that can bind to a single operator site,
there are two states, designated either bound or unbound (Figure 3.8).
a) Unbound State
Operator Promoter
Coding Sequence
b) Bound State
TF
Figure 3.8: Transcription Factor (TF) Bound and Unbound States.
If the operator site can enhance RNA polymerase binding then the bound
state is considered the active state and the unbound state the inactive state.
If the operator is an inhibitory site then the bound state is the inactive state
and the unbound state the active state.
Some bacterial transcription factors such as the lactose repressor (LacI) are
present at very low levels, on the order of 5 to 10 copies per cell [?, ?]. It is
therefore appropriate to consider the probability that a given transcription
factor is bound to an operator site. The state of an operator site can be
described in terms of this probability. These probabilities are influenced by
the association constant of binding, the availability of transcription factors,
3.4. GENE REGULATORY RATE LAWS
75
and other regulators.
Gene Activation
Gene Repression
Multiple Control
Gene Cascade
Auto-Regulation
Regulation by Small
Molecule
~P
Regulation by
Phosphorylation
Figure 3.9: Various Simple Gene Regulatory Motifs.
Once bound, the transcription factor influences the probability of RNA
polymerase binding to the promoter site. There are many mechanisms by
which transcription factors can influence RNA polymerase. One of the
simplest is for a transcription factor to bind to the promoter site itself,
and by an act of exclusion, prevent the RNA polymerase from binding.
Such transcription factors act as repressors. A similar effect occurs if a
transcription factor binds downstream of the promoter site (closer to the
start of the coding sequence). This prevents the RNA polymerase from
moving into the coding sequence by either physical obstruction or because
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CHAPTER 3. BASIC ENZYME KINETICS
the transcription factor has formed DNA loops.
a) Downstream Obstruction
RNA Pol
TF
Promoter
Operator
Coding Sequence
b) Promoter Obstruction
TF
c) Sequestration of an activator resulting in inhibition
TF
TF
RNA Pol
RNA Pol RNA Polymerase
TF
“Activator”
Repressing Transcription Factor
Figure 3.10: Obstruction, exclusion and sequestration models for repressing gene expression.
Examples of downstream obstruction include the galR and galS operators,
where both operators are located beyond the promoter site [?]. LacI is
a good example of promoter exclusion although the LacI repressor only
overlaps about 40% of the promoter.
Another mechanism for repression is by sequestration. This is rarer but one
example is CytR repressed promoters. The CytR protein can form a dimer
with CRP (which itself is a transcriptional activator). Once the dimer is
formed, CRP is unable to bind, therefore inhibiting expression [?].
Activation by transcription factors is more subtle. One mechanism is for a
transcription factor to bind upstream, close to the promoter site. In this in-
3.4. GENE REGULATORY RATE LAWS
77
stance the transcription factor can offer a suitable but weak molecular face
for the RNA polymerase to bind (Figure 3.11). This allows RNA polymerase to stay on the promoter longer and therefore increases the probability of transcription. For example, weak binding may occur between
hydrophobic areas on both proteins. An example of an activating TF is
CRP on the lac operon. The CRP binding site is located only 15 bases upstream from the lacI promoter (Figure ??). Binding of CRP to its binding
site allows the flexible RNA polymerase domains, ˛C TD and ˛N TD to
bind to CRP, thereby increasing the likelihood of RNA successfully binding to the promoter site.
Transcription factors themselves can be controlled by other transcription
factors binding to operator sites. Control can also be accomplished by
other proteins binding to the transcription factor or by small molecules,
called inducers, that bind to the transcription factor and alter the operator
binding affinity. LacI is an example of a transcription factor where the
inducer molecule allolactose can bind, thereby altering the binding affinity
of LacI. CI from the virus, lambda phage is an example of a transcription
factor where control is exerted by influencing its production rate.
3.4.3
Fractional Occupancy
One of the most important concepts to consider when quantifying how
transcription factors influence gene expression is the fractional occupancy
or degree of saturation at the operator site. This quantity expresses the
probability of a particular occupancy relative to the total of all occupancy
states. A simple example best describes this concept.
Transcriptional Activation
Consider a single operator site upstream of a promoter (Figure 3.11 and
3.12). The operator site binds a single monomeric transcription factor,
A. Assume that when the transcription factor binds to the operator, the
RNA polymerase has a higher probability of binding to the promoter site
by virtue of complementary patches on the RNA polymerase and transcription factor. If we assume the rate of gene expression is proportional
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CHAPTER 3. BASIC ENZYME KINETICS
a) Activation by RNA polymerase requitment
TF
RNA Pol
Operator Promoter
Coding Sequence
b)Sequestration of a repressor resulting in activation
TF
RNA Pol
TF
TF
Transcription Factor
RNA Pol RNA Polymerase
Figure 3.11: Gene regulation by an activating transcription factor. a)
The operator site is upstream of the promoter, binding of the transcription factor increases the likelihood of RNA polymerase binding by way
of weak interactions between the transcription factor and RNA polymerase. Alternatively, b) an activator can sequester a repressor transcription factor.
to the probability of bound RNA polymerase, and that RNA polymerase
has a constant concentration and activity in the cell, then we can assume
the fractional occupancy of the transcription factor is proportional to gene
expression.
Let us designate the concentration of the unbound operator site by the symbol U , the bound operator site by the symbol AU and the free transcription
factor by A as shown in Figure 3.12. The fractional occupancy of the operator site is then given by the degree of bound operator relative to the total
of all occupancy states, that is:
f D
AU
U C AU
If we assume the rate of binding and unbinding of transcription factor to
the operator site is much faster than transcription, then we can also assume
3.4. GENE REGULATORY RATE LAWS
79
U
Operator
AU
Coding Sequence
A
A Transcription Factor
Figure 3.12: Bound (AU) and unbound (U) states for a simple transcriptional activation model.
the binding and unbinding process is at equilibrium. That is, the following
process is at equilibrium:
U C A AU
We can express the equilibrium condition using the association constant,
Ka , where:
AU
Ka D
U A
Given this information we can express AU in terms U :
f D
Ka U A
U C Ka U A
(3.4)
The unbound state, U , can now be eliminated to yield:
f D
Ka A
1 C Ka A
(3.5)
We have seen this same approach when using the rapid equilibrium assumption from enzyme kinetics. Much of the following should therefore
be familiar. Relation (3.5) yields a value between zero and one. Zero
indicates an unbound state, and one indicates the operator site is fully occupied. To obtain the actual rate of expression, assume the rate is linearly
proportional to the fractional occupancy, so that:
v D Vm
Ka A
1 C Ka A
(3.6)
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CHAPTER 3. BASIC ENZYME KINETICS
where Vm is the maximal rate of gene expression (Figure 3.13). Equation (3.6) yields a familiar hyperbolic plot.
Gene Expression Rate, v
1
0:8
0:6
0:4
0:2
0
0
2
4
6
8
10
Transcription Factor Concentration
Figure 3.13: Gene expression rate as a function of a monomeric transcription factor that activates gene expression. The association constant,
Ka , has a value of 1. The reaction rate is normalized by Vm .
If the association constant Ka is substituted by the dissociation constant
(Ka D 1=Kd ), then we obtain:
v D Vm
A
Kd C A
(3.7)
At half saturation it is easy to show that Kd D A. This result provides a
simple way to estimate the Kd from a binding curve by locating the halfsaturation point and then reading the corresponding transcription factor
concentration.
Transcriptional Repression
Repression can be handled in a similar manner. In this case we note that
the active state is now the unbound state, U , so the fractional occupancy is
3.4. GENE REGULATORY RATE LAWS
81
given by:
f D
U
U C AU
Using the same equilibrium relation as before, we obtain (Figure 3.14):
v D Vm
1
1 C Ka A
(3.8)
Gene Expression Rate, v
1
0:8
0:6
0:4
0:2
0
0
2
4
6
8
10
Transcription Factor Concentration
Figure 3.14: Gene expression rate as a function of a monomeric transcription factor that represses gene expression.
As with the activation example in the last section, the dissociation constant, Kd , is equal to the transcription factor concentration at half saturation. The companion book, Enzyme Kinetics for Systems Biology covers
additional topics such as multi-transcriptional control and cooperativity in
gene regulation.
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CHAPTER 3. BASIC ENZYME KINETICS
3.5 Generalized Rate Laws
3.5.1
Power Laws
There are a number of approximate rate laws that have been used in past
models. The simplest approximation is the power law which takes the
form:
vi D ˛i
Y
Sj "ji
The " term is the kinetic order or elasticity and can and often is a noninteger. Negative values for " can be used to indicate inhibition. The advantage of the power law equation over the simpler linear rate law is that
it shows a curvature reminiscent of an enzyme kinetic response. However
the function does not saturate and this is one of its main drawbacks. It has
found extensive use in Biochemical Systems Theory which was developed
by Michael Savageau.
3.5.2
Linear-Logarithmic Rate Laws
An improved approximation over the power law is the linear-logarithmic
approximation (or linlog for short). One of the main limitations of the
power law approximation is that there is no saturation effect at high reactant concentration whereas lin-log kinetics will show some degree of
saturation. The general form of the equation is given by:
v D vo
e
eo
X
S
1C
" ln
So
where S is the reactant concentration and " the elasticity. The rate law
is always defined around some reference state, a reference rate, vo and a
reference reactant concentration, so . The utility of this approximation is
that the values of the elasticities (kinetic orders) can to some extent be estimated from the known thermodynamic properties of the reaction. The
values of the elasticities will be a function of the reference state. If no
thermodynamic information is available then the elasticities can even be
3.5. GENERALIZED RATE LAWS
83
set to the stoichiometries of the respective reactants. In either case it is
important that the lin-log approximation is only valid around the chosen
reference state and also depends on how much the reactant levels diverge
from the references during a simulation and the degree to which the elasticities are affected. A sensitivity analysis can be made to ascertain these
details. An example of how to set a lin-log rate law is given in a subsequent
section on elasticities.
The companion book, Enzyme Kinetics for Systems Biology has a much
more extensive discussion of rates including additional sections on other
generalized rate laws.
3.5.3
Choosing a suitable Rate Law
Given the huge range of possible rate laws, the novice modeler might seem
at a loss to know which rate law to select for a given reaction step. However, before a model is built, its purpose should be clearly understood because that will help decided on the types of rate laws to employ. Ultimately
models only have two functions: 1) Describe known observations and 2)
Make new non-trivial predictions. So long as these requirements are satisfied the model is useful. It is often the case that novice modelers will
feel it necessary to add every small detail into a model when in fact much
of the detail can be dispensed. A model is a simplification of reality not
a replica and the art of building models is knowing what details can be
left out and what details are necessary. The question whether a particular
reaction should use a specific pion-pong based rate law or a generalized
rate law depends on how this choice determines the behavior of the model
particularly within the constraints of measurements. A useful strategy is
to carry out a sensitivity analysis to determine how much influence parameters or particular rate laws have on the dynamics of a model. If a
particular parameter has little influence then there is no need to obtain a
precise value for it while if a particular ate law has little influence than
a simpler rate laws can be used instead which will often have much few
parameters to set. It might be possible to use lin-log rate laws at many reaction steps while certain steps require a much more detailed description.
As more detailed measurements become available it might be found that
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CHAPTER 3. BASIC ENZYME KINETICS
some of the lin-log approximations are too approximate and subsequent
experimental efforts can focus on the descriptions at those particular steps.
Exercises
1. In the steady state derived Michaelis-Menthen equation, what units
does the Km have?
2. What is the concentration of substrate that yields half the reaction
velocity for an irreversible Michaelis-Menten rate law?
3. An enzyme has a Vm of 10 mmols 1 mg 1 . The substrate Km is 0.5
mM. What is the initial rate when the substrate concentration is 0.5
mM and 5 mM?
4. At low substrate concentration is the order of the reaction, zero, first
or second order?
5. Do enzymes change the equilibrium constant for a reaction?
6. List the assumptions made when the Michaelis-Menten equation is
derived using the steady state assumption.
7. Using the irreversible Hill equation, show that thepsubstrate concentration at half the maximal velocity is given by n Kd where Kd is
the dissociation constant and n the Hill coefficient.
8. Show that the reversible Hill equation reduces the the irreversible
Hill equation when the product P is set to zero.