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3Basic Enzyme Kinetics

Wednesday 7th March, 2012 at 12:13 Noon Control

Theory for Biologists, Draft 0.81 www.sys-bio.org

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:

E C S k1*)k1

ESk2! E C P (3.1)

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

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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:

E C S k1*)k1

ESk2*)k2

E C P

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 discus-

sion 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 un-

less it is absolutely necessary in order to capture a particular type of dy-namical 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 ac-

tion:

E C S k1*)k1

ESk2! E C P

Two different assumptions have been employed to reduce this scheme toa 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 sec-

ond 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 as-

sumed that the enzyme substrate complex rapidly reached steady state.

This was less restrictive that the rapid equilibrium assumption. Enzymerate laws are often are derived using the steady state assumption, however
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3.1. ENZYME KINETICS 57

0 0:2 0:4 0:6 0:8 1 1:2 1:40

2

4

6

8

10

P

E

S

ES

Time

Concentration

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 ofES and E (dES=dt 0). At thesame time, the rate of change of S and P are constant over this period.

k1 D 20; k1 D 1; k2 D 10, that is: E C S20

*)1

ES10

! E C P

because the mathematics can become complicated, many complex mech-

anisms, 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 constantfor binding:

Ks DE : S

ES

and noting that the total concentration of enzyme, Et , is the sum of free

enzyme, E and enzyme substrate complex, ES: Et D E C ES, it is easyto show that the equilibrium concentration ofES is given by:

ESD Et : SKs C S
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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 withthe previous relation for ES, yields our result:

v D Et : k2 SKs 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 enzy-

matic species. Note that the concentration of the enzyme substrate com-

plex 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 sub-

strate complex (??) can be written down using the laws of mass-action:

dES

dtD k1 E : S k1 ES k2 ES

The concentration of enzyme substrate complex is assumed to rapidlyreach steady-state (Fig. ??) so that the above equation can be set to zero:

0 D k1 E : S k1 ES k2 ES

We also note that the total concentration of enzyme, Et , is the sum of free

enzyme, E and enzyme substrate complex, ES:

EtD

E

CES

From these relationships, the steady-state concentration of enzyme sub-

strate complex can be derived:

ESD Et : S.k1 C k2/=k1 C S

By assuming that the rate of reaction is given by v D k2 ES, we obtain:

v D Et k2 S.k1 C k2/=k1 C S

(3.2)
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The Vmax can be expressed as the total enzyme concentration times the

rate constant for the product formation, Et k2. We can also combine the

constants .k1 C k2/=k1 into a single new constant called the Michaelisconstant, or Km.

0 5 10 15 20 25 300

0:2

0:4

0:6

0:8

1Vmax

Km Substrate Concentration

React

ionRate

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.

v DVmax 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 (Fig-

ure. 3.2).

Reversible Michaelis-Menten Rate law

The derivation of the irreversible Michaelis-Menten is an instructive exer-cise, however it is not a particularly realistic model to use in models be-
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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 k2 E : P

The expression that describes the steady-state concentration of the enzyme

substrate complex also has an additional term from the product binding

(k2 EP). Taking these into consideration leads to the general reversible

rate expression (See Appendix B for a full derivation):

v D Vf S=KS Vr P =KP1C 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 DPeq

SeqD Vf KP

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 toeliminate one of the kinetic constants and substitute the equilibrium con-

stants in its place. This is useful because equilibrium constants tend to be

better known that kinetic constants. Incorporating the Haldane relationship

yields the equation

v D Vf=KS .S P =Keq/1C 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


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following expression:

v D .S P =Keq/Vf=KS

1C S=KS C P =KPThe fact that the equilibrium constant appears are a constant factor in the

expression suggests that enzymes do notchange 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=KPwhere Seq and Peq represent the equilibrium concentrations for substrate

and product. Rearrangement yields

Keq D PeqSeq

D Vf KPVr 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 equi-

librium constants in its place. This is useful because equilibrium constants

tend to be better known that kinetic constants. Incorporating the Haldanerelationship yields the equation

v D Vf=KS .S P =Keq/1C 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 thefollowing expression:

v D .S P =Keq/Vf=KS

1C S=KS C P =KP
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The fact that the equilibrium constant appears are a constant factor in the

expression suggests that enzymes do notchange 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 im-

portant industrial example of this is the conversion of lactose to galactose

by the enzyme galactosidase where galactose will compete with lactoseand thereby slow the forward rate (Gekas and Lopex-Leiva, 1985). The re-

versible 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:

v D VmSSCKm

1C P =Kp

Further discussion on this is given in more detail in section ?? when dis-

cussing competitive inhibition.

The steady-state approximation that allows us to derive convenient aggre-

gate 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 concentra-

tions. Models that employ the Michaelis-Menten laws compared to ex-

plicit mass-action models can exhibit changes in their behavior. In par-

ticular 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 behav-

ior will be largely unaffected except in some cases where the dynamic

stability might change, for example leading to the onset of oscillatory be-

havior. Ideally one should check whether in a particular model the use


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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 aggre-

gate 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 concentra-

tions. Models that employ the Michaelis-Menten laws compared to ex-

plicit mass-action models can exhibit changes in their behavior. In par-

ticular 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 behav-

ior will be largely unaffected except in some cases where the dynamic

stability might change, for example leading to the onset of oscillatory be-

havior. 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, phosphofructok-

inase (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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other subunits is termed cooperative binding. If ligand binding increases

the affinity of subsequent ligand binding, then it is termed positive cooper-

ativity, 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.

0 0:5 1 1:5 2 2:5 30

0:2

0:4

0:6

0:8

1

Substrate Concentration

ReactionRate

Figure 3.3: Plot comparing positive cooperativity to a hyperbolic re-

sponse.

Hill Equation

The Hill equation was originally derived empirically to describe the sig-

moid 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 thatall the remaining ligands also bind simultaneously, then we can represent


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this situation as follows:

E C n S! ESAssuming the rapid equilibrium assumption we can write:

KD ESE : Sn

where K is the association constant for ligand binding. Using the conser-

vation relation Et D E C ES, the relative saturation can be shown to be

given by: ES

EtD S

n

1=KC Sn DSn

KdC SnThis is the Hill equation where Kd is the dissociation constant. Often the

Hill equation is represented in the following way in the literature:

v D Vmax Sn

KdC Sn

where Kd is the dissociation constant and h the Hill coefficient. Some-

times 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 np

Kd

That is np

Kd

is the half-maximal activity value, or KH Dnp

Kd

, that is

KnHD Kd. We can therefore write the Hill equation in an alternative form

as:

v D Vmax Sn

KnHC Sn D

Vmax .S=KH/n

1C

SKH

n

In the literature both forms are presented but they all have the same be-

havior. The equation in terms of the half-maximal activity has advantagesbecause half-maximal activity can be measured directly from experiments.


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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. How-

ever, fitting the Hill equation to real data rarely gives integer estimates ton suggesting that the model is not a faithful representation of any real sys-

tem. The utility of the Hill equation however lies in its ability to represent

sigmoid behavior for simple cooperative systems such as transcription fac-

tor 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.

0 0:5 1 1:5 2 2:5 30

0:2

0:4

0:6

0:8

1

n D 8 n D 4n D 2


ReactionRate

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


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reaction. In recent years, Hofmeyr and Cornish-Bowden published a de-

scription of the reversible Hill equation with modifiers. The general form

of the reversible Hill equation without modifiers is given by:

v DVf

S

Ks

1

Keq

S

KsC P

Kp

h1

1C

S

KsC P

Kp

h

Figure 3.5 illustrates the sigmoid behavior with respect to the substrate

concentration. The K constants in the equation are the half saturation con-

stants. is the mass-action ratio and Keq the equilibrium constant for

the reaction. What is significant about this formulation is that the ther-

modynamic 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 re-duced to the non-cooperative reversible Michaelis rate law and of course

if reversibility is removed as well the equation reduces to the simple irre-

versible 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 denomina-

tor. 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 actsas an activator.

v DVf

S

Ks

1

Keq

S

KsC P

Kp

h1

1C MKm

h

1C M

KmhC

S

KsC P

Ks

h
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< 1 inhibitor > 1 activator

0 2 4 6 8

0

0:2

0:4

0:6

0:8

1

Ks = 1 2.4 4.0


ReactionRa

te

Figure 3.5: Plot showing the response of the reaction rate for a re-

versible 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 D4:85; Ke D 2:75; D 105, 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 uni-

uni reversible Hill equation that incorporated multiple modulators:


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0 2 4 6 80

0:1

0:2

0:3

0:4

0:5

Ke = 1.2 2.0 4.2

Inhibitor Concentration

ReactionRate

Figure 3.6: Plot showing the response of the reaction rate for a re-

versible Hill model with respect to the inhibitor concentrations as a

function of the inhibitor Michaelis constant. KsD

2; S

D1, all other

parameter were identical to the previous figure.

v DVf

1C

Keq

. C /h1

QnmiD1

1Ch

i

1Cihi

C . C /h

To simplify the notation in the above equation, D S=Ks , D P =Kpand D M=Km. is the modifier factor that determines whether themodifier is an activator (> 1) or an inhibitor. Kx are the Michaelian con-

stants, 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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3.3 Multiple Substrate Enzymes

It is probably fair to say that most enzyme catalyzed reactions involve twosubstrates. For example, all oxidoreductases involve two substrates, one an

oxidant and the other a reductant. Even apparently single substrate reac-

tions 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 sin-

gle 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 compulsory-

order, when one substrate must bind before the other, random-order where

substrates can bind in any order and double-displacement where one sub-

strate 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 dif-

ference 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 im-

portant. 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 ratelaws was introduced by Alberty in 1953:

v D VmABKBACMAB C AB CKiAKB

where KA and KB are Michaelian constants and KiA is a dissociation con-

stant.

A useful reversible rate law is given by:


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3.3. MULTIPLE SUBSTRATE ENZYMES 71

Reaction Scheme Rate Law

A$ B Vf Vr 1C C

AC B $ C Vf Vr1C C C C

AC B $ CCD Vf Vr 1C

C

C

C

C

C

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.

v D

1 P Q

Keq A B

Vm A B

KAKB1C A

KAC Q

KQ

1C B

KBC P

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 kineticswhich is a further generalization that includes a range of different stoichio-

metric reaction schemes some of which are given in the table below.

Of more interest is the reversible Hill equation described in the last sec-

tion. The reversible Hill equations can be generalized to accommodate

many different possibilities, including multi-substrate, multi-modulators

and irreversibility.


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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 re-

gions. 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 transcrib-

ing the DNA coding sequence into messenger RNA (mRNA). The bindingof proteins called transcription factors (TF) to the operator sites are re-

sponsible for influencing the binding of RNA polymerase and thus can

modulate mRNA production.

......

One or More Coding SequencesPromoter

OperatorsOperators

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 pro-

moter. 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 ri-

bosome binding site (RBS) and the terminator. The RBS is often a six to

seven base nucleotide base sequence located about eight nucleotides up-

stream from the coding sequence start codon and is used by the ribosome
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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 ofgene transcription. Multiple transcription factors may also interact to con-

trol the expression of a single operon. These interactions can emulate sim-

ple logical functions (such as AND, OR, etc.) or more elaborate compu-

tations. Gene regulatory networks range from a single controlled gene to

hundreds of genes interlinked with transcription factors forming a com-

plex, decision making network.

Different classes of transcription factors also exist. For example, the bind-

ing of some transcription factors is modulated by small molecules, a well

known example being the binding of allolactose (a disaccharide very sim-

ilar to lactose) to the lac repressor or cAMP to the catabolite activator

protein (CAP), also known as the cAMP receptor protein (CRP). Alter-

natively, a transcription factor may be expressed by one gene and either

directly modulate a second gene (which could be itself) or via other tran-

scription factors. Additionally, some transcription factors only become

active when phosphorylated or unphosphorylated by protein kinases andphosphatases (Figure 3.9).

The size of gene regulatory networks vary from organism to organism. The

genome ofE. 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 isa 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 bytranscription factors

, RNA poly-merase, and proteins called factors. factors are transcriptional initia-
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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 oftranscription factors. These proteins either enhance or reduce the ability of

RNA polymerase to bind to the promoter region and commence transcrip-

tion. 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 pro-

moter.

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 in-

fluence 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).

Coding SequencePromoterOperator

a) Unbound State

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 boundstate 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 bythe association constant of binding, the availability of transcription factors,
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and other regulators.

Gene Activation

Gene Cascade

Auto-Regulation

Gene Repression

Multiple Control

Regulation by Small

Molecule

Regulation by

Phosphorylation

~P

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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the transcription factor has formed DNA loops.

Coding SequenceOperatorPromoter

Repressing Transcription Factor

RNA Polymerase

Downstream Obstructiona)

Promoter Obstructionb)

TF

TF

RNA Pol

TF

RNA Pol

c) Sequestration of an activator resulting in inhibition

TF

TF

Activator

RNA Pol

Figure 3.10: Obstruction, exclusion and sequestration models for re-

pressing gene expression.

Examples of downstream obstruction include the galR and galSoperators,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-


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stance the transcription factor can offer a suitable but weak molecular face

for the RNA polymerase to bind (Figure 3.11). This allows RNA poly-

merase to stay on the promoter longer and therefore increases the prob-ability 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 up-

stream from the lacI promoter (Figure ??). Binding of CRP to its binding

site allows the flexible RNA polymerase domains, CTD and NTD to

bind to CRP, thereby increasing the likelihood of RNA successfully bind-

ing 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 tran-scription factor. If we assume the rate of gene expression is proportional
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Coding SequencePromoterOperator

RNA Polymerase

b)Sequestration of a repressor resulting in activation

TF

TF

RNA Pol

a)Activation by RNA polymerase requitment

TF

TF Transcription Factor

RNA Pol

RNA Pol

Figure 3.11: Gene regulation by an activating transcription factor. a)

The operator site is upstream of the promoter, binding of the transcrip-

tion factor increases the likelihood of RNA polymerase binding by way

of weak interactions between the transcription factor and RNA poly-

merase. Alternatively, b) an activator can sequester a repressor tran-

scription 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 sym-bol 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 oper-

ator site is then given by the degree of bound operator relative to the total

of all occupancy states, that is:

f D AUUC 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
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AU

Coding Sequence

U

Transcription Factor

Operator

A

A

Figure 3.12: Bound (AU) and unbound (U) states for a simple transcrip-

tional activation model.

the binding and unbinding process is at equilibrium. That is, the following

process is at equilibrium:

UCA AU

We can express the equilibrium condition using the association constant,

Ka, where:

Ka D AUU A

Given this information we can express AU in terms U:

f D Ka U AUCKa U A

(3.4)

The unbound state, U, can now be eliminated to yield:

f DKa

A

1CKaA (3.5)

We have seen this same approach when using the rapid equilibrium as-

sumption 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 oc-

cupied. To obtain the actual rate of expression, assume the rate is linearly

proportional to the fractional occupancy, so that:

v D VmKaA

1CKaA(3.6)
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where Vm is the maximal rate of gene expression (Figure 3.13). Equa-

tion (3.6) yields a familiar hyperbolic plot.

0 2 4 6 8 100

0:2

0:4

0:6

0:8

1

Transcription Factor Concentration

Gene

ExpressionRate,

v

Figure 3.13: Gene expression rate as a function of a monomeric tran-

scription factor that activates gene expression. The association constant,

Ka, has a value of1. 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 VmA

KdC A(3.7)

At half saturation it is easy to show that Kd D A. This result provides asimple way to estimate the Kd from a binding curve by locating the half-

saturation 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
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given by:

f

DU

UC AUUsing the same equilibrium relation as before, we obtain (Figure 3.14):

v D Vm1

1CKaA(3.8)

0 2 4 6 8 100

0:2

0:4

0:6

0:8

1

Transcription Factor Concentration

GeneExpressionRate,

v

Figure 3.14: Gene expression rate as a function of a monomeric tran-

scription factor that represses gene expression.

As with the activation example in the last section, the dissociation con-

stant, Kd, is equal to the transcription factor concentration at half satura-tion. 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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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 iY

Sj"ij

The " term is the kinetic order or elasticity and can and often is a non-

integer. Negative values for " can be used to indicate inhibition. The ad-

vantage 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 re-

actant 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

1C

X" ln

S

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 es-

timated 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


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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 divergefrom the references during a simulation and the degree to which the elas-

ticities 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. How-

ever, before a model is built, its purpose should be clearly understood be-

cause 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 sat-

isfied 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 modelparticularly within the constraints of measurements. A useful strategy is

to carry out a sensitivity analysis to determine how much influence pa-

rameters 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 re-

action 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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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 mmols1mg1. 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 the substrate concen-

tration at half the maximal velocity is given by np

Kd where Kd is

the dissociation constant and n the Hill coefficient.

8. Show that the reversible Hill equation reduces the the irreversibleHill equation when the product P is set to zero.

cb chapter3

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