Untangling Complex Systems, page 35
3
2
0.5
Amplitude
Amplitude 1
0.0
0
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Frequency (years−1)
Frequency (years−1)
FIGURE 6.13 Fourier spectra of the percent annual GDP growth for the USA (on the left) and China (on
the right).
has been proposed by the Soviet economist Nikolai Kondratiev (1935) and was attributed to
fundamental technological innovations that spark revolutions and create leading industrial
or commercial sectors. Many cycle theorists (Korotayev and Tsirel, 2010) have proposed
five cycles so far:
1. The Industrial Revolution: steam engines and industrialization (~1780 ~1830).
2. The Age of steel, railways, heavy engineering (~1830 ~1880).
3. Electricity and chemical industry (~1880 ~1930).
4. The petrochemical industry, automobile, mass production (~1930 ~1970).
5. Information technology (~1970 ~2010).
Now, we are undertaking a new cycle. Its features are still in fieri, but they are sometimes
connected to nano- and bio-technologies.
The Emergence of Temporal
7 Order within a Living Being
A living being is like “a house with clocks in every room and every corner, yet in one way or
another, they work in an organized way.”
Derk-Jan Dijk (1958 AD)
7.1 INTRODUCTION
A cell is the basic unit of any living being. Within a biological cell, there are tens of thousands of
different biochemical species (Reactome website). These species interact in a highly organized
manner, both spatially and temporally. A cell looks like a “molecular ecosystem.” Within this
“microscopic ecosystem,” proteins play important roles. In fact, the term protein derives from the
Greek “πρωτείος” meaning “holding the first place” (Vickery 1950). There are thousands of dif-
ferent kinds of proteins in a human cell. Proteins are the workhorses of the cell (Lodish et al.
2000). Distinct sets of proteins characterize cells belonging to different tissues. Many of the pro-
teins within cells are enzymes, which are the catalysts of the biochemical reactions.1 Other proteins allow cells to move and do work, maintain internal cell rigidity, and transport molecules across
membranes. Reflecting their numerous functions, proteins come in many shapes and sizes. Proteins
are formed from only 20 different L- α-amino-acids. The definition of a protein structure requires
the specification of hierarchical features. First, it is necessary to know the sequence of amino acids
(the so-called primary structure). Then, we need to specify the presence of alpha helices and/or beta
sheets (the secondary structure). Furthermore, it is necessary to describe the folding of alpha helices
and/or beta sheets (the tertiary structure) forming globules. Finally, we define the arrangement of
the globules or subunits (the quaternary structure).2 This chapter shows examples of the roles played by the proteins in metabolic, signaling, and epigenetic cellular events. We will discover that proteins
can participate in phenomena of temporal self-organization.
7.2 METABOLIC EVENTS
Every second, an astonishing number of chemical reactions proceeds inside the cells. Many of these
reactions provide the energy for vital processes and produce the substances necessary for growth,
internal structuring and reproduction. They constitute the ensemble of processes known as metabo-
lism. In the relatively low-temperature environment of the cell, most of the reactions are unlikely
to occur. Fortunately, there are the enzymes ( E). Enzymes are highly specific catalysts, due to their
structure. The many enzymatic reactions occurring in a living cell must be balanced to keep the cell
healthy. The regulation of the enzymatic reaction ultimately requires the control of the enzymatic
activity and the control of the enzyme amount.
The mechanism of action of an enzyme is usually described by either the Michaelis-Menten
or the Hill equation. These equations look like the Type II and Type III functional responses,
1 The term enzyme was coined by the German physiologist Wilhelm Kühne from the Greek “ένζυμον” which means “in leaven” when he discovered the possibility of inducing the process of fermentation also in the absence of living beings.
2 You can consult any book on biochemistry to see a picture of the hierarchical structure of a protein.
167
168
Untangling Complex Systems
respectively, which we encountered in Chapter 5. In fact, often, proteins play the roles of predators for their substrates that behave as preys. 3
7.2.1 michaelis-menTen kineTics
Usually, proteins have one site where they host selectively a substrate S, forming a complex ( ES):
kc
E S
ES
kh
+ →
( )
E + P
←
→
. [7.1]
k− c
Note the formal link between equation [7.1] and the mechanism modeling the Type II functional
response (equation [5.17]). S represents the prey. The complex (ES) transforms S into the product P.
The rate of product formation is
d [ P]
= k ( )
h
ES
dt
. [7.2]
The rate of change in the concentration of the complex ( ES) is
d ( ES)
= k [ ][ ] − ( )
c E
S − k c ES kh ES . [7.3]
dt
− (
)
If we indicate with C the analytical concentration of E (i.e., the sum [ E]
E
+[( ES)]), and we apply the
steady-state condition for [( ES)], we obtain
C [ ]
[ ]
E S
CE S
( ES) =
. [7.4]
kh k
[ S
− c
[ S]+ KM
]+ +
=
kc
The quantity K is called the Michaelis constant, after the German enzymologist Leonor Michaelis
M
(1875–1949). Introducing the steady-state concentration of [( ES)] (equation [7.4]) into equation [7.2],
we achieve the well-known Michaelis-Menten formula, representing the rate of product formation
as a function of the analytical concentration of E, K , and [ S]:
M
d [ P] k C [ ]
h E S
=
. [7.5]
dt
[ S]+ KM
The amount ( /
1 CE )( d[ P]/ dt) = ( kh[ S] / ([ S] + KM )) is formally equivalent to the Type II functional response of equation [5.22]. The rate of product formation (equation [7.5]) is a hyperbolic function
of [ S]. When [ S] is small compared to K , the rate is a linear function of [ S]. On the other hand, M
when [ S] is much larger than K , the rate becomes independent of [ S] and reaches its maximum value, M
which is given by ( k C ). The rate of the reaction cannot be any faster than when every enzyme mol-
h
E
ecule is in a complex with a substrate molecule, i.e., when CE = [( ES)]. The ratio of the maximum
rate of reaction to the enzyme analytical concentr ation is known as the turnover number k , i.e., the
h
number of substrate’s moles converted to product per unit of time.
3 An excellent example is in our immune system wherein antibodies are predators and antigens are preys.
The Emergence of Temporal Order within a Living Being
169
7.2.2 hill kineTics
There exist also enzymes having more than one binding site. Their mechanism of action is like that
of equation [7.6]:
E nS
kc
ES
kh
+
→
(
) n
ESn−1 + P
←
→
. [7.6]
k− c
Note the formal link between equation [7.6] and the mechanism modeling the Type III functional
response (equation 5.27).
The rate of change for [( ES )] is
n
d ( ES )
n = k
n
[ ][ ] ( −
)( )
c E
S − k c + kh
ES . [7.7]
dt
n
If C is the analytical concentration of the enzyme (i.e., C
)]), applying the steady state
E
E = [ E] + [( ESn
approximation to ES , we obtain:
n
k
n
[ ] (
) ) ( −
)( )
c S
CE − ( ESn = k c + kh
ESn
. [7.8]
If we rearrange equation [7.8], we achieve
[ S n] C
n
[ ]
E
S C
( ES
E
) n =
. [7.9]
n
k
n
h
k
[ S
− c
[ S] + K
] + +
=
kc
The ratio [ ES ]/ C is known as Hill equation (Haynie 2001). The rate of product P formation will be n
E
d [ P] k C
n
[ ]
h E S
=
. [7.10]
dt
[ S n] + K
The amount ( /
1 C
n
n
E )( d[ P]/ dt ) = ( kh[ S ] /([ S ] + K )) is formally equivalent to the Type III functional response of equation [5.31].
The enzymes having more than one site may have an exciting feature: that of showing indirect
interactions between the distinct binding sites. In such cases, the enzymes are defined allosteric.
This term was introduced by Jacques Monod and François Jacob to characterize the end-product
(L-isoleucine) inhibition of the enzyme L-threonine deaminase (Cui and Karplus 2008). L-isoleucine
does not compete with the reactant L-threonine in binding at the catalytic site; it instead binds at a
regulatory site, inhibiting the reaction. The term “allosteric” comes from the Greek αλλος- στερεος
meaning “other-space” and refers to the influence the binding at one site has on the binding at a
remote location in the same macromolecule. The ligand that brings about the allosteric regulation
of the binding of another ligand is called effector, or modulator.4 If the effectors are identical to the substrate molecule, we speak about “homotropic interactions” between the ligands and the enzyme.
If the binding of a substrate to the first site of the protein facilitates binding to the second, and so
on, the interaction is defined as “positive cooperativity.” On the other hand, if the binding of the
4 The control of enzyme activity can be achieved by modification of the enzyme molecules through either covalent change (such as phosphorylation and hydrolysis) or non-covalent conformational change.
170
Untangling Complex Systems
With activator
kh
Without effector
V
With inhibitor
(a) 0
[ S]
With activator
Without effector
V
With inhibitor
(b) 0
[ S]
FIGURE 7.1 Rate profiles for allosteric enzymes in case of competitive ligands (a) and uncompetitive
ligands (b).
substrate molecule to the first site inhibits the binding to the second, and so on, the interaction is
defined as “negative cooperativity.” When the effectors are ligands having a structure different from
that of the substrate, we speak about “heterotropic interactions.” In heterotropic interactions, the
effectors can play as either activators or inhibitors of enzymatic activity. Depending on how ligands
affect the enzymatic activity, their actions are also typified as “competitive” or “uncompetitive”
concerning the substrate binding to the enzyme. Ligands are defined as competitive when their
binding changes the K constant of equation [7.10] (in fact, they are also named as “K systems;” see
Figure 7.1a). When the binding of a ligand changes the maximum velocity of the enzymatic reaction (i.e., k in equation [7.10]), they are referred to as uncompetitive or “V systems” (where V stands for
h
Velocity; see Figure 7.1b) (Hammes and Wu 1974).
7.2.3 The nonlineariTy of allosTeric enzymes
Many proteins turn their activities on and off by the nonlinear allosteric effects. Examples of het-
erotropic interactions are offered by the allosteric enzyme Aspartate Transcarbamylase (ATCase)
catalyzing the reaction shown in Figure 7.2, which is the first step in pyrimidine biosynthesis.
ATCase has catalytic and regulatory sites. The aspartic acid (R ) binds to the catalytic site of
2
ATCase and in the presence of a saturating concentration of carbamoyl phosphate (R ) produces
1
N-carbamoyl aspartate (P ) and phosphate (P ). This reaction is the first step in the biosynthesis of
1
2
O
O–
O
H
ATCase
NH
+
–
2C
OP O– + NH
CH
NH
3
2C
N CH COO–
COO–
+ H2PO4
O
CH
CH
2
2
COO–
COO–
(R1)
(R2)
(P1)
(P2)
FIGURE 7.2 The first step of pyrimidine synthesis.
The Emergence of Temporal Order within a Living Being
171
NH2
NH2
N
O
O
O
N
O
O
O
N
O–
P O
P
O P O CH
N
2
N
O–
P O
P
O P O CH
O
2
N
O
O
O–
O–
O–
O–
O–
O–
OH
OH
OH
OH
CTP
ATP
FIGURE 7.3 Structures of cytidine triphosphate (CTP) and adenosine triphosphate (ATP).
molecules such as cytosine, thymine, and uracil, which contain the pyrimidine unit and are some of
the nitrogenous bases of DNA and RNA. The velocity of the enzymatic reaction as a function of the
concentration of aspartate shows a sigmoidal dependence (Hammes and Wu 1974). Along the syn-
thetic pathway leading to pyrimidines as final products, cytidine triphosphate (CTP) is produced.
CTP competes with the structurally similar adenosine triphosphate (ATP) (see Figure 7.3) in bind-
ing to the regulatory sites of ATCase.
When the concentration of CTP is high, it overcomes ATP, and it preferentially associates with
ATCase exerting an inhibiting effect on its catalytic activity, such as negative feedback in the syn-
thesis of N-carbamoyl aspartate (P ). On the other hand, when the concentration of CTP is not high,
1
ATP wins the competition in binding to the regulatory sites of ATCase. When it is ATP to bind to
ATCase, the final effect is the opposite: ATP activates the catalytic power of ATCase. CTP and ATP
are K systems (see Figure 7.1a), because they alter the value of K and not that of k . h An example of an allosteric protein exhibiting positive cooperativity in homotropic interactions is
the blood protein hemoglobin (Hb). Hb plays a vital role in the transfer of O from the lungs to other
2
cells of the body where it is combustive and participates in reactions releasing thermal energy. Hb is
a tetramer, consisting of four subunits each having an oxygen-binding site, due to the presence of iron
atom coordinated with a porphyrin ring. When an oxygen molecule binds to a site of hemoglobin, it
increases the affinity of the other sites for oxygen (Haynie 2001). In 1965, Monod along with Wyman
and Changeux proposed a model (known as MWC model) for allostery based on a few assumptions.
Allosteric proteins are oligomers made of identical monomers in a symmetric arrangement. Each
monomer has two folded tertiary conformations ( T and R, indicated by squares and circles, respec-
tively, in Figure 7.4). All monomers are in either one conformation or the other. When a ligand binds to a monomer of the protein, all monomers undergo the R-to- T or the T-to- R transition simultaneously. This model is represented by the structures enclosed by dashed lines. There are no “hybrid”
quaternary structures with some subunits in R state and some in T state. There exists another model
2
0.5
Amplitude
Amplitude 1
0.0
0
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Frequency (years−1)
Frequency (years−1)
FIGURE 6.13 Fourier spectra of the percent annual GDP growth for the USA (on the left) and China (on
the right).
has been proposed by the Soviet economist Nikolai Kondratiev (1935) and was attributed to
fundamental technological innovations that spark revolutions and create leading industrial
or commercial sectors. Many cycle theorists (Korotayev and Tsirel, 2010) have proposed
five cycles so far:
1. The Industrial Revolution: steam engines and industrialization (~1780 ~1830).
2. The Age of steel, railways, heavy engineering (~1830 ~1880).
3. Electricity and chemical industry (~1880 ~1930).
4. The petrochemical industry, automobile, mass production (~1930 ~1970).
5. Information technology (~1970 ~2010).
Now, we are undertaking a new cycle. Its features are still in fieri, but they are sometimes
connected to nano- and bio-technologies.
The Emergence of Temporal
7 Order within a Living Being
A living being is like “a house with clocks in every room and every corner, yet in one way or
another, they work in an organized way.”
Derk-Jan Dijk (1958 AD)
7.1 INTRODUCTION
A cell is the basic unit of any living being. Within a biological cell, there are tens of thousands of
different biochemical species (Reactome website). These species interact in a highly organized
manner, both spatially and temporally. A cell looks like a “molecular ecosystem.” Within this
“microscopic ecosystem,” proteins play important roles. In fact, the term protein derives from the
Greek “πρωτείος” meaning “holding the first place” (Vickery 1950). There are thousands of dif-
ferent kinds of proteins in a human cell. Proteins are the workhorses of the cell (Lodish et al.
2000). Distinct sets of proteins characterize cells belonging to different tissues. Many of the pro-
teins within cells are enzymes, which are the catalysts of the biochemical reactions.1 Other proteins allow cells to move and do work, maintain internal cell rigidity, and transport molecules across
membranes. Reflecting their numerous functions, proteins come in many shapes and sizes. Proteins
are formed from only 20 different L- α-amino-acids. The definition of a protein structure requires
the specification of hierarchical features. First, it is necessary to know the sequence of amino acids
(the so-called primary structure). Then, we need to specify the presence of alpha helices and/or beta
sheets (the secondary structure). Furthermore, it is necessary to describe the folding of alpha helices
and/or beta sheets (the tertiary structure) forming globules. Finally, we define the arrangement of
the globules or subunits (the quaternary structure).2 This chapter shows examples of the roles played by the proteins in metabolic, signaling, and epigenetic cellular events. We will discover that proteins
can participate in phenomena of temporal self-organization.
7.2 METABOLIC EVENTS
Every second, an astonishing number of chemical reactions proceeds inside the cells. Many of these
reactions provide the energy for vital processes and produce the substances necessary for growth,
internal structuring and reproduction. They constitute the ensemble of processes known as metabo-
lism. In the relatively low-temperature environment of the cell, most of the reactions are unlikely
to occur. Fortunately, there are the enzymes ( E). Enzymes are highly specific catalysts, due to their
structure. The many enzymatic reactions occurring in a living cell must be balanced to keep the cell
healthy. The regulation of the enzymatic reaction ultimately requires the control of the enzymatic
activity and the control of the enzyme amount.
The mechanism of action of an enzyme is usually described by either the Michaelis-Menten
or the Hill equation. These equations look like the Type II and Type III functional responses,
1 The term enzyme was coined by the German physiologist Wilhelm Kühne from the Greek “ένζυμον” which means “in leaven” when he discovered the possibility of inducing the process of fermentation also in the absence of living beings.
2 You can consult any book on biochemistry to see a picture of the hierarchical structure of a protein.
167
168
Untangling Complex Systems
respectively, which we encountered in Chapter 5. In fact, often, proteins play the roles of predators for their substrates that behave as preys. 3
7.2.1 michaelis-menTen kineTics
Usually, proteins have one site where they host selectively a substrate S, forming a complex ( ES):
kc
E S
ES
kh
+ →
( )
E + P
←
→
. [7.1]
k− c
Note the formal link between equation [7.1] and the mechanism modeling the Type II functional
response (equation [5.17]). S represents the prey. The complex (ES) transforms S into the product P.
The rate of product formation is
d [ P]
= k ( )
h
ES
dt
. [7.2]
The rate of change in the concentration of the complex ( ES) is
d ( ES)
= k [ ][ ] − ( )
c E
S − k c ES kh ES . [7.3]
dt
− (
)
If we indicate with C the analytical concentration of E (i.e., the sum [ E]
E
+[( ES)]), and we apply the
steady-state condition for [( ES)], we obtain
C [ ]
[ ]
E S
CE S
( ES) =
. [7.4]
kh k
[ S
− c
[ S]+ KM
]+ +
=
kc
The quantity K is called the Michaelis constant, after the German enzymologist Leonor Michaelis
M
(1875–1949). Introducing the steady-state concentration of [( ES)] (equation [7.4]) into equation [7.2],
we achieve the well-known Michaelis-Menten formula, representing the rate of product formation
as a function of the analytical concentration of E, K , and [ S]:
M
d [ P] k C [ ]
h E S
=
. [7.5]
dt
[ S]+ KM
The amount ( /
1 CE )( d[ P]/ dt) = ( kh[ S] / ([ S] + KM )) is formally equivalent to the Type II functional response of equation [5.22]. The rate of product formation (equation [7.5]) is a hyperbolic function
of [ S]. When [ S] is small compared to K , the rate is a linear function of [ S]. On the other hand, M
when [ S] is much larger than K , the rate becomes independent of [ S] and reaches its maximum value, M
which is given by ( k C ). The rate of the reaction cannot be any faster than when every enzyme mol-
h
E
ecule is in a complex with a substrate molecule, i.e., when CE = [( ES)]. The ratio of the maximum
rate of reaction to the enzyme analytical concentr ation is known as the turnover number k , i.e., the
h
number of substrate’s moles converted to product per unit of time.
3 An excellent example is in our immune system wherein antibodies are predators and antigens are preys.
The Emergence of Temporal Order within a Living Being
169
7.2.2 hill kineTics
There exist also enzymes having more than one binding site. Their mechanism of action is like that
of equation [7.6]:
E nS
kc
ES
kh
+
→
(
) n
ESn−1 + P
←
→
. [7.6]
k− c
Note the formal link between equation [7.6] and the mechanism modeling the Type III functional
response (equation 5.27).
The rate of change for [( ES )] is
n
d ( ES )
n = k
n
[ ][ ] ( −
)( )
c E
S − k c + kh
ES . [7.7]
dt
n
If C is the analytical concentration of the enzyme (i.e., C
)]), applying the steady state
E
E = [ E] + [( ESn
approximation to ES , we obtain:
n
k
n
[ ] (
) ) ( −
)( )
c S
CE − ( ESn = k c + kh
ESn
. [7.8]
If we rearrange equation [7.8], we achieve
[ S n] C
n
[ ]
E
S C
( ES
E
) n =
. [7.9]
n
k
n
h
k
[ S
− c
[ S] + K
] + +
=
kc
The ratio [ ES ]/ C is known as Hill equation (Haynie 2001). The rate of product P formation will be n
E
d [ P] k C
n
[ ]
h E S
=
. [7.10]
dt
[ S n] + K
The amount ( /
1 C
n
n
E )( d[ P]/ dt ) = ( kh[ S ] /([ S ] + K )) is formally equivalent to the Type III functional response of equation [5.31].
The enzymes having more than one site may have an exciting feature: that of showing indirect
interactions between the distinct binding sites. In such cases, the enzymes are defined allosteric.
This term was introduced by Jacques Monod and François Jacob to characterize the end-product
(L-isoleucine) inhibition of the enzyme L-threonine deaminase (Cui and Karplus 2008). L-isoleucine
does not compete with the reactant L-threonine in binding at the catalytic site; it instead binds at a
regulatory site, inhibiting the reaction. The term “allosteric” comes from the Greek αλλος- στερεος
meaning “other-space” and refers to the influence the binding at one site has on the binding at a
remote location in the same macromolecule. The ligand that brings about the allosteric regulation
of the binding of another ligand is called effector, or modulator.4 If the effectors are identical to the substrate molecule, we speak about “homotropic interactions” between the ligands and the enzyme.
If the binding of a substrate to the first site of the protein facilitates binding to the second, and so
on, the interaction is defined as “positive cooperativity.” On the other hand, if the binding of the
4 The control of enzyme activity can be achieved by modification of the enzyme molecules through either covalent change (such as phosphorylation and hydrolysis) or non-covalent conformational change.
170
Untangling Complex Systems
With activator
kh
Without effector
V
With inhibitor
(a) 0
[ S]
With activator
Without effector
V
With inhibitor
(b) 0
[ S]
FIGURE 7.1 Rate profiles for allosteric enzymes in case of competitive ligands (a) and uncompetitive
ligands (b).
substrate molecule to the first site inhibits the binding to the second, and so on, the interaction is
defined as “negative cooperativity.” When the effectors are ligands having a structure different from
that of the substrate, we speak about “heterotropic interactions.” In heterotropic interactions, the
effectors can play as either activators or inhibitors of enzymatic activity. Depending on how ligands
affect the enzymatic activity, their actions are also typified as “competitive” or “uncompetitive”
concerning the substrate binding to the enzyme. Ligands are defined as competitive when their
binding changes the K constant of equation [7.10] (in fact, they are also named as “K systems;” see
Figure 7.1a). When the binding of a ligand changes the maximum velocity of the enzymatic reaction (i.e., k in equation [7.10]), they are referred to as uncompetitive or “V systems” (where V stands for
h
Velocity; see Figure 7.1b) (Hammes and Wu 1974).
7.2.3 The nonlineariTy of allosTeric enzymes
Many proteins turn their activities on and off by the nonlinear allosteric effects. Examples of het-
erotropic interactions are offered by the allosteric enzyme Aspartate Transcarbamylase (ATCase)
catalyzing the reaction shown in Figure 7.2, which is the first step in pyrimidine biosynthesis.
ATCase has catalytic and regulatory sites. The aspartic acid (R ) binds to the catalytic site of
2
ATCase and in the presence of a saturating concentration of carbamoyl phosphate (R ) produces
1
N-carbamoyl aspartate (P ) and phosphate (P ). This reaction is the first step in the biosynthesis of
1
2
O
O–
O
H
ATCase
NH
+
–
2C
OP O– + NH
CH
NH
3
2C
N CH COO–
COO–
+ H2PO4
O
CH
CH
2
2
COO–
COO–
(R1)
(R2)
(P1)
(P2)
FIGURE 7.2 The first step of pyrimidine synthesis.
The Emergence of Temporal Order within a Living Being
171
NH2
NH2
N
O
O
O
N
O
O
O
N
O–
P O
P
O P O CH
N
2
N
O–
P O
P
O P O CH
O
2
N
O
O
O–
O–
O–
O–
O–
O–
OH
OH
OH
OH
CTP
ATP
FIGURE 7.3 Structures of cytidine triphosphate (CTP) and adenosine triphosphate (ATP).
molecules such as cytosine, thymine, and uracil, which contain the pyrimidine unit and are some of
the nitrogenous bases of DNA and RNA. The velocity of the enzymatic reaction as a function of the
concentration of aspartate shows a sigmoidal dependence (Hammes and Wu 1974). Along the syn-
thetic pathway leading to pyrimidines as final products, cytidine triphosphate (CTP) is produced.
CTP competes with the structurally similar adenosine triphosphate (ATP) (see Figure 7.3) in bind-
ing to the regulatory sites of ATCase.
When the concentration of CTP is high, it overcomes ATP, and it preferentially associates with
ATCase exerting an inhibiting effect on its catalytic activity, such as negative feedback in the syn-
thesis of N-carbamoyl aspartate (P ). On the other hand, when the concentration of CTP is not high,
1
ATP wins the competition in binding to the regulatory sites of ATCase. When it is ATP to bind to
ATCase, the final effect is the opposite: ATP activates the catalytic power of ATCase. CTP and ATP
are K systems (see Figure 7.1a), because they alter the value of K and not that of k . h An example of an allosteric protein exhibiting positive cooperativity in homotropic interactions is
the blood protein hemoglobin (Hb). Hb plays a vital role in the transfer of O from the lungs to other
2
cells of the body where it is combustive and participates in reactions releasing thermal energy. Hb is
a tetramer, consisting of four subunits each having an oxygen-binding site, due to the presence of iron
atom coordinated with a porphyrin ring. When an oxygen molecule binds to a site of hemoglobin, it
increases the affinity of the other sites for oxygen (Haynie 2001). In 1965, Monod along with Wyman
and Changeux proposed a model (known as MWC model) for allostery based on a few assumptions.
Allosteric proteins are oligomers made of identical monomers in a symmetric arrangement. Each
monomer has two folded tertiary conformations ( T and R, indicated by squares and circles, respec-
tively, in Figure 7.4). All monomers are in either one conformation or the other. When a ligand binds to a monomer of the protein, all monomers undergo the R-to- T or the T-to- R transition simultaneously. This model is represented by the structures enclosed by dashed lines. There are no “hybrid”
quaternary structures with some subunits in R state and some in T state. There exists another model
