Bioinspired Self-Assembly II: Principles of Cooperativity in Bioinspired Self-Assembling Systems
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BIOINSPIRED SELF-ASSEMBLY II: PRINCIPLES OF COOPERATIVITY
the following three-step procedure: (1) evaluation of the isolated binding events;∗8
(2) development of a model (the noncooperative model) where each binding event
of the system behaves as if it were isolated from the others; and (3) testing of the
behavior of the real system against that predicted by the noncooperative model:
any deviation, positive or negative, of the real system is by deﬁnition the unambiguous mark of cooperativity. This procedure will be followed to analyze the
three basic types of cooperativity mechanisms: (1) allosteric cooperativity, arising
from the interplay of intermolecular binding events; (2) chelate cooperativity, due
to the mere presence of one or more intramolecular binding interactions; and (3)
interannular cooperativity, occurring when intramolecular binding events are not
independent of other.7
The structure of this chapter is as follows. In Section 3.2, a detailed
derivation and explanation of statistical factors is presented, since they are
essential components in describing the equilibria of multivalent interactions. In
Section 3.3, allosteric cooperativity, which is the type of cooperativity exhibited
by hemoglobin, is brieﬂy discussed, mainly to illustrate the above three-step
procedure. In Section 3.4, the concept of effective molarity (EM ) is introduced, as
this physicochemical parameter is of fundamental importance for the understanding
and discussion of any intramolecular interaction. In Sections 3.5 and 3.6, the two
mechanisms of chelate and interannular cooperativity, respectively, are presented.
Finally, in Section 3.7, the stability of a self-assembling system is discussed, taking
into account inter- and intramolecular interactions and their possible interplay.
3.2
STATISTICAL FACTORS IN SELF-ASSEMBLY
An accurate and consistent evaluation of statistical factors in self-assembly processes is crucial to predict the expected stability constant in the absence of cooperative effects and, therefore, to spotlight the emergence of cooperativity. The
evaluation of statistical factors has recently been critically reexamined.9 Two methods are most useful: the symmetry number method and the direct count method.
The two methods if properly applied give the same results; however, the symmetry number method is generally of easier and faster application, and therefore we
recommend its use. Only in case of doubt, we recommend the use of the direct
count method as an independent check. For brevity reasons, we only present here
an outline of the symmetry number method.
The observed equilibrium constant, Kobs , of a generic equilibrium (Eq. 3.1) can
be regarded as given by the product of a microscopic or “chemical” constant K
and a statistical factor, Kσ .
Kobs =Kσ K
aA + bB −−
−−
−−
−−
−−
−−
−− cC + dD
(3.1)
∗ In the case of multiple binding of the same type of ligand (homotropic cooperativity), only a
single reference interaction is needed, whereas in the case of binding of different types of ligands
(heterotropic cooperativity), more than a single reference interaction must be deﬁned. The binding
of oxygen to hemoglobin is the classical example of homotropic cooperativity. For a recent example
of heterotropic cooperativity.
STATISTICAL FACTORS IN SELF-ASSEMBLY
49
According to the symmetry number method, pioneered by Benson,10 Kσ is given
by the ratio of symmetry numbers for reactant and product species in equilibrium
(Eq. 3.2).
Kσ =
σ aσ b
σreactants
= Ac Bd
σproducts
σC σD
(3.2)
The factor σ is the product of the external, σext , and internal, σint , symmetry
numbers. The external symmetry number is deﬁned as the number of different but
indistinguishable atomic arrangements that can be obtained by rotating a given
molecule as a whole. It is found, in practice, by multiplying the order of the
independent simple rotational axes of the point group to which the molecule belongs
(axes of inﬁnite order are not considered because they do not generate different
atomic arrangements). External symmetry numbers for the several point groups are
shown in Table 3.1.
The internal symmetry number is deﬁned as the number of different but indistinguishable atomic arrangements that can be obtained by internal rotations around
single bonds, or, in the case of ﬂuxional molecules, by inversion, pseudorotation,
or other intramolecular processes. It is implied that the processes giving rise to the
internal symmetry number are fast with respect to the time scale in which the equilibrium in Eq. 3.1 is attained and measured. For example, ethane has σ = 18 that
is the product of σext = 6 because of D3d symmetry (in this point group there are
a threefold axis and an independent twofold axis) and σint = 3, due to the internal
rotation of one methyl group with respect to the other; ammonia has σ = 6 that is
the product of σext = 3 because of C3v symmetry (only a threefold rotational axis)
and σint = 2, due to the process of pyramidal inversion. Equation 3.2 has its physical basis on the fact that the symmetry number of a molecule affects its rotational
entropy by a factor −R ln σ . A different type of correction accompanies a chiral
molecule present at equilibrium as a racemic mixture; its symmetry number must
be divided by 2 to account for the entropy of mixing of the two enantiomers.9, 10
A complication arises from ﬂexible molecules in which internal rotations
produce distinct conformations of different energy; in these cases the only viable
and consistent approximation is one that considers all the internal rotations as
TABLE 3.1 External Symmetry Numbers for Various
Point Groups
Point Group
σext
C1 , Ci , Cs , C∞v , R3
D∞h
Cn , Cnv , Cnh
Dn , Dnd , Dnh
Sn (n even)
Td
Oh
Ih
1
2
n
2n
n/2
12
24
60
50
BIOINSPIRED SELF-ASSEMBLY II: PRINCIPLES OF COOPERATIVITY
free, implying that all torsion angles have equal probability of occurrence. Under
this approximation, it can be demonstrated that the external symmetry number of
a molecule is equal to that of the most symmetrical of its conformations.9 For
example, in the case of a normal alkane, the most symmetrical conformation is
the extended one (all dihedrals in anti) whose point group is C2h (σext = 2). Since
σint = 32 , because of the internal rotations of the two terminal methyl groups,
the overall symmetry number of a normal alkane is σ = 18. In the case of a
cycloalkane with n carbons, the most symmetrical conformation is the planar one
(Dnh symmetry) for which σ = σext = 2n, even though this conformation does not
even correspond to a minimum. It goes without saying that the symmetry number
obtained under this approximation has no bearing on the rotational entropy of the
molecule.
3.3
ALLOSTERIC COOPERATIVITY
Allosteric cooperativity arises from the interplay of intermolecular binding interactions. It can be homotropic or heterotropic, depending on whether the binding to
a multivalent receptor involves the same or different types of ligands. Homotropic
cooperativity, exempliﬁed by the binding of oxygen to hemoglobin,1 is the most
interesting and the most difﬁcult to realize in artiﬁcial receptors; thus, we will limit
the discussion to this type of cooperativity.
Consider, for example, the stepwise binding of a monovalent ligand to a divalent
receptor as depicted in Figure 3.1. The observed stepwise constants are given by
the two microscopic association constants K1 and K2 multiplied by the statistical
factors 2 and 12 , respectively. The statistical factors are easily understood considering that the unbound receptor has σ = 2, the half-bound receptor has σ = 1, and
the fully bound receptor has σ = 2. In general, for the interaction of an n-valent
receptor with a monovalent ligand, the statistical factor for the ith stepwise equilibrium is given by (n − i + 1)/ i. The reference constant K can be evaluated by
studying the binding of the monovalent ligand B to a monovalent model, A, of
the receptor, or alternatively, directly taking the value of the constant K1 as the
reference constant. Thus, in the absence of cooperativity, all the microscopic stepwise equilibrium constants must be equal to the reference value K (for the case in
Figure 3.1, K2 = K1 = K). A way to quantify allosteric cooperativity is to evaluate
the cooperativity factor α given by the ratio of the overall experimental constant
to the hypothetical overall noncooperative constant. For the general case of the
2 K1
+ 2
AA
Figure 3.1
B
1/2 K2
+
AA·B
B
AA·B2
Stepwise binding of a monovalent ligand B to a divalent receptor AA.
ALLOSTERIC COOPERATIVITY
51
interaction of a monovalent ligand with an n-valent receptor, α is given by Eq. 3.3:
i
α=
Ki
Kn
(3.3)
The factor α is a dimensionless constant larger than 1 in the case of positive
cooperativity, equal to 1 in the case of noncooperativity, and smaller than 1 in the
case of negative cooperativity. It can be viewed as the equilibrium constant for
the conversion of the hypothetical noncooperative complex (independent binding
sites) into the cooperative complex (interacting binding sites) (Figure 3.2).
There are other equivalent tests to assess allosteric cooperativity, mainly graphical ones, based on the calculation of the occupancy, r, that is to say the average
number of occupied sites of the n-valent receptor.2a,11 It is easy to show that the
occupancy is given by Eq. 3.4:
r=
1
n
j
j =1 jβj [B]
+ nj=1 βj [B]j
(3.4)
where βj are the cumulative binding constants given by Eq. 3.5 and [B] is the free
ligand concentration.
j
βj =
i=1
(n − i + 1)
Ki
i
(3.5)
If the binding is noncooperative (Ki = K for all i values), it can be demonstrated
that Eq. 3.4 reduces to the binding isotherm shown in Eq. 3.6, where y(= r/n) is
the degree of saturation of the receptor.2a,11
y=
K[B]
1 + K[B]
(3.6)
Plots that deviate from the above equation are diagnostic for cooperativity. In
particular, a sigmoid plot is clear-cut evidence of marked positive cooperativity.
However, such deviations are not always easily recognizable. A much better diagnostic can be obtained by putting Eq. 3.6 into linear form because deviations from a
α
Figure 3.2 The cooperative factor α is the equilibrium constant for the conversion of
the hypothetical noncooperative complex into the cooperative complex. The ﬁgure depicts
the case of a saturated divalent receptor.
52
BIOINSPIRED SELF-ASSEMBLY II: PRINCIPLES OF COOPERATIVITY
straight line are more easily detectable. Two popular linear forms are the Scatchard
equation (Eq. 3.7) and the Hill equation (Eq. 3.8):
r
= −Kr + nK
[B]
y
= log[B] + log K
log
1−y
(3.7)
(3.8)
A Scatchard plot is a plot of r/[B] as a function of r; it is linear in the case of noncooperativity, whereas it presents a concave downward curve in the case of positive
cooperativity, or a concave upward curve in the case of negative cooperativity. A
Hill plot is a plot of log y/(1 − y) versus log [B]; apart from the case of noncooperativity that is evidenced by a straight line of unit slope, cooperativity manifests
itself as two lines of unit slope connected by an S-shaped curve. The value of the
slope in the central region of the curve is called the Hill coefﬁcient (nH ). It can
vary between 0 and n; values larger than 1 are diagnostic for positive cooperativity
whereas values lower than 1 are diagnostic for negative cooperativity.2a,11
It is useful to remark that all of these methods to assess cooperativity provide
meaningful results only for a collection of intermolecular binding events.12
3.4
EFFECTIVE MOLARITY
Intramolecular reactions are often more favored than analogous intermolecular
reactions. This advantage, known as the proximity effect or the chelate effect,
is measured by the effective molarity (EM ), a physicochemical parameter that has
units, as the name implies, of mol L−1 .13, 14 To illustrate the concept of EM , let
us envisage a solution of a chain molecule with two end groups –A and –B, as
schematically illustrated in Figure 3.3. The functional group –A can react reversibly
with the functional group –B to form a new bond AB; if –A and –B belong to
the same chain the reaction is intramolecular and leads to the formation of a ring,
whereas if the two groups belong to different chains the reaction is intermolecular. The equilibrium constants for the two processes are, respectively, Kintra that is
dimensionless, and K that has units of mol−1 L. There are a number of equivalent
deﬁnitions of EM ; the ﬁrst in order of time, and somewhat outdated, considers the
EM as the molar concentration of one chain end experienced by the other end of
the same chain. Although formally correct, this deﬁnition is not physically reliable
for short chains because the EM can assume unreasonably high values such as 108
mol L−1 . However, this deﬁnition is a useful starting point to obtain other more
physically accurate deﬁnitions. On the basis of Figure 3.3, it appears that if the
concentration of the bifunctional chain A–B is equal to the EM , the intermolecular
process proceeds with the same extent of reaction of the intramolecular process.
This observation is very important because the EM can now be viewed as the limit
concentration of the chain A–B below which intramolecular processes are more
favored than intermolecular processes. Under the condition that [A–B] = EM,
EFFECTIVE MOLARITY
53
A
B
A
B
Figure 3.3 Schematic representation of the competition between an intramolecular process and an intermolecular process aimed at illustrating the concept of effective molarity.
the intramolecular constant Kintra is equal to the apparent intermolecular constant
K[A–B] = K ·EM, from which an operative deﬁnition of EM is immediately
obtained (Eq. 3.9):
EM = Kintra /K
(3.9)
The constant Kintra can thus be viewed as the product of K representing the inherent chemical reactivity of end groups, and EM representing a connection factor
that accounts for the fact that the two reactive groups are connected to each other.
The EM has both an enthalpic and an entropic component (EM = EMH EMS ).
In most of the cases the enthalpic component only depends on the strain energy of
the ring, so EMH is lower than 1 unless a strainless ring is formed, in which case,
it is equal to 1. For a strainless ring the EM is solely dependent on entropy. The
value of EMS decreases on increasing the number of rotatable bonds of the chain
connecting the end groups because internal rotations become more restricted upon
ring closure. Thus, the maximum EM value can be reached when the end groups
are connected by a rigid structure, preorganized to form a strainless ring. The
maximum EM value has been estimated by Page and Jencks.15 According to their
classical analysis, bringing two molecules together is accompanied by a negative
change in entropy because of the reduced volume of space available to the
reactants. Mechanically, a free molecule has three degrees of translational freedom
and three degrees of overall rotational freedom. When two molecules condense
to form one, three degrees of each are converted into six degrees of vibrational
freedom that have a lower entropy content. The more severely the geometrical
relationship between the reactant molecules is deﬁned the greater is the loss of
entropy. For bond-forming reactions commonly encountered in organic chemistry,
this entropy change is about −150 J K−1 mol−1 at a standard state of 1 mol L−1 ; it
makes intermolecular reaction equilibria unfavorable by a factor of about 108 mol
L−1 . The formation of weaker bonds such as hydrogen bonds and metal–ligand
54
BIOINSPIRED SELF-ASSEMBLY II: PRINCIPLES OF COOPERATIVITY
bonds, often encountered in self-assembling systems, involves a smaller entropy
change (about −40 J K−1 mol−1 at a standard state of 1 mol L−1 ) that makes
intermolecular reaction equilibria unfavorable by a factor of about 102 mol L−1 .
These changes do not occur in intramolecular reactions and thus 108 mol L−1 and
102 mol L−1 are the maximum EM values that can be found for intramolecular
processes involving the formation of tight and loose bonds, respectively. Introduction of rotors in the chain connecting the end groups reduces the value of the EM
from these maxima. In Figure 3.4 are reported the EM values for covalent13b–d
and noncovalent16 cyclization processes versus the number of rotors in the linking
chain (r). The limits to which the experimental results tend in the absence of
rotors are in substantial agreement with the analysis of Page and Jencks.15
The theory for predicting the EM is well developed for long ﬂexible chains, say
more than 25–30 skeletal bonds, yielding strainless rings; according to the theory,
the EM of an r-meric chain is proportional to r −3/2 .17 This factor is related to the
probability that a Gaussian chain of r repeating units has its ends coincident. Thus,
for long ﬂexible chains, the original deﬁnition of the EM as the molar concentration
107
106
105
EMs [mol L−1]
104
Covalent
103
102
101
100
Noncovalent
10−1
10−2
10−3
1
2
3
4
5
6 7 8 9 10
20
r
Figure 3.4 Log–log plot of the entropic component of the effective molarity (EMS ) for
covalent (dashed line) and noncovalent (solid line) cyclization processes and the number
of rotors in the linking chain (r). The covalent EMS values are from Refs. 13b–d;
the noncovalent EMS values, according to Ref. 16, satisfy the linear relationship log
EMS ≈ 10 − 32 log r.
CHELATE COOPERATIVITY
55
of one chain end experienced by the other end of the same chain makes physical
sense. Hunter and co-workers have recently shown that the same relationship also
holds for cyclizations of shorter chains provided that the newly formed intramolecular bond is noncovalent.16 This ﬁnding is evidenced in Figure 3.4 by the straight
line of slope − 32 observed for noncovalent cyclizations. Rather interestingly, the
covalent curve approaches the noncovalent one when the orientational correlations
between the end groups are less severe because of the increased chain length.17c
From the analysis of Page and Jencks,15 as well as from the experimental data
of Hunter and co-workers,16 it appears that noncovalent EM s never reach the
very high values observed for covalent processes, which places limitations on the
magnitudes of the effects that one is likely to achieve through the use of chelate
interactions in supramolecular assembly. On the other hand, the decrease in EM
due to the introduction of conformational ﬂexibility is less dramatic than one might
expect based on the behavior of covalent systems, which limits the losses in binding
afﬁnity caused by poor preorganization of the interaction sites.
3.5
CHELATE COOPERATIVITY
Chelate cooperativity arises from the presence of one or more intramolecular
binding interactions, that is, as a consequence of the chelate effect, also called
multivalency. While allosteric cooperativity is well recognized, the assessment of
chelate cooperativity is still the object of debate.3 – 7 Let us consider the archetypal
case that involves the binding of a divalent ligand BB to a divalent receptor AA
(Figure 3.5). The ligand is present in a large excess relative to the receptor so that
complexes involving more than one receptor molecule can be neglected, and α = 1
to exclude allosteric cooperativity. Under the given conditions there are only four
states for the receptor: free AA, the partially bound 1:1 open complex o-AA·BB,
the fully bound 1:1 cyclic complex c-AA·BB, and the 1:2 complex AA·(BB)2 . The
intramolecular binding constant is expressed as the product 12 K EM , where 12 is
the statistical factor for the cyclization process, K is the microscopic intermolecular constant expressing the strength of the binding interaction, and EM is the
microscopic effective molarity.∗
An important characteristic of chelate cooperativity is that, in contrast with
allosteric cooperativity, it depends on ligand concentration. To illustrate this behavior, in Figure 3.6 are reported the speciation proﬁles for the equilibria shown
in Figure 3.5 assuming K· EM = 50. Positive cooperativity is characterized by
a low concentration of partially bound species. In the most extreme cases only
the unbound and fully bound receptor are signiﬁcantly populated and the system
exhibits an “all-or-none” behavior. From Figure 3.6 it appears that the presence
of the chelate interaction leads to a low concentration of the partially bound intermediate complex o-AA·BB favoring the fully bound cyclic complex c-AA·BB.
∗
Sometimes the statistical factor for the cyclization process is incorporated into the value of the EM .
We use the term microscopic effective molarity to indicate that the symbol EM does not account
for the statistical factor.
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BIOINSPIRED SELF-ASSEMBLY II: PRINCIPLES OF COOPERATIVITY
1/2 K EM
4K
+
AA
o-AA·BB
+
BB
BB
c-AA·BB
+
BB
1/2 EM
K
AA·(BB)2
Figure 3.5 Binding of a divalent ligand BB to a divalent receptor AA, assuming
[BB]0
[AA]0 and α = 1.
1.0
0.8
Fraction
0.6
0.4
0.2
0.0
–4
–2
0
2
4
log K [BB]0
Figure 3.6 Speciation proﬁles for the equilibria shown in Figure 3.5 in the case where
K · EM = 50. The concentration scale on the abscissa is normalized by multiplying by
K. Symbol legend: o-AA·BB (ﬁlled circles); AA·(BB)2 (squares); c-AA·BB (triangles);
degree of saturation of the receptor AA (empty circles).
CHELATE COOPERATIVITY
57
However, although the overall degree of saturation of the receptor tends to an
“all-or-none” process, the speciation proﬁle of the chelate complex c-AA·BB is bellshaped, suggesting that the intramolecular process can be more properly regarded
as “none-all-none.” This is due to the fact that at low concentration the cyclic
complex is disfavored with respect to the unbound receptor, whereas at high concentration it suffers from the competition with the fully bound 1:2 open complex.
The ligand concentration at which the switch between the cyclic complex and the
fully bound open complex occurs is easily obtained by considering the equilibrium between these two species illustrated in Figure 3.5 and the corresponding
equilibrium constant deﬁned by Eq. 3.10:
[c − AA · BB][BB]
EM
=
2
[AA · (BB)2 ]
(3.10)
When [BB] = EM/2 the concentrations of the two species are identical. Thus, it
appears that EM is the threshold concentration of ligand binding groups above
which the intramolecular process loses the competition with the intermolecular
one. The conclusion is that the advantage provided by the chelate interaction is
dissipated at high ligand concentrations. The dependency of chelate cooperativity
on ligand concentration has been often overlooked, leading to inconsistencies. For
example, it has been advocated that chelate cooperativity manifests itself when the
intramolecular constant ( 12 K · EM in Figure 3.5) is larger than the intermolecular
constant (4K in Figure 3.5)∗ .5 This comparison does not make sense because the
intramolecular and the intermolecular constants have different units, speciﬁcally,
the former is unitless whereas the latter has units of mol−1 L. Another inconsistency has been pointed out by Jencks18 with reference to the binding of a divalent
asymmetric ligand AB to a complementary receptor as illustrated in Figure 3.7:
a common pitfall is to consider that in the absence of chelate cooperativity the
observed free energies of binding for the individual parts A and B are additive
in the molecule AB, so that GAB ◦ = GA ◦ + GB ◦ . Jencks pointed out that
there is no basis for this assumption: the addition of Gibbs energies is equivalent
to the multiplication of binding constants and, if KA , KB , and KAB are measured
in mol−1 L, the equation KAB (mol−1 L) = KA KB (mol−1 L)2 is meaningless. The
correct way to address the problem of additivity of binding energies is to add
KAB
+
AB
Figure 3.7
Binding of a divalent asymmetric ligand AB to a complementary receptor.
∗
The concept was expressed in terms of dissociation constants, that is, Kd2 < Kd1 , where Kd2 =
2(K · EM)−1 and Kd1 = (4K)−1 . See the caption of Figure 1(b) of Ref. 5.
58
BIOINSPIRED SELF-ASSEMBLY II: PRINCIPLES OF COOPERATIVITY
a connection Gibbs energy that represents the change in the probability of binding that results from the connection of A and B in AB. This connection Gibbs
energy is in fact the free energy associated to the EM , GEM ◦ . Accordingly,
GAB ◦ = GA ◦ + GB ◦ + GEM ◦ , that when translated into binding constants
gives a dimensionally correct equation KAB (mol−1 L) = KA KB ·EM (mol−1 L).
In analogy with the deﬁnition of the allosteric cooperativity factor α, it is tempting
to assume the EM as the measure of chelate cooperativity; however, this assumption
is inconsistent because, in contrast with the factor α which is dimensionless, the
EM has units of concentration, and its numerical value depends on the choice of
the standard state. Moreover, as pointed out above, the ligand concentration must
also be taken into account for the correct assessment of chelate cooperativity.
Recently, it has been suggested the product K · EM is a measure for chelate
cooperativity.6 Again, this factor does not account for the dependency of chelate
cooperativity on ligand concentration. Moreover, the product K ·EM tends to zero
in the absence of cooperativity, whereas a consistent cooperativity factor must
tend to 1. One of us has previously shown that the product K ·EM is related
to the maximum amount of a cyclic or multicyclic supramolecular assembly in
solution.14, 19 However, although K ·EM is the driving force for self-assembly, it
has no bearing on chelate cooperativity.
To assess this type of cooperativity, we follow the three-step procedure detailed
in the introduction. In the absence of chelate cooperativity, the overall constant for
the fully bound receptor, as given by Eq. 3.11, is used as the reference intermolecular binding:
4K 2 =
[AA · (BB)2 ]
[AA][BB]2
(3.11)
In the presence of chelate cooperativity, the overall apparent constant for the fully
saturated receptor is given by Eq. 3.12:
EM
([AA · (BB)2 ] + [c − AA · BB])
= 4K 2 + 2K 2
2
[AA][BB]
[BB]
(3.12)
The two terms on the right-hand side of Eq. 3.12 represent the contributions of
intermolecular and intramolecular binding, respectively. The chelate cooperativity
factor, β, is the ratio of the intramolecular binding to the reference intermolecular
binding as given by Eq. 3.13:
β=
EM
2[BB]
(3.13)
It represents the apparent equilibrium constant for the conversion of the complex
AA·(BB)2 into the cyclic complex c-AA·BB as shown in Figure 3.8. The value
of β depends on ligand concentration. When β = 1, [BB] = EM/2; at this ligand
concentration, the fully bound open complex and the cyclic complex are equally
populated. At lower ligand concentrations, the cyclic complex is dominant (β > 1,