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H + G H G
H S (H G) SG S+
in reality
K
[HG]
[H][G]Ka = [HG]
[H][G]
Kd =(M-1
) (M)
SS+
explicit solvent is not used because G for the association constant
reflects the stability of solvated H and G relative to solvated HG and released solvent
G= -RTln(Ka)assuming activity = concentra\ tion
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Kd (1/Ka) is a convenient reference point for estimating the amount of complex
Ka= 100 M-1
Kd= 1x10-2M = 10 mM
course
0,1 0,3 0,5 0,7 0,9Total concentration of A
0
20
40
60
80
100
%f
ormationrelativetoA
[H]0= [G]0= 20 mM [H] = [G] = 10 mM and [HG] = 10 mM
At Kd, [H] = [G] = [HG] = Kd (!! only in case [H]0= [G]0!!)
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G= -RTln(Ka)
Ka(M-1) G0(kJ.mol-1)
1 0
10 -5.8
100 -11.5
1000 -17.3
10000 -23.0
100000 -28.8
at T = 301 K (R = 8.314 J.mol-1.K-1)
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
-G(kJmol-1)
log Ka
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H + G H GKa
how to determine the concentration of each species ?
if H, G, and HG are all known, then K is easily calculated
[HG]
[H][G]Ka =
regrettable this is hardly ever the case
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determination of the binding constant Ka
almost all experimental methods to measure binding constants rely on the analysis
of a binding isotherm.
A binding isotherm is the theoretical change in the concentration of one component
as a function of the concentration of another component at constant temperature.
The concentrations are experimentally determined (e.g. NMR, UV/vis, etc) and fitted to
the theoretical binding isotherm.
0,1
0 0,2 0,4 0,6 0,8 1,0Total concentration of A
0
20
40
60
80
100
%f
ormationrelativetoB Ka= 100 M
-1
[H]0
= 0.01Mexample
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Titration experiments
Typically a titration is performed holding the concentration of one species (H) constant
while varying the concentration of the other (G).
During the course of this titration, the physical changes in the system are monitored,usually spectroscopically, and this change is then plotted as a function of guest added
to host (equivalent guests).
The mathematical model used to obtain the association constant is usually developed
from realising that the physical change (Y, e.g., a NMR shift or a change in UV-Visabsorbance) observed is correlated to the concentration of the complex [HG] as Y =
f[HG], or in some cases, the free hostf[H] or the free guestf[G].
The physical change (Y) being monitored can usually be described as the aggregate of the
individual components according to eqn (7) as a function of concentration (e.g., for UV-Visspectroscopy) or eqn (8) as a function of mole fractions fX(fXdefined as: fX= [X]/[X]0) in
the special case of NMR.
Y = YH[H] + YG[G] + YHG[HG]
Y = YHfH+ YGfG+ YHGfHG
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[H] = [H]0 [HG]
[G] = [G]0 [HG]
[HG]
([H]0 [HG])([G]0 [HG])Ka =
[HG]
[H][G]K
a
=
Example
only HG has a particular absorption band in the UV/vis spectrum.
Thus, A = HG[HG]
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[HG]
([H]0 [HG])([G]0 [HG])Ka =
[HG]
[H]0[G]0 [HG][G]0- [HG][H]0+[HG]2
Ka =
Ka([H]0[G]0 [HG][G]0- [HG][H]0+[HG]2
) = [HG]
Ka[HG]2(Ka[G]0-Ka[H]0+1)[HG]+Ka[H]0[G]0= 0
thus
this expresses [HG] as a function of Ka, which is the only unknown !!
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Example, only HG has a particular absorption band in the UV/vis spectrum.
Since, A = HG[HG]
and
Then A =f(HG
, Ka
)
The power of this equation should not be understated as we can now start to developsolutions that require only the knowledge of the total (or initial) concentrations of the
host and guest ([H]0and [G]0) in addition to the association constant (Ka) and the physical
properties (Y) that are changing (Y) during the course of the titration.
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Y = YHfH+ YGfG+ YHGfHG
If we further assume that one of the components is silent e.g., a non-absorbing freeguest [G], we can simplify the equation to
which, since fHG = [HG]/[H]0 and fH= 1- fHG, can be further simplified to
Y=YH+([HG]/[H]0)(YHG- YH)
and, finally
Y = YHfH+ YHGfHG
in which Y = Y-YHand YHG= YHG-YH
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(Old-fashioned) Shortcuts to the binding constant
Older references and textbooks are full of examples on how some of the above
expressions and equations can be simplified or transformed to linear equations(y = a + bx) which could then be plotted by hand to obtain the Kaand other parameters
of interest by inspection of the slope and intercepts.
BenesiHildebrand plot
(determination of binding constants based on absorbance)
assumingthat [G]0>> [H]0(and thus AH> [H]0one can assume that [G]=[G]0and this
with = HG- G
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[H] = [H]0 [HG]
[HG][H][G]
Ka = [HG]
[G]([H]0-[HG])Ka = and Ka([G]([H]0-[HG]))-[HG]=0
which is
Ka[G][H]0-Ka[G][HG]-[HG]=0
or
The binding isotherm can be rewritten as
Together this gives
and finally
(assuming that [G] = [G]0)
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intercept
slope1
b[H]0Ka
Other examples include Lineweaver-Burke plots, Scatchard plots, etc.
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There are two key problems associated with using these linear transformations that
make their use highly questionable:
(i) they violate some of the fundamental assumption of linear regression by
distorting the experimental error
(ii) they frequently involve assumptions and shortcuts (such as assuming that [G]0
>> [H]0 or that YHG= Y at the end of titration (i.e., the complex is fully formed atthe end of titration - which would then help to give YHG). These assumptions
are often not valid and distort the results.
The non-linear regression approach with exact solutions of the quadratic equation
(see before) produces the most accurate results. This approach is not difficult with
modern computer technology and there is no real excuse for using old-fashion lineartransformations anymore!
Limitations
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Scientist
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0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02
Example: NMR titration
[G0]
o
bs
(ppm)
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6.8
7
7.2
7.4
7.6
7.8
8
8.2
0 0.005 0.01 0.015 0.02
Serie1
Serie2
K 1938.702 78
DH 7.00624 0.004DHG 8.000867 6.00E-03
M-1
ppmppm
[G0]
obs
(ppm)
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Experimental conditions
When a supramolecular titration study is carried out one has to first make a decisionon what technique is going to be used to follow the physical changes (Y) in the
system during the course of experiment. The two key concerns here should be:
i. The expected association constant(s).
ii. The expected physical changes (Y) upon association.
The expected association constant determines what concentration should be chosen for the
host system which in turn will have an influence on the choice of technique.
Wilcox, using a parameter defined as probability of binding (p), showed that it is vital to
collect as many data point as possible within the range: 0.2
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[HG]
([H]0 [HG])([G]0 [HG])Ka =
gives
Ka =p
[G]0 ([H]0+ [G]0)p+[H]0p2
p
for [H]0=[G]0=0.001 M
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nearp =0 and p = 1, small errors inp (experimental error in determination of the
concentrations !!= results in large variations in Ka.
The best results are obtained in the region 0.2
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EXAMPLE
p
for [H]0=0.001 M
Ka= 2000 M-1
0.00
0.20
0.40
0.60
0.80
1.00
1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00
log [G]0
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0.00
0.20
0.40
0.60
0.80
1.00
0.00E+00 2.00E-03 4.00E-03 6.00E-03 8.00E-03 1.00E-02
[G]0
for [H]0=0.001 M
Ka= 2000 M-1
p
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Using
it is possible to calculatepfor a range of [H]0, [G]0and Kavalues.When the results are plotted for a fixed [H]0concentration (here 10
-5M) as a
function of Kaand [G]0/[H]0 (equivalents of guest added) typically employed in
UV-Vis spectroscopy studies a revealing pattern appears with the shaded areas
indicatingpin the range of 0.20.8 (note that this applies only to 1 : 1 binding
systems).
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if Kd> [H]0(hence Kafairly low) then a relatively large excess of [G]0is required to obtain
goodp-values. In this situation it would be advisable to collect several data points in the
range of 150 equivalents of G added.
If Kd> [H]0(hence Kafairly high) the only data points with goodp-values are within the
range of [G]0o [H]0. In other words, it is essential to obtain as many points as possible
between 01 equivalents of G added.
If Kd [H]0, goodp-values are obtained almost anywhere within the range of 0 to >10
equivalent of G added. Note that when Kd= [H]0= [G]0, then p = 0.38.
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The fourth scenario to consider is when Kd1000 , it is clear that there is very little
information content in the isotherms.
When binding occurs under saturation
conditions, this implies that the experimental
conditions are NOT adequate for
determination of K.
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NMR concentrations UV-Vis concentrations
fluorescence concentrations
The suitable analytical technique fordetermination of Kadepends on its
value.
limit: [H]0/Kd> 100
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analytical techniques: scope and limitations
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The most informative technique in most situations is 1H NMR. Other forms (13C, 19F etc.)
of NMR are also applicable. Apart from the quantitative information that an
NMR titration can yield, the relative shifts and changes in symmetry can often give
valuable information about how the host and guest(s) are interacting and thestoichiometry of interaction. This information can be of significant benefit even
in situations where complete quantitative data cannot be obtained from the NMR
titration.
Classical approaches for data analysis of NMR titrations assume that the resonance () of
interest is the weighted average of the free host (H) and the bound host in the complex(HG) in the experiment for a simple 1 : 1 system
obs= HH+ HGHG
since H = 1 - HG
obs- H= HG(HG HG)
NMR spectroscopy
since
this gives
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NMR spectroscopy
With modern NMR instruments it is possible to obtain good quality spectra with sub-
millimolar concentrations (routinely now as low as 10-4M), suggesting that NMR is
suitable for Kaup to and even above 106M-1. Many literature references will statethat 105M-1is the limit for NMR titration experiments.
With NMR, one has also to take into account the relative exchange rates within the
hostguest the relationship between the equilibrium association constant and the
kinetics on/off rates (Ka=k1/k-1) and the timescale of the NMR experiment. The reallimiting factor for NMR titrations is therefore whether the system of interest is in the
fast or slow exchange region under the conditions used.
It may be tempting to think that in the (very) slow exchange region of NMR, one could
obtain an association constant directly from the relative ratios of the free and bound
host, however, can be difficult in practice due to complications that arise in theintermediate-to-slow region with the size (amplitude) of the observed resonances and
the usual limitation of obtaining accurate (quantitative) integration from NMR
experiments.
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UV-Vis spectroscopy
The second most common method for supramolecular titration experiment is probably
UV-Vis spectroscopy. With the right chromophore, host concentration in the sub-
micromolar (10-7
M) can be applied, making the determination of association constantsas high as 109M-1in simple 1 : 1 systems possible (albeit difficult) with Kd/[H]0= 100 as
discussed above.
advantages
rapid. absorption and related phenomenon (fluorescence) occur on ps time scale or
faster in all cases faster than complex dissociation rate slow exchange
straightforward to correlate signal intensity to concentration (linear regime;
Lambert-Beer)
sensitive
disadvantage titration by UV-Vis spectroscopy is particularly vulnerable to dilution and temperature
effects (all supramolecular titration experiments need some temperature control) and
the presence of impurities in either host or guest solutions.
requires chromophoric hosts or guests
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fluorescence spectroscopy
The phenomenal sensitivity of this technique makes routine measurements in the sub-micromolar,
even nanomolar (nM) range possible and hence, fluorescence spectroscopy is ideal for the
determination of very large association constants (Ka> 1010M-1).
Fluorescence is a particularly useful technique in the case when only one of the species in solution
is fluorescently active, i.e. when either the free host or guest is fluorescent silent or inactive and
the fluorescence of the remaining species is either turned off (quenched) or on upon
complexation.
If quenching plays a role, it is necessary to differentiate between static and dynamic (collisional)quenching, with only the former of real significance for supramolecular binding studies.
Dynamic quenching is usually measured by plotting the ratio of the initial (F0) and measured (F)
fluorescence intensity ratio (F0/F) against the concentration of the quencher [Q] according to the
SternVolmer relation F0/F = 1 + KSV[Q], with KSV= the SternVolmer constant.
Unfortunately, pure 1 : 1 static quenching follows a nearly identical relation: F0/F = 1 + Ka[Q], with [Q]
= the free concentration of the quencher (guest) and Kais the association constant of interest in
supramolecular binding studies.
In many cases the observed quenching is a mixture of both static and dynamic quenching which can
lead to some complication in the analysis of the titration data.
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Swager et al.J. Am. Chem. Soc., 1995, 7017-7018
control
Stern-Volmer plots (F0/F=1+K[PQ2+] )
Small molecule sensing
2
Mean molecular weight (PDI)
6: 31 100 (1.6)
7: 65 400 (1.6)
8: 122 500 (1.8)
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+
+
++
+
++
+
+
++
+
+
++
+
- --2
0
250
500
750
1000
0.0 1.0 2.0 3.0 4.0 5.0
FI(a.u.)
[ATPF] (M)
ATPF(ex= 305 nm, em= 370 nm)
2.5 M
~ 18
[TACN Zn(II)] = 10 M
pH = 7.0, T = 25 C.
[TACN Zn(II)] = 10 M
pH = 7.0, T = 25 C.
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G= -RTln(Ka)
From K to G and H and S (Van t Hoff analysis)
G= H- TSln(Ka) = -H/RT+ S/R
slope = -H/R
intercept = + S/R
problem: often small temperature interval possible + inversed temperature
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Isothermal calorimetry (ITC)
Q = VH[HG]
or
dq= VH [HG]
fitting provides K and H
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Determination of stoichiometry
The other aim of a supramolecular titration experiment is the determination of the
stoichiometry of the system.
Methods
(i) The method of continuous variations (Jobs method).
(ii) Consistency with the host structure and available information on the host guest complex structure.
(iii) Specific experimental evidence such as isosbestic point(s).
(iv) Constancy of stability concentration as the concentration is varied, that is, the
success of a stoichiometric model to account for the data.
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Jobs method
The idea behind it is simple; the concentration of a HmGn([HmGn]) complex is at
maximum when the [H]/[G] ratio is equal to m/n.
To do this, the mole fraction (fG) of the guest is varied while keeping the total
concentration of the host and guest constant ([H]0+[G]0=constant). The concentration of
the hostguest complex [HmGn] is then plotted against the mole fractionfGyielding a
curve with a maxima atfG= n/(m + n), which in the case of m = n (e.g., 1 : 1) appears at fG
= 0.5
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When there is more than one complex present, the Jobs method becomes
unreliable. This includes many situations with m/n = 1 : 2 or 2 : 1 as these usuallyinclude two forms of complexes (e.g., HG and HG2) that have different physical
properties, hence the assumption that the physical property of interest (e.g., obs)
is linearly dependent may not be valid. For similar reasons, the Jobs method is
likely to fail when either the host or guest aggregates in solution.
Limitations
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This method is perhaps the simplest but often the most effective of all the approachesavailable to determine the stoichiometry in hostguest complexes. In modern
supramolecular chemistry it is now rare not to have detailed information through X-ray
crystallography, 2D-NMR and Molecular Modelling about the structure of the host and
guest and, in some cases, even the hostguest complex itself. This structural information
can make the prediction of stoichiometry quite straightforward and accurate.
(ii) Consistency with the host structure and available information on the
hostguest complex structure.
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This relies on specific evidence such as isosbestic points which can be used to confirm
that more than one type of complex is present and hence that simple 1 : 1 complexationis not appropriate to describe the system if more than one isosbestic point is observed.
The converse is not necessarily true, i.e. the absence of more than one isosbestic point
cannot be used to rule out more complex stoichiometry such as 1 : 2 complex formation,
especially in cases where the cooperative (positive or negative) processes play a
significant role.
(iii) Specific experimental evidence such as isosbestic point(s).
(i ) C t f t bilit t ti th t ti i i d th t i th
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(iv) Constancy of stability concentration as the concentration is varied, that is, the
success of a stoichiometric model to account for the data.
This method is probably the most generally applicable method for determining
stoichiometry.
Firstly, if anything other than 1 : 1 stoichiometry is suspected, the data should be fitted
to other plausible models (e.g., 1 : 2) and the quality of fit of the different models
compared in details, taking into account factors such as the increase in parameters in
the fitting process.
Secondly, and more importantly, it is strongly advisable to carry out the titration at
different concentrations and even with different techniques (e.g., NMR and UV-Vis). If a
particular model is successful at explaining the data at different concentrations then it
can be taken as very strong evidence for that model.
8.2
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6.8
7
7.2
7.4
7.6
7.8
8
0 0.005 0.01 0.015 0.02
Serie1
Serie2
[G0]
obs
(ppm)
1:1 model
K 1938.702 78
DH 7.00624 0.004
DHG 8.000867 6.00E-03
M-1ppmppm
6.8
7
7.2
7.4
7.6
7.8
8
8.2
0 0.005 0.01 0.015 0.02
Serie1
Serie2
1:2 model
[G0]
obs(p
pm)
K 2615630 5.00E+05
DH 6.98578 0.02
DHG 8.169739 0.04
M-1ppmppm
[G0]
obs
(ppm)
2:1 modelHHG
K 3566244 1.20E+06
DH 7.05578 0.02
DHG 7.92317 0.03
M-1ppmppm
6.8
7
7.2
7.4
7.6
7.8
8
8.2
0 0.005 0.01 0.015 0.02
S
S
HGG
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Setting up a titration experiment
weigh H stocksolution H
weigh G
(concentrated solution)
use to fill
stocksolution G
(contains H at the same concentration !)
1) measure initial spectrum
2) add aliquots of G3) measure after each addition
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