Download SUPPORTING INFORMATION Obtaining equation 8 Consider a

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SUPPORTING INFORMATION
Obtaining equation 8
Consider a solution containing two (or more) receptors that can interact with a
fluorophore that can be quenched by a given species (quencher) present in the solution.
These receptors in the present case are cyclodextrins, but the following treatment can be
applied to other receptors, such as surfactant monomers and micelles, for example.
In the absence of a quencher, the intensity of emission of the fluorophore, in the
presence of two cyclodextrins, CD1 and CD2, would be:
(I em ) 0 
(I em ) f  (I em )1 K 1 [CD1 ]  (I em ) 2 K 2 [CD 2 ]
1  K 1 [CD1 ]  K 2 [CD 2 ]
S-1
In this equation (Iem)f corresponds to the emission intensity of the probe in the absence
of receptors, (Iem)1 to the situation in which all the probe molecules are bound to CD1
and (Iem)2 to CD2. This equation follows from the fact that the concentrations of free
fluorophore and fluorophore bound to CD1 and CD2 are given by:
In the presence
[R f ] 
1
[R ]
1  K 1 [CD1 ]  K 2 [CD 2 ]
[R 1 ] 
K 1 [CD1 ]
[R ]
1  K 1 [CD1 ]  K 2 [CD 2 ]
[R 2 ] 
K 2 [CD 2 ]
[R ]
1  K 1 [CD1 ]  K 2 [CD 2 ]
of a quencher,
consequently,
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S-2
(I em ) Q  (I em ) Qf  (I em ) Q1  (I em ) Q 2 and
(I em ) 0
(I )  (I em ) 01  (I em ) 02
 em 0 f
(I em ) Q (I em ) Qf  (I em ) Q1  (I em ) Q 2
S-3
In this equation (Iem)0f, (Iem)01 and (Iem)02 represent the emission intensities of Rf, R1 and
R2 in the absence of the quencher and (Iem)Qf, (Iem)Q1 and (Iem)Q2 in the presence of the
quencher.
Of course,
(I em ) 0i
 1  (K SV ) i [Q]
(I em ) Qi
i  f , 1, 2
S-4
Thus
(I em ) 0i  (I em ) Qi  (I em ) Qi (K SV ) i [Q]
i  f , 1, 2
S-5
Introducing equation S-5 in equation S-3,
(I em ) Qf (K SV ) f  (I em ) Q1 (K SV )1  (I em ) Q 2 (K SV ) 2
(I em ) 0
 1
[Q]
(I em ) Q
(I em ) Qf  (I em ) Q1  (I em ) Q 2
S-6
Now, (Iem)Qi can be written as:
(I em ) Qi   i l i 0 [R i ]  i
S-7
In this equation I is the molar extinction coefficient of species i, l is the light path
length of the cuvette, i0 is the intensity of the incident beam on the sample and I is the
quantum yield for the emission of species i, in the presence of the quencher. Of course,
[Ri] are given by equations S-2.
Introducing equations S-2 and S-7 in equation S-6 one obtains:
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l f ε f i 0 (K SV ) f
l ε i (K ) K [CD1 ] l 2 ε 2 i 0 (K SV ) 2 K 2 [CD 2 ]
 1 1 0 SV 1 1

(I em ) 0
1  K 1 [CD1 ]  K 2 [CD 2 ] 1  K 1 [CD1 ]  K 2 [CD 2 ] 1  K 1 [CD1 ]  K 2 [CD 2 ]
 1
[Q] 
l f ε f i 0
l1ε 1i 0 K 1 [CD1 ]
l 2 ε 2 i 0 K 2 [CD 2 ]
(I em ) Q


1  K 1 [CD1 ]  K 2 [CD 2 ] 1  K 1 [CD1 ]  K 2 [CD 2 ] 1  K 1 [CD1 ]  K 2 [CD 2 ]
a 
a 
(K SV ) f   1 (K SV )1 K 1 [CD1 ]   2 (K SV ) 2 K 2 [CD 2 ]
 af 
 af 
 1
[Q]
 a1 
 a2 
1   K 1 [CD1 ]   K 2 [CD 2 ]
 af 
 af 
S-8
with ai= ii.
Thus, as
(I em ) 0
 1  (K SV ) obs [Q]
(I em ) Q
S-9
it results in
(K SV ) obs
a
(K SV ) f   1
 af


a 
(K SV )1 K 1 [CD1 ]   2 (K SV ) 2 K 2 [CD 2 ]

 af 
a 
a 
1   1 K 1 [CD1 ]   2 K 2 [CD 2 ]
 af 
 af 
S-10
or
(K SV ) obs 
(K SV ) f  (K SV )1 (K app )1 [CD1 ]  (K SV ) 2 (K app ) 2 [CD 2 ]
1  (K app )1 [CD1 ]  (K app ) 2 [CD 2 ]

a  
 (K app ) i   i  K i 
a  

 f 

This is equation 8.
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S-11