STUDIA UNIVERSITATIS BABEŞ-BOLYAI, PHYSICA, SPECIAL ISSUE, 2003
NMR STUDY OF HOMO- AND
HETEROASSOCIATION OF AROMATIC
MOLECULES IN AQUEOUS SOLUTION.
NUMERICAL SIMULATIONS
Diana Bogdan1, C. Morari1, M. Bogdan1
1
National Institute for Research and Development of
Isotopic and Molecular Technologies, P.O. Box 700,
Donath Str. # 71-103, 400293 Cluj-Napoca 5,
Romania
Tel. +4-0264-584037, Fax: +4-0264-420042
Abstract. Investigation of the homo- and heteroassociation of
different molecules and their competitive binding to receptors deal with
some important aspects of molecular interactions. Although there have
been a number of attempts, to develop models of homo- and
heteroassociation of aromatic molecules, there are often limitations in
their use because rather approximate expressions for equilibrium
concentrations are used. In this report, we analyse different models in
which indefinite aggregates exist for both self- and heteroassociation
between molecules in solution. Using these models based on numerical
simulations a NMR analysis was developed in order to determine the
structural and thermodynamic parameters of molecular homo- and
heteroassociation in solution using the chemical shifts variation of
different protons as a function of concentration.
Introduction
Investigation of the homo- and heteroassociation of different molecules
and their competitive binding to receptors deal with some important aspects of
molecular interactions. From a pharmacological point of view, association
complexes and competitive binding may influence the activity of drugs when used
in combination or in the interaction of drugs with aromatic molecules from food
sources.
Although there have been a number of attempts, in recent years, to
develop models and analyses of homo- and heteroassociation of aromatic
DIANA BOGDAN, C. MORARI, M. BOGDAN
molecules, there are often limitations in their use. The model developed by Baxter
et al. [1] for the heteroassociation of aromatic molecules is not applicable to the
general case because rather approximate expressions are used for equilibrium
monomeric concentrations of one of the components in the mixed solution. Models
used for the interpretation of optical spectroscopic data of the association of
aromatic molecules either to not take into account the formation of n-mer
aggregated for all the components present in the mixed solution or only consider
formation of a 1:1 heterocomplex without taking into account the self-association
of aromatic molecules in solution [2].
For these reasons, in the present paper we intend to present a general
model for analysis of indefinite association of molecules in solution, based on
NMR spectroscopy.
NMR spectroscopy has some advantages over optical investigations of
molecular complexation because it can be used to determine both the equilibrium
and structural details of multicomponent complex formation in solution [3]. The
NMR analysis has been developed based on chemical shift measurements of one
species as a function of concentration. The analytical expression was obtained
without any approximations or restrictive hypotheses.
Theory
Isodesmic model
The isodesmic model – also known as the indefinite non-cooperative
model – is based on the assumption that solute molecules associate to form stacks
(superior order oligomers), An, n  ( 2,  ) , where the value of equilibrium
constant, Ka, for each step are assumed to be equal: K = K2 = K3 = …= Kn, (n→∞).
In this case, based on superior order oligomers formation reactions, it is
possible to express the concentration of the n-mer as a function of association
constant, K, and the concentration of monomer, [A], as follows
[An] = Kn-1·[A]n.
Defining the total concentration, [A0], as
(1)
BIMODAL MOLEC. ENCAPSULATION OF MEFENAMIC ACID BY β-CD IN SOLUT. AND SOLID STATE
[A0] = [A] + 2 [A2] + 3 [A3] + … + i [Ai] + … =
A
.
1  K A2
(2)
we obtain for [A] the following expression
A 
1
2K
2
A 0 
2K A 0   1 

4 K A 0   1 .
(3)
The general accepted hypothesis in the case of isodesmic non-cooperative
model considers that the chemical shifts for each nucleus observed by NMR and
belonging to a molecule at the end of the oligomer (δe) is the arithmetical average
of the chemical shifts corresponding to the monomer (δm) and to the molecule
intercalated into the n-mer (δi). Hence
e 
m  i
.
(4)
2
Therefore:
 obs   m   i   m 
2 K A 0   1  4 K A 0   1
2 K A 0 
.
(5)
It is worth of mentioning that the isodesmic non-cooperative model cannot
distinguish between the dimerisation process and the indefinite association model.
The only differences being that K = 2K2 and δi – δm = δdimer - δm.
Eq. (4) defines a parameter f  1
2
in the following equivalent equation
δe = (1 - f) δm + f δi.
(6)
In the general case, the parameter f from eq. (7) can be defined as
0≤f≤1
(8)
In a molecule, the environment of nuclei is different, so that they may have
different values for the f parameter. In this case, besides the parameters K and δi,
the equation which describes the observed chemical shift variation as a function of
DIANA BOGDAN, C. MORARI, M. BOGDAN
concentration will also contain f as a fit parameter. By performing a simple
mathematical calculus, the following expression for δobs can be
 obs 

2f









1

K
A
1

2
f


K
A

m


A 0 1  K A 

A
 K A2 f  1  
i  . (9)
 
1  K A
 
The total concentration, [A0], is given by eq. (2), yielding the solution (3).
In this case, the final expression of the observed chemical shift may be written as:
δobs = (1–K [A]) {1–K [A] (2f–1)} δm + K [A]{2f–K [A] (2f–1)} δi.
(10)
where:
K A  
2 K A 0   1  4 K A 0   1
2 K A 0 
.
(11)
In the case when f  1 , the eq. (10) come down to eq. (5), which is
2
characteristic for the isodesmic usual accepted model.
Heteroassociation
The heteroassociation study was performed using two different molecules
A and B. We considered that the concentration of A molecules is very low (aprox.
1 mM) and at this concentration the A molecules are essentially monomeric. Even
so, there are many different species present in solution, of which the only
components that have been considered are B molecules stacks containing one A
molecule either at the end or intercalated at some point within the stack.
The B molecule is assumed to stack according to the isodesmic model with
association constant Km. These interactions are assumed to be unaffected by the
presence of A molecules elsewhere in the stack.
Thus:
B  B  B2
B  AB  B2 A
B  BAB  B2 AB
all have the same equilibrium constant, Km.
(12)
BIMODAL MOLEC. ENCAPSULATION OF MEFENAMIC ACID BY β-CD IN SOLUT. AND SOLID STATE
On addition of B molecules to a solution of monomeric A molecules, the
A molecules may associate at the ends of B stacks with association constant, Ke
B  A  BA
B  A  B2 A
2
(13)
B  AB A
n
Ke 
n
AB B2 A Bn A


.
BA B2 A Bn A
(14)
or A molecules may intercalate into B stacks with association constant, Ks
B  A  BAB
2
B  A  B2 AB or BAB
3
2
B  AB
n
Ks 
n 1
AB or B
n-2
(15)
AB
2
or BAB
n 1
Bn1 AB
BAB B2 AB


.
B2 A B3 A
Bn A
(16)
For these expressions, equations for the concentrations of each species can
be derived in terms of the concentrations of monomers. For example
Bn   K mn1 Bn
BnC   K mn1 K e M n A
(17)
As is usual in the isodesmic model, it is assumed that the change in
chemical shift of an A molecule at the end of a stack is only half that for
intercalation into a stack. Hence
 
m
e  i
.
(18)
2
where
δe – the proton chemical shift of A molecule associated at the end of a B stack
δi – the proton chemical shift of A molecule intercalated into a B stack
DIANA BOGDAN, C. MORARI, M. BOGDAN
δm – the chemical shift of monomeric A
Following the procedure outlined by Baxter et al. [1], we obtain for the
observed chemical shift of A the expression
 
m

A  A0
AB i   m K   K K B .



m
m
A 0 
A 0 1  Km B  2 e i s m 
(19)
To complete the derivation, the concentration of monomeric A and B
molecules must be calculated.
The concentration of monomeric A molecule is given by
A 
A0 1  Km B
.
2
1  K m B  Ke B  K s K m B
(20)
For the monomeric B molecule, Baxter et al. [1] made an approximation
considering the concentration of monomeric B in the absence of A, namely

B  B0 
1 
2


2

12
4 K B 0  1 

m


(21)
We have calculated the exact expression for [B] and obtained
1  Ke  A  Ks Km  AB2  Km B
B0  B
1  Km B2
Experimental
We used the foloowing values:
Km = 100 M-1
[A0] = 1 mM
Ks = 4 M-1
δm = 2.5 ppm
Ke = 8 M-1
δi = 3.3 ppm
.
(22)
BIMODAL MOLEC. ENCAPSULATION OF MEFENAMIC ACID BY β-CD IN SOLUT. AND SOLID STATE
3.3
without approximations
with approximations, [1]
3.2
3.1
obs
3.0
2.9
2.8
2.7
2.6
2.5
2.4
0.00
0.01
0.02
0.03
0.04
0.05
[B]0
References
1.
N.J. Baxter, M.P. Williamson, T.H. Lilley, E. Haslam, J. Chem. Soc. Faraday
Trans., 92, 231 (1996).
2.
R.W. Larsen, R. Jasuja, R. Hetzler, P.T. Muraoka, V.D. Andrada, D.M. Jameson,
Biophys. J., 70, 443 (1996)
3.
D.B. Davies, A.N. Veselkov, J. Chem. Soc. Faraday Trans., 92, 3545 (1996)