The two-state dimer
receptor model. A general
model for receptor dimers
Supplementary material
Rafael Franco, Vicent Casadó, Josefa Mallol, Carla Ferrada, Sergi Ferré, Kjell Fuxe,
Antoni Cortés, Francisco Ciruela, Carmen Lluis and Enric I. Canela
Correspondence address:
Enric I. Canela
Dept. Bioquimica i Biologia Molecular
Universitat de Barcelona
Av. Diagonal, 645. 08028 Barcelona. Spain
[email protected]
Scheme of the model
Inactive
Vacant
A+A+(RR)
K
Constitutive
aK
A+A(RR)*
Occupied
A(RR)A
ba L
aL
L
A+A+(RR)*
A+A(RR)
mK
mbK
A(RR)*A
Active
Constants
List of the equilibrium constants for the two-state dimer model
Parameter
Description
K
Equilibrium association constant of A to
RR
L
Equilibrium receptor isomerization
constant
a
b
m
Intrinsic efficacy of A in relation to the
binding to unoccupied receptors
Intrinsic efficacy of A in relation to the
binding to single-occupied receptors
Binding cooperativity between first and
second A molecule: ratio of affinity of A
for A(RR) and (RR) [µK being the
equilibrium association constant of A to
A(RR)]
Definition&
A(RR)
A (RR)
(RR) 
*
(RR) 
A(RR)  (RR)
(RR)  A(RR)
A(RR) A A(RR)
A(RR)A A(RR) 
*
*
*
*
A(RR)A (RR)
A(RR) A(RR)
& In this symmetric dimer model [A(RR)] refers to the concentration of dimer with A bound, i.e irrespective of whether
A is bound to one site or the other.
Ligand binding
I. Functions
The ligand binding function
ABound
RT 
K  (1  aL )  A   2K 2 m  (1  abL )  A 
2

1  L  K  (1  aL )  A   K 2 m  (1  abL )  A 
2
The saturation function
is a 2:2 function
2
2
 1   K A   (1  aL )  2K m A   (1  abL ) 
Y   
2
 2   1  L  K A   (1  aL )  K 2 m A   (1  abL ) 
Ligand binding
II. Cooperativity analysis
_



  1   K A  (1  aL)  2K m A   (1  abL)   K A  

Y  Y      
2
_

2


  2   1  L  K A   (1  aL)  K m A   (1  abL)   1  K A 


2
_
Saturation function
2
Reference saturation function
It corresponds to a theoretical
noncooperative binding of A to a dimer
_
K is the average association constant K 
K  1  La 
2  1  L 
Ligand binding
III. Cooperativity analysis
If
_

Y  Y   1




Positive cooperativity
If
_

Y  Y   1




Negative cooperativity
If
_

Y  Y   1




2

1  La 
Noncooperativity. Occurs when m 
4  1  L   1  Lab 
Ligand binding
IV. Fitting data to the model
ABound
RT 
K  (1  aL )  A   2K 2 m  (1  abL )  A 
2

1  L  K  (1  aL )  A   K 2 m  (1  abL )  A 
2
Rearranging and defining c1 and c2
ABound
.
RT c1A   2  RT A 
2

c2  c1A   A 
2
being
A50 
c2
where

1  aL 
c1 
K  m  1  abL 

1 L
c2  2
K m  1  abL 

c2 gives an idea of the affinity



positive cooperativity occurs when c1 < 2·[A]50
negative cooperativity occurs when c1 > 2·[A]50
c1 = 2·[A]50 gives noncooperativity
Scheme of the model for two competing ligands
A
+
A
+
B+B+(R2)*
A
+
B+(R2)*B
q M
A
+
A
+
B+B+(R2)
ad K
M
A
+
B+(R2)*A
K
qd M
bm K
(R2)*A2
qw L
A
+
B+(R2)B
gK
(R2)*AB
aL
aqd/g L
A
+
B+(R2)A
mK
ba L
(R2)A2
(R2)*B2
qL
L
aK
fw M
g M
(R2)AB
fM
(R2)B2
Constants for competing agent
List of the equilibrium constants for the two-state dimer model
Parameter
Description
Definition&
B(RR) 
B (RR) 
M
Equilibrium association constant of B to RR
g
Binding cooperativity between A and B: ratio of affinity of
A for B(RR) and R or of B for A(RR) and RR
A(RR)B  (RR) 
A(RR)  B(RR) 
q
Intrinsic efficacy of B in relation to the binding to
unoccupied receptors
B(RR)  (RR) 
(RR)  B(RR) 
w
Intrinsic efficacy of B in relation to the binding to singleoccupied receptors
*
*
B(RR) B B(RR) 
B(RR)B  B(RR) 
*
*
f
Binding cooperativity between first and second B molecule:
ratio of affinity of B for A(RR) and (RR) [fM being the
equilibrium association constant of B to B(RR)]
B(RR)B  (RR) 
B(RR)  B(RR) 
d
Activation cooperativity between A and B: ratio of affinity
of A for B(RR)* and B(RR) or of B for A(RR)* and A(RR)
(RR) * A(RR)  B(RR ) A(RR) *
(RR)  A(RR) *B(RR) * A(RR)B 
& In this symmetric dimer model [B(RR)] refers to the concentration of dimer with B bound, i.e irrespective of whether
B is bound to one site or the other.
Competition experiments
I. Function
The ligand binding function
A Bound
RT 
K A   1  aL   2  mK 2 A   1  abL   KM A B  g  aqdL 

2
1  L  K A   1  aL   mK 2 A   1  abL   M B  1  qL  
2
fM 2 B  1  qwL   KM A B  g  aqdL 
2
Being A the initial bound ligand and B the competing ligand
Competition experiments
II. Fitting data to the model
defining c3, c4 and c5
c3 
M  1  qL 
K 2  m  1  abL 
M 2f  1  qwL 
c4  2
K m  1  abL 
M  g  aqdL 
c5 
K  m  1  abL 
Competition experiments
III. Fitting data to the model
Substituting and rearranging
A Bound
ABound
c1A   2  A   c5 A B
2
 RT
c2  c1A   A   c3 B  c4 B  c5 A B
2
2
c1 A   2  A   c 5 A  B
2
 RT
c
2

 c1 A   A   c 3  c 5 A  B  c 4 B
2
2
Competition experiments
IV. Fitting data to the model
Simplifying when A binding is noncooperative
A Bound
c1 A   2  A   c 5 A   B
2

c
2
2
 4  c1 A   A    c 3  c 5 A   B  c 4 B


2
1
or
ABound
2  A   c1  A   c5 A   B
 2


RT
2
c1  A   c  c A   B  c B2
3
5
4
 2

RT