#### Transcript Binding Polynomials

```Adsorption Chromatography
 v  f mvm  f s vs  f mvm
fm 
cmVm
cmV m csVs
fm 
1
1  K
fs  1 fm 
K
1  K
vm
 v 
1  K
K
1 vm   v 
 v
Binding Polynomials, Q
Ki
P  iX 
PX i
t
Q   Ki xi
Ki 
i 0
[ P]  [ PX1 ]  [ PX 2 ]  ...  [ PX t ]
[ P]  [ P]K1 x  [ P]K 2 x 2  [ P]K 3 x 3  ...  [ P]K t x t

[ P] 1  K1 x  K 2 x 2  K 3 x 3  ...  K t x t

[ P]Q

Q  1  K1 x  K 2 x 2  K 3 x 3  ...  K t x t
t
Q   Ki xi
i 0

[ PX i ]
[ P]x i
Relative Population of Ligation States

Q  1  K1 x  K 2 x 2  K 3 x 3  ...  K t x t

# Ligands Bound
Relative Population
0
1/Q
1
K1x/Q
2
K2x2/Q
t
Ktxt/Q
Average number of ligands bound per P molecule,
v=<i>
xdQ d ln Q
v

Qdx d ln x
t
v  i   ip (i )
i 0
K x  2K x  3K x  ...  tK x 
v
1  K x  K x  K x  ...  K x 
2
1
3
2
3
2
1
t
v
i
iK
x
 i
i 0
Q
2
t
t
3
3
t
t
Average number of ligands bound per P
molecule, v=<i>
Binding Polynomials
Binding Polynomials
Two Binding Sites: Cooperativity
Ka
P  X 
Pa X
Kb
P  X 
Pb X
Kc
P  2 X 
Pab X 2
• Average number of ligands per P molecule,
K a x  Kb x  2Kc x 2
v
1  K a x  Kb x  Kc x 2
Two Binding Sites: Cooperativity
K c  K a K b c
• Independent sites: Non-cooperative binding
Kc  K a Kb
v
Ka x
Kb x

1  K a x 1  Kb x
• Not independent sites:
Kc  K a Kb
– Cooperative binding
– Anti-cooperative binding
Kc  K a Kb
Kc  K a Kb
Two Binding Sites: Stoichiometric Approach
K1
P  X 
PX1
K2
P  X 1 
PX 2
K1 x  2 K1 K 2 x 2
v
1  K1 x  K1 K 2 x 2
K1 x
v
1 K1 x
K2 x
v 1 
1 K2 x
Two Binding Sites in Glycine
Two Binding Sites: Stoichiometric Approach
K1 x  2 K1 K 2 x 2
v
1  K1 x  K1 K 2 x 2
K1 x  2 K1 xK2 x
v
1  K1 x  K1 xK2 x
v
K1 x(1  2 K 2 x)
1  K1 x(1  K 2 x)
1
v  1  K1 
2
xmid
K1 x
v
1  K1 x
K2 x  1
Two Binding Sites: Stoichiometric Approach
K1 x  2 K1K 2 x 2  Q
v 1 
Q
K1 x  2 K1 K 2 x 2  (1  K1 x  K1 K 2 x 2 )
v 1 
(1  K1 x  K1 K 2 x 2 )
K1 K 2 x 2  1
v 1 
(1  K1 x  K1 K 2 x 2 )
v 1 
K1 xK2 x  1
1  K1 x(1  K 2 x)
1
v 1  1  K2 
2
x'mid
K2 x
v 1 
1 K2 x
K2 x  1
K1 x  1
Ligation States for Two Binding Sites

Q  1  K1x  K1K 2 x 2
Relative Population of 0 ligand
bound: 1/Q
Relative Population of 1 ligand
bound: K1x/Q
Relative Population of 2 ligand
bound: K1K2x2 /Q

x  xmid
Site-model and stoichiometric model binding constants
n Independent Sites with Identical Affinities
Q  1  Kx1 1  Kx2 ...1  Kxn
Q  1  Kx 
n
v
nKx
1  Kx
v
Kx

n 1  Kx
n Independent Sites with Identical Affinities
Scatchard Plot
v
Kx

n 1  Kx
v
 nK  vK
x
n Independent Sites with Identical Affinities
Scatchard Plot
v
Kx

n 1  Kx
v
 nK  vK
x
Scatchard Plot for an Antibody
Cooperative Binding
K
nX  P 
PX n
K
[ PX n ]
[ P]x n


Q  1  Kxn
xdQ d ln Q
v

Qdx d ln x
nKx n
v
1  Kx n
Cooperative Binding
The Hill Plot
nKx n
v
1  Kx n
Kx n

1  Kx n
  (1   ) Kx

1
Kx n
1  1
1  Kx n
1  Kx n  Kx n
1 
1  Kx n
1
1 
1  Kx n
n
 Kx n
  
ln 
  ln K  n ln x
1 
Cooperative Binding
The Hill Plot
Cooperative Binding
The Hill Plot
Cooperative Binding
The Hill Plot
Aggregation and Micellization are Cooperative
Processes
K
nA1 

An
[ An ]
K
[ A1 ]n
x  [ A1 ]
[ A1 ]  n[ An ]
v
[ A1 ]  [ An ]
[ A1 ]  nK [ A1 ]n [ A1 ](1  nK [ A1 ]n 1 )
v

n
[ A1 ]  K [ A1 ]
[ A1 ](1  K [ A1 ]n 1 )
1  nKx n 1
v
1  Kx n 1
Critical Micelle Concentration, cmc
v
1 n
 x n 1  1
K
2
Micelle Well-Defined Sizes
The BET Adsorption Model of Layers of Ligands
Q  1  K1 X  K1K 2 x 2  K1K 22 x3  K1K 23 x 4  ...
Q  1
K1 X
1 K2 x
Pr ovided
K2 x  1
v
K1 x
(1  K 2 x)[1  ( K 2  K1 ) x]
```