Binding Polynomials
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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
Adsorption Chromatography
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 Kx1 1 Kx2 ...1 Kxn
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]