1
In the name of
GOD
2
Zeinab Mokhtari
06-Jan-2010
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06-Jan-2010
Macromolecules
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06-Jan-2010
DNA
a lattice-like macromolecule : such as a nucleic acid or a carbohydrate
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06-Jan-2010
N : potential binding sites per macromolecule
l : the length of the ligand
Thermodynamic parameters (k , H, …)
T
B
[M]
[L]
governersof the behavior of the system
Macromolecules
The number of ligand molecules bound per macromolecule
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06-Jan-2010
ν versus [L] : hyperbolic representation, requires nonlinear regression analysis
Scatchard plot is a transformed representation.
v /[L] as a function of ν
N equivalent and independent binding sites and a ligand with size l = 1
a linear plot
Several classes of binding sites,or interacting binding sites,or a ligand with size l > 1
a curved plot
),,;(][
klNfL
Scatchard plot
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06-Jan-2010
),,,;(][
klNf
L
Valid for infinite homogeneous lattices with:any ligand size, lany ligand dissociation constant, kany level of cooperativity interaction between ligands bound contiguously,ω
Extensions of this theory for :finite lattices heterogeneous systems ( the binding sites are not homogeneously distributed throughout the macromolecule)different classes of binding sites present in the macromoleculedifferent ligand binding modes a mixture of different ligands
McGhee and von Hippel
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06-Jan-2010
Case I :non-cooperative (independent)
ligand binding
Case II :cooperative ligand
binding
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06-Jan-2010
An excellent tool for studying
biomolecular reactions
ITC : Isothermal titration calorimetry
The heat associated with a binding process
is measured.
Injections of a ligand solution from a computer-controlled syringe into a macromolecule solution placed in a thermostatized cell.
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06-Jan-2010
V
viMM iT 1][][ 0,
V
viLL iT 11][][ 0,
The syringe rotates in place during the ITC experiment. The end of the syringe has been adapted to provide continuous mixing in the ITC cell. The plunger is computer-controlled and injects precise volumes of ligand.
The total concentration of each reactant in the cell after injection i :
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06-Jan-2010
Case ILineal lattice-like macromolecule with noncooperative ligand binding: macromolecule with N equivalent and
independent binding sites and ligand size l ≥ 1
),,;(][
klNfL
Case I
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06-Jan-2010
As binding proceeds to saturation, it is more difficult to find l free consecutive binding sites when l is large. Then, the larger the ligand size l, the larger such
effect.1
)1(][][
l
TT lN
lN
k
lN
ML
1
)1(][
l
lN
lN
k
lN
L
linear in ν only if l = 1
01][][ 1 llTT lNklNML
an (l + 1)-order polynomial equation
11,, ])[1(][ iiTiiTi M
V
vMHVq
A nonlinear regression :the optimal values of N, l, k, and H from the experimental data.
Case I
13Fig. 1. Simulated calorimetric titration
The larger the ligand size, the larger the difference between the apparent and the actual maximal binding numbers.
06-Jan-2010
Therefore, if the data analysis is performed with a model for a ligand with size l = 1, then, both the maximal binding number and the binding affinity will be estimated with a significant error.
l > 1 : entropic resistance to saturation
Case I
N = 100l = 3
14Fig. 2. Effect of the ligand size on the calorimetric titration06-Jan-2010
N = 100l = 1 N/l = 100
The ratio N/l does not define univocally the geometric features of the system but also that the values of N and l by themselves are important.
Therefore, fixing N to different values in the fitting procedure will give different sets of estimated thermodynamic parameters.
Case I
N = 200l = 2
N = 1000l = 10
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06-Jan-2010
Case IILineal lattice-like macromolecule with cooperative
ligand binding: macromolecule with N equivalent and dependent binding sites and ligand size l ≥ 1
),,,;(][
klNf
L
Case II
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06-Jan-2010
21
)(2
)1(
))(1(2
))(12(
][
lN
RlN
lN
RlN
k
lN
L
l
2/12 )(4))1(( lNlNR
21
)(2
)1(
))(1(2
))(12(
][][
lN
RlN
lN
RlN
k
lN
ML
l
TT
0))(2(
)))(1(2())1((
)))(12)(()(][]([
2
12
1
lN
lNkRlN
RlNlNMLl
lTT
an (l + 3)-order polynomial equation
Case II
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06-Jan-2010
ITC allows discriminating between different cases.
Interaction between bound ligands : an additional contribution to the overall enthalpy of binding(The interaction or cooperativity enthalpy h)
][
][
separated
sidebyside
1 ≤ ω < ∞
ω = 1
0 ≤ ω < 1 Negative cooperativity
Positive cooperativity
No cooperation
It is obvious that at a given value of the ligand size l, there will be a value of the interaction parameter ω that almost compensates the negative entropic effect from l, resulting in a nearly linear plot.Difficult discriminating between different situations
Case II
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06-Jan-2010
))])[1(])([(
)])[1(])([2
(
)])[1(]([(
1,1,,,
1,1,,,
1,1,,,
idciTidciT
isciTisciT
iisoliTiisoliTi
MV
vMhH
MV
vM
hH
MV
vMHVq
h : the enthalpy associated with the interaction of nearest neighbor bound ligandsνisol : the partial number of ligand molecules
bound isolatedνsc : the partial number of ligand molecules
bound with only one nearest neighbor (singly contiguous)
νdc : the partial number of ligand molecules bound with two nearest neighbors (doubly
contiguous)
Case II
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06-Jan-2010
1
))(1(2
))(12()][]([
l
TTisol lN
RlN
k
lNML
l
TTsc lN
RlN
k
RNlML
))(1(2
))(12()1(
1)][]([
1
22
))(1(2
))(12(
)(
))1((
)1(2)][]([
l
TTdc
lN
RlN
lNk
RNlML
dcscisol
Case II
20
06-Jan-2010Fig. 5. Effect of the cooperativity parameter on the evolution of the partial binding numbers along the titration
cooperativity parameter :ω = 0 (dashed-dotted line)ω = 0.1 (dotted line)ω = 1 (continuous line)ω = 10 (dashed line)
N = 100l = 4
Case II
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06-Jan-2010
Case II
Fig. 6. Effect of the interaction enthalpy on the calorimetric titration
ω < 1 ω > 1
h+0-
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06-Jan-2010
Case II
Fig. 7. Comparison between direct titrations (L into M)
and reverse titrations (M into L).
h
direct reverse
+0
23Thanks
24
A day with
out laughte
r is
a day waste
d.
Charlie Chaplin