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