1 In the name of GOD

2 Zeinab Mokhtari 06-Jan-2010

3 Macromolecules

4 06-Jan-2010 DNA a lattice-like macromolecule : such as a nucleic acid or a carbohydrate

5 06-Jan-2010 N : potential binding sites per macromolecule l : the length of the ligand Thermodynamic parameters (k,  H, …) governers of the behavior of the system Macromolecules The number of ligand molecules bound per macromolecule

6 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 Scatchard plot

7 06-Jan-2010 Valid for infinite homogeneous lattices with: any ligand size, l any ligand dissociation constant, k any 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 macromolecule different ligand binding modes a mixture of different ligands McGhee and von Hippel

8 06-Jan-2010 Case I : non-cooperative (independent) ligand binding Case II : cooperative ligand binding

9 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.

10 06-Jan-2010 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 :

11 06-Jan-2010 Case I Lineal lattice-like macromolecule with noncooperative ligand binding: macromolecule with N equivalent and independent binding sites and ligand size l ≥ 1 Case I

12 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. linear in ν only if l = 1 an (l + 1)-order polynomial equation A nonlinear regression : the optimal values of N, l, k, and  H from the experimental data. Case I

13 Fig. 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 = 100 l = 3

14 Fig. 2. Effect of the ligand size on the calorimetric titration 06-Jan-2010 N = 100 l = 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 = 200 l = 2 N = 1000 l = 10

15 06-Jan-2010 Case II Lineal lattice-like macromolecule with cooperative ligand binding: macromolecule with N equivalent and dependent binding sites and ligand size l ≥ 1 Case II

16 06-Jan-2010 an (l + 3)-order polynomial equation Case II

17 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) 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

18 06-Jan-2010 h h : the enthalpy associated with the interaction of nearest neighbor bound ligands ν isol : the partial number of ligand molecules bound isolated ν sc : the partial number of ligand molecules bound with only one nearest neighbor (singly contiguous) ν dc : the partial number of ligand molecules bound with two nearest neighbors (doubly contiguous) Case II

19 06-Jan-2010 Case II

20 06-Jan-2010 Fig. 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 = 100 l = 4 Case II

21 06-Jan-2010 Case II Fig. 6. Effect of the interaction enthalpy on the calorimetric titration ω < 1ω > 1 hh

22 06-Jan-2010 Case II Fig. 7. Comparison between direct titrations (L into M) and reverse titrations (M into L). hh direct reverse +0+0

23 Thanks

24 A day without laughter is a day wasted. Charlie Chaplin