Single Site Binding Model
[M] = free protein [L] = free ligand [ML] = protein ligand complex
Assumptions of Binding
- reversible
- ligand availability is limited only by diffusion
- measurements taken when thermodynamic eq. is reached
- assumptions supported with structural data about number of ligand binding sites / conformational changes on binding
Rates of Binding
forward rate : kon [M][L] forward rate : koff [ML] - at equilibrium the rates are equal Kon / Koff = Ka (association constant) - Kd is Koff / Kon
Association Constant
- high affinity binding: large Ka
- low affinity binding: small Ka
Thermodynamics of Single Site Binding
- change in standard Gibbs free energy of association
- Gibbs free energy links free energy changes with binding constants
G = -RTlnKa
Fractional Saturation
- fraction of protein saturated with ligand
- 0-1
Y = [ML] / [M][L] - Y easily determined from quantity measured in the experiment
- but by rearranging we can express Y in terms of ligand concentration and not protein ligand complex concentration
Y = [L] / Kd [L]
Langmuir Isotherm
- hyperbolic binding curve
- [L] vs. Y
- Kd is the ligand concentration at which binding site is half saturated (Y = 0.5)
Semi-log Plot
- plot binding curve on a semi-log plot
- changes in affinity shifts the curve to the right and Kd comparison is easier
Experimental Errors
- systematic errors: incorrect binding model, aggregation, inactive protein, etc
- random errors: random experimental errors
Scatchard Equation
The Scatchard equation is an equation used in molecular biology to calculate the affinity and number of binding sites of a receptor for a ligand
* see equation *
Scatchard Plot
- Scatchard plot shows that the slope equals to -1/Kd while the x-intercept equals the number of ligand binding sites n.
- linearising the data by transforming the expression for Y
- plot Y/[L] vs Y
- higher affinity has a steeper gradient
- easier to see deviations from a straight line but transformation multiplies errors
Scatchard Analysis
- mix solution of retinoic acid receptor and retinoic acid that is radiolabelled
- add charcoal pellets to bind free ligand and remove it
- then analyse the receptor and bound ligand
Kd when receptor concentration is not known
** see notes
Multiple Sites of equal affinity
- two binding constants (K)
- macroscopic eq. association constants for binding of the 1st and 2nd ligands respectively
- independent binding to multiple sites can be analyzed as for single site binding except Y replaced by v
- ** see notes!!!!!
v
[M] of bound protein / total [M] of protein
- varies from 0 to n
v = 2[L] / Kd + [L]
Plotting Multiple Binding Sites
- scatchard plot of v vs v/[L]
- intercept of n/Kd
- gradient of -1/Kd
- model is similar to normal Scatchard equation except Y is replaced by v
Cooperative Binding
- binding of first ligand changes affinity for 2nd ligand
- positive cooperativity: binding enhances affinity
- shape goes in a N shape as low affinity goes to high affinity back to lower
- negative cooperativity: binding reduces affinity
- gives a parabolic Scatchard plot (N or U shape)
Hill Plot
- quantitative analysis of cooperativity
The Hill plot is the rearrangement of the Hill–Langmuir Equation into a straight line.
A slope greater than one thus indicates positively cooperative binding between the receptor and the ligand, while a slope less than one indicates negatively cooperative binding.
Calcuation and Fitting of Binding Data
** SEE NOTEs **
