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Scatchard Equation

The scatchard equation is used in calculating the dissociation constant ($$K_d$$) of a ligand with a protein.

$$\frac {[LP]}{[L]} = \frac{n[L_o]}{[K_d]} -\frac{[LP]}{K_d} $$

[L]=Concentration of unbound ligand [LP]=Concentration of AB n=number of ligand binding sites $$K_d$$=Dissociation constant $$L_0$$=Total concentration of P at time=0, representing both bound & unbound P.

The Scatchard Plot
Sepearative methods --such as Frontal affinity chromatography, equilibrium dialysis and gel shift assay-- are used in determining free and bound ligand concentrations. The ligand concentration is varied, whilst the protein's concentration is maintained to a constant concentration

Deriving the Scatchard Equation
A simple reversible protein-ligand interaction can be shown as: [Equation 1]    P + L $$\Longleftrightarrow$$  PL

Where P=Protein, L=ligand, and PL=the protein-ligand complex.

At equilibrium the forward rate of reaction is equal to the reverse rate of reaction. It follows, then, that [Equation 2] $$R_1$$[P][L]= $$R_{-1}$$[PL]

Where $$R_1$$ =the forward rate constant,  $$R_{-1}$$=the reverse rate constant, [P]=concentration of protein, [L]=concentration of ligand and [PL]=concentration of protein-ligand complex.

This can be re-arranged, giving the standard dissociation constant equation:

[Equation 3] $$ \frac{K_1}{K_{-1}} = \frac{[P][L]}{[PL]}$$

By the dissociation constant's definition, it follows that since [Equation 4] $$\frac{K_1}{K_{-1}} = K_d$$

then

[Equation 5] $$ K_d = \frac{[P][L]}{[PL]}$$

At equilibrium the concentration of unbound ligand [L] is equal to it's initial concentration $$L_0$$, minus the concentration of bound ligand [LP]; Or, algebraically,

[Equation 6] [L]= [$$L_0$$]-[LP]

Substituting equation 6 into equation 5 gives:

[Equation 7] $$ K_d = \frac{[P]L_0]-[PL}{[PL]}$$

Multiplying both sides by [PL] gives: [Equation 8] $$ K_d [PL]= [P]L_0]-[PL$$ Dividing both sides by $$K_d$$ gives: [Equation 9] $$[PL]= \frac{[P]L_0]-[PL}{K_d}$$

Nultiplying out the numerator gives: [Equation 10] $$[PL]= \frac{[P][L_0]-[P][PL]]}{K_d}$$

Dividing both sides by [P], and spliting apart the numerator into two fractions gives the scatchard equation for a one-to-one interaction between ligand and protein: [Equation 11] $$\frac{[PL]}{[P]}= \frac{[L_0]}{K_d}-\frac{[PL]}{K_d}$$

It follows that for a many-to-one interaction, the stoichometric coefficent "n" is introduced:

$$\frac {[LP]}{[L]} = \frac{n[L_o]}{[K_d]} -\frac{[LP]}{K_d} $$

Wrong......
Multiplying out the numerator gives:

[Equation 8] $$ K_d = \frac{[P][L_0]-[P][PL]]}{[PL]}$$

Splitting the numerator into its two components gives:

[Equation 9] $$ K_d = \frac{[P][L_0]}{[PL]} -\frac{[P][PL]}{[PL]}$$

[PL] is present in both the numerator and denominator within the second fraction, so it can be similified further to:

[Equation 10] $$ K_d = \frac{[P][L_0]}{[PL]} -[P]$$ [P] is brought over to R.H.S [Equation 11] $$ K_d + [P]= \frac{[P][L_0]}{[PL]}$$ Both sides are multiplied by [PL] [Equation 12] $$ [K_d + [P]][PL]= [P][L_0]$$ Both sides are divided by [$$K_d$$ +[P]], giving [Equation 13] $$ \frac{[K_d + [P]][PL]}{K_d + [P]}= \frac{[P][L_0]}{K_d +[P]}$$ Simplifing gives: [Equation 14] $$ [PL]= \frac{[P][L_0]}{K_d +[P]}$$ Notice the similarity between Eq14 and the michellis menten equation.

The Scatchard equation as a model for protein-ligand interactions
At hight concentrations of ligand: At low concentrations of ligand: When the ligand concentration=$$K_d$$

Links
Scatchard plot http://www.graphpad.com/curvefit/scatchard_plots.htm