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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