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Scatchard Equation

The scatchard equation is used in calculating the dissociation constant (Kd) 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

Kd=Dissociation constant

L0=Total concentration of P at time=0, representing both bound & unbound P.

Contents

[edit] 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


[edit] 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] R1[P][L]= R − 1[PL] Where R1 =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 L0, minus the concentration of bound ligand [LP]; Or, algebraically,

[Equation 6] [L]= [L0]-[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] Kd[PL] = [P][[L0] − [PL]]

Dividing both sides by Kd 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}

[edit] 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] [Kd + [P]][PL] = [P][L0]

Both sides are divided by [Kd +[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.

[edit] The Scatchard equation as a model for protein-ligand interactions

At hight concentrations of ligand: At low concentrations of ligand: When the ligand concentration=Kd


[edit] Links

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