The Scatchard equation is an equation used in

molecular biology Molecular biology is the branch of biology that seeks to understand the molecular basis of biological activity in and between cells, including biomolecular synthesis, modification, mechanisms, and interactions. The study of chemical and phys ...

to calculate the

affinity Affinity may refer to: Commerce, finance and law * Affinity (law), kinship by marriage * Affinity analysis, a market research and business management technique * Affinity Credit Union, a Saskatchewan-based credit union * Affinity Equity Part ...

and number of binding sites of a receptor for a

ligand In coordination chemistry, a ligand is an ion or molecule ( functional group) that binds to a central metal atom to form a coordination complex. The bonding with the metal generally involves formal donation of one or more of the ligand's ele ...

. It is named after the American chemist George Scatchard.

Equation

Throughout this article, 'RL''denotes the concentration of a receptor-ligand complex, 'R''the concentration of free receptor, and 'L''the concentration of free ligand (so that the total concentration of the receptor and ligand are 'R'' 'RL''and 'L'' 'RL'' respectively). Let ''n'' be the number of binding sites for ligand on each receptor molecule, and let represent the average number of ligands bound to a receptor. Let ''K_d'' denote the

dissociation constant In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex ...

between the ligand and receptor. The Scatchard equation is given by :

\frac = \frac - \frac

By plotting / 'L''versus , the Scatchard plot shows that the slope equals to -1/''K_d'' while the x-intercept equals the number of ligand binding sites ''n''.

Derivation

''n''=1 Ligand

When each receptor has a single ligand binding site, the system is described by :

+ \underset L

with an on-rate (''k''_on) and off-rate (''k''_off) related to the dissociation constant through ''K_d''=''k''_off/''k''_on. When the system equilibrates, :

k_= k_L

so that the average number of ligands bound to each receptor is given by :

\bar = \frac = \frac = (1 - \bar) \frac

which is the Scatchard equation for ''n''=1.

''n''=2 Ligands

When each receptor has two ligand binding sites, the system is governed by :

+ \underset L

L + \underset L_2

At equilibrium, the average number of ligands bound to each receptor is given by :

\bar = \frac = \frac = \frac = (2 - \bar) \frac

which is equivalent to the Scatchard equation.

General Case of ''n'' Ligands

For a receptor with ''n'' binding sites that independently bind to the ligand, each binding site will have an average occupancy of 'L''(''K_d'' + 'L''. Hence, by considering all ''n'' binding sites, there will :

\bar = n \frac = (n - \bar) \frac.

ligands bound to each receptor on average, from which the Scatchard equation follows.

Problems with the method

The Scatchard method is less used nowadays because of the availability of computer programs that directly fit parameters to binding data. Mathematically, the Scatchard equation is related to Eadie-Hofstee method, which is used to infer kinetic properties from enzyme reaction data. Many modern methods for measuring binding such as

surface plasmon resonance Surface plasmon resonance (SPR) is the resonant oscillation of conduction electrons at the interface between negative and positive permittivity material in a particle stimulated by incident light. SPR is the basis of many standard tools for measu ...

and

isothermal titration calorimetry Isothermal titration calorimetry (ITC) is a physical technique used to determine the thermodynamic parameters of interactions in solution. It is most often used to study the binding of small molecules (such as medicinal compounds) to larger macro ...

provide additional binding parameters that are globally fit by computer-based iterative methods.

References

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