Scatchard equation

Key takeaways

Scatchard equation belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Scatchard equation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Scatchard equation from memory before moving on to harder problems.

Reference excerpt

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. 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 n represent the average number of ligands bound to a receptor. Let Kd denote the dissociation constant between the ligand and receptor. The Scatchard equation is given by

n ¯ [ L ] = n K d − n ¯ K d {\displaystyle {\frac {\bar {n}}{[L]}}={\frac {n}{K_{d}}}-{\frac {\bar {n}}{K_{d}}}}

By plotting n/[L] versus n, the Scatchard plot shows that the slope equals to -1/Kd 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

[ R ] + [ L ] ⇌ k on k off [ R L ] {\displaystyle [R]+[L]{\underset {k_{\text{off}}}{\overset {k_{\text{on}}}{\rightleftharpoons }}}[RL]}

with an on-rate (kon) and off-rate (koff) related to the dissociation constant through Kd=koff/kon. When the system equilibrates,

k on [ R ] [ L ] = k off [ R L ] {\displaystyle k_{\text{on}}[R][L]=k_{\text{off}}[RL]}

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

n ¯ = [ R L ] [ R ] + [ R L ] = [ L ] K d + [ L ] = ( 1 − n ¯ ) [ L ] K d {\displaystyle {\bar {n}}={\frac {[RL]}{[R]+[RL]}}={\frac {[L]}{K_{d}+[L]}}=(1-{\bar {n}}){\frac {[L]}{K_{d}}}}

which is the Scatchard equation for n=1.

n=2 Ligands When each receptor has two ligand binding sites, the system is governed by

[ R ] + [ L ] ⇌ 2 k on k off [ R L ] {\displaystyle [R]+[L]{\underset {k_{\text{off}}}{\overset {2k_{\text{on}}}{\rightleftharpoons }}}[RL]}

[ R L ] + [ L ] ⇌ k on 2 k off [ R L 2 ] . {\displaystyle [RL]+[L]{\underset {2k_{\text{off}}}{\overset {k_{\text{on}}}{\rightleftharpoons }}}[RL_{2}].}

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

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Scatchard equation

Start with the simplest possible case. Write down what Scatchard equation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Scatchard equation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Scatchard equation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Scatchard equation

In research: Scatchard equation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Scatchard equation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Scatchard equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Biochemistry methods, Proteins, so understanding it makes those chapters shorter.
In everyday life: Look for Scatchard equation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Scatchard equation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Scatchard equation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Scatchard equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Scatchard equation in simple terms?

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. It is named after the American chemist George Scatchard.

Why does Scatchard equation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Scatchard equation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Scatchard equation.