AFFINITY

How tightly does it bind?

K_D = k_off / k_on

The dissociation constant K_D is the concentration at which half the receptors hold a target. It equals the off rate divided by the on rate, so a smaller K_D means tighter binding. Antibody–antigen pairs often have K_D values in the nanomolar range.

Small K_D, strong binding.

THE LANGMUIR ISOTHERM

Fraction bound versus concentration

θ = c / (c + K_D)

At equilibrium, the fraction of receptors occupied is θ = c/(c + K_D), where c is the target concentration. At c = K_D, half are bound. At c = 10 K_D, about 91% are bound. At c = K_D/10, only about 9%.

Concentration relative to K_D sets the response.

THE BINDING CURVE

A curve that saturates

The response climbs steeply at low concentration and flattens at high concentration. Most useful measurements fall between about 0.1 and 10 times K_D. Far below that, there is too little signal, and far above it the receptors are nearly full.

The sensor works best near its K_D.

SPEED

How fast does it settle?

τ = 1 / (k_on·c + k_off)

The signal approaches equilibrium exponentially with time constant τ = 1/(k_on·c + k_off). At low concentration τ is roughly 1/k_off, which can be minutes or hours for tight binders. So measuring very small concentrations takes patience.

Low concentration means a slow response.

WORKED EXAMPLE

How much is bound?

c = 5 nM, K_D = 1 nM: θ ≈ 0.83

With c = 5 nM and K_D = 1 nM, θ = 5/(5 + 1) ≈ 0.83, so 83% of the receptors hold a target. If a full layer would shift Vth by 60 mV, then 83% coverage gives about 50 mV.

The response scales with the fraction of receptors occupied.

UNIT 18 STUDY COMPLETE

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Ready for the Fab Challenge?

You've covered how much target binds at a given concentration, and how fast.