Binding and thermodynamics

Why it matters

Almost everything in biology works by molecules binding each other: hormones to receptors, antibodies to antigens, transcription factors to DNA, drugs to targets. The strength of that binding, the affinity, decides how much complex forms at a given concentration. Drug programs rank compounds by affinity, antibody engineers optimise it, and biophysicists ask what drives it: is the binding powered by favourable contacts (enthalpy), or by released water and flexibility (entropy)? This lesson covers the dissociation constant Kd, how to estimate it from a binding curve, and how it links to thermodynamics.

The dissociation constant

For a protein P binding a ligand L to form a complex PL:

P + L  ⇌  PL          Kd = [P][L] / [PL]

Kd has units of concentration. A smaller Kd means tighter binding: an antibody might have Kd around 1 nM, a weak fragment hit around 1 mM. Kd is also the ratio of rate constants, Kd = k_off / k_on, so tight binders usually dissociate slowly.

When the ligand is in large excess over the protein (so binding does not deplete free ligand), the fraction of protein bound follows a hyperbola, like Michaelis–Menten:

θ = [L] / (Kd + [L])

At [L] = Kd, half the protein is bound. To go from 10 % to 90 % bound, [L] must rise about 81-fold (from Kd/9 to 9·Kd), so binding curves are usually plotted against log [L], where they look sigmoidal.

Measuring binding

• Fluorescence (intrinsic tryptophan fluorescence, fluorescence anisotropy with a labelled ligand), surface plasmon resonance (SPR), biolayer interferometry, microscale thermophoresis and isothermal titration calorimetry (ITC) all give binding curves.
• SPR also measures k_on and k_off directly.
• ITC measures the heat released or absorbed at each injection of ligand into protein. One experiment gives Kd, the binding enthalpy ΔH and the stoichiometry n, with no labels needed.

Worked example: fitting Kd

A fluorescence titration gives fraction bound at several ligand concentrations (µM), with the protein concentration far below Kd:

[L] (µM):  0.05   0.1    0.2    0.5    1      2      5      10     20
θ:         0.060  0.124  0.191  0.395  0.553  0.712  0.881  0.928  0.961

Half-binding falls between 0.5 and 1 µM. A least-squares fit of θ = [L]/(Kd + [L]) gives Kd ≈ 0.79 µM. Check: at 1 µM, the model predicts 1/(0.79 + 1) ≈ 0.56, close to the measured 0.553.

From Kd to free energy

The standard free energy of binding uses Kd relative to a 1 M standard state:

ΔG° = RT · ln(Kd / 1 M)

with R = 8.314 J/(mol·K) and T in kelvin. For Kd = 0.79 µM at 25 °C (298.15 K):

ΔG° = 8.314 × 298.15 × ln(0.79 × 10⁻⁶) ≈ −34.8 kJ/mol

A negative ΔG° means binding is favourable. Every tenfold improvement in Kd at 25 °C is worth about 5.7 kJ/mol (RT·ln 10).

ITC also gives ΔH directly, so the entropy term follows:

ΔG° = ΔH° − T·ΔS°        →   −T·ΔS° = ΔG° − ΔH°

If ΔH° = −50 kJ/mol, then −TΔS° = −34.8 − (−50) = +15.2 kJ/mol: binding is enthalpy driven and pays an entropy cost (for example, from freezing flexible groups). Two compounds with the same Kd can have very different enthalpy/entropy balances, which medicinal chemists use to guide optimisation.

Common mistakes

• Ignoring ligand depletion. If the protein concentration is near or above Kd, free [L] is not the total [L] added, and the simple hyperbola gives the wrong Kd. Use the quadratic binding equation instead.
• Not reaching saturation. Without points well above Kd, the fit cannot pin down the plateau.
• Mixing units in the ΔG° formula: Kd must be in molar (M) units there.
• Using °C for T. Use kelvin.
• Confusing Kd and Ka. The association constant Ka = 1/Kd; larger Ka means tighter binding.

Recap

• Kd = [P][L]/[PL]; smaller Kd is tighter binding; θ = [L]/(Kd + [L]) when ligand is in excess.
• Fit Kd by least squares on the untransformed curve.
• ΔG° = RT ln(Kd) (Kd in M); ITC adds ΔH and hence TΔS.

Try it

On the Do tab, write the fraction-bound equation in Python, fit Kd to the titration data and convert it to ΔG°.