From interaction to number: free energy and Kd

You now know what holds a ligand in a pocket. This module turns that into a number, and then tells you exactly what a docking score is and is not comparable to. It is the point where the chemistry of Part 4 becomes something you can put on an axis.

Binding is favourable when the free energy change is negative:

ΔG° = ΔH° − TΔS°

ΔH°, the enthalpy, is the interactions: hydrogen bonds, van der Waals contacts, electrostatics — minus the desolvation those cost. ΔS°, the entropy, is freedom: the ligand and protein lose translational, rotational and conformational freedom on binding, which opposes it, while released water gains entropy, which favours it.

Every term you met in modules 14 to 18 lands in one of those two buckets, and several land in both.

The link between the energy and the measurement is:

ΔG° = RT ln K_d

with K_d in molar. R is 1.987 × 10⁻³ kcal mol⁻¹ K⁻¹, and at 298 K, RT is about 0.593 kcal/mol. A tighter binder has a smaller K_d and a more negative ΔG°.

One log unit of affinity is 2.303 RT, which at room temperature is 1.36 kcal/mol. Learn this number; it is the exchange rate between the two currencies of this course.

Its implications are sobering, and they explain a great deal about why drug discovery is hard.

• A ten-fold gain in potency is 1.36 kcal/mol — less than a single hydrogen bond's gas-phase strength, and roughly what a couple of good van der Waals contacts give you.
• Going from 1 µM to 1 nM — a thousand-fold, a whole medicinal chemistry programme — is about 4 kcal/mol. That is a very small energy: about the strength of one hydrogen bond in the gas phase, or roughly the error bar on a good free energy calculation.
• Conversely, a small modelling error is a large potency error. Get the energy wrong by 2.7 kcal/mol and you have mispredicted potency by a hundredfold.

This is why scoring functions struggle. They are being asked to resolve differences of one or two kcal/mol between large numbers — the total interaction and desolvation energies are each tens of kcal/mol — and the quantity of interest is the small difference between them.

These are not interchangeable, and confusing them is one of the commonest errors in the literature.

• K_d — the dissociation constant for a binding equilibrium. A thermodynamic property of the ligand-protein pair. Measured directly by calorimetry or surface plasmon resonance.
• K_i — the inhibition constant. Essentially a K_d measured in a functional assay, corrected for the assay conditions.
• IC50 — the concentration giving 50% inhibition in a particular assay. It is not a property of the molecule alone.
• EC50 — the concentration giving 50% of maximal effect in a functional or cellular assay. Even further from a binding constant, since it folds in permeability, efflux and the shape of the response.

The critical distinction: an IC50 depends on the assay and a K_i does not. Run the same competitive inhibitor against a higher substrate concentration and the IC50 rises, because the inhibitor is competing harder. The K_i is unchanged, because it describes the inhibitor and the enzyme.

The correction that converts one into the other, for a competitive inhibitor:

K_i = IC50 / (1 + [S]/K_m)

So at a substrate concentration equal to K_m, the K_i is half the IC50. At ten times K_m, it is a eleventh. This is why comparing IC50 values between papers is unsafe unless both used the same substrate concentration — and very often the papers do not say.

An awkward empirical observation: when you improve the enthalpy of binding, the entropy often gets worse by a similar amount, so ΔG barely moves.

The physical intuition is reasonable. A tighter, more specific interaction holds the ligand more rigidly, which costs conformational entropy. Add a hydrogen bond and you gain enthalpy and lose freedom.

Some of the reported compensation is real and some is an artefact of how ΔH and ΔS are extracted from the same measurement, with correlated errors. Both views have serious support. The practical lesson survives either way: optimising a single thermodynamic term rarely delivers what you expect, and ΔG is the quantity that matters.

K_d is a ratio of rates: K_d = k_off / k_on. The same K_d can come from fast-on-fast-off or slow-on-slow-off, and those are not the same drug.

Residence time, 1/k_off, is how long the complex survives. In the body, drug concentration falls as the drug is cleared — so a slow off-rate can keep the target occupied long after the plasma concentration has dropped. For targets where duration of effect matters, residence time can matter more than K_d.

This is now routinely measured, and it is one of the clearest cases where a single affinity number hides something clinically important.

Everything above has been building to this.

A docking score is an empirical function, fitted to reproduce a ranking on a training set of complexes. It is typically a weighted sum of terms — some hydrogen bond counting, a hydrophobic contact term, a torsion penalty, sometimes an electrostatic term — with the weights regressed against known affinities.

It is reported in energy-like units, often kcal/mol. It is not a binding free energy. Specifically:

• It usually does not include ligand strain — the module 8 penalty.
• It handles desolvation crudely or not at all — so modules 14, 16 and 17 are all approximated badly.
• It treats entropy with a rotatable bond count, at best.
• It has no water in it, usually.
• It cannot represent a sigma hole, as module 18 showed.

So what is it for? Enrichment. A docking score is reasonably good at separating things that might bind from things that will not, which is exactly what a virtual screen needs. It is poor at ranking actives against each other, and it should never be reported as a predicted affinity.

The honest way to speak: "this compound docks with a score of −9.2, which places it in the top 2% of the library" — not "this compound is predicted to bind with ΔG of −9.2 kcal/mol". The first is a claim about a ranking. The second is a claim about thermodynamics, and the score does not support it.