Van der Waals contacts and shape complementarity

Hydrogen bonds get the attention. Van der Waals contacts do the work. Each one is worth a fraction of a kcal/mol, and a well-fitting ligand makes dozens — which is why potency tracks buried surface area more closely than it tracks any specific polar interaction.

The van der Waals interaction between two atoms is the sum of an attraction and a repulsion, and they operate over different ranges.

London dispersion is the attraction. Electrons move; at any instant an atom has a small, fluctuating dipole; that dipole induces a matching one in a neighbouring atom; the two attract. It is present between all atoms, polar or not, and it is the only attractive force available between two non-polar groups. It falls off as 1/r⁶ — very steeply, so it is a contact interaction with essentially no reach.

Pauli repulsion is the barrier. Two filled electron clouds cannot occupy the same space. It rises as 1/r¹² in the usual approximation — even more steeply — so it is negligible until the atoms touch and then it is overwhelming.

Put them together and you get the Lennard-Jones form: a shallow well at the contact distance, with a soft outer wall and an almost vertical inner one. That asymmetry matters for docking. Being slightly too far apart costs little; being slightly too close costs enormously.

Every atom has a van der Waals radius — the distance at which repulsion starts to bite. Carbon is about 1.7 Å, nitrogen 1.55, oxygen 1.52, sulfur 1.8, chlorine 1.75.

A good contact is at roughly the sum of the two radii: carbon against carbon at about 3.4 to 4.0 Å. Meaningfully closer than the sum is a clash, and by convention a clash is flagged when the distance falls more than about 0.4 to 0.5 Å below the sum.

This is worth internalising as a number you can check by eye in a viewer. Two carbons at 3.6 Å are packed nicely. At 3.0 Å something is wrong with the structure or the pose.

The natural measure of how well two things pack is how much surface stops being exposed to water when they come together.

Solvent accessible surface area is traced by rolling a water-sized probe, 1.4 Å in radius, over the molecule. Compute it for the free ligand, for the free protein, and for the complex; the amount that disappears is the buried surface area.

For a typical drug-sized ligand this is a few hundred square ångströms. The empirical relationship — and it is empirical, not derived — is roughly 25 to 30 cal/mol per Ų of buried non-polar surface. Bury 300 Ų of hydrophobic surface and you have collected something like 7 to 9 kcal/mol, which is a very substantial fraction of any drug's binding energy.

Shape complementarity asks not just how much surface is buried but how well the two surfaces match. A ligand that occupies the middle of a pocket while leaving gaps around it buries surface without making good contacts — and worse, the gaps are usually left with trapped, badly-solvated water.

This is the difference between a ligand that fills a pocket and one that merely sits in it. Scoring functions capture it poorly, because a pairwise-additive term counts contacts and does not notice a void. When you inspect a pose, look for gaps.

Here is the arithmetic that makes this module matter.

A single van der Waals contact is worth perhaps 0.1 to 0.5 kcal/mol. That is nothing. But a drug-sized ligand in a well-fitting pocket makes dozens of them, and unlike hydrogen bonds they carry almost no desolvation penalty — stripping water off a non-polar surface is cheap, and in fact releasing that water is favourable, which is module 17's subject.

So twenty contacts at 0.3 kcal/mol each is 6 kcal/mol, net, with no offset. That comfortably exceeds what any single hydrogen bond will net you after desolvation.

Two consequences for design, both of which run against intuition:

• Potency often tracks buried surface area more closely than any specific polar contact. If you plot affinity against buried non-polar surface for a congeneric series, you frequently get a decent line. Plot it against hydrogen bond count and you usually do not.
• Small hydrophobic additions are reliable. Adding a methyl into a pocket that has room for it typically gains a few-fold in potency. This is the famous "magic methyl", and most of the time it is not magic at all — it is dispersion, in a place with space for it.

The caution, which module 22 will make quantitative: this is also exactly how the lipophilicity trap operates. Adding hydrophobic bulk reliably improves potency and reliably degrades everything else. The van der Waals argument explains why the trap is so effective, not why you should walk into it.