Kd(AB) = Kd(A) * Kd(B) * E
Kd(AB) is the dissociation constant for the linked molecule AB
Kd(A) is the dissociation constant for fragment A
Kd(B) is the dissociation constant for fragment B
E is a “linking coefficient”, reflecting the costs and benefits of linking
The lower the Kd the better, so ideally E < 1, though in practice finding a suitable linker can be tricky and all too often E > 1 (sometimes >> 1). But how low can E go? How much of a boost can you get by linking two fragments? Claudio Luchinat and colleagues at the University of Florence looked at this question experimentally in a recent paper in J. Med. Chem.
The researchers took PMAHA, a known inhibitor of the matrix metalloproteinase MMP-12, and dissected it into two fragments, AHA and PMS, by conceptually “cleaving” the bond connecting them (see figure). This simplifies analysis: since the two fragments are almost identical to the linked molecule, there are no concerns that atoms in the linker interact with the protein.
The crystal structure of PMAHA bound to MMP-12 had been previously reported, but Luchinat and co-workers solved the co-crystal structure of AHA and PMS bound simultaneously to MMP-12. The two fragments overlay fairly well with the parent molecule: AHA binds to the catalytic zinc, while PMS binds in the S1’ pocket. The AHA fragment is rotated with respect to its position in PMAHA, though it makes the same interactions in both structures.
Thermodynamic binding parameters for the three molecules were determined (see figure). As expected, PMAHA binds considerably more tightly than the product of the affinities of the two fragments: E << 1 (in fact, about 0.0021). And in nice accord with theory, this enhanced affinity is entropic: both fragments bind with favorable enthalpy and unfavorable entropy, while the linked molecule has both favorable enthalpy and entropy. In other words, the salutary effect of linking these two fragments does seem to come entirely from entropic effects.
One of the more interesting lessons from this paper is a sense of how much of a boost in potency you can expect if fragment linking goes well: about 500-fold. In theory you could do better, but in practice you should expect much more modest benefits: a prominent success of SAR by NMR on a different metalloproteinase reported a 14-fold boost in affinity. But just like the lottery, the hope of a big payout will continue to attract people to the linking game.