Kd ↔ ΔG + ligand efficiency

The relationship between K_d and ΔG

For a 1:1 complex, the standard binding free energy is related to the dissociation constant by

ΔG°bind = RT ln(Kd / c°)

where R = 8.314462618 J mol⁻¹ K⁻¹, T is the absolute temperature, and c° = 1 M is the standard concentration. The division by c° is often left out of textbook formulas. It matters because it makes the logarithm dimensionless, and it means every ΔG° you calculate refers to a 1 M standard state. A different standard state shifts all values by a constant (Gilson et al., 1997). Because K_d is below 1 M for any real binder, ΔG° is negative, and tighter binding means more negative ΔG°.

A handy rule of thumb: at 25 °C, RT ln 10 = 1.364 kcal/mol (5.708 kJ/mol). Every tenfold improvement in affinity is therefore worth about 1.36 kcal/mol. One kcal/mol is worth a factor of about 5.4 in K_d. Keep this in mind when you read docking or MM-PBSA results: a reported "improvement" of 0.5 kcal/mol corresponds to less than a factor of 2.5 in affinity, which is well inside the error of most methods.

Ligand efficiency metrics

• Ligand efficiency (LE) = −ΔG° / N_heavy, in kcal/mol per heavy atom (Hopkins et al., 2004). It normalises affinity by size. Values around 0.3 kcal/mol/HA or higher are commonly treated as a good starting point for a lead, and fragments often start higher. LE depends on size, so compare it only between compounds of similar size.
• Lipophilic ligand efficiency (LLE or LipE) = pK_d (or pIC50) − cLogP (Leeson & Springthorpe, 2007). It asks whether potency is being bought with lipophilicity, which tends to hurt solubility, selectivity and safety. LLE ≥ 5 is often quoted as desirable for oral drug candidates.
• Binding efficiency index (BEI) = pK_d / MW(kDa) (Abad-Zapatero & Metz, 2005). It is similar in spirit to LE but uses molecular weight.

LLE and BEI are computed from pK_d, which is −log₁₀(K_d/M). In ΔG → K_d mode, the K_d is first calculated from your ΔG.

Worked example

A compound with K_d = 10 nM, measured at 25 °C, with 25 heavy atoms, cLogP 3 and MW 350 Da:

• ΔG° = (0.0019872 kcal mol⁻¹ K⁻¹)(298.15 K) ln(1 × 10⁻⁸) = −10.91 kcal/mol (−45.66 kJ/mol).
• At 37 °C, the same K_d gives −11.35 kcal/mol. The temperature enters through RT, so always state the temperature, and avoid mixing data measured at 25 °C and 37 °C.
• LE = 10.91 / 25 = 0.44 kcal/mol/HA.
• pK_d = 8, so LLE = 8 − 3 = 5.0 and BEI = 8 / 0.350 = 22.9.

These are the defaults loaded in the calculator above. Change any value and the results update immediately.

Assumptions and caveats

• K_i and IC50 are not K_d. A K_i from a properly analysed competitive assay approximates K_d. An IC50 depends on the assay conditions, so convert it first with the IC50 → Ki converter. Free energies computed from raw IC50 values are, at best, "apparent".
• 1:1 binding at equilibrium. The formula assumes a single site and reversible binding. It does not apply to covalent inhibitors or cooperative systems.
• ΔG is not ΔH. The free energy mixes enthalpy and entropy. To split them you need ITC or a van 't Hoff analysis. Do not compare a docking score with ΔH.
• Comparing with calculations. Docking scores are in "kcal/mol" units, but they are empirical scoring functions, not free energies. MM-PBSA values usually lack a full entropy term and a standard-state correction, so they are often far more negative than experiment. Compare rankings and correlations, not absolute numbers. See the gmx_MMPBSA tutorial.
• Heavy-atom counts must be whole, positive numbers. Count every non-hydrogen atom, including halogens.

References

1. Gilson MK, Given JA, Bush BL, McCammon JA. The statistical-thermodynamic basis for computation of binding affinities: a critical review. Biophys J 1997;72:1047–1069. doi:10.1016/S0006-3495(97)78756-3
2. Hopkins AL, Groom CR, Alex A. Ligand efficiency: a useful metric for lead selection. Drug Discov Today 2004;9:430–431. doi:10.1016/S1359-6446(04)03069-7
3. Leeson PD, Springthorpe B. The influence of drug-like concepts on decision-making in medicinal chemistry. Nat Rev Drug Discov 2007;6:881–890. doi:10.1038/nrd2445
4. Abad-Zapatero C, Metz JT. Ligand efficiency indices as guideposts for drug discovery. Drug Discov Today 2005;10:464–469. doi:10.1016/S1359-6446(05)03386-6
5. Hopkins AL, Keserü GM, Leeson PD, Rees DC, Reynolds CH. The role of ligand efficiency metrics in drug discovery. Nat Rev Drug Discov 2014;13:105–121. doi:10.1038/nrd4163

Results are for research and education and are not a substitute for checking against the primary literature. See the disclaimer. Found a bug? Report it.