In silico prediction of solubility: Solid progress but no solution? Dr John Mitchell University of St Andrews

Given accurately measured solubilities of 100 molecules, can you predict the solubilities of 32 similar ones?

For this study Toni Llinàs measured 132 solubilities using the CheqSol method. He used a Sirius glpKa instrument

K0K0 KaKa Intrinsic solubility- Of an ionisable compound is the thermodynamic solubility of the free acid or base form (Horter, D, Dressman, J. B., Adv. Drug Deliv. Rev., 1997, 25, 3-14) A NaA - ………. Na + AH S 0 is essentially the solubility of the neutral form only.

Diclofenac In Solution Powder ● We continue “Chasing equilibrium” until a specified number of crossing points have been reached ● A crossing point represents the moment when the solution switches from a saturated solution to a subsaturated solution; no change in pH, gradient zero, no re-dissolving nor precipitating…. SOLUTION IS IN EQUILIBRIUM Random error less than 0.05 log units !!!! Supersaturated Solution Subsaturated Solution 8 Intrinsic solubility values * A. Llinàs, J. C. Burley, K. J. Box, R. C. Glen and J. M. Goodman. Diclofenac solubility: independent determination of the intrinsic solubility of three crystal forms. J. Med. Chem. 2007, 50(5), ● First precipitation – Kinetic Solubility (Not in Equilibrium) ● Thermodynamic Solubility through “Chasing Equilibrium”- Intrinsic Solubility (In Equilibrium) Supersaturation Factor SSF = S kin – S 0 “CheqSol”

Caveat: the official results are used in the following slides, but most of the interpretation is my own.

A prediction was considered correct if it was within 0.5 log units

Not a very generous margin of error!

A “null prediction” based on predicting everything to have the mean training set solubility would have got 9/32 correct

Using an R 2 threshold of 0.500, only 18/99 entries were good

GOOD BAD

3 “WINNERS” 3 Pareto optimal entries which I think of as “winners”. These combine best R 2 with most correct predictions.

Some molecules proved much harder to predict than others – the most insoluble were amongst the most difficult.

My opinion is that the overall standard was rather poor; It’s obvious that some entries were much better than others; But entries were anonymous; So we can’t judge between either specific researchers or between their methods; We can only rely on the “official” summary … Conclusions from Solubility Challenge

We can only rely on the “official” summary … … “a variety of methods and combinations of methods all perform about equally well.” Conclusions from Solubility Challenge

How should we approach the prediction/estimation/calculation of the aqueous solubility of druglike molecules? Two (apparently) fundamentally different approaches: theoretical chemistry & informatics.

What Error is Acceptable? For typically diverse sets of druglike molecules, a “good” QSPR will have an RMSE ≈ 0.7 logS units. A RMSE > 1.0 logS unit is probably unacceptable. This corresponds to an error range of 4.0 to 5.7 kJ/mol in  G sol.

What Error is Acceptable? A useless model would have an RMSE close to the SD of the test set logS values: ~ 1.4 logS units; The best possible model would have an RMSE close to the SD resulting from the experimental error in the underlying data: ~ 0.5 logS units?

Theoretical Approaches

Theoretical Chemistry “The problem is difficult, but by making suitable approximations we can solve it at reasonable cost based on our understanding of physics and chemistry”

Theoretical Chemistry Calculations and simulations based on real physics. Calculations are either quantum mechanical or use parameters derived from quantum mechanics. Attempt to model or simulate reality. Usually Low Throughput.

Drug Disc.Today, 10 (4), 289 (2005)

Thermodynamic Cycle

Crystal Gas Solution

Sublimation Free Energy Crystal Gas

Sublimation Free Energy Crystal Gas

Sublimation Free Energy Crystal Gas

Sublimation Free Energy Crystal Gas Calculating  G sub is a standard procedure in crystal structure prediction

Crystal Structure Prediction Given the structural diagram of an organic molecule, predict the 3D crystal structure. Slide after SL Price, Int. Sch. Crystallography, Erice, 2004

CSP Methodology Based around minimising lattice energy of trial structures. Enthalpy comes from lattice energy and intramolecular energy (DFT), which need to be well calibrated against each other: trade-off of lattice vs conformational energy. Entropy comes from phonon modes (crystal vibrations); can get Free Energy.

CSP Methodology DFT calculation on monomer to obtain DMA electrostatics. Generate many plausible crystal structures using different space groups. Minimise lattice energy using DMA + repulsion-dispersion potential. Many structures may have similar energies.

34 These methods can get relative lattice energies of different structures correct, probably to within a few kJ/mol. Absolute energies harder.

35 Additional possible benefit for solubility: if we don’t know the crystal structure, we could reasonably use best structure from CSP.

Other approaches to Lattice Energy Periodic DFT calculations on a lattice are an alternative to the model potential approach. Advantageous to optimise intra- and intermolecular energies simultaneously using the same method. Disadvantage: it’s hard to get accurate dispersion.

Empirical routes to  G sub Alternatively one could estimate sublimation energy from QSPR (no crystal structure needed, but no obvious benefit over direct informatics approach to solubility).

Thermodynamic Cycle Crystal Gas Solution

Hydration Free Energy

We expect that hydration will be harder to model than sublimation, because the solution has an inexactly known and dynamic structure, both solute and solvent are important etc.

Simulation: MD/FEP Parameterised continuum models

… and of course the parameterised RISM work of our hosts. Quoted RMS error ~5kJ/mol or 0.9 log units.

… and this one both calculates solubility directly and is simulation based: FEP or Monte Carlo.

Luder et al.’s results correspond to an RMS error of about 6kJ/mol, or 1 logS unit, but only when an empirical “correction” is applied ….

… their uncorrected results are less impressive.

Hydration Energy Our currrent methodology here is just to try the various different PCM continuum models available in Gaussian.

We observe than our TD cycle method based on lattice energy minimisation for sublimation and a PCM continuum model of hydration correlates reasonably with experiment, but is not quantitatively predictive (at least without arbitrary correction). Caveat: currently only a small sample of molecules.

An alternative route is via octanol, then using logP.

Theoretical  Approaches ish So really these are

Using a training-test set split to optimise parameters & validate: RMSE(te)=0.71 r 2 (te)=0.77 N train = 34; N test = 26

Informatics Approaches

Informatics “The problem is too difficult to solve using physics and chemistry, so we will design a black box to link structure and solubility”

Informatics and Empirical Models In general, Informatics methods represent phenomena mathematically, but not in a physics-based way. Inputs and output model are based on an empirically parameterised equation or more elaborate mathematical model. Do not attempt to simulate reality. Usually High Throughput.

Machine Learning Method Random Forest

Random Forest: Solubility Results RMSE(te)=0.69 r 2 (te)=0.89 Bias(te)=-0.04 RMSE(tr)=0.27 r 2 (tr)=0.98 Bias(tr)=0.005 RMSE(oob)=0.68 r 2 (oob)=0.90 Bias(oob)=0.01 DS Palmer et al., J. Chem. Inf. Model., 47, (2007) N train = 658; N test = 300

Random Forest: Replicating Solubility Challenge (post hoc) RMSE(te)=1.09 r 2 (te)= /32 correct within 0.5 logS units N train  100; N test  32 CDK descriptors

Support Vector Machine

SVM: Solubility Results et al., N train = ; N test = 87 RMSE(te)=0.94 r 2 (te)=0.79

What can we Learn from Informatics?

What Descriptors Correlate with logS? …amongst the solubility challenge training set, once intercorrelated descriptors with R 2 > 0.64 are removed?

The first 21 are all negatively correlated with logS … … things that reduce solubility.

The first 21 are all negatively correlated with logS … … things that reduce solubility. Some of this is meaningful: aromatic groups reduce solubility. Some is accidental: logP happens to be defined as octanol:water, rather than water:octanol.

Future Work Explore different models of hydration: PCM, simulation (MD/FEP), RISM … Route: Direct to water or via octanol? Machine Learning (Random Forest, SVM etc.) for hybrid experimental/parameterised models. Consistent training and validation sets and methodologies to compare methods: e.g., solubility challenge {100+32}.

Conclusions thus far…

Solubility has proved a difficult property to calculate. It involves different phases (solid & solution) and different substances (solute and solvent), and both enthalpy & entropy are important. The theoretical approaches are generally based around thermodynamic cycles and involve some empirical element.

Thanks Pfizer & PIPMS Gates Cambridge Trust SULSA Dr Dave Palmer, Laura Hughes, Dr Toni Llinas James McDonagh, Dr Tanja van Mourik James Taylor, Simon Hogan, Gregor McInnes, Callum Kirk, William Walton (U/G project)