Quantum Chemical and Machine Learning Calculations of the Intrinsic Aqueous Solubility of Druglike Molecules Dr John Mitchell University of St Andrews

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

The Two Faces of Computational Chemistry Theoretical Chemistry Informatics

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)

Existing Theoretical Approaches Thus far, although theoretical methods have shown promise, they have not matched the accuracy of QSPR. There is no theoretical method that deals directly with solubility, so the problem has to be broken down into parts. There are several different ways of doing this.

Our First Principles Method We present one such approach and believe this to be the world’s most cost- effective first principles solubility method.

Thermodynamic Cycle

 G sub from lattice energy minimisation  G hydr from Reference Interaction Site Model (RISM) Different kinds of theoretical method are used for each part

 G sub from lattice energy & a phonon entropy term; DMACRYS using B3LYP/6-31G(d,p) multipoles and FIT repulsion-dispersion potential.  G hydr from Reference Interaction Site Model with Universal Correction (RISM/UC). Different kinds of theoretical method are used for each part

OUR DATASET (25 molecules)

We have experimental logS for all 25 molecules, but can only subdivide into ΔG sub and ΔG hydr for 10 of them.

Thermodynamic Cycle Crystal Gas Solution

Sublimation Free Energy Crystal Gas

Sublimation Free Energy Crystal Gas

Sublimation Free Energy Crystal Gas Calculating Δ 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.

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

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

Lattice energies from DMACRYS with FIT atom-atom model potential and B3LYP/6-31G(d,p) distributed multipoles. Results for ΔG sub

Reasonable prediction of ΔG sub, but small number of molecules. Results for ΔG sub

To see the trends in errors, we need to look at more molecules. RMSE = 20.4 kJ/mol (46 molecules)

The 46 compound set shown here has a larger error, mostly due to some large outliers. Error statistics vary with dataset. RMSE = 20.4 kJ/mol (46 molecules)

RMSE = 22.4 kJ/mol (46 molecules)

The predicted ΔH sub is much better correlated with experiment than is TΔS sub. However, ΔH sub has a much larger range of values and contributes more to the RMS error.

Thermodynamic Cycle Crystal Gas Solution

Hydration Free Energy We expected that hydration would be harder to model than sublimation, because the solution has an inexactly known and dynamic structure, both solute and solvent are important etc.

Reference Interaction Site Model (RISM) Combines features of explicit and implicit solvent models. Solvent density is modelled, but no explicit molecular coordinates or dynamics. ~45 CPU mins per compound

RISM

Reference Interaction Site Model (RISM) Palmer, D.S., et al., Accurate calculations of the hydration free energies of druglike molecules using the reference interaction site model. The Journal of Chemical Physics, (4): p

Perhaps surprisingly, error in  G hyd is smaller than in  G sub. Results for ΔG hyd

Other Hydration Energy Approaches An alternative methodology here is just to try the various different continuum solvent models available in Gaussian.

logS from Thermodynamic Cycle Crystal Gas Solution Add the two terms to get ΔG sol and hence logS.

Results for ΔG sol

Conclusions: Theory Must calculate  G sub &  G hyd separately; Expt data sparse and errors may be large; RISM is efficient & fairly accurate for  G hyd ; Dataset size and composition make comparisons of methods hard; Not yet matched accuracy of informatics.

Informatics Approaches “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.

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?

Machine Learning Method Random Forest

Random Forest: Solubility Results RMSE(te)=0.69 r 2 (te)=0.89 Bias(te)=-0.04 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

Support Vector Machine

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

100 Compound Cross-Validation Theoretical energies don’t seem to improve descriptor models.

100 Compound Cross-Validation McDonagh et al., J Chem Inf Model, 54, 844 (2014)

Replicating Solubility Challenge (post hoc) McDonagh et al., J Chem Inf Model, 54, 844 (2014)

Replicating Solubility Challenge (post hoc) RMSE(te)=1.00; 0.89; 1.08 r 2 (te)= 0.49; 0.58; ; 12; 13/28 correct within 0.5 logS units N train = 94; N test = 28 CDK descriptors: RF, PLS, SVM

Replicating Solubility Challenge (post hoc) N train = 94; N test = 28 CDK descriptors: RF, PLS, SVM Although the test dataset is small, it is a standard set.

Conclusions: Informatics Expt data: errors unknown, but limit possible accuracy of models; CheqSol - step in right direction; Dataset size and composition hinder comparisons of methods; Solubility Challenge – step in right direction.

Thanks SULSA James McDonagh, Dr Tanja van Mourik, Neetika Nath (St Andrews) Prof. Maxim Fedorov, Dr Dave Palmer (Strathclyde) Laura Hughes, Dr Toni Llinas James Taylor, Simon Hogan, Gregor McInnes, Callum Kirk, William Walton (U/G project)