Quantum Chemical and Machine Learning Calculations of the Intrinsic Aqueous

Solubility of Druglike Molecules

Dr John MitchellUniversity 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

TheoreticalChemistryInformatics

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

Gsub from lattice energy minimisation

Ghydr from Reference Interaction Site Model (RISM)

Different kinds of theoretical method are used for each part

Gsub from lattice energy & a phonon entropy term;DMACRYS using B3LYP/6-31G(d,p) multipoles and FIT repulsion-dispersion potential.

Ghydr 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 ΔGsub and ΔGhydr 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 ΔGsub is a standard procedure in crystal structure prediction

Crystal Structure Prediction

• Given the structural diagram of an organic molecule, predict the 3D crystal structure.

S NBr

OO

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.

-74

-73

-72

-71

-70

-69

149 150 151 152 153 154 155

Volume/molecule (Å3)

Lat

tice

Ene

rgy

(kJ/

mol

)

P1 P_1P21 P21/cCc C2C2/c PmP2/c P21/mP21212 PcP212121 Pca21Pna21 PbcnPbca Pmn21Pma21 ALPHABETA GAMMA

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

-74

-73

-72

-71

-70

-69

149 150 151 152 153 154 155

Volume/molecule (Å3)

Lat

tice

Ene

rgy

(kJ/

mol

)

P1 P_1P21 P21/cCc C2C2/c PmP2/c P21/mP21212 PcP212121 Pca21Pna21 PbcnPbca Pmn21Pma21 ALPHABETA GAMMA

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 ΔGsub

Reasonable prediction of ΔGsub, but small number of molecules.

Results for ΔGsub

80.00 90.00 100.00 110.00 120.00 130.00 140.00 150.00 160.0080.00

100.00

120.00

140.00

160.00

180.00

200.00

f(x) = 1.07431033938862 x + 3.60559227465819R² = 0.531284409642467

ΔH sub experimental v ΔH sub predicted

dH Vs exp dH

Linear (dH Vs exp dH)

Experimental ΔH sub kJ mol-1

Pre

dic

ted

ΔH

su

b k

J m

ol-

1

To see the trends in errors, we need to look at more molecules.

RMSE = 20.4 kJ/mol(46 molecules)

80.00 90.00 100.00 110.00 120.00 130.00 140.00 150.00 160.0080.00

100.00

120.00

140.00

160.00

180.00

200.00

f(x) = 1.07431033938862 x + 3.60559227465819R² = 0.531284409642467

ΔH sub experimental v ΔH sub predicted

dH Vs exp dH

Linear (dH Vs exp dH)

Experimental ΔH sub kJ mol-1

Pre

dic

ted

ΔH

su

b k

J m

ol-

1

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)

30 35 40 45 50 55 60 65 70 75 8050.00

52.00

54.00

56.00

58.00

60.00

62.00

64.00

66.00

68.00

f(x) = 0.0857428957234206 x + 56.0784056418822R² = 0.0940590648113177

TΔS sub experimental v TΔS sub predicted

TdS Vs exp TdS

Linear (TdS Vs exp TdS)

TΔS experimental kJ mol-1

TΔ

S p

red

icte

d k

J m

ol-

1

RMSE = 9.3 kJ/mol(46 molecules)

30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 110.00 120.000.00

20.00

40.00

60.00

80.00

100.00

120.00

f(x) = 0.742207217400145 x + 28.1100057140445R² = 0.312904269021572

ΔG sub experimental v ΔG sub predicted

G Experimental Vs Predicted

Linear (G Experimental Vs Predicted)

ΔG sub Experimental

ΔG

su

b P

red

icte

d

RMSE = 22.4 kJ/mol(46 molecules)

• The predicted ΔHsub is much better correlated with experiment than is TΔSsub.

• However, ΔHsub 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 minsper 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, 2010. 133(4): p. 044104-11.

Perhaps surprisingly, error in Ghyd is smaller than in Gsub.

Results for ΔGhyd

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 ΔGsol and hence logS.

Results for ΔGsol

Conclusions: Theory

• Must calculate Gsub & Ghyd separately;

• Expt data sparse and errors may be large;• RISM is efficient & fairly accurate for Ghyd;

• 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 structureand 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 Gsol.

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)=0.69r2(te)=0.89Bias(te)=-0.04

RMSE(oob)=0.68r2(oob)=0.90Bias(oob)=0.01

DS Palmer et al., J. Chem. Inf. Model., 47, 150-158 (2007) Ntrain = 658; Ntest = 300

Support Vector Machine

SVM: Solubility Results

et al.,

Ntrain = 150 + 50; Ntest = 87RMSE(te)=0.94r2(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; 1.08r2(te)= 0.49; 0.58; 0.41

12; 12; 13/28 correct within 0.5 logS units

Ntrain = 94; Ntest = 28

CDK descriptors: RF, PLS, SVM

Replicating Solubility Challenge (post hoc)

Ntrain = 94; Ntest = 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)