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
Informatics
Theoretical
Chemistry
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 costeffective first principles solubility method.
Thermodynamic Cycle
Different kinds of theoretical method are
used for each part
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).
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
Gas
Crystal
Solution
Sublimation Free Energy
Gas
Crystal
Sublimation Free Energy
Gas
Crystal
Sublimation Free Energy
Gas
Crystal
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.
Br
N
S
O
O
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.
Volume/molecule (Å3)
149
150
151
152
153
154
155
-69
Lattice Energy (kJ/mol)
-70
-71
-72
-73
-74
P1
P21
Cc
C2/c
P2/c
P21212
P212121
Pna21
Pbca
Pma21
BETA
P_1
P21/c
C2
Pm
P21/m
Pc
Pca21
Pbcn
Pmn21
ALPHA
GAMMA
Additional possible benefit for solubility: if we don’t know the
crystal structure, we could reasonably use best structure from
crystal structure prediction.
Volume/molecule (Å3)
149
150
151
152
153
154
155
-69
Lattice Energy (kJ/mol)
-70
-71
-72
-73
-74
P1
P21
Cc
C2/c
P2/c
P21212
P212121
Pna21
Pbca
Pma21
BETA
P_1
P21/c
C2
Pm
P21/m
Pc
Pca21
Pbcn
Pmn21
ALPHA
GAMMA
Results for ΔGsub
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.
To see the trends in errors, we need to look at more molecules.
ΔH sub experimental v ΔH sub predicted
200.00
y = 1.0743x + 3.6056
R² = 0.5313
Predicted ΔH sub kJ mol-1
180.00
160.00
140.00
dH Vs exp dH
Linear (dH Vs exp dH)
120.00
RMSE = 20.4 kJ/mol
(46 molecules)
100.00
80.00
80.00
90.00
100.00
110.00
120.00
130.00
Experimental ΔH sub kJ mol-1
140.00
150.00
160.00
The 46 compound set shown here has a larger error, mostly
due to some large outliers. Error statistics vary with dataset.
ΔH sub experimental v ΔH sub predicted
200.00
y = 1.0743x + 3.6056
R² = 0.5313
Predicted ΔH sub kJ mol-1
180.00
160.00
140.00
dH Vs exp dH
Linear (dH Vs exp dH)
120.00
RMSE = 20.4 kJ/mol
(46 molecules)
100.00
80.00
80.00
90.00
100.00
110.00
120.00
130.00
Experimental ΔH sub kJ mol-1
140.00
150.00
160.00
TΔS sub experimental v TΔS sub predicted
68.00
66.00
TΔS predicted kJ mol-1
64.00
y = 0.0857x + 56.078
R² = 0.0941
62.00
TdS Vs exp TdS
Linear (TdS Vs exp TdS)
60.00
58.00
RMSE = 9.3 kJ/mol
(46 molecules)
56.00
54.00
0
10
20
30
40
50
TΔS experimental kJ mol-1
60
70
80
90
ΔG sub experimental v ΔG sub predicted
120.00
y = 0.7422x + 28.11
R² = 0.3129
100.00
ΔG sub Predicted
80.00
60.00
G Experimental Vs Predicted
Linear (G Experimental Vs Predicted)
40.00
RMSE = 22.4 kJ/mol
(46 molecules)
20.00
0.00
0.00
20.00
40.00
60.00
ΔG sub Experimental
80.00
100.00
120.00
• 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
Gas
Crystal
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, 2010. 133(4): p. 044104-11.
Results for ΔGhyd
Perhaps surprisingly, error in Ghyd is smaller than in Gsub.
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
Gas
Crystal
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 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 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?
Random Forest
Machine Learning Method
Random Forest: Solubility Results
RMSE(oob)=0.68
r2(oob)=0.90
Bias(oob)=0.01
RMSE(te)=0.69
r2(te)=0.89
Bias(te)=-0.04
DS Palmer et al., J. Chem. Inf. Model., 47, 150-158 (2007)
Ntrain = 658; Ntest = 300
Support Vector Machine
SVM: Solubility Results
et al.,
RMSE(te)=0.94
r2(te)=0.79
Ntrain = 150 + 50; Ntest = 87
100 Compound Cross-Validation
Theoretical energies don’t seem to improve descriptor models.
Replicating Solubility Challenge (post hoc)
CDK descriptors: RF, RF, PLS, SVM
RMSE(te)=1.09; 1.00; 0.89; 1.08
r2(te)=0.39; 0.49; 0.58; 0.41
10; 12; 12; 13/28 correct within 0.5 logS units
Ntrain  94; Ntest  28
Replicating Solubility Challenge (post hoc)
CDK descriptors: RF, RF, PLS, SVM
Although the test dataset is small, it is a standard set.
Ntrain  94; Ntest  28
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)