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
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- ……….Na+
A Na
Ka
AH
K0
AH
Solubility Versus pH Profile
-1
[ H  ][ A ] [ H  ][ A ]
Ka 

[ AH ]Solution
S0
-2
logS
-3
St  [ AH ]  [ A ]  S0 
-4
-5
2
3
4
5
6
7
pH (Concentration scale)
8
9
K 0  S0 
[ AH ]Solution
[ AH ]Solid

K a S0
Ka 

S
1

0
 
[H  ]
[
H
]

S0 is essentially the solubility of the neutral form only.
Diclofenac
O
Cl
Cl
HN
O
Cl
Cl
ONa
+
Na
HN
O
● First precipitation –
Kinetic Solubility (Not in Equilibrium)
● Thermodynamic Solubility through
“Chasing Equilibrium”Intrinsic Solubility (In Equilibrium)
dpH/dt Versus Time
0.008
O
Supersaturation Factor
SSF = Skin – S0
Cl
Cl
HN
OH
0.004
Supersaturated Solution
In Solution
Powder
dpH/dt
0.000
O
Cl
Cl
HN
OH
-0.004
8 Intrinsic solubility values
Subsaturated Solution
-0.008
20
25
30
Time (minutes)
35
40
45
Random error less than 0.05 log units !!!!
“CheqSol”
● 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
* 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), 979-983
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!
Solubility Challenge Results
14
12
Number of Entries
10
8
6
4
2
0
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Number of Correct Predictions out of 32 Molecules per Entry
A “null prediction” based on predicting everything to have the
mean training set solubility would have got 9/32 correct
21
Correlation Achieved by Entrants
30
25
Number of Entries
20
15
10
5
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
R squared
Using an R2 threshold of 0.500, only 18/99 entries were good
0.9
Number of Correct Predictions vs R squared
0.700
0.600
R squared
0.500
0.400
0.300
0.200
0.100
0.000
4
6
8
10
12
14
Number of Correct Predictions
16
18
20
22
Number of Correct Predictions vs R squared
0.700
GOOD
0.600
R squared
0.500
0.400
0.300
0.200
BAD
0.100
0.000
4
6
8
10
12
14
Number of Correct Predictions
16
18
20
22
Number of Correct Predictions vs R squared
0.700
0.600
3 “WINNERS”
R squared
0.500
0.400
0.300
0.200
0.100
0.000
4
6
8
10
12
14
16
18
20
Number of Correct Predictions
3 Pareto optimal entries which I think of as “winners”. These
combine best R2 with most correct predictions.
22
Some molecules proved much harder to predict than others – the
most insoluble were amongst the most difficult.
Conclusions from Solubility Challenge
• 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.”
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 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?
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
Thermodynamic Cycle
Gas
Crystal
Solution
Sublimation Free Energy
Gas
Crystal
Sublimation Free Energy
Gas
Crystal
Sublimation Free Energy
Gas
Crystal
Sublimation Free Energy
Gas
Crystal
Calculating Gsub 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.
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.
These methods can get relative lattice energies of different
structures correct, probably to within a few kJ/mol. Absolute
energies harder.
Volume/molecule (Å3)
149
150
151
152
153
154
155
-69
Lattice Energy (kJ/mol)
-70
-71
-72
-73
-74
34
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
CSP.
Volume/molecule (Å3)
149
150
151
152
153
154
155
-69
Lattice Energy (kJ/mol)
-70
-71
-72
-73
-74
35
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
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 Gsub
• Alternatively one could estimate sublimation energy from
QSPR (no crystal structure needed, but no obvious
benefit over direct informatics approach to solubility).
Thermodynamic Cycle
Gas
Crystal
Solution
Hydration Free Energy
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.
Predicted solubility FIT nosymm IEFPCM UFF
1
1E-15
1E-13
1E-11
1E-09
0.0000001
0.00001
0.001
0.1
0.1
y = 12.297x0.4774
R² = 0.8466
Experimental solubility (mol/L)
0.01
0.001
Entropy from phonon modes
Power (Entropy from phonon modes)
0.0001
0.00001
0.000001
Predicted solubility (mol/L)
0.0000001
An alternative route is via octanol, then using logP.
So really these are
ish
Theoretical Approaches
Using a training-test set split to optimise parameters & validate:
RMSE(te)=0.71
r2(te)=0.77
Ntrain = 34; Ntest = 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.
Random Forest
Machine Learning Method
Random Forest: Solubility Results
RMSE(tr)=0.27
r2(tr)=0.98
Bias(tr)=0.005
RMSE(oob)=0.68
r2(oob)=0.90
Bias(oob)=0.01
DS Palmer et al., J. Chem. Inf. Model., 47, 150-158 (2007)
RMSE(te)=0.69
r2(te)=0.89
Bias(te)=-0.04
Ntrain = 658; Ntest = 300
Random Forest:
Replicating Solubility Challenge (post hoc)
CDK descriptors
RMSE(te)=1.09
r2(te)=0.39
10/32 correct within 0.5 logS units
Ntrain  100; Ntest  32
Support Vector Machine
SVM: Solubility Results
et al.,
RMSE(te)=0.94
r2(te)=0.79
Ntrain = 150 + 50; Ntest = 87
What can we Learn from Informatics?
What Descriptors Correlate with logS?
…amongst the solubility challenge training set, once
intercorrelated descriptors with R2 > 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)