In silico prediction of solubility:

Solid progress but no solution?

Dr John MitchellUniversity 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

K0

Solid

Solution

AH

AHS

][

][K 00

Ka

0a

]][[

][

]][[K

S

AH

AH

AH

Solution

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+ AHAH

][1

][][][S 0

00t H

KS

H

SKSAAH aa

Solubility Versus pH Profile

logS

pH (Concentration scale)

-5

-4

-3

-2

-1

2 3 4 5 6 7 8 9 S0 is essentially the solubility of the neutral form only.

DiclofenacDiclofenac

NH

O

Cl Cl

ONa NH

OCl Cl

O

Na+

NH

OCl Cl

OH

NH

OCl Cl

OH

In SolutionPowder

● 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 !!!!

dpH/dt Versus Time

dpH

/dt

Time (minutes)

-0.008

-0.004

0.000

0.004

0.008

20 25 30 35 40 45

Supersaturated Solution

Subsaturated Solution8 Intrinsic solubility values8 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), 979-983

● First precipitation – Kinetic Solubility (Not in Equilibrium)● Thermodynamic Solubility through “Chasing Equilibrium”- Intrinsic Solubility (In Equilibrium)

Supersaturation Factor SSF = Skin – S0

“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!

Solubility Challenge Results

0

2

4

6

8

10

12

14

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Number of Correct Predictions out of 32 Molecules per Entry

Nu

mb

er o

f E

ntr

ies

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

Correlation Achieved by Entrants

0

5

10

15

20

25

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

R squared

Nu

mb

er o

f E

ntr

ies

Using an R2 threshold of 0.500, only 18/99 entries were good

Number of Correct Predictions vs R squared

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

4 6 8 10 12 14 16 18 20 22

Number of Correct Predictions

R s

qu

ared

Number of Correct Predictions vs R squared

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

4 6 8 10 12 14 16 18 20 22

Number of Correct Predictions

R s

qu

ared

GOOD

BAD

Number of Correct Predictions vs R squared

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

4 6 8 10 12 14 16 18 20 22

Number of Correct Predictions

R s

qu

ared

3 “WINNERS”

3 Pareto optimal entries which I think of as “winners”. These combine best R2 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 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

Crystal

Gas

Solution

Sublimation Free Energy

Crystal

Gas

Sublimation Free Energy

Crystal

Gas

Sublimation Free Energy

Crystal

Gas

Sublimation Free Energy

Crystal

Gas

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.

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.

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.

-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

34

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

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 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

Crystal

Gas

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.

An alternative route is via octanol, then using logP.

Theoretical Approachesish

So really these are

Using a training-test set split to optimise parameters & validate:

RMSE(te)=0.71r2(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 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.

Machine Learning Method

Random Forest

Random Forest: Solubility Results

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

RMSE(tr)=0.27r2(tr)=0.98Bias(tr)=0.005

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

Random Forest: Replicating Solubility Challenge (post hoc)

RMSE(te)=1.09r2(te)=0.3910/32 correct within 0.5 logS units Ntrain 100; Ntest 32

CDK descriptors

Support Vector Machine

SVM: Solubility Results

et al.,

Ntrain = 150 + 50; Ntest = 87RMSE(te)=0.94r2(te)=0.79

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)