The document discusses using quasi-Monte Carlo integration to generate molecular shape fingerprints for efficient comparison of molecular shapes. Shape fingerprints represent the shape of a molecule as a bit string based on whether quasi-random points fall inside or outside the van der Waals surface. This allows for fast generation and comparison of shape fingerprints to enable large-scale virtual screening and clustering based on molecular shape. Examples are given analyzing shape clusters of ROCK2 inhibitors from PubChem. Future work may explore different shape representations and point distributions.

1. Quasi-Monte Carlo integration for
the fast and effective generation of
molecular shape fingerprints
John D. MacCuish
Norah E. MacCuish,
Michael Hawrylycz, and Mitch Chapman
ACS San Francisco 2010
COMP Drug Discovery
john.maccuish@mesaac.com

2. Outline
• Why Shape? Why Shape Fingerprints?
• Prior Work
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• quasi-Monte Carlo Integration
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• Shape Fingerprint Generation
• Performance, Examples
• Future Work

3. Why Shape?
• Shape is a component in binding...
• 3D QSAR
• similarity searching
• compound acquisition (shape diversity)
• virtual screening...
• Can be confounding -- multiple
binding sites, surface binding...
• One more tool in the toolbox...
• Boost to 2D
• Pluses and minuses: Molecular Shape and Medicinal Chemistry:
A Perspective, JMedChem, Feb. 2010, Nicholls, et al

4. Why Shape Fingerprints?
• Compute shape comparisons
efficiently on a large scale
• Efficient storage
• Simple.

5. Prior Work
• Shape as a mixture of spherical
gaussians (atoms) (Grant and
Pickup...’96)
• Fingerprints composed of key
shapes, given the above method
(Haigh, et al...’05).
• Ultrafast Shape Recognition:
Ballester, Richards 2007
• ShaEP, Vainio, et al 2009

6. Monte Carlo Integration
• Approximate an integral (e.g., a 3D
volume) by random sampling
• Approximate volume of odd shapes or
manifolds that are analytically difficult
or have no closed form solution.
• Better error convergence than grid
sampling (Metropolis).

7. quasi-Monte Carlo
Integration (QMC)
• In practice, quasi-randomly generated
points have best error convergence in
low dimensions.
• Beats uniform random sampling (e.g.,
pseudo, dart throwing, etc.)
• QMC became popular in early-90s in
computer graphics (Shirley, ’91) mid-90s
in finance - options/futures pricing
(Morokoff, Caflisch, ’95)

8. Approximate Volume
quasi-random
grid pseudo-random (low discrepancy) P
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Error related to the Error related to just Error related to just
number of points number of points number of points
and the dimension

9. Approximation Error
• grid -- for error ε, need 1/εd grid points
-- error convergence exponential in
the dimension
• pseudo -- 1⁄√n convergence
• quasi -- 1/n convergence in practice
in low dimensions
• quasi-random fewest points needed
for equivalent approximation error.

10. Approximating the
Volume of a molecule
• E.g., CPK with van der Waals radii --
Set of intersecting spheres
• Find a suitable bounding region for
molecules
• Point sample the bounding region
and tally up the points either inside
or outside of the atom spheres.

11. Approximating the
Volume of a molecule
• Bounding Volume times (Points
Inside)/(All Points) ~= molecular
volume.
• Sphere or scalene ellipsoid as a
bounding region reduces total overall
points.

12. Volume Percent Error
95% below 95% below
4961 low energy
4.5% error 3.6% error
conformations
1357 Molecules
100-550 MW
flexible, inflexible
balloon
conformations
95% below 95% below 1-15 conformations
2% error 1.3% error

13. Fingerprint Generation
Preliminaries
• Need a bounding region that covers the
conformational space of the database of
small molecule conformations.
• Cube? Sphere? Scalene Ellipsoid.
• Need orientation and alignment.

14. Fewer points
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Don’t need these points Don’t need these points
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15. Fingerprint Generation
1. Generate a sample quasi-point set bounding
region centered and fixed at (0,0,0); Sphere of
~11 Å radius. This is fixed, done once.
2. Mean center atom centers point set of a
conformation to be fingerprinted.
3. Find sample points in the atom spheres (or
function or your choice, e.g., solvent accessible
surface ...)
4. Find the principal axes for a shape using the
sampled points and the atom centers with SVD.
Adjust for SVD sign ambiguity (Bro, et al, 2008).

16. Fingerprint Generation
5. Rotate atom centers point set to Principal
Axes (PA)
6. Find points in PA configuration and
create fingerprint with 4 orientations --
points in molecule ‘1’, points out ‘0’
7. Subfingerprints, each the length of the
number of points.

17. Molecule in a Volume
of quasi-Random points

18. Four configurations

19. Alignment Overlay

20. Alignment Overlay

21. Shape Fingerprints
• 4 fingerprints per shape. e.g., 10,240
X 4 bits = 5.12 Kbytes.
• Number and density of points
determines resolution, speed of
comparison, and storage size.
• Number of bits on is small, typically
3-10% with CPK model. Fingerprints
compress significantly.

22. Fingerprint Similarity
• Choose 1 subFP of Shape FP A and compare it
with all 4 subFPs of Shape FP B.
• The largest similarity comparison is chosen.
• Bit difference per subFP is small in a given FP

23. Fingerprint Similarity
• Inherent slight asymmetry of alignment of
point sets of different sizes.
• Aligning B to A’s principal axes, vs aligning A
to B’s principal axes.
• Try this on your favorite method...
• Use bit string similarity measure of your
choice... Tanimoto, Ochai (Cosine), ...,
Tversky,... Baroni-Urbani/Bush...

24. Performance
• With ~10K points and including IO:
• Fingerprint generation: ~500K
conformations per hour
• Fingerprint Tanimoto comparisons:
130KX130K matrix in 24 hours
• Large scale similarity searching trivial,
e.g., 10X1M < 10 minutes
• Numbers estimated with 2009 iMac, with an Intel Core i7 2.8GHz processor
and 4GB RAM, running Mac OS X 10.6.2; small to large compounds;

25. Space!
• 5.12 Kbytes per fingerprint -- ~90%
compression
• 2 million conformations = 1+ GBs
• Space and CPU improving all of the
time...

26. FP Experiments
Typical all-pairs
Tanimoto similarity
distribution
1357, 2D 768 MACCS
Keys fingerprints
Where the action is:
Similarity searching,
Clustering, etc.
above 0.7

27. FP Experiments
All-pairs ShapeFingerprints
4961 confs.
Where the action is:
Similarity searching,
Still values above 0.7
Clustering,
Alignment, etc.
above ~0.65

28. Examples
• Multi-conformation set of actives in a
secondary screen in ROCK2 from
PubChem
• Xray ROCK2 ligand
• Fingerprint shape cluster
• Analyze shape cluster that contains the
xray structure with ChemTattoo 3D
“Fragment database analysis using molecular shape fingerprints”,
ACS San Francisco, Wednesday 8:40 , CINF 106

29. Shape Clustering
Example - ROCK2

30. Future Work
• Other shape functions
• Other low discrepancy point sets in
different distributions -- density
where it is needed.
• Speed (time) and compression
(space)
• More practical applications

31. Acknowledgments
• JMol
• Balloon
• OpenBabel
• R
• PubChem
• PDB
john.maccuish@mesaac.com