1. Chemical Diversity
Qualify and/or quantify the extent of variety within a set of
compounds. Try to define the extent of chemical space.
In combinatorial chemistry, we are interested in the
diversity of a library.

2. Example 1: here we are looking at compounds that can possess up
to 2 functional groups. How do we define libraries that have
different numbers of these cells occupied? How do we quantify
those that have duplicates within cells?

3. Chemical Diversity based on properties
Example 2: We can try to define the diversity based on properties of the compounds. For example, we could look at the naturally occurring amino acids and span the space define by their pI. This gives a poor spread, so try pI and MW. Could go to higher dimensions by also looking at the number of H-bonds they make, the number of OH groups, their dipole moment, etc.

4. Why is Diversity Important?
Similar Property Principle
Structurally similar compounds will exhibit similar physicochemical and biological properties
Test only representative compounds, eliminate redundancies
For lead discovery want a diverse space to locate all possible hits (actives) – called a diverse library
For refining a lead into a drug (lead optimization), want to survey a range of similar compounds – called a focused library
Diversity hypothesis
Diverse reactants will lead to diverse products
Potentially useful for library design
Quantify whether a library can be supplemented by additions of other compounds, other libraries
Beno, Drug Discovery Today, 2001, 6, Brown, JCICS, 1996, 36, 572
Gillet, JCICS, 1997, 37, 731

5. Types of Diversity A library with members that
sample chemical space evenly –
an ideal situation for lead
discovery
A library that covers the same
chemical space but the
compounds cluster and leave
large holes.
A library with even sampling of
space, but only with limited
diversity – useful for modification
of a lead.
From Rose, Drug Discovery Today, 2002, 7, 133.

6. Quantifying Diversity
Need to define how similar (or dissimilar) two compounds are from each other
Similarity indices
Then need to determine the spread of the compounds throughout space
Distance-based
Cell-based partitioning
Clustering
Agrafiotis, Mol. Diversity, 1999, 4, 1

7. Defining Similarity Descriptors Structural keys Fingerprints
Property-based
Structure-based 2D 3D
Pharmacophore
Structural keys
Fingerprints
Similarity/Distance Coefficients
Beno, Drug Discovery Today, 2001, 6, 251
Willett, Curr. Opin. Biotechnology, 2000, 11, 85
Willett, JCICS, 1998, 38, 983
Daylight,

8. Structural Keys
Boolean array expressing whether a pattern in present (TRUE) or not (FALSE) within a molecule
This array is usually represented as a string of 1s (TRUE) or 0s (FALSE) – a bitmap
So create a list of structural features and then set the corresponding bit to 1 if the feature is present
Martin, J. Med. Chem., 1995, 38, 1431
Flower, JCICS, 1998, 38, 379

9. Fingerprints Problems with structural keys Solution – Fingerprint
Lack of generality
Choice of structural keys is arbitrary and may not be appropriate for the search or question at hand
List of structural keys can be very long and unwieldy to generate and test
Solution – Fingerprint
Also a bitmap but NO assigned meaning to any particular bit!
Your fingerprint is characteristic of you, but there is no meaning to any particular fragment of it
Generate patterns from the molecule itself, such as a pattern for
Each atom
Each atom with nearest neighbors
Each group of atoms and bonds connected by up to 2 bonds long
Continuing with paths up to 3, 4, 5, 6, and 7 bonds long (seven seems to be the longest typically employed)
This list of patterns is exhaustive, meaning all are generated for every molecule

10. Fingerprints. II.
Since the number of patterns is huge, not possible to assign a particular bit to each pattern
Instead, each pattern is the input into a hash function that creates a number of set bits (typically 4-5 bits). These set bits are then added (with logical OR) to the fingerprint.
Note that bit sets for different patterns may have some bits in common
This conflict is not a problem since every bit set from some pattern (substructure) will be set in the molecule’s fingerprint.
Each pattern (substructure) generates its particular set of bits, and it is unlikely that another pattern will set those exact same bits. So a search for that substructure simply means looking to see if those bits have been set.
Fingerprint advantages
No predefined set of patterns (structural keys)
Structural keys are usually quite sparse, fingerprints are much more dense

11. Similarity Coefficients
a = S xjA number on bits in A
b = S xjb number on bits in B
c = S xjA xjB number on bits in both A and B
D(A,B) is similarity of A and B using bits
S(A,B) is similarity of A and B using
continuous variables
Euclidean Distance
Tanimoto Coefficient
Cosine Coefficient
D(A,B) = [a + b – 2c]1/2 range 0 to n bits
S(A,B) = [S (xjA – xjB)2 ]1/2 range 0 to infinity
D(A,B) = c/[a + b – c] range 0 to 1
S(A,B) = S xjAxjB / [S xjA2 + S xjB2 + S xjAxjB] range to 1
D(A,B) = c/[ab]1/2 range 0 to 1
S(A,B) = S xjAxjB / [S xjA2 S xjB2 ]1/2 range –1 to 1
Willett, JCICS, 1998, 38, 983

12. Example: Bitmap for 2,2-dimethylbutane 1111011000000 a = 6
Ethylcyclobutane b = 9
c = 5
Euclid distance = (6+9-10)1/2 = 2.24
Tanimoto coefficient = 5/(6+9-5) = 0.5
Cosine coefficient = 5/(6*9)1/2 = 0.68

13. Problems with Tanimoto and related similarity indices
Flower, JCICS, 1998, 38, 379

14. Quantifying Diversity Rules for a diversity function
adding redundant molecules does not change the value of the diversity
Adding non-redundant molecules always increases the value of the diversity
Space-filling behavior should be preferred
Perfect filling of space gives a finite value of the diversity
As dissimilarity of a pair of compounds increases, the diversity should increase asymptotically
Waldman, J. Mol. Graph. Model., 2000, 18, 412

15. Diversity definition 1 Where SIM(J,K) is some similarity measurement
between compounds A and B.
Can use this to build up a compound selection procedure
for creating the sublibrary with maximal diversity
Find similarities of all compounds in the library
Select compound that is most dissimilar from all other
Select 2nd compound that is most dissimilar from the first
Select 3rd compound that is most dissimilar from first 2
Continue until you have selected as many
compounds as you desire

16. Cell-based Partitioning
Divide each dimension into a number of parts
These divisions are called cells or bins
Place compounds into appropriate bin based on the value of its properties and/or descriptors
Can now create a sublibrary by choosing one compound from each bin, usually the one nearest the center of the bin
Schematic representation of different sampling of diversity space
(a) Maximize Euclidean distance to create maximum diversity
(b) cell-based selection, choosing compound nearest center of each cell
From Rose, Drug Discovery Today, 2002, 7, 133

17. Diversity definition 2 and 3
Suppose 10 molecules divided into 2 cells.
Distribution 1: (5,5) – Dc2 = 0
Distribution 2: (7,3) - Dc2 = -8
So the more even distribution is scored as being more diverse.
But this may actually go too far –
Dc2(2,2,2) > Dc2 (4,1,1) = Dc2 (3,3,0)
Makes these last two equivalent, but the (4,1,1) appears to
be intuitively more diverse.
This entropy-like definition ranks the three sets
Dentropy(2,2,2) > Dentropy(4,1,1) > Dentropy(3,3,0)
Waldman, J. Mol. Graph. Model., 2000, 18, 412

18. Clustering. I. Hierarchical clusters
Small clusters within larger clusters
Typically some relationship between clusters
Two procedures
Agglomerative
Start with singletons and move upwards
Calculate all similarities of all pairs
Merge two most similar into a cluster
Continue until all only one cluster remains
Divisive
Start with one cluster and break into smaller clusters
Calculate all dissimilarities of all pairs
Take the pair of most dissimilar structures and assign all other structures to the least dissimilar of these initial cluster centers.
Recursively select the cluster with the largest diameter and partition it intow two such that largest resulting cluster has the smallest diameter
Repeat step (c) for a maximum of n-1 times
Brown, JCICS, 1996, 36, 572

19. Clustering. II. Nonhierarchical clusters No relation between clusters
Jarvis-Patrick method
calculate similarities of all pairs
Record top n most similar structures to each structure (nearest-neighbor list)
Assign compounds to clusters. A and B are in the same cluster if:
A is in the top K nearest-neighbor list of B
B is in the top K nearest-neighbor list of A
A and B have at least Kmin of their top K nearest-neighbors in common
Tends to produce lots of small clusters (singletons) under strict conditions or a few very large clusters under less strict conditions
Brown, JCICS, 1996, 36, 572

20. Goals for Diversity Metrics
Insure the exploratory libraries are broad enough to locate active molecules
Insure that focused (directed) libraries are both broad enough to sample space but compact enough to maintain activity
Need to keep libraries small enough to readily manage – so want to insure that sublibraries separate actives from inactives

21. Other Diversity Comments
Krchnak, Mol. Diversity, 1996, 1, 193 (
General comments of combinatorial methods and diversity
Good, JCICS, 1997, 40, 3926
Use of 3d pharmacophores demands selection of products not reagents, since they are not additive
Martin, J. Comb. Chem., 1999, 1, 32
Beyond diversity, library construction should include MW, lipophilicity, ease of synthesis, pharmacophore features, reagent cost, solubility, complementarity to other libraries.
Distance measures assess redundancy, coverage of space is better assessed with maps or binning procedures
Diversity functions often overweight edges
Oprea, J. Comb. Chem., 2001, 3, 157
Big numbers (lots of compounds) and serendipity are not enough
Martin, J. Comb. Chem., 2001, 3, 231
Chemical similarity not always good predictor of bioproperties
Unlikely that a few thousand compounds can span all of chemical space
Just how much diversity is enough?