1. Similarity and Diversity Alexandre Varnek, University of Strasbourg, France

2. What is similar?

3. Different „spaces“, classified by:
Shape
Colour
Size
Pattern

4. 16 diverse aldehydes...

5. ...sorted by common scaffold

6. ...sorted by functional groups

7. The „Similarity Principle“ :
Structurally similar molecules are assumed to have similar biological properties
Compounds active as opioid receptors

8. Structural Spectrum of Thrombin Inhibitors
structural similarity “fading away” …
reference compounds
0.56
0.72
0.53
0.84
0.67
0.52
0.82
0.64
0.39

9. Key features in similarity/diversity calculations:
Properties to describe elements (descriptors, fingerprints)
Distance measure („metrics“)

10. N-Dimensional Descriptor Space
descriptorn
Each chosen descriptor adds a dimension to the reference space
Calculation of n descriptor values produces an n-dimensional coordinate vector in descriptor space that determines the position of a molecule
molecule Mi
= (descriptor1(i), descriptor2(i), …, descriptorn(i))
descriptor2
descriptor1
descriptor3

11. Chemical Reference Space
Distance in chemical space is used as a measure of molecular “similarity“ and “dissimilarity“
“Molecular similarity“ covers only chemical similarity but also property similarity including biological activity
descriptor3
descriptor2
descriptor1
descriptorn
DAB B A

12. Distance Metrics in n-D Space
If two molecules have comparable values in all the n descriptors in the space, they are located close to each other in the n-D space.
how to define “closeness“ in space as a measure of molecular similarity?
distance metrics

13. Descriptor-based Similarity
When two molecules A and B are projected into an n-D space, two vectors, A and B, represent their descriptor values, respectively.
A = (a1,a2,...an)
B = (b1,b2,...bn)
The similarity between A and B, SAB, is negatively correlated with the distance DAB
shorter distance ~ more similar molecules
in the case of normalized distance (within value range [0,1]), similarity = 1 – distance
descriptor3
descriptor2
descriptor1
descriptorn B
DAB
DBC C A
DAB>DBC  SAB<SBC

14. Metrics Properties The distance values dAB 0; dAA= dBB= 0
Symmetry properties: dAB= dBA
Triangle inequality: dAB dAC+ dBC
descriptor3
descriptor2
descriptor1
descriptorn B
DAB
DBC C A

15. Euclidean Distance in n-D Space
Given two n-dimensional vectors, A and B
A = (a1,a2,...an)
B = (b1,b2,...bn)
Euclidean distance DAB is defined as:
Example:
A = (3,0,1); B = (5,2,0)
DAB = = 3
descriptor3
descriptor2
descriptor1
descriptorn
DAB A B

16. Manhattan Distance in n-D Space
Given two n-dimensional vectors, A and B
A = (a1,a2,...an)
B = (b1,b2,...bn)
Manhattan distance DAB is defined as:
Example:
A = (3,0,1); B = (5,2,0)
DAB = = 5
descriptor3
descriptor2
descriptor1
descriptorn
DAB A B

17. Distance Measures („Metrics“):
Euclidian distance:
[(x11 - x21) 2 + (x12 - x22)2] 1/2 =
= ( )1/2 = 4.472
Manhattan (Hamming) distance:
|x11 - x21| + |x12 - x22| = = 6
Sup distance:
Max (|x11 - x21|, |x12 - x22|) =
= Max (4, 2) = 4

18. Binary Fingerprint

19. Popular Similarity/Distance Coefficients
Similarity metrics:
Tanimoto coefficient
Dice coefficient
Cosine coefficient
Distance metrics:
Euclidean distance
Hamming distance
Soergel distance

20. Tanimoto Coefficient (Tc)
Definition:
value range: [0,1]
Tc is also known as Jaccard coefficient
Tc is the most popular similarity coefficient A B C

21. Example Tc Calculation
a = 4, b = 4, c = 2 A B
binary

22. Dice Coefficient Definition: value range: [0,1]
monotonic with the Tanimoto coefficient

23. Cosine Coefficient Definition: Properties: value range: [0,1]
correlated with the Tanimoto coefficient but not strictly monotonic with it

24. Hamming Distance Definition:
value range: [0,N] (N, length of the fingerprint)
also called Manhattan/City Block distance

25. Soergel Distance Definition: Properties: value range: [0,1]
equivalent to (1 – Tc) for binary fingerprints

28. Similarity coefficients

29. Properties of Similarlity and Distance Coefficients
Metric Properties
The distance values dAB 0; dAA= dBB= 0
Symmetry properties: dAB= dBA
Triangle inequality: dAB dAC+ dBC
Properties of Similarlity and Distance Coefficients
The Euclidean and Hamming distances and the Tanimoto coefficients (dichotomous variables) obey all properties.
The Tanimoto, Dice and Cosine coefficients do not obey inequality (3).
Coefficients are monotonic if they produce the same similarlity ranking

30. Similarity search
Using bit strings to encode molecular size. A biphenyl query is compared to a series of analogues of increasing size. The Tanimoto coefficient, which is shown next to the corresponding structure, decreases with increasing size, until a limiting value is
reached.
D.R. Flower, J. Chem. Inf. Comput. Sci., Vol. 38, No. 3, 1998, pp

31. Similarity search
Molecular similarity at a range of Tanimoto coefficient values
D.R. Flower, J. Chem. Inf. Comput. Sci., Vol. 38, No. 3, 1998, pp

32. Similarity search
The distribution of Tanimoto coefficient values found in database searches with a range of query molecules of increasing size and complexity
D.R. Flower, J. Chem. Inf. Comput. Sci., Vol. 38, No. 3, 1998, pp

33. Molecular Similarity
A comparison of the Soergel and Hamming distance values for two pairs of structures to illustrate the effect of molecular size
A R. Leach and V. J. Gillet "An Introduction to Chemoinformatics" , Kluwer Academic Publisher, 2003

34. Molecular Similarity Similarity = Nbonds(MCS) / Nbonds(query)
The maximum common subgraph (MCS) between the two molecules is in bold
Similarity = Nbonds(MCS) / Nbonds(query)
A R. Leach and V. J. Gillet "An Introduction to Chemoinformatics" , Kluwer Academic Publisher, 2003

35. Activity landscape

36. How important is a choice of descriptors ?
Inhibitors of acyl-CoA:cholesterol acyltransferase represented with MACCS (a), TGT (b), and Molprint2D (c) fingerprints.

37. R. Guha et al. J.Chem.Inf.Mod., 2008, 48, 646
discontinuous SARs
continuous SARs
gradual changes in structure result in moderate changes in activity
“rolling hills” (G. Maggiora)
small changes in structure have dramatic effects on activity
“cliffs” in activity landscapes
Structure-Activity Landscape Index: SALIij = DAij / DSij
DAij (DSij ) is the difference between activities (similarities) of molecules i and j
R. Guha et al. J.Chem.Inf.Mod., 2008, 48, 646

38. discontinuous SARs VEGFR-2 tyrosine kinase inhibitors
MACCSTc: 1.00
Analog
6 nM
2390 nM
small changes in structure have dramatic effects on activity
“cliffs” in activity landscapes
lead optimization, QSAR
bad news for molecular similarity analysis...

39. Example of a “Classical” Discontinuous SAR
Any similarity method must recognize these
compounds as being “similar“ ...
(MACCS Tanimoto similarity)
Adenosine deaminase inhibitors

40. Libraries design
Goal: to select a representative subset from a large database

41. Chemical Space
Overlapping similarity radii  Redundancy
„Void“ regions  Lack of information

42. Chemical Space
„Void“ regions  Lack of information

43. Chemical Space
No redundancy, no „voids“
 Optimally diverse compound library

44. Subset selection from the libraries
Clustering
Dissimilarity-based methods
Cell-based methods
Optimisation techniques

45. Clustering in chemistry

46. What is clustering?
Clustering is the separation of a set of objects into groups such that items in one group are more like each other than items in a different group
A technique to understand, simplify and interpret large amounts of multidimensional data
Classification without labels (“unsupervised learning”)

47. Where clustering is used ?
General:
data mining, statistical data analysis, data compression, image segmentation, document classification (information retrieval)
Chemical:
representative sample,
subsets selection,
classification of new compounds

48. Overall strategy Select descriptors Generate descriptors for all items
Scale descriptors
Define similarity measure (« metrics »)
Apply appropriate clustering method to group the items on basis of chosen descriptors and similarity measure
Analyse results

49. Data Presentation Pattern matrix Proximity matrix dii = 0; dij = dji
descriptors
molecules
molecules
molecules
Pattern matrix
Proximity matrix
Library contains n molecules,
each molecule is described
by p descriptors
dii = 0; dij = dji

50. Clustering methods Hierarchical Non-hierarchical Agglomerative
Single Link
Complete Link
Agglomerative
Divisive
Group Average
Hierarchical
Weighted Gr Av
Monothetic
Centroid
Polythetic
Median
Single Pass
Ward
Jarvis-Patrick
Nearest Neighbour
Non-hierarchical
Mixture Model
Relocation
Topographic
Others

51. Hierarchical Clustering
divisive
agglomerative
A dendrogram representing an hierarchical clustering of 7 compounds

52. Sequential Agglomerative Hierarchical Non-overlapping (SAHN) methods
Simple link
Group average
Complete link
In the Single Link method, the intercluster distance is equal to the minimum distance between any two compounds, one from each cluster.
In the Complete Link method, the intercluster distance is equal to the furthest distance between any two compounds, one from each cluster.
The Group Average method measures intercluster distance as the average of the distances between all compounds in the two clusters.

53. Hierarchical Clustering: Johnson’s method
The algorithm is an agglomerative scheme that erases rows and columns in the proximity matrix as old clusters are merged into new ones.
Step 1. Group the pair of objects into a cluster
Step 2. Update the proximity matrix
Single-link
Complete-link

54. Hierarchical Clustering: single link

55. Hierarchical Clustering: complete link

56. Hierarchical Clustering: single vs complete link

57. Non-Hierarchical Clustering: the Jarvis-Patrick method
At the first step, all nearest neighbours of each compound are found by calculating of all paiwise similarities and sorting according to this similarlity.
Then, two compounds are placed into the same cluster if:
They are in each other’s list of m nearest neighbours.
They have p (where p< m) nearest neighbours in common. Typical values: m = 14 ; p = 8.
Pb: too many singletons.

58. Non-Hierarchical Clustering: the relocation methods
Relocation algorithms involve an initial assignment of compounds to clusters, which is then iteratively refined by moving (relocating) certain compounds from one cluster to another.
Example: the K-means method
Random choise of c « seed » compounds. Other compounds are assigned to the nearest seed resulting in an initial set of c clusters.
The centroides of cluster are calculated. The objects are re-assigned to the nearest cluster centroid.
Pb: the method is dependent upon the initial set of cluster centroids.

59. Efficiency of Clustering Methods
Storage space
Time
Hierarchical
(general)
O (N2)
O (N3)
(Ward’s method + RNN)
O (N)
Non-Hierarchical
O (MN)
(Jarvis-Patric method)
N is the number of compounds and M is the number of clusters

60. Validity of clustering
How many clusters are in the data ?
Does partitioning match the categories ?
Where should be the dendrogram be cut ?
Which of two partitions fit the data better ?

61. Dissimilarity-Based Compound Selection (DBCS)
4 steps basic algorithm for DBCS:
Select a compound and place it in the subset.
Calculate the dissimilarity between each compound remaining in the data set and the compounds in the subset.
Choose the next compound as that which is most dissimilar to the compounds in subset.
If n < n0 (n0 being the desired size number of compounds in the final subset), return to step 2.

62. Dissimilarity-Based Compound Selection (DBCS)
Basic algorithm for DBCS:
1st step – selection of the initial compound
Select it at random;
Choose the molecules which is « most representative » (e.g., has the largest sum of similarlities to other molecules);
Choose the molecules which is « most dissimilar » (e.g., has the smallest sum of similarlities to other molecules).

63. Dissimilarity-Based Compound Selection (DBCS)
Basic algorithm for DBCS:
2nd step – calculation of dissimilarity
Dissimilarity is the opposite of similarity
(Dissimilarity)i,j = 1 – (Similarity )i,j
(where « Similarity » is Tanimoto, or Dice, or Cosine, … coefficients)

64. Diversity
Diversity characterises a set of molecules

65. Dissimilarity-Based Compound Selection (DBCS)
Basic algorithm for DBCS:
3nd step – selection the most dissimilar compound
There are several methods to select a diversed subset containing m compounds
1). MaxSum method selects the compound i that has the maximum sum of distances to all molecules in the subset
2). MaxMin method selects the compound i with the maximum distance to its closest neighbour in the subset

66. Basic algorithm for DBCS:
3nd step – selection the most dissimilar compound
3). The Sphere Exclusion Algorithm
Define a threshold dissimilarity, t
Select a compound and place it in the subset.
Remove all molecules from the data set that have a dissimilarity to the selected molecule of less than t
Return to step 2 if there are molecules remaining in the data set.
The next compound can be selected
randomly;
using MinMax-like method

67. DBCS : Subset selection from the libraries

68. Cell-based methods
Cell-based or Partitioning methods operated within a predefined low-dimentional chemical space.
If there are K axes (properties) and each is devided into bi bins, then the number of cells Ncells in the multidimentianal space is

69. Cell-based methods The construction of 2-dimentional chemical space.
LogP bins: <0, 0-0.3, 3-7 and >7
MW bins: 0-250, , , > 750.

70. Cell-based methods
A key feature of Cell-based methods is that they do not requere the calculation of paiwise distances Di,j between compounds; the chemical space is defined independently of the molecules that are positioned within it.
Advantages of Cell-based methods
Empty cells (voids) or cells with low ocupancy can be easily identified.
The diversity of different subsets can be easily compared by examining the overlap in the cells occupied by each subset.
Main pb: Cell-based methods are restricted to relatively low-dimentional space

71. Optimisation techniques
DBCS methods prepare a diverse subset selecting interatively ONE molecule a time.
Optimisation techniques provide an efficient ways of sampling large search spaces and selection of diversed subsets

72. Optimisation techniques
Example: Monte-Carlo search
Random selection of an initial subset and calculation of its diversity D.
A new subset is generated from the first by replacing some of its compounds with other randomly selected.
The diversity of the new subset Di+1 is compared with Di
if DD = Di+1 - Di > 0, the new set is accepted
if DD < 0, the probability of acceptence depends on the Metropolis condition, exp(- DD / kT).

73. Scaffolds and Frameworks

74. Frameworks
Bemis, G.W.; Murcko, M.A. J.Med.Chem 1996, 39,

75. Frameworks
Dissection of a molecule according to Bemis and Murcko. Diazepam contains three sidechains and one framework with two ring systems and a zero-atom linker.
G. Schneider, P. Schneider, S. Renner, QSAR Comb.Sci. 25, 2006, No.12, 1162 – 1171

76. Graph Frameworks for Compounds in the CMC Database
(Numbers Indicate Frequency of Occurrence)
Bemis, G.W.; Murcko, M.A. J.Med.Chem 1996, 39,

77. Scaffolds et Frameworks
L’algorithme de Bemis et Murcko de génération de framework :
les hydrogènes sont supprimés,
les atomes avec une seule liaison sont supprimés successivement,
le scaffold est obtenu,
tous les types d’atomes sont définis en tant que C et tous les types de liaisons
sont définis en tant que simples liaisons, ce qui permet d’obtenir le framework.
Bemis, G.W.; Murcko, M.A. J.Med.Chem 1996, 39,
Contrairement à la méthode de Bemis et Murcko, A. Monge a proposé de distinguer les liaisons aromatiques et non aromatiques (thèse de doctorat, Univ. Orléans, 2007)

78. Scaffold-Hopping: How Far Can You Jump?
G. Schneider, P. Schneider, S. Renner, QSAR Comb.Sci. 25, 2006, No.12, 1162 – 1171

79. The Scaffold Tree − Visualization of the Scaffold Universe by Hierarchical Scaffold Classification
A. Schuffenhauer, P. Ertl, S. Roggo, S. Wetzel, M. A. Koch, and H.Waldmann
J. Chem. Inf. Model., 2007, 47 (1), 47-58

80. Scaffold tree for the results of pyruvate kinase assay
Scaffold tree for the results of pyruvate kinase assay. Color intensity represents the ratio of active and inactive molecules with these scaffolds.
A. Schuffenhauer, P. Ertl, S. Roggo, S. Wetzel, M. A. Koch, and H.Waldmann J. Chem. Inf. Model., 2007, 47 (1), 47-58