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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5884_Библиотеки_им_академика_М_И_Перельмана.pdf
5 Clustering of Small Molecules 111
may be highly impractical. Moreover, some similarities between objects may not be
obvious from a fast visual inspection.
Computer-aided clustering of chemical entities therefore has three potential
advantages: (a) automation, (b) high-throughput, and (c) the possibility of detecting
clustering patterns that exceed prior knowledge and prior judgments regarding
which molecular features are relevant for meaningful and informative clustering.
1.1 Applications in Drug Discovery
The clustering of small molecules is of substantial importance in the field of
computer-guided drug discovery and in conducting virtual and wet screening experiments. Depending on the pursued goals, the structure of chemical data in suitable
subspaces of the chemical space can be relevant to sample chemical data sets in a
representative or biased manner.
First, the stratified split of data sets into representative training and validation sets
is regularly performed when building quantitative structure–activity relationships
(QSAR) and, in general, for any kind of machine-learning application. Such a
practice can respond to two different (and, in a certain sense, opposite) purposes:
(a) ensure that the training data present maximum coverage of the chemical space
covered by the data set, in order to expand the generalizability and the applicability
domain of the inferred models [21, 30], and (b) avoid compo und biases and
overoptimistic validation output by allocating very similar compounds in both the
training and validation sets ([32, 58]). For more details, see Chap. 6.
Second, structuring data in the chemical space can be useful in investigating
unexplored regions for novel bioactive compounds [16, 52]. On the contrary,
focusing on specific already-explored regions that congregate active compounds
may be of interest to generate preferred screening collections or focused compound
libraries ([6, 59]), or simply to increase the probability of finding a bioactive
compound (at the expense of chemical novelty).
Third, the clustering of small molecules c an be useful to guide the selection of
subsets of in silico hits that will be submitted for experimental confirmation
[38]. Because of funding limitations, the number of virtual hits that arise from in
silico screening experiments most often exceeds the number that can be physically
accessed (synthesized, isolated, or acquired from vendors) and evaluated experimentally. Clustering and subsequent representative sampling can guarantee the
coverage of as much chemical diversity as possible with those relatively scarce in
silico hits submitted to wet assays. Later, those experimentally confirmed hits can be
enriched by testing other compounds present in the library with the same or similar
scaffolds. Likewise, the same principle can be used to expedite structure-based
screening without exhaustive docking of every molecule in ultra-large chemical
libraries [54, 55].
112 A. Talevi et al.
1.2 Classification
Broadly speaking, clustering approaches in general (and also in the particular case of
small molecule clustering) can be classified as hierarchical or nonhierarchical
methods.
In hierarchical clustering, the data are partitioned successively, in a nested way
[18]. Bottom-up or agglomerative approaches proceed by a series of consecutive
fusions of the N objects into groups (eventually merging them all together in the last
step of the procedure). Top-down or divisive approaches, in contrast, serially divide
the N objects into smaller groups (eventually resulting in isolated elements). Hierarchical clustering is most often graphically represented as a dendrogram or tree
(a representation of clusters at multiple levels). A disadvantage of these approaches
is that once the individual elements have been merged (into supra-clusters, agglom-
erative approaches) or segregated (into subclusters, divisive approaches), this
cannot be reverted or revised in subsequent steps, that is, they cannot be optimized.
Typical examples of hierarchical methods are single, complete, and Ward linkage,
with the latter being frequently used in the context of chemical library clustering
[37]. Another example is the Maximal Common Substructure (MCS) approach,
which groups compounds based on their common subgraph of the great est cardinality. It has been implemented at Chemaxon ’ s Jchem (https://chemaxon.com/).
Nonhierarchical methods, for their part, do not resort to a tree-like structure, and
usually require, on the part of the user, to predefine the number of clusters that they
wish to obtain or the initial data points that will be assigned to each cluster, which
generally requires prior knowledge regarding the structure of the data. Moreover, as
hierarchical methods decide locally, step by step, which objects should be merged
into the same clusters or split into different clusters, they avoid hard combinatorial
optimization problems. In contrast, optimization approaches, such as those used in
nonhierarchical methods, often involve iterative processes to minimize an objective
function, which can be computationally intensive. The clusters are iteratively refined
by either minimizing or maximizing some numerical criterion, as in the popular
K-means [60] (where each cluster is represented by its centroid or center of gravity,
which in general does not match actual data point) or K-medoids algorithms [26]
(where each cluster is represented by actual data points that minimize the sum of
distances to other members of the same cluster). Both K-means and K-medoids
algorithms can be regarded as prototype-based clustering, as the data are arranged
around representative points for each partition. Sphere-exclusion algorithms, a type
of contiguity-based clustering, as in the popular method by Butina [10], represent
another example of a frequently used nonhierarchical approach.
The preceding classification is, however, extremely basic, and only one of many
other possible classification schemes [1, 14, 46, 47, 56]. For instance, we may
consider density-based algorithms (which construct density functions that reflect
the spatial distribution of the data points: each cluster corresponds to a high-density
region separated from other high-density regions by sparsely populated regions of
the space), distribution- or model-based clustering (in which the algorithms assume
5 Clustering of Small Molecules 113
Fig. 5.2 Schematic representation of some of the types of clustering mentioned in the main text
that the data are generated from a mixture of underlying probability distributions,
e.g., Gaussian); grid-based algorithms (which quantize the space into a finite
number of cells for the sake of efficiency), fuzzy or soft clustering (which, in line
with fuzzy logic, considers that a given object belongs to multiple clusters, in
contrast to traditional hard or exclusive clustering, in which each object exclusively
corresponds to a single cluster), or partial clustering (which considers that some
object may be outliers that may not belong to any cluster, in contrast with complete
clustering), among other possible classifications. Some of these approximations are
graphically schematized in Fig. 5.2.
114 A. Talevi et al.
2 Validations and Controls
A common issue in the applications of clustering techniques in the field of small
molecules is the frequent omission of validation procedures and validity indexes
(or evaluation measures) to establish the goodness of the results obtained or,
occasionally, to compare the performance of different clustering methods or parameters. The problem of the lack of adequate validation in a large number of applications cuts across many domains of cheminformatics and bioinform atics (e.g., very
frequently molecular docking applications do not resort to any sort of scoring power
validation). An exception is the field of quantitative structure–activity relationships,
where recognized experts and international organizations have long advocated the
use of well-defined techniques and metrics to evaluate the generalizability and
robustness of models [20, 48–50], which have even been used with regulatory
scope in the field of toxicology.
Cluster vali dation should be part of any cluster analysis because clustering
approaches will generally find clusters in any given data set, even if it does not
present a “natural” cluster structure.
We can mention three general approaches to validate the result of a clustering
experiment, which are [8, 47] internal, external,andrelative cluster validation.
These will be next described in separate subsections.
2.1 Internal Validation
Internal clustering validation measures how good and stable the proposed clustering
structure is, without resorting to any criterion external to the data set. Internal
validation is possibly the most objective (or “pure”) way to judge the goodness of
a given clustering structure since it does not obey any external criteria unrelated
(or diffusely connect ed) to the structure of the data. It should be noted that, in
principle, any external criterion could respond to a subjective interest of the
researcher and, therefore, bias what is considered a good result toward one that
confirms a previous hypothesis about the data. In fact, cluster techniques are usually
described as unsupervised machine-learning techniques, precisely to highlight that
they are not based on any type of prior judgment, information, or assumptions about
the data. Therefore, it sounds quite reasonable to resort to unsupervised methods to
evaluate the validity of an unsupervised method . However, as discussed extensively
in Sect. 3, the product of a clust er analysis is usually rather evaluated in practical
terms, based on its usefulness or functionality for specific objectives.
An ideal clustering structure is usually recognized as one that provides highly
cohesive clusters (i.e., with small within-cluster or intra-cluster distances) well
separated in space from each other (i.e., with large intercluster or between-cluster
distances). Connectedness, that is, to what extent an object is assigned to the same
cluster as its nearest neighbor in the data space, is also sometimes assessed. On the
5 Clustering of Small Molecules 115
other hand, a stable clustering structure would be one that is not substantially
modified when the data set is subjected to perturbation (which would suggest that
the structure of the data that has been found obeys a natural underlying structure, if
such a structure exists).
To begin with, we could point out that cohesion can be calculated based on the
sum of distances or the sum of proximities between elements assigned to the same
cluster Ci, usually weighting it by the size of the corresponding cluster, while
separation could be calculated in an analogous way but considering the distances
or proximities of elements assigned to different clusters Ci and Cj [47], where x and
y are elements of clusters:
Cohesion C
Separation C
ðÞ=
i
, C
i
proximity x, yðÞ ð5:1Þ
xϵC
i
yϵC
i
=
j
proximity x, yðÞ ð5:2Þ
xϵC
i
yϵC
j
Eventually, cohesion can also be calculated as the sum of distances between an
object and the centroid or medoid of the corresponding cluster, whereas separation
can be estimated as the distance between centroids or medoids of different clusters,
or the minim al distance between elements from different clusters.
The metrics most used to evaluate the goodness of the clustering structure
produced by a certain method simultaneously reflect, in a single variable, the
cohesion and separation of the clustering, which makes sense since cohesion and
separation show opposite trends versus the number of clusters (cohesion increases
with the number of clusters considered, while separation decreases). The Dunn index
[17] and the silhouet te width [43] are two popular examples of internal validity
measures used to validate nonhierarchical clustering approaches.
The Dunn index D is computed as the ratio between the minimal between-cluster
distance to the maximal within-cluster distance (which is also regarded as the
maximal diameter):
separation
D =
diameter
ð5:3Þ
The Dunn index can assume values between zero and infinity and must be
maximized.
The silhouette width or silhouette index can be calculated as the average of the
silhouette value S(i) across all the elements of the clustered data set.
If we call a(i) the average distance of the ith element of the data set with respect to
all the other elements of its own cluster, and if we call b(i) the minimum average
116 A. Talevi et al.
distance between the ith element and the elements of any other cluster than its own
(in other words, the average distance between element i and those elements assigned
to the close st foreign cluster), then S(i ) is calculated as
SiðÞ=
aiðÞ- biðÞ
max aiðÞ, biðÞ
fg
ð5:4Þ
A negative S(i) indicates that, probably, the ith element has been assigned to an
incorrect cluster since it is on average closer to the objects of a foreign cluster than to
those of its own. An S(i) close to zero suggests that element i is positioned
equidistantly between two clusters. The closer S(i) is to 1, the more confidence
there will be that the cluster assignment has been correct. The silhouette width ranges
from -1 to 1, and a value equal to 1 reflects ideal clustering: dense and well-spaced
clusters, with a correct assignment of all elements to their respective clust ers.
Other internal validation measures that combine within- and between-cluster
distances are the Calinski and Harabasz index, also known as the variance ratio
criterion [ 11], McClain and Rao index [33], the Gamma index [5], the Davies–
Bouldin index [15], and the SD and SDbw indexes [14], among others. The Frey
index [61], for its part, also combines measures of cohesion and separation, but can
only be used in the context of hierarchical clustering techniques. This index calculates the ratio of difference scores from two successive levels in the hierarchic tree
(level j and level j + 1). The difference between the average between-cluster distance
for each of the two hierarchical levels is placed in the numerator, while the difference
between the average within-cluster distance for the two successive levels is placed in
the denominator:
d
- d
K =
b
d
w
jþ1
jþ1
- d
b
j
w
j
ð5:5Þ
where
dbrepresents the mean between-cluster distance and dwrepresents the mean
within-cluster distance. Its authors proposed using a ratio score of 1.00 to identify the
correct cluster level. Other validation metrics commonly used in the context of
hierarchical clustering, such as the cophenetic correlation coefficient, measure the
degree of agreement between the cophenetic matrix and the proximity matrix. The
cophenetic distance between two objects, in the framework of an agglomerative
clustering approach, is the distance at which the algorithm merges those objects in
the same cluster at the same time. In the cophenetic matrix, the entries are the
cophenetic distances betw een the different pairs of objects. Further information on
these and other internal measures can be found in [14].
Stability measures, on the other hand, compare the results of clustering when it is
performed using the entire data set or after excluding one or more data points (i.e.,
they are based on a re-sample and re-cluster strategy to assess the significance of the
cluster derived from the original data set) [8, 22]. In some ways, the procedure used
to assess stability is like cross-validation procedures used in the context of
5 Clustering of Small Molecules 117
supervised machine learning techniques (such as leave-one-out or leave-some-out
cross-validation), only that in this case other numerical measures to quantitatively
evaluate the robustness of the originally generated partitions [23]. For example, the
Average Proportion of Nonoverlap measures the average proportion of observations
not placed in the same cluster by clustering based on the complete data set and
clustering based on the fraction of the data set that remains after the removal of one
or more data points. Other options include calculating and comparing the mean
within- and between-cluster distances before and after data removal. Naturally, the
greater the perturbation that a data structure resists, without substantial modifications, the more stable such a data structure will be considered. In this case, the size of
the perturbation would be related, fundamentally, to the proportion of data that are
removed.
Cluster replicability is another similar approach that can be used to assess the
robustness of a given data structure. This may be done, for instance, by separately
clustering two subsamples of a dataset and comparing the resulting clusters (e.g., in
the case of prototype-based clustering, by measuring the distances between the
corresponding centroids or medoids). Alternately, after clustering the first subsample, the objects of the second subsample can be assigned to the cluster represented by
the nearest centroid and, later, used to conduct a second independent clustering
analysis on the second subsam ple and compare the degree of agreement between the
“predicted” and “actual” cluster structures through a quantitative measure, such as
the Rand index or a variant of it [29]. Replication of the results not only across
entities but also across techniques and molecular features [40] would be indicative of
high-order stability. Importantly, Krieger and Green [29] showed that under certain
circumstances (large sample sizes and highly correlated clustering variables), internal validation may fail as criteria of goodness of clustering. Moreover, using
synthetic data, it was shown that replication experiments tend to provide high
Rand indexes at low numbers of clusters (below 5) even if the “true” number of
clusters in the data is higher [7].
2.2 External Validation
External validation uses an external criterion from outside the clustering process
(a label or dependent variable that is not used to cluster the data) to verify the ability
of the cluster to produce meaningful results [8]. But first, it is necessary to understand what is considered significant. When working with synthetic data, it is
relatively easy to establish an underlying, “true” structure of the data. On the other
hand, when working with real data, defining a truth criterion can be more complicated and, as we discuss in the next section, subject to the researcher’s personal
interests or ideas. As Milligan [36] accurately stated several decades ago: “... the
validity of the external criterion variable in an applied situation is usually open to
question. Failure to find significant group differences may be due to either a lack of
cluster structure or an invalid criterion variable.”
118 A. Talevi et al.
For example, in the field of drug discovery, the external data label may be related
to a biological activity or other pharmaceutically relevant property. Such properties
are undeniably related to the molecular structure of the data set instances; however,
the inability of a clustering algorithm to replicate the structure imposed by the
external criterion does not necessarily mean that the clustering algorithm is failing.
It is possible that the algorithm has detected a “true” cluster structure unrelated with
the chosen external criterion, or that the molecular characteristics used to partition
the data do not correlate well with the chosen class label.
Classification and regression metrics for model validation in the field of supervised machine learning have been reviewed in Chap. 6.
2.3 Relative Cluster Validation
As many clustering algorithms (e.g., the K-means and K-medoids algorithms)
require input from the user on the number of clusters to consider, a frequent
incognita when using these approaches is how to decide on such a parameter (and,
eventually, on other parameters used as input). This is often done in a rather
systematic way, by computing the value of a validity measure (e.g., an internal
validity index) across a range of values of the parameter to optimize (e.g., the
number of partitions to consider). Therefore, relative validation does not involve a
particular type of measure but uses supervised or unsupervised measures commented
on in the previous subsections for a specific purpose [47]: to optimize the value of a
parameter or hyperparameter.
Occasionally, more than one complementary measure of validity can be considered, with the subsequent visualization of solutions in two- or three-dimensional
space. This can be particularly useful for the visualization of the results obtained
using conflicting measures [22].
The elbow method is an example of a commonly used graphical approach to
decide on the number of clusters. It is usually performed in the context of the
K-means algorithm, by plotting the within-cluster-sum-of-squares (or other measure
of goodness) versus different possible values of K. The optimal K value is the point
at which an “elbow” or “knee” (i.e., a relatively sharp drop in variance) is formed in
the graph. The elbow method (like other visual approaches) has often been criticized
because of its subjective nature [62] as, at times, a clear elbow may not appear in the
plot (suggesting no “natural” group structure in the data?) or more than one elbow
may be observed (suggesting more than one “ natural” cluster structure fits the data);
even if the optimal value of the parameter under optimization is found by more
numerical (rather than just visual means), such as slope analysis, the result can be
sensitive to parameter range and pre-processing (e.g., scaling) [44]. Depending on
the criterion of goodness under use, the relationship between the validity criterion
and the number of clusters may not increase or decrease monotonically, and a
“peak,” instead of a sharp variance drop, could be found at the optimal value.
Various studies comparing the capacity of various clustering validity indexes to
5 Clustering of Small Molecules 119
identify the ground-truth cluster structure of synthetic and real-world data sets have
been published. In general, validity indices that combine some measure of withincluster and between-cluster distances are among the best-performing indices. The
seminal work of Milligan and Cooper (1985) compared the ability of 30 clustering
validity measures available at the time to identify the true structure of 108 synthetic
data sets. The Calinski and Harabasz index was the one that demonstrated the best
performance (it is worth noting that at that time the silhouet te index had not yet been
reported). A much more recent similar study [3] also compared 30 clustering validity
indexes, in this case against a much broader spectrum of data that included 720 synthetic data sets and 20 real data sets. The silhouette index obtained the best results,
closely followed by the Davies–Bouldin and Calinski–Harabasz indexes. Despite the
temporal distance between them, both studies agreed on the relatively few studies in
which the results of a clustering experiment are evaluated using statistical tests
(which is also commonplace in the field of cheminformatics), a situation that
contrasts with what occurs in the field of supervised machine learning. It should
be noted that Arbelaiz et al. observ ed that the performance of validity metrics to
identify data structure dropped substantially when noisy data sets or data sets with
overlapping clusters were analyzed. Surprisingly, introducing just 10 percent of
noisy data dropped the average value of the indices by two-thirds.
Two recently reported small molecular clustering approaches, iRaPCA and deep
clustering, use visual inspection of the silhouette width versus K to select the
adequate number of clusters [21, 38] (Fig. 5.3).
Another way in which the optimal K can be selected as the initial input for
algorithms that require pre-definition of the number of clusters involves a two-step
hybrid hierarchical plus nonhierarchical approach in which the hierarchical algorithm is used to define the number of clusters and provides an initial guess of cluster
centroids (a stochastically or systematically chosen seed for each cluster, taken from
Fig. 5.3 Silhouette width versus K plots from iRaPCA. One plot is generated per random subspace.
In the example, subset 2 (red plot) provides the best cluster structure according to the silhouette
index (about 0.9) for K = 4
120 A. Talevi et al.
the partitions of the dendrogram); these data points then serve as the starting points
for subsequent nonhierarchical clustering [18, 28]. It has been known for decades
that randomly selected seeds typically lead to poorer recoveries of known partitions
than seed selection obtained via initial hierarchical clustering [35].
A selection of freely available resources to perform small molecule clustering is
included in Table 5.1, with the authors’ appraisal of their advantages and disadvantages. Some questions to answer when planning a clustering study are listed in
Table 5.2.
3 Challenges and Perspectives
Suppose that we were assigned the “simple” task of clustering the following three
objects: a goldfish, a dolphin, and a human. Some people may argue that the goldfish
and the dolphin should be grouped together, based on criteria of shape, living
environment, external organs, etc. Other people may suggest that dolphins and
humans are more similar based, for instance, on the similarities between their
internal organs (lungs, brains, etc.). Both arguments are sound. The previous example illustrates that the cluster structure of a dataset strongly depends on the features
that have been used to partition the data: if different sets of features are considered,
we will possibly obtain different results.
It is commonly argued, on the other hand, that the cluster structures that are most
interesting are those that are “meaningful,” a concept that is usually used as a
synonym for “useful.” For instance, it could be said that a classification of books
based on subject matter, literary genre, or geographical origin of the author(s) would
be of much wider use than a classification based on the color of the book’s binding
[18]. However, if we Google “noir books,” we may be surprised to see that the colors
red and black predominate in the cover images that the search returns. Another
curious fact that contributes to this line of argument: in Italy, the books belonging to
the genre of crime fiction are known as libri gialli (literally, “yellow books”),
because back in the 1920s the prestigious publishing house Mondadori characterized
a popula r series of crime fiction with yellow covers.
This second example illustrates two other important points. The first is that the
features that a researcher chooses based on the “natural” structure that he subjectively anticipates in the data can omit valuable (but unexpected) features that could,
in turn, lead to a useful or meaningful cluster structure (e.g., cover colors as a
possible feature to group books) and that, at first glance, would surely be discarded
in favor of more “reasonable” features. Second, if the structure of the data depends
on the variables used to partition it, if unsuspected variables can lead to valuable
clusters, and if the value criterion varies according to the objective of the clustering
experiment, it is possible that there is no natural underlying structure (ground truth)
to structure the data.
Although the existence of such ground truth is assured when working with
synthetic data, it is not so evident when working with real data. The personal view