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228 Handbook of Chemoinformatics Algorithms

02468

Observed activities

Predicted activities

y = 0.91x + 0.62

= 0.80

FIGURE 7.3 A typical example when one outlier increases the prediction accuracy. Without

the outlier, R

= 0.13.

obtained (see Section 6.4) and each descriptor matrix is processed by some algorithms

(depending on the user’s decision and types of descriptors and optimization proce-

dures selected) described in Section 6.5 and possibly other procedures not mentioned

here. If necessary, a diverse subset of compounds is selected from the entire modeling

set (see Section 6.6). Possible structural and activity outliers are removed from the

modeling set (see Section 6.7). Additional procedures may be required for classifica-

tion and category QSAR modeling, especially, if a dataset is imbalanced or large (see

Section 6.8). Each dataset should be divided into modeling and external evaluation

sets (see Sections 6.9 and 6.10). For each descriptor set, splitting a modeling set into

multiple training and test sets is carried out using a sphere-exclusion algorithm (see

Section 6.10). Then the models are developed (see Section 7.1) for all combinations

of pairs (descriptor collection, method); appropriate target function and criteria of

predictivity are applied (see Section 7.2) based on the nature of the response variable

(continuous, classification, or category). The models with acceptable statistics for

both training and test sets (for the test set prediction, the respective ADs are used—

see Section 7.5), if any, should be rigorously validated by procedures described in

Section 6.9 and Sections 7.3 and 7.4, and some other procedures and their ADs should

be established (see Section 7.5). Finally, the results of QSAR modeling will be ready

to be used for chemical database or virtual library mining.

With more than 40 years of history behind it, QSAR modeling is a well-established

research field that (as perhaps with any scientific area) had its ups and downs. There

were several recent publications that criticized the current state of the field. Thus, a

recent editorial published by the leading chemoinformatics Journal of Chemical Infor-

mation and Modeling (JCIM; also reproduced by the Journal of Medicinal Chemistry)

introduced severe limitations on the level and quality of QSAR papers to be considered

acceptable [37]. Another recent editorial opinion by G. Maggiora [38] outlined lim-

itations and some reasons for failures of QSAR modeling that relate to the so-called

activity cliffs. In another recent important paper, T. Stouch addressed the question as

Predictive Quantitative Structure–Activity Relationships Modeling 229

to why in silico ADME/Tox models fail [39]. These examples naturally lead to an

important and perhaps critical question as to whether there is any room for further

advancement of the field via innovative methodologies and important applications.

Our previous and ongoing research in the area of QSAR suggests that the answer

is a resounding “yes.” We believe strongly that many examples of low impact QSAR

research are due to frequent exploration of datasets of limited size with little attention

paid to model external validation. This limitation leads to models having questionable

“mechanistic” explanatory power but perhaps little if any forecasting ability outside

of the training sets used for model development. We believe that the latter ability

along with the capabilities of QSAR models to explore chemically diverse datasets

with complex biological properties should become the chief focus of QSAR studies.

This focus requires the re-evaluation of the success criteria for the modeling as well

as the development of novel chemical data mining algorithms and model validation

approaches. In fact, we think that the most interesting era in QSAR modeling is just

beginning with the rapid growth of the experimental SAR data space [40].

In the last 15 years, innovative technologies that enable rapid synthesis and high-

throughput screening of large libraries of compounds have been adopted in almost

all major pharmaceutical and biotech companies. As a result, there has been a huge

increase in the number of compounds available on a routine basis to quickly screen for

novel drug candidates against new targets or pathways. In contrast, such technologies

have rarely become available to the academic research community, thus limiting its

ability to conduct large-scale chemical genetics or chemical genomics research. The

NIH Molecular Libraries Roadmap Initiative has changed this situation by forming

the national Molecular Library Screening Centers Network (MLSCN) [41] with the

results of screening assays made publicly available via PubChem [42]. These efforts

have already led to the unprecedented growth of available databases of biologically

tested compounds (cf. our recent review where we list about 20 available databases

of compounds with known bioactivity [40]). This growth creates new challenges for

QSAR modeling such as developing novel approaches for the analysis and visual-

ization of large databases of screening data, novel biologically relevant chemical

diversity or similarity measures, and novel tools for virtual screening of compound

libraries to ensure high expected hit rates. Due to the significant increase in recent

years of the number of publicly available datasets of biologically active compounds

and the critical need to improve the hit rate of experimental compound screening,

there is a strong need in developing widely accessible and reliable computational

QSAR modeling techniques and specific end-point predictors.

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232 Handbook of Chemoinformatics Algorithms

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Structure Enumeration

and Sampling

Markus Meringer

CONTENTS

8.1 Isomer Counting................................................................234

8.1.1 Counting Permutational Isomers ......................................235

8.1.2 Counting Isomers of Acyclic Structures and Other Compound

Classes..................................................................239

8.2 Isomer Enumeration: Deterministic Structure Generation ...................241

8.2.1 Early Cyclic and Acyclic Structure Generators.......................241

8.2.1.1 Acyclic Structure Generators ...............................241

8.2.1.2 Cyclic Structure Generator ..................................242

8.2.2 Orderly Generation ....................................................246

8.2.2.1 Enumerating Labeled Graphs ...............................246

8.2.2.2 Enumerating Unlabeled Graphs.............................247

8.2.2.3 Introducing Constraints .....................................248

8.2.2.4 Variations and Refinements .................................249

8.2.2.5 From Simple Graphs to Molecular Graphs .................250

8.2.3 Beyond Orderly Generation ...........................................252

8.3 Isomer Sampling: Stochastic Structure Generation...........................253

8.3.1 Uniformly Distributed Random Sampling ............................253

8.3.2 Monte Carlo and Simulated Annealing ...............................254

8.3.3 Genetic Algorithms ....................................................257

8.4 Beyond Isomer Enumeration ..................................................259

8.4.1 Virtual Chemical Space................................................259

8.4.2 Combinatorial Libraries ...............................................261

8.4.2.1 Counting Combinatorial Libraries ..........................261

8.4.2.2 Generating Combinatorial Libraries ........................263

Acknowledgment .....................................................................264

References ............................................................................264

Chemical structure enumeration and sampling have been studied by mathematicians,

computer scientists, and chemists for quite a long time. Given a molecular formula

plus, optionally, a list of structural constraints, the typical questions are: (1) How

233

234 Handbook of Chemoinformatics Algorithms

many isomers exist? (2) Which are they? And, especially if (2) cannot be answered

completely: (3) How to get a sample?

In this chapter, we describe algorithms for solving these problems. The techniques

are based on the representation of chemical compounds as molecular graphs (see

Chapter 2), that is, they are mainly applied to constitutional isomers. The major

problem is that in silico molecular graphs have to be represented as labeled structures,

while in chemical compounds, the atoms are not labeled. The mathematical concept

for approaching this problem is to consider orbits of labeled molecular graphs under

the operation of the symmetric group. We have to solve the so-called isomorphism

problem.

According to our introductory questions, we distinguish several disciplines: count-

ing, enumerating, and sampling isomers. While counting only delivers the number of

isomers, the remaining disciplines refer to constructive methods. Enumeration typi-

cally encompasses exhaustive and non-redundant methods, while sampling typically

lacks these characteristics. However, sampling methods are sometimes better suited

to solve real-world problems.

There is a wide range of applications where counting, enumeration, and sampling

techniques are helpful or even essential. Some of these applications are closely linked

to other chapters of this book. Counting techniques deliverpure chemical information;

they can help to estimate or even determine sizes of chemical databases or compound

libraries obtained from combinatorial chemistry.

Constructive methods are essential to structure elucidation systems (see Chap-

ter 9). They are used to generate structures that fulfill structural restrictions obtained

from spectroscopy in a pregeneration step, while in a postgeneration step, virtual

spectra of the generated structures can be computed and compared with the measured

data in order to determine which of them achieves the best fit.

Other applications use structure enumeration algorithms in order to produce

candidate structures for virtual screening (see Chapter 5). Structure–activity and

structure–property relationships, as introduced in Chapter 6, can be used in com-

bination with structure enumeration or sampling as rudimentary approaches toward

inverse QSAR (see Chapter 10) and de novo design algorithms often have their roots

in conventional structure generation.

The nonquantitative aspects of reaction network generation (see Chapter 11) are

also based on methods similar to those used for isomer enumeration.

8.1 ISOMER COUNTING

Counting means that only the number of structures is calculated and the structures

themselves are not produced by the algorithm. The most powerful counting technique

available to chemists is Pólya’s theorem [1], see also Refs. [2,3]. There are, of course,

various predecessors, for example a paper by Lunn and Senior [4], who were the first

to note that group theory plays a role, and a paper by Redfield [5] that contained even

better results. However, Pólya’s paper gave rise to the development of a whole theory

that is nowadays called Pólya’s Theory of Counting. Typical applications are counting

of permutational isomers and acyclic compounds.

Structure Enumeration and Sampling 235

8.1.1 COUNTING PERMUTATIONAL ISOMERS

Pólya’s approach to the enumeration of molecules with a given molecular formula is to

subdividethe molecule in question into a skeleton and a set of univalent substituents.It

leads to the following challenge: Evaluate the set of essentially different distributions

of the substituents over the sites of the skeleton with respect to the given symmetry

group of the skeleton.

In mathematical terms, the symmetry group G acts on the set of mappings m

from

the n sites of the skeleton onto the m available substituents. The set of orbits under this

group operation, m

// G, is in a one-to-one relation with the different constitutions.

If the skeleton shows no symmetries, that is, if G is of order 1, then it is clear that

there are m

different substitutions. Note that m

has two different meanings, one for

denoting the set of mappings and another for its cardinality. If the order of G is larger

than 1, the situation is more interesting.

The resulting isomers are called permutational or substitutional isomers. For

example the 22 permutational isomers of dioxin (tetrachlorodibenzo-p-dioxin) are

the essentially different distributions of four hydrogen and four chlorine atoms over

the eight sites of the skeleton depicted on the left:

Counting these isomers is described in detail in Kerber’s comprehensive book [6]

on finite group actions. In this section we will discuss the example of permutational

isomers of dichlorobenzene, which are based on the benzene skeleton sketched on

the right.

First we will try to describe Pólya’s approach in general. The symmetry group G

of the skeleton with respect to the n binding sites is required as input. The procedure

works with the topological symmetry group as well as with the geometrical symmetry

group. Of course, the results might differ. For ways to compute a molecule’s symmetry

group, see Chapter 2. A suitable data structure for representing groups is described

by Sims [7].

The reader should be familiar with the cycle notation of permutations, which is

briefly described in Example 8.1.1. At this point, it is useful to know that every

permutation has a unique decomposition into disjoint cycles. For more details of

permutations and cycles, the reader is referred to Ref. [6].

The cycle index of a permutation g ∈ G is a monomial in variablesz

. It is definedas

Z(g) =

k=1

(g)

, (8.1)

236 Handbook of Chemoinformatics Algorithms

where c

(g) is the number of cycles of g having length k. The cycle index Z(G) of G

is the averaged sum of cycle indices of group elements:

Z(G) =

|G|



g∈G

k=1

(g)

. (8.2)

In order to obtain a counting series from the cycle index, a so-called generating

function has to be inserted. Having m > 1 different chemical elements to choose from,

a suitable generating function is y

+···+y

. Inserting the generation function into

the cycle index means that every occurrence of z

in Z(G) is replaced by y

+···+

.Ifm = 2 the easier generation function 1 +x can be used instead, and during

insertion, z

is replaced by 1 +x

The coefficient of the monomial

i=1

of the counting series equals the number

of isomers with j

substituents of type i. In the case where we have just two different

elements to substitute, say H and Cl, the coefficient of x

equals the number of isomers

with j hydrogen atoms. To summarize, we can formulate the following.

ALGORITHM 8.1.1 COUNTING PERMUTATIONAL ISOMERS BY

PÓLYA’S THEOREM

1. Calculate the cycle index of the skeleton with respect to the binding sites

2. Insert the generation function into the cycle index

3. Evaluate the coefficients of the counting series

This formal description of how to solve the counting problem in general needs to be

illustrated by an example. Counting C

constitutions with a benzene skeleton

will be explained step by step.

Example 8.1.1: Counting Isomers of Dichlorobenzene

Below we see the benzene skeleton together with its symmetry axes (1), ..., (6).

(1)

(2)

(3)

(4)

(6) (5)

Besides six axis reﬂections, the benzene skeleton allows some more symmetry

operations, namely ﬁve proper rotations.Table 8.1 lists all the symmetry operations

and the according permutations of benzene’s symmetry group D

. E denotes the

identity, C

+/−

represents a rotation by 360/i degrees, where the sign describes

the direction of the rotation, and σ

( j)

represents a reﬂection at axis ( j).

Structure Enumeration and Sampling 237

TABLE 8.1

Permutations of the Automorphism Group D

of Benzene

List Cycle Cycle

Operation Representation Representation Index

E [123456] (1)(2)(3)(4)(5)(6) z

[234561] (123456) z

−

[612345] (654321) z

[345612] (135)(2 4 6) z

−

[561234] (531)(6 4 2) z

[456123] (14)(2 5)(3 6) z

(1)

[165432] (1)(4)(2 6)(3 5) z

(2)

[543216] (3)(6)(1 5)(2 4) z

(3)

[321654] (2)(5)(1 3)(4 6) z

(4)

[654321] (16)(2 5)(3 4) z

(5)

[432165] (14)(2 3)(5 6) z

(6)

[216543] (12)(3 6)(4 5) z

Permutations are given in two notations. The list representation might appear

more straightforward to the reader, because for a permutation π the ith component

in the list simply deﬁnes the image of i, that is the list representation of π is

[π(1), ..., π(n)].

The cycle notation is a little more difﬁcult to understand, but gives direct access

to the cycle index, which is needed to compute counting series. The cycle repre-

sentation consists of 1 to n cycles, which are enclosed by round brackets. Each

cycle itself consists of 1 to n elements. Cycles of only one element (i) show that i

is ﬁxed under the permutation, that is, π(i) = i. Sometimes such cycles are even

suppressed in cycle notations. Cycles (i

, ..., i

) with more than one element indi-

cate that i

is mapped onto i

, i

is mapped onto i

, and so on. The length l of

the cycle is determined by the minimum number of applications of π that map i

again onto i, that is, l = min{h : π

(i) = i}. In particular, this means that the last

element of the cycle, i

, is mapped onto i

, and generally the cycle of i

can be

written as [i

, π(i

), π

), ..., π

l−1

)].

Finally, the cycle indices of the elements of the automorphism group can be

derived directly from the cycle notations using Equation 8.1. This results in the

cycle index

Z(D

) =

+4z

+2z

+3z

)

for benzene’s automorphism group D

. Inserting the generating function 1 +x

leads to the counting series

C(D

) =

((1 + x)

+4(1 +x

)

+2(1 +x

)

+ 2(1 + x

) + 3(1 +x)

(1 + x

)

= 1 +x + 3x

+3x