Backbone diversity analysis in catalyst design

Article abstract of DOI:10.1002/adsc.200800574

We present a computer-based heuristic framework for designing libraries of homogeneous catalysts. In this approach, a set of given bidentate ligand-metal complexes is disassembled into key substructures ("building blocks"). These include metal atoms, ligating groups, backbone groups, and residue groups. The computer then rearranges these building blocks into a new library of virtual catalysts. We then tackle the practical problem of choosing a diverse subset of catalysts from this library for actual synthesis and testing. This is not trivial, since 'catalyst diversity' itself is a vague concept. Thus, we first define and quantify this diversity as the difference between key structural parameters (descriptors) of the catalysts, for the specific reaction at hand. Subsequently, we propose a method for choosing diverse sets of catalysts based on catalyst backbone selection, using weighted D-optimal design. The computer selects catalysts with different backbones, where the difference is measured as a distance in the descriptors space. We show that choosing such a D-optimal subset of backbones gives more diversity than a simple random sampling. The results are demonstrated experimentally in the nickel-catalysed hydrocyanation of 3-pentenenitrile to adiponitrile. Finally, the connection between backbone diversity and catalyst diversity, and the implications towards in silico catalysis design are discussed.

Full text of DOI:10.1002/adsc.200800574

FULL PAPERS
DOI: 10.1002/adsc.200800574
Backbone Diversity Analysis in Catalyst Design
a
b
c
Ana G. Maldonado, Jos A. Hageman, Sergio Mastroianni,
a,
and Gadi Rothenberg *
Vanꢀt Hoff Institute of Molecular Sciences, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam,
The Netherlands
a
Fax : (+31)-(0)20-525-5604; phone: (+31)-(0)20-525-6963
g.rothenberg@ uva.nl
b
c
Biometris, Wageningen University & Research Centre, Bornsesteeg 47, 6708 PD Wageningen, The Netherlands
Rhodia CRL, 85 Av Frꢁres Perret, 69190 St Fons, France
Received: September 17, 2008; Revised: December 18, 2008; Published online: February 4, 2009
Abstract: We present a computer-based heuristic catalysts based on catalyst backbone selection, using
framework for designing libraries of homogeneous weighted D-optimal design. The computer selects
catalysts. In this approach, a set of given bidentate catalysts with different backbones, where the differ-
ligand-metal complexes is disassembled into key sub- ence is measured as a distance in the descriptors
structures (“building blocks”). These include metal space. We show that choosing such a D-optimal
atoms, ligating groups, backbone groups, and residue subset of backbones gives more diversity than a
groups. The computer then rearranges these building simple random sampling. The results are demonstrat-
blocks into a new library of virtual catalysts. We then ed experimentally in the nickel-catalysed hydrocya-
tackle the practical problem of choosing a diverse nation of 3-pentenenitrile to adiponitrile. Finally, the
subset of catalysts from this library for actual synthe- connection between backbone diversity and catalyst
sis and testing. This is not trivial, since ꢂcatalyst di- diversity, and the implications towards in silico catal-
versityꢀ itself is a vague concept. Thus, we first define ysis design are discussed.
and quantify this diversity as the difference between
key structural parameters (descriptors) of the cata- Keywords: catalysis descriptors; combinatorial
lysts, for the specific reaction at hand. Subsequently, chemistry; homogeneous catalysis; ligand design;
we propose a method for choosing diverse sets of structure-activity relationship
Introduction
lem. Generating millions of structures via computer is
easy, but how should one choose the candidates for
Sustainability and sustainable development are high synthesis and testing? Moreover, how should one
st
on our agenda in the 21 century. The tools of green select the right components for assembling these vir-
chemistry and catalysis play a key role here, as they tual libraries?
can provide mankind with bona fide solutions to envi-
The pharmaceutical industry has solved an analo-
[
1,2]
ronmental problems.
Importantly, chemists now gous problem in the field of drug design. They used
also have suitable experimental tools for approaching topological modelling, as well as selection and classifi-
[
3,4]
these problems.
combination of laboratory automation,
modelling and data mining algorithms puts us on the groups of features that embody drug efficiency. Such
In homogeneous catalysis, the cation methods for screening large virtual libraries of
[
5–7]
[11]
advanced drug candidates.
These methods aim to identify
[
8]
brink of in silico catalyst design. Much progress has two-dimensional descriptor models are less common
been made since the 1990s in quantitative structure/ in catalysis, where mechanistic studies rely primarily
[
12]
activity relationships (QSAR) and quantitative struc- on quantum mechanical (QM) calculations.
ture/property relationship (QSPR) models.
The
[
9,10]
To re- reason for this is simple. You get what you pay for,
alise the goal of true in silico catalyst design, however, and topological modelling is computationally cheap.
we must be able to assemble and screen virtual libra- Thus, the results of QSAR/QSPR models based on
ries of ligands and ligand-metal complexes (Figure 1). topological descriptors are less accurate than those
This new mode of operation requires also a new based on QM ones. Nevertheless, unlike QM calcula-
mode of thinking, especially where large libraries are tions, topological descriptors give a realistic approach
concerned. Here, catalyst selection is the major prob- for modelling large catalyst libraries. Recently, we
Adv. Synth. Catal. 2009, 351, 387 – 396
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Moreover, the bidentate structure offers ample oppor-
tunities for fine-tuning and optimising a variety of ste-
[
19,20]
reoelectronic effects.
clude Noyoriꢀs BINAP ligands, the Josiphos ligands
Some notable examples in-
[21]
[
22]
used in the Ciba-Geigy metolachlor process, Jacob-
[
23,24]
senꢀs catalyst,
and the Keim complexes used in
[
25]
the Shell higher olefins process (SHOP). Although
there are several important studies on bidentate
[
26,27]
ligand descriptors,
there is only scant information
regarding the problem of ligand diversity for library
[
15]
assembly and catalyst selection. Here, we examine
measures of diversity for bidentate ligands, with a spe-
cific emphasis on the ligand backbone. As an exam-
ple, we study the nickel-catalysed hydrocyanation of
3-pentenenitrile to adiponitrile. We also discuss how
these measures can help chemists take better deci-
sions when applying in silico catalyst design.
Results and Discussion
Relating Backbone Diversity and the Catalyst Space
Figure 1. Flow chart showing the screening of virtual libra-
ries and subsequent iterative modelling and laboratory vali-
dation. In each iteration, promising candidates are synthes-
ised and tested, and results are fed back to the model. Note
that the sets of catalysts that are actually synthesised are
One can envisage diversity in homogeneous catalysis
as a spread of ligand-metal complexes over a given
catalyst space, or as a lack of similarity within a set of
much smaller than the virtual library. The primary challenge, catalysts. Saying that “library I is more diverse than li-
therefore, is choosing the right candidates for building the li- brary II” can mean that library I covers a larger
brary and validating the model.
volume of the catalyst space. Alternatively, it can
mean that the catalysts in I are better spread in the
same space. To illustrate this, we divide the system in
showed that by combining chemical principles, topo- three spaces (Figure 2). The first space, A, is the cata-
logical descriptors, and statistical analysis, one can lyst space. It is a grid containing a large number of
[
13]
predict important parameters in such libraries.
Here we apply the same principles to the complex notes one catalyst that has been tested experimental-
problem of ligand library diversity analysis. ly. Space B, or descriptor space, contains the values of
metal-ligand complexes. Each point in space A de-
Practically speaking, a library of catalysts should be the catalystsꢀ descriptors (internal parameters such as
as diverse as possible, while still maintaining a realis- backbone flexibility, partial charge on the metal atom,
[14]
tic size. Unfortunately, although the term “diversi- polarity or lipophilicity) as well as the reaction condi-
ty” is widely used in homogeneous catalysis, it is ill- tions (external parameters such as temperature, pres-
defined. Everyone agrees that a diverse library of li-
gands is ꢂone that gives a good spread of resultsꢀ. In
practice, however, diversity is problem-related, since
different chemical reactions are often sensitive to dif-
ferent parameters (see below for details). When deal-
ing with large libraries (thousands of candidates and
more), one cannot estimate the diversity by simply
looking at the structures. One way to solve this prob-
lem is by defining diversity indices, that can be calcu-
[
15]
lated automatically.
This requires descriptors that
quantify the ligandsꢀ structure-activity relationships.
These descriptors should be chemically relevant, and
simple and easy to calculate.
Figure 2. Simplified representation of the multi-dimensional
spaces A, B, and C, containing the catalysts, the molecular
descriptor values, and the figures of merit, respectively. For
clarity, we show only three axes per space. In reality, the
spaces are multi-dimensional, with each dimension repre-
[
13,16]
Organometallic complexes with bidentate ligands
are well established as catalysts in academia and in-
[17,18]
dustry.
Thanks to the chelate effect, they are senting a structural feature, a descriptor or a figure of merit,
more stable and robust than monodentate complexes. respectively.
388
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Backbone Diversity Analysis in Catalyst Design
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sure, and solvent type). All of these parameters may
influence the reaction outcome, and a model may in-
clude both internal and external descriptors. Finally,
C is the space of the figures of merit (FOM, e.g., the
TON, TOF, product selectivity, price, and so forth).
Note that while space A is a discrete grid, B and C
are continuous, and are arranged such that each di-
mension represents one property. Essentially, spaces
B and C are connected via structure/activity and
structure/property relationships.
Traditionally, chemists define diversity in space A,
by comparing molecular structures and especially
functional groups. However, relevant differences be-
tween catalysts are much better represented by dis-
tances in space B, the descriptor space. Thus, instead
of evaluating molecules based on their ꢂChemdrawꢄ
structuresꢀ, we should compare them based on their
descriptor values. Choosing the right descriptors is a
crucial step, because diversity is problem-related. One
reaction can be controlled by the bite angle, while for
another, the reaction temperature and pressure may
be the key parameters (broader discussions on de-
Figure 3. The nickel-catalyzed hydrocyanation of butadiene
to 3-pentenenitrile (top) and the subsequent hydrocyanation
[28]
[27,29]
scriptor types and descriptor selection
are pub-
lished elsewhere). Here, we focus on the backbone di- of 3-pentenenitrile to adiponitrile (bottom).
versity of bidentate ligands. The reason for this is that
the ligand backbone often dictates the final size, flexi-
bility, and electronic properties of the complex. These
are directly related to the reaction pocket environ-
[
27]
ment and the catalytic performance. Moreover, the
concept of derivative synthesis is based on a common
backbone, and indeed different ligand types are often
referred to by their backbone designation (e.g., Salen
or BINAP catalysts). Importantly, ligand libraries are
often prepared in practice via derivative synthesis,
i.e., by assembling various ligating and residue groups
[
30]
onto a small number of scaffolds. This means that
backbone diversity holds the key to catalyst diversity.
Case Study: Ni-Catalysed Hydrocyanation
With the above set-up in mind, we turned to a real- Figure 4. Example of a biphosphine-nickel complex, one of
life example of a homogeneously catalysed reaction: the initial set of 115 catalysts (top); cartoon showing the
formal division into ligating (L), backbone (B), and residue
groups (R, bottom).
the nickel-catalysed hydrocyanation of 1,3-butadiene
[
31]
to adiponitrile (Figure 3). This important industrial
process forms the basis of Nylon 6.6 and Nylon 6 pro-
duction. Despite the fact that this is a well known
[
32,33]
[8]
process, catalyst optimisation remains a challenge.
We focus here on the second step, the hydrocyanation 115 ligands reduces to 25 ligating groups, 60 residue
of 3-pentenenitrile to adiponitrile (Figure 3, bottom). groups and 42 backbones. The backbone structures
the different building blocks. In total, our dataset of
For our initial library (space A), we selected 115 bi- are shown in Figure 5. Even with such small numbers,
phosphite and biphosphine ligand-nickel complexes assessing the backbone diversity by simply looking at
[
34]
from over 200 articles and patents.
Each catalyst the structures is difficult. The task becomes impossi-
was divided into two ligating groups, a backbone, and ble when one moves from backbones to actual cata-
up to four residue groups. This division gives a stan- lysts, since the 115 catalysts by no means cover the
dard framework for backbone comparison. Figure 4 entire combination space. The entire catalyst space
6
0
[35]
shows an example of such catalyst structures, denoting contains over 10 compounds. Even the number of
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Figure 5. Structures of the 42 backbones used in this study. The backbones belonging to the D-10 subset (vide infra) are
framed.
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Table 1. Examples of typical molecular descriptors.
main attributes: size, flexibility, and polarity. This
gives a matrix, where each row corresponds to a back-
bone and each column to a descriptor. Plotting these
attributes gives a 3D backbone “map”. Such maps
allow us to compare the diversity of backbone sets,
and ultimately of ligands and catalysts libraries.
Descriptor Typical representation
type
Examples
1D
C H NiOP
2
Molecular weight;
atom counts
3
9
32
Figure 6 shows the three-dimensional “map” for the
42 backbones of the bidentate ligand-Ni complexes.
The two-dimensional projections on the different de-
scriptor axes are shown in Figure 7. These maps tell
us how “different” the backbones are from each
other. If two dots are far away from each other (e.g.,
Fragment counts;
topological indices;
connectivity
2D
19 and 20), the backbones are different. Conversely, if
the dots are close to each other (e.g., 35 and 36), the
backbones are similar. Looking at Figure 6, we see
four main clusters. Some of these are easily explained.
Backbones 35, 36 and 37, all contain a very bulky sub-
stituted para-benzene moiety. Ferrocene, (backbone
Molecular surface;
molecular volume;
interaction energies
3D
31) due to its particular electronic and steric proper-
ties, is an outlier. The two large clusters are clearly in-
fluenced by the size parameter, as shown in Figure 7
(
highlighted in gray in the middle and bottom
graphs).
To demonstrate the experimental validity of our hy-
building block permutations is an overwhelming 3.4ꢅ pothesis, we chose three catalysts, derived from the
9
1
0 . Such numbers of catalysts are beyond computa- backbones shown in Figure 5. Two figures of merit
tional treatment, let alone synthesis. Thus, we limit were measured experimentally for each catalyst: 3-
our diversity analysis to the backbones only. Our ob- pentenenitrile conversion and product selectivity (see
jective is to see whether this analysis can help us the Experimental Section for details). The three
select a representative subset of backbones, which ligand-Ni complexes (structures 43–45 in Figure 8) are
would then be used for synthesising a representative based on backbones 15, 6, and 31, respectively.
test set of catalysts.
Our hypothesis states that if the catalyst backbones
For all the backbones, we calculate a set of selected are close to each other in the descriptor space, B,
descriptors. Choosing relevant descriptors is a chal- then these catalysts will show similar figures of merit.
[16,36–38]
lenge.
Descriptors can be determined either Comparing the FOM values in Figure 8, with the
from the molecular structure, or the physical proper- three-dimensional descriptor plot in Figure 6 we see
[14]
ties. Table 1 shows some typical examples of molec- that 43 and 44, belong to the same backbone cluster.
ular descriptors for the complex shown in Figure 4. The structures may differ, but their properties are
Depending on cost limits and the problem at hand, similar. Happily, both ligands indeed give similar se-
one can use 2D (ꢂquick and dirtyꢀ) descriptors and/or lectivity for adiponitrile (73.0% and 73.5%, respec-
3D (more accurate, but also more costly) descriptors. tively). The conversion rates are less similar, but still
Drug design companies use different strategies for within 10%. Conversely, complex 45 is based on a
guiding descriptor selection. One is selecting a partic- very different backbone, when comparing descriptor
ular set of descriptors that show good performance on values, and gives very different figures of merit.
[
11,39]
a given problem.
culate a large number of descriptors and then remove
Alternatively, you can first cal-
[
40,41]
any correlated ones.
Another different approach Choosing the “Right” Backbones for Diversity
is letting the computer choose the optimal combina-
tion of descriptors by testing all the possible combina- All the possible combinations of a set of backbones,
tions. In our previous studies of bidentate ligand- ligands and residues represent on one hand a small
nickel complexes, we showed that the most important part of the whole chemical space, and on the other
topological descriptors were related to the bite angle hand an impossible large set for synthesis and testing.
[
13]
and flexibility,
while the most relevant electronic One must choose a small catalysts subset, and this is
[
27]
descriptors were related to charge densities. Here, precisely where the usefulness of our analysis comes
we calculated a total of 168 topological, constitution- in. Analysing the graphs shown in Figure 6 and
al, geometric and electrostatic descriptors for all the Figure 7, we can select a representative subset of
42 backbones. This set was then ranked down to three backbones, and from these generate a diverse set of
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Figure 6. Three-dimensional descriptor space plot of the full set of 42 backbone structures (top), highlighting the four main
clusters. The zoom-in view shows the distribution of the structures in the main cluster (bottom).
new ligands for synthesis purposes. This can be done, scriptor space is short. For example, if synthesising a
[
42,43]
for example, using D-optimal design
. We parti- complex with backbone 36 is difficult, we could prob-
tion the space into n equal partitions, choosing from ably use backbone 35 instead (see Figure 6).
each partition a number of backbones that depends
on the backbone density in that partition. Such a
weighted selection gives each partition the correct in- Exploring Diversity
fluence on the results (a good analogy is the US presi-
dential elections, where each state has a number of To analyse the backbone distribution in the descriptor
electoral votes that is proportional to the number of space, we used the average distance of each backbone
[
15]
its inhabitants).
to all other backbones. This is the mean intermolec-
Such a selection gives an optimal coverage of the ular distance, daverage, given by Eq. (1) (where d is the
ij
catalyst space for a given number of experiments.
There is less chance of missing a good candidate,
compared with a random sampling of the catalyst
space. Importantly, the distances in the descriptor
space reflect the backbonesꢀ similarity. This means
that if synthesising the combination {L ACHTUNGTRENNNU(G R ,R )-B-L -
1 1 2 2
ACHTUNGTRENNUNG( R ,R )} is difficult, you can use an alternative back- distance between points i and j, and N is the number
3 4
bone, B’, providing that the distance BB’ in the de- of backbones). If daverage is large, the backbones in the
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Figure 8. Catalyst structures used in the experimental dem-
onstration, with their corresponding figures of merit.
(
where dmin,i is the distance between point i and its
nearest neighbour). dmin gives an idea of the clustering
of the backbones in space B. If dmin is small, each
backbone has at least one other backbone close by.
Conversely, if dmin is large, the backbones are on aver-
age far apart.
A diverse library should be large, with the back-
bones distributed well over the space. Thus, the prod-
uct daverage ꢅdmin is a good measure of library diversity.
Detailed discussions of daverage and dmin, complete with
[1,15]
worked-out examples, are published elsewhere.
Table 2 compares the daverage ꢅdmin values for three
Figure 7. Two-dimensional projections of the 3D plot shown
in Figure 6, on the MW/flexibility (top), flexibility/polarity backbone sets: The full set of 42 backbones, a ran-
medium) and polarity/MW (bottom) axes. The shaded areas domly chosen set of 10 backbones (R-10), and a set of
represent the main backbone cluster {1, 3, 5, 6, 9, 10, 11, 12,
3, 14, 15, 20, 21, 22, 23,24, 25, 26, 27, 29, 33, 34, 39, 42}.
(
1
0 backbones chosen using weighted D-optimal
1
design in the descriptor space (D-10). Note that se-
lecting the latter two subsets requires multiple sam-
pling (see the Experimental Section for details). The
library are further apart. This means that the library
covers a larger volume of space B.
However, daverage does not tell us how the backbones
are distributed. For this, we must also calculate the
average distance from each backbone to its nearest
neighbour. This parameter, dmin, is defined by Eq. (2)
Table 2. Diversity indices for various backbone sets.
Dataset
Full set
daverage
dmin
daverage ꢀdmin
118.0
140.6
139.4
7.2
858
[a]
R-10
D-10
20.1
30.9
2829
4308
[a]
Representative average set from a population of 500
random subsets, see Experimental Section for details.
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space can be tackled by isolating the main backbone
motifs from ligand libraries, creating a representative
set of manageable proportions. This set can then be
modeled using 2D and 3D descriptors. However, for
synthesis purposes, a subset of backbones must be
chosen. This choice is not trivial, but it can be done
by selecting structures that are far away from each
other in the descriptor space. As shown here, this is
easily done using weighted D-optimal design. This se-
lection technique gives more diverse subsets than
random sampling. The final backbone set can then
be combined with selected ligating and residue
Figure 9. Backbone three-dimensional descriptor space for
the selected subsets: R-10 (left) and D-10 (right).
daverage ꢅdmin value for the full set of 42 backbones is groups, generating a diverse library of homogeneous
low, mainly due to the lower value of the dmin parame- catalysts.
ter in comparison with those of R-10 and D-10. If dmin
is small, there is a high clustering of the data. This is
shown in Figure 6 and Figure 7 for the set of 42 back- Experimental Section
bones. The main clusters are in the lower polarity (0–
0.3) and lower flexibility (0–0.2) regimes. Indeed, the Computational Methods
high average daverage ꢅdmin for the R-10 backbone
subset is an evidence of the poor diversity of the 42 Ligand Construction and Optimisation
backbones subset, since any subset of 10 random
backbones will represent better the structural and The 42 backbone structures used in the analysis were chosen
manually from a dataset of 115 biphosphite and biphosphine
ligand-nickel complexes, taken from papers and patents
on the Ni-catalysed hydrocyanation of butadiene. Each
structure was decomposed following the building block
scheme described in Figure 4. Ligand geometry optimisation
properties features of the data. The R-10 and D-10
[34]
daverage ꢅdmin values are very different. The D-10 subset
is more diverse. Interestingly, the main difference is
in dmin, while daverage remains almost constant for both
subsets.
When studying the random subsets we observed a
pattern for some subsets that showed high diversity.
(
for calculating the 3D descriptors) was performed using
[44]
Hyperchem. We used the MM+ force field in combina-
tion with a conjugate gradient optimisation method (Polak–
All the subsets with daverage ꢅdmin values>4000 con- Ribiere). The backbone descriptors size, flexibility and po-
tained a density weighted selection of backbones larity (see Table 3) were computed with the Codessa soft-
[45]
[46]
from the four clusters shown in Figure 9. This con- ware package and analysed using Matlab scripts.
firms that density-weighted D-optimal design in the
descriptor space is indeed an optimal backbone selec-
tion method.
Table 4. Diversity indexes for five runs for 100 different
combinations of 10 random backbones.
Conclusions
We have shown that in silico catalysis design can R-10 (100 times)
1
2
3
4
5
Average
assist chemists in the difficult task of ligand selection.
The huge diversity of the multi-dimensional ligand
daverage ꢅdmin
2877 2634 2840 2976 2677 2800
Table 3. The three influential backbone descriptors.
Descriptor
type
Name (units)
Description
À1
Constitutional Molecular weight (gmol )
Sum of the individual isotopic masses of all the atoms of a molecule.
1
2
[47]
kÁ k
Topological
Kier flexibility index (no units,
normalised descriptor)
F=
NSA
1
2
where k and k are the Kier shape indices (a function of non-hydrogen
atom counts and the number of paths of length n in the molecular
graph) and NSA the number of non-hydrogen atoms in the molecule.
[48]
Electrostatic Polarity parameter (electron
P=QmaxÀQ
min
charge density e, normalised)
where Q is the most positive and Q the most negative atomic par-
max
min
[48]
tial charge in the molecule, respectively.
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Backbone Diversity Analysis in Catalyst Design
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Procedure for Choosing a Representative Random
Subset of Backbones
Procedure for Catalytic Hydrocyanation of
3-Pentenenitrile
To obtain a statistically consistent measure for the R-10 da-
taset, a single combination of backbones is not sufficient. In-
stead, we first compute the average daverage ꢅdmin for 100 ran-
domly selected different combinations of 10 backbones. We
repeat the calculation five times, and then compute the aver-
age value as shown on Table 4. After studying the composi-
tions of some random sets of 10 backbones out of the 100
sets, we chose a group of 10 random backbones with 2700<
daverage ꢅdmin <2900.
The catalyst precursor, Ni ACHTUNGRETNNUNG( COD) (0.5 mmol, 138 mg), the
2
ligand (2.5 mmol), 3-pentenenitrile (15 mmol, 1.21 g) and
ZnCl (0.5 mmol, 68 mg) were added to a 60-mL Schlenk
2
glass tube equipped with a septum stopper. The mixture was
stirred and heated to 708C. Acetone cyanohydrin (previ-
ously dried over molecular sieves) was fed to the mixture by
À1
a pressure syringe (0.45 mLh ). The addition was stopped
after 3 h. The mixture was cooled to 258C, diluted with ace-
tone and analysed by GC. HCN levels were permanently
monitored in the laboratory and reaction hood and kept
under 2 ppm. Excess acetone cyanohydrin was disposed of
by slowly adding to aqueous sodium hypochloride.
Procedure for Selecting a Weighted D-Optimal
Subset of Backbones
Acknowledgements
For the D-optimal subset selection, we partitioned the back-
bone descriptor space in nine rectangles. However, due to
the uneven distribution of the backbones in this space, pick-
ing one backbone from each rectangle yields relatively low
diversity indices (daverage ꢅdmin ꢀ3000). To avoid this, we se-
We thank Rhodia for financial support.
lected from every rectangle a number of backbones that is References
proportional to the number of total backbones in that rec-
tangle. Subsequently, we chose which backbones should be
selected by searching for backbone combinations that in-
crease the diversity of the subset. This is done by studying
the compositions of top-performing random subsets of 10
backbones out of 100 sets. We observe that all the random
high-scoring subsets have always some backbones from se-
lected clusters. In this case, the 14 highest-performing
random sets (4200<daverage ꢅdmin <5800) have always one or
two backbones of the cluster {35, 36, 37}. Other backbones
that appear often are: 24 (7 times in the top 14 sets), 21 (7
times), 29 (6 times), and 8 (5 times).
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