Topology in Chemistry: Discrete Mathematics of Molecules

400 Pages·2002·22.91 MB·English

by  D.H. Rouvray

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Discrete Mathematics of Molecules
“Tallring of education, people have now a-days (said he) got a strange
opinion that every thing should be taughtb y lectures. Now, I cannot see that
lectures can do so much good as reading the books fiom which the lectures
are taken. I know nothing that can be best taught by lectures, expect where
experiments are to be shewn. You may teach chymestry by lectures. - You
might teach making of shoes by lectures!”
James Boswell: Life of Samuel Johnson 1766
(1 709- 1784)
- -
“Every aspect of the world today even politics and international relations
is affected by chemistry.”
Linus Pauling, Nobel Prize winner for Chemistry, 1954, and
Nobel Peace Pke 1962
Biographical Sketches
DENNIS H. ROUVRAY
Dennis H. Rouvray was born in Rochford, Essex, U.K. in 1938 and studied at Imperial
College (University of London) where he was awarded the B.Sc.(Hons) degree in 1961
and thePh. D. degree in 1964. He then became a traveling academic and held posts at
Dalhousie University, Halifax, Nova Scotia, Canada; the University of Liverpool, U.K.;
the University of the Witwatersrand, Johannesburg, South Africa; the University of
Oxford, U.K. (in the C. A. Coulson group); and the Max-Planck-Institut fur
Strahlenchemie, Mulheim an der Ruhr, Germany. He moved to the University of
Georgia where he is currently Adjunct Senior Research Scientist. For the past 25 years
he has worked in various areas of mathematical chemistry and has played an active role
in promoting this discipline. In 1987 he founded the Journal of Mathematical Chemistry
and has co-organized three international conferences on mathematical chemistry in 1987
(Athens, Georgia), 1995 (Pitlochry, Scotland), and 2001 (Athens, Georgia). He has
some 250 publications of which over 30 are books, his most well-known being the best-
selling Fuzzy Logic in Chemistry. In addition to scientific papers he has written
numerous articles for magazines such as Scientific American and New Scientist. His
hobbies include listening to high-volume Mozart, cooking spicy vegetarian food, and
hiking in the Swiss mountains. He is married and has three married sons and four
grandchildren.
R. BRUCE KING
R. Bruce King was born in Rochester, New Hampshire in 1938, attended Oberlin
College (B. A. 1957), and was an NSF Predoctoral Fellow with Prof. F. G. A. Stone at
Harvard University (Ph. D. 1961). After a year at du Pont and 4 years at the Mellon
Institute he joined the faculty of the University of Georgia where he has been Regents’
Professor of Chemistry since 1973. His research interests have ranged from synthetic
organometallic and organophosphorus chemistry to applications of topology and graph
theory in inorganic chemistry and the inorganic chemistry of nuclear waste treatment.
Prof. King was the American Regional Editor of the J. Organometal. Chem. from 1981
to 1998 as well as Editor-in-Chief of the Encyclopedia of Inorganic Chemistry published
in 1994. He is the recipient of American Chemical Society Awards in Pure Chemistry
(1971) and Inorganic Chemistry (1991). Prof. King has approximately 600 research
publications. During the past decade he has authored books entitled Applications of
Graph Theory and Topology in Inorganic Cluster and Coordination Chemistry (1993).
Inorganic Main Group Element Chemistry (1994), and Beyond the Quartic Equation
(1996). Prof. King is married and the father of two grown sons; he also has a young
grandson. His hobbies include contract bridge, music, and travel.
Topology in Chemistry:
Discrete Mathematics of
Molecules
D. H. Rouvray and R. B. King, Editors
Horwood Publishing Limited
HORWOOD PUBLISHING LIMITED
International Publishers
Coll House, Westergate, Chichester, West Sussex, PO20 3QL
England
First published in 2002
Reprinted 2003
COPYRIGHT NOTICE
All Rights Reserved. No part of this publication may be produced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise, without the permission of Horwood
Publishing, International Publishers, Coll House, Westergate, Chichester, West
Sussex, England
0 Dennis Rouvray and R. Bruce King 2002
British Library Cataloguing In Publication Data
A catalogue record of this book is available from the British Library
ISBN 1-898563-76-4
Printed and bound in Great Britain by Antony Rowe Ltd
The volume before you covers a number of areas within the broad discipline of
mathematical chemistry. Its prime focus is on the applications of graph-theoretical and
topological methods to chemical problems. The majority of its chapters are devoted to
discussion of the use of topological indices for the prediction of the properties of diverse
chemical substances. This area has grown by leaps and bounds over the past decade, so
much so that topological methods are now able to predict virtually any property of any
chemical substance. These exciting developments mean that not only can we now
predict the physical, chemical, biological, and toxicological properties of almost all
materials but that these methods can also be applied in a host of diverse fields, including
the design of new drugs, the evaluation of environmental pollutants, the development of
new agrochemicals. and the determination of the toxicity of newly synthesized chemical
species. It seemed to us a good time to take stock of what has been achieved to date, i.e.,
to consider how far we have come along the road of applying topological methods to
chemistry.
The great pioneer of this field was the late Harry Wiener (1924-1998) and this volume
has been prepared with the aim of commemorating his life and work and, in particular,
his seminal contributions to mathematical chemistry. Wiener was the first to introduce
topological indices into chemistry. He called his two indices the polarity number and the
path number. The latter index has become by far the more famous of the two and is
nowadays much more commonly called the Wiener number or the Wiener index.
Wiener was able to demonstrate that these two simple descriptors were sufficient to
predict a whole range of properties for members of the alkane series. Since Wiener’s
pioneering endeavors, this topology-based approach has been extended to virtually every
other material within the pantheon of chemical substances.
For some time it had seemed to us that the time was ripe to organize an international
conference for the purpose of reviewing the current capabilities and future potential of
topological methods in chemistry. The opportunity presented itself when the Wiener
family contacted one of us (DHR) to inform us of the death of Harry Wiener. They also
indicated at the time that they were willing to promote the broad area of mathematical
chemistry and finance its further development. We felt that the best use of their
unexpected generosity would be to set up an international conference that would
consider the current status of the field initiated by Wiener some 55 years ago.
Accordingly, we organized an international conference with the theme The Role of
Topology in Chemistry. Our conference took place during the period March 20-24.
2001, at the University of Georgia Center for Continuing Education. Participants from a
dozen different countries attended and presented papers on various aspects of topological
chemistry.
The complete plenary lectures appear in somewhat extended form as the chapters of this
monograph. Although the majority of the chapters herein review various advanced
states of the development of the concept of topological indices, a number of other topics
are also addressed. Rouvray’s opening chapter takes a look at the life and times of Harry
Wiener and describes how Wiener came to put forward his two indices. Mezey
discusses the information content of the molecular electron density cloud and shows how
it leads to a quantitative shape-activity relationship that is useful in the prediction of
molecular properties. Lukovits presents a method for the exhaustive generation of non-
redundant sets of structural formulas. King examines the use of topology-based ideas
for the elucidation of the structure and bonding in an important class of inorganic
compounds, namely the boranes.
We would like to take this opportunity to thank a number of individuals who helped us
make our international conference a reality. First and foremost, we would express our
appreciation to the Wiener family, namely Harry Wiener’s stepbrother Dr. Alfred
Wingharn, and Dr. Wingham’s son, Mr. Mike Wingham, who provided the financial and
moral support without which our conference would not have been possible. We also
thank Allen Henderson of the Georgia Center for Continuing Education for making
many of the arrangements to host our conference. Finally, we would like to thank Ms.
Trudy Galynker, a long-term colleague of Dr. Wiener at Pfizer, Inc., in New York City,
who kindly provided some of the background material for Chapter 1 and who came to
our conference and delivered a speech full of insightful reminiscences about the man that
our conference was designed to commemorate-Harry Wiener.
Dennis H. Rouvray
R. Bruce King
Athens, Georgia, USA
July, 2001
Table of Contents
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1 Harry in the Limelight: The Life and Times of Harry Wiener
D .H . Rouvray
1.1 Background Information ............................................................................................. i
1.2 The Early Years ........................................................................................................... 3
1.3 The American Debut ................................................................................................... 4
1.4 The Medical Man ........................................................................................................ 6
1.5 The Corporate Executive ............................................................................................ 8
1.6 The Human Being ..................................................................................................... 11
1.7 Concluding Remarks ................................................................................................ 13
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2 The Rich Legacy of Half a Century of the Wiener Index
D .H . Rouvray
2.1 Setting the Scene ....................................................................................................... 16
2.2 Raising the Curtain ................................................................................................... 18
2.3 Blazing the Trail ....................................................................................................... 21
2.4 Extending the Approach ........................................................................................... 23
2.5 Explaining the Success ............................................................................................. 26
2.6 Introducing the Matrix .............................................................................................. 29
2.7 Probing the Index ...................................................................................................... 31
2.8 Fulfilling the Promise ............................................................................................... 34
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3 Mathematical and Chemical Analysis of Wiener’s Polarity Number
H. Hosoya and Y.-DG. ao
3.1 Introduction ............................................................................................................... 38
3.2 Definitions of Topological Indices ........................................................................... 41
3.3 Interrelations among Various Topological Indices ................................................. 42
3.4 Correlation of p with other Topological Indices ...................................................... 43
3.5 Correlation ofp with Liquid Density ....................................................................... 44
3.6 Microscopic Interpretation of High p-d Correlation ................................................ 45
Appendix: Rotational Polynomial ........................................................................... 50
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4 The Wiener Number: Some Applications and New Developments
D . Bonchev
4.1 Introduction ............................................................................................................... 58
4.2 Molecular Branching ................................................................................................. 59
4.3 The Transformability (Comparability)G raph and Molecular Properties ...............6 3
4.4 Molecular Cyclicity ................................................................................................... 65
4.5 The Graph Center Concept ....................................................................................... 67
4.6 Information-TheoreticA nalogues of the Wiener Index .......................................... 69
4.7 The Wiener Number in Crystal Studies .................................................................... 71
4.7.1 Modelling of crystal growth ........................................................................ 71
4.7.2 Modelling of crystal vacancies and defect atoms ...................................... 71
4.7.3 The Wiener number as criterion for stability of clusters of atoms ............ 73
4.8 The Wiener Number in Polymer Studies ................................................................. 73
4.8.1 The TEMPO method ................................................................................... 73
4.8.2 General approach to applying the Wiener number to polymers ................ 75
4.8.3 The Wiener number direct link to the radius of gyration
and viscosity of polymer melts and solutions ........................................... 76
4.9 The Overall Wiener Index ........................................................................................ 79
4.10 Theorems for the Wiener Number of a Composite Graph Constructed
by Combining Several Smaller Graphs ................................................................... 83
4.1 1 Conclusion and Questions ......................................................................................... 83
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5 A Comparison between Various Topological Indices. Particularly
between the Index J and Wiener’s Index W
A.T. Balaban
5.1 Introduction ............................................................................................................... 89
5.2 Topological Indices and a Comparison between some of them .............................. 91
5.2. I The simplest comparisons between topological indices ............................ 92
5.2.2 Variable cluster analysis ............................................................................. 93
5.2.3 Ordering of alkanes ..................................................................................... 94
5.3 Refinements in Computing Topological Indices based on Topological
Distances according to the Parity of such Distances ............................................. 101
5.3.1 Wiener index extension according to evedodd distances ....................... 101
5.3.2 Future work ................................................................................................ 108
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6 Applications of Topological Indices in the Property/Bioactivity/
Toxicity Prediction of Chemicals
S.C.B asak. D. Mills. B.D. Cute. G.D. Crunwald and A.T. Balaban
6 .I Introduction ............................................................................................................. 113
6.2 Topological Indices ................................................................................................. 115
6.2.1 Graph-theoretic definitions and calculation methods .............................. 115
6.3 Too Many Topological Indices? ............................................................................. 126
6.4 Intercorrelation of Topological Indices .................................................................. 128
6.5 Characterization of Molecular Structure using Topological Indices ..................... 132
6.6 Use of Topological Indices in QSAIUQSPR of Congeners .................................. 138
6.7 QSAWQSPR Studies with a Combination of Topological Indices
and Physicochemical Properties ............................................................................ 141
6.8 Diverse Structures need Diverse Molecular Descriptors ....................................... 143
6.8.1 QSARs with diverse topological indices .................................................. 143
6.8.2 Hierarchical QSAR (Hi-QSAR) for structurally heterogeneous
databases using diverse TIs ....................................................................... 146
6.8.2.1 The hierarchical approach ......................................................... 146
6.8.2.2 Results of hierarchical studies ................................................... 146
6.8.3 Quantification of molecular similarity using diverse TIs ........................ 153
6.8.3.1 Development of structure spaces to quantify molecular
similarity .................................................................................... 154
6.8.3.2 Selection of mutually different molecular similarity
methods ...................................................................................... 156
6.8.3.3 Selection of analogs using molecular similarity methods ......... 157
6.8.3.4 Comparison of spaces derived from measured
physicochemical properties vis-a-vis topological
descriptors ................................................................................... 157
6.8.3.5 Estimation of properties of chemicals using the KNN
method ......................................................................................... 160
6.8.3.6 Estimation of toxic modes of action from the MOA of
neighboring chemicals ................................................................ 162
6.8.4 Molecular dissimilarity in clustering of databases ................................... 162
6.9 Comparison of QSAWQSPR and QMSA Methods in the Estimation of
Properties ................................................................................................................. 164
6.10 Integrated QSAR: The New Modelling Approach for the Twenty-first
Century .................................................................................................................... 165
6.1 1 Discussion ................................................................................................................ 167
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7 The Wiener Number in the Context of Generalized Topological
Indices
E . Estrada
7. I Introduction ............................................................................................................. 185
7.2 Some Classical Topological Indices ....................................................................... 187
7.3 The Vector-Matrix-Vector Multiplication Procedure ............................................ 189
7.4 Generalized Topological Indices ............................................................................ 190
7.5 Multidimensional Representations ......................................................................... 192
7.6 Optimization of the Wiener W Index ...................................................................... 193
7.6.1 Octane boiling points ................................................................................ 196
7.6.2 C,-C alkane boiling points ...................................................................... 199
7.7 Structural Interpretation of W(xp. ) Indices ............................................................. 201
7.8 Conclusions .............................................................................................................. 203
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8 Mixed Higher-Order Connectivity-Pseudoconnectivity Terms
L Pogliani
8.1 Introduction ............................................................................................................. 208
8.1 .1 General considerations .............................................................................. 208
8.1.2 Molecular connectivity indices ................................................................. 209
8.1.3 Molecular pseudoconnectivity indices ..................................................... 210
8.1.4 Molecular connectivity and pseudoconnectivity terms ............................ 211
8.2 Method and Algorithms .......................................................................................... 212
8.2.1 The molecular connectivity and pseudoconnectivity indices ..................2 12
8.2.2 Molecular connectivity and pseudoconnectivity terms ............................ 215
8.3 Results and Discussion ............................................................................................ 217
8.3.1 Amino acids ............................................................................................... 217