Graph theory and topology in chemistry

GRAPH THEORY AND
TOPOLOGY IN CHEMISTRY
Edited by

R.B. King and D.H. Rouvray


www.pdfgrip.com
studies in physical and theoretical chemistry 51

GRAPH THEORY AND TOPOLOGY
IN CHEMISTRY


www.pdfgrip.com
studies in physical and theoretical chemistry 51

GRAPH THEORY
AND TOPOLOGY
IN CHEMISTRY
A Collection of Papers Presented
at an International Conference
held at the University of Georgia,
Athens, Georgia, U.S.A., 1 6—20 March 1987

Edited by

R.B. KING and D.H. ROUVRAY
Department o f Chemistry, University o f Georgia
Athens, Georgia 30602, U.S.A.

ELSEVIER
Amsterdam — Oxford — New York — Tokyo 1987


www.pdfgrip.com
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat 25
P.O. Box 2 1 1 ,1 0 0 0 AE Amsterdam, The Netherlands

Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
52, Vanderbilt Avenue
New York, NY 10017, U.S.A.

ISBN 0-4 4 4 -4 2 8 8 2 -8 (Vol. 51)
ISBN 0 -4 4 4 -4 1 6 9 9 -4 (Series)

Printed in The Netherlands


www.pdfgrip.com
studies in physical and theoretical chemistry
Other titles in this series
1 Association Theory: The Phases of Matter and Their Transformations by R.
Ginell
2 Statistical Thermodynamics of Simple Liquids and Their Mixtures by T.
Boublik, I. Nezbeda and K. Hlavaty
3 Weak Intermolecular Interactions in Chemistry and Biology by P. Hobza and
R. Zahradnik
4 Biomolecular Information Theory by S. Fraga, K.M.S. Saxena and M. Torres

5 Mossbauer Spectroscopy by A. Vertes, L. Korecz and K. Burger
6 Radiation Biology and Chemistry: Research Developments edited by H E.
Edwards, S. Navaratnam, B.J. Parsons and G.O. Phillips
7 Origins of Optical Activity in Nature edited by D C. Walker
8 Spectroscopy in Chemistry and Physics: Modern Trends edited by F.J. Comes,
A. Muller and W.J. Orville-Thomas
9 Dielectric Physics by A. Chetkowski
10 Structure and Properties of Amorphous Polymers edited by A.G. Walton
11 Electrodes of Conductive Metallic Oxides. Part A edited by S. Trasatti
Electrodes of Conductive Metallic Oxides. Part B edited by S. Trasatti
12 Ionic Hydration in Chemistry and Biophysics by B E. Conway
13 Diffraction Studies on Non-Crystalline Substances edited by I. Hargittai and
W.J. Orville-Thomas
14 Radiation Chemistry of Hydrocarbons by G. Foldiak
1 5 Progress in Electrochemistry edited by D. A.J. Rand, G.P. Power and I.M. Ritchie
1 6 Data Processing in Chemistry edited by Z. Hippe
1 7 Molecular Vibrational-Rotational Spectra by D. Papousek and M R. Aliev
18 Steric Effects in Biomolecules edited by G. Naray-Szabo
19 Field Theoretical Methods in Chemical Physics by R. Paul
20 Vibrational Intensities in Infrared and Raman Spectroscopy edited by W.B.
Person and G. Zerbi
21 Current Aspects of Quantum Chemistry 1981 edited by R. Carbo
22 Spin Polarization and Magnetic Effects in Radical Reactions edited by Yu.N.
Molin
23 Symmetries and Properties of Non-Rigid Molecules: A Comprehensive
Survey edited by J. Maruani and J. Serre
24 Physical Chemistry of Transmembrane Ion Motions edited by G. Spach
25 Advances in Mossbauer Spectroscopy: Applications to Physics, Chemistry
and Biology edited by B.V. Thosar and P.K. Iyengar
26 Aggregation Processes in Solution edited by E. Wyn-Jones and J. Gormally

27 Ions and Molecules in Solution edited by N. Tanaka, H. Ohtaki and R. Tamamushi
28 Chemical Applications of Topology and Graph Theory edited by R.B. King
29 Electronic and MoleculaKStructure of Electrode-Electrolyte Interfaces edited
by W.N. Hansen, D.M. Kolb and D.W. Lynch
30 Fourier Transform NMR Spectroscopy (second edition) by D. Shaw
31 Hot Atom Chemistry: Recent Trends and Applications in the Physical and
Life Sciences and Technology edited by T. Matsuura
32 Physical Chemistry of the Solid State: Applications to Metals and their
Compounds edited by P. Lacombe


www.pdfgrip.com
33 Inorganic Electronic Spectroscopy (second edition) by A.B.P. Lever
34 Electrochemistry: The Interfacing Science edited by D A J . Rand and A M.
Bond
35 Photophysics and Photochemistry above 6 eV edited by F. Lahmani
36 Biomolecules: Electronic Aspects edited by C. Nagata, M. Hatano, J. Tanaka
and H. Suzuki
37 Topics in Molecular Interactions edited by W.J. Orville-Thomas, H. Ratajczak
and C.N.R. Rao
38 The Chemical Physics of Solvation. Part A. Theory of Solvation edited by R.R.
Dogonadze, E. Kalman, A.A. Kornyshev and J. Ulstrup
The Chemical Physics of Solvation. Part B. Spectroscopy of Solvation edited
by R.R. Dogonadze, E. Kalman, A.A. Kornyshev and J. Ulstrup
39 Industrial Application of Radioisotopes edited by G. Foldiak
40 Stable Gas-in-Liquid Emulsions: Production in Natural Waters and Artificial
Media by J.S. D'Arrigo
41 Theoretical Chemistry of Biological Systems edited by G. Naray-Szabo
41 Theory of Molecular Interactions by I.G. Kaplan
43 Fluctuations, Diffusion and Spin Relaxation by R. Lenk

4 4 The Unitary Group in Quantum Chemistry by F A. Matsen and R. Pauncz
45 Laser Scattering Spectroscopy of Biologial Objects edited by J. Stepanek, P.
Anzenbacher and B. Sedlacek
46 Dynamics of Molecular Crystals edited by J. Lascombe
47 Kinetics of Electrochemical Metal Dissolution by L. Kiss
48 Fundamentals of Diffusion Bonding edited by Y. Ishida
49 Metallic Superlattices: Artificially Structured Materials by T. Shinjo and T.
Takada
50 Photoelectrochemical Solar Cells edited by K.S.V. Santhanam and M. Sharon
51 Graph Theory and Topology in Chemistry edited by R.B. King and D.H. Rouvray
52 Intermolecular Complexes by P. Hobza and R. Zahradnik
53 Potential Energy Hypersurfaces by P.G. Mezey


www.pdfgrip.com

V II

CONTENTS
Preface
List o f Authors

Xl
X III

Section A: Knot Theory and R eaction Topology

1

K n o ts, M a c ro m o le c u le s and C h e m ic a l D yn a m ics

D.W . Sum ners

3

T o p o lo g ic a l S te re o c h e m is try : K n o t T h e o ry o f M o le c u la r G raphs
D . M . W alba

23

A T o p o lo g ic a l A p p ro a c h to th e S te re o c h e m is try o f N o n rig id M o le cu le s
J. Sim on

43

C h ir a lity o f N o n -S ta n d a rd ly Em bedded M dbius L a d de rs
E. F lapan

76

E x trin s ic T o p o lo g ic a l C h ir a lit y In d ice s o f M o le c u la r G raphs
D .P . Jonish and K .C . M ille t t

82

New D e v e lo p m e n ts in R e a c tio n T op o lo g y
P.G. M eze y

91

An O u tlin e f o r a C o v a ria n t T h e o ry o f C o n s e rv a tiv e K in e tic F orce s

L. Peusner

106

T o p o lo g ic a l C o n trib u tio n s to th e C h e m is try o f L iv in g S ystem s
D .C . M ik u le c k y

115

Section B: M ole c u la r C o m p lex ity , System S im ila rity , and Topological Indices

125

On th e T o p o lo g ic a l C o m p le x ity o f C h e m ic a l S ystem s
D. Bonchev and O .E. P olansky

126

N u m e ric a l M o d e llin g o f C h e m ic a l S tru c tu re s : L o c a l G raph In v a ria n ts
and T o p o lo g ic a l In d ice s
A .T . Balaban
The F ra c ta l N a tu re o f A lk a n e P h y sico ch e m ica l B e h a v io r
D .H . R o u vra y

159

177

The C o rre la tio n b e tw e e n P h ysica l P ro p e rtie s and T o p o lo g ic a l Indices
o f N—A lk a n e s

v /
N. A d le r and L. K o v a c ic -B e c k

194

The Use o f T o p o lo g ic a l In d ice s to E s tim a te th e M e ltin g P o in ts o f
O rg a n ic M o le c u le s
M . P. Hanson and D .H . R q u vra y

201

Some R e la tio n s h ip s b e tw e e n th e W ie n e r N u m be r and th e N u m b e r o f S e lfR e tu rn in g W alks in C h e m ic a l G raphs
D. Bonchev, O. M ekenyan and O .E. Polansky

209


www.pdfgrip.com
VIII
U nique M a th e m a tic a l F e a tu re s o f th e S u b s tru c tu re M e tr ic A p p ro a ch
to Q u a n tita tiv e M o le c u la r S im ila r it y A n a ly s is
M . Johnson, M . N a im , V. N ich o lso n and C .-C . Tsai

219

A Subgraph Iso m o rp h ism T he o re m fo r M o le c u la r G raphs
V. N ich o lso n , C .-C . T sai, M . Johnson and M . N a im

226

A T o p o lo g ic a l A p p ro a ch to M o le c u la r - S im ila rity A n a ly s is and its A p p lic a tio n
C . - C . T sai, M . Johnson, V. N ich o lso n and M . N a im

231

Section C: Polyhedra, Clusters and the Solid S ta te

237

P e rm u ta tio n a l D e s c rip tio n o f th e D yn a m ics o f O c ta c o o rd in a te P o lyh e d ra
J. Brocas

239

S y m m e try P ro p e rtie s o f C h e m ic a l G raphs X. R e a rra n g e m e n t o f A x ia lly
D is to rte d O cta h e d ra
M . R andi£, D .J . K le in , V. K a to v ic , D .O . O akland, W .A . S e itz and A .T . Balaban

266

G raphs fo r C h e m ic a l R e a c tio n N e tw o rk s : A p p lic a tio n s to th e Is o m e riz a tio n s
A m o n g th e C a rbo ra n e s
B .M . G im a rc and J .J . O tt

285

T o p o lo g y and th e S tru c tu re s o f M o le cu le s and Solids
J . K. B u rd e tt

302

T o p o lo g ic a l A sp e cts o f In fin ite M e ta l C lu s te rs and S u p e rco n d u cto rs
R .B. K in g

325

T h e rm o d y n a m ic s o f Phase T ra n s itio n s in M e ta l C lu s te r S ystem s
M .H . Lee

344

Random G raph M odels fo r P hysical S ystem s
K . T. B a lin ska and L .V . Q u in tas

349

F ro m Gaussian S u b c ritic a l to H o lts m a rk (3/2 - L e vy S ta b le ) S u p e rc ritic a l A s y m p to tic
B e h a v io r in "R in g s F o rb id d e n " F lo ry -S to c k m a y e r M odel o f P o ly m e riz a tio n
B. P itte l, W .A . W o yczyn ski and J .A . Mann

362

Section D: Eigenvalues, Conjugated Systems, and Resonance

371

G ro u n d -S ta te M u ltip lic it ie s o f O rg a n ic D i- and M u lti-R a d ic a ls
M . Shen and 0 . S in a n o g lu

373

O rg a n ic P o ly ra d ic a ls , H ig h -S p in H yd roca rb o n s, and O rg a n ic F e rro m a g n e ts
D . J. K le in and S.A. A le x a n d e r

404

G round S ta te P ro p e rtie s o f C o n ju g a te d System s in a S im p le Bond O rb ita l
Resonance T h e o ry (B O R T )
T .P . ^ iv k o v ic

420

The C o n ju g a te d C ir c u its M od e l: On th e S e le c tio n o f th e P a ra m e te rs fo r
C o m p u tin g th e Resonance E nergies
M . R a ndic, S. N ik o lic , and N. T rin a js tic

429

S im p le E s tim a te s o f th e T o ta l and th e R e fe re n ce s P i-E le c tro n E nergy
o f C o n ju g a te d H y d roca rb o n s
A . G ra o va c, D. B abic, and K . K o v a c e v ic

448


www.pdfgrip.com
IX

Resonance in P o ly -P o ly p h e n a n th re n e s : A T ra n s fe r M a tr ix A p p ro a ch
W .A. S e itz , G .E. H ite , T .G . S chm alz and D .J . K le in

R apid C o m p u ta tio n o f th e E ig e nvalues o f S m a ll H e te ro c y c le s using a
F u n c tio n a l G ro u p -lik e C o n ce p t
J . R. Dias

458

466

On K e k u le S tru c tu re and P-V P ath M eth o d
H. W enjie and H. W enchen

476

O n e -to -O n e C o rre sp o n d en ce b e tw e e n K e ku le and S e x te t P a tte rn s
H. W enchen and H. W enjie

484

Section E: Coding, Enum eration and D a ta Reduction

489

P e rim e te r Codes f o r B e nzenoid A r o m a tic H yd ro ca rb o n s
W .C. H erndon and A .J . B ruce

491

C o m p u ta tio n a l G raph T h e o ry
K. B a la su b ram a n ia n

514

C oding and F a c to ris a tio n o f P o ly c y c lic C h e m ic a l G raphs
E.C . K irb y

529

R e d fie ld E n u m e ra tio n A p p lie d to C h e m ic a l P ro b le m s
E .K . L lo y d

537

C o u n tin g th e Spanning T rees o f L a b e lle d , P la n a r M o le c u la r G raphs
Em bedded on th e S u rfa ce o f a Sphere
B. O 'L e a ry and R.B. M a llio n

544

The E n u m e ra tio n o f K e k u le S tru c tu re s o f R e cta n g le -S h a p e d Benzenoids
C. Rong-si

552

Graphs o f C h e m is try and Physics: On D a ta R e d u c tio n o f C h e m ic a l In fo rm a tio n
S. E l-B a s il

557

Index

565

s
\


www.pdfgrip.com
XI

P R E FA C E

The b u rg e o n in g g ro w th o f c h e m ic a l graph th e o ry and r e la te d areas in re c e n t ye a rs
has g e n e ra te d

th e

need

m a th e m a tic a l c h e m is try .

fo r

in c re a s in g ly

fre q u e n t

c o n fe re n c e s

c o v e rin g

th e

area o f

T his book c o n ta in s th e papers p re se n te d a t th e In te rn a tio n a l

C o n fe re n c e on G raph T h e o ry and T o p o lo g y in C h e m is try h e ld a t th e U n iv e r s ity o f G e o rg ia ,
A th e n s, G e o rg ia ,

U .S .A ., d u rin g th e p e rio d M a rc h

76-20,

1987.

This C o n fe re n c e was

in m any ways a sequel to a sym posium held a t o ur u n iv e rs ity in A p r il, 1983, th e papers
fro m

w h ich w e re also p u b lish e d by E ls e v ie r in a sp e cia l sym po siu m v o lu m e .

B o th o f

these m e e tin g s w ere sponsored by th e U.S. O ffic e o f N a va l R esearch.
The p rin c ip a l goal o f o u r C o n fe re n c e

was to

p ro v id e

a fo ru m

f o r c h e m is ts and

m a th e m a tic ia n s to in te ra c t to g e th e r and to becom e b e tte r in fo rm e d on c u rre n t a c tiv itie s
and new d e v e lo p m e n ts in th e b ro a d areas o f c h e m ic a l to p o lo g y and c h e m ic a l g ra p h th e o ry .
The purpose o f th is book is to m ake a v a ila b le to a w id e r a u d ie nce a p e rm a n e n t re c o rd
o f th e papers p re se n te d a t th e C o n fe re n c e .

The 41 papers c o n ta in e d h e re in span a w id e

range o f to p ic s , and f o r th e co n ve n ie n ce o f th e re a d e r have been grouped in to fiv e m a jo r
se ctio n s.

A lth o u g h

p re s e n ta tio n s

w ill

we
a lw a ys

a p p re c ia te

th a t

any

such

be so m e w h a t a r b itr a r y ,

s u b d ivisio n

of

we hope th a t

th e

C o n fe re n c e

g ro u p in g th e papers

in th is w ay w ill h elp th e re a d e r to lo c a te those papers o f p a r tic u la r p e rsonal in te re s t
w ith g re a te r f a c ilit y .
O ur C o n fe re n c e also p ro v id e d an ideal s e ttin g f o r la u n c h in g th e n e w ly e sta b lish e d
Journal

o f M a th e m a tic a l

C h e m is try , e d ite d by D r.

D .H .

R o u vra y.

A c o m p lim e n ta ry

copy o f th e f ir s t issue o f th is jo u rn a l was d is trib u te d to e v e ry C o n fe re n c e p a r tic ip a n t.
As p a rtic ip a n ts

cam e

fro m

te n d if f e r e n t

(People's R e p u b lic ), E g y p t, G re a t

Britain,

c o u n trie s ,

n a m e ly

B u lg a ria ,

Canada, C hina

In d ia , Japan, M e x ic o , th e U n ite d S ta te s, and

Y u g o sla via , a w ide c ir c u la tio n o f th e new jo u rn a l was assured.

D u rin g th e C o n fe re n c e

the fle d g lin g In te rn a tio n a l S o c ie ty f o r M a th e m a tic a l C h e m is try was also discussed and
several d e cisio n s ta k e n .

Thus,

in a d d itio n

to p u re ly s c ie n t if ic

m a tte rs , a n u m b e r o f

o th e r issues w ere addressed by o u r C o n fe re n c e .
The C o n fe re n c e c o u ld n o t have been th e success it was w ith o u t th e s u p p o rt o f a
n u m ber o f o rg a n iz a tio n s and in d iv id u a ls whom we should lik e to th a n k p u b lic ly here.

r"

We are in d e b te d to th e U.S. O ffic e |>f N aval R esearch f o r th e m a jo r fin a n c ia l su p p ort
th a t

m ade

our

C o n fe re n c e

po ssib le .

L o ca l

su p p ort

fro m

th e

U n iv e rs ity

o f G e o rg ia


www.pdfgrip.com
X II
R esearch

F o u n d a tio n

also a ckn o w le d g e d .
Payne o f th e

and th e

U n iv e rs ity o f G e o rg ia

School o f C h e m ic a l

Sciences is

M e n tio n m ust also be m ade o f th e s te rlin g e f f o r t s o f M r. D a vid

G e o rg ia

C e n te r f o r C o n tin u in g E d u c a tio n in c o o rd in a tin g a rra n g e m e n ts

fo r th e C o n fe re n c e , and o f th e q u ie t e f f ic ie n c y o f o u r s e c re ta ry , Ms.

Ann Low e , who

k e p t tr a c k o f num erous a d m in is tr a tiv e d e ta ils and who assisted g re a tly in th e p ro d u c tio n
o f th is C o n fe re n c e v o lu m e .

U n iv e rs ity o f G e o rg ia

R. Bruce K in g

A th e n s, G e o rg ia

D ennis H. R o u vra y

June, 1987


www.pdfgrip.com
XIII

LIST OF A U T H O R S
N.

A D L E R , F a c u lty o f T ech n o lo g y, The U n iv e rs ity o f Zagreb, P.O. Box 177, 41001 Zagreb,
C ro a tia , Y u g o sla via

S.A.

ALEXAND ER,
3261 1, U .S .A .

Q ua ntu m

T h e o ry

P ro je c t,

U n iv e r s ity

of

F lo rid a ,

G a in e s v ille ,

FL

D.

B A B IC , In s titu te fo r M e d ic a l Research and O c c u p a tio n a l H e a lth , Y U -4 1 0 0 1, Zagreb,
POB 291 Y u g o sla via .

K.

B A L A S U B R A M A N IA N , D e p a rtm e n t o f C h e m is try , A riz o n a S ta te U n iv e rs ity , T em pe,
AZ 85287, U .S .A .

A .T .

B A L A B A N , P o ly te c h n ic In s titu te , D e p a rtm e n t
In d ep e n d e n tei 313, 76206, B u ch a re st, R oum ania

of

O rg a n ic

C h e m is try ,

S p la iu l

K .T .

B A L IN S K A , The T e c h n ic a l U n iv e rs ity o f Poznan, PI. M. S k lo d o w s k a -C u rie 5, 60-965,
Poznan, Poland.

D. B O N C H E V , H ig h e r School o f C h e m ic a l T e ch n o lo g y, B U -8 0 1 0 Burgas, B u lg a ria
J. B R O C A S, C h im ie O rga n iq u e Physique, U n iv e rs ite L ib re de B ru x e lle s , Brussels, B e lg iu m .
A.

J.

B R U C E , D e p a rtm e n t o f C h e m is try , U n iv e r s ity o f Texas a t El Paso, El Paso, TX
79912, U .S .A .

J .K .

B U R D E T T , C h e m is try D e p a rtm e n t, th e U n iv e r s ity o f C h ica g o , C h ica g o , IL 60637,

U .S .A .

J.R .

D IAS , D e p a rtm e n t o f C h e m is try , U n iv e rs ity o f M is s o u ri, Kansas C ity , M O , U .S.A

S. E L -B A S IL , F a c u lty o f P h a rm a cy, K asr E l- A in i S tre e t, C a iro , E g yp t.
E. F L A P A N , D e p a rtm e n t o f M a th e m a tic s , Pom ona C o lle g e , C la re m o n t, C A 91711, U .S .A .
B.

M . G IM A R C , D e p a rtm e n t
SC 29208, U .S .A .

A. G R A O V A C , R uder B o sko vic
Y u g o s la v ia .
M .P . H A N S O N ,
U .S .A .

of

C h e m is try ,

In s titu te ,

Y U -4 1 0 0 1,

of

South

Zagreb,

C a ro lin a ,

POB

1016,

C o lu m b ia ,

C ro a tia ,

D e p a rtm e n t o f C h e m is try , A u g u sta n a C o lle g e , Sioux F a lls , SD 57197,

W .C. H E R N D O N , D e p a rtm e n t o f C h e m is try ,
TX 79912, U .S .A .
G .E.

U n iv e r s ity

H IT E , D e p a rtm e n t o f M a rin e
G a lv e s to n , T X 7755$, U .S .A .

U n iv e r s ity o f Texas a t El Paso, El Paso,

Sciences,

Texas

ASM

U n iv e rs ity

at

G a lve sto n ,

I
M. JO H N S O N , C o m p u ta tio n a l C h e m is try , The U pjohn C o m pa n y, K a la m a zo o , M l, U .S .A .
D .P . JO N N IS H , M a th e m a tic s
C A 93106, U .S .A .

D e p a rtm e n t,

U n iv e r s ity

of

C a lifo r n ia ,

Santa

B a rb a ra ,


www.pdfgrip.com
XIV
V.

K A T O V IC ,

U .S .A .

D e p a rtm e n t o f C h e m is try , W rig h t S ta te

U n iv e rs ity , D a y to n , OH 45435,

R .B. K IN G , D e p a rtm e n t o f C h e m is try , U n iv e rs ity o f G e o rg ia , A th e n s, G A 30602, U .S .A .
E.C .

D.

K IR B Y , Resource Use In s titu te ,
5DS, S c o tla n d , U .K .
J. K L E IN , D e p a rtm e n t o f M a rin e
G a lv e s to n , TX 77553, U .S .A .

K . K O V A C E V IC , R uder B o sko vic
Y u g o sla via .
L. KO V A C IC -B E C K ,
Y u g o sla via .

14 L o w e r O a k fie ld ,

Sciences,

In s titu te ,

IN A -P e tro le u m

T exas

Y U -4 1 0 0 1,

In d u s try ,

P itlo c h ry ,

A&M

U n iv e rs ity

Z agreb,

P.O . Box

P e rth s h ire

POB

1014, 41001

at

PH16

G a lve sto n ,

1016,

C ro a tia ,

Zagreb,

C ro a tia ,

M.

H. LEE , D e p a rtm e n t o f Physics, U n iv e rs ity o f G e o rg ia , A th e n s, G A 30602, U .S .A .

E.

K . L L O Y D , F a c u lty o f M a th e m a tic a l S tudies, The U n iv e rs ity , S o u th a m p to n , S09 5N H ,
U .K .

R.

B. M A L L IO N , The K in g 's S chool, C a n te rb u ry , CT1 2ES, U .K .

J.

A . M A N N , C h e m ic a l E n g in e e rin g
C le v e la n d , OH 44106, U .S .A .

D e p a rtm e n t,

Case

W estern

R eserve

U n iv e rs ity ,

O. M E K E N Y A N , H ig h e r School o f C h e m ic a l T e ch n o lo g y, BU -8010 Burgas, B u lg a ria .
P.

G. M E Z E Y , D e p a rtm e n t o f C h e m is try and D e p a rtm e n t o f M a th e m a tic s ,
o f S askatchew an, S askatoon, Canada S7N 0W0.

D .C . M IK U L E C K Y ,
D e p a rtm e n t
of
P h ysio lo g y,
M e d ic a l
C o lle g e
C o m m o n w e a lth U n iv e rs ity , R ich m o nd , VA 23298-0001, U .S .A .
K.

C . M IL L E T T , M a th e m a tic s
C A 93106, U .S .A .

D e p a rtm e n t,

U n iv e r s ity

of

C a lifo r n ia ,

U n iv e rs ity

of

V irg in ia

Santa

B a rb a ra ,

M . N A IM , M a th e m a tic s Sciences D e p a rtm e n t, K e n t S ta te U n iv e rs ity , K e n t, OH, U .S .A .
V.

N IC H O L S O N , M a th e m a tic s Sciences D e p a rtm e n t, K e n t S ta te U n iv e rs ity , K e n t, OH,
U .S .A .

S. Nl K O L IC , The R u g je r B o sko vic
Y u g o sla via .

In s titu te ,

P.O .B.

1016,

41001

Zagreb,

C ro a tia ,

D .O . O A K L A N D , D e p a rtm e n t o f M a th e m a tic s and C o m p u te r S cience, D ra ke U n iv e rs ity ,
Des M oines, Iowa 5031 1, U .S .A .
B.

O' L E A R Y , D e p a rtm e n t o f C h e m is try ,
B irm in g h a m , A L 35294, U .S .A .

U n iv e r s ity

of

A la b a m a

at

B irm in g h a m ,

J .J . O TT , D e p a rtm e n t o f C h e m is try , F urm an U n iv e rs ity , G re e n v ille , SC 29613, U .S .A .
L. P E U S N E R , L e o na rd o
04101, U .S .A .
B. P IT T E L , M a th e m a tic s
U .S .A .

Peusner A sso cia te s,

D e p a rtm e n t,

O hio

In c.,

S ta te

181

S ta te

U n iv e rs ity ,

S tre e t,

P o rtla n d ,

C olum bus,

OH

M aine

43210,


www.pdfgrip.com
XV
q #E#

L.

P O LA N S K Y, M a x - P la n c k - ln s titu te fYJr S tra h le n c h e m ie , D -4330 M tilh e im a.d.
F ed e ra l R e p u b lic o f G e rm a n y.

V.

Q U IN T A S , M a th e m a tic s D e p a rtm e n t, Pace U n iv e rs ity , New Y o rk, N Y 10038, U .S .A .

M . R A N D IC , D e p a rtm e n t o f M a th e m a tic s and C o m p u te r S cience, D ra ke
Des M oines, Iowa 5031 1, and A m es L a b o ra to ry -D .O .E ., Iowa S ta te
A m es, Iowa 5001 1, U .S .A .
C. R O N G -S I, C o lle g e o f F in a n ce and E co n o m ics,
The People's R e p u b lic o f C h in a .
D.

R uhr,

F uzhou U n iv e rs ity ,

U n iv e rs ity ,
U n iv e rs ity ,

F uzhou,

F u jia n ,

H. R O U V R A Y , D e p a rtm e n t o f C h e m is try , U n iv e r s ity o f G e o rg ia , A th e n s, G A 30602,
U .S .A .

T .G . S C H M A L Z , D e p a rtm e n t o f M a rin e S ciences, Texas A S M U n iv e rs ity a t G a lve sto n ,
G a lv e s to n , TX 77553, U .S .A .
W .A. S E ITZ, D e p a rtm e n t o f M a rin e
G a lv e s to n , TX 77553, U .S .A .

Sciences,

T exas

ASM

U n iv e rs ity

at

G a lve sto n ,

M. SH E N , S te rlin g C h e m is try L a b o ra to ry , Y a le U n iv e rs ity , P.O . Box 6666, New H aven,
CT 0651 1, U .S .A .
J. S IM O N , D e p a rtm e n t
U .S .A .

o f M a th e m a tic s ,

U n iv e r s ity

O. S IN A N O G L U , S te rlin g C h e m is try L a b o ra to ry ,
Haven, C T 0651 1, U .S .A .
D.W . SU M NERS, D e p a rtm e n t
F L 32306, U .S .A .

of

M a th e m a tic s ,

N. TR IN A JSTIC , The R u g je r B o sko vic
Y u g o s la v ia .

In s titu te ,

of

Iow a, Iowa C it y ,

Iowa

52242,

Y ale U n iv e rs ity , P.O. Box 6666, New

F lo rid a

POB

S ta te

1016,

U n iv e rs ity ,

41001

T allahassee,

Z agreb,

C.

-C . T S A I, C h e m is try D e p a rtm e n t, K e n t S ta te U n iv e r s ity , K e n t, OH, U .S .A .

D.

M. W A L B A , D e p a rtm e n t o f C h e m is try and
C o lo ra d o , B o u ld e r, CO 80309-0215, U .S .A .

B io c h e m is try ,

Box

215,

C ro a tia ,

U n iv e r s ity

HE W E N C H E N , Hebei C h e m ic a l E n g in e e rin g In s titu te , S h ijia zh u a n g , The People's R e p u b lic
o f C h in a .
HE W E N JIE , H ebei A c a d e m y o f Sciences, S h ijia zh u a n g , The People's R e p u b lic o f C h in a .
W .A . W O Y C Z Y N S K I, M a th e m a tic s and S ta tis tic s
U n iv e rs ity , C le v e la n d , OH 44106, U .S .A .
V

T.P .

s

M

D e p a rtm e n t, Case W e ste rn

R eserve

s

Z IV K O V IC , The In s titu te R u d je r B o sko vic, 41001 Z agreb, C ro a tia , Y u g o sla via .

of


www.pdfgrip.com

SECTION A

Knot Theory

and

Rea c tio n Topology


www.pdfgrip.com
Graph Theory and Topology in Chemistry, A Collection of Papers Presented at an
International Conference held at the University of Georgia, Athens, Georgia, U.S.A.,
16-20 March 1987, R.B. King and D.H. Rouvray (Eds)

Studies in Physical and Theoretical Chemistry, Volume 51, pages 3-22
© 1987 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands

3

KNOTS, MACROMOLECULES AND CHEMICAL DYNAMICS
D. W. Sumners1
1 Department of Mathematics, Florida State University, Tallahassee, Florida 32306
ABSTRACT
Knot theory is the mathematical study of placement of flexible graphs in 3-space.
Configurations of macromolecules(such as p o l y e t h y l e n e and DNA) can be
analyzed(both quantitatively and qualitatively) by means of knot theory. These large
molecules are very flexible, and can present themselves in 3-space in topologically
interesting ways. For example, in DNA research, various enzymes{topoisomerases and
recombinases) exist which, when reacted with unknotted closed circular DNA, produce
enzyme-specific characteristic families of knots and catenanes. One studies these
experimentally produced characteristic geometric forms in order to deduce enzyme
mechanism and substrate conformation. This particular application is an interesting mix
of knot theory and the statistical mechanics of molecular configurations. This paper will
give a brief overview of knot theory and DNA, and will discuss a new topological model
for site-specific recombination..
Another interesting application of knot theory and differential topology arises in the
topological description of propogating waves in excitable media. For example, in a thin
layer(a 2-dimensional medium), the Belusov - Zhabotinsky reaction produces a beautiful
pattern of spiral wave forms which rotate about a number of central rotor points. The
waves represent points which are in phase with respect to the reaction, and the rotor
points are the phaseless points-the organizing center of the reaction. In a 3-dimensional
medium, a knot or catenane can form an organizing center for a reaction. These
characteristic spiral rotating waves are seen in many biological and chemical contexts.
This paper will discuss a topological model for wave patterns in 2 and 3 dimensions

which relates wave patterns to a phase map. In the context of this model, a quantization
condition conjectured by Winfree and Strogatz can be shown to be a necessary and
sufficient condition for the mathematical existence of a spiral rotating wave pattern.

KNOT THEORY

In Euclidean geometry, two objects in Euclidean space are equivalent if there is some
rigid motion of space which superimposes one object on the other.

If, however, one

wishes to model systems or objects which allow deformation, one must introduce a more
flexible notion of equivalence.

The mathematical science of topology is the study of

equivalence of objects with various degrees of relaxation of the rigidity condition. In its
most relaxed version, two topological spaces {X,Y} are homeomorphic if there is a
function h:X ~> Y such that h is 1-1, onto, and both h and h '1 are continuous. Such a


www.pdfgrip.com
4

function h is called a homeomorphism

, and is a very general notion of intrinsic

equivalence. Intuitively, one thinks of a homeomorphism as an elastic deformation which
transforms one object into another.

During the deformation, any possible stretching,

shrinking, twisting, etc. is allowed-moves which are not allowed include cutting or
breaking an object and later reassembling it, and passing one part of an object through
another.

If one desires a theory with discriminatory powers, one cannot allow the

unrestricted cutting apart and reassembling of a space. After all, any two brick buildings
start out as a pile of bricks. It turns out, however, that the controlled cutting apart and
reassembling of a space has great utility, both within mathematics and in applications of
mathematics to other disciplines. One case of interest will be discussed below, the case
of site-specific recombination , where an enzyme(called recombinase ) breaks apart and
recombines DNA in a controlled way.
In chemistry, one often models molecules by means of the molecular graph , in which
the vertices represent atoms, and the edges represent covalent bonds between atoms.
Macromolecules

are molecules of large molecular weight, such as synthetic

polymers( p o l y e t h y l e n e ) and biopolymers(DNA). While one imagines small bits(a few
atoms bonded together) of these molecules as being somewhat rigid, when one
concatenates long strings of these bits, the resulting molecules can be very flexible
indeed.
Knot theory is the study of the placement of flexible graphs in Euclidean 3-space.

If

G is a finite graph, a given placement(or positioning) of the graph in 3-space is called an

embedding of the graph. Any given graph admits infinitely many "different" positions,
many of which are intuitively "the same"-those differing by a translation or rotation, for
example. We shall regard graphs as completely flexible, and any two placements of a
graph will be equivalent if there is an elastic motion of 3-space which transforms one
position to the other-that is, one placement gets superimposed on, or made congruent to,
the other. Moreover, we do not necessarily insist that the congruence take vertices to
vertices. The motion of 3-space which moves one position to the other may introduce
any possible stretching, shrinking, or twisting of the graph-it may not, however, break and
then reconnect the graph in any way. We also do not allow the motion to pull knots
infinitely tight so as to make them disappear-we wish to model molecules which have a
definite thickness to them. For a fixed graph G, an equivalence class of such embeddings
is called a knot type , or just knot for short. A particular embedding in an equivalence


www.pdfgrip.com
class is called a representative of that equivalence class. We often abuse language by
calling a representative by the name "knot". We trust that the context will make it clear
whether we are speaking of the equivalence class or a representative of it.
It is clear that the above definition of equivalence of embeddings of a graph is
physically unrealistic-one cannot stretch or shrink molecules at will, nor can one forget
where the atoms are! Nevertheless, the definition is, on the one hand, broad enough to
generate a body of mathematical know!edge(ref. 1,2,3,4), and, on the other hand, precise
enough to place useful and computable limits on the physically possible motions and
configuration changes of molecules(ref. 5,6,7,8,9).
For the remainder of this paper, we will only consider graphs which are collections of
disjoint circles and arcs. In order to study embeddings of graphs in 3-space, one draws
planar pictures of them, called projections . A projection of an embedded graph in
3-space is a shadow cast by the configuration on a plane, with the light source far away.
A crossover is a place in the projection where 2 or more strings cross. It is clear that, by
rotating the configuration slightly, we can arrange that no more than 2 strings meet at any

crossover, and that they meet transversely. If the ends of an arc in 3-space can move
freely, the arc cannot contribute to knotting, because the free ends can pe pulled through
to undo any possible entanglement, either with itself or with any other graph components
which happen to be present. In order to achieve knotting, either the ends of the arc must
be somehow constrained, or joined together to form a circle, which admits lots of
knots(ref. 10,11). When considering a family of \i circles, the unknot or trivial knot is the
equivalence class of any planar embedding of the ji disjoint circles. For any configuration
of p circles in 3-space, the crossover number is the minimum number of crossovers
possible for that equivalence class of embeddings-minimized over all representatives of
the equivalence class, and all projections of each representative. If ji> 2, an equivalence
class of embeddings for which no subcollection of circles can be removed from the others
by elastic spatial deformation is called a catenane in chemistry, and is an example of a
link in mathematics. Chemically, a catenane corresponds to topological bonding at work
to hold the disjoint circular parts of the molecule together(ref. 12).

DNA

The DNA molecule is a biopolymer which is long and threadlike, and often naturally


www.pdfgrip.com
6

occurs in closed circular form. Knot theory has been brought to bear on the study of the
geometric action of various naturally occurring enzymes(called topoisomerases ) which
alter the way in which the DNA is embedded in the cell(refs. 7,13).

In the cell,

topoisomerases are believed to facilitate the central genetic events of replication,

transcription and recombination via geometric manipulation of the DNA.

This

manipulation includes promoting writhing (coiling up) of the molecule, passing one strand
of the molecule through an enzyme-bridged break in another strand, and breaking a pair
of strands and rejoining them to different ends(a move performed by recombinant
enzymes).

The strategy is to use knot theory to deduce enzyme mechanism and

substrate configuration from changes in DNA topology effected by an enzyme reaction. In
order to understand the action of these enzymes on linear(and circular) DNA in vivo (in
the cell), reaction experiments are done on circular DNA in vitro (in the lab). This is
because the changes in topology(creation of knots and catenanes) due to enzyme action
can be captured in circular DNA, but would be lost in linear DNA during workup of
reaction products for analysis by gel electrophoresis and electron microscopy.

The

experimental technique is to react closed circular DNA substrate(usually unknotted) with
an enzyme, and then to separate the reaction products by agarose gel electrophoresis.
The experimental result here is that each enzyme produces a characteristic family of
knots and catenanes. At the most fundamental level of analysis, the family of reaction
products forms a signature for the enzyme; the ultimate goal is to use careful topological
analysis of the reaction products to extract precise information about exactly what each
enzyme is doing. It turns out that the gel mobility of the reaction products is determined by
the crossover number of each configuration-the higher the crossover number, the more
compact the molecule, and the greater its gel mobility.

Configurations with the same

crossover number migrate to approximately the same postion in the gel. Gel
electrophoresis yields a ladder of gel bands, and comparison with a reference knot
ladder(where adjacent bands correspond to a difference of one in crossover number)
determines the difference in crossover number represented by adjacent bands(ref. 14, 15,
16)).

The DNA can be removed from the gel, and to greatly enhance resolution for

electron microscopy, the molecules are coated with recA

protein(ref.17). This coating

thickens the DNA strands from about 10& to about 100&, simultaneously affording
unambiguous determination of the crossovers, and fewer extraneous crossovers. It is in
fact this recA coating technique which has opened the door for the active involvement of


www.pdfgrip.com
knot theory in the analysis of DNA enzyme mechanism.

SITE-SPECIFIC RECOMBINATION
We will now consider the situation of site-specific recombination enzymes operating on
closed circular duplex DNA. Duplex DNA consists of two linear backbones of sugar and
phosphorus. Attached to each sugar is one of the four bases:A = adenine, T = thymine, C
= cytosine, G = guanine. A ladder is formed by hydrogen bonding between base pairs,
where A binds with T, and C binds with G. In the classical Crick-Watson model for DNA,
the ladder is twisted in a right-hand helical fashion, with a relaxed-state pitch of
approximately 10.5 base pairs per full helical twist. Duplex DNA can exist in closed

circular form, where the rungs of the ladder form a twisted cylinder(instead of a twisted
Mobius band).

In certain closed circular duplex DNA, there exist two short identical

sequences of base pairs, called recombination sites

for the recombinant enzyme.

Because of the base pair sequencing, the recombination sites can be locally oriented
(reading the sequence from right to left is different from reading it left to right). If one then
orients the circular DNA(puts an arrow on it), there is induced a local orientation on each
site. If the local orientations agree, this is the case of direct

repeats , and if the local

orientations disagree, this is the case of inverted repeats .

The recombinase

nonspecifically attaches to the molecule, and then the sites are aligned(brought close
together), either through enzyme manipulation or random thermal motion(or both), and
both sites are then bound by the enzyme. This stage of the reaction is called synapsis ,
and the complex formed by the substrate together with the bound enzyme is called the
synaptic complex. In a single recombination event, the enzyme then performs two
double-stranded breaks at the sites, and recombines the ends in an enzyme-specific
manner(see Fig. 1).
We call the molecule before recombination takes place the substrate, and after
recombination takes place, the p ro d u c t. If the substrate is a single circle with direct
repeats, the product is a pair of circles, with one site each, and can form a DNA catenane.

If the substrate is a pair of circles with one site each, the product is a single circle with two
sites. If the substrate is a single circle with inverted repeats, the product is a single circle,
and can form a DNA knot(see Fig. 2).


www.pdfgrip.com

8

Sites Aligned

Duplex Strands
Broken

Ends Recombined

Fig. 1. A Single Recombination Event

Wr ith ing

R ecombination

Fig. 2. Hypothetical Recombination Knot Synthesis(lnverted Repeats)

THE TOPOLOGICAL MODEL
In site-specific recombination, two kinds of geometric manipulation of the DNA occurs.
The first is a global move, in which the sites are juxtaposed, either through enzyme action
or random collision(or a combination of these two processes).

After synapsis is

achieved,the next move is local, and entirely due to enzyme action. Within the region
bound by the enzyme, the molecule is broken in two places, and the ends recombined.


www.pdfgrip.com

9

We will model this local move. We model the enzyme itself as being homeomorphic to
the solid ball B3, where B3 is the set of all points in Euclidean 3-space of distance < 1
from the origin.

The recombination sites(and some contiguous DNA bound by the

enzyme) form a configuration of two arcs in the enzyme ball, known mathematically as a
tangle. During the local phase of recombination, we assume that the action takes place
entirely within the interior of the enzyme ball, and that the substrate configuration outside
the ball remains fixed while the strands are being broken and recombined.

After

recombination takes place, the molecule is released by the enzyme, and moves around
under chemical and thermal influences.
For symmetry of mathematical exposition, we take the point of view that the reaction is
taking place in the 3-sphere S3 , the set of all points distance 1 from the origin in
Eucildean 4-space. S3 can be viewed as R3(Euclidean 3-space) closed up with a point
at infinity, in the same way that the Euclidean plane(R2) can be closed up to give the
2-sphere S2, the set of all points distance 1 from the origin in R3. Every reaction in R3
can be viewed as a reaction in S3, and vice versa. The reason for viewing the reaction as

being in S3 ( instead of R3 ) is that the boundary of the recombination ball is
homeomorphic to S2, and this enzyme S2 functions as an equator in S3, dividing S3 into
two complimentary 3-balls, glued together along their common boundary to yield S3. In
Fig. 2, the dotted circle represents an equatorial circle on the enzyme S2. The enzyme
S2 in fact divides the substrate into two complimentary tangles, the substrate tangle S,
and the site tangle T . The local effect of recombination is to delete tangle T from the
synaptic complex, and replace it with the recombinant tangle R . As in Fig. 3, the knot type
of the substrate and product each yield an equation in the variables S, T and R.
Specifically, if we start with unknotted substrate, we have the equation

S # T = Unknot

(1)

After recombination, we have the equation
S # R = Product Knot (Catenane)

(2)

In the above equations, the symbol # denotes that that tangles are to be identified along
their common boundary, a 2-sphere with 4 distinguished points(the endpoints of the DNA
arcs). Ideally, we would like to treat each of R, S, and T in equations (1,2) as unknowns,


www.pdfgrip.com
10

or recombination variables , and to solve these equations for these unknowns. Since a
single recombinant event yields only 2 equations for 3 unknowns, the best we can hope
for, given only this information, is to solve for 2 of them in terms of a third. Although it is

indeed possible to make substantial progress on the problem as posed in this
generality(ref.18, 19, 20), the analysis is greatly simplified by making some biologically
reasonable assumptions.

One such assumption is, for example, that T and R are

enzyme-determined constants, independent of the variable geometry of the substrate(the
tangle S).

s # R

Product(Torus Catenane)

Fig. 3. Tangle Equations Posed by Recombination(Direct Repeats)

THE MATHEMATICS OF TANGLES
Consider the standard 3-ball B3 in R3. Orient(put an arrow on) the equator of S2 =
0B3(the boundary). Select 4 points on the equator(called NW, SW, SE, NE ), cyclically
arranged so that one encounters them in the order named upon traversing the equator in
the direction specified by the chosen orientation. This copy of the S2 with 4 distinguished
equatorial points will be called the standard tangle boundary. A 2-string tangle , or just
tangle for short, will denote any 3-ball with a configuration of 2 arcs in it, satisfying the
following conditions: (i) the arcs meet the boundary of the 3-ball in endpoints, and all 4


www.pdfgrip.com
11

endpoints are in the boundary, and (ii) there is a fixed homeomorphism from the
boundary of the 3-ball to the standard tangle boundary, which takes the endpoints to the 4

distinguished points {NW, SW, SE, NE}. This fixed homeomorphism is called a boundary
parameterization (refs. 11, 21, 22, 23). By means of this boundary parameterization, we
can regard the boundaries of any two tangles as being identical.

( 2 , 1 , 3 )** 1 1 / 3
3 + 1 / ( 1 + 1/ 2 ) = 1 1 / 3
Fig. 4. Rational Tangles

Fig. 5.

2-bridge(4-plat) Knots and Catenanes

Two tangles are is o m o rp h ic if it is possible to superimpose the arcs of one upon the
arcs of the other, by means of moving the arcs around in the interior of the 3-balls, leaving
their common boundary pointwise fixed. Mathematically, there is a well-understood class
of tangles which look like DNA micrographs, and which are created by twisting strands


www.pdfgrip.com
12

about each other. These tangles are called rational tangles , and have been completely
classified up to isomorphism by Conway(ref. 21). There is a canonical form for rational
tangles, and when written in canonical form, these tangles are classified by a vector with
integer entries, each entry corresponding to a number of half-twists. The entries of the
classifying vector likewise determine via a continued fraction calculation a rational
number which itself classifies the tangle(hence the terminology)(see Fig. 4).

Closely associated with rational tangles is a large class of knots and catenanes known
as 2-bridge, or 4-plats .

Like rational tangles, these knots and catenanes admit a

canonical form and classifying vector(ref. 4).

Fig. 5 shows some rational tangles and

2-bridge knots and catenanes in canonical form, and their classifying vectors.

One

relationship enjoyed by rational tangles and 2-bridge knots is the following: if A and B are
rational tangles, then A # B is 2-bridge.

The salient point here is that this class of

configurations is not only biologically reasonable, but is also computationally
manageable, in which one can solve tangle equations posed by experimental results. In
fact, as we shall see later, the experimental results often force the tangles to be rational,
providing mathematical proof of structure!
Phaae Lambda Int
Bacteriophage I is a virus which attacks bacteria, inserting its own genetic material into
that of the host, eventually turning the host into a virus factory. The genetic insertion
mechanism is site-specific recombinaton by the enzyme Int . When reacted with
unknotted closed circular duplex substrate in vitro , the Int reaction products are V ' torus
knots and catenanes of type (2,k)-2 strands twisted about each other, with k right-hand
half-twists. If k is odd, we obtain a V torus knot, and if k is even, we obtain a "+" torus
catenane. These reaction products form a special subclass of the set of all 2-bridge knots
and catenanes. Fig. 6 shows two remarkable electron micrographs of Int products which
appear in Spengler et a!.(ref. 16).