Graph theory and topology in chemistry

This paper will give a brief overview of knot theory and DNA, and will discuss a new topological model for site-specific recombination... In a 3-dimensional medium, a knot or catenane c

GRAPH THEORY AND TOPOLOGY IN CHEMISTRY

Edited by R.B King and D.H Rouvray

studies in physical and theoretical chemistry 51

GRAPH THEORY AND TOPOLOGY

IN CHEMISTRY

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

ELSEVIER SCIENCE PUBLISHERS B.V.

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P.O Box 2 1 1 ,1 0 0 0 AE Am sterdam , The Netherlands

Distributors for the United States and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY INC

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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)

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 T rasatti

Electrodes of Conductive Metallic Oxides Part B edited by S T rasatti

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

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

E x tr in 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 82

D P Jonish and K C M ille t t

New D e v e lo p m e n ts in R e a c tio n T o p o lo g y 91 P.G M e z e y

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 t ic F o rce s 106

D Bonchev and O E P o la n sky

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 rap h In v a ria n ts

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 t r ic A p p ro a c h

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 219

M Johnson, M N a im , V N ic h o ls o n and C -C Tsai

A Subgraph Iso m o rp h ism T h e o re m f o r M o le c u la r G raphs 226

V N ic h o ls o n , C -C T sai, M Johnson and M N a im

A T o p o lo g ic a l A p p ro a c h to M o le c u la r - S im ila r ity A n a ly s is and its A p p lic a tio n 231

C - C T sai, M Johnson, V N ic h o ls o n and M N a im

Section C: Polyhedra, C lusters and th e 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 y n a m ic s o f O c ta c o o rd in a te P o ly h e d ra 239

J B rocas

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

M R andi£, D J K le in , V K a to v ic , D O O a kla n d, W A S e itz and A T Balaban

G raphs f o 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

K T B a lin ska and L V Q u in ta s

F ro m Gaussian S u b c ritic a l to H o lts m a rk (3/2 - L e v y S ta b le ) S u p e r c 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 362

B P it t e l, W A W o yczyn ski and J A Mann

Section D: Eigenvalues, C onjugated 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 373

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 t r ix A p p ro a c h

W A S e itz , G E H ite , T G S ch m a lz and D J K le in

R apid C o m p u ta tio n o f th e E ig e n va lu e s 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 c e p t

J R D ias

On K e k u le S tr u c tu re and P-V P ath M e th o d

H W e n jie and H W enchen

O n e -to -O n e C o rre s p o n d e n c e b e tw e e n K e k u le and S e x te t P a tte rn s

H W enchen and H W e n jie

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

P e rim e te r Codes f o r B e n ze n o id A r o m a tic H y d ro c a rb o n s

W C H e rndon and A J B ruce

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 c e o f a Sphere

X I

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 g ra p h th e o ry and r e la te d areas in re c e n t y e a rs has g e n e ra te d th e need f o r in c re a s in g ly fr e 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

m a th e m a tic a l c h e m is try T his book c o n ta in s th e pa p ers p re s e 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 7 6 -2 0 , 1987 This C o n fe re n c e was

in m any ways a sequel to a s y m po siu m h eld a t o u r u n iv e r s ity in A p r il, 1983, th e papers fro m w h ic h w e re also p u b lis h e d by E ls e v ie r in a s p e c ia l sym p o siu m v o lu m e B o th o f these m e e tin g s w e re sponsored b y th e U.S O ffic e o f N a va l R e se a rch

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 r a c t to g e th e r and to b e co m e b e t t e r in fo rm e d on c u rr e n t a c t iv it ie 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 n c e a p e rm a n e n t re c o rd

o f th e papers p re s e n te d a t th e C o n fe re n c e The 41 pa p ers 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 c o n v e n ie n c e o f th e re a d e r have been g ro u p e d in to f iv e m a jo r

s e c tio n s A lth o u g h we a p p re c ia te th a t any such s u b d iv is io n o f th e C o n fe re n c e

p re s e n ta tio n s w ill a lw a y s be s o m e w h a t a r b it r a r y , we hope th a t g ro u p in g th e papers

in th is w ay w ill h e lp th e re a d e r to lo c a te those pa p ers o f p a r t ic u la r p e rso n a l 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 a lso 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 s ta b lis h e d

J o u rn a l 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 v ra 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 tr ib 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 fr o m te n d if f e r e n t c o u n trie s , n a m e ly B u lg a ria , C anada, C h in a (People's R e p u b lic ), E g y p t, G re a t Britain, 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 id e 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

th e 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 tr y was also discussed and

se ve ra l d e c is io 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 r s , a n u m b e r o f

o th e r issues w e re addressed b y 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 i t was w ith o u t th e s u p p o rt o f a

n u m b e r 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 h e re

r"

We are in d e b te d to th e U.S O f f ic e |>f N a va l R e se a rch f o r th e m a jo r fin a n c ia l s u p p o rt

th a t m ade o u r C o n fe re n c e p o s s ib le L o c a l su p p o rt fr o m th e U n iv e r s ity o f G e o rg ia

R esearch F o u n d a tio n and th e U n iv e r s ity o f G e o rg ia School o f C h e m ic a l S ciences is also a c k n o w le d g e d 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 v id Payne o f th e 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

f o 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 L o w e , who

k e p t t r a c k o f num e ro u s a d m in is tr a tiv e d e ta ils and w ho assiste d g r e a tly in th e p ro d u c tio n

LIST OF A U T H O R S

N A D L E R , F a c u lty o f T e c h n o lo g y , The U n iv e r s ity o f Z ag re b , P.O Box 177, 41001 Z agreb,

C r o a tia , Y u g o sla via

S.A A L E X A N D E R , Q u a n tu m T h e o ry P ro je c t, U n iv e r s ity o f F lo rid a , G a in e s v ille , F L

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 c h 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 P hysique, 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, T X

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 o f C h e m is try , U n iv e r s ity o f South C a ro lin a , C o lu m b ia ,

M J O H N S O N , C o m p u ta tio n a l C h e m is try , The U p jo h n C o m pa n y, K a la m a z o o , M l, U S A

D P JO N N IS H , M a th e m a tic s D e p a rtm e n t, U n iv e r s ity o f C a lifo r n ia , Santa B a rb a ra ,

V K A T O V IC , 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 r s ity , D a y to n , OH 45435,

M H LEE , D e p a rtm e n t o f P hysics, 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

E K L L O Y D , F a c u lty o f M a th e m a tic a l S tu d ie s, The U n iv e r s 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 D e p a rtm e n t, Case W e ste rn R eserve U n iv e r s ity ,

C le v e la n d , OH 44106, U S A

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 c h n o lo g y , B U -8 0 1 0 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 , U n iv e r s ity

o f S aska tch e w a n , S askatoon, Canada S7N 0W0.

D C M IK U L E C K Y , D e p a rtm e n t o f P h y s io lo g y , M e d ic a l C o lle g e o f V irg in ia

C o m m o n w e a lth U n iv e r s ity , R ic h m o n d , VA 232 98 -0 0 0 1, U S A

K C M IL L E T T , M a th e m a tic s D e p a rtm e n t, U n iv e r s ity o f C a lifo r n ia , Santa B a rb a ra ,

C A 93106, U S A

M N A IM , M a th e m a tic s S ciences 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, O H, U S A

V N IC H O L S O N , M a th e m a tic s S ciences 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, O H,

J J O T T , D e p a rtm e n t o f C h e m is try , F u rm a n U n iv e r s ity , G re e n v ille , SC 29613, U S A

L P E U S N E R , L e o n a rd o Peusner A s s o c ia te s , In c , 181 S ta te S tr e e t, P o rtla n d , M ain e

04101, U S A

B P IT T E L , M a th e m a tic s D e p a rtm e n t, O h io S ta te U n iv e r s ity , C o lu m bu s, OH 43210,

X V

q #E# P O L A 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 -4 33 0 M tilh e im a d R uhr,

F e d e ra l R e p u b lic o f G e rm a n y

L 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 r s ity , N ew 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 cie n ce , D ra k e U n iv e r s ity , Des M oine s, Iow a 5031 1, and A m es L a b o r a to ry - D O E , Iowa S ta te U n iv e r s ity ,

HE W E N JIE , H ebei A c a d e m y o f S ciences, S h ijia z h 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 D e p a rtm e n t, Case W e s te rn R e se rve

U n iv e r s ity , C le v e la n d , OH 44106, U S A

T P Z IV K O V IC , The In s titu te R u d je r B o s k o v ic , 41001 Z ag re b , C r o a tia , Y u g o s la v ia

SECTION A

K not T heory and R ea c tio n T opology

Graph Theory and Topology in Chemistry, A Collection of Papers Presented at an 3International 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

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

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

M acromolecules 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

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

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

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)

Sites Aligned Duplex Strands Ends Recombined

Broken

Fig 1 A Single Recombination Event

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

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

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,

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

can regard the boundaries of any two tangles as being identical.

Fig 4 Rational Tangles

(2,1,3)** 1 1 / 3

3 + 1 / ( 1 + 1/2) = 1 1 / 3

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

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)

Int Torus Knot Ll3l

Int Torus Catenane [4]

Fig 6 Electron Micrographs of Int Knots and Catenanes(from ref 16).

For the sake of exposition, let us now assume that all tangles {R,S,T} are rational In this case, we have the following theorem:

THEOREM 1: Suppose that S and T are rational, and that S#T = unknot If

R = (0)(sites aligned in parallel), then S = (n, 0)(see Fig 7) That is, S is a

plectonemically interwound tangle, with n half twists These half-twists

may be either right-handed(n > 0) or left-handed(n < 0)

Fig 7 R = (0), S = (n, 0)

Consider now the case of the Int reaction on unknotted substrate with inverted repeats

We have 12 different reaction products, the torus knots {[2k+1]} 0 < k < 11 (see Fig 8)

We assume that Int is doing the same thing for all the different substrates; that is, that T

and R are enzyme-specific constant tangles This means that we must have at least 12 different substrate tangles For each of the product knot types {[2k+1]}, 0 < k < 11, select a tangle = (n^,0) such that S^# R = [2k+1].

Fig 8 Int Knots(lnverted Repeats).

Theorem 2 :For the recombinant enzyme (inverted repeats), if R,S and T are

rational and T = (0),then R = (r, 0), and n^ + r = 2k+1, 0 < k < 1 1

That is, R is a p le c to n e m ic a lly interwound tangle with r right(left)-hand half-twists, and n^ is uniquely determined The analogous result holds for the case of direct repeats

The proofs of Theorems 1 and 2 can be accomplished by rational tangle calculus,

where one manipulates the classifying symbols to solve the experimentally imposed equations One can try other scenarios in the model In the above scenario(Theorems 1 and 2), we assumed a specific form for the site tangle T, namely T = (0) One can, for example, think of one of the two constant tangles {R,T} as a parameter, and then solve the equations for the rest of the recombination variables in terms of that parameter Although

we assumed that the tangles {R,S,T} were all rational, it is possible to relax this assumption In complete generality, assuming nothing about {R,S,T}, one can prove the following:

THEOREM 3: For the recombinant enzyme In t, the site tangle T and the

recombinant tangle R must be rational tangles

The proof of Theorem 3 involves heavy use of the theory of 3-manifolds , 2-fold branched cyclic covers, Dehn surgery on Seifert Fiber Spaces, and the recently proved

cyclic surgery theorem (ref 24) The proofs of theorems 1-3 will appear elsewhere(ref

20)

CHEMICAL DYNAMICS

Another interesting application of knot theory occurs in the arena of nonlinear wave phenomena in excitable media In their study of propagating wave patterns in excitable biological and chemical media, Winfree and Strogatz(ref 25, 26, 27, 28, 29) produced beautiful pictures and on-target intuition concerning the topological description and mathematical quantization of these patterns Consider the following thought-experiment: grass fires on a large prairie The fire is a propogating wave, and when two fires collide, they annihilate each other The brown grass immediately in front(in the direction of propogation) of the wave is excitable, and an individual blade of grass is stimulated to burn when its neighbors catch fire After burning, the grass is in a refractory state, unable

to transmit pulses of fire But the rains come, the grass grows green and turns to brown, and the cycle repeats^ The characteristic rotating spiral wave patterns turn up in many different contexts: chemical(the Belusov-Zhabotinsky(BZ) reaction and lamellar growth spirals in synthetic semicrystalline polymers(ref 30)), biological(AMP pulses in slime mold colonies), and are believed to be useful in modelling heart fibrillation and neural

networks Fig 9(from ref 29) shows a photograph of the 2-dimensional BZ reaction-the reaction is taking place in a thin layer in a Petri dish In 2 dimensions, the characteristic wave forms are expanding rings(target patterns), and spiral waves rotating about

organizing centers The 2-dimensional wave forms can be thought of as cross-sections of 3-dimensional wave patterns-expanding 2-manifolds and scroll waves (fig 9) The

points on the wavefront are precisely those in phase with respect to the reaction, and (away from tangential intersection of wavefronts) form a codimension one submanifold of the reaction medium The organizing center of the pattern are the points about which the spiral waves rotate, and have no phase with respect to the cyclic reaction The organizing center is a codimension 2 submanifold of the reaction medium The direction of wave

p ro p a g a tio n forms a normal vector field to the wavefront submanifold That is, the

wavefront, together with its vector field, forms a codimension one framed submanifold of

the medium The wavefronts come into the organizing center like the leaves of a book, with the organizing center forming the binding of the book(Fig 10) There is a well-known relationship between codimension one framed submanifolds of a space and maps to

S ^ re f 27) It is this relationship which we will exploit to provide a necessary and sufficient algebraic condition for the existence of wave patterns

Fig 9 The BZ Reaction in 2-dimensions; a 3-dimensional Scroll Wave

Organizing Center Geometry

For the sake of mathematical exposition, we will describe a simplified version of the wave pattern with no tangential wave intersections and no corners formed by intersecting mutually annihilating waves We will also assume that the reaction is taking place in either or S ^ -o r equivalently, the reaction is a local disturbance in a large excitable medium, and none of the wavefronts hit the boundary Corners, tangential intersections,

Fig 10 Wavefronts Impinging on an Organizing Center

and waves intersecting the boundary of the medium are technicalities which can be mathematically dealt with, but with which we will not be presently concerned In this

situation, the wavefront is a smooth codimension one framed submanifold, and the

organizing center is a smooth codimension two framed submanifold In the 2-dimensional case, the organizing center is a collection of rotor points {Pj}, 1 < i < n, with

Nj arms(wavefronts) rotating in tandem around each Pj Fig 10 shows the geometry of 3 arms impinging on a rotor point The arrows in Fig 10 indicate counterclockwise wave rotation around the rotor point For each i, let ej denote the sign of the rotation directionaround Pj-the sign is +1 if the rotation direction is counterclockwise, and -1 if the rotation

direction is clockwise In the 3-dimensional case, the organizing center is a framed link(catenane) in 3-space, a collection of circular components {C,}, 1<i<n, with Njsheets(wavefronts) coming in at each Cj Fig 10 shows 3 sheets coming into a circular component, and the rotation of the wavefront around the curve Cj induces an orientation(arrow) on Cj by the right-hand rule(Fig 10) The 3-dimensional analogue of

the 2-dimensional rotation sign is the linking number of two oriented circles in 3-space

The linking number of two oriented circles can be computed from any projection of the configuration Given a crossover between oriented circular components in any projection, one uses the right-hand rule to assign a sign to the crossover, as shown in Fig 11

Fig 11 Crossover Sign Convention

The linking number Ljj between curves Cj and Cj(i * j) is then the sum of the signed crossovers between Cj and Cj in a given projection, divided by two Self-crossings ofeach curve are ignored It can be shown that the number so calculated is independent of projection, and is a topological invariant of the pair of oriented curves The linking number of two curves is a measure of how much the two curves wind around each other One can also use the wavefronts coming into a given circular component of the

organizing center to define a self-linking number Ljj as follows: thinking of Cj as a

boundary component of a wave sheet, let Cj* be a close parallel copy of Cj which lies onone of the wave sheets coming into Cj Orient Cj* in parallel with Cj We then define Ljj

as the linking number between Cj and Cj* Ljj is a measure of how much the wavefronttwists around Cj

The Phase Map

Let X denote the set of all points in the medium which have a definite phase with respect to the ongoing reaction That is, X is the complement of the organizing center

Assignment of phase determines a map F: X —> s \ where S1 denotes the phase circle Let Q denote the basepoint of the circle(corresponding to the phase angle 0 = 2 k ) The

point Q comes equipped with a normal vector, and Q together with its normal vector forms

a codimension 1 framed submanifold of the phase circle Let W denote the wavefront corresponding to phase angle 0 Then F(W) = Q, and F matches the normal vectors to W with the normal vector to Q In fact, any codimension 1 framed submanifold W' can be

used to construct a phase map F' by the Thom-Pontryagin Construction(rei 31) One

maps W to Q, the normal lines near W' are wrapped around S1 from -tc to +tc, and the rest

of the space is mapped to n Conversely, (nearly) every map F:X > S1 determines a

codimension one framed submanifold W = F-1(Q) That is, Q pulls back to W, and the normal vector to Q pulls back to a normal vector field on W It can be shown that, up to very small deformation, every map does this(ref 31) So, roughly speaking, wavefronts correspond to and characterize phase maps This correspondence can be made

mathematically precise by introducing the ideas of homotopic phase maps and framed cobordism of framed submanifolds.

The Theorems

Fix attention on the 2-dimensional case The question is the following: Given an arbitrary set of local data {Pj, Nj, ej}, 1<i<n, is there a wave pattern in S^(or, equivalently a phase map F:X2 > S1) which satisfies this data?

Theorem 4: A phase map F:X^ ~> S1 exists if and only if

Theorem 5: A phase map F:X3 ~> S1 exists if and only if

i L N =

ij J 0 , 1 < i < n

The proof that the algebraic conditions in Theorems 4 and 5 are necessary involve calculation of winding numbers in dimension 2, and linking numbers in dimension 3 The proof that these same algebraic conditions are sufficient for the existemce of a phase map

involves obstruction theory The local information is in fact the description of a map

fiBX-^S1 One desires to extend this map over all of X The obstruction to extension can

be identified with the summation Since it equals zero in each case, the map extends Details of the proof will appear elsewhere(ref 32) 0

21

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Graph Theory and Topology in Chemistry, A Collection of Papers Presented at an 23International 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 23-42

© 1987 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands

TOPOLOGICAL STEREOCHEMISTRY: KNO T THEORY OF M O LECULAR GRAPHS

D AV ID M W A LB A

Department o f Chemistry and Biochem istry, Box 215, U niversity o f Colorado, Boulder, CO80309-0215

ABSTRACT

Chemists have always been intrigued and stimulated by consideration o f the structural causes

o f isomerism (the phenomenon where two chemical compounds w ith the same number and kind o f atoms exist as distinct, isolable entities) Wasserman firs t suggested a connection between

isomerism and low dimensional topological properties o f molecular graphs, and we have formalized the interface between stereochemistry and low dimensional topology by defining topological stereoisomers as molecules w ith molecular graphs which are homeomorphic, but not homeotopic (the term homeotopic describes graphs which are "interconvertable by continuous deformation in 3- space" and seems preferable to "isotopic" due to the lack o f conflict w ith terms in common chemical usage) Synthesis o f molecules capable o f exhibiting topological stereoisomerism is one focus o f our work resulting in the preparation o f the firs t molecular Mobius strip— a 3-rung Mobius ladder

w ith "colored" rungs The novel topology and dynamics o f the 3-rung Mobius ladder molecule w ill

be discussed Consideration o f the topological and chemical properties o f this and other

topologically interesting molecules has led to the application o f the techniques o f knot theory to graphs In fact, we have shown that in order to understand simple chemical properties o f certain easily conceivable molecules, the techniques o f low-dimensional topology must be used The topological approach to describing molecular symmetry has also led to the discovery o f an

interesting new class o f knots and graphs termed topological rubber gloves These concepts w ill be discussed, as w ell as our most recent studies directed towards the synthesis o f the firs t non-DNA tre fo il knot by cutting "in h a lf' o f a 3 half-tw ist Mobius ladder Finally, the novel "hook and ladder" approach to the synthesis o f molecular knots, which could lead to the preparation o f a figure-8 knot and a topological rubber glove molecule, w ill be oudined

INTRO DUCTION

Topological stereochemistry

A connection between topology and stereochemistry was firs t made by Wasserman almost

30 years ago (ref 1) As discussed in detail in several recent papers (ref 2), we have form alized the interface between lqw dimensional topology and stereochemistry by the follow ing definitions:

• Isomers: compounds w ith the same number and kind o f atoms which may be separated and isolated at room temperature;

• M olecular graph: the graph defined by the structure o f a molecule where the atomic nuclei are represented by vertices and covalent bonds are represented by edges;

• Constitutional isomers: isomers possessing non-homeomorphic molecular graphs;

s: isomers possessing homeomorphic molecular graphs;

£: interconvertable by continuous deformation in 3-space (ref 3);

Topological stereoisomers: stereoisomers possessing non-homeotopic molecular graphs;

• Topological enantiomers: topological stereoisomers which may achieve m irror image

As a part o f stereochemical theory, topological stereochemistry follow s very naturally from standard stereochemical thinking since both are heavily rooted in the concept o f the molecular graph, and both involve the concept o f fle x ib ility o f graphs However, chemists often consider molecular graphs first as rig id Euclidean constructions, then add onto this picture the concept o f molecular fle x ib ility as needed in order to deal w ith the time-average properties o f non-rigid molecules Topological stereochemical thinking takes an alternative approach to stereochemistry wherein one starts w ith the molecular graph as a topological object embedded in 3-space, then adds metric properties as needed This leads naturally to some interesting notions regarding the classical approach to solution o f fundamental problems in chemistry dealing w ith phenomena such as isomerism and topicity o f atoms, especially when dealing w ith large, very flexible, topologically non-trivial molecules

The two aspects o f topological stereochemistry are intim ately related; synthesis o f the molecular Mobius strip led to the firs t form alization o f topological stereochemistry, while the theory continues to suggest interesting new targets for total synthesis

The TH YM E "Mobius strip approach" to the synthesis o f a trefoil knot

The chemistry o f the TH Y M E polyethers provides the foundation o f this work Our version

o f Wasserman's "M obius strip approach" to the synthesis o f a tre fo il knot is illustrated in Scheme 1 Thus, a ladder shaped diol-ditosylate o f type 1, composed o f a series o f crown ether rings fused by the tetrahydroxymethylethylene (TH YM E) unit, is cyclized to give Mobius ladders and prisms The strategy is based upon the premise that i f n is large enough, then products w ith 0, 1, 2 and 3 half­twists (2, 3, 6 and 7, respectively) w ill be formed Cleavage o f the double bond rungs o f the 2

and 3 half-tw ist compounds would give the lin k 8 and tre fo il 9 as shown It is hoped that this two- step process fo r synthesis o f a molecular tre fo il w ill be more easily realized than spontaneous knotting o f a single chain upon cyclization, since the latter seems highly disfavored based upon the fact that, while many such cyclizations have been carried out, no molecular knotted ring has yet been isolated

In this paper are presented our most recent results in continuing studies o f the chemical properties and dynamics o f the 3-rung TH Y M E molecules, a discussion o f some novel chirality concepts which grew out o f consideration o f the topology o f the 3-rung Mobius ladder molecule, results obtained w ith the 4-rung TH Y M E system, and fin a lly a brief discussion o f the interesting

"hook and ladder" approach to the synthesis o f molecular knots

STUDIES ON THE 3-RUNG TH Y M E POLYETHERS

Breaking the rungs o f a M5bius ladder

As we reported some time ago (ref 5), when the 3-rung diol-ditosylate 1 (n=2) undergoes macrocyclization, two products, the TH Y M E cylinder or prism 2 (n=2) and the TH Y M E Mobius ladder 3 (n=2) w ith 0 and 1 half-tw ist, respectively, are produced This was expected based upon examination o f space fillin g molecular models The structures o f these products were established by

a combination o f X-ray crystallography on the crystalline prism 2 (n=2), and by *H and NMR and mass spectrometry studies on both materials From the beginning, however, it was envisioned that the classic operation o f cutting the molecular strips "in h a lf' by cleavage o f the double bond rungs could be achieved Anticipation o f the feasibility o f this process is, indeed, one o f the major attractions o f the TH Y M E system Analysis o f the products o f the cleavage reaction in the 3-rung system provides corroboration o f the structural assignments, and o f course, an efficient cleavage is necessary if a tre fo il knot is to be synthesized by this strategy

SCHEME 2

As shown in Scheme 2, the double bonds o f the 3-rung TH Y M E cylinder 2 (n=2) and the Mobius ladder 3 (n=2) can, in fact, be cleaved by ozonolysis to give the 30-crown-9 triketone 10 and the 60-crown-18 hexaketone 12, respectively These crown ether ketones may be characterized

by gel permeation chromatography, mass spectrometry and *H and 13C NMR spectroscopy However, compounds 10 and 12 are unstable, making a complete characterization o f the pure materials impossible (ref 2c) The stability problem apparently arises from a tendency fo r the carbonyl groupings to react intram olecularly in an aldol condensation to afford products such as 14 Thus, the "unknot" products o f the cleavage reaction spontaneously change topology upon standing!

This is a nuisance in the 3-rung system It was anticipated, however, that this type o f reactivity would prove a serious problem in the scenario leading to the synthesis o f a trefoil knot,

since in the knotted polyketone product the carbonyl groupings would be held very close together in space, accelerating the aldol reaction and making purification o f the simple knotted ring

problematical

A method was therefore sought fo r blocking the carbonyl reactivity o f the ozonized products

to afford stable materials which could be fu lly characterized A fter considerable experimentation, it was found that W ittig methyleneation o f the carbonyl groupings o f polyketones 10 and 12 could be achieved in good yield by treatment o f the ketones w ith lithium -free methylenetriphenylphosphorane

in tetrahydrofuran (THF) This reaction gave the cycles 11 and 13, respectively, which were stable, and could be obtained in very pure form and fu lly characterized

Note that fo r these crown ethers, the and 13C NMR spectra are particularly diagnostic o f the structures This point was a key factor in design o f the carbonyl blocking group chosen The final proof o f the Mobius strip structure o f compound 3 (n=2) is exhibited in Figure 1, showing the fast atom bombardment mass spectra o f the cycles obtained from the cleavage reaction o f

compounds 2 (n=2) and 3 (n=2) follow ed by W itting methyleneation Thus, as clearly shown, the prism 2 (n=2) is cut in half by the cleavage reaction, while the Mobius ladder 3 (n=2) remains in one piece

FIG 1 The top diagram shows the FAB mass spectrum o f compound 11 (m/e 475 (M + l)+) using

a glycerol m atrix w ith a trace o f FICL The bottom spectrum, taken under identical conditions, is from compound 13 (m/e 949 (M + l)+ ) Peaks at m/e 645, 553, 461, and 369 are due to protonated glycerol oligomers