6 1 
liiform.itici st. 3 letnik 1978 
mathematical 
nnodels far computer-
assisted solutions 
of non - nunnerical 
problems in chennistry 
UDK 681.3:54 
b.jerman 
blažič 
J. Štefan Institute, Universit/ of Ljubljana, 
61000 Ljubljana, Vugoslavia 
MATEMATIČNI MODHLI ZA REŠEVANJE NENUMERIČNIH PROBLEMOV IZ KEMIJE S POMOČJO RAČUNALNIKA. V članku je 
dan pregled uporabe molekularne topologije pri računalniškem reševanju nekaterih nenumeričnih problemov iz kemije. Obdelani 
so naslednji problemi: identifikacija spojin v sistemih zo iskanje in shranjevanje informacij, enumerocijo in generiranje struktur­
nih izomerov s ciljem izločevanja molekularnih struktur na podlagi eksperimentalnih podatkov, predstavitev molekularnih struktur 
in procesa načrtovanja sintez v računalniškem načrtovanju organskih sintez. 
The articie gives o revievv of the applications of the moleculor topolog/ in computer-aided solving of some non-numerical chemi-
col problems. The problems are: Identification of a compound in chemical information retrievol s/stems, enumeration and gene-
ration of itructural isomers for the purpose of speciol chemical studies and in computer-aided elucidation of moleculor structure 
on the basis of experimenfal dota, representotion of moleculor structures and s/nthesis-design process in computer-oided planning 
of organi C syntheses. 
1 . Introduction 
There seems to be hordi/ any concept in natural sciences 
which is closer to the notion of agraph thon the moleculor 
structure of chemical compounds. A moleculor structure may 
be viewed as o groph composed of nodes (otoms) linked 
with edges (chemical bonds). In foct there is no essential 
difference betv/een a graph and a structural formula. A graph 
is a mathematical structure which con be used to represent 
the topolog/ of given molecule. The advdntage of using ; 
grophs in the representotion of moleculor structure lies in 
the possibilit/ of applying directl/ the mathemotical op-
poratus of the graph theor/ for solving speciol chemical 
problems. The idea that metric choracteristics of the mole-
cules (that is bond lengths ond bond ongles) con be 
neglected in chemical studies is more and more popular.The 
moleculor topology allow5 the nori-metric relotionships of 
the moleculor structures ond the totolit/ of information 
cohtained in the moleculor grophs to be investigoted ond 
opplied in o very simple monner. -
Concepts of topology ond groph theory though not always 
recognized as such, are nowdays analysed ond opplied to 
vorious bronches of chemical science: photochemistry (1), 
stereochemistr/ (2), transition metals chemistry (3), boron 
hydrid chemistry (4), saturofed (5) and unsaturated (6) 
hydrocarbon chemistry, etc. Furthermore, basic concepts of 
chemistry such os configuration, isomerism, valency etc, are 
shown to hdve a topological basis (7). 
In the present orticle we wish to review moinl/ the ap-
plication of the moleculor topology qnd the topology of a 
space of molecules in computer-aided solving of some non-
numerical chemical problems. The problems are: Identification 
of o compounds in chemical information retrievol systems, 
enumeration ond generotion of structural isomers for the 
purpoše of speciol chemical studies ond in computer-aided 
elucidation of moleculor structure on the basis of experi-
mental doto, representotion of moleculor structures arid 
synthesis design process in computer-aided planning of 
orgonic syntheses. 
Let us ot first to define a chemical graph and the 
ossociated notions: A chemical groph is (8) a graph con-
sisting of nodes ossociated with atom names, ond edges which 
correspond to chemical bonds. The degree of a node in the 
chemical graph hos its usuol meoning, i.e. the number of 
(non-hydrogen) edges connected to it. The valence of eoch 
atom determines its maximum degree in the groph. A speciol 
kind of chemical grophs are vertex-graphs. V.ertex grdphs are 
cyclic chemicol grophs (8), from which nodes of degree less 
than three hove been deleted. 
2. Identification of o structure of chemical compound in on 
information retrievol system 
There is probably no science in greoter need of mechanized 
information retrievol thon chemistry. Millions of chemical 
compounds are known; new ones ore produced at on even 
foster rate. The chemist hos two moin problems: first, he 
vvonts to find out whether the substance in his test tube is 
already known; second, given a substance, he wants to know 
the properties of similor substonces. 
Both problems con be reduced to o matching process: o 
description of the given compound hos to be matched ogainst 
descriptions of the compounds that moke up the doto bose of 
the retrievol system. To ossure a complete identificotion of 
the compound structure, a detoiled atom-by-atom comporison 
is usuall/ needed between the compound in the query ond the 
compounds in the doto base. If chemical compounds are re-
presented os chemical grophs, the problem of matching the 
query item with the libror/ item becomes identical to the 
problem of isomorphism of grophs, considerably simplified by 
the lobels carried by the chemical graph nodes. 
The problem of isomorphism of grophs received litHe ottention 
in the literature until late 1950's (9). Research into thit 
problem was stimulated by the development of chemical 
Information retrievol systems, wlth chemical structure re­
presentotion in the system's files. An opproach that was 
Implemented in severa! computer programs was the procedure 
of a node-to-node matching in seorch of coincidence. The 
nodes of two chemical grophs are matched one at tirne unfii 
either o volid correspondence is found or unfll lncompatibillfy 
arises (10). In the later, it is necessary to bocktrack to a 
point of former coincidence ond start ogdin with o different 

62 
choice of nodes. Large amount of backtracking is requ!red 
In this technique due to the lock of any criterla In fhe 
decision-making step. From the computational point of view, 
the unavoidable bocktracklng is time wasting as oni/ In 
rore instance is the correct choice made ot each point of 
decision. Thls technique aiso requires Information to be 
saved in order to restart from the last point of ogreement. 
The use of a standard numbering procedure for the nodes in 
the chemical grophs mokes the problem of establishing Iso-
morphism in graphs trivial. Man/ attempts hove been mode 
in order to develop standard numbering procedures. Bouman 
(11) hcs suggested ordering of the nodes in chemical graphs 
besed on the examination of the degree of o node and the 
degree of the nodes to which it is connected. Randi£ (12) 
proposed o ver/ interesting procedure for labelling the 
atoms in the groph b/ considering the rows of the adjacent/ 
matrix of the groph os digits coded in a linear code. The 
seorch for motrices corresponding to a complete graph or to 
a fragment of a groph Is to be carried out b/ ordering of 
the matrices occording to decreasing volues of the numbers 
representing the rov/ vectors. A /ear later, the same auihor 
suggested another solution to the unique labelllng of the 
groph nodes. The sequenclng of otoms is performed occording 
to the relatlve magnitudes of the coefficient of the largest 
eigenvector of the adjacenc/ matric (13). Similar opproach 
to the Bouman scheme is the Morgan algorithm (15) which 
exploits the concept of extended connectivit/. The Morgan'j 
algorithm was implemented in the Information s/stem of 
Chemical Abstroct Service. The classlficatlon of the atoms 
is obtained b/ odding the inltiol connectivit/ volues of 
neorest neighbours and ossignlng the sum to the node con-
sidered. As (16) recognized, this method does not alwa/s 
allow the maxlmum possible dlfferentiotion, although it 
generali/ allows the atoms to be divided into several closses 
depending on the number of non-h/drogen attachements to 
each atom. 
Bort and Giordano proposed (14) a new graph motchlng pro­
cedure in which for the one-to-one motching the Fourler maxlma 
of specifle entities of the knovvn chemical structure was used. 
A completel/ differeni and most computer-oriented opproach 
to groph Identification was odvonced b/ Sussenguth (17). 
The procedure that he suggested is bosed on two principles. 
First, If graphs G and G* are isomorphic, then the subset 
of nodes of G that exhlbit some propert/ must correspond 
to the subset of nodes of G* that exhiblt thls same propert/. 
Second, if the subsets of the nodes of G and G* that are 
chorocterlzed b/ some propert/ do not hove the same number 
of elements, then the two graphs cannot be Isomorphic. The 
motching procedure storts wlth generotlon of subsets of 
nodes that represent the same type of atom. 
The purpose of generotion of subsets of nodes is to reduce 
the number of nodes of G* to vvhich a node of G con 
correspond. The purpose is ochieved b/ toking intersection 
ofthe subsets and b/ motching the resultlng nodes. If some 
nodes are not matched, new subsets must be generoted. The 
algorithm terminotes when every node In G Is paired off 
with a node In G*, or when two corresponding subsets of 
nodes of G and G* are found to differ in the number of 
nodes the/ contoln. If the former Is the čase, graphs G and 
G* are isomorphic, if the later, Isomorphism Is impossible. 
Occasionall/, the algorithm exhausts oll subset generating 
propertles before one of the two conditions is sotisfied. 
This happens when more than one isomorphism is possible 
befween the two graphs, or when the subset generating 
propertles are Incomplete in the sense that some propert/ 
that would establlsh isomorphism or the lock of it has been 
neglected In the design of the algorithm. An improvement 
of the described algorithm was suggested b/ Ming and 
Tauber (18). The/ seporated the stnjcture search and sub-
structure search into a distinct part of the algorithm and 
included first order degree (17) and second order degree in 
the control vector for use In structure search. A short cut 
of the atom-b/-atom search technique and set generotion 
procedure Is the connectivit/ code developed b/ Penn/ (19). 
The connectivit/ code although it is not o solution in itself, 
con be o useful lool when used In conjunction with the two 
general techniques (atom-by-atom search ond the set gene­
rotion algorithm) as it Is done In the compuler program of 
Tauber and Ming (18). 
3. Computer-aided generotion and enumeration of structurol 
isomers 
Problems of structurol isomerism in chemistr/ have received 
much ottention for a long time, but oni/ occasionol attempts 
hove been mode toward o s/stemotic solution of the 
underl/ing groph theoreticol problems of structurol isomerism. 
Groph theoreticions hove frequently considered vorious 
aspects of this topic, but not necessoril/ in the context of 
orgonic molecules. Pol/o prescnted a theorem (20) vvhich 
permits colculotlon of the number of structurol isomers for o 
giveh ring s/stem. Hill (21) ond Taylor (22) pointed out 
thot Pol/a's theorem permitted enumeration of geometrical 
ond optical Isomers In oddition to structurol isomers. More 
recent formulos for the enumeration of isomers of monoc/clic 
oromatic compounds bosed on the graph theor/, permutation 
groups ond Pol/a's theorem were presented (23). Although 
the number of Isomers mo/ be interesting, these methods do 
not displo/ the structure of each isomer. Even in simple 
coses, the task of specif/ing each structure b/ hond vvithout 
duplicotion is on enormous one. BaJoban publlshed o series of 
popers (16) oddressed in part to the problem of specification 
of isomeric structures. Although his method represents on 
important contributlon to the problem of Isomeric structures, 
it does not contoln o mechanism for ovoiding o duplicote 
structures. Most successful in solvlng this problem V/ere the 
vvorks based on the Dendrol algorithm (24). The algorithm 
permits on enumerotion ond representotion of oll possible 
moleculor structures with o given empiricol formula, i.e. o 
glven set of atoms. Chemical structures of ali possible isomers 
are obtained b/ molhemotical permutation of oc/clic and 
c/clic graphs representing oppropriote ring s/stems ond 
ottoched oc/clic cholns of otoms. The Dendral olgorithms was 
implemented in o computer program colled Structure Generator 
(8). The list of the structurol isomers generoted b/ the 
progrom is in the form of a special kind of groph -AND/OR 
tree (25). 
The ring s/stems in the program are constructed from vertex 
graphs (8), vvhich are defined in o given problem b/ a 
series of calculotlons. The first level of the tree, ofter the 
specification of the initiol collection of atoms, is the set of 
oll possible partitlons of the initiol set of nodes. Each 
portitlon consists of the c/clic subunit ond the remaining set 
of nodes. The c/clic subunits are o collection of atoms from 
vvhich oll possible ring s/stems con be constructed on the 
bosis of the oppropriote vertex graph. The otoms in the 
remaining set form oc/clic parts of finol structures, combined 
In ali possible wa/s wlth the ring structures from the 
corresponding initial portitlon. The second level of the tree 
specifies aH possible ring s/stems that con be constructed from 
the vertex graph corresponding to the c/clic subunit in the 
first level of the tree. The next level of the tree just be/ond 
the node specif/ing o possible ring s/stem, specifies the 
possible wo/s In vvhich the remoining atoms can be linked to 
the unfilled llnks of the s/stem. After the three first levels of 
the tree generotion, the program becomes recursive. Each set 
of unstructured nodes is token up as a fresh problem until 
there are no more unstructured nodes. The Structure Generator 
represents o part of a ver/ complex and sophlsticoted computer 
program - Heuristic Dendral Program (25,8) for elucidotion of 
moleculor structure based on structurol feotures of unknovvn 
molecules derived from chemical, ph/sical and spectroscopic 
doto. 

63 
RecentI/, a similar approoch to the problem of exhaustive 
enumerotion ond generotion of chemicol structures WQS 
published by Sasoki and Kudo (26). Their s/stem successfull/ 
deduce oll logicollv valid structures, oc/clic and c/clic on 
the bosis of previousi/ settled proposition occording to the 
input Information obout the structure, of a given compound. 
4. Mathematical models in the computer-oided-plonning of 
orgonic syntheses 
The problems of isomorphism of chemicol grophs ond gene­
rotion and enumerotion of structurol isomers are closely 
related to the vvorks connected with the design of chemicol 
structure Information systems. The justiflcotlon of the 
chemicol structure information systems is the assistance in 
the research process. It happens very often thot It is not 
only the structure of the compound whlch intersets the 
chemist, but olso the properties of the compound whlch the 
structure represents. For instance, the chemist may be 
interested in the synthe5is of the compound, in some of its 
physicarproperties. In Its behavior in a living system etc. 
For these reosons greoter ottentlon should he pold to the 
problem of collection, evaluotion ond correlotion of the 
dota ossocioted wlth a porticular compound. Closer to these 
desired objectives are the studies carrled out in the field of 
the appllcatlon of mochlne computotlon to the generotion of 
chemicol pathways for the synthesis of complex organic 
molecules. Mathematical models in the form of grophs and 
the ossocioted theory in the computer-alded design of 
chemicol synthesis are Involved in two ways: first as a tool 
in the representotion of structures stored into the progroms; 
ond second, os o model In the computer representotion of 
synthesis pathwoys. Synthesls pathwoys con be viewed as o 
tree (27) In vvhich the root is the synthetic torget, the 
Intermediates in the synthesls process are the nodes, and the 
chemicol reoctlons are the edges linking the nodes of the 
tree. 
Severol olternotlves to the computer-alded design have been 
ottempted. The strongest ottentlon from the chemists hos 
evinced the work of the groups of Princeton and Horvard 
(28). Their vvorks were bosed on the Interactive compu-
tation with on on-IIne guidance by a chemist. This feature 
hos enobled them to solve Interesting chemicol problems by 
pooling the resources of a computer wlth a chemist os o 
source of information on reoction ond on strategic design 
decisions. 
The other most significant approoch Is bosed on the Heuristlc 
Search Paradigm of Artlflciol Intelligence (30) research. 
The program developed ot Stony Brook (29) designes 
synthesls vvithout the chemist's intervention using the re­
octlons from the program reoction llbrary and progrommed 
design strotegies. 
A similar opproach can be found in the vvorks of VVhitlock 
on the heuristic solutlons of the functlonol group synthesl5 
problem (32). The reoction library In his program represents 
the implementation of the tronsitlon graph of a finite 
outomoto, vvherein the nodes are functlonol groups ond edges 
ore the reoctlons thot transform one functlonol group into 
another. 
The most mothematical scheme of synthesis-plonning problem 
was suggested by o group of prominent chemists and mathe-
maticions (31). The scheme is bosed on the recognition 
thot ali chemicol reoctlons correspond to Interconversions of 
isomeric ensembles of molecules (lEM) within a family of 
Isomeric ensembles of molecules (FIEM) (31). Distlnguishable 
lEM of FIEM Is represented by a famlly of be-motrlces 
(bond and electron motrlces) F = (Mg, M] ..., Mf). The 
be-matrix Mi of on ensemble of molecules EMj consisting 
of a set A, vvhich contoln n otoms, A= (A^, ..., Ap) Is 
an nxn matrlx as shovvn below: 
M. 
'1 = 12 
°21 °22 
In 
2n 
nI nn; 
were the entries OJ; (i / j) ore the formal bond orders of 
the bonds betvveen polrs of atoms A; ond A:, the diagonal 
entries oij are numerically equlvalent to the number of free 
volence electrons belonging to atom A| in EM|. 
A sllghtl/ modified version of this mathematical model of 
chemicol systems ond their relotlons was used as a bosis for 
the constructlon of algorithm, whlch generotes multistep 
syntheses of a given chemicol compound. The olgorithm and 
the former mothematlcol model were Implemented In an 
organic-synthesIs-plannIng program colled HEDOS (33). The 
program is confined to systems consisting of the benzene ring 
ond functlonol groups attoched to it. Specially designed 
heuristic rules governing the generotion of the best synthesis 
were incorporated Into the program in view of the complexIty 
of the synthesis pothwoy ond the number of steps involved. 
In this approoch, the be-motrices of o FIEM defining metric 
topology, were embedded os elements of stote space, thot we 
called the stote spoce of ensembles of molecules (33). The 
assoclated^set of operotors of the stote spoce was defined os 
o set of reoction matrices D(n); D(n) = l - R| R is the re­
oction matrix which fits (31) the elements of FIEM|. The 
set of fitting matrices F(n) for some stote EM| In the FIEM 
represented by a be-matrlx B| is obtained by the mapping y: 
y: D(n)xFlEM-» F(n); y(D(n),B,)= |F F - B, < O I 
The torget molecule Z is contoined in an initlol EM, denoted 
vvith EM^ • Flnol stotes are oll possible EM ossigned as, EML 
provided thot the chemicol species in EM|_ are contoined In 
the list of avallable chemicol compounds i>, meoning that 
they con be easily synthetislzed or found in the commerciol 
catalogue of the v/orld-known suppliers of fine chemicols. The 
molecules In EML are storting moterlols for the synthesis of 
molecule Z ond compose the list L, L c. f . Thus, the 
problem of synthesis design for a compound Z can be reduced 
to the problem of finding a path K, K = | R^ , ..., Rp I 
into a spoce of FIEM whlch transforms EM^ in EML- The 
seorch for o path through o stote space Is equivalent to the 
travel through o directed graph, in which the nodes correspond 
to varlous EM from FIEM ond edges correspond to the set of 
reoctlons D(n). The root in the directed graph which is o 
tree in this cose is the ensemble EM^. The implemented 
synthesIs-plannIng-algorithm in the program HEDOS generotes 
the spoce of FIEM and seorches for o mlnlmol length path 
vvhich leods from EM to some EMi_. This minimol path meets 
some prescribed criteria, vvhich garantee the feasibility of the 
proposed reoctlons and the valld!ty of generoted structures. 
Information contoined in the be-matrices of the EM Is usually 
insufficient for the evaluotion of the proposed reoctlons ond 
intermedlate structures, so additlonal information concerning 
other moleculor properties was stored in the second symmetrlc 
triangle of the be-matrix. The new matrices were defined os 
bel matrices: 
M, = 
0. . for i < i 
M 
1. . for i > i additlonal Information 
1,1 ' 
The program is not Interactive, i.e. the chemist cannot 
interrupt the program to ossist in the search for synthetic 
intermediates or in the evaluotion of the synthetlc path. The 
program must make ali decisions by itself ond is strictly 
experimental. It wa5 designed for the purpose of developing 
and testing artiflclal intelligence mechanism wlth the old of 
a strongly defined topology of moleculor structures and 
related reactiora. 

64 
5. Conclusion 
The recent approaches to computer-oided solving of non-
numerical chemicol problems have been reviewed and ihe 
merihs and drav/bock of implemenfed mathemaMcal models 
outlined. It seems that the use of graphs as mathematical 
structure In the representation of chemical compounds, os 
they provide o form sultable for computer manipulation, 
becomes more and more popular. Best results in this field 
was achieved in application of groph theory and permutotion 
groups in computer programs which generate ond enumerate 
ali possible structural isomers of a given set of atoms. Thus, 
the problem of exhau5tive isomer generation can in general 
be considered os solved. 
The other problem, Identification of chemical compounds in 
the Information retrieval system, for the solution of which 
the same mathematical model vvas used, was not so success-
fully solved. The majority of Information chemical systems 
stili performe the structure and substructure seorches by 
using logical combinations of the structure fragments, as the 
compounds in the system's files are presented in one of the 
linear notations (WLN, JUPAC-DYSON) or by different 
fragment codes (Mechanical Chemical Code, KWIC indexes). 
Both forms of representations are simple to operate. As the 
volume and the interdisciplinary needs of chemistry, especial-
ly in the research process have increased, and the need for 
fully explicit structure representation of molecules becomes 
essential, various chemical Information systems (CAS, DARC, 
TOSAR) have included graphs as a form of representation 
of chemical notions, but only as a supplement to the files 
with standard records. The great deficiency of this form of 
representation is the tirne - consuming Identification of com­
pounds. The chemist and Information scientists stili work on 
the development of fast and effective groph matching 
techniques, as the problem is not only o chemical problem, 
but aiso a computing problem. The task is large and 
difficult and should require common efforts for its solution. 
The computer-assisted plonning of orgonic syntheses is just 
beginning. The applied mathematical models and artificiol 
intelligence methods have exhibited many deficiences, but 
they con be overcome. The success of the implemented 
computer programs justifies the expectations that the use of 
the computer-assisted-synthesi5 analysis will become o 
routine in the near future. The described mathematical model 
of the space of ensembles of molecules, provides o useful 
basis for the construction of a synthesis-planning algorithm. 
At the same tirne it offers possibilities for further ušes, as 
it is the first ottempt toward o systematization of procedures 
for storing and handling of the vost quantlty of chemical 
information that is currenfly available. 
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