A graph theoretical approach to SPR

8/20/2019 A graph theoretical approach to SPR

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a

~m po siu m n G ra ph Theory in Chemistry

6 10

2 4

Graph-Theoretical Approach

to Structure-Property Relationships

Zlatko MlhaliC

Faculty of Science and Mathematics, The University of Zagreb, Strossmayerov trg 14.41000 Zagreb, The Republic of Croatia

Nenad TrinajstlC

The Rugjer BoSkoviC Institute,

P.O.B.

1016,41001 Zagreb,The Republic of Croatia

A

fundamental concept of chemistry is that the struc-

tural characteristics of a molecule are responsible for its

pmperties

1).

This was pointed out in the middle of the

last century by Crum Brown and Fraser

2 )

who had also

devised one of the first structure-property models. How-

ever, the earliest work in which this relationship was ob-

served (the toxicity of methyl and amyl alcohols)was a the-

sis by Cms in 1863 (3).

A

Topological Model of Matter

The origin of the structure-property concept can be

traced (4) to the work of the Croatian Jesuit priest, scien-

tist, and philosopher Rugjer Josip BoGkoviC

5)

who intro-

duced the idea of representing atoms as points in space (6).

(His major work was the theory of a single law of forces.)

By allowing the point atoms to assume a variety of differ-

ent arrangements, BogkoviC was able to account for the ex-

istence of different substances.

In this way the BobkoviC model may be considered as the

forerunner of a topological model for the structure of mat-

ter. BoBkovib's fundamental idea, which is of the greatest

importance in chemistry, was that substances have differ-

ent properties because they have different structures. This

idea was used, for example, by Davy to rationalize the dif-

ference between diamond and graphite (4,

7 .

Table 1. List of Selected Topological Indices

Topological Standard Structural interpretationa Author (Year)

index symbol

Wiener

W

Sum o distances

n

a Weiner (1947)

number molecular graph

Hosoya Sum o countsof non- Hosoya (1971)

index adjacent edges

n

a

molecular graph

RandiC Sum of weighted edges RandiC (1975)

index

n

a molecular graph

Balaban

J

Sum of weighted Balaban (1982)

index distances

n

a molecular

graph

Schultz

MTI

Sum of elements of the Schultz (1989)

index structural row matrix

v[A

+bD]of a molecular

graph

Haraty H Sum of squares of PlavSiC, NikoliiC

number reciprocal distances

n

a TrinajstiC (1991)

molecular araoh

Table 2 List of Properties that

Are

Deslrable for

Topological Indices a s Proposed by RandiC 18)

1

Direct structural interpretation

2

Good correlation with at least one molecular property

3 Good discrimination of isomers

4

Locally defined

5 Generalizable

Linearly independent

Simplicty

Not based on physical or chemical properties

Not trivially related to other indices

Effidencyo construction

Based on familiar structural concepts

Correct size dependence

Gradual change with gradual change in structures

QSPR

The structure-~m~ertvelationships want ifv the con-

nection between ihe structure and p pekies o


2/12

liferation will stop in the near future. Here we will review

onlv several selected touoloeical indices. Table 1 ists six

topblogical indices thatAwill-be considered in this report.

Table

2

gives a list of useful properties that are desirable

for topological indices 18).

The desirable properties proposed by RandiL

18)

epre-

sent the very high level of sophistication that a topological

index should achieve. All six indices listed in Table 1 ap-

proach this ideal. Their weakest point is the discrimina-

tion of isomers. This narticular urouertv is rather low for

all topological indices'considerei here except the Balaban

index (19-22). However, this is the weak point of most to-

pological indices, except for molecular identification num-

bers (23). Nonetheless. the low discriminatorv Dower of

many indices does not prevent them from being useful de-

scriptors in structure property activity modelling.

In the next section we will give a brief survey of elemen-

tary (chemical) grapb-theoretical concepts. This section

will be followed by a section containing definitions of the

six selected to~oloeicalndices. In the fourth section a de-

sign of the ~ t ~ c t - ~ m ~ e r t ~elationships will be deline-

ated. Then a didactic example will be presented.

Elementary Graph Theoretical Concepts

We will cover only those graph-theoretical concepts that

will be used in this report. In doing so, we will follow the

book Graph Theory by FrankHarary(24) and both editions

of our book Chemical Graph Theoq (8,16,25).

Graph theory is a branch of discrete mathematics, re-

lated to topology and wmbinatorics. It deals with the way

objects a re connected and with all the consequences of the

connectivity. The connectivity in a system is, thus, a funda-

mental quality of graph theory.

Chemical graph theory is a branch of mathematical

chemistry, and consequently of theoretical chemistry. I t is

concerned with handling chemical graphs, tha t is, graphs

that represent chemical systems. Hence, chemical graph

theory deals with analyses of all consequences of connec-

tivity in a chemical system. In other words, chemical graph

theory is concerned with all aspects of the application of

graph theory to chemistry.

The Concept of a Graph in Graph Theory

The central concept in graph theory is tha t of a graph.

For a graph theorist, a graph is the application of a set on

itself, tha t is, a collection of elements of the set and of bi-

nary relations between these elements. Graphs are one-di-

mensional objects, but they can be embedded or realized in

spaces of higher dimensions.

For a chemist, the two-dimensional realization of a

graph is more appealing, tha t is, a set of vertices (points)

and of edges (lines) oining these vertices. Agraph G can be

visualized by a diagram when the vertices are drawn as

small circles or dots, and the edges as lines or curves con-

Figure

1.

Adiagram of a labeled (numbered)graph, showing vertices

as circles and edges as lines. The graph is aclualy aone-dimensional

entity, by

it

can be realized

in

two dimensions, as shown here.

necting the appropriate circles. Because a diagram of a

graph completely describes the graph, it is customary and

convenient to refer to the diagram of the graph as the

graph itself.

Mainly due to their diagrammatic representation,

graphs have appeal as structural models in science, in gen-

eral, and in chemistry, in particular (26,271.As an exam-

ple, Figure

1

hows a diagram of a labelled graph. Agraph

is called labeled when a specific numbering of the its verti-

ces is introduced.

The Concept of a Graph in Chemistry

In chemistry, graphs can be used to represent a variety

of chemical objects such as molecules, reactions, crystals,

polymers, and clusters. The common feature of chemical

systems is the presence of sites and connections between

them. Sites can be atoms, electrons, molecules, molecular

fragments, intermediates, ete., while the connections be-

tween sites can represent bonds, reaction steps, van der

Waals forces, etc. Chemical systems can be represented by

chemical graphs using a simple conversion rule: Sites are

replaced by vertices and wnnections by edges.

Molecular Gmphs

A special class of chemical graphs a re molecular graphs.

Molecular graphs (or constitutional graphs) are chemical

graphs tha t represent the constitution of molecules. In

these graphs vertices correspond to individual atoms, and

edges correspond to the bonds between them. An interest-

ing historical detail

i.;

related to the concept of the molecu-

lar graph: The term graph was introduced by English

mathematician Sylvester (28) in 1878 on the basis of the

constitutional formulas used by the chemists of his day.

To simplify the manipulation of molecular graphs, hy-

drogen-depleted graphs are often used. Such graphs repre-

sent only the molecular skeletons, omitting hydrogen

atoms and their bonds. As an example, Figure gives a

labeled molecular hydrogen-depleted graph that depicts

the carbon skeleton of 2,3,4-trimetbylhexane.

Figure

2.

A aoelea, hydrogen-depleted,molecular graph correspond-

mg lo tne carbon skeleton of 2.3.4-trimethylhexane.Tne vertices cor-

respono to aroms, an0 the edges correspono to chem.ca wnos.

Analyzing and Comparing Graphs

Two graphs GI and Gz are isomorphic if there exists a

one-to-one correspondence between their vertex sets V(GJ

and V(G2), which induces a one-to-one correspondence be-

tween their edge sets E(GJ and E(G2).

n

invariant of a

graph G is a quantity associated with G tha t has the same

value for any graph that is isomorphic with G. I t should be

noted that topological indices are graph invariants.

Two vertices i and of a graph G are adjacent if there is

an edge joining them; the vertices i and are then incident

to such an edge. Similarly, two edges of G are adjacent if

they have a vertex in common. The valency of a vertex i of

G is the number of edges incident to i. This is denoted by

dm.

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A walk of a graph G is an alternating sequence of vertices

and edges, beginning and ending with vertices, in which

each edge is incident with the two vertices immediately

preceding and following it. A path is a walk in which no

vertex occurs more than once. The distance between two

vertices is the number of edges in the shortest path that

joins the two vertices. Agraph G is connected if every pair

of its vertices is joined by a path. Otherwise, a graph is

considered disconnected.

A graph whose vertices all have the same valence is

called a regular graph. If all vertices in a regular graph

have a valence of 2, then the graph is called a cycle. A tree

is a connected acyclic graph. The molecular graph in Fig-

ure 2

is

an example of a tree. A graph is acyclic

if

it has no

cycles.

Associating Graphs with Matrices

A labeled (chemical) graph may be associated with sev-

eral matrices. Two very important graph-theoretical ma-

trices a re the vertex-adjacency matrix and the distance

matrix.

The vertex-adjacency matrix, A = A(G), of a labeled con-

nected graph G with N vertices, is a square symmetric ma-

trix of orderN. It is commonly called the adjacency matrix.

It is defined below.

1 if

vertices

i

and are adjacent

1)

The distance matrix, D = D G), f a labeled connected

graph G with N vertices is a square symmetric matrix of

order N. It is defined below.

where l j is the length of the shortest path (i.e., the dis-

tance) between the vertices and in G.

Very often the distance matrix of a graph G can be gen-

erated using powers of the corresponding adjacency matrix

of G 29).Table 3 gives the adjacency matrix and the dis-

tance matrix that correspond

t

the molecular graph in

Figure 2.

Table3 The Adjacency Matrix and the Distance Matrix

of the Molecular Graph in Figure2

Definitions of the Selected Topological Indices

Wiener Number

The Wiener number, W = W(G) of G, wasintroduced by

Wiener

in

1947 as the path number 30).This topological

index is defined as the half-sum of the elements of the dis-

tance matrix 15).

Table 4 gives an example for computing he Wiener num-

ber.

Table

4

The Computation of the Wiener Number for a

Tree TDepicting the Carbon Skeleton

of BMethylbutane.

(a)A labeled tree 7

b)

The distance matrix of T

(c) The Wiener number of T

W n =-( I+ 8.2+ 4.3) 18

2

Table5 The Computation of the Hosoya Index for a

Tree TRepresenting the Carbon Skeleton

of 2 3-Dimethylpentane.

(a)A tree

T

(b)

The count of the ~(Tfiquantities

n T

i)

p T;O)

= 1

ii)

p T;I)

=

6

i i i ) p(T;2)

=

8

iv)

p(T:3)

= 2

(c)The Hosoya index of

T

Z n

=

p(T;O) p(T;l)

p(T;2)

p(T;3) 17

Volume 69 Number 9 September 1992

7 3


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Table6. The Edge Weights of 10 Edge Types

Which Appear in Graphs Corresponding to the Carbon

Skeletons of Hydrocarbons

Table8. The Computation of the Balaban lndex

for a Labeled Tree TRepresenting the Carbon Skeleton

of 2,3-Dimethylpentane

(a)A labeld tree 7

I ? 1

1 2 0.7071

1,3 0.5774

1,4 0.5

2 2 0.5

2 3 0.4082

2,4 0.3536

3,3 0.3333

3,4 0.2887

4,4 0.25

Table 7. The Computation of RandiC lndex for a Tree T

Depicting the Carbon Skeleton

of 4-Ethyl-2-methylheptane

(a)

A

tree T

b) Count o the edge-types (the numbers at the vertices

represent their valencies)

4 2 = 2

4 3 = 2

bL2= 1

b 3 = 4

c)

The Randit index o

x q 2.0.7071 2.0.5774 0.5 4.0.4082 4.7018

Hosoya lndex

The Hosoya index, Z

=

Z(G), was introduced by Hosoya

in 1971 as the Z index 15).This index is defined below.

wherep(G; i) is the number of selections of i mutually non-

adjacent edges in G.

By definition, p(G; 0) = 1, and p(G; 1) s the number of

edges in

G.

Table 5 gives an example of computing the

Hosoya index.

RandiC lndex

The Randid index,

=

x(G) of G was introduced by

RandiC in 1975

as

the connectivity index (31). This is one

of the most widely used topological indices in QSPR (32-

c) The Balaban index o

b)The distance sums

34) (and also in quantitative structure-reactivity relation-

ship ( SARI 35)).

The Randid index is defined as

D( z)=

where d(i) and d j)are the valencies of the vertices

i

and

tha t define the edge ij.

For saturated hydrocarbons, eq 5 may be givenin closed

form. In molecular graphs that depict the carbon skeletons

of hydrocarbons, only four types of vertices with respect to

their valencies appear, tha t is, vertices with

d

=

1,2,

3 4.

These give rise to 10 types of edges whose weights are

given in Table 6.

If the number of each edge type is denoted by

0 1 2 3 4 2 3

1 0 1 2 3 1 2

2 1 0 1 2 2 1

3 2 1 0 1 3 2

4 3 2 1 0 4 3

2 1 2 3 4 0 3

3 2 1 2 3 3 0

bg

where i

= 1, ... 4

j

i ... 4

and if the edge weights from Table 6 are used, then eq 5

becomes the following.

This expression reveals tha t the Randid indices of hydm-

carbons are fully determined by the counts of the edge

types in the corresponding hydrogen-depleted graphs.

Table

7

gives an example of computing he RandiC index by

means of eq 6.

Balaban lndex

The ~a l a banndex,

J

= J(G) of G, was introduced by

Balaban in 1982 as the average-distance sum connectivity

(36). It is defined as

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Table

10.

The Com~utation f the Haraw Number for a

able 9. The Computationof the Schultz lndex for a

Tree TDe~ictinahe Carbon Skeleton

a)A labeled tree

T

b)The adjacency matrix of

T

ic) The distance matrix of

d) The adjacency-plus-distance matrix of T

I l

e)The valence row matrix of T

v T ) = [ 1 3 2 2 1 11

1)

The v[A Dl row matrix

v[A D] T) [22 15 16 16 25 221

(g)

The Schukz index of T

MTZ T)

= 2.22 15 16 18 25= 118

where

M

is the number of edges in G; v is the eyelomatic

number of G;and D) i s the distance sum where

i

= 1,2,

...,

N

The cyclomatic number

= p G)

of a polycydic graph

G

is equal to the minimum number of edges that must be

removed from

G

to transform it to the related acyclic

graph. For trees,

= 0;

for monocycles,

v =

1

The distance sum Dlifor

a

vertex i of G represents a sum

of all entries

in

the corresponding row of the distance ma-

trix.

Clearly the Wiener number can also be expressed in

terms of the distance sums.

Table 8 gives an example of computing the Balaban index.

Tree T~eljictin~he Carbon skeleton

of 2,3-Dimethylhexane

a)

A

labeled tree T

b)

The distance matrix

of 7

c)The D- matrix of T

I

.2 0.25

0.33 0.5 1

0

0.2 0.25

0.5 1

0.5 0.33 0.25

0.2

0 0.33

0.33 0.5 1

0.5 0.33 0.25

0.33

0

d)The

D-

matrix o

T

I

1 0.25 0.11 0.06

0.04 0.25

0.11

1

0 1 0.25

0.11 0.06 1

0.25

0.25

1

0

1

0.25 0.11 0.25

1

e)The Harary number of T

H T)

= 14.1 16.0.25 14.0.11

8.0.06 +40.04) 10.10

Schultz lndex

The Schultz index, MTI = MTI G) of G , was introduced

by Schultz in

989

as the molecular topological index 3 7 ) .

This index is defined below 21,371.

MTI = i

i = l

10)

where the ezs i

=

1,2, ...,N represent the elements of the

following row matrix of order

N

where

v

s the valency row matrix,

A

is the adjacency ma-

trix, and is the distance matrix. Table

9

gives an exam-

ple of computing the Schultz index.

araiy Number

The Harary number, H = H G ) of G , was introduced by

PlavSiC et al. 3 8 ) n 99 n honor of Professor Frank Har-

ary on his 70th birthday He greatly influenced the devel-

opment of graph theory and chemical graph theory. This

index is defined below.

Volume

69

Number

9

September

992

7 5


6/12

Table

11.

The Wiener

Numbers

IWI.

Hosova

Indices a.andic indices

irl

Balaban lndices

(4,

chultz indic M T ~ arary ~u ni be rsH) and Boili

Points (bp In 'C) of Alkanes with Up to 1 Carbon Atoms

Alkane

W Z

J

MTI

H

p

methane 0

ethane 1

propane 4

2-methylpropane 9

butane 10

2,2-dimethylpropane 16

2-methylbutane 18

pentane 20

2.2-dimethyl butane 28

2.3-dimethyl 29

butane

2-methylpentane 32

3-methylpentane 31

hexane 35

2,2,3-trimethylbutane 42

2,2-dimethylpentane 46

3,3-dimethylpentane 44

29-dimethylpentane 46

2,4-dimethylpentane

48

2-methylhexane 52

3-methylhexane 50

3-ethylpentane 48

heptane 56

2,2,3,3-tetramethyl- 58

butane

2,2,3-trimethyl pentane 63



2,2-dimethyl hexane 71

3,3-dimethylhexane 67

3-ethyl-3-methyl- 64

pentane

2,3.4-trimethylpentme 65


3-ethyl-2- 67

methyipentane



2,s-dimethylhexane 74

2methylheptane 79

3-methylheptane 76

4methylheptane 75

3-ethylhexane 72

octane 84

2,2,3,3- 82

tetramethylpentane

2,2,3,4- 86

tetramethylpentane

2,2,3-trimethylhexane 92

2.2-dimethyl-3- 88

ethylpentane

3.34-trimethylhexane 88

2,3,3,4- 84

tetramethylpentane

2,3,3-trimethylhexane 90

2,3-dimethyl-3- 86

ethylpentane

2,2,4,4- 88

tetramethylpentane

where V s th e mat rix whose ele-

ments ar e th e squares of the reciprocal

distances in

G.

TheD matrix may be considered as

the distance matrix of a class of spe-

cially weighted graphs in which

weights between vertices in

G

mimic

the Coulomb law between the sites in

the corresponding structure. Table 10

eives a n examole of comoutine the

ka ra ry number:

Table 11 eives the Wiener and Har

ry numbers, and the Hosoya, RandiC,

Balaban, and Schultz indices for al-

kanes with up to 10 carbon atoms.

Designing QSPR Models

There a re several ways to design

QSPR models 39-44).Here we outline

one possible strategy. Figure 3 con-

tains a flow diagram of the steps in-

volved in the design of a QSPR model.

This is an iterative approach.

Step

1 Get a reliable source of experi-

mental data for a given set

of

molecules.

This initial set of molecules is sometimes

called the training set

45).

The data in this

set must be reliable and accurate. The qual-

ity of the selected data is important because

it will affect all the following steps.

Step 2 The topological index is selected

and computed. This is also an important

step because selecting the appropriate topo-

logical index (or indices) can facilitate find-

ing the most accurate model.

Step 3 The two sets of numbers are then

statistically analyzed using a suitable alge-

braic expression.

The QSPR model is t hus

a

regression

model, and one must be careful about

its statistical stability. Chance factors

could yield spuriously accurate corre-

lations (4648). The quali ty of the

QSPR models can be conveniently

measured by the correlation coefficient

r and the s tandard deviations. Agood

QSPR model must have > 0.99, while

depends on the property. For exam-

ple, for boiling points,

s c

5 C. There-

fore, Step 3 is a central step in the de-

sign of the structure-property models.

Step 4 Predictions are made for the val-

ues of the molecular property for species

that are not part of the training set

via

the

obtained initial

QSPR

model. The unknown

molecules are ~ t ~ ~ t u r d l yelatedto the ini-

tial set

of

compounds.

Step 5

The predictions are tested with

unknown molecules by experimental deter-

mination of the predicted properties. This

step is rather involved because it requires

acquiring or preparing the test molecules.

Step

6. If the tests support the predic-

tions, one presents the

QS R

model in its

final form with all necessary statistical

characteristics.

If the te sts do not support the initial

QSPR model, it must be revised and

7 6

Journal of Chemical Education


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Table 11 Continued

Alkane

2 2 4trimethylhexane

2 4 4trimethylhexane

2 2 5-trimethylhexane

22-dimethyiheptane

3 bdimethylheptane

44-dimethylheptane

3-ethyi-3-methylhexane

3 bdiethylpentane

23.4-trimethylhexane

2 4-dimethyl-3-ethyipentane

2 3 5-trimethylhexane

2 3-dimethylheptane

3-ethyl-2-methylhexane

3 4-dimethylheptane

3-ethyl-4methylhexane

2 4-dimethylheptane


3.5-dimethyiheptane

2 5-dimethylheptane

2 6-dimethyiheptane

2-methyloctane

3-methyioctane

4-methyloctane

Sethylheptane

4-ethylheptane

nonane

2 2 3 3 4-pentamethylpentane

2 2 3 3-tetramethylhexane

3-ethyl-22.3-trimethylpentane

3 3.4 4-tetramethylhexane

2 2 3 4 4-pentamethylpentane


3-ethyl-2 2 44rimethylpentane

2 3 4 4tetramethyihexane

2 2 3 5tetramethylhexane

2 2 3-trimethylheptane

2 2dimethyl-3-ethylhexane

3 3 4trimethylheptane

3.3-dimethyl-4-ethylhexane



3 4-dimethyl-3-ethylhexane

3-ethyl-234-lrimethylpentane

2 3 3 54etramethylhexane


2.3-dimethyl-3-ethylhexane

33diethyl-2-methylpentane


2 2 5-trimethylheplane

2 5 54rimethylheptane

2 2 6-trimethyiheptane

2 2-dimethyloctane

3 3-dimethyloctane

4 4-dimethyloctane

3-ethyl-3-methylheptane

4-ethyl-4-melhylheptane

3 3-diethylhexane

MT


121 58 4.4641 3.8140 436 13.9933 161

Volume 69 Number

9

September

1992

707


8/12

Table

11

Continued

Alkane

2,3.4-trimethylheptane

2,3-dimethyi-4-ethylhexane

2,3-dimethyl-4-ethylhexane

2,4-dimethyl-3-ethyihexane

3,4,5-trimethyiheptane

2,4-dimethyl-3-isopropylpentane

3-isopropyl-2-methylhexane

2,35trimethylheptane

2,5-dimethyl-3-ethylhexane



2,3-dimethyloctane


3.4-dimethyloctane

4-isopropylheptane


43-dimethyloctane


3.4-diethylhexane

2,4,6-trimethylheptane

2,4-dimethyloctane


3,5-dimethyloctane


2,5-dimethyloctane


3.6-dimethyloctane

2.6-dimethyioctane

2.7-dimethyloctane

2-methylnonane

3-methylnonane

4-methylnonane

3-ethyloctane

5-methylnonane

4-ethyloctane

4-propylheptane

decane 165 89 4.9142

the procedure repeated. The

QS R

model thu s estab-

lished, even for a narrow c lass of compounds, is a very use-

ful

tool for predicting t he properties of hypothetical com-

pounds a nd for the s earch for new compounds with

programmed properties 12).

An Instructive Example

We will apply the procedure from the preceding section,

to give an instru ctive example of the design of the

QSPR

model for predicting th e boiling points of alkanes. As the

initial set we will consider alkanes with up t

8

carbon

atoms (40 molecules).

Step

The boiling points ( C) of the alkanes are taken from the

CRC Handbook of Chemistry and Physics

49)

and Beil-

stein

50).

Step

2

We will consider at th is s tage

all

six topological indices

discussed i n this report.

3.5833

3.7561

3.7561

3.7979

3.6854

3.9835

3.7280

3.4617

3.6033

3.5027

3.3014

3.1296

3.3978

3.3088

3.4999

3.5637

3.3759

3.5299

3.6982

3.3374

3.1600

3.3908

3.2686

3.4123

3.1244

3.2555

3.1682

3.0333

2.9095

2.7732

2.8862

2.9680

3.0869

2.9984

3.2055

3.2951

2.6476

Step

MTI

find

The following structure-property models ar e th e most

successful for each index considered:

p 77.93 (M.97) ~30899 0 0137 - (3.35 .02)10~

-164.24 (i4.99) (13)

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Journal of Chem ical Education


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Table

12.

The Predicted Values of Boiling Points

( C)

of Nonanes

Figure 3. A flow diagram of the steps involved

n

the design of a

QSPR model.

1 Source of experimental data. 2: Seledion of

the

topological index.

3:

Statistical work and senino uo the QSPR model. 4 Predictions.

~ .~ .

r

~

~

5: Test ng the predictions.

6.

The final foml ofthe OSPR model. S:

Tests confirmea he nit:al model. Tne model appears to be satlsfac-

lory for f~rtherwork. hS: Tens rejected the nit al model as not sat~ s-

factory. Tne model mJst be rev,seo and the proced~reepeateo

~ n t i l

the satisfactory model is obtained

The most accurate models ar e those based on in

Z

eq 14)

and eq 15). They

will

be used in th e next step.

Step 4

We use eqs

14

and

15

o predict t he boiling points of non-

anes 35 molecules) see Table 12).

Step 5

We compare the predicted and experimental values of

the nonane boiling ~ o i n t ssee Table 13).

Both models have problems with some members of the

nonane series. However. when S t e ~i s r e ~ e a t e dsine the

boiling points of all alkan es with

up

o

9

Ar ba n atom; the

QSPR models based on

n Z

an d did not improve. The

slight improvement happened only when a hiparametric

model with and N is th e number of carbon atoms in alk-

ane) was used.

This model

is

given by

predicted boiling point

Nonane ~q 14 ~q 15

2,2,3,3-letramethylpenlane

119.26 119.40

2,2,3,4-tetramethylpentane

2,2,3-trimethylhexane

2,2-dimethyl-3-ethylpentane



233-trimethylhexane



2,2,Plrimethylhexane

2.4,Ptrirnethylhexane

2.2,5-lrimethylhexane

22-dimelhylheptane

3.3-dimethylheptane

4.4-dimethylheptane


3.3-diethylpentane

2,3,Ptrimethylhexane


2,3,5trimethylhexane

2,3-dimethylheptane


3,4dimelhylheptane

3-ethyl-Pmethylhexane

2,4-dimethylheplane


3,5-dimelhylheptane

2,5-dimethylheptane

2,6-dimethylheptane

2-methyloctane

3-methyloctane

4-methyloctane

3-elhylheptane

4-ethylheptane

nonane

The procedure may be repeated, a nd we will eventually

arrive a t the best possible QS R model for predicting the

boiling points of alkanes.

Step

6

All thre e models expressed

as

14, 15, and 19 may serve

as reliable models for predicting th e alk ane boiling points.

Plots of

p

vs in

Z

p

vs

X

and

p

vs

X

nd th e accompa-

nying statis tical da ta a re given, respectively, in Figures

4-

6.

The boiling points of alkane s have been predicted many

times 8,13,15 ,3037 ,40,51 ). Althoughmost of the QSPR

models produced are very accurate

r > 0.998, s <

2

W

they suffer from several shortcomings.

i. Methane was not considered.

In

some cases other lighter

alkanes, such as ethane and propane, were also eliminated

from the study.

ii. Models were built for a limited set of alkanes, usually for

C4-C7 families.

iii. The complexity of some of the accurate QSPR models n

the l iterature is forbidding.

For

example, one

of

the most

a m -

rate QSPR models for predicting boiling points of

alkanes

is

the following 40 ) .

All

alkanes with up to

9

carbon atoms have

been considered but methsne.)

Volume

69

Number

9

Sevternber

1992

709


10/12

Table 13 Comparison between Predicted (Two Models) and Experimental Values of Boiling Points ( C) of Nonanes

Nonane (bp)exp Model Model Nonane (bp) Model Model

(14) (15) (14) (15)


2.2,3,4-tetramethylpentane

2.2,3-trimethylhexane

2,2-dimethyl-3-

ethylpentane



2,3,3-trlmethylhexane

2.3-dimethyl-3-

ethylpentane

2,2,4. tetramethylpentane

2,2+trimethylhexane

2,4, trimethylhexane

2.23-trimethylhexane

2,2-dlmethylheptane

3,3-dimethylheptane

4,4-dimethylheptane


32-diethylpentane


2,4-dimethyl-3-

ethylpentane


2.3-dimethylheptane

3-ethyl-2methylhexane

3,4dimethylheptane


2,4dmethylheptane


3,bdimethyiheptane

2,5dimethylheptane

2,6dimethylheptane

2-methyloctane

3-methyloctane

4-methyloctane

3-ethylheptane

4-ethylheptane

nonane

2M

0 W

0 50

1 w 1 50

2w

25 0 3 w 3 50

In

gure 4. plot of p

vs

In Zfor the first 40 alkanes.

710 Journal of Chemical Education


11/12

Figure 5 Aplot of bpv s for the first

40

alkanes

Figure 6 A plot of bpvs y or the first

75

alkanes

Volume 69 Number 9 September 199 2

711


12/12

The

in eq 20

are

defined Figure 7. Examples

of

a path (3rd order), a cluster (3rd order) and a pathcluster 4th order) for a

as follows.

tree Tcorresponding to 3-methylpentane.

The

extended connectivity index

m ~ =[d(i) dm

...

d(m l)la5

(21)

where m represents the order of possible fragments. When

m = 1. framnents are edges which lead to the f int-order

connekivitY ndex

x.

-

The

zero-connectivity index

u

where nl,

n 2 ,n3,

and

n4

are the numbers of vertices with

valencies 1,2,3, and 4, respectively

Connectivity indices ' x of order m and type t can be ob-

tained by summing analogous terms over subgraphs in-

volving paths (t = p ,clusters (t= c), or path-cluster ( t =pc)

combinations ofm edges. Examples of a path, a cluster and

a path-cluster are given in Figure 7.

To conclude this section we stress that there is no simple

QSPR model for predicting boiling points over a wide

range of alkanes. However, if we limit ourselves to a simple

family of alkanes (especially with less than 10 carbon

atoms), then simple aceurate models are possible

34).

Conclusions

In this report we presented a strategy for designing the

quantitative structure-property relationships based on to-

pological indices. The instructive example was directed to

the design of the structure-property model for predicting

the boiling points of alkanes. Six selected topological indi-

ces were tested. The most accurate QSPR models for alk-

ane boiline ~oi n ts re based on ln

2.

and

Nu.

The accu-

~ ~~

racy of t h l bodel was judged according to thLcorrelation

coefficient and the standard error. The umer limits for the

accurate models were set a t

r

> 0.995

z s

5

T

We conclude that there is no simple single-parameter

QSPR model for predicting the boiling points over a wide

range of alkanes due to the great diversity among experi-

mental values. Multivariate regression models appear to

be verv accurate due to a varietv ~arametersnvolved in

the correlation. Each of these p&.meters takes care of a

certain structural detail of a large alkane. When all di-

verse structural features of alkanes are considered, the

model usually gives extremely good agreement between

the experimental and calculated boiling points.

Acknowledaement

We are thankful to the Ministry of Science, Technology,

and Informatics of the Republic of Cmatia for support.

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A graph theoretical approach to SPR

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