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A topological characterization for simple molecular surfaces
Journal of Molecular Structure ( Theochem), 166 (1988) 11-16
11
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
A TOPOLOGICAL
CHARACTERIZATION
FOR SIMPLE MOLECULAR SURFACES
Gustavo A. ARTECA1 and Paul G. NEZEY1'2 Department of Chemistry I and Department Saskatchewan,
Saskatoon,
of Mathematics 2, University of
Saskatchewan,
Canada S7N 0WO.
SUMMARY A method for the shape characterization of some molecular surfaces is proposed. The technique consists of defining a rigorous domain partitioning of the surface, followed by the construction of a hierarchy of topological objects, obtained by truncation of a number of domains according to their areas. Finally, the series of truncated objects is characterized by several topological invariants. The procedure is illustrated for the case of van der Waals surfaces. The description provides a tool which is complementary to the usual solvent-accessibility surface defined over the van der Waals surface.
INTRODUCTION The three-dimensional
representation
of a molecular surface has become
nowadays a very useful tool, widely employed in different areas of both applied and theoretical
chemistry and biology.
valuable when describing molecular Different surfaces can be used.
Appropriate molecular surfaces may be
reactivity,
as well as molecular shape.
Among them, one can mention isodensity
contours or surfaces with constant value of molecular electrostatic (See, for instance,
refs.
1,2, and others quoted therein.)
Volume and size effects can be better described surfaces such as van der Waals surfaces accessibility
surface.(For
The systematic
example,
(VDWS's),
in terms of molecular or the corresponding
use of any of the above molecular
surfaces requires
of their shape features.
important
that involve the comparison
of molecules. drug design.
of surfaces for a number
An example is found in the standard research in computer-aided According to the usual procedure,
series of molecules.
alternatives
one relies on a graphic display among the shapes of an
The method is largely subjective,
and several
have been proposed to provide a concise numerical quantification
of some shape features.(For
example,
refs. 4-10.)
another possible approach to characterize In order to illustrate
the method,
In this work we discuss
a family of surfaces.
this article will be restricted to
surfaces built from superposition
of spheres,
the technique can be reformulated
for dealing with more complicated
0166-1280/88/$03.50
i
:
:
a
This is particularly
in a computer terminal in order to perceive similarities extensive
solvent-
Ref. 3 and others cited therein.)
rigorous characterization in applications
potential.
such as VDWS's.
Notwithstanding, surfaces.
© 1988 Elsevier Science Publishers B.V.
i
ii!i!
~i
¸
i
i
/
/
12 Our proposal can be stated as follows.
Firstly,
of domains on the surface is introduced.
a criterion to define a number
Next, a number of topological
are derived by truncation of original domains accordin~ to their area. the family of truncated objects is characterized invariants available
by a number of topological
from general results in algebraic
The idea of using an area-dependent accessibility
surface.
objects Finally,
truncation
topology./11/
is inspired by the solvent-
In this case, a number of regions are found on the
VDWS, that cannot be touched by the solvent molecular surface. similar domains on the surface,
We define here
but without actually testing the solvent.
The
only intuitive notion involved is that the smaller is the area of one of the spherical
pieces forming the VDWS, the more confined is that piece by other
neighbouring
ones, and, consequently,
METHOD OF SHAPE
less accessible
it is to the solvent.
CHARACTERIZATION
A VDWS for N-atomic molecules
is completely determined
by the set
XN }, containing the N position vectors for the nuclei, and the Cx:{~ I , ~2' set Cp={p I , P2' "''' PN }' corresponding to the N respective atomic van der Waals radii. However, it is not obvious from this information which are the features that characterize
the overal shape of the molecular envelope.
Formally one may define the molecular set of an appropriate etc) in 3-space, elsewhere.
G = {~c~:
function
surface as the boundary of the level
(charge density,
electrostatic
HOMO,
Without any loss of generality we can take a=1: /9/
f(~)=1}
(1)
As it is known, G (the VDWS) is the envelope surface obtained interpenetration spheres.
potential,
say f(E), that takes a constant value a on the VDWS, and zero
of N spheres.
Any point on the VDWS may belong simultaneously
one spheres.
from the
This surface G is actually formed by pieces of to one or more than
In this latter case, the points belong to the boundaries
interpenetrating
between
spheres, which appear as circular arcs on the VDWS.
The spherical pieces on the VDWS can be written as subsets d i of G.
A set
d. contains points lying on the atomic sphere for the i-th atom, 1 d i = { weG
:
jw_xiJ=o i ,
PiCCp,
xicC ~ }
(2)
Observe that two sets d i and dj, i~j, may or may not have points in common. Furthermore,
a set d. may be formed by more than a single piece. This is l sometimes the case when one atomic sphere is strongly interpenetrated simultaneously
by two or more of others.
These pieces can be represented
by a
13
number of subsets of d i, {di(J), j=1,2,
..., ni}.
These subsets are maximum connected components of d i, in the sense that any two points belonging to two sets di(J) and di(k), k~j, cannot be connected by a path on the VDWS totally contained in d i.
Notice, however, that d i might also
be empty, if the i-th atomic sphere is totally contained inside the VDWS. The set D={ di(J), j=1,2, the VDWS
..., hi; i=1,2,
..., N } of all spherical faces of
provides a natural partitioning for G.
an element of D will
be indicated as D
For the sake of simplicity
(which coincides evidently with some
q set d.(J)). l Each of the faces Dq is characterized by an area, that we indicate A(Dq).
From the practical point of view, the determination of both the spherical faces and their corresponding areas is not a forbiddingly difficult task. This computation is usually a feature available through the programs used to obtain the VDWS. Our aim is to characterize the surface G.
To that purpose we introduce a
family of surfaces {Gi}, obtained by truncation from the VDWS.
The truncation
is performed according to the area of the spherical faces in G. Let A(Dq) be the area associated to the face DqeD.
We introduce a set of
real positive numbers B={B I, B 2, ..., Bm}, with the condition Bi
D (i)
:
{ DqcG :
This defines the set D(i):
A(Dq)
(3)
In order to describe the shape features of G, we will study the surface G i that is left
G i : G \ ~ Dq
after
,
eliminating all
f a c e s i n D( i )
f o r a g i v e n v a l u e Bi
Dq eD (i)
(4)
This surface is not generally closed, and it has some boundary.
The set of
topological objects {G i} can be characterized by a number of topological invariants.
The homology groups of algebraic topology HP(Gi), p:O, 1, 2, /11/
are appropriate, simple invariants.
In turn, they are characterized by their
ranks, the Betti numbers bp(i)=bp(HP(Gi)),
and the Euler-Poincar~ characteris-
tic, defined as follows
X i = bo(i) - b1(i) + b2(i)
(5)
14 The computation
of Betti numbers and Euler-Poincar4
standard problem in algebraic easily.(See
refs. 7 and 8.)
the characteristics As a result,
topology and it can be accomplished For our present purposes,
of those spherical
faces with area below a given fixed
..., m, will uncover relevant information
shape, which can be expressed
way by listing the Euler-Poincar4
in a concise,
(6)
...,
ILLUSTRATIVE
EXAMPLE OF APPLICATION OF THE METHOD
Xm}
In order to show how the method is actually applied,
corresponds
compact algebraic
characteristics:
X = {X 1, X2,
cular VDWS.
rather
it is enough to employ
Xi.
A scanning over Bi, i=1,2,
on the molecular
is a
some shape features of a VDWS are revealed by the relative
spatial arrangements value B i.
characteristics
One of its views is displayed to the isoarecolone
molecular
in Figure
I.
we consider a partiThe molecule
cation at its equilibrium geometry.
The compound is in fact of some interest due to its role as a nicotinic agonist./12/ The analysis of the whole VDWS reveals the presence of 27 spherical from spherical triangles spheres for carbon, atoms.
to spherical heptagons.
one corresponds
to carbon is smaller than the area
to any other atom.
Front view Fig.1
Ten of the faces are atomic
to the oxygen, and the rest to hydrogen
The area of the faces corresponding
of the faces corresponding
faces,
Back view
Two views of the VDWS for the isoarecolone
cation
[(CH3CO)C5H7N(CH3)2 ]+.
The acetyl group is at the top, and in the so-called oxygen atom points to the viewer.
front view the
15
As an illustration,
consider
the smallest area of a hydrogenic
face.
If we
take the value B I as equal to this area, all carbon faces must be truncated. One notices that the ten carbon faces can be grouped into three sets. first group there are five isolated some boundary points in common, common boundary points.
Another group of three faces have
and the last group consisting of two faces with
Accordingly,
a 2-sphere with seven holes,
×I
faces.
In the
G I is an object which is homeomorphic
characterized
to
as follows:
: - 5
(7)
The procedure can be applied to any other B. value without any further i The sequence of Euler-Poincar4 characteristics for a sequence
modification.
of reference values B. provides a simple description for some features of the i molecular shape, which can replace, up to a certain extent, the classification of the surface by visual inspection
of a graphical
graphs,
to apply the method.
as Fig. l, are not essential
The procedure
We notice that
can be useful as a tool for studying the shape of molecular
surfacesl3 Its generalization difficult.
display.
to surfaces more complicated
than VDWS's is not
The simplest method would consist of introducing a partitioning
the molecular
surface by following a curvature criterion,
as explained
of
in
refs. 7-10.
ACKNOWLEDGMENT This work was supported Engineering
by a research grant from the Natural Sciences and
Research Council
(NSERC) of Canada.
REFERENCES I. 2. 34. 5. 6. 7.
P. Politzer and D.G. Truhlar, Eds., Chemical Applications of Atomic and Molecular Electrostatic Potentials, Plenum, New York, 1981. G.D. Purvis III and C. Culberson, Int. J. Quantum Chem., Quantum Biol. Symp., 13 (1986) 261. M.L. Connolly, Science, 221 (1983) 709. R. Carb6, L. Leyda, and M. Arnau, Int. J. Quantum Chem., 17 (1980) 1185. P.E. Bowen-Jenkins, D.L. Cooper, and W.G. Richards, J. phys. Chem., 89 (1985) 2195. A.M. Richard and J.R. Rabinowitz, Int. J. Quantum Chem., 31 (1987) 309. P.G. Mezey, Int. J. Quantum Chem., Quantum Biol. Symp., 12 (1986) 113.
18 8. 9. 10. 11.
12.
13.
P.G. Mezey, J. Comput. Chem., 8 (1987) 462. P.G. Mezey, Int. J. Quantum Chem., Quantum Biol. Symp., in press. G.A. Arteca and P.G. Mezey, Int. J. Quantum Chem., Quantum Biol. Symp., in press. E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966; J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Menlo Park, 1984. M.J. Ross, M.W. Klymkowsky, D.A. Agard, and R.M. Stroud, J. Mol. Biol. 120 (1986) 127; J.A. Waters, C.E. Spivak, M.A. Hermsmeier, J.S. Yadav, R.F. Liang, and T.M. Gund, J. Med. Chem., in press; J.S. Yadav, M.A. Hermsmeier, and T.M. Gund, Int. J. Quantum Chem., Quantum Biol. Symp., in press. For related applications see: G.A. Arteca, V.B. Jammal, P.G. Mezey, J.S. Yadav, M.A. Hermsmeier, and T.M. Gund, J. Mol. Graph., in press.
11
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
A TOPOLOGICAL
CHARACTERIZATION
FOR SIMPLE MOLECULAR SURFACES
Gustavo A. ARTECA1 and Paul G. NEZEY1'2 Department of Chemistry I and Department Saskatchewan,
Saskatoon,
of Mathematics 2, University of
Saskatchewan,
Canada S7N 0WO.
SUMMARY A method for the shape characterization of some molecular surfaces is proposed. The technique consists of defining a rigorous domain partitioning of the surface, followed by the construction of a hierarchy of topological objects, obtained by truncation of a number of domains according to their areas. Finally, the series of truncated objects is characterized by several topological invariants. The procedure is illustrated for the case of van der Waals surfaces. The description provides a tool which is complementary to the usual solvent-accessibility surface defined over the van der Waals surface.
INTRODUCTION The three-dimensional
representation
of a molecular surface has become
nowadays a very useful tool, widely employed in different areas of both applied and theoretical
chemistry and biology.
valuable when describing molecular Different surfaces can be used.
Appropriate molecular surfaces may be
reactivity,
as well as molecular shape.
Among them, one can mention isodensity
contours or surfaces with constant value of molecular electrostatic (See, for instance,
refs.
1,2, and others quoted therein.)
Volume and size effects can be better described surfaces such as van der Waals surfaces accessibility
surface.(For
The systematic
example,
(VDWS's),
in terms of molecular or the corresponding
use of any of the above molecular
surfaces requires
of their shape features.
important
that involve the comparison
of molecules. drug design.
of surfaces for a number
An example is found in the standard research in computer-aided According to the usual procedure,
series of molecules.
alternatives
one relies on a graphic display among the shapes of an
The method is largely subjective,
and several
have been proposed to provide a concise numerical quantification
of some shape features.(For
example,
refs. 4-10.)
another possible approach to characterize In order to illustrate
the method,
In this work we discuss
a family of surfaces.
this article will be restricted to
surfaces built from superposition
of spheres,
the technique can be reformulated
for dealing with more complicated
0166-1280/88/$03.50
i
:
:
a
This is particularly
in a computer terminal in order to perceive similarities extensive
solvent-
Ref. 3 and others cited therein.)
rigorous characterization in applications
potential.
such as VDWS's.
Notwithstanding, surfaces.
© 1988 Elsevier Science Publishers B.V.
i
ii!i!
~i
¸
i
i
/
/
12 Our proposal can be stated as follows.
Firstly,
of domains on the surface is introduced.
a criterion to define a number
Next, a number of topological
are derived by truncation of original domains accordin~ to their area. the family of truncated objects is characterized invariants available
by a number of topological
from general results in algebraic
The idea of using an area-dependent accessibility
surface.
objects Finally,
truncation
topology./11/
is inspired by the solvent-
In this case, a number of regions are found on the
VDWS, that cannot be touched by the solvent molecular surface. similar domains on the surface,
We define here
but without actually testing the solvent.
The
only intuitive notion involved is that the smaller is the area of one of the spherical
pieces forming the VDWS, the more confined is that piece by other
neighbouring
ones, and, consequently,
METHOD OF SHAPE
less accessible
it is to the solvent.
CHARACTERIZATION
A VDWS for N-atomic molecules
is completely determined
by the set
XN }, containing the N position vectors for the nuclei, and the Cx:{~ I , ~2' set Cp={p I , P2' "''' PN }' corresponding to the N respective atomic van der Waals radii. However, it is not obvious from this information which are the features that characterize
the overal shape of the molecular envelope.
Formally one may define the molecular set of an appropriate etc) in 3-space, elsewhere.
G = {~c~:
function
surface as the boundary of the level
(charge density,
electrostatic
HOMO,
Without any loss of generality we can take a=1: /9/
f(~)=1}
(1)
As it is known, G (the VDWS) is the envelope surface obtained interpenetration spheres.
potential,
say f(E), that takes a constant value a on the VDWS, and zero
of N spheres.
Any point on the VDWS may belong simultaneously
one spheres.
from the
This surface G is actually formed by pieces of to one or more than
In this latter case, the points belong to the boundaries
interpenetrating
between
spheres, which appear as circular arcs on the VDWS.
The spherical pieces on the VDWS can be written as subsets d i of G.
A set
d. contains points lying on the atomic sphere for the i-th atom, 1 d i = { weG
:
jw_xiJ=o i ,
PiCCp,
xicC ~ }
(2)
Observe that two sets d i and dj, i~j, may or may not have points in common. Furthermore,
a set d. may be formed by more than a single piece. This is l sometimes the case when one atomic sphere is strongly interpenetrated simultaneously
by two or more of others.
These pieces can be represented
by a
13
number of subsets of d i, {di(J), j=1,2,
..., ni}.
These subsets are maximum connected components of d i, in the sense that any two points belonging to two sets di(J) and di(k), k~j, cannot be connected by a path on the VDWS totally contained in d i.
Notice, however, that d i might also
be empty, if the i-th atomic sphere is totally contained inside the VDWS. The set D={ di(J), j=1,2, the VDWS
..., hi; i=1,2,
..., N } of all spherical faces of
provides a natural partitioning for G.
an element of D will
be indicated as D
For the sake of simplicity
(which coincides evidently with some
q set d.(J)). l Each of the faces Dq is characterized by an area, that we indicate A(Dq).
From the practical point of view, the determination of both the spherical faces and their corresponding areas is not a forbiddingly difficult task. This computation is usually a feature available through the programs used to obtain the VDWS. Our aim is to characterize the surface G.
To that purpose we introduce a
family of surfaces {Gi}, obtained by truncation from the VDWS.
The truncation
is performed according to the area of the spherical faces in G. Let A(Dq) be the area associated to the face DqeD.
We introduce a set of
real positive numbers B={B I, B 2, ..., Bm}, with the condition Bi
D (i)
:
{ DqcG :
This defines the set D(i):
A(Dq)
(3)
In order to describe the shape features of G, we will study the surface G i that is left
G i : G \ ~ Dq
after
,
eliminating all
f a c e s i n D( i )
f o r a g i v e n v a l u e Bi
Dq eD (i)
(4)
This surface is not generally closed, and it has some boundary.
The set of
topological objects {G i} can be characterized by a number of topological invariants.
The homology groups of algebraic topology HP(Gi), p:O, 1, 2, /11/
are appropriate, simple invariants.
In turn, they are characterized by their
ranks, the Betti numbers bp(i)=bp(HP(Gi)),
and the Euler-Poincar~ characteris-
tic, defined as follows
X i = bo(i) - b1(i) + b2(i)
(5)
14 The computation
of Betti numbers and Euler-Poincar4
standard problem in algebraic easily.(See
refs. 7 and 8.)
the characteristics As a result,
topology and it can be accomplished For our present purposes,
of those spherical
faces with area below a given fixed
..., m, will uncover relevant information
shape, which can be expressed
way by listing the Euler-Poincar4
in a concise,
(6)
...,
ILLUSTRATIVE
EXAMPLE OF APPLICATION OF THE METHOD
Xm}
In order to show how the method is actually applied,
corresponds
compact algebraic
characteristics:
X = {X 1, X2,
cular VDWS.
rather
it is enough to employ
Xi.
A scanning over Bi, i=1,2,
on the molecular
is a
some shape features of a VDWS are revealed by the relative
spatial arrangements value B i.
characteristics
One of its views is displayed to the isoarecolone
molecular
in Figure
I.
we consider a partiThe molecule
cation at its equilibrium geometry.
The compound is in fact of some interest due to its role as a nicotinic agonist./12/ The analysis of the whole VDWS reveals the presence of 27 spherical from spherical triangles spheres for carbon, atoms.
to spherical heptagons.
one corresponds
to carbon is smaller than the area
to any other atom.
Front view Fig.1
Ten of the faces are atomic
to the oxygen, and the rest to hydrogen
The area of the faces corresponding
of the faces corresponding
faces,
Back view
Two views of the VDWS for the isoarecolone
cation
[(CH3CO)C5H7N(CH3)2 ]+.
The acetyl group is at the top, and in the so-called oxygen atom points to the viewer.
front view the
15
As an illustration,
consider
the smallest area of a hydrogenic
face.
If we
take the value B I as equal to this area, all carbon faces must be truncated. One notices that the ten carbon faces can be grouped into three sets. first group there are five isolated some boundary points in common, common boundary points.
Another group of three faces have
and the last group consisting of two faces with
Accordingly,
a 2-sphere with seven holes,
×I
faces.
In the
G I is an object which is homeomorphic
characterized
to
as follows:
: - 5
(7)
The procedure can be applied to any other B. value without any further i The sequence of Euler-Poincar4 characteristics for a sequence
modification.
of reference values B. provides a simple description for some features of the i molecular shape, which can replace, up to a certain extent, the classification of the surface by visual inspection
of a graphical
graphs,
to apply the method.
as Fig. l, are not essential
The procedure
We notice that
can be useful as a tool for studying the shape of molecular
surfacesl3 Its generalization difficult.
display.
to surfaces more complicated
than VDWS's is not
The simplest method would consist of introducing a partitioning
the molecular
surface by following a curvature criterion,
as explained
of
in
refs. 7-10.
ACKNOWLEDGMENT This work was supported Engineering
by a research grant from the Natural Sciences and
Research Council
(NSERC) of Canada.
REFERENCES I. 2. 34. 5. 6. 7.
P. Politzer and D.G. Truhlar, Eds., Chemical Applications of Atomic and Molecular Electrostatic Potentials, Plenum, New York, 1981. G.D. Purvis III and C. Culberson, Int. J. Quantum Chem., Quantum Biol. Symp., 13 (1986) 261. M.L. Connolly, Science, 221 (1983) 709. R. Carb6, L. Leyda, and M. Arnau, Int. J. Quantum Chem., 17 (1980) 1185. P.E. Bowen-Jenkins, D.L. Cooper, and W.G. Richards, J. phys. Chem., 89 (1985) 2195. A.M. Richard and J.R. Rabinowitz, Int. J. Quantum Chem., 31 (1987) 309. P.G. Mezey, Int. J. Quantum Chem., Quantum Biol. Symp., 12 (1986) 113.
18 8. 9. 10. 11.
12.
13.
P.G. Mezey, J. Comput. Chem., 8 (1987) 462. P.G. Mezey, Int. J. Quantum Chem., Quantum Biol. Symp., in press. G.A. Arteca and P.G. Mezey, Int. J. Quantum Chem., Quantum Biol. Symp., in press. E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966; J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Menlo Park, 1984. M.J. Ross, M.W. Klymkowsky, D.A. Agard, and R.M. Stroud, J. Mol. Biol. 120 (1986) 127; J.A. Waters, C.E. Spivak, M.A. Hermsmeier, J.S. Yadav, R.F. Liang, and T.M. Gund, J. Med. Chem., in press; J.S. Yadav, M.A. Hermsmeier, and T.M. Gund, Int. J. Quantum Chem., Quantum Biol. Symp., in press. For related applications see: G.A. Arteca, V.B. Jammal, P.G. Mezey, J.S. Yadav, M.A. Hermsmeier, and T.M. Gund, J. Mol. Graph., in press.