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New Shape Descriptors for Quantitative Treatment of Steric Effects. III. A New Globularity Measure for QSPR/QSAR Studies

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Buletinul
Ş
tiin
ţ
ific al Universit
ăţ
ii “Politehnica” din Timisoara, ROMÂNIA Seria CHIMIE
Ş
I INGINERIA MEDIULUI
Chem. Bull. "POLITEHNICA" Univ. (Timi
ş
oara) Volume 51(65), 1-2, 2006
9
New Shape Descriptors for Quantitative Treatment of Steric Effects. III. A New Globularity Measure for QSPR/QSAR Studies
Mihai Medeleanu*, Dan Ciubotariu**, Ciprian Ciubotariu***
*
POLITEHNICA University of Timi
ş
oara, Applied Chemistry and Organic and Natural Compounds Engineering Department, Timi
ş
oara, C. Telbisz 6, Romania, e-mail: mihai.medeleanu@chim.upt.ro
**
University of Medicine and Pharmacy, Organic Chemistry Department, Timi
ş
oara, P-ta. E. Murgu 2, Romania
***
POLITEHNICA University of Timi
ş
oara, Computers Department, Timi
ş
oara, Bd. Vasile Pârvan 2, Romania
Abstract:
Size and shape of the organic molecules are useful descriptors in QSPR/QSAR studies. Usual characterization of molecular size is based on steric parameters like van der Waals, STERIMOL, Charton or Taft types. The main problem with the second molecular property is the quantification of shape. In this paper we propose a new shape descriptor based on van der Waals parameters like volume and surface. The computation algorithm and a set of molecular examples are depicted.
1. Introduction
The modern studies concerning the quantitative structure-properties relationships (QSPR) and those concerning the quantitative structure-activity relationships (QSAR) are focused on the quantification of both the molecular structure and the physical, chemical properties or biological activity. This procedure requires a separation of the effects involved in chemical or biological interactions, an independent quantification of these as electronic, steric, hydrophobic, or other types of factors and the quantitative treatment of reactivity or biological activity by means of the correlation analysis models. The use of the extrathermodynamic formalism made possible the theoretical validation of this approach [1]. It is known that van der Waals parameters are useful for modeling the steric effects of organic molecules. The van der Waals volume and surface are largely used in studies like packing density of molecules[2], protein folding[3], quantification of biological activity[4-6]. Another derived parameter was presented in an early stage of this study and defined as “molecular globularity”[7], a useful shape descriptor of molecular structure. In this paper, a new and very simple globularity parameter will be defined together with some applications and a comparison with the former one.
2. Computational Methods and Results
Various definitions of the globularity can be found in the literature, mainly starting from two basic van der Waals descriptors: the van de Waals volume and the van der Waals surface. The volume can be computed by analytical methods[8] or by Monte Carlo algorithms[9,10]. In this way a molecular van der Waals envelope,
Γ
, can be defined as the external surface resulted following the interaction of all vdW spheres, corresponding to the atoms of the molecule M. The points (x,y,z) inside the envelope satisfy at least one of the following inequalities:
222w2iiii
(X-x)+(Y-y)+(Z-z)(r) i=1,m
≤
wM
M
(1) where m represents the number of atoms of M. Consequently the total volume from inside the envelope represents the molecular vdW volume (
V
) of the molecule M. Let be a function F=f(x,y,z), continuous in a bordered and closed domain M. The integral:
I=f(x, y, z)×dx×dy×dz
∫∫∫
w M
V=dV dV=dx×dy×dz
∫∫∫
(2) can be interpreted as a weight, f(x,y,z) being the repartition of density in space M. If we consider the function f(x,y,z) (the density) is identical equal with 1, than the following integral:
(3) can be intuitively justified as a volume. This assumption is natural because the properties of molecular vdW space can be considered independent from the nature of the atoms, even in the case of intersection domains between the vdW atomic spheres. One of the main problem is to find out the volume of the intersection of the spheres in order to avoid multiple computations of the same volume. To estimate the integral (3), the spheres that models the molecule are inserted into a parallelepiped with the volume V
p
. The random numbers are generated into the domain M, which include the parallelepiped. If n
t
is the total number of generated points and n
s
the number of points that satisfies the equations (1), than the van der Waals volume is:
ws pt
n×V n
M
V=
(4)
Chem. Bull. "POLITEHNICA" Univ. (Timi
ş
oara) Volume 51(65), 1-2, 2006
A similar method is used for the van der Waals surface. The algorithm implies the generation of a uniform grid[11] on each sphere of the molecule, followed by the detection from the number of all points generated on the surface (n
t
) of those (n
e
) that satisfies the
equality
in equations (1). For every “hard sphere” i, one can compute the relation:
weiit
(n)S=4(r)n
π
w2ik
(5) The final surface is computed after the elimination of overlays ():
k
S
(6)
mwwii=1k
S=S-S
∑∑
The full algorithms and results are described in some previous papers on this subject [9-11]. The molecular globularity is defined as the ratio between the surface of an equivalent sphere (the sphere with the volume equal to molecular volume) and the molecular surface [12], or the inverse of the above ratio [13]. A value of 1.0 means a shape of a perfect sphere. With the van der Waals volume and surface computed, two other parameters can be defined. The first is the ratio between the volume and the surface of the molecule, which has the dimensions of a length:
www
V L=S
(7) The second one, defined only for acyclic molecules is named the globularity measure (G
m
) and is given by the relation:
wms
LG= R
(8) where L
w
is defined by relation (7) and R
s
represents the ratio between the volume and the surface of an equivalent sphere which surround the molecule, with the radius equal to the half of the longest dimension of the parallelepiped which circumscribes the molecule. For cyclic molecules, the above relation cannot be used because the volume of the equivalent sphere includes the internal empty space (for instance a cavity), which is not included in the van der Waals volume. The results for the globularity of such molecules are far from reality (for C60 fullerene G
m
=0.686). These two parameters can be used to describe the shape of the acyclic molecules. The globularity measure decreases with the growth of the linear chains and increases toward unity when the molecule is highly branched or compacted. In Table 1 the van der Waals volumes, surfaces, L
w
and G
m
parameters are listed for C2-C8 alkanes (M represents the methyl group, and the numbers before reflect th position of branching).
TABLE 1.
The van der Waals parameters for C2-C8 alkanes
As expected, the longer the chain, the smaller the globularity measure. This measure increases for alkanes such as 2,2- and 2,3-dimethylbutane, 2,2,3-trimethylbutane, trimethylpentanes, ethyl-methyl-pentanes or tetrametrhylbutanes. An useful example which describes the utility of the globularity measure is in the case of molecules with long chain, which can have different conformations. For instance, let be the normal C16 alkane, with two conformations presented below (figures 1 and 2).
Alcane
V
w
S
w
L
w
G
m
C2 44.205 70.920 0.623 0.653 C3 60.234 93.026 0.647 0.554 C4 77.845 115.315 0.675 0.516 2M-C3 77.275 114.675 0.674 0.558 C5 94.749 138.127 0.686 0.451 2M-C4 94.047 136.241 0.690 0.522 22MM-C3 92.368 135.225 0.683 0.527 C6 110.242 159.338 0.692 0.417 2M-C5 109.970 158.477 0.694 0.465 3M-C5 109.676 156.114 0.703 0.453 22MM-C4 110.161 155.015 0.711 0.537 23MM-C4 110.215 155.033 0.711 0.496 C7 127.399 181.834 0.701 0.375 2M-C6 127.317 179.742 0.708 0.429 3M-C6 126.956 178.780 0.710 0.448 3E-C5 125.874 176.827 0.712 0.458 22MM-C5 126.822 176.663 0.718 0.486 33MM-C5 125.592 174.301 0.721 0.451 24MM-C5 127.320 178.291 0.714 0.462 23MM-C5 127.492 174.419 0.731 0.472 223MMM-C4 126.616 172.912 0.732 0.511 C8 142.613 203.964 0.699 0.347 2M-C7 143.144 201.813 0.709 0.388 3M-C7 143.774 200.463 0.717 0.373 4M-C7 143.925 199.927 0.720 0.401 3E-C6 141.739 198.490 0.714 0.422 22MM-C6 142.733 198.900 0.718 0.443 33MM-C6 142.566 195.458 0.729 0.410 23MM-C6 142.990 195.654 0.731 0.429 24MM-C6 144.186 197.986 0.728 0.429 25MM-C6 143.512 198.892 0.722 0.396 34MM-C6 142.278 195.908 0.726 0.456 234MMM-C5 142.593 196.954 0.724 0.465 223MMM-C5 142.505 193.130 0.738 0.482 224MMM-C5 142.473 197.110 0.723 0.478 233MMM-C5 141.973 191.844 0.740 0.468 3E-2M-C5 142.069 195.215 0.728 0.481 3E-3M-C5 142.811 192.962 0.740 0.473 2233MMMM-C4 142.466 189.633 0.751 0.522
10
Chem. Bull. "POLITEHNICA" Univ. (Timi
ş
oara) Volume 51(65), 1-2, 2006
Fig. 1. The linear conformation of C16 alkane Fig. 2. A twisted conformation of C16 alkane
TABLE 2. van der Waals parameters for the two conformations of C16 alkane
The described parameters could be used in the globularity studies of macromolecules like polypeptides, proteines and others macromolecules. In order to avoid the problems raised by the described globularity measure (G
m
) and to include the cavities volume for molecules with this kind of structure, another molecular descriptor has been defined: the ellipsoidal globularity G
e
. For the computation of this molecular descriptor the molecule is surrounded by an ellipsoid tangent to the extremities. The ellipsoidal shape is a better approach than the parallelepiped use in former computations. Considering the longest half axis of the ellipsoid, the corresponding sphere can be setup. The ratio between the volume of the ellipsoid and the volume of the sphere defines the ellipsoidal globularity:
elels
V GV
=
(9) Knowing the volume of a sphere:
3
43
s
VR
π
=⋅
(10) and that of an ellipsoid:
43
el
Vabc
π
=
(11) where
R
is the sphere radius,
a
,
b
,
c
the half of the ellipsoidal axes, and
max(,,)
Rabc
=
(12) It is obvious that the ellipsoidal globularity is much easier to compute. The van der Waals volume is not necessary.
TABLE 3. The globularity measures
In Table 3 are listed the spherical and ellipsoidal volumes together with the corresponding globularities. As it can be seen the ellipsoidal globularity gives a much better shape description for molecules such as fullerenes, but also for highly branched alkanes. From the same table one can observe that the aromatic molecules have a lower globularity, in agreement with their real planar structure.
3. Conclusions
Two new methods to setup molecular descriptors for the molecular shape have been proposed. Their results are the globularity measure G
m
and the ellipsoidal globularity G
el
. The former is useful in shape studies for acyclic
Alcane
V
w
S
w
L
w
G
m
C16 normal
283.63 381.51 0.743 0.214
C16 twisted
244.04 249.15 0.980 0.448
molecule
V
S
V
el
G
m
G
el
C2 130.30 114.65 0.613 0.880 C3 226.23 160.41 0.534 0.709 C4 310.33 190.72 0.494 0.615 C5 469.06 247.29 0.438 0.527 C6 603.19 284.80 0.408 0.472 C7 843.12 352.73 0.368 0.418 C8 1038.19 397.35 0.346 0.383 C9 1376.35 476.58 0.317 0.346 C10 1644.28 528.17 0.300 0.321 C15 4215.23 958.81 0.223 0.227 C20-fullerene 251.93 222.63 0.805 0.884 22MM-C3 303.30 254.81 0.513 0.840 2M-C3 246.28 217.79 0.536 0.884 22MM-C4 318.27 311.76 0.518 0.980 23MM-C4 382.21 332.28 0.488 0.869 2M-C4 321.10 264.91 0.504 0.825 22MM-C5 462.81 390.69 0.461 0.844 23MM-C5 480.68 397.14 0.461 0.826 24MM-C5 532.12 363.28 0.438 0.683 2M-C5 466.43 339.18 0.449 0.727 33MM-C5 499.17 373.21 0.457 0.748 3E-C5 502.22 390.36 0.450 0.777 3M-C5 492.65 318.16 0.446 0.646 2M-C6 610.12 389.06 0.414 0.638 3M-C6 641.43 369.41 0.411 0.576 C60 681.62 665.85 0.686 0.977 Anthracene 1197.43 293.22 0.382 0.245 Benzene 371.61 152.67 0.512 0.411 Naphthalene 709.72 218.38 0.440 0.308 Phenanthrene 1217.99 261.84 0.384 0.215 Pyrene 1195.52 292.85 0.396 0.245
11
Chem. Bull. "POLITEHNICA" Univ. (Timi
ş
oara) Volume 51(65), 1-2, 2006
molecules (including macromolecules) and the later, a more general descriptor can be used in the shape description of any kind of molecules, including ones with cavities Acknowledgments The authors acknowledge Prof. Dr. Mircea Mracec from the Chemical Institute of the Romanian Academy in Timi
ş
oara for providing the HyperChem[14] package for molecular optimizations of the studied structures.
References
1. Ciubotariu, D.; Deretey, E.; Medeleanu, M.; Gogonea, V.; Iorga, I. Chem. Bull. PIT,
1990
, 35, (49), 83 2. Kitaigorodski, A.I. “Molecular Crystals and Molecules”, Academic Press, New York,
1973
3. Lee, B.; Richards, F.M. J.Mol.Biol.,
1971
, 55, 379;
1974
, 82, 1 4. Msrcuchi, I.; Kanada, Y.; Komatsu, K. Chem.Pharm.Bull.,
1976
, 24, 1799 5. Tichy, M. Int.J.Quantum Chem.,
1979
, 16, 509 6. Palmer, R.A.; Tickle, J.H.; Tickle, J.L. J.Mol.Graphics,
1983
, 1, 94 7. Medeleanu, M. “Structure-Properties Correlations by Topological Methods”, PhD. Thesis, Univ. “Politehnica”, Bucure
ş
ti,
1997
8. Gogonea, V.; Osawa, E. J.Comput.Chem.,
1995
, 16, 817 9. Mure
ş
an, S.; Bologa, C.; Medeleanu, M.; Gogonea, V.; Drago
ş
, D.; Ciubotariu, D.; Chem.Bull.Tech. Univ. (Timi
ş
oara),
1994
, 39, 47 10. Ciubotariu,D.; Medeleanu,M.; Iorga,I.; Deretey,E.; Bologa,C.; Mure
ş
an,S., Chem.Bull.Tech.Univ.(Timi
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oara),
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, 38, 83 11. Gogonea, V.; Ciubotariu, D.; Deretey, E.; Popescu, M.; Iorga, I.; Medeleanu, M. Rev. Roum. Chim.,
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, 36, 465 12. Filliponi, E.; Cruciani, G.; Tabarrini, O.; Cecchetti, V.; Fravolini, A. J. Comput.-Aided Mol. Design
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, 2, 72-88 14. Hypercube, Inc. HyperChem®Computational Chemistry,
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12

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