Bonchev D., Rouvray D.H. (editors) Complexity in Chemistry, Biology, and Ecology
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40 Chapter 1
A
B
Figure 1.10. (A) Stereo-view of an achiral zigzag nanotube with open ends having t
+
= 5,
t
−
=0, with open ends. It has 20 benzenoid rings, diameter 3.9 Å, and strain energy (MM2)
292.89 kcal/mol (14.64 kcal/mol per hexagon). (B) Stereo-view of an achiral armchair
nanotube having t
+
= t
−
= 3, with open ends. It has 21 benzenoid rings, diameter 4.05 Å,
and strain energy 237.03 kcal/mol (11.29 kcal/mol per hexagon).(cont.)
On the Complexity of Fullerenes and Nanotubes 41
C
Figure 1.10. (Cont.) (C) Stereo-view of a chiral nanotube having t
+
=4, t
−
=2, with open
ends. It has 24 benzenoid rings, diameter 4.13 Å, and strain energy (MM2) 245.92 kcal/mol
(10.25 kcal/mol per hexagon).
capped end. One may also speculate that heteroatoms might simplify these
ends of nanotubes [84, 85]. Also, nanotubes will be considered to be con-
stituted exclusively of benzenoid rings, without any kinks caused by 5- or
7-membered rings.
In principle, as discussed earlier in this review, higher symmetry implies
smaller complexity. One may argue therefore that the longer the nanotube,
the higher its symmetry. On the other hand, the complexity of molecules
increases with its number of atoms. Thus, the lengthening of the nanotube
can both increase the complexity by adding carbon atoms, and decrease the
complexity by enhancing the symmetry in a more subtle fashion. There-
fore we will consider only the complexity derived from the parameters
t
+
and t
−
, which define both the helicity (Eq. 10.2) and the diameter
(Eq. 10.1).
42 Chapter 1
Figure 1.11. Stereo-view of a zigzag nanotube with one capped end, having t
+
= 5, t
−
=
0.
No single parameter seems to be “definitory” for the complexity, for
instance by compensating for increased complexity due to a larger diameter
by the decreased complexity due to lower or zero helicity.
The complexity of a nanotube will depend on several parameters: length,
diameter, helicity. The difference t
+
− t
−
that is involved in Eq. (10.2)
provides a measure of helicity. The values for the ratio t
−
/t
+
range from
zero (for zigzag nanotubes) through 0.5 (for the skew nanotube shown
in Fig. 10C) to 1 for t
−
= t
+
(for armchair nanotubes), and consequently
the values for the helicity H range from zero (for zigzag and armchair
nanotubes) to 1 for t
−
/t
+
=0.5. It seems that the highest possible complexity
due to helicity as indicated by Eq. (10.1) should be assigned to the case
when t
−
/t
+
= 0.5 (Fig. 10C), at equal “informational distance” from the
two achiral nanotubes illustrated in Figs. 10A and 10B. The second variable
On the Complexity of Fullerenes and Nanotubes 43
determining the complexity may be considered to be the nanotube diameter,
given by Eq. (10.1), so that the two parameters (t
+
and t
−
) are sufficient to
define both the diameter and the helicity. The same two parameters (actually
the difference t
+
− t
−,
namely if it is or is not divisible by 3) determine the
HOMO–LUMO gap in nanotubes; this gap in turn influences their electrical
conductivity [86]. A discussion on the aromaticity of armchair nanotubes
may be found in [87].
M. V. Diudea, T. S. Balaban, E. C. Kirby and A. Graovac have recently
published a study on how to derive single-walled nanotubes from cylin-
drical surfaces generated from a square net by deleting bonds in order to
change squares into hexagons [88]. Their end result is the prediction of
metallic or semiconductor behavior of the nanotube in terms of character-
istics of the parent square lattice. This study contains further references to
papers by Diudea and Kirby on this topic, including ones that allow coding
of the nanotube or carbon torus structure [89-92].
Since the length, the possible capping, bending, and presence of multi-
walled nanotubes will further complicate the picture, we prefer to terminate
here the discussion of nanotube complexity.
Discussing the complexity of few other types of carbon nanostruc-
tures has not been included in the present chapter, but leading refer-
ences will be included in the following. For carbon tori, in addition to
the literature cited above [89-92], it must be mentioned that they were
discussed in the literature [93-95] before being obtained experimentally
along with nanotubes and initially called “crop circles” [96]. Coiled
nanotubes [97-99], hyperfullerenes with negative curvature [100], and
Y-junctions of carbon nanotubes [101] are closely related to nanotubes.
However, graphitic cones are quite different, and were predicted [76]
before they were observed for the first time [77]. Subsequently, they were
found to occur naturally [102], and were also obtained synthetically [103].
Some of their topological characteristics have also been discussed [104-
105].
Acknowledgements
ATB thanks Professor D. J. Klein for discussions that will form the basis of a future joint
paper on the complexity of nanotubes. MR would like to thank Professor Jure Zupan for
kind hospitality during his visit to the National Institute of Chemistry, Ljubljana, Slovenia,
and XG is grateful to National Science Foundation of China, which in part supported the
work on this Project.
44 Chapter 1
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Chapter 2
COMPLEXITY AND SELF-ORGANIZATION
IN BIOLOGICAL DEVELOPMENT
AND EVOLUTION
Stuart A. Newman
New York Medical College, Valhalla, NY 10595
Gabor Forgacs
University of Missouri, Columbia, MO 65211
1. Introduction: Complex Chemical Systems in
Biological Development and Evolution
The field of developmental biology has as its major concern embryoge-
nesis: the generation of fully-formed organisms from a fertilized egg, the
zygote. Other issues in this field, organ regeneration and tissue repair
in organisms that have already passed through the embryonic stages,
have in common with embryogenesis three interrelated phenomena: cell
differentiation, the production of distinct cell types, cell pattern formation,
the generation of specific spatial arrangements of cells of different types,
and morphogenesis, the molding and shaping of tissues [1]. The cells
involved in these developmental processes and outcomes generally have
the same genetic information encoded in their DNA, the genome of the
organism, so that the different cell behaviors are largely assocated with
differential gene expression.
Because of the ubiquity and importance of differential gene expression
during development, and the fact that each type of organism has its own
unique genome, a highly gene-centered view of development prevailed
for several decades after the discovery of DNA’s capacity to encode in-
formation. In particular, development was held to be the unfolding of a
“genetic program” specific to each kind of organism and organ, based
on differential gene expression. This view became standard despite the
fact that no convincing models had ever been presented for how genes
or their products (proteins and RNA molecules) could alone build three
49