Molecular Nanotechnology: Golden Mean as a Driving Force of Self-Assembly

Abstract

In this paper we are considering self-assembly approach from nanobiology to molecular nanotechnology. Because the genetic code, as a most complex biomolecular system, is determined by Golden mean, we used it knowledge to study approaches to nanotechnology. Understanding protein self-assembly driving force and structure of clathrin and microtubules based on Golden mean we have found that fullerene C60 and nanotubes could be very useful materials for molecular nanotechnology. One possible solution as a sample is given.

Full text

Molecular Nanotechnology:
Golden Mean as a Driving Force of Self-Assembly
DJURO KORUGA
1
,
JOVANA SIMIC-KRSTIC
2
, LIDIJA MATIJA
3
, LJUBISA
PETROV
1
AND ZELJKO RATKAJ
1
1
Molecular Machines Research Center, Faculty of Mechanical Engineering, University of
Belgrade, 27.Marta br.80, 11120 Belgrade, SERBIA
2
Institute for Chemical Power Sources, Batajnicki put 27,11080 Belgrade, SERBIA
3
Mechanical Textile College, Starine Novaka 23,11000 Belgrade, SERBIA
Abstract: In this paper we are considering self-assembly approach from nanobiology to molecular
nanotechnology. Because the genetic code, as a most complex biomolecular system, is
determined by Golden mean, we used it knowledge to study approaches to nanotechnology.
Understanding protein self-assembly driving force and structure of clathrin and microtubules
based on Golden mean we have found that fullerene C
60
and nanotubes could be very useful
materials for molecular nanotechnology. One possible solution as a sample is given.
Key-words: Golden mean, nanobiology, clathrin, microtubules, molecule C
60
, nanotubes,
molecular nanotechnology
1 Introduction
Beauty is a conspicuous element in
the abstract completeness aimed at in the
higher mathematics. Also, trough the
symmetry law it is the goal of physics as it
seeks to construe the order both elementary
particules and the Universe. However, it
ought to be the inspiration of genetic and
molecular study of life.
Beauty is a word which has defied
the efforts of philosophers to define in a way
that commands general agreement. In
mathematics it is mean ratio, golden triangle
or Fibonacci golden series. In material
science it is crystal state based on number of
transformations of point symmetry groups.
In molecular machines technology it could
be driving force of self-assembly based on
structure, energy and environmental
properties.
2 Problem Formulation
Human organism is the most
complex local Universe technology. In spite
of that, structural information code (DNA) is
based on only four elements, in form as an
aperiodic crystal. [1] However, it has been
shown that the genetic code is a Golden
mean determined system. [2] On the other
hand, the Golden mean is driving force of
protein biomolecular machinery [3] For
nannotechnology it is important to find
biomolecules (proteins) as a systems which
structure and energy properties are based on
Golden mean. Based on nanobiology
knowledge we can develop new materials
with self-assembles properties for molecular
nanotechnology. [4]

2.1 Golden Mean in Nanobiology:
Clathrin and microtubules are two
main proteins which structure and energy
properties are based on Golden mean.
Clathrin was discovered in 1969 by two
Japanise scientists, Kanaseki and Kadota [5],
while microtubules at the first time was
predicted and published by a 24-year-old
Austrian cell biologist Sigmund Freud (well
known psychologist) [6].
2.1.1 Clathrin
Clathrin- coated vesicles are cell
structures found in all nucleated cells, from
yeast to human. It is a protein with a
molecular weight of 180,000 daltons
(dalton: “D” is the equivalent of the weight
of one hydrogen atom). Clathrins are the
major components of coated vesicles,
important organelles for intracellular
material transfer including synaptic
neurotransmitter release. Based on
molecular weights, isoelectric points and
antigenic determinants, two proteins,
α
- and
β
-tubulin subunits, have been found to be
associated with coated vesicles in both
bovine brain and chicken liver.
The basic form of clathrin is a
trimer (three subunits), but the basic form of
assembly protein, which associates with
clathrin, is a dimer. Clathrin is highly
conserved in evolution and is composed of
three large polypeptide chains and three
smaller polypeptide chains that together
form a triskelion. A different number of
triskelions assemble into a basket-like
network with Golden mean properties based
on 12 pentagons and various numbers of
hexagons.
Clathrin as a truncated icosahedron
is dominant in neurons, while clathrin with
30 hexagons is found in other types of cells
(e.g. liver, etc.). Large clathrin of 60
hexagons with a diameter of about 120 [nm]
have been found in fibroblasts. The number
of amino acids with negative charges
(aspartic acid and glutamic acid) is 378 per
clathrin, while the number of amino acids
with positive charges (histidine, lysine and
arginine) is 201. This means that there are
Figure 1: Clathrin self-assembly: Clathrin triskelions are the assembly units of the sphere lattice
on the surface of coated pits and coated vesicles [4]

77 more electrons than positive charge per
molecule, or per “cage” there are about
8.316 more electrons than positive charges.
Figure 2: Neural cells (neurons) contain CV and clathrin with 12 pentagons and 20 hexagons with
diameters of 70-80 nm [4]
2.1.2 Microtubules
Microtubules are biopolymers
present in all eucariotic cells. They are
involved in several specialized functions,
including cell shape regulation, mitosis,
intracellular translocation, cell motility, and
secretion. Structurally, microtubules are
hollow cylinders 25 nm in diameter which
are composed of 13 linear chains
(protofilaments) which consist of proten
subunits called tubulin. Microtubules are
composed of equimolar amount of the two
globular 50 kD subunits, alfa and beta
tubulin, each having a similar amino acid
sequence and a similar overall shape.
Tubulin is a globular (spherical)
protein composed of amino acids. α -tubulin
subunits usually contain 450 amino acids,
while β -subunits contain about 445 amino
acids. There are about two times more
negative charges (Asp and G1u) than
positive charges (Lys, Arg and His). The
diameter of tubulin subunits is about 4 nm.
This heterodimer is the main building
element of microtubules. Microtubules may
exist from 7-17protofilaments, but usually
(85%) a microtubule contains 13
protofilaments.
Figure 3: Most neuronal CV and clathrin are
concentrated in synapses (where
consciousness arise), some associated with
microtubules (usually there are five CV per
one microtubule). [4]
The subunits of tubulin molecules
are assembled into long tubular structures
with an average exterior diameter of 25-30
nm, capable of changes of length by

assembly or disassembly of their subunits.
Assembly/disassembly is sensitive to cold,
high hydrostatic pressure, several specific
chemicals such as colchicine and
vinblastine, and other factors.
Since some experimental results link
tubulin and microtubules to bioinformation
processes such as memory, learning and
consciousness microtubules have become
the subject of intensive research.
[
7-9
]
2.3 Golden Mean in Nanomaterials
2.3.1 Molecule C
60
The C
60
molecule is the third known
pure crystal form of carbon, in addition to
graphite and diamond. It is predicted in 1970
by Japanise scientist Osawa, and synthesis in
1985 by Kroto/Smalley research team. The
electronic structure is a complex, “many
body” problem, because there are 360
electrons. Conversely, the C
60
molecule has
attributes of a “big atom”, because it has a
close spherical electronic shell and possesses
unique icosahedral symmetry properties. In
Figure 5: Nanotubes as a double-helix
(proposed by Osawa [10]) could be very
important basic element for molecular
nanotechnology based on self-assembly.
the truncated icosahedral structure there are
two characteristic C-C bond lengths: C
5
-C
5
in pentagons, C
5
-C
6
double bonds in
hexagons (or link between two pentagons).
There are sixty carbon p
z
orbitals,
each pointed along radial axes. If
interactions among p
z
orbitals belonging to
carbon atoms on a certain pentagon are
considered and interactions among orbitals
located on different pentagons (there are 12
such pentagons) are neglected, then the five
eigenstates based on K
h
symmetry (spherical
harmonics).
Figure 4: Golden mean surface energy state
of molecule C
60
[4]
There are three sets of orbitals
which occur grouped together: ψ
0
, ψ
1(+,-)
,
and ψ
2(+,-)
. Interaction among 12 pentagons
will split the twelve ψ
0
orbitals to A
g
+ H
g
+
T
1u
+ T
2u
, while the 24 ψ
1(+,-)
orbitals, two
per pentagon, will split into T
1g
+ G
g
+ H
g
+
T
1u
+ G
u
+ H
u
, irreducible representations (or
symmetries). The final 24 ψ
2(+,-)
orbitals,
with the highest energy, will be split into
T
2g
, G
u
, G
g
, H
u
, T
2u
and H
g
. Irreducible
representations T
1g
, T
2g
,T
1u
and T
2u
for
symmetry elements C
5
and S
10
possess
Golden mean properties.[4]
2.3.2 Nanotubes
Nanotubes are similar to
microtubules like C
60
molecule is similar to

clathrin. Nanotubes are quasi-1D structure,
while C
60
is quasi-0D entity; first crystallize
around axis (1-dimension) and second
crystallize around point (0-dimension).
There are single-wall and multi-wall
nanotubes with different physical properties.
All types of nanotubes are intersting for
nanotechnology, but helical one could be
very intersting for molecular
nanotechnology based on self-assembly.
3 Problem solution
We are proposing one possible approach to
problem solution of molecular
nanotechnology based on Golden mean.
Nanomaterials will be fullerene molecule
C
60
with intelligent solvent and magnetic
field with density of magnetic flux by
Golden mean (Figure 6).
We plan to use knowledge from
nanobiology (self assembly of clathrins and
microtubules) to molecular nanotechnology
(C
60
and nanotubes) in sense like
relationship between the flight of a bird and
the flight of an aircraft: both use wings but
in different manners. [11] However,
molecular coding chain of spiral nanotubes,
with C
60
and endohedral M@C60 as basic
elements, should exist.
Figure 6: One possible approach of molecular nanotechnology based on Golden mean and
symmetry reduction (from
∞
trough 48 to 9 and from
∞
trough 120, 60, 24 to 12 - adapt from
reference [4]).

4 Conclusion
Golden mean is very powerful force
of self-assembly in molecular biology (DNA
and proteins). It is possible to take this
knowledge from nanobiology and
implement into molecular nanotechnology.
Very promising nanomaterials for self-
assembly approach based on Golden mean
in molecular nanotechnology are fullerenes
and carbon nanotunbes. Environment for
self –assembly process should be intelligent
solvent and internal pulsing magnetic field
by Golden mean.
References
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and Metter, Cambridge University
Press, 1967
[2] Rakocevic, M., The genetic code as a
Golden mean determined system,
BioSystems, 46: pp283-291,1998
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of amino acids by module 3/2, Archive
of Oncology, Vol.5, No.3, pp.137-139,
1997
[4] Koruga, D., et al, Fullerene C
60
:
History, Physics, Nanobiology,
Nanotechnology, North-Holland,
Amsterdam, 1993
[5] Kaneski, T. Kadota,K., Structure of
Clathrins, J. Cell Biology, 20:pp.202-
220,1969
[6] Freud, S. Sitzungsber. d. Wiener
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Press, Oxford 1994
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Fullerene C
60
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th
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