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Part II Chapter 4 Some big ones
Hardy non-uniform volume changes in Herpes capsid
We repeat the disk concept up to the big virus capsids.
The Hermite functions are used for an accurate description of capsid structures.
The plural structure of the Herpes capsid is discussed in connection with the effect of
pressure in a non-uniform pentagonal space in biology.
4.1 Structure and some ‘big ones’
We conclude with some examples of what we have found to be important in this study of
virus capsid structures.
A. The chemical geometry on molecular level: The disk and the description of capsid
structures with the Bragg morphotropic principles(ref 10).
B. The nature of mathematics of the virus capsid. The mathematics of the exponential
scale, finite and periodic GD functions. New mathematical methods for the description of
capsid structures.
C. Many new polyhedra found. A generalization of the polyhedron concept is given.
Polyhedra in virus space have rectangular and triangular faces instead of square and
hexagonal faces. Still in 53 or 53m symmetry.
The important bilateral or reflective operation has been used to describe the majority of
the pT=3, T=3, and T=4 capsid structures. The two enantiomorphs are intergrown in the
capsid structure. Our mathematical descriptions in fig 3.3.4 and 3.3.11 from equations
3.3.2and 3.3.3 are in perfect agreement with corresponding virus structures. We say that
this is an example of a bilateral or reflective operation.
D. Duals and plurals. The volume changes.
It is the disk concept from what we call chemical geometry that unifies the whole subject
as described in part 1 of this essay. It goes straight through the extended polyhedral
concept and the expansion of duality into plurality. It also makes it clear how to describe
many very big ones. It is remarkable how more and more different disks form bigger and
bigger asymmetric units. We are doing finite periodicity – or local Hardy translation in
pentagonal space using four different Alhamra tilings that form the asymmetric unit of
the Rice dwarf or Blue tongue capsid structures! Until the biggest of them all – the
gigantic PBCV-1 by Rossman et al - with one kind of disk that is repeated with finite
periodicity in pentagonal space, or local Hardy translation, to build blocks of a structure
that is similar to other capsid structure types. This occurs on an icosahedral triangle to
form the gigantic structure that can be described after the Bragg morphotropic
principles(10).
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We show below a short review that ends with these big structures. We repeat again some
important parts – this is to make it convenient for the reader. And the important parts are
the interwoven ensembles of asymmetric units and disks.
We start with the BPMV disk in pT=3 and the 3 fold axis is obvious.
Fig.4.1.1 BPMV pT=3
We continue with BMV from T=3, and again the disk has the 3 fold center of the capsid
structure. Its swollen form is shown to be very similar to the disk of Bacteriophage Ga.
Fig.4.1.2a BMV disk T=3
b Swollen disk of BMV
c Bacteriophage Ga disk T=3
Coming to T=4 the disks are in the 2 fold centers of the capsid structure.
Fig.4.1.3 Nudaurelia, T=4
Human Hepatitis T=4
Semliki T=4
For T=7 the asymmetric unit has become the disk which is then without symmetry.
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Fig.4.1.4 Asymmetric unit and disk of
the Simian capsid. T=7
Four different disks build the asymmetric
unit of the Human Adenovirus. T=13
Four different triangular disks in fig 4.1.5 build the capsids for T13 IBDV and Rice dwarf
or Blue tongue.
Fig 4.1.5 Asymmetric unit for T13 IBDV
Asymmetric unit for Rice dwarf or Blue tongue
The asymmetric unit for Rice dwarf or Blue tongue forms a formidable net known as the
Alhambra tilings, here in Hardy translation in pentagonal space net in fig 4.1.6a. Another
model for this is the local hexagonal structure from Human Adenovirus in fig 4.1.6b
below.
Fig 4.1.6a The Alhambra net in
Rice dwarf or Blue tongue
b Human Adenovirus Two icosahedral triangles of almost
crystallographic symmetry
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In the gigantic PBCV-1 the disk as in Fig 4.1.7 is compared with the asymmetric unit in
the Human Adeno virus. Such disks build triangular icosahedral surfaces after Hardy
translation which are repeated after pentagonal symmetry to build the capsid. The
detailed triangular icosahedral surface is organized after morphotropic methods using
simpler units from a Human Adeno virus structure.
Fig 4.1.7 a
Disk in PBCV-1
b Part of asymmetric unit
in the Human Adeno virus
c Full asymmetric unit
in the Human Adeno virus
4.2 Capsid structures related to the Human Adeno virus structure. The Hermites.
With the new exponential mathematics,with the disk concept, with many new polyhedra,
with topological relations between capsids, with common structure building principles
that go from the smallest to the biggest capsids, with the plural concept and the volume
variations, we have a good understanding of the intrinsic structure of viruses.
The mathematical methods used have been extended into a new application of the
Hermite functions as given here in equation 4.2.1. The harmonic oscillator is composed
of a Gauss distribution function(damping), and a product of its n derivatives. To its nature
the oscillator is a finite product and similar to our exponential functions. Physically it
describes oscillation which of course contain finite periodicity. We introduce the
dodecahedral symmetry and our ‘m’ below is just the n:th derivative of the Hermite
function. And we shall see that ’the end justifies the means’: We obtain a most
remarkable and accurate description of capsids as seen in figs 4.2.1a-j with the
Bean Pod Mottle, Bacteriophage 174, Cosackie, Human Hepatitis and Human Adeno
viruses in fig 4.2.1:
(Ne
N=
!
2
2 2 2
"1/2[(2# + 2)[x + y + z ]
1
2
2m (m ! )3
2
H[m,(# x + y)] H[m,(x + # y)] H[m,(y + # z)] H[m,(-# x + y)] H[m,(x - # y)] H[m,(y - # z)]) = const
4.2.1
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Fig 4.2.1 a-j
a m=1 icosahedron b Bean Pod Mottle
e m=3 trunc icosahedron
i m=5 new polyhedron
f Coxsackie
c m=2 icosidodecahedron d Bact phi 174
g m=4 Tri-pent-hex I
h Herpes
j Human Adenovirus Type 5
Figs 4.2.1a-j. The constants in the calculations are .05(m=5), .05(m=4), .05(m=3),
.05(m=2) and .001(m=1).
In the figures above we have used a single building element. A truncated triangle that
grows in size with m.
To describe the results of this beautiful Hermite mathematics in detail we need the
generalization of the polyhedron concept we have found and described above. We also
need the volume variation and the plurality.
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We show the plurality for m=4 in fig 4.2.2a. The polyhedra are icosahedron(white),
dodecahedron(yellow), icosidodecahedron(blue), the great rhombicosidodecahedron(red),
Tri-pent-hex I(green).The red plural net is described by BPMV in chapter 3. Of course the
volume change is also present.
The volume change is in fig 4.2.2b for m=2 and the icosidodecahedron, and is very
obvious for the plurals in red and blue, or the rhombicosidodecahedron and truncated
dodecahedron.
Fig 4.2.2a m=4 Plurality
b m=2 const =.012 Volume change
m=3 gives an important plural structure in fig 4.2.3 that agrees very well with the
Bacteriophage alfa 3 in fig 4.2.4b. Some rotation and volume change are needed but
nevertheless the agreement is impressive.
Fig 4.2.3 m=3
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Another important comparison is with the Bacteriophage KH 97 structure in fig 4.2.4a.
The difference with the calculated picture from the Hermite function m=3, and the
Bacteriophage alfa 3 structure is shown by the insertion of one more plural in
Bacteriophage KH 97. This is marked with a truncated dodecahedron in green, and
clearly, this accounts for the increase of size in capsid volume for the Bacteriophage KH
97. And we have again a splendid structural relationship between the two bacteriophages.
Rotation and volume changes are there of course.
Fig 4.2.4a Bacteriophage KH 97
b Bacteriophage alfa 3
4.3 Herpes and non-uniform volume changes
The volume change for a Platonic solid is uniform. For an Archimedean polyhedron like
the truncated icosahedron it can occur in two different manners, or steps as described
above. One is the uniform change as for a Platonic solid. The other, we simply call it nonuniform, is simply related to truncation. As described above in 1.2.
For bigger virus structures with more edges it turns complicated. We go directly to a
virus like Herpes (ref 12) and study then the plural net as shown in fig 4.3.1a which
contains five independent edges to vary (The plural net is n=7 in the Blue tongue series,
Blue tongue itself is n=6). To this we add three edges from the Tri-pent-hex I polyhedron
description of the virus capsid. Many are the different non-uniform capsid structures that
can be imagined for complicated virus structures. (T=16 for Herpes).
This means that there are great many possibilities for variations of capsid volumes.
Small changes in the genome could be picked up by a plural as non-uniform changes.
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Fig 4.3.1a Herpes capsid with the
Blue Tongue net as plural
b Calculated Hermite picture of Herpes
after eq 4.2.1 m=4
Notice the structurally differences at different parts of the surfaces, due to non-uniform volume changes.
4.3.1c const=.1
d const =.012
With a given capsid is there any respond for a change in external pressure? Probably not
for moderate changes. The capsid structure is relatively rigid.
But a different thing is a change of pressure when the replication is on. ‘The growth cycle
of poliovirus is extremely brief and extraordinarily efficient, the entire process is
complete within 8 h, and yields in excess of 100,000 particles per cell are not
uncommon’(ref 13). This is chemically and physically a very delicate process.
I have discussed with Kåre Larsson the physical consequences of a change of pressure
would have on a biological system. We propose that even a small change in pressure
would have dramatic effects. During the replication, the plural structure, as the more
sensitive, would respond to an instant change in volume in a non-uniform way.
In 4.3.1a there is the normal Herpes structure with the plural in white hexagons. We have
shown in fig 4.3.1b-d, which are Hermite calculations and which are in an astonishing
agreement with the observed in white hexagons. There is however structural changes in
the plural structure as caused by a change in the constant. Which is understood as a
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sudden change in volume or pressure. From a detailed comparison in fig 4.3.1c-d it is
obvious that the non-uniform volume changes vary in detailed parts of the surface-this is
obviously due to structurally differences at different points. Another kind of Herpes virus
– normally to the worse for the production of ordinary Herpes virus capsid – but to the
better for us, has been formed due to a sudden change in pressure.
An interesting aspect is that a small change of the volume of the genome could be
transferred into the capsid structure as a metric change in the distance between the
spikes(white hexagon net, with green spikes). With the finer details of structure using the
plural concept for Herpes as an example, very small changes of volumes could give a
great number of varieties. As it is natural to assume that the metric changes would occur
ordered on the capsid surface.
So why is the finite pentagonal symmetry so dominating?
An obvious answer rests in the density difference between arrangements of bodies with
pentagonal and crystallographic symmetry obvious from comparing the cube octahedron
and the icosahedron. Their edge/inter-radius ratios are 2/ 3 and 5 -1. The replication
occurs in a cell and the finite volume would have elevated pressure. This would favor
pentagonal symmetry. So the answer is chemistry.
! of the
! structural variations with
An important advantage might be that the number
pentagonal symmetry is greatly superior. As beautifully demonstrated with the stellations,
the compounds, the truncations, the non-convex polyhedra, see ref 11. And also the
extension of the polyhedra and the plural concepts add to the richness of variations for
pentagonal symmetry.
We have the parallels in the crystallographic world with the snub cube, the small and the
great rhombicuboctahedra, etc. But the manifold does not seem to be there with the
stellations etc.