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Image processing for cryo
microscopy
Lecture 5E.Orlova
Point group symmetries
1 - 11 September 2015
Practical Course Birkbeck College London
Symmetry in planeSymmetry in 2D spaceSymmetry in 3D space
Point group symmetries
Symmetry in space –
symmetry in projections
WHAT IS
SYMMETRY
The term
Symmetry
has Greek origins. For the
Greeks it meant the harmony of parts , proportionsand rhythm.Another more mathematical way, that is related toour image analysis and crystallography is:An object can be divided by a point, or line, orradiation lines or planes into to or more parts EXACTLY SIMILAR
in size and shape, and in
position relative to the dividing element. Repetitionof exactly similar parts facing each other or a centre.
Why is it Important?
We can classify molecules (biocomplexes) according to thelevel of their symmetry elements, so they can be groupedtogether having the same set of symmetry elements.This classification is very important, because it allows tomake some general conclusions about molecular propertieswithout extra calculation. On the atomic level it helps toreveal the molecular properties without any calculations.On atomic level, we will be able to decide if a molecule has adipole moment or not , and to know how these property arereflected on their surfaces.What sort of interaction hold biomolecules together in hugebiocomlexes such as viruses, secretion systems, etc.
Symmetry around us
We say that such an object is
symmetric
with
respect to a given
operation
if this operation,
when applied to the object, does not appearto change it.Symmetry may depend on the propertiesunder consideration:for
an image
we may consider just the
shapes, or also the colors;for
an object (3D),
we may additionally
consider density, chemical composition,contacts between domains, etc.
Biological objects The person - hereafter referredto as ‘it’ has a mirror plane,provided it stands straight (andthat we ignore its internalorgans). Each half of the figure isan asymmetric unit.
Moving an arm or leg destroysthe symmetry and the wholefigure can then be treated as anasymmetric unit.This little ‘object’, both with andwithout its mirror plane will beused to illustrate furthersymmetry elements, and tobuild more complicated groups.
1D
2D
3D
Operations with Multimeric Functions
One dimensional function -> a curve 1
5
2
3
4
6
10
7
8
9
11
12
14 13
15
1
5
2
3
4
6
10
7
8
9
11
12
14 13
15
Int Int
X X X
There are two types operations with functions inone-dimensional space.
1. Translation
1D
2D
3D
) a x ( g ) x ( g
a
g(x)
g(x+a)
a
g(x+2a)
Operations with Multimeric Functions
Reflection = mirror
) r a ( g ) r ( g
^
There is a plain between objects. So the distance from any pointof the object and the plain is the same as the distance betweenthe same point of the mirrored object and the plain
Operations with Multimeric Functions
T
T
Mirror-image symmetry
This is the most familiar andconventionally taught type ofsymmetry. It applies for instancefor the letter
T
: when this letter is
reflected along a vertical axis, itappears the same.
T
has a
vertical symmetry axis.A reflection "flips" an object over aline (in 2D) or plane (in 3D), invertingit to its mirror image, as if in a mirror.If the result is the same then wehave mirror-image symmetry (alsoknown in the terminology of modernphysics as P-symmetry).
T
Operations with Multimeric Functions
Reflection (mirror operation)
Translation
Shift and reflection
Operations with Multimeric Functions
o
Rotation in plane
Operations with Multimeric Functions
Inversion
g(r)
=g(
‐r)
Operations with Multimeric Functions There is a point between objects. So the distances from anypoint of the object and the point are the same as the distancesbetween the similar points of the inversed object and the point
Rotational symmetry Cn A rotation rotates an object about a point (in 3D: about anaxis). Rotational symmetry of order
n
, also called
n
-fold
rotational symmetry, with respect to a particular point or axismeans that rotation by an angle of 360
°/n does not change the
object.
°^ …..
etc )
For each point or axis of symmetry the symmetry group isisometric with the
cyclic group
C
n^
of order
n
. The
fundamental domain is a sector of 360
°/n.
C
2,^
C
3,^
C
4,^
C
5,^
C
6,^
C
7,^
C
8,^
C
9
One point remains unmoved, which is the rotationpoint
Rotational symmetry
2-fold
3-fold
5-fold
6-fold
C
C
C
C
Symmetry combinations More complex symmetries are combinations ofreflectional, rotational, translational, and glide reflectionsymmetry.Mirror-image symmetry in combination with
n
-fold
rotational symmetry, with the point of symmetry on the lineof symmetry, implies mirror-image symmetry with respectto lines of reflection rotated by multiples of 180
°/ n
, i.e. n
reflection lines which are radially spaced evenly (for odd nthis already follows from applying the rotational symmetryto a single reflection axis, but it also holds for even n).
Combination of symmetries in plane
Proteins don’t do this – pack by translations
Symmetry in 3D spaceThree types operations with objects in spaceTranslation in three directionsReflection (mirror)
in space
Rotation
(Inversion )
Rotations can be combined with translationsand reflections, however they cannot becombined in an arbitrary way.Symmetry operators impose constrains.Combination of operators can generateinfinite lattices as we can see that in crystals.
Basic symmetry operations in space
Translation in three directions
Unit
cell
a
A B
C
Crystall
Symmetry in 3D space
A space relationship One point remains unchanged.There are no translational operatorsCombination of rotation, mirror and inversiongives 32 combinationsBut for the proteins we will have only 11combinations: no inversion or mirror
between elements in each
oligomeric molecule can be described by a setsymmetry operations that describes the overallmolecular
symmetry.
This
combination
of
operations
define
the
POINT
GROUP
of
the
molecule.
Basic symmetry operations in space
Mirror
plane
Mirror
plane,
shown
as
dashed
line
Combination
of
two
‐fold
axis
with
mirror
planesMirror
plane
Reflection (mirror)in space
Rotations
axis
Basic symmetry operations in space
Rotation
in space
Mirror
plane Mirror
plane
Basic symmetry operations in space
The
group
does
not
form
a^ mirror
plane
Reflection (mirror)
in space
Rotation
in space
C
Cm
symmetry
C
Rotational symmetry is Cn, if the object has several elements, that arrangedin a circular system. The number of elements determines the order ofsymmetry.
Projections
Symmetry in 3D space
symmetry
C
CC^23
Projections
of
the
object
rotational
symmetry
and
their
symmetry.
Symmetry in 3D space
a-Latrotoxin520kDa
Ca release channel2.4 Mda
C
C
C
Portal protein SPP
C
Rotational symmetry
Symmetry in 3D space
symmetry
D
Symmetry in 3D space
Dihedral point group symmetry Dn are a combination of cyclicalsymmetries with a two-fold axis, which is perpendicular to the axisof rotation.
D
D
52 D
Symmetry in 3D space
D
Palinirus elephas hemocyanin(75kDa x 6)
D
Keyhole Limpet Hemocyeanin
Plato
and
Aristotle
Platonic and
Archimedean Polyhedra
The Platonic Solids, discovered by the Pythagoreans butdescribed by Plato (in the
Timaeus
) and used by him for his
theory of the 4 elements, consist of surfaces of a single kindof regular polygon, with identical vertices.The Archimedean Solids, consist of surfaces of more than asingle kind of regular polygon, with identical vertices andidentical arrangements of polygons around each polygon.
Cubic pointgroup symmetries
T -
Tetrahedral symmetry requeires a minimum of 12 identical subunits
O -
432 Octahedral point grouop symmetry,
needs 24 subunits
I -
532 Icosahedral symmetry, 60 subunits
Cube
Octahedron
Combination of symmetries in 3D
Combination of symmetries in 3D
Dodecahedron 2
Icosahedron
Combination of symmetries in 3D
http://www.staff.ncl.ac.uk/j.p.goss/symmetry/
Icosahedrael symmetry
Dodecahedron
Icosahedron
RCNMV-
Red
clover necroticmosaic virus
Herpesvirus
Tetrahedron
Octahedron
Cube
Dodecahedron
Icosahedron
GraphicsFaces
4 triangles
8 triangles
6 squares
12 pentagons
20 triangles
Vertices
4
6
8
20
12
Edges
6
12
12
30
30
Point Group
Td
Oh
Oh
Ih^
Ih
For each of the point groups
T d
,^ O
, andh
I h^
there exists sub-groups
T ,
O
, and
I
which contain all
C
symmetry elements, but none of the n^
S
operations (including n
inversion and reflection). Adding a
σh
mirror plane or an inversion center to the
T
group yields
T h
.
The high-symmetry point groups in which more than one
C^
axis with n
n^
≥^ 3 is
present are best visualized by the five regular polyhedra (
Platonic solids
) as
shown below. In these objects, all faces, vertices, and edges are symmetryrelated and thus equivalent. The
octahedron
and the
cube
are close related to
each other as they contain the same symmetry elements, but in differentorientations. The same applies to the
dodecahedron
and
icosahedron
.
