Lecture 3'  Introduction to Molecular Symmetry
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Molecular
Symmetry
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Biological Systems tend to beSymmetric
At various levels:organism shape:
e.g., bilateral, spiral, radialsymmetry.
shape of molecularstructures:
e.g., helix (polypeptides),Bhelix (DNA).
even though monomers areasymmetric (chiral).
Here, we focus onsymmetry in biologicalmacromolecules.
the types of symmetry.
developing a mathematicaldescription.
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Why Model Symmetry?
A model provides a compact, simplifieddescription of a complex structure.
which retains important details in minimal form.
Simplifies many problems in PhysicalBiochemistry:
structural prediction:e.g. : helices describe the most likely local secondary
structures (key to protein folding).help predict the likely outcomes of monomervariations.
structural determination:helps interpret results from Xray diffraction, electrondiffraction, etc.
image reconstruction:
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Defining Symmetry
Symmetry refers to:a correspondence in system properties on oppositesides of a dividing line or median plane.
shape, composition, or relative position of parts.
A Symmetric object will be reducible to a set of copies of an elementary object
each approximately identical.this unique, elementary object is called a motif, m.
The choice of motif depends upon the structure.e.g., starfish has 5 identical armseach is a motif.
Symmetry implies an orderly arrangement of thecopies to make the whole object.
starfish = 5 arms, arranged by rotation about a point.thenobject = motif details + arrangement details.
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The Symmetry Operator
A model of symmetry:models the structure of the overall object, in terms of the arrangement of the motif copies.
Copies arranged about a point, line, or plane of symmetry.
Operator Model: m repeatedly copied aboutthe axis of symmetry
by applying a symmetry operator,O
on m to give arelated motif (copy), m : O (m) = m .
In general, application of O may result in:a translated, reflected, or rotated copy of m(biopolymers).
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Point Symmetry
The simplest type of symmetry is pointsymmetry:
motifs are arranged about a point.then O (m) implements a rotation and/or a reflection.the complete set of motifs generated by O is calleda point group.
There are two types of point symmetry:mirror symmetry motifs related by reflection .rotational symmetry motifs related by rotation .
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The Types of Point Symmetry
Mirror Symmetry relates two motifs on oppositesides of a line or plane.e.g.: the Human Body.
motif = body.two halves related by reflection.
Rotational Symmetry relates motifs distributed abouta point or axis.Radial symmetry about a point:
motifs related by rotation.e.g.: diatoms.
Screw symmetry about an axis:motifs also translated down theaxis.e.g.: spiral seashell.
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Conventions
We will discuss each type of symmetry in somedetail.
but first, some conventions:
Each atom in our molecule is placed ata unique set of coordinates, (x,y,z).
we adopt a righthanded, Cartesiancoordinate system.
x x y = zpositive rotations: righthand rule.
Any rotation is multivalued...e.g., +90 o is also 270 o, +450 o, etc
Convention: Rotations are singlevalued and right
handed.rotation described by the (smallest) positive value of the
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The Symmetry Operator, O
Application of symmetry operator, O to motif,m, generates a second motif, m.
the coordinates of corresponding points in m (x,y,z)and m (x,y,z) are related by the transformation
equations:a 1x + b 1y + c 1 z = x
a 2x + b 2y + c 2 z = y
a 3x + b 3y + c 3 z = z
or, in matrix form:
this model correponds to O (m) = m
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Mirror Symmetry
Left and right hands are related by mirrorsymmetry.
about a plane passing through the center of the body.
Consider the structure formed by 2 facinghands:
1 on either side of the xz plane let m = right hand.
m = left hand.
corresponding pts in m and mrelated by the mirror operator,
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Mirror Symmetry (cont.)
The mirror operator, i :expresses the mathematical relationship between anypair of motifs that are exact mirror images
about an appropriately defined plane of symmetry.includes the stereoisomers of chiral monomers:
L and D forms of the amino acid residues.
PseudosymmetryAnother instance of mirror symmetry: the Humanbody.
where each half is a motif.
however, the symmetry is only approximate.e.g.: the heart is not in the center, but displaced.
Approximate symmetry is called pseudo symmetry.
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Rotational Symmetry
Symmetry about a point or axis is rotationalsymmetry.
no inversion of a motif.instead: reorientation in space, about the center of
mass (axis).Consider the structure formed by 2 right hands: placed in the 1 st and 3 rd quadrants
let m = hand in the 1 st quadrant.m = hand in the 3 rd quadrant.
corresponding pts in m and mrelated by a rotation operator,

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2fold Rotational Symmetry
Two applications of c causes a full 360 o rotation.
mathematically, c 2 = I, the identity matrix.c is thus identified as C 2 :
the 2fold rotation operator.said to produce a 2fold rotation.
and our objects symmetry axisthe zaxisis called a 2fold rotational axis
of symmetry.as denoted by the symbol at the origin.
C2 symmetry is also called Dyad symmetry.
and the axis the dyad axis.
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nfold Rotational Symmetry
An object with n motifs, related by rotation, =360 o/n about an axis of symmetry
is said to have nfold rotational, or C n symmetry.
The general operator for rotationabout the z axis by is:
for = 180 o , c reduces to C 2 .
For = 360 o/n, c = C n , the nfold rotational symmetryoperator.note: C 1 applies to nonsymmetric ( i.e., chiral)
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nfold Rotational Symmetry(cont.)
Some examples of nfold rotational symmetry:
each symmetry axis denoted by an nsided figure. note: in each case, O n = I , the identity matrix
n x sized rotations visit all motifs, and return us back
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Higher SymmetrySymmetry operators does not neet berestricted:
to nfold rotations, or rotations about the same axisor point.
Multiple symmetry elements may be combined toproduce higher symmetries.
In Biopolymers, such multiple sets relateidentical subunits,
organized at the level of quaternary structure.
examples discussed next lecture.Common symmetry groups in biopolymers:
Cn rotational symmetry about a point, or axis.D dihedral symmetry.
T tetragonal symmetry.O octahedral symmetry.
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Dihedral Symmetry
The most elementary higher symmetry isDihedral.
nfold Dihedral symmetry (D n) combines: nfold rotational symmetry about one axis;n C 2 axes, each perpendicular to the nfold axis.total number of motifs = 2n.
Example: D4 symmetry combines 1 C 4 and 4 C 2 axes ( eachdegenerate ):
(*) Note the alternate view:
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mhedral Symmetry
Point groups that combine multiple rotationalaxes describe mhedral symmetry .
m = the number of faces on the solid shape.m also indicates the total # of C 3 axes.
Axes may pass through faces or corners
N = the number of repeating motifs = 3m.3 motifs/face or 3 motifs/corner
Objects with mhedral symmetry will also exhibit
point symmetries for n < m. N must be divisible by n for C n axes to be present.
if C n axes present N = n x m,determines number of C n symmetry axes (m = N/n).
C3 symmetry always present;
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Screw Symmetry
For radial symmetry:motifs periodically arranged about a point or axis.360 o rotation returns us to the starting position.
e.g.: for C n , we require that On = I.
If (only) requirement 2 is broken i.e., 360 o rotation also causes a translation P,
down the axis of symmetry.
Then: the object has Screw symmetry.
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Example of Screw Sym