Lecture 3' - Introduction to Molecular Symmetry

8/14/2019 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),B-helix (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 X-ray 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 right-handed, Cartesiancoordinate system.

x x y = zpositive rotations: right-hand rule.

Any rotation is multi-valued...e.g., +90 o is also 270 o, +450 o, etc

Convention: Rotations are single-valued 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.

Pseudo-symmetryAnother 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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2-fold 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 2-fold rotation operator.said to produce a 2-fold rotation.

and our objects symmetry axisthe z-axisis called a 2-fold 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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n-fold Rotational Symmetry

An object with n motifs, related by rotation, =360 o/n about an axis of symmetry

is said to have n-fold 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 n-fold rotational symmetryoperator.note: C 1 applies to non-symmetric ( i.e., chiral)


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n-fold Rotational Symmetry(cont.)

Some examples of n-fold rotational symmetry:

each symmetry axis denoted by an n-sided 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 n-fold 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.

n-fold Dihedral symmetry (D n) combines: n-fold rotational symmetry about one axis;n C 2 axes, each perpendicular to the n-fold 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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m-hedral Symmetry

Point groups that combine multiple rotationalaxes describe m-hedral 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 m-hedral 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 Symmetry

The spiral staircase:Has steps symmetric about the z-axis.

This sounds like point symmetry

But, a 360 o rotation also causes a

translation (movement) down the z-axis. This translation, P is called the pitch.

This is generally called Screwsymmetry .

May also have a scaling of motifs.Examples: spiral seashell, a screw

The staircase is actually a specialcase:

symmetry only at discrete points aboutthe axis.


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n-fold Screw Symmetry

For an object with n-fold screw symmetry:n equal rotations of = 360 o/n generate:

1 turn of the helix.a translation, P down the axis (assume, z-axis).

corresponding points related by:

(x,y,z) = C n (x,y,z) + T ;

Cn is the corresponding point group.

T = translation operator, (0, 0, P/n).

Screw symmetry can be either:right-handed:

T z > 0 for CW (+) rotation.

left-handed: T z < 0 for CCW (-) rotation.


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Conclusion

In this Lecture, we have discussed: The use of Symmetry in simplifying the description of macromolecular structure;

Various types of simple Symmetry.

In the next Lecture, we begin our discussion of biopolymer structure:

With a description of the typical folded structures of proteins and polypeptides .

Lecture 3' - Introduction to Molecular Symmetry

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