Lecture 3' - Introduction To Molecular Symmetry

Lecture 3 – Introduction to Molecular Symmetry

Biological Systems tend to be Symmetric At various levels: 

organism shape:  e.g., bilateral, spiral, radial

symmetry.

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shape of molecular structures:  e.g., α-helix (polypeptides),

B-helix (DNA).

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even though monomers are asymmetric (chiral).

Here, we focus on symmetry in biological macromolecules.  

the types of symmetry. developing a mathematical description.

Why Model Symmetry? A model provides a compact, simplified description of a complex structure. 

which retains important details in minimal form.

Simplifies many problems in Physical Biochemistry: 

structural prediction:  e.g.: helices describe the most likely local secondary

structures (key to protein folding).  help predict the likely outcomes of monomer variations. 

structural determination:  helps interpret results from X-ray diffraction, electron

diffraction, etc. 

image reconstruction:

Defining Symmetry ‘Symmetry’ refers to: 

a correspondence in system properties on opposite sides 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’ arms…each is a motif.



Symmetry implies an orderly arrangement of the copies to make the whole object.  starfish = 5 arms, arranged by rotation about a point.  then…object = ‘motif details’ + ‘arrangement details’.

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 about the axis of symmetry… 

by applying a symmetry operator, O on m to give a related motif (copy), m’ : O (m) = m’.



In general, application of O may result in:  a translated, reflected, or rotated copy of m

(biopolymers). 

Point Symmetry The simplest type of symmetry is point symmetry:   

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 called a point group.

There are two types of point symmetry:  

mirror symmetry – motifs related by reflection. rotational symmetry – motifs related by rotation.

The Types of Point Symmetry Mirror Symmetry – 



relates two motifs on opposite sides of a line or plane. e.g.: the Human Body.  motif = ½ body.  two halves related by reflection.

Rotational Symmetry – 



relates motifs distributed about a point or axis. Radial symmetry about a point:  motifs related by rotation.  e.g.: diatoms.



Screw symmetry about an axis:  motifs also translated down the

axis.  e.g.: spiral seashell.

Conventions We will discuss each type of symmetry in some detail. 

but first, some conventions:

Each atom in our molecule is placed at a unique set of coordinates, (x,y,z). 

we adopt a right-handed, Cartesian coordinate system. xxy=z  positive rotations: right-hand rule.



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

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Convention: Rotations are single-valued and righthanded.  rotation described by the (smallest) positive value of the

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: a1x + b1y + c1 z = x’ a2x + b2y + c2 z = y’ a3x + b3y + c3 z = z’

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or, in matrix form:

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this model correponds to O(m) = m’ …

Mirror Symmetry Left and right hands are related by mirror symmetry. 

about a plane passing through the center of the body.

Consider the ‘structure’ formed by 2 facing hands: 

1 on either side of the xz plane…  let m = right hand.

m’ = left hand. 

corresponding pts in m and m’ related by the mirror operator,

Mirror Symmetry (cont.) The mirror operator, i : 

expresses the mathematical relationship between any pair of motifs that are exact mirror images…  about an appropriately defined plane of symmetry.

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includes the stereoisomers of chiral monomers:  L- and D- forms of the amino acid residues.

Pseudo-symmetry 

Another instance of mirror symmetry: the Human body.  where each half is a motif.

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however, the symmetry is only approximate.  e.g.: the heart is not in the center, but displaced.

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Approximate symmetry is called pseudo-symmetry.

Rotational Symmetry Symmetry about a point or axis is rotational symmetry.  

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 1st and 3rd quadrants…  let m = hand in the 1st quadrant.

m’ = hand in the 3rd quadrant. 



corresponding pts in m and m’ related by a rotation operator,

2-fold Rotational Symmetry Two applications of c causes a full 360o rotation. 

mathematically, c2 = I, the identity matrix.

c is thus identified as C2 : 

the 2-fold rotation operator.  said to produce a 2-fold rotation.



and our object’s symmetry axis  the z-axis…  is 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.

n-fold Rotational Symmetry An object with n motifs, related by rotation, Θ =360o/n about an axis of symmetry… 

is said to have n-fold rotational, or Cn symmetry.

The general operator for rotation about the z axis by Θ is:

 



for Θ = 180o, c reduces to C2. For Θ= 360o/n, c = Cn, the n-fold rotational symmetry operator. note: C1 applies to non-symmetric (i.e., chiral)

n-fold Rotational Symmetry (cont.) Some examples of n-fold rotational symmetry:

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each symmetry axis denoted by an n-sided figure.

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note: in each case, O

n

= I, the identity matrix…

 n x Θ-sized rotations visit all motifs, and return us back

Higher Symmetry Symmetry operators does not neet be restricted: 



to n-fold rotations, or rotations about the same axis or point. Multiple symmetry elements may be combined to produce ‘higher’ symmetries.

In Biopolymers, such multiple sets relate identical 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.

Dihedral Symmetry The most elementary ‘higher’ symmetry is Dihedral. 

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n-fold Dihedral symmetry (Dn) combines:  n-fold rotational symmetry about one axis;  n C2 axes, each perpendicular to the n-fold axis.  total number of motifs = 2n. Example:  D4 symmetry combines 1 C4 and 4 C2 axes (each

degenerate):

 (*) Note the alternate view:

m-hedral Symmetry Point groups that combine multiple rotational axes describe m-hedral symmetry. 

m = the number of faces on the solid shape.  m also indicates the total # of C3 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 Cn axes to be present.

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if Cn axes present… N = n x m,  determines number of Cn symmetry axes (m = N/n).

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C3 symmetry always present;

m-hedral Symmetry m=4: Tetrahedral symmetry (T).  12 motifs = (3/face)(4 faces) 

C2 and C3 symmetry:  4 C3 axes; 6 C2 axes.

m=8: Octahedral symmetry (O).  

Octahedron (or cube). 24 motifs… C2, C3, and C4 symmetry.  3/face and 3/corner, respectively.  6 C4 axes; 8 C3 axes; 12 C2 axes.

m=20: Icosahedral symmetry (I).  60 motifs = (3/face)(20 faces) 

C2, C3, and C5 symmetry.

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C6 not present...

Screw Symmetry For radial symmetry: 

motifs periodically arranged about a point or axis.

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360o rotation returns us to the starting position.  e.g.: for Cn , we require that On = I.

If (only) requirement 2 is broken… 

i.e., 360o rotation also causes a translation P,  down the axis of symmetry.



Then: the object has Screw symmetry.

Example of Screw Symmetry The spiral staircase: 

Has steps symmetric about the z-axis.  This sounds like point symmetry…



But, a 360o rotation also causes a translation (movement) down the zaxis.  This translation, P is called the pitch.

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This is generally called Screw symmetry.  May also have a scaling of motifs.  Examples: spiral seashell, a screw

The staircase is actually a special case:  symmetry only at discrete points about

the axis.

n-fold Screw Symmetry For an object with n-fold screw symmetry: 

n equal rotations of Θ = 360o/n generate:  1 turn of the helix.  a translation, P down the axis (assume, z-axis).

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corresponding points related by: (x’,y’,z’) = Cn (x,y,z) + T ;  Cn is the corresponding point group.  T = translation operator, (0, 0, P/n).

Screw symmetry can be either: 

right-handed:  Tz > 0 for CW (+) rotation.



left-handed:  Tz < 0 for CCW (-) rotation.

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.