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 Sym