Structural Representation

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Structural Representation
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http//www.math.iastate.edu/wu/Math597HW0000/index
.htm

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

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Structural Representation
Algebraic Representation Cartesian coordinates
(3n)
Geometric Representation inter-atomic distances
n(n-1)/2
Trigonometric Representation dihedral angles (
amino acids)
Each of representation formats may have its own
advantages in representing certain types of
structural data or computing the structure with
certain types of methods.
Probabilistic Representation Electron Density
Distribution (8)
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Algebraic Representation
It is the most straightforward way to represent a
molecular structure in Cartesian coordinates.
Potential energy functions can be expressed,
evaluated, or minimized more easily when defined
in terms of Cartesian coordinates.
Example Energy Function
Structural changes or dynamics are described in
terms of the coordinates of the atoms as well.
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Geometric Representation
In geometry, we learned that a triangle is
determined if the three sides are given. In other
words, three points are determined if the
distances in between them are known.
The structure is determined by the distances.
In three-dimensional space, can we determine the
positions for n points given all distances
between every pairs of points?
Inter-atomic distances are known for certain
pairs of atoms, such as the bond lengths in
between the bonded atoms, the distances between
hydrogen atoms that can be detected by NMR.
Answer Yes! A molecular structure is
determined if the distances between all pairs of
atoms are given.
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Trigonometric Representation
dihedral angle
The distances and the angle determine the
structure.
a
b
c
c
N
a
C
N
Ca
b
bond angle
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Dihedral Angles
dihedral angle
aijkl
a
b
c
a xj xi b xk xj c xl xk
xi
xk
aijk
xl
ajkl
xj
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Probabilistic Representation

? (x)
electron density distribution function
FH
X-ray
structure factors
X-ray Diffraction
X-ray Crystallography
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Electron Density Distribution
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Transformation of Coordinate Systems
Cartesian Coordinates lt gt Atomic Distances
Cartesian Coordinates lt gt Dihedral Angles
Cartesian Coordinates lt gt Electron
Density Distribution