cartesian coordinates

1. Cartesian coordinate

2. Cartesian coordinate
● A Cartesian coordinate system is a coordinate
system that
– specifies each point uniquely
– in a plane
– by a pair of numerical coordinates,
– which are the signed distances from the point
– to two fixed perpendicular directed lines,
– measured in the same unit of length.

4. ● The first step in the molecular modeling is visualizing a biological molecule.
● For achieving this it is necessary to define a coordinate frame of reference.
● Generally ‘Cartesian coordinate system’ is used.
● In this, there are three mutually perpendicular axes (OX, OY and OZ)
passing through a point O (the origin).
● To get X, Y, Z coordinates of point P in space, first a perpendicular (PM) is
drawn from a point P on the XY plane.
● From point M perpendiculars (ML and MN) are drawn on axes OX and OY.
● Distance OL, LM and PM respectively describe the Cartesian coordinates (X,
Y and Z) of the atom located at P.

6. ● The Cartesian coordinates of any a molecule
cannot be fed directly to the computer for
displaying on the screen.
● use of graphics visualization software.
● RasMol (http://www.umass.edu/microbio/rasmol/),
● MolMol (http://www.tucows.com/preview/9805),
● Swiss-PDB viewer (http://www.expasy.ch/spdbv/).

7. ● Each graphic package uses a specific FORMAT for supplying
Cartesian coordinates.
● Most popular is PDB FORMAT.
● The coordinates of a molecule are given in 3 formats
1) ‘crystal internal frame of reference’ attached to the unit cell
of the crystal,
2) ‘internal coordinates’, viz. bond lengths, bond angles and
torsional angles, related to three preceding atoms,
3) ‘Cylindrical polar coordinates’ (r, θ , z), as these are more
relevant from crystallographic or chemical point of view.

8. ●
Coordinates for DNA are usually given in the ‘Cylindrical polar
coordinate system’ because of its helical symmetry.
● In such cases, one has to first transform these coordinates into
Cartesian coordinate system (X,Y, Z) and then only molecular
graphics software can be used to visualize these molecules.
●
Another necessary information for molecular visualization is
chemical connectivity table.
● Majority of the graphics packages compute connectivity or supply
this information on the basis of chemical formulas.

9. DNA

10. ●
For generation of standard DNA structures, with the helical
symmetry, only two parameters are needed,
1) the base pair rise tr- which is the distance between the
successive base pairs along the helical axis
2) and the twist tw- which is the angle of rotation of the following
base pair with respect to existing one, are needed.
●
If one of the Cartesian axes (in general Z- axis) coincides with the
helical axis of the molecule, we can generate the DNA polymer
using set of rules.

11. Set of rules
● Suppose xi , yi and zi are the coordinates of the
ith atom in a building block, the coordinates of
the same atom in the nth residue can be
obtained by:
● Rotating coordinates of all the atoms in a block
by angle (n-1).tw by a rotational transformation,
● Followed by translation of the unit along the
helix axis by an amount (n-1).tr.

12. Proteins
● In contrast to DNA, proteins have a wide range of
structures, which primarily depend upon their amino acid
sequences, though not necessarily.
● Predicting 3D-structure of a protein, with several hundred
amino acids, purely on theoretical basis, with no structural
or chemical information available, is still a dream.
● Computer modeling can be used to simulate and energy
minimize small peptidic fragments.
● There are several algorithms to predict secondary structural
elements in proteins using statistical, chemical, homology
based and ‘threading the sequence through structural
motifs’ techniques.
● Several such tools are available at ExPASy

13. ● The starting point in protein structure simulation is the structural
information on 20 commonly occurring amino acids.
● Since, there are various possible side chains and secondary
structures, it is difficult to generate a polypeptide chain using
‘Cylindrical polar’ or ‘Cartesian coordinates’ building blocks.
● It is easier to use ‘Internal coordinate building blocks’ for each
amino acid and assign torsional angles to all the backbone
atoms, on the basis of its secondary structure (α, β, turn (T)and
coil(C)) given in Table.

14. Table