1. PRESENTATION Presentation of Coordinate system

2. APPLICATION OF COORDINATE SYSTEM Modeling small molecules building 3D- structures of small molecules of biological relevance RasMol (http://www.umass.edu/microbio/rasmol/), MolMol (http://www.tucows.com/preview/9805), Swiss-PDB viewer (http://www.expasy.ch/spdbv/).

3. 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.

4. Cartesian coordinate system A Cartesian coordinate system 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.

5. Cartesian coordinate system

6. The Euclidean distance between two points of the plane with Cartesian coordinates (x 1,y 1 ) and (x 2,y 2 ) is This is the Cartesian version of Pythagoras' theorem. In three dimensional space, the distance between points (x 1,y 1,z 1 ) and (x 2,y 2,z 2 ) is

7. Internal coordinate system Small molecules can be built using internal coordinates based on chemical parameters. For any atom j these are defined in terms of coordinates of the three preceding atoms: j' to which atom j is connected, j" to which atom j' is connected and j"' to which atom j'' is connected.

8. The bond length R J is defined as the distance between the atoms j and j'. The bond angle α j is defined as the angle between the atoms j-j'-j". The torsional angle ß j is the angle between the images of bonds j-j' and j"-j"’ on the plane, perpendicular to the bond j'-j". The clockwise rotation of the bond j-j' about the bond j'-j" to bring j-j' in the direction of j"-j'" is taken as the positive angle.

9. Polar coordinate system The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole with the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.

10. A cylindrical coordinate system is a three-dimensional coordinate system, where each point is specified by the two polar coordinates of its perpendicular projection onto some fixed plane, and by its (signed) distance from that plane.

11. Cylindrical polar coordinate system

12. The three coordinates (ρ, φ, z) of a point P are defined as: The radial distance ρ is the Euclidean distance from the z axis to the point P. The azimuth φ is the angle between the reference direction on the chosen plane and the line from the origin to the projection of P on the plane. The height z is the signed distance from the chosen plane to the point P.

13. Coordinate system conversions