13.cartesian coordinates
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The document discusses Cartesian coordinate systems and their use in molecular modeling. It describes how Cartesian coordinates are used to specify unique points in space using distances from perpendicular axes. Molecular graphics software uses Cartesian coordinates to visualize biological molecules by defining an coordinate frame of reference and assigning X, Y, Z coordinates. While Cartesian coordinates are used for modeling, other coordinate systems like cylindrical polar coordinates are sometimes used instead, especially for modeling helical molecules like DNA, and these first need transforming into Cartesian coordinates for visualization.
13.cartesian coordinates
- 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