Choice of coordinate systems

One of the common questions often encountered in molecular modelling relates to the choice of representation between Cartesian  coordinate and internal coordinate.

Cartesian coordinate:  The position of atomic nuclei is described in three dimensional space using three variables x, y and z.

Cartesian coordinate of acetaldehyde can be presented as follows:

Internal (Z-matrix) : When talking of Z-matrix, we keep track of relative position of atoms. Z-matrix is composed of atom number, bond length, bond angle and dihedral angle.
Z-matrix for acetaldehyde:

Therefore, one can say that the basic difference between the two representations is that the Cartesian coordinate is an absolute way of describing position, whereas Z-matrix is a relative.

When is Z-matrix better than Cartesian?

Z-matrix is very useful while optimizing a structure by keeping some restraints. For example, optimizing the structure for minimum energy by keeping a particular dihedral angle within certain range of value. Z-matrix is also useful when dealing with molecules like water (C_2v symmetry) and ammonia (C_3v symmetry). In ammonia three hydrogen atoms are pyramidal and equivalent. Using z-matrix you can tell the program about the fact that ammonia is C_3v symmetric by giving proper constraints and the molecule will indeed be optimized accordingly.
However, Cartesian coordinates should generally be preferred when dealing with non-symmetric molecule or very large molecule (to reduce error due to rounding off).

How to convert Cartesian to Z-matrix?

The recipe from converting Cartesian coordinates into Z-matrix consists following  two elements.

1. Bonds which are calculated as Euclidean distances.
2. Angles that are determined using the following equation:

${\displaystyle \cos \varphi =\frac{\left(x_0-x_1\right).\left(x_2-x_1\right)}{\left|x_0-x_1\right|\left|x_2-x_1\right|} \; }{\displaystyle }$

In the above equation, torsional angle is the angle made by two planes. One comprising of points X₀, X₁ and X₂ and other comprising of points X₁, X₂ and X₃ respectively. These two planes can be imagined as in the figure below.

Dihedral angle formed by two plane as described above

Some programs that are used for such conversions include OPEN-BABEL and newzmat (in Gaussian).

Which coordinate system do you prefer? Participate in the poll below.

Which coordinate system do you prefer?

When do you use Z-matrix and when do you prefer Cartesian? Leave a comment and share the rationale behind your choice of coordinate system.

Golden Jubilee of Ramachandran Plot

Exactly fifty years from now i.e. in the year 1963, G. N. Ramachandran et. al published breakthrough original research in Journal of Molecular Biology. Ramachandran plot still remains a touchstone for protein form and structure (for example, its use in validating homology models). This plot is remarkable because it came ahead of time. It was proposed in 1963 when there were no computers, mechanical calculators were the cutting edge of technology!

Backbone conformations of: (A) glycine and (B) pre-proline. Backbone schematic and observed Ramachandran plot of (A) glycine and (B) pre-proline. Taken from the data-set of Lovell et al. (2003). The clustered regions are labeled on the Ramachandran plots.

Late Prof. Ramachandran (affectionately called GNR) established Molecular Biophysics Unit (MBU) at Indian Institute of Science (IISc), Banglore in 1971. Following the footsteps of his mentor Dr. C.V. Raman, G. N. Ramachandran carried out most of his research in India. To mark the celebration of 50 years of Ramachandran map MBU organized "International Conference on Biomolecular Forms and Fuctions" from 8^th - 11^th January 2013. (official website: http://icbff2013.com/)

Several eminent scientists including Professor Tom Blundell and Professor J. Andrew McCammon delivered their talk during the conference.

Some interesting references: 

1. Kleywegt GJ, Jones TA. Phi/psi-chology: Ramachandran revisited. Structure. 1996 Dec 15;4(12):1395-400. PubMed PMID: 8994966. 

 2. Lovell SC, Davis IW, Arendall WB 3rd, de Bakker PI, Word JM, Prisant MG, Richardson JS, Richardson DC. Structure validation by Calpha geometry: phi,psi and Cbeta deviation. Proteins. 2003 Feb 15;50(3):437-50. PubMed PMID: 12557186.