Cyclohexane.world File Reference

A cycloalkane with a 6 carbon loop. More...

Detailed Description

Introduction

Cyclohexane is a cycloalkane with the molecular formula C₆H₁₂. Cyclohexane is used as a nonpolar solvent for the chemical industry, and also as a raw material for the industrial production of adipic acid and caprolactam, both of which are intermediates used in the production of nylon.

The purpose of this benchmark is to show how, using CuikSuite, one can compute the set of cyclohexane conformations that are compatible with its rigid-geometry model. This model assumes that all bond lengths and angles are kept fixed, and only the torsion angles can vary. Conformations that satisfy such constraints are among the most stable ones. In the robotics language, this is equivalent to finding the possible configurations of a 6R loop: the links are the carbon atoms, and the revolutes are the free torsions around each C-C bond. This loop, observe, has a special geometry, as the two axis of each link are intersecting. This particularity makes the linkage overconstrained because, despite involving six bodies, it exhibits a finite self-motion (the skew boat form). The loop can also be assembled into a rigid conformation though (the chair form).

The same benchmark is used in [Emiris et al. 2006, Manocha et al. 1995, Manocha et al. 1994] to illustrate the performance of resultant-based techniques on this problem. We note that while on such works the authors must perturb the loop geometry a bit to make the solution space 0-dimensional, we need not resort to such artifacts in CUIK, obtaining a complete map with all conformations - including the flexible ones.

Note that the definition we provide allows to reconstruct the position of all cyclohexane atoms (even the hydrogens rigidily attached to the carbons). If we are only interested in the loop of carbons a simpler formulation is possible: a unitary vector along each CC bond, a fixed dot product (i.e., fixed angle) between those vectors and a sum of all vertors set to 0.

References

D. Manocha, Y. Zhu, W. Wright, "Conformational Analysis of Molecular Chains using Nano-Kinematics", Computer Application of Biological Sciences (CABIOS), 11(1), pp. 71-86, 1995.
D. Manocha, Y. Zhu, "Kinematic Manipulation of Molecular Chains Subject to Rigid Constraints", Proc. 2nd Symposium on Molecular Biology , pp. 285-294, 1994.
I. Emiris, E. Fritzilas, D. Manocha, "Algebraic Algorithms for Structure Determination in Biological Chemistry", International Journal of Quantum Chemistry, 2006

Definition in file Cyclohexane.world.