Cyclohexane.world File Reference

Detailed Description

[Introduction] [Process] [Statistics] [Results] [References]

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.

Process

This example is treated following this steps (from the main CuikSuite folder):

  • Generate the equations: Execute
  • Solve the positional analysis problem: Execute
    • bin/cuik examples/Cycloalkanes/Cyclohexane
  • Sort the solutions:
  • Animate the solutions (the flexible and the isolated):
    • scripts/cuikplayer examples/Cycloalkanes/Cyclohexane examples/Cycloalkanes/Cyclohexane_1
    • scripts/cuikplayer examples/Cycloalkanes/Cyclohexane examples/Cycloalkanes/Cyclohexane_isol
  • Plot and visualize the configuration space (you will see the one-dim solution and the isolated small boxes):
    • bin/cuikplot3d examples/Cycloalkanes/Cyclohexane 6 5 17 0.01 c6.gcl
    • geomview c6.gcl

Statistics

Characteristics of the problems:

Nr. of loops 1
Nr. of links 6
Nr. of joints 6
Nr. of equations (in the simplified system) 11
Nr. of variables (in the simplified system) 11

Here you have the statistics about the execution (on an Intel Core i7 at 2.9 Ghz).

Nr. of empty boxes 17
Nr. of solution boxes 170
Solver time (s) 12

Results

The following figure shows the output boxes returned by the algorithm on the cyclohexane loop. The problem was with SIGMA set to 0.1. As expected, the conformational space is formed by two isolated points plus a closed curve of mobile conformations, corresponding to the chair and skew boat forms of this molecule, respectively.

Note that the two chair conformations are essentially the same up to a symmetry. The formulation used here, though, does not allow to account for the symmetries and identify the two conformations separately.

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.