The cyclohexane

                                                     cyclohexan [1]

Conformations specify the 3-dimensional structure of the molecule. It has been argued by Gō and Scherga [3] that energy minima can be approximated by allowing only the dihedral angles to vary, while keeping bond lengths and bond angles fixed. At a first level of approximation, therefore, solving for the dihedral angles under the assumption of rigid geometry provides information for the energetically favorable configurations (see also [4]).

The equations generated below, are obtained by fixing the length between the consecutive molecules and the angles betweenv the links and by computing the the relations between the tangents of half the "flip" angles. The general equations are in ℝ_{∗}[t_{1..3}] if the angles and length are given by approximate coefficients.

\{ \begin{matrix}−13 − t_{2}^{2} − t_{3}^{2} + 24 t_{2}t_{3} − t_{2}^{2}t_{3}^{2} \\ −13 − t_{3}^{2} − t_{1}^{2} + 24 t_{1}t_{3} − t_{3}^{2}t_{1}^{2} \\ −13 − t_{1}^{2} − t_{2}^{2} + 24 t_{1}t_{2} − t_{1}^{2}t_{2}^{2} \end{matrix}  (1)

Approximation of the solutions (which are all reals for this system) are given in this table:

\begin{matrix}t_{1} & t_{2} & t_{3} \\ 10.85770360 & 0.7795480449 & 0.7795480451 \\ - 10.85770360 & - 0.7795480449 & - 0.7795480451 \\ 0.3320730984 & 4.625181601 & 4.625181601 \\ - 0.3320730984 & - 4.625181601 & - 4.625181601 \\ 0.7795480449 & 0.7795480449 & 0.7795480451 \\ 0.7795480449 & 10.85770360 & 0.7795480451 \\ 0.7795480440 & 0.7795480457 & 10.85770360 \\ 4.625181601 & 4.625181601 & 4.625181601 \\ 4.625181601 & 0.3320730984 & 4.625181601 \\ 4.625181600 & 4.625181613 & 0.3320730984 \\ - 0.7795480440 & - 0.7795480457 & - 10.85770360 \\ - 0.7795480449 & - 0.7795480449 & - 0.7795480451 \\ - 0.7795480449 & - 10.85770360 & - 0.7795480451 \\ - 4.625181600 & - 4.625181613 & - 0.3320730984 \\ - 4.625181601 & - 4.625181601 & - 4.625181601 \\ - 4.625181601 & - 0.3320730984 & - 4.625181601 \end{matrix}

Characteristics:

• Number of variables: 3.

• Dimension: 0.

• Degree: 16.

• Complete intersection in P^{1} × P^{1} × P^{1}.

• It can have up to 16 real solutions (for instance, the system (1)).

Example 1:

[-310+959*x[2]^2+774*x[3]^2+1389*x[2]*x[3]+1313*x[2]^2*x[3]^2,
-365+755*x[3]^2+917*x[1]^2+1451*x[1]*x[3]+1269*x[3]^2*x[1]^2,
-413+837*x[1]^2+838*x[2]^2+1655*x[1]*x[2]+1352*x[1]^2*x[2]^2];

Example 2:

[.86605*x[2]^2*x[3]^2+.50000*x[2]^2+2.*x[2]*x[3]+.50000*x[3]^2-.86605,
.86605*x[3]^2*x[1]^2+.50000*x[1]^2+2.*x[1]*x[3]+.50000*x[3]^2-.86605,
.86605*x[1]^2*x[2]^2+.50000*x[1]^2+2.*x[1]*x[2]+.50000*x[2]^2-.86605];

Example 3:

[ 1.334568086*x[2]^2*x[3]^2+.8485720485*x[2]^2+1.302855903*x[2]*x[3]+
.8485720484*x[3]^2-.2888519406,
1.334568086*x[1]^2*x[3]^2+.8485720484*x[1]^2+1.302855903*x[1]*x[3]+
.8485720485*x[3]^2-.2888519406,
1.334568086*x[1]^2*x[2]^2+.8485720485*x[1]^2+1.302855903*x[1]*x[2]+
.8485720484*x[2]^2-.2888519406];

Example 4:

parameter:= a;
[a*x[2]^2*x[3]^2+1/2*x[2]^2+2*x[2]*x[3]+1/2*x[3]^2-a,
 a*x[3]^2*x[1]^2+1/2*x[1]^2+2*x[1]*x[3]+1/2*x[3]^2-a,
 a*x[1]^2*x[2]^2+1/2*x[1]^2+2*x[1]*x[2]+1/2*x[2]^2-a,4*a^2-3];

Problem: Find the roots of the systems.

• Solution by I.Z. Emiris and B. Mourrain, INRIA, Projet SAFIR, 2004 Route des Lucioles,BP 93, 06902 Sophia-Antipolis (France),

mourrain@sophia.inria.fr, April 1996.

Several methods have been compared on 3 systems of this type (see the source code), using Bezoutian (BZ), Sylvester’s resultant in cascade (RC). These computations have been done in maple with 15 digits, on a Dec. Alpha Station 200 - 96M .

S1  S2  S3
T1  T2  T3  ϵ  T1  T2  T3  ϵ  T1  T2  T3  ϵ
BZ  0.18  2.02  0.50  3.1 10−11  0.35  0.68  1.87  3.0 10−12  0.20  25.38  1.02  1.48 10−4
RC  0.60  0.10  0.30  4.8 10−5  0.18  0.13  1.4  2.2 10−9  0.58  0.82  0.28  1.5 10−4

where T_{1} is the time for constructing the matrix(ces), T_{2} is the time for solving the univariate polynomial (in this case with fsolve), T_{3} is the time for computing the kernel or the other coordinnates of the roots, ϵ is the maximal absolute value of the input polynomials at the computed roots. See [2] or here for more information

• Solution by P. Zimmerman, INRIA Lorraine, Technopole de Nancy-Brabois, 615 rue du Jardin Botanique, BP 101, F-54600 Villers-les-Nancy.

Paul.Zimmermann@loria.fr December 1996

Computation done on a SGI R10000.

   *----*    MuPAD 1.3  ---  Multi Processing Algebra Data Tool
  /|   /|
 *----* |    Copyright (c) 1992-96 by B. Fuchssteiner, Automath
 | *--|-*    University of Paderborn.  All rights reserved.
 |/   |/
 *----*      Demo version, please register with
             MuPAD-distribution@uni-paderborn.de
>> sys:={-y^2*z^2-y^2+24*y*z-z^2-13,-x^2*z^2-x^2+24*x*z-z^2-13,
&> -x^2*y^2-x^2+24*x*y-y^2-13}:
>> time((sol:=solve(sys)));

                                   3 s
>> lprint(sol);
{[x = z, y = z*43/3 + z^3*(-2/3), z = -(3^(1/2)*(-6) + 11)^(1/2)], [x = z,\
 y = z*43/3 + z^3*(-2/3), z = (3^(1/2)*(-6) + 11)^(1/2)], [x = z, y = z*43\
/3 + z^3*(-2/3), z = (3^(1/2)*6 + 11)^(1/2)], [x = z, y = z*43/3 + z^3*(-2\
/3), z = -(3^(1/2)*6 + 11)^(1/2)], [x = z*23/3 + z^3*(-1/3) + (z^2*532 + z\
^4*(-46) + z^6-78)^(1/2)*(-1/3), y = z, z = -(3^(1/2)*(-6) + 11)^(1/2)], [\
x = z*23/3 + z^3*(-1/3) + (z^2*532 + z^4*(-46) + z^6-78)^(1/2)*(-1/3), y =\
 z, z = (3^(1/2)*(-6) + 11)^(1/2)], [x = z*23/3 + z^3*(-1/3) + (z^2*532 + \
z^4*(-46) + z^6-78)^(1/2)*(-1/3), y = z, z = (3^(1/2)*6 + 11)^(1/2)], [x =\
 z*23/3 + z^3*(-1/3) + (z^2*532 + z^4*(-46) + z^6-78)^(1/2)*(-1/3), y = z,\
 z = -(3^(1/2)*6 + 11)^(1/2)], [x = z*23/3 + z^3*(-1/3) + (z^2*532 + z^4*(\
-46) + z^6-78)^(1/2)*1/3, y = z, z = -(3^(1/2)*(-6) + 11)^(1/2)], [x = z*2\
3/3 + z^3*(-1/3) + (z^2*532 + z^4*(-46) + z^6-78)^(1/2)*1/3, y = z, z = (3\
^(1/2)*(-6) + 11)^(1/2)], [x = z*23/3 + z^3*(-1/3) + (z^2*532 + z^4*(-46) \
+ z^6-78)^(1/2)*1/3, y = z, z = (3^(1/2)*6 + 11)^(1/2)], [x = z*23/3 + z^3\
*(-1/3) + (z^2*532 + z^4*(-46) + z^6-78)^(1/2)*1/3, y = z, z = -(3^(1/2)*6\
 + 11)^(1/2)], [x = z*237/17 + z^3*(-2/17), y = z*237/17 + z^3*(-2/17), z \
= -(3^(1/2)*(-34) + 59)^(1/2)], [x = z*237/17 + z^3*(-2/17), y = z*237/17 \
+ z^3*(-2/17), z = (3^(1/2)*(-34) + 59)^(1/2)], [x = z*237/17 + z^3*(-2/17\
), y = z*237/17 + z^3*(-2/17), z = (3^(1/2)*34 + 59)^(1/2)], [x = z*237/17\
 + z^3*(-2/17), y = z*237/17 + z^3*(-2/17), z = -(3^(1/2)*34 + 59)^(1/2)]}
>> lprint(allvalues(sol));
{[x = -0.77954804500, y = -0.77954804500, z = -10.857703590], [x = 0.77954\
804500, y = 0.77954804500, z = 10.857703590], [x = 4.6251816010, y = 4.625\
1816010, z = 0.33207309830], [x = -4.6251816010, y = -4.6251816010, z = -0\
.33207309830], [x = -0.33207309830, y = -4.6251816010, z = -4.6251816010],\
 [x = 4.6251816010, y = 4.6251816010, z = 4.6251816010], [x = 10.857703590\
, y = 0.77954804500, z = 0.77954804500], [x = -0.77954804500, y = -0.77954\
804500, z = -0.77954804500], [x = -4.6251816010, y = -4.6251816010, z = -4\
.6251816010], [x = 0.33207309830, y = 4.6251816010, z = 4.6251816010], [x \
= 0.77954804500, y = 0.77954804500, z = 0.77954804500], [x = -10.857703590\
, y = -0.77954804500, z = -0.77954804500], [x = -4.6251816010, y = -0.3320\
7309830, z = -4.6251816010], [x = 4.6251816010, y = 0.33207309830, z = 4.6\
251816010], [x = 0.77954804500, y = 10.857703590, z = 0.77954804500], [x =\
 -0.77954804500, y = -10.857703590, z = -0.77954804500]}

• Solution by C. Traverso, Dipartimento di Matematica, via Buonarroti 2, 56127 Pisa, ITALY, traverso@posso.dm.unipi.it, December 1996.

Computation done with the Posso-library on an old home 486:

Computation required:
Problem         User time System  Real   Memory PageFault       Host
                0.21      0.01    0.26    134552        0       nello
Computed Pairs Summary:
        Total   Useful  Useless Discarded
        32      19      13      0
@
#0 z8-140z6+2622z4-1820z2+169
#1 17yz4-374yz2+221y+2z7-281z5+5240z3-3081z
#2 204y2+136yz3-3128yz+z6-149z4+2739z2+117
#3 17xz4-374xz2+221x+2z7-281z5+5240z3-3081z
#4 136xy-136xz-136yz+2z6-281z4+5376z2-3081
#5 204x2+136xz3-3128xz+z6-149z4+2739z2+117

Trying to triangularize, one remarks that eq. 1 is

17y(z^{4} − 22z^{2} + 13) + ...

hence z^{4} − 22z^{2} + 13 divides z^{8} − 140z^{6} + 2622z^{4} − 1820z^{4}2 + 169 Adding the equation z^{4}4 − 22z^{4}2 + 13, the lex basis is (in 0.06")

#0 z4-22z2+13
#1 3y2+2yz3-46yz-z2+26
#2 xy-xz-yz+z2
#3 3x2+2xz3-46xz-z2+26

eq. 2 is x(y − z) + ..., hence we add y − z: (in 0.01")

#0 z4-22z2+13
#1 y-z
#2 3x2+2xz3-46xz-z2+26

or divide by y − z: (in 0.02")

#0 z4-22z2+13
#1 3y+2z3-43z
#2 x-z

or divide the original eq. by z^{4} − 22z^{2} + 13: (in 0.22")

#0 z4-118z2+13
#1 17y+2z3-237z
#2 17x+2z3-237z

Hence reducing to three triangular systems solvable through square roots.

References

[1] D. Bini and B. Mourrain. Polynomial test suite. 1996.

[2] I. Z. Emiris and B. Mourrain. Polynomial system solving; the case of a 6-atom molecule. Rapport de Recherche 3075, INRIA, Dec. 1996.

[3] N. Gō and H.A. Scheraga. Ring closure and local conformational deformations of chain molecules. Macromolecules, 3(2):178–187, 1970.

[4] D. Parsons and J. Canny. Geometric problems in molecular biology and robotics. In Proc. 2nd Intern. Conf. on Intelligent Systems for Molecular Biology, pages 322–330, Palo Alto, CA, August 1994.