CPHF PVED calculations for deoxyribose
1
Coupled-perturbed Hartree-Fock calculations of the parityviolating energy difference of deoxyribose
Brandan R. Robertson
Department of Physics, School of Science and Computer Engineering
University of Houston-Clear Lake
2700 Bay Area Boulevard, Houston, Texas 77058
Abstract Homochirality of certain key biochemical enantiomers is generally held to
have been necessary for the development of life on Earth.
One of the proposed
mechanisms for the creation of this enantiomeric excess is the small energy difference
between enantiomers due to the parity-violating weak force. The parity-violating energy
difference (PVED) of a molecule can be calculated ab initio with computer programs. If
the enantiomer calculated to have lower energy using this method matches the naturally
abundant enantiomer, the weak force theory of enantioselection is supported. Previous
energy calculations of deoxyribose by Tranter (1992) used uncoupled-perturbed HartreeFock theory.
New calculations for the PVED of deoxyribose using updated bond
parameters and an updated computer code that incorporates coupled-perturbed HartreeFock theory are presented. The results show a negative Boltzmann-averaged parity-
CPHF PVED calculations for deoxyribose
violating energy shift (PVES) for both C2-endo and C3-endo conformations of
2
D-
deoxyribose, with the D- form C2-endo conformation most negative. These PVED values
are larger than those previously published.
Keywords Coupled-perturbed Hartree-Fock · Deoxyribose · Enantioselection · Origin of
homochirality · Parity-violating energy difference
1. Introduction
The establishment of homochirality of certain chemical enantiomers is thought to have
been a prerequisite for the development of life on Earth. How this enantiomeric excess
could have come to be is unknown, but various theoretical mechanisms have been
proposed including magnetic fields, circularly polarized sunlight, and irradiation of presolar clouds by rotating neutron stars, as reviewed by MacDermott (2002). Most of these
mechanisms are all either the result of providential local circumstances, as in the case of
polarized sunlight and pre-solar irradiation, or not truly chiral, as in the case of magnetic
fields. One proposed mechanism that does not suffer from these limits is the influence of
the weak force.
The weak force is the only one of the four forces of nature that violates parity.
The parity violation is manifested in the unequal participation of left-handed and righthanded forms of fermions in weak interactions, via W bosons and Z bosons, the weak
force mediating particles. The parity-violating W interactions of interest all involve -
CPHF PVED calculations for deoxyribose
3
decay; however the short range of some of these emitters and the scarcity of others makes
radiolysis by -electrons an unlikely source of biological homochirality and no positive
experimental results involving -radiolysis have yet been obtained (MacDermott, 2002).
Z bosons mediating weak neutral current interactions interact preferentially with
left-handed electrons, making an equal mixture of right- and left-handed electrons behave
as if it were predominantly left-handed.
The parity violation of these interactions
between a molecule’s electrons and its chiral nuclear frame generates a parity-violating
energy shift in the energy of two chemical enantiomers (which are actually
diastereoisomers—a true enantiomer would be made of antimatter). The parity-violating
energy difference between the enantiomers is equal to twice the PVES of the two
enantiomers. Various amplification methods have been proposed to turn the resulting
small enantiomeric excesses into fully homochiral solutions, though at present the
Kondepudi mechanism (1987) remains the only method not subsequently discredited.
An expression for the PVED of a given molecule can be calculated given the
Hamiltonian for the parity-violating weak neutral current interaction between electrons
and neutrons. This expression can then be calculated ab initio with computer programs
written for this purpose using the Hartree-Fock self-consistent field method. If the
enantiomer calculated to have the lowest energy using this method matches the
enantiomer that is more prevalent in nature, this supports the theory that the weak force is
responsible for enantioselection.
Such calculations have been carried out for modern amino acids and regular
oligopeptide conformations, and have shown the naturally abundant forms to be PVEDfavored (Mason and Tranter, 1983, 1984, 1985; Tranter, 1985a, b 1986, 1987a, b). Past
CPHF PVED calculations for deoxyribose
4
calculations for biochemical sugars have indicated that different conformations have
different stable enantiomers, namely that while C2-endo ribose and deoxyribose are more
stable in the D- form, the C3-endo conformations are more stable in the L- form (Tranter
et al., 1992). However, there is uncertainty regarding what sugars formed the basis of
primitive replicating nucleic acids. Thus this discrepancy may serve as evidence against
the weak force theory of selection, or contrarily it may provide insight into the structure
of primitive replicators.
These prior calculations were performed with uncoupled-perturbed Hartree-Fock
methods, which do not adequately account for electron-electron interactions.
The
timescales and sizes of the solutions necessary for amplification of the tiny initial
enantiomeric excesses due to the PVED are very sensitive to the magnitude of the PVED
(Kondepudi, 1987) so it is important to obtain accurate magnitudes for the PVED in the
calculations. This paper presents new calculations for the PVED of deoxyribose using
updated deoxyribose molecular parameters and a suite of subroutines in the Cambridge
Analytic Derivatives Package software package (CADPAC) that were written expressly
to calculate the PVES of a molecule using coupled-perturbed Hartree-Fock theory. The
results show a negative Boltzmann-averaged PVES for both conformations of
D-
deoxyribose using CPHF calculations, with the PVES more negative in the C2-endo
conformation as compared with the C3-endo conformation. In addition to providing
more insight into which enantiomer is PVED-favored and providing a more accurate
value for the PVED of deoxyribose for use in amplification calculations, the results help
to quantify the degree of sensitivity of the PVED to bond parameters and the theory used
in the calculation.
CPHF PVED calculations for deoxyribose
5
2. Methods
The parity-violating Hamiltonian for electron-neutron interactions is
G  1 


 
 N a s (i )  p(i ),  3 ( ri  ra )
Hˆ pv   F 
2 2  me c  i a

where GF is the Fermi coupling constant, me is the mass of the electron, Na is the neutron


number of nucleus a, s (i ) is the spin angular momentum of electron i, p (i ) is the linear

momentum of the electron i, r is the position vector of the electron or nucleus under
consideration, and the braces denote the anticommutator. The PVES is the expectation
value of the Hamiltonian; using first-order perturbation theory and including spin-orbit
coupling effects, one obtains the expression
E pv
  l Vˆ  l  s s  s  l Vˆ  l  s s  s
j
pv
k
j
k
k
so
j
k
j
 2 o u Re 

E j  Ek
j
k





for the PVES, where Vˆpv and Vˆso are the respective parity-violating and spin-orbit
potentials, j and k the energies of the wave functions, and
 o and  u
j
k
the respective
CPHF PVED calculations for deoxyribose
6
sums over all occupied and unoccupied molecular orbitals in the ground state
(MacDermott and Tranter, 1990).
The Hartree-Fock self-consistent field method can be employed to obtain
numerical values for the PVES. Using the restricted formalism, summarized nicely in
Atkins and Friedman (2005) and the coupled-perturbed corrections one obtains for the
PVES the expression
E pv   o u
j
k

  
a ,b , , , , 
j* k
k*
j
a a  b  b 
c c c c
 N aU jk

b   b lb b 
where the coefficients c are undetermined coefficients,  is the spin-orbital, ξb is the spin
orbit coupling constant for the given electron in the field of nucleus b, lb is the orbital
angular momentum of the given electron if the field of nucleus b, the sums are over all
nuclei a, b, and c and over all types of atomic orbital  on each nucleus, and Ujk is the
coupled-perturbed parity-violating matrix element (MacDermott et al., submitted 2007).
updated deoxyribose molecular parameters and a suite of subroutines in the Cambridge
Analytic Derivatives Package software package (CADPAC) that were written expressly
to calculate the PVES of a molecule using coupled-perturbed Hartree-Fock theory.
To calculate the PVES of D-deoxyribose, the above calculations were performed
using the Cambridge Analytic Derivatives Package (CADPAC) software with a suite of
built-in subroutines that were written expressly to calculate the PVES of a molecule using
coupled-perturbed Hartree-Fock theory. The calculations were performed using a 6-31G
CPHF PVED calculations for deoxyribose
7
basis set. In addition to using a more complex theory in the calculation, updated furanose
bond parameters were used (Gelbin et al., 1996) so before exploring the effect of the
coupled-perturbed method on the PVED of deoxyribose, the effect of the new bond
parameters was explored.
To probe the sensitivity of the calculations, a CADPAC input file for
D-
deoxyribose in the C2-endo ap conformation that was used in the original uncoupled
calculations of Tranter et al. (1992) was run using the coupled method, leaving the
original bond parameters intact. The same conformation was then run again through the
coupled calculations with the updated bond parameters. In both cases, the standard bond
lengths for carbon-hydrogen bonds and oxygen-hydrogen bonds found in Saenger (1984)
were used.
The switch to the coupled method from the uncoupled method had a
significant effect in both cases, increasing the magnitude of the PVES by roughly a factor
of two over the original calculation. The new bond parameters then appeared to have a
moderating effect on the increase, in both cases scaling the magnitude of the PVES back
slightly over the coupled calculations of the original parameters.
The results are
summarized in Table 1.
With an understanding of the possible origins of the differences between the
coupled and uncoupled results that may arise, the deoxyribose calculations commenced
using the new bond parameters. Due to the limitations of x-ray techniques in determining
hydrogen bond parameters, these bond parameters are not well known. Tranter et al.
chose to examine three orientations of the O1 hydrogen atom because of the large
contribution from the atoms of this group to the PVES, and adopted a cis- orientation for
the others. For the new coupled analysis, the orientation of the O1 hydrogen atom was
CPHF PVED calculations for deoxyribose
8
examined in greater detail in order to find the orientation with the minimum selfconsistent field (SCF) energy. The coupled calculations were initially run using tendegree increments of the hydrogen bond torsion angle, and both the resulting PVES and
overall SCF energy were calculated to ensure that the most stable orientation was used in
calculation of the PVES. An example of the data analysis used in determination of the
optimal angular region for the O1 hydrogen is shown in Figure 1. Once the angular
region of minimum SCF energy was identified for the O1 H atom, the minimum-energy
angle was used in a similar examination of the O3 H atom, and these two minimums were
likewise used in a parametric examination of the O5 H atom orientation. The process
was repeated until the solutions converged on the minimum-energy torsion angles.
In order to determine a suitable PVES for the molecule, a Boltzmann average
must be carried out over all molecular energies. The Boltzmann average over three
independent variables  is given by
E pv 


E pve

[ E scf ( , , )  E scf min ]
, ,
kT
Z
where Z is the total Boltzmann partition function.
Because energies more than about 3kT (approximately .003 hartree) above the
minimum SCF energy do not contribute significantly to the Boltzmann average PVES, a
nested series of calculations was performed over the angle ranges of each torsion angle
encompassing the 3kT range on the converged optimization plots. For the C2-endo
CPHF PVED calculations for deoxyribose
9
conformation, the energies were calculated at ten-degree increments, resulting in 2584
data points. The values produced by these analyses were then used to compute the
Boltzmann average at a range of temperatures. However, the considerable computational
time involved with this series of calculations made a larger angle increment desirable. To
gauge the sensitivity of the average PVES to the number of data points, the data points
for every odd increment were removed for each torsion angle, resulting in a data set of
about one-eighth the size of the original, and the resulting new average PVES values
were calculated.
magnitude.
The PVES using the reduced data set was only 1.4% lower in
Because this difference was very small, the C3-endo energies were
calculated at twenty-degree increments, and Boltzmann averages computed.
3. Results
The minimum energy torsion angles for the C2-endo conformation were found to be
66.0° for the O1 H atom, 52.5° for the O3 H atom, and 305.0° for the O5 H atom. For the
C3-endo conformation, the minimum energy torsion angles were 73.0° for the O1 H
atom, 315.0° for the O3 H atom, and 305.0° for the O5 H atom. The Boltzmann averaged
PVES values for D-deoxyribose C2-endo ap and C3-endo ap are shown in Figures 2 and
3, respectively. The Boltzmann-averaged PVES was firmly negative; the average PVES
was not strongly dependent on temperature over the range studied, with the change in
PVES averaging about 7  10 24 hartree per degree Kelvin, the PVES increasing with
increasing temperature.
CPHF PVED calculations for deoxyribose
10
4. Discussion
The negative values for the PVES indicate that the D- form of deoxyribose is the more
stable of the enantiomers, in line with the results from the UPHF calculations of Tranter
et al. (1992), though larger in magnitude. The relative values of the PVES for the C2endo and C3-endo conformations also generally agree with the Tranter observations, with
the C2-endo conformation having a more negative PVES than the C3-endo configuration.
However, whereas the C3-endo conformation in the study of Tranter et al. had a positive
PVES for all configurations studied, the Boltzmann averaged PVES for the C3-endo
conformation in this study was negative.
The results of the sensitivity studies indicate that while the updated molecular
parameters (on the order of a degree or two for the bond angles, and tenths of an
angstrom for bond lengths) can have a noticeable effect on the PVES, the switch to CPHF
is more important.
The appreciable sizes of the differences demonstrate that it is
important that continued updates of the PVED of biomolecules are made as the precision
of calculation methods and physical parameters increase.
Based on these latest computations, the D- form of deoxyribose does appear to be
the energetically favored form for both the C2- and C3-endo conformations, supporting
the theory of weak force enantioselection. In addition, the PVED calculated in this study
was substantially larger than the PVED calculated in previous studies.
This study did not examine the dependence of the PVED on the orientation of the
O5 structure, in that only the orientation associated with the ap confirmation of the base
in the updated bond parameters was examined and not +/- sc conformations. It would be
CPHF PVED calculations for deoxyribose
11
interesting to include this torsion angle in future energy minimization studies. Another
parameter this study left unexplored was the sensitivity to the furanose ring torsion
angles. Because the values of the torsion angles in this study are experimental in origin,
the theoretically computed minimum-energy torsion angles are not likely to be
significantly different. However, the sensitivity to the torsion angle is unknown and if
the sensitivity is high, the contributions of nearby energy states may have a significant
effect on the Boltzmann averaged value of the PVED. Thus, it may be a useful exercise
to examine the effect of the furanose ring torsion angles in a future study. Lastly, the
effect of the new parameters and methods on the conformation of deoxyribose that were
studied indicates a similar change in PVED magnitude is likely for ribose, and this should
be examined.
Acknowledgments The author would like to thank Dr. Alexandra MacDermott for her
help in providing the in-depth theoretical background for this study and guidance along
the way. The author would also like to thank Roy Nakatsuka and Terry Fu for their
previous work in creating the useful data analysis tools that were used in performance of
the analysis and preparation of the results.
References
Amos RD et al (1987) Efficient calculation of vibrational magnetic dipole transition
moments and rotational strengths. Chem Phys Lett 133:21-26
CPHF PVED calculations for deoxyribose
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Atkins P, Friedman R (2005) Molecular Quantum Mechanics, 4th ed. Oxford University
Press, Oxford
Gelbin A, et al. (1996) Geometric parameters in nucleic acids: sugar and phosphate
constituents. J Am Chem Soc, 118:519-529.
Kondepudi DK (1987) Selection of molecular chirality by extremely weak chiral
interactions under far-from-equilibrium conditions. BioSystems 20:75-83
MacDermott AJ (2002) The Origin of Biomolecular Chirality. In: Lough WJ, Wainer IW
(eds) Chirality in Natural and Applied Science. Blackwell Science, Oxford, pp 23-52
MacDermott AJ, Hyde GO, Cohen AJ. Evaluation of coupled perturbed and density
functional methods of computing the parity-violating energy difference between
enantiomers. Submitted for review to Orig Life Evol Biosph, November 2007)
MacDermott AJ, Tranter GE (1990) Biomolecular handedness and the weak interaction.
In Gruber B, Yopp JH (eds) Symmetries in Science IV: Biological and Biophysical
Systems.Plenum, New York, pp 67-124
Mason SF, Tranter GE (1983) Energy inequivalence of peptide enantiomers from parity
non-conservation. J Chem Soc Chem Commun (3):117-119
CPHF PVED calculations for deoxyribose
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Mason SF, Tranter GE (1984) The parity-violating energy difference between
enantiomeric molecules. Mol Phys 53:1091-1111
Mason SF, Tranter GE (1985) The electroweak origin of biomolecular handedness. Proc
R Soc Lond A 397:45-65
Saenger W (1984) Principles of Nucleic Acid Structure. Springer, Berlin
Tranter GE (1985a) The parity-violating energy differences between the enantiomers of
-amino acids. Chem Phys Lett 130:93-96
Tranter GE (1985b) Parity-violating energy differences of chiral minerals and the origin
of biomolecular homochirality. Nature 318:172-173
Tranter GE (1986) Parity-violating energy differences and the origin of biomolecular
homochirality. J Theor Biol 119:467-479
Tranter GE (1987a) Parity violation and the origins of biomolecular handedness.
BioSystems 20:37-48
Tranter GE (1987b) The enantio-preferential stabilization of D-ribose from parity
violation. Chem Phys Lett 135:279-282
CPHF PVED calculations for deoxyribose
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Tranter GE, MacDermott AJ, Overill RE et al (1992) Computational studies of the
electroweak origin of biomolecular handedness in natural sugars. Proc R Soc Lond A
436:603-615
CPHF PVED calculations for deoxyribose
15
Table 1 Results of the sensitivity studies of bond parameters and calculation
method on D-deoxyribose C2-endo ap.
Calculation method
Original parameters using
UPHF1
Original parameters using
CPHF
New parameters using CPHF
1
PVES with O1 H @ -60°
PVES with O1 H @ 180°
(hartree)
(hartree)
 2.20  10 20
 1.30  10 20
 6.12  10 20
 2.49  10 20
 4.95  10 20
 1.88  10 20
From Tranter et al. (1992)
Figure 1 The parity-violating energy and self-consistent field energy for D-deoxyribose
C2-endo ap as a function of the torsion angle of the O1 hydrogen bond. The O3 and O5
CPHF PVED calculations for deoxyribose
16
torsion angles are held constant at 52 degrees and 305 degrees, respectively. Zero
degrees represents cis- geometry and 180 degrees represents trans-.
Figure 2 The Boltzmann-averaged PVED for the minimum-energy D-deoxyribose C2endo ap orientation as a function of temperature.
CPHF PVED calculations for deoxyribose
17
Figure 3 The Boltzmann-averaged PVED for the minimum-energy D-deoxyribose C3endo ap orientation as a function of temperature.