Parity violation and chiral symmetry breaking of a racemic mixture

BioSystems, 20 (1987) 49--56 Elsevier Scientific Publishers Ireland Ltd.

49

PARITY VIOLATION AND C H I R A L SYMMETRY BREAKING OF A RACEMIC M I X T U R E

ROGER A. HEGSTROM

Department of Chemistry, Wake Forest University, Winston-Salem, North Carolina (U.S.A.) The chiral symmetry breaking of a racemic mixture by the parity violating weak interaction is considered. Particular attention is given to a mechanism recently proposed by Mason and Tranter whereby the weak neutral current interaction in chiral molecules leads to the differential absorption of unpolarized light by D vs. L enantiomers. After extending the usual theory of optical activity to include weak neutral currents, it is found that for spin-allowed transitions in typical organic molecules the weak photoabsorption asymmetry is much smaller than the value obtained using the reasoning of Mason and Tranter. Upon making a comparison with other mechanisms, it is concluded that differential radiolysis by beta electrons is likely to produce the largest symmetry breaking effect by the weak interaction.

Keywords: Origin of biomolecular chirality ; Weak interaction; Chiral symmetry breaking; Asymmetric absorption of light.

The chiral symmetry breaking of a racemic mixture, leading to an excess of L or D isomer, is very likely the first step of the process which led to the biomolecular homochirality presently observed in nature (L amino acids and D sugars). A possibility which has been considered since 1957 is whether the weak interaction, which violates parity and thereby universally distinguishes between right and left, could have broken the initial racemic symmetry (Vester et al., 1959). The interest in this possibility has been high because of the fundamental and universal nature of the handedness of the weak interaction and because of its potential capability of explaining the sign of the observed biomolecular handedness (e.g. the presence o f D rather than L sugars in nucleic acids) as well as its ~zistence (for a recent review, see Mason, 1984). Recent w o r k has shown that although the enantiomeric excesses that can be produced by the weak interaction alone are likely to be very small (~<10-9% for organic compounds), they can nevertheless under certain conditions be larger than values produced by statistical fluctuations (Hegstrom et aL, 1985). Coupled with molecular kinetic amplification mechanisms in

chemical systems that seem natural for primitive biochemical processes, the weak interaction may produce much larger enantiomeric excesses (Kondepudi and Nelson, 1985; Hegstrom, 1985). There are two types of weak interaction: a charged current interaction, which is responsible for radioactive beta decay and similar processes; and a weak neutral current (WNC) interaction, which is similar to the electromagnetic interaction except that it is much weaker and of much shorter range (for a review, see Abers and Lee, 1973). Both of these types of weak interaction violate parity. The WNC interaction between electrons and nuclei produces an energy difference between mirror image molecules (Rein, 1974), and hence gives different rate constants for mirror image kinetic chemical processes. This energy difference AE has been calculated to be of order 10 -3s J/molecule for mirror image organic c o m p o u n d s (Hegstrom et al., 1980; Mason and Tranter, 1985; Tranter, 1985}, with L amino acids having lower energy than the corresponding D isomers. The resulting relative difference or asymmetry Ak in the rate constants is calculated to be of order A E / k T 10 -17 near room temperature.

0303-2647/87/$03.50 © 1987 Elsevier Scientific Publishers Ireland Ltd. Published and Printed in Ire]and

~

50 The charged current interaction produces, in radioactive beta decay, beta electrons which are left-handed, that is, the electron spin is directed opposite to its momentum. These chiral electrons may then, via electromagnetic interactions, radiolyze (excite or ionize) mirror image molecules at different rates (Vester et al., 1959; see also Garay and Hrasko, 1975). Approximate calculations (Hegstrom, 1982) predict a relative difference or asymmetry A R in the rate of radiolysis of D vs. L isomers which is of order 10 -1~. Larger values of A k and A R are predicted for molecules containing heavy atoms; all numerical values given in this paper refer to typical organic molecules (whose atoms have atomic numbers Z ~ 6) unless otherwise indicated. Both the neutral and charged current processes can produce an enantiomeric excess (also called chiral polarization) in an initially racemic mixture. In the absence of amplification mechanisms, 77, defined as ~?

nL-

nD

-

(1)

nL + nD

where nL and nD are the number of L and D isomers present, attains a maximum value given by (Hegstrom, 1984; Hegstrom et al., 1985) eSAR

kAk

k + eS

k + eS

rl ~ - - +

(2)

where S is the strength of the radioactive beta source (number of beta particles per second), eS is the mean rate constant for primary beta radiolysis, and k is the mean rate constant of the process (e.g. racemization) affected by the WNC interaction. The first term in Eqn. (2) depends u p o n the temperature as well as upon the beta source strength and has been estimated to vary from less than 10 -19 for weak beta sources and high temperatures to a maximum value near 10 -~1 for strong beta sources and low temper-

atures. It has been suggested that values of which approach this maximum may well have been present in localized regions, against a background of smaller ~ values, in the prebiotic era (Hegstrom et al., 1985). The second term in Eqn. (2), due to the WNC interaction, has been calculated to be of order 10 -17 and rather insensitive to temperature. The small asymmetries produced b y both kinds of weak interaction can be greatly increased by amplification mechanisms as mentioned above. Recently it has been suggested (Mason, 1984; Mason and Tranter, 1985) that the chiral symmetry of a racemic mixture can be broken by another weak process: due to the WNC interaction in chiral molecules, mirror image substances absorb unpolarized light at different rates. Consequently, one of the enantiomers may react faster in photochemical reactions. Assuming a Z 6 dependence, Mason and Tranter (1985} estimate the size of the absorption asymmetry A~ for organic molecules by extrapolating the Kuhn anisotropy factor gw ~ 10 -7 observed (Sandars, 1980; Emmons et al., 1984) for magnetic dipole transitions in heavy atoms (Bi, Z = 83). They obtain the value A~ ~ gw ~ 10 -14. However, the correct Z-dependence of gw for this t y p e of transition is Z 3 (Bouchiat and Bouchiat, 1974), which, if the same reasoning is followed, gives a much larger value A 3 , ' - ' g w ~ 10 -11. This latter value is the same order of magnitude as that predicted for the radiolysis a s y m m e t r y A R and, if correct, would be the most likely candidate for a prebiotic chiral symmetry breaking process by weak interactions. In what follows, it is shown that a careful theoretical treatment of this weak t)hotoabsorption asymmetry leads to an estimated value of A~ which is much smaller than 10 -11 for typical organic molecules. For strongly allowed molecular electronic transitions, which are expected to dominate the photoabsorption from sunlight, for example, A~ is estimated to be of order 10 -16 or less. According to the quantum theory of the interaction between matter and electromag-

51 netic radiation (Condon, 1937), the extinction coefficient er, l for the absorption of right (r) or left (l) handed circularly polarized light near an electronic transition a-* b is proportionai to the relative transition probability I(bl ~ ( P i ~- k S i ) • Ar, l exp(ik • xi)la>I~4 v

(3)

Atomic units ( a . u . ) a r e used: h / 2 u = m e = lel = 1 , c = a -~, m e and e are the electron mass and charge, c is the speed of light, and = 1/137 is t:he fine structure constant. In Eqn. (3) a denotes the ground state, b the excited state, :Pi is the m o m e n t u m operator for electron /, xi is the spatial coordinate operator for i, Si is the spin operator for i, Ar, l = A ( I +- i ] ) / , j ~ -

(4)

is the magnetic vector potential associated with the electromagnetic radiation, k is the wave vector, which has a magnitude k = w / c = a¢o, and ¢o is the angular frequency. The unit vectors ~i, ], a n d [¢= k / k are fixed in the laboratory frame of reference. The symbol " A v " denotes an average over all molecular orientations. Denoting by X (= L or D) the handedness of the chiral raolecule, the mean absorption coefficient is defined as g(x) = &¢~(xl, + fief(x)

(5a)

To calculate A~, the exponential in Eqn. (3) is expanded in powers of k • x i ~ a¢o. For ordinary absorption ~¢o ~ 1 and terms in the expansion of order (a¢o)2 and higher may be neglected. This corresponds to retaining the usual electric dipole (El), magnetic dipole (M1) and electric quadrupole (E2) transition matrix elements and neglecting higher multipole terms. It can be shown that this approximation is valid for the purposes of this paper (see Appendix A). The result is that the mean extinction coefficient is given by e ( X ) = c o n s t [ S b a ( X ) -- 2P~ (co/¢Oba)Rba (X)]

(7) where Sba, the strength of the transition a -* b, is given by S b a ( X ) = Idba(X)12 + (co / a~ba)2 I m b a ( X ) l 2 + (3/40)(aw)2lQba(X)l 2

(8)

and where Rba, the rotational strength of the transition, is given by Rba(X) = Im[dab(X)

• mba(X)]

(9)

In Eqns. (7)--(9) Wba = E b -- Ea where Ea,b are the energies of the states a,b; and dba, tuba, Qba are, respectively, the electric dipole, magnetic dipole, and electric quadrupole matrix elements, with dab given by

or

d a b ( X ) = 7(X)

=

'

(10)

"

~[er(X) + el(X)] + "{P.1[er(X) with -

el(X)]

(5b) d = -/~xi

where fr, l are the fractions of r and 1 circularly polarized photons present in the electromagnetic radiation and where P v = fr - ft is the circular polarization (for unpolarized light fr = fl = 1/2, Pv = 0). The photoabsorption asymmetry Av is now defined as e ( L ) - }.'-(D) A'r - e(L) + i.:-(D)

(6)

(11)

and with similar definitions for mba and Qba. The effect of the WNC interaction H w = (G~/2~f2) ~ . Q w [ S i "

+ 63(xi -- x~)Si" Pi]

Pi53(xi - x~)

(12)

(see for example Hegstrom et al., 1980} is

52

now considered. Here G = 2.19 × 10 -14 a.u. is the Fermi weak coupling constant, Q~ ~ N " where N ~ is the neutron n u m b e r of nucleus a, and the summation is over all electrons i and nuclei a in the molecule. The effect of H w is to perturb the electronic states a, b:

R ~ a ( X ) = I m [deb(X) • m ~ ( X )

+ d~b(X) - m~a(X)]

(16b)

In these equations the E1 matrix elements are defined as deab(X) = ( a e ( X ) fdi be(X))

(17a)

(13)

la) = la e ) + [aw )

daW b (X) = (where " e " denotes electromagnetic " w " denotes weak), where

and

la w > = ~_~ (Ea - En)-~[n e) (n eIHwlae> (14a) n

-_ ~ l n

(14b)

e) e na w

n

It is known that ewa is of order G a Z 3 (Bouchiat and Bouchiat, 1974) so that for carbon, ewa ~ 10 -16 a.u. When the El, M1, and E2 matrix elements are evaluated with their WNC corrections and substituted into Eqns. (8) and (9) one finds

Sba(X) = S a(X) + sy

(x)

Rba(X) = R~a(X ) + R~a(X)

(15a) (15b)

with S~a(X) and R ~ a ( X ) given by Eqns. (8) and (9), respectively, but with the purely electromagnetic matrix elements d~a, m~a, and Q~a replacing the weakly perturbed ones dba , mba , and Qba. We call S~a and R~a the electromagnetic strength and electromagnetic rotational strength, respectively. The weak strength and weak rotational strength S~a and RbWa are given by S ~ a ( X ) = 2Re[deab(X) • d~'a(X) + (¢O/Wba): meb(X) " m~'a(X)

+ (3/40)(aw):Qaeb(X) • Q~a(X)] (16a)

+ ( a W ( X ) l d l be(X))

(17b)

with similar definitions for the M1 and E2 matrix elements. S i n c e d~b and ena w are parity-odd (change sign upon inversion of the coordinate system), and since m~b and Qaeb are parity-even (no sign change upon inversion), it follows that S~a and R~a are parity-even and R~a and S ~ are parity-odd. Consequently we may write Seba(D) = S~a(L ) =- S~a

(18a)

R~a(D ) = R ~ a ( L ) = R~a

(18b)

R~a(D ) = - R ~ a ( L ) - - R ~ a

(18c)

S~a(D ) = - S~a( L ) - - S ~ a

(18d)

Substitution of these values into Eqns. (15), (7), and (6) then leads to the following result for the photoabsorption asymmetry: S~a -- 2P~ ( w / W b a ) n ~ a A.y = S~a _ 2p.y(cO/Wba)R~ a

(19)

which is one of the main results of this paper. Note that if the weak interactions are neglected and if Pv = ±1, Eqn. (19) reduces to the usual expression for the absorption of circularly polarized light. Equation (19) also provides the criterion for the weak photoabsorption symmetry to dominate the ordinary electromagnetic absorption asymmetry, namely l P~ < -~(¢Oba/~ ) (Sba/R w eba )

(20)

53 From now on we will consider only the weak p h o t o a b s o r p t i o n asymmetry, A,~ = Sba/SbaW e

(21)

which is equal to the total photoabsorption asymmetry for unpolarized light, and estimate its order of magnitude. We initially consider only spin-allowed (AS = 0) transitions. First consider S~a. It follows from Eqn. (8), given the orders of magnitude of the transition matrix elements d ~ ~ 1, m ~ a ". o~, Q~,a "" 1, that SI~a is of order 1 for strongly allowed E1 transitions. For weakly allowed E1 transitions d b a ~ oL a n d S~,, is of order ~2. Next consider S~a and Av. It follows from Eqns (17b) and ( 1 4 b ) t h a t formally d ~ ~ G a Z 3, and similarly rnaWb " G a " Z 3 and Q Wb "-. G a Z 3, so that according to Eqn. (16a) S~a appears to be of order G a Z 3 for strongly allowed E1 transitions and of order G a 2 Z 3 for weakly allowed E1 transitions. It then follows from Eqn. (21) that, formally, A~ ~ G a Z 3 ~ 10 -~4 fo:r strongly allowed E1 transitions and A . r ". G Z 3--, 10 -12 for weakly allowed transitions. However, it is shown in Appendix A that these conclusions are valid only for molecules with orbitally degenerate states. For molecules with orbitally nondegenerate states, the values of S ~ and A~ given above are generally reduced by a factor of ( a Z ) 2. This is because the quantity inside the curly brackets in Eqn. (16a) is pure imaginary in the absence of corrections to the wave functions arising from spin-dependent terms in ~he molecular Hamiltonian, and consequently S ~ - - - 0 without such corrections. Inclusion of spin-orbit coupling corrections to the wave function then gives A.y "-" Go~3Z s ~ 1O'-'~ for strongly allowed E1 transitions and A ~ r ~ G a 2 Z s ~ 10 -~4 for weakly allowed transitions between singlet states. For weakly allowed transitions between non-singlet states A.~ ~ Go~Z s ~ 10 -12 (see Appendix A). The above treatment considers only spinallowed transitions. For spin-forbidden transitions (e.g. singlet-triplet) the main difference

is that the matrix elements d~b, M~b, and Q~b are all identically zero in the absence of spin-orbit coupling corrections to the wave function (Hameka, 1965) and consequently the absorption is extremely weak. When the spin-orbit corrections are included, d~b ~ (aZ) 2, S~a ~ ( a Z ) 4, and S~a ~ V o t 3 Z s so that the asymmetry can be relatively large, A~ ~ G Z / a ~ 10 -11. A few remarks are necessary at this point. First, none of the above numerical estimates of A7 take into account molecular electronic structure effects, which are expected to reduce the value of AT. A reduction by a factor of order 10-3--10 .4 can probably be expected for orbitally non-degenerate molecules; this estimate is based upon experience with molecular orbital calculations of the WNC energy difference between organic enantiomers (Hegstrom et al., 1980; Mason and Tranter, 1985; Tranter, 1985). Second, orbitally degenerate chiral molecules, for which relatively large values of A~ are possible, seem unlikely to have played any role in molecular evolution since organic molecules of this type are rare; on the other hand orbitally degenerate high Z chiral molecules or spin-forbidden transitions in orbitally non-degenerate ones are expected to be the best candidates for an experimental detection of A~. Finally, since strongly allowed transitions are expected to dominate the absorption from any source of light which emits a continuous distribution of wavelengths (e.g. sunlight), it is expected that, after an integration over wavelengths, these transitions will dominate the value of the photoabsorption asymmetry A7 for prebiotic chiral molecules. An exception would occur if only weakly allowed or spin-forbidden transitions were important in molecular evolution. Hence it is concluded that, for prebiotic organic molecules, the weak photoabsorption asymmetry A~ is expected to be less than 10 -~6. A more realistic estimate is probably A~ ~ 10-19--10 -2°. This value of A~ is smaller than the predicted rate constant asymmetry Ak ~ising from the WNC energy difference

54 between enantiomers and much smaller than the predicted rate constant asymmetry A R for beta radiolysis. Finally, it is possible to extend the theory in a similar manner in order to consider the effect of the WNC interaction u p o n the circular dichroism (CD) ratio A e / [ ( X ) = 2 e l ( X ) - er(X)

(22)

el(X) + er(X) This is done in Appendix B, where it is found that, contrary to the purely electromagnetic case, the CD values for L and D isomers are no longer equal in magnitude, as pointed o u t by Mason and Tranter (1985). The magnitudes are found to differ b y the amount 8(¢o/¢Oba)g~a, where g~a is the weak interaction contribution to the Kuhn anisotropy factor gba (X) = R ba (X)/Sba (X)

(23)

Hence, in comparing the present work with the work of Mason and Tranter (1985), it is seen that the weak correction g~a to the anisotropy factor affects the circular dichroism but, contrary to what was suggested by the above-mentioned authors, it does n o t affect the weak photoabsorption asymmetry. The latter is determined not by the weak contribution to the rotational strength R ba, b u t rather b y the weak contribution to the line strength Sba. Appendix A Consider the quantity ~ b " ¢1~ appearing on the right hand side of Eqn. (16a). It follows from Eqns. (17) and (14) that this can be written

bative corrections to the wave functions arising from spin-dependent terms in the molecular Hamiltonian have been neglected. Application of Wigner-Eckart theorem (see for example Edmonds, 1974), in the spin space, to the matrix elements appearing in Eqn. (A1) gives the equalities Sa = Sb = Sn, Ma = Mb = Mn for the spin q u a n t u m numbers of the respective electronic states a,b,n. It then follows from the time reversal symmetry and the spin dependence of the operators d and H w that dab and dbn are pure real numbers and e w is pure imaginary. Hence the quantity d~b • ¢1~ is a pure imaginary number. In the same way it follows that the remaining t w o terms on the right hand side of Eqn. (16a) are pure imaginary (in the case of spin degeneracy a summation over the degenerate sublevels is required) and hence that S~a = 0. Next, the effect of including spin-orbit coupling corrections as well as WNC corrections to the wave functions is considered. Equation (13) is replaced by

l a ) = la e> + la s ) + la w) with

a similar

w + dab e . elan d e ) • d~ n e na

for

Ib>, where

]a s ) = ~ ( E a - En) -1 In e) (nelt'lSlae)

(A3)

rl

= ~

Ine>eSna

(A4)

n

where/_/s is the spin-orbit coupling operator (see for example Hegstrom et al., 1980). It is k n o w n (Hegstrom et al., 1980} that e~a is roughly of order (aZ) 2 a.u. Equation (16a) n o w contains cross terms between the WNC interaction and the spin-orbit interaction, for example d~b. d~a contains the term

~ deb • d~a= ~ ( d e b

expression

(A2)

n

eSarn en bw d~ a . dmne

(A5)

m

r~

(A1) We assume initially that the electronic states are orbitally non-degenerate and that pertur-

for which e~m and eWb are pure imaginary, and cl~a and ¢l~nn are pure real. Hence the term (A5) is pure real and gives a non-vanishing contribution to S~'a which is of order (~Z) 2-

55

(GaZ s) for strongly allowed transitions and of order a(c~Z)2(GaZ 3) for weakly allowed transitions. In addition to the t y p e of term shown in (A5), the inclusion of H s also leads to terms in Eqn. (16a) of the type ~ n

eSmb e w e • d~n na dam

(A6)

where

gL(x) = R

(B4)

(x)/sL(x)

and

S o(X) g~a(X) - S~a(X)

g~a(X) S~a(X)

(B5)

m

which are formally of order (aZ)2(GaZ 3) for both strongly and weakly allowed transitions. In the case of transitions between singlet states, however, these terms vanish since, because of the spin selection rules for the electric dipole matrix elements, the spins are zero for all states in the summations in (A6) and consequently eSb = eiiaW = O. Finally, exm~ination of the terms higher than E l , E2, and M1 in the multipole expansion o f Eqn. (3) reveals that these terms give contributions to Av which are at most of order Gc~3Z3, and hence negligible.

Appendix B The effect of the WNC interaction on circular dichroism and other optical properties of chiral molecules can be f o u n d in a manner similar to that shown for the photoabsorption asymmetry. If follows from Eqn. (22) and from the theoretical expression for the extinction coefficients

el, r(X) = const, [Sba(X) +- 2 (¢o/¢Oba)Rba(X)]

(m) that the CD ratio is given b y A e/~'(X) = 4(~/~ba)gba (X)

(B2)

where gba(X) is defined by Eqn. (23). Evaluating the rotational strength Rba as a sum of electromagnetic and weak contributions as was done for the line strength Sba then gives the result

gba(X) = g L ( Z ) + g~a(X)

(B3)

The parities are

geba(D ) = --geba(L ) =- _ geba

(B6)

g~a(D) = g~a(L) - g~a

(B7)

and hence, from Eqn. (B2) it follows that

A e / e ( L ) = 4(~/COba ) (geba + g~a)

(B8)

Ae/e(D) = 4(c~/Wba) (--g~,a + g~a)

(B9)

These results also apply to atoms and achiral molecules, for which ~ a = 0. By a procedure analogous to the one described for A~, the order of magnitude of g~ba can be estimated. For the case of orbitally non-degenerate molecules, gUbb a is found to be of order Gc~4Z s for strongly allowed E1 transitions, of order Ga2Z s for weakly allowed transitions between singlet states, and of order Gc~Z s for weakly allowed transitions between non-singlet states. For the case of orbitally degenerate molecules, gUbb a may be as large as G Z 3 for weakly allowed transitions (this is the case for atomic Bi, where g~ba ~ 10-7 )" References Abers, E.S. and Lee, B.W., 1973, Phys. Reports (Phys. Lett.) 9C, 1--141. Bouchiat, M.A. and Bouchiat, C., 1974, J. Phys. (Paris) 35, 899--927. Condon, E.U., 1937, Rev. Mod. Phys. 9, 432--457. Edmonds, A.R., 1974, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton), p. 75. Emmons, T.P., Reeves, J.M. and Fortson, E.N., 1984, Phys. Rev. Lett. 52, 86. Hameka, H.F., 1965, Advanced Quantum Chemistry (Addison-Wesley, Reading, Massachusetts), Chapter 10.

56

Garay, A.S. and Hrasko, P.J., 1975, J. Mol. Evol. 6, 77--~9. Hegstrom, R.A., 1985, Nature 316, 749--750. Hegstrom, R.A., 1984, Origins Life 14, 405--411. Hegstrom, R.A., 1982, Nature 297,643--647. Hegstrom, R.A., Rein, D.W. and Sandars, P.G,H., 1980, J. Chem. Phys. 73, 2329--2341. Hegstrom, R.A., Rich, A. and Van House, J., 1985, Nature 313, 391--392. Kondepudi, D.K. and Nelson, G.W., 1985, Nature 314, 438--441.

Mason, S.F., 1984, Nature 311, 19--23. Mason, S.F. and Tranter, G.E., 1985, Proc. R. Soc. London A 397, 45--65. Rein, D.W., 1974, J. Mol. Evol. 4, 15--22. Sandars, P.G.H., 1980, in: Fundamental Interactions

and the Structure of Matter, Vol. 57, K. Crowe, J. Duclos, G. Fiorentini and G. Torelli (eds.), (Plenum, New York). Tranter, G.E., 1985, Mol. Phys. 56, 825--838. Vester, F., Ulbricht, T.L.V. and Krauss, H., 1959, Naturwissenschaften 46, 68--69.