Progress in Biological Chirality G. Palyi, C. Zucchi and L. Caglioti (Editors) © 2004 Elsevier Ltd. All rights reserved.

Chapter 32

Origin of Biological Chirality

Tetsuyuki Yukawa Coordination Center for Research and Education, The Graduate University for Advanced Studies, Hayama, Miura, Kanagawa 240-0193, JAPAN yukawa@soken. ac.jp

1. Introduction Asymmetries appear at various stages in the biological evolution. Among them the most

fundamental asymmetry is the molecular asymmetry known as the chirality in proteins, RNA's, and DNA's. Proteins of biological origin are known to be formed by left-handed L-amino acids, except glycine which is chirality neutral, while sugars in RNA's and DNA's are selectively right-handed D-ribose. At present it is a complete mystery when, where, and how this asymmetry arose, but I believe it can be understood scientifically as a result of the natural law. Under the circumstance that we do not yet have any reliable scenario of the origin of life, we inevitably give some plausible assumptions at various stages of evolution, especially to the initial conditions. We assume that life began on the Earth within its 4.6 billion years history, and at an early stage of the history when the surface temperature was still very high, chemical evolution started to synthesize amino acids.

Origin of the biological chiral asymmetry has been searched for either in natural environments or in the elementary process. I prefer to choose the cause of asymmetry in the elementary process, which I mean the parity violation of weak neutral current, mainly because of its universality. There have been many arguments for and against the parity violation theory as a possible explanation of the asymmetry. The energy difference between parity doublets due to the parity violating interaction (PVED) has been calculated [1], which has given us some hope to explain the preference of L-amino acid and D-ribose because of their energetical stability, although energy differences have been so tiny. It has then been claimed that the self-catalyzing process would enhance small difference of the initial population exponentially large [2, 3]. However, we will see in the following that the parity violation alone is not enough for producing the asymmetry and existence of the self-catalysts does not mean the homo-chirality.

Another problem, which has not been taken in consideration so seriously in the asymmetry production, is the second law of thermodynamics, i.e. the entropy increase. Although the self-catalytic process enhances the asymmetry, the racemization process tends to draw the system back to the state of maximum entropy, namely the thermal equilibrium. It looks true that in living organisms the entropy decreases by expelling high entropy wastes out of the system,

398 Progress in Biological Chirality

and we tend to ignore the entropy increase in biological processes. However, if we are dealing with the chemical evolution prior to the biological evolution, we have to respect the second law of thermodynamics. I would like to give a resolution to this problem by employing the scenario which the particle physics has developed in explaining particle dominance of the universe.

In the followings I shall firstly discuss the problem of the parity violation scenario in the origin of chiral-asymmetry, and the resolution on the bases of quantum mechanical description of the racemization process. It is then extended to a system in the thermal environment. Secondly, I shall introduce three necessary conditions for the chiral asymmetry of bio-molecules on the Earth, by rewriting the conditions which have been given originally by Sakharov [4] in order to explain particle asymmetry of the universe. Finally, I shall present a possible scenario of the homo-chirality of proteins.

2. Parity Violation Revisited

2.1 Energy difference due to the parity violating force Let us review here the quantum mechanical energies of an amino acid for states with

different chirality due to the symmetry violating force. We write {| 1 >, | d >} being chiral pair states related through the parity operation, P, as

\l>P\d>, \d>P\l>, (2.1)

where P^ = 1. A Hamiltonian can be split into two parts, H = Hi + H2 , where Hi commutes and H2 anti-commutes with P, respectively;

[ P , H i ] = 0, { P , H 2 } = 0 . (2.2)

The second term, H2 , is responsible to the symmetry breaking. They can be written explicitly as,

H,^-{H^PHP), H^=-iH-PHP) .

Then, the energy expectation values for chiral pair states are given by

e, = < 11 Hi 11 > + < 11 H211 > = ei + e2, ed = < d | H i | d > + < d | H 2 | d > = < l | PHiP 11 > + < 11 PH2P 11 > (2.3) = < l | H , | l > - < l | H 2 | l > = e i - e 2 .

The energy differences between chiral pairs have been calculated for various amino acids [1], which turned out to be about

Ae=ei-ed~-10-^^eV (2.4)

with the correct sign in favour of L-amino acids energetically, but much too small to expect

Origin of Biological Chirality 399

any physical relevance. One may hope that the amplifying mechanism of self-catalytic process will save this

difficulty through exponential growth of the population difference [2, 3], AN(t)-[N,(0)-Nd(0)]e^^ . (2.5)

However, this process creates both types of chiral molecules exponentially, and as time goes by the larger component, Ni (t), stops growing, because of the shortage of L-amino acids. If both types of amino acids are initially created equally, Nd (t) will catch up eventually. Even though the initial population of L-amino acids happen to be larger due to certain natural environments, the racemization process will take over to diminish the asymmetry and bring the system into the equilibrium distribution. In fact the energy is only meaningful in the equilibrium to give the Boltzmann factors:

N,/N,^e ^2p&e

This reminds us that the direction to homo-chirality is opposite to the thermal equilibrium.

2.2 Quantum motion of an amino acid From the considerations in the last sub-section we find what we should study is not the

equilibrium state, but the non-equilibrium process where the asymmetry increases against the racemization. It requires us to describe the system dynamically. To begin with let us consider the motion of an amino acid quantum mechanically. The chiral pair states make the racemization transition by tunnelling, and thus a general state of an amino acid is a linear superposition of the chiral pair states;

|^(0>c,(0|/>+c,(0l^> (2.6)

Motion of the wave fiinction is governed by the Schrodinger equation (with the convention

where

.dc ,^ / — - h e

dt

c,{t) h =

K Kdj

(2.7)

(2.8)

The Hamiltonian matrix elements are written as

hii = ei + e2 , hid = vi + iv2 , hdi == vi - iv2 , hdd = ei - e2 , (2.9)

with defining vi = < 1 | Hi | d > and iv2 = < 1 | H2 | d > . The off-diagonal components are responsible to the racemization.

The equation can be solved analytically, where the explicit form of the solution is given in Appendix 1. In order to relate a solution to the observation process the density matrix is often introduced by

P^it) = c^{t)c,*{t) , (2.10)

A solution of the Schrodinger equation can be expressed by p(t)^K{t)p{())K*{t) (2.11)

400 Progress in Biological Chirality

where K (t) is the unitary evolution matrix whose components are also found in Appendix 1. If we give the initial distribution symmetric, i.e. equal amount of L- and D-amino acids, by

P(0)-, ifi o^

,0 1, (2.12)

the density matrix does not change at anytime, because of unitary nature of the time evolution matrix: K (t) K*(t) = 1. This shows that the parity violating interaction does not induce asymmetry as far as the system starts symmetrically, and evolves unitarily, i.e. conserving the total probability.

2.3 Source of the asymmetry We shall look for the origin of asymmetry in the wave function of a peptide. A peptide is a

linear molecule of n-amino acids {ai, a2, .., an}, formed by the dehydration. In general the amino acid ai of the i-th position can have either the L- or the D- chirality except glycine. We shall write the peptide wave function whose amino acid at the i-th position happen to be an L-amino acid li by,

|ai*a2*....l,*....an*> | L > . (2.13)

Here we put * to each amino acid to indicate that it is the residue after dehydration. We also write its chiral conjugate by | D > = Pi| L > , or more explicitly by

| D > = |ai*a2*....di*....a„*>. (2.14)

Now we assume that the overlap integral of states | L > and | D > does not vanish;

^ , = < / ) | L > - < L | / ^ | L > . (2.15)

There are at least two reasons to support this assumption: the wave ftinction of an amino residue in a peptide is extended, and it is not necessarily the parity eigen-state.

When the overlap integral does not vanish there should be a slight modification to the equation of the motion. Writing the overlap matrix as

n = \ , (2.16) {e \)

where we keep eyes only on the racemization of the i-th residue, and have suppressed the i-dependence of the overlap integral. In general racemization takes place at any position at the same time. The Schrodinger equation now becomes

;4 [«" ' c (0 ] = //[/7"^c(0], (2.17) at

where the Hamiltonian matrix also suffers a modification; H = n "̂^̂ h n •̂ ^̂ . This equation can also be solved analytically (see Appendix 1), and the solution in the density matrix form is given by

p(t) - n'''^K{t)Yn'''^p{^)n''-\K * {ty'"^ . (2.18) It is clear that even if we choose the initial density matrix symmetric, it evolves in time.

When we define the asymmetry function by A(t) - p^^it) - p^jit), we have

Origin of Biological Chirality 401

A(t) = -eM- -^(sm2dtsm7] + 2-^sin dtcosrf) , d d

(2.19)

where we have defined d ^ = e2̂ + vi^ + W , and cosrj = vJ{vl-\-vl)'''^. It has the non-vanishing time average:

A=-sA\~ -y cos 77 (2.20)

It shows that the asymmetry remains finite at least when 8e2COS77 does not vanish. It is interesting to notice that the asymmetry is proportional to the parity violating energy difference e2, and thus the L-amino acid is favoured if the overlap integral ^cos;; is positive.

Since the transition is considered to take place in the high temperature environment of the early Earth, we need to make sure that the asymmetry will not be washed out by the heat noise. In Appendix 2 we give the quantum statistical equation under the Gaussian white noise for the case where COST; = 1 (v2=0). The asymmetry fimction averaged over the ensemble is obtained as

- . 2 ^

< A{t) >= -s V, V,'-^X

{\-e ^(cosvj/-—sinvj/)} (2.21)

with X being the coupling strength to the heat bath. It shows that the asymmetric function converges to an asymptotic value.

<.4(0>- -^-s-V, v.'+je

(2.22)

I should notice that the asymptotic form (2.22) is valid only in the case 62 is negligible comparing to A as is discussed in Appendix 2. In Figure 1 I show <A(t)> assuming

Figure 1. Time behaviour of the asymmetric fimction.

3. Scenario of the Biological Asymmetry

3. J Cosmological scenario of the particle asymmetry In the last section we find that the parity violating force together with the asymmetry of

wave functions creates non-vanishing chiral asymmetry of peptides. However, the asymmetry is very small as we can expect fi-om eq.(2.22) and eventually the distribution will reach to the equilibrium. We need something very special to happen so that the asymmetry becomes more significant, while the system tends to the maximum entropy state.

It is a common belief of particle physicists such an extraordinary event has happened at the

402 Progress in Biological Chirality

period of creation of the universe. As far as we can observe there are no sign of stars made of anti-matter, although the standard particle theory predicts creation of the equal amounts of particles and anti-particles at and just after the big bang. It was in 1967 Sakharov [4] proposed three conditions necessary to create the cosmological asymmetry; namely, (i) baryon number (here we mean net number of protons and neutrons) should not be conserved. It looks trivial since we find no anti-particle, which is assigned negative baryon number, in nature. The second condition is (ii) the asymmetry of transitions between particle and anti-particle. If they are the same, two transitions will compensate each other and result no net creation of asymmetry. Thirdly, (iii) the universe should be in the non-equilibrium state. Otherwise, the asymmetry is of the same order as the symmetry breaking energy which is negligible. It has been neglected for some time till the C (the charge conjugation) and the CP breaking interaction has been discovered. Together with the baryon number breaking of the grand unified theory (GUT) and the non-equilibrium state of expanding universe, there are possible scenarios for the cosmological asymmetry [5].

Let us illustrate the cosmological scenario briefly leaving the details to literatures which depend much on several uncertain facts in the evolution of the universe. About 13.7 billion years ago just after the Big Bang the universe was filled with high energy photons within a compact space. During expansion of the space photons with energy high enough to create baryon anti-baryon pairs were in the equilibrium with matter and anti-matter. As the universe continued to expand adiabatically, the temperature dropped down, and when high energy photons disappeared, the equilibrium condition was not maintained any more, and the matter and anti-matter distributions have decoupled and temporally fi^ozen. After this period, whenever a baryon met its partner anti-baryon, they annihilate each other to photons, which were not active enough to revive to matter and anti-mater anymore, and became to so-called relic photons in the cosmic background radiation. If there exists the baryon number asymmetry, there should be excess matter which was leftover of the annihilation process. They form stars in this universe.

3.2 Biological scenario for the chiral asymmetry By modifying the cosmological scenario of the matter asymmetric universe we will try to

fit for the asymmetry of bio-molecules. The three necessary conditions seem to be readily ftilfilled: (i) the racemization allows the change of numbers of L- and D-molecules; (ii) the parity violation with overlapping wave fiinctions creates asymmetric transition between L-and D-chirality states, while (iii) the thermal history of the Earth satisfies the non-equilibrium condition. Let us illustrate a possible scenario of the biological asymmetry under these conditions.

At the beginning of the Earth it was covered by the magma ocean. During cooling down by radiation emission the chemical evolution synthesize amino acids either in the super critical atmosphere. The creation occurred through the dominantly parity conserving electro-magnetic force, and the products of L- and D-amino acids are almost the same amount (Figure 2(a)). As temperature went down the rain belt covering the hot atmosphere lowered, and finally reached to Earth's surface. Then the concentration of amino acids got high, and they started interacting each other. At high temperature both polymerization and racemization proceeded quickly, and all the possible racemic mixtures were almost in the equilibrium distribution. As the temperature continued to get lower the racemic transition became very slow because it occurred through tunnelling process, and the distribution was temporary fi*ozen (Figure 2(b)).

Origin of Biological Chirality 403

tzm

Figure 2. Evolution of L- and D-amino residues in pure L-peptides (L), pure D-peptides (D) and racemic compounds (R) at three stages: (a) the creation of amino acids, (b) the frozen distribution, and (c) the LD

araiihilation and excess L

The number distribution of peptides with 1 L-amino residues and d D-amino residues in the equilibrium is given by

A^(/,^)c (l + d)\ l\d\

df-d (3.1)

Here K is the rate of polymerization and Q is the suppression factor relating to the asymmetry parameter given by eq. (2.22) through

c-1 - < yl(QO) >

1+ < ^(oo) > (3.2)

which is obtained by requiring that the numbers of excess amino residues is equal to the asymmetry parameter times total amino residues.

It is a common belief that racemates, which I mean a polymer with any mixture of L- and D-amino residues, are inactive in polymerization [6], and only pure L- or D-peptides will enter to play important roles in the biological evolution. In this way the dominant part of entropy is exhausted by racemates. During peptides evolution whenever a L-peptide and a D-peptide ftised to a racemic compound it became inactive and disappear from the scene of the biological evolution. We make a simplified scenario that an L-peptide of the length n, L(n), and a D-peptide D(n) makes the chemical evolution,

L(n) + L(m) «-> L(n+m) D(n) + D(m) <-> D(n+m) L(n) + D(m) ~> R(n+m) ,

where R(n+m) represents a racemic compound with n+m amino residues. A simple numerical simulation shows the tendency to the mono-chirality as we expect (Figure 3). The excess L-peptides are of course due to the asymmetry of racemic transition. This concludes the scenario of the origin of biological asymmetry (Figure 2(c)).

404 Progress in Biological Chiralit\

time

Figure 3. Numbers of L(D)-amino residues in L(D)-peptides

4. Discussion We have shown a possibility of explaining the biological asymmetry of chiral-molecules

by making use of the cosmological scenario of the matter asymmetric universe. One of the

important points which have to be clarified is the appropriateness of parameters to be used for the theory to apply as the scenario on the Earth. The overlap integral 6-of eq.(2.15) is one of them. Another important problem is to make sure the role played by racemic compounds as the entropy carrier. We emphasize that these quantities and properties are basically approachable either numerically or experimentally.

5. Acknowledgement The author thanks Dr. Mitsuzawa for giving him many useful information and comments.

He also thanks to the organizers of the international meeting on the Biological Chirality for giving him a chance to present his work.

6. References [1] (a) D. Rein, ATOMKIKozl. Suppl. 16 (1974) 185-193. (b) S.F. Mason, Nature (London) 311 (1984) 19-23.

(c) S.F. Mason and G.E. Tranter, Mot. Phys. 53 (1984) 1091-1111. (d) V.I. Goldanskii and V.V. Kuz'min, Sov. Phys. UspeJchi 32 (1989) 1. (e) W. A. Bonner, Orig. Life Evol. Biosphere 21 (1991) 59-111. (f) J.L. Bada, Nature 374 (1995) 594-595. (g) A. MacDermott, Orig. Life Evol. Biosphere 25 (1995) 191-199. (h) P. Lazzeretti and R. Zanasi, Chem. Phys. Lett 279 (1997) 349-354. (i) O. Kikuchi, / Mot. Struct. (Theochem) 589-590 (2002) 183-193.^

[2] F.C. Frank, Biochem. Biophys. Acta 11 (1953) 459-463. [3] (a) D.K. Kondepudi and G.W. Nelson, Nature 314 (1985) 438-441. (b) A.J. Salam,A/o/. Evol. 33 (1991)

105-113. [4] A.D. Sakharov, Zh. Eksp. Teor. Fiz.Pis'ma 5 (1%7) 32. [5J (a) Ya. Zeldovich, Zh. Eksp Teor. Phys. 67 (1974) 2357. (b) M. Yoshimura, Phys. Rev. Lett. 41 (1978)

281-284. (c) S. Dimopoulos and L. Susskind, Phys. Rev. D18 (1978) 4500-4509; (d) idem, Phys. Lett. 81B (1979) 416. (e) D. Toussaint, S.B. Treiman, F. Wilcek and A. Zee, Phys. Rev. D19 (1979) 1036-1045. (f) S. Weinberg, Phys. Rev. Lett. 42 (1979) 850-853. (g) J. Ellis. M.K. Gaillard and D.V. Nanopoulos, Phys. Lett. 80B (1979) 360.

[6] J. Sarfati. TJ 12 (1998) 281-284.