The Chiral Puzzle of Life
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THE CHIRAL PUZZLE OF LIFE
Noemie Globus and Roger D. Blandford Center for Cosmology & Particle Physics, New-York University, New-York, NY10003, USA. E-mail: [email protected] Center for Computational Astrophysics, Flatiron Institute, Simons Foundation, New-York, NY10003, USA Kavli Institute for Particle Astrophysics & Cosmology, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected]
ABSTRACTBiological molecules chose one of two structurally, chiral systems which are related by reflection in amirror. It is proposed that this choice was made, causally, by cosmic rays, which are known to havea large role in mutagenesis. It is shown that magnetically-polarized cosmic rays, that dominate atground level today, can impose a small, but persistent, chiral bias in the rate at which they inducestructural changes in simple, chiral monomers that are the building blocks of biopolymers. A muchlarger effect should be present with helical biopolymers, in particular, those that may have been theprogenitors of RNA and DNA. It is shown that the interaction can be both electrostatic, just involvingthe molecular electric field, and electromagnetic, also involving a magnetic field. It is argued thatthis bias can lead to the emergence of a single, chiral life form over an evolutionary timescale. If thismechanism dominates, then the handedness of living systems should be universal. Experiments areproposed to assess the efficacy of this process.INTRODUCTIONLiving organisms comprise a system of molecules orga-nized with specific handedness. Handedness - or chiral-ity - is, following Kelvin’s original definition, the geomet-ric property of an object that cannot be superimposedon its mirror image (Lord Kelvin 1894). In chemistry,mirror images of the same chiral molecule are calledenantiomers . Both share the same chemical charac-teristics.The ribonucleic and deoxyribonucleic acids (RNA andDNA), responsible for the replication and storage of ge-netic information, are made up of linear sequences ofbuilding blocks with the same handedness, called nu-cleotides, whose arrangement is neither periodic norrandom and contains the genetic information neededto sustain life (Schr¨odinger 1944; Shannon 1948; Wat-son & Crick 1953; Shinitzky et al. 2007). The chiralityof the nucleotides confers helical structure on nucleicacids. Nucleic acids are very large molecules and thetorsional angles between the chiral units vary system-atically, as exhibited by the Ramachandran plot (Keat-ing et al. 2011) which demonstrates that even a flex-ible biopolymer retains chirality. As RNA and DNAare made of D-sugars (right-handed, by human conven-tion), the more stable conformation is a right-handedhelix (see Figure 1). The homochirality of the sugars from the Greek (cid:15) χθρ ´o ς , ”enemy” or ”opposite”. has important consequences for the stability of the he-lix, and hence, on the fidelity or error control of thegenetic code. All the twenty encoded amino acids areleft-handed (again by human convention). Sometimes,both enantiomers of the same molecule are used by liv-ing organisms, but not in the same quantity and theyperform different tasks.While DNA/RNA-based life, as observed so far, hasclearly chosen one functional chirality, which we call“live”, the alternative choice, which we call “evil”,could have developed along a separate, synchronizedpath making similar evolutionary choices in response tochanges in common environments, except for very smalleffects which are the main topic of this Letter. However,a precise equilibrium between the two choices seemsquite unlikely given the high replication rate. Thereis a small entropic price, but this is surely paid by thegreater facility of storing information and the higher re-liability of the replication (Schr¨odinger 1944).For DNA today, radiation increases the frequency ofgene mutations; this has been known since the pioneer-ing work of (Muller 1927) that showed that the mutationrate is proportional to the radiation dose, much of it at-tributable to ionization by cosmic rays. The muon com-ponent dominates the flux of particles on the ground atenergies above 100 MeV, contributing 85% of the ra-diation dose from cosmic rays (Atri & Melott 2011).Muons have an energy sufficient to penetrate consid-erable depths, and they are, on average, spin-polarized. a r X i v : . [ q - b i o . O T ] M a y base PO D-riboseL-ribosePO base D-unit
D-RNA (live system)
L-RNA (evil system)
L-unit Right-handed helix (side view)
Left-handed helix (side view) base PD-ribose
C1'C2' C3'C4' C5' x x x Q Q Q monomer (D-unit) z r r r r z (top view) basessugar-phosphate backbone (top view) Figure 1 . The figure shows the 3D structure of the RNA molecule and its mirror image. The direction of the helical conformationof the nucleic acids derives from the underlying chemical chirality of the sugar backbone. The nucleic acids contain only right-handedsugars (D-ribose in RNA, D-deoxyribose in DNA), shown in the right-hand side of the figure. They naturally assume a right-handedhelical conformation. In the mirror world (left-hand side in the figure), the nucleic acids would contain only left-handed sugars (L-riboseor L-deoxyribose) and would assume a predominantly left-handed helical conformation.
Ionization by spin-polarized radiation could be enantios-elective (Zel’dovich et al. 1977). Therefore, we arguethat the mutation rate of live and evil organisms wouldbe different. As there could be billions or even trillionsof generation of the earliest and simplest life forms, asmall difference in the mutation rate could easily sus-tain one of the two early, chiral choices.When Pasteur discovered biological homochirality, herecognized it as a consequence of some asymmetry inthe laws of nature: ”If the foundations of life are dis-symmetric, then because of dissymmetric cosmic forcesoperating at their origin; this, I think, is one of the linksbetween the life on this earth and the cosmos, that is thetotality of forces in the universe” (Pasteur 1848; Quack1989). Had Pasteur been alive a century later, the dis-covery of parity violation in the weak interaction (Lee& Yang 1956; Wu et al. 1957) would have strengthenedhis view. An object exhibits physical chirality when itsmirror image does not exist in Nature, as a consequenceof parity violation in the weak interaction. The resultof applying the parity operation on an elementary weakprocess, such as the decay, π + → µ + + ν µ , is not foundin Nature because neutrinos are chiral particles. In thelanguage of quantum mechanics, the basic Hamiltonianof a chiral molecule does not commute with the parityoperator and, if we include weak neutral currents, therewill be a parity-violating energy difference (PVED) be-tween the two enantiomers (Yamagata 1966). Howeverit is extremely small, ∼ − kT in water (Salam 1991)and larger consequences of chirality must be sought.While the effectivity of PVED in generating biologicalhomochirality is still under debate, some authors haveattempted to work with this small PVED and showedthat it may theoretically suffice to bring strong chiral se-lectivity (Kondepudi & Nelson 1984). An enantiomericexcess due to neutral weak currents has been reported incrystalline materials (Szab´o-Nagy & Keszthelyi 1999).In a beautiful paper, Pierre Curie addressed the ques-tion of chirality transfer from light to molecules, specif-ically involving circular polarisation (Curie 1894). Thesense of the rotation reflects the underlying chirality ofthe molecules, though the relationship is not simple anddepends upon the wavelength of the light (Optical Ro-tatory Dispersion). This rotation can be accompaniedby a difference in the absorption (Circular Dichroism),consistent with the Kramers-Kronig relations (Kramers1927; Kronig 1926). On this basis, it has been suggestedthat a specific source of circularly polarized light (CPL)might favor one set of enantiomers over the other (Baileyet al. 1998).Laboratory experiments have demonstrated that itis possible to induce an enantiomeric excess of aminoacids by irradiation of interstellar ice analogs withUV CPL (de Marcellus et al. 2011). However, this raises two problems. Firstly, Circular Dichroism is alsowavelength-, pH- and molecule-specific (D’hendecourtet al. 2019). It is hard to see how one sense of circularpolarization can enforce a consistent chiral bias, giventhe large range of environments in which the moleculesare found. Secondly, it is often supposed that astro-nomical sources supply the polarization. However, opti-cal polarimetry within the Galaxy reveals no consistentsense of circular polarisation and the observed degrees ofpolarization in the UV are generally quite small (Bailey2001).If we seek a universal, chiral light source, that con-sistently emits one polarization over another, then weare again drawn to the weak interaction in order to ac-count for a universal asymmetry. One option is to invokespin-polarized particles, which can radiate one senseof circular polarization through ˇCerenkov radiation orbremsstrahlung and can preferentially photolyze chiralmolecules of one handedness (Vester et al. 1959; Lahoti& Takwale 1977; Gusev & Guseva 2019). Another op-tion is to invoke supernova neutrinos (Boyd et al. 2018).However, the small chiral bias is unlikely to lead to a ho-mochiral state and some pre-biotic amplification mecha-nism is still required. This suggests considering, instead,enantioselective bias in the evolution of the two livingsystems.MOLECULAR CHIRALITY OF BIOMOLECULESConsider, first, a model of a small chiral molecule,part of a larger helical polymer (see Fig.1), which weidealize as an unequal tripod. There is a vertex or “tar-get” at the origin and three distinguishable atoms orgroups with position vectors x , x , x . There is a clas-sical electrostatic field associated with the point chargesat these four sites. It is helpful to introduce a pseu-doscalar (changes sign under reflection) “molecular chi-rality”, M , which has to change sign upon reflection anda clear choice is M tripod = ˆ x × ˆ x · ˆ x .A second simple, semi-classical model has electricalcharge and current confined to the surface of a spheresurrounding a central, nuclear charge which is canceledby the net charge on the sphere. The current allows foran electromagnetic chirality, with the simplest expres-sion M em = ˆ d · ˆ m .These two models are appropriate forsmall molecules or monomers that are the constituentsof naturally helical biopolymers.Our third, simple model involves a cylindrical, electro-static potential, like a “Barber pole”, (cf. Wagner et al.1997), Φ = R ( r ) + R ( r ) cos ( k z − m φ ) with k >
0. Inthis case, the molecular chirality M can be chosen as M helix = m = + (− ) for a live (evil) molecule. Thesedefinitions of M are illustrated in Fig. 1. COSMIC-RAY LODACITYCharged, cosmic ray protons, with energies just abovethe threshold for pion production, collide with nitrogenand oxygen nuclei in the upper atmosphere to create π + , π − (Gaisser 2012). The π + decay within a few me-ters into µ + with half life ∼ µ s, which decay, in turninto e + . As pions are spinless and the decays are weak,the µ + and e + , spin directions ˆ s µ, e , are preferentiallyanti-aligned with their direction of motion, ˆ v in orderto balance the antiparallel spins of the accompanyingneutrinos (Fig. 2). The associated magnetic dipole mo-ments are given by µ µ, e = e (cid:126) ˆ s µ, e / m µ, e . The π − decayinto µ − , e − with preferentially aligned spins but withmagnetic moments also anti-aligned with ˆ v (see Fig. 2and appendix A).We introduce a pseudoscalar quantity, “lodacity” (af-ter lodestone) to express the physical chirality of thecosmic rays. This is defined by L i ( T ) = ˆ µ · ˆ v , (1)where we average over all cosmic rays of type i and ki-netic energy T . Well above threshold, L ∼ − µ , and L ∼ − . e , in the pion restframe (see supplementary materials). L will be furtherdegraded as the cosmic rays lose energy through scat-tering electrons with L ∝ v roughly. In addition the sec-ondary electrons will be mostly unpolarized and furtherdiminish the lodacity of the cosmic rays that irradiatethe molecules.At sea-level today, most cosmic rays are muons withan average flux ∼
160 m − s − (Lipari 1993). However,the flux and the atmosphere could have been quite dif-ferent; the young sun and its wind are likely to havebeen much more active. The protobiological site, whichwe call the “fount”, may have been below rock, water orice which can change the shower properties and lodacity.ENANTIOSELECTIVE INTERACTIONWe now turn to the interaction that couples the cos-mic ray lodacity L to the molecular chirality M . Weseek an effect that is proportional to the product LM which will distinguish live and evil molecules exposed tothe same cosmic ray flux and will be unchanged uponreflection - a chiral bias. This effect must be translatedinto a difference in the ultimate mutation rate, a path-way that is poorly understood even in contemporary bi-ology. A high energy particle can excite an electronlocally (Rosenfeld 1928). Typically, the de-excitation isfast and radiationless and involves vibrational and ro-tational modes. This “internal conversion” can there-fore cause local structural change in the molecule. Cos-mic radiation also induce ionization which introduceschanges in the electronic structure of the biomolecules and can lead to mutations. DNA, today, is presum-ably a far less error-prone copier than the first geneticbiopolymer. Repair may also have been a factor whenlife began.The cosmic rays themselves are supposed here tobe spatially homogeneous and isotropically distributedwith respect to the molecules. Their cosmic ray-averaged magnetic moment is also presumed to bestrictly antiparallel to v although scattering processes,or an external magnetic field, can introduce an anglebetween v and µ . This is important because the chi-ral part of the electrostatic interaction involves a forcegiven by v · µ × ∇ E which vanishes unless the velocity isperturbed. Details of the calculation of the chiral bias δ ln P are presented in appendix B.We start with the unequal tripod (as illustrated byFig. B1 in the Supplements). Consider a cosmic raywith charge qe mass Mm e , subrelativistic velocity v and impact parameter vector with respect to the tar-get at the origin given by b . The trajectory will belinearly perturbed by the Coulomb force due to the thecharge Q e at x . This will cause a velocity pertur-bation δ v , which creates a second order chiral forcein combination with the electric field from the secondatom. This produces a displacement at the target andthe gradient of the displacement is equivalent to a chiralchange in the particle flux. However, this change van-ishes after we average over ˆ v . This is expected becausewe have only involved two of the atoms. We have togo to third order perturbations to get an average chi-ral difference. Furthermore, the chiral bias vanishesif two of the tripod legs are of equal length. This isalso be expected because the charges Q i are multiplica-tive and if the bonds are of equal length then the ge-ometrical structure by itself is not chiral. If the prob-ability of a mutation is P and the difference betweenthis probability for live and evil molecules is δ P , then δ ln P ∼ α LM tripod q Q M − ( c / v ) (derived in the Sup-plements), where α ∼ . δ ln P ∼ α LM em qM − ( c / v ) . Asimilar conclusion was reached through a quite differentargument, by Zel’dovich et al. (1977). However, thereseems to be no good reason why M em should be non-zero.The third, electrostatic helical model is also chiral (asillustrated by Fig. B3 in the Supplements). We in-voke an isotropic “mutability”, κ , which is the probabil- Figure 2 . The chiral quantity associated with cosmic-ray showers is the lodacity L = ˆ µ · ˆ v . Spin-polarized muons (respectively antimuons)and their daughter electrons (respectively positrons) are produced in air showers mainly from charged pion decay. They are indicated incolor in the sketch (including the parent charged pions) while the non spin-polarized (electromagnetic and nucleonic) components are ingrey. The muon component dominates at ground level but will slow and decay into electrons and positrons below ground level. The chargeparity (CP) invariance leads to an universal sign of the cosmic-ray lodacity L < ity per unit length of cosmic ray trajectory through themolecule that a significant mutation will result. We sup-pose that the mutability κ has a both a radial and a he-lical component, like the electrostatic potential. We findthat the third order chiral bias comprises a sum of termsthat contain two helical factors and one radial factor. Ifthe structure is at all similar to RNA then it is likelythat the bias is dominated by terms with an axisymmet-ric mutability, κ . We find that the original cosmic raypositrons, which outnumber the electrons, are deflectedradially inward when encountering a live molecule andoutward with an evil molecule. This implies that inter-actions with the nucleobases must cause more mutationsthan those with the suger-phosphate backbone. Theoverall chiral bias is given by δ ln P ∼ α LM qM − ( c / v ) .Finally, we consider an electromagnetic helical modelwhere we suppose that if the individual monomers carrymagnetic dipoles as well as electric dipoles. Then, al-though the magnets do not line up as in a ferromag-net, there may be enough near neighbor correlation forthere to be an electromagnetic, chiral bias which couldbe ∼ ( v / α c ) times the electrostatic bias.The bias δ ln P ( ∼ − for keV electrons, times the lodacity and the fractional difference between positiveand negative charges in the barber pole model) is therelative difference in the mutation rate between live andevil organisms. In appendix C, we use the the logis-tic equations to model the population growth. Startingwith a racemic mix the enantiomeric excess is e . e . = tanh ( δ ln P T / ) where T is the time multiplied by thegrowth rate. If we add a balanced, live-evil “conflict”then homochiralization is speeded up. Either way, asmall bias in the mutation rate can achieve this on anevolutionary timescale.DISCUSSIONIn this paper, we have proposed that homochiralityis a deterministic consequence of the weak interaction,expressed by cosmic irradiation of helical biopolymerswhich may have affected the way they fold or assemble tomake the first living organisms. This is the consequenceof the coupling ML that can lead to symmetry-breakingas anticipated by Pasteur. The choice that was madeis then traceable to the preponderance of baryons overantibaryons, established in the early universe and ulti-mately to the symmetries of fundamental particle inter-actions presenting requirements (including leptonic CPviolation) as first elucidated by (Sakharov 1967). Wehave also demonstrated how this quite small chiral biascan lead to homochirality after sufficient generations ofself-replicating molecules and shown how conflict canspeed up this Manichean struggle.Much more study is needed to determine if these pro-cesses suffice to account for homochirality. In particular,it will be necessary to investigate different shower mod-els to understand the evolution of the lodacity and todevelop a quantum mechanical model of collisional ex-citation. There are many other effects to explore. Forexample, it has been shown that the adsorption of chiralmolecules on specific surfaces can enhance the optical ac-tivity by several orders of magnitude because of the elec-tric dipole - electric quadrupole interaction (Wu et al.2017). The interaction investigated may have analogs inbiological environments.A key issue facing astrobiology is assessing what sub-set of environments are necessary for the emergenceof life. Good candidate environments for the poly-merization of meteorite-delivered nucleobases are small,warm ponds produced by hydrothermal conditions as-sociated with volcanic activity on early Earth (Pearceet al. 2017). Their wet and dry cycles have been shownto promote the polymerization of nucleotides into longchains (Da Silva et al. 2015). Any rocky planet, withactive geological processes and water, has the potentialto create life, because it is likely to support considerableenvironmental diversity; in particular, surface-based lo-cales, including beaches and sea-ice interfaces (St¨uekenet al. 2013). Lingam & Loeb (2018) outlined the biolog-ical consequences of tides in producing wet-dry cycles orproviding biological rhythms in environments where thelight-dark cycle is absent. Irradiation by polarized ra-diation can only lead to small enantiomeric excess, andcannot explain the large excesses (15%) found in me-teorites and amplification mechanisms must be sought(Glavin et al. 2019). Should the amino acids found onmeteorites be biogenic, they would have existed long be-fore the appearance of life on Earth.Future space missions will return to Earth with sam-ples collected on asteroids and on the martian sub-surface (Lauretta et al. 2017; Yamaguchi et al. 2018;Vago et al. 2017). This will provide insight on the na-ture of the organic molecules and their chirality. As cosmic rays provide a natural connection between theweak interaction and living systems, we predict that, ifever indigenous biopolymers are found ( i .e. traces ofliving systems), they will have the same handedness aslife on Earth. (Similar remarks apply to future samplesreturned from deep subterranean sites.) Where life ap-peared first and if cosmic rays played a role in its forma-tion, are still open questions; but the important evolu-tionary consequences of spin-polarized cosmic radiation,which we propose here, is testable experimentally.A prediction of our model is that the mutation rateis dependent upon the spin-polarization of the radia-tion. A possible experiment would be to measure themutation rate of two cultures of bacteria under spin-polarized radiation (either e ± or µ ± ) of different lodac-ity with energy above the threshold necessary to inducedouble strand breaks in DNA ( ∼
50 eV). If the couplingbetween lodacity and molecular chirality is efficient inintroducing a chiral bias, one of the two cultures shouldexhibit a much lower mutation rate. We emphasize thatmuch can be learned experimentally from the compar-ison of chiral molecules involved in biology and usingboth signs of lodacity which can be created at accel-erators. It is not necessary to create “mirror life” toproceed. Once the dominant processes are identified,we can have confidence in our understanding of par-ticle physics and quantum chemistry to draw the nec-essary conclusions. Additional experiments, relevant toour electromagnetic models, involve measuring the mag-netic structure and properties of biopolymers.If these experiments show that the evolution of bac-teria is sensitive to polarization, this will be a good in-dication that magnetically-polarized cosmic rays are animportant piece of the chiral puzzle of life.ACKNOWLEDGEMENTSThe research of NG is supported by the Koret Founda-tion, New York University and the Simons Foundation.NG thanks Louis d’Hendecourt for discussions that in-spired this work. The hospitality of the astrophysicistsat RIKEN (ABBL, iTHEMS, r-EMU) during part ofthis work, is gratefully acknowledged. We thank DavidAvnir, David Deamer, David Eichler, Anatoli Fedynitch,Peter Graham, Andrei Gruzinov, Ralph Pudritz, StuartReynolds, Jack Szostak and Jonas Lundeby Willadsenfor helpful and instructive comments.REFERENCES
Atri, D., & Melott, A. L. 2011, Geophysical Research Letters, 38,https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2011GL049027Bailey, J. 2001, Origins of Life and Evolution of the Biosphere,31, 167Bailey, J., Chrysostomou, A., Hough, J. H., et al. 1998, Science,281, 672 Biondi, E., Branciamore, S., Maurel, M.-C., & Gallori, E. 2007,BMC evolutionary biology, 7, S2Boyd, R. N., Famiano, M. A., Onaka, T., & Kajino, T. 2018,The Astrophysical Journal, 856, 26Breslow, R., & Cheng, Z.-L. 2010, Proceedings of the NationalAcademy of Sciences, 107, 5723
Castillo, H., Schoderbek, D., Dulal, S., et al. 2015, Internationaljournal of radiation biology, 91, 749Curie, P. 1894, Journal de physique th´eorique et appliqu´ee, 3, 393Da Silva, L., Maurel, M.-C., & Deamer, D. 2015, Journal ofmolecular evolution, 80, 86de Marcellus, P., Meinert, C., Nuevo, M., et al. 2011,Astrophysical Journal Letters, 727, L27D’hendecourt, L., Modica, P., Meinert, C., Nahon, L., &Meierhenrich, U. 2019, arXiv e-prints, arXiv:1902.04575Frank, F. C. 1953, Biochimica et biophysica acta, 11, 459Gaisser, T. K. 2012, Astroparticle Physics, 35, 801Glavin, D. P., Burton, A. S., Elsila, J. E., Aponte, J. C., &Dworkin, J. P. 2019, Chemical reviewsGusev, G., & Guseva, Z. 2019, Bulletin of the Lebedev PhysicsInstitute, 46, 54Guzm´an-Marmolejo, A., Ramos-Bernal, S., & Negr´on-Mendoza,A. 2009, in Bioastronomy 2007: Molecules, Microbes andExtraterrestrial Life, Vol. 420, 229Hazen, R. M., & Sverjensky, D. A. 2010, Cold Spring Harborperspectives in biology, 2, a002162Keating, K. S., Humphris, E. L., & Pyle, A. M. 2011, QuarterlyReviews of Biophysics, 44, 433466Kim, Y.-K., & Desclaux, J.-P. 2002, Phys. Rev. A, 66, 012708Kondepudi, D. K., & Nelson, G. W. 1984, Physica A: StatisticalMechanics and its Applications, 125, 465Kramers, H. A. 1927, in Atti Cong. Intern. Fisica (Transactionsof Volta Centenary Congress) Como, Vol. 2, 545–557Kronig, R. d. L. 1926, Josa, 12, 547Lahoti, S. S., & Takwale, R. G. 1977, Pramana, 9, 163Lauretta, D., Balram-Knutson, S., Beshore, E., et al. 2017, SpaceScience Reviews, 212, 925Lee, T.-D., & Yang, C.-N. 1956, Physical Review, 104, 254Lingam, M., & Loeb, A. 2018, Astrobiology, 18, 967Lipari, P. 1993, Astroparticle Physics, 1, 195Lord Kelvin. 1894, The Molecular Tactics of a Crystal(Clarendon Press)McVoy, K. W. 1957, Physical Review, 106, 828Messiah, A. 1981, Quantum mechanics, Vol. 2 (Elsevier)Muller, H. J. 1927, Science, 66, 84Orgel, L. E. 2004, Critical reviews in biochemistry and molecularbiology, 39, 99 Pasteur, L. 1848, M´emoire Sur La Relation Qui Peut ExisterEntr La Forme Cristalline et La Composition Chimique, et SurLa Cause de La Polarization Rotatoire, Acad´emie des SciencesPearce, B. K., Pudritz, R. E., Semenov, D. A., & Henning, T. K.2017, Proceedings of the National Academy of Sciences, 114,11327Quack, M. 1989, Angewandte Chemie International Edition inEnglish, 28, 571Rosenfeld, L. 1928, Zeitschrift f¨ur Physik, 52, 161Sakharov, A. D. 1967, JETP lett., 5, 24Salam, A. 1991, Journal of Molecular Evolution, 33, 105Schr¨odinger, E. 1944, What is life? The physical aspect of theliving cell and mind (Cambridge University Press Cambridge)Shannon, C. E. 1948, Bell system technical journal, 27, 379Shinitzky, M., Shvalb, A., Elitzur, A. C., & Mastai, Y. 2007, JPhys Chem B, 111, 11004Soai, K., Shibata, T., Morioka, H., & Choji, K. 1995, Nature,378, 767Sokolov, A. 1940, Doklady Phys, 28, 415St¨ueken, E., Anderson, R., Bowman, J., et al. 2013, Geobiology,11, 101Svensmark, H. 2006, Astronomische Nachrichten, 327, 871Szab´o-Nagy, A., & Keszthelyi, L. 1999, Proceedings of theNational Academy of Sciences, 96, 4252Thorne, K. S., & Blandford, R. D. 2017, Modern ClassicalPhysics: Optics, Fluids, Plasmas, Elasticity, Relativity, andStatistical PhysicsVago, J. L., Westall, F., Coates, A. J., et al. 2017, Astrobiology,17, 471Vester, F., Ulbricht, T., & Krauch, H. 1959,Naturwissenschaften, 46, 68Wagner, K., Keyes, E., Kephart, T. W., & Edwards, G. 1997,Biophysical journal, 73, 21Watson, J. D., & Crick, F. H. C. 1953, Nature, 171, 737Wu, C.-S., Ambler, E., Hayward, R., Hoppes, D., & Hudson,R. P. 1957, Physical review, 105, 1413Wu, T., Zhang, W., Wang, R., & Zhang, X. 2017, Nanoscale, 9,5110Yamagata, Y. 1966, Journal of Theoretical Biology, 11, 495Yamaguchi, T., Saiki, T., Tanaka, S., et al. 2018, ActaAstronautica, 151, 217Zel’dovich, B., Saakyan, D., & Sobel’man, I. 1977, JETP Lett,25, 94
APPENDIX A. AIR SHOWER ASYMMETRIESA.1.
Charge ratio
Primary cosmic rays comprise mostly positive nucleons. This excess is transmitted via nuclear interactions to pionsand then, on to muons. The muon charge ratio is R µ ∼ .
25 below 1 TeV and increases to above ∼ . µ ± produced from decaying pions and kaonsare on average spin-polarized. (The dominant contribution is from pion decay.) Their daughter electrons and positronsare also, on average, spin-polarized. The spin-polarized cosmic-rays can also produce UV CPL when propagating inthe medium through emitting ˇCerenkov radiation and bremsstrahlung.A.2. Spin-polarized secondary particles
The spinless charged pion with a lifetime of 26 ns decays at rest into a left-handed muon neutrino and a muon: π − → µ ¯ ν µ (and π + → µ + ν µ respectively). The pion has a mass of m π = 140 MeV/c , the muon has a mass m µ = 106MeV/c and the neutrino is effectively massless. We define r π = ( m µ / m π ) . In the pion rest frame (denited by ∗ ), themomentum of the muon is | p ∗ µ | = | p ∗ ν | = m π c ( − r π ) ∼ . / c (A1)and E ∗ µ = (cid:113) p ∗ µ c + m µ c ∼ . v π / c = β π e z . Defining θ ∗ ,the angle of emission of the muon in the pion rest frame, we have the following relations for the muon momentum,energy, helicity ( h = ˆ s · p /| p | ) and angle of emission in the lab rest frame (Lipari 1993): p µ = γ π p ∗ µ cos θ ∗ + β π γ π E ∗ µ , (A2) E µ = γ π E ∗ µ + β π γ π p ∗ µ cos θ ∗ , (A3) h ( β π , θ ∗ ) = β µ (cid:20) − r π + ( + r π ) cos θ ∗ β π + r π + ( − r π ) cos θ ∗ β π (cid:21) , (A4)tan θ = β ∗ µ sin θ ∗ γ π ( β π + β ∗ µ cos θ ∗ ) . (A5)In the limit β π =
0, the velocity of the muon is: β µ = ( − r π )/( + r π ) ≡ β ∗ µ ∼ .
27 and we have h = + 1 independentof the angle of emission of the muon. The polarization of the positive muon flux at sea level varies between ∼ ∼ µ − → e − ν µ ¯ ν e ( µ + → e + ν e ¯ ν µ ). The decay probability of a positron is W ( θ ) = ( + a cos θ )/( πτ µ ) where θ isthe angle between the spin direction and the positron trajectory, τ µ ∼ . µ s is the mean lifetime, and the asymmetryterm a is a direct consequence that the muon decay is governed by the weak interaction, and depends on the positronenergy, so the positron angular distribution is d Γ / d cos θ = W ( θ ) . The maximum and mean positron energies resultingfrom the three body decay are given by: E e + max = ( m µ + m e ) c /( m µ ) = .
82 MeV and ¯ E e + = . E e + max , we have the maximum asymmetry a =
1. When averaged over all positronenergies, a = /
3. A.3.
Circularly polarized radiation ˇCerenkov radiation has a small degree of circular polarization which is dependent on the orientation of the spinof the initial particle. This is a purely relativistic quantum effect (Sokolov 1940). In the following we consider thedifference between the number of left-handed photons and right-handed photons emitted from an electron of helicity1 / i.e. spin along its direction of motion, denoted by ” ”.Defining the ratio of the photon to the electron energies, ξ = (cid:126) ω /( E e ) , the velocity and Lorentz factor of the electron β = v / c , γ = ( − β ) − / , the ˇCerenkov angle cos θ c = [ + ξ ( n − )]/( n β ) , and the function F = cos χ ( cos θ c − n β ) + γ − sin χ cos φ sin θ c where the angles are α = π / χ = φ = N , + (respectively left-handed N , − ) is (Lahoti & Takwale 1977) N , ± ∝ . ( β sin θ c ) + ξ ( n − ) ± . ( β sin θ c ) cos ( α ) ∓ sin ( α ) ξ F . (A6)As an example, the ratio ( N , + − N , − )/( N , + + N , − ) emitted by an electron of energy ∼ . β = . ∼ . − at a wavelength of 206 nm. (If the electron has helicity -1/2, ( N , + − N , − )/( N , + + N , − ) has thesame magnitude, but opposite sign). For muons at the same velocity ( ∼
166 MeV), the ratio is 1 . − at the samewavelength.Longitudinally polarized β -radiation gives rise to circularly polarized bremsstrahlung. Using the Born approximation,McVoy (1957) derived the following formula for circular polarization in the limit where E e ∼ h ν and the emission angleof the photon θ = ◦ : P γ P e = (cid:18) + ( − β )( E e + mc )( − β ) E e (cid:19) − . (A7)Here P e is the polarization of the electron. The polarization transfer drops rapidly at electron energies E e below ∼ ∼ − in the example above, it could still be largeenough to impose homochirality if there is a general reason to couple this physical chirality to the geometrical chiralityof biological molecules. This matter deserves further consideration and, perhaps, experimental investigation.A.4. Lodacity Evolution
As discussed previously, cosmic rays are preferentially positively-charged and they create µ + and e + with lodacity L i = ˆ µ · ˆ v < B ext2 ; the second is deflection of the particle momentum in a Coulomb interaction whileleaving the direction of the magnetic moment unchanged. This also leads to energy loss. The third is the dilution ofthe lodacity by unpolarized, “knock on” electrons. These are created during ionization loss when polarized cosmic rayscollide with electrons in atoms. We consider these effects, in turn, for antimuons/muons and for positrons/electrons.The Larmor radii of muons exceed their decay lengths so long as B ext (cid:46) ∼ ( p /( m e c ))( B ext / ) − m, where p is the electron momentum. This is quite likely to besmall compared with their ranges and so the positrons and electrons will be channelled by an ordered magnetic field.Under these circumstances, the equation of motion for a positron is d p dt = e γ m e p × B ext , (A8)where γ = ( − p / m c ) − / . The magnetic moment will also precess about the magnetic field according to d µ dt = e γ m e µ × B ext . (A9)In the non-relativistic limit, which concerns us most, these equations then imply that p and µ precess about B ext witha common angular velocity − eB ext / m e . We expect the spin-polarized daughter positrons to outnumber spin-polarizedelectrons of similar momenta and to be created with a momentum distribution that is axisymmetric about a downwarddirection, ˆ g ≡ g ˆ e z . Furthermore, for each p , the distribution of µ will be axisymmetric about B ext . For a given magneticfield direction, this can lead to an average spin/magnetic moment polarization projected perpendicular to the velocity.However, in this case, it is only L that has the required pseudoscalar form and the perpendicular component leads tono bias after full averaging. The precession contributes modest degradation of the mean polarization.Now turn to the cumulative effect of the deflections during Coulomb interactions. These are dominated by distantencounters and therefore are mostly small (e.g. Thorne & Blandford 2017). A cosmic ray with momentum p exchangestransverse momentum ∆ p ⊥ with an individual electron effectively at rest, will lose kinetic energy ∆ T = ∆ p ⊥ / m e , andbe deflected through an angle ∆ θ = ∆ p ⊥ / p . Averaging over all encounters, we obtain (cid:28) d ( ∆ θ ) d ∆ T (cid:29) = − m e p , (A10)for the sum of the mean square deflections along along two axes perpendicular to the momentum. It is helpful tointroduce a quantity τ = ln ( + m e c / T ) which satisfies d τ / dT = − m e / p . The mean square deflection angles addstochastically and so we can approximate individual deflections as small and write (cid:28) d ( ∆ θ ) d τ (cid:29) = . (A11)In addition, < d ∆ θ / d τ > = P ( θ , τ ) relative to the initialdirection and the initial mean magnetic moment. It will satisfy a Fokker-Planck equation (e.g. Thorne & Blandford2017) with τ replacing time. ∂ P ∂τ =
14 sin θ ∂∂θ sin θ ∂ P ∂θ . (A12)Multiplying this equation by cos θ and integrating over solid angle, we obtain d < cos θ > d τ = ∫ π d θ cos θ ∂∂θ sin θ ∂ P ∂θ (A13)After integrating twice by parts, we obtain d < cos θ > d τ = − < cos θ >, (A14)so that < µ z > ∝ τ − / . Precession within the molecule is ignorable. Figure B1 . Example of electric chirality (tripod model). The electric charge distribution between the three components of nucleic acids(base, sugar, phosphate) of a nucleotide is chiral and the sign of the electric chirality is given by M tripod = ˆ x × ˆ x · ˆ x . The electrondensity (colored by electrostatic potential) is shown on the right; for our simple model we consider the charge distribution of the threegroups (base, sugar, phosphate) separately and study their combined action to the ionization probability of the electrons located aroundthe chiral carbon C (cid:48) (which we consider to be the target in our simple model, as it is located between the base - here adenine - and thesugar-phosphate backbone). Now consider the evolution of the mean spin polarization of all secondary cosmic-rays of same mass m e . If thecosmic-rays are created relativistically with lodacity L , then, when they are nonrelativistic, L( T ) ∼ L (cid:18) T m e c (cid:19) / . (A15)The third effect — lodacity dilution by energetic knock on electrons — is quite sensitive to the fount, the materialthat lies above it and the energies of particles that are most effective in bringing about mutation. The only particlesthat are of interest are those that are created close to the fount. An interesting complication for the electrons is that,as they are identical to particles with which they are colliding their total wave functions should be antisymmetric.This can introduce a spin-dependent interference term into the collision cross section (Messiah 1981). This effect canalso be a factor in the quantum mechanical treatment of the direct interaction of an electron with the molecule.A proper understanding of lodacity dilution requires shower simulations and a quantum chemistry treatment. Theseare underway. B. CHIRAL TRANSFER FROM MAGNETIZED COSMIC RAYS TO BIOMOLECULESThe evolution of living organisms is influenced by mutations, which can be caused by cosmic rays which can changethe electronic structure of biomolecules, mostly through ionization. Magnetized cosmic rays (i.e. cosmic rays withnon-zero lodacity) can affect live and evil molecules slightly differently through the coupling of the lodacity L tothe molecular chirality, M , which is a pseudoscalar chosen to describe the geometric structure of the molecule, with |M| ≤ Chiral monomer
B.1.1.
Electric chirality (tripod model)
We first consider a simple model of a small chiral molecule. We suppose that there is a single target site, O , wherethere is an energy-dependent cross section for inducing a mutation. We then suppose that O is bound to three,non-coplanar, charged sites, representing three atoms or groups at different distances from O. This is the minimumnecessary to exhibit chirality. Figure B1 shows that this chiral unit can be a simple model of a nucleobase. In thisexample, O is identified with the atom C1’ which bonds the backbone to the base. The chiral bias will then be given1by the relative difference in the mutation rate from that for an evil molecule due to the combined action of the threeneighboring non-coplanar with 0 charges (labeled Q , Q , Q on the figure). Electromagnetic force — For a non-relativistic cosmic ray of charge q , mass M , magnetic moment µ and velocity v ,moving through an electric field E , and magnetic field B , the electromagnetic force can be written as F = q ( E + v × B ) + ∇ (cid:104) µ · (cid:16) B − v c × E (cid:17)(cid:105) . (B16)This force is responsible for three types of perturbations which we consider in turn. Electric deflection — Let one of these atoms have charge Q and be located at x , relative to O . The impact parameterof the passing cosmic ray is b = x · ˆ v ˆ v − x . We measure distance along the trajectory in the direction of motion fromclosest approach by z . The first order, perpendicular velocity perturbation due to the Coulomb electric field is thengiven by δ v ⊥ ( z ) = qQ b π(cid:15) M v ∫ z −∞ dz (cid:48) ( b + z (cid:48) ) / . (B17)Henceforth, we will measure all lengths in terms of the Bohr radius a = π(cid:15) (cid:126) / m e e , all masses in units of m e , allspeeds in units of c , accelerations in c / a , and all charges in units of e . Evaluating the integral, we obtain δ v ⊥ ( z ) = α qQMb v (cid:18) + z ( b + z ) / (cid:19) b , (B18)where α = e / π(cid:15) (cid:126) c ≈ . Magnetic displacement — In addition, to the electric field, there will be a magnetic field B (cid:48) = − v × E in the frameof the cosmic-ray and this can interact with its intrinsic magnetic moment µ = qe (cid:126) ˆ µ /( Mm e ) . This magnetic force,reminiscent of spin-orbit coupling, is given by ∇( µ · B (cid:48) ) = v × µ · ∇ E . (The left hand side of this equation is familiar froma Hamiltonian formalism and the right hand side results from recognizing that the field is solenoidal with significantlocal current.) Evaluating the electric gradient along the trajectory, the transverse acceleration is δ a ⊥ ( z ) = α qQ ( b + z ) / (cid:18) v ( z ) × ˆ µ − v ( z ) × ˆ µ · b b + z b (cid:19) , (B19)where v ( z ) includes the unperturbed and the perturbed velocity.The associated transverse displacement at O is given by δ r O ⊥ = − v ∫ − x · ˆ v −∞ dz ( x · ˆ v + z ) δ a ⊥ ( z ) , (B20) Energy shift — If the cosmic ray undergoes a transverse displacement at O , it will acquire a perturbation to its kineticenergy due to the electric potential from an atomic site δ ln T = − α Q v x · δ r O ⊥ , (B21) Chiral bias — Now combine the perturbations. We first suppose that the positrons have a fixed lodacity L = ˆ µ · ˆ v and that there is no average perpendicular magnetization, as discussed in Appendix A. This means that the averagemagnetic moment is along the initial direction of motion. We must apply an electric deflection for there to be anaverage coupling to the cosmic ray dipole moment. However, this is insufficient for a chiral difference. If we substitute δ v ⊥ ( z ) associated with one atomic site in the expression for δ r O ⊥ (Eq. (B20) from another site, we obtain a second orderdisplacement at O which is a function of the impact parameter, b , at the first atomic site at x and, implicitly, theimpact parameter at the second site at x and also of the initial velocity direction ˆv . The cosmic rays are focused at O and there is a fractional difference in the mutation rate between live and evil molecules given by the relative changein the cosmic ray flux δ ln P = − ∂ b · δ r O ⊥ ∝ L α M − v − < ˆ v · ˆ x × ˆ x > . We call δ ln P , the chiral bias. If we substitutethe evil molecule, the effect has the opposite sign and this, clearly, survives averaging over b . (When evaluated strictlyclassically with point charges, δ ln P is logarithmically divergent. The divergence is removed if we associate finite deBroglie wavelengths with the particles.)If we now average over ˆv - it suffices to change its sign - this chiral bias vanishes. This is entirely consistent with ourexpectation. If we only consider second order perturbations, involving two atomic sites in addition to O , the interactioncannot be geometrically chiral. So, if the fount is completely isotropic, we need to consider third order perturbations,2involving three distinct atomic sites, in addition to O , to have the possibility of a chiral coupling. Furthermore, it isapparent that the strength of the perturbations associated with each site is proportional to the scalar charges Q , anda third order perturbation to the mutation rate will be simply proportional to the product of these charges, which isunchanged on inversion. It is only their relative locations that matter.There are several types of third order perturbation. For example, we can use the first order displacement due tothe first charge to evaluate the electric field due to the second charge along the perturbed trajectory and computea second order velocity perturbation to calculate the third order magnetic displacement at O . Alternatively, we cantake the second order displacement at O and combine this with the electric field due to a third charge to calculate athird order change in the kinetic energy of the cosmic ray at O . We must then sum over all permutations of charge.If the mutation cross section is energy-dependent then there will be an additional chiral bias. All of these terms havethe same order, δ ln P ∼ α LM tripod M − v − , where M tripod = ˆ x × ˆ x · ˆ x . In addition it is found that the coefficientvanishes if two of the bond lengths x are equal and the bonds are no longer chiral. Of course, any of the perturbingcharges could also be a target and there could be many more atomic sites.The effect that we have estimated is manifestly too small ( ∼ − for relativistic electrons and ∼ − for relativisticmuons) to be of interest in the current context. In addition, it demonstrates that if we were to apply it to a varietyof pre-biotic molecules then the signs of the individual biass in a set of chemically compatible enantiomers are notguaranteed to be the same and the net chiral bias could be reduced even further. However, the tripod model is valuablebecause it directly and explicitly couples the physical chirality of the cosmic ray to the geometrical chirality of themolecule and is highly instructive. B.1.2. Electromagnetic chirality
In this second, idealized model, we suppose that, instead of concentrating the charge at three or more points, wedistribute electrons smoothly on the surface of a sphere of radius R . There may also be a central charge at the origin.The model is semi-classical in the sense that the distribution can be considered as a representation of the expectation ofthe quantum mechanical charge density. The restriction to the surface of a sphere is not essential — we could integrateover R — but it simplifies calculating an actual, chiral bias and suffices to demonstrate electromagnetic chirality andto make some more key points.We expand the surface charge density for an (arbitrarily assigned) live molecule in terms of multipole moments,chosen to be chiral in combination (Fig.B2) and calculate the electric field inside and outside the sphere. We thenconsider a cosmic ray with velocity v and impact parameter with respect to the origin of the sphere b . The linear,transverse displacement at ingress and egress can be expressed as a sum over electric multipoles. We next supposethat the ionization/mutation probability for a cosmic ray traversing the sphere is proportional to the electron densitieson the sphere at the actual points of ingress and egress, (designated − and + , respectively), correcting for the densitygradient. We average over b and v assuming that the cosmic ray flux is uniform and the molecules are isotropicallyoriented. Next, we repeat the exercise for the evil counterpart molecule and calculated the chiral difference.The net result is that there is no purely electrostatic chiral difference for an arbitrary sum of multipoles. This can beunderstood on quite general grounds because there is no way to combine electric multipole moments electrostaticallyto form a pseudoscalar. As we demonstrated with the unequal tripod, it is necessary to employ the magnetic field thatis created following a frame transformation and this involves the Levi-Civita tensor and, consequently, chirality. Putanother way, just because a molecule is geometrically chiral does not ensure a chiral bias; it is necessary to invoke aninteraction that couples the molecular chirality to the physical chirality of the cosmic ray.To this end, we now add magnetic multipole moments. In the spirit of our semi-classical approach, we associatethese with surface current flowing in the sphere due to electron orbital angular momentum. Of course, atomic andmolecular magnetism is associated more with electron spin but this is unimportant here. It is easy to see that thesimplest and strongest molecular chirality combines the electric, d , and magnetic, m , dipole moments. We call thiselectromagnetic chirality and it has the simplest definition M em = < ˆ d · ˆ m > . We therefore just concentrate on thesetwo multipoles. Charge density and magnetic field — The time-averaged surface charge density, Σ , on the sphere surrounding a pointcharge Ze at the origin, O , is Σ = π R (cid:18) − Ze + d · r R + . . . (cid:19) , for | r | = R . (B22)The magnetic field B due to the motion of the electrons on the sphere is B = µ m π R + . . . , r < R , = µ π (cid:18) ( m · r ) r r − m r + . . . (cid:19) , r > R . (B23)3 Figure B2 . Example of electromagnetic chirality. Left: Example of an electric charge distribution (projected on a sphere) of twobiomolecules of opposite chirality, as seen from left and from right. This simply combines an electric dipole and an electric quadrupole. Weneed to reflect and rotate by 180 degrees the live molecule to obtain the evil one. The model we discuss in Section B.1.2 is even simpler,combining electric and magnetic dipoles, M = ˆ d . ˆ m . The magnetic dipole moment m is invariant under parity transformation. Right:Unpertubed vs. pertubed magnetically polarized cosmic ray trajectories through a chiral unit. The unpertubed trajectory is along z . Theperturbed trajectory due to the chiral molecular field B is shown. The perturbed cosmic-ray therefore experience a slightly different chargedistribution which would lead to a difference in the ionization rate between the two enantiomers. Cosmic Ray Path — We now consider the path of a single cosmic ray with impact parameter b with respect to thecenter O of a live molecule, as seen in Fig.B2. The (classical) force acting on the cosmic ray is given by Eq. B16 and weneed the transverse displacement as the cosmic ray enters and leaves the sphere. We introduce the coordinate z = r · ˆ v ,so that ingress and egress are at r ∓ = b + z ∓ ˆ v with z ∓ = ∓( R − b ) / . We continue to assume that µ is on averageantiparallel to ˆ v . The largest chiral interaction is with B and there is a linear perturbative force of µ z ∇ B z . Prior toingress, we use ∇ × B = δ v = µ z B / M v .We can now calculate the displacement perpendicular to the unperturbed path at ingress. (This is not same as thedisplacement at a fixed time.) This is given to first order by δ r −⊥ = µ z T ∫ z − −∞ dz ( B − ( ˆ v · B ) ˆ v ) , = µ µ z π T R (cid:20)(cid:18) − ( + η )( − η ) / η (cid:19) ( m · ˆ b ) ˆ b − η / ( m · ˆ v ) ˆ b − (cid:18) − ( − η ) / η (cid:19) ( m · ˆ v × ˆ b ) ˆ v × ˆ b (cid:21) . (B24)where T = M v / η = b / R .The velocity perturbation immediately after ingress has to take account of the impulse due to the current flowingon the surface of the sphere. However, there is no additional force as the interior magnetic field is uniform. The chiralpart of the transverse displacement at egress can then be shown to be δ r + ⊥ = δ r −⊥ + ( z + − z − ) δ v + ⊥ v = µ µ z π T R (cid:20)(cid:18) − ( + η )( − η ) / η (cid:19) ( m · ˆ b ) ˆ b − η / ( m · ˆ v ) ˆ b − (cid:18) − ( + η )( − η ) / + η ( − η ) η (cid:19) ( m · ˆ v × ˆ b ) ˆ v × ˆ b (cid:21) . (B25) Ionization Rate — The classical ionization cross section is σ ion = Z π a (cid:18) I H I (cid:19) (cid:18) I H T (cid:19) ∼ . × − T − m . (B26)4The probability that a cosmic ray incident upon an atom, idealized as a charged sphere, will create an ionization istherefore P ion ∼ σ ion / π R ∼ . T − . Direct measurements below ∼ T increases as well as quantum mechanical effects. Again, this is unimportant forour limited purpose. In addition to its transverse displacement, a cosmic ray will have a slightly different energy andcross section as it crosses the sphere and there is an associated chiral bias. This turns out to be subdominant in ourmodel and we ignore it although it is likely to be significant in a more realistic description. Chiral bias — We have computed the first order deflection at ingress and egress. By itself, this leads to no net changein the ionization rate. However, the deflection results in the cosmic ray encountering a slightly different surface densityof electrons due to the gradient in the electron density within the sphere. The second order change in the relativeionization rate, the chiral bias, is then given by δ ln P = −( δ r −⊥ · ∇ ⊥ ln Σ − + δ r + ⊥ · ∇ ⊥ ln Σ + ) , (B27)where the perpendicular, logarithmic gradient in the relative surface charge density at ingress and at egress is ∇ ⊥ ln Σ ∓ = − Z R (cid:16) ( − η ) / η ( d · ˆ b ) ˆ b ± η / ( d · ˆ v ) ˆ b + d · ˆ v × ˆ b ˆ v × ˆ b (cid:17) , (B28)and we have used Eq. (B22). (There is no gradient in the monopolar surface density and any quadrupolar term doesnot survive averaging.)So far, we have considered one molecule and one cosmic ray. We must now average over direction. The simplestassumption to make is that the cosmic ray flux is isotropic with respect to the molecule. This still allows the cosmicrays to be anisotropic if, as is likely, the molecules are randomly oriented, for example in water. (We note that thereare circumstances when orientation biases may be present and these could lead to a larger chiral bias.) In order tocarry out the angle average, we first note that any term contributing to δ ln P that is odd in ˆ v can be dropped as itwill be canceled by the effect of a cosmic ray with the opposite velocity or impact parameter. We then average theremaining terms over azimuth, perpendicular to v using < m · ˆ b d · ˆ b > = m · d / η over a unitcircle. The final step is to average ˆ v over the surface of a unit sphere. Averages of the form < u · ˆ v w · ˆ v > become u · w /
3. After performing these integrals and averages, we obtain a chiral bias δ ln P = − . α Z − dm LM M − v − (B29)where the dipole moment of the chiral unit, d , is measured in units of D ≡ . ea and the magnetic moment, m , is in µ B . Magnetic moment — Electromagnetic chirality can lead to a relatively large chiral bias (cf. Zel’dovich et al. 1977). How-ever it requires the magnetic moment in a small chiral unit to be aligned systematically with the dipole moment. Thepermanent magnetic moments of small biological molecules have been studied less than their electric dipole moments.They should exist when there are unpaired electron spins and they contribute to the paramagnetic susceptibility. How-ever, there does not seem to be a good atomic physics reason why they should align with the electric dipole moment.Another way of expressing this is to say that the volume integral of the relativistic invariant E · B over all space is − Z d · m / π R , where Z is the impedance of free space and a reason has to be found why this should be non-zero.B.2. Helical biopolymer
We now turn to an idealization of a biopolymer. We hypothesize that single-stranded, naturally twisted biopolymers,that had some limited capacity to replicate, albeit with a high frequency of errors/mutations, were the prime geneticagent when the transition to life occurred. We argue that, if this were the case, these long, helical molecules representa more likely candidate for an explanation of homochirality than the much smaller molecules that were present duringa pre-biotic epoch. These simpler and more primitive helical precursors contrast with the highly-evolved DNA andRNA of today, which replicate relatively efficiently and with much greater fidelity. An early, twisted biopolymershould exhibit no chemical preference for handedness. However, the twist is likely to exhibit long range order alongthe polymer (Keating et al. 2011). A single cosmic ray will only interact with a short segment of the molecule and itis the generic, geometrical disposition of the electrical charge and field within this segment that confers the sign of thechiral preference. This should be common to most helical biopolymers and leads to a much larger chiral bias than thetripod model.5
Figure B3 . Example of electric chirality (barber pole model). The electric charge distribution of two biopolymers of opposite chirality,( Σ ( r , ϕ ) projected onto a cylinder) is shown, together with the unperturbed vs. perturbed trajectory of a magnetically polarized cosmic rayinteracting with the molecule. B.2.1.
Electric chirality (barber pole model)
A RNA-like molecule comprises a chain of nucleotides that spirals like a barber pole and can be modeled by anelectrostatic potential, Φ = R ( r ) + R ( r ) cos [ k z − m ( ϕ − ϕ )] , with radial and helical terms (cf. Wagner et al. 1997). Asthe helix is assumed to be single-stranded, we set the azimuthal quantum number to m = m = − | m | = k ∼ . Φ ismeasured in units of e /( π(cid:15) a ) .) One way to show formally that this model is chiral is to suppose that R ( r ) has amaximum at r max . Construct a radius vector r from the axis to this maximum at z =
0. Displace the origin of thisvector a distance z along the axis until the radius vector, now r , has turned though ± π /
2. The quantity ˆr × ˆr · ˆz involves the Levi-Civita tensor, and is therefore chiral. It is unchanged under the transformation ˆz → − ˆz . Note thatif we regard the two radius vectors as defining a tetrahedron, the equal edges do not share a common vertex. Thisdifferentiates the barber pole from the equal tripod which is non-chiral. Electric deflection — Consider a cosmic ray with charge qe , mass Mm e and velocity v (measured in units of c) makingan acute angle Ψ with ˆz , as shown in Fig. B3. Let the cosmic ray have impact parameter b lying in the z = ϕ =
0. Its unperturbed trajectory is s = b + z sec Ψ ˆv .The perpendicular velocity perturbation due to the Coulomb electric field is given by δ v ⊥ = − q α M v cos Ψ ∫ z −∞ dz (cid:48) (∇ Φ ) ⊥ . (B30) Magnetic displacement — Just as with the tripod model, we can calculate the second order perturbation to the transversemagnetic acceleration in direct analogy to Eq. (B19) The magnetic force is still given by ∇( µ · B (cid:48) ) = v × µ ·∇ E . Evaluatingthe electric gradient along the trajectory, the transverse acceleration is δ a ⊥ = (cid:18) q α ˆ µ × v ( z ) M · ∇ (cid:19) ∇ Φ . (B31)Hence, using our definition of lodacity, δ r ⊥ = ( v cos Ψ ) ∫ z −∞ dz (cid:48) ( z − z (cid:48) ) δ a ⊥ = − q α L ( M v cos Ψ ) ∫ z −∞ dz (cid:48) ∫ z (cid:48) −∞ dz (cid:48)(cid:48) ( z − z (cid:48)(cid:48) )( ˆv · ∇ Φ ( z (cid:48) ) × ∇)∇ Φ ( z (cid:48)(cid:48) ) (B32)where the bar denotes an average over the cosmic ray magnetic moment. Chiral bias — We now make a change from the assumption that we made when we considered electromagnetic chirality.Instead of assuming that the density of ionizable electrons is essentially that of the perturbing electrical charge, for ahelix, we suppose, instead, that the effective density is distinct, while sharing the same symmetry as the electrostatic6potential. The reason for doing this in the context of RNA and its possible progenitors is that we do not understandthe path from ionization to mutation. It may be more important to break bonds in the central bases. Alternatively,the outer sugar backbone may be more relevant.In order to include this freedom, we introduce a ”mutability”, κ , (probability per unit length of cosmic ray trajectoryfor mutation). The lodal contribution to the probability of mutation is then δ P = ∫ ∞−∞ dz δ r ⊥ ( z ) · ∇ κ ( z ) = − q α L ( M v cos Ψ ) ∫ ∞−∞ dz ∫ z −∞ dz (cid:48) ∫ z (cid:48) −∞ dz (cid:48)(cid:48) ( z − z (cid:48)(cid:48) ) ˆv · ∇ Φ ( z (cid:48) ) × ∇∇ Φ ( z (cid:48)(cid:48) ) · ∇ κ ( z ) (B33)where the only derivatives we need are perpendicular to ˆv along ˆb , ˆh . Note that the effect has opposite sign for thetwo signs of cosmic ray charges and the same negative sense of lodacity.We adopt a general, separable potential of the form Φ = R ( r ) + R ( r ) cos [ k z − m ( ϕ − ϕ )] , with radial and helicalterms. The associated charge density contributed by the combination of the nuclei, inner shell electrons and bindingelectrons is given by ρ = − (cid:15) ∇ Φ and should represent the actual distribution in a realistic model. Likewise, we assume κ = K ( r ) + K ( r ) cos [ k z − m ( ϕ − ϕ )] . δ P can be considered as a triple integral over z , z (cid:48) , z (cid:48)(cid:48) , proportional to a sum overterms containing factors describing the mutability gradient, the electric field and the electric field gradient contributingto the pseudoscalar F = ˆv · ∇ Φ ( z (cid:48) ) × ∇∇ Φ ( z (cid:48)(cid:48) ) · ∇ κ ( z ) .Next, expanding all the trigonometric factors in the expression for δ P live , ( m = F involve productsof the radial functions R ( r ) , R ( r ) , K ( r ) , K ( r ) and derivatives. (There are nearly 50,000 terms!) Likewise, for the evilmolecule with m =-1. We then subtract the evil from the live terms. The next step is to average over ϕ and thenaverage the velocity with inclination Ψ over the surface of a sphere. This is made difficult because Ψ appears in thearguments of the K and R functions. We must also average the impact parameter, b over the circumference of a circle.The integrals over z , z (cid:48) , z (cid:48)(cid:48) require substituting r = ( b + z tan Ψ ) / , ϕ = tan − (( z tan Ψ )/ b ) , etc. (In principle,derivatives R ( r ) , R ( r ) , K ( r ) , K ( r ) of can be removed through integration by parts.) In order to complete the calcu-lation we would have to perform a five dimensional integration for each specific choice of the R and K functions. Animportant general conclusion is that, after performing the averages, all of the chiral terms combine three factors withone factor being a radial function, either R ( r ) or K ( r ) , and two of them being helical functions chosen from R ( r ) , K ( r ) . In practice it is better to perform these calculations by imitating the cosmic rays and performing a Monte Carlosampling of isotropic cosmic ray trajectories. However, the analysis so far suffices to demonstrate that there is a finitechiral bias and what it depends upon.It is not obvious which handedness is preferred, but it is probably generic, depending mostly on the gross chargedistribution. It appears that the dominant chiral combination of functions is R R K and derivatives. Note that it isnot, in practice, necessary for the mutability to be helical. If this is indeed the case, then it appears to be genericallytrue that the live chiral bias follows from the mutability being associated with ionizing the central bases instead of thesugar-phosphate backbone. This is under investigation.The best estimate for the chiral bias for mutation is δ ln P ∼ α LM M − v − . Several comments can be made. For agiven speed and lodacity, muons are ∼ − times as effective as positrons or electrons and we therefore emphasize thelatter. Clearly lower speed particles are also more effective. However, the lodacity of the initial cosmic rays is L ∝ v so the chiral bias of the primaries is ∼ α v − . This can be as large as ∼ − L .Secondary electrons will generally be unpolarized (although spin-dependent interference terms in the scattering crosssection of identical electrons (Messiah 1981) might confer some persistence of the lodacity). To address this requiresa more careful shower simulation. Also it should be emphasized that a classical approach surely fails when v declinestowards α , the characteristic speed of a molecular electron. Under these circumstances, a better approach is to solvefor the electron orbitals in a simple idealization of the barber pole and to compute the matrix elements and transitionprobabilities for collisional excitation and ionization. This should exhibit a qualitatively similar chiral bias to thesemi-classical calculation we have sketched here.B.2.2. Electromagnetic chirality
We have not considered, yet, the possibility of electromagnetic chirality of helical biopolymers. External electricand magnetic fields would influence the conformational flexibility of nucleic acids and hence affect the molecular chiralquantity M . Although it has been shown that RNA and DNA are ferroelectric, nucleic acids do not show strongpermanent magnetic moments (at least in their neutral form). Local correlations between neighboring bases could, inprinciple, cause local ˆ d · ˆ m but this is far from certain. However, when a magnetic field is applied, nucleic acids becomemagnetic. This is due to the aromatic rings in the bases. In the presence of a magnetic field, magnetic moments7 Figure B4 . Solutions of Eqs.C34-C35 for x = x = (thin line). For x = , the homochiralization time scale dividedby ∼
10. We start at T = perpendicular to the plane of the bases are induced by the so-called ring currents. These induced magnetic momentswould be localized towards the center of the helix, where the bases are located. It would surely induce a preferredorientation for the biopolymers (likely the same effect as the influence of a ferromagnetic fount) and this could changethe strength of the molecular chirality, but not its sign which is related to the handedness of the helix.We emphasize that it is the geometrical, helical structure of the biopolymers that appears to have the durability andstrength to define and sustain a universal handedness. The electromagnetic chirality ˆ d · ˆ m of a monomer can dependon the pH of the solvent and cannot lead to a universal sign. It is of interest to investigate the effect of magneticmoments in an helical configuration (they would also spiral around the helix axis), superimposed with the electricmolecular chirality in the barber pole model. C. BREAKING THE BIOLOGICAL MIRRORWe have shown that there are interactions that can couple the lodacity of cosmic rays to the molecular chiralityof the first, vital molecules. The biases we have estimated are all very small, ∼ − for keV electrons (times thelodacity and the fractional difference between positive and negative charges in the barber pole model), although largerbiases may be found through including other factors in the interaction. We now turn to considering how this smallenantioselectivity might evolve over time to homochirality.Enantioselective auto-catalysis, where molecules of the same chirality catalyze their own production while inhibitingthe formation of their mirror-image (Frank 1953), has served as an important model because it exhibit some of thefeatures of life, i .e., self-replication. However, living organisms have the ability not only to self-replicate but to increasetheir complexity, whenever complexity is beneficial to their survival. This must be the case for any early life form notyet well adapted to its environment. The evolutionary change is based on the accumulation of many mutations withsmall effects.The way a biopolymer folds into space determines its biological function. Clay minerals, present in the fount, mayhave catalyzed the polymerization of the first biopolymers; they can also protect the bases adsorbed on their surfacefrom radiation (Biondi et al. 2007). They, too, could play a role in the homochiralisation process, enhancing the chiralselection (see e.g. Hazen & Sverjensky 2010, for a review). It has also been shown that clay minerals can protect thebuilding blocks of biomolecules against the effects of high energy radiation (Guzm´an-Marmolejo et al. 2009).We do not deal here with the chemical pathway needed for the assembly of the first polymers (Orgel 2004). In theabsence of a chiral driving force, a prebiotic chemical reaction necessarily yields a racemic state; prebiotic pathwaysleading to a non-racemic (but not necessarily homochiral) state have been explored invoking chiral catalysts, that needto be be present in the fount to induce a bias (Soai et al. 1995; Breslow & Cheng 2010). Instead we propose thatprebiotic chemistry produces both chiral versions of the molecular ingredients of life ( i.e. helical polymers capable ofself-replication), and that at some stage in the earliest development of biomolecules, a small difference in the mutationrate, attributable to the lodacity of the cosmic rays, gives a chiral bias to the live genetic polymers over their evilcounterparts.Cosmic rays are generally recognized as agents of natural selection. At modest intensity, which interests us here,8they promote mutation and natural exploration of biochemical and evolutionary pathways; when the intensity is high,they will be destructive and will create sterile environments. It seems that cosmic radiation also affects the growthrate of the living organisms; for example, it has been reported that during episodes of high cosmic-ray flux and coldclimate there is an enhancement of biological productivity (Svensmark 2006) although this relationship is controversial.Conversely, radiation deprivation has been reported to inhibit bacterial growth (e.g. Castillo et al. 2015).Growth rates and mutation rates are correlated functions. When the growth rate is low, the probability to accumulatean adaptive mutation is strongly limited. To demonstrate the effect we therefore assume a simple relation between thegrowth rate and the rate of mutation in the nucleobase sequence, g ( M ) = C σ M F , where C is a positive constant, σ M is the mutation cross section (which depends on ionization and excitation of the nucleobases) and F the cosmic rayflux. While it is reasonable to postulate that cosmic rays increased the rate of genetic mutations in proto-lifeforms, theexact relationship between the radiation dose and the mutation rate is unknown. For sake of simplicity, we assumeda linear dose response. Our argument that a difference in the mutation rate of live and evil organisms would lead tohomochirality is not premised on the linear proportionality of g on the cosmic ray flux F . However, the timescale atwhich homochiralization occurs, does depend on the relationship between g and F , so the reader should keep in mindthat our simple, working hypothesis might be inadequate to estimate the homochiralization timescale.When considering living organisms, the definition of ”enantiomeric excess” is more subtle, because the living or-ganisms are never the same molecular entities at a given time. However, even if the genetic information evolves, thechirality of the nucleotides (which is related to the handedness of the sugar) is maintained. We denote by N live and N evil the number of live and evil molecules, respectively. The evolution of the two populations is given by: d ln N live dT = + δ M − xN evil , (C34) d ln N evil dT = − δ M − xN live . (C35)where • T ( σ M , C , F ) is the time multiplied by the growth rate; • x is the antagonism rate divided by the growth rate; • δ M (L , M) ∼ δ ln P is the relative difference in the mutation rates estimated in the previous section.With zero conflict, i.e. x =
0, the enantiomeric excess is e . e . = ( N live − N evil )/( N live + N evil ) = − [ + N live , / N evil , exp ( δ MT )] − . If we start with a racemic mixture (which is likely to be the case as shown by labora-tory experiments) then N live , / N evil , = e . e . = tanh ( δ MT / ) . Homochiralisation occurs when T ∼ δ M −1