of sequence-independent asymmetric cooperativity because of the anti-parallel strand orientations of the two strands.

Due to the asymmetric nature of the nucleotides, the extant informational biomolecule, DNA, is constrained to replicate unidirectionally on a template. As a product of molecular evolution that sought to maximize replicative potential, DNA's unidirectional replication poses a mystery since symmetric bidirectional self-replicators obviously would replicate faster than unidirectional self-replicators and hence would have been evolutionarily more successful. Here we carefully examine the physico-chemical requirements for evolutionarily successful primordial self-replicators and theoretically show that at low monomer concentrations that possibly prevailed in the primordial oceans, asymmetric unidirectional self-replicators would have an evolutionary advantage over bidirectional self-replicators. The competing requirements of low and high kinetic barriers for formation and long lifetime of inter-strand bonds respectively are simultaneously satisfied through asymmetric kinetic influence of inter-strand bonds, resulting in evolutionarily successful unidirectional self-replicators. Within our model, circular strands, the configuration prefered by primitive life forms, have higher replicative potential compared to linear strands.

Kaplansky [4] proved the now classical result that a compact Hausdorff space is characterised by its lattice of real continuous functions. It is the purpose of this paper to show that the same arguments can be extended to prove the generalization: a lattice isomorphism between two f-semisimple ;e-tings induces a homeomorphism between the structure spaces of their maximal f-ideals. This is in a direction different from that of Shirota [5]. It is not known whether the main result of [5] is a special case of our theorem.We shall concern ourselves only with commutative rings with unit element. Recall that an E-ideal of a lattice-ordered ring is the kernel of a lattice-ringhomomorphism onto a lattice-ordered ring. It may be characterized as a ring ideal I satisfying the property: lY[ < Ixl, x~I =~y~I . Zorn's lemma is used to show that any g-ideal is contained in a maximal f-ideal. A latticeordered ring is said to be f-semisimple if the intersection of all its maximal f-ideals is the zero ideal. For details on lattice-ordered tings, we refer to [1].The structure space of a lattice-ordered ring is defined as the set of all its maximal E-ideals endowed with the hull-kernel topology. A maximal E-ideal M belongs to the closure of a set {M,} of maximal f-ideals if M contains ~M~. That the hull-kernel operator is indeed a Kuratowski's closure operator for a topology is verified here easily because of the fact that the sum of f-ideals is again an (-ideal [1]. A convenient subbase for closed sets will then be sets of maximal ~-ideals containing singletons of the lattice-ordered ring (in which txyt = lxt lyt is true). Obviously, the structure space of a lattice-ordered ring would not be altered if it is made Y-semisimple by factoring out the fradical from it. Henceforth, therefore, we shall consider only f-semisimple lattice-ordered rings. The structure space of a lattice-ordered ring is always T~ and also compact. Compactness is shown from the fact that the sum of any family of f-ideals is proper provided the sum of any finite subfamily is proper.An ;e-ring is defined as a lattice-ordered ring that is also a subdirect union of totally ordered rings. Thus any identity valid in all totally ordered rings carries over to all f-rings; and an identity is valid in a particular ;e-ring if and only if it is valid in all totally ordered homomorphic images. From this point of view, we shall freely use the well-known properties of ;e-rings without reference to any source or explanation. In an ;e-ring, the maximal f-ideals are prime [I].Theorem. If two ;e-rings are lattice-isomorphic, then their structure spaces of maximal ~-ideals are homeomorphic.

Background We hypothesize prebiotic evolution of self-replicating macro-molecules (Alberts, Molecular biology of the cell, 2015; Orgel, Crit Rev Biochem Mol Biol 39:99-123, 2004; Hud, Nat Commun 9:5171) favoured the constituent nucleotides and biophysical properties observed in the RNA and DNA of modern organisms. Assumed initial conditions are a shallow tide pool, containing a racemic mix of diverse nucleotide monomers (Barks et al., Chembiochem 11:1240-1243, 2010; Krishnamurthy, Nat Commun 9:5175, 2018; Hirao, Curr Opin Chem Biol 10:622-627), subject to day/night thermal fluctuations (Piccirilli et al., Nature 343:33-37, 1990). Self-replication, like Polymerase Chain Reactions, followed as higher daytime thermal energy “melted” inter-strand hydrogen bonds causing strand separation while solar UV radiation increased prebiotic nucleobase formation (Szathmary, Proc Biol Sci 245:91-99, 1991; Materese et al., Astrobiology 17:761-770, 2017; Bera et al., Astrobiology 17:771-785, 2017). Lower night energies allowed free monomers to form hydrogen bonds with their template counterparts leading to daughter strand synthesis (Hirao, Biotechniques 40:711, 2006). Results Evolutionary selection favoured increasing strand length to maximize auto-catalytic function in RNA and polymer stability in double stranded DNA (Krishnamurthy, Chemistry 24:16708-16715, 2018; Szathmary, Nat Rev Genet 4:995-1001, 2003). However, synthesis of the full daughter strand before daytime temperatures produced strand separation, longer polymer length required increased speed of self-replication. Computer simulations demonstrate optimal polynucleotide autocatalytic speed is achieved when the constituent nucleotides possess a left-right asymmetry that decreases the hydrogen bond kinetic barrier for the free nucleotide attachment to the template on one side and increases bond barrier on the other side preventing it from releasing prior to covalent bond formation. This phenomenon is similar to asymmetric kinetics observed during polymerization of the front and the back ends of linear cytoskeletal proteins such as actin and microtubules (Orgel, Nature 343:18-20, 1990; Henry, Curr Opin Chem Biol 7:727-733, 2003; Walker et al., J Cell Biol 108:931-937, 1989; Crevenna et al., J Biol Chem 288:12102-12113, 2013). Since rotation of the nucleotide would disrupt the asymmetry, the optimal nucleotides must form two or more hydrogen bonds with their counterpart on the template strand. All nucleotides in modern RNA and DNA have these predicted properties. Our models demonstrate these constraints on the properties of constituent monomers result in biophysical properties found in modern DNA and RNA including strand directionality, anti-parallel strand orientation, homochirality, quadruplet alphabet, and complementary base pairing. Furthermore, competition between RNA and DNA auto-replicators for 3 nucleotides in common permit states coexistence and possible cooperative interactions that could be incorporated into nascent living systems. Conclusion Our findings demonstrate the molecular properties of DNA/RNA could have emerged from Darwinian competition among macromolecular replicators that selected nucleotide monomers that maximized the speed of autocatalysis.