The Hypercycle

A Proposal for the RNAome at the Dawn of the Last Universal Common Ancestor

From the most ancient RNAs, which followed an RNY pattern and folded into small hairpins, modern RNA molecules evolved by two different pathways, dubbed Extended Genetic Code 1 and 2, finally conforming to the current standard genetic code. Herein, we describe the evolutionary path of the RNAome based on these evolutionary routes. In general, all the RNA molecules analysed contain portions encoded by both genetic codes, but crucial features seem to be better recovered by Extended 2 triplets. In particular, the whole Peptidyl Transferase Centre, anti-Shine-Dalgarno motif, and a characteristic quadruplet of the RNA moiety of RNAse-P are clearly unveiled. Differences between bacteria and archaea are also detected; in most cases, the biological sequences are more stable than their controls. We then describe an evolutionary trajectory of the RNAome formation, based on two complementary evolutionary routes: one leading to the formation of essentials, while the other complemented the molecules, with the cooperative assembly of their constituents giving rise to modern RNAs.

Primordial aminoacyl-tRNA synthetases preferred minihelices to full-length tRNA

Aminoacyl-tRNA synthetases (AARS) and tRNAs translate the genetic code in all living cells. Little is known about how their molecular ancestors began to enforce the coding rules for the expression of their own genes. Schimmel et al. proposed in 1993 that AARS catalytic domains began by reading an 'operational' code in the acceptor stems of tRNA minihelices. We show here that the enzymology of an AARS urzyme•TΨC-minihelix cognate pair is a rich in vitro realization of that idea. The TΨC-minihelixLeu is a very poor substrate for full-length Leucyl-tRNA synthetase. It is a superior RNA substrate for the corresponding urzyme, LeuAC. LeuAC active-site mutations shift the choice of both amino acid and RNA substrates. AARS urzyme•minihelix cognate pairs are thus small, pliant models for the ancestral decoding hardware. They are thus an ideal platform for detailed experimental study of the operational RNA code.

Circular code in introns

A massive statistical analysis based on the autocorrelation function of the circular code X observed in genes is performed on the (eukaryotic) introns. Surprisingly, a circular code periodicity 0 modulo 3 is identified in 5 groups of introns: birds, ascomycetes, basidiomycetes, green algae and land plants. This circular code periodicity, which is a property of retrieving the reading frame in (protein coding) genes, may suggest that these introns have a coding property. In a well-known way, a periodicity 1 modulo 2 is observed in 6 groups of introns: amphibians, fishes, mammals, other animals, reptiles and apicomplexans. A mixed periodicity modulo 2 and 3 is found in the introns of insects. Astonishing, a subperiodicity 3 modulo 6 is a common statistical property in these 3 classes of introns. When the particular trinucleotides N1N2N1 of the circular code X are not considered, the circular code periodicity 0 modulo 3, hidden by the periodicity 1 modulo 2, is now retrieved in 5 groups of introns: amphibians, fishes, other animals, reptiles and insects. Thus, 10 groups of introns, taxonomically different, out of 12 have a coding property related to the reading frame retrieval. The trinucleotides N1N2N1 are analysed in the 216 maximal C3 self-complementary trinucleotide circular codes. A hexanucleotide code (words of 6 letters) is proposed to explain the periodicity 3 modulo 6. It could be a trace of more general circular codes at the origin of the circular code X.

Accurate phenotypic self-replication as a necessary cause for biological evolution

Since the Origin of Species, it has been known that evolution depends on what Darwin called the "strong principle of inheritance." Highly accurate replication of cellular phenotype is a universal phenomenon in all of life since LUCA and is often taken for granted as a constant in evolutionary theory. It is not known how self-replication arose during the origin of life. In this report I use the simple mathematics of evolutionary theory to investigate the dynamics of self-replication accuracy and allelic selection. Results indicate that the degree of self-replication accuracy must be greater than a threshold related to the selection coefficients of the alleles in a population in order for evolution to occur. Accurate replication of cellular phenotype and of the molecules involved in genotype/phenotype linkage is necessary for the origin of evolution and may be considered the fundamental principle of life.

A Proposal of the Ur-RNAome

It is widely accepted that the earliest RNA molecules were folded into hairpins or mini-helixes. Herein, we depict the 2D and 3D conformations of those earliest RNA molecules with only RNY triplets, which Eigen proposed as the primeval genetic code. We selected 26 species (13 bacteria and 13 archaea). We found that the free energy of RNY hairpins was consistently lower than that of their corresponding shuffled controls. We found traces of the three ribosomal RNAs (16S, 23S, and 5S), tRNAs, 6S RNA, and the RNA moieties of RNase P and the signal recognition particle. Nevertheless, at this stage of evolution there was no genetic code (as seen in the absence of the peptidyl transferase centre and any vestiges of the anti-Shine-Dalgarno sequence). Interestingly, we detected the anticodons of both glycine (GCC) and threonine (GGU) in the hairpins of proto-tRNA.

An Analysis of Nucleotide-Amyloid Interactions Reveals Selective Binding to Codon-Sized RNA

Interactions between RNA and proteins are the cornerstone of many important biological processes from transcription and translation to gene regulation, yet little is known about the ancient origin of said interactions. We hypothesized that peptide amyloids played a role in the origin of life and that their repetitive structure lends itself to building interfaces with other polymers through avidity. Here, we report that short RNA with a minimum length of three nucleotides binds in a sequence-dependent manner to peptide amyloids. The 3'-5' linked RNA backbone appears to be well-suited to support these interactions, with the phosphodiester backbone and nucleobases both contributing to the affinity. Sequence-specific RNA-peptide interactions of the kind identified here may provide a path to understanding one of the great mysteries rooted in the origin of life: the origin of the genetic code.

Circular mixed sets

In this article, we introduce the new mathematical concept of circular mixed sets of words over an arbitrary finite alphabet. These circular mixed sets may not be codes in the classical sense and hence allow a higher amount of information to be encoded. After describing their basic properties, we generalize a recent graph theoretical approach for circularity and apply it to distinguish codes from sets (i.e. non-codes). Moreover, several methods are given to construct circular mixed sets. Finally, this approach allows us to propose a new evolution model of the present genetic code that could have evolved from a dinucleotide world to a trinucleotide world via circular mixed sets of dinucleotides and trinucleotides.

From vesicles toward protocells and minimal cells

In contrast to ordinary condensed matter systems, "living systems" are unique. They are based on molecular compartments that reproduce themselves through (i) an uptake of ingredients and energy from the environment, and (ii) spatially and timely coordinated internal chemical transformations. These occur on the basis of instructions encoded in information molecules (DNAs). Life originated on Earth about 4 billion years ago as self-organised systems of inorganic compounds and organic molecules including macromolecules (e.g. nucleic acids and proteins) and low molar mass amphiphiles (lipids). Before the first living systems emerged from non-living forms of matter, functional molecules and dynamic molecular assemblies must have been formed as prebiotic soft matter systems. These hypothetical cell-like compartment systems often are called "protocells". Other systems that are considered as bridging units between non-living and living systems are called "minimal cells". They are synthetic, autonomous and sustainable reproducing compartment systems, but their constituents are not limited to prebiotic substances. In this review, we focus on both membrane-bounded (vesicular) protocells and minimal cells, and provide a membrane physics background which helps to understand how morphological transformations of vesicle systems might have happened and how vesicle reproduction might be coupled with metabolic reactions and information molecules. This research, which bridges matter and life, is a great challenge in which soft matter physics, systems chemistry, and synthetic biology must take joined efforts to better understand how the transformation of protocells into living systems might have occurred at the origin of life.

RNP-world: The ultimate essence of life is a ribonucleoprotein process

The fundamental essence of life is based on process of interaction between nucleic acids and proteins. In a prebiotic world, amino acids, peptides, ions, and other metabolites acted in protobiotic routes at the same time on which RNAs performed catalysis and self-replication. Nevertheless, it was only when nucleic acids and peptides started to interact together in an organized process that life emerged. First, the ignition was sparked with the formation of a Peptidyl Transferase Center (PTC), possibly by concatenation of proto-tRNAs. This molecule that would become the catalytic site of ribosomes started a process of self-organization that gave origin to a protoorganism named FUCA, a ribonucleic ribosomal-like apparatus capable to polymerize amino acids. In that sense, we review hypotheses about the origin and early evolution of the genetic code. Next, populations of open biological systems named progenotes were capable of accumulating and exchanging genetic material, producing the first genomes. Progenotes then evolved in two paths: some presented their own ribosomes and others used available ribosomes in the medium to translate their encoded information. At some point, two different types of organisms emerged from populations of progenotes: the ribosome-encoding organisms (cells) and the capsid-encoding organisms (viruses).

The Roots of Genetic Coding in Aminoacyl-tRNA Synthetase Duality

Codon-dependent translation underlies genetics and phylogenetic inferences, but its origins pose two challenges. Prevailing narratives cannot account for the fact that aminoacyl-tRNA synthetases (aaRSs), which translate the genetic code, must collectively enforce the rules used to assemble themselves. Nor can they explain how specific assignments arose from rudimentary differentiation between ancestral aaRSs and corresponding transfer RNAs (tRNAs). Experimental deconstruction of the two aaRS superfamilies created new experimental tools with which to analyze the emergence of the code. Amino acid and tRNA substrate recognition are linked to phase transfer free energies of amino acids and arise largely from aaRS class-specific differences in secondary structure. Sensitivity to protein folding rules endowed ancestral aaRS-tRNA pairs with the feedback necessary to rapidly compare alternative genetic codes and coding sequences. These and other experimental data suggest that the aaRS bidirectional genetic ancestry stabilized the differentiation and interdependence required to initiate and elaborate the genetic coding table.

Infinite combinatorics in mathematical biology

Is it possible to apply infinite combinatorics and (infinite) set theory in theoretical biology? We do not know the answer yet but in this article we try to present some techniques from infinite combinatorics and set theory that have been used over the last decades in order to prove existence results and independence theorems in algebra and that might have the flexibility and generality to be also used in theoretical biology. In particular, we will introduce the theory of forcing and an algebraic construction technique based on trees and forests using infinite binary sequences. We will also present an overview of the theory of circular codes. Such codes had been found in the genetic information and are assumed to play an important role in error detecting and error correcting mechanisms during the process of translation. Finally, examples and constructions of infinite mixed circular codes using binary sequences hopefully show some similarity between these theories - a starting point for future applications.

Reciprocally-Coupled Gating: Strange Loops in Bioenergetics, Genetics, and Catalysis

Bioenergetics, genetic coding, and catalysis are all difficult to imagine emerging without pre-existing historical context. That context is often posed as a "Chicken and Egg" problem; its resolution is concisely described by de Grasse Tyson: "The egg was laid by a bird that was not a chicken". The concision and generality of that answer furnish no details-only an appropriate framework from which to examine detailed paradigms that might illuminate paradoxes underlying these three life-defining biomolecular processes. We examine experimental aspects here of five examples that all conform to the same paradigm. In each example, a paradox is resolved by coupling "if, and only if" conditions for reciprocal transitions between levels, such that the consequent of the first test is the antecedent for the second. Each condition thus restricts fluxes through, or "gates" the other. Reciprocally-coupled gating, in which two gated processes constrain one another, is self-referential, hence maps onto the formal structure of "strange loops". That mapping uncovers two different kinds of forces that may help unite the axioms underlying three phenomena that distinguish biology from chemistry. As a physical analog for Gödel's logic, biomolecular strange-loops provide a natural metaphor around which to organize a large body of experimental data, linking biology to information, free energy, and the second law of thermodynamics.

Footprints of a Singular 22-Nucleotide RNA Ring at the Origin of Life

(1) Background: Previous experimental observations and theoretical hypotheses have been providing insight into a hypothetical world where an RNA hairpin or ring may have debuted as the primary informational and functional molecule. We propose a model revisiting the architecture of RNA-peptide interactions at the origin of life through the evolutionary dynamics of RNA populations. (2) Methods: By performing a step-by-step computation of the smallest possible hairpin/ring RNA sequences compatible with building up a variety of peptides of the primitive network, we inferred the sequence of a singular docosameric RNA molecule, we call the ALPHA sequence. Then, we searched for any relics of the peptides made from ALPHA in sequences deposited in the different public databases. (3) Results: Sequence matching between ALPHA and sequences from organisms among the earliest forms of life on Earth were found at high statistical relevance. We hypothesize that the frequency of appearance of relics from ALPHA sequence in present genomes has a functional necessity. (4) Conclusions: Given the fitness of ALPHA as a supportive sequence of the framework of all existing theories, and the evolution of Archaea and giant viruses, it is anticipated that the unique properties of this singular archetypal ALPHA sequence should prove useful as a model matrix for future applications, ranging from synthetic biology to DNA computing.

Transfer RNA: The molecular demiurge in the origin of biological systems

Herein, we review recent works on the role that the tRNA molecule played in the early origins of biological systems. tRNAs gave origin to the first genes (mRNA), the peptidyl transferase center (PTC), the 16S ribosomal molecule, proto-tRNAs were at the core of a proto-translation system, and the anticodon and operational codes appeared in tRNAs molecules. Metabolic pathways emerged from evolutionary pressures of the decoding systems. The transitions from the RNA world to the ribonucleoprotein world to modern biological systems were driven by two kinds of tRNAs transitions, to wit, tRNAs leading to both mRNA and rRNA.

Viral Symbiosis in the Origins and Evolution of Life with a Particular Focus on the Placental Mammals

Advances in understanding over the last decade or so highlight the need for a reappraisal of the role of viruses in relation to the origins and evolution of cellular life, as well as in the homeostasis of the biosphere on which all of life depends. The relevant advances have, in particular, revealed an important contribution of viruses to the evolution of the placental mammals, while also contributing key roles to mammalian embryogenesis, genomic evolution, and physiology. Part of this reappraisal will include the origins of viruses, a redefinition of their quintessential nature, and a suggestion as to how we might view viruses in relation to the tree of life.

Ribosome Evolution and Structural Capacitance

In addition to the canonical loss-of-function mutations, mutations in proteins may additionally result in gain-of-function through the binary activation of cryptic "structural capacitance elements." Our previous bioinformatic analysis allowed us to propose a new mechanism of protein evolution - structural capacitance - that arises via the generation of new elements of microstructure upon mutations that cause a disorder-to-order (D→O) transition in previously disordered regions of proteins. Here we propose that the D→O transition is a necessary follow-on from expected early codon-anticodon and tRNA acceptor stem-amino acid usage, via the accumulation of structural capacitance elements - reservoirs of disorder in proteins. We develop this argument further to posit that structural capacitance is an inherent consequence of the evolution of the genetic code.

Circular code motifs in the ribosome: a missing link in the evolution of translation?

The origin of the genetic code remains enigmatic five decades after it was elucidated, although there is growing evidence that the code coevolved progressively with the ribosome. A number of primordial codes were proposed as ancestors of the modern genetic code, including comma-free codes such as the RRY, RNY, or GNC codes (R = G or A, Y = C or T, N = any nucleotide), and the X circular code, an error-correcting code that also allows identification and maintenance of the reading frame. It was demonstrated previously that motifs of the X circular code are significantly enriched in the protein-coding genes of most organisms, from bacteria to eukaryotes. Here, we show that imprints of this code also exist in the ribosomal RNA (rRNA). In a large-scale study involving 133 organisms representative of the three domains of life, we identified 32 universal X motifs that are conserved in the rRNA of >90% of the organisms. Intriguingly, most of the universal X motifs are located in rRNA regions involved in important ribosome functions, notably in the peptidyl transferase center and the decoding center that form the original "proto-ribosome." Building on the existing accretion models for ribosome evolution, we propose that error-correcting circular codes represented an important step in the emergence of the modern genetic code. Thus, circular codes would have allowed the simultaneous coding of amino acids and synchronization of the reading frame in primitive translation systems, prior to the emergence of more sophisticated start codon recognition and translation initiation mechanisms.

Bias for 3'-Dominant Codon Directional Asymmetry in Theoretical Minimal RNA Rings

Aminoacyl tRNA synthetases ligate tRNAs specifically with their cognate amino acid. These synthetases are among life's earliest proteins, class II tRNA synthetases (cognates A, D, F, G, H, K, N, P, S, and T) presumably preceding class I tRNA synthetases (cognates C, E, I, L, M, Q, R, V, W, and Y). Classification of codons into palindromic (structure XYX), 5'-dominant (YXX), and 3'-dominant (XXY) (Codon Directional Asymmetry [CDA]) shows that class II tRNA synthetases aminoacylate amino acids associated with XXY. Our working hypothesis expects bias for XXY codons in primordial RNAs, such as theoretical minimal RNA rings, designed in silico to mimic life's earliest RNAs. Twenty-five RNA rings have been computed, which code over a minimal length (22 nucleotides) for a start codon, stop codon, and one and only one codon for each of the 20 amino acids, and form stem-loop hairpins preventing degradation; these 25 minimal RNAs are the only ones matching these constraints and they seem homologous to consensus tRNA sequences. This similarity defined candidate RNA ring anticodons and corresponding cognate amino acids. Here, analyses of RNA ring codon contents confirm bias for XXY codons in 13 among 14 RNA rings with unequal XXY and YXX codon numbers. This bias increases with the genetic code integration order of the RNA ring's cognate amino acid across and within tRNA synthetase classes, suggesting that evolutionary processes, and not physicochemical constraints, produced the association between CDA and tRNA synthetase classes. The self-referential hypothesis for genetic code origin, a very complete genetic code evolutionary hypothesis integrating many translational machinery components, predicts best among genetic code evolutionary hypotheses CDA biases in RNA rings. The RNA rings' simple design inadvertently reproduces CDAs predicted by the genetic code's structure, confirming theoretical minimal RNA rings as good proxies for life's earliest RNAs.

Genetic Code Evolution Investigated through the Synthesis and Characterisation of Proteins from Reduced-Alphabet Libraries

The universal genetic code of 20 amino acids is the product of evolution. It is believed that earlier versions of the code had fewer residues. Many theories for the order in which amino acids were integrated into the code have been proposed, considering factors ranging from prebiotic chemistry to codon capture. Several meta-analyses combined these theories to yield a feasible consensus chronology of the genetic code's evolution, but there is a dearth of experimental data to test the hypothesised order. We used combinatorial chemistry to synthesise libraries of random polypeptides that were based on different subsets of the 20 standard amino acids, thus representing different stages of a plausible history of the alphabet. Four libraries were comprised of the five, nine, and 16 most ancient amino acids, and all 20 extant residues for a direct side-by-side comparison. We characterised numerous variants from each library for their solubility and propensity to form secondary, tertiary or quaternary structures. Proteins from the two most ancient libraries were more likely to be soluble than those from the extant library. Several individual protein variants exhibited inducible protein folding and other traits typical of intrinsically disordered proteins. From these libraries, we can infer how primordial protein structure and function might have evolved with the genetic code.

The evolutionary dynamics of metabolic protocells

Protocell multilevel selection models have been proposed to study the evolutionary dynamics of vesicles encapsulating a set of replicating, competing and mutating sequences. The frequency of the different sequence types determines protocell survival through a fitness function. One of the defining features of these models is the genetic load generated when the protocell divides and its sequences are assorted between the offspring vesicles. However, these stochastic assortment effects disappear when the redundancy of each sequence type is sufficiently high. The fitness dependence of the vesicle with its sequence content is usually defined without considering a realistic account on how the lower level dynamics would specify the protocell fitness. Here, we present a protocell model with a fitness function determined by the output flux of a simple metabolic network with the aim of understanding how the evolution of both kinetic and topological features of metabolism would have been constrained by the particularities of the protocell evolutionary dynamics. In our model, the sequences inside the vesicle are both the carriers of information and Michaelis-Menten catalysts exhibiting saturation. We found that the saturation of the catalysts controlling the metabolic fluxes, achievable by modifying the kinetic or stoichiometric parameters, provides a mechanism to ameliorate the assortment load by increasing the redundancy of the catalytic sequences required to achieve the maximum flux. Regarding the network architecture, we conclude that combinations of parallel network motifs and bimolecular catalysts are a robust way to increase the complexity of the metabolism enclosed by the protocell.

The protocell hypothesis conjectures the existence of a vesicle containing catalytic and replicating sequences as the primordial cellular organization during the early stages of the evolution of life. Mathematical models of protocells traditionally consider RNA sequences being encapsulated and having both an informational and a catalytic role in the same molecule. Because of this dual function, the protocell sequences are evolutionarily constrained. Mathematical models have been extensively used to study the evolutionary dynamics of protocells with a focus on the processes, like mutation or stochastic sequence assortment upon division, that affect the protocell information capacity in terms of the coexistence of different sequence types. Here we introduce a simple model of metabolic networks whose output determines the survival of the protocell with the aim of studying the effect of modifying the kinetic and architectural properties of the network on sequence coexistence. We find that stochastic assortment and mutation limit the architectures able to be encapsulated by the protocell with a given fraction of the population harbouring all possible sequence types.

Biological meaning of "habitable zone" in nucleotide composition space

Organisms generally display two contrasting properties: large biodiversity and a uniform state of "life". In this study, we focused on the question of how genome sequences describe "life" where a large number of biomolecules are harmonized. We analyzed the whole genome sequence of 2664 organisms, paying attention to the nucleotide composition which is an intensive parameter from the genome sequence. The results showed that all organisms were plotted in narrow regions of the nucleotide composition space of the first and second letters of the codon. Since all genome sequences overlap irrespective of the living environment, it can be called a "habitable zone". The habitable zone deviates by 500 times the standard deviation from the nucleotide composition expected from the random sequence, indicating that unexpectedly rare sequences are realized. Furthermore, we found that the habitable zones at the first and second letters of the codon serve as the background mechanisms for the functional network of biological systems. The habitable zone at the second letter of the codon controls the formation of transmembrane regions and the habitable zone at the first letter controls the formation of molecular recognition unit. These analyses showed that the habitable zone of the nucleotide composition space and the exquisite arrangement of amino acids in the codon table are conjugated to form biological systems. Finally, we discussed the evolution of the higher order of genome sequences.

Reaction kinetics in open reactors and serial transfers between closed reactors

Kinetic theory and thermodynamics of reaction networks are extended to the out-of-equilibrium dynamics of continuous-flow stirred tank reactors (CSTR) and serial transfers. On the basis of their stoichiometry matrix, the conservation laws and the cycles of the network are determined for both dynamics. It is shown that the CSTR and serial transfer dynamics are equivalent in the limit where the time interval between the transfers tends to zero proportionally to the ratio of the fractions of fresh to transferred solutions. These results are illustrated with a finite cross-catalytic reaction network and an infinite reaction network describing mass exchange between polymers. Serial transfer dynamics is typically used in molecular evolution experiments in the context of research on the origins of life. The present study is shedding a new light on the role played by serial transfer parameters in these experiments.

A Proposal of the Ur-proteome

Herein we outline a plausible proteome, encoded by assuming a primeval RNY genetic code. We unveil the primeval phenotype by using only the RNA genotype; it means that we recovered the most ancestral proteome, mostly made of the 8 amino acids encoded by RNY triplets. By looking at those fragments, it is noticeable that they are positioned, not at catalytic sites, but in the cofactor binding sites. It implies that the stabilization of a molecule appeared long before its catalytic activity, and therefore the Ur-proteome comprised a set of proteins modules that corresponded to Cofactor Stabilizing Binding Sites (CSBSs), which we call the primitive bindome. With our method, we reconstructed the structures of the "first protein modules" that Sobolevsky and Trifonov (2006) found by using only RMSD. We also examine the probable cofactors that bound to them. We discuss the notion of CSBSs as the first proteins modules in progenotes in the context of several proposals about the primitive forms of life.

Molecular evolution between chemistry and biology : The interplay of competition, cooperation, and mutation

Biological evolution is reduced to three fundamental processes in the spirit of a minimal model: (i) Competition caused by differential fitness, (ii) cooperation of competitors in the sense of symbiosis, and (iii) variation introduced by mutation understood as error-prone reproduction. The three combinations of two fundamental processes each, ([Formula: see text]) competition and mutation, ([Formula: see text]) cooperation and competition, and ([Formula: see text]) cooperation and mutation, are analyzed. Changes in population dynamics that are induced by bifurcations and threshold phenomena are discussed.

Self-complementary circular codes in coding theory

Self-complementary circular codes are involved in pairing genetic processes. A maximal [Formula: see text] self-complementary circular code X of trinucleotides was identified in genes of bacteria, archaea, eukaryotes, plasmids and viruses (Michel in Life 7(20):1-16 2017, J Theor Biol 380:156-177, 2015; Arquès and Michel in J Theor Biol 182:45-58 1996). In this paper, self-complementary circular codes are investigated using the graph theory approach recently formulated in Fimmel et al. (Philos Trans R Soc A 374:20150058, 2016). A directed graph [Formula: see text] associated with any code X mirrors the properties of the code. In the present paper, we demonstrate a necessary condition for the self-complementarity of an arbitrary code X in terms of the graph theory. The same condition has been proven to be sufficient for codes which are circular and of large size [Formula: see text] trinucleotides, in particular for maximal circular codes ([Formula: see text] trinucleotides). For codes of small-size [Formula: see text] trinucleotides, some very rare counterexamples have been constructed. Furthermore, the length and the structure of the longest paths in the graphs associated with the self-complementary circular codes are investigated. It has been proven that the longest paths in such graphs determine the reading frame for the self-complementary circular codes. By applying this result, the reading frame in any arbitrary sequence of trinucleotides is retrieved after at most 15 nucleotides, i.e., 5 consecutive trinucleotides, from the circular code X identified in genes. Thus, an X motif of a length of at least 15 nucleotides in an arbitrary sequence of trinucleotides (not necessarily all of them belonging to X) uniquely defines the reading (correct) frame, an important criterion for analyzing the X motifs in genes in the future.

From quasispecies to quasispaces: coding and cooperation in chemical and electronic systems

This contribution addresses the physical roles of spatial structures, either externally imposed or generated through self-assembly, either passive or active, on the physical chemistry of evolution. Starting with simple diffusion in closed capillaries, a one-dimensional space, it covers eight aspects of experimental and theoretical research into the interaction of evolution with spatial structures: in various dimensions, including hitherto unexplored ones, spanning from externally defined physical spaces to actively tailored spaces, assembled by the evolving components themselves. As such, it contains some original research by the author as well as tracing how other insights grew over three decades out of the mentorship of Manfred Eigen in the 1980s. Much of the early interest in spatial structures centres on its role in stabilizing higher order cooperative structures involving the coevolution of different molecules, as the genetic coding system exemplifies. Modern nanotechnology enables the design and construction of genetically encoded variants of smart components that can actively control both the proliferation of molecules and the structuring of space. A key role for this article is to show the continuity in this line of enquiry, beginning with quasispecies and projecting to autonomous microparticles with electronic genomes able to form programmable quasispaces.

Nonequilibrium Entropic Bounds for Darwinian Replicators

Life evolved on our planet by means of a combination of Darwinian selection and innovations leading to higher levels of complexity. The emergence and selection of replicating entities is a central problem in prebiotic evolution. Theoretical models have shown how populations of different types of replicating entities exclude or coexist with other classes of replicators. Models are typically kinetic, based on standard replicator equations. On the other hand, the presence of thermodynamical constraints for these systems remain an open question. This is largely due to the lack of a general theory of statistical methods for systems far from equilibrium. Nonetheless, a first approach to this problem has been put forward in a series of novel developements falling under the rubric of the extended second law of thermodynamics. The work presented here is twofold: firstly, we review this theoretical framework and provide a brief description of the three fundamental replicator types in prebiotic evolution: parabolic, malthusian and hyperbolic. Secondly, we employ these previously mentioned techinques to explore how replicators are constrained by thermodynamics. Finally, we comment and discuss where further research should be focused on.

Enrichment of Circular Code Motifs in the Genes of the Yeast Saccharomyces cerevisiae

A set X of 20 trinucleotides has been found to have the highest average occurrence in the reading frame, compared to the two shifted frames, of genes of bacteria, archaea, eukaryotes, plasmids and viruses. This set X has an interesting mathematical property, since X is a maximal C3 self-complementary trinucleotide circular code. Furthermore, any motif obtained from this circular code X has the capacity to retrieve, maintain and synchronize the original (reading) frame. Since 1996, the theory of circular codes in genes has mainly been developed by analysing the properties of the 20 trinucleotides of X, using combinatorics and statistical approaches. For the first time, we test this theory by analysing the X motifs, i.e., motifs from the circular code X, in the complete genome of the yeast Saccharomyces cerevisiae. Several properties of X motifs are identified by basic statistics (at the frequency level), and evaluated by comparison to R motifs, i.e., random motifs generated from 30 different random codes R. We first show that the frequency of X motifs is significantly greater than that of R motifs in the genome of S. cerevisiae. We then verify that no significant difference is observed between the frequencies of X and R motifs in the non-coding regions of S. cerevisiae, but that the occurrence number of X motifs is significantly higher than R motifs in the genes (protein-coding regions). This property is true for all cardinalities of X motifs (from 4 to 20) and for all 16 chromosomes. We further investigate the distribution of X motifs in the three frames of S. cerevisiae genes and show that they occur more frequently in the reading frame, regardless of their cardinality or their length. Finally, the ratio of X genes, i.e., genes with at least one X motif, to non-X genes, in the set of verified genes is significantly different to that observed in the set of putative or dubious genes with no experimental evidence. These results, taken together, represent the first evidence for a significant enrichment of X motifs in the genes of an extant organism. They raise two hypotheses: the X motifs may be evolutionary relics of the primitive codes used for translation, or they may continue to play a functional role in the complex processes of genome decoding and protein synthesis.

Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems

As of today, the most credible scientific paradigm pertaining to the origin of life on Earth is undoubtedly the RNA World scenario. It is built on the assumption that catalytically active replicators (most probably RNA-like macromolecules) may have been responsible for booting up life almost four billion years ago. The many different incarnations of nucleotide sequence (string) replicator models proposed recently are all attempts to explain on this basis how the genetic information transfer and the functional diversity of prebiotic replicator systems may have emerged, persisted and evolved into the first living cell. We have postulated three necessary conditions for an RNA World model system to be a dynamically feasible representation of prebiotic chemical evolution: (1) it must maintain and transfer a sufficient diversity of information reliably and indefinitely, (2) it must be ecologically stable and (3) it must be evolutionarily stable. In this review, we discuss the best-known prebiotic scenarios and the corresponding models of string-replicator dynamics and assess them against these criteria. We suggest that the most popular of prebiotic replicator systems, the hypercycle, is probably the worst performer in almost all of these respects, whereas a few other model concepts (parabolic replicator, open chaotic flows, stochastic corrector, metabolically coupled replicator system) are promising candidates for development into coherent models that may become experimentally accessible in the future.

Strong Comma-Free Codes in Genetic Information

Comma-free codes constitute a class of circular codes, which has been widely studied, in particular by Golomb et al. (Biologiske Meddelelser, Kongelige Danske Videnskabernes Selskab 23:1-34, 1958a, Can J Math 10:202-209, 1958b), Michel et al. (Comput Math Appl 55:989-996, 2008a, Theor Comput Sci 401:17-26, 2008b, Inf Comput 212:55-63, 2012), Michel and Pirillo (Int J Comb 2011:659567, 2011), and Fimmel and Strüngmann (J Theor Biol 389:206-213, 2016). Based on a recent approach using graph theory to study circular codes Fimmel et al. (Philos Trans R Soc 374:20150058, 2016), a new class of circular codes, called strong comma-free codes, is identified. These codes detect a frameshift during the translation process immediately after a reading window of at most two nucleotides. We describe several combinatorial properties of strong comma-free codes: enumeration, maximality, self-complementarity and [Formula: see text]-property (comma-free property in all the three possible frames). These combinatorial results also highlight some new properties of the genetic code and its evolution. Each amino acid in the standard genetic code is coded by at least one strong comma-free code of size 1. There are 9 amino acids [Formula: see text] among 20 such that for each amino acid from S, its synonymous trinucleotide set (excluding the necessary periodic trinucleotides [Formula: see text]) is a strong comma-free code. The primeval comma-free RNY code of Eigen and Schuster (Naturwissenschaften 65:341-369, 1978) is a self-complementary [Formula: see text]-code of size 16. Furthermore, it is the union of two strong comma-free codes of size 8 which are complementary to each other.

A unified model of the standard genetic code

The Rodin-Ohno (RO) and the Delarue models divide the table of the genetic code into two classes of aminoacyl-tRNA synthetases (aaRSs I and II) with recognition from the minor or major groove sides of the tRNA acceptor stem, respectively. These models are asymmetric but they are biologically meaningful. On the other hand, the standard genetic code (SGC) can be derived from the primeval RNY code (R stands for purines, Y for pyrimidines and N any of them). In this work, the RO-model is derived by means of group actions, namely, symmetries represented by automorphisms, assuming that the SGC originated from a primeval RNY code. It turns out that the RO-model is symmetric in a six-dimensional (6D) hypercube. Conversely, using the same automorphisms, we show that the RO-model can lead to the SGC. In addition, the asymmetric Delarue model becomes symmetric by means of quotient group operations. We formulate isometric functions that convert the class aaRS I into the class aaRS II and vice versa. We show that the four polar requirement categories display a symmetrical arrangement in our 6D hypercube. Altogether these results cannot be attained, neither in two nor in three dimensions. We discuss the present unified 6D algebraic model, which is compatible with both the SGC (based upon the primeval RNY code) and the RO-model.

Kinetics and thermodynamics of living copolymerization processes

Theoretical advances are reported on the kinetics and thermodynamics of free and template-directed living copolymerizations. Until recently, the kinetic theory of these processes had only been established in the fully irreversible regime, in which the attachment rates are only considered. However, the entropy production is infinite in this regime and the approach to thermodynamic equilibrium cannot be investigated. For this purpose, the detachment rates should also be included. Inspite of this complication, the kinetics can be exactly solved in the regimes of steady growth and depolymerization. In this way, analytical expressions are obtained for the mean growth velocity, the statistical properties of the copolymer sequences, as well as the thermodynamic entropy production. The results apply to DNA replication, transcription and translation, allowing us to understand important aspects of molecular evolution.This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.

Recent advances in inferring viral diversity from high-throughput sequencing data

Rapidly evolving RNA viruses prevail within a host as a collection of closely related variants, referred to as viral quasispecies. Advances in high-throughput sequencing (HTS) technologies have facilitated the assessment of the genetic diversity of such virus populations at an unprecedented level of detail. However, analysis of HTS data from virus populations is challenging due to short, error-prone reads. In order to account for uncertainties originating from these limitations, several computational and statistical methods have been developed for studying the genetic heterogeneity of virus population. Here, we review methods for the analysis of HTS reads, including approaches to local diversity estimation and global haplotype reconstruction. Challenges posed by aligning reads, as well as the impact of reference biases on diversity estimates are also discussed. In addition, we address some of the experimental approaches designed to improve the biological signal-to-noise ratio. In the future, computational methods for the analysis of heterogeneous virus populations are likely to continue being complemented by technological developments.

Some mechanistic requirements for major transitions

Major transitions in nature and human society are accompanied by a substantial change towards higher complexity in the core of the evolving system. New features are established, novel hierarchies emerge, new regulatory mechanisms are required and so on. An obvious way to achieve higher complexity is integration of autonomous elements into new organized systems whereby the previously independent units give up their autonomy at least in part. In this contribution, we reconsider the more than 40 years old hypercycle model and analyse it by the tools of stochastic chemical kinetics. An open system is implemented in the form of a flow reactor. The formation of new dynamically organized units through integration of competitors is identified with transcritical bifurcations. In the stochastic model, the fully organized state is quasi-stationary whereas the unorganized state corresponds to a population with natural selection. The stability of the organized state depends strongly on the number of individual subspecies, n, that have to be integrated: two and three classes of individuals, [Formula: see text] and [Formula: see text], readily form quasi-stationary states. The four-membered deterministic dynamical system, [Formula: see text], is stable but in the stochastic approach self-enhancing fluctuations drive it into extinction. In systems with five and more classes of individuals, [Formula: see text], the state of cooperation is unstable and the solutions of the deterministic ODEs exhibit large amplitude oscillations. In the stochastic system self-enhancing fluctuations lead to extinction as observed with [Formula: see text] Interestingly, cooperative systems in nature are commonly two-membered as shown by numerous examples of binary symbiosis. A few cases of symbiosis of three partners, called three-way symbiosis, have been found and were analysed within the past decade. Four-way symbiosis is rather rare but was reported to occur in fungus-growing ants. The model reported here can be used to illustrate the interplay between competition and cooperation whereby we obtain a hint on the role that resources play in major transitions. Abundance of resources seems to be an indispensable prerequisite of radical innovation that apparently needs substantial investments. Economists often claim that scarcity is driving innovation. Our model sheds some light on this apparent contradiction. In a nutshell, the answer is: scarcity drives optimization and increase in efficiency but abundance is required for radical novelty and the development of new features.This article is part of the themed issue 'The major synthetic evolutionary transitions'.

Anticodon Modifications in the tRNA Set of LUCA and the Fundamental Regularity in the Standard Genetic Code

Based on (i) an analysis of the regularities in the standard genetic code and (ii) comparative genomics of the anticodon modification machinery in the three branches of life, we derive the tRNA set and its anticodon modifications as it was present in LUCA. Previously we proposed that an early ancestor of LUCA contained a set of 23 tRNAs with unmodified anticodons that was capable of translating all 20 amino acids while reading 55 of the 61 sense codons of the standard genetic code (SGC). Here we use biochemical and genomic evidence to derive that LUCA contained a set of 44 or 45 tRNAs containing 2 or 3 modifications while reading 59 or 60 of the 61 sense codons. Subsequent tRNA modifications occurred independently in the Bacteria and Eucarya, while the Archaea have remained quite close to the tRNA set as it was present in LUCA.

Kinetics and thermodynamics of DNA polymerases with exonuclease proofreading

Kinetic theory and thermodynamics are applied to DNA polymerases with exonuclease activity, taking into account the dependence of the rates on the previously incorporated nucleotide. The replication fidelity is shown to increase significantly thanks to this dependence at the basis of the mechanism of exonuclease proofreading. In particular, this dependence can provide up to a 100-fold lowering of the error probability under physiological conditions. Theory is compared with numerical simulations for the DNA polymerases of T7 viruses and human mitochondria.

tRNA Core Hypothesis for the Transition from the RNA World to the Ribonucleoprotein World

Herein we present the tRNA core hypothesis, which emphasizes the central role of tRNAs molecules in the origin and evolution of fundamental biological processes. tRNAs gave origin to the first genes (mRNA) and the peptidyl transferase center (rRNA), proto-tRNAs were at the core of a proto-translation system, and the anticodon and operational codes then arose in tRNAs molecules. Metabolic pathways emerged from evolutionary pressures of the decoding systems. The transitions from the RNA world to the ribonucleoprotein world to modern biological systems were driven by three kinds of tRNAs transitions, to wit, tRNAs leading to both mRNA and rRNA.

Codon Distribution in Error-Detecting Circular Codes

In 1957, Francis Crick et al. suggested an ingenious explanation for the process of frame maintenance. The idea was based on the notion of comma-free codes. Although Crick’s hypothesis proved to be wrong, in 1996, Arquès and Michel discovered the existence of a weaker version of such codes in eukaryote and prokaryote genomes, namely the so-called circular codes. Since then, circular code theory has invariably evoked great interest and made significant progress. In this article, the codon distributions in maximal comma-free, maximal self-complementary C3 and maximal self-complementary circular codes are discussed, i.e., we investigate in how many of such codes a given codon participates. As the main (and surprising) result, it is shown that the codons can be separated into very few classes (three, or five, or six) with respect to their frequency. Moreover, the distribution classes can be hierarchically ordered as refinements from maximal comma-free codes via maximal self-complementary C3 codes to maximal self-complementary circular codes.

n-Nucleotide circular codes in graph theory

The circular code theory proposes that genes are constituted of two trinucleotide codes: the classical genetic code with 61 trinucleotides for coding the 20 amino acids (except the three stop codons {TAA,TAG,TGA}) and a circular code based on 20 trinucleotides for retrieving, maintaining and synchronizing the reading frame. It relies on two main results: the identification of a maximal C(3) self-complementary trinucleotide circular code X in genes of bacteria, eukaryotes, plasmids and viruses (Michel 2015 J. Theor. Biol. 380, 156-177. (doi:10.1016/j.jtbi.2015.04.009); Arquès & Michel 1996 J. Theor. Biol. 182, 45-58. (doi:10.1006/jtbi.1996.0142)) and the finding of X circular code motifs in tRNAs and rRNAs, in particular in the ribosome decoding centre (Michel 2012 Comput. Biol. Chem. 37, 24-37. (doi:10.1016/j.compbiolchem.2011.10.002); El Soufi & Michel 2014 Comput. Biol. Chem. 52, 9-17. (doi:10.1016/j.compbiolchem.2014.08.001)). The univerally conserved nucleotides A1492 and A1493 and the conserved nucleotide G530 are included in X circular code motifs. Recently, dinucleotide circular codes were also investigated (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. (doi:10.1155/2013/538631); Fimmel et al. 2015 J. Theor. Biol. 386, 159-165. (doi:10.1016/j.jtbi.2015.08.034)). As the genetic motifs of different lengths are ubiquitous in genes and genomes, we introduce a new approach based on graph theory to study in full generality n-nucleotide circular codes X, i.e. of length 2 (dinucleotide), 3 (trinucleotide), 4 (tetranucleotide), etc. Indeed, we prove that an n-nucleotide code X is circular if and only if the corresponding graph [Formula: see text] is acyclic. Moreover, the maximal length of a path in [Formula: see text] corresponds to the window of nucleotides in a sequence for detecting the correct reading frame. Finally, the graph theory of tournaments is applied to the study of dinucleotide circular codes. It has full equivalence between the combinatorics theory (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. (doi:10.1155/2013/538631)) and the group theory (Fimmel et al. 2015 J. Theor. Biol. 386, 159-165. (doi:10.1016/j.jtbi.2015.08.034)) of dinucleotide circular codes while its mathematical approach is simpler.