The probability of the evolution of a character depends on two factors: the probability of moving from one character state to another character state and the probability of the new character state fixation. More the evolution of a character is probable more convergent evolution will be witnessed, consequently, convergent evolution could mean that the convergent character evolution result as a combination of these two factors. We investigate this phenomenon by studying the convergent evolution of biochemical functions. We use for the investigation the case of β-lactamases. β-lactamases hydrolyzes β-lactams which are antimicrobials able to block the DD-peptidases involved in bacterial cell wall synthesis. β-lactamase activity is present in two different superfamilies: the metallo-β-lactamase and the serine β-lactamase superfamily. The mechanism used to hydrolyze the β-lactam is different for the two superfamilies. We named this kind of evolution an allo-convergent evolution. We further show that the β-lactamase activity evolved several times within each superfamily, a convergent evolution type that we named iso-convergent evolution. Both types of convergent evolution can be explained by the two evolutionary mechanisms discussed above. The probability of moving from one state to another is explaining the promiscuous β-lactamase activity present in the ancestral sequences of each superfamily, while the probability of fixation is explained in part, by positive selection as the organisms having β-lactamase activity allows them to resist to organism secreting β-lactams. Indeed a mutation increasing the β-lactamases activity will be selected as the organisms having this activity will have an advantage over the others.

A surprising insight from the advent of genomic sequencing was that many genes are deeply conserved during evolution. With a particular focus on genes that interact with light in animals, I explore the metaphor of genetic toolkits, which can be operationalized as lists of genes involved in a trait of interest. A fascinating observation is that genes of a toolkit are often used again and again during convergent evolution, sometimes across vast phylogenetic distances. Such a pattern in the evolution of toolkits requires three different stages: origin, maintenance, and redeployment of the genes. The functional origins of toolkit genes might often be rooted in interactions with external environments. The origins of light interacting genes in particular may be tied to ancient responses to photo-oxidative stress, inspiring questions about the extent to which the evolution of other toolkits were also impacted by stress. Maintenance of genetic toolkits over long evolutionary timescales requires gene multifunctionality to prevent gene loss when a trait of interest is absent. Finally, the deployment of toolkit genes in convergently evolved traits like eyes sometimes involves the repeated use of similar, ancient genes but other times involves different genes specific to each convergent origin. How often a particular gene family is used time and again for the same function may depend on how many possible biological solutions are available. When few solutions exist and are maintained, evolution is constrained to use the same genes over and over. However, when many different solutions are possible, the innovative possibilities of evolution are often on display. Therefore, a focus on genetic toolkits highlights the combination of legacy plus innovation that drives the evolution of biological diversity.

There are no theorems (proven theories) in the biological sciences. We propose that the 3 31 nt minihelix tRNA evolution theorem be universally accepted as one. The 3 31 nt minihelix theorem completely describes evolution of type I and type II tRNAs from ordered precursors (RNA repeats and inverted repeats). Despite diversification of tRNAome sequences, statistical tests overwhelmingly support the theorem. Furthermore, the theorem relates the dominant pathway for the origin of life on Earth, specifically, how tRNAomes and the genetic code may have coevolved. Alternate models for tRNA evolution (i.e., 2 minihelix, convergent and accretion models) are falsified. In the context of the pre-life world, tRNA was a molecule that via mutation could modify anticodon sequences and teach itself to code. We relate the clearest history to date of the chemical evolution of life on Earth. From analysis of tRNA evolution, ribozyme-mediated RNA ligation was a primary driving force in evolution of complexity during the pre-life to life transition on Earth.

A fundamental question in biology is whether phenotypes can be predicted by ecological or genomic rules. At least five cases of convergent evolution of the crab-like body plan (with a wide and flattened shape, and a bent abdomen) are known in decapod crustaceans, and have, for over 140 years, been known as ‘carcinization’. The repeated loss of this body plan has been identified as ‘decarcinization’. In reviewing the field, we offer phylogenetic strategies to include poorly known groups, and direct evidence from fossils, that will resolve the history of crab evolution and the degree of phenotypic variation within crabs. Proposed ecological advantages of the crab body are summarized into a hypothesis of phenotypic integration suggesting correlated evolution of the carapace shape and abdomen. Our premise provides fertile ground for future studies of the genomic and developmental basis, and the predictability, of the crab-like body form.

Polar auxin transport (PAT) and auxin efflux carriers are known components controlling leaf complexity and venation patterns in some model plant species. Evidence indicates that PAT generates auxin converge points (CPs) that in turn lead to local leaf formation and internally into major vein formation. However, the role of PAT in more diverse leaf arrangements and vein patterns is largely unknown. We used pharmacological inhibition of PAT in developing pinnate tomato, trifoliate clover, palmate lupin, and bipinnate carrot leaves and observed dosage-dependent reduction to simple leaves in these eudicots. Leaf venation patterns changed from craspedodromous (clover, carrot), semi-craspedodromous (tomato) and brochidodromous (lupin) to more parallel patterning with PAT inhibition. The visualization of auxin responses in transgenic tomato plants showed that discrete and separate CPs in control plants were replaced by diffuse convergence areas near the margin. These effects indicate that PAT plays a universal role in the formation of different leaf and vein patterns in eudicot species via a mechanism that depends on the generation as well as separation of auxin CPs. Computer simulations indicate that variations in PAT can alter the number of CPs, corresponding leaf lobe formation and position of major leaf veins along the leaf margin in support of experimental results.

The evolution of divaricate plants in New Zealand has been the subject of long-running debate among botanists and ecologists. Hypotheses about this remarkable case of convergent evolution have focused mainly on two different types of selective pressures: the Plio-Pleistocene advent of cool, dry climates, or browsing by now-extinct moa. Here, we review the scientific literature relating to the New Zealand divaricates, and present a list of 81 taxa whose architectures fall on the divaricate habit spectrum. We recommend a series of standardised terms to facilitate clear communication about these species. We identify potentially informative areas of research yet to be explored, such as the genetics underlying the establishment and control of this habit. We also review work about similar plants overseas, proposing a list of 47 such species as a first step towards more comprehensive inventories; these may motivate further studies of the ecology, morphology and evolutionary history of these overseas plants which could help shed light on the evolution of their New Zealand counterparts. Finally, we compile published divergence dates between divaricate species and their non-divaricate relatives, which suggest that the divaricate habit is fairly recent (< 10 My) in most cases.

This research not only dedicated a less restrictive method of iteration-varying function for a learning control law to design a controller but also synchronize two nonlinear systems with free time-delay. In addition, the mathematical theory of system synchronization has proved rigorously and the theory verified through an example to demonstrate the behavior of each parameter in the theory. The design of a controller using the iterative learning control law is significant for robotic tracking. The controller in this research generates a feed-forward control input using the error dynamics among the drive-response systems. The error dynamics satisfies the Lyapunov function and the combination of output errors, which respectively represented relative estimated differences of the drive-response systems. The iterative learning control rule serves the function of a filter adding previous control error after the end of each iteration. The numerical example of a synchronous system is given a Lorenz system for driving and another with the iterative learning control law for response under different initial condition. The results verify and demonstrate the proposed mathematical theory. The simulation exhibits consistency in the behavior of each parameter to match mathematical theory.

In genetic conflicts, driver and killer elements achieve biased survival, replication, or transmission over sensitive and targeted elements through a wide range of molecular mechanisms, including mimicry. Driving mechanisms manifest at all organismal levels, from the biased propagation of individual genes, as demonstrated by transposable elements, to the biased transmission of genomes, as illustrated by viruses, to the biased transmission of cell lineages, as in cancer. Targeted genomes are vulnerable to molecular mimicry through the conserved motifs they use for their own signaling and regulation. Mimicking these motifs enables a selfish element to control core target processes, and can occur at the sequence, structure, or functional level. Molecular mimicry was first appreciated as an important phenomenon more than twenty years ago. Modern genomics technologies, databases, and machine learning approaches offer tremendous potential to study the distribution of molecular mimicry across genetic conflicts in nature. Here, we explore the theoretical expectations for molecular mimicry between conflicting genomes, the trends in molecular mimicry mechanisms across known genetic conflicts, and outline how new examples can be gleaned from population genomic datasets. We discuss how mimics involving short sequence-based motifs or gene duplications can evolve convergently from new mutations. Whereas, processes that involve divergent domains or fully-folded structures occur among genomes by horizontal gene transfer. These trends are largely based on a small number of organisms and should be reevaluated in a general, phylogenetically independent framework. Currently, publicly available databases can be mined for genotypes driving non-mendelian inheritance patterns, epistatic interactions, and convergent protein structures. A subset of these conflicting elements may be molecular mimics. We propose approaches for detecting genetic conflict and molecular mimicry from these datasets.

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.

Reptilians or reptoids are staple characters of Hollywood cinema, science fiction literature, ufology and conspiracy theories. Despite its popularity, the reptoid has been dismissed by the scientific community as a figment of the imagination; born out of a speculation, paranoia and wanton anthropocentricism. In this paper, however, the Life Sequence Conjecture is advanced as a supportive framework for the notion of reptilian extraterrestrials. This paper is not an argument in support of the existence of extraterrestrials. Rather, it proposes that if extraterrestrials do exist, then a reptilian morphology is not untenable.