Genomic analysis indicates that SARS-CoV-2 is most related to RaTG13, a beta corona virus derived from bats by 96% ¹. At present, RaTG13 is only available on the public database in the form of a genome sequence. The genome of RaTG13 (MN996532.1) was sequenced from the RNA of a bat faecal swab collected in 2013 from Yunnan, China, however the exact location is not mentioned. Since RaTG13 is one of the main supports for SARS-CoV-2 to have a natural origin, it is of utmost importance to understand the sample location. RNA dependent RNA polymerase (RdRp) sequence of RaTG13 shows that it is 100% similar to that of bat corona virus BtCoV/4991 and 98.7-98.9% similar to SARS-CoV-2 RdRp ². BtCoV/4991 was described to be a SARS-like (SL-) corona virus from bat faeces sampled in an abandoned mine from Mojiang ². Both the publications ^1,2 are authored by Dr. Zheng-li Shi (Z-L Shi), who is described as the bat woman of China ³. However, BtCoV/4991 has not been mentioned by Zhou et al 2020 ¹ where novel corona virus was first described. Based on the RdRp sequence similarities, similarities in sample collection dates, sample locations, and the fact that RaTG13 is mentioned synonymous to BtCoV/4991 on the Chinese bat database, it is predicted that RaTG13 and BtCoV/4991 originate from the same sample. The sample, bat faecal swab was collected in 2013 from an abandoned mineshaft in Mojiang by Dr. Shi and her work group. In 2012, in a Mojiang mineshaft, six mine workers suffered from atypical pneumonia and three of them died. These workers were engaged in the work of clearing debris from a mineshaft which had a lot of bats and bat faeces ^3,4. A detailed health investigation indicated that the miners suffered from atypical pneumonia mostly of the viral origin ⁴. Therefore, in the light of the present Covid-19 caused by SARS-CoV-2, the fact that its phylogenetic neighbour RaTG13 originated from bat faeces collected from a mineshaft, which was also the origin of pneumonia-like disease in miners in 2012, should be noted.

Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing like the fast Fourier transform, Ramanujan sum signal processing and many other techniques. For space-time topological objects in physics and biology, we propose a type of algebraic processing based on schemes in which the discrimination of singularities within objects is based on the space-time-spin group SL(2,C). Such topological objects possess an homotopy structure encoded in their fundamental group and the related SL(2,C) multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to an Akbulut cork in exotic R4, to an hyperbolic model of topological quantum computing based on algebraic surfaces and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum.

We define a spinor-Minkowski metric for SL(4,C). It is not a trivial generalization of the SL(2,C) metric and it involves the Minkowskian one. We define 4x4 version of the Pauli matrices and eight 4-component associated generalized eigenvectors that can be regarded as undotted covariant spinors. The 4-component spinors can be grouped into four categories. Each category transforms in its own way. The outer products of pairwise combinations of 4-component spinors can be associated with 4-vectors. Including the dotted covariant, undotted and dotted contravariant forms totally we have sixteen pairs of spinors. Eight of them live in the conjugate space which has no countepart in SL(2,C).

We reelaborate on the basic properties of PT symmetry from a geometrical Perspective. The transfer matrix associated with these systems induces a Möbius transformation in the complex plane. The trace of this matrix classifies the actions into three types that represent rotations, translations, and parallel displacements. We find that a PT invariant system can be pictured as a complex conjugation followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations and link them with measurable properties of the system.

Leishmania major is the main causative agent of cutaneous leishmaniasis in humans. The Friedlin strain of this species (LmjF) was chosen when a multi-laboratory consortium undertook the objective of deciphering the first genome sequence for a parasite of the genus Leishmania. The objective was successfully attained in 2005, and this represented a milestone for Leishmania molecular biology studies around the world. Although the LmjF genome sequence was done following a shotgun strategy and using classical Sanger sequencing, the results were excellent and this genome assembly served as the reference for subsequent genome assemblies in other Leishmania species. Here, we present a new assembly for the genome of this strain (named LMJFC for clarity), generated by the combination of two high throughput sequencing platforms, Illumina short-read sequencing and PacBio Single Molecular Real-Time (SMRT) sequencing, which provides long-read sequences. Apart from resolving uncertain nucleotide positions, several genomic regions have been reorganized and a more precise composition of tandemly repeated gene loci was attained. Additionally, the genome annotation has been improved by adding 542 genes and more accurate coding-sequences defined for around two hundred genes, based on the transcriptome delimitation also carried out in this work. As a result, we are providing gene models (including untranslated regions and introns) for 11,238 genes. Genomic information ultimately determines the biology of every organism; therefore, our understanding of molecular mechanisms will depend on the availability of precise genome sequences and accurate gene annotations. In this regards, this work is providing an improved genome sequence and updated transcriptome annotations for the reference L. major Friedlin strain.

We recently proposed that topological quantum computing might be based on $SL(2,\mathbb{C})$ representations of the fundamental group $\pi_1(S^3\setminus K)$ for the complement of a link $K$ in the $3$-sphere. The restriction to links whose associated $SL(2,\mathbb{C})$ character variety $\mathcal{V}$ contains a Fricke surface $\kappa_d=xyz -x^2-y^2-z^2+d$ is desirable due to the connection of Fricke spaces to elementary topology. Taking $K$ as the Hopf link $L2a1$, one of the three arithmetic two-bridge links [the Whitehead link $5_1^2$, the Berge link $6_2^2$, the double-eight link $6_3^2$] or the link $7_3^2$, the $\mathcal{V}$ for those links contains the reducible component $\kappa_4$, the so-called Cayley cubic. In addition, the $\mathcal{V}$ for the later two links contains the irreducible component $\kappa_3$, or $\kappa_2$, respectively. Taking $\rho$ to be a representation with character $\kappa_d$ ($d<4$), with $|x|,|y|,|z| \le 2$, then $\rho(\pi_1)$ fixes a unique point in the hyperbolic space $\mathcal{H}_3$ and is conjugate to a $SU(2)$ representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the $4$-punctured sphere and Painlev\'e VI equation. The $0$-surgery on the $3$ circles of the Borromean rings L6a4 is taken as an example.

Our recent work has focused on the application of infinite group theory and related algebraic geometric tools in the context of transcription factors and microRNAs. We were able to differentiate between “healthy" nucleotide sequences and disrupted sequences that may be associated with various diseases. In this paper, we extend our efforts to the study of messenger RNA metabolism, showcasing the power of our approach. We investigate (i) mRNA translation in prokaryotes and eukaryotes, (ii) polyadenylation in eukaryotes, which is crucial for nuclear export, translation, stability, and splicing of mRNA, (iii) miRNAs involved in RNA silencing and post-transcriptional regulation of gene expression, and (iv) the identification of disrupted sequences that could lead to potential illnesses. To achieve this, we employ (a) infinite (finitely generated) groups fp, with generators representing the r+1 distinct nucleotides and a relation between them [e.g., the consensus sequence in the mRNA translation (i), the poly(A) tail in item (ii), and the miRNA seed in item (iii)], (b) aperiodicity theory, which connects “healthy groups" fp to free groups Fr of rank r and their profinite completion F^r, and (c) the representation theory of groups fp over the space-time-spin group SL2(C), highlighting the role of surfaces with isolated singularities in the character variety. Our approach could potentially contribute to the understanding of the molecular mechanisms underlying various diseases and help develop new diagnostic or therapeutic strategies.

The symmetries of a Riemann surface $\Sigma \setminus \{a_i\}$ with $n$ punctures $a_i$ are encoded in its fundamental group $\pi_1(\Sigma)$. Further structure may be described through representations (homomorphisms) of $\pi_1$ over a Lie group $G$ as globalized by the character variety $\mathcal{C}=\mbox{Hom} (\pi_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the $4$-punctured Riemann sphere $\Sigma=S_2^{(4)}$ and the \lq space-time-spin' group $G=SL_2(\mathbb{C})$. In such a situation, $\mathcal{C}$ possesses remarkable properties (i) a representation is described by a $3$-dimensional cubic surface $V_{a,b,c,d}(x,y,z)$ with $3$ variables and $4$ parameters, (ii) the automorphisms of the surface satisfy the dynamical (non linear and transcendental) Painlev\'e VI equation (or $P_{VI}$), (iii) there exists a finite set of $1$ (\mbox{Cayley-Picard})+$3$ (\mbox{continuous platonic})+$45$ (\mbox{icosahedral}) solutions of $P_{VI}$. In this paper we feature on the parametric representation of some solutions of $P_{VI}$, (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups $f_p$ encountered in TQC or DNA/RNA sequences are proposed.

This paper presents the symmetries of differential equations associated with one-variable and Bivariate Hermite polynomials by proposing a representation of Lie algebra for these differential operators. Applying the Baker-Campbell-Hausdorff formula to these algebras, results in new relations and generating functions in one-variable and Bivariate Hermite polynomials. A general form of representation for other orthogonal polynomials such as Laguerre polynomials is introduced.

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. In this paper, we describe the algebraic geometry of both TFs and miRNAs thanks to group theory. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group $\pi$, we compute the $SL(2,\mathbb{C})$ character variety, where $SL(2,\mathbb{C})$ is simultaneously a \lq space-time' (a Lorentz group) and a \lq quantum' (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when $\pi$ looks like a free group $F_r$ ($r =1$ to $3$) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis $\mathcal{G}$ of the variety. A surface with simple singularities (like the well known Cayley cubic) within $\mathcal{G}$ is a signature of a potential disease even when $G$ looks like a free group $F_r$ in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in $\{A,T/U,G,C\}$. Several human TFs and miRNAs are investigated in detail thanks to our approach.

As a matter of course, the first gene must be formed in the absence of gene. On the other hand, many biopolymers including gene are produced under the genetic system in extant organisms. Thus far, no idea explaining how the first gene was formed in the absence of gene, has been proposed except the idea based on the GADV hypothesis. GADV means four amino acids; Gly [G], Ala [A], Asp [D] and Val [V]. In this article, a reliable answer to the question, how the first gene was generated, is provided. The idea is as follows. The first gene (genetic information) was formed by random joining of anticodons, GNCs, which were carried by primeval anticodon stem-loop (AntiC-SL) tRNAs produced during repeated cycles of random joining of nucleotides and degradation of oligonucleotides. However, it might be difficult for many persons to accept the idea, because tRNAs and metabolic pathways using various proteins, which are produced under genetic functions, must be used to form the first gene. Then, it is reconsidered based on definition of genetic information and minimum necessary things to generate the first gene in this article, how the first gene was generated. Consequently, it could be reconfirmed that five prototypes of members (genetic code, tRNA, metabolism, cell structure and protein) composing the fundamental life system, except gene, are certainly necessary to generate the first gene. Namely, it can be concluded that the first gene was formed through nucleotide metabolism with immature [GADV]-proteins, primeval AntiC-SL tRNA formation with immature [GADV]-proteins, formations of single-stranded (GNC)n codon sequence by random joining of anticodons, GNCs, carried by AntiC-SL tRNAs, double-stranded (ds)-(GNC)n RNA and maturation of an immature [GADV]-protein, which was produced from one strand of the ds-RNA. Thus, the first gene was formed using the five primitive or prototypes of members, which were produced in the absence of gene. There would be no other way for generating the first gene.