Exact properties of Frobenius numbers and fraction of the symmetric semigroups in the weak limit for n=3

We generalize and prove a hypothesis by V. Arnold on the parity of Frobenius number. For the case of symmetric semigroups with three generators of Frobenius numbers we found an exact formula, which in a sense is the sum of two Sylvester's formulaes. We prove that the fraction of the symmetric semigroups is vanishing in the weak limit

Elevation as a selective force on mitochondrial respiratory chain complexes of the Phrynocephalus lizards in the Tibetan plateau

Animals living in extremely high elevations have to adapt to low temperatures and low oxygen availability (hypoxia), but the underlying genetic mechanisms associated with these adaptations are still unclear. The mitochondrial respiratory chain can provide >95% of the ATP in animal cells, and its efficiency is influenced by temperature and oxygen availability. Therefore, the respiratory chain complexes (RCCs) could be important molecular targets for positive selection associated with respiratory adaptation in high-altitude environments. Here, we investigated positive selection in 5 RCCs and their assembly factors by analyzing sequences of 106 genes obtained through RNA-seq of all 15 Chinese Phrynocephalus lizard species, which are distributed from lowlands to the Tibetan plateau (average elevation >4,500 m). Our results indicate that evidence of positive selection on RCC genes is not significantly different from assembly factors, and we found no difference in selective pressures among the 5 complexes. We specifically looked for positive selection in lineages where changes in habitat elevation happened. The group of lineages evolving from low to high altitude show stronger signals of positive selection than lineages evolving from high to low elevations. Lineages evolving from low to high elevation also have more shared codons under positive selection, though the changes are not equivalent at the amino acid level. This study advances our understanding of the genetic basis of animal respiratory metabolism evolution in extreme high environments and provides candidate genes for further confirmation with functional analyses

Detecting archaic introgression using an unadmixed outgroup.

Human populations outside of Africa have experienced at least two bouts of introgression from archaic humans, from Neanderthals and Denisovans. In Papuans there is prior evidence of both these introgressions. Here we present a new approach to detect segments of individual genomes of archaic origin without using an archaic reference genome. The approach is based on a hidden Markov model that identifies genomic regions with a high density of single nucleotide variants (SNVs) not seen in unadmixed populations. We show using simulations that this provides a powerful approach to identifying segments of archaic introgression with a low rate of false detection, given data from a suitable outgroup population is available, without the archaic introgression but containing a majority of the variation that arose since initial separation from the archaic lineage. Furthermore our approach is able to infer admixture proportions and the times both of admixture and of initial divergence between the human and archaic populations. We apply the model to detect archaic introgression in 89 Papuans and show how the identified segments can be assigned to likely Neanderthal or Denisovan origin. We report more Denisovan admixture than previous studies and find a shift in size distribution of fragments of Neanderthal and Denisovan origin that is compatible with a difference in admixture time. Furthermore, we identify small amounts of Denisova ancestry in South East Asians and South Asians

Quasi-isométries entre espaces métriques hyperboliques, aspects quantitatifs

In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ``quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, L^p-cohomology spaces provides useful quasi-isometry invariants and Poincaré constants of balls are their quantitative incarnation. We study how Poincaré constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincaré constants for locally homogeneous metrics of the form dt²+∑_{i}e^{2µ_{i}t}dx²_{i}, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).Dans cette thèse, nous considérons les chemins possibles pour donner une mesure quantitative du fait que deux espaces ne sont pas quasi-isométriques. De ce point de vue quantitatif, on reprend la définition de quasi-isométrie et on propose une notion de “croissance de distorsion quasi-isométrique” entre deux espaces métriques. Nous révisons notre article [32] où une borne supérieure optimale pour le lemme de Morse est donnée, avec la variante duale que nous appelons Anti-Morse Lemma, et leurs applications.Ensuite, nous nous concentrons sur des bornes inférieures sur la croissance de distorsion quasi-isométrique pour des espaces métriques hyperboliques. Dans cette classe, les espaces de L^p-cohomologie fournissent des invariants de quasi-isométrie utiles et les constantes de Poincaré des boules sont leur incarnation quantitative. Nous étudions comment les constantes de Poincaré sont transportées par quasi-isométries. Dans ce but, nous introduisons la notion de transnoyau. Nous calculons les constantes de Poincaré pour les métriques localement homogènes de la forme dt²+∑_{i}e^{2µ_{i}t}dx²_{i}, et donnons une borne inférieure sur la croissance de distorsion quasi-isométrique entre ces espaces.Cela nous permet de donner des exemples présentant différents type de croissance de distorsion quasi-isométrique, y compris un exemple sous-linéaire (logarithmique)

VGsim: scalable viral genealogy simulator for global pandemic

As an effort to help contain the COVID-19 pandemic, large numbers of SARS-CoV-2 genomes have been sequenced from all continents. More than one million viral sequences are publicly available as of April 2021. Many studies estimate viral genealogies from these sequences, as these can provide valuable information about the spread of the pandemic across time and space. Additionally such data are a rich source of information about molecular evolutionary processes including natural selection, for example allowing the identification and investigating the spread of new variants conferring transmissibility and immunity evasion advantages to the virus. To validate new methods and to verify results resulting from these vast datasets, one needs an efficient simulator able to simulate the pandemic to approximate world-scale scenarios and generate viral genealogies of millions of samples. Here, we introduce a new fast simulator VGsim which addresses this problem. The simulation process is split into two phases. During the forward run the algorithm generates a chain of events reflecting the dynamics of the pandemic using an hierarchical version of the Gillespie algorithm. During the backward run a coalescent-like approach generates a tree genealogy of samples conditioning on the events chain generated during the forward run. Our software can model complex population structure, epistasis and immunity escape. The code is freely available at https://github.com/Genomics-HSE/VGsim