Related papers: Ultrametrics in the genetic code and the genome
We study the properties of ultrametric matrices aiming to design methods for fast ultrametric matrix-vector multiplication. We show how to encode such a matrix as a tree structure in quadratic time and demonstrate how to use the resulting…
DNA strings and their properties are widely studied since last 20 years due to its applications in DNA computing. In this area, one designs a set of DNA strings (called DNA code) which satisfies certain thermodynamic and combinatorial…
The genetic code is the function from the set of codons to the set of amino acids by which a DNA sequence encodes proteins. Since the codons also influence the shape of the DNA molecule itself, the same sequence that encodes a protein also…
Using concepts of physics of elementary particles concerning the breaking of symmetry and grannd unified theory we propose to study with the algebraic approximation the degeneracy finded in the genetic code with the incorporation of a…
High dimensional, sparsely populated data spaces have been characterized in terms of ultrametric topology. This implies that there are natural, not necessarily unique, tree or hierarchy structures defined by the ultrametric topology. In…
An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a natural hierarchical embedding. Such hierarchical structure can be global in the data…
We introduce genetic algorithms as a means to analyze supernovae type Ia data and extract model-independent constraints on the evolution of the Dark Energy equation of state. Specifically, we will give a brief introduction to the genetic…
The ongoing explosion of genome sequence data is transforming how we reconstruct and understand the histories of biological systems. Across biological scales, from individual cells to populations and species, trees-based models provide a…
While we once thought of cancer as single monolithic diseases affecting a specific organ site, we now understand that there are many subtypes of cancer defined by unique patterns of gene mutations. These gene mutational data, which can be…
New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths.…
A heuristic diagram of the evolution of the standard genetic code is presented. It incorporates, in a way that resembles the energy levels of an atom, the physical notion of broken symmetry and it is consistent with original ideas by Crick…
We perform geometrization of genetics by representing genetic information by points of the 4-adic {\it information space.} By well known theorem of number theory this space can also be represented as the 2-adic space. The process of…
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…
The depinning of an elastic line interacting with a quenched disorder is studied for long range interactions, applicable to crack propagation or wetting. An ultrametric distance is introduced instead of the Euclidean distance, allowing for…
An effective $p$-adic encoding of dendrograms is presented through an explicit embedding into the Bruhat-Tits tree for a $p$-adic number field. This field depends on the number of children of a vertex and is a finite extension of the field…
Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. Families of genetic (4*4)- and (8*8)-matrices show unexpected connections of the genetic system…
Genetic sequencing has become an increasingly affordable and accessible source of genomic data in computational biology. This data is often represented as $k$-mers, i.e., strings of some fixed length $k$ with symbols chosen from a reference…
It has been proposed that the degeneracy of the genetic code,i.e., the phenomenon that different codons (base triplets) of DNA are transcribed into the same amino acid, may be interpreted as the result of a symmetry breaking process. In the…
We present a version of Smale's $\alpha$-theory for ultrametric fields, such as the $p$-adics and their extensions, which gives us a multivariate version of Hensel's lemma.
The algebraic structures of the genetic code are most important to obtain additional information about the semantic code and its applications. In this paper we define two dual Boolean codon lattices of the genetic code using hydrogen bond…