On the lexicographic representation of numbers

It is proven that, contrarily to the common belief, the notion of zero is not necessary for having positional representations of numbers. Namely, for any positive integer $k$, a positional representation with the symbols for $1, 2, \ldots, k$ is given that retains all the essential properties of the usual positional representation of base $k$ (over symbols for $0, 1, 2 \ldots, k-1$). Moreover, in this zero-free representation, a sequence of symbols identifies the number that corresponds to the order number that the sequence has in the ordering where shorter sequences precede the longer ones, and among sequences of the same length the usual lexicographic ordering of dictionaries is considered. The main properties of this lexicographic representation are proven and conversion algorithms between lexicographic and classical positional representations are given. Zero-free positional representations are relevantt in the perspective of the history of mathematics, as well as, in the perspective of emergent computation models, and of unconventional representations of genomes.Comment: 15 page

Metabolic computing

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Informational laws of genome structures

In recent years, the analysis of genomes by means of strings of length k occurring in the genomes, called k-mers, has provided important insights into the basic mechanisms and design principles of genome structures. In the present study, we focus on the proper choice of the value of k for applying information theoretic concepts that express intrinsic aspects of genomes. The value k\u2009=\u2009lg2(n), where n is the genome length, is determined to be the best choice in the definition of some genomic informational indexes that are studied and computed for seventy genomes. These indexes, which are based on information entropies and on suitable comparisons with random genomes, suggest five informational laws, to which all of the considered genomes obey. Moreover, an informational genome complexity measure is proposed, which is a generalized logistic map that balances entropic and anti-entropic components of genomes and is related to their evolutionary dynamics. Finally, applications to computational synthetic biology are briefly outlined

A word recurrence based algorithm to extract genomic dictionaries

Genomes may be analyzed from an information viewpoint as very long strings, containing functional elements of variable length, which have been assembled by evolution. In this work an innovative information theory based algorithm is proposed, to extract significant (relatively small) dictionaries of genomic words. Namely, conceptual analyses are here combined with empirical studies, to open up a methodology for the extraction of variable length dictionaries from genomic sequences, based on the information content of some factors. Its application to human chromosomes highlights an original inter-chromosomal similarity in terms of factor distributions

Inverse Dynamical Problems: An Algebraic Formulation Via MP Grammars

Metabolic P grammars are a particular class of multiset rewriting grammars introduced in the MP systems' theory for modelling metabolic processes. In this paper, a new algebraic formulation of inverse dynamical problems, based on MP grammars and Kronecker product, is given, for further motivating the correctness of the LGSS (Log-gain Stoichiometric Stepwise) algorithm, introduced in 2010s for solving dynamical inverse problems in the MP framework. At the end of the paper, a section is included that introduces the problem of multicollinearity, which could arise during the execution of LGSS, and that de nes an algorithm, based on a hierarchical clustering technique, that solves it in a suitable way

The Discovery of Initial Fluxes of Metabolic P Systems

A central issue in systems biology is the study of efficient methods to infer fluxes of biological reactions starting from experimental data. Among the different techniques proposed in the last years, in the theory of Metabolic P systems Log-Gain principles have been introduced, which prove to be helpful for deducing biological fluxes from temporal series of observed dynamics. However, crucial tasks remain to be performed for a complete suitable application of these principles. In particular the algebraic systems introduced by the Log-Gain principles require the knowledge of the initial fluxes associated with a set of biochemical reactions. In this paper we propose an algorithm for estimating initial fluxes, which is tested in two case studies

Discrete Solution of Differential Equations by P Metabolic Algorithm

The relationships existing between MP graphs, metabolic P systems, and ODE systems are investigated. Formal results show that every MP system, once derived by its MP graph, results in an ODE system whose solution equals, in the limit, the solution obtained by a non-cooperative MP system that is ODE equivalent to the original one. The freedom of choice of the ODE equivalent from the original MP system resembles the same freedom which is left in the choice and optimization of a numerical scheme while computing the solution of an ODE system

Toward a Self-replicating Metabolic P System

This work concerns the synthesis of a "minimal cell' by means of a P system, which is a distributed rewriting system inspired by the structure and the functioning of the biological cell. Specifically, we aim to define a dynamical system which exhibits a steady metabolic evolution, resulting in self-maintenance and self-reproduction. Metabolic P systems represent a class of P systems particularly promising to model a minimal cell in discrete terms, since they have already successfully modeled several metabolisms. The main further step is thus to find a simple way to obtain Metabolic P system self-replication. This paper deals with ideas presented at the BWMC11 (held in Seville, Feb 2011) and opens a new trend in membrane computing, based on computational synthetic biology oriented applications of P systems modeling. The framework is here outlined, and some problems to tackle the synthesis of a minimal cell are discussed. Moreover, an overview of literature and a list of appealing research directions is given, along with several references

MP Modeling of Glucose-Insulin Interactions in the Intravenous Glucose Tolerance Test

The Intra Venous Glucose Tolerance Test (IVGTT) is an experimental pro- cedure in which a challenge bolus of glucose is administered intra-venously and plasma glucose and insulin concentrations are then frequently sampled. An open problem is to construct a model representing simultaneously the entire control system. In the last three decades, several models appeared in the literature. One of the mostly used one is known as the minimal model, which has been challenged by the dynamical model. However, both the models have not escape from criticisms and drawbacks. In this paper we apply Metabolic P systems theory for developing new physiologically based models of the glucose-insulin system which can be applied to the Intra Venous Glucose Tolerance Test. We considered ten data-sets obtained from literature and for each of them we found an MP model which ts the data and explains the regulations of the dynamics. Finally, further analysis are planned in order to de ne common patterns which explain, in general, the action of the glucose-insulin control system

Metabolic Algorithm with Time-varying Reaction Maps

A symbolic-based approach to modelling biochemical processes and cellular dynamics is likely to turn useful in computational biology, where attempts to represent the cell as a huge, complex dynamic system must trade with the linguistic nature of the DNA and the individual behavior of the organelles living within. The early version of the metabolic algorithm gave a first answer to the problem of representing oscillatory biological phenomena, so far being treated with traditional (differential) mathematical tools, in terms of rewriting systems. We are now working on a further version of this algorithm, in which the rule application is tuned by reaction maps depending on the specific phenomenon under consideration. Successful simulations of the Brusselator, the Lotka-Volterra population dynamics and the PKC activation foster potential applications of the algorithm in systems biology