Modelling quarantine effects on SARS-CoV-2 epidemiological dynamics in Chilean communes and their relationship with the Social Priority Index.

Background: An epidemiological model (susceptible, un-quarantined infected, quarantined infected, confirmed infected (SUQC)) was previously developed and applied to incorporate quarantine measures and calculate COVID-19 contagion dynamics and pandemic control in some Chinese regions. Here, we generalized this model to incorporate the disease recovery rate and applied our model to records of the total number of confirmed cases of people infected with the SARS-CoV-2 virus in some Chilean communes. Methods: In each commune, two consecutive stages were considered: a stage without quarantine and an immediately subsequent quarantine stage imposed by the Ministry of Health. To adjust the model, typical epidemiological parameters were determined, such as the confirmation rate and the quarantine rate. The latter allowed us to calculate the reproduction number. Results: The mathematical model adequately reproduced the data, indicating a higher quarantine rate when quarantine was imposed by the health authority, with a corresponding decrease in the reproduction number of the virus down to values that prevent or decrease its exponential spread. In general, during this second stage, the communes with the lowest social priority indices had the highest quarantine rates, and therefore, the lowest effective viral reproduction numbers. This study provides useful evidence to address the health inequity of pandemics. The mathematical model applied here can be used in other regions or easily modified for other cases of infectious disease control by quarantine.

Average and Standard Deviation of the Error Function for Random Genetic Codes with Standard Stop Codons.

The origin of the genetic code has been attributed in part to an accidental assignment of codons to amino acids. Although several lines of evidence indicate the subsequent expansion and improvement of the genetic code, the hypothesis of Francis Crick concerning a frozen accident occurring at the early stage of genetic code evolution is still widely accepted. Considering Crick's hypothesis, mathematical descriptions of hypothetical scenarios involving a huge number of possible coexisting random genetic codes could be very important to explain the origin and evolution of a selected genetic code. This work aims to contribute in this regard, that is, it provides a theoretical framework in which statistical parameters of error functions are calculated. Given a genetic code and an amino acid property, the functional code robustness is estimated by means of a known error function. In this work, using analytical calculations, general expressions for the average and standard deviation of the error function distributions of completely random codes with standard stop codons were obtained. As a possible biological application of these results, any set of amino acids and any pure or mixed amino acid properties can be used in the calculations, such that, in case of having to select a set of amino acids to create a genetic code, possible advantages of natural selection of the genetic codes could be discussed.

Exploring the structure-function relationship in a hypothetical membrane protein from a nucleotide sequence.

An educational activity is proposed that uses software for proteomic analysis and databases available for free on the Internet to estimate the structure and function of a hypothetical protein from its coding nucleotide sequence. This bioinformatics activity is recommended for integrated introductory courses that address the structure function relationship in proteins.

Experimental induction and mathematical modeling of Ca2+ dynamics in rat round spermatids.

Cytosolic Ca2+ concentration ([Ca2+ ]) has an important role in spermatozoa and hence it regulates fertilization. In male germinal cells, there are indirect evidences that this ion could regulate physiological processes in spermatogenesis. Since little is known about Ca2+ homeostasis in spermatogenic cells, in this work we propose a mathematical model that accounts for experimental [Ca2+ ] dynamics triggered by blockade of the SERCA transport ATPase with thapsigargin in round rat spermatids, without external Ca2+ and with different extracellular lactate concentrations. The model included three homogeneous calcium compartments and Ca2+-ATPase activities sensitive and insensitive to thapsigargin, and it adjusted satisfactorily the experimental calcium dynamic data. Moreover, an extended version of the model satisfactorily adjusted the stationary states of calcium modulated by extracellular lactate, which is consistent with the participation of a low affinity lactate transporter and further lactate metabolism in these cells. Further studies and modeling would be necessary to shed some light into the relation between Ca2+-lactate-ATP homeostasis and cell-cell interactions in the seminiferous tubules that are expected to modulate Ca2+ dynamics by hormonal factors or energetic substrates in meiotic and postmeiotic spermatogenic cells.

Local conditions for global stability in the space of codons of the genetic code.

The polar requirement is an attribute of amino acids that is a major determinant of the structure and function of the proteins, and it plays a role in the flexibility and robustness of the genetic code. The viability of an organism depends on flexibility, which allows the exploration of new functions. However, robustness is necessary to protect the organism from deleterious changes derived from misreading errors and single-point mutations. Compared with random codes, the standard genetic code is one of the most robust against such errors. Here, using analytical and numerical calculations and the set of amino acid-encoding codons, we have proposed some local conditions that are necessary for the optimal robustness of the genetic code, and we explored the association between the local conditions and the robustness. The localness of the proposed conditions and the underlying evolutionary mechanism, which begins with a random code and progresses toward more efficient codes (e.g., the standard code), might be biologically plausible.

Flux theory for Poisson distributed pores with Gaussian permeability.

The mean of the solute flux through membrane pores depends on the random distribution and permeability of the pores. Mathematical models including such randomness factors make it possible to obtain statistical parameters for pore characterization. Here, assuming that pores follow a Poisson distribution in the lipid phase and that their permeabilities follow a Gaussian distribution, a mathematical model for solute dynamics is obtained by applying a general result from a previous work regarding any number of different kinds of randomly distributed pores. The new proposed theory is studied using experimental parameters obtained elsewhere, and a method for finding the mean single pore flux rate from liposome flux assays is suggested. This method is useful for pores without requiring studies by patch-clamp in single cells or single-channel recordings. However, it does not apply in the case of ion-selective channels, in which a more complex flux law combining the concentration and electrical gradient is required.

Learning about solubility.

Qualitative questions are proposed to assess the understanding of solubility and some of its applications. To improve those results, a simple quantitative problem on the precipitation of proteins is proposed.

Fluxes theory in experiments with random distributed channels on vesicles.

When channels are randomly distributed in a population of vesicles, disregarding the number of channels per vesicle, these channels follow a Poisson distribution. This has been verified in many cases, determining the average of channels per vesicle. However, to determine kinetic parameters in population studies, a mathematical expression for the mean flux of solute through channels per vesicle is necessary. Hence, here, this mean flux is calculated, assuming Poisson distributed channels in a population of vesicle. Moreover, this result has been generalized to any number of different kinds of channels (i.e., channels with different permeabilities). These results, useful for in vitro experiments with mixed both channels and vesicles, can be supplemented with those from other techniques, in order to understanding how the nature of the lipid membrane affects kinetic parameters of channel.

Probable relationship between partitions of the set of codons and the origin of the genetic code.

Here we study the distribution of randomly generated partitions of the set of amino acid-coding codons. Some results are an application from a previous work, about the Stirling numbers of the second kind and triplet codes, both to the cases of triplet codes having four stop codons, as in mammalian mitochondrial genetic code, and hypothetical doublet codes. Extending previous results, in this work it is found that the most probable number of blocks of synonymous codons, in a genetic code, is similar to the number of amino acids when there are four stop codons, as well as it could be for a primigenious doublet code. Also it is studied the integer partitions associated to patterns of synonymous codons and it is shown, for the canonical code, that the standard deviation inside an integer partition is one of the most probable. We think that, in some early epoch, the genetic code might have had a maximum of the disorder or entropy, independent of the assignment between codons and amino acids, reaching a state similar to "code freeze" proposed by Francis Crick. In later stages, maybe deterministic rules have reassigned codons to amino acids, forming the natural codes, such as the canonical code, but keeping the numerical features describing the set partitions and the integer partitions, like a "fossil numbers"; both kinds of partitions about the set of amino acid-coding codons.

The most probable number of blocks for the partitions of the set of codons could have determined the number of standard amino acids.

Given a genetic code formed by 64 codons, we calculate the number of partitions of the set of encoding amino acid codons. When there are 0-3 stop codons, the results indicate that the most probable number of partitions is 19 and/or 20. Then, assuming that in the early evolution the genetic code could have had random variations, we suggest that the most probable number of partitions of the set of encoding amino acid codons determined the actual number 20 of standard amino acids.

Lipid metabolizing enzyme activities modulated by phospholipid substrate lateral distribution.

Biological membranes contain many domains enriched in phospholipid lipids and there is not yet clear explanation about how these domains can control the activity of phospholipid metabolizing enzymes. Here we used the surface dilution kinetic theory to derive general equations describing how complex substrate distributions affect the activity of enzymes following either the phospholipid binding kinetic model (which assumes that the enzyme molecules directly bind the phospholipid substrate molecules), or the surface-binding kinetic model (which assumes that the enzyme molecules bind to the membrane before binding the phospholipid substrate). Our results strongly suggest that, if the enzyme follows the phospholipid binding kinetic model, any substrate redistribution would increase the enzyme activity over than observed for a homogeneous distribution of substrate. Besides, enzymes following the surface-binding model would be independent of the substrate distribution. Given that the distribution of substrate in a population of micelles (each of them a lipid domain) should follow a Poisson law, we demonstrate that the general equations give an excellent fit to experimental data of lipases acting on micelles, providing reasonable values for kinetic parameters--without invoking special effects such as cooperative phenomena. Our theory will allow a better understanding of the cellular-metabolism control in membranes, as well as a more simple analysis of the mechanisms of membrane acting enzymes.

Changes of enzyme activity in lipid signaling pathways related to substrate reordering.

The static fluid mosaic model of biological membranes has been progressively complemented by a dynamic membrane model that includes phospholipid reordering in domains that are proposed to extend from nanometers to microns. Kinetic models for lipolytic enzymes have only been developed for homogeneous lipid phases. In this work, we develop a generalization of the well-known surface dilution kinetic theory to cases where, in a same lipid phase, both domain and nondomain phases coexist. Our model also allows understanding the changes in enzymatic activity due to a decrease of free substrate concentration when domains are induced by peptides. This lipid reordering and domain dynamics can affect the activity of lipolytic enzymes, and can provide a simple explanation for how basic peptides, with a strong direct interaction with acidic phospholipids (such as beta-amyloid peptide), may cause a complex modulation of the activities of many important enzymes in lipid signaling pathways.