Entropy

In a previous study, the authors utilized a single dimensional operationalization of species density that at least partially demonstrated dynamic system behavior. For completeness, a theory needs to be developed related to homology/cohomology, induction of the time dimension, and system hierarchies. The topological nature of the system is carefully examined and for testing purposes, species density data for a wild Dictyostelia community data are used in conjunction with data derived from liquid-chromatography mass spectrometry of proteins. Utilizing a Clifford algebra, a congruent zeta function, and a Weierstra{beta}[weierp] function in conjunction with a type VI Painleve equation, we confirmed the induction of hierarchy and time through one-dimensional probability space with certain topologies. This process also served to provide information concerning interactions in the model. The previously developed \"small s\" metric can characterize dynamical system hierarchy and interactions, using only abundance data along time development.

Understanding the differences between genomic sequences of different lives is crucial for biological classification and phylogeny. Here, we downloaded all the reliable sequences of the seven kingdoms and determined the dimensions of the genome space embedded in the Euclidean space, along with the corresponding Natural Metrics. The concept of the Grand Biological Universe is further proposed. In the grand universe, the convex hulls formed by the universes of seven kingdoms are mutually disjoint, and the convex hulls formed by different biological groups within each kingdom are mutually disjoint. This study provides a novel geometric perspective for studying molecular biology and also offers an accurate way for large-scale sequence comparison in a real-time manner. Most importantly, this study shows that, due to the space-time distortion in the biological genome space similar to Einsteins theory, it is futile to look for a single metric to measure different biological universes, as previous studies have done.

In this study, with the use of the information theory, we have proposed and proved a mathematical theorem by which we argue the reason for the existence of human diseases. To introduce our theoretical frame of reference, first, we put forward a modification of Shannons entropy, computed for all available proteomes, as a tool to compare systems complexity and distinguish between the several levels of biological organizations. We establish a new approach to differentiate between several taxa and corroborate our findings through the latest tree of life. Furthermore, we found that human proteins with higher mutual information, derived from our theorem, are more prone to be involved in human diseases. We further discuss the dynamics of protein network stability and offer probable scenarios for the existence of human diseases and their varying occurrence rates. Moreover, we account for the reasoning behind our mathematical theorem and its biological inferences.

Triplets, as minimal informative rooted trees, are fundamental units of information in phylogenetics. Their importance for phylogenetic reconstruction, cladistic biogeography, or supertree methods relies on the fact that any rooted tree can be decomposed into a set of triplets. In order to formalize the tree building from a consistent triplet set, several k -adic rules of inference, i.e., rules that allow us to deduce at least one new triplet from exactly k other ones, have been identified. However, it remains unclear whether it is possible to reduce all the possible k -adic rules to a finite set of basic properties. In order to solve this problem, we propose here to define triplets in terms of degree of equivalence relations. Given the axiomatic definition of the latter, we establish a list of the most basic properties for triplets. With such an approach, we finally prove that the closure of any coherent triplet set can be computed uniquely from these basic properties.

We propose a novel mutation mechanism for points and ordinary or palindromic sequences of DNA and RNA. We adopted non-Hermitian approaches based on quantum mechanics. Hermiticity is in the limelight of any physical structure with quantum character, like DNA, or RNA, as it creates quantum stability in that it yields real eigenvalues and orthonormal states. We show that, through the mutation mechanism we constructed based on non-Hermitian physics, the deterioration of the Hermitian character of the original DNA states, nucleotides, does not create a stability problem. We show that Weyls perturbation theory helps us determine the stability of mutated DNA or RNA. We prove that mutations made in the laboratory with conventional nucleotides using non-Hermitian physics methods are not different from mutations that occur spontaneously in nature. This result may help to reveal the quantum nature of genetic diseases in the near future and may shape the molecular approaches.

The concept of an autocatalytic network of reactions that can form and persist, starting from just an available food source, has been formalised by the notion of a Reflexively-Autocatalytic and Food generated (RAF) set. The theory and algorithmic results concerning RAFs have been applied to a range of settings, from metabolic questions arising at the origin of life, to ecological networks, and cognitive models in cultural evolution. In this paper, we present new structural and algorithmic results concerning RAF sets, by studying more complex modes of catalysis that allow certain reactions to require multiple catalysts (or to not require catalysis at all), and discuss the differing ways catalysis has been viewed in the literature. We then focus on the structure and analysis of minimal RAFs, and derive structural results and polynomial-time algorithms, with applications to metabolic network data described briefly.

Network alignment aims to uncover topologically similar regions in the protein-protein interaction (PPI) networks of two or more species under the assumption that topologically similar regions perform similar functions. Although there exist a plethora of both network alignment algorithms and measures of topological similarity, currently no "gold standard" exists for evaluating how well either is able to uncover functionally similar regions. Here we propose a formal, mathematically and statistically rigorous method for evaluating the statistical significance of shared GO terms in a global, 1-to-1 alignment between two PPI networks. We use combinatorics to precisely count the number of possible network alignments in which k proteins share a particular GO term. When divided by the number of all possible network alignments, this provides an explicit, exact p-value for a network alignment with respect to a particular GO term.

Using a notably large amount of data in investigating physical and chemical phenomena demands new statistical and computational approaches; besides, the cross-validations require well-established theoretical frameworks. This study aims to validate the statistical efficiency of alternative definitions for the information-theoretic measures, such as transfer entropy, using the so-called (, q)-framework. The primary goal is to find measurements of high-order correlations that preserve information-theoretic properties of information transfer between the components of a dynamical system (such as a protein) due to local operations. Besides, this study aims to decode the information contained in the amino acid sequence establishing a three-dimensional protein structure by comparing the amino acids physical-chemical properties with their ranked role in the protein interaction network topology using new graph-theoretic measures based on the constructed digraph models of (, q) information transfer within a heat flow kernel embedding framework. Moreover, this study aims to use the Deep Graph Convolution Neural Networks for classifying the role of each amino acid in a protein trained upon short equilibrium structure fluctuations at sub-nanosecond time scales. In particular, this study examines the influence of disulphide bridges on the three-dimensional structure of the Bovine Pancreatic Trypsin Inhibitor wild type and mutated analogue protein.

Three-item statements, as minimal informative rooted binary phylogenetic trees on three items, are the minimal units of cladistic information. Their importance for phylogenetic reconstruction, consensus and supertree methods relies on both (i) the fact that any cladistic tree can always be decomposed into a set of three-item statements, and (ii) the possibility, at least under some conditions, to build a new cladistic tree by combining all or part of the three-item statements deduced from several prior cladistic trees. In order to formalise such procedures, several k-adic rules of inference, i.e., rules that allow us to deduce at least one new three-item statement from exactly k other ones, have been identified. However, no axiomatic background has been proposed, and it remains unknown if a particular k-adic rule of inference can be reduced to more basic rules. In order to solve this problem, we propose here to define three-item statements in terms of degree of equivalence relations. Given both the axiomatic definition of the latter and their strong connection to hierarchical classifications, we establish a list of the most basic properties for three-item statements. With such an approach, we show that it is possible to combine five three-item statements from basic rules although they are not combinable only from dyadic rules. Such a result suggests that all higher k-adic rules are well reducible to a finite set of simpler rules.

Ever since the publication of Shannons article about information theory, there have been many attempts to apply information theory to the neuroscience field. Meanwhile, the Weber- Fechner law of psychophysics states that the magnitude of a subjective sensation of a person increases in proportion to the logarithm of the intensity of the external physical-stimulus. It is hardly surprising that we assign the amount of information to the response in the Weber- Fechner law. But, to date no one has succeeded in applying information theory directly to that law: the direct links between information theory and that response in the Weber-Fechner law have not yet been found. The proposed theory unveils a link between information theory and that response, and differs subtly from the field such as neural coding that involves complicated calculations and models. Because my theory targets the Weber-Fechner law which is a macroscopic phenomenon, this theory does not involve complicated calculations. My theory is expected to mark a new era in the fields of sensory perception research. My theory must be studied in parallel with the fields of microscopic scale such as neural coding. This article ultimately aims to provide the fundamental concepts and their applications so that a new field of research on stimuli and responses can be created.

The present work shows that the maximum-entropy method can be applied to a sample of neuronal recordings along two different routes: (1) apply to the sample; or (2) apply to a larger, unsampled neuronal population from which the sample is drawn, and then marginalize to the sample. These two routes give inequivalent results. The second route can be further generalized to the case where the size of the larger population is unknown. Which route should be chosen? Some arguments are presented in favour of the second. This work also presents and discusses probability formulae that relate states of knowledge about a population and its samples, and that may be useful for sampling problems in neuroscience.

Three nucleotides per codon had been determined before deciphering the genetic code. However, it is still a mystery why there are three nucleotides per codon. This is a deceptively simple problem, which need first to clarify the prebiotic picture that has been in debate for decades. The triplet nature of life has been observed not only in the triplet codons but also in the universal 3-base periodicity in genome sequences. Here, a statistical picture on the prebiotic sequence evolution has been proposed by ascertaining the profound relationship between the evolution of the genetic code and the diversification of life. There are indications that the triplet nature of the genetic code is due to a mixture of the periods in the superhelical structures of bent DNAs.

The analysis of computer simulation data requires efficient statistical and computational approaches, based on well-established theoretical frameworks. This study aims to introduce such approaches for topological data analysis within the persistent homology framework and to describe the manifold of the protein structure dynamics within the differential geometry of the directed graphs framework. Furthermore, the asymmetric kernel-directed graphs determined by the transfer entropy will describe the information flow in this manifold. The primary goal is to characterise changes in the topology of the protein structure due to the mutations. Moreover, this study aims to define the embedded manifold of dimension m of the amino acid sequence interaction network using the graphs Laplacian matrix for determining the local embedded vector fields and coordinate vectors in this manifold for each amino acid as the vertices of either a directed or undirected graph. Furthermore, this study strives to show that encoding the amino acid sequence information in an m-dimensional manifold is statistically efficient by decoding that information in a much lower-dimensional space. Then, using the topological data analysis, we can observe protein structure dynamics changes in a multidimensional manifold, for example, due to amino acid mutations. The analysis showed that short equilibrium structure fluctuations at a few nanoseconds enable the construction of such a manifold. As a case study, the influence of the mutation of the two disulphide bridges on the three-dimensional structure of the Bovine Pancreatic Trypsin Inhibitor protein is investigated.

1Today we are able to describe genome evolution but, there are still several open questions on this area such as: Which is the probability that a mutation occurs at a nucleotide level? Is it possible to predict the evolution of a particular genome?, or talking about preservation, is there a way to simulate the genetic diversity for endangered species? In this paper it is shown that it is possible to make a mathematical model not only of mutations on the genome of species, but of evolution itself, including factors such as artificial and natural selection. It is also presented the algorithm to obtain the probabilities of mutation for each specific part of the genome and for each specie.\n\nEven more, it is presented a mathematical method to estimate the amount of generations between two related genomes and a function capable of predict the amount of mutations through time a genome will suffer.\n\nThe potential of having this tool is giantic going from genetic engineering applied to medicine to filling up blank spaces in phylogenetic studies or preservation of endangered species due to genetic diversity.

Regulated biological networks are commonly represented as logical diagrams, in which the exact interactions between the elements remain out of sight. Here we propose a new type of excitation-inhibition graph based on Boolean logic, which we name "logical directed graph or simply, logical digraph of the biological system". Such logical digraph allows the representation of every possible regulatory interaction among elements, based on Boolean interactions. The logical digraph contains information about the connectivity, dynamics, limit cycles, and attractors of the network. As proof of the application, the logical digraph was applied to analyze the functioning of the well-known neural network that produces oscillatory swimming in the mollusk Tritonia. Our method permits to transit from a regulatory network to its logical digraph and vice versa. In addition, we show that the spectral properties of the so-called state matrix provide mathematical evidence about why the elements in the attractors and limit cycles contain information about the dynamics of the biological system. Open software routines are provided for the calculations of the components of the network and the attractors and limit cycles. This approach offers new possibilities to visualize and analyze regulatory networks in biology.

Each amino acid in a polypeptide chain has a distinctive R-group associated with it. We report here a novel method of species characterization based upon the order of these R-group classified amino acids in the linear sequence of the side chains associated with the codon triplets. In an otherwise pseudo-random sequence, we search for forbidden combinations of kth order. We applied this method to analyze the available protein sequences of various viruses including SARS-CoV-2. We found that these ubiquitous forbidden orders (UFO) are unique to each of the viruses we analyzed. This unique structure of the viruses may provide an insight into viruses chemical behavior and the folding patterns of the proteins. This finding may have a broad significance for the analysis of coding sequences of species in general.

A qualitative mathematical model of the notion of immunocompetence is developed, based on the formalism of Memory Evolutive Systems (MES), from which, immunocompetence is defined as an emergent structure of a higher order arising from the signal networks that are established between effector cells and molecules of the immune response in the presence of a given antigen. In addition, a possible mechanism of functorial nature is proposed, which may explain how immunocompetence is achieved in an organism endowed with innate and adaptive components of its immune system. Finally, a practical method to measure the immunocompetence status is established, using elements of the theory of small random graphs and taking into account the characteristics of the immune networks, established through transcriptional studies, of patients with severe COVID-19 and healthy patients, assuming that both types of patients were vaccinated with an effective biological against SARS-CoV-2.

From Synthesis perspective, whether Logic Synthesis, Physical Synthesis, Chemical Synthesis, or Biological Synthesis, Physical Geometry such as Universal Geometry and Quantum Geometry, and Biological Geometry like Conformal Geometry supported by Tensors and Manifolds, are the outcome of physical laws and biological laws in modeling non-linear physical and biological dynamics as opposed to traditional partial differential/difference equation way. We discover that Multiversal SpaceTime instead of Neural Network, governing physical and biological world at macroscopic and microscopic level, is the ultimate source of intelligence. With that we propose Multiversal Synthesis-based Artificial Design Automation (ADA), a bio-physical inspired model based on Multiverse in Darwin Dynamics, Generalized Quantum Mechanics, and Extended General Relativity, for Artificial Super Intelligence (ASI) implementation. Based on Schrodinger Equation of Quantum Mechanics, we generalize the 4-Dimensional Hilbert Space based Discrete Quantum SpaceTime to N-Dimensional (1 << N < M, with M is limited by Planck Length) Hilbert Space based Discrete MSpaceTime as part of MSpaceTime, in modeling both Micro-Environment Intelligence and Micro-Agent Intelligence of ASI; likewise based on Einstein Equations of General Relativity, we make a T-Symmetry extension first, and then extend the 4-Dimensional Pseudo-Riemannian Manifold based Continuous Curved SpaceTime as part of MSpaceTime to N-Dimensional (1 << N < {infty}) Pseudo-Riemannian Manifold based Continuous MSpaceTime extension, in modeling both Macro-Environment Intelligence and Macro-Agent Intelligence of ASI. Our discovery only solves the black box puzzle of AI, but also paves the way in achieving ASI through ADA. Of course, our Multiverse Endeavor will never stop from there.

Centered kernel alignment (CKA) and representational similarity analysis (RSA) of dissimilarity matrices are two popular methods for comparing neural systems in terms of representational geometry. Although they follow a conceptually similar approach, typical implementations of CKA and RSA tend to result in numerically different outcomes. Here, I show that these two approaches are largely equivalent once one incorporates a mean-centering step into RSA. This equivalence holds for both linear and nonlinear variants of these methods. These connections are simple to derive, but appear to have been thus far overlooked in the context of comparing neural representations. By unifying these measures, this paper hopes to simplify a complex and fragmented literature on this subject.

Carcinogenesis is one example of transition between two biological states involving strong gene expression rearrangements [1]. Normal and tumor states are understood as two different states of a Gene Regulatory Network [2]. In terms of the normal state characteristics, the transition may be thought as a deregulation process [3,4,5]. According to a very abstract model for the transition, the available data on cancer risk support the idea of a single big jump in expression space [6], which may be associated to a deregulation cascade. Here, we use the measured frequency distribution of gene deregulations, and concepts from the probabilistic theory of causation [7] in order to infer causal relations between pairs of genes and construct a deregulation causal network [8]. Then, the deregulated genes in each sample are organized according to the network in such a way that they show the deregulation cascade behind the transition from the normal to the tumor state. The explicit analysis for prostate adenocarcinoma is given. In most cases, the cascade happens to be unique and initiated by a single deregulation event. Using results from a companion paper [9], we show that cascades conform classes which may be labeled by the deregulations in a predefined panel of 15 genes. The results of the paper may be checked in experiments with cellular lines or in animal models, and could have an impact on personalized cancer therapy.