# BIOLOGY

**Book: the multidimensional world**

Recently introduced versions of the Borsuk-Ulam theorem (BUT) reveal that a feature vector on a n-manifold projects two feature vectors (matching descriptions of a single object) onto an n+1 manifold. Starting from this rather simple, yet far-reaching, computational topology observation, we build __a fruitful general framework, able to elucidate disparate__ “real” physical and biological phenomena, from quantum entanglement to gauge theories. Summarizing this novel topological approach, we take into account projections among functional or real dimensions. __We achieve a system of mappings __that fit very well with experimental results, making it possible to assess countless systems in far-flung scientific branches. This book __highlights the computational character of matching descriptions__ (arising from descriptively proximal objects) that display a widest range of possible uses. Such observations point to BUT not just from the standpoint of a novel interpretation of almost all the biological and physical phenomena, but also as suitable tools in evaluating the slight (objective and subjective) differences that make our world an astonishing realm of rich heterogeneity.

**A novel theorem**

**Gravitational lensing**

**Pauli exclusion principle**

**Small world networks**

**Ergodicity**

**Group theory**

**Thermodynamic entropy: the arrow of time**

**Shannon, Rényi entropy**

**The dimensions of living beings**

**Natural projections and evolution**

**How to detect hidden dimensions?**

**Chaotic 4D paths & Feigenbaum constant in a multidimensional world**

__The possible presence of further dimensions hidden in our three-dimensional-plus time__ world might help to elucidate countless physical and biological systems’ behaviors, from quantum entanglement to brain function. Nevertheless, suggestions concerning multidimensional arrangement of physical and biological systems do not deserve the role of scientific claims, unless the suggested additional dimensions can be verified via empirically testable hypotheses and experimental apparatus. Here we suggest that the widespread nonlinear dynamics and chaotic behavior of physical and biological collective systems might mirror further dimensions hidden in our world. Indeed, bringing together disparate knowledge from seemingly unrelated fields (brane cosmology, fluid dynamics, algebraic topology, computational topology, dynamic systems theory, logic and statistical mechanics), we show how, in logistic maps derived from nonlinear dynamical equations, __the typical bifurcation diagrams might arise from linear flow paths, that intersect largesized hidden dimensions at the canonical phase parameter’s__ values between three and four. Therefore, chaotic dynamics suggests the existence of a further hidden dimension in our Universe. We also provide a thermodynamic framework which suggests that the cosmic entropy is encompassed in a multidimensional manifold. PDF

**Dynamic systems theory and evolution (my comment published by Nature)**

Tozzi A. 2014. Evolution: Networks and Energy Count. Nature 515: 343. doi:10.1038/515343c.

Lysenko,suggested the heritability of acquired characteristics. His heretic ideas were dismissed with disgust in favor of the “post-Darwinist” standard evolution theory (SET), one of the most pervasive paradigms of the modern science. However, after half a century of oblivion, the debate is once again an hot topic of current research. In particular, __the possible epigenetic inheritance within organisms have been suggested as neo-Lamarckian__ in nature and talks about a picture different from SET, despite Wray’s skeptical claims. If we examine the problem from the novel perspective of the supramolecular chemistry, we notice that the epigenetic information involves the storage of information at the molecular level and its retrieval, transfer and processing at the supramolecular level, via transitory processes that are self-organized, self-assembled and dynamic. __SET does not keep into account that the complexity of adaptive evolving systems__ (including species, niches and environment) is best understood as dynamic networks of relationships, aiming to decrease their free energy via entropy transfer. The DNA is just one of the countless functional tasks of interest in the study of evolution: __changes propagate through interlinked levels of organization__, inducing connectivity and interaction on all scales of the multilevel system, with no preferred level of granularity. __Models of fitness attractors intended to capture the process of natural selection__ are starting to be developed, taking into account power laws, non-equilibrium steady-state at the edge of the chaos and energetic landscapes made of basins, valleys, floors, ridges and saddle points. In conclusion, it would be useful to investigate SET in the framework of dynamical system theories. PDF

**Building fractals from noise **

Scale-free dynamics are an intrinsic feature of a large class of natural models, from earthquakes to brain activity. Assessing a geometrical/mathematical model of synthetic power law oscillations, we noticed that __a wave containing a ____fractal-like structure can be produced by summing a random oscillation to a carefully chosen one__. This observation gives rise to countless applications: a “hidden” oscillation may cause a scale-free behavior in a random noise; a fractal system can be produced by simply choosing the appropriate oscillation to bring in; if power laws are involved in random walks, phase transitions and self-organized criticality, then __the ____superimposition ____of ____a carefully chosen ____oscillation may lead to systems of increased complexity__; “nested” waves from the central nervous system’s spontaneous networks may be the source of the scale-free dynamics seen in EEG and fMRI; in the event of brain 1/f scaling disruption caused by illnesses such as Alzheimer’s disease, __an external wave - for instance, via transcranial stimulation - could restore__ the broken symmetry. PDF

**A Gauge theory for living cells**

Tozzi A, Peters JF, Navarro J, Kun W, Lin B, Marijuán PC. 2017. Cellular Gauge Symmetry and the Li Organization Principle. Progress in Biophysics and Molecular Biology. https://doi.org/10.1016/j.pbiomolbio.2017.06.004.

Based on novel topological considerations, we postulate a gauge symmetry for living cells and proceed to interpret it from a consistent Eastern perspective: the li organization principle. Gauge theories had a tremendous impact in particle physics and have been recently proposed in order to assess nervous activity too. Herein, taking into account novel claims from topology, the mathematical branch that allows the investigation of the most general systems activity, we aim to sketch a gauge theory addressed to the fundamentals of cellular organization. In our framework, __the reference system is the living cell__, equipped with general __symmetries and energetic constraints standing for the intertwined biochemical, metabolic and signaling pathways__ that allow the global homeostasis of the system. Abstractly, these functional movements would follow donut-like trajectories. __Environmental stimuli stand for forces able to locally break__ the symmetry of metabolic and signaling pathways, while __the species-specific DNA is the gauge field__ that restores the global homeostasis after external perturbations. We show how the Borsuk-Ulam Theorem (BUT), which states that a single point on a circumference maps two points on a sphere, allows an inquiry on the evolution from inorganic to organic structures as well as the comparison between prokaryotic and eukaryotic metabolisms and modes of organization. Furthermore, using recently developed BUT variants, we operationalize __a methodology for the description of cellular activity in terms of topology/gauge fields__ and discuss about the experimental implications and feasible applications. We converge on the strategic role that second messengers have played regarding the emergence of such a unitary gauge field for the cell, and the subsequent evolutionary implications for multicellulars. A new avenue for a deeper investigation of biological complexity looms. Philosophically, along this overall exploration of cellular dynamics and biological complexity, we might be reminded of the duality between two essential concepts proposed by the great Chinese synthesizer Zhu Xi (in the XIII Century). His explanatory scheme epitomizes a feasible philosophical interpretation of the present proposal: on the one side, the li organization principle, which may be taken as equivalent to the dynamic interplay between symmetry and information; and on the other side, the qi principle, which can be interpreted as the energy participating in the process, and which always appears as interlinked with the former. In contemporary terms, it would mean the required interconnection between information and energy, and at the same time it would be pointing at essential interpretive principles of information philosophy. PDF

**Geometric curves underlying physical and biological dynamics**

The present Addendum complements the accompanying paper “Cellular Gauge Symmetry and the Li Organization Principle”; it illustrates a recently-developed geometrical physical model able to assess electronic movements and energetic paths in atomic shells. The model describes a multi-level system of circular, wavy and zigzag paths which can be projected onto a horizontal tape. This model ushers in a visual interpretation of the distribution of atomic electrons’ energy levels and the corresponding quantum numbers through rather simple tools, such as compasses, rulers and straightforward calculations. Here we show how this geometrical model, with the due corrections, among them __the use of geodetic curves__, might be able to describe and quantify the structure and the temporal development of countless physical and biological systems, __from Langevin equations for random paths, to symmetry breaks occurring ubiquitously in physical and biological phenomena, to the relationships among different frequencies of EEG electric spike__s. Therefore, in our work we explore the possible association of binomial distribution and geodetic curves configuring a uniform approach for the research of natural phenomena, in biology, medicine or the neurosciences. PDF

**A Timeless biology; when time does not count**

__Contrary to claims that physics is timeless while biology is time-dependent, we take the opposite standpoint__: physical systems’ dynamics are constrained by the arrow of time, while living assemblies are time-independent. Indeed, the concepts of “constraints” and “displacements” shed new light on the role of continuous time flow in life evolution, allowing us to sketch a physical gauge theory for biological systems in long timescales. In the very short timescales of biological systems’ individual lives, time looks like “frozen” and “fixed”, so that the second law of thermodynamics is momentarily wrecked. __The global symmetries (standing for biological constrained trajectories__, i.e. the energetic gradient flows dictated by the second law of thermodynamics in long timescales) are __broken by local “displacements” where time is held constant__, i.e., modifications occurring in living systems. Such displacements stand for brief local forces, able to temporarily “break” the cosmic increase in entropy. The force able to restore the symmetries (called “__gauge field”) stands for the very long timescales of biological evolution__. Therefore, at the very low speeds of life evolution, time is no longer one of the four phase space coordinates of a spacetime Universe: but __it becomes just a gauge field superimposed to three-dimensional biological systems__. We discuss the implications in biology: when assessing living beings, __the underrated role of isolated “spatial” modifications__ needs to be emphasized, living apart the evolutionary role of time. PDF

**Time-reversal entropy: an underrated actor**

Tozzi A. Peters JF. 2017. Critique of pure free energy principle: Comment on “Answering Schrödinger's question: A free-energy formulation” by Maxwell James DésormeauRamstead et al. Physics of Life Reviews.DOI: 10.1016/j.plrev.2017.10.003.

The paper by Ramstead et al. reminds us the efforts of eminent scientists such as Whitehead and Godel. After having produced influential manuscripts, they turned to more philosophical issues, understanding the need for a larger formalization of their bounteous scientific results. In a similar way, the successful free-energy principle has been generalized, in order to encompass not only the brain activity of the original formulation, but also the whole spectrum of life. Here we go through philosophical (the principle of identity) and physical (__temperature, Pandemonium architecture, time reversal entropy__) issues that might be correlated with the free energy principle. PDF

**Curvatures in biology: when the misused "analogy" is still helpful in scientific affairs**

Geometry is correlated with both analogical thinking and physical/biological observables. Indeed, __naïve, common-sense descriptions of objects’ shapes and biological trajectories in geometric phase spaces may help experimental investigation.__ For example, different biological dynamics, such as the developmental growth patterns of the oldest known animal (the extinct __Dickinsonia) and the human brain electric oscillations__, __display striking analogies__: when encompassed in abstract spaces, their paths describe the same changes in curvature, from convex, to flat, to concave and vice versa. Such dynamical behavior, anticipated by Nicholas de Cusa in his 1440 __analogic account of “coincidentia oppositorum__”, can be used to describe widespread biological paths in terms of concave, flat and convex curves on a donut-like structure. Every one of the achieved trajectories on the torus can be located, through a topological technique called Hopf fibration, into a four-dimensional space with positive curvature. We discuss how __the correlation between Hopf fibration and Navier-Stokes equation__ allows us to treat biological and neuroscientific issues in terms of flows taking place in a viscous fluid medium which can be experimentally assessed and quantified.

**Alexander horned sphere: a toroidal phase space for both linear and nonlinear dynamics**

Linear dynamics of physical and biological systems can be described in terms of trajectories taking place inside manageable donut-like manifolds equipped with genus one. In turn, widespread nonlinear dynamics, such as paths at the edge of chaos, power law and free scale phenomena, systems’ bifurcations, iterative maps, ordinary differential equations, Bayesian issues, are difficult to map and locate inside toroidal phase spaces. Here we ask: is it feasible to build a single donut-like phase space, where both linear and non-linear dynamics might be mapped? The answer is positive, if we consider a torus equipped with an Alexander horn. Each path is arranged inside a phase space consisting of an horned ring, i.e., a modification of the pathological object termed Alexander horned sphere. This relatively simple description of both linear and nonlinear trajectories taking place inside an horned torus-shaped manifold allows to standardize the assessment of a wide range of natural and artificial paths, from time series analysis to iterative maps and population dynamics.

**Horizontal transmission modifies our concept of species: another case of decision limit problems**

__Demarcation among objects and things is somewhat arbitrary__, because our mind tends to exclude the continuity among hidden or unknown structures of the world. We are used to draw lines of separation among things that we judge different, arbitrarily excluding or including issues in our description, to achieve positive demarcations that allow a pragmatic treatment of the world based on regularity and uniformity. In touch with set theory, __observers tend to spatially and temporally split the set of the entire world in different, arbitrary, fictious subsets__ that could not really stand for different objects or events. There’s a plenty of boundaries in math, physics and biology. In these disciplines, __the concept of boundary is grounded on the presence of internal and external surfaces__. This concept seems straightforward in math, and in particular in topology, where we are in front of planes split into an “interior” region bounded by curves and an “exterior” one, containing all of the nearby and far away exterior points.

Concerning the biophysical realm, many accounts of our real world are used to think at living being as individual, self-preserving, unique subjects equipped with borders, such as cellular membranes, envelopments, Markov blankets, and so on. The canonical illustration of biological organisms __depicts “something” equipped with some peculiar activity, confined into itself, __which struggles against (and cooperates with) the external world, in order to keep its entropy as low as possible and devoted to self-preservation. However, __the demarcation among objects, things and living beings could be somewhat arbitrary__. The same limitation holds for the very concept of evolution, which is based on species. __Our concept of species is very arbitrary,__ because it throws (not experimentally demonstrated) borders among living beings. Thinking, e.g., to __different populations of Homo__, despite we are used to consider Sapiens, Neandertals, Denisovians, Floresians as different species, genetic studies undoubtedly point towards their ancient ibridation, which occurred more than once in different prehistorical contexts (Slon et al., 2018). __Concerning our DNA__, the delimitation of a species from another is sometimes difficult, due, e.g., to the presence in animal genome of viral and bacterial sequences. To make an example, __the horizontal transmission__ and lateral transduction of genetic material described by Chen et al. (2018) __is able to overtake the so-called canonical “species-specific” barriers__, making sometimes difficult to clearly encompass individual living beings in a given species.

**Facial muscles of extinct hominids**

Tozzi A. 2016. Muscles of facial expression in extinct species of the genus Homo. bioRxiv 072884; doi: https://dx.doi.org/10.1101/072884.

We present a detailed description of mimetic muscles in extinct human species, framed in comparative and phylogenetic contexts. Using known facial landmarks, we assessed the arrangement of muscles of facial expression in Homo sapiens, neanderthalensis, erectus, heidelbergensis and ergaster. __In modern humans, several perioral muscles are proportionally smaller in size__ (levator labii superioris, zygomaticus minor, zygomaticus major and triangularis) __and/or located more medially__ (levator labii superioris, zygomaticus minor and quadratus labii inferioris) than in other human species. As mimetic musculature is examined in the most ancient specimens up to the most recent, there is __a general trend towards an increase in size of corrugator supercillii and triangularis__. Homo ergaster’s mimetic musculature closely resembles modern Homo, both in size and in location; furthermore, Homo erectus and Homo neanderthalensis share many muscular features. The extinct human species had an elaborate and highly graded facial communication system, but it remained __qualitatively different from that reported in modern Homo__. Compared with other human species, __Homo sapiens exhibits a lower degree of facial expression__, possibly correlated with more sophisticated social behaviours and with enhanced speech capabilities. The presence of anatomical variation among species of the genus Homo raises important questions about the possible taxonomic value of mimetic muscles. PDF