Noname manuscript No. (will be inserted by the editor) Exploring group theory and topology for analyzing the structure of biology Shun Adachi Received: date / Accepted: date Abstract The concepts of population and species play a fundamental role in biology. The existence and precise definition of higher-order hierarchies, such as division into species, is open to debate among biologists. First, we seek to show a fractal structure of species. We are able to define a species as a p-Sylow subgroup of a particular community in a single niche, confirmed by topological analysis. We named this model the patch with zeta dominance (PzDom) model. Next, the topological nature of the system is carefully examined and for test- ing purposes, species density data are used in conjunction with data derived from liquid-chromatography mass spectrometry of proteins. We confirm the induction of hierarchy and time through a one-dimensional probability space with certain topologies. For further clarification of induced fractals including the relation to renormal- ization in physics, a theoretical development is proposed based on a newly identified fact, namely that scaling parameters for magnetization exactly cor- respond to imaginary parts of the Riemann zeta function’s nontrivial zeros. A master torus and a Lagrangian/Hamiltonian are derived expressing fractal structures as a solution for diminishing divergent terms in renormalization. We will also focus on an application of our developed model. We extend current PzDom model to the so-called exPzDom model to qualify population dynamics as a topological matter as a whole, not focusing on hierarchy. The indicators in the exPzDom model adhere well to the empirical dynamics of SARS-CoV-2 infected people and align appropriately with actual policies instituted by the S. Adachi Department of Microbiology, Kansai Medical University, 2-5-1 Shin-machi, Hirakata, Osaka 573-1010, JAPAN Tel.: +81-72-804-2380 arXiv:1603.00959v9 [q-bio.PE] 16 Nov 2020 Fax: +81-72-804-2389 E-mail: [email protected] ORCID: 0000-0001-7555-8373 2 Shun Adachi Japanese government. In our patch with zeta dominance (PzDom) model or its extended version (exPzDom), calculations only require knowledge of the density of individuals over time. Keywords Species Hierarchy Group theory Topology Riemann zeta function Selberg zeta· function · · · · 1 Definitions To apply logical relations of topics of interests similar to the mathematical ter- minology, we introduce following terms and their definitions in this manuscript. Please note the definitions are only applicable to this manuscript; they have nothing to do with corresponding mathematical terms in a direct sense. Axiom A prerequisite for a logical derivation, either an experimental fact or an as- sumption. Theorem A discussion of an interest. Lemma A discussion that is not on a main line of logic, but interesting. Conclusion A conclusion. Problem A problem still not solved. Scholium A miscellaneous for a certain topic. Group theory and topology for biology 3 Case A distinctive case for an experimental system. Note Included axioms are only five, and the reader can easily acquire the under- lying basic assumptions. If the reader would like to grasp ideas in brief, the reader can only go through axioms, theorems and conclusions. Lemmas are for advanced discussions. Scholia are for additional information. The reader can interpret the data in the various cases. 2 Introduction Problem 1: an introduction to the species problem Living organisms encompass several levels of scaling and hierarchy. Inside cells, protein molecules are in the order of nanometers, and they cooperate or com- pete by activation or inhibition of specific biological activities. Normal eukary- otic cells are in the order of 10 µm, and their activities are the consequence of interactions between the molecules inside of them. In multicellular systems, each cell has its own role, and these combine to determine the interactions between the cells in the system and between the systems and its environment. In nature, the number of individuals in a population increases or decreases fol- lowing the particular dynamics of that population, as characterized by their intrinsic physiology and the interactions between individuals within the popu- lation and with those in other populations. Molecular biology and ecology have already elucidated certain roles for the hierarchies that are inherent in living organisms. Therefore, to compare the dynamics of communities of various bi- ological taxa, it is important to have a common definition of species. However, there are various ways to define a species, and none of these is uniformly applicable to phenomena considered in the context of biology. For example, biological species concept is stated as “groups of actually or potentially in- terbreeding natural populations, which are reproductively isolated from other such groups”. Although this species concept is experimentally quantifiable and the most well-defined concept compared to others, it cannot be applied to asex- ual living organisms that do not mate each other. For others, morphological species concept, “a group of organisms in which individuals conform to cer- tain fixed properties”, can be applied only when the characteristics of interests have discrete natures so as to distinguish a pair of species well. Evolutionary species concept, “an entity composed of organisms which maintains its identity from other such entities through time and over space, and which has its own independent evolutionary fate and historical tendencies”, has arbitrary nature because it is not clear what extent of divergence and distant-related phylogeny can necessarily and sufficiently cause the difference of species level. 4 Shun Adachi Problem 2: species problem regarded as indicator values and morphisms, es- pecially via group theory The situations is complex and it is very likely that we need a simpler definition than the ones currently in use for a pragmatic analyses of species. Let us start from simple philosophical beginnings. When we regard a certain level of hier- archy, the level has to possess identical homeostasis. To maintain its identity requires adaptation to the environment in natural systems. The biological term ‘adaptation’ is thus integrated as a basic idea in evolvable systems with hierar- chy. If we start from a certain level of hierarchy where adaptation is obviously applicable (such as the individual level, or, in our case, metapopulation: a set of local populations that are linked by dispersal; metapopulation is introduced here to grasp population dynamics linked to actual space, or a sum of patch in the environment, to function as a unit of dynamic behavior that obeys a rule of heat bath), by utilizing morphisms between subsequent levels of the hierarchy, reoccurrence of a certain indicator value in an adaptive hierarchy different from a level to another level among the sequence of the morphisms indicates there is another level of hierarchies with adaptation, proving the ex- istence of the hierarchy. Therefore, if we select a proper set of indicator value and morphism, we can recognize the actual hierarchy in nature with adapta- tion. The existences of such morphisms and indicator values are guaranteed by believing that there is a discrete nature of species in biological observations, empirically supported by clustering of genetic distances along phylogenetical tree/web. If there is no such set of morphisms and indicators, the identity of hierarchy collapsed and the clustering of phylogeny would vanish. We shall conclude to exist later in the text, as in 4.4. Based on the theories later in this manuscript, this section proposes a particular example of such a set of morphisms and indicators as a p-Sylow subgroup, which we will introduce in the following texts. For a trial, we introduce group theory to our model as an example of such a morphism. If you read this manuscript till 4.4, you can grasp this simple introduction may sufficient to select the group theory as a theory for the morphisms. A set G can be regarded as a group if G is accompanied by an operation (group law) that combines any two elements of G and satisfies the following axioms. (i) closure: a result of an operation is also an element in G; (ii) associativity: for all aG, bG and cG in G, (aGbG)cG = aG(bGcG); (iii) identity element: for all elements aG in G, there exists an element 1G in G such that 1GaG = aG1G = aG holds; (iv) inverse element: for all aG in 1 1 1 G, there exists an element aG− such that aGaG− = aG− aG = 1G. Next, we move forward to the nilpotent group. If a group G has a lower central series of subgroups terminating in the trivial subgroup after finitely many steps: G = G0 G1 Gj = 1G where at all i = 1, 2,...,j, Gi 1/Gi [a ⊃ ⊃···⊃ { } − ⊂ center of G/Gi] then G is called a nilpotent group. If a group is a nilpotent group, after finitely many steps (or, in other words, finitely many time steps), it can converge to a certain identity as a unique trivial subgroup. In biology, an adaptation process to a particular environment with time series is likely Group theory and topology for biology 5 to fulfill the condition here, to reach a certain goal for a particular living organism. One line for the motivation for this theory is theoretically all the p- group must be nilpotent group (from textbooks for algebra), and if the group is not a nilpotent, it must not be a p-Sylow subgroup. If it is not, we cannot proceed to the next step. The other line is also notable that 1G status is a status in equilibrium in a particular niche, as its multiplication does not alter the status of a particular niche. For the ring structure as it proposed in 4.4, 0G appears and addition of a different niche to a particular community in equilibrium does not alter the remaining niche in equilibrium.