CONCEPT PAPER | doi:10.20944/preprints201911.0372.v1
Subject: Biology, Plant Sciences Keywords: arabinogalactan proteins; phyllotaxis; Hechtian Oscillator; calcium homeostasis; auxin
Online: 29 November 2019 (08:22:04 CET)
Sixty years ago in the lab adjacent to Fred Sanger (1958 Nobel Prize for protein chemistry), I discovered the cell surface hydroxyproline-rich glycoproteins. Nature keeps some of her secrets longer than others. It has taken many years to dissect the molecular function and biological role of extensins and arabinogalactan proteins (AGPs). Extensins template the formation of new cell walls. AGPs remained baffling and enigmatic until a Eureka moment when computer prediction of AGP calcium binding depicted paired glucuronic acid residues and thus the likely role of a cell surface AGP-Ca2+capacitor: In conjunction with the auxin-activated proton pump that releases bound Ca2+ it led us to formulate the Hechtian Growth Oscillator as A Global Paradigm with a pivotal role in Ca2+ homeostasis. The ramifications are profound. They cannot be shrugged off with sceptical disdain but demand critical reappraisal of current dogma. Phyllotaxis is an ancient problem; it involves an essential role for auxin and the auxin efflux “PIN” proteins together with mechanotransduction of stress-strain as phyllotactic determinants. However, a general explanation remains elusive despite much effort, particularly by mathematicians. Here we propose a novel biochemical algorithm: Hechtian oscillator transduction of cell wall stress generates phyllotactic patterns quite independent of a mathematical approach. Plants simply use different rules and follow a different route.
ARTICLE | doi:10.20944/preprints201908.0284.v4
Subject: Life Sciences, Genetics Keywords: hyperbolic numbers; matrix; eigenvectors; genetics; Punnett squares; Fibonacci numbers; phyllotaxis; music harmony; literary texts; doubly stochastic matrices
Online: 13 April 2020 (11:04:05 CEST)
The article is devoted to applications of 2-dimensional hyperbolic numbers and their algebraic 2n-dimensional extensions in modeling some genetic and cultural phenomena. Mathematical properties of hyperbolic numbers and their bisymmetric matrix representations are described in a connection with their application to analyze the following structures: alphabets of DNA nucleobases; inherited phyllotaxis phenomena; Punnett squares in Mendelian genetics; the psychophysical Weber-Fechner law; long literary Russian texts (in their special binary representations). New methods of algebraic analysis of the harmony of musical works are proposed, taking into account the innate predisposition of people to music. The hypothesis is put forward that sets of eigenvectors of matrix representations of basis units of 2n-dimensional hyperbolic numbers play an important role in transmitting biological information. A general hyperbolic rule regarding the oligomer cooperative organization of different genomes is described jointly with its quantum-information model. Besides, the hypothesis about some analog of the Weber-Fechner law for sequences of spikes in single nerve fibers is formulated. The proposed algebraic approach is connected with the theme of the grammar of biology and applications of bisymmetric doubly stochastic matrices. Applications of hyperbolic numbers reveal hidden interrelations between structures of different biological and physical phenomena. They lead to new approaches in mathematical modeling genetic phenomena and innate biological structures.
ARTICLE | doi:10.20944/preprints202205.0207.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: prime numbers; galactic spiral equations; ROTASE model; phyllotaxis; sunflower seed head
Online: 16 May 2022 (11:04:46 CEST)
In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. For the spirals only produced by the prime numbers in the 6-spiral pattern plotting, the spiral 2 and spiral 3 each has only one spiral point produced by prime number 2 and 3, respectively, all other spiral points produced by other prime numbers are located on the spiral 1 and spiral 5. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P1 = 1 + 6 * n (n > 0) P5 = 5 + 6 * n (n ≥ 0) In other words, every prime number greater than 3 is either a P1 prime number or a P5 prime number, no exception. Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Hope such sufficient condition can be found in the future. The number of P1 prime numbers roughly equal the number of P5 prime number in the first 2 billion prime numbers. The galactic spiral equations with golden angle can duplicate Vogel’s result for the simulation of sunflower seed head pattern, and a pinwheel pattern can be produced also with galactic spiral equations and 1 degree more than golden angle.