Fibonacci Sequences, Symmetry and Order in Biological Patterns, Their Sources, Information Origin and the Landauer Principle

Abstract

Physical roots, exemplifications and consequences of periodic and aperiodic ordering (represented by Fibonacci series) in biological systems are discussed. The physical and biological roots and role of symmetry and asymmetry appearing in biological patterns are addressed. A generalization of the Curie–Neumann principle as applied to biological objects is presented, briefly summarized as: “asymmetry is what creates a biological phenomenon”. The “top-down” and “bottom-up” approaches to the explanation of symmetry in organisms are presented and discussed in detail. The “top-down” approach implies that the symmetry of the biological structure follows the symmetry of the media in which this structure is functioning; the “bottom-up” approach, in turn, accepts that the symmetry of biological structures emerges from the symmetry of molecules constituting the structure. A diversity of mathematical measures applicable for quantification of order in biological patterns is introduced. The continuous, Shannon and Voronoi measures of symmetry/ordering and their application to biological objects are addressed. The fine structure of the notion of “order” is discussed. Informational/algorithmic roots of order inherent in the biological systems are considered. Ordered/symmetrical patterns provide an economy of biological information, necessary for the algorithmic description of a biological entity. The application of the Landauer principle bridging physics and theory of information to the biological systems is discussed.

Bias in the arrival of variation can dominate over natural selection in Richard Dawkins’s biomorphs

“…Less information is required to describe bonding patterns that lead to higher symmetry, and thus such phenotypes have a higher probability of appearing upon random mutations [ 3 ]. One could imagine extending this preference for symmetry, modulated by processes such as symmetry breaking [ 71 ], to larger-scale developmental processes (see [ 72 , 73 ] for a discussion). In other cases, including the RNA secondary structures and branching morphologies (see ref [ 74 ]), different signatures of simplicity need to be employed to identify processes that can be described by shorter algorithms, which should be easier to find through random mutations.…”

Biological hypercrystals

“…A Voronoi pattern results from the slicing of a plane with convex polygons so that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other one. Therefore, the partitioning of an infine plane into regions based on the distance to a specified discrete set of points (called seeds or nuclei) constructs the Voronoi tessellation [12,13]. The pattern stemming from the cells' membranes in the epithelial tissue shown in Figure 1A reasonably corresponds to the Voronoi tessellation arising from to the cells' nuclei distribution throughout the space.…”

“…In this diagram the projection window W defines the so-called acceptance domain, since only those lattice points inside the strip of width W are projected onto E║ (dashed lines). (B) A spiral pattern with phyllotactic index (8,13). The spiral lattice points contained inside the concentric annulus (circular dashed and dotted lines) are radially projected onto the center (dotted radial lines), thereby intersecting the inner circle in a series of long (A) and short (B) arcs arranged according to the Fibonacci sequence (Reprinted from E Maciá, Aperiodic crystals in biology, J.…”

Inclusion of mathematics in biological concepts at the senior secondary level in Indian education system

“…Leonardo Pisano, who was later known as Fibonacci presented the Fibonacci sequence, in which every number (after the first two) is the sum of the two preceding numbers, for example: 0, 1, 1, 2, 3,5,8,13,21,34,55,89,144, and so on. Fibonacci numbers appear in various biological forms like the shape of sunflower head, shape of mollusc shell, human anatomy (cochlea, phalanges of hand), leaves and cones of plants, ciliary rows in eukaryotic protozoans, body plans in arthropods as well as microscopic structures like the DNA double helix, and many more (Figure 1) (Bormashenko, 2022;Sinha, 2019;Persaud-Sharma & Leary, 2015). The students at the senior secondary level study about this important sequence only in Mathematics (this concept has been included under chapter 'sequences and series' in National Council of Educational Research and Training (NCERT) Mathematics Textbook of Class XI).…”