Ring shape Golden Ratio multicellular structures are algebraically afforded by asymmetric mitosis and one to one cell adhesion

Abstract

AbstractGolden Ratio proportions are found throughout the world of multicellular organisms but the underlying mechanisms behind their appearance and their adaptive value if any remain unknown. The Golden Ratio is a real-valued number but cell population counts are whole numbered. Binet's formula connects the Golden Ratio to the whole numbered Fibonacci sequence (fn+1 = fn + fn–1 where f1 = 1 and f2 = 2), so we seek a cellular mechanism that yields Fibonacci cell kinetics. Drawing on Fibonacci’s description of growth patterns in rabbits, we develop a matrix model of Fibonacci cell kinetics based on an asymmetric pause between mitoses by daughter cells. We list candidate molecular mechanisms for asymmetric mitosis such as epigenetically asymmetric chromosomal sorting at anaphase due to cytosine-DNA methylation. A collection of Fibonacci-sized cell groups produced each by mitosis needs to assemble into a larger multicellular structure. We find that the mathematics for this assembly are afforded by a simple molecular cell surface configuration where each cell in each group has four cell to cell adhesion slots. Two slots internally cohere a cell group and two adhere to cells in other cell groups. We provide a notation for expressing each cell’s participation in dual Fibonacci recurrence relations. We find that single class of cell to cell adhesion molecules suffices to hold together a large assembly of chained Fibonacci groups having Golden Ratio patterns. Specialized bindings between components of various sizes are not required. Furthermore, the notation describes circumstances where chained Fibonacci-sized cell groups may leave adhesion slots unoccupied unless the chained groups anneal into a ring. This unexpected result suggests a role for Fibonacci cell kinetics in the formation of multicellular ring forms such as hollow and tubular structures. In this analysis, a complex molecular pattern behind asymmetric mitosis coordinates with a simple molecular cell adhesion pattern to generate useful multicellular assemblies. Furthermore, this reductively unifies two of the hypothesized evolutionary steps: multicellularity and cellular eusociality.