Abstract
Asymmetric cell division is one of the fundamental processes to create cell diversity in the early stage of embryonic development. During this process, the polarity formation in the cell membrane has been considered as a key process by which the entire polarity formation in the cytosol is controlled, and it has been extensively studied in both experiments and mathematical models. Nonetheless, a mathematically rigorous analysis of the polarity formation in the asymmetric cell division has been little explored, particularly for bulk-surface models. In this article, we deal with polarity models proposed for describing the PAR polarity formation in the asymmetric cell division of a C. elegans embryo. Using a simpler but mathematically consistent model, we exhibit the long time behavior of the polarity formation of a bulk-surface cell. Moreover, we mathematically prove the existence of stable polarity solutions of the model equation in an arbitrary high-dimensional domain and analyse how the boundary position of polarity domain is determined. Our results propose that the existence and dynamics of the polarity in the asymmetric cell division can be understood universally in terms of basic mathematical structures.
Highlights
The polarity formation emerging during the early stage of embryonic development is a spectacular mechanism wherein a single mother cell of fertilised egg creates completely different daughter cells through asymmetric cell divisions (Campanale et al 2017; Gönczy 2005)
After the symmetry is broken by sperm entry, the posterior PARs (pPAR) spreads from the posterior pole, and the growth of the domain of pPAR polarity stops at approximately half the egg length with the formation of the two segregated polarity domains of anterior PARs (aPAR) and pPAR
Our results suggest that the existence and dynamics of the PAR polarity during asymmetric cell division can be understood based on a basic mathematical structure, which should be held universally without dependence on a specific choice of parameter values
Summary
The polarity formation emerging during the early stage of embryonic development is a spectacular mechanism wherein a single mother cell of fertilised egg creates completely different daughter cells through asymmetric cell divisions (Campanale et al 2017; Gönczy 2005). Few studies on the PAR polarity in terms of high-dimensional models reflecting the cell geometry of a bulk cytosol space and a surface cell membrane have been conducted numerically or mathematically. Related to bulk cytosol-surface models, we refer to studies on Turing-type instability (Levine and Rappel 2005; Morita and Sakamoto 2018, 2020; Rätz and Röger 2012, 2014), the existence and stability of a polarized solution reduced on the sphere (Diegmiller et al 2018), and polarized patterns numerically shown in 3dimensional domains with complex geometries (Cusseddu et al 2019), where different types of reaction–diffusion systems are investigated. We consider the aPAR-pPAR polarity models suggested by Seirin-Lee and Shibata (2015) and Goehring et al (2011b), and extend these models to include a high-dimensional case with a cell geometry composed of a bulk cytosol space and a cell membrane. We consider two types of off-rate functions, which have been
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