Long time behavior and stable patterns in high-dimensional polarity models of asymmetric cell division

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.

Delineating the mechanisms and design principles of Caenorhabditis elegans embryogenesis using in toto high-resolution imaging data and computational modeling

“…The cell size was also determined to be crucial for the establishment of polarity, due to the reaction–diffusion system describing the interplay between aPAR and pPAR not being linearly scalable; this explains why the asymmetric division doesn't take place in the last germline precursor cell, P4 [163] . In theory, the location of the transition region between anterior and posterior domains is sensitive to changes in cell shape when a cell is relatively small, which implies that additional programmed regulation might be employed to ensure the precision and robustness of cell polarization [186] .…”

Segregation pattern in a four-component reaction-diffusion system with mass conservation

“…A model describing this polarity was proposed in [13] using a four-component reactiondiffusion system, which numerically demonstrates a segregation pattern corresponding to the polarity. However, due to the complexity of the four-component system, a mathematical theory for the model system had not been well studied until [18]. In [18], through reasonable modifications to the nonlinear terms, they discovered a favorable variational structure for the model system and mathematically showed the existence of a stable nonconstant equilibrium solution for a specific parameter regime.…”

“…In this article, we investigate the system proposed in [18] and prove the existence of a solution representing the segregation pattern. The system which we deal with is written as…”

“…In [18] u 1 and v 1 represent the density of anterior GPA in the membrane and the cytosol respectively, while u 2 and v 2 represent the density of posterior GPA in the membrane and cytosol respectively. The modeling naturally assumes…”

“…Proposition 1.1. ( [18]) Let (u * 1 , u * 2 ) be a local minimizer of (10) and let (v * 1 , v * 2 ) be defined as in (9). Then, given ε > 0, there exists δ > 0 such that if the solution (u 1 (x, t), v 1 (x, t), u 2 (x, t), v 2 (x, t)) to (1) with (2) and (3) satisfies…”