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.

Keywords: ..

Fig. 1

PARs polarity in asymmetric cell division and schematic diagram of model. a PAR polarity process in a C. elegans embryo is divided into three phases: symmetry breaking during the initial phase, the patterning(establishment) phase of an emerging pattern, and the maintenance phase of the stationary state of polarity. A, P indicate the points of anterior and posterior poles, respectively. b Schematic diagram of model reductions is shown. Gray-coloured regions imply cytoplasm and black-colored thick line region implies cell membrane. $D_{\epsilon }$ and $D_{\epsilon }^{{}^{\prime }}$ are the cross-sectional areas of membrane and cytoplasmic spaces which are separated by the inner boundary region ($\mathrm{\Gamma }$) between cell membrane and cytosol. ${\mathrm{\Omega }}^{\prime }$ indicates bulk cytosol space in ${\mathbb{R}}^N$, and $\mathrm{\Omega }{{}^{\prime }}_{\epsilon }$ is the cytosol region around the cell membrane region, ${\mathrm{\Gamma }}_{\epsilon }^{{}^{\prime }}$. c The domain shapes of $\mathrm{\Omega }=(0,L)\times D$ are shown with respect to Neumann and periodic boundary conditions. D is defined by the edge points of line, the vertical line, and cross-sectional area in one, two, and three dimensional spaces, respectively (color figure online)

Fig. 2

Numerical simulations and polarity solutions. a Numerical results for bulk-surface model (1.2) are shown. The color maps indicate the concentrations of aPAR and pPAR on the cytosol and membrane in each region. The detailed parameter values are as follows: $D_m=7.2\times {10}^{-7},{\overline{D}}_m=1.652\times {10}^{-6},D_c={\overline{D}}_c=3.6\times {10}^{-4},\gamma =\overline{\gamma }=0.3,\alpha =\overline{\alpha }=0.06,K_1={\overline{K}}_1=0.4,K=\overline{K}=0.05$ for model (1.2) with (1.3), and ${\gamma }_1={\gamma }_2=0.3,{\alpha }_1={\alpha }_2=0.06,\kappa =0.4$ for model (1.2) with (1.4). b The concentrations of aPAR and pPAR on the cell circumference calculated in (a) are plotted. c Stable nonconstant equilibrium solutions of the cell membrane periphery model (2.3)–(2.6) on $\mathrm{\Omega }=L\times D$, where $L=(0,2)$ and $D=[0,0.2]$. The detailed parameter values are given as $d_1=7.2\times {10}^{-6},d_2=1.652\times {10}^{-5},D_1=D_2=3.6\times {10}^{-3},{\gamma }_1=2.6,{\gamma }_2=2.0,{\alpha }_1={\alpha }_2=0.06,\kappa =0.4,\tau =1.0$, $m_1=0.737115$, and $m_2=1.0293325$. (D) Stable nonconstant equilibrium solutions of the shadow system (2.17)–(2.18) on $\mathrm{\Omega }=[0,2]$ under the Neumann boundary conditions and the periodic boundary conditions. The detailed parameters are given to $d_1=7.2\times {10}^{-6},d_2=1.652\times {10}^{-5}$, $\tau =1.0$, ${\alpha }_1={\alpha }_2=0.6$, ${\gamma }_1=2.6,{\gamma }_2=2.0$, $M_1=0.737115$, and $M_2=1.0293325$ (color figure online)

Fig. 3

Stable nonconstant equilibrium solutions and minimal energy at the boundary of polarity domains. a The solutions to the full system (2.3)–(2.6) shown in Fig. 2c are plotted at $x^{\prime }=0.1$. With the same parameter values, the stationary solutions of the transformed system (3.13) in a one-dimensional space are plotted. The solutions for both systems mostly overlapped. b Energy function (3.24). The red point indicates the location of minimal energy and is at (0.750762, 0.083651). In addition, ${\ell }_b$ is the location where the energy function (3.24) is at minimum, i.e. ${\ell }_b=0.750762$. c Approximate solution given in (3.21) is plotted using $\ell ={\ell }_b$. The detailed parameter values are the same for a–c and are given as $d_1=7.2\times {10}^{-6},d_2=1.652\times {10}^{-5},D_1=D_2=3.6\times {10}^{-3},{\gamma }_1=2.6,{\gamma }_2=2.0,{\alpha }_1={\alpha }_2=0.06,\kappa =0.4,\tau =1.0$, $m_1=0.737115$, and $m_2=1.0293325$ (color figure online)

Fig. 4

The size of polarity domain and the effect of cell shape. a The effect of the cell membrane length on the size of the pPAR domain is plotted by the energy functionals ${\mathcal{E}}_s$, ${\tilde{E}}_s$, and $\tilde{e}$ ((3.24), (3.25), and (3.27)). Here, $L_D=L/|D|$ where |D| is fixed. In addition, ${\ell }_b$ is the length of the pPAR domain corresponding to the value of $\ell$ where the energy functionals (3.24), (3.25), and (3.27) have minimal energy, respectively. b–c Schematic figures for the effect of the cell shape to the length scale (boundary position) of the polarity domain. From a, the ratio of the pPAR (aPAR) domain to the cell membrane length becomes constant. $V_S$ and V are the cell volumes and $Vs\ll V$. $L_D^{\mathit{name}}$ is the length of cell circumference. d The effect of the total mass on the size of the pPAR domain is plotted by the energy functionals. E The effects of both the cell membrane length and the total mass. The same parameter values as in Fig. 2 are chosen for a, d, and e

Fig. 5

Conservation of total mass in phase-field-combinded model. The concentration of total mass of aPAR and pPAR is plotted for the model (1.2) with (1.4) calculated in Fig. 2a. The average of numerical data of the total mass is 0.90319596, and the standard derivation is 0.00474245