Abstract
In the process of asymmetric cell division, the mother cell induces polarity in both the membrane and the cytosol by distributing substrates and components asymmetrically. Such polarity formation results from the harmonization of the upstream and downstream polarities between the cell membrane and the cytosol. MEX5/6 is a wellinvestigated downstream cytoplasmic protein, which is deeply involved in the membrane polarity of the upstream transmembrane protein PAR in the Caenorhabditis elegans embryo. In contrast to the extensive exploration of membrane PAR polarity, cytoplasmic polarity is poorly understood, and the precise contribution of cytoplasmic polarity to the membrane PAR polarity remains largely unknown. In this study, we explored the interplay between the cytoplasmic MEX5/6 polarity and the membrane PAR polarity by developing a mathematical model that integrates the dynamics of PAR and MEX5/6 and reflects the cell geometry. Our investigations show that the downstream cytoplasmic protein MEX5/6 plays an indispensable role in causing a robust upstream PAR polarity, and the integrated understanding of their interplay, including the effect of the cell geometry, is essential for the study of polarity formation in asymmetric cell division.
Introduction
Asymmetric cell division is an elegant developmental process that creates cell diversity (Campanale et al. 2017; Knoblich 2008; Gönczy 2005). A mother cell distributes substrates and components asymmetrically before cell division and transfers them to two daughter cells, asymmetrically. Ultimately, this leads to two daughter cells with different functions and sizes. One representative experimental model of asymmetric cell division is the fertilized egg cell of Caenorhabditis elegans (Cuenca et al. 2002; Gönczy 2005; Motegi and Seydoux 2013). With the entry of sperm, a fertilized egg cell undergoes symmetry breaking in the posterior pole site (Fig. 1a). Concurrent with the symmetry breaking, the actomyosin network in the cell cortex begins contracting from the site of symmetry breaking and stops contracting in the middle of the cell (Nishikawa et al. 2017; Niwayama et al. 2011). It is known that actomyosin contraction causes cortical flow directed from the posterior to the anterior side of the cell, and cytoplasmic flow directed from the anterior to the posterior side in the center of the cell but directed from the posterior to the anterior side in the periphery of the cell membrane (Gönczy 2005; Goehring et al. 2011b; Niwayama et al. 2011) (Fig. 1a, blue arrows).
Initially, PAR6, PAR3, and PKC3, a group known as anterior proteins (aPAR), are homogeneously distributed in the membrane and cytosol, while PAR2 and PAR1, a group known as posterior proteins (pPAR), are homogeneously distributed in the cytosol. However, once symmetry breaking occurs, these protein groups begin to form exclusive polarity domains in the membrane (Fig. 1a). pPAR relocates to the site of symmetry breaking, and aPAR relocates to the opposite site. The location of the polarity domain of these protein groups determines the anterior–posterior axis of the mother cell, and the boundary of the two exclusive polarity domains in the membrane is maintained for approximately 16 min (Gönczy 2005) after the establishment phase of the polarity, which is observed to be approximately 6–8 min (Cowan and Hyman 2004).
PAR polarity in the membrane is considered to play the central role in regulating the entire process of asymmetric cell division; therefore, both experimental and theoretical approaches to elucidate the mechanism of PAR polarity formation have been extensively studied (Cortes et al. 2018; Hoege and Hyman 2013; Lang and Munro 2017; Motegi and Seydoux 2013; Rappel and Levine 2017; SeirinLee 2020; SeirinLee et al. 2020a; Small and Dawes 2017; Zonies et al. 2010). The formation of an exclusive domain is underlined by the mutual inhibition dynamics between the anterior and posterior protein groups in which the aPAR/pPAR protein transmits pPAR/aPAR protein from the membrane to the cytosol. Theoretically, it has been demonstrated that bistability, due to mutual inhibition dynamics, and mass conservation are the basic mechanisms of polarity formation (Kuwamura et al. 2018; SeirinLee et al. 2020b; Trong et al. 2014).
Interestingly, similar polarity dynamics are also observed for cytoplasmic proteins (Cuenca et al. 2002; Daniels et al. 2010). The cytoplasmic MEX5/6 protein simultaneously creates a polarity in the cytosol with PAR polarity formation in the membrane (Fig. 1a). The cytoplasmic MEX5/6 protein, distributed homogeneously in the cytosol before symmetry breaking, becomes polarized to the anterior side, and the boundary of the MEX5/6 polarity domain is observed in a location similar to the boundary of the anterior and posterior polarity domains (Cuenca et al. 2002; Schubert et al. 2000). Unlike the mechanism of PAR polarity formation in the membrane, it was found that MEX5/6 has two different diffusive types: slowdiffusing and fastdiffusing, and it was suggested that MEX5/6 creates polarity using the conversion dynamics of mobility. In the early stages of MEX5/6 polarity studies, it had been hypothesized that the conversion dynamics of mobility is regulated by the phosphorylation and dephosphorylation cycle directly controlled by the membrane pPAR and aPAR proteins (Daniels et al. 2010). However, Griffin et al. (2011) suggested that pPAR (PAR1) plays a key role and promotes the conversion from slowdiffusing MEX5/6 to fastdiffusing MEX5/6, but that aPAR does not play a direct role in the conversion of MEX5/6 diffusivity. Furthermore, they hypothesized that the phosphatase PP2A antagonizes PAR1dependent phosphorylation of MEX5, returning MEX5 to the slowdiffusing state. However, aPAR (PKC3) has been found to be significantly involved in regulating the conversion dynamics of the fast diffusive type of MEX5/6 to the slow diffusive type (Wu et al. 2018), though it was supposed that the regulation of the conversion dynamics is likely to be indirectly regulated by aPAR proteins (Griffin et al. 2011).
While the mechanism underlying cytoplasmic polarity MEX5/6 has been well investigated experimentally at a molecular level, a theoretical approach that integrates experimental observations is lacking. Moreover, it is not clear how MEX5/6 polarity formation is related to the dynamics of the PARs. In particular, there has not been a study that explores how the cortical and cytoplasmic flows interact with MEX5/6 when realistic cell geometry is included. Thus, in this study, we focus on three issues: Firstly, we formulate the MEX5/6 model by combining it with the upstream PAR dynamics. Secondly, we explore how MEX5/6 polarity in the cytosol affects the spatial and temporal dynamics of membrane PAR polarity. Finally, we investigate the effect of the flow dynamics and cell geometry. We explore how these two factors affect the dynamics of the cytoplasmic proteins and, consequently, the formation of membrane PAR polarity. In this study, we also introduce a general method, using phasefield modeling, to combine cell geometry with a convection–reaction–diffusion system. This method will present an easy numerical technique to simulate convection–reaction–diffusion equations on a higherdimensional bulksurface domain of various cell shapes.
This study suggests that it is not only the upstream polarity of PARs that dominates the downstream polarity of MEX5/6, but also that the downstream polarity of MEX5/6 can critically affect both the spatial and temporal dynamics of PAR polarity, and that the interaction between membrane PAR polarity and cytoplasmic MEX5/6 polarity is vital for inducing robust cell polarity during asymmetric cell division.
Model Development
PARs Model
Mathematical models for the polarity formation of PAR dynamics have been proposed in several studies (Dawes and Munro 2011; Goehring et al. 2011b; SeirinLee and Shibata 2015; Tostevin and Howard 2008; Trong et al. 2014). All these models suggested similar mathematical structures based on bistability and mass conservation for the creation of polarity. Therefore, we adopt the standard model for PAR dynamics suggested by SeirinLee and Shibata (2015), and extend it to a higherdimensional bulksurface model. Let us define the cytosol by \(\Omega \subset {\mathbb {R}}^{N}\) and the membrane by \(\partial \Omega ~(\equiv \Gamma )\), where \(\Omega \) is an open subset of \({\mathbb {R}}^{N}\) such that \({\bar{\Omega }}\) represents a cell (Fig. 1b). We also define the concentrations of anterior proteins (aPAR) in the membrane and cytosol by \([A_{m}]({\mathbf {x}},t)\) and \([A_{c}]({\mathbf {x}},t)\), respectively, and the concentrations of posterior proteins (pPAR) in the membrane and cytosol by \([P_{m}]({\mathbf {x}},t)\) and \([P_{c}]({\mathbf {x}},t)\), respectively, where \({\mathbf {x}}\in {\mathbb {R}}^{N}\) and \(t\in [0, \infty )\). Then, the PAR polarity model is given by
where \({\mathbf {n}}\) is the inner normal vector on \(\partial \Omega \). Here, \({\mathbf {v}}_{m}\) and \({\mathbf {v}}_{c}\) are cortical and cytoplasmic flow velocity functions, respectively, \(D_m^A\) and \(D_m^P\) are the diffusion rates of aPAR and pPAR in the membrane, respectively, and \(D_c^A\) and \(D_c^P\) are the diffusion rates of aPAR and pPAR in the cytosol, respectively. \(F_{\text {on}}^{A}\) and \(F_{\text {on}}^{P}\) are the onrate functions of aPAR and pPAR from the cytosol to the membrane, respectively, and \(F_{\text {off}}^{A}\) and \(F_{\text {off}}^{P}\) are the offrate functions of aPAR and pPAR, respectively. Note that \(D_c^A>D_m^A\) and \(D_c^P>D_m^P\) because diffusion in the cytosol is faster than that in the membrane (Kuhn et al. 2011; Goehring et al. 2011b). We define the detailed form of the flow velocity functions in Sect. 2.5.
The offrate functions reflect the effect of the mutual inhibition of aPAR and pPAR. aPAR/pPAR transports pPAR/aPAR from the membrane to the cytosol (Fig. 1c), and we select the functional forms suggested in SeirinLee and Shibata (2015):
where \(\alpha _{1}\) and \(\alpha _{2}\) are basal offrates, and \(K_{1},K_{2}, {\overline{K}}_2, K_{3}, K_{4}\) and \({\overline{K}}_{4}\) are positive constants determining the offrates. \(n(>1)\) is the Hill coefficient, and we select \(n=2\) for the simulations. The onrate functions are given by
where \(\gamma _1\) and \(\gamma _2\) are the onrates of aPAR and pPAR, respectively.
MEX5/6 Model
It is well known that MEX5/6 has both slow and fast diffusion types in the cytosol, and that the conversion of one diffusion type to the other is regulated by PAR proteins (Daniels et al. 2010; Griffin et al. 2011; Wu et al. 2018). To develop a MEX5/6 model combined with the PARs dynamics, we first formulate a general conversion model of MEX5/6 diffusion. Defining the concentrations of the fast diffusive type of MEX5/6 and slow diffusive type of MEX5/6 by \([M_{f}]({\mathbf {x}},t)\) and \([M_{s}]({\mathbf {x}},t)\), respectively, the general MEX5/6 conversion model is given by
where \(D_s\) and \(D_f\) are the diffusion coefficients for slow and fast diffusive types of MEX5/6, respectively, with \(D_f>D_s\). \(G_{S\rightarrow F}^{\ell }({\mathbf {x}},t)\) and \(G_{F\rightarrow S}^{\ell }({\mathbf {x}},t)\) are conversion functions from the slow diffusive type to the fast diffusive type, and from the fast diffusive type to the slow diffusive type, respectively, where \(\ell \) denotes either cytosol (C) or membrane (M).
Next, we derive forms for \(G_{S\rightarrow F}^{\ell }({\mathbf {x}},t)\) and \(G_{F\rightarrow S}^{\ell }({\mathbf {x}},t)\). In the wild type of C. elegans, the diffusion rate of MEX5/6 in the posterior side is notably higher than in the anterior side. The studies by Daniels et al. (2010) and Griffin et al. (2011) suggest that pPAR (PAR1) regulates the slow type of MEX5/6, and the diffusion rate of MEX5/6 of the posterior side in the PAR1 mutant cell is significantly decreased compared to the wild type. This result indicates that PAR1 promotes the conversion dynamics of the slow type of MEX5/6 to the fast type. Thus, we suppose that the conversion rate from the slow type to the fast type has a positive correlation with pPAR concentration, and we propose that
where \(\mu _1\) and \(\mu _3\) are positive correlation constants reflecting the effective strength of pPAR on the conversion rate around the cell membrane and within the bulk cytosol, respectively.
The study by Wu et al. (2018) suggests that the diffusion rate of MEX5/6 in the anterior side of the aPAR(PKC3) mutant cell is significantly increased compared to the wild type. This result suggests two hypotheses: either PKC3 promotes the conversion dynamics of the fast type of MEX5/6 to the slow type, or PKC3 plays a role in inhibiting the conversion dynamics from the slow type to the fast type of MEX5/6. However, the diffusion rates of both PKC3 and PAR1 mutant cells did not show a notable difference from that of only PKC3 mutant cell. This indicates that PKC3 does not play the latter role, and it may promote the conversion of fast type to slow type of MEX5/6. Thus, we suppose that the conversion rate from the fast type to the slow type has a positive correlation with aPAR concentration, and this leads us to define
where \(\mu _2\) and \(\mu _4\) are positive correlation constants reflecting the effective strength of aPAR on the conversion rate in the cell membrane and bulk cytosol, respectively.
MEX5/6CombinedPARs Model
Experimental observations of the C. elegans embryo suggest that MEX5/6 regulates the expansion of the pPAR domain by helping to exclude the aPAR domain, rather than directly promoting pPAR localization (Cuenca et al. 2002; Schubert et al. 2000), which suggests the possibility that MEX5/6 may directly regulate the translocation dynamics of aPAR between the membrane and the cytosol. On the other hand, the detailed molecular mechanism of the interaction between MEX5/6 and the PARs is still unclear. Thus, we consider two possible assumptions. We suppose that the experimental observation of Cuenca et al. (2002) is related to either the onrate or the offrate of aPAR in the model (1). Thus, we assume that in one model, MEX5/6 inhibits the recruitment of aPAR from the cytosol to the membrane (i.e., the onrate of aPAR), and in the other model, we assume that MEX5/6 promotes the transport of aPAR from the membrane to the cytosol (i.e., the offrate of aPAR). We call these models H1 and H2, respectively (Fig. 1c). We formulate the simplest type of model as follows:
where \([M]({\mathbf {x}},t)=[M_{f}]({\mathbf {x}},t)+[M_{s}]({\mathbf {x}},t)\), and \(\mu _{0}\) is either the inhibition rate of aPAR recruitment or the promotion rate of aPAR transport from the membrane to cytosol by MEX5/6. Note that \(\mu _0=0\) recovers the original model (1) without the effect of MEX5/6. We name the combination of model (1) including (5), with model (3), the MEX5/6combinedPARs Model (Fig. 1c).
MEX5/6CombinedSelfRecruitment pPAR Model
The selfrecruitment model of PAR dynamics was first suggested in SeirinLee and Shibata (2015), in which the aPARpPAR model is reduced to either an aPAR alone or a pPAR alone model, and the effect of the offrate (by the mutual inhibition) is replaced by a selfrecruitment effect resulting from either aPAR or pPAR itself. By applying this reduction to the selfrecruitment model, we can study the interaction of PAR polarity dynamics with pPAR alone. This gives us more precise information on how pPAR is directly, or indirectly, involved in MEX5/6 dynamics (Fig. 1d, e). Thus, we here reduce the MEX5/6combinedPARs Model (1)–(3) to the selfrecruitment form and show that the conversion model of MEX5/6 by aPAR and pPAR given in (3) is essentially equivalent to the conversion model through pPAR alone.
The polarity of the PARs is formed only in the membrane. We find that the interface between the aPAR and pPAR domains in the membrane is sufficiently narrow and that the concentration of pPAR in the membrane is very low where the concentration of aPAR is high (Fig. 3a, middle panel). Thus, we approximate the effect of the offrate of aPAR (\(F_{\text {off}}^A\)) under the condition that \([P_{m}]\ll 1\) by Taylor expansion :
We can easily calculate \(F'(0)=0\) for \(n\ge 2\), \(F''(0)=2K_1K_2^{1}\) for \(n=2\) and \(F''(0)=0\) for \(n\ge 3\). Assuming \(n=2\), we obtain
We assume that the fast diffusion of aPAR in the cytosol leads to a wellmixed state and that the concentration of aPAR in the cytosol quickly approaches an equilibrium state, namely \(A_c^*=(1/\Omega )\int _{\Omega }[A_c]({\mathbf {x}},t)d{\mathbf {x}}\). This leads to
With the approximation (6) and the onrate function \(F^{A}_{\text {on}}\) (2), we have the same approximation for both H1 and H2, such that
where \(\delta _1=K_1/(K_2\alpha _1)\) and \(\delta _2=\gamma _1 A_c^*/\alpha _1\). Thus, both the H1 and H2 models are essentially the same, and the effect of MEX5/6 on aPAR dynamics is likely to decrease the concentration of aPAR in the membrane.
Substituting the approximate equation (7) for \([A_m]\) into the offrate function \(F_{\text {off}}^P\) of pPAR, (2), we obtain
where \(\beta _1=\overline{K}_4 \delta _2^2, ~\beta _2=K_4, ~\beta _3=2K_4 \delta _1\), and \(\beta _4=K_3 \delta _2^2\). Finally, \(G^C_{F\rightarrow S}({\mathbf {x}},t)\) and \(G^M_{F\rightarrow S}({\mathbf {x}},t)\), given in (4), are transformed to
from equation (7).
Combining Eqs. (6) and (9) with the pPAR equations of the model (1) and the MEX5/6 model (3), we obtain the MEX5/6combinedselfrecruitment pPAR Model as follows:
and
where we have replaced \(A_c^*\) by a positive parameter, \(\delta _3\), without loss of generality. The model is a conservation system and the total mass of pPAR and MEX5/6 is conserved.
From a direct comparison between the MEX5/6combinedPARs Model (1)–(3) and MEX5/6combinedselfrecruitment pPAR Model (10)–(11), we find that the regulation network involving aPAR, pPAR, and MEX5/6 can be reduced to the network of pPARalone conversion control in the MEX5/6 dynamics (Fig. 1e). We can interpret the model (11) such that conversion from the fast diffusive type to slow diffusive type of MEX5/6 is promoted constantly (the terms \(\mu _4\delta _3\) and \(\mu _2\delta _2\)) by a substrate and it is simultaneously downregulated by pPAR in the membrane. The substrate may be considered to be phosphatase PP2A, and the inhibition by pPAR on the membrane may be considered as an indirect role of aPAR on the conversion of MEX5/6, as suggested in Griffin et al. (2011) and Wu et al. (2018). Our model reduction suggests that the direct conversion model involving aPAR and pPAR is essentially the same as the pPARalone conversion model. Indeed, we confirm that the two models essentially show similar dynamics (see Fig. 3). In this paper, we explore our results on the MEX5/6combinedselfrecruitment pPAR Model (10)–(11).
Flow Velocity Function
After sperm entry, the actomyosin network located in the cell cortex begins contracting toward the anterior side from the posterior side (Gönczy 2005) (Fig. 1a), causing a cortical flow in the same direction as the actomyosin contraction, and a cytoplasmic flow in the opposite direction of the center of the cytosol and in the same direction as the cortical flow in the periphery of the membrane (Niwayama et al. 2011) (Fig. 2a). The velocity of the flows has been investigated in detail experimentally, revealing that the maximal velocity (approximately 0.156±0.044 \(\mu m/s\)) is reached during the establishment phase and the velocity goes to zero around the time that the establishment phase terminates (Goehring et al. 2011b; Niwayama et al. 2011). In this study, we explicitly formulate the flow velocity function, \({\mathbf {v}}({\mathbf {x}},t)=(v^{x}({\mathbf {x}},t), v^{y}({\mathbf {x}},t))\), defined for the entire cell region (\({\bar{\Omega }}\)) based on the experimental data in Niwayama et al. (2011).
The flow velocity function is given by
where
and \(c_i(i=1\ldots 9)\) are positive constants. \(T_{0}\) is the temporal point at which the velocity is maximum. We select the parameter values so that the flow velocity function approaches the maximal velocity of about 0.12 \(\upmu \) m/s at 3.8 min, and the flow ceases around 8 min (Fig. 2b, c, Movie S1). The cortical flow velocity function, \({\mathbf {v}}_{m}({\mathbf {x}},t)\), is given by the value of \((v^{x}({\mathbf {x}},t), v^{y}({\mathbf {x}},t))\) on the domain \(\partial \Omega \), and the cytoplasmic flow velocity function, \({\mathbf {v}}_{c}({\mathbf {x}},t)\), is given by the value of \((v^{x}({\mathbf {x}},t), v^{y}({\mathbf {x}},t))\) on the domain \(\Omega \). Detailed temporal and spatial data for the cortical and cytoplasmic flows used in our simulations are shown in Fig. 2b, c. Note that the flow functions in a cell satisfy incompressibility almost everywhere (Fig. 2d and Fig. S1)
Model Incorporated with the BulkSurface Cellular Geometry
Here, we introduce a method to combine the phasefield function with the bulksurface system (10)–(11). The method allows for a simple numerical technique to solve a convection–reaction–diffusion model on any highdimensional cellular shape, and we can also simply include the flow dynamics in the model system. Let us express a fixed cell domain using a phasefield function for some time \(t^*\), namely \(\phi ({\mathbf {x}},t^*)\), as follows (Fig. 1b):
We now explain a general method to make a phasefield function of a cell. Let us first define the free energy function, \(E_0\), of Ginzburg–Landau type for a cell such that
where A denotes the area of the system in which \(\phi \) is defined, \(\varepsilon (>0)\) is a sufficiently small constant that defines the thickness of the cell membrane, and \(g(\phi )=\frac{1}{4}\phi ^2(1\phi )^2\). Here, the symmetric potential \(g(\phi )\) is used for setting the local minima at \(\phi =0\) and \(\phi =1\).
Next, we define the energy function which determines the volume of the cell such that
where \(\alpha (>0)\) is the intensity constant of the energy for cell volume, \(\overline{V}\) is the target volume of the cell, and \(h(\phi )=\phi ^3(1015\phi +6\phi ^2)\), which is used for the induction of an energetic asymmetry between \(\phi =0\) and \(\phi =1\), driving the interface while keeping \(\phi =0\) and \(\phi =1\) as local minima of the energy function (see Appendix of SeirinLee et al. (2017) for more detail). Finally, we define a time evolution equation for the total energy of the cell, satisfying
where \(\mu (>0)\) is the constant defining the mobility of the interface. Substituting for \(E_0\) and \(E_1\) into the above equation, we arrive at the equation
By providing the target cell volume (\({\bar{V}}\)) and initial conditions, we can readily generate cells that have different shapes and sizes. We generate the C. elegans embryo by setting \(\overline{V}\) as the actual size (the area in twodimensional simulations) of the embryo and the initial condition to be an ellipse with the embryo scale of short and long axes.
With the cell phasefield function \(\phi \), we rewrite the MEX5/6combinedselfrecruitment pPAR model (10)–(11) in a form in which the cell geometry is reflected (SeirinLee 2016; Teigen et al. 2009; Wang et al. 2017). The model system, combined with the phasefield function that we used for numerical simulations, is given by
for \({\mathbf {x}}\in A(\equiv [0, L_x]\times [0, L_y])\), where \(B(\phi )=\nu \phi ^{2}(1\phi )^2~(\nu >0)\), a function defining the membrane region (Fig. 1b). The cortical flow velocity function, \({\mathbf {v}}_{m}({\mathbf {x}},t)\), is given by \({\mathbf {v}}_{m}({\mathbf {x}},t)=B(\phi ({\mathbf {x}},t)){\mathbf {v}}({\mathbf {x}},t)\), and the cytoplasmic flow velocity function, \({\mathbf {v}}_{c}({\mathbf {x}},t)\), is given by \({\mathbf {v}}_{c}({\mathbf {x}},t)=\phi ({\mathbf {x}},t){\mathbf {v}}({\mathbf {x}},t)\).
One can confirm that a sharp interface limit recovers the boundary conditions in the cytosol equations of the phasefield combined model (see “Appendix A” for more detail). Note that we can numerically solve the bulksurface model (10)–(11) using a standard finite difference method on a square. The details of the initial conditions and parameter values are given in “Appendix B.”
Results
Regeneration of PAR and MEX5/6 Polarities
We first confirm that the MEX5/6combinedPARs model (1)–(3) and the MEX5/6combinedselfrecruitment pPAR model (10)–(11) are essentially the same (Fig. 3), and there are no qualitative differences in the model dynamics, suggesting that the two different conversion dynamics suggested by Daniels et al. (2010) and Griffin et al. (2011) can be reconsidered by our mathematical models, and they have essentially the same mathematical structure. This implies that our model is integrating all relevant molecular dynamics observed in the previous experiments and is thus a general model to capture the dynamics of both MEX5/6 and PAR, simultaneously.
In our model, we confirmed that PAR polarity in the membrane and MEX5/6 polarity in the cytosol are simultaneously generated (Fig. 3b, Movie S2, S3), as observed experimentally (Cuenca et al. 2002). The simulations showed that the establishment phase finishes at approximately 6–7 min, and the boundary of the pPAR domain stops at around the middle of the cell, for a representative parameter set. With a small initial stimulus at the posterior polar site (Fig. 3b, \(t=0\) panel), the pPAR domain begins to emerge toward the anterior from the posterior, and simultaneously MEX5/6 generates polarity in the same direction on the emergence of pPAR. The domain boundary of MEX5/6 is always determined in a location similar to the domain boundary of pPAR in the periphery of the membrane (Fig. 3c, upper panel). This result suggests that polarity formation of MEX5/6 and PAR is very interactive, both temporally and spatially.
On the other hand, the MEX5/6 concentration profile in the bulk region of the cytosol shows that the distribution of MEX5/6 is not clearly distinguished by the two domains of different concentration levels (Fig. 3c, lower panel). This is likely to be a consequence of the homogeneity of the pPAR concentration in the cytosol (SeirinLee et al. 2020b). We further found that the slow diffusive MEX5/6 has a similar distribution shape to that of the total MEX5/6, whereas the fast diffusive MEX5/6 is almost homogeneous. This indicates that the conversion of MEX5/6 to a slow diffusive type is essential to create the MEX5/6 polarity, and that the inhibition/activation role of pPAR on the conversion of MEX5/6 to a slow/fast diffusive type is critical. Our bulksurface model proposes that the polarity of MEX5/6 is mainly formed in the periphery of the cell membrane rather than in the bulk space of the cytosol, and that the heterogeneity of pPAR polarity in the membrane plays an important role in generating MEX5/6 polarity.
Role of MEX5/6 Polarity on PAR Polarity Formation
How the upstream PAR proteins, and their polarity, influence formation of MEX5/6 have been investigated in detail, both experimentally and mathematically (Wu et al. 2018; SeirinLee et al. 2020b). However, it is still unknown how the downstream MEX5/6 polarity influences the upstream PAR polarity. Thus, we explore here the biochemical roles of MEX5/6 on PAR polarity formation by investigating how MEX5/6 affects PAR polarity formation, spatially and temporally, with respect to the symmetry breaking, establishment, and maintenance phases, without flow effects. To see the influence on the symmetry breaking phase, we investigated whether the polarity pattern can emerge for the cases when the effect of MEX5/6 is absent (\(\mu _0=0\)) or present (\(\mu _0>0\)). We found that MEX5/6 can promote symmetry breaking (Fig. 4a), implying that MEX5/6 supports pPAR invasion into the membrane by suppressing aPAR. To see how MEX5/6 controls pPAR recruitment, we analyzed how the property of the bistability of pPAR dynamics can be affected by the parameter \(\mu _0\) (see “Appendix D”). The analysis showed that \(\mu _0\) leads to a wider parameter region for the bistability of pPAR dynamics (Fig. S2), suggesting that MEX5/6 plays an important supporting role in the formation of pPAR polarity.
Next, we investigated the emerging speed of polarity pattern in the establishment phase (Fig. 4b). We found that the speed is almost constant (Fig. 4b, left panel) before it enters the maintenance phase, in which the speed should be slower in order to halt the speed of establishment polarity pattern. Thus, we explored how the emerging speed is affected by MEX5/6, namely \(\mu _0\), during the early stage of the establishment phase (Fig. 4b, right panel). The results show that the emerging speed is highly affected by changes in \(\mu _0\). The emerging speed increased by more than double its value when \(\mu _0\) doubled in magnitude, indicating that MEX5/6 plays a critical role in regulating the temporal dynamics of PAR polarity formation.
Finally, to see the effect of MEX5/6 in the maintenance phase, we focused on two phenomena: one is the length scale of the pPAR polarity domain (\(L_p=\) [Length of polarity domain]/[Length of cell circumference]) and the other is the location of pPAR polarity. We first found that the length scale of the pPAR domain can be affected by MEX5/6 (Fig. 4c). As \(\mu _0\) increased, the length of the pPAR domain increased. The total mass of proteins is conserved; therefore, it is likely that the MEX5/6 redistributed the pPAR protein between the membrane and cytosol through helping pPAR stay in the membrane. However, the parameter range of \(\mu _0\) where pPAR forms a stationary polarity pattern was approximately within 8% variation of the length scale, and there existed two threshold values of \(\mu _0\) at which pPAR either fails to invade, or spreads throughout the whole cell membrane. This indicates that the effect of MEX5/6 on the length scale of the pPAR domain may be negligible, but the maintenance of pPAR polarity may be tightly regulated by MEX5/6.
We also found that the asymmetry of MEX5/6 is critical to maintain the pPAR domain. To test this, we set \(D_f=D_s\) and controlled MEX5/6 to be spatially homogeneous (Fig. 4d, left panels). The result showed that pPAR fails to maintain polarity and the polarity domain spread throughout the whole cell membrane. On the other hand, we found that the location of MEX5/6 polarity does not affect pPAR polarity (Fig. 4d, right panels). To see how the location of MEX5/6 polarity affects pPAR polarity, we switched the conversion roles of MEX5/6 artificially in the MEX5/6 model (11) such that
Using this model, we controlled the location of MEX5/6 polarity so that it had a high concentration in the posterior side. Unexpectedly, the simulation result showed that pPAR polarity is maintained robustly, even though the MEX5/6 polarity is formed on the opposite site to the wildtype case. Taking these results together, we conclude that the location of MEX5/6 polarity is not essential, but the asymmetry of MEX5/6 distribution is indispensable for PAR polarity maintenance.
Interplay with the Flows and Cell Geometry
Cortical and cytoplasmic flows induced by actomyosin contraction have been considered as critical factors in the patterning phase of PAR polarity formation (Goehring et al. 2011b). Nevertheless, it has not yet been explored how these flows can affect cytoplasmic polarity. To see how the cytoplasmic protein MEX5/6 interacts with the flow dynamics and, consequently, affects the polarity dynamics, we have explored three cases: wildtype case, flow absent case, and flow present case only for MEX5/6, in which the advection terms in the pPAR model (10) are removed. We first compared the temporal dynamics of polarity pattern for the aforementioned three cases (Fig. 5A). We found that the flows around the membrane can speed up the patterning time of pPAR, although the cytoplasmic flow in the bulk cytosol space has the opposite direction to that of the cortical flow (black dots and white dots in Fig. 5A). This indicates that the temporal dynamics of pPAR are affected strongly by the cortical flow, rather than the cytoplasmic flow.
On the other hand, we found that the interplay of MEX5/6 and flows can slow down pattern emergence and negatively affect the temporal dynamics of patterning (red dots and white dots in Fig. 5A). This supposes that the cytoplasmic flows around the membrane transport the slow diffusive type of MEX5/6 from the posterior pole to the anterior pole, resulting in lower MEX5/6 concentration, and a weakening of the positive effect of MEX5/6 for the pPAR to stay on the membrane (namely, either the inhibition effect on aPAR recruitment, or the activation effect on aPAR transmembrane offrate). This result indicates that the flow dynamics do not always play a role in promoting PAR polarity, but can affect it negatively via the interplay with MEX5/6. Nevertheless, such a negative effect is likely to be eliminated by the positive effect of the cortical flow on pPAR.
Next, we explored how the directions of flow interplay with MEX5/6 dynamics and, consequently, influence the dynamics of PAR polarity. For this, we artificially imposed an opposite direction for the flows to the wild type only in MEX5/6, with no flow in pPAR (Fig. 5B(b1),(b2)). We found that the flow directions greatly affected the dynamics of MEX5/6, leading to completely different PAR polarity patterns. In contrast, there was no difference in the final polarity pattern (stationary steady state) in the case that both pPAR and MEX5/6 are simultaneously affected by oppositely directed flows, although the temporal dynamics of patterning was affected when the velocity of flows was increased (Fig. 5B(b3)). This result indicates that when the flows affect pPAR and MEX5/6 simultaneously, the influence of flow direction on the spatial dynamics of polarity can be negligible. To confirm this, we investigated the effect of the spatial position of symmetry breaking (Fig. 5C). We set the symmetry breaking position of pPAR polarity to be in a perpendicular location to that for the wildtype case but with the wildtype flow dynamics. This setting gives us the situation in which both pPAR and MEX5/6 are strongly perturbed by flow. Nevertheless, we found that the final polarity domain is formed robustly, in the middle portion of the cell, even though the patterning phase is strongly affected by the flow directions (Fig. 5C). This result suggests that the MEX5/6 is affected by the direction of flows, but its influence on spatial patterning can be greatly restricted by interaction with the PAR dynamics.
On the other hand, we found a very intriguing result by comparing Fig. 5B(b1) and D. In Fig. 5D, we considered only the flow effect for MEX5/6 to be the same as Fig. 5B(b1) but with a different symmetry breaking position. That is, in these two numerical experiments, MEX5/6 and pPAR are undergoing the same biochemical interactions under the same effect of cytoplasmic flows. However, the final polarity patterns are very different. Thus, we hypothesize that cell geometry may affect the state of biochemical interaction of cytoplasmic proteins and, consequently, results in different patterns. To confirm this, we investigated the effect of different symmetric breaking positions without flows (Fig. 5E, F). We found that this cell geometry effect is not seen in the PAR alone model, namely when we removed the effect of MEX5/6 with \(\mu _0=0\) (Fig. 5E), but the polarity pattern dynamics were dramatically changed, both temporally and spatially with respect to the cell geometry, when the effect of MEX5/6 was included (Fig. 5F). This result proposes that the dynamics of cytoplasmic protein can be strongly affected by cell geometry and, consequently, the interplay between MEX5/6 and cell geometry can lead to a dramatical change in PAR polarity dynamics. Summarizing the results, the effect of the interplay of MEX5/6 with flows on the spatial dynamics of PAR polarity may be negligible, but the effect of cell geometry can be critical.
Discussion
For the last ten years, the polarity phenomenon in asymmetric cell division has been extensively studied by both experimental and theoretical approaches. In particular, PAR polarity in the membrane has been found to be the most upstream regulator which controls the dynamics of cytoplasmic polarity. However, how downstream polarity in the cytosol affects PAR polarity, and how they interact, is still unknown. In this study, we revisited the question of polarity formation of the cytoplasmic protein MEX5/6 by combining the dynamics of the upstream protein PAR and explored how the cytoplasmic polarity of MEX5/6 interacts with the membrane PAR polarity in the highdimensional bulksurface model.
The mechanism for generating MEX5/6 polarity has been studied in MEX5/6 alone models (Daniels et al. 2010; Wu et al. 2018), which assumed that the conversion rate of MEX5/6 diffusion is spatially heterogeneous, and did not include PAR dynamics. By contrast, we have developed a conversion model in which the conversion rate functions of the two diffusion types of MEX5/6 are determined by PAR proteins in a concentrationdependent manner, so that the model formulation directly includes the dynamics of PAR proteins. In our model, we supposed that the conversion rate function from fast type to slow type depends on the concentration of aPAR. However, this term was converted to a suppression term by pPAR concentration in the selfrecruitment model (Sect. 2.4). The formulation of our selfrecruitment model implicitly includes the dynamics of PAR1, which acts repressively on the phosphorylated substance PP2A, converting the fast type of MEX5/6 to the slow type (Griffin et al. 2011). In fact, for the conversion dynamics of MEX5/6 diffusion type, Daniels et al. (2010) assumed a direct conversion from fast type to slow type by aPAR. In contrast, Griffin et al. (2011) proposed that pPAR represses the conversion of MEX5/6 from fast diffusive type to the slow type. However, our model formulation and analysis showed that such apparently contrasting propositions are essentially the same. This suggests that the conversion assumptions of our model capture an essential mechanism of interplay between cytoplasmic protein MEX5/6 and PAR proteins, which may suggest a general mathematical structure for pattern formation in cytoplasmic proteins.
With our MEX5/6combinedpPAR model, we explored the specific role of MEX5/6 on PAR polarity formation. We found that MEX5/6 can play a critical role in inducing the symmetry breaking. Motegi et al. (2011) showed that the recruitment of PAR2 around the posterior pole by microtubule transport may promote symmetry breaking. Our study suggests that symmetry breaking can also be promoted by the repression of aPAR recruitment around the posterior pole by MEX5/6, implying that the selfrecruitment of pPAR is indirectly promoted. These results propose that the first stage of symmetry breaking in asymmetric division may be regulated by the synergistic effect of multiple positive feedbacks of pPAR recruitment from cytosol to membrane. We also found that the length of the pPAR domain tends to be shorter as the regulation effect (\(\mu _0\)) of MEX5/6 on PAR decreases. This is consistent with the dynamics observed in previous experiments with the mex5(RNAi) and mex6(RNAi) embryos of C. elegans, where the length of the PAR2 domain was shorter than that of the wild type (Cuenca et al. 2002; Schubert et al. 2000). However, we also found that the length scale of the pPAR polarity domain is not sensitive to the regulation effect of MEX5/6, indicating that the length of the PAR domain may be robustly regulated by other factors, such as the total mass of PAR proteins (SeirinLee and Shibata 2015; Goehring et al. 2011b). Our study suggests that the upstream polarity of the PARs, and the downstream polarity of MEX5/6, significantly regulate each other with respect to both spatial and temporal dynamics in polarity formation. Even if cytoplasmic polarity serves as a downstream regulator, cytoplasmic polarity can play a critical role in upstream PAR polarity, and the balance of their bidirectional regulation is important for generating robust polarity.
In the history of the study of pattern formation, the effect of domain geometry has been considered as an important factor that can regulate spatial patterning (Crampin et al. 1999; Dawes and Iron 2013; Murray 1993; SeirinLee 2017), and there is biological evidence supporting the hypothesis that the shape, or size, of domain is likely to play a critical role in determining cell function, via regulation of pattern formation (Kondo and Asai 1995; SeirinLee et al. 2019). In this study, we found that cell geometry may play an important role in the dynamics of cytoplasmic protein in polarity formation. Many cell polarity studies using mathematical models have been focused on the dynamics of membrane polarity in simplified onedimensional domains, neglecting cell geometry. In general, the fast diffusion in cytosol, and the homogeneity of cytosol concentration, have validated this model simplification. However, our study suggests that the effect of cell geometry on the cytoplasmic protein, which creates a spatial heterogeneity in the bulk cytosol space, can play a critical role in the dynamics of polarity patterning in both membrane and cytosol, and that cell geometry should not be neglected. Furthermore, the flow dynamics is likely to be affected by cell geometry (Mittasch et al. 2018), which may consequently affect PAR polarity formation.
In this study, we presented simulation results for representative parameter sets. However, a rigorous mathematical analysis of the highdimensional bulksurface model of PARs alone proves that the polarity pattern exists within a large parameter range (Morita and SeirinLee 2020). Furthermore, our bistability analysis shows that the parameter region can be extended as the effect of MEX5/6 increases (Fig. S2B), implying that we could have a polarity pattern in the MEX5/6combinedPAR model which is robust to the values of the kinetic parameters. However, it is a mathematical challenge to analyze the bifurcation structure, existence of the polarity solution, and the details of polarity dynamics, in the highdimensional bulksurface MEX5/6combinedPARs system.
Finally, our study proposes that to understand the whole process of cell polarity in asymmetric cell division, it is vital to integrate biochemical interaction, biophysical dynamics, and cell geometry.
References
Campanale JP, Sun TY, Montell DJ (2017) Development and dynamics of cell polarity at a glance. J Cell Sci 130:1201–1207
Cortes DB, Dawes A, Liu J, Nickaeen M, Strychalski W, Maddox AS (2018) Unite to dividehow models and biological experimentation have come together to reveal mechanisms of cytokinesis. J Cell Sci 131:1–10
Cowan CR, Hyman AA (2004) Asymmetric cell division in C. elegans: cortical polarity and spindle positioning. Annu Rev Cell Dev Biol 20:427–453
Crampin EJ, Gaffney EA, Maini PK (1999) Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull Math Biol 61:1093–1120
Cuenca AA, Schetter A, Aceto D, Kemphues K, Seydoux G (2002) Polarization of the C. elegans zygote proceeds via distinct establishment and maintenance phases. Development 130:1255–1265
Daniels BR, Dobrowsky TM, Perkins EM, Sun SX, Wirtz D (2010) Mex5 enrichment in the C. elegans early embryo mediated by differential diffusion. Development 137:2579–2585
Dawes AT, Iron D (2013) Cortical geometry may influence placement of interface between par protein domains in early Caenorhabditis elegans embryos. J Theor Biol 333:27–37
Dawes AT, Munro EM (2011) PAR3 oligomerization may provide an actinindependent mechanism to maintain distinct Par protein domains in the early Caenorhabditis elegans embryo. Biophys J 101:1412–1422
Goehring NW, Hoege C, Grill SW, Hyman AA (2011a) PAR proteins diffuse freely across the anteriorposterior boundary in polarized C. elegans embryos. J Cell Biol 193(3):583–594
Goehring NW, Trong PK, Bois JS, Chowdhury D, Nicola EM, Hyman AA, Grill SW (2011b) Polarization of PAR proteins by advective triggering of a patternforming system. Science 334(6059):1137–1141
Gönczy P (2005) Asymmetric cell division and axis formation in the embryo. WormBook.org. https://doi.org/10.1895/wormbook.1.30.1
Griffin EE, Odde DJ, Seydoux G (2011) Regulation of the MEX5 gradient by a spatially segregated kinase/phosphatase cycle. Cell 146:955–968
Hoege C, Hyman AA (2013) Principles of PAR polarity in Caenorhabditis elegans embryos. Mol Cell Biol 14:315–322
Knoblich JA (2008) Mechanisms of asymmetric stem cell division. Cell 132:583–597
Kondo S, Asai R (1995) A reactiondiffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376:765–768
Kuhn T, Ihalainen TO, Hyvaluoma J, Dross N, Willman SF, Langowski J, VihinenRanta M, Timonen J (2011) Protein diffusion in mammalian cell cytoplasm. PLoS ONE 6(8):e22962
Kuwamura M, SeirinLee S, Ei SI (2018) Dynamics of localized unimodal patterns in reactiondiffusion systems related to cell polarization by extracellular signaling. SIAM J Appl Math 78(6):3238–3257
Lang CF, Munro E (2017) The PAR proteins: from molecular circuits to dynamic selfstabilizing cell polarity. Development 144:3405–3416
Mittasch M, Gross P, Nestler M, Fritsch AW, Iserman C, Kar M, Munder M, Voigt A, Alberti S, Grill SW, Kreysing M (2018) Noninvasive perturbations of intracellular flow reveal physical principles of cell organization. Nat Cell Biol 20:344–351
Morita Y, SeirinLee S (2020) Long time behaviour and stable pattern in the systems of cell polarity model. Preprint
Motegi F, Seydoux G (2013) The PAR network: redundancy and robustness in a symmetrybreaking system. Philos Trans R Soc B 368:20130010
Motegi F, Zonies S, Hao Y, Cuenca AA, Griffin E, Seydoux G (2011) Microtubules induce selforganization of polarized PAR domains in Caenorhabditis elegans zygotes. Nat Cell Biol 13(11):1361–1367
Murray JD (1993) Mathematical biology II: spatial models and biomedical applications, 3rd edn. Springer, Berlin
Nishikawa M, Naganathan SR, Jülicher F, Grill SW (2017) Symmetry breaking in a bulk surface reaction diffusion model for signalling networks. eLife 6:e19595
Niwayama R, Shinohara K, Kimura A (2011) Hydrodynamic property of the cytoplasm is sufficient to mediate cytoplasmic streaming in the Caenorhabiditis elegans embryo. PNAS 108(29):11900–11905
Rappel WJ, Levine H (2017) Mechanisms of cell polarization. Curr Opin Syst Biol 3:43–53
Rose LS, Gönczy P (2014) Polarity establishment, asymmetric division and segregation of fate determinants in early C. elegans embryo. WormBook
Schubert CM, Lin R, de Vries CJ, Plasterk RHA, Priess JR (2000) MEX5 and MEX6 unction to establish soma/germline asymmetry in early C. elegans embryo. Mol Cell 5:671–682
SeirinLee S (2016) Lateral inhibitioninduced pattern formation controlled by the size and geometry of the cell. J Theor Biol 404:51–65
SeirinLee S (2017) The role of domain in pattern formation. Dev Growth Differ 59:396–404
SeirinLee S (2020) From a cell to cells in asymmetric cell division and polarity formation?: Shape, length, and location of par polarity. Dev Growth Differ 62:188–195
SeirinLee S, Shibata T (2015) Selforganization and advective transport in the cell polarity formation for asymmetric cell division. J Theor Biol 382:1–14
SeirinLee S, Tashiro S, Awazu A, Kobayashi R (2017) A new application of the phasefield method for understanding the reorganization mechanisms of nuclear architecture. J Math Biol 74:333–354
SeirinLee S, Osakada F, Takeda J, Tashiro S, Kobayashi R, Yamamoto T, Ochiai H (2019) Role of dynamic nuclear deformation on genomic architecture reorganization. PLOS Comput Biol 15(8):e1007289
SeirinLee S, Gaffney EA, Dawes AT (2020a) CDC42 interactions with Par proteins are critical for proper patterning in polarization. Cells 9:2036
SeirinLee S, Sukekawa T, Nakahara T, Ishii H, Ei SI (2020b) Transitions to slow or fast diffusions provide a general property for inphase or antiphase polarity in a cell. J Math Biol 80:1885–1917
Small LE, Dawes AT (2017) PAR proteins regulate maintenancephase myosin dynamics during Caenorhabditis elegans zygote polarization. Mol Biol Cell 28:2220–2231
Teigen KE, Li X, Lowengrub J, Wang F, Voigt A (2009) A diffuseinterface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Commun Math Sci 4(7):1009–1037
Tostevin F, Howard M (2008) Modeling the establishment of PAR protein polarity in the onecell C. elegans embryo. Biophys J 95:4512–4522
Trong PK, Nicola EM, Goehring NW, Kumar KV, Grill SW (2014) Parameterspace topology of models for cell polarity. New J Phys 16:065009
Wang W, Tao K, Wang J, Yang G, Ouyang Q, Wang Y, Zhang L, Liu F (2017) Exploring the inhibitory effect of membrane tension on cell polarization. PLOS Comput Biol 13(1):e1005354
Wu Y, Han B, Li Y, Munro E, Odde DJ, Griffin EE (2018) Rapid diffusionstate switching underlies stable cytoplasmic gradients in Caenorhabditis elegans zygote. PNAS 115(36):E8440–E8449
Zonies S, Motegi F, Hao Y, Seydoux G (2010) Symmetry breaking and polarization of the C. elegans zygote by the polarity protein PAR2. Development 137:1669–1677
Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers JP19H01805 and JP17KK0094, and by the JSPS A3 Foresight Program.
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Supplementary material 1 (mov 302 KB)
Supplementary material 2 (mov 80 KB)
Supplementary material 3 (mov 74 KB)
Appendices
Sharp Interface Limit and ZeroFlux Boundary Condition
We show that a sharp interface limit recovers the boundary conditions in the cytosol equations of the phasefield combined model (14). The general form of the cytoplasmic protein dynamics in our model is given by
where u is the concentration of cytoplasmic protein, D is the diffusion coefficient, \({\mathbf {v}}\) is the flow velocity, and \({\mathbf {n}}\) is the normal vector on \(\Gamma (\equiv \partial \Omega )\). F(u) is a function describing a reaction on \(\Gamma \) and it is not necessarily only a function of u. The cytoplasmic model equation incorporating cell shape is given by
where \(\phi \) is the phasefield function at a fixed time, i.e., \(\phi =\phi ({\mathbf {x}})\), defined by
In what follows, we show that Eq. (17) recovers the boundary condition (16) in the sharp interface. Let us define \(T_\xi \) to be the interface region of \(\phi \) with thickness \(\xi \). That is,
Integrating Eq. (17) over the interface yields
On the other hand, substituting the original Eq. (15) into the lefthand side of equation (18) gives
Thus, from (18) and (19), we obtain
Since \(\nabla \phi \ne 0\) as \(\xi \rightarrow 0\),
should hold.
Initial Conditions and Parameter Values
Before fertilization, aPAR is homogeneously distributed in the membrane, and pPAR and MEX5/6 are homogeneously distributed in the cytosol. The core mechanism for inducing symmetry breaking remains unclear, as does the reason why the pPAR domain begins around the posterior polar site. On the other hand, it is known that some signals from the centrosome and its microtubule asters help recruit pPAR into the membrane around the posterior polar site (Rose and Gönczy 2014; Motegi et al. 2011; Motegi and Seydoux 2013). Thus, we define the initial conditions such that all proteins are at homogeneous steady state, and some small stimulus of pPAR is added in a small region of the membrane.
The detailed form of the initial conditions are :
where \(\sigma \) is the strength of the signal, and \(\omega \) is the sufficiently small region in which the signal is imposed. \(\delta \) is a positive constant and has a very small value. Typically, we set \(\delta =0.02\). \(\psi ({\mathbf {X}})\) is a random function with uniform distribution, which takes values in the range \([0.5,0.5]\). \([A_{m}]_{0}\), \([A_{c}]_{0}\), \([P_{m}]_{0}\), \([P_{c}]_{0}\), \([M_{f}]_{0}\) and \([M_{s}]_{0}\) are equilibrium concentrations. We set \([A_{m}]_{0}, [P_{m}]_0\), \([M_s]_{0}\) and \([M_f]_{0}\), directly, where \([A_{c}]_{0}\) and \([P_{c}]_{0}\) are calculated as the equilibrium values in model (1)–(3), or (10)–(11) when \(\mu _0=0\).
We selected the parameter values based on experimental data for the C. elegans embryo. A fertilized C. elegans egg is usually elliptical, and the radii of the long and short axes are observed typically in the range of \(27.0\pm 1.7 ~\upmu \mathrm{m}\) and \(14.8\pm 1.0 ~\upmu \mathrm{m}\), respectively (Goehring et al. 2011a). The diffusion rates of aPAR and pPAR in the membrane were selected based on the diffusion rate of PAR6 and PAR2, respectively (Goehring et al. 2011a). The cytoplasmic diffusion rates of aPAR and pPAR were chosen from the data in Kuhn et al. (2011). For the diffusion rates of MEX5/6, we used data from Daniels et al. (2010) and Wu et al. (2018), and for the flow velocity, we used data from Goehring et al. (2011a) and Niwayama et al. (2011). We selected the time scale to coincide with the quantitative data and the qualitative dynamics of PAR polarity. The unit time in the nondimensionalized system corresponds to the dimensional time, 0.3375 s. In the simulations, we used the nondimensionalized model, combined with the phasefield functions, and solved the equations on the nondimensional region \(1\times 1\), where the dimensional space \(L_x\times L_y\) has been scaled by \(L_x=L_y=75~\upmu \mathrm{m}\) (Fig. 1a). Since the phasefield function is timeindependent and is used to define the cell region, we assume the parameters used in the phasefield function as dimensionless quantities. The parameter values are listed in Table 1, and the detailed parameter values used in the flow functions, phasefield function, and the figures are given in the following. For the other figures, the parameter values in Table 1 were used.

Flow functions, (12): \( c_1 = 0.001, ~c_2 = 400.0, ~c_3 = 0.02, ~c_4 = 4.2, ~c_5 = 0.01,~c_6 = 0.00001,~c_7 = 3.5,~c_8 = 0.0021,~c_9 = 0.00005, ~T_0 = 3.3375 ~\text {min}\)

Phasefield function, (13): \(\alpha =120,~\overline{V}=0.2828, \mu =1.0,~\varepsilon =2.0\times 10^{3}, ~\nu =16.0\)

Figure 1a: \(D^{A}_{m}=0.000001652, ~D^{A}_{c}=0.00036, ~D^{P}_{m}=0.00000072,~D^{P}_{c}=0.00036, \gamma _1=\gamma _2=0.3, ~\alpha _1=\alpha _2=0.06,~K_1=K_3=0.4, ~K_2=K_4=1.0,~\overline{K}_2=\overline{K}_4=0.05,~\mu _0=0.01,~\mu _1=0.05,~\mu _2=0.01,~\mu _3=\mu _4=0.005\).

Figure 4a: \(\beta _3=0.5\), \(\mu _0=0\) or 0.2, and the other parameters are given in Table 1.

Figure 4b: \(\mu _0=0.2\) and the other parameters are given in Table 1.
Incompressibility
The detailed figure of incompressibility is shown in Fig. S1.
Supporting Role of MEX5/6 on pPAR Patterning
Here, we show that MEX5/6 can play a supporting role in inducing pPAR polarity patterning by analyzing a parameter space where bistability is possible. Let us assume that the concentration of MEX5/6 is given by a constant, namely, \(M^*\). Then, from Eq. (10), the equilibrium state of pPAR can be written as
where \(P_{c}^*\) and \(P_m^*\) are the equilibrium concentrations of pPAR in the cytosol and the membrane, respectively. Note that
where \(P_\mathrm{tot}=\int _{\Omega }P_{c}+\int _{\partial \Omega }P_{m}\), \(P_\mathrm{tot}^*=P_\mathrm{tot}/\Omega \), and \(A=\partial \Omega /\Omega \). Substituting Eq. (21) into Eq. (20), we obtain
For two positive stable states, the equation \(G'(p_m^*)=0\) should have two positive solutions (Fig. S2A). Denoting the two solutions by \(P_1\) and \(P_2\), the following property is satisfied :
and
We can easily check that the conditions (22) are satisfied for all positive parameter values. Thus, we compare the parameter region of \(\gamma _2\) (onrate) and \(\beta _4\) (magnitude of offrate) for bistability with respect to the cases \(\mu _0=0\) and \(\mu _0=0.2>0\). The result shows that the case of \(\mu _0>0\) gives a wider parameter space for bistability than the case of \(\mu _0=0\).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
SeirinLee, S. The Role of Cytoplasmic MEX5/6 Polarity in Asymmetric Cell Division. Bull Math Biol 83, 29 (2021). https://doi.org/10.1007/s11538021008600
Received:
Accepted:
Published:
Keywords
 Pattern formation
 Cell polarity