Actomyosin Regulation and Symmetry Breaking in a Model of Polarization in the Early Caenorhabditis elegans Embryo

Abstract

Polarization, whereby a cell defines a spatial axis by segregating specific determinants to distinct regions, is an essential and highly conserved biological process. The process of polarization is initiated by a cue that breaks an initially symmetric distribution of determinants, allowing for a spatially asymmetric redistribution. The nature of this cue is currently not well understood. Utilizing the conservation of polarization process and its determinants, we theoretically investigate the nature of the cue and the regulation of contractility that enables the establishment of polarity in early embryos of the nematode worm Caenorhabditis elegans (C. elegans). Our biologically based model, which consists of coupled partial differential equations, suggests that a biochemical but not mechanical cue is sufficient for symmetry breaking, and inhibition of contractile elements by specific determinants is needed for sustained spatial redistribution.

Appendices

Appendix 1: Model Nondimensionalization

We base the model used here on previous Par proteins models (Dawes and Munro 2011; Dawes and Iron 2013). We assume that ParA can dimerize, and that ParA and ParP interact through mutual phosphorylation. We assume that ParA can enhance actomyosin assembly while ParP inhibits it. We assume that all proteins diffuse in the plane of the cortex and that actomyosin can be modeled as a viscoelastic substance where local tension gradients induce advective movement, making advection an emergent property of the model. Please see Dawes and Iron (2013) for further details on the kinetic and spatial terms. Incorporating diffusion and advection, the full dimensional model studied here is:

$$\begin{aligned} \frac{\partial A_{1}}{\partial t}&= k_{\mathrm{on}}^{A}A_y - k_{\mathrm{off}}^{A} A_1 - 2 k_{d}^{+} A_{1}^{2} +2 k_{d}^{-} A_{11} - k_{d}^{+}A_y A_1 + k_{d}^{-} A_{10}\nonumber \\&-\,r_{A} P \cdot A_{1} + D_a\Delta A_1 - \nu \nabla \cdot (A_1 \nabla (M)) \end{aligned}$$

(14)

$$\begin{aligned} \frac{\partial A_{10}}{\partial t}&= k_{\mathrm{on}}^{A_{10}}A_{2y} - k_{\mathrm{off}}^{A} A_{10} - k_{\mathrm{on}}^{A_{11}} A_{10} +k_{\mathrm{off}}^{A} A_{11} - k_{d}^{-}A_{10} + k_{d}^{+}A_y A_{1}\nonumber \\&-\,r_{A} P \cdot A_{10} + D_a\Delta A_{10} - \nu \nabla \cdot (A_{10} \nabla (M)) \end{aligned}$$

(15)

$$\begin{aligned} \frac{\partial A_{11}}{\partial t}&= k_{d}^{+} A_{1}^{2} - k_{d}^{-} A_{11} - k_{\mathrm{off}}^{A}A_{11} + k_{\mathrm{on}}^{A_{11}} A_{10} - 2 r_{A} P \cdot A_{11}\nonumber \\&+\,D_a\Delta A_{11} - \nu \nabla \cdot (A_{11} \nabla (M)) \end{aligned}$$

(16)

$$\begin{aligned} \frac{\partial P}{\partial t}&= k_{\mathrm{on}}^{P}P_y - k_{\mathrm{off}}^{P} P - r_{P} (A_1+A_{10}+2 A_{11}) \cdot P\nonumber \\&+\,D_p\Delta P - \nu \nabla \cdot (P \nabla (M)) \end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial M}{\partial t}&= k_{\mathrm{on}}^M M_y\left( 1- \frac{P}{\kappa _P+P}+k_a\frac{A}{\kappa _A+A}\right) - k_{\mathrm{off}}^M M +D_m\Delta M - \nu \nabla \cdot (M \nabla M) \end{aligned}$$

(18)

As discussed in Sect. 2.2, we do not know many of the kinetic rates associated with the interactions modeled by our equations. As a result, we non-dimensionalize the equations so that our variables can only take on values between 0 and 1, and our parameters can be expressed in terms of groupings of rate constants. We make the following substitution for our dependent and independent variables: $C=\bar{C}C^*$ where $C=A,P,M,t,x$, and $\bar{C}$ is the characteristic (dimensional) scale for that variable, and $C^*$ is the dimensionless variable. We define the following characteristic scales:

$$\begin{aligned} \bar{A_1} = \bar{A_{10}} = \bar{A_{11}}= A_y,\, \bar{P}=P_y,\, \bar{M}=M_y,\, \tau =\frac{1}{k_{\mathrm{off}}^A},\, \bar{x}=L \end{aligned}$$

After substitution and simplification (and dropping the $^*$), we have Eqs. 1–5 with the parameter groupings given in Table 1.

Appendix 2: Simplification of Anterior Par Protein Model

Anterior Par proteins have been shown to form higher-order complexes, and this oligomer formation is required for proper polarization in a variety of cell types (Joberty et al. 2000; Lin et al. 2000; Yamanaka et al. 2001; Izumi et al. 1998; Etienne-Manneville and Hall 2003; Suzuki et al. 2001; Li et al. 2010). In our previous models of Par protein dynamics, we explicitly modeled the binding and interconversion of the various possible states of the anterior Par proteins (Dawes and Munro 2011; Dawes and Iron 2013). By distilling down the interaction network as described below, we can reduce the dynamics of the anterior Par proteins to one variable.

The original anterior Par protein model consists of three equations, tracking the change over time of cortically bound ParA monomers, singly bound ParA dimers (where only one part of the dimer is bound to the cortex) and doubly bound dimers (both parts of the dimer are bound to the cortex). Please see Dawes and Munro (2011), Dawes and Iron (2013) for a more detailed discussion of the motivation, assumptions and equations behind this model. Here, we distill that model to its fundamental core: two ParA monomers in the cytoplasm bind to a cortical site, forming a dimer. We can express those interactions as the following arrow diagram:

$$\begin{aligned} 2A_y + B \mathop {\mathop {\rightleftharpoons }\limits _{k^A_{\mathrm{on}}}}^{k^A_{\mathrm{off}}} A \end{aligned}$$

(19)

where $A_y$ is the number of cytoplasmic ParA monomers, $B$ is the number of cortical binding sites, and $A$ is the number of cortically bound ParA dimers. The number of cortical binding sites was implicitly included in the previous model, by having a maximum number of monomers and dimers that could bind to the cortex at any time. This is a biologically reasonable assumption, since steric interactions will prevent an unlimited number of ParA dimers to bind to the cortex, and suggests the first conservation law: The total number of cortical binding sites is constant. In other words,

$$\begin{aligned} B+A=B_T \end{aligned}$$

(20)

Similarly, by having a maximum number of ParA monomers and dimers bound to the cortex, this suggests the total number of ParA monomers and dimers is conserved. That is,

$$\begin{aligned} 2A_y+A=A_T \end{aligned}$$

(21)

Note here we are considering the number of monomers and dimers, and not concentrations. This simplifies the model by not having to include a volume conversion term for converting between the cortical and cytoplasmic concentrations.

Assuming mass-action kinetics and using conservation laws 20 and 21, the simplified one variable equation for $A$ is:

$$\begin{aligned} \frac{\hbox {d}A}{\hbox {d}t}&= k^A_{\mathrm{on}}B\cdot 2A_y^2 - k^A_{\mathrm{off}}A \end{aligned}$$

(22)

$$\begin{aligned} \Rightarrow \frac{\hbox {d}A}{\hbox {d}t}&= k^A_{\mathrm{on}}(B_T-A)\cdot 2(A_T-A)^2 - k^A_{\mathrm{off}}A \end{aligned}$$

(23)

However, we are also interested in spatial dynamics in addition to the kinetics. We observed that the actomyosin profile strongly mirrors the ParA profile during simulations. Thus, for this simplified model, we assume that actomyosin is proportional to ParA, allowing us to write the resulting one variable spatial model for $A$ as:

$$\begin{aligned} \frac{\partial A}{\partial t} = k^A_{\mathrm{on}}(B_T-A)\cdot 2(A_T-A)^2 - k^A_{\mathrm{off}}A+D_1\nabla \cdot (\nabla A) - \nu \nabla \cdot (A \nabla A) \end{aligned}$$

(24)

After non-dimensionalizing as in Appendix 1, using $\bar{A}=A_T,\,\tau =\frac{1}{k_{\mathrm{off}}^A},\, \bar{x}=L$, we have

$$\begin{aligned} \frac{\partial A}{\partial t}=\gamma _1(\gamma _2-A)(1-A)^2-A + D_A \nabla \cdot (\nabla A) -\mu \nabla \cdot (A \nabla A) \end{aligned}$$

(25)

where

$$\begin{aligned} \gamma _1&= \frac{k^A_{\mathrm{on}}A_T^2}{k^A_{\mathrm{off}}}\\ \gamma _2&= \frac{B_T}{A_T}\\ \mu&= \frac{\nu A_T}{k^A_{\mathrm{off}}L^2}\\ D_A&= \frac{D_1}{k^A_{\mathrm{off}} L^2} \end{aligned}$$