Modelling contractile ring formation and division to daughter cells for simulating proliferative multicellular dynamics

Abstract

Cell proliferation is a fundamental process underlying embryogenesis, homeostasis, wound healing, and cancer. The process involves multiple events during each cell cycle, such as cell growth, contractile ring formation, and division to daughter cells, which affect the surrounding cell population geometrically and mechanically. However, existing methods do not comprehensively describe the dynamics of multicellular structures involving cell proliferation at a subcellular resolution. In this study, we present a novel model for proliferative multicellular dynamics at the subcellular level by building upon the nonconservative fluid membrane (NCF) model that we developed in earlier research. The NCF model utilizes a dynamically-rearranging closed triangular mesh to depict the shape of each cell, enabling us to analyze cell dynamics over extended periods beyond each cell cycle, during which cell surface components undergo dynamic turnover. The proposed model represents the process of cell proliferation by incorporating cell volume growth and contractile ring formation through an energy function and topologically dividing each cell at the cleavage furrow formed by the ring. Numerical simulations demonstrated that the model recapitulated the process of cell proliferation at subcellular resolution, including cell volume growth, cleavage furrow formation, and division to daughter cells. Further analyses suggested that the orientation of actomyosin stress in the contractile ring plays a crucial role in the cleavage furrow formation, i.e., circumferential orientation can form a cleavage furrow but isotropic orientation cannot. Furthermore, the model replicated tissue-scale multicellular dynamics, where the successive proliferation of adhesive cells led to the formation of a cell sheet and stratification on the substrate. Overall, the proposed model provides a basis for analyzing proliferative multicellular dynamics at subcellular resolution.

Graphical abstract

Appendix 1. Energy function for inducing anisotropic stress on triangular surface

Anisotropic stress exerted on an arbitrary triangle in the ${{xyz}}$-orthogonal coordinates is considered. The triangle is composed of vertices 0, 1, and 2 (Fig. 11). Normal vector of the triangle is represented by ${\varvec{n}}$ in the ${{xyz}}$-orthogonal coordinates. In addition, the $x^{\prime}y^{\prime}$ orthogonal coordinates can be locally defined in the plane of the triangle, whose basis vectors are represented by ${\varvec{e}}_{{x^{\prime}}}$ and ${\varvec{e}}_{{y^{\prime}}}$ ($= {\varvec{n}} \times {\varvec{e}}_{{x^{\prime}}}$). Using ${\varvec{e}}_{{x^{\prime}}}$ and ${\varvec{e}}_{{y^{\prime}}}$, the area of the triangle, represented by $A$, is given by

$$ A = \frac{1}{2}\left( {\left( {{\varvec{d}}_{01} \cdot {\varvec{e}}_{{x^{\prime}}} } \right)\left( {{\varvec{d}}_{20} \cdot {\varvec{e}}_{{y^{\prime}}} } \right) - \left( {{\varvec{d}}_{20} \cdot {\varvec{e}}_{{x^{\prime}}} } \right)\left( {{\varvec{d}}_{01} \cdot {\varvec{e}}_{{y^{\prime}}} } \right)} \right), $$

(A1)

where ${\varvec{d}}_{\alpha \beta }$ is the distance vector from the $\alpha$- to $\beta$- vertices.

Fig. 11

Description of anisotropic stress on a triangular element. Anisotropic stress acting on a triangle is expressed as a set of forces applied to vertices in two mutually orthogonal directions

To introduce the anisotropic stress on the membrane, we redefine the area of the triangle as a function dependent on direction. This is represented by $A_{{x^{\prime}}}$ and $A_{{y^{\prime}}}$, where ${\varvec{\alpha}}$ is a directional vector. Both $A_{{x^{\prime}}}$ and $A_{{y^{\prime}}}$ have the same value as $A$. The partial derivatives of $A_{{x^{\prime}}}$ and $A_{{y^{\prime}}}$ with respect to the displacement of a vertex, denoted by $dA_{{x^{\prime}}}$ and $dA_{{y^{\prime}}}$, respectively, correspond to the changes in area when the displacement has only $x^{\prime}$ and $y^{\prime}$ components. By regarding the components of vertex locations along ${\varvec{e}}_{{y^{\prime}}}$ as constants in Eq. (A1), Eq. (A1) is rewritten as follows

$$ \begin{gathered} A_{{x^{\prime}}} = \frac{1}{2}\left( {c_{{y^{\prime}1}} \left( {{\varvec{d}}_{01} \cdot {\varvec{e}}_{{x^{\prime}}} } \right) - c_{{y^{\prime}2}} \left( {{\varvec{d}}_{20} \cdot {\varvec{e}}_{{x^{\prime}}} } \right)} \right), \hfill \\ c_{{y^{\prime}1}} = {\varvec{d}}_{20} \cdot {\varvec{e}}_{{y^{\prime}}} ,\hfill \\ c_{{x^{\prime}2}} = {\varvec{d}}_{20} \cdot {\varvec{e}}_{{x^{\prime}}} , \hfill \\ \end{gathered} $$

(A2)

where $c_{{y^{\prime}1}}$ and $c_{{y^{\prime}2}}$ are regarded as constants. Similarly, by regarding the components of vertex locations along ${\varvec{e}}_{{x^{\prime}}}$ as constants in (A2), Eq. (A1) is given by

$$ \begin{gathered} A_{{y^{\prime}}} = \frac{1}{2}\left( {c_{{x^{\prime}1}} \left( {{\varvec{d}}_{20} \cdot {\varvec{e}}_{{y^{\prime}}} } \right) - c_{{x^{\prime}2}} \left( {{\varvec{d}}_{01} \cdot {\varvec{e}}_{{y^{\prime}}} } \right)} \right), \hfill \\ c_{{x^{\prime}1}} = {\varvec{d}}_{01} \cdot {\varvec{e}}_{{x^{\prime}}} , \hfill \\ c_{{x^{\prime}1}} = {\varvec{d}}_{20} \cdot {\varvec{e}}_{{x^{\prime}}}, \hfill \\ \end{gathered} $$

(A3)

where $c_{{x^{\prime}1}}$ and $c_{{x^{\prime}2}}$ are regarded as constants.

Using Eqs. (A2) and (A3), the energy of anisotropic stress in the triangle surface, represented by $u$, can be given by

$$ u \equiv \sigma_{{x^{\prime}}} A_{{x^{\prime}}} + \sigma_{{y^{\prime}}} A_{{y^{\prime}}} , $$

(A4)

where the constants $\sigma_{{x^{\prime}}}$ and $\sigma_{{y^{\prime}}}$ are the exerted stress along ${\varvec{e}}_{{x^{\prime}}}$ and ${\varvec{e}}_{{y^{\prime}}}$, respectively. When the exerted tension is isotropic ($\sigma_{{x^{\prime}}} = \sigma_{{y^{\prime}}} = \sigma /2$), the energy function becomes $u = \sigma A$.

By calculating the partial derivatives of Eq. (A4), the forces acting on vertices 0, 1, and 2, represented by ${\varvec{f}}_{0}$, ${\varvec{f}}_{1}$, and ${\varvec{f}}_{2}$, respectively, were given by

$$ \begin{gathered} f_{0} = - \frac{1}{2}\sigma_{{x^{\prime}}} \left( {d_{12} \cdot e_{{y^{\prime}}} } \right)e_{{x^{\prime}}} - \frac{1}{2}\sigma_{{y^{\prime}}} \left( {d_{12} \cdot e_{{x^{\prime}}} } \right)e_{{y^{\prime}}} , \hfill \\ f_{1} = - \frac{1}{2}\sigma_{{x^{\prime}}} \left( {d_{20} \cdot e_{{y^{\prime}}} } \right)e_{{x^{\prime}}} - \frac{1}{2}\sigma_{{y^{\prime}}} \left( {d_{20} \cdot e_{{x^{\prime}}} } \right)e_{{y^{\prime}}} , \hfill \\ f_{2} = - \frac{1}{2}\sigma_{{x^{\prime}}} \left( {d_{01} \cdot e_{{y^{\prime}}} } \right)e_{{x^{\prime}}} - \frac{1}{2}\sigma_{{y^{\prime}}} \left( {d_{01} \cdot e_{{x^{\prime}}} } \right)e_{{y^{\prime}}} , \hfill \\ \end{gathered} $$

(A5)

respectively. The sum of each of the forces and moments in Eq. (A5) becomes zero. According to the above formulation, the area of the jth triangle in the ith cell is rewritten as a direction-dependent function, where the components of vertex position vectors in the direction normal to ${\varvec{\alpha}}$ is regarded as constants, $A_{j\left( i \right)}^{\parallel } \left( {\varvec{\alpha}} \right)$.