Vertex dynamics simulations of viscosity-dependent deformation during tissue morphogenesis

Abstract

In biological development, multiple cells cooperate to form tissue morphologies based on their mechanical interactions; namely active force generation and passive viscoelastic response. In particular, the dynamic processes of tissue deformations are governed by the viscous properties of the tissues. These properties are spatially inhomogeneous because they depend on the tissue constituents, such as cytoplasm, cytoskeleton, basement membrane and extracellular matrix. The multicellular mechanics of tissue morphogenesis have been investigated in vertex dynamics models. However, conventional models are applicable only to quasi-static deformation processes, which do not account for tissue viscosities. We propose a vertex dynamics model that simulates the viscosity-dependent dynamic deformation processes during tissue morphogenesis. By incorporating local velocity fields into the governing equation of vertex movements, the model turns Galilean invariant. In addition, the viscous properties of tissue components are newly expressed by formulating friction forces on vertices as functions of the relative velocities among the vertices. The advantages of the proposed model are examined by epithelial growth simulations under the employed condition for quasi-static processes. As a result, the epithelial vesicle simulated by the proposed model is linearly elongated with nearly free stress, while that simulated by the conventional model is undulated with compressive residual stress. Therefore, the proposed model is able to reflect the timescale of deformations by satisfying Galilean invariance. Next, the applicability of the proposed model is assessed in epithelial growth simulations of viscous extracellular materials. In this test, the epithelial vesicles are deformed into tubular shapes by oriented cell divisions, and their morphologies are extremely sensitive to extracellular viscosity. Therefore, the dynamic deformations in the proposed model depend on the viscous properties of tissue components. The proposed model will be useful for simulating dynamic deformation processes of tissue morphogenesis depending on viscous properties of various tissue components.

Appendix

1.1 Balance of friction forces among the vertices of a geometrical element

Friction forces are exerted between multiple bodies moving at different velocities. In the proposed model, viscous frictions are defined at the several vertices comprising each geometrical element.

From Eq. (4), the viscous friction force exerted on the ${i^\text {c}}$th vertex, represented by $\varvec{f}^\text {v}_{i^\text {v}}$, is given by

$$\begin{aligned} \varvec{f}^\text {v}_{i^\text {v}}&= \sum ^\text {element}_{j^\text {e}} \eta ^\text {e}_{j^\text {e}} \left( \varvec{v}_{i^\text {v}} - \varvec{v}_{j^\text {e}} \right) \delta _{i^\text {v} ; j^\text {e}} \end{aligned}$$

(18)

$$\begin{aligned}&= \sum ^\text {element}_{j^\text {e}} \eta ^\text {e}_{j^\text {e}} \left( \varvec{v}_{i^\text {v}} - \frac{1}{n^\text {v}_{j^\text {e}}} \sum ^\text {vertex}_{k^\text {v}} \varvec{v}_{k^\text {v}} \delta _{k^\text {v} ; j^\text {e}} \right) \delta _{i^\text {v} ; j^\text {e}} \end{aligned}$$

(19)

$$\begin{aligned}&= \sum ^\text {vertex}_{k^\text {v}} \sum ^\text {element}_{j^\text {e}} \frac{\eta ^\text {e}_{j^\text {e}}}{n^\text {v}_{j^\text {e}}} \delta _{k^\text {v} ; j^\text {e}} \delta _{i^\text {v} ; j^\text {e}} \left( \varvec{v}_{i^\text {v}} - \varvec{v}_{k^\text {v}} \right) , \end{aligned}$$

(20)

where $\varvec{v}_{i^\text {v}}$ is the displacement velocity of the $i^\text {v}$th vertex, $\hbox {d} \varvec{r}_{i^\text {v}} / \hbox {d} t$.

To derive the viscous friction force exerted on the ${i^\text {v}}$th vertex by the ${l^\text {v}}$th vertex, Eq. (20) is multiplied by the following equation.

$$\begin{aligned} 1=\delta _{k^\text {v} l^\text {v}}+(1-\delta _{k^\text {v} l^\text {v}}), \end{aligned}$$

(21)

where $\delta _{k^\text {v} l^\text {v}}$ is the Kronecker delta function. The force $\varvec{f}^\text {v}_{i^\text {v}}$ can now be separated into two forces in terms of $k^\text {v}=l^\text {v}$ and $k^\text {v} \ne l^\text {v}$ as follows.

$$\begin{aligned} \varvec{f}^\text {v}_{i^\text {v}}&= \sum ^\text {vertex}_{k^\text {v}} \sum ^\text {element}_{j^\text {e}} \frac{\eta ^\text {e}_{j^\text {e}}}{n^\text {v}_{j^\text {e}}} \delta _{k^\text {v} ; j^\text {e}} \delta _{i^\text {v} ; j^\text {e}} \left( \varvec{v}_{i^\text {v}} - \varvec{v}_{k^\text {v}} \right) \delta _{k^\text {v} l^\text {v}} \nonumber \\&+ \sum ^\text {vertex}_{k^\text {v}} \sum ^\text {element}_{j^\text {e}} \frac{\eta ^\text {e}_{j^\text {e}}}{n^\text {v}_{j^\text {e}}} \delta _{k^\text {v} ; j^\text {e}} \delta _{i^\text {v} ; j^\text {e}} \left( \varvec{v}_{i^\text {v}} - \varvec{v}_{k^\text {v}} \right) (1-\delta _{k^\text {v} l^\text {v}}).\nonumber \\ \end{aligned}$$

(22)

Hence, the viscous friction force $\varvec{f}^\text {vv}_{i^\text {v} l^\text {v}}$, exerted on the ${i^\text {v}}$th vertex by the ${l^\text {v}}$th vertex, is the first term involving $\delta _{k^\text {v} l^\text {v}}$ in Eq. (22):

$$\begin{aligned} \varvec{f}^{\text {vv}}_{i^\text {v} l^\text {v}}&= \sum ^\text {element}_{j^\text {e}} \frac{ \eta ^\text {e}_{j^\text {e}} }{ n^\text {v}_{j^\text {e}} } \delta _{i^\text {v} ; j^\text {e}} \delta _{l^\text {v} ; j^\text {e}} \left( \varvec{v}_{i^\text {v}} - \varvec{v}_{l^\text {v}} \right) . \end{aligned}$$

(23)

According to Eq. (23), $\varvec{f}^{\text {vv}}_{i^\text {v} l^\text {v}}$ is a linear function of the relative velocity. Therefore, the viscous friction coefficient of the ${i^\text {v}}$th vertex introduced by the ${l^\text {v}}$th vertex given by $\eta ^\text {vv}_{i^\text {v} l^\text {v}}$ can be written as

$$\begin{aligned} \eta ^\text {vv}_{i^\text {v} l^\text {v}} =\sum ^\text {element}_{j^\text {e}} \frac{ \eta ^\text {e}_{j^\text {e}} }{ n^\text {v}_{j^\text {e}} } \delta _{i^\text {v} ; j^\text {e}} \delta _{l^\text {v} ; j^\text {e}}. \end{aligned}$$

(24)

The viscous friction coefficients between the ${i^\text {v}}$th and the ${l^\text {v}}$th vertices are then related by

$$\begin{aligned} \eta ^\text {vv}_{i^\text {v} l^\text {v}} =\eta ^\text {vv}_{l^\text {v} i^\text {v}}. \end{aligned}$$

(25)

Thus, the viscous friction coefficients are symmetric for any pair of vertices.

Using Eq. (23), the sum of viscous friction forces between a pair of vertices is given by

$$\begin{aligned} \varvec{f}^{\text {vv}}_{i^\text {v} l^\text {v}} + \varvec{f}^{\text {vv}}_{l^\text {v} i^\text {v}} = \varvec{0}. \end{aligned}$$

(26)

Thus, the viscous friction forces exerted between any two vertices are balanced.

1.2 Galilean invariance of vertex dynamics model

The following theoretical analysis clarifies whether the vertex dynamics in the proposed and conventional models are invariant under the Galilean transformation. The Galilean transformation is

$$\begin{aligned} \tilde{\varvec{r}}_{i^\text {v}} = \varvec{r}_{i^\text {v}} + \varvec{V} t, \end{aligned}$$

(27)

where $\varvec{V}$ is an arbitrary velocity vector. Substituting Eq. (27) in Eq. (1), we retrieve Eq. (1). Hence, the proposed model is invariant under the Galilean transformation.

The conventional models (Honda et al. 2004; Okuda et al. 2013a, b, c) neglect the local velocity vectors (i.e., $\varvec{v}^\text {f}_i=\varvec{0}$ in Eq. (1)). Hence, the movement of the $i^\text {v}$th vertex is given by

$$\begin{aligned} \eta ^\text {v}_i \frac{d \varvec{r}_{i^\text {v}}}{d t} = - \nabla _{i^\text {v}} U. \end{aligned}$$

(28)

Substituting Eq. (27) into Eq. (28), we obtain

$$\begin{aligned} \eta ^\text {v}_i \frac{d \tilde{\varvec{r}}_{i^\text {v}}}{d t} = - \nabla _{i^\text {v}} U - \eta ^\text {v}_0 \varvec{V}. \end{aligned}$$

(29)

Compared to Eq. (29), Eq. (28) contains an additional force term $-\eta ^\text {v}_0 \varvec{V}$. Thus, the conventional models described by Eq. (28) violate Galilean invariance unless the additional force can be ignored.

1.3 Cell rearrangement and division behaviors

Cell rearrangement is expressed by reconnecting local network patterns using the reversible network reconnection (RNR) model (Okuda et al. 2013a). The RNR model generates continuous cell rearrangements that are geometrically, energetically and topologically reversible. Cell proliferation is accomplished by a cell proliferation model (Okuda et al. 2013b), in which cells divide (increase their number) and grow (increase their volume). In particular, cell division is simulated by dividing a single polyhedron along a plane.

When the number of molecules in the ${i^\text {c}}$th cell $n^\text {cm}_{i^\text {c}}$ increases to $\left( 4/3 \right) n^\text {cm}_{0}$, the ${i^\text {c}}$th cell divides into two daughter cells. The dividing plane is regulated to be normal to the plane of the cell sheet and globally oriented along the growth direction, as shown in Fig. 2a. The normal direction of the cell plane is specified as a vector pointing from the center of an outside polygonal face of each cell to the center of that cell. Details of the cell division are similar to those of the global regulation used in our previous study (Okuda et al. 2013c).

1.4 Numerical implementation

Equations (1) and (28) were numerically time-integrated using the Euler method with a time step of $\Delta t$. Vertex velocities in Eq. (1) were iteratively solved by convergent calculations. Convergence was reached when the mean residual error was below the threshold $RE_\text {th}$, as follows.

$$\begin{aligned} \overline{ \left| \eta ^\text {v}_{i^\text {v}} \left( \frac{\mathrm{d} \varvec{r}_{i^\text {v}}}{\mathrm{d} t} - \varvec{v}^\text {f}_{i^\text {v}} \right) + \nabla _{i^\text {v}} U \right| } \le RE_\text {th} \end{aligned}$$

(30)

Local network patterns were reconnected when each edge included in a local pattern turned shorter than a specified threshold $\Delta l_\text {th}$. The reconnection rule was trialed at each edge and each trigonal face during each time interval $\Delta t_r$. Numerical parameters are listed in Table 2.

Table 2 Numerical parameters used in the model

All experiments were performed on a cluster computer comprising 12 nodes with 2.9 GHz Intel Xeon dual processors and 64 GB RAM (Visual Technology Co., Japan).

1.5 Stress tensor estimation

To analyze the deformation mechanics within the tissues, stress tensors were estimated from the locations and forces of vertices according to (Hardy 1982).

The stress tensor $\varvec{\sigma }_{\text {v}}$ over the whole epithelial vesicle is estimated as

$$\begin{aligned} \varvec{\sigma }_{\text {v}}=- \frac{1}{ v_{\text {t}} } \sum ^\text {vertex}_{j^\text {v}} \left( \varvec{r}_{j^\text {v}} \otimes \frac{\partial U}{\partial \varvec{r}_{j^\text {v}} } \right) , \end{aligned}$$

(31)

where the scalar $v_{\text {t}}$ is the volume of the epithelial vesicle, and $\sum _{j^\text {v}}^\text {vertex}$ denotes the summation overall vertices. The stress tensor $\varvec{\sigma }_{i^\text {c}}$ at the ${i^\text {c}}$th cell is estimated as

$$\begin{aligned} \varvec{\sigma }_{i^\text {c}} = - \frac{1}{ v_{i^\text {c}} } \sum ^\text {vertex}_{j^\text {v}} \left( \varvec{r}_{j^\text {v}} \otimes \varvec{f}^\text {cv}_{i^\text {c} j^\text {v}} \right) \delta _{j^\text {v} ; i^\text {c}}, \end{aligned}$$

(32)

where the scalar $v_{i^\text {c}}$ is the volume of the $i^\text {v}$th cell, and $\sum _{j^\text {v}}^\text {vertex}$ sums overall vertices. Vector $\varvec{f}^\text {cv}_{i^\text {c} j^\text {v}}$ denotes the interior force of the ${i^\text {c}}$th cell exerted on the $j^\text {v}$th vertex:

$$\begin{aligned} \varvec{f}^\text {cv}_{i^\text {c} j^\text {v}} = - \nabla _{j^\text {v}} \left( u^{\text {cv}}_{i^\text {c}} + u^{\text {cs}}_{i^\text {c}} \right) . \end{aligned}$$

(33)