Reaction–Diffusion Finite Element Model of Lateral Line Primordium Migration to Explore Cell Leadership

Abstract

Collective cell migration plays a fundamental role in many biological phenomena such as immune response, embryogenesis and tumorigenesis. In the present work, we propose a reaction–diffusion finite element model of the lateral line primordium migration in zebrafish. The population is modelled as a continuum with embedded discrete motile cells, which are assumed to be viscoelastic and able to undergo large deformations. The Wnt/ß-catenin–FGF and cxcr4b–cxcr7b signalling pathways inside the cohort regulating the migration are described through coupled reaction–diffusion equations. The coupling between mechanics and the molecular scenario occurs in two ways. Firstly, the intensity of the protrusion–contraction movement of the cells depends on the cxcr4b concentration. Secondly, the intra-synchronization between the active deformations and the adhesion forces inside each cell is triggered by the cxcr4b–cxcr7b polarity. This influences the inter-synchronization between the cells and results in two main modes of migration: uncoordinated and coordinated. The main objectives of the work were (i) to validate our assumptions with respect to the experimental observations and (ii) to decipher the mechanical conditions leading to efficient migration of the primordium. To achieve the second goal, we will specifically focus on the role of the leader cells and their position inside the population.

Appendix

1.1 LLP Geometry

The cell network $\Omega _n$is defined by a characteristic function $h_n \left( {{\varvec{p}}} \right)$ as follows:

$$\begin{aligned} h_n \left( {{\varvec{p}}} \right) =\left\{ {{\begin{array}{ll} 1&{} { \mathrm{if} \left\| {{{\varvec{p}}}-2\cdot r_\mathrm{c} \cdot round\left( {p_x } \right) {{\varvec{i}}}_x -2 \cdot r_\mathrm{c} \cdot round\left( {p_y } \right) {{\varvec{i}}}_y } \right\| <r_c^2 } \\ 0&{} \mathrm{otherwise} \\ \end{array} }} \right. . \end{aligned}$$

(8)

with round being the classical integer function and ${{\varvec{p}}}=\left( {p_x ,p_y } \right)$ the initial position of any particle of the system .

The ECM domain $\Omega _\mathrm{ECM}$ is identified by the characteristic function $h_\mathrm{ECM} \left( {{\varvec{p}}} \right)$ which reads

$$\begin{aligned} h_\mathrm{ECM} \left( {{\varvec{p}}} \right) =1-h_n \left( {{\varvec{p}}} \right) . \end{aligned}$$

(9)

Each cell inside the population is denoted by $c(i,j)$ where the indices $i$ and $j$ vary as follows:

$$\begin{aligned} \left\{ {\begin{array}{l} 1\le i\le N_\mathrm{c} =i_\mathrm{max } \\ 1\le j\le n_\mathrm{c} \left( i \right) =n_\mathrm{{c,\max }} \sqrt{1-\left( {\frac{2i-1}{N_\mathrm{c} }} \right) ^{2}} \\ \end{array}} \right. \end{aligned}$$

(10)

with $N_\mathrm{c} =\frac{L}{r_\mathrm{c}}, i_\mathrm{max } =18$ and $n_\mathrm{{c,\max }} =\frac{l}{r_\mathrm{c} }$ being the number of cells along the two axes of the ellipse (Fig. 2b, c).

The domain $\Omega _{\mathrm{c}_{i,j} }$ occupied by each cell $c(i,j)$ is defined through a characteristic function as follows

$$\begin{aligned} h_{\mathrm{c}_{i,j} } \left( {{\varvec{p}}} \right) =\left\{ {\begin{array}{l} 1\quad \mathrm{if}\left\| {{{\varvec{p}}}-{{\varvec{c}}}_{i,j} } \right\| <r_c^2 \\ 0\quad \mathrm{otherwise} \\ \end{array}} \right. \end{aligned}$$

(11)

Each cell is equipped with a frontal $\partial \Omega _{\mathrm{sf}_{i,j} }$ and a rear $\partial \Omega _{\mathrm{sr}_{i,j}}$ adhesion region (Fig. 2d) described, respectively, by two characteristic functions

$$\begin{aligned} h_{\mathrm{sf}_{i,j} } \left( {{\varvec{p}}} \right)&= \left\{ {\begin{array}{ll} 1&{} \quad \mathrm{if}\left( {{{\varvec{p}}}-{{\varvec{c}}}_{i,j} ,{{\varvec{i}}}_x } \right) >l_\mathrm{f} \\ 0&{}\quad \mathrm{otherwise} \\ \end{array}} \right. \nonumber \\ h_{\mathrm{sr}_{i,j} } \left( {{\varvec{p}}} \right)&= \left\{ {\begin{array}{ll} 1&{} \quad \mathrm{if}\left( {{{\varvec{p}}}-{{\varvec{c}}}_{i,j} ,{{\varvec{i}}}_x } \right) <-l_r \\ 0&{}\quad \mathrm{otherwise} \\ \end{array}} \right. \end{aligned}$$

(12)

where $\left( {{{\varvec{a}}},{{\varvec{b}}}} \right)$ defines the scalar product and $l_\mathrm{f}$ and $l_\mathrm{r}$ are the distances of ${{\varvec{c}}}_{i,j}$ from the frontal and rear adhesion surfaces, respectively.

The ellipse is divided into cell rows $r(i)$ (Fig. 2b), which are numbered, similarly to the single cells, from the “stern” (left) to the “bow” (right) of the ellipse ($1\le i\le N_\mathrm{c} =i_\mathrm{max }$) (Fig. 2c) and are defined through a characteristic function as

$$\begin{aligned} h_{r_i } \left( {{\varvec{p}}} \right) =h_n \left( {{\varvec{p}}} \right) \left\{ {\begin{array}{ll} 1&{} \quad \mathrm{if}\left( {p_x -c_{i,j_x } } \right) <r_c \\ 0&{} \quad \mathrm{otherwise}. \\ \end{array}} \right. \end{aligned}$$

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1.2 Leading and Trailing Edge of the LLP

The Wnt/ß-catenin–FGF network is mainly based on the spatial polarization of the LLP. We define the leading, $\Omega _\mathrm{front}$, and the trailing, $\Omega _\mathrm{rear}$, edges of the LLP through the characteristic functions $h_\mathrm{front}$ and $h_\mathrm{rear}$, respectively, as follows:

$$\begin{aligned} h_\mathrm{front} =\left\{ {\begin{array}{ll} 1&{} \quad \mathrm{if}\, p_x >p_{x0} \\ 0&{} \quad \mathrm{otherwise} \\ \end{array}} \right. \nonumber \\ h_\mathrm{rear} =\left\{ {\begin{array}{ll} 1&{} \quad \mathrm{if}\, p_x <p_{x0} \\ 0&{} \quad \mathrm{otherwise} \\ \end{array}} \right. \end{aligned}$$

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where $p_{x0}$ is the axial coordinate defining the boundary between the leading and the trailing edges.

1.3 Description of Mutants

In the following, we define the reaction–diffusion equations that have been used to describe the molecular and chemokine patterns specific to each mutant as mentioned in Sect. 2.2.

• apc embryo

  $$\begin{aligned} \frac{\partial \left[ W \right] }{\partial t}&= \underbrace{D_a \nabla ^{2}\left[ W \right] }_\mathrm{diffusion}+\underbrace{S_a \left[ W \right] \left( {1-\left[ W \right] } \right) }_{signalling}-\underbrace{R_\mathrm{a} \left[ W \right] \left[ F \right] }_\mathrm{reaction\, by\, dkk1}\end{aligned}$$

  (15)
  $$\begin{aligned} \frac{\partial \left[ F \right] }{\partial t}&= \underbrace{D_\mathrm{b} \nabla ^{2}\left[ F \right] }_\mathrm{diffusion}+\underbrace{P_\mathrm{b} \left[ W \right] \left( {1-\left[ F \right] } \right) }_\mathrm{production}-\underbrace{R_\mathrm{b} \left[ F \right] \left[ W \right] }_\mathrm{reaction\, by\, sef}\end{aligned}$$

  (16)
  $$\begin{aligned} \frac{\partial \left[ {c_4 } \right] }{\partial t}&= \underbrace{P_\mathrm{c} \left[ {c_4 } \right] \left( {1-\left[ {c_4 } \right] } \right) }_\mathrm{production}-\underbrace{R_\mathrm{c} \left[ {c_4 } \right] \left[ F \right] }_\mathrm{reaction\, by\, Fgf}\end{aligned}$$

  (17)
  $$\begin{aligned} \frac{\partial \left[ {c_7 } \right] }{\partial t}&= \underbrace{P_\mathrm{d} \left[ {c_7 } \right] \left( {1-\left[ {c_7 } \right] } \right) }_\mathrm{production}-\underbrace{R_\mathrm{d} \left[ {c_7 } \right] \left[ W \right] }_\mathrm{reaction\, by\, Wnt} \end{aligned}$$

  (18)

• SU5402 embryo

  $$\begin{aligned} \frac{\partial \left[ W \right] }{\partial t}&= \underbrace{D_\mathrm{a} \nabla ^{2}\left[ W \right] }_\mathrm{diffusion}+\underbrace{S_\mathrm{a} \left[ W \right] \left( {1-\left[ W \right] } \right) h_\mathrm{front} }_\mathrm{signalling}-\underbrace{R_\mathrm{a} \left[ W \right] \left[ F \right] h_\mathrm{rear} }_\mathrm{reaction\, by\, dkk1}\end{aligned}$$

  (19)
  $$\begin{aligned} \frac{\partial \left[ F \right] }{\partial t}&= \underbrace{D_\mathrm{b} \nabla ^{2}\left[ F \right] }_\mathrm{diffusion}-\underbrace{R_\mathrm{b} \left[ F \right] \left[ W \right] h_\mathrm{front} }_\mathrm{reaction\, by\, sef}\end{aligned}$$

  (20)
  $$\begin{aligned} \frac{\partial \left[ {c_4 } \right] }{\partial t}&= \underbrace{P_\mathrm{c} \left[ {c_4 } \right] \left( {1-\left[ {c_4 } \right] } \right) }_\mathrm{production}-\underbrace{R_\mathrm{c} \left[ {c_4 } \right] \left[ F \right] }_{\mathrm{reaction}\, {by}\, Fgf}\end{aligned}$$

  (21)
  $$\begin{aligned} \frac{\partial \left[ {c_7 } \right] }{\partial t}&= \underbrace{P_\mathrm{d} \left[ {c_7 } \right] \left( {1-\left[ {c_7 } \right] } \right) }_\mathrm{production}-\underbrace{R_\mathrm{d} \left[ {c_7 } \right] \left[ W \right] }_{\mathrm{reaction\, by}\, Wnt} \end{aligned}$$

  (22)

• dkk1 embryo

  $$\begin{aligned} \frac{\partial \left[ W \right] }{\partial t}&= \underbrace{D_\mathrm{a} \nabla ^{2}\left[ W \right] }_\mathrm{diffusion}+\underbrace{S_\mathrm{a} \left[ W \right] \left( {1-\left[ W \right] } \right) h_\mathrm{front} }_\mathrm{signalling}-\underbrace{R_\mathrm{a} \left[ W \right] \left[ F \right] }_{\mathrm{reaction}\, \mathrm{by}\, dkk1}\end{aligned}$$

  (23)
  $$\begin{aligned} \frac{\partial \left[ F \right] }{\partial t}&= \underbrace{D_\mathrm{b} \nabla ^{2}\left[ F \right] }_\mathrm{diffusion}+P_\mathrm{b} \left[ W \right] \left( {1-\left[ F \right] } \right) -\underbrace{R_\mathrm{b} \left[ F \right] \left[ W \right] h_\mathrm{front} }_{\mathrm{reaction\, by}\, sef}\end{aligned}$$

  (24)
  $$\begin{aligned} \frac{\partial \left[ {c_4 } \right] }{\partial t}&= \underbrace{P_\mathrm{c} \left[ {c_4 } \right] \left( {1-\left[ {c_4 } \right] } \right) }_\mathrm{production}-\underbrace{R_\mathrm{c} \left[ {c_4 } \right] \left[ F \right] }_{\mathrm{reaction\, by}\, Fgf}\end{aligned}$$

  (25)
  $$\begin{aligned} \frac{\partial \left[ {c_7 } \right] }{\partial t}&= \underbrace{P_\mathrm{d} \left[ {c_7 } \right] \left( {1-\left[ {c_7 } \right] } \right) }_\mathrm{production}-\underbrace{R_\mathrm{d} \left[ {c_7 } \right] \left[ W \right] }_{\mathrm{reaction\, by}\, Wnt} \end{aligned}$$

  (26)

1.4 Constitutive Model

As mentioned in Sect. 2.3, the behaviour of the cells is described through a generalized viscoelastic 2D Maxwell model (Allena 2013; Allena and Aubry 2012).

The Cauchy stress, $\varvec{\sigma }$, is assumed to be the sum of the solid ($\varvec{\sigma }_\mathrm{s}$) and the fluid ($\varvec{\sigma }_\mathrm{f}$) Cauchy stresses, while the deformation gradient ${{\varvec{F}}}$ is equal to the solid (${{\varvec{F}}}_\mathrm{s}$) and the fluid (${{\varvec{F}}}_\mathrm{f}$) deformation gradients.

The decomposition of the deformation gradient (Allena et al. 2010; Lubarda 2004) is used to describe the solid deformation tensor, ${{\varvec{F}}}_\mathrm{s}$, which is then given by

$$\begin{aligned} {{\varvec{F}}}_\mathrm{s} ={{\varvec{F}}}_\mathrm{se} {{\varvec{F}}}_\mathrm{sa} \end{aligned}$$

(27)

where ${{\varvec{F}}}_\mathrm{se}$ is the elastic deformation tensor responsible for the stress generation and ${{\varvec{F}}}_\mathrm{sa}$ is the active deformation tensor responsible for the pulsating movement (protrusion–contraction) of each cell. Similarly, the fluid deformation tensor ${{\varvec{F}}}_\mathrm{f}$ is the multiplicative decomposition of the fluid elastic (${{\varvec{F}}}_\mathrm{fe}$) and the fluid viscoelastic (${{\varvec{F}}}_\mathrm{fv}$) gradients.

Both the solid $\varvec{\sigma }_\mathrm{se}$ and the fluid elastic $\varvec{\sigma }_\mathrm{fe}$ Cauchy stresses are given by isotropic hyperelastic models $\bar{{\varvec{\sigma }}}_\mathrm{se}$ and $\bar{{\varvec{\sigma }}}_\mathrm{fe}$, respectively, as

$$\begin{aligned} \varvec{\sigma }_\mathrm{se}&= \bar{{\varvec{\sigma }}}_\mathrm{se} \left( {{{\varvec{e}}}_\mathrm{se} } \right) \nonumber \\ \varvec{\sigma }_\mathrm{fe}&= \bar{{\varvec{\sigma }}}_\mathrm{fe} \left( {{{\varvec{e}}}_\mathrm{fe} } \right) \end{aligned}$$

(28)

with ${{\varvec{e}}}_\mathrm{se}$ and ${{\varvec{e}}}_\mathrm{fe}$ the Euler–Almansi deformation tensors for the solid elastic and the fluid elastic phases, respectively. Additionally, $\varvec{\sigma }_\mathrm{fe}$ has to be expressed in the actual configuration according to the multiplicative decomposition described above. Finally, the deformation rate $\dot{{{\varvec{e}}}}_\mathrm{fv}$ is related to the deviator part of the fluid viscous stress $\varvec{\sigma }_\mathrm{fv}^D$ as follows:

$$\begin{aligned} \dot{{{\varvec{e}}}}_\mathrm{fv} =\frac{\varvec{\sigma }_\mathrm{fv}^D }{{\mu }_\mathrm{fv} } \end{aligned}$$

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where $\mu _\mathrm{fv}$ is the viscosity and the dot is the derivative with respect to time.

1.5 Coordinated and Uncoordinated Migration

The characteristic functions $h_\mathrm{c}$ and $h_\mathrm{uc}$ are expressed as follows

$$\begin{aligned} h_\mathrm{c}&= \left\{ {\begin{array}{ll} 1&{} \mathrm{if} \left\{ {\begin{array}{c} \left( {\left[ {c_4 } \right] >\left[ {c_\mathrm{max } } \right] } \right) h_\mathrm{front} \wedge \left( {\left[ {c_4 } \right] <\left[ {c_\mathrm{min } } \right] } \right) h_\mathrm{rear} \\ \wedge \\ \left( {\left[ {c_7 } \right] >\left[ {c_\mathrm{max } } \right] } \right) h_\mathrm{rear} \wedge \left( {\left[ {c_7 } \right] <\left[ {c_\mathrm{min } } \right] } \right) h_\mathrm{front} \\ \end{array}} \right. \\ 0&{} \mathrm{otherwise} \\ \end{array}} \right. \nonumber \\ h_\mathrm{uc}&= \left\{ {\begin{array}{ll} 1&{} \mathrm{if} \left\{ {\begin{array}{c} \left( {\left[ {c_4 } \right] <\left[ {c_\mathrm{max } } \right] } \right) h_\mathrm{front} \wedge \left( {\left[ {c_4 } \right] >\left[ {c_\mathrm{max } } \right] } \right) h_\mathrm{rear} \\ \wedge \\ \left( {\left[ {c_7 } \right] <\left[ {c_\mathrm{max } } \right] } \right) h_\mathrm{rear} \wedge \left( {\left[ {c_7 } \right] >\left[ {c_\mathrm{min } } \right] } \right) h_\mathrm{front} \\ \end{array}} \right. \\ 0&{} \mathrm{otherwise} \\ \end{array}} \right. \end{aligned}$$

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with $\wedge$ being the Boolean operator AND and $c_\mathrm{max}$ and $c_\mathrm{min}$ being two thresholds fixed here to 0.9 and 0.2, respectively.

The terms $e_{a,c}$ and $e_{a,uc}$ describe the cyclic deformation of protrusion–contraction, and they read

$$\begin{aligned} e_{a,c}&= \frac{\left[ {c_4 } \right] }{\alpha _c }\sin \left( {2\pi \frac{t-\frac{T}{2}\left( {i_\mathrm{max } -i} \right) }{T}} \right) h_{r_i } \left( {{\varvec{p}}} \right) h_{wave} \left( {{{\varvec{p}}},t} \right) \nonumber \\ e_{a,uc}&= \frac{\left[ {c_4 } \right] }{\alpha _{uc_{ij} } }\sin \left( {2\pi \frac{t}{T_{uc_{ij} } }} \right) \end{aligned}$$

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where $t$ is time.

For the coordinated migration, $\alpha _c$ is set to 2 and $T$ indicates the duration of a migration period which has been fixed here to 60 s (Allena and Aubry 2012; Dong et al. 2002). Additionally, a wave progressively covers the LLP from the “bow” to the “stern” to activate, one by one, the cell row $r(i)$ with a velocity equal to $\frac{2t}{T}$. The wave is expressed by the characteristic function $h_{wave} \left( {{{\varvec{p}}},t} \right)$ as follows:

$$\begin{aligned} h_{wave} \left( {{{\varvec{p}}},t} \right) =\left\{ {\begin{array}{ll} 1&{} \mathrm{if}\left( {2L-r_c } \right) -p_x -2r_c \frac{2t}{T}<0 \\ 0&{} \mathrm{otherwise}. \\ \end{array}} \right. \end{aligned}$$

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For the uncoordinated migration, $\alpha _{uc_{ij}}$ and $T_{uc_{ij}}$ may vary between 0 and 1 and between 60 and 120 s, respectively, for each cell $c(i,j)$.