On the Mechanical Interplay Between Intra- and Inter-Synchronization During Collective Cell Migration: A Numerical Investigation

Abstract

Collective cell migration is a fundamental process that takes place during several biological phenomena such as embryogenesis, immunity response, and tumorogenesis, but the mechanisms that regulate it are still unclear. Similarly to collective animal behavior, cells receive feedbacks in space and time, which control the direction of the migration and the synergy between the cells of the population, respectively. While in single cell migration intra-synchronization (i.e. the synchronization between the protrusion-contraction movement of the cell and the adhesion forces exerted by the cell to move forward) is a sufficient condition for an efficient migration, in collective cell migration the cells must communicate and coordinate their movement between each other in order to be as efficient as possible (i.e. inter-synchronization). Here, we propose a 2D mechanical model of a cell population, which is described as a continuum with embedded discrete cells with or without motility phenotype. The decomposition of the deformation gradient is employed to reproduce the cyclic active strains of each single cell (i.e. protrusion and contraction). We explore different modes of collective migration to investigate the mechanical interplay between intra- and inter-synchronization. The main objective of the paper is to evaluate the efficiency of the cell population in terms of covered distance and how the stress distribution inside the cohort and the single cells may in turn provide insights regarding such efficiency.

Appendix

1.1 A.1 Geometry of the cell population

The cells network Ω _n is described through a characteristic function h _n (p), which reads

$$ h_n (\boldsymbol{p})= \left \{\begin{array}{l@{\quad}l} 1 & \mbox{if}\ \| \boldsymbol{p} - 2 \cdot r_c \cdot \mathit{round}(\boldsymbol{p}_x)\boldsymbol{i}_x- 2\cdot r_c \cdot \mathit{round} (\boldsymbol{p}_y)\boldsymbol{i}_y\|< r_c^2 \\ 0 & \mbox{otherwise} \end{array} \right . $$

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with round being the classical integer function and p(p _x ,p _y ) the initial position of any particle of the system.

Consequently, the ECM domain Ω _ECM is defined by a characteristic function as follows:

$$ h_{\mathrm{ECM}} ( \boldsymbol{p} ) = 1 - h_{n} ( \boldsymbol{p} ) $$

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

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

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with $N_{c} = \frac{L}{r_{c}}$ and $n_{c,\max} = \frac{l}{r_{c}}$ being the number of cells along the two axes of the ellipse (Fig. 1c).

The domain $\varOmega_{c_{i,j}}$ of each cell c(i,j) is defined through a characteristic function as follows:

$$ h_{c_{i,j}} ( \boldsymbol{p} ) = \left \{ \begin{array}{l@{\quad}l} 1 & \mbox{if}\ \Vert \boldsymbol{p} - \boldsymbol{c}_{i,j} \Vert < r_{c}^{2} \\ 0& \mbox{otherwise} \end{array} \right . $$

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Each cell is equipped with a frontal $\partial \varOmega_{sf_{i,j}}$ and a rear $\partial \varOmega_{sr_{i,j}}$ adhesion region (Fig. 1d) described by two characteristic functions as

$$\begin{aligned} \begin{aligned} h_{sf_{i,j}} ( \boldsymbol{p} ) &= \left \{ \begin{array}{l@{\quad}l} 1 & \mbox{if}\ ( \boldsymbol{p} - \boldsymbol{c}_{i,j},\boldsymbol{i}_{x} ) > l_{f} \\ 0& \mbox{otherwise} \end{array} \right . \\ h_{sr_{i,j}} ( \boldsymbol{p} ) &= \left \{ \begin{array}{l@{\quad}l} 1 & \mbox{if}\ ( \boldsymbol{p} - \boldsymbol{c}_{i,j},\boldsymbol{i}_{x} ) < l_{r} \\ 0 & \mbox{otherwise} \end{array} \right . \end{aligned} \end{aligned}$$

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where (a,b) defines the scalar product between two vectors, l _f and l _r are the distances of c _i,j from the frontal and rear adhesion surfaces, respectively.

The ellipse is divided into cell rows r(i) (Fig. 1b), which are numbered, similarly to the single cells, from the stern (left) to the bow (right) of the ellipse (1≤i≤N _c =i _max) (Fig. 1c) and are defined through a characteristic function as

$$ h_{r_{i}} ( \boldsymbol{p} ) = h_{n} ( \boldsymbol{p} ) \left \{ \begin{array}{l@{\quad}l} 1 & \mbox{if}\ ( \boldsymbol{p}_{x} - \boldsymbol{c}_{i,j_{x}} ) < r_{c} \\ 0 & \mbox{otherwise} \end{array} \right . $$

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1.2 A.2 Constitutive Model of the Cells

As mentioned in Sect. 2.2, the behavior of the active and quiescent cells is described through a generalized viscoelastic 2D Maxwell model (Allena 2013; Allena and Aubry 2012). Since the cells within the cohort may undergo large rotations and deformations during their locomotion, a fully non-linear tensorial approach is required.

For the active cells, the Cauchy stress σ _a is assumed to be the sum of the solid (σ _a,s ) and the fluid (σ _a,f ) Cauchy stresses, while the deformation gradient F _a is equal to the solid (F _a,s ) and the fluid (F _a,f ) deformation gradients.

The decomposition of the deformation gradient (Allena et al. 2010; Lubarda 2004) is used to describe the solid deformation tensor F _a,s which is then equal to

$$ \boldsymbol{F}_{a,s} = \boldsymbol{F}_{a,se} \boldsymbol{F}_{a,sa} $$

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where F _a,se is the elastic strain tensor responsible for the stress generation and F _a,sa is the active strain tensor responsible for the pulsating movement (protrusion-contraction) of each cell. Similarly, the fluid deformation tensor F _a,f is the multiplicative decomposition of the fluid-elastic (F _a,fe ) and the fluid-viscoelastic (F _a,fv ) gradients.

Both the solid σ _a,se and the fluid-elastic σ _a,fe Cauchy’s stresses are given by isotropic hyperelastic models $\bar{\sigma}_{a,se}$ and $\bar{\sigma}_{a,fe}$, respectively, as

$$ \begin{aligned} &\sigma_{a,se} = \bar{\sigma}_{a,se} ( \boldsymbol{e}_{a,se} ) \\ &\sigma_{a,fe} = \bar{\sigma}_{a,fe} ( \boldsymbol{e}_{a,fe} ) \end{aligned} $$

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

$$ \dot{\boldsymbol{e}}_{a,fv} = \frac{\sigma_{a,fv}^{D}}{\mu_{a,fv}} $$

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where μ _a,fv is the viscosity.

For the quiescent cells, the same equations can be applied but one has to notice that the solid deformation gradient can now be written as

$$ \boldsymbol{F}_{q,s} = \boldsymbol{F}_{q,se} \boldsymbol{F}_{q,sa} = \boldsymbol{F}_{q,se}\boldsymbol{I} $$

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since no active strains take place in these cells (Sect. 2.2).

1.3 A.3 Numerical Implementation of the Constitutive Law

In this section, we provide the general framework of the numerical approach. For further details, we refer the reader to similar works and applications proposed by Glowinski and Pan (1992) and Vennat et al. (2010).

The cell population is modeled as a continuum (the ellipse). Each one of the three internal regions (active cells Ω _a,k , quiescent cells Ω _q,k , and ECM Ω _ECM) is represented by a level-set function (h _a,k ,h _q,k , and h _ECM, respectively). The constitutive behavior of the active and the quiescent cells is described through a 2D generalized Maxwell model, while the ECM is described by a viscoelastic material. In the finite element formulation, the Cauchy stress σ and the viscous strain rate $\dot{\boldsymbol{e}}_{fv}$ are computed at each point p of the continuum taking into account the contributions of the three regions as follows:

$$ \begin{aligned} &\sigma ( \boldsymbol{p} ) = h_{\mathrm{ECM}} ( \boldsymbol{p} ) \sigma_{\mathrm{ECM}} ( \boldsymbol{p} ) + \sum h_{a,k} ( \boldsymbol{p} )\sigma_{a,k} ( \boldsymbol{p} ) + \sum h_{q,k} ( \boldsymbol{p} )\sigma_{q,k} ( \boldsymbol{p} ) \\ &\dot{\boldsymbol{e}}_{fv} ( \boldsymbol{p} ) = h_{\mathrm{ECM}} ( \boldsymbol{p} )\dot{\boldsymbol{e}}_{\mathrm{ECM}_{fv}} ( \boldsymbol{p} ) + \sum h_{a,k} ( \boldsymbol{p} )\dot{\boldsymbol{e}}_{a,k_{fv}} ( \boldsymbol{p} ) + \sum h_{q,k} ( \boldsymbol{p} )\dot{ \boldsymbol{e}}_{q,k_{fv}} ( \boldsymbol{p} ) \end{aligned} $$

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where $\dot{\boldsymbol{e}}_{\mathrm{ECM}_{fv}} ( \boldsymbol{p} )$, $\dot{\boldsymbol{e}}_{a,k_{fv}} ( \boldsymbol{p} )$, and $\dot{\boldsymbol{e}}_{q,k_{fv}} ( \boldsymbol{p} )$ are the viscous strain rates for the correspondent domains. Thus, the finite element mesh is not adapted to each sub-region of the continuum, but everything is handled via the level set functions, which allow localizing the mechanical behavior.

Then such a constitutive behavior is implemented in the dynamics equation (Eq. (1)), which involves the aforementioned stress, the displacement acceleration a and the adhesion forces f _adh. This equation is first transformed into the weak form of the problem (i.e. principle of the virtual works) and it is then discretized by finite elements. Accordingly, cells/ECM or cells/cells mutual forces are automatically equilibrated in a weak sense although in general, the cell boundaries intersect the finite elements edges. In fact, the level set functions describing the sub-regions of the system are defined independently from the finite element mesh.

1.4 A.4 Traveling Wave with Pulse Signal (or Worm-Like Migration)

In this mode of migration, a traveling wave spans the cell population and successively activate and de-activate the cell rows r(i). Thus, only one row is active at the time and the spatial coordinate $c_{i, j_{x}}$ defining its position changes every migration cycle T (Fig. 2), so that the active cell network h _a,2(p,t) is defined as

$$ h_{a,2} ( \boldsymbol{p},t ) = h_{n} ( \boldsymbol{p} ) \cdot \left \{ \begin{array}{l@{\quad}l} 1 & \mbox{if}\ ( \vert \boldsymbol{p}_{x} - c_{i,j_{x}} ( t ) \vert < r_{c} ) \\ 0 & \mbox{otherwise} \end{array} \right . $$

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where $c_{i, j_{x}}(t)$ reads

$$ c_{i,j_{t}} ( t ) = ( 2L - r_{c} ) - 2r_{c} \cdot \mathit{round} \biggl( \frac{t}{T} - 0.5 \biggr) + 2L \cdot \mathit{round} \biggl( \frac{t}{T \cdot N_{c}} - 0.5 \biggr) $$

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In this case, also the quiescent domain h _q,2 varies in space and time as h _q,2(p,t)=h _n (p)−h _a,2(p,t).

Fig. 2

Trend of the traveling wave, which defines the spatial coordinate $c_{i,j_{x}}(t)$ of the temporary active row for the worm-like migration (only 1 h is represented)

Fig. 3

Random distribution of the active cells network h _a,3(p,t) between 0–6 h (a), 6 h–12 h (b), 12 h–18 h (c) and 18 h–24 h (d) for the tsunami-like migration. In red the active cells (τ _i,j =1) and in blue the quiescent cells (τ _i,j =0) (Color figure online)

Fig. 4

Random values of the cyclic component e _a0,ij of the active strain e _a0 (a) and of the migration period T _ij (b) for the random migration

Fig. 5

The distance covered by the cell population after 24 h for the different modes of migration: chemoattractant migration (blue), worm-like migration (red), tsunami-like migration (green), chemoattractant tsunami-like migration (purple), random migration (pink) (Color figure online)

1.5 A.5 Traveling Wave with Random Unit Step Signal (or Tsunami-Like Migration)

The characteristic function h _a,3(p,t) defining the active cells network for this mode of migration reads

$$ h_{a,3} ( \boldsymbol{p},t ) = \tau_{i,j} \cdot h_{r_{i}} ( \boldsymbol{p} ) \cdot h_{sw} ( \boldsymbol{p},t ) $$

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where h _sw (p,t) describes the progressive wave, which gradually covers the population with a velocity equal to $\frac{2t}{T}$ and is expressed as

$$ h_{sw} ( \boldsymbol{p},t ) = \left \{ \begin{array}{l@{\quad}l} 1 & \mbox{if}\ ( 2L - r_{c} ) - \boldsymbol{p}_{x} - 2r_{c}\frac{2t}{T} < 0 \\ 0 & \mbox{otherwise} \end{array} \right . $$

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Similarly to the previous case, the quiescent cells domain h _q,3 reads h _q,3(p,t)=h _n (p)−h _a,3(p,t).