The cell normal (tensional/compressive) stress is responsible for changes in the cell packing density, while the shear stress has no impact on the cell packing density. Cell tensional stress causes a decrease in the cell packing density, while compressive stress induces an increase in the cell packing density over a time-scale of hours. An increase in cell packing density intensifies cell-cell interactions, which have a feedback impact on cell cohesion and adhesion energies. The probability of a cell-cell interaction is proportional to the cell volume fraction φ. Cell mechanical stress is caused by in-plane cell strain generated during epithelial monolayer inhomogeneous wetting/de-wetting. The in-plane cell strain can be uni-axial or biaxial depending on cell-cell interactions. Consequently, cell compressive, tensional, and shear stress components are generated during cell rearrangement as follows:

When a cell monolayer undergoes anisotropic active wetting, then extension in the direction of cell movement (i.e., active wetting) leads to compression in the direction perpendicular to cell movement (i.e., passive de-wetting) in order to maintain the monolayer’s structural integrity.

Anisotropic compression in the direction of cell movement (i.e., active de-wetting) results in a generation of cell tensional stress in the direction perpendicular to the cell movement (i.e., passive wetting).

Some multicellular domains perform more intensive wetting than surrounding domains and compress them. Pérez-González et al. [11] confirmed this experimentally. The wetting of the monolayer’s central region was more intensive than the wetting of its peripheral region, caused by the radial distribution of epithelial surface tension accompanied by a concentration of E-cadherin [11].

Some multicellular domains undergo active wetting, while others undergo active de-wetting. The consequence of the existence of the wetting and de-wetting domains can be the generation of forwards and backwards flows [1]. Collisions between these flows can generate additional compressive stress.

Some multicellular domains undergo inhomogeneous de-wetting, which causes an inhomogeneous accumulation of the cell compressive stress and can induce the formation of holes within the monolayer [21].

Cell shear stress can be generated along the borders between multicellular domains depending on their velocities. When the cell packing density is lower than or equal to \(\:_\) (where \(\:_\) is the cell packing density in the confluent state), local cell shear stress generation has been recorded within wetting MDCK cell monolayers [1, 6].

These scenarios demonstrate that cell compressive stress can be accumulated locally even when the epithelial monolayers undergo wetting, while the de-wetting of the monolayers results in an intense accumulation of compressive stress. It is consistent with the experimental observation of cell jamming domains as an indicator of cell compressive stress, by Serra-Picamal et al. [1], Nnetu et al. [32], Tlili et al. [30], and by many others who have considered active wetting of epithelial monolayers. The maximum compressive stress generated during the rearrangement of confluent MDCK cell monolayers, and the maximum tensional stress caused by the wetting of MDCK cell monolayers, were 300 Pa [1, 2]. The cell shear stress generated during the wetting of MDCK epithelial monolayers is a few tens of Pa [1, 6]. In the next section, the cell mechanical stress will be discussed depending on the viscoelasticity of multicellular systems and the cell-matrix interfacial tension.

Cell mechanical stress generation caused by collective cell migration

The cell mechanical stress generated during collective cell migration is influenced by the viscoelasticity of epithelial monolayers and by the cell-matrix interfacial tension [38]. The viscoelasticity of epithelial monolayers and cell-matrix interfacial tension depend on the strength of cell-cell and cell-matrix adhesion contacts, and cell contractility. Both types of adhesion contacts are influenced by the stiffness of the substrate matrix.

The cell-matrix interfacial tension depends on the epithelial surface tension, matrix surface tension, and cell-matrix adhesion energy. This physical parameter is time-dependent and can be expressed as:

$$\:_\left(r,\tau\:\right)=_\left(r,\tau\:\right)+_\left(r,\tau\:\right)-_\left(r,\tau\:\right)$$

(1)

where the cell-matrix adhesion energy \(\:_\) is released when two surfaces come into contact. The interfacial tension decreases with the strength of FAs. The equilibrium (static) tissue surface tension measured after uni-axial compression of cell aggregates is: (1) \(\:4.5\:\frac}}\) for F9 WT cell aggregates [39]; (2) \(\:1.6\pm\:0.6\:\frac}}\) to \(\:4.0\pm\:1.0\:\frac}}\) within 9 days for embryonic neural retina aggregates [40]; and (3) \(\:22.8\pm\:3\:\frac}}\) for aggregates of CHO cells [41]. The static surface tension of collagen I matrix decreases from \(\:62\:\frac}}\) to \(\:57\:\frac}}\) at \(\:21\:^}\text\) when the concentration of collagen increases from \(\:1\:\frac}}\) to \(\:4\:\frac}}\) (in experiments without cells) [42]. The inhomogeneous distribution of the strength of cell-cell and cell-matrix adhesion contacts, as well as the surface rearrangement of the substrate matrix, caused by cell tractions, lead to an inhomogeneous distribution of the interfacial tension. An inhomogeneous distribution of the epithelial surface tension causes hole formation during passive de-wetting of murine sarcoma (S-180) cell monolayers on a non-adhesive substrate matrix [21]. Pérez-González et al. [11] observed a radial distribution of E-cadherin concentration, and consequently, the epithelial surface tension within the monolayers. An inhomogeneous distribution of the matrix surface tension can be induced by rearrangement of the polymer matrix caused by cell tractions [38]. Clark et al. [26] considered the movement of A431 cell clusters on the collagen I matrix and revealed that the distribution of collagen concentration around the cell cluster is asymmetric, such that the collagen concentration near the front of the cluster is ~ 30% lower than that near its rear. The change in collagen in-plane concentration causes the establishment of a matrix surface tension gradient, which has a feedback impact on the directional migration of the cell cluster [38]. The strength of the cell FAs, as well as cell traction forces varies along the cell monolayers [43]. Strong cell-cell adhesion contacts within keratinocyte monolayers localize the traction forces to the colony periphery [43]. The main characteristic of migrating epithelial collectives is the inhomogeneous distribution of cell tractions, cell packing density, velocity, and accumulated stress [1, 30, 32]. From Eq. 1 the interfacial tension gradient can be expressed as: \(\:\gamma\:}_=\overrightarrow_+\overrightarrow_-\overrightarrow_\).

Consequently, both the interfacial tension and its gradient influence the generation of the cell residual stress, i.e., the stress, that remains in the cellular systems during collective cell migration and changes on a time scale of hours [7]. The cell residual stress can have both normal (tensional/compressive) and shear components. All components of the cell stress have been measured within migrating epithelial monolayers [1, 2, 6]. The cell normal residual stress includes isotropic and deviatoric parts. The isotropic part of the cell normal residual stress is induced by the work of the epithelial-matrix interfacial tension in decreasing the biointerface area expressed by the Young-Laplace equation. The deviatoric part of the cell normal stress is the viscoelastic normal stress attributed to collective cell migration. It is in accordance with fact that migrating cell groups perform directional migration which can be perturbed during inhomogeneous wetting/de-wetting. Consequently, the cumulative effects of cell-matrix interactions lead to generation of the isotropic part of the cell normal residual stress, while the deviatoric part the normal residual stress is generated internally within multicellular systems. Consequently, the cell normal residual stress can be expressed as:

$$\:}}_\varvec\varvec}=\pm\:\varDelta\:_\stackrel}+}}_\varvec\varvec}}^\varvec\varvec\:}$$

(2)

where \(\:}}_\varvec\varvec}\) is the cell normal residual stress part, which includes \(\:_\) and \(\:_\) components, \(\:\stackrel}\) is the unity tensor, \(\:\varDelta\:_\) is the isotropic part of the cell normal stress equal to \(\:\varDelta\:_=-_\left(\overrightarrow\cdot\:\overrightarrow}\right)\), \(\:\overrightarrow}\) is the normal vector of the cell-matrix biointerface, and \(\:}}_\varvec\varvec}}^\varvec\varvec\:}\) is the deviatoric part of the cell normal residual stress with the components \(\:_}^\) and \(\:_}^\). The positive and negative signs of the isotropic stress part indicate tension and compression, respectively. The deviatoric part of normal stress depends on the viscoelasticity of epithelial monolayers. While passive wetting/de-wetting generates an isotropic contribution to the cell normal residual stress (caused by effects along the epithelial-matrix biointerface), collective cell migration during active wetting/de-wetting generates an anisotropic (i.e., deviatoric) contribution to the normal residual stress. The viscoelasticity further depends on the cell packing density and the strength of the cell-cell adhesion contacts, which will be discussed in more detail.

The inhomogeneous distribution of the cell normal stress components, generated during collective cell migration, causes an inhomogeneous distribution of cell packing density within monolayers. Three subpopulations can be distinguished:

1.

A migratory, proliferative subpopulation with the cell packing density \(\:_<_\) (where \(\:_\) is the cell packing density at homeostasis and \(\:_\) is the epithelial packing density);

2.

A homeostatic cell subpopulation with cell packing density \(\:_\sim_\), which satisfies the condition that proliferation is inhibited; and

3.

A jamming cell subpopulation with cell packing density \(\:_\sim_\), (where \(\:_\) is the cell packing density at the jamming state), which satisfies the condition that proliferation and locomotion are inhibited.

The existence of subpopulations 2 and 3 is related primarily to the accumulation of cell compressive stress [44]. The cell packing density of the jamming subpopulation is lower than that of the migratory and homeostatic subpopulations, i.e. \(\:_<_\) [44]. This phenomenon, observed by Kaliman et al. [44], has not yet been explained. We will offer an explanation from the standpoint of physics in the next section. The cell packing densities, characteristic for subpopulations 2 and 3, depend on the cell type and matrix stiffness [44]. The dynamical interrelation between subpopulations is presented schematically in Fig. 1:

Fig. 1

Schematic presentation of the interrelation between the three types of subpopulations within migrating epithelial monolayers

An increase in compressive stress drives the forwards transition from subpopulation 1 to subpopulation 2, while cell extrusion induces the transition backwards from subpopulation 2 to subpopulation 1. The transition from subpopulation 2 is also possible to subpopulation 3 and vice versa during cell jamming/unjamming. A special interest here is to understand the main properties of the cell-cell interactions, which lead to the transition from subpopulation 2 to subpopulations 1 and 3.

Tlili et al. [30] considered the active wetting of MDCK epithelial monolayers and revealed that cell packing density varies from \(\:^\:\frac}}^}\) to \(\:^\:\frac}}^}\). An increase in cell packing density from \(\:^\:\frac}}^}\) to \(\:^\:\frac}}^}\) resulted in a decrease in cell velocity from \(\:0.8\:\frac\text}}\) to zero [30]. Nnetu et al. [32] pointed out that the velocity of epithelial MCF-10 A cells drops to zero at a cell packing density of \(\:\sim^\:\frac}}^}\), corresponding to the cell jamming. Petitjean et al. [45] revealed that the MDCK cell monolayers reached the confluence for a cell packing density of \(\:_\sim2.5x^\:\frac}}^}\) and a cell velocity of \(\:\sim0.14\:\frac\text}}\).

External compression of confluent MDCK cell monolayers with 28% strain caused an increase in cell packing density to \(\:1.39x_\), which stimulated the extrusion of live cells [4]. In this case, the corresponding fraction of extruded cells reached 6% [4]. Consequently, the cell packing density under jamming is higher than or equal to the cell packing density for the cell extrusion. A detailed description of the underlying physical mechanisms will be discussed in the next two sections.

The cell shear residual stress includes two parts. One part is generated by natural convection as a consequence of the existence of the interfacial tension gradient \(\:\overrightarrow_\), while the other part is generated by forced convection (i.e., by collective cell migration). The cell active/passive extension from the multicellular domains of lower interfacial tension towards the domains of higher interfacial tension is part of the Marangoni effect [46]. The phenomenon of cell movement along multicellular surfaces caused by the surface tension gradient has been confirmed experimentally by Gsell et al. [47]. The Marangoni effect has also been recognized in various soft matter systems under temperature or concentration gradients [48].

Consequently, the cell shear residual stress can be expressed as:

$$\:\overrightarrow}\cdot\:}}_\varvec\varvec}\cdot\:\overrightarrow}=\overrightarrow_\cdot\:\overrightarrow}+\overrightarrow}\cdot\:}}_\varvec\varvec}}^\cdot\:\overrightarrow}$$

(3)

where \(\:}}_\varvec\varvec}\) is the cell shear residual stress component, which is symmetric and satisfies the condition that the corresponding components are \(\:_=_\), \(\:}}_\varvec\varvec}}^\) is the cell shear residual stress generated by collective cell migration with the component \(\:_}^\), and \(\:\overrightarrow}\) is the tangent vector of the cell-matrix biointerface. The gradient of interfacial tension can be expressed as \(\:\frac_}\) (where \(\:\varDelta\:_\) is the interfacial tension difference and \(\:\varDelta\:L\) is the distance in which this gradient exist). If it is supposed that the interfacial tension difference corresponds to only \(\:\varDelta\:_\approx\:1\:\frac}}\) and the distance is \(\:\varDelta\:L\approx\:100\:\text\) (i.e., an order of magnitude larger than the size of single cell), this gradient of the interfacial tension corresponds to a cell shear stress part of \(\:\sim10\:\text\). This is a relatively large value, bearing in mind that a cell shear stress of a few tens of Pa can induce inflammation of epithelial cells [49].

The cell shear/normal residual stress caused by collective cell migration depends on the mechanism of cell migration, and on that basis, it depends on the cell packing density. Epithelial cell migration occurs via: (1) the convective mechanism for the cell packing density \(\:_\le\:_\), (2) the diffusion mechanism for the cell packing density \(\:_<n}_<_\), and (3) the sub-diffusion mechanism for the cell packing density \(\:_\sim_\) [3, 50]. Constitutive models proposed for various modes of epithelial cell migration are shown in Table 1.

Table 1 Some constitutive models proposed for various modes of epithelial cell migration

Serra-Picamal et al. [1] and Notbohm et al. [2] considered the rearrangement of MDCK cell monolayers with the cell packing density \(\:_\le\:_\) and revealed that the long-term cell stress (i.e., the cell residual stress) correlates with the corresponding strain, pointing out the viscoelastic solid behaviour. It is in accordance with the fact that epithelial cells establish strong E-cadherin mediated cell-cell adhesion contacts. Another important behaviour of epithelial monolayers, characteristic for this regime of cell packing density, is the ability of cell stress to relax towards the cell residual stress. Khalilgharibi et al. [52] reported that the stress relaxation time corresponds to a time-scale of minutes, while the cell residual stress accumulation occurs on a time-scale of hours [7]. The stress relaxation ability caused by uni-axial compression of cell aggregates was observed by Marmottant et al. [53]. Based on these findings, Pajic-Lijakovic and Milivojevic [7] concluded that cell stress change occurs through many short-time stress relaxation cycles, while cell strain (induced by cell movement) and corresponding cell residual stress change over a time scale of hours. A suitable constitutive model, satisfying the conditions (1) that the stress relaxes exponentially on a time scale of minutes and (2) that the cell residual stress correlates with the corresponding strain, pointing to long-term elastic behaviour, could be the Zener model presented in Table 1 [18]. In this case, energy dissipation, characteristic of the viscoelastic behaviour of multicellular systems, occurs on a time scale of minutes as a consequence of the remodelling of cell-cell adhesion contacts [3]. The cell stress relaxes towards the elastic cell residual stress. Cell residual stress, cell velocity and corresponding strain, oscillate on a time scale of hours, which has been discussed in the context of mechanical waves [1, 2, 7]. In this case, the cell actual stress can be expressed as: \(\:}}_\varvec}\left(r,t,\tau\:\right)=}}_\varvec\varvec}\left(r,\tau\:\right)+\varDelta\:}}_\varvec}}^\varvec\varvec\:}\left(r,t,\tau\:\right)\) (where \(\:i\equiv\:V,S\) is the subscript in Eqs. 2 and 3, which indicates normal and shear stress components, \(\:}}_\varvec\varvec}\left(r,\tau\:\right)\) is the cell residual stress (normal and shear) expressed by Eqs. 2 and 3, \(\:\varDelta\:}}_\varvec}}^\varvec\varvec\:}\) is an increment of the actual cell stress change during a single short-time stress relaxation cycle equal to \(\:\varDelta\:}}_\varvec}}^\varvec\varvec\:}\left(r,t,\tau\:\right)=}}_\varvec}}^\varvec\varvec\:}\left(r,t,\tau\:\right)-}}_\varvec\varvec}\left(r,\tau\:\right)\) and \(\:}}_\varvec}}^\varvec\varvec\:}\left(r,t,\tau\:\right)\) is the part of actual stress caused by collective cell migration, which is expressed by the Zener model and presented in Table 1 for the cell packing density \(\:_\le\:_\)).

Further increase in the cell packing density, in the range of \(\:_<n}_<_\), results in suppression of the cell stress relaxation. Cell-cell frictional effects, characteristic of higher cell packing densities, lead to a long term dissipation of energy during cell rearrangement. In accordance with the fact that a linear, diffusion mechanism underlies cell movement, the corresponding constitutive model should also be linear. Pajic-Lijakovic and Milivojevic [18] proposed the Kelvin-Voigt constitutive model for this regime (Table 1). Corresponding long-term changes in cell stress account for both the elastic and viscous contributions. In this case, cell actual stress is equal to cell residual stress.

While the viscoelasticity of epithelial monolayers shows linear behaviour for cell packing densities \(\:_<_\), the damped cell movement, described by the sub-diffusion mechanism, induces nonlinearity in the viscoelastic behaviour. For describing the damped movement of cells at homeostasis, it is necessary to use fractional derivatives. Pajic-Lijakovic and Milivojevic [50] proposed the fractional stress-strain model for this regime of viscoelasticity (Table 1). In this case too, cell actual stress is equal to cell residual stress.

Cell actomyosin contractility has two main effects on cellular behaviour. First, it enhances the strength of E-cadherin adhesion contacts, which in turn affects the surface tension of the epithelial monolayer. Secondly, it induces cell tractions, which then influence the surface tension of the extracellular matrix and the energy of epithelial-matrix adhesion. As a result, cell contractility plays a crucial role in determining the interfacial tension between the epithelial monolayer and the matrix, as well as the gradient of this tension. This, in turn, affects the cell mechanical stress. The impact of cell contractility on the strength of cell-cell adhesion contacts also has implications for the viscoelasticity of the epithelial monolayer. Active (contractile) cells exhibit greater stiffness compared to non-contractile cells, primarily due to the accumulation of contractile energy. Research by Schulze et al. [54] has shown that the Young’s modulus of contractile MDCK cell monolayers is approximately 33.0 ± 3.0 kPa, while non-contractile cells have a modulus that is roughly half of this value. Furthermore, active wetting and de-wetting processes lead to the generation of higher cell residual stress compared to passive wetting and de-wetting under the same strain conditions. It is important to note that cell contractility affects all physical parameters involved in the generation of cell mechanical stress, making it impossible to separate the stress into active and passive contributions.

In summary, cell actomyosin contractility plays a significant role in modulating the behaviour of epithelial cells. Its effects on adhesion contacts, surface tension, and mechanical stress have important implications for the overall mechanical properties and behaviour of epithelial monolayers. Cell mechanical stress, generated during cell active wetting/de-wetting, can induce the formation of the topological defects in cell alignment which, has a feedback impact on cell rearrangement.