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Edited by: Aljaz Godec, Max Planck Institute for Biophysical Chemistry, Germany

Reviewed by: Claudia Tanja Mierke, Leipzig University, Germany; Andrew Clark, Institut Curie, France

This article was submitted to Biophysics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Various types of mechanical waves, such as propagative waves and standing waves, are observed during 2D collective cell migration. Propagative waves are generated during monolayer free expansion, whereas standing waves are generated during swirling motion of a confluent monolayer. Significant attempts have been made to describe the main characteristics of mechanical waves obtained within various experimental systems. However, much less attention is paid to correlate the viscoelasticity with the generated oscillatory instabilities. Mechanical waves have recognized during flow of various viscoelastic systems under low Reynolds number and called “the elastic turbulence.” In addition to Reynolds number, Weissenberg number is needed for characterizing the elastic turbulence. The viscoelastic resistive force generated during collective cell migration caused by a residual stress accumulation is capable of inducing apparent inertial effects by balancing with other forces such as the surface tension force, the traction force, and the resultant force responsible for cell migration. The resultant force represents a product of various biochemical processes such as cell signaling and gene expression. The force balance induces (1) forward flow and backward flow in the direction of cell migration as characteristics of the propagative waves and (2) inflow and outflow perpendicular to the direction of migration as characteristics of the standing waves. The apparent inertial effects are essential for appearing the elastic turbulence and represent the characteristic of (1) the backward flow during the monolayer free expansion and (2) the inflow during the cell swirling motion within a confluent monolayer.

Collective cell migration within a monolayer induces spontaneous generation of mechanical waves [

Oscillator, wave-like motion of multicellular systems has been related to long-time effective inertia [

Significant attempts have been made to describe the main characteristics of mechanical waves by considering various types of experimental systems. The main characteristics of standing waves are (1) the radial velocity and cell tractions are uncorrelated; (2) radial stress component σ_{crr} and the corresponding strain rate _{cxx} and corresponding strain rate _{e} number, i.e., high velocity and low viscosity. However, viscoelastic systems have a few properties that distinguish them from Newtonian fluids [_{e} → 0, i.e., low velocity and high viscosity [_{R} is the stress relaxation time). For the case of polymer solutions, this elastic turbulence is accompanied by stretching of polymer chains resulting in significant system stiffening. The system stiffening is caused by residual stress accumulation, which leads to sharp growth of the flow resistance. Groisman and Steinberg [_{e} number. Steinberg [_{i}> 1 and vanishingly small _{e}≪1 number. Larson et al. [_{R} and the viscous diffusion time _{v}, i.e., _{e} → 0 which lead to inflow and outflow. The inflow for _{r}< 0 (where _{r} is the radial velocity component) corresponds to the compression and the outflow for _{r}> 0 corresponds to extension. Multicellular systems are much complex than polymer solutions, but their viscoelastic nature significantly influences cell rearrangement and should not be neglected [_{e} number flow. However, apparent inertial effects could represent a consequence of inflow during cell swirling motion and backward flow during monolayer expansion caused by residual stress accumulation. The aim of this work is to relate viscoelastic nature of cell monolayer during collective cell migration with generated standing and propagative mechanical waves. Consequently, it is necessary to (1) postulate viscoelastic constitutive model for cell monolayer and extracellular matrix; (2) describe cell packing density change, matrix density change, and their interrelation (needed for the description of volume force balance); and (3) formulate the volume force balance that drives cell rearrangement by accounting for two time scales, i.e., a time scale of minutes (for the stress relaxation) and time scale of hours (for collective cell migration, strain change, and residual stress accumulation).

Flow of viscoelastic systems should be characterized by two dimensionless numbers: (1) Reynolds number _{e} and Weissenberg number _{i} as well as (2) their ratio _{i}> 1 and vanishingly small _{e} → 0 number [_{e} and _{i} can be estimated based on experimental data from the literature such as (1) the cell velocity _{x}Δτ (where Δτ is the period of oscillation equal to Δτ ≈ 4 − 6 ^{5} _{R} = 3 − 14 min [_{i}, we formulated the effective value of the Weissenberg number, which accounts for the characteristic time for the residual stress accumulation equal to _{i eff}~ 0.3, while ^{14}. Groisman and Steinberg [_{e} = 0.3 and _{i} = 3.5. Generation of oscillatory instabilities in 2D collective cell migration has been experimentally confirmed [

Cell long-time rearrangement caused by collective cell migration should be discussed based on formulated interrelations between various variables such as (1) cell velocity _{st} and (2) cell packing density

The schematic representation of interrelations between main model parameters that influence the generation of mechanical waves during (1) monolayer free expansion and (2) cell swirling motion within a confluent monolayer.

Consequently, the modeling consideration accounts for the following steps: (1) the expression of cell velocity

Viscoelasticity of multicellular systems caused by collective cell migration has been considered on two time scales [

Stress relaxation is primarily induced by adaptation of adhesion contacts and cell shapes [

Cell velocity

where

where

Stress relaxation ability under constant strain condition represents the characteristic of the viscoelastic solid rather than viscoelastic liquid. The Maxwell model suitable for viscoelastic liquid describes stress relaxation under constant strain rate [

where _{c} is the elastic modulus, and η is the viscosity. The relaxation of stress under constant strain condition

where

Notbohm et al. [_{crrR}. They pointed out that the long-time change of residual stress

Various hydrogel matrices have been used as a substrate for cell migration. The rheological behavior of hydrogels frequently corresponds to a poroviscoelasticity [

where _{m} is the free energy function that accounts for cell–matrix mechanical and electrostatic interactions, and

where

Cell migration speed, cell packing density, and correlation of cell migration depend on cell–matrix mechanical and electrostatic interactions, which influence the state of cell–matrix adhesion contacts and on that basis the state of single cells. Viscoelasticity of matrix influences accumulated stress within a monolayer and on that basis the correlation of cell migration [

where _{d} represents a measure of cell–matrix mechanical interactions, which influence the matrix displacement field _{sm} is the elastic shear modulus of a matrix. Besides of the matrix viscoelasticity, the matrix density is also influenced by cell–matrix interaction. Long-time change of the matrix density has the feedback impact on the packing density of cells as well. The phenomenon can be expressed in the form of haptotaxis flux [

where _{h} is the measure of cell–matrix interactions which influences the matrix density ρ. Change of the matrix density ρ caused by cell tractions has been described by Murray et al. [

where

where _{g} is the measure of cell–matrix electrostatic interactions, and ϕ_{e} is the local electrostatic potential.

Cell packing density change

where _{eff} is the effective diffusion coefficient, and _{e} is the electrostatic potential for the galvanotaxis, ϕ ≡ _{sm} is the local shear modulus of a matrix for the durotaxis, ϕ ≡ _{sc} is the local shear modulus of cells for plitotaxis, whereas _{i} is the model parameter that accounts for various types of interactions such as mechanical, electrostatic, or chemical. Conductive flux accounts for cell response to a local variation of cell density. Haptotaxis, durotaxis, and galvanotaxis fluxes account for cell–matrix mechanical and electrostatic interactions. The tractions exerted by cells on the matrix generate gradients in (1) the matrix density and correspondingly the haptotaxis flux, (2) the electrostatic potential and the galvanotaxis flux, and (3) the matrix stiffness and the durotaxis flux [

if

if

where _{C} is the period of long-time oscillations equal to _{C} ≈ 4 − 6 _{max} is the maximum velocity correlation length. As the cell packing density increases and cells become more dense and slow down their movement, the correlation length first increases to ~10-cell lengths or _{max} ~ 150 μ

Cell long-time rearrangement is described by interrelation between the following variables such as (1) the displacement of cells

The force balance is responsible for oscillatory patterning the cell long-time rearrangement. Murray et al. [_{e} number is low. The force balance proposed by Murray et al. [

where _{st} is the surface tension,

where

The free expansion of a cell monolayer has been considered in 2D by using Cartesian coordinates such that _{x} and _{y} are the velocity components) [

For

For

The monolayer expansion occurs in two opposite directions, i.e., _{cxx} and corresponding strain rate

The schematic presentation of

Serra-Picamal et al. [_{cx} is the x-component of cell velocity) [

The initial unlimited forward flow of cellular domains leads to their extension. When the extension becomes significant, it induces (1) an increase in the resistive force

This state corresponds to the limited forward flow. Cell domains that correspond to the unlimited forward flow conditions are softer than those related to the limited forward flow conditions due to an accumulation of the cell residual stress. The local stiffening of the monolayer, which corresponds to the limited forward flow regime, is induced by the extension of adhesion contacts and force-induced repolarization (FIR) [

Collective cell migration within a confluent monolayer leads to cell swirling motion [_{cr} is the radial component and _{cθ} is the azimuthal component of velocity) [_{cr} > 0. The centrifugal force decreases with an increase in

The corresponding momentum balance can be expressed as follows:

For

For θ-direction:

where the internal centrifugal is equal to _{CL} acts to reinforce the radial flow [

The standing waves represent a characteristic of local cell rearrangement, which leads to swirling motion. The main characteristics of standing waves are (1) the radial velocity and cell tractions are uncorrelated, (2) normal stress component σ_{crr} and the corresponding strain rate

Notbohm et al. [_{cr} simultaneously changes a direction every ~ 4_{cr} is approximately constant within domains Δ_{crr} ≈

Radial extension of swirl parts (the cell outflow for _{cr} > 0 and _{cθ}, as well as a decrease in the centrifugal force, which causes the cell radial inflow and consequently the local compression of swirl parts. The inflow is characterized by the radial component of velocity such that _{cr} < 0 and _{cr} > 0. The centrifugal force is larger in the swirl core region in comparison with the peripheral region. Consequently, the inflow and outflow events are more intensive in the core region as was experimentally observed by Notbohm et al. [

The aim of this theoretical consideration is to emphasize the role of viscoelasticity in provoking apparent inertial effects and generating the oscillatory mechanical instabilities in the form of standing waves and propagative waves within multicellular systems caused by collective cell migration. The propagative waves are generated during monolayer expansion, while the standing waves are generated during the cell swirling motion within a confluent monolayer. These flow instabilities represent a characteristic of the elastic turbulence that occurs under low _{e} number. The phenomenon has been experimentally confirmed during flow of various viscoelastic systems such as polymer liquids [

Multicellular systems are much complex than polymer liquids, but their viscoelastic nature significantly influences characteristics of collective cell migration [

Generation of mechanical waves caused by collective cell migration accounts for cause–consequence relations between (1) cell packing density

A long-time cell rearrangement during monolayer expansion is accomplished by local forward flow and backward flow. The forward flow is divided into two regimens unlimited forward flow and limited forward flow. The limitations come from the action of the viscoelastic force against migration. The forward flow induces an accumulation of the extensional residual stress and an increase in the resistive viscoelastic force, which leads to the stiffening of monolayer parts and suppresses cell migration. The forward flow also induces an increase in cell displacement field and on that basis an increase in the surface tension force. Once the forward flow is suppressed, the surface tension force induces the backward flow, which leads to (1) a decrease in the surface tension force itself and (2) the softening of the monolayer part. The backward flow decreases rapidly because of (1) a decrease in the surface tension force and (2) collisions with surrounding domains under forward flow. This softening results in a decrease in the viscoelastic force. Lower values of the viscoelastic force as well as the surface tension force induce forward flow of the monolayer again. Those long-time cycles repeat many times in the form of the propagative waves [

A long-time cell rearrangement during the cell swirling motion (within a confluent monolayer) should be considered in the context of cell radial inflow and outflow. The confluence induces reduction of cell polarity alignment, which is essential for appearing cell swirls [

Both types of waves represent a consequence of apparent inertial effects. The apparent inertial effects are related to the periodic generation of (1) the backward flow during monolayer expansion and (2) the inflow during a cell swirling motion. The maximum velocity for (1) inflow and outflow is ^{2} [

The main difference between propagative waves generated during monolayer expansion and standing waves generated during cell swirling motion within a confluent monolayer is related to oscillatory stress change. Generated propagative waves induce damped oscillatory change of the extensional residual stress, whereas the compressive stress is not generated based on the experimental data by Serra-Picamal et al. [

Model developed can be applied to describe generation of mechanical waves within 3D multicellular systems. Standing waves as a characteristic of the cell swirling motion (1) can be generated during migration of strongly connected cell clusters through dense environment made by cells in passive (resting) state during a tissue development [

The rheological behavior of a matrix influences cell–cell and cell–matrix interactions and through the viscoelastic force influences the rate of cell spreading, as well as characteristics of generated mechanical waves.

Oscillatory instabilities in the form of mechanical waves are generated during collective cell migration such as propagative waves and standing waves. The propagative waves are generated during monolayer free expansion, whereas standing waves are generated during cell swirling motion of a confluent monolayer. Significant attempts have been made to describe the main characteristics of mechanic waves. The main characteristics of standing waves are (1) the radial velocity and cell tractions are uncorrelated; (2) radial stress component σ_{crr} and the corresponding strain rate _{cxx} and corresponding strain rate _{e} number. The phenomenon is called the elastic turbulence. The elastic turbulence has been quantified by the ratio between two dimensionless parameters such Weissenberg number and Reynolds number. These oscillatory flow instabilities have also been monitored experimentally during 2D collective cell migration.

Propagative waves represent the consequence of cell forward flow and cell backward flow during monolayer expansion driven by interrelation between forces such as the viscoelastic force, the traction force, and the surface tension force. These forces influence the rate of change of momentum and lead to periodic extensions in the direction of flow. Standing waves represent the consequence of cell radial inflow and outflow during swirling motion driven by the interrelation between the centrifugal force, the viscoelastic force, and the traction force, while the influence of the surface tension force can be neglected. This force balance leads to the periodic extension and compression in the direction perpendicular to flow. The apparent inertial effects represent the characteristic of (1) the backward flow during monolayer free expansion and (2) the inflow during the cell swirling motion within a confluent monolayer.

Additional experiments are necessary in order to determine a long-time constitutive model for 2D multicellular systems caused by collective cell migration and correlate the migrating patterns with the residual stress distribution and the rate of its change. Cell long-time rearrangement can be controlled by matrix viscoelasticity. This theoretical consideration could help in deeper understanding of various biological processes by which an organism develops its shape and heals wounds in the context of the mechanism underpinning the epithelial expansion.

IP-L: conceptualization, methodology, and writing—original draft preparation. MM: illustrations and writing—review and editing. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Authors would like to thank Dr. Giovanni Cappello for useful discussion which inspired this work.