Mechanical Oscillations in 2D Collective Cell Migration: The Elastic Turbulence

Abstract

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.

Viscoelasticity and cell jamming state transition

“…Notbohm et al (2016) considered 2D cell swirling motion within a confluent monolayer by monitoring the maximum radial velocity of . They reported that the long-time change of radial cell residual stress component correlated well with the long-time strain change during the time period of 24 h. This result points out to the Zener constitutive model, suitable for viscoelastic solid, which was elaborated rheologicaly by Pajic-Lijakovic and Milivojevic (2020c).…”

“…A long-time change of cell packing density was expressed by modifying the model proposed by Murray et al (1988): where τ is the long-time scale, is the flux of cells equal to , such that is the convective flux, is cell velocity, is the conductive flux, D eff is the effective diffusion coefficient, while are the sum of various fluxes such as haptotaxis, galvanotaxis, chemotaxis, durotaxis and plithotaxis (Murray et al, 1988; Pajic-Lijakovic and Milivojevic, 2020c). All mentioned fluxes influences the rate .…”

“…The cell velocity at a mesoscopic level represents a rate of change the cell displacement field during CCM. For convective and conductive regimes it can be expressed as (Pajic-Lijakovic and Milivojevic, 2020c;2020b): Where is convective velocity or conductive velocity and is the local cell displacement field equal to (where is the displacement field of the j-th cell). The cell velocity for the damped-conductive regime can be expressed as: Where is the fractional derivative described by eq.…”

“…Consequently, five viscoelastic states gained within three viscoelastic regimes will be elaborated rheologicaly (Figure 3) by discussing the corresponding constitutive models such as: (1) the Maxwell model for a viscoelastic liquid (Guevorkian et al, 2011; Lee and Wolgemuth, 2011; Garcia et al, 2015) (for the convective regime:), (2) the Zener model for a viscoelastic solid (Pajic-Lijakovic and Milivojevic, 2020c) (for the convective regime:), and (3) the Kelvin-Voigt model constitutive models for a viscoelastic solid (Liu et al, 2017) (for the conductive regime:), and (4) the fractional constitutive model for transient and jamming viscoelastic states (Pajic-Lijakovic and Milivojevic, 2019c) (for the damped-conductive regime:).…”

“…Stress relaxation under constant strain condition per single short-time relaxation cycle can be expressed starting from the initial condition as: where is the residual shear or normal stress equal to . Notbohm et al (2016) reported that the long-time change of radial cell residual stress component correlated well with the long-time strain change which represents the confirmation of the viscoelastic solid behavior described by the Zener constitutive model σ (Pajic-Lijakovic and Milivojevic, 2020c). Complex modulus G * ( ω ) can be formulated after Fourier transform of eq.…”