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Table 1 Viscoelasticity of migrating epithelial collectives

From: Role of viscoelasticity in the appearance of low-Reynolds turbulence: considerations for modelling

Characteristics of cell movement

Cell packing density

Cell speed

Constitutive model

Convective cell migration

\({n}_{e}\le {n}_{conf}\)

\(0.1 <\Vert {\overrightarrow{{\varvec{v}}}}_{{\varvec{e}}}\Vert <\sim 1 \frac{\mu m}{min}\)

The Zener model for viscoelastic solids:

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,t,\tau \right)+{\tau }_{Rck} {{\dot{\widetilde{{\varvec{\sigma}}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}={E}_{ck}{\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{e}}{\varvec{k}}}\left(r,\tau \right)+{\eta }_{ck}{\dot{\widetilde{{\varvec{\varepsilon}}}}}_{{\varvec{e}}{\varvec{k}}}\)

Stress relaxation under constant strain condition \({\widetilde{{\varvec{\varepsilon}}}}_{0{\varvec{c}}{\varvec{k}}}\) per single short-time relaxation cycle:

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,t,\tau \right)={\widetilde{{\varvec{\sigma}}}}_{0{\varvec{e}}{\varvec{k}}}{e}^{-\frac{t}{{\tau }_{Rck}}}+{{\widetilde{{\varvec{\sigma}}}}_{{\varvec{r}}{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,\tau \right)\left(1-{e}^{-\frac{t}{{\tau }_{Rck}}}\right)\)

Cell residual stress is elastic.

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{r}}{\varvec{e}}{\varvec{k}}}}^{CCM}={E}_{ck} {\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{e}}{\varvec{k}}}\)

Conductive (diffusion) cell migration

\({n}_{j}>{n}_{e}>{n}_{conf}\)

\({n}_{j}\) is the cell packing density at the jamming state

\(\Vert {\overrightarrow{{\varvec{v}}}}_{{\varvec{e}}}\Vert \sim {10}^{-3}-{10}^{-2}\frac{\mu m}{min}\)

The Kelvin-Voigt model for viscoelastic solids:

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,\tau \right)={E}_{ck}{\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{e}}{\varvec{k}}}+{{\eta }_{ck} \dot{\widetilde{{\varvec{\varepsilon}}}}}_{{\varvec{e}}{\varvec{k}}}\)

The stress cannot relax.

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}={{\widetilde{{\varvec{\sigma}}}}_{{\varvec{r}}{\varvec{e}}{\varvec{k}}}}^{CCM}\)

A long-time change of the stress accounts for elastic and viscous contributions.

Damped conductive (sub-diffusion) cell migration near the cell jamming

\({n}_{e}\to {n}_{j}\)

\(\Vert {\overrightarrow{{\varvec{v}}}}_{{\varvec{e}}}\Vert \to 0\)

The Fraction model for the jamming state for viscoelastic solids:

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,\tau \right)={\upeta }_{\alpha k}{D}^{\alpha }\left({\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{e}}{\varvec{k}}}\right)\)

For \(0<\alpha <1/2\)

The stress cannot relax.

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}={{\widetilde{{\varvec{\sigma}}}}_{{\varvec{r}}{\varvec{e}}{\varvec{k}}}}^{CCM}\)

Convective cell migration after epithelial-to-mesenchymal cell state transition

\({n}_{e}\le {n}_{conf}\)

\(\Vert {\overrightarrow{{\varvec{v}}}}_{{\varvec{e}}}\Vert \ge 1 \frac{\mu m}{min}\)

The Maxwell model for viscoelastic liquids:

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,t,\tau \right)+{\tau }_{Rck}{{\dot{\widetilde{{\varvec{\sigma}}}}}_{{\varvec{e}}{\varvec{k}}}}^{{\varvec{C}}{\varvec{C}}{\varvec{M}}}={\upeta }_{ck}{\dot{\widetilde{{\varvec{\varepsilon}}}}}_{{\varvec{e}}{\varvec{k}}}\left(r,\tau \right)\)

Stress relaxation under constant strain rate \({\dot{\widetilde{{\varvec{\varepsilon}}}}}_{0{\varvec{c}}{\varvec{k}}}\) per single short-time relaxation cycle:

\({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,t,\tau \right)={\widetilde{{\varvec{\sigma}}}}_{0{\varvec{e}}{\varvec{k}}}{e}^{-\frac{t}{{\tau }_{Rck}}}+{{\widetilde{{\varvec{\sigma}}}}_{{\varvec{r}}{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,\tau \right)\left(1-{e}^{-\frac{t}{{\tau }_{Rck}}}\right)\)

Cell residual stress is purely dissipative.

\({{{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}=\eta }_{{\varvec{c}}{\varvec{k}}}{\dot{\widetilde{{\varvec{\varepsilon}}}}}_{{\varvec{e}}{\varvec{k}}}\)

  1. where \(k\equiv S,V\), \(S\) is shear, \(V\) is volumetric, \({\tau }_{Rck}\) is the cell stress relaxation time, \({E}_{ck}\) is the elastic modulus, \({\upeta }_{ck}\) is the cell viscosity (shear or bulk), \(r\) is the space coordinate, \(t\) is a short-time scale (i.e. minutes), \(\tau\) is a long-time-scale (i.e. hours), \(\Vert {\overrightarrow{{\varvec{v}}}}_{{\varvec{e}}}\Vert\) is the cell speed, \({\overrightarrow{{\varvec{v}}}}_{{\varvec{e}}}\) is the cell velocity equal to \({\overrightarrow{{\varvec{v}}}}_{{\varvec{e}}}=\frac{{\varvec{d}}\overrightarrow{{\varvec{u}}}}{{\varvec{d}}{\varvec{\tau}}}\), \(\overrightarrow{{\varvec{u}}}\left(r,\tau \right)\) is the cell local displacement field, \({{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\left(r,t,\tau \right)\) is the cell stress (normal or shear), \({{\dot{\widetilde{{\varvec{\sigma}}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}\) is the rate of stress change \({{\dot{\widetilde{{\varvec{\sigma}}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}=\frac{d{{\widetilde{{\varvec{\sigma}}}}_{{\varvec{e}}{\varvec{k}}}}^{CCM}}{dt}\) caused by the stress relaxation, \({\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{c}}{\varvec{k}}}\) is the cell strain such that the volumetric strain is equal to \({\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{e}}{\varvec{V}}}\left(r,\tau \right)=\overrightarrow{(\nabla }\cdot \overrightarrow{{\varvec{u}}})\widetilde{{\varvec{I}}}\), \(\widetilde{{\varvec{I}}}\) is the unit tensor, and the shear strain \({\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{e}}{\varvec{S}}}\left(r,\tau \right)=\frac{1}{2}\left(\overrightarrow{\nabla }\overrightarrow{{\varvec{u}}}+{\overrightarrow{\nabla }\overrightarrow{{\varvec{u}}}}^{{\varvec{T}}}\right)\), \({\dot{\widetilde{{\varvec{\varepsilon}}}}}_{{\varvec{c}}{\varvec{k}}}\) is the corresponding strain rate equal to \({\dot{\widetilde{{\varvec{\varepsilon}}}}}_{{\varvec{e}}{\varvec{k}}}=\frac{d{\widetilde{{\varvec{\varepsilon}}}}_{{\varvec{e}}{\varvec{k}}}}{d\tau }\), \({\upeta }_{\alpha k}\) is the effective modulus, \({D}^{\alpha } \widetilde{{\varvec{\varepsilon}}}\left(r,\tau \right)=\frac{{d}^{\alpha }\widetilde{{\varvec{\varepsilon}}}\left(r,\tau \right)}{d{\tau }^{\alpha }}\) is the fractional derivative, and \(\mathrm{\alpha }\) gives the order of fractional derivatives (the damping coefficient). Caputo’s definition of the fractional derivative of a function \(\widetilde{{\varvec{\varepsilon}}}\left(r,\tau \right)\) is used and expressed as: \({D}^{\alpha }\widetilde{{\varvec{\varepsilon}}}=\frac{1}{\Gamma\left(1-\alpha \right)}\frac{d}{dt}{\int }_{0}^{t}\frac{\widetilde{{\varvec{\varepsilon}}}\left({r,\tau }{\prime}\right)}{{\left(\tau -\tau {\prime}\right)}^{\alpha }}d\tau {\prime}\) (where Г \(\left(1-\alpha \right)\) is a gamma function) [77]
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Journal of Biological Engineering

ISSN: 1754-1611