Onset of Mechanochemical Pattern Formation in Poroviscoelastic Models of Active Cytoplasm

Abstract

The cytoplasm of living cells is a complex structure formed by a fluid phase composed by water and small molecules and a second phase composed by the filaments of the cytoskeleton forming a viscoelastic gel. The interaction between the two phases gives rise to a poroviscoelastic structure which combines elastic and viscous responses to external stimuli. On the other hand, the cytoplasm is active, and molecular motors perform active stress. Different molecules regulate the activity of the motors. In the cytoplasm the biochemistry and the mechanics are interconnected, while motors and biochemical regulators are transported by flows in the two phases, the motors produce active stresses into the cytoskeleton and generate active flows. Here, we compare two active poroviscoelastic models with different viscoelastic properties, which can produce oscillations and the polarization of a living cell. The main features of the different mechanisms of pattern formation are studied by linear stability analysis of the homogeneous steady state and by numerical simulations.

Appendix

For linear viscoelastic models the strain $\varepsilon$ is the deformation by unit of length of the material and the stress $\sigma$ is the tension of the material under deformation. For a solid, both quantities are linearly related $\sigma _s = E \varepsilon _s$, by the elastic modulus E. Such dependence is equivalent to the Hooke law, therefore, the spring is a good phenomenological model for a solid. For a fluid the stress is only proportional to the velocity and, therefore, to the rate of change of the strain $\sigma _{f} = \eta \partial _t \varepsilon _f$, where the constant of proportionality is the viscosity or viscous damp $\eta$.

The strain $\varepsilon$ corresponds to an infinitesimal change of the deformation field u and therefore $\varepsilon = \partial _x u$ in the continuous models used in the chapter.

1.1 Stress-Strain Relation for the Kelvin–Voigt Model

For the Kelvin–Voigt model, a spring and a dampshot are coupled in parallel, see Fig. 2a. Strains of both branches are the same $\varepsilon = \varepsilon _s = \varepsilon _f$ and the total stress $\sigma$ is the addition of both stresses:

$$\begin{aligned} \sigma =\sigma _{s} + \sigma _{f} = E \varepsilon + \eta \partial _t \varepsilon ; \end{aligned}$$

(24)

which, for constant stress $\sigma =\sigma _o$ gives rise to a saturation of the strain with a characteristic time $\tau =\eta /E$:

$$\begin{aligned} \varepsilon (t) = \frac{\sigma _o}{E} \left( 1- e^{-t/\tau } \right) ; \end{aligned}$$

(25)

which produces to the dynamics shown in Fig. 2a.

1.2 Stress-Strain Relation for the Maxwell Model

For the Maxwell model, a spring and a dampshot are coupled in series, see Fig. 2b. While the stress is the same along the branch $\sigma = \sigma _s = \sigma _{f}$, the individual displacements of the spring and the dashpot may be different. The total strain is the addition of both strains $\varepsilon =\varepsilon _s + \varepsilon _f$, and its derivative gives rise to the relation $\partial _t \varepsilon =\partial _t \varepsilon _s + \partial _t \varepsilon _f$. The stress-strain relation can be then obtained:

$$\begin{aligned} \frac{1}{\eta } \sigma + \frac{1}{E} \partial _t \sigma = \partial _t \varepsilon ; \end{aligned}$$

(26)

which, in the case of continuous rate of change of strain $\partial _t \varepsilon ={\dot{\varepsilon }}_o$ produces a saturation of the stress with a characteristic time $\tau =\eta /E$:

$$\begin{aligned} \sigma (t) = \eta {\dot{\varepsilon }}_o \left( 1- e^{-t/\tau } \right) ; \end{aligned}$$

(27)

which produces to the dynamics shown in Fig. 2b.