An interpenetrating-network theory of cytoplasm

Abstract

Under many physiological and pathological conditions such as division and migration, cells undergo dramatic deformations, under which their mechanical integrity is supported by cytoskeletal networks (i.e. intermediate filaments, F-actin, and microtubules). Recent observations of cytoplasmic microstructure indicate interpenetration among different cytoskeletal networks, and micromechanical experiments have shown evidence of complex characteristics in the mechanical response of the interpenetrating cytoplasmic networks of living cells, including viscoelastic, nonlinear stiffening, microdamage, and healing characteristics. However, a theoretical framework describing such a response is missing, and thus it is not clear how different cytoskeletal networks with distinct mechanical properties come together to build the overall complex mechanical features of cytoplasm. In this work, we address this gap by developing a finite-deformation continuum-mechanical theory with a multi-branch visco-hyperelastic constitutive relation coupled with phase-field damage and healing. The proposed interpenetrating-network model elucidates the coupling among interpenetrating cytoskeletal components, and the roles of finite elasticity, viscoelastic relaxation, damage, and healing in the experimentally-observed mechanical response of interpenetrating-network eukaryotic cytoplasm.

FIG. B1.

(a) The computational domain and mesh used in our simulations, where the dotted line indicates the axis of rotational symmetry. (b) A cutaway view of the full 3-D domain, formed by a 180-degree rotation of the mesh about the axis of symmetry. A rigid displacement $U$ is applied to the micro-particle in the ${\mathit{e}}_2$-direction, and the resulting reaction force is measured.

FIG. 1.

Cytoplasm is an interpenetrating network. Confocal image of (a) vimentin intermediate filament, (b) microtubule, (c) F-actin, and (d) overlay of a mouse embryonic fibroblast [2]. (Scale bar, 5 μm). Schematics: (e) A cell. (f) The cytoplasm is supported by interpenetrating cytoskeletal fibers, which are intermediate filaments (green), F-actin (red), and microtubule (yellow). In a typical optical-tweezers measurement, a micro-size particle (grey) is perturbed within the cytoplasm. (g) Typical cyclic force-displacement response of vim only, vim−/−, and WT. Multiple loading-unloading cycles are applied by a laser-trapped particle. After 10 cycles, the particle stays in its original position for 10 minutes before another loading is applied.

FIG. 2.

(a) The force-displacement relation in the vimentin-only cells is elastic, while the force-displacement relation in the vim−/− cells is dissipative. Experimental data from [2]. (b)The displacement-magnitude field at the maximum bead displacement $(u=0.8\mu \text{m})$. The deformation is more diffuse in the interpenetrating network compared to a single non-equilibrium network. Nonlinear-stiffening network transduces long-ranged mechanical deformation inside the cytoplasm. The color map indicates the magnitude of displacement in the deformed configuration.

FIG. 3.

Damage and healing of interpenetrating-network cytoplasm in the WT cells the cytoplasm in the vimentin knock-out cells. Cyclic loading damages the viscoelastic network and reveals the elastic-stiffening network, in both (a) experiment and (b) theoretical prediction. The non-equilibrium network is damaged with 10 cycles of loading, while it almost fully heals in 10 minutes, in both (a) experiment and (b) theoretical prediction. 1st, 2nd, 3rd: the first, second and third loading-unloading cycle; 10th: the tenth loading; 10 min: loading in 10 minutes after the initial ten cycles are finished. Multiple-cycle damage of vim−/− cytoplasm in (c) experiment and (d) theoretical prediction. Experimental data from [2].