Active matter in a viscoelastic environment

Active matter systems such as eukaryotic cells and bacteria continuously transform chemical energy to motion. Hence living systems exert active stresses on the complex environments in which they reside. One recurring aspect of this complexity is the viscoelasticity of the medium surrounding living systems: Bacteria secrete their own viscoelastic extracellular matrix, and cells constantly deform, proliferate, and self-propel within viscoelastic networks of collagen. It is therefore imperative to understand how active matter modifies, and gets modified by, viscoelastic fluids. Here we present a two-phase model of active nematic matter that dynamically interacts with a passive viscoelastic polymeric phase and perform numerical simulations in two dimensions to illustrate its applicability. Motivated by recent experiments we first study the suppression of cell division by a viscoelastic medium surrounding the cell. We further show that the self-propulsion of a model keratocyte cell is modified by the polymer relaxation of the surrounding viscoelastic fluid in a nonuniform manner and find that increasing polymer viscosity effectively suppresses the cell motility. Last, we explore the hampering impact of the viscoelastic medium on the generic hydrodynamic instabilities of active nematics by simulating the dynamics of an active stripe within a polymeric fluid. The model presented here can provide a framework for investigating more complex dynamics such as the interaction of multicellular growing systems with viscoelastic environments.

Figure 1

A schematic of various interpretations of the two-phase active-viscoelastic model.

Figure 2

Simulation snapshots of a cell undergoing division for the ${\tau }_C$ case (first two columns), the no-polymer case (third column), and the ${\nu }_C$ case (last three columns).

Figure 3

Typical flow field of a cell elongating during the initial steps of cell division. Velocity vectors are shown as red arrows, and polymer deformations $\mathbf{C}-\mathbf{I}$ are shown as blue line segments. Inset: Zoom of the lower left part of the cell, showing polymers aligning with the stretching direction. Polymer deformations and velocities are rescaled by 150% for visibility.

Figure 4

Total time for cell division in a viscoelastic fluid $t_c$, normalized by the total time to divide in the no-polymer case $t_0$, for (a) the ${\tau }_C$ case and (b) the ${\nu }_C$ case.

Figure 5

Simulation snapshots of a moving model keratocyte cell for the ${\tau }_C$ case (first two columns), the no-polymer case (third column), and the ${\nu }_C$ case (last three columns). Snapshots were taken at times $t=0$ (top, very light blue), $t=3.5\times {10}^5$ (middle, light blue), and $t=7\times {10}^5$ (bottom, blue) after initialization.

Figure 6

Mean vertical velocity of the center of mass of the model keratocyte cell in a viscoelastic fluid $v$, normalized by the mean velocity in a Newtonian fluid $v_0$ for (a) the ${\tau }_C$ case and (b) the ${\nu }_C$ case. Error bars represent 1 standard deviation of averaged velocities sampled 60 times over $1.2\times {10}^6$ LB time steps.

Figure 7

Mean magnitude of the principal polymer stress for a model keratocyte cell propelling through a viscoelastic fluid for (a) the ${\tau }_C$ case and (b) the ${\nu }_C$ case. Error bars represent 1 standard deviation calculated as in Fig. 6.

Figure 8

Average polymer extension around the active phase, quantified via averaging $Tr\left[\mathbf{C}-\mathbf{I}\right]$ over time and space, divided by the polymer relaxation time ${\tau }_C$, as a function of ${\tau }_C$.

Figure 9

Simulation snapshots of an active stripe developing deformations for the ${\tau }_C$ case (first two columns), the no-polymer case (third column), and the ${\nu }_C$ case (last three columns).

Figure 10

Perimeter of the active stripe in a viscoelastic fluid $P_C$, normalized by the perimeter in the no-polymer case $P_0$ at the same time for (a) the ${\tau }_C$ case and (b) the ${\nu }_C$ case. Different curves refer to different points in time $t$.

Figure 11

Time series of the average polymer stress for an active stripe within a viscoelastic fluid for different values of polymer viscosity ${\nu }_C$.