Maximal Fluctuations of Confined Actomyosin Gels: Dynamics of the Cell Nucleus

We investigate the effect of stress fluctuations on the stochastic dynamics of an inclusion embedded in a viscous gel. We show that, in nonequilibrium systems, stress fluctuations give rise to an effective attraction towards the boundaries of the confining domain, which is reminiscent of an active Casimir effect. We apply this generic result to the dynamics of deformations of the cell nucleus, and we demonstrate the appearance of a fluctuation maximum at a critical level of activity, in agreement with recent experiments [E. Makhija, D. S. Jokhun, and G. V. Shivashankar, Proc. Natl. Acad. Sci. U.S.A. 113, E32 (2016)].

Figure 1

(a),(b) Projected intensity of actin (phalloidin labeling) and of the nucleus (DNA staining using DAPI labeling) of cells placed on (a) circular and (b) rectangular patches of fibronectin (scale bar, $10\mu \mathrm{m}$-excerpt from Ref. [7]). (c),(d) Schematic representation of the cortex (green) within cells placed on patches either (c) circular ($500\mu {\mathrm{m}}^2$) or (d) rectangular ($1800\mu {\mathrm{m}}^2$). Physical links (blue and red) connect the nucleus (gray) to the actin meshwork (green). In (d), the nucleus is compressed by actin fibers (solid green line) anchored to the substrate through focal adhesions (brown triangles). (e),(f) Models considered here, with (e) a restoring force provided by the geometry of the active segments on an elastic inclusion maintained at a distance from the base and (f) a linear restoring force on a rigid inclusion.

Figure 2

(a) Nuclear fluctuations of the projected area as a function of the cell actin concentration (phalloidin label) normalized to 1 in the rectangular control case, estimated over time windows. The actin-disruption drug cytochalasin-D (cytoD) reduces the actin concentration. (b) Finite-time variance in the width of an elastic inclusion [defined in Eq. (13)] as a function of the cortical activity parameter $Z$, for two values of the normalized nuclear elasticity $B=2$ (blue circles) and $B=4$ (red triangles) with a model nuclear height $L_n=0.2L$. The observation time $\mathrm{\Delta }t=5t_u=5\times {10}^3\mathrm{s}$ is comparable to the duration of experiments in Ref. [7] (see Supplemental Material [22]). We model the effect of the drug cytoD though a reduction in the normalized activity parameter $Z$.

Figure 3

Probability distribution (a) for an active noise (solid blue line) and for a thermal noise (dashed red line) in the absence of an external force; (b) for an active noise and for various strengths of the restoring force $k=0.1$ (blue curve), $k=1$ (red curve), and $k=3$ (orange curve). (b, $xy$ plane) Typical trajectories over $\mathrm{\Delta }t=2$: For $k=0.1$ (blue), the inclusion is confined to the edges, while for a large restoring force $k=3$ (orange), it is confined to the center. For $k=1$ (red), the particle exhibits the largest displacements over the observation time $\mathrm{\Delta }t$.

Figure 4

(a) Probability map $P(X_1,X_2)$ for the geometry depicted in Fig. 1; the active stress is $Z=3.5$, the elastic modulus is $B=5$, and the nuclear height is $L_n=0$ (respectively, 0.2) in the upper (respectively, lower) half. Activity tends to confine the nucleus to the edges of the gel, while the nuclear height focuses the nucleus to the cell center. (b) Surface of the optimal activity level $Z_{\mathrm{opt}}$, colored by the value of the maximal $\sqrt{A_{\mathrm{\Delta }t}(Y)}$. The red line corresponds to the set of parameters used in Fig. 2.