Soft inclusion in a confined fluctuating active gel

We study stochastic dynamics of a point and extended inclusion within a one-dimensional confined active viscoelastic gel. We show that the dynamics of a point inclusion can be described by a Langevin equation with a confining potential and multiplicative noise. Using a systematic adiabatic elimination over the fast variables, we arrive at an overdamped equation with a proper definition of the multiplicative noise. To highlight various features and to appeal to different biological contexts, we treat the inclusion in turn as a rigid extended element, an elastic element, and a viscoelastic (Kelvin-Voigt) element. The dynamics for the shape and position of the extended inclusion can be described by coupled Langevin equations. Deriving exact expressions for the corresponding steady-state probability distributions, we find that the active noise induces an attraction to the edges of the confining domain. In the presence of a competing centering force, we find that the shape of the probability distribution exhibits a sharp transition upon varying the amplitude of the active noise. Our results could help understanding the positioning and deformability of biological inclusions, e.g., organelles in cells, or nucleus and cells within tissues.

Figure 1

(a) Schematic of the cell nucleus embedded in an active cytoplasm and confined by actin stress fibers. Following Ref. [21], we model the nucleus as an inclusion embedded in an active fluid confined by the cell boundary. To address various possible contexts, we treat the inclusion in turn as (b) a passive rigid element (Sec. 4), (c) an elastic element (Sec. 5), and (d) a viscoelastic (Kelvin-Voigt) element (Sec. 6).

Figure 2

Steady-state probability distribution $P\left(x_{\text{cm}}\right)$ for the inclusion center of mass (COM) $x_{\text{cm}}$ as a function of (a) the dimensionless stiffness of the confining potential $\left(k{L}^2{\eta }_c\right)/\left(2{\mathrm{\Lambda }}_c\right)$ and (b) of the dimensionless boundary flow $\left(v_0{\eta }_c^2\right)/\left(2{\mathrm{\Lambda }}_c\right)$. (a) For an active noise $(\alpha =1/2$; indicated by a solid red line), we distinguish between three regimes of (i) boundary adhesion, (ii) boundary adhesion with metastable centering, and (iii) stable centering. The existence of the boundary adhesion phase in the active situation contrasts with the two phases found in the thermal situation $(\alpha =1$, dashed black line): a uniform distribution and a stable centering. (b) The active case, $\alpha =1/2$ (solid red line), shows a transition from (i) boundary adhesion to (iii) stable centering via (ii) a uniform distribution at 0.5.

Figure 3

Variance of the dimensionless COM position $\sqrt{\langle x_{\text{cm}}^2\rangle }$ for thermal (dashed black line) and active (solid red line) noises, with $v_0=0$ as a function of (a) system size $L$, where it goes from being strongly affected by the boundary, when $L$ is small, to reaching a limiting value determined by the confining strength for large $L$; and (b) confining strength $\left(k{L}^2{\eta }_c\right)/\left(2{\mathrm{\Lambda }}_c\right)$, where it asymptotes to a fixed value for small stiffness and goes to zero for large stiffness.

Figure 4

Marginal distribution of the center of mass (COM, $z)$ and phase plot of most likely COM position, as a function of the dimensionless elasticity of the inclusion $\left(b=2lB{\eta }_c/{\mathrm{\Lambda }}_c\right)$ and the strength of the confinement $\left(k{\eta }_c/4{\mathrm{\Lambda }}_c\right)$, for active and thermal fluctuations and parameter values $2L=3,2l=1$, and $\zeta \mathrm{\Delta }\mu c_0{\eta }_c/{\mathrm{\Lambda }}_c=-1$. (a–c) Marginal distribution $P\left(z\right)$ of the COM, for active (solid red lines), and thermal (dotted black lines), for $2lB{\eta }_c/{\mathrm{\Lambda }}_c=50$ in the decentered phase I (a), in the boundary phase (b), and in the centered phase II (c). (d) Phase plot showing the most likely COM position for active fluctuations in the $(B,k)$ parameter plane. Different phases correspond to the inclusion being decentred and extended $\left(\langle y\rangle >2l\right)$ (I), centered and compressed $\left(y<2l\right)$(II), and centered and extended (III). Contrast this with the (e) phase plot of the most likely COM position for thermal fluctuations, which has only two phases, where the inclusion is centered and compressed $\left(y<2l\right)$(II) and centered and extended (III).

Figure 5

Moments as a function of the normalized elastic modulus $b$: (a) mean and (b) standard deviation of the inclusion length $y$, and (c) variance of the inclusion COM position $z$. Plotted for two sets of parameters $L/l=1.6$ (black curve), and $L/l=3$ (red curve), where we have fixed $y_0=2l=1$, for both thermal fluctuation (dotted line), and active fluctuation (solid line). Thermal fluctuations exhibit reduced moments compared to the active, for both sets of parameters. For large $b$, we recover the rigid inclusion moments (see Sec. 4), while for small $b$ the moments converge to the calculated asymptotic limit (see Sec. 5).

Figure 6

(a) Contour plot of the mean inclusion size $\langle y\rangle$ as function of the stress ratio $b$ and the viscosity ratio $r$, keeping the other parameters fixed (see text). (b, c) Mean inclusion size $\langle y\rangle$ versus $r$ and $b$, respectively.

Figure 7

(a) Contour plot of the standard deviation of inclusion size $\sqrt{\langle y^2\rangle -{\langle y\rangle }^2}$ as function of the stress ratio $b$ and the viscosity ratio $r$, all other parameters are fixed. (b, c) Standard deviation of inclusion size $y$ versus $r$ and $b$, respectively.

Figure 8

(a) Contour plot of the standard deviation of center of mass position of the inclusion $\sqrt{\langle z^2\rangle }$ as function of the stress ratio $b$ and the viscosity ratio $r$, all other parameters fixed. (b, c) Mean center of mass position $\sqrt{\langle z^2\rangle }$ versus $r$ and $b$, respectively.

Figure 9

Average stress for an elastic inclusion embedded in a viscous fluid subject to thermal (dotted line) and active (solid line) noise, for $2l=1,2L=2$ (black), and $2L=3$ (red) as function of $b$, the ratio of the elastic to the fluctuating stress. In (a) the stress is expressed in units of the elastic stress $\langle \sigma \rangle /2lB$ and plotted as a function of ${\eta }_c/{\mathrm{\Lambda }}_c$, for fixed $B$, while in (b) the stress is in units of the fluctuation stress $\langle \sigma \rangle {\eta }_c/{\mathrm{\Lambda }}_c$ and plotted as a function of $B$, for fixed ${\eta }_c/{\mathrm{\Lambda }}_c$.