Maximal fluctuations of confined actomyosin gels: dynamics of the cell nucleus

TL;DR

This paper explores how stress fluctuations influence the stochastic movement of the cell nucleus within a viscous gel, revealing an active Casimir-like attraction and a fluctuation maximum at a critical activity level, aligning with experimental observations.

Contribution

It demonstrates that stress fluctuations induce boundary attraction in non-equilibrium gels and identifies a fluctuation maximum in nuclear deformations at a specific activity level.

Findings

Stress fluctuations cause an effective attraction to boundaries.

A fluctuation maximum occurs at a critical activity level.

Results agree with recent experimental data.

Abstract

We investigate the effect of stress fluctuations on the stochastic dynamics of an inclusion embedded in a viscous gel. We show that, in non-equilibrium 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)].

Tables

Table 1. Table 1: List of notations and parameters estimate.

Quantity	Meaning	Value
$L$	Radius of the cell ($x$ direction)	$10\mu m$
$L_n$	Apical to basal height of the nucleus (circular patch - $z$ direction)	$3\mu m$
$l_n$	Transversal width of the nucleus (circular patch – $y$ direction)	$3\mu m$
$e_0$	Width of the cortex	$100\mathrm{nm}$
${\eta }_0$	Bulk viscosity of the cortex	${10}^5\mathrm{Pa}.s$ Joanny and Prost (2009)
$\zeta \Delta {\mu }_b$	Cortical active stress	${10}^3\mathrm{Pa}$ Joanny and Prost (2009)
${\tau }_A$	Correlation time, active fluctuations	${10}^{2}s$ Li et al. (2014)
${\tau }_v$	Maxwell time	$5s$ Wottawah et al. (2005)Saha et al. (2016)
$\eta =e_0{L}_n\left({\eta }_0-{\tau }_A\zeta \Delta {\mu }_b\right)$	One-dimensional viscosity in the presence of activity	$5\cdot {10}^{-7}N.s$
$\zeta \Delta \mu =\zeta \Delta {\mu }_b{e}_0{L}_n$	Active force	$0.5\mathrm{nN}$
$\delta e_0$	Typical fluctuating height of the apical cortex	$20\mathrm{nm}$
$\zeta \Delta {\mu }_b\delta e_0{L}_n$	Typical fluctuating active tension	$1\cdot {10}^{-10}N$
$\lambda$	Correlation length of the active noise	$100\mathrm{nm}$
${\Lambda }_A={\left(\zeta \Delta {\mu }_b\delta e_0{L}_n\right)}^2{\tau }_v\lambda$	Variance of the active noise	${10}^{-23}{N}^{2}ms$
$t_u={\eta }^{2}L/{\Lambda }_A$	Time unit; characteristic displacement time of the nucleus	${10}^{3}s\approx 15\min$
$D_{\mathrm{nucleus}}={\Lambda }_{A}L/{\eta }^2$	Typical diffusion coefficient of the nucleus	${10}^{-3}\mu m{s}^{-1}$ Radhakrishnan et al. (2017)
$E_0$	Nuclear elastic modulus probed at $1$Hz.	${10}^3\mathrm{Pa}$ Guilak et al. (2000); Lammerding (2011)
$E\left(10min\right)$	Nuclear elastic modulus probed over $10\min$ timescale	${10}^2\mathrm{Pa}$ Dahl et al. (2005)
$K$	Spring constant for the elastic centering force	${10}^{-4}N.m^{-1}$
$K{L}_n$	Elastic centering force	$0.5\mathrm{nN}$
$k=\left(KL{t}_u\right)/\left(2\eta \right)$	Elastic centering force (dimensionless)	$2$
$B=EL{L}_n{t}_u/\eta$	Elastic spring force (dimensionless)	$5$
$Z=\zeta \Delta \mu t_u/\eta$	Normalized activity (dimensionless)	$1$

Equations

$$({\tau }_v{\partial }_t+1)\sigma (x,t)=\eta {\partial }_{x}v(x,t)+\zeta \Delta \mu +{\theta }_A+{\varphi }_T,$$

$$({\tau }_v{\partial }_t+1){\sigma }^{(L)}$$

$${\tau }_v{\ddot{v}}_t+{\dot{v}}_t+\frac{\lambda (x_t)}{m}{v}_t=\frac{f+{\tau }_v\dot{f}+\mu (x_t)\left[{\Theta }_A+{\Phi }_T\right]}{m},$$

$${\mu }^2(x)=\frac{2L}{L^2-x^2}=\frac{\lambda (x)}{\eta }.$$

$$mv(t)={\int }_{-\infty }^{t}ds{\chi }_{x_t}^{(t-s)}\left\{f+\mu (x)\left[{\Theta }_A+{\Phi }_T\right]\right\},$$

$${\chi }_{x_t}^{(t-s)}=\sqrt{\frac{{\tau }_m}{{\tau }_v}}\sin \left[\frac{t-s}{\sqrt{{\tau }_m{\tau }_v}}\right]\exp \left(-\frac{t-s}{2{\tau }_v}\right).$$

$$\mu (x_s)$$

$$v_t=\frac{{\tau }_m}{m}\left\{f+\mu \left[{\Theta }_A+{\Theta }_T\right]\right\}+{\Omega }_T,$$

$${\Omega }_T$$

$${\dot{x}}_t=\frac{f}{\lambda }-\frac{{\Lambda }_T\mu {\mu }^{\prime }}{2{\lambda }^2}+\frac{\mu \left[{\Theta }_A+{\Theta }_T\right]}{\lambda },$$

$${\partial }_{t}P$$

$$P(x)=\frac{\sqrt{\pi }\Gamma \left(k+\frac{1}{2}\right)}{L\Gamma (k+1)}(1-(x/L{)}^2{)}^{-1/2+k},$$

$$A_{\Delta t}(x)=⟨\overline{x_t^2}-{\overline{x_t}}^2{⟩}_s,$$

$$A_{\Delta t}(x)$$

$$\mathbf{A}=\left[\begin{array}{c}\frac{B(X_2-X_1)l^3}{(1+X_1{)}^2}-\frac{Z{l}^2}{1+X_1}\\ \frac{B(X_1-X_2)r^3}{(1-X_2{)}^2}+\frac{Z{r}^2}{1-X_2}\end{array}\right],\mathbf{D}=\left[\begin{array}{cc}\frac{l^{3/2}}{1+X_1}& 0\\ 0& \frac{r^{3/2}}{1-X_2}\end{array}\right],$$

$$P(x_1,x_2)=\frac{N(lr{)}^{3/2}{e}^{-B(x_2-x_1{)}^2-2Z(l+r)}}{(1-x_2)(1+x_1)},$$

$$(1+{\tau }_v{\partial }_t){\sigma }_b={\eta }_b\nabla v+\zeta \Delta {\mu }_b+{\phi }_{b,T},$$

$$\frac{\partial e}{\partial t}+\mathrm{div}\left[e(x)v(x)\right]=\frac{e(x)-e_s}{{\tau }_A},$$

$$\delta e(x)=-e_0{\tau }_A{\partial }_{x}v+{\vartheta }_A.$$

$$(1+{\tau }_v{\partial }_t)\sigma$$

$${\lambda }_{cor}\left(v_{cor}-v\right)$$

$${\lambda }_{cyt}\left(v_{cyt}-v\right)$$

$$l_{cor}^2=\frac{{\eta }_{cor}}{{\lambda }_{cor}}\text{mbox}and{l}_{cyt}^2=\frac{{\eta }_{cyt}}{{\lambda }_{cyt}}.$$

$${\sigma }_{cyt}^L-{\sigma }_{cyt}^R={\eta }_{\mathrm{cyt}}\left(\frac{{\dot{V}}_{\mathrm{L}}}{V_{\mathrm{L}}}-\frac{{\dot{V}}_{\mathrm{R}}}{V_{\mathrm{R}}}\right),$$

$${\sigma }_{cyt}^L-{\sigma }_{cyt}^R$$

$$E(t)=E_0{\left(\frac{t}{1\mathrm{s}}\right)}^{-\alpha },$$

$${⟨I_1(t)⟩}_{x_0}={\int }_0^{t}d{t}^{\prime }x(t^{\prime })\mathrm{and}{⟨I_2(t)⟩}_{x_0}={\int }_0^{t}d{t}^{\prime }{x}^2(t^{\prime }).$$

$$A_{\Delta t}(X)=⟨\overline{X^2}-{\overline{X}}^2{⟩}_s=⟨I_2/t-I_1^2/t^2{⟩}_s,$$

$${\partial }_{t}P={\partial }_x\left((k+D)x+D{\partial }_x(1-x^2)\right)P.$$

$$⟨x^r{⟩}_{x_0}(t)={\int }_{-1}^1\mathrm{d}x{x}^{r}P(x,t|x_0,0),$$

Full text

There has been growing interest in the role of intracellular mechanical fluctuations on cell behavior, with several recent studies showing that stem cells and cancer cells display higher fluctuation levels than normal differentiated cells Talwar et al. (2013); Guo et al. (2014); Mandal et al. (2016). The corresponding physiological role of such fluctuations remains unclear. Contrary to expectations, recent experiments have shown that fluctuations of nuclear components (eg. membrane or DNA loci) are not governed by intra-nuclear activity, but rather by the extra-nuclear actomyosin contractility Weber et al. (2012); Hameed et al. (2012); Versaevel et al. (2012); Makhija et al. (2015); Ramdas and Shivashankar (2015); Radhakrishnan et al. (2017).

In this Letter, we introduce a simple model based on active gel theory Kruse et al. (2005); Marchetti et al. (2013); Prost et al. (2015) which illustrates how variation of active stress noise can induce a maximum in the fluctuation spectrum of the nuclear shape. We discuss the rather unexpected experimental result of Ref. Makhija et al. (2015) which, combining drug treatments and geometric constraints, shows that the nuclear area fluctuations are maximal for an intermediate level of contractility. Treatment by Cytochalasin-D – an actin depolymerazing agent – reduces the amplitude of nuclear fluctuations in cells placed on small circular patches (ie. with low level of contractile filamentous actin) while the same drug increases nuclear fluctuations in cells on large rectangular patches (ie. with a high level of contractile filamentous actin).

Though we focus here on the cell nucleus, our findings are also applicable to many other situations in which an active medium drives fluctuations of an inclusion, e.g. tri-cellular junctions in a tissues Curran et al. (2017) or colloids in active suspensions Lau and Lubensky (2009); Vizsnyiczai et al. (2017); Marchetti et al. (2013).

We describe the dynamics of an inclusion subject to stress fluctuations arising from the surrounding actomyosin fluid confined within $x\in\left[-L,L\right]$ (for simplicity, we consider a one dimensional situation). We successively investigate the cases of a rigid inclusion (eg. a stiff nucleus) and of an elastic inclusion (eg. soft nucleus) that is embedded in a confined active gel of fluctuating activity; further geometries are considered in Ref. Singh et al. (2017). We find that, in a non-equilibrium situation, stress fluctuations generate an effective attraction towards the edges of the confining domain. Based on a Maxwell description of the gel, we check that this effect does not appear for thermal fluctuations. The attraction reported here is distinct from the well-known accumulation of dry active particles close to walls Solon et al. (2015); Elgeti and Gompper (2015), as we consider a passive inclusion and hydrodynamics interactions; it is rather reminiscent of a Casimir effect, since the fluctuation-induced force on the inclusion originates from its surrounding active medium Bartolo et al. (2003); Aminov et al. (2015). We finally predict the existence of an optimal contractility level that maximizes the amplitude of fluctuations of an elastic inclusion, providing a rationale for the unexpected experimental results of Makhija et al. (2015).

We propose an original model for the active stress fluctuations felt by the inclusion (whether rigid or elastic). We expect cortical actomyosin fluctuations to be the leading contribution to the nuclear dynamics since cortical dissipation is significantly larger than cytoplasmic dissipation at relevant length and time scales of the cell Turlier et al. (2014); Rupprecht . Integrating over the apical cortex thickness, we find that the tension $\sigma(x,t)$ along the gel coordinate $x$ reads:

[TABLE]

where $\tau_{v}$ is the Maxwell relaxation time of the gel; $\eta$ is an effective one dimensional viscosity and $v(x)$ is the gel velocity; $\zeta\Delta\mu$ models the medium contractility, e.g. the tension generated by contractile motors Kruse et al. (2005); Prost et al. (2015); $\phi_{T}$ and $\theta_{A}$ are fluctuating tensions of thermal and active origins, respectively. In the cell context, we expect those active fluctuations to result from fluctuations in the cortical thickness (see SM Rupprecht ). We assume that these noise sources are mutually independent and spatially uncorrelated. Notice that we consider a constant viscosity in Eq. (1); as a consequence of the fluctuation-dissipation theorem, the thermal noise $\phi_{T}$ is to be considered as delta-correlated in time: $\langle\phi_{T}(x,t)\phi_{T}(x^{\prime},t^{\prime})\rangle=2\Lambda_{T}\delta_{x,x^{\prime}}\delta_{t,t^{\prime}}$ Mori (1965); Kubo (1966). In contrast, we assume that the active noise $\theta_{A}$ displays significant time correlation, which we choose to be exponential: $\langle\theta_{A}(x,t)\theta_{A}(x^{\prime},t^{\prime})\rangle=\Lambda_{A}\delta_{x,x^{\prime}}\exp(-\lvert t-t^{\prime}\lvert/\tau_{A})/\tau_{A}$. Similar assumptions on the active noise have been introduced to study the diffusion of DNA loci in the presence of intra-nuclear remodeling processes Ghosh and Gov (2014); Vandebroek and Vanderzande (2015); Osmanovic and Rabin (2016). Here, activity is related to cortical remodeling processes; we estimate that $\tau_{A}\approx 60\,\mathrm{s}$ based on images of the dynamics of apical actin foci Li et al. (2014).

We now consider a rigid inclusion at the position $x_{t}$ (see Fig. 1f). At the cell scale, low Reynolds number dynamics holds and forces should be balanced at any time step. In the absence of external friction, this implies that tension should be constant in both segments to the left and right of the inclusion, e.g. $\sigma(x,t)=\sigma^{(L)}(t)$ for $x\in\left[-L,x_{t}\right]$ (conversely $\sigma^{(R)}$ on the right). The existence of a fluctuating gel velocity field leads (by continuity) to the random motion of the inclusion. Averaging Eq. (1) over the segment $\left[-L,x_{t}\right]$ leads to

[TABLE]

where we assumed a zero velocity on the edge $-L$. We derive a similar equation on $\sigma^{(R)}$.

Combining Eq. (2) with the force balance equation on the inclusion $\sigma^{(L)}-\sigma^{(R)}=f(x_{t})-m\dot{v}_{t}$, where $f$ is an externally applied force and $m$ is the mass of the inclusion, leads to the following dynamics on $v_{t}=\dot{x}_{t}$:

[TABLE]

where $\Theta_{A}$ is an exponentially correlated noise ($\left\langle\Theta_{A}(t)\Theta_{A}(s)\right\rangle=\Lambda_{A}\exp(-\lvert t-s\lvert/\tau_{A})/\tau_{A}$) and $\Phi_{T}$ is a Gaussian white noise of variance $2\Lambda_{T}$; the friction term reads $\lambda(x)=(2\eta L)/(L^{2}-x^{2})$ and

[TABLE]

Here, we derive the Fokker-Planck equation corresponding to Eq. (3) through successive adiabatic elimination of the time scales $\tau_{m}=m/\lambda(x)\ll\tau_{v}$ and $\tau_{v}\ll t_{u}$, where $t_{u}=\eta^{2}L/\max(\Lambda_{A},\Lambda_{T})$ is a characteristic displacement time of the nucleus. Our method generalizes Ref. Sancho et al. (1982), which did not include the fluid relaxation time. We first notice that the long-time solution of Eq. (3) can be expressed in terms of a Green function $\chi^{(t-s)}_{x_{t}}$ as

[TABLE]

where the term $\dot{f}=f^{\prime}v_{t}$ is included in the friction term. At first order in $\tau_{m}\ll\tau_{v}$, we find that:

[TABLE]

The integral on the continuous functions $f$ and $\mu\Theta_{A}$ can be simplified thanks to the relation $\chi^{(t-s)}_{x_{t}}\sim\tau_{m}\delta_{t,s}$ in the limit $\sqrt{\tau_{m}\tau_{v}}\ll t-s$. However, this relation does not hold over the discontinuous white noise $\mu\Phi_{T}$. Following Sancho et al. (1982), we consider the Taylor expansion of the noise amplitude $\mu(x)$ to obtain:

[TABLE]

which holds up to $\tau^{1/2}_{m}$ terms. From Eqs. (5) and (7), we obtain the following Langevin equation that is valid at $\tau^{3/2}_{m}$ order:

[TABLE]

where $\Theta_{T}=(1/\tau_{m})\int^{t}_{-\infty}ds\,\chi^{(t-s)}_{x_{t}}\Phi_{T}(s)$ is an exponentially correlated noise and

[TABLE]

After some algebra, we find that $\left\langle\Omega_{T}\right\rangle=-\tau^{2}_{m}/(2m^{2})\mu^{\prime}\mu$ and $\left\langle\Omega_{T}(t)\Omega_{T}(s)\right\rangle=0$ at order $\tau^{3/2}_{m}$. This leads to the following dynamics on $\dot{x}_{t}=v(t)$ at order $\tau^{3/2}_{m}$

[TABLE]

where $\Theta_{A/T}$ are exponentially correlated noise of variance $\Lambda_{A/T}$. In the limit $\tau_{v}\ll\tau_{A}\ll t_{u}$, both $\Theta_{T}$ and $\Theta_{A}$ are to be interpreted in the Stratonovich convention Wong and Zakai (1965); Gardiner (2009) and the Fokker-Planck equation associated to Eq. (9) reads:

[TABLE]

where $\Lambda=(\Lambda^{2}_{T}+\Lambda^{2}_{A})^{\nicefrac{{1}}{{2}}}$.

In the absence of an active noise ($\Lambda_{A}=0$), we find that the steady state solution of Eq. (10) is the Boltzmann distribution $P\propto\exp(-U(x)/k_{b}T)$, where $f=-\partial_{x}U$ and $k_{b}T=\Lambda_{T}/\eta$. The dynamics $\dot{x}_{t}=f/\lambda+(\mu/\lambda)\Theta_{T}$ is consistent with thermodynamics only when the Gaussian white noise is interpreted with the Hanggi-Klimontovich convention Lau and Lubensky (2007); Sokolov (2010).

Whenever $\Lambda_{A}>0$, the steady state of Eq. (10) is not Boltzmann distributed and there is no effective temperature such that $P\propto\exp(-U(x)/k_{b}T_{\mathrm{eff}})$. Remarkably, Eq. (10) shows that, in general, the coexistence of thermal and active noise sources cannot be described by any $\alpha$-convention Lau and Lubensky (2007), e.g. neither by the Stratonovich ($\alpha=1/2$) nor by the Hanggi-Klimontovich ($\alpha=1$) conventions.

When the thermal noise is negligible ($\Lambda_{T}=0$), Eq. (10) amounts to

[TABLE]

which corresponds to the Stratonovich convention of Eq. (9). This result originates from our assumption that the active noise is correlated over a long time scale compared to the viscous memory time ($\tau_{A}\gg\tau_{v}$).

We now apply Eq. (11) in the case of a harmonic restoring force $f(x)=-Kx$; additionally, we assume a constant friction $\Gamma(x)=L/(2\eta)$ for illustration purposes (see Singh et al. (2017) for the complete case). Then, the stationary probability distribution reads

[TABLE]

where $k=(KL^{2}\eta)/(2\Lambda_{A})$ is a dimensionless parameter. From Eq. (12), we observe a significant change in the shape of the probability distribution at $k=k_{c}=1/2$. For large restoring forces $k>1/2$, the inclusion is localized around the center. In contrast, for small restoring forces $k<1/2$ the probability distribution diverges on the edges, indicating the inclusion is attracted by the boundaries (see Fig. 3). This divergence can be traced back to the fixed velocity boundary condition at the edges which leads to a diverging friction term (see Eq. (4)).

Finite-time variance

Even though the shape of the probability distribution undergoes a dramatic change at $k=k_{c}$, the variance $\mathrm{Var}(x_{t})$ of the inclusion position exhibits no singularity as a function of $k$ but rather decreases monotonically as $1/k$. However, we show that fluctuations measured on sufficiently short finite-time windows $\Delta t$ are maximal at an optimal restoring force parameter $k_{\mathrm{opt}}$. We define the following measure of the amplitude of fluctuations, called finite-time variance:

[TABLE]

where $\overline{x}(t)=\int^{\Delta t}_{0}dt^{\prime}\,x_{t^{\prime}}$ refers to a running average over the time window $\Delta t$ while $\left\langle.\right\rangle_{s}$ corresponds to a steady-state average of the initial position $x_{0}$. For the dynamics that corresponds to Eq. (12), we find that

[TABLE]

which holds for arbitrary values of $\Delta t$ Rupprecht . From Eq. (14), we find that $A_{\Delta t}$ exhibits a maximum at an optimal restoring force parameter $k_{\mathrm{opt}}$ as soon as $\Delta t<\Delta t_{c}\approx 6.1t_{u}$, and that $k_{\mathrm{opt}}\approx k_{c}$ for $\Delta t=t_{u}$ Rupprecht . This optimization of fluctuations is illustrated on individual trajectories in Fig. 3b. For a long observation time $\Delta t\gg t_{u}$ there is no optimal restoring force, since the finite-time variance converges to $\mathrm{Var}(x_{t})$, which decreases monotonically with $k$.

Geometry-induced centering

As visible on Fig. 1b, cells placed on rectangular patches display actin fibers on top of the nucleus. We model the nucleus as an elastic inclusion under a contractile compressive loading. We assume that that the two ends of the inclusion remain at a fixed height $z=L_{n}$. Mechanical equilibrium requires the identity of the projected tensions: $\sigma_{L}(1+X_{1})/l_{t}=\sigma_{R}(1-X_{2})/r_{t}=B(X_{2}-X_{1})$, where $X_{i}$ are the coordinates of the two edges of the inclusion (in cell length $L$ unit) while $l=\sqrt{(1+X_{1})^{2}+L^{2}_{n}}$ and $r=\sqrt{(1-X_{2})^{2}+L^{2}_{n}}$ are the left and right actin fibers lengths (see Fig. 1e). Applying the same method as in our first calculation, we find that the dynamics of $\bm{X}=(X_{1},X_{2})$ reads $\dot{\bm{X}}=\bm{A}+\bm{D}\cdot\overline{\Theta}_{A}$,

[TABLE]

where (i) time is expressed in units of $t_{u}$, (ii) $B=EL^{2}t_{u}/\eta$ is a normalized elastic modulus, (iii) $Z=\zeta\Delta\mu t_{u}/\eta$, is a normalized apical cortex activity and (iv) $\overline{\Theta}_{A}=(\Theta_{1},\Theta_{2})$ is an uncorrelated Gaussian white noise vector with $\left\langle\overline{\Theta}_{i}(t)\overline{\Theta}_{j}(t^{\prime})\right\rangle=\delta_{ij}\delta(t-t^{\prime})$ Rupprecht .

We interpret Eq. (15) in the Stratonovich convention under which the Fokker-Planck equation reads: $\partial_{t}P=\partial_{i}\left(A_{i}P+D_{ik}\partial_{j}D_{jk}P\right)$, with summed indices Gardiner (2009). The corresponding stationary probability distribution reads:

[TABLE]

where $N$ is a constant Rupprecht . The distribution $P(x_{1},x_{2})$ is peaked at the edges (where fluctuations are minimal) for large value of $Z$ while it is peaked at the center (where fluctuations are maximal) for large values of $B$ (see Fig. 4a); therefore, we expect fluctuations to be maximal for intermediate values of $B$ and $Z$. For a rigid inclusion ($B=\infty$) and in a flat configuration ($L_{n}=0$), Eq. (16) reduces to Eq. (12).

As a proxy for the nuclear area, we consider the length of the inclusion $Y=X_{2}-X_{1}$. We evaluate $A_{\Delta t}(Y)$ by Monte-Carlo sampling and we find that, for any given value of $B$ and $L_{n}$, $A_{\Delta t}(Y)$ is maximal for an optimal activity $Z_{\mathrm{opt}}$ – and this even for remarkably large values of the observation time window $\Delta t=5t_{u}$. Simulations further indicate the finite time variance at optimality is an increasing function of both the elastic strength $B$ and of the nucleus height $L_{n}$; the latter modulates the contribution of the cortical activity to the centering force.

Comparison to experiments

Our theoretical model explains the counter-intuitive experimental observation that the same biochemical perturbation can either reduce or increase the amplitude of fluctuations, depending on the cell geometry (Fig. 2a). We expect the cell contractility to be higher in rectangular cells (i.e. large $Z$) than in circular cells (i.e. low $Z$). Indeed, actin filaments appear disorganized in circular confinement while cell-size linear actin structures (called stress fibers) appear in rectangular confinement (see Fig. 1b). The ordering of myosin motors within stress fibers is known to be similar to that of muscle sarcomeres, which are designed to produce large contractile forces Burridge and Wittchen (2013). Conversely, the cell cortex produces lower overall contractions since these result from an average over the random organization of acto-myosin structures Liverpool et al. (2009); Lenz (2014). Experiments show that the stress exerted by these fibers lies in $10\,\mathrm{kPa}$ range Plotnikov et al. (2012); Trichet et al. (2012), which leads to a total tension in the $10\,\mathrm{nN}$ range; in contrast, the apical cortex contractile stress is of the order of $\zeta\Delta\mu_{b}=1\,\mathrm{kPa}$ Joanny and Prost (2009), which leads to a tension in the $1\,\mathrm{nN}$ range (corresponding to $Z=1$ Rupprecht ). Such increase of the cell contractility due to stress fibers is further supported by the observation that, in rectangular confinement, the nucleus is strongly compressed vertically Versaevel et al. (2012); Makhija et al. (2015).

By varying the contractility $Z$, we reproduce the observation of a maximum in the amplitude of fluctuations in the inclusion size. The experiments done on rectangular patches with untreated cells correspond to a point on the theoretical curve that is on the right of the maximum, since the contractility is high; reducing contractility by CytoD treatment drives the cell in the direction of the maximum and increases fluctuations (see Fig. 2a). Conversely, the case of untreated cells on circular patches corresponds to a point close to the maximum; CytoD treatment drives the cell away from the maximum and decreases fluctuations. We illustrate this on Fig. 4b representing the optimal activity maximizing $A_{\Delta t}(Y)$ as a function of the nucleus elastic strength and height. We set $B=5$, which corresponds to the elastic modulus of the nucleus when probed the $t_{u}=10^{3}\,\mathrm{s}$ timescale, in the range $E=10^{2}\,\mathrm{Pa}$ Dahl et al. (2005).

Conclusion

In this Letter, we show that the incorporation of an out-of-equilibrium fluctuating noise in generalized hydrodynamics equations leads to the attraction of a confined inclusion to the boundaries. In particular, we have shown how active fluctuations lead to steady states that are qualitatively different from any Boltzmann distribution, hence hampering the definition of an effective temperature. This effect explains an experimental observations on the nuclear area fluctuations Makhija et al. (2015).

The fluctuation-induced force described here is generic to any active materials with space-dependent friction and diffusivity. Spatially varying friction and diffusion coefficients are commonly encountered in thermal systems that break translational invariance Reimann (2002), e.g. thermal colloids diffusing near walls Lau and Lubensky (2007); Faucheux and Libchaber (1994). We expect our result to hold in such two or three dimensional media since the multiplicative nature of the noise results from an integration over space of field fluctuations. The fluctuation-induced force described here can affect the interpretation of particle tracking experiments, e.g. micro-rheology based on correlations between tracers Levine and Lubensky (2000); Hameed et al. (2012) or Bayesian inference of binding potentials Beheiry et al. (2015). Finally, active fluctuations in tissues may provide unsuspected driving forces for cell neighbour exchange Curran et al. (2017). These topics will be the subject of our future research work.

Acknowledgements.

We thank E. Makhija, D. S. Jokhun and A. V. Radhakrishnan for helpful discussions and comments on the manuscript, as well as F. Julicher, K. Vijaykumar and S. Ramaswamy for fruitful discussions on the thermodynamic limit. J-F. R. and AS thank NCBS Bangalore and MBI Singapore, respectively, for hospitality. J.-F. R., G. V. and J. P. are supported by the National Research Foundation, Prime Minister’s Office, Singapore and the Ministry of Education under the Research Centres of Excellence programm.

Appendix A Applicability of the model to the dynamics of the nucleus

A.0.1 Derivation of the constitutive equation

Here, we detail our model assumptions that lead to Eq. (1) presented in the main text. We assume that fluctuations in the height of the cortex give rise to an active noise term that is correlated over a long time scale. We model the cortex as a slab of length $-L<x<L$, width $0<y<l_{n}$ and height $0<z<e(x)=e_{0}+\delta e$, where $\delta e\ll e_{0}$ (see Fig. S1). We then consider an active gel equation for the stress density $\sigma_{b}$ within an elementary volume of the cortex Prost et al. (2015); Kruse et al. (2005):

[TABLE]

where $\tau_{v}$ is the gel Maxwell time; $\eta_{b}$ is the bulk viscosity of the cortex; $\zeta\Delta\mu_{b}$ is a bulk activity parameter and $\vartheta_{b,T}$ refers to thermal Gaussian white noise of variance $\lambda_{T}$.

We consider a weakly perturbed flow conservation equation on a slab of gel of height $e(x)$:

[TABLE]

where $\tau_{A}$ is a characteristic relaxation time; $e_{s}=e_{0}+\vartheta_{A}$ is a fluctuating target cortex width, where $\vartheta_{A}$ is a Gaussian noise that is exponentially correlated in time: $\left\langle\vartheta_{A}(x,t)\vartheta_{A}(x^{\prime},t^{\prime})\right\rangle=\lambda_{A}\delta(x-x^{\prime})\exp(-\lvert t-t^{\prime}\lvert/\tau_{A})/\tau_{A}$. We focus on the one-dimensional geometry where $\nabla v=\partial_{x}v(x)$. Neglecting the time derivative in Eq. (S2) (fast cortex relaxation time) and considering the limit of small perturbations $\delta e=e(x)-e_{0}\ll e_{0}$, we obtain the following height fluctuations equation:

[TABLE]

The tension exerted on the elementary slab at $x$ of volume $dx\times L_{n}\times e(x)$ reads $\sigma=\int^{l_{n}}_{0}\!dy\int^{e(x)}_{0}\!dz\sigma_{b}(x,y,z)$, which, combined with Eq. (S1) and Eq. (S3) leads to

[TABLE]

where $\phi_{T}\approx\int^{l_{n}}_{0}\!dy\int^{e_{0}}_{0}\!dz\varphi_{b,T}$ is approximated as Gaussian white noise of variance $\Lambda_{T}=e_{0}l_{n}\lambda_{T}$; similarly, we define $\Lambda_{A}=e_{0}l_{n}\lambda_{A}$ such that $\theta_{A}$ converges to a Gaussian white noise of variance $\Lambda_{A}$ in the limit $\tau_{A}\ll t$. Equation S4 corresponds to Eq. (1) in the main text. Notice that we neglect second order terms in $\delta e(x)\ll e_{0}$ and that we have defined the following one-dimensional effective (i) activity $\zeta\Delta\mu=e_{0}l_{n}\zeta\Delta\mu_{b}$, (ii) noise term $\theta_{T}=e_{0}l_{n}\vartheta_{b,T}$ and (iii) viscosity $\eta=e_{0}l_{n}(\eta_{0}-\zeta\Delta\mu_{b}\tau_{A})$, which we expect to be lower than the bare viscosity $e_{0}\eta$ as $\zeta\Delta\mu>0$ for actomyosin systems Hatwalne et al. (2004)).

A.0.2 Cortical effects versus bulk effects

We explain why the contact of the nucleus with the cortex, though relatively thin, should be the leading contribution to the nuclear fluctuations, dominating over contributions from the bulk of the cell. We first show that (1) cortical and cytoplasmic friction forces are small compared to the corresponding viscous contribution, and (2) the viscous contribution of the cytoplasm is negligible compared to the cortex. For this, we consider a viscous two-fluid description of the cytoplasm and cortex. This is valid, since as shown in Moeendarbary et al. (2013), elastic terms are important at short times scales, of order $10\mathrm{s}$, which is significantly smaller than the timescale of interest, $t>1\,\mathrm{min}$. Indeed, we expect the Maxwell time of the cortex to be significantly smaller than the cortex turnover time which is also estimated to be $\tau\sim 1\,\mathrm{min}$ (see Salbreux et al. (2012)).

A general two-fluid (gel+solvent) Stokes description for both the cortex (cor) and cytoplasm (cyt) is described by,

[TABLE]

where $v_{cor}$ and $v_{cyt}$ are the gel velocities of the cortex and cytoplasm respectively and $v$ is the fluid velocity. The $\lambda$’s and $\eta$’s are the corresponding permeation coefficients and viscosities of the gel fractions. The rest of the contributions to the local stress density are clubbed into the $\sigma$’s, written as a sum of an isotropic pressure and active stress, the former getting contributions from both the gel and the fluid.

To show that gel viscous dissipation dominates over gel friction, we need to show that the length scales of interest are smaller than the hydrodynamic screening lengths,

[TABLE]

The friction coefficient $\lambda$ can be estimated by equating it to $\eta/a^{2}$, where $\eta=10^{-3}\,\mathrm{Pa}.\mathrm{s}$ is the solvent viscosity and $a$, the gel mesh size. For the cortex, $a_{\mathrm{cor}}\approx 10\,\mathrm{nm}$, thus with an $\eta_{\mathrm{cor}}=10^{5}\,\mathrm{Pa}.\mathrm{s}$, we find that $l_{cor}=100\,{\mu}\mathrm{m}$. Comparing this to the scale of the nucleus $\approx 5\,{\mu}\mathrm{m}$, we conclude that the contribution of the friction of the fluid on the cortex to dynamics of the nucleus, is entirely negligible. For the cytoplasm, if we take $a_{\mathrm{cyt}}\approx 10^{2}-10^{3}\,\mathrm{nm}$, then with an $\eta_{\mathrm{cyt}}\approx 1\,\mathrm{Pa}\,\mathrm{s}$, we find that $l_{cyt}=10\,{\mu}\mathrm{m}$. However, as we show next, cytoplasmic viscous effects have a negligible contribution to the nucleus dynamics at the timescales of interest. Our numerical estimates above, allows us to drop the friction term in Eq. S5. We now show that the contribution of the cytoplasm to the dynamics of the nucleus is negligible compared to the cortex. Consider the nucleus as an inclusion separating the cell into two compartments - left (L)/ right (R) - with length scales $l_{\mathrm{L}}$ and $l_{\mathrm{R}}$, respectively. In the absence of friction, Eq. S5 for each compartment reduces to, $\sigma_{cyt}^{\mathrm{L}}\approx\eta_{\mathrm{cyto}}\dot{l}_{\mathrm{L}}/l_{\mathrm{L}}=\eta_{\mathrm{cyto}}\dot{V}_{\mathrm{L}}/V_{\mathrm{L}}$ where $V_{\mathrm{L}}$ refers to the volume on the left hand side of the inclusion (similar relation for $R$). Evaluating this at either boundary of the inclusion (assuming continuity of stress density), we obtain,

[TABLE]

which corresponds to a dissipation stress due to cytoplasmic flow. In terms of the position of the inclusion $X_{t}$, the volumes can be expressed as $V_{\mathrm{L}}\approx lL_{n}(L+X_{t})$ and $V_{\mathrm{R}}\approx lL_{n}(L-X_{t})$, where $L_{n}$ and $l$ are the $z$ (height) and $y$ (transverse) extensions of the nucleus. After integrating the stress Eq. (S8) over the transverse area of the nucleus, the corresponding force contribution from the cytoplasm reads

[TABLE]

To this we add the contribution coming from the cortex of width $e_{0}$ (see, main text) to obtain the total stress imbalance from L to R. We then obtain a similar stress equation that is identical to Eq. (2) in the main text, but with a new one dimensional viscosity that is $\eta_{\mathrm{new}}=\eta\left[1+\eta_{\mathrm{cyt}}lL_{n}/\eta\right]$. Thus the effect of the pressure difference appears as a simple increase in the value of the dissipation coefficient. However this enhancement of the effective dissipation is small, since $\eta_{cor}e_{0}l\gg\eta_{cyt}L_{n}l$, due to the large value of the cortical viscosity: with $\eta_{\mathrm{cor}}\approx 10^{5}\,\mathrm{Pa}.\mathrm{s}$, $e_{0}\approx 0.2\mu\mathrm{m}$, $l=5\mu\mathrm{m}$, we get $\eta_{cor}e_{0}l\approx 10^{-7}\,\mathrm{N}.\mathrm{s}$, whereas with a cytoplasm viscosity $\eta_{\mathrm{cyt}}\approx 1\,\mathrm{Pa}\,\mathrm{s}$ and $L_{n}\approx l$, we get $\eta_{\mathrm{cyt}}L_{n}l\approx 2.10^{-11}\,\mathrm{N}.\mathrm{s}$. This conclusion is consistent with previous analysis, see Turlier et al. (2014).

A.0.3 Estimate of the parameters value

Correlation time

We identify the active noise correlation time to a typical remodeling time of the cortex. Confocal imaging shows that the apical cortex exhibits stronger fluctuations: local kinks within the apical cortex appear and vanish within $\tau_{A}\approx 10^{2}\,\mathrm{s}$ Li et al. (2014). In comparison, the nucleus moves by around $1\mu m$ within $15$ minutes (see Radhakrishnan et al. (2017)). This separation of time scales supports the hypothesis that the correlation time of the noise can be described through a Stratonovich interpretation of the Langevin equation. The active correlation time is comparable to the actin cortex turnover time Salbreux et al. (2012), which is necessarily significantly larger than the gel Maxwell time $\tau_{v}$ ($\approx 5\,\mathrm{s}$ in Wottawah et al. (2005)).

Amplitude of the active noise

We estimate the mean cortical tension to be of the order of $\zeta\Delta\mu_{b}\delta e_{0}l_{n}$, where $\zeta\Delta\mu_{b}$ is bulk active stress $\zeta\Delta\mu_{b}$ and $\delta e_{0}l_{n}$ is the cortical cross-section. Estimating a typical fluctuation of the cortex width to be $\delta e_{0}=20\,\mathrm{nm}$, we expect that the tension fluctuation amplitude should read $\Lambda_{A}=(\zeta\Delta\mu_{b}\delta e_{0}l_{n})^{2}\tau_{A}\lambda$, where we assume that the spatial correlation length is approximatively $\lambda=100\,\mathrm{nm}$. This leads to a typical diffusion coefficient of the nucleus $D_{\mathrm{nucleus}}=\Lambda_{A}L/\eta^{2}\approx 10^{-3}\,\mu\mathrm{m}^{2}.\mathrm{s}$ that is consistent with the values measured in Radhakrishnan et al. (2017) over the range $t\approx 10^{2}\,\mathrm{s}$.

Nuclear elasticity

The order of the elastic modulus of the cell nucleus is typically $E=1\,\mathrm{kPa}$ when probed over the range of seconds Guilak et al. (2000); Lammerding (2011). However, when probed at a longer time scale, the cell nucleus appears softer due to the long-time viscous behavior of some of the elastic structures. In particular, Dahl et al. (2005) reports that the nuclear elasticity scales with the timescale of the deformation $t$ as:

[TABLE]

which is indeed a decreasing function of time as $\alpha\approx 0.30\pm 0.02>0$. This result was obtained through micropipette aspiration experiments performed on nuclei that were extracted outside of the cell and over a time range spanning from $1\,\mathrm{s}$ to $10^{3}\,\mathrm{s}$. Assuming that $E_{0}=10^{3}\,\mathrm{Pa}$, we find that $E(t=t_{u}=15\,\mathrm{min})=10^{2}\,\mathrm{Pa}$.

Appendix B Rate of convergence of the finite time variance

Here, we derive analytical expressions for the rate of convergence of the finite time variance in two generic cases: the classical Ornstein-Uhlenbeck process with additive noise and the rigid inclusion case considered in Eq. (3) of the main text, which corresponds to an Ornstein-Uhlenbeck process with a particular choice of a multiplicative noise.

We evaluate the evolution the first and second moments in the particle position at the time $t$ after initialization at a position $x_{0}$, denoted $\left\langle x(t)\right\rangle_{x_{0}}$ and $\left\langle x^{2}(t)\right\rangle_{x_{0}}$. Extending the method presented in Dechant et al. (2011), we show that these two first moments differ from the following running averages $\overline{x}=\left\langle I_{1}(t)\right\rangle_{x_{0}}/t$ and $\overline{x^{2}}=\left\langle I_{2}(t)\right\rangle_{x_{0}}/t$, where:

[TABLE]

In terms of Eq. (S10), the finite-time variance defined in Eq. (7) in the main text reads:

[TABLE]

where $\left\langle.\right\rangle_{s}$ refers to an average over initial positions $x_{0}$ chosen according to the steady state distribution.

Moments

We consider a dynamics in the form $\dot{x}=-kx+\sqrt{2D(1-x^{2})}\,\zeta$, where $\zeta$ is a Gaussian white noise interpreted in Stratonovich convention. We consider the probability $P=P(x,t\lvert x_{0},0)$ to reach the point $x$ at the time $t$ from the point position $x_{0}$ at time $t=0$. The forward Fokker-Planck associated to $P$ reads

[TABLE]

We consider the following averaged over paths that originates from $x_{0}$.

[TABLE]

which we call first ($r=1$) and second moments ($r=2)$. Averaging Eq. (S12), we obtain that:

[TABLE]

The last term in the r.h.s of Eq. (B) can be neglected since $P(x)(1-x^{2})$ tends to zero at $x=\pm 1$. Therefore, we obtain:

[TABLE]

With the initial condition $\langle x(0)\rangle_{x_{0}}=x_{0}$, the solutions of these equations are

[TABLE]

We now define the operator $\left\langle\right\rangle_{s}$ corresponding to an average over all possible initial position $x_{0}$ weighted according to the steady state distribution of S12. Importantly, we notice that $\left\langle x^{2}_{0}\right\rangle_{s}=D/(k+2D)$. A subtle point is that the following measure of the variance:

[TABLE]

does not correspond to the finite time variance defined in Eq. (S11). This is further discussed in the next paragraph.

Finite time variance

Based on the definition Eq. (S10), we solve the equation $\partial_{t}{\langle I_{2}(t)\rangle}_{x_{0}}=\left\langle\partial_{t}I_{2}(t)\right\rangle_{x_{0}}=\left\langle x^{2}(t)\right\rangle_{x_{0}}$ to obtain:

[TABLE]

Similarly, we find that the following evolution for the mean running average

[TABLE]

which, as expected, converges to $x_{0}$ when $t\ll 1$.

Obtaining the evolution of $I_{1}(t)$ is more complex. Here, adapting the method of Dechant et al. (2011), we notice that the vector $(I_{1},x)$ satisfies the following Langevin equation: $\partial_{t}(I_{1},x)=(x,-kx+\sqrt{2D(1-x^{2})}\zeta)$, which leads to the corresponding following Fokker-Planck equation

[TABLE]

where $P$ is the probability $P=P(x,I_{1},t\lvert x_{0},0,0)$ to reach a point $(x,I_{1})$ after a time $t$ after initialization at the point $(x_{0},0)$. Similarly to Eq. (S13), we define the average operator

[TABLE]

Integrating the Fokker-Planck Eq. (S19), we obtain the following closed set of equations for the two moments:

[TABLE]

In the rest of the calculation, we set $D=1/2$ (main text convention). With the initial condition $\langle x(0)\rangle_{x_{0}}=x_{0}$, the solutions of Eq. S21 read

[TABLE]

Combining Eq. (S17) and Eq. (S24), we derive an expression of the finite time variance from an initial point $x_{0}$. The average over steady-state distributed initial position leads to:

[TABLE]

where $t=\Delta t$ is the observation time. Equation (S25) corresponds to Eq. (7) in the main text.

We analyze the existence of an optimal restoring force in two specific regimes

when the observation time is comparable to $\Delta t\approx t_{u}<\Delta t_{c}=6.20t_{u}$. In this case, fluctuations measured are maximal at an optimal restoring force parameter $k_{\mathrm{opt}}$ close to the value of $k_{c}$. Considering the derivative of Eq. (S25) with respect to $k$, we find that the optimal restoring force is a solution of the equation:

[TABLE]

By considering the limit $k\rightarrow 0$ of Eq. (S26), we obtain the following equation on the critical observation time $\Delta t_{c}$ beyond which there is no optimum

[TABLE]

This condition is not exactly solvable, but we find an approximate solution is $\Delta t_{c}\approx 6.20$ though a series expansion around $\Delta t_{c}=6$. 2. 2.

in the limit $\Delta t\ll t_{u}$, the existence of an optimum originates specifically from the multiplicative nature of the noise. We find that $k_{\mathrm{opt}}$ continues to decay as $\Delta t$ is increased. From the Eq. (S26), we obtain the following scaling relation for the optimal restoring spring constant

[TABLE]

which holds in the limit of short observation times $\Delta t\ll 1$. In that regime $\Delta t\ll t_{u}$, the origin of the optimality lies in the fact that the centering force tends to localize the particle towards a region where the amplitude of fluctuations is maximal. Over very short time scales $\Delta t\ll 1$, we expect the fluctuations to be governed by the local diffusion coefficient $D(x_{0})=\sqrt{1-x_{0}^{2}}$ for a starting position $x_{0}$; as the centering force is increased (i.e. increasing $k$), the initial position $x_{0}$ is (as averaged over the steady state distribution) decreased and $D(x_{0})$ increased. Therefore, we intuitively expect that an optimum restoring force always exists even for a large restoring force provided that the observation time windows is sufficiently small.

Ornstein-Uhlenbeck process

To conclude, we check that there is no optimum of the finite time variance in the case of an Ornstein-Uhlenbeck process with additive noise: $\dot{X}_{t}=-kX_{t}+\sqrt{2}\zeta$ with $\zeta$ a standard white Gaussian noise. Following the method defined in the previous paragraph, we find that the finite state variance reads:

[TABLE]

In contrast to the rigid inclusion case, in the Ornstein-Uhlenbeck case there cannot be any optimum of the finite time variance as a function of $k$.