Energetics of active fluctuations in living cells

The nonequilibrium activity taking place in a living cell can be monitored with a tracer embedded in the medium. While microrheology experiments based on optical manipulation of such probes have become increasingly standard, we put forward a number of experiments with alternative protocols that, we claim, will provide insight into the energetics of active fluctuations. These are based on either performing thermodynamiclike cycles in control-parameter space or determining response to external perturbations of the confining trap beyond simple translation. We illustrate our proposals on an active itinerant Brownian oscillator modeling the dynamics of a probe embedded in a living medium.

Figure 1

(a) Schematic representation of the energetic landscape rearrangement due to motor activity and its modeling using the active burst applied on the local minimum. In the passive case without motors, the tracer is confined within a harmonic potential. When motors are introduced, their activity modifies the network structure, thus leading to a displacement $v_{\mathrm{A}}\tau$ of the potential local minimum $x_0$. (b) Example trajectory of the active burst projection $v_{\mathrm{A}}$. It equals zero over a random duration of average ${\tau }_0$ and is a random value between $-v$ and $v$ during a random time of order $\tau$.

Figure 2

(a) Mean square displacement as a function of the scaled time $t/\tau$ for active (red solid line) and passive (blue dot-dashed line) systems. (b) Mean square displacement as a function of the scaled time $t/\tau$ with (green dashed line) and without (red solid line) an external potential, in this case a harmonic optical trap. The evolution is qualitatively similar for time scale smaller than ${\tau }_{\mathrm{t}}/\varepsilon$. At large time scale, there is a plateau due to the confinement within the optical trap, which value diverges with $\varepsilon$. $\{T,\varepsilon ,k,{\tau }_{\mathrm{d}},{\tau }_{\mathrm{t}},T_{\mathrm{A}},\tau \}=\{{10}^2,{10}^{-4},2,{10}^{-2},2\times {10}^2,1,1\}$.

Figure 3

Schematic representation of the energetic landscape when a quadratic optical trap is applied on the tracers, in addition to the harmonic confinement potential.

Figure 4

Study of the influence of the $O\left(\varepsilon \right)$ correction term in Eq. (9). The quasistatic work $W_{\mathrm{H}}$ is obtained numerically from simulations of the dynamics in Eq. (3), where $\varepsilon =\{0.3,0.25,0.2,0.15\}$. We extract the correction term as $W_{\mathrm{H}}-W_{\mathrm{H}}^{(-1)}-W_{\mathrm{H}}^{\left(0\right)}$, where the expression of $W_{\mathrm{H}}^{\left(n\right)}=O\left({\varepsilon }^n\right)$ is given by Eq. (9). The analytic expression of the $O\left(\varepsilon \right)$ correction term is plotted in a cyan dotted line as a function of $k_{\mathrm{f}}$, and it agrees with numerical simulations for $\varepsilon =0.15$. For larger values of $\varepsilon$, the next order terms should be taken into account to explain the deviation of the simulated quasistatic work from the prediction in Eq. (9). $\{T,k,k_{\mathrm{i}},\gamma ,{\tau }_0,\tau ,v\}=\{0,1,1,1,5,0.6,4\}$.

Figure 5

Schematic representation of cycle $\mathcal{C}$. The optical trap parameters undergo the transformations $\left(A\right)$ to $\left(D\right)$: $\{b_{\mathrm{i}},k_{\mathrm{i}}\}\stackrel{\left(A\right)}{}\{b_{\mathrm{i}},k_{\mathrm{f}}\}\stackrel{\left(B\right)}{}\{b_{\mathrm{f}},k_{\mathrm{f}}\}\stackrel{\left(C\right)}{}\{{\beta }_{\mathrm{f}},k_{\mathrm{i}}\}\stackrel{\left(D\right)}{}\{b_{\mathrm{i}},k_{\mathrm{i}}\}$. The shape of the external potential tuned by the operator is depicted as a function of the position for the four parameter sets, where $k_{\mathrm{f}}=2k_{\mathrm{i}}<0$ and $b_{\mathrm{f}}=2b_{\mathrm{i}}$.

Figure 6

(a) Effective temperature as a function of the scaled frequency $\omega \tau$. The plateau value at low frequency equals $T+T_{\mathrm{A}}/\varepsilon$ (red solid line), and it equals $T$ at high frequency as for the passive case (blue dot-dashed line). Between the two saturations, it scales successively like $1/{\omega }^2$ and $1/{\omega }^4$, provided the time scales $\tau$ and ${\tau }_{\mathrm{d}}/\sqrt{\varepsilon }$ are well separated. (b) Evolution of the force-position correlation function with the scaled time $t/\tau$ in the passive (blue dot-dashed line) and active (red solid line) cases. The correlation function is negative at short time scale with an initial value $-T$. It remains negative in the passive case. There is a linear growth regime in the active case (black dashed line), and the correlation function saturates to a plateau value $T_{\mathrm{A}}$. (a) $\{T,\varepsilon ,{\tau }_{\mathrm{d}},T_{\mathrm{A}},\tau \}=\{1,{10}^{-8},1,{10}^2,{10}^2\}$. (b) $\{T,{\tau }_{\mathrm{d}},T_{\mathrm{A}},\tau \}=\{5,0.2,10,5\times {10}^2\}$.

Figure 7

(a) Evolution of the Fourier transform of the spectral density of the energy dissipation rate with the scaled frequency $\omega \tau$ when $\tau \gg {\tau }_{\mathrm{d}}$. The plateau value at low frequency equals $2T_{\mathrm{A}}$. It scales like $1/{\omega }^4$ at high frequency, and there is a crossover regime $1/{\tau }_{\mathrm{d}}\ll \omega \ll 1/\tau$ with another power law $1/{\omega }^2$. (b) Antisymmetric force-position correlation function as a function of the scaled time $t/\tau$. It is linear in time at short time scale with a growth coefficient $J$ and saturates to the value $T_{\mathrm{A}}$ at large time scale. $\{T_{\mathrm{A}},\tau ,{\tau }_{\mathrm{d}}\}=\{2,10,0.1\}$.