Time reversal breaking of colloidal particles in cells

Abstract

We investigate signatures of broken time reversal symmetry in stochastic trajectory data, employing the previously introduced three point correlation called mean back relaxation. We specifically investigate data from a simple driven model, as well as from colloidal particles within living or passivated biological cells. Both in the model as well as in cell data, MBR detects broken time reversal symmetry, and furthermore, allows to determine relevant time and length scales of activity. For the cells, we show, by applying various drugs, that it is predominantly the presence of microtubules which is needed for a time reversal symmetry breaking. We employ a bound for entropy production, finding that it is in striking relation to previously determined active energies that quantify violation of the fluctuation dissipation theorem.

Figures (11)

• Figure 1: (a) Sketch of the dRHC model of \ref{['eq:dRHC']}. A Brownian particle is trapped in a harmonic potential whose center performs a discrete random walk with time intervals $\cal{T}$ and step length $\Delta q$. (b) Shows an example trajectory of $x$ and $q$ for $\Delta q = 3.0 \sqrt{\frac{k_B T}{k]}}$ and $\mathcal{T} = 5.0 \frac{\gamma}{k}$. This illustrates that $q$ performs discrete steps in space and time, while $x$ follows with a delay.
• Figure 2: (a) MBR for the dRHC model of \ref{['eq:dRHC']} as a function of time $t$ for $\tau = \frac{\gamma}{k}$, $l= 0.01 \sqrt{\frac{k_B T}{k}}$ and $(\Delta q, {\cal T}) \in \left\lbrace (0,2), (3,5), (3,2), (5,2) \right\rbrace$ with $\Delta q$ in units of $\sqrt{\frac{k_B T}{k}}$ and ${\cal T}$ in units of $\frac{\gamma}{k}$. Solid (dashed) lines: MBR from forward (backward) trajectories. Differences between solid and dashed signal time reversal symmetry breaking. Blue lines show the case of $\Delta q=0$ (equilibrium), and solid dashed are on top of each other. (b) $\text{MBR}_\text{anti}$ of Eq. \ref{['eq:MBR_anti']} for the same parameters, i.e., difference between solid and dashed lines of panel a).
• Figure 3: Anti-symmetric part of MBR, i.e., $\text{MBR}_\text{anti}$ of Eq. \ref{['eq:MBR_anti']}, for trajectories from dRHC model of \ref{['eq:dRHC']} as functions of $\tau$ and $l$. Model parameters are $\Delta q = 3 \sqrt{\frac{k_B T}{k}}$ and ${\cal T} = 2.0 \frac{\gamma}{k}$. Gray lines are guides to the eye, and they follow \ref{['eq:kp-slope_grid']}. Horizontal distance between neighboring dashed lines is ${\cal T}$, vertical distance is $\Delta q$. This allows extraction of the active time and length scale. Inset shows $\text{MBR}_\text{anti}$ as a function of $\tau$ for $l=0.45 \sqrt{\frac{k_B T}{k}}$ (indicated by dash-dotted line in main graph), illustrating the oscillatory nature.
• Figure 4: Anti-symmetric part of MBR, i.e., $\text{MBR}_\text{anti}$ of Eq. \ref{['eq:MBR_anti']}, for dRHC of \ref{['eq:dRHC']}, with all parameters identical to \ref{['fig:kp-LT']}, but with Gaussian distributed step intervals with standard deviation $\sigma_t/{\cal T} = 1/4$. Gray lines are guides to the eye, and they follow \ref{['eq:kp-slope_grid']}, i.e., horizontal distance between neighboring dashed lines is ${\cal T}$, vertical distance is $\Delta q$. Inset shows $\text{MBR}_\text{anti}$ as a function of $\tau$ for $l=0.45 \sqrt{\frac{k_B T}{k}}$ (indicated by dash-dotted line in main graph), illustrating the decaying oscillatory nature.
• Figure 5: (a) MBR for various cell types, as labelled, for $\tau = 0.047s$ and $l = 0.002µm$. Solid lines show MBR evaluated for time-forward trajectories and dotted lines for time-backward trajectories. For the wild-type cells, the solid and dotted lines disagree, indicating a breakage of time reversal symmetry. For the drug-treated HeLa cells, there is no significant difference between forward and backward curves. (b) Anti-symmetric part of MBR of Eq. \ref{['eq:MBR_anti']}, i.e., difference between respective dashed and solid lines in (a). Colored envelopes give the standard deviation of a bootstrap distribution (4000 samples). a) and b) also show MBR from trajectories of a probe particle in the purely passive medium of aragose, as a reference.