Non-Markovian non-equilibrium modeling of experimental cell-motion trajectories reveals dependence of propulsion-force correlations on solvent viscosity

TL;DR

This work tackles how non-Markovian propulsion forces and hydrodynamic memory shape single-cell motility in non-equilibrium environments. They develop a data-driven non-equilibrium generalized Langevin equation that combines hydrodynamic friction with a propulsion-force correlation $\Gamma_R(t)$ extracted from trajectories of $Salmonella\ typhimurium$ and $Chlamydomonas\ reinhardtii$, avoiding predefined kernels. The recovered $\Gamma_R(t)$ shows a double-exponential form for salmonella and an oscillatory double-exponential form for algae, enabling accurate predictions of the MSD and long-time diffusivity $D$; they observe a non-monotonic dependence of $D$ on solvent viscosity $\eta$, with a maximum at intermediate $\eta$, and report single-cell power outputs of order $a\mathrm{W}$ and $f\mathrm{W}$. This data-driven non-Markovian framework provides a pathway to infer propulsion mechanisms from trajectories and can be extended to study collective effects in dense suspensions, with implications for infection, ecology, and active-matter physics.

Abstract

Cell motility underlies many biological processes, including cancer metastasis, bacterial infection, and evolutionary adaptation. We introduce a non-equilibrium single-cell motility model inspired by the generalized Langevin equation, which accounts for hydrodynamic friction and correlated propulsion force. From video microscopy of Chlamydomonas reinhardtii algae and Salmonella typhimurium bacteria we extract the propulsion-force dynamics on the single-cell level, which we find to exhibit multi-exponential correlations, not captured by literature non-equilibrium cell-motility models. Based on our data-driven model, we predict the effective cell diffusivities beyond experimentally resolved timescales and demonstrate a diffusivity maximum at intermediate solvent viscosity for both cell types. This means that cells adapt their propulsion-force characteristics according to the solvent viscosity. In addition, our model predicts the power output of single cells, which is on the order of aW for the salmonella and fW for the algae.

Figures (3)

• Figure 1: Microscopy image of (a) a single salmonella (e) a single algal cell, where the major axis is indicated by a blue line and the minor axis by a red line. The corresponding distributions of cell diameters measured in buffer solution of viscosity $\eta=0.89\,\rm{mPas}$ for (b) salmonella, (f) algae. Exemplary trajectory of a single (c) salmonella cell, (d) algal cell, extracted from video microscopy, where the insets show enlarged details of the respective trajectory. The total trajectory length is $2.09\,\rm{s}$ for the salmonella and $2.272\,\rm{s}$ for the algal cell (see SI for trajectory-length distributions). Distribution of inertial times $\tau_m$ for (d) salmonella, (h) algae.
• Figure 2: Propulsion-force correlation $\Gamma_R(t)$ in buffer solution with $\eta=0.89\,\rm{mPas}$ extracted via Eq. \ref{['eq_random_force_extraction']} (blue lines) is compared to $C_{vv}(t)/\tau_m^2$ (cyan dotted lines) and the respective fits of Eqs. \ref{['eq_noneq_kern_salmo']},\ref{['eq_noneq_kern_cr']} (red dotted lines) are shown for (a) a single salmonella, (b) a single algal cell. The insets show the plots on larger time scales with logarithmic y-axis. Fitting details are explained in the SI. (c), (d) MSD of the respective single cells are shown as blue dots, where the red line is the prediction from the respective fit as explained in the SI. The diffusivities predicted from the fits are $D=105\,\mu\rm{m}^2/\rm{s}$ for the salmonella and $D=9.6\times 10^4\,\mu\rm{m}^2/\rm{s}$ for the algal cell. Black dashed lines indicate scaling behavior.
• Figure 3: The ensemble average over all individual time-averaged MSDs is shown at different viscosities and the prediction by the median of all single-cell fit parameters as dotted lines for (a) salmonella, (b) algae. (c), (d) Boxplots of the long-time effective diffusivities that are predicted from the single-cell fits of Eqs. \ref{['eq_noneq_kern_salmo']},\ref{['eq_noneq_kern_cr']} respectively, where red lines denote the median and blue dots denote the mean of the distribution. Boxes represent the range from the first to the third quartile and whiskers represent values within 1.5 times this interquartile range from the box edges, while black circles mark outliers beyond this range.