Detecting active Lévy particles using differential dynamic microscopy

TL;DR

This work extends differential dynamic microscopy (DDM) to detect active Lévy particles by analyzing the intermediate scattering function (ISF) across scales. It derives the ISF for the simplest active Lévy particle model and reveals distinct large-scale asymptotics: for 2<μ<3 the ISF scales as $1/(s+K_μ k^{μ-1})$, while for μ>3 it scales as $1/(s+ k^2/[d(μ-3)])$, enabling discrimination from run-and-tumble particles (RTPs). The authors validate the protocol with simulations and apply it to experimental data from E. coli and Euglena gracilis, finding RTP-like dynamics in E. coli and active Lévy behavior in E. gracilis under illumination. This provides a high-throughput, scale-spanning framework to identify Lévy walks in microorganisms and to extract kinetic parameters such as μ and τ_R from ISF measurements. The approach highlights the necessity of sampling a broad range of length scales (∼1–10 ℓ_p) to capture the hallmark ALP signatures and demonstrates practical applicability to real biological systems.

Abstract

Detecting Lévy flights of cells has been a challenging problem in experiments. The challenge lies in accessing data in spatiotemporal scales across orders of magnitude, which is necessary for reliably extracting a power-law scaling. Differential dynamic microscopy has been shown to be a powerful method that allows one to acquire statistics of cell motion across scales, which is a potentially versatile method for detecting Lévy walks in biological systems. In this article, we extend the differential dynamic microscopy method to self-propelled Lévy particles, whose run-time distribution has a algebraic tail. We validate our protocol using synthetic imaging data and show that a reliable detection of active Lévy particles requires accessing length scales of one order of magnitude larger than its persistence length. Applying the protocol to experimental data of E. coli and E. gracilis, we find that E. coli does not exhibit a signature of Lévy walks, while E. gracilis is better described as active Lévy particles.

Figures (6)

• Figure 1: (a) A paradigmatic model of run-and-tumble-like particles. A particle switches between run and tumble states. The probability density functions, or the propagators, that a running (tumbling) particle travels a distant $\bm{\mathrm{r}}$ after time $\tau$ are denoted as $\mathbb{P}_R(\bm{\mathrm{r}},t)$ ($\mathbb{P}_T(\bm{\mathrm{r}},t)$). The run and tumble time distributions are $\varphi_R(\tau)$ and $\varphi_T(\tau)$, respectively. (b,c) Typical trajectories of (b) a run-and-tumble particle and (c) an active Lévy particle with exponent $\mu=2.5$. The RTP and the ALP have the same persistence length.
• Figure 2: The intermediate scattering functions (ISFs) of active Lévy particles (ALPs) and their asymptotic behavior. We consider the simplest case $D=0$, $\sigma_v=0$, and $\tau_T=0$. (a-d) ISFs of ALPs (a-c) and RTPs (d) with fixed persistence length $\ell_p=1$ and varying wavenumber $k$. For ALPs, we use $\mu=2.2$ (a), 2.8 (b), 3.5 (c). Color encodes wavenumber $k$ normalized by the persistence length $\ell_p:=v_0\tau_R$. Circles represents ISFs measured from particle simulations using Eq. \ref{['eqn_isf_def']}, and solid lines shows the theoretical prediction calculated by numerically inverse Laplace transformation of Eq. \ref{['eqn_fks_alp_simplest']} and \ref{['eqn_fks_rtp']}. The red dashed lines show the swimming propagator \ref{['eqn_swim_propagator_2d']} in 2D with $D=0$ and $\sigma_v=0$ for $2\pi/k=\ell_p$. The black dashed lines shows the asymptotic function \ref{['eqn_asym_alp']} and \ref{['eqn_asym_rtp']} for $2\pi/k=16\ell_p$, $32\ell_p$, $64\ell_p$, $128\ell_p$. (e) The decaying rate $\gamma(k)$ of ISFs of ALPs as a function of $k$, where we fit the ISFs of ALPs by an exponential function $\exp(-\gamma(k)\tau)$. The dots represent $\gamma(k)$ measured from numerical ISFs calculated using Eq. \ref{['eqn_isf_def']}. Color encodes $\mu$. The dashed lines shows the asymptotic decaying rate shown in Eq. \ref{['eqn_asym_alp']}. We fix $\tau_0=1$ in this panel. Parameters: $v_0=1$.
• Figure 3: The mean-squared displacement (MSD) $d\langle\Delta x^2(t)\rangle$ of active Lévy particles (ALPs) with varying exponent $\mu$. Dots represents MSD measured from particle simulations, and solid lines shows the theoretical prediction \ref{['eqn_alp_msd']}. The theoretical prediction \ref{['eqn_alp_msd']} is exact for all time $t$ and exponent $\mu>2$.
• Figure 4: Validation of the fitting protocol using simulations. The ISFs calculated from the synthetic images are shown in (a,b,c,e,f,g) for ALPs and in (d,h) for RTPs. The ISFs are shifted vertically for better visualization. Symbols represent the ISFs measured from simulations, and lines are fits using the ALP model (a-d) and the RTP model (e-g). Colors encode wavenumber $k$. (i) Comparison between the fitted exponent $\mu_{\rm fit}$ and the ground truth $\mu_{\rm real}$. The dashed blue line shows the position for the perfect fit $\mu_{\rm fit}=\mu_{\rm real}$. (j) The estimates of mean run time $\tau_R$ of ALPs from the ALP model and the RTP model. The dashed green line represents the ground truth. (k) The estimates of $\bar{v}$ and $\sigma_v$ of ALPs using both models. The green dashed lines show the ground truth of both parameters. (l) The residue of the fitting of ISFs of ALPs using both models. The residue is defined as $\sum |f_{\rm fit}(k,\tau)-f_{\rm data}(k,\tau)|^2$ where the summation is over all points considered in fitting. The correct model generally fits better the synthetic data and provide estimates close to the ground truth.
• Figure 5: Fitting results of the ISFs of E. coli cells using the active Lévy particle model. (a-b) Comparison of the ISFs of E. coli strain NZ1 with IPTG concentration 150 $\upmu$M fitted by (a) the ALP model and (b) the RTP model, correspondingly. The ISFs are shifted vertically and gray dashed lines correspond to $f=0$. Symbols represent experimental data and lines are fits to the theory. Colors encode wavenumber $k$. (c) Comparison of fitted persistence length $\ell_p$ with respect to different IPTG concentration from the ALP model (solid symbols) and the RTP model (open symbols). (d) The fitted exponent $\mu$ in the ALP model as a function of IPTG concentration. The ISF data and fits of the RTP model are taken from Ref. kurzthaler2024characterization. Red and blue symbols in panels c, d correspond to two biological replicates. (e) The residue of fitting of all samples. The residue is defined as $\sum |f_{\rm fit}(k,\tau)-f_{\rm data}(k,\tau)|^2$ where the summation is over all points considered in fitting.