Likelihood-Based Heterogeneity Inference Reveals Non-Stationary Effects in Biohybrid Cell-Cargo Transport

TL;DR

This work tackles how heterogeneity in a population of active cells affects the motion of attached passive beads by using a likelihood-based hierarchical framework. Each bead n is modeled as Brownian with a per-trajectory diffusion strength $\sigma_n^2$ drawn from a Gamma distribution $\Gamma(\alpha,\beta)$, and the authors derive an analytic per-trajectory likelihood $\mathcal{L}_n(\alpha,\beta)$ involving the modified Bessel function $K_{(b_n-\alpha)}(2\sqrt{a_n\beta})$, enabling a full data likelihood $\mathcal{L}(\alpha,\beta)=\sum_n\mathcal{L}_n(\alpha,\beta)$ to obtain the MLE $\hat{(\alpha,\beta)}$ and its uncertainty via the Hessian. Compared to a two-step approach, the full likelihood approach more accurately captures heterogeneity, especially with limited trajectory data. Time-resolved analysis reveals non-stationarity: the heterogeneity mean and variance decay in the first two hours and then stabilize into a quasi-stationary state, likely due to cell-bead adhesion dynamics and possible quorum-sensing–driven changes in cell motility. Overall, the method provides robust, uncertainty-aware inference of time-dependent heterogeneity with potential applicability to more complex active-passive systems.

Abstract

Variability of motility behavior in populations of microbiological agents is a ubiquitous phenomenon even in the case of genetically identical cells. Accordingly, passive objects introduced into such biological systems and driven by them will also exhibit heterogeneous motion patterns. Here, we study a biohybrid system of passive beads driven by active ameboid cells and use a likelihood approach to estimate the heterogeneity of the bead dynamics from their discretely sampled trajectories. We showcase how this approach can deal with information-scarce situations and provides natural uncertainty bounds for heterogeneity estimates. Using these advantages we particularly uncover that the heterogeneity in the system is time-dependent.

Figures (4)

• Figure 1: Statistical characterization of bead trajectories driven by an active cell bath. (a) Mean-squared displacement. The blue and the orange lines show eamsd and eatamsd, respectively. The transparent gray lines show the tamsd of the individual trajectories (see Eq. \ref{['eq:data_tamsd']}). The dotted black line shows the theoretical tamsd of a long trajectory of a purely diffusive particle, which is proportional to $t$. (b) Rescaled and normalized displacement autocorrelation functions. The black line indicates the theoretical result for a purely diffusive particle. (c) Step-size distributions in the $x$-direction. The blue curves are histograms calculated from the dataset. Each cross indicates the center of a bin interval. The red dashed lines are the predictions from the heterogeneity inference (see Sec. \ref{['sec:Inference']}).
• Figure 2: Heterogeneity inference on bead trajectories driven by an active cell bath. A sparse version of each trajectory with $\Delta t= \text{2~min}$ is considered to ensure that the dynamics are in the diffusive regime. (a) Log-likelihood of the noise strength given the trajectories of the beads (see Eq. \ref{['eq:Inf_noisestrengthllh']}). The trajectories are sorted by their maximum likelihood estimate for the noise strength. (b) Inferred probability density function of the noise strengths. The green curve represents the result of the full likelihood approach, while the dashed orange curve is the result from a two-step approach. The histogram shows the MLEs of the noise strengths based on the functions shown in panel (a). (c) Log-likelihood $\mathcal{L}(\boldsymbol{ \theta })$ of the heterogeneity parameters with respect to the sparse dataset. The green dot denotes the MLE $\hat{\boldsymbol{ \theta }}$ that maximizes $\mathcal{L}(\boldsymbol{ \theta })$. The green ellipse is the uncertainty estimate calculated from the Hessian matrix at $\boldsymbol{ \theta }$. The orange dot denotes the estimate obtained from a two-step inference approach.
• Figure 3: Time dependence of heterogeneity inference. Inference was performed on reduced datasets containing only data points within a time window of 16 min. The start time $t_{\text{start},j}$ of this time window was shifted over the observation time of the experiment. (a) Inferred heterogeneity parameters $\hat{\boldsymbol{ \theta }}_j$. The black cross denotes the result of the inference from the complete dataset. Note that the error bars appear non perpendicular due to the skewed aspect ratio. (b) Inferred heterogeneity distribution. The black dashed line denotes the result of the inference from the complete dataset, reflecting an average behavior. (c) Mean and variance of the inferred distribution over time. The dashed line denotes the result of the inference from the complete dataset. The plotted error bounds correspond to the highest and lowest values of mean and variance within the 1$\sigma$ uncertainty bounds in $\boldsymbol{ \theta }$ space. In panel (c), the plotted $t_{\text{start},j}$ are spaced 8 min apart, which means that the windows partially overlap. Note that in panels (a) and (b) only every second point is plotted for better readability, leading to a spacing of 16 min.
• Figure 4: Time dependence of heterogeneity inference using a two-step approach with $\hat{\sigma}^2_n$ obtained from single trajectory MLEs. Inference was performed on reduced datasets containing only data points within a time window of 16 min. The start time $t_{\text{start},j}$ of this time window was shifted over the observation time of the experiment. (a) Inferred heterogeneity parameters $\hat{\boldsymbol{ \theta }}_j$. (b) Inferred heterogeneity distribution. As a reference, the black cross and the black dashed line denote the result of the likelihood inference from the complete dataset. (c) Mean and variance of the inferred distribution over time. The dashed line denotes the result of the likelihood inference from the complete dataset.