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

This study introduces a non-Markovian, non-equilibrium model based on generalized Langevin equations to analyze experimental trajectories of *Chlamydomonas reinhardtii* and *Salmonella typhimurium*, revealing that cells adapt their multi-exponential propulsion-force correlations to solvent viscosity, resulting in a maximum effective diffusivity at intermediate viscosities and enabling the prediction of single-cell power outputs.

Original authors: Anton Klimek, Prince V. Baruah, Prerna Sharma, Roland R. Netz

Published 2026-01-27

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CC BY 4.0

Original authors: Anton Klimek, Prince V. Baruah, Prerna Sharma, Roland R. Netz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine tiny biological swimmers—like microscopic bacteria and algae—navigating a world that feels like thick honey or thin water. For a long time, scientists tried to predict how these cells move using simple rules, kind of like assuming a car always drives at the same speed regardless of the road conditions. But this paper argues that these cells are much smarter and more complex than those simple models suggest.

Here is the story of what the researchers discovered, explained in everyday terms:

The Problem: The "Simple Model" Didn't Fit

Scientists have been trying to understand how single cells move for years. They usually use a "toy model" (like the Active Ornstein-Uhlenbeck process) that assumes a cell pushes itself forward with a force that fades away quickly and randomly.

Think of it like a person trying to walk through a crowd while holding a balloon. If the balloon pops instantly, the person stumbles forward and then stops. The old models assumed cells were like this: they push, the push fades, and they drift.

However, when the researchers looked closely at real video footage of Salmonella bacteria and Chlamydomonas algae, they saw something different. The cells didn't just push and forget; they seemed to have a "memory." Their pushing force didn't fade away instantly; it lingered, wobbled, and changed in complex ways that the old simple models couldn't catch.

The New Tool: A "Memory" Equation

To fix this, the team built a new, more sophisticated mathematical tool based on a "Generalized Langevin Equation."

Imagine you are trying to figure out how hard a swimmer is kicking by watching them move through water.

The Old Way: You guessed the kick strength based on a standard rule.
The New Way: The researchers used a "reverse-engineering" trick. They knew how water resists movement (friction). By watching exactly how the cell moved, they could mathematically "work backward" to calculate exactly how hard the cell was pushing at every single moment.

This allowed them to see the "propulsion force" (the kick) in high definition. They found that:

Salmonella push with a force that fades away in two distinct stages (like a double sigh).
Algae push with a force that wobbles back and forth (like a rhythmic breaststroke) before fading away.

The Big Surprise: The "Goldilocks" Viscosity

The researchers didn't just watch the cells in plain water. They added a thickening agent (PEG) to the water to make it stickier, simulating environments like mucus in the human body.

They expected that as the water got thicker (more viscous), the cells would get slower and slower, just like a car struggling in deep mud.

But that's not what happened.

Instead, they found a "Goldilocks" zone:

Too thin: The cells moved, but not super efficiently.
Too thick: The cells struggled and slowed down.
Just right (Intermediate viscosity): The cells actually moved fastest and covered the most ground.

It's as if the cells sensed the thickening water and decided, "Hey, the road is getting slippery; I need to kick harder!" They adapted their swimming style to match the environment. This adaptation meant their ability to spread out (diffusivity) peaked at a medium thickness, rather than just dropping steadily as the water got thicker.

Why This Matters (According to the Paper)

The paper highlights a few key takeaways without making up future scenarios:

Cells are Adaptable: These tiny organisms aren't just passive objects being pushed around; they actively change how they push based on how thick their environment is.
Power Output: The researchers calculated how much energy these cells burn to swim. They found that Salmonella burn energy at the level of attowatts (one quintillionth of a watt) and algae at femtowatts (one quadrillionth of a watt). Interestingly, like their speed, their energy output also peaked at that "just right" thickness.
Better Predictions: Because their new model captures these complex "memories" of the cells, it can predict how far a cell will travel over a long time (hours or days) based on just a few seconds of video. The old models failed at this long-term prediction.

The Bottom Line

This study shows that cells are like skilled drivers who adjust their engine and steering based on the road conditions, rather than just following a fixed script. By using a new, data-driven method, the researchers proved that these microscopic swimmers have a complex "memory" in their movements and that they perform best in environments that are neither too runny nor too thick. This helps us understand how they navigate the real world, including the thick mucus layers inside our bodies.

Technical Summary: Non-Markovian Non-Equilibrium Modeling of Experimental Cell-Motion Trajectories

Problem Statement
Cell motility is a fundamental non-equilibrium process critical to biological functions such as infection, metastasis, and adaptation. While various modeling approaches exist (e.g., active Ornstein-Uhlenbeck processes, active Brownian motion), they often rely on coarse-grained assumptions that fail to capture the specific non-equilibrium character of propulsion forces or cannot be applied to systems with non-Gaussian statistics. A significant challenge in analyzing single-cell trajectories is the inability to simultaneously deduce both the hydrodynamic friction kernel ( $\Gamma_v(t)$ ) and the propulsion-force correlation ( $\Gamma_R(t)$ ) from a single trajectory measurement without prior structural assumptions. Furthermore, existing models often struggle to resolve short-time dynamics while simultaneously predicting long-time diffusivity, and the dependence of propulsion characteristics on solvent viscosity remains poorly understood at the single-cell level.

Methodology
The authors introduce a data-driven, non-equilibrium single-cell motility model based on the Generalized Langevin Equation (GLE). The approach involves the following steps:

Experimental Data: High-resolution video microscopy was performed on individual Chlamydomonas reinhardtii (algae) and Salmonella typhimurium (bacteria) cells. Trajectories were recorded in dilute suspensions confined between non-sticky surfaces to ensure focus stability and minimize wall friction.
Hydrodynamic Assumption: Based on theoretical predictions for spherical/ellipsoidal objects in water and the observed cell speeds/shapes, the authors assume the friction kernel is frequency-independent on the observation timescale. This simplifies the GLE friction term to a delta function, $\Gamma_v(t) = 2\delta(t)/\tau_m$ , where $\tau_m$ is the inertial time.
Force Extraction: By inverting the GLE, the authors extract the propulsion force $F_R(t)$ directly from the measured trajectories using discrete derivatives.
Model Fitting: The extracted force correlations are fitted to specific functional forms without assuming a priori models for the propulsion mechanism.
- For Salmonella, a double-exponential decay model is used.
- For Algae, an oscillatory double-exponential model is used to account for flagellar beating frequencies.
Viscosity Variation: The solvent viscosity was systematically increased using Polyethylene Glycol (PEG) concentrations (0–10 wt%) to study the adaptation of propulsion characteristics.
Prediction: The fitted models are used to analytically calculate the Mean Squared Displacement (MSD) and the long-time effective diffusivity ( $D$ ) via the integral of the velocity autocorrelation function, extending predictions beyond the experimental observation window.

Key Contributions

Non-Markovian Force Correlation: The study successfully extracts the actual non-equilibrium propulsion-force correlation $\Gamma_R(t)$ from experimental data, revealing multi-exponential correlations that are not captured by standard literature models.
Data-Driven Modeling: The approach avoids assuming a priori models for propulsion forces; instead, the data dictates the model structure (e.g., distinguishing between standard active OU behavior for bacteria and oscillatory behavior for algae).
Viscosity Adaptation: The model demonstrates that cells adapt their propulsion-force characteristics to solvent viscosity, evidenced by a non-monotonic dependence of diffusivity on viscosity.
Power Output Estimation: The framework allows for the estimation of single-cell power output (energy dissipation) based on the extracted force correlations and speeds.

Results

Force Correlations: The extracted propulsion-force correlations exhibit long memory (decay over seconds), significantly longer than the flagellar oscillation periods (~0.01–0.02 s). Salmonella correlations fit a double-exponential decay, while Algae show strong short-time oscillations superimposed on exponential decay.
Diffusivity Maximum: Contrary to the expectation that increasing viscosity monotonically decreases cell speed and diffusivity (assuming constant propulsion force), the study finds a maximum in effective diffusivity ( $D$ ) at intermediate solvent viscosities for both cell types.
Power Output: The estimated power output is on the order of attowatts (aW) for Salmonella and femtowatts (fW) for algae. Similar to diffusivity, the power output exhibits a maximum at intermediate viscosity, suggesting cells invest more energy into propulsion when sensing higher viscosity.
Model Validation: The data-driven models accurately predict the MSD of individual cells and the ensemble average, matching experimental data and extending to long timescales where direct measurement is impossible. The predicted long-time diffusivity for algae agrees with literature values derived from collective cell density expansion.

Significance and Claims
The paper claims that its data-driven modeling approach provides a robust method to infer intricate details of single-cell propulsion mechanisms solely from spatial trajectories, without assuming specific propulsion-force characteristics. The discovery of a diffusivity maximum at intermediate viscosity implies that these cells actively adapt their propulsion-force characteristics to their environment, a finding crucial for understanding how pathogens like Salmonella navigate viscous biological barriers (e.g., mucus layers in the gut). The authors suggest that while the mechanism of this adaptation (active sensory response vs. mechanical consequence) requires further investigation, the non-linear viscosity dependence is highly relevant for biological function. Finally, the model's ability to handle both Gaussian and non-Gaussian systems and incorporate localization noise makes it applicable to a wide range of organisms and potentially useful for simulating collective effects in dense suspensions in future work.