🔬 condensed matter

Simulation of Flagellated Bacteria Near a Solid Surface: Effects of Flagellar Morphology and Ionic Strength

This study systematically investigates how flagellar morphology and ionic strength govern the three-stage surface entrapment process of swimming bacteria, revealing that these factors collectively determine the stable height, inclination angle, and curvature of their near-surface circular trajectories.

Original authors: Baopi Liu, Bowen Jin, Ning An

Published 2026-08-26

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CC BY 4.0

Original authors: Baopi Liu, Bowen Jin, Ning An

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

Microscopic swimmers, such as the common bacterium E. coli, navigate their world by spinning a tiny, corkscrew-like tail called a flagellum. This rotation propels them through liquid, but when they drift near a solid boundary like a glass slide or a cell wall, the rules of their movement change dramatically. The liquid right next to the solid surface cannot slip past it, creating a friction that drags on the swimming cell. This interaction, combined with invisible electrical and atomic forces that act over tiny distances, traps the bacteria against the wall. Once caught, they do not simply stick; they begin to swim in circles. Understanding exactly how high these bacteria hover above the surface and how tightly they turn is crucial for science. These behaviors dictate how bacteria stick to medical devices to form slimy, hard-to-remove layers known as biofilms, which are responsible for many persistent infections and contamination issues.

A team of researchers has now built a detailed computer simulation to map out exactly how these bacteria get trapped and how they behave once they are there. They focused on a specific type of bacteria that uses a single, rigid, spiral tail. By creating a mathematical model of this swimmer, they were able to watch, in slow motion, the entire process of a bacterium approaching a wall, adjusting its angle, and settling into a stable circular path. The study reveals that this journey happens in three distinct phases. First, the bacterium swims straight toward the surface at a speed determined by how fast its motor spins. Second, it reorients itself, tilting its body until it is nearly parallel to the wall. Finally, it finds a stable spot where it can swim in a perfect circle without drifting away or crashing into the surface.

The researchers found that the final position of the bacterium is a delicate balance between the push of its spinning tail and the pull of the surface. The tail acts like a propeller that tries to push the cell away, while the friction of the water against the wall tries to rotate the cell body until it lies flat. At the same time, invisible forces described by a theory known as DLVO come into play. These forces include a weak attraction that pulls the cell close and an electrical repulsion that pushes it away, depending on the chemistry of the water. The simulation showed that the bacteria settle at a specific height and a specific tilt angle where all these forces cancel each other out. For bacteria with a left-handed spiral tail, this balance results in a clockwise circular motion along the surface.

One of the most significant discoveries concerns the water itself. The researchers tested how the "ionic strength," or the amount of dissolved salt in the water, changed the bacteria's behavior. They found that in water with less salt, the bacteria swim much higher above the surface, sometimes hovering more than one hundred nanometers up, whereas in saltier water, they stay much closer, sometimes just a few nanometers away. This variation in height helps explain why different experiments in the past have reported such widely different distances for how close bacteria swim to walls. The study also looked at the shape of the tail. They discovered that longer tails cause the bacteria to tilt less steeply and swim in wider circles, while shorter tails make them point more sharply toward the surface and turn in tighter loops.

The size of these circular paths is also directly linked to the saltiness of the water. In low-salt conditions, the bacteria swim in circles that can be tens of micrometers wide, whereas in high-salt conditions, the circles are much smaller. The researchers noted that the speed of the motor spinning the tail had very little effect on the final height or the angle of the tilt, suggesting that the shape of the tail and the chemistry of the water are the dominant factors. The simulation confirmed that the bacteria always find a single, stable spot to settle, regardless of where they started. This work provides a clear, unified picture of why bacteria behave the way they do near surfaces, offering a potential key to understanding and perhaps controlling how they form the biofilms that plague medical equipment and industrial systems.

Technical Summary: Simulation of Flagellated Bacteria Near a Solid Surface

Problem Statement
The dynamics of bacteria swimming near solid surfaces are critical to understanding biofilm formation, bacterial adhesion, and various biomedical and environmental processes. While experimental observations confirm that bacteria undergo a three-stage entrapment process (approach, reorientation, and surface swimming) and exhibit stable circular motion, the mechanisms governing the dispersion of observed stable heights remain unclear. Reported stable heights vary significantly, ranging from a few nanometers to over one hundred nanometers. Furthermore, it is not fully understood how flagellar morphology and the ionic strength of the electrolyte solution collaboratively influence the stable height, inclination angle, and the radius of curvature of these circular trajectories. Hydrodynamic interactions alone are insufficient to explain these behaviors; the interplay with Derjaguin–Landau–Verwey–Overbeek (DLVO) forces is essential.

Methodology
The authors develop a mathematical model and numerical simulation framework for a monotrichous bacterium (modeled as a rigid spherical cell body with a single rigid helical flagellum) swimming near a solid surface under low Reynolds number conditions.

Hydrodynamics: The system is treated as a chiral two-body model. The resistance matrices for the cell body and the flagellum are calculated separately using Resistive Force Theory (RFT) for the flagellum and analytical solutions for a sphere near a wall. Hydrodynamic interactions between the flagellum and the surface, as well as between the body and flagellum, are neglected, justified by the typical separation distances being larger than the filament radius.
Forces: The model incorporates near-field hydrodynamic forces and DLVO forces (van der Waals and electrostatic). The DLVO interaction is modeled using standard expressions for a sphere near a plane, with the Debye length ( $\kappa^{-1}$ ) serving as the primary parameter for ionic strength.
Simulation: The equations of motion are solved iteratively. The authors perform phase averaging over one flagellar rotation period to eliminate rapid oscillations. The simulation tracks the time evolution of the bacterium's height ( $h$ ) and the inclination angle of the flagellar axis ( $\Psi$ ) relative to the surface.
Parameters: Simulations vary the motor rotation frequency, flagellar contour length ( $\Lambda$ ), and Debye length (representing ionic strength). Zeta potentials are set to $-20$ mV for both the bacterium and the surface to isolate the effect of ionic strength.

Key Contributions and Results
The study systematically characterizes the three-stage process of surface entrapment and quantifies the stable fixed point $(h^*, \Psi^*)$ and the radius of curvature ( $r$ ) of the resulting circular motion.

Three-Stage Entrapment Dynamics:
- Approach: Bacteria swim toward the surface with a velocity approximately proportional to the motor rotation frequency.
- Reorientation: The height remains roughly constant while the inclination angle evolves. The cotangent of the inclination angle ( $\cot \Psi$ ) decays exponentially with the product of motor frequency and time ($ft$).
- Surface Swimming: Bacteria reach a stable fixed point where hydrodynamic torques and DLVO forces balance, resulting in stable circular motion. Left-handed chiral flagella result in clockwise circular trajectories.
Influence of Ionic Strength (Debye Length):
- Decreasing the ionic strength (increasing the Debye length) significantly increases the stable height ( $h^*$ ) and the radius of curvature ( $r$ ).
- Conversely, decreasing ionic strength reduces the stable inclination angle ( $\Psi^*$ ).
- The stable height ranges from several nanometers to over one hundred nanometers depending on the ionic strength, which aligns with the wide dispersion observed in experiments.
Influence of Flagellar Morphology:
- Contour Length ( $\Lambda$ ): Increasing the flagellar contour length reduces the stable inclination angle and increases the radius of curvature. However, the stable height is found to be largely insensitive to changes in contour length within the tested parameter range.
- Inclination Angle Scaling: An empirical power law is identified relating the stable inclination angle to the distance ( $l$ ) between the centers of the cell body and the flagellum: $-\cos \Psi^* \propto l^{-2.5}$ . Shorter flagella tend to orient the bacterium more vertically ("nose down").
Motor Frequency:
- The motor rotation frequency has a negligible effect on the stable fixed point (height and inclination angle) and the radius of curvature, as the relevant torques and forces scale proportionally with frequency.

Significance and Claims
The paper claims that its work provides a unified explanation for the experimentally observed dispersion of stable heights for bacteria near solid surfaces. By demonstrating that the stable height is highly sensitive to the ionic strength of the solution (via the Debye length) but relatively insensitive to flagellar contour length and motor frequency, the authors suggest that variations in environmental ionic conditions are a primary driver of the reported height variability.

Furthermore, the study elucidates the specific roles of flagellar morphology and ionic strength in determining the geometric parameters of surface swimming (inclination angle and curvature radius). The findings offer a theoretical basis for understanding how bacteria interact with surfaces under varying physicochemical conditions, which is fundamental for manipulating biofilm formation and bacterial adhesion in applications such as wastewater treatment, bioremediation, and biomedical engineering. The authors note that while zeta potential also affects DLVO forces, their current focus on Debye length highlights the critical role of ionic strength in modulating bacterial motility near interfaces.

Technical Summary: Simulation of Flagellated Bacteria Near a Solid Surface

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