Hydrodynamics can determine the optimal route for microswimmer navigation

The problem of optimal navigation—finding the most efficient path to a target—is well-studied for macroscopic agents like airplanes and spacecraft, often using optimal control theory. However, the optimal navigation of microswimmers, which experience hydrodynamic interactions with their environment, is far less understood. Microswimmers, self-propelling agents in low-Reynolds-number fluids, are increasingly important in nanomedicine, potentially enabling targeted drug delivery. Their navigation presents unique challenges due to overdamped dynamics, thermal fluctuations, and long-range hydrodynamic interactions with walls and obstacles. Previous research has explored optimal navigation for dry active particles (neglecting hydrodynamics) and microswimmers in simplified environments (e.g., mazes) using reinforcement learning or analytical approaches. This study addresses the significant gap in understanding the impact of hydrodynamic interactions with walls and obstacles on microswimmer navigation strategies, aiming to provide insights into the optimal routes for microswimmers in more realistic environments.

Existing literature extensively covers optimal navigation for macroscopic systems using optimal control theory. Studies on biological systems include animal foraging and T-cell target search. Recent work has focused on optimal navigation of microswimmers and dry active particles, often employing reinforcement learning to find optimal steering strategies in various scenarios, including point-to-point navigation and navigation in external flow fields. Analytical approaches have also been developed to complement these studies. While these works have advanced the field, the crucial role of hydrodynamic interactions between microswimmers and boundaries (walls, obstacles) in shaping optimal navigation has remained largely unaddressed. This study aims to fill this gap by explicitly considering these interactions.

The authors investigate the optimal microswimmer navigation problem through a combination of analytical and numerical approaches. They model a self-propelling active particle interacting with a 3D fluctuating environment, where the particle's velocity depends on its swimming direction and hydrodynamic interactions with walls or obstacles. The authors focus on minimizing travel time to a predefined target, considering the swimmer's ability to freely control its swimming direction but not its speed. The model incorporates different types of hydrodynamic interactions: source dipole, force dipole, and force quadrupole, representing various types of microswimmers. For source dipole swimmers, they eliminate the steering angle from the equations of motion to derive a Lagrangian, which is then numerically solved to find the optimal path. For force dipole and force quadrupole swimmers, similar Lagrangians are derived and solved. For fluctuating environments, they compare two navigation strategies: 'straight swimming' (always heading directly toward the target) and 'optimal swimming' (recalculating the optimal path at each time step). Finally, the authors explore time-dependent microswimmers with oscillatory flow fields, leveraging Pontryagin's maximum principle from optimal control theory to determine necessary conditions for optimal trajectories.

The key findings demonstrate that the shortest path is not always the fastest for microswimmers due to hydrodynamic interactions. Even in the absence of external fields, microswimmers often take detours to reach their targets faster. The shape and nature of these detours are highly dependent on the type of hydrodynamic interaction (source dipole, force dipole, force quadrupole) and the sign of the relevant singularity coefficients. Source dipole swimmers with positive singularity coefficients (σ > 0) avoid walls, while those with negative coefficients (σ < 0) approach them. Surprisingly, force dipole swimmers (pushers and pullers) show identical optimal trajectories, although their steering strategies differ. In fluctuating environments, the 'optimal swimming' strategy significantly outperforms the 'straight swimming' strategy across a range of noise levels, highlighting the importance of strategically using hydrodynamic interactions to mitigate the effects of fluctuations. Analysis of time-dependent microswimmers reveals complex trajectories with step-plateau-like structures, whose travel times depend non-monotonically on frequency and amplitude parameters. Parameter regime analysis suggests that the observed effects are relevant for typical microswimmers, even at moderate distances from walls.

This study's findings directly address the limitations of previous research by demonstrating that hydrodynamic interactions substantially alter optimal navigation strategies for microswimmers. The results challenge the intuition that the shortest path is always the fastest, emphasizing the importance of considering the microswimmer's interaction with its environment. The significant improvement of the 'optimal swimming' strategy in fluctuating environments suggests that microorganisms may have evolved strategies to exploit hydrodynamic interactions to enhance their navigation efficiency and survival. The model's versatility, encompassing different types of hydrodynamic interactions and time-dependent scenarios, provides a robust framework for further research into microswimmer navigation.

This research establishes that hydrodynamic interactions fundamentally change optimal microswimmer navigation compared to dry active particles or macroagents. Microswimmers strategically utilize detours to reach targets faster, an effect amplified by fluctuations. This suggests microorganisms may exploit wall proximity for navigation efficiency. Future work could explore more complex environments (penetrable boundaries, viscosity gradients) and investigate optimal strategies in partially known environments using machine-learning techniques.

The study's model employs certain simplifications, such as neglecting rotational diffusion and assuming constant swimming speed. While the superposition approximation used for multiple obstacles is common, its accuracy could be limited in certain configurations. The analysis of time-dependent swimmers focuses on a specific example, and a more comprehensive exploration of various time-dependent scenarios may be valuable.