Extracting hydrodynamic properties of microscale helical filament from Brownian motion using light-sheet microscopy

A motile microscopic object, such as a swimming bacteria, experiences both Stokes flow and thermal fluctuations. While both physical processes occur simultaneously, they are often studied separately. Using the fluctuation-dissipation theorem, here we show how an analytical method combined with experimental results can determine the propulsion matrix of a single rigid microscale filament, namely E. Coli flagellum, using the Brownian motion analysis. A high resolution, single-objective light-sheet microscopy technique capable of recording a 3D movie at up to 26 volumes/sec, was used to measure the translational and rotational diffusion of an isolated individual fluorescent flagellum extracted from E. coli. Our measurements show that the propulsion matrix elements derived from analyzing the flagellum are two orders of magnitude smaller than those obtained by measurement of trapped living E. coli by optical tweezers. In addition, our preliminary data for 3D diffusion measurement revealed translational diffusion close to the surface is modified by hydrodynamic interactions between the helical filament and the boundary. At a distance of 15±3 μm from the surface, the flagellum with an average length of 7 μm diffuse at a similar rate (p-value=0.37, N=7) on its major axis (D_‖=0.21±0.04 μm²/sec) and the two orthogonal axes (D_⊥=0.20±0.03 μm²/sec). Doubling the distance to 30±3 μm from the surface leads to flagella diffusing nearly twice as fast (p-value=0.004, N=10) along its major axis (D_‖=0.30±0.05 μm²/sec) compared to the two orthogonal axes (D_⊥=0.16±0.02 μm²/sec). Our work shows that beyond measuring diffusion constants, thermal fluctuation can provide information about the propulsion matrix of microscale objects. Broadly, our findings show that the hydrodynamic evaluation of organisms at low Reynolds numbers in crowded environments can be determined from their Brownian motion without enforcing any fluid flow, which is impractical in a complex biological milieu.