  The fluctuation-dissipation theorem describes the intimate connection between
the Brownian diffusion of thermal particles and their drag coefficients. In the
simple case of spherical particles, it takes the form of the Stokes-Einstein
relationship that links the particle geometry, fluid viscosity, and diffusive
behavior. However, studying the fundamental properties of microscopic
asymmetric particles, such as the helical-shaped propeller used by $\textit{E.
coli}$, has remained out of reach for experimental approaches due to the need
to quantify correlated translation and rotation simultaneously with sufficient
spatial and temporal resolution. To solve this outstanding problem, we
generated volumetric movies of fluorophore-labeled, freely diffusing, isolated
$\textit{E. Coli}$ flagella using oblique plane microscopy. From these movies,
we extracted trajectories and determined the hydrodynamic propulsion matrix
directly from the diffusion of flagella via a generalized Einstein relation.
Our results validate prior proposals, based on macroscopic wire helices and low
Reynolds number scaling laws, that the average flagellum is a highly
inefficient propeller. Specifically, we found the maximum propulsion efficiency
of flagella is less than 5%. Beyond extending Brownian motion analysis to
asymmetric 3D particles, our approach opens new avenues to study the propulsion
matrix of particles in complex environments where direct hydrodynamic
approaches are not feasible.
