Tuning the torque-speed characteristics of the bacterial flagellar motor to enhance swimming speed

In a classic paper, Purcell [Proc. Natl. Acad. Sci. U. S. A. 94, 11307 (1997)10.1073/pnas.94.21.11307] analyzed the dynamics of flagellated bacterial swimmers and derived a geometrical relationship which maximizes the propulsion efficiency. Experimental measurements for wild-type bacterial species E. coli have revealed that they closely satisfy this geometric optimality. However, dependence of the flagellar motor speed on the load and more generally the role of the torque-speed characteristics of the flagellar motor are not considered in Purcell's original analysis. Here we derive a tuned condition representing a match between the flagella geometry and the torque-speed characteristics of the flagellar motor to maximize the bacterial swimming speed for a given load. This condition is independent of the geometric optimality condition derived by Purcell. Interestingly, this condition is not satisfied by wild-type E. coli which swims 2-3 times slower than the maximum possible speed given the amount of available motor torque. Finally, we present experimental data on swimming dynamics of a cargo laden bacterial system which follows our analytical model. Our analysis also reveals the existence of an anomalous propulsion regime where the swim speed increases with increasing load (drag).