In a classic paper, Edward Purcell analysed the dynamics of flagellated bacterial swimmers and derived a geometrical relationship which optimizes the propulsion efficiency. Experimental measurements for wild-type bacterial species E. coli have revealed that they closely satisfy this geometric optimality. However, the dependence of the flagellar motor speed on the load and more generally the role of the torque-speed characteristics of the flagellar motor is not considered in Purcells 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 and interestingly this condition is not satisfied by wild-type E. coli which swim 2-3 times slower than the maximum possible speed given the amount of available motor torque. Our analysis also reveals the existence of an anomalous propulsion regime, where the swim speed increases with increasing load (drag). Finally, we present experimental data which supports our analysis.
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