Swimming strategies of microorganisms must conform to the principles of self-propulsion at low Reynolds numbers. Here we relate the translational and rotational speeds to the surface motions of a swimmer and, for spheres, make evident novel constraints on mechanisms for propulsion. The results are applied to a cyanobacterium, an organism whose motile mechanism is unknown, by considering incompressible streaming of the cell surface and oscillatory, tangential surface deformations. Finally, swimming efficiency using tangential motions is related to the surface velocities and a bound on the efficiency is obtained.

Measurements of the variance in rotation period of tethered cells as a function of mean rotation rate have shown that the flagellar motor of Escherichia coli is a stepping motor. Here, by measurement of the variance in rotation period as a function of the number of active torque-generating units, it is shown that each unit steps independently.

Bacteria that swim without the benefit of flagella might do so by generating longitudinal or transverse surface waves. For example, swimming speeds of order 25 microns/s are expected for a spherical cell propagating longitudinal waves of 0.2 micron length, 0.02 micron amplitude, and 160 microns/s speed. This problem was solved earlier by mathematicians who were interested in the locomotion of ciliates and who considered the undulations of the envelope swept out by ciliary tips. A new solution is given for spheres propagating sinusoidal waveforms rather than Legendre polynomials. The earlier work is reviewed and possible experimental tests are suggested.