Undulating surface model for the motility of bacteria gliding on a layer of non-Newtonian slime

Gliding bacteria are a taxonomically heterogeneous group of rod-shaped prokaryotes that adhere to and translocate (parallel to their longitudinal axis) on certain substrata by mechanism(s) that are unknown. Organelles of motility, such as flagella, have not been observed on these bacteria. Furthermore, their rigidified cell walls preclude amoeboid-type locomotion. Here, we examine a hydrodynamical model of motility involving an undulating cell surface which transmits stresses through a layer of exuded slime to the substratum. It is shown that this mechanism generates a force that can propel the glider at a realistic speed and which requires an output of power that is much less than the organism's metabolic rate of energy production. For the case of a slime that is Newtonian, we find that the lift force on the glider is zero. Therefore, there is one degree of freedom among the dimensionless variables formed from the quantities: speed of the glider; amplitude, wavelength and phase speed of the undulations, and the thickness and viscosity of the secreted slime. Optimization schemes obtain a relation among the dimensionless groups formed from these variables. However for non-Newtonian slime, represented by the model of the third-order approximation of a simple fluid, a lift force is generated due to the normal stresses. Also for this shear-thinning viscoelastic fluid, the power required for translation is reduced.