* Corresponding author. Fax: #1-717-771-8404. E-mail address: ams5@psu.edu (A.M. Siddiqui). q Deceased. International Journal of Non-Linear Mechanics 36 (2001) 743}761 An undulating surface model for the motility of bacteria gliding on a layer of non-Newtonian slime A.M. Siddiqui*, R.P. Burchard, W.H. Schwarz q Department of Mathematics, Pennsylvania State University York Campus, York, PA 17403, USA Department of Biological Sciences, University of Maryland Baltimore County, Baltimore, MD 21228, USA Department of Chemical Engineering, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA Received 6 July 1999; accepted 24 March 2000 Abstract 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 #agella, have not been observed on these bacteria. Furthermore, their rigidi"ed 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 "nd 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 #uid, a lift force is generated due to the normal stresses. Also for this shear-thinning viscoelastic #uid, the power required for translation is reduced.  2001 Elsevier Science Ltd. All rights reserved. 1. Introduction Gliding bacteria are ubiquitous, rod-shaped microorganisms that are only motile when adher- ent to di!erent kinds of solid or a semi-solid surfa- ces [1}4]. Organelles of motility, such as #agella, have not been observed on these bacteria. Electron microscopy has demonstrated that they have a rigidi"ed multi-layered wall characteristic of Gram- negative bacteria that precludes amoeboid-type locomotion. Nearly all gliding bacteria produce extracellular polymers and leave a trail of slime or mucilage [1,3]. A variety of mechanisms for motil- ity have been proposed: osmotic forces, electrokin- esis, surface tension gradients, propulsion by the secretion of slime, directional propagation of undu- lating waves (along the cell surface), rotating cell surface discs, mobile cell surface adhesion sites, and contractile components [1}7]. There is no conclus- ive evidence in support of any mechanism of motility. Since this is a taxonomically highly 0020-7462/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 2 8 - 7