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Bacterial Flagellar Microhydrodynamics: Laminar Flow over Complex Flagellar Filaments, Analog Archimedean Screws and Cylinders, and Its Perturbations

Shlomo Trachtenberg ^*, Dalia Fishelov  and Matania Ben-Artzi 

^* Department of Membrane and Ultrastructure Research, The Hebrew University-Hadassah Medical School, Jerusalem 91120, Israel; Tel Aviv Academic College of Engineering, Tel Aviv 69107, Israel, and School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; and Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Correspondence: Address reprint requests to Shlomo Trachtenberg, Tel.: 972-2-675-8166; Fax: 972-2-678-4010; E-mail: shlomot{at}cc.huji.ac.il.

The flagellar filament, the bacterial organelle of motility, is the smallest rotary propeller known. It consists of 1), a basal body (part of which is the proton driven rotary motor), 2), a hook (universal joint—allowing for off-axial transmission of rotary motion), and 3), a filament (propeller—a long, rigid, supercoiled helical assembly allowing for the conversion of rotary motion into linear thrust). Helically perturbed (so-called "complex") filaments have a coarse surface composed of deep grooves and ridges following the three-start helical lines. These surface structures, reminiscent of a turbine or Archimedean screw, originate from symmetry reduction along the six-start helical lines due to dimerization of the flagellin monomers from which the filament self assembles. Using high-resolution electron microscopy and helical image reconstruction methods, we calculated three-dimensional density maps of the complex filament of Rhizobium lupini H13-3 and determined its surface pattern and boundaries. The helical symmetry of the filament allows viewing it as a stack of identical slices spaced axially and rotated by constant increments. Here we use the closed outlines of these slices to explore, in two dimensions, the hydrodynamic effect of the turbine-like boundaries of the flagellar filament. In particular, we try to determine if, and under what conditions, transitions from laminar to turbulent flow (or perturbations of the laminar flow) may occur on or near the surface of the bacterial propeller. To address these questions, we apply the boundary element method in a manner allowing the handling of convoluted boundaries. We tested the method on several simple, well-characterized cylindrical structures before applying it to real, highly convoluted biological surfaces and to simplified mechanical analogs. Our results indicate that under extreme structural and functional conditions, and at low Reynolds numbers, a deviation from laminar flow might occur on the flagellar surface. These transitions, and the conditions enabling them, may affect flagellar polymorphism and the formation and dispersion of flagellar bundles—factors important in the chemotactic response.

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