Vorticity dynamics and numerical Resolution of Navier-Stokes Equations

We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data for some test cases to which we apply the computational scheme

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

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

Geometric Parameters of <i>Spiroplasma</i> Cells.

<p>Geometric parameters of <i>Spiroplasma</i> cells were measured directly from high-intensity, dark-field light microscopy and cryoelectron microscopy. Using the helical symmetry of the cell, parameters were extrapolated to entire cells. STEM mass data, obtained per unit length or area, were similarly extrapolated to whole cells. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0087921#pone-0087921-g001" target="_blank"><b>Fig. 1</b></a> illustrates diagramatically several of these parameters.</p

Biochemical Analysis of Whole <i>Spiroplasma</i> Cells.

<p>Total dry mass of analyzed cells was 1,022 µg. Individual values are given as absolute weights (in µg) within this total amount, as well as in percent of the total dry mass. <i>n</i> = 3 analyses per sample.</p>1<p>Total protein, lipid, and carbohydrate content are the sum of all subtypes.</p>2<p>Unconjugated protein is defined as pure protein or peptide free of detectable carbohydrate or lipid; it is derived by subtracting the sum of lipoprotein and glycoprotein from total protein.</p>3<p>Free lipid is defined as lipids with no detectable carbohydrate or protein; it is derived by subtracting the sum of lipid subtypes from total lipid.</p>4<p>Free carbohydrate is defined as carbohydrates with no detectable lipid or protein.</p