Curvature and torsion in growing actin networks

Intracellular pathogens such as Listeria monocytogenes and Rickettsia rickettsii move within a host cell by polymerizing a comet-tail of actin fibers that ultimately pushes the cell forward. This dense network of cross-linked actin polymers typically exhibits a striking curvature that causes bacteria to move in gently looping paths. Theoretically, tail curvature has been linked to details of motility by considering force and torque balances from a finite number of polymerizing filaments. Here we track beads coated with a prokaryotic activator of actin polymerization in three dimensions to directly quantify the curvature and torsion of bead motility paths. We find that bead paths are more likely to have low rather than high curvature at any given time. Furthermore, path curvature changes very slowly in time, with an autocorrelation decay time of 200 seconds. Paths with a small radius of curvature, therefore, remain so for an extended period resulting in loops when confined to two dimensions. When allowed to explore a 3D space, path loops are less evident. Finally, we quantify the torsion in the bead paths and show that beads do not exhibit a significant left- or right-handed bias to their motion in 3D. These results suggest that paths of actin-propelled objects may be attributed to slow changes in curvature rather than a fixed torque

Superattracting fixed points of quasiregular mappings

We investigate the rate of convergence of the iterates of an n-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity

Fast, cheap and somewhat in control

Efforts to manipulate living organisms have raised the question of whether engineering principles of hierarchy, abstraction and design can be applied to biological systems. Here, we consider the practical challenges to controlling living organisms that must be surmounted, or at least managed, if synthetic biology and cellular bioengineering are to be productive

Iteration of quasiregular tangent functions in three dimensions

We define a new quasiregular mapping T in three dimensions that generalizes the tangent function on the complex plane and shares a number of its geometric properties. We investigate the dynamics of the family \lambda T for \lambda>0, establishing results analogous to those of Devaney and Keen for the meromorphic family \lambda tan z, \lambda>0, although the methods used are necessarily original.Comment: 24 pages, 3 figure

Microscale Fluid Behavior during Cryo-EM Sample Blotting

Blotting has been the standard technique for preparing aqueous samples for single-particle electron cryo-microscopy for over three decades. This technique removes the excess solution from a transmission electron microscope grid by pressing absorbent filter paper against the specimen before vitrification. However, this standard technique produces vitreous ice with inconsistent thickness from specimen to specimen and from region to region within the same specimen, the reasons for which are not understood. Here, high-speed interference contrast microscopy is used to demonstrate that the irregular pattern of fibers in the filter paper imposes tortuous, highly variable boundaries during the removal of excess liquid from a flat, hydrophilic surface. As a result, aqueous films of nonuniform thickness are formed while the filter paper is pressed against the substrate. This pattern of nonuniform liquid thickness changes again after the filter paper is pulled away, but the thickness still does not become completely uniform. We suggest that similar topographical features of the liquid film are produced during the standard technique used to blot EM grids and that these manifest in nonuniform ice after vitrification. These observations suggest that alternative thinning techniques, which do not rely on direct contact between the filter paper and the grid, may result in more repeatable and uniform sample thicknesses

The limits of filopodium stability

Filopodia are long, finger-like membrane tubes supported by cytoskeletal filaments. Their shape is determined by the stiffness of the actin filament bundles found inside them and by the interplay between the surface tension and bending rigidity of the membrane. Although one might expect the Euler buckling instability to limit the length of filopodia, we show through simple energetic considerations that this is in general not the case. By further analyzing the statics of filaments inside membrane tubes, and through computer simulations that capture membrane and filament fluctuations, we show under which conditions filopodia of arbitrary lengths are stable. We discuss several in vitro experiments where this kind of stability has already been observed. Furthermore, we predict that the filaments in long, stable filopodia adopt a helical shape