A Proof of the Hairy Ball Theorem

Eisenberg, Murray; Guy, Robert D.

doi:10.2307/2320587

Cited by 67 publications

(61 citation statements)

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“…The resulting structures are intrinsically three-dimensional (3D), in strong contrast to the more twodimensional (2D) aggregates reported previously, such as rings formed in fascin-bundled actomyosin systems (40) and vortices formed in actomyosin networks on surfaces (42). Although planar ring structures might relate to the vortex-like states predicted by active gel theories (29,31,43), such features are intrinsically 2D patterns and are topologically not possible in 3D (44). Thus, the nonplanar, shell-like structures we observe must form by another mechanism.…”

Section: Discussionmentioning

confidence: 76%

Active multistage coarsening of actin networks driven by myosin motors

Silva

Depken

Stuhrmann

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

209

174

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In cells, many vital processes involve myosin-driven motility that actively remodels the actin cytoskeleton and changes cell shape. Here we study how the collective action of myosin motors organizes actin filaments into contractile structures in a simplified model system devoid of biochemical regulation. We show that this self-organization occurs through an active multistage coarsening process. First, motors form dense foci by moving along the actin network structure followed by coalescence. Then the foci accumulate actin filaments in a shell around them. These actomyosin condensates eventually cluster due to motor-driven coalescence. We propose that the physical origin of this multistage aggregation is the highly asymmetric load response of actin filaments: they can support large tensions but buckle easily under piconewton compressive loads. Because the motor-generated forces well exceed this threshold, buckling is induced on the connected actin network that resists motor-driven filament sliding. We show how this buckling can give rise to the accumulation of actin shells around myosin foci and subsequent coalescence of foci into superaggregates. This new physical mechanism provides an explanation for the formation and contractile dynamics of disordered condensed actomyosin states observed in vivo.active gels | molecular motors | nonequilibrium | soft condensed matter C ells undergo dramatic changes in shape and internal organization during vital processes such as migration and division. These changes involve remodeling of the cytoskeleton partly driven by collective physical interactions between molecular motors and cytoskeletal filaments. The motors use adenosine triphosphate (ATP) as fuel and hydrolyze it to actively generate forces and move along filaments (1). Multiheaded motors or complexes of motors may cross-link neighboring filaments and generate relative motion between them. Kinesin and dynein motors interact with microtubules to form the mitotic spindle, which is responsible for chromosome segregation (2). Myosin motors interact with filamentous actin (F-actin) to form complex arrays such as the contractile ring driving cell division (3, 4) and contractile networks that drive cell migration (4) and polarizing cortical flows (5, 6).To identify the biophysical processes underlying cytoskeletal organization, many in vitro model systems of purified motors and filaments that lack biochemical regulation have been recently developed. It is known that kinesins can organize microtubules into polarity sorted asters, as well as vortex or bundle states (7-10). These structures resemble physiological arrays such as the mitotic spindle (11). In contrast to microtubules, purified F-actin does not form well-defined structures when motors are added. Actin-myosin II solutions remain disordered at high levels of ATP (12-14) and generate dense condensates that appear internally unstructured if the ATP level is lowered or when the actin filaments are cross-linked (15-18). Interestingly, similar dense condensates appear in...

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Section: Discussionmentioning

confidence: 76%

Active multistage coarsening of actin networks driven by myosin motors

Silva

Depken

Stuhrmann

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

209

174

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“…Because of the curvature, the ordered state is forced to be inhomogeneous, in general. On a sphere, polar order additionally requires topological defects or vortices (the "hairy-ball theorem" [41]) in order to satisfy the constraints imposed by the global topology of the surface. With a wellmotivated ansatz, we show that the covariant hydrodynamic model is capable of predicting generic inhomogeneous steady ordered phases that accommodate the curvature and topology of the underlying surface in a natural fashion.…”

Section: Introductionmentioning

confidence: 99%

Topological Sound and Flocking on Curved Surfaces

2017

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Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature, orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state because of the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow, the system additionally supports long-wavelength propagating sound modes that get gapped by the curvature of the underlying substrate. We analytically compute the steady-state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry-protected topological modes that get localized to special geodesics on the surface (the equator or the neck, respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.

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“…Consequently, analogous to the Hairy-Ball theorem [64], it is geometrically impossible to construct a continuous distribution of anisotropy at all interfaces between the PVs and the atrium. This problem is similar to the mathematical construction of a continuous and differentiable vector field on a connected curved surface with n holes, for example, eight holes for the human atrium.…”

Section: Geometric Characteristics Of the Pvmentioning

confidence: 99%

A mathematical model of the unidirectional block caused by the pulmonary veins for anatomically induced atrial reentry

Chun

2014

J Biol Phys

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It is widely believed that the pulmonary veins (PVs) of the left atrium play the central role in the generation of anatomically induced atrial reentry but its mechanism has not been analytically explained. To understand this mechanism, a new analytic approach is proposed by adapting the geometric relative acceleration analysis from spacetime physics based on the hypothesis that a large relative acceleration can translate to a dramatic increase in the curvature of a wavefront and subsequently to conduction failure. By verifying the strong dependency of the propagational direction and the magnitude of anisotropy for conduction failure, this analytic method reveals that a unidirectional block can be generated by asymmetric propagation toward the PVs. This model is validated by computational tests in a T-shaped domain, computational simulations for three-dimensional atrial reentry and previous in-silico reports for anatomically induced atrial reentry.

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A Proof of the Hairy Ball Theorem

Cited by 67 publications

References 0 publications

Active multistage coarsening of actin networks driven by myosin motors

Active multistage coarsening of actin networks driven by myosin motors

Topological Sound and Flocking on Curved Surfaces

A mathematical model of the unidirectional block caused by the pulmonary veins for anatomically induced atrial reentry

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