We propose a model for the self-propulsion of the marine bacterium Synechococcus utilizing a cont... more We propose a model for the self-propulsion of the marine bacterium Synechococcus utilizing a continuous looped helical track analogous to that found in Myxobacteria [1]. In our model cargo-carrying protein motors, driven by proton-motive force, move along a continuous looped helical track. The movement of the cargo creates surface distortions in the form of small amplitude traveling ridges along the S-layer above the helical track. The resulting fluid motion adjacent to the helical ribbon provides the propulsive thrust. A variation on the helical rotor model of [1] allows the motors to be anchored to the peptidoglycan layer, where they drive rotation of the track creating traveling helical waves along the S-layer. We derive expressions relating the swimming speed to the amplitude, wavelength, and velocity of the surface waves induced by the helical rotor, and show that they fall in reasonable ranges to explain the velocity and rotation rate of swimming Synechococcus.
Geometric equivalence on nonholonomic three-manifolds
We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional... more We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional contact manifolds. Our main result is to derive the differential invariants for these structures and give geometric interpretations. We show that the symmetry group of such a structure has dimension at most four. Our motivation is to study the geometry associated with classical mechanical systems with nonholonomic constraints.
Micro-engineering pumping devices without mechani- cal parts appeared "way back" in the... more Micro-engineering pumping devices without mechani- cal parts appeared "way back" in the early 1990's. The working prin- ciple is acoustic streaming. Has Nature "rediscovered" this invention 2.7 Gyr ago? Strands of marine cyanobacteria Synechococcus swim 25 diameters per second without any visible means of propulsion. We show that nanoscale amplitude vibrations on the S-layer (a crys- talline shell outside the outer membrane present in motile strands) and frequencies of the order of 0.5-1.5 MHz (achievable by molecu- lar motors), could produce steady streaming slip velocities outside a (Stokes) boundary layer. Inside this boundary layer the flow pattern is rotational (hence biologically advantageous). In addition to this purported "swimming by singing", we also indicate other possible instantiations of acoustic streaming. Sir James Lighthill has propo- sed that acoustic streaming occurs in the cochlear dynamics, and new findings on the outer hair cel...
In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of... more In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of equivalence? can provide the invariants of nonholonomic systems on a manifold ?? with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ?? ? ???? preserve the nonholonomic connection ?????? = Proj?? ?????, ??,?? ? ??, where ????? is the Levi-Civita connection on ?? and Proj?? is the orthogonal projection over ??. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30,31,62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: ?????? = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide i...
Certain cyanobacteria, such as open ocean strains of Synechococcus, are able to swim at speeds up... more Certain cyanobacteria, such as open ocean strains of Synechococcus, are able to swim at speeds up to 25 diameters per second, without flagella or visible changes in shape. The means by which Synechococcus generates thrust for self-propulsion is unknown. The only mechanism that has not been ruled out employs tangential waves of surface deformations. In Ehlers et al, the average swimming velocity for this mechanism was estimated using the methods inaugurated by Taylor and Lighthill in the 1950's and revisited in differential geometric language by Shapere and Wilczek in 1989. In this article we propose an entirely different physical principle self propulsion based on acoustic streaming (AS). Micro-pumps in silicon chips, based on AS, have been constructed by engineers since the 1990's, but to the best of our knowledge acoustic streaming as a means of microorganisms locomotion has not been proposed before. Our hypothesis is supported by two recent discoveries: (1) In Samuel, et ...
Dedicated to Alan Weinstein on his 60th Birthday * The authors thank the Brazilian funding agenci... more Dedicated to Alan Weinstein on his 60th Birthday * The authors thank the Brazilian funding agencies CNPq and FAPERJ: a CNPq research fellowship (JK), a CNPq postdoctoral fellowship at Berkeley (PMR), a FAPERJ visiting fellowship to Rio de Janeiro (KE). (JK) thanks the E. Schrödinger Institute, Vienna, for financial support during Alanfest and the Poisson Geometry Program, August 2003.
Hamiltonian Systems and Celestial Mechanics (HAMSYS-98) - Proceedings of the III International Symposium, 2000
We have been interested in nonholonomic motion and in microswimming since two of us (JK and RM) l... more We have been interested in nonholonomic motion and in microswimming since two of us (JK and RM) listened an inspiring talk by Frank Wilczek at Cornell, about ten years ago. Our general program was presented in 3 and in particular, we described the collective" N-body ...
Rubber Rolling: Geometry and Dynamics of 2-3-5 Distributions
IUTAM Bookseries
... Montgomery and Bor give an explicit description of the infinitesimal action of G∈ on the ball... more ... Montgomery and Bor give an explicit description of the infinitesimal action of G∈ on the ball-ball system and Zelenko gave a geometric interpretation of the fundamental differential invariant constructed by Cartan. ... SERGEI B. MEDVEDEV1, VLADIMIR ZEITLIN2 ...
Spectral methods for Stokes flows: the Lorentz operator
Page 1. . Page 2. Spectral methods for Stokes flows: the Lorentz operator ∗ Kurt M.Ehlers†,Joaqui... more Page 1. . Page 2. Spectral methods for Stokes flows: the Lorentz operator ∗ Kurt M.Ehlers†,Joaquin Delgado Fernandez‡, Jair Koiller and Marco Antonio Raupp Laboratório Nacional de Computac˜ao Cientıfica R.Lauro Muller 455, Rio de Janeiro, 22290-160, RJ, Brazil and ...
Cartan's moving frames method is a standard tool in riemannian geometry. We set up the machinery ... more Cartan's moving frames method is a standard tool in riemannian geometry. We set up the machinery for applying moving frames to cotangent bundles and its sub-bundles defined by non-holonomic constraints.
Micro-swimming without flagella: Propulsion by internal structures
Regular and Chaotic Dynamics, 2011
... Meadows Community College 7000 Dandini Blvd, Reno, NV 89512 USA 2Escola de Matemática Aplicad... more ... Meadows Community College 7000 Dandini Blvd, Reno, NV 89512 USA 2Escola de Matemática Aplicada, Fundaç˜ao Getulio Vargas Praia ... gliding motility on surfaces by individual cells of Myxocococcus xanthus is associated with rotation of a continuous looped helical rotor. ...
Rubber" coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by "marble"... more Rubber" coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by "marble" coated bodies [28]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at the corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2-3-5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G 2). The 2-3-5 nonholonomic geometries are classified in a companion paper [29] via Cartan's equivalence method [19]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [21, 57, 35, 27, 39] with SO(3) symmetry group, total space Q = SO(3) × S 2 and base S 2 , that can be reduced to an almost Hamiltonian system in T * S 2 with a non-closed 2-form ω NH. In this paper we present some basic results on the spheresphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ω NH is conformally symplectic [59]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1 2 (b a − 1). Using sphero-conical coordinates we verify the results by Borisov and Mamaev [12, 13] that the system is integrable for p = −1/2 (ball over a plane) and p = −3/2 (rolling ball with twice the radius of a fixed internal ball).
Proceedings of the National Academy of Sciences, 1996
Bacteria that swim without the benefit of flagella might do so by generating longitudinal or tran... more Bacteria that swim without the benefit of flagella might do so by generating longitudinal or transverse surface waves. For example, swimming speeds of order 25 microns/s are expected for a spherical cell propagating longitudinal waves of 0.2 micron length, 0.02 micron amplitude, and 160 microns/s speed. This problem was solved earlier by mathematicians who were interested in the locomotion of ciliates and who considered the undulations of the envelope swept out by ciliary tips. A new solution is given for spheres propagating sinusoidal waveforms rather than Legendre polynomials. The earlier work is reviewed and possible experimental tests are suggested.
Sir James Lighthill proposed in 1992 that acoustic streaming occurs in the inner ear, as part of ... more Sir James Lighthill proposed in 1992 that acoustic streaming occurs in the inner ear, as part of the cochlear amplifier mechanism. Here we hypothesize that some of the most ancient organisms use acoustic streaming not only for self-propulsion but also to enhance their nutrient uptake. We focus on a motile strain of Synechococcus, a cyanobacteria whose mechanism for self-propulsion is not known. Molecular motors could work like piezoelectric transducers acting on the crystalline structure surrounding the outer cell membrane. Our calculations show that a traveling surface acoustic wave (SAW) could account for the observed velocities. These SAW waves will also produce a non-negligible Stokes layer surrounding the cell: motion within this region being essentially chaotic. Therefore, an AS mechanism would be biologically advantageous, enhancing localized diffusion processes and consequently, chemical reactions. We believe that acoustic streaming, produced by nanometer scale membrane vibrations could be widespread in cell biology. Other possible instances are yeast cells and erythrocytes. Flows generated by acoustic streaming may also be produced by silica coated diatoms along their raphe. We note that microelectromechanical (MEMS) acoustic streaming devices were first introduced in the 1990's. Nature may have preceded this invention by 2.7 Gyr.
Stokesian swimming is a geometric exercise, a collective game. In Part I, we review Shapere and W... more Stokesian swimming is a geometric exercise, a collective game. In Part I, we review Shapere and Wilczek's gauge-theoretical approach for a single organism. We estimate the speeds of organisms moving by propagating small amplitude waves, and we make a conjecture regarding a new inequality for the Stokes' curvature. In Part II, we extend the gauge theory to collective motions. We advocate the influx of nonlinear control theory and subriemannian geometry. Computationally, parallel algorithms are natural, each microorganism representing a separate processor 9 In the final section, open questions motivated by biology are presented 9
The concept of wavelength-dependent absorp-tionÅngström coefficients (AACs) is discussed and clar... more The concept of wavelength-dependent absorp-tionÅngström coefficients (AACs) is discussed and clarified for both single and two-wavelengths AACs and guidance for their implementation with noisy absorption spectra is provided. This discussion is followed by application of the concept to models for brown carbon bulk absorption spectra including the damped simple harmonic oscillator model, its Lorentzian approximation, and the band-gap model with and without Urbach tail. We show that the band-gap model with Urbach tail always has an unphysical discontinuity in the first derivative of the AAC at the band-gap-Urbach-tail matching wavelength. Complex refractive indices obtained from the bulk damped simple harmonic oscillator model are used to calculate absorption spectra for spherical particles, followed by a discussion of their features. For bulk material and small particles, this model predicts a monotonic decrease of the AAC with wavelength well above the resonance wavelength; the model predicts a monotonic increase for large particles.
Los mecanismos de bombeo en microingeniería aparecieron al principio de la década de los 90. El p... more Los mecanismos de bombeo en microingeniería aparecieron al principio de la década de los 90. El principio detrás de esto es el de flujo acústico. ¿Ha descubierto la Naturaleza este invento de hace 2.700 millones de años? Algunas cianobacterias marinas de la especie Synechococcus nadan 25 diámetros por segundo sin ningún medio visible de propulsión. Especulamos en este artículo que vibraciones de amplitud de nanoescala del estrato S (una cáscara cristalina que cubre las membranas exteriores en las cepas móviles) y con frecuencias del orden de 0,5-1,5 MHz (y esto es factible por los motores moleculares), podrían producir velocidades de deslizamiento del fluido, en el exterior de la frontera de la región Stokes. Dentro de esta capa límite (que para nuestra sorpresa resulta ser relativamente ancha) el comportamiento del flujo es rotacional (y en consecuencia, ventajoso desde el punto de vista biológico). Adicionalmente a este supuesto mecanismo que se podria llamar "nadando cantando", mostramos otros posibles ejemplos biológicos de corrientes acústicas. Sir James Lighthill ha sugerido que el flujo acústico también se da en la cóclea del oído de los mamíferos, y son muy sugerentes los nuevos hallazgos en las células ciliadas externas. Otras posibilidades son flujos acústicos producidos por vibraciones de las membranas en células de levadura, mejorando su química (¡cerveza y pan!), el contoneo de los glóbulos rojos en los tubos capilares y el bombeo de fluido producido por las diatomeas.
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