Self-Organized Beating and Swimming of Internally Driven Filaments

Abstract

We study a simple two-dimensional model for motion of an elastic filament subject to internally generated stresses and show that wavelike propagating shapes which can propel the filament can be induced by a self-organized mechanism via a dynamic instability. The resulting patterns of motion do not depend on the microscopic mechanism of the instability but only of the filament rigidity and hydrodynamic friction. Our results suggest that simplified systems, consisting only of molecular motors and filaments, could be able to show beating motion and self-propulsion. [S0031-9007(99)08456-2] PACS numbers: 87.10. + e, 02.30.Jr, 46.25.Cc, 47.15.Gf Cilia and flagella are hairlike appendages of many cells which generate motion and are used for self-propulsion and to stir the surrounding fluid. They all share the characteristic architecture of their core structure, the axoneme, a common structural motive that was developed early in evolution. It is characterized by nine parallel pairs of microtubules, which are long and rigid protein filaments, that are arranged in a circular fashion together with a large number of dynein molecular motors [1]. In the presence of adenosine triphosphate (ATP) which is a fuel, the dynein motors attached to the microtubules generate relative forces while acting on neighboring microtubules; the resulting internal stresses induce relative sliding motion of filaments which leads to the propagation of bending waves [1,2].These biological systems are complex; they consist of a large number of different components and various patterns of motion have been observed. Attempts to model their behavior are either based on the assumption that some unknown control system generates oscillatory motor activity [3] or that a self-organized mechanism is at work [4,5]. Generically, the latter involves a dynamical instability. Theoretical studies of simple models for collective action of molecular motors have demonstrated the possibility of such instabilities [4,[6][7][8]. Several examples of oscillatory motion of biological many-motor systems are known. Recently, it was suggested that spontaneous oscillations observed in muscles could be a property of the motor-filament system alone [7,9]. This idea is supported by the fact that the oscillations continue to exist after all regulatory systems are removed [9] but also by the observation that an in vitro motor-filament system shows the signature of a dynamic transition [10]. Furthermore, the observations that flagellar dyneins are able to generate oscillatory motion on microtubules [11] and that isolated and demembranated flagella in solution containing ATP above a threshold concentration swim with a simple wavelike motion [12] support the idea that basic types of flagellar beating could result from a dynamic instability. Eventually, the beating motion of flagella such as those of sea urchin sperms is planar, which suggests that basic properties can already be captured in a two-dimensional description [2].In this article, we introduce a simple two-dimensional mode...

Floppy swimming: Viscous locomotion of actuated elastica

“…This equation was also derived in earlier work by Machin in the context of wave propagation in the flagella of swimming microorganisms [16,17]. A similar theoretical treatment was proposed as a simple model of the sliding filament model of eukaryotic axonemal beating by coupling the elasto-hydrodynamics problem with models for the behavior of active molecular motors [18,19].…”

“…This equation was also derived in earlier work by Machin in the context of wave propagation in the flagella of swimming microorganisms [16,17]. A similar theoretical treatment was proposed as a simple model of the sliding filament model of eukaryotic axonemal beating by coupling the elasto-hydrodynamics problem with models for the behavior of active molecular motors [18,19].The main features of this problem have been successfully exploited experimentally to measure the bending modulus of biopolymers (actin filaments and microtubules), either using thermal fluctuations [20] or using an active actuation [21,22]. Related studies include the dynamics of magnetic filaments [23][24][25], the three-dimensional actuation and *…”

Twirling and Whirling: Viscous Dynamics of Rotating Elastic Filaments

“…We study competition between three processes: twist injection at the rotated end, twist diffusion, and writhing. Analytical and numerical methods reveal two dynamical regimes of motion: twirling, in which the straight but twisted rod rotates about its centerline, and whirling, in which the centerline of the rod writhes and crankshafts around the rotation axis in a steady state.This work is a natural outgrowth of recent studies of forced elastica in the plane [10,11], and dynamic twistbend coupling [12][13][14]. The balance considered between elastic and viscous stresses complements that between elasticity and inertia in the inviscid limit (as in whirling shafts [15,16]), where twist waves propagate [15,17].…”

“…This work is a natural outgrowth of recent studies of forced elastica in the plane [10,11], and dynamic twistbend coupling [12][13][14]. The balance considered between elastic and viscous stresses complements that between elasticity and inertia in the inviscid limit (as in whirling shafts [15,16]), where twist waves propagate [15,17].…”

Colloidal Microworms Propelling via a Cooperative Hydrodynamic Conveyor Belt

“…From a fundamental perspective, the transport of microscopic objects in a liquid medium poses the appealing challenge to find an adequate swimming strategy due to the negligible role of inertial force compared to viscous one. At low Reynolds number the Navier-Stokes equations become time reversible [1], and any strategy based on reciprocal motion, i.e., a motion composed by symmetric backward and forward displacements, will fail to produce net propulsion [2].Facing this challenge, the last few years have witnessed the theoretical propositions of several suitable geometries and procedures to propel micromachines in viscous fluids [3][4][5][6][7][8][9]. Parallel advances in miniaturization have led to the generation of new classes of chemically powered [10,11] or externally actuated [12][13][14][15] prototypes with exciting applications in emerging fields such as microsurgery [16,17] or lab-on-a-chip technology [18,19].…”