We discuss a two-dimensional model for the dynamics of axonemal deformations driven by internally generated forces of molecular motors. Our model consists of an elastic filament pair connected by active elements. We derive the dynamic equations for this system in presence of internal forces. In the limit of small deformations, a perturbative approach allows us to calculate filament shapes and the tension profile. We demonstrate that periodic filament motion can be generated via a self-organization of elastic filaments and molecular motors. Oscillatory motion and the propagation of bending waves can occur for an initially non-moving state via an instability termed Hopf bifurcation. Close to this instability, the behavior of the system is shown to be independent of microscopic details of the axoneme and the force-generating mechanism. The oscillation frequency however does depend on properties of the molecular motors. We calculate the oscillation frequency at the bifurcation point and show that a large frequency range is accessible by varying the axonemal length between 1 and 50µm. We calculate the velocity of swimming of a flagellum and discuss the effects of boundary conditions and externally applied forces on the axonemal oscillations.

We introduce the concept of self-tuned criticality as a general mechanism for signal detection in sensory systems. In the case of hearing, we argue that active amplification of faint sounds is provided by a dynamical system that is maintained at the threshold of an oscillatory instability. This concept can account for the exquisite sensitivity of the auditory system and its wide dynamic range as well as its capacity to respond selectively to different frequencies. A specific model of sound detection by the hair cells of the inner ear is discussed. We show that a collection of motor proteins within a hair bundle can generate oscillations at a frequency that depends on the elastic properties of the bundle. Simple variation of bundle geometry gives rise to hair cells with characteristic frequencies that span the range of audibility. Tension-gated transduction channels, which primarily serve to detect the motion of a hair bundle, also tune each cell by admitting ions that regulate the motor protein activity. By controlling the bundle's propensity to oscillate, this feedback automatically maintains the system in the operating regime where it is most sensitive to sinusoidal stimuli. The model explains how hair cells can detect sounds that carry less energy than the background noise. Detecting the sounds of the outside world imposes stringent demands on the design of the inner ear, where the transduction of acoustic stimuli to electrical signals takes place (1). Each of the hair cells within the cochlea, which act as mechanosensors, must be responsive to a particular frequency component of the auditory input. Moreover, these sensors need the utmost sensitivity, because the weakest audible sounds impart an energy, per cycle of oscillation, which is no greater than that of thermal noise (2). At the same time, they must operate over a wide range of volumes, responding and adapting to intensities that vary by many orders of magnitude. Clearly, some form of nonlinear amplification is necessary in sound detection. The familiar resonant gain of a passive elastic system is far from sufficient for the required demands because of the heavy viscous damping at microscopic scales (3). Instead, the cochlea has developed active amplificatory processes, whose precise nature remains to be discovered.There is strong evidence that the cochlea contains forcegenerating dynamical systems that are capable of executing oscillations of a characteristic frequency (4-10). In general, such a system exhibits a Hopf bifurcation (11): as the value of a control parameter is varied, the behavior abruptly changes from a quiescent state to self-sustained oscillations. When the system is in the immediate vicinity of the bifurcation, it can act as a nonlinear amplifier for sinusoidal stimuli close to the characteristic frequency. That such a phenomenon might occur in hearing was first proposed by Gold (3) more than 50 years ago. The idea was recently revived by Choe, Magnasco, and Hudspeth (12) in the context of a specific model of the hair cell. No gene...

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...