Systems in thermodynamic equilibrium are not only characterized by time-independent macroscopic properties, but also satisfy the principle of detailed balance in the transitions between microscopic configurations. Living systems function out of equilibrium and are characterized by directed fluxes through chemical states, which violate detailed balance at the molecular scale. Here we introduce a method to probe for broken detailed balance and demonstrate how such nonequilibrium dynamics are manifest at the mesosopic scale. The periodic beating of an isolated flagellum from Chlamydomonas reinhardtii exhibits probability flux in the phase space of shapes. With a model, we show how the breaking of detailed balance can also be quantified in stationary, nonequilibrium stochastic systems in the absence of periodic motion. We further demonstrate such broken detailed balance in the nonperiodic fluctuations of primary cilia of epithelial cells. Our analysis provides a general tool to identify nonequilibrium dynamics in cells and tissues.

Cilia and flagella are model systems for studying how mechanical forces control morphology. The periodic bending motion of cilia and flagella is thought to arise from mechanical feedback: dynein motors generate sliding forces that bend the flagellum, and bending leads to deformations and stresses, which feed back and regulate the motors. Three alternative feedback mechanisms have been proposed: regulation by the sliding forces, regulation by the curvature of the flagellum, and regulation by the normal forces that deform the cross-section of the flagellum. In this work, we combined theoretical and experimental approaches to show that the curvature control mechanism is the one that accords best with the bending waveforms of Chlamydomonas flagella. We make the surprising prediction that the motors respond to the time derivative of curvature, rather than curvature itself, hinting at an adaptation mechanism controlling the flagellar beat.DOI: http://dx.doi.org/10.7554/eLife.13258.001

The unicellular green alga Chlamydomonas swims with two flagella that can synchronize their beat. Synchronized beating is required to swim both fast and straight. A long-standing hypothesis proposes that synchronization of flagella results from hydrodynamic coupling, but the details are not understood. Here, we present realistic hydrodynamic computations and high-speed tracking experiments of swimming cells that show how a perturbation from the synchronized state causes rotational motion of the cell body. This rotation feeds back on the flagellar dynamics via hydrodynamic friction forces and rapidly restores the synchronized state in our theory. We calculate that this "cell-body rocking" provides the dominant contribution to synchronization in swimming cells, whereas direct hydrodynamic interactions between the flagella contribute negligibly. We experimentally confirmed the two-way coupling between flagellar beating and cell-body rocking predicted by our theory. flagellar force-velocity relation | low-Reynolds-number hydrodynamics E ukaryotic cilia and flagella are long, slender cell appendages that can bend rhythmically and thus present a prime example of a biological oscillator (1). The flagellar beat is driven by the collective action of dynein molecular motors, which are distributed along the length of the flagellum. The beat of flagella, with typical frequencies ranging from 20-60 Hz, pumps fluids, for example, mucus in mammalian airways (2), and propels unicellular microswimmers such as Paramecia, spermatozoa, and algae (3). The coordinated beating of collections of flagella is important for efficient fluid transport (2, 4, 5) and fast swimming (6). This coordinated beating represents a striking example for the synchronization of oscillators, prompting the question of how flagella couple their beat. Identifying the specific mechanism of synchronization can be difficult because synchronization may occur even for weak coupling (7). Further, the effect of the coupling is difficult to detect once the synchronized state has been reached.Hydrodynamic forces were suggested to play a significant role for flagellar synchronization already in 1951 by Taylor (8). Since then, direct hydrodynamic interactions between flagella were studied theoretically as a possible mechanism for flagellar synchronization (9-12). Another synchronization mechanism that is independent of hydrodynamic interactions was recently described in the context of a minimal model swimmer (13)(14)(15). This mechanism crucially relies on the interplay of swimming motion and flagellar beating.Here, we address the hydrodynamic coupling between the two flagella in a model organism for flagellar coordination (16-19), the unicellular green alga Chlamydomonas reinhardtii. Chlamydomonas propels its ellipsoidal cell body, which has typical diameter of 10 μm, using a pair of flagella, whose lengths are about 10 μm (16). The two flagella beat approximately in a common plane, which is collinear with the long axis of the cell body. In that plane, the two beat pa...

When the green alga Chlamydomonas reinhardtii swims, it uses the breaststroke beat of its two flagella to pull itself forward [1]. The flagellar waveform can be decomposed into a static component, corresponding to an asymmetric time-averaged shape, and a dynamic component, corresponding to the time-varying wave [2]. Extreme lightening conditions photoshock the cell, converting the breaststroke beat into a symmetric sperm-like beat, which causes a reversal of the direction of swimming [3]. Waveform conversion is achieved by a reduction in magnitude of the static component, whereas the dynamic component remains unchanged [2]. The coupling between static and dynamic components, however, is poorly understood, and it is not known whether the static component requires the dynamic component or whether it can exist independently. We used isolated and reactivated axonemes [4] to investigate the relation between the two beat components. We discovered that, when reactivated in the presence of low ATP concentrations, axonemes displayed the static beat component in absence of the dynamic component. Furthermore, we found that the amplitudes of the two components depend on ATP in qualitatively different ways. These results show that the decomposition into static and dynamic components is not just a mathematical concept but that the two components can independently control different aspects of cell motility: the static component controls swimming direction, whereas the dynamic component provides propulsion.

The bending of cilia and flagella is driven by forces generated by dynein motor proteins. These forces slide adjacent microtubule doublets within the axoneme, the motile cytoskeletal structure. To create regular, oscillatory beating patterns, the activities of the axonemal dyneins must be coordinated both spatially and temporally. It is thought that coordination is mediated by stresses or strains, which build up within the moving axoneme, and somehow regulate dynein activity. During experimentation with axonemes subjected to mild proteolysis, we observed pairs of doublets associating with each other and forming bends with almost constant curvature. By modeling the statics of a pair of filaments, we show that the activity of the motors concentrates at the distal tips of the doublets. Furthermore, we show that this distribution of motor activity accords with models in which curvature, or curvature-induced normal forces, regulates the activity of the motors. These observations, together with our theoretical analysis, provide evidence that dynein activity can be regulated by curvature or normal forces, which may, therefore, play a role in coordinating the beating of cilia and flagella.