Veikko F. Geyer

Broken detailed balance at mesoscopic scales in active biological systems

Identifying nonequilibrium dynamics Living systems clearly operate out of thermodynamic equilibrium at the molecular scale. How these activities are manifest at the cellular scale, however, has been unclear. Battle et al. use video microscopy together with statistical thermodynamics to unambiguously identify which random fluctuations at the cellular scale are out of equilibrium (see the Perspective by Rupprecht and Prost). Transitions between states obey a detailed balance in equilibrium, whereas imbalanced transitions point to nonequilibrium dynamics. For instance, nonequilibrium dynamics can be identified in the periodic beating of a flagellum and in the nonperiodic fluctuations of primary cilia. Science, this issue p. 604; see also p. 514 Nonequilibrium dynamics can be identified in randomly fluctuating mesoscopic systems. 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.

Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga

Significance The eukaryotic flagellum is a best-seller of nature: These slender cell appendages propel sperm and many other microswimmers, including disease-causing protists. In mammalian airways or the oviduct, collections of flagella beat in synchrony to pump fluids efficiently. Flagellar synchronization was proposed to rely on mechanical feedback by hydrodynamic forces, but the details are not well understood. Here, we used theory and experiment to elucidate a mechanism of synchronization in the model organism Chlamydomonas, a green algal cell that swims with two flagella like a breaststroke swimmer. Our analysis shows how synchronization arises by a coupling of swimming and flagellar beating and characterizes an exemplary force–velocity relationship of the flagellar beat. 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.

Dynamic curvature regulation accounts for the symmetric and asymmetric beats of Chlamydomonas flagella

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

Motor Regulation Results in Distal Forces that Bend Partially Disintegrated Chlamydomonas Axonemes into Circular Arcs

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.

Reconstitution of Flagellar Sliding

The motile structure within eukaryotic cilia and flagella is the axoneme. This structure typically consists of nine doublet microtubules arranged around a pair of singlet microtubules. The axoneme contains more than 650 different proteins that have structural, force-generating, and regulatory functions. Early studies on sea urchin sperm identified the force-generating components, the dynein motors. It was shown that dynein can slide adjacent doublet microtubules in the presence of ATP. How this sliding gives rise to the beating of the axoneme is still unknown. Reconstitution assays provide a clean system, free from cellular effects, to elucidate the underlying beating mechanisms. These assays can be used to identify the components that are both necessary and sufficient for the generation of flagellar beating.

Automatic optimal filament segmentation with sub-pixel accuracy using generalized linear models and B-spline level-sets

Biological filaments, such as actin filaments, microtubules, and cilia, are often imaged using different light-microscopy techniques. Reconstructing the filament curve from the acquired images constitutes the filament segmentation problem. Since filaments have lower dimensionality than the image itself, there is an inherent trade-off between tracing the filament with sub-pixel accuracy and avoiding noise artifacts. Here, we present a globally optimal filament segmentation method based on B-spline vector level-sets and a generalized linear model for the pixel intensity statistics. We show that the resulting optimization problem is convex and can hence be solved with global optimality. We introduce a simple and efficient algorithm to compute such optimal filament segmentations, and provide an open-source implementation as an ImageJ/Fiji plugin. We further derive an information-theoretic lower bound on the filament segmentation error, quantifying how well an algorithm could possibly do given the information in the image. We show that our algorithm asymptotically reaches this bound in the spline coefficients. We validate our method in comprehensive benchmarks, compare with other methods, and show applications from fluorescence, phase-contrast, and dark-field microscopy.

Curvature regulation of the ciliary beat through axonemal twist.

Cilia and flagella are hairlike organelles that propel cells through fluid. The active motion of the axoneme, the motile structure inside cilia and flagella, is powered by molecular motors of the dynein family. These motors generate forces and torques that slide and bend the microtubule doublets within the axoneme. To create regular waveforms the activities of the dyneins must be coordinated. It is thought that coordination is mediated by stresses due to radial, transverse, or sliding deformations, that build up within the moving axoneme. However, which particular component of the stress regulates the motors to produce the observed flagellar waveforms remains an open question. To address this question, we describe the axoneme as a three-dimensional bundle of filaments and characterize its mechanics. We show that regulation of the motors by radial and transverse stresses can lead to a coordinated flagellar motion only in the presence of twist. By comparison, regulation by shear stress is possible without twist. We calculate emergent beating patterns in twisted axonemes resulting from regulation by transverse stresses. The waveforms are similar to those observed in flagella of Chlamydomonas and sperm. Due to the twist, the waveform has non-planar components, which result in swimming trajectories such as twisted ribbons and helices that agree with observations.

Characterization of the Beat of Chlamydomonas Axonemes

The axoneme is an evolutionarily conserved mechanical apparatus within cilia and flagella made up of microtubules, several different axonemal dynein heavy chains, and accessory proteins. The aim of this study was to determine how the shape of the axonemal beat depends on the dynein composition and the chemical and mechanical properties of the axoneme.We used high-speed microscopy to record the shapes of beating Chlamydomonas axonemes. Through image analysis we measured the amplitudes of the Fourier modes that characterize the shapes of regularly beating axonemes. We then used the Fourier description to compare the waveforms of wildtype and mutant axonemes. By changing chemical conditions (e.g. ATP concentration) we determined how the shape correlates with the beat frequency and by comparing the waveforms of intact cells and isolated, reactivated axonemes, we determined how the shape correlates with hydrodynamic loading and boundary conditions.We anticipate that these data will provide insight into how the axonemal beat depends on the chemomechanical interactions between the dynein molecules.

Direct Measurement of the Force-Velocity Relationship for Multiple Kinesin-1 Motors by Magnetic Tweezers

Transport of intracellular cargo is known to be achieved by the concerted operation of multiple motor proteins. However, force generation by multiple motors remains a matter of debate even though the load-bearing properties of single motors have intensively been characterized by various in vitro assays. Here, we report a novel assay to study cooperative transport in the presence of external load. In particular, we designed a magnetic-tweezers setup that is capable of exerting horizontal forces of up to 100 pN on superparamagnetic beads that are attached to microtubules gliding on a surface coated with kinesin-1 motors. Dependent on the magnitude and direction of the applied load, we demonstrate the redirection, stalling and backward slipping of moving microtubules. Moreover, for transport events involving less than 10 motors, we precisely determine the force-dependent gliding velocities from the fluorescent signal of the magnetic bead using an automated tracking algorithm. At constant load, we observe velocity-steps which we hypothesize to result from transitions in the number of engaged motors. At variable load, i.e. by monotonically increasing or decreasing the magnetic force, we directly measure force-velocity curves during multi-motor transport. Our method, which allows the characterization of a dynamic multi-motor system in terms of forces and velocities, is expected to elucidate general properties of intracellular cargo transport by a small number of kinesin-1 or other microtubule motors.