Flagella-like Beating of a Single Microtubule

Wave-like oscillations of clamped microtubules driven by collective dynein transport

Microtubules (MTs) are observed to move and buckle driven by ATP-dependent molecular motors in both mitotic and interphasic eukaryotic cells as well as in specialized structures such as flagella and cilia with a stereotypical geometry. In previous work, clamped MTs driven by a few kinesin motors were seen to buckle and occasionally flap in what was referred to as flagella-like motion. Theoretical models of active-filament dynamics and a following force have predicted that, with sufficient force and binding-unbinding, such clamped filaments should spontaneously undergo periodic buckling oscillations. However, a systematic experimental test of the theory and reconciliation to a model was lacking. Here, we have engineered a minimal system of MTs clamped at their plus ends and transported by a sheet of dynein motors that demonstrate the emergence of spontaneous traveling-wave oscillations along single filaments. The frequencies of tip oscillations are in the millihertz range and are statistically indistinguishable in the onset and recovery phases. We develop a 2D computational model of clamped MTs binding and unbinding stochastically to motors in a "gliding-assay" geometry. The simulated MTs oscillate with a frequency comparable to experiment. The model predicts the effect of MT length and motor density on qualitative transitions between distinct phases of flapping, regular oscillations, and looping. We develop an effective "order parameter" based on the relative deflection along the filament and orthogonal to it. The transitions predicted in simulations are validated by experimental data. These results demonstrate a role for geometry, MT buckling, and collective molecular motor activity in the emergence of oscillatory dynamics.

Building on-chip cytoskeletal circuits via branched microtubule networks

Controllable platforms to engineer robust cytoskeletal scaffolds have the potential to create novel on-chip nanotechnologies. Inspired by axons, we combined the branching microtubule (MT) nucleation pathway with microfabrication to develop "cytoskeletal circuits." This active matter platform allows control over the adaptive self-organization of uniformly polarized MT arrays via geometric features of microstructures designed within a microfluidic confinement. We build and characterize basic elements, including turns and divisions, as well as complex regulatory elements, such as biased division and MT diodes, to construct various MT architectures on a chip. Our platform could be used in diverse applications, ranging from efficient on-chip molecular transport to mechanical nano-actuators. Further, cytoskeletal circuits can serve as a tool to study how the physical environment contributes to MT architecture in living cells.

Active Bending of Disordered Microtubule Bundles by Kinesin Motors

Active networks of biopolymers and motor proteins in vitro self-organize and exhibit dynamic structures on length scales much larger than the interacting individual components of which they consist. How the dynamics is related across the range of length scales is still an open question. Here, we experimentally characterize and quantify the dynamic behavior of isolated microtubule bundles that bend due to the activity of motor proteins. At the motor level, we track and describe the motion features of kinesin-1 clusters stepping within the bending bundles. We find that there is a separation of length scales by at least 1 order of magnitude. At a run length of <1 μm, kinesin-1 activity leads to a bundle curvature in the range of tens of micrometers. We propose that the distribution of microtubule polarity plays a crucial role in the bending dynamics that we observe at both the bundle and motor levels. Our results contribute to the understanding of fundamental principles of vital intracellular processes by disentangling the multiscale dynamics in out-of-equilibrium active networks composed of cytoskeletal elements.

Conformation and dynamics of a tethered active polymer chain

The conformational and dynamical properties of a tethered semiflexible polymer chain under tangential active force (f_{a}) are studied by using the Langevin dynamics simulation method. The head of the polymer is fixed near an infinite flat surface at z=0. The polymer is equilibrated first at f_{a}=0 and then subjected to the active force. Under the influence of the active force, the polymer is gradually compressed. Specially, for large f_{a} and large bending rigidity (k_{b}), the polymer is buckled into a quasihelical structure rotating around the z axis at the steady state. It is found that both the radius of the quasihelical structure (R) and the angular velocity of the rotation (ω) are nearly independent of the polymer length (N), but show scaling relations with f_{a} and k_{b}, i.e., R∝f_{a}^{-1/3}k_{b}^{1/3} and ω∝f_{a}^{4/3}k_{b}^{-1/3}, which are explained by simple dynamical models. Before reaching the steady state, it is further found that the buckling velocity of the polymer is proportional to f_{a} but roughly independent of k_{b} and N, then the buckling time (t_{b}) can be described by a scaling relation t_{b}∝Nf_{a}^{-1}. The underlying mechanism of the buckling process is revealed.

A Synthetic Minimal Beating Axoneme

Cilia and flagella are beating rod-like organelles that enable the directional movement of microorganisms in fluids and fluid transport along the surface of biological organisms or inside organs. The molecular motor axonemal dynein drives their beating by interacting with microtubules. Constructing synthetic beating systems with axonemal dynein capable of mimicking ciliary beating still represents a major challenge. Here, the bottom-up engineering of a sustained beating synthoneme consisting of a pair of microtubules connected by a series of periodic arrays of approximately eight axonemal dyneins is reported. A model leads to the understanding of the motion through the cooperative, cyclic association-dissociation of the molecular motor from the microtubules. The synthoneme represents a bottom-up self-organized bio-molecular machine at the nanoscale with cilia-like properties.

Active beating modes of two clamped filaments driven by molecular motors

Biological cilia pump the surrounding fluid by asymmetric beating that is driven by dynein motors between sliding microtubule doublets. The complexity of biological cilia raises the question about minimal systems that can re-create similar patterns of motion. One such system consists of a pair of microtubules that are clamped at the proximal end. They interact through dynein motors that cover one of the filaments and pull against the other one. Here, we study theoretically the static shapes and the active dynamics of such a system. Using the theory of elastica, we analyse the shapes of two filaments of different lengths with clamped ends. Starting from equal lengths, we observe a transition similar to Euler buckling leading to a planar shape. When further increasing the length ratio, the system assumes a non-planar shape with spontaneously broken chiral symmetry after a secondary bifurcation and then transitions to planar again. The predicted curves agree with experimentally observed shapes of microtubule pairs. The dynamical system can have a stable fixed point, with either bent or straight filaments, or limit cycle oscillations. The latter match many properties of ciliary motility, demonstrating that a two-filament system can serve as a minimal actively beating model.

Spontaneously Beating Biomimetic Structures

The propulsion of motile cells such as sperms and the transport of fluids on cell surfaces rely on oscillatory bending of cellular appendages that can perform periodic oscillations. These structures are flagella and cilia. Their beating is driven by the interaction between microtubules and motor proteins and the mechanism regulating this is still a puzzle. One approach to address this issue is the assembling of synthetic minimal systems by using natural building blocks, e.g., microtubules and kinesin motors, which undergo persistent oscillation in the presence of ATP. An example of an autonomous molecular system is reported in this chapter. It dynamically self-organizes through its elasticity and the interaction with the environment represented by the active forces exerted by motor proteins. The resulting motion resembles the beating of sperm flagella. Assembling such minimal systems able to mimic the behavior of complex biological structures might help to unveil basic mechanisms underlying the beating of natural cilia and flagella.

Buckling instabilities and spatio-temporal dynamics of active elastic filaments

Biological filaments driven by molecular motors tend to experience tangential propulsive forces also known as active follower forces. When such a filament encounters an obstacle, it deforms, which reorients its follower forces and alters its entire motion. If the filament pushes a cargo, the friction on the cargo can be enough to deform the filament, thus affecting the transport properties of the cargo. Motivated by cytoskeletal filament motility assays, we study the dynamic buckling instabilities of a two-dimensional slender elastic filament driven through a dissipative medium by tangential propulsive forces in the presence of obstacles or cargo. We observe two distinct instabilities. When the filament's head is pinned or experiences significant translational but little rotational drag from its cargo, it buckles into a steadily rotating coiled state. When it is clamped or experiences both significant translational and rotational drag from its cargo, it buckles into a periodically beating, overall translating state. Using minimal analytically tractable models, linear stability theory and fully nonlinear computations, we study the onset of each buckling instability, characterize each buckled state, and map out the phase diagram of the system. Finally, we use particle-based Brownian dynamics simulations to show our main results are robust to moderate noise and steric repulsion. Overall, our results provide a unified framework to understand the dynamics of tangentially propelled filaments and filament-cargo assemblies.

Hydrodynamic Synchronization of Spontaneously Beating Filaments

Using a geometric feedback model of the flagellar axoneme accounting for dynein motor kinetics, we study elastohydrodynamic phase synchronization in a pair of spontaneously beating filaments with waveforms ranging from sperm to cilia and Chlamydomonas. Our computations reveal that both in-phase and antiphase synchrony can emerge for asymmetric beats while symmetric waveforms go in phase, and elucidate the mechanism for phase slips due to biochemical noise. Model predictions agree with recent experiments and illuminate the crucial roles of hydrodynamics and mechanochemical feedback in synchronization.

Beating to rotational transition of a clamped active ribbon-like filament

We present a detailed study of a clamped ribbon-like filament under a compressive active force using Brownian dynamics simulations. We show that a clamped ribbon-like filament is able to capture beating as well as rotational motion under the compressive force. The nature of oscillation is governed by the torsional rigidity of the filament. The frequency of oscillation is almost independent of the torsional rigidity. The beating of the filament gives a butterfly-shaped trajectory of the free-end monomer, whereas rotational motion yields a circular trajectory on a plane. The binormal correlation and the principal component analysis reveal the butterfly, elliptical, and circular trajectories of the free end monomer. We present a phase diagram for different kinds of motion in the parameter regime of compressive force and torsional rigidity.