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Cilia and flagella are highly conserved slender organelles that exhibit a variety of rhythmic beating patterns from non-planar cone-like motions to planar wave-like deformations. Although their internal structure, composed of a microtubule-based axoneme driven by dynein motors, is known, the mechanism responsible for these beating patterns remains elusive. Existing theories suggest that the dynein activity is dynamically regulated, via a geometric feedback from the cilium's mechanical deformation to the dynein force. An alternative, open-loop mechanism based on a 'flutter' instability was recently proven to lead to planar oscillations of elastic filaments under follower forces. Here, we show that an elastic filament in viscous fluid, clamped at one end and acted on by an external distribution of compressive axial forces, exhibits a Hopf bifurcation that leads to non-planar spinning of the buckled filament at a locked curvature. We also show the existence of a second bifurcation, at larger force values, that induces a transition from non-planar spinning to planar wave-like oscillations. We elucidate the nature of these instabilities using a combination of nonlinear numerical analysis, linear stability theory and low-order bead-spring models. Our results show that, away from the transition thresholds, these beating patterns are robust to perturbations in the distribution of axial forces and in the filament configuration. These findings support the theory that an open-loop, instability-driven mechanism could explain both the sustained oscillations and the wide variety of periodic beating patterns observed in cilia and flagella.

Motile cilia are used by many eukaryotic cells to transport flow. Cilia-driven flows are important to many physiological functions, yet a deep understanding of the interplay between the mechanical structure of cilia and their physiological functions in healthy and diseased conditions remains elusive. To develop such an understanding, one needs a quantitative framework to assess cilia performance and robustness when subject to perturbations in the cilia apparatus. Here we link cilia design (beating patterns) to function (flow transport) in the context of experimentally and theoretically derived cilia models. We particularly examine the optimality and robustness of cilia design. Optimality refers to efficiency of flow transport, while robustness is defined as low sensitivity to variations in the design parameters. We find that suboptimal designs can be more robust than optimal ones. That is, designing for the most efficient cilium does not guarantee robustness. These findings have significant implications on the understanding of cilia design in artificial and biological systems.

The temporal order of cancer gene mutations in tumors is essential for understanding and treating the disease. Existing methods are unable to infer the order of mutations that are identified at the same time in individual tumor samples, leaving the heterogeneity of the order unknown. Here, we show that through a complex network-based approach, which is based on the newly defined statistic -carcinogenesis information conductivity (CIC), the temporal order in individual samples can be effectively inferred. The results suggest that tumor-suppressor genes might more frequently initiate the order of mutations than oncogenes, and every type of cancer might have its own unique order of mutations. The initial mutations appear to be dedicated to acquiring the function of evading apoptosis, and some order constraints might reflect potential regularities. Our approach is completely data-driven without any parameter settings and can be expected to become more effective as more data will become available.