Mechanical Properties of a Primary Cilium from the Stochastic Motions of the Cilium Tip

Abstract

The stochastic tip dynamics of a primary cilium held within an optical trap is quantified by combining experimental, analytical and computational tools. Primary cilia are cellular organelles, present on most vertebrate cells, hypothesized to function as a fluid flow sensor. The mechanical properties of a cilium remain incompletely characterized. We measured the fluctuating position of an optically trapped cilium tip under untreated, Taxol-treated, and HIF-stabilized conditions. We applied analytical modeling to derive the mean-squared displacement of the trapped tip of a cilium and compared the results with experimental measurements. We provide, for the first time, evidence that the effective flexural rigidity of a ciliary axoneme is length-dependent, and longer cilia are stiffer than shorter cilia. We then provide a rational explanation for both effects. We demonstrate that the apparent length-dependent flexural rigidity can be understood by a combination of modeling axonemal microtubules orthotropic elastic shells and including (actin-driven) active stochastic basal body motion. It is hoped that our improved characterization of cilia will result in deeper understanding of the biological function of cellular flow sensing by this organelle. Our model could be profitably applied to motile cilia and our results also demonstrate the possibility of using easily observable ciliary dynamics to probe interior cytoskeletal dynamics. Experimental RationaleHere, we consider the primary cilium as a passive mechanical beam capable of transducing kinetic energy of flowing fluid into elastic strain (bending) energy. Bending the cilium seems to be required to initiate calcium signaling (2), and the amount of bending in response to applied flow can be characterized by the flexural rigidity (equivalently, bending modulus) (14) 'EI'. Even so, the mechanical response of primary cilia to external loading remains incompletely characterized (15). Most prior measurements of the ciliary bending modulus fall within the range of 10-50 pN*µm 2 (14), (16), but there are also measurements that report significantly higher bending modulus, as high as 300 pN*µm 2 (17). As we will show and discuss, our results potentially resolve this discrepancy but also raise important new questions. It is essential to note that prior to this report, all reported measurements of the ciliary bending modulus, with the excepton of (15), calculate the bending modulus based on fitting an image of a bent cilium to the equation for a homogeneous cantilevered beam (discussed below, in 'Analysis'). Our approach is qualitatively different and does not rely on imaging the cilium.Measuring the strained position of the cilium tip provides information about the mechanical response of the cilium to an applied mechanical load. For example, subjecting a uniform cantilever of length 'L' to a static load 'q' localized to the free end results in a static tip displacement y = q L 3 /3EI. When both L and y are known, the expression can be inverted to calculate an 'effective' bendin...