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Axons are linear processes of nerve cells that can range from a few tens of micrometers up to meters in length. In addition to external cues, the length of an axon is also regulated by unknown internal mechanisms. Molecular motors have been suggested to generate oscillations with an axon length-dependent frequency that could be used to measure an axons extension. Here, we present a mechanism that depends on the spectral decomposition of the oscillatory signal to determine the axon length.
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Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution (Hornos et al, Phys. Rev. E {bf 72}, 051907 (2005) and subsequent studies). We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.
Axonal growth and guidance at the ventral floor plate is here followed $textit{in vivo}$ in real time at high resolution by light-sheet microscopy along several hundred micrometers of the zebrafish spinal cord. The recordings show the strikingly stereotyped spatio-temporal control that governs midline crossing. Commissural axons are observed crossing the ventral floor plate midline perpendicularly at about 20 microns/h, in a manner dependent on the Robo3 receptor and with a growth rate minimum around the midline, confirming previous observations. At guidance points, commissural axons are seen to decrease their growth rate and growth cones increase in size. Commissural filopodia appear to interact with the nascent neural network, and thereby trigger immediate plastic and reversible sinusoidal-shaped bending movements of neighboring commissural shafts. Ipsilateral axons extend concurrently, but straight and without bends, at three to six times higher growth rates than commissurals, indicating they project their path on a substrate-bound surface rather than relying on diffusible guidance cues. Growing axons appeared to be under stretch, an observation that is of relevance for tension-based models of cortical morphogenesis. The textit{in vivo} observations provide for a discussion of the current distinction between substrate-bound and diffusible guidance cues. The study applies the transparent zebrafish model that provides an experimental model system to explore further the cellular, molecular and physical mechanisms involved during axonal growth, guidance and midline crossing through a combination of $textit{in vitro}$ and $textit{in vivo}$ approaches.
Filopodia are long, finger-like membrane tubes supported by cytoskeletal filaments. Their shape is determined by the stiffness of the actin filament bundles found inside them and by the interplay between the surface tension and bending rigidity of the membrane. Although one might expect the Euler buckling instability to limit the length of filopodia, we show through simple energetic considerations that this is in general not the case. By further analyzing the statics of filaments inside membrane tubes, and through computer simulations that capture membrane and filament fluctuations, we show under which conditions filopodia of arbitrary lengths are stable. We discuss several in vitro experiments where this kind of stability has already been observed. Furthermore, we predict that the filaments in long, stable filopodia adopt a helical shape.
The viscous liquid surrounding a hair bundle dissipates energy and dampens oscillations, which poses a fundamental physical challenge to the high sensitivity and sharp frequency selectivity of hearing. To identify the mechanical forces at play, we constructed a detailed finite-element model of the hair bundle. Based on data from the hair bundle of the bullfrogs sacculus, this model treats the interaction of stereocilia both with the surrounding liquid and with the liquid in the narrow gaps between the individual stereocilia. The investigation revealed that grouping stereocilia in a bundle dramatically reduces the total drag. During hair-bundle deflections, the tip links potentially induce drag by causing small but very dissipative relative motions between stereocilia; this effect is offset by the horizontal top connectors that restrain such relative movements at low frequencies. For higher frequencies the coupling liquid is sufficient to assure that the hair bundle moves as a unit with a low total drag. This work reveals the mechanical characteristics originating from hair-bundle morphology and shows quantitatively how a hair bundle is adapted for sensitive mechanotransduction.
Motivated by the formation of ring-like filament structures in the cortex of plant and animal cells, we study the dynamics of a two-dimensional layer of cytoskeletal filaments and motor proteins near a surface by a general continuum theory. As a result of active processes, dynamic patterns of filament orientation and density emerge via instabilities. We show that self-organization phenomena can lead to the formation of stationary and oscillating rings. We present state diagrams which reveal a rich scenario of asymptotic behaviors and discuss the role of boundary conditions.
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