original posting: February 4, 2014
Hydrodynamics of Arthropod Cuticular Setae in Water
Daniel K. Hartline and Petra H. Lenz
Békésy Laboratory of Neurobiology
Pacific Biosciences Research Center
University of Hawaii at Manoa
Honolulu, Hawaii 96822
Predictions of the Humphrey et al model of movements of a cylindrical seta subject to sinusoidal water movement.
Arthropods use mechanosensory innnervation of fine hairs or "setae" attached to their body surfaces as a primary means for detecting motion of the surrounding air or water that they inhabit. The properties of the physical interaction of the setae with the surrounding medium shapes the sensory detection capabilities of the receptors. The setae act like an extended mass attached to the animal's body by a springy hinge that tends to restore the setal position after it has been deflected. In air, the setae of arthropods such as insects, spiders and scorpions, are tuned to a particular frequency component of the air movement, owing to the resonance properties of the mechanical system. The exact frequency to which they are tuned depends on the length of the seta and the spring constant of the setal hinge. The mathematics of this movement has been studied carefully in a series of classical papers from the laboratory of Friedrich Barth and several of his collaborators, in particular the late J. A. C. Humphrey (see review by Barth 2004). However, this same math applies equally well to setae of aquatic arthropods, provided the proper adjustments are made in the physical constants inserted into the model. The figure below shows model predictions for this situation. Most notably, the sharp resonance observed in setae in air is absent. Instead, the hair acts like a high-pass filter, insensitive to low frequency oscillations in the surrounding water, but responsive to frequencies (or frequency components in a comlex time-varying signal) above a "corner frequency."
Above: Predictions Humphrey et al. (1993) model. (A) Predictions of the equations generating Figure 4a of Devarakonda et al. (1996) for the steady-state motion of a cylindrical seta anchored to an infinite plane surface reacting to sinusoidal movements of sea water with a fixed peak velocity of 5 mm s-1. The solid line shows the maximum predicted angular displacement (in degrees) for a seta of length 500 μm and diameter 7 μm having no intrinsic damping (R = 0) and a spring constant (restoring force) of 4 x 10-12 N m rad-1 (left hand abscissa; solid line connecting open squares). The right abscissa (filled diamonds) plots the water displacement at the same frequencies. Note the close match between water and seta displacement, indicating that in this frequency range, the seta is closely locked to water movement, typical of the high-pass filter characteristics of a seta in water (phase angles relative to displacement, close to 0ş; not shown). Other parameters taken from Fig. 4 in Devarakonda et al. (1996), but using density of sea water (1035 kg m-3). Inset: plot of the parameters c1 and c2 of Eq (4) and (5) of PDF document against frequency (Hz) for the seta used in the main plot. Note the relative (but not perfect) constancy for frequencies above 75 Hz, which partially validates the analytically exact, though physically approximate, results described by Humphrey et al. (2001). For t = 0, initial water velocity (and seta movement) is in the negative direction. (B) Bode plots (frequency responses on a log-log scale) of Humphrey et al (Humphrey et al. 1993, Humphrey et al. 2001) model showing setal response properties characteristic of a high-pass filter. Plots show angular displacement amplitudes for setae of six different lengths to sinusoidal water displacement of fixed amplitude (1 μm). Inset: log-log plots of corner frequency (fc) and maximum seta deflection (theta-plat) against seta length (μm). For full details, download Hartline&Lenz_2014_Setal-hydrodynamics.pdf.
We are very grateful to Dr. Freiedrich Barth of Universität Wien for his constructive suggestions to improve the discussions presented here.
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