Finite element modeling of arachnid slit sensilla—I. The mechanical significance of different slit arrays

Abstract

Arachnid strain sensitive slit sensilla are elongated openings in the cuticle with aspect ratios (slit length l / slit width b) of up to 100. Planar Finite Element (FE) models are used to calculate the relative slit face displacements, D _c, at the centers of single slits and of arrangements of mechanically interacting slits under uni-axial compressive far-field loads. Our main objective is to quantitatively study the role of the following geometrical parameters in stimulus transformation: aspect ratio, slit shape, geometry of the slits‘ centerlines, load direction, lateral distance S, longitudinal shift λ, and difference in slit length Δl between neighboring slits. Slit face displacements are primarily sensitive to slit length and load direction but little affected by aspect ratios between 20 and 100. In stacks of five parallel slits at lateral distances typical of lyriform organs (S = 0.03 l) the longitudinal shift λ substantially influences slit compression. A change of λ from 0 to 0.85 l causes changes of up to 420% in D _c. Even minor morphological variations in the arrangements can substantially influence the stimulus transformation. The site of transduction in real slit sensilla does not always coincide with the position of maximum slit compression predicted by simplified models.

Appendix

Within the linear regime of the material the face displacements at the center of a given slit, D ^* _c , can be evaluated as a function of the slit length l ₀ and the far-field strain ɛ _a as

$$D^{*}_{{\rm c}} = \frac{{D_{{\rm c}}}}{{D_{{{\rm sc}}}}} \cdot \frac{{D_{{{\rm sc}}}}}{{l_{0}}} \cdot l_{0} \cdot \frac{{\varepsilon _{{\rm a}}}}{{\varepsilon _{{\rm a,r}}}}.$$

The value for D _c /D _sc depends on the geometrical configuration and is obtained from one of the diagrams in Figs. 5, 6, 8–11 for the required aspect ratio l ₀ /b. The “reference values” for the single slit under normal loading, D _sc / l ₀ and ɛ _a,r, are taken from Table 1. In case the far-field stress σ_a is given rather than the far-field strain, the relationship

$$ D_{{\rm c}}^{*} = \frac{{D_{{\rm c}} }} {{D_{{{\rm sc}}} }} \cdot \frac{{D_{{\rm sc}} }} {{l_0 }} \cdot l_0 \cdot \frac{{E_{{\rm r}} }} {E} \cdot \frac{{\sigma _{{\rm a}} }} {{\sigma _{{\rm a,r}} }} $$

must be used, in which the quotient E _r /E accounts for the effects of the Young’s modulus.

For example at the mid-length position of the central slit (slit 3) in the oblique bar formation (Fig. 3f) in a disc with Young’s modulus E = 17 GPa, with a length of l ₀ = 100 μm, a width of b = 1 μm, a lateral spacing ofs S/l ₀ =  0.04, and a longitudinal shift of λ / l ₀ =  0.85, which is loaded by a compressive far-field stress of σ=−500 kPa, this procedure gives

$$D^{*}_{{\rm c}} = 4.2 \times 5.034 \cdot 10^{{- 5}} \cdot 100 \times 10^{{- 6}} \cdot \frac{{18 \times 10^{9}}}{{17 \times 10^{9}}} \cdot \frac{{- 500 \times 10^{3}}}{{- 0.45 \times 10^{6}}}{\hbox{m}} = 24.9\,{\hbox{nm}},$$

where D _c / D _sc ≈ 4.2 is taken from Fig. 6b.