Chapter 4 . Biologically Realistic Bundles

Chapter 4 . Biologically realistic bundles

Data representing three actual hair cells has been obtained and will be used to analyze their mechanics. All three cells are found in the vestibular system of the turtle, Trachemys (Pseudemys) scripta. Two of the cells are found in the crista of the posterior semicircular canal, and one on medial side of the utricle.

Common Characteristics

All stereocilia within these bundles are assumed to have a Young’s modulus of 3×109 N/m2 and a shear modulus of 1×106 N/m2. Each stereocilium was given a maximum element size of one micron. The tapered base was approximated by three to five elements of increasing diameter.

Tip links are assumed to follow loose packing description of Bagger-Sjoback [1988], that is each stereocilium extends a single tip link towards the stereocilium immediately next to it in the excitatory direction of the bundle. Tip link diameter is assumed to be 5 nm and the link is assumed to rise 45 degrees to the taller stereocilium in the undeformed state. The Young’s modulus of the tip links is assumed to be 3×106 N/m2, which corresponds to the stiffness of a protein such as elastin.

Side links (also called subapical bands or lateral links) are assumed to be present over the upper two microns of the shorter stereocilium of each linked stereocilium pair. They are spaced every quarter micron, for a total of nine links. They were given a diameter of 9 nm. Although in reality, the links may be thinner in diameter, the excess diameter in the modeled side links was chosen to give the links stiffness reflective of having several links (specifically three of 5 nm each) at each point. The exact number of three was chosen to give similarity with the earlier work of Duncan[1993]. The side link’s Young’s modulus was set to 3×106 N/m2. Switches were set in the bmod program to allow link buckling and radial extension.

The force step size was chosen small enough to capture the functional relationship between force and stiffness, as well as to limit the number of links buckling each step, and was verified by reducing the force step size and observing convergence. At each step, the solution accuracy was checked by examining the condition number of the global stiffness matrix.

Crista Type I (large) cell

The first cell studied will be taken from the crista of the posterior semicircular canal. The crista of the turtle contains both of the two types of hair cells. Type I cells are characterized by a large bulb shaped cell body. The bundles of type I cells are longer, thicker and contain more cilia than type II cells [Peterson, et. al., 1996].

This cell is found in what is referred to as the central area. There are two types of type I cells found in this area [ibid.]. In the large cells, there are more stereocilia and the bundle is

60 longer and wider. There are also small cells, so called because their bundle is shorter and narrower.

Model creation

For our idealized type I (large) cell, the geometric distribution of cilia across the cuticular plane is as seen below. There are 81 cilia distributed over 11 columns, ranging from 3 to 11 cilia per column. 1 23 465 78910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 34 3536 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 7273 74 75 76 77 78 79 80 81

Figure 4-1. Top view showing ciliary layout of Type I large cell.

The ciliary heights are distributed as seen below in figure 4.2, ranging from 30 to 4 microns tall.

25 Ciliary Height (µm) 20

10 1 0.6 0.2 5 -0.2 0 -0.6 0 0.24 -1 0.48 0.72 0.96 1.2 1.44 1.68 1.92 2.16

Figure 4-2. Height distribution of Type I (large) cell

61 For modeling purposes, the stereocilia all have a thickness of 0.20 µm, which tapers to a diameter at insertion of 0.12 µm. The height of the taper section will be 1.0 µm. The center-to- center spacing of neighboring cilia the bundle is 0.24 µm.

Results

The deformed shape of the bundle under a force of 200 pN is shown in figure below.

Figure 4-3. Deformed shape of the Type I (large) cell of the posterior semicircular canal.

62 The initial stiffness of this cell was determined to be 82.2 pN/µm. Using the nonlinear analysis techniques described in Chapter 2, stiffness could be seen to increase 24 percent as the deflection increased from 0 to 100nm, as seen in figure 4-4.

125.00

120.00

115.00

110.00 m) µ 105.00

100.00

95.00 Stiffness (Pn/

90.00

85.00

80.00 0.01 0.1Deflection (µm) 1 10

Figure 4-4. Stiffness as a function of deflection for the Type I (large) cell of the posterior semicircular canal.

For an applied force of 200 pN, the tension of the first link (between cilia numbers 1 and 5 in figure 4-1) was 10.57 pN. Relative values of tip link tensions of the cell are depicted below in figure 4-5.

63 1.00 0.95 0.95 0.52 0.70 0.52 0.27 0.44 0.44 0.27 0.34 0.38 0.34 0.29 0.31 0.31 0.29 0.26 0.27 0.26 0.27 0.24 0.24 0.27 0.11 0.21 0.21 0.21 0.11 0.16 0.19 0.19 0.16 0.12 0.16 0.17 0.16 0.12 0.04 0.13 0.14 0.14 0.13 0.04 0.08 0.11 0.12 0.11 0.08 0.03 0.07 0.09 0.09 0.07 0.03 0.04 0.07 0.07 0.07 0.04 0.04 0.06 0.06 0.04 0.04 0.04 0.04 0.03 0.03 0.02

Figure 4-5. Relative tip link tensions across a Type I (large) cell of the posterior semicircular canal under a load of 200 pN. Gray shading of cilia is reflective of relative tip link tension above it.

Crista Type II cell

The second cell is also taken from the crista of the posterior semicircular canal. However, it is a type II cell, and its ciliary bundle is shorter in length and width than the type I cell. Also the ciliary heights drop more rapidly along the length of the column.

Model Creation The geometric distribution of cilia cell across the cuticular plane in the type II is as seen below in figure 4-6. There are 39 cilia distributed over 7 columns of 5 to 7 cilia per column. The ciliary heights are distributed as seen below in figure 4-7, ranging from 30 to 4 microns tall.

64 1 23 465 78910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3233 34 35 36 37 38 39

Figure 4-6. Top view showing the ciliary location of the type II cell.

Cilary height 15 (µm)

10 0.6 0.4 0.2 5 0 -0.2 -0.4 0 -0.6 0 0.6 0.12 0.24 0.36 1.2 0.48 0.72 0.84 0.96 1.08 1.32 1.44

Figure 4-7. Height distribution of type II cell.

As with the large Type I cell, the stereocilia all have a thickness of 0.20 µm, which tapers to a diameter at insertion of 0.12 µm. The height of the taper section will be 1.0 µm. The center- to-center spacing of cilia in the bundle is 0.24 µm.

Results

The deformed shape of the bundle under a force of 200 pN is seen below in figure 4-8.

65 Figure 4-8. Deformed shape of the Type II cell of the posterior semicircular canal.

The initial stiffness of this cell was determined to be 54.1 pN/µm. As the cell is deformed, its stiffness increases 41 percent over a deflection range of 0 to 100 nm, as seen in the figure 4-9 below.

66 100.00

90.00

m) 80.00 µ

70.00

60.00 Stiffness (pN/ 50.00

40.00 0.001 0.01 0.1 1 10 Deflection (µm)

Figure 4-9. Stiffness as a function of deflection for the type II cell of the posterior semicircular canal.

For an applied force of 200 pN, the tension of the first link (between cilia numbers 1 and 5 in figure 4-1) was 13.04 pN. Relative values of tip link tensions of the cell are depicted below in figure 4-10.

1.00 0.89 0.89 0.49 0.45 0.49 0.33 0.45 0.45 0.33 0.31 0.29 0.31 0.21 0.29 0.29 0.21 0.21 0.20 0.21 0.14 0.18 0.18 0.14 0.11 0.11 0.11 0.07 0.08 0.08 0.07 0.05

Figure 4-10. Tip link tension magnitudes for Type II cell from posterior semicircular canal.

Utricular cell

For the third cell, we move to the utricle. The cell is found on the medial side of the striolar region. This cell is a type II cell also, but is quite different from the previous cell. It has a shorter kinocilium than previous cells (14 µm), has very short stereocilia (all less than 4 µm), has fewer columns, and is quite long.

67 Model Creation The geometric distribution of cilia across the cuticular plate for the utricular cell is idealized as seen in figure 4-11 below. There are 44 cilia distributed over 5 columns with 8 to 10 cilia per column.

1 23 465 7 8 910 11 12 13 14 15 16 17 18 19 20 21 22 23 2425 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Figure 4-11. Top view showing ciliary distribution of utricular hair cell.

Ciliary heights were assigned as seen in figure 4-12.

Ciliary height 8 (µm) 6

2 2 1 0 0 -1 1 2 3

4 -2 5 6 7 8

Figure 4-12. Height distribution of medial utricular hair cell.

The data collection for this cell included the addition of a kinocilium to this bundle, which is the forced cilia (numbered 1 in figure 4-11). The kinocilium has a diameter of 0.35 µm, and was modeled as being isotropic, with a Young’s modulus of 1.3 x 108 N/m2. To reflect biologic observations, the kinocilum was not tapered. The remainder of cilia were stereocilia. Stereocilia

68 were modeled as having a diameter of 0.2 µm, a base diameter of 0.1 µm, and a taper height of 0.9 µm. Stereocilia are spaced at 0.3 µm.

Results

The deformed shape of the utricular hair cell bundle is seen in figure 4-13 below.

Figure 4-13 Deformed shape of the utricular hair cell bundle.

69 The initial stiffness of this cell was determined to be 150 pN/µm. Interestingly enough, the stiffness of the kinocilium alone is 105 pN/µm. When nonlinear techniques were added to the simulation of this bundle, stiffness increased slightly as seen in figure 4-14. The small dip in stiffness results in the loss of energy from side links buckling.

158.00

156.00 m) µ 154.00

152.00 Stiffness (Pn/ 150.00

148.00 0.01 0.1 1 10 Deflection (µm)

Figure 4-14. Stiffness as a function of deflection for the utricular hair cell bundle.

For an applied force of 200 pN, the tension of the first link (between cilia numbers 1 and 5 in figure 4-1) was 8.86 pN. Relative values of tip link tensions of the cell are depicted below in figure 4-15.

1.00 0.79 0.79 0.12 0.42 0.12 0.42 0.42 0.10 0.19 0.10 0.22 0.22 0.07 0.08 0.07 0.11 0.11 0.11 0.04 0.11 0.05 0.05 0.01 0.02 0.01 0.02 0.02 0.01 0.01 0.01 0.01 0.01

Figure 4-15. Tip link tensions in the utricular hair cell.

Discussion

70 Experimental stiffness measurements have been in the range of 10-4 to 10-2 N/m [Szymko, 1992]. The tallest cilia in these cells are roughly 4 to 10 µm. With a stiffness of 1.5×10-4 N/m, the utricular cell is within the range of reported values. Both of the canal cell’s stiffnesses are slightly below the reported range (0.82 N/m and 0.54 N/m) but with maximum heights of 30 µm, this result is not surprising.

Comparing bundles, the taller bundles of the semicircular canal are more compliant than the utricular cell bundle, which can be attributed to both the shorter utricular height, and the stiffer structure (larger diameter and no taper) of the forced kinocilium in the utricle. Type I cells are stiffer than Type II because successive stereocilia are taller and brace the first cilium. The additional cilia at the end of the type I cell are not significant in the additional stiffness based on the tests done on column length in the previous chapter as well as earlier work [Peterson, et. al., 1996].

This leads us to ask, why are there more little stereocilia in the Type I bundle? One possible answer might be to provide more tip links, and therefore more ion channels. The structure and distribution of tip link tensions seem to indicate these extra channels would open when the bundle deflects large amounts. Hence the type I cell could possibly have a wider sensitivity, be more responsive at larger deflections, or both.

Bundles from the semicircular canal exhibit significant increases in bundle stiffness as the applied force increases. In contrast, the utricular cell exhibits a nearly constant stiffness. Whether or not this nonlinear stiffness serves an actual purpose or is merely the inevitable byproduct of the structure is not known.

In all bundles, tip link tensions dropped quickly as distance from the kinocilium increased. Then the tension drop decreases. In all bundles, the tension never drops below one percent of the largest link tension.

Conclusions

It may be misleading to draw conclusions about hair cell bundles in general from these three bundles in particular. However, there seem to be several similarities between the three bundles worth noting.

As theorized, when a certain minimal tension is achieved in the tip link, the ion channel opens. This minimal tension is called the gating tension. It would seem reasonable that this gating tension is constant throughout the bundle, or even across all hair cell bundles. Since tensions are greatest nearer the kinocilium, the ion channels near the bundle’s tall end will be the first to open. Furthermore, if all links are open when the cell is displaced to saturation, then the gating tension could be estimated by the smallest tension at this deflection. For our three bundles, it seems the gating tension would need to be less than 0.1 pN.

From these three bundles, this methodology of modeling the bundle structure has been demonstrated. An interesting contrast between this technique and the experimental methods can

71 be drawn. Because of the small and delicate nature of hair cell bundles, biologic observations are limited to applied force, deflection, cell response (depolarization), and qualitative observations about deformation shape. Furthermore, the expense and time required in obtaining, preparing, and testing actual hair cell bundles greatly limits the sample size of biologic observations.

In contrast, modeling gives us a richness of data. We have complete access to the deformed position of each cilium and link tensions. We can be extremely precise in the location and amount of the applied load. Furthermore, we can test putative variations in structure and material that lead to potentially an infinite amount of data to analyze.