Pergamon

Adv. Space Res. Vol. 14, No. 8, pp. (8)121-(8)124, 1994 Cop)right O 1994 COSPAR

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MECHANICAL ANALYSIS OF STATOLITH ACTION IN ROOTS AND RHIZOIDS

Paul Todd

University of Colorado, Campus Box 424, Boulder, CO 80309-0424, U.S.A.

ABSTRACT

Published observations on the response times following gravistimulation (horizontal positioning) of Chara rhizoids and developing roots of vascular plants with normal and "starchless" amyloplasts were reviewed and compared. Statolith motion was found to be consistent with gravitational sedimentation opposed by elastic deformation of an intracellular material. The time required for a statolith to sediment to equilibrium was calculated on the basis of its buoyant density and compared with observed sedimentation times. In the examples chosen, the response time following gravistimulation (from horizontal positioning to the return of downward growth) could be related to the statolith sedimentation time. Such a relationship implies that the tran~duction step is rapid in comparison with the perception step following gravistimulation of rhizoids and developing roots.

INTRODUCTION

Three categories of graviresponsive statoliths can be characterized and related to one another: Chara rhizoids and developing roots of vascular plants with normal and "starchless" amyloplasts. Gravitational biology in space has yielded compelling evidence that statoliths of columella cells and rhizoids are moved and positioned by the gravity vector/1/. In this report, an attempt is made to utilize these valuable space-based findings for the quantitative mechanical characterization of statolith dynamics and statics. In particular, it is found that relative response times to shifts in inertial vector can be predicted, and that the eytoskeleton surrounding statoliths is of extremely low elastic modulus. A modified version of a common paradigm/2/ for mechanical analysis of organdies is used in this study by considering the elastic deformation of the cytoskeleton.

FORCE BALANCE

The statolith is considered as a solid sphere with radius a and density p sedimenting under the acceleration of gravity g through a medium with viscosity rl and into an elastic medium

(8)121

(8)122 P. Todd

(cytoskeleton) with elastic modulus E. The force balance at constant sedimentation velocity is

F g - F b - F a - F e = 0, (i)

where the forces are, respectively, gravitational, buoyant, drag (Stokes) and elastic (eytoskeleton), as indicated schematically in Figure 1. Substituting measurable variables and solving for the terminal velocity of the statolith yields

v = [4a2(p - Po)g- 3aE]/[18nl (2)

The elastic force EA is the forced required to displace the cytoskeleton a length At, assuming that the cytoskeleton undergoes elastic deformation according to Hooke's law. A is the cross-sectional area, and E is the elastic modulus and is calculated below.

CELL

Fb ; d

Statolith

q~ytoakeleton

Figure 1. Balance of forces on a statolith sedimenting downward under the acceleration of gravity. Opposing forces are the buoyant force, drag force, and elastic deformation of the cytoskeleton.

ELASTIC MODULUS OF THE CYTOSKELETON

The elastic modulus E of a material is defined as the stress (force on a unit area) F / A divided by the strain (fractional change in length) A~/t when an axial force is applied to a one-dimensional system such as a string, wire, rope, bar or spring. It is difficult to evaluate E for the cell eytoskeleton, and E may have specific values in different kinds of cells. However, thanks to a series of sounding-rocket experiments in which Sievers and co-workers /1 /de te rmined the positions of statoliths during low gravity, it is possible to solve equation (2) for E by setting g = 0. This approximation is justified by the very low inertial acceleration that existed during the free fall phase of the rocket flight. Rearranging equation (2) gives

E = 6nv/a, (3)

which depends on knowing the viscosity of the eytosol. For convenience of calculations, a

value of 0.5 Poise was used, although, ideally, accurately measured values should be used for each individual case.

Analysis of Sta~lith Action (8)123

It was assumed that, in 6 rain of space flight, stress on the cytoskeleton due to statoliths is completely relieved, and the microfilaments are at force equilibrium. In other words, the cytoskeleton has returned to its normal geometry in the absence of the stress of the statolith, so a measurement of the statolith displacement in low g is a measure of the distance traveled in less than 6 min. This interpretation is consistent with statolith displacements measured in cress roots in space shuttle flights /1/. Therefore v was estimated in equation (3) by dividing the measured displacement by 6 rain.

The Chara statolith can be considered as an aggregated mass, about 10 tim in radius or as single independent particles 1.5 , m in radius. The latter was assumed, because a single microfilament is very small compared the to diameter of the organelle. For cress statoliths, a = 1.5 urn. Densities of 3.1 and 1.5 g/cm a were assumed for Chara and cress rhizoids, respectively. Using these values in equation (3), the following values are found:

E(Chara) = 0.1 dyn/cm 2

E(cress) = 0.03 dyn/cm 2.

These are very low values for polymeric systems, and it appears that Chara statoliths are supported by a stiffer eytoskeleton than are cress statoliths.

RELATIVE RATES OF MOVEMENT

With an assumed viscosity and a calculated elastic modulus, it is possible to calculate relative velocities of different statoliths using equation (2). The term "relative" is used, since absolute velocities will depend on knowing viscosities for each system. Nevertheless, within the constraints of relative calculations, the following questions can be addressed: What is a reasonable cytosolic viscosity if we know the velocity of sedimentation of a statolith during gravistimulation? How do the velocities of statoliths in Chara rhizoids, normal plant roots, and "starchless" plant roots compare with each other? and How does the motion of "starchless" amyloplasts depend on their density?

30

25

20

TIM E,min 15

10

5

0 Chera cress starch(-)

Figure 2. Time required for three categories of statolith to sediment 10 #m. The "starchless" statolith was assigned a density of 1.25 for convenience of display on this scale.

(8)124 P. Todd

These questions are addressed in Table 1, which lists, in the final column, the time required for a statolith with the stated properties to sediment 10 #m. It can be seen that equation (2) correctly predicts that Chara statoliths sediment faster than cress statoliths, that an arbitrary "starchless" amyloplast sediments much more slowly, owing to its reduced density, and, at sufficiently reduced density, not at all (last line). The bar graph in Figure 2 expresses these findings. The choice of 0.5 Poise for viscosity yields values of v and t that are similar to those that have been observed.

Table 1. Calculated rates of statolith movement, using equation 2 and the values listed in the table, t is the time in min for the statolith to sediment 10 am.

CELL ~(P) a(Bm) E(dyn/cm z) p(g/cm 3) v(~m/min) t(min)

Chara O. 5 i. 5 O. i0 3.50 ii. 4 1 cress O. 5 i. 5 O. 03 i. 50 i. 8 6 starch " " " i. 25 0.3 30

less " " " 1.22 0.1 i00 " " " 1.21 0.04 250 ,t . " 1 • 20 < 0

DISCUSSION

These calculations revealed surprisingly low elastic modulus values for the cytoskeleton surrounding statoliths. These are inconsistent with expected values for muscle, for example, and even natural rubber, which has an elastic modulus at least 10,000 times that found here. This finding stimulates further questioning about a possible special nature of the eytoskeleton of columella cells and rhizoids; does its elastic modulus differ from that of all or most other cells?

Lateral roots of Phaseolus vulgaris have been shown not to be noticeably geotropic/3/ . One possible explanation is an increased elastic modulus (more nearly normal) in the columella cells of lateral, vs. primary, roots.

The above calculations were designed to determine whether or not the relative geotropic responses of three categories of statoliths could be predicted on the basis of mechanics without arguing that the statolith plays no role in the geotropic response. It was found that observed relative geotropic response rates could be predicted by the mechanics of the respective statoliths of the organisms.

REFERENCES

1. A. Sievers. Gravity sensing mechanisms in plant cells. Amer. Soc. Grav. Space Biol. Bull. 4, 43-50 (1991).

2. P. Todd. Essay: Gravity dependent phenomena at the scale of the single cell. Amer. Soc. Gray. Space Biol. Bull. 2, 95-113 (1991).

3. J. S. Ransom and R. Moore. Geoperception in primary and lateral roots of Phaseolus vulgaris (Fabacae). I. Structure of columella cells. Amer. J. Bet. 70, 1048-1056 (1983).