PLANT BIOPHYSICS - Y. I. Posudin - 2004

I. PHYSICAL PROPERTIES OF PLANT CELLS AND TISSUES

1. MECHANICAL PROPERTIES

1.4. ELASTIC PROPERTIES OF THE PLANT CELL

Spherical Cell. Let us determine The Cell wall thickness for a spherical cell of radius r (Fig. 1.4). The force balance acting on The Cell wall is expressed as:

Class="center">

where p is the intracellular pressure; σ is the stress generated within the cell wall; Δr is the cell wall thickness.

Hence:

The cell wall thickness can be found using the following expression:

Fig. 1.4. Force balance acting on the wall of a spherical cell of radius r.

Substituting typical values of mechanical parameters for a spherical cell: r = 1 µm; p = 20 atm = 2·10⁶ N·m^-2; σ = 2·10⁷ N·m^-2 into the final expression yields Δr = 50 nm.

Cylindrical cell. Cells of this shape, characteristic of Algae such as Nitella or Chara, experience mechanical stresses in their cell walls. If the radius of the cylinder is r, and the force generated by the internal hydrostatic (turgor) pressure is F_p = p · S = p · πr², this force is counterbalanced by the force resulting from the longitudinal stress σ_L developed in the cell wall (Fig. 1.5, a). The cross-sectional area subject to the longitudinal stress is S_L = 2πr · Δr, where Δr is the cell wall thickness. Consequently, the force generated in the cell wall is F_L = σ_L · 2πr · Δr. From the force balance acting on the wall (F_p = F_L), the stress in the cell wall can be determined:

whence:

The longitudinal stress acts parallel to the cylinder axis and counteracts cell elongation.

In addition, a tangential stress σ_T arises within the cell, restricting radial expansion caused by internal pressure. Let us consider a cylindrical cell cut in half (Fig. 1.5, b). The internal pressure p acts on a rectangular area of 2rL, generating a force F = p · 2rL. This force is counterbalanced by the force resulting from the tangential stress σ_T developed in the cell wall. The area subject to the tangential stress is S_T = 2rL, where L is the length of the cell wall (Fig. 1.5, b). Therefore, from the force balance (F_p = F_T), the tangential stress can be found:

whence:

Fig. 1.5. Mechanical stresses in the cell wall:

a – longitudinal stress σ_L acting parallel to the axis of a cylindrical cell and counteracting cell tensional deformation;

b – tangential stress σ_t arising in the cell wall [Nobel, 1973]. Here p is the pressure acting on the cell, r is the cell radius, dr is the cell wall thickness, and L is the cell length.