Growth of Walled Cells: From Shells to Vesicles

The growth of isolated walled cells is investigated. Examples of such cells range from bacteria to giant algae, and include cochlear hair, plant root hair, fungi, and yeast cells. They are modeled as elastic shells containing a liquid. Cell growth is driven by fluid pressure and is is similar to a plastic deformation of the wall. The requirement of mechanical equilibrium leads to two new scaling laws for cell size that are in quantitative agreement with the compiled biological data. Given these results, possible shapes for growing cells are computed by analogy with those of vesicle membranes.

Figure 1

Physical picture of cell growth. (a) Schematic of a model cell with radius of curvature $R$ and wall thickness $h$, whose growth is driven by the inner fluid pressure $P$. (b) Stress-strain $\sigma (a)$ (solid line) and growth rate $G(a)$ (dashed line) curves for the cell wall, which is assumed to be a perfectly plastic material yielding only in extension. The wall is elastic (with modulus $E$) for a strain $a$ smaller then the yield threshold $a_y$ and it grows above. If the stress is decreased, the released elastic energy (the shaded area) is $1/2\times E{a}_y^2$.

Figure 2

Test of the scaling law Eq. (4): cell radius $R$ as a function of $h(E/P{)}^{1/3}$ ($h$ is the wall thickness, $E$ its elastic modulus, and $P$ the inner pressure). The data were compiled from the literature. When not listed, the lengths were measured from the published photographs. All these cells are cylindrical except the spherical yeast. $R$ is the radius of the cylindrical part of the cell or the radius of the sphere. When $E$ or $P$ was not available, they were replaced by their values for similar species. Alga cells: Chara corallina (□) [15, 16], Nitella (▽) [17, 18, 19], and Acetabularia acetabulum (◇) [15, 16, 18, 19, 20]. Root hair: Arabidopis thaliana (△) [15, 18, 21, 22]. Yeast: Saccharomyces cerevisiae (⊞) [23]. Fungi: Phycomyces (⊠) [24, 25, 26] and Saprolegnia ferax (⊙) [24, 27, 28]. Guinea pig cochlear outer hair cell (●) [29]. Bacteria: Magnetospirillum gryphiswaldense (■) [30], Bacillus subtilis (◆) [2, 9, 31], Escherichia coli (▲) [32], Pseudomonas aeruginosa (▼) [32], and Saccharopolyspora erythraea (⊡) [30, 33]. Gas vesicles (included for comparison): Microcystis spirillum (+) [34] and Anabeana flos-aquae (×) [34] are small crystalline vesicles found in some bacteria. Solid line: best fit to Eq. (4) of all data except filled symbols and gas vesicles.

Figure 3

Experimental testing of the scaling laws Eqs. (4, 5) for cell radii. Cell aspect ratio $R/h$ as a function of the modulus to pressure ratio $E/P$. Same symbols as in Fig. 2. Solid lines: best fit of filled symbols to Eq. (5) and of all other symbols except gas vesicles to Eq. (4).

Figure 4

Shape $z(r)$ of a tip growing cell according to Eqs. (10, 11). The surface is axisymmetric with respect to the $z$ axis and matches onto a cylinder. Lengths are nondimensionalized by the radius of the cylinder. (a) 3D view. (b) Cut along a plane of symmetry.