Physical Limits and Design Principles for Plant and Fungal Movements

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Science  27 May 2005:
Vol. 308, Issue 5726, pp. 1308-1310
DOI: 10.1126/science.1107976


The typical scales for plant and fungal movements vary over many orders of magnitude in time and length, but they are ultimately based on hydraulics and mechanics. We show that quantification of the length and time scales involved in plant and fungal motions leads to a natural classification, whose physical basis can be understood through an analysis of the mechanics of water transport through an elastic tissue. Our study also suggests a design principle for nonmuscular hydraulically actuated structures: Rapid actuation requires either small size or the enhancement of motion on large scales via elastic instabilities.

From the twirling circumnutation of growing tendrils to the opening and closing of stomata to the growth of fungal hyphae (1), plants and fungi are moving all of the time, often too slowly to notice. Rapid movements, though rarer, are used by many plants in essential functions such as seed or sporangium dispersal (Dwarf mistletoe, Hura crepitans, and the fungus Pilobulus); pollen emplacement (Catasetum orchids and Stylidium triggerplants); defense (Mimosa); and nutrition (Venus flytrap, carnivorous fungi). The mechanisms involved in these movements are varied: Hura crepitans (2) uses explosive fractures to disperse seeds at speeds as great as 70 m/s, the Venus flytrap (3) uses an elastic buckling instability to catch insects in 0.1 s, and the noose-like carnivorous fungus Dactylaria brochophaga (4) traps nematodes in less than 0.1 s by swelling rapidly.

The diversity of these nonmuscular hydraulic movements, often referred to as nastic movements, raises two related questions: Is there a physical basis for their classification? What, if any, are the principles underlying the biological designs for rapid movements in plants and fungi? To address these, we note that plants and fungi have a common feature that allows us to consider them together here: a cell wall that allows their cells to sustain a large internal (turgor) pressure of up to 10 atmospheres that can be harnessed for growth and motion. Indeed, movements are eventually driven by differential turgor, which may be regulated actively [e.g., by osmotic control as in stomata (5)] or passively [e.g., by differential drying as in Hura crepitans (2)]. In either case, the speed is limited by the rate of fluid transport. Thus, a biophysical characterization of these movements requires knowledge of both the duration of the movement τ and the distance through which the fluid is transported L, which is usually the smallest macroscopic dimension of the moving part. In Fig. 1, we plot τ vs. L and see two categories of movements dominated either by swelling or by elastic instabilities, separated by a dashed line.

Fig. 1.

Classification of plant and fungal movements. The duration of the movement τ is plotted as a function of L, the smallest macroscopic dimension of the moving part (see SOM for detailed references). The dashed line τp = 1.6 L2 s/mm2 characterizes the poroelastic time, whereas the solid line τi = 10-5 L s/mm characterizes the inertial time. These lines set performance limits on plant and fungal movements while classifying them into two categories: those limited by fluid transport, i.e., τp < τ, and those that use elastic instabilities to go beyond, eventually limited by inertia, i.e., τi < τ < τp. The elastic instabilities can be further categorized as either snap-buckling or explosive fracture. Both types of instabilities rely on geometries capable of gradually storing elastic energy and suddenly releasing it. The difference between the two groups is a matter of how the energy is released: Snap-buckling is associated with a rapid geometric change of a thin shell that does not rupture; explosive fracture involves a rapid geometric change from tissue tearing. The order of the labels in the figure legend coincides with their order in the figure from top to bottom.

To understand this boundary, we recall the physics of water movements through a porous elastic material such as plant tissue. One limit to the speed of movement is determined by the time taken to transport water across the tissue, of characteristic thickness L. Because the fluid and the tissue material are approximately incompressible, the movement of water is compensated by that of the tissue relative to it. Consequently, a flow across a tissue will expand the cells on one side and contract the cells on the other, thereby creating a differential strain. If the typical tissue displacement is denoted by u, the typical fluid velocity field is denoted by v, and the volume fraction of fluid is denoted by φ, the continuity relation embodied in the previous statement reads as φv = -(1 - φ)∂tu ∼ - [(1 - φ)u]/τp, where τp is the characteristic time for this movement, which is to be determined. Furthermore, the velocity v of the interstitial fluid of viscosity μ in the porous tissue with hydraulic permeability k obeys Darcy's law (6), which states that the fluid velocity relative to the medium is proportional to the fluid pressure gradient. Then, if the pressure in the fluid (7) p varies over a characteristic length L, we may write φ[v - (up)] ∼ kpL. This flow is coupled to the tissue elastic stress so that a local balance of forces in the fluid-infiltrated medium yields Eu/L ∼ φp, where E is the elastic modulus of the tissue (8). Substituting the latter expression for the pressure in the expression for the fluid velocity and using the continuity relation written earlier, we find a time scale known as the poroelastic time (9, 10) Math(1)

The poroelastic time characterizes the time for the diffusive equilibration of pressure via fluid transport in soft, wet tissues and is thus of crucial importance in determining the rate of hydraulic actuation in these systems. In Fig. 1, the dotted line corresponds to τp/L2 = 1.6 s/mm2, consistent with typical values for soft plant tissue (3, 11). This line separates all naturally occurring movements into two categories: slow movements (τ > τp), whose duration is limited by fluid transport; and rapid movements (τ < τp), which rely on elastic instabilities to cross the boundary τ = τp.

The elastic instabilities used by plants and fungi can be divided into two broad categories: snap-buckling and explosive fracture. To sufficiently understand the instability mechanism a particular plant uses requires a detailed study of its geometry [for example (3)]. The main difference between these two instabilities is the release mechanism: Snap-buckling involves the rapid geometric changes associated with the buckling of thin shells, whereas explosive fracture involves the rapid geometric changes due to tearing the plant tissue.

The boundary τ = τp separating the two categories of movement is clearly length-scale dependent and provides a significant barrier for rapid large-scale movements. Elastic instabilities provide the only way to cross the line τ = τp. To illustrate this, we compare Aldrovanda (12) (circled 1 in Fig. 1) with the closely related Venus flytrap (circled 2 in Fig. 1), both of which close their leaves rapidly to capture prey; Aldrovanda closes its leaves in ∼0.02 s, whereas the Venus flytrap closes in ∼0.2 s. However, although the leaves of the Venus flytrap snap by reversing their curvature, Aldrovanda's leaves are already initially curved inward so that closure does not produce a snap. To understand how the snapless Aldrovanda can be more rapid than the snapping Venus flytrap, we use Eq. (1) for the poroelastic time: for a Venus flytrap leaf of typical thickness L = 0.5 mm, τp = 1.6 s/mm2 (0.5 mm)2 ∼ 0.4 s, whereas for an Aldrovanda L = 0.05 mm, the value of τp is ∼0.004 s. Because an Aldrovanda leaf is about 1/10th the size of the Venus flytrap, it can be actuated 100 times more rapidly and does not require an elastic instability to catch prey, whereas the Venus flytrap does, which is consistent with our classification.

The absolute physical limit of motion in self-actuated mechanical systems is determined by the speed of elastic waves in them, which propagate at a speed that scales as √(E/ρ). This yields an estimate for the inertial time given by Math(2)

The inertial time scale characterizes the time for wave propagation in mechanical signalling in systems and must be less than τ, the time scale of the motion. In Fig. 1, the solid line denotes τi/L ∼ 10-5 s/mm, consistent with typical values for soft plant tissue (3, 11), beyond which there can be no natural nastic movements. For the fungus Pilobolus's sporangium discharge (13), L ∼ 0.05 mm, so that τi ∼ 10-7 s < 10-5 s ∼ τ, whereas for fruit of Hura crepitans (2), L ∼ 5 mm, so that τi ∼ 10-5 s < 10-4 s ∼ τ, as shown in Fig. 1. These explosive movements characterize Nature's best attempts to reach the physical limits of autonomous motion in elastic tissues.

In conclusion, we see that the size-dependent inertial-elastic time τi and the poroelastic time τp given by Eqs. (1) and (2) provide us with a physical basis for the classification of the hydraulic movements in plants and fungi and yield limits on their performance. This implies that the engineering of soft, nonmuscular hydraulically actuated systems for rapid movement requires either small size or the enhancement of motion on large scales via elastic instabilities. Nature has already implemented many such designs exquisitely; we simply need to follow her lead.

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