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| Fig. 1: A photo of a venus flytrap, highlighting the main leaf in an open state. (Courtesy of Noah Elhardt.). |
Venus Fly Traps have long caught the public imagination as among the few plants in the world that ingest animals (insects, in particular) as part of their natural existence. Part of the fascination here is that, generally speaking, plants operate on much slower timescales in terms of their external actions, relative to humans and animals. One generally, for example, does not perceive a trees roots extending to a new water source while one can easily see an elephant or insect searching out new sources of nutrition. With a snapping time of 100ms, it is indeed the fastest known action in the plant kingdom, which Darwin himself remarked was "one of the most wonderful in the world" [1]. In this report we summarize recent findings that fully model and describe the snapping action of the venus flytrap in terms of a biochemically triggered change in the trap's curvature, which provides a mechanism for the storing and rapid release of elastic energy. We do not focus on the microscopic, biochemical causes for the configuration but accept it as a given -- instead we focus on a rather ingenious mechanistic approach that successfully models and describes the characteristics of how a venus fly trap snaps.
To motivate the model described later on this page, let's briefly consider some of the experimental evidence and analysis of venus flytrap-snapping used by the authors of this paper. To study how leaf geometry evolves during the snap, the researchers painted small ultraviolet-fluorescent dots on the surface of the leaves and recorded snapping using a high-speed video camera. Their observations identified three distinct phases during snapping:
Two key parameters considered are the averaged mean curvature, κm and spatially average gaussian curvature, κg. Both are invariant under rigid body motions and are excellent proxies for understanding changes in shape. The Gaussian curvature does, indeed, change during the course of the snap. In a manner similar to these three phases, this curvature does change slowly, then rapidly as it passes a minimum and then slowly again. Since κg is correlated with stretching in the mid-plane of the leaf, the authors hypothesized that the first phase corresponds to a slow storage of elastic energy, followed by an intermediate stage where this elastic energy is rapidly released.
The central component of this mechanistic description of how a venus flytrap snaps is a comprehensive model of the leaf of the venus flytrap itself. Foreterre et al. [1] model it as a thin, shallow shell with uniform thickness h, Young's modulus E and natural curvature κxn and κyn = κ. Instead of a complete theory that captures complex contraction and extension effects, a simple energetic approach is employed in this model. The leaf shape can be well approximated by a paraboloid defined as
If we now define the following dimensionless curvature values:
The total energy due to bending and stretching can then be written as
Note here that the parameter α numerically defines the coupling between bending and stretching deformations in terms of leaf size and thickness parameters and the observed curvature of the open leaf:
A small α value would thus imply that it is relatively easy to stretch the mid-plane of the leaf by changing its curvature. As it turns out, we can separate these α values into two regions by introducing a critical alpha value, αc and defining
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| Fig. 2: This plot highlights the transition between open (positive Kxn) and closed (negative Kxn) of the venus flytrap leaf. For small values of the geometrical parameter α there is a smooth transition. At α = αc the system begins to approach a region of bistability before rapid snapping (note the steep slope). At greater α values the bistability becomes more apparent. |
Experimental observations yield constraints on the curvature values; namely that the spatially homogenous curvatures in the x- and y-directions are equal to a consistent value at t = 0 (prior to snapping). The observations also tells us that the natural curvature in the y-direction is constant, ie: κyn = κy. So, summarizing: κx(t=0) = κy(t=0) = κyn(t) = κ. If we now minimize U with respect to Kx and Ky for different values of the parameter alpha, we can obtain the change in leaf shape as a function of curvature in the x-direction:
This, then, reduces to solving for Kx and Ky in the following two coupled equations:
Solving for Kx and Ky, and expressing them as the mean curvature, Km = (Kx + Ky)/2 as a function of the parameter Kxn, we obtain the graph in Fig. 2. As we approach αc ~ 0.8, we see that the energy function, U, goes from having one minimum, to two minima, for α > αc. Thus, in the strong bending-stretching coupling regime, the leaf passes through a region of bistability in which both states of the leaf (open and closed) do not fluctuate for small perturbations. This strong coupling regime corresponds to the snapping transition, while the weak coupling corresponds to a smooth transition from open to closed states. The authors verified this bistable configuration experimentally by noting that leaves often begin to close when stimulated but do not snap or close fully. However, it was possible, on these same leaves to induce snapping mechanically by squeezing the leaves.
Fig. 2 was generated with the following code.
We do not delve into the details of the dynamics of snapping in our exposition, but do highlight the key features in the model proposed by [1]. If the leaf is treated as a poroelastic material, then curvature changes during snapping cause interstitial water to move relative to the tissue. This causes energy dissipation, and by balancing the elastic power due to curvature changes with viscous dissipation rate due to fluid flow, the following energy-balance equation can be obtained [2]:
Here, we use a scaled time T = t/τp, where τp, the poroelastic time, is defined as
where ψ is the fluid volume fraction for an impermeable poroelastic plate [2]. By using measured values for the Young's modulus, E and other parameters, a poroelastic time of τp = 0.1s is found, consistent with observed snapping times of 100ms. This quantitative model, when solved, also captures the three phases of snapping detailed in the experimental observations section, quite successfully.
By making detailed observations of the venus flytrap snapping, two key components of the snap were identified. First, a biochemical stimulation causes the plant to actively alter its natural curvature, κxn. However, we focused here on the second component, where after the leaf geometry changes it provides a deft mechanism by which elastic potential energy is both stored and released. Depending on the value of a key geometrical parameter, α, the leaf either snaps or closes smoothly. A static model captures this element quite well, with a complementary dynamic model identifying the presence of fluids as a means by which the leaf is able to snap as rapidly as it does. As is often the case, nature has yielded a remarkable and clever mechanism by which even a "slow" plants can capture "fast" insects; a mechanism built on sound physics!
© 2007 Aaswath P. Raman. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] Y. Forterre, J. M. Skotheim, J. Dumais and L. Mahadevan, "How the Venus Flytrap Snaps," Nature 433, 421 (2005).
[2] J. M. Skotheim and L. Mahadevan, "Dynamics of Poroelastic Filaments," Proc. R. Soc. Lond. A 460, 1995 (2004).
[3] O. Stuhlman and E. B. Darder, "The Action Potentials Obtained from Venus's-Flytrap," Science 111, 492 (1950).
