December 7, 2007

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:

- slow, initial phase (20% of total displacement in 0.33s)
- rapid, intermediate phase (60% of total displacement in 0.1s)
- slow, second phase (20% of total displacement in 0.33s)

Two key parameters considered are the averaged mean
curvature, κ_{m} and spatially average gaussian
curvature, κ_{g}. 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 κ_{g} 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 κ_{xn} and κ_{yn} =
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,
α_{c} and defining

**weak**bending-stretching coupling (α < α_{c})**strong**bending-stretching coupling (α < α_{c})

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:
κ_{yn} = κ_{y}. So,
summarizing: κ_{x}(t=0) = κ_{y}(t=0) =
κ_{yn}(t) = κ. If we now minimize U with
respect to K_{x} and K_{y} 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 K_{x} and
K_{y} in the following two coupled equations:

Solving for K_{x} and K_{y}, and
expressing them as the mean curvature, K_{m} = (K_{x} +
K_{y})/2 as a function of the parameter K_{xn}, we
obtain the graph in Fig. 2. As we approach α_{c} ~ 0.8, we
see that the energy function, U, goes from having one minimum, to two
minima, for α > α_{c}. 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/τ_{p}, where τ_{p}, 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
τ_{p} = 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, κ_{xn}. 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).