98
Fig. 4.8. The red seaweed Chondrus
crispus
Fig. 4.9. Pythagorous tree model of the
red seaweed Chondrus crispus
4. SIMULATING GROWTH AND FORM
in various other seaweeds, see for example Figs. i.jb-d and 2.8. Some of
the variation in morphology has been shown to be due to plasticity of the
growth process in response to changing environmental conditions such as
light supply, hydrodynamics, or temperature (Kiibler and Dudgeon 1996).
Such changes in morphology can compensate for lower nutrient availability,
increased metabolic costs related to temperature, or decreased light intensity
by increasing the fractal dimension of fronds and thereby the ratio of reactive
surface available for exchanges with the environment to the volume of living
tissue.
A simple, iterative fractal called the Pythagoras tree was used as the
basis of a model of the growth of Chondrus crispus in two dimensions. In this
species, macroscopic branches represent the continued growth of bunches of
microscopic filaments until the next time that a "decision" is made to branch.
So a model which introduces entire branches punctuated by branch points,
while very simplistic, is reasonable for examining the effects of changes in
the timing of that branching decision relative to other growth processes. The
model starts with a box, adds the hypotenuse of a right triangle to the top of
the box, then smaller boxes to each of the free sides of the triangle, and iterates
this pattern as illustrated in Fig. 4.9. Each block in the model represents the
pseudoparenchymatous bundle of filaments that make up one macroscopic
branch, and the apex of each triangle represents the point where inhibition of
growth of one filament divides the branch. Changing the dimensions of the
boxes or the ratio of the sides of the triangles results in different overall shapes
of the simulated objects after n iterations or after the accumulation of some
total surface area . Built-in randomization was added to provide a population
of related forms for each set of initial parameters. Data for the rates of linear
extension and area-specific growth rates at a range of temperatures were used
to set the initial parameters of the model.
The model was tested for two different cases: plasticity due to temperature change in a single population, and geographic variation in morphology
between populations. The model produced shorter branches at high temperature and therefore, a branchier, higher fractal dimension structure in
a given increment of linear growth. This is in agreement with the results of
a growth experiment in which Chondrus crispus was grown at two temperatures. Apical segments of the branches are shown in Fig. 4.10, illustrating the
increased branching frequency at the higher temperature.
When individuals from two distinct populations were simultaneously
grown at two different temperatures, the above pattern held, overall, but
the individuals from the northern population better fit a model with more
scope for random variation. The comparison of this very simplistic model
to actual growth experiments gives some insight into the potential mechanisms of two types of morphological variation in this seaweed. In the first
case, plasticity within the lifetime of a simple organism is, in part at least,
SOC
Fig.4.10. Silhouettes of apices of Chon- ~
drus crispus grown at two different
r
temperatures
20 °C
Fig. 4.8. The red seaweed Chondrus
crispus
Fig. 4.9. Pythagorous tree model of the
red seaweed Chondrus crispus
4. SIMULATING GROWTH AND FORM
in various other seaweeds, see for example Figs. i.jb-d and 2.8. Some of
the variation in morphology has been shown to be due to plasticity of the
growth process in response to changing environmental conditions such as
light supply, hydrodynamics, or temperature (Kiibler and Dudgeon 1996).
Such changes in morphology can compensate for lower nutrient availability,
increased metabolic costs related to temperature, or decreased light intensity
by increasing the fractal dimension of fronds and thereby the ratio of reactive
surface available for exchanges with the environment to the volume of living
tissue.
A simple, iterative fractal called the Pythagoras tree was used as the
basis of a model of the growth of Chondrus crispus in two dimensions. In this
species, macroscopic branches represent the continued growth of bunches of
microscopic filaments until the next time that a "decision" is made to branch.
So a model which introduces entire branches punctuated by branch points,
while very simplistic, is reasonable for examining the effects of changes in
the timing of that branching decision relative to other growth processes. The
model starts with a box, adds the hypotenuse of a right triangle to the top of
the box, then smaller boxes to each of the free sides of the triangle, and iterates
this pattern as illustrated in Fig. 4.9. Each block in the model represents the
pseudoparenchymatous bundle of filaments that make up one macroscopic
branch, and the apex of each triangle represents the point where inhibition of
growth of one filament divides the branch. Changing the dimensions of the
boxes or the ratio of the sides of the triangles results in different overall shapes
of the simulated objects after n iterations or after the accumulation of some
total surface area . Built-in randomization was added to provide a population
of related forms for each set of initial parameters. Data for the rates of linear
extension and area-specific growth rates at a range of temperatures were used
to set the initial parameters of the model.
The model was tested for two different cases: plasticity due to temperature change in a single population, and geographic variation in morphology
between populations. The model produced shorter branches at high temperature and therefore, a branchier, higher fractal dimension structure in
a given increment of linear growth. This is in agreement with the results of
a growth experiment in which Chondrus crispus was grown at two temperatures. Apical segments of the branches are shown in Fig. 4.10, illustrating the
increased branching frequency at the higher temperature.
When individuals from two distinct populations were simultaneously
grown at two different temperatures, the above pattern held, overall, but
the individuals from the northern population better fit a model with more
scope for random variation. The comparison of this very simplistic model
to actual growth experiments gives some insight into the potential mechanisms of two types of morphological variation in this seaweed. In the first
case, plasticity within the lifetime of a simple organism is, in part at least,
SOC
Fig.4.10. Silhouettes of apices of Chon- ~
drus crispus grown at two different
r
temperatures
20 °C