92
4. SIMULATING GROWTH AND FORM
of the modules is not predefined by the formalism, they often represent individual cells. Each module is characterized by its type and, possibly, one or
more numerical parameters (Prusinkiewicz and Lindenmayer 1990), which
collectively determine the module's state. The development of a structure is
described in terms of rewriting rulesor productions that replace a module by
zero, one, or more new modules, or change the module's state. Productions
are applied in parallel in order to capture the simultaneous development of
different parts of the organism.
One of the simplest biologically relevant examples of L-systems is the
model of the filamentous bacteria Anabaena catenula (Lindenmayer 1987,
Lindenmayer and [iirgensen 1992, Prusinkiewicz and Lindenmayer 1990),
which formalizes the developmental rules first formulated by Mitchison and
Wilcox (1972). The model describes the development of a so-called vegetative
segment of Anabaena using productions that operate on two types of cells:
large cells L and small cells S. Each cell is assumed to have one of two
.... t
........
possible polarities, indicated by superscript arrows: L, , and S,S. During the
development, cells S elongate and change their state to L, while cells L divide,
producing a cell L and a cell S. Taking the polarities into account, this process
is captured by the following productions:
l~rs
r~sr
The development of the filament is simulated as a sequence of stages, with
the next stage obtained by applying appropriate productions simultaneously
to all cells of the previous stage (Fig. 4.1).
In spite of its simplicity, this example reflects the essential features of
the L-system formalism. The model is discrete in three senses: "the state
transformations are defined on discrete subunits (cells); each subunit may be
present in one of a finite set of states; and the transformations are performed
in discrete time steps" (Lindenmayer and [iirgensen 1992). The arrangement
of cells in the filament is determined by the pattern of cell division. No
mechanism exists to rearrange a set of existing cells, since in algae and plants
the cells are tightly cemented together.
The above model implies that the time between the formation and division of a small cell is twice as long as the time between the formation and
division of a large cell. In reality, a large cell takes only about 20% longer to
divide than a small cell. This behavior can be incorporated into the model
by assuming that the cells undergo sequences of state transitions before they
divide, as represented by the following L-system (Lindenmayer 1978).
The same model can be described more concisely using a parametric
L-system (Prusinkiewicz and Hanan 1990, Prusinkiewicz and Lindenmayer 1990). In that case, modules are associated with numerically-valued
parameters, and productions have the form:
predecessor: condition ~ successor.
Fig. 4.1. Visualization of the L-system
modelofAnabaena catenula. Themodel
capturesthe arrangement ofshorter and
longercells in avegetative segmentofthe
filament.
4. SIMULATING GROWTH AND FORM
of the modules is not predefined by the formalism, they often represent individual cells. Each module is characterized by its type and, possibly, one or
more numerical parameters (Prusinkiewicz and Lindenmayer 1990), which
collectively determine the module's state. The development of a structure is
described in terms of rewriting rulesor productions that replace a module by
zero, one, or more new modules, or change the module's state. Productions
are applied in parallel in order to capture the simultaneous development of
different parts of the organism.
One of the simplest biologically relevant examples of L-systems is the
model of the filamentous bacteria Anabaena catenula (Lindenmayer 1987,
Lindenmayer and [iirgensen 1992, Prusinkiewicz and Lindenmayer 1990),
which formalizes the developmental rules first formulated by Mitchison and
Wilcox (1972). The model describes the development of a so-called vegetative
segment of Anabaena using productions that operate on two types of cells:
large cells L and small cells S. Each cell is assumed to have one of two
.... t
........
possible polarities, indicated by superscript arrows: L, , and S,S. During the
development, cells S elongate and change their state to L, while cells L divide,
producing a cell L and a cell S. Taking the polarities into account, this process
is captured by the following productions:
l~rs
r~sr
The development of the filament is simulated as a sequence of stages, with
the next stage obtained by applying appropriate productions simultaneously
to all cells of the previous stage (Fig. 4.1).
In spite of its simplicity, this example reflects the essential features of
the L-system formalism. The model is discrete in three senses: "the state
transformations are defined on discrete subunits (cells); each subunit may be
present in one of a finite set of states; and the transformations are performed
in discrete time steps" (Lindenmayer and [iirgensen 1992). The arrangement
of cells in the filament is determined by the pattern of cell division. No
mechanism exists to rearrange a set of existing cells, since in algae and plants
the cells are tightly cemented together.
The above model implies that the time between the formation and division of a small cell is twice as long as the time between the formation and
division of a large cell. In reality, a large cell takes only about 20% longer to
divide than a small cell. This behavior can be incorporated into the model
by assuming that the cells undergo sequences of state transitions before they
divide, as represented by the following L-system (Lindenmayer 1978).
The same model can be described more concisely using a parametric
L-system (Prusinkiewicz and Hanan 1990, Prusinkiewicz and Lindenmayer 1990). In that case, modules are associated with numerically-valued
parameters, and productions have the form:
predecessor: condition ~ successor.
Fig. 4.1. Visualization of the L-system
modelofAnabaena catenula. Themodel
capturesthe arrangement ofshorter and
longercells in avegetative segmentofthe
filament.