| |
ISSN 1320-0682 | |||
| Volume 02 | April 1995 | |||
Fran Soddell, Robert Seviour, Jacques Soddell
Department of Computing and Information Science
Biotechnology Research Centre
La Trobe University, Bendigo
Bendigo Vic 3550 Australia
Email: fsoddell@redgum.ucnv.edu.au
in the early stages
of growth, and changes in
direction of growing hyphae may be significant. These have not
been considered before as mechanisms for producing circular
colonies. It was found that L systems are not only alternatives
to conventional models but also excellent experimental tools.
Filamentous fungi are well adapted for growth on solid substrates
where they grow
from single spores to a mature mycelium made up of branched
filaments, or hyphae. Germ tubes
extend to form leading hyphae that produce primary branches that
in turn produce secondary
branches and so on [1]. Hyphae grow by extending at
their
tip which provides a way of
exploring new regions for fresh nutrients, and branching allows
full use of the medium already
colonised [2]. Despite any initial asymmetry following
spore germination, mycelia
develop into regular macroscopic colonies having both circular
outline and even distribution
of hyphae through 
. We used Lindenmayer systems (L
systems) to investigate how these
colonies may become circular.
L systems are string rewriting mechanisms that Lindenmayer introduced to model the development of multicellular organisms [3,4]. Smith [5] proposed using them to create computer-generated images of plants, and Prusinkiewicz and Lindenmayer [6] perfected systems that produce realistic images of trees, bushes and flowers. Each system is made up of a finite set of symbols called an alphabet, a start symbol or string of symbols called an axiom that represents the start state of a growing organism, and a finite set of rules called productions. The rules are applied in parallel to all symbols in the axiom to give a new string of symbols representing the state of the organism at the end of the first time step. This process is repeated to generate a sequence of strings representing the developmental steps of a growing organism. These strings must be interpreted in a meaningful way, and this study employed the turtle graphics interpretation derived from the idea of a turtle that traces out a sequence of steps to draw an image on a computer screen.
We used bracketed 0L systems, one of which is deterministic and parametric, one stochastic, and the others parametric with stochastic rules. Lindenmayer [7] described context-free 0L systems that ignore cellular interaction and environmental input. In deterministic systems, only one production rule applies to a symbol at a given time, so they produce a single infinite derivation sequence [8]. Brackets are used as special symbols to represent the development of branching structures, and, for 0L systems, it is assumed that each segment grows and branches according to its own set of rules, without input from other branches [9]. Deterministic systems produce identical strings of symbols and, therefore, identical images, so that combining images in one picture produces "a striking artificial regularity" [6]. Stochastic systems allow differences in both image topology and geometry through the stochastic application of production rules. In parametric systems, if a numerical parameter is associated with a turtle graphics symbol, it controls the turtle's movements [6], and the system controls factors like the size of branching angles and the length of turtle steps.
L systems have not often been applied to fungi, although Liddell and Hansen [10] designed one to represent fungal hyphae forming circular colonies. However, this allowed interbranch extension of filaments and this is unknown in fungal hyphae that extend only at the tip. We designed systems based on known characteristics of the growth and morphology of filamentous fungi to investigate mechanisms that may be involved in the formation of circular colonies. Hutchinson et al. [11] used conventional modelling with the fungus Mucor hiemalis, and we incorporated their published data into L systems models. Measurements of Mucor M41 in the very early stages of growth were used as parameters in stochastic systems to generate images representing this fungus at a later growth stage.
The turtle graphics L systems interpreter Lsys (Leech 1990,
CB#3175UNC-CH, Chapel
Hill, NC USA, email: leech@cs.unc.edu) was used under SunOS
Release 4.1.3 on a SUN
SPARC2 computer. The output PostScript files were viewed on a
486-PC using
Ghostscript v2.5 (Aladdin Enterprises, distributed by the
Free software Foundation, Inc,
675 Mass Ave, Cambridge, USA). L systems in this study used the
turtle graphics symbol
F to draw a line of one unit in the forward direction,
+ to turn left by a specified
angle, - to turn right by a specified angle, [ to save
the current position, and ]
to return to the saved position. If there was no rule to specify
how to rewrite a symbol, it was
rewritten without change. Numerical parameters were associated
with symbols so that, for
example, +(31.5) was interpreted as turn left by
,
as turn right by 
, and
F(1.5) as draw a line 1.5 units long. Parameters were also
associated with symbols to ensure that
different rules applied under different conditions. For example,
the following two rules both
apply to the user-defined symbol B associated with the variable
parameter x.
Starting with the axiom B(0), the condition x=0 is true so the rule p1 will be applied and B(0) is rewritten as FB(1) so that, in the next iteration, x=1 is true and the rule p2 will be applied to B. This ensures that the rules will be applied alternately from iteration to iteration. More than one parameter was often associated with one symbol, and some conditions used combinations of logical operators.
Figure 1: Images of simulated filamentous fungi growing on solid
substrate. (a)-(d) were generated by the deterministic system in
Table 1. (e) was generated by the stochastic system in Table 2
that allows variation in hyphal orientation and branching angles.
(f) was generated hy the stochastic parametric system in Table
3 that allows variation in hyphal orientation but the branching
angle is constant at 
. (h)-(i) were generated using
published data describing M. hiemalis. (h) has variation
in branching angles, interbranch distances, and growth rates; (i)
has variation in branching angles and hyphal orientation. (j)-(l)
were generated from measurements of early growth of Mucor
M41. (j) is the growth phase in which measurements were taken,
(k) and (l) are at a later stage of growth, and (l) has a
constant branching angle of 
.
| #define a 0
#define b 30 START: B(0) |
p1 :
B(t) : t=0 -> [+(a)FB(1)B(2)] |
p2 :
B(t) : t=1 -> [-(a)FB(0)B(3)] |
p3 :
B(t) : t=2 -> [+(b)FB(3)B(0)] |
p4 :
B(t) : t=3 -> [(b)FB(2)B(1)] |
Table 1: Parametric deterministic 0L system for dichotomous and lateral branching.
There are two predefined constants, a and b, that specify the size of branching angles. For lateral branching, a was set to 0 to simulate forward growth and b was set to the size of the branching angle. To simulate dichotomous branching, a and b were set to the same size, so the angle between branches was their sum. The symbol B is associated with the variable parameter t so that the value of t determines which rule is applied when rewriting B.
By examining simulated images, we noted the effect on colony morphology of different types of branching and different branching angles, and of variation in hyphal orientation and/or branching angle sizes. We re-examined previous work on M. hiemalis, and present here preliminary results on an investigation of Mucor M41.
Lateral branching occurs when new branches form to the side of
leading hyphae that continue
to grow in the same direction (Fig.1(b), (d)), while dichotomous
branching occurs when new
branches and leading hyphae move in opposite directions
(Fig.1(a), (c)). The system in Table
1 can be used to model both types of branching and forces hyphae
to branch first left and then
right. Simulations of branching at 
show that the outline
with dichotomous branching is
slightly more circular (Fig.1(a)) than with lateral (Fig.1(b)).
Although the outline is circular,
hyphae do not spread evenly through 
. For branching angles
of 
, hyphae are
distributed evenly at an early stage but have an unrealistically
regular outline (Fig.1(c), (d)).
This suggests that circular colonies are more likely to form if
branching angles are close to
in an early stage of development. Images of Coprinus
sterquilinus [12]
show early branching at close to 
but this has not been
suggested before as a mechanism
for producing circular colonies.
Growing hyphae wander from side to side and, by including
changes of direction in computer
simulations, Yang et al. [13] produced realistic images
of mycelial pellets growing in
liquid media. We found that using stochastic rules to allow
hyphae to change direction (Table
2, Table 3), not only generated realistic images but also
generated circular colonies (Fig.1(e))
even when all branching angles were 
(Fig.1(f)).
However,
because rule p18 in Table 3
is stochastic, hyphae simulated by this system occasionally
reverted to almost straight paths
resulting in an unrealistically regular outline (Fig.1(g)).
Table 2: Stochastic 0L system with variations in hyphal
orientation and size of branching angle
The stochastic rules p1 and p2 allow one of the alternatives to be applied according to the specified probabilities. This is a nonparametric system so different angle sizes must be entered as multiples of the predefined angle delta (15).
A major contribution to our understanding of how fungal mycelia achieve circular morphology was made by Hutchinson et al. [11] who produced approximately circular images from simulations of M. hiemalis. They hypothesised that a circular colony is the most likely shape to result from observed statistical variation in hyphal extension rate, internode length and branching angle size. We designed a parametric stochastic 0L system based on their data and, on many runs, this also generated a circular shape (Fig.1(h)). The main difference between images of simulated colonies and tracings of observed colonies published by Hutchinson et al. [11] is that observed hyphae are slightly wavy, whereas simulated hyphae are straight. So we incorporated only their measurements of branching angles into a system that allows hyphae to wander from side to side. This produced images that were both circular and realistic (Fig.1(i)), challenging their conclusion that growth rates and distances between branches contribute more to colony shape than hyphae wandering from a straight path.
We are currently investigating the growth of Mucor M41 by
entering data
collected during early colony development
into stochastic parametric 0L systems. Simulations of this stage
of growth produced realistic
images like that in Fig.1(j). Continuing iterations to a later
stage
generates a circular colony
(Fig.1(k)) even when a constant branching angle size of
is substituted for observed
variations (Fig.1(l)). This suggests that early behaviour of an
organism may greatly affect the eventual
shape of a colony, and shows that variation in the sizes of
branching
angles is not necessary for the
formation of circular colonies.
This study showed that L systems can be used to experiment with factors that may contribute to the development of circular colonies. Although they produced unrealistically regular images, deterministic 0L systems were valuable for showing that different angle sizes and different types of branching affect the shape of a colony. Stochastic systems produced realistic images and were needed to test the effect of varying the size of branching angles and of allowing hyphae to wander from side to side. Parametric systems were valuable for applying different rules to the same symbol at different times, and for entering observed values as numerical parameters. L systems are not only alternatives to conventional models but, because of the ease of changing probabilities and numerical parameters, are also excellent experimental tools. Using them has increased our understanding of how filamentous fungi may attain circular morphology and has led us to identify areas for further study.
| Rule p1 controls direction of hyphae | Rule p2 controls size of the branching angle |
|
#define delta 15
START: Fg p1: g -> (0.2)bFg ->(0.2)+bFg ->(0.2)++bFg ->(0.2)-bFg ->(0.2)--bFg |
p2: b -> (0.1)[++Fg]
->(0.1)[+++Fg] ->(0.1)[++++Fg] ->(0.1)[+++++Fg] ->(0.1)[++++++Fg] ->(0.1)[--Fg] ->(0.1)[---Fg] ->(0.1)[----Fg] ->(0.1)[-----Fg] ->(0.1)[------Fg] |
Table 3: L system with parametric stochastic rules allowing variation in hyphal
orientation. Parametric L systems provide a way of controlling growth rates. Rules p1 to
p17 are
based on the kinetics of growth of individual hyphae that accelerate towards a maximum rate and
then grow linearly (p17). The values of parameters associated with the symbol C
determine which
rule applies according to the specified conditions. The predefined constants i, k and
m can be predefined to change the growth rate for specific organisms. Rule p18 allows
hyphae
to wander randomly from a straight line path by a maximum of
with equal probability of left or right movement. Rule p20 controls whether
a branch occurs to the right or left or not at all. Rules p21 and p22 control left
and right branching, with the branching angle size set to the predefined constant b, which
was 
.
Note
Some L systems used in this study are not presented here because
of their complexity but can
be obtained by contacting the first named author
(fsoddell@redgum.ucnv.edu.au).
Acknowledgments
Peter Goddard, Dept. of Computing and Information Science, La
Trobe University, Bendigo.
The ARC Small Grants Scheme partially funded this study.
Using Lindenmayer Systems to Investigate How Filamentous Fungi May Produce Round Colonies
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