Phyllotaxis in Bryophyllum tubiflorum

Ada Bot. Neerl.
23(4), August 1974,
Phyllotaxis
in
tubiflorum:
P.H.
Bryophyllum
Morphogenetic
computer
and
473-492.
p.
studies
simulations
and A. Lindenmayer
Hellendoorn
Afdeling Theoretische Biologie and
Botanisch
Laboratorium, Rijksuniversiteit Utrecht
SUMMARY
imperfect pairs,
were
introduced
either to
Observations
and
1.
transition
and
on
the
in
also confirmed
of
phyllotaxis
tricussate
model
that these
of an inhibitor
production
and
on the
pseudo-bijugate,
patterns:
inverted
helix.
in
changes
Assumptions
phyllotaxis
apical surface,
or
due
are
alternatively to
of
structure
the
the latter
supported
apex
was
found
assumption.
for the
necessary
Mirror
asym-
computer simulation,
observation.
by
INTRODUCTION
there has been much discussion about theories of
Although
place
where
leaves
proposed
mechanical
a
in
a
new
leaf
nearest
are
theory
1907
van
Iterson
treatment
regular
of
the
primordia,
like
provided by
circles
Erickson
cylinder
The radii of
cylinders,
Each
are
of
the
two
or
Schwendener
a
primordium.
new
in which
it
applied
cones,
etc.,
recent
This
phyllotaxis.
on
to
as
he gave
phyllotaxis.
well
resume
mikroskopischa
He described
different
as
of
mathematical
van
of
shapes
Iterson’s
ideas
is
(1972).
surfaces
newly
mordia which
position
1878
“Mathematische und
and
and folioids. A
Van Iterson assumed the
on
his
of
position
(1966)
Blattstellungen”,
“regular point systems”
point systems
by
the
almost
that
(1868), namely
leaf. In
new
phyllotaxis,
which stated that fields of tension arise around
published
on
is fixed
arises,
to
control the
turn
anatomische Studien fiber
circles
axiom of Hofmeister
is also the basis of Schuepp’s view
theory
In
that
and these
primordia
the
accepted
the apex,
on
three older
2.
a computer
of the first two leaves
all botanists have
1.
kinds
imperfect trios,
size.
position
the
was
of
several
helix,
computed
changing rates
changing apical
metry in
exhibits
tubiflorum
Bryophyllum
following properties
(see fig.
inserted
for
regular systems
of
touching
1):
primordia
the
are
same
as
those of young
pri-
already present;
primordium
is in
contact
with
least
at
two
older
members
(rule
of
Hofmeister);
3.
The
4.
Small
primordia
are
irregularities
laid down in
like the
eliminated during the continuous
The ideas
of
van
Iterson
are
the
shape
largest
and
growth
found
gaps between
position
of
new
te
previous
primordia
ones;
can
be
of the system.
again
in somewhat modified form in the
474
P.
“first available
cording
this
to
space” theory
apex above and between the
that
cycle,
are
gaps
leaves
In
was
occupied
by
1913 Schoute
by
Schoute,
the
shape,
Starting
the
1.
Each
spheres
primordia
apex contains
threshold,
Each leaf
1.
mordia
Fig.
a new
centre
2.
Example
ona
contact
on
a
Ac-
1947).
on
the
members in the top
width and
necessary
a
sequence, in which the several
the
position
and
shapes
of those
the diffusion
theory,
and
like
hypotheses
at
might
first
be
develop directly
which
theory
later
Richards & Schwabe (1969).
those
new
of
organs
Iterson
van
never
touching. By starting
in their advanced
and
touch each
from
some
shape.
right,
Schoute stated
:
particular
substance that inhibits the
of
development
of
a
of
cylindrical
tor, produced by
the
a
apex,
where the
primordium
goes
directly
will
into
an
influence of this inhibitor reaches
a
arise;
inhibitor
producing state,
the inhibitor
surrounding tissue;
1+2+3 phyllotaxis pattern following
cylindrical
Example
on a
a
of the
place
each
diffuses into the
dia
to
1934,
leaves;
low
Fig.
other
The
(1951)
because
of influence
have
1933,
from the fact that the axiom of Hofmeister is
On
3.
or
depends
another
rejected
following theory (fig. 2)
new
2.
their
leaves
Richards
contact
Schwendener should be
other, only
(1931,
LINDENMAYER
A.
gap.
proposed
further elaborated
to
& Snow
AND
leaf arises in the first available
space
growing-point.
leaves,
new
which border the
According
HELLENDOORN
in the first space which attains both
means
only
new
existing
distance below the
necessary
of Snow
each
hypothesis
H.
van
Iterson.
The circles
are
pri-
apex.
1+2+3 phyllotaxis
apex, while the
primordia.
pattern
following
circles represent
the
Schoute. The
threshold
points
concentration
are
of
primor-
an
inhibi-
4.
From
the
leaves
can
cells
topmost
arise
in
only
Schoute, however,
computer simulation is
work
out
found
in
kinds of
able
more
work
to
theory
some
tip
pseudo-bijugate,
the
on
of this
theory
so
new
geometrical
of the
have
(1973)
theory
attempted
computer.
theory
the
to
constant
are
phyllotaxis
for
at
least
helix,
patterns
has
tubiflorum
transition
imperfect pairs,
basis of
on
consequences
Bryophyllum
each of which
well,
as
of the apex.
his
out
Harv..
tubiflorum
phyllotaxis patterns,
of nodes:
inhibitor diffuses
an
precise and realistic
of this
applies
Bryophyllum
apex
under the
needed. Veen & Lindenmayer
consequences
The present paper
of the
region
a
not
was
constructions. To obtain
to
475
BRYOPHYLLUM TUBIFLORUM
PHYLLOTAXISIN
some
several
number
imperfect trios,
tricussate, and inverted helix (see fig. 3 and 4).
“Pseudo-bijugate” designates
normal
bijugate
of about
angle
whorl
are
In
a
very few
starting
Because
deviates
a
in
somewhat from
helix with
a
in which
pattern
other and
is
which
a
the three leaves
each
helix is
the
divergence
whorl
of
characterized
a
60°
turns
by
its
the direction of the transition helix.
to
the
cases
which
whorl. The inverted
previous
opposite
becomes
phyllotaxis
tetracussate
a
pattern
directly
decussate pattern.
of the
needed for
is
120° from each
the
to
direction which is
after the
pattern
135°. “Tricussate”
located
with respect
a
The transition pattern
pattern.
different
phyllotaxis
Bryophyllum tubiflorum
is
an
and
patterns
propagation (vegetative embryos
of the rather
produced
are
on
short
the adult
interesting and useful plant for work
on
time
leaves)
changing
phyllotaxis patterns (see fig. 5).
From
the
computer simulations (Veen and Lindenmayer)
described
transitions
can
of the inhibitor
production
increase of apex size
be
(the
about
brought
size of the apex
during development by
a
by
a
it
was
predicted
10-fold decrease
remaining constant),
factor of about
3/2 (the
or
that
in
the
by
an
inhibitor
production remaining constant).
Out of symmetry considerations
which
Lindenmayer
two
had
leaves in order that the
ated.
Namely,
wise,
all
one
needs
subsequent
bijugate
different levels
always
mirror symmetry
Accordingly,
a.
b.
c.
the
by
will
be
be mirror
Two leaves
on
mirror
formulated
by
Veen and
without mirror symmetry.
symmetric
pattern.
If
symmetric.
the
was
initial arrangement of the
first
and then the tricussate patterns be gener-
of different sizes and 180°
or
6).
symmetry
requirement
starting condition
a
stages
with the observed helical transition
situation would
a
be fulfilled
to
same
one
If
and this
would
one starts
opposite they
level,
start
but
not
with
with
will
180°
does
one
two
Other-
not
agree
leafthe
leaves
on
still have mirror
opposite,
also have
{fig. 6).
one
needs
as
minimum starting conditions:
Two initial leaves,
The leaves
must
production
rates
The leaves
The
leaves
two
are
be
must not
leaves
not
on
on
for the
in
the
different levels
or
on
the
same
level
with different
inhibitor,
be 180°
figure
same
opposite.
6c
just
fulfil these
level and
are
not
requirements:
180°
opposite.
the
two
initial
Alternately, they
476
P.
H. HELLENDOORN
Fig.
3.
Cortex
stripe
of
Bryophyllum tubiflorum (tricussate).
Fig.
4.
Cortex
stripe
of
Bryophyllum tubiflorum (inverted helix).
AND
A. LINDENMAYER
PHYLLOTAXIS IN
Fig.
5.
Plantlets
could be
In
on
this
477
BRYOPHYLLUM TUBIFLORUM
on a
the
fully
same
research
grown leaf of
and abaxial
Bryophyllum tubiflorum (adaxial
view).
level but have differentrates of inhibitor production.
the
starting
patterns have been studied
on
conditions for
this
plant,
the
and the
of
generation
phyllotactic
following questions
have been
asked :
Are the first
leaves
two
different in age;
or
different
on
the leaves
are
on
levels;
the
the influences of the leaves different (is
fig. 5).
areas
Are the first
of the apex in
in order
of about
2.
to
3
find
/2
out
leaves
plants
180°
opposite
with different
the
bigger
not?
or
same
same
than the
but
age
other)? (See
Furthermore,
phyllotaxis patterns
level but
are
also
the surface
have been studied
during development
with
a
factor
phases.
METHODS
Organisms
Bryophyllum tubiflorum
house of the Botanical Garden in
South Africa,
experiments
*
leaf
whether the apex increases
AND
The crassulacean
in
one
on
level and of the
between the decussate (bijugate) and tricussate
MATERIALS
2.1.
two
the leaves
are
same
The valid
and
was
Kalanchoe
name
of this
used for
most
daigremontiana
species
of physiological literature
Utrecht,
use
the
is Kalanchoe
synonym B.
Harv.*
from
a
of the
was
12)
experiments.
R. Hamet
et
we
collected in 1939
For
one
of the
Perr. de la Bathie, obtained
tubiflora (Harv.)
tubiflorum
obtained from the green-
clone (Bt
R.
Hamet.
also followed
Since
most
that usage.
authors
478
Fig.
P.
6.
Starting conditions
a.
Two leaves
on different
b.
Two leaves
on
c.
Two leaves not
from the
the
same
the
HELLENDOORN AND
A.
LINDENMAYER
for the first two leaves.
levels and 180°
same
on
H.
level
same
opposite,
but not 180°
level
greenhouse
and not
was
opposite,
opposite.
180°
taken.
This
species
has
decussate
constant
a
phyllotaxis.
For the
(Bt 7)
in
Spain,
shape
same
was
the
experiment
and Kalanchoe
used. This
of Kalanchoe
hybrid (B
species
between
7)
x
daigremonticma (Bd
Bryophyllum tubiflorum
obtained from C.
1),
Gomez-Campo
has the property that the leaves have
and
daigremontiana
the
exactly
nearly
phyllotaxis
same
the
Bry-
as
ophyllum tubiflorum.
2.2.
was
cortex
possible
stripes
the
of
Preparation
From
as
to see
made
are
the
Elongation
complete
9b).
of leaves
position
graph
on
with
plantlets
days.
After 7
vertical line and
a
(fig. 7,
paper
3 and
4).
Cortex
plant.
by separating
The
scars
were
position
of
them.
by pin pricks through
visible second leaf
a
days
of sections
the
through
Kalanchoe
plantlets
basal
was
fixed in FAA
The sections
were
plate
and
one
were
were
pair
were
observed
plate
and
made of the
and
daigremontiana
put in
under
first
a
inter-
(according
to
their
Jensen
plantlets
hybrid
1962)
on
of
a
of the
were
photographed.
first internode
are
One
shown here
Bry-
micro-
and embedded
stained in orcein that contained acetic acid
orcein. Afterwards the sections
the basal
along
it
Gomez-Campo
the whole
vascular bundles in the leaf
sections of 10 micron thickness
The material
paraffin.
2%
practically
for internode elongation.
ophyllum tubiflorum,
and
stem
method of C.
the
plantlets
Transverse
tome.
frozen
a
The central
and
darkness for 7
of
to
of leaves of
of internodes
Preparations
node
stripes
recorded
was
plantlets
stereomicroscope
in
stem.
references for the
Rather young
2.4.
position
by cutting
each central bundle
2.3.
cortex
prepared according
from the
cortex
taken
stripes
(45%)
photograph
of
(figs. 8, 9a and
PHYLLOTAXIS
IN
479
BRYOPHYLLUM TUBIFLORUM
Fig.
7.
Detail of a cortex
Fig.
8.
Plantlet
Fig.
9a.
Transverse section
through the
Fig.
9b.
Transverse section
through
stripe.
with basal
plate (BP)
and first intemode
first intemode
the basal
plate
of
(FI)
of Bryophyllum
tubiflorum.
Bryophyllum tubiflorum.
480
P.
Geometrical
10.
Fig.
2.5
construction
Determination of
To estimate the
of the leaves
of the first
the middle-lines
this
project
pair
into
two
this
pair
The
leaf
pairs.
tubiflorum
from
of
plants
the
Zenker
of
in
the
fixation
graphed.
the
same
is the
the
through
From this
centre.
it is
the
possible
of the
centre
by drawing
Now it is
with
plate
centre
medium
orcein
stains
made of
transition
(fig. 11a, b,
zone
The sections
nuclear
d).
were
apices
zone,
to
to
possible
four
vascular
measure
the
apices
for
was
tricussate
was
live
by
were
hr.
measured with
a
the
taken
mea-
fixed
and
stained in orcein and
material. Afterwards
primordia
Bryophyl-
ascertained
The
1962)
from
in the
The material
was
primordia.
Jensen
to
c,
age. This
plastochron
(according
paraffin (52°C).
The
were
in the
of the youngest visible
between the youngest visible
apical
in
were
photosurface
planimeter.
RESULTS
3.1.
Observations
the
cortex
meaningfully
may be
(angles)
the
the
section of the first internode
bijugate phase,
nearly
lengths
embedded in
On
references for the
as
find
can
find the
to
of the different ages
sections
apical
and in the inverted helix
suring
3.
one
pairs.
between the first four leaves.
Preparation
phase
LINDENMAYER
needs
one
used). If,
be
taken,
Transverse sections of 10 micron thickness
lum
A.
primordia
growth,
pairs (because
crossing point
section
of
phase
cannot
are
transverse
the
of
angles
in the first
bundles
in the
(Jig. JO).
centre
bundles of the first
angles
pair
the vascular
position
second and third
2.6.
divergence
centre
HELLENDOORN AND
of the centre of the first two leaf
between the second and the third leaf
centre
leaf
apical
for the estimation
H.
on
stripes
horizontal
measured because in the
irregularly elongated.
were
3, 4, 7 and 12)
cortex
only
plotted,
bijugate phase
it will
From
distances
case
fig.
(divergence
12,
in
which
be apparent that there is
into the transition
zone
angles)
can
be
of vertical distances the internodes
(see
a
the horizontal distances
gradual
transition from
also Gomez-Campo
1974).
PHYLLOTAXISIN
BRYOPHYLLUM
481
TUBIFLORUM
Fig.
11a.
Transverse section
through a bijugate apex
Fig.
11b.
Transverse section
through a
tricussate
Fig.
11c.
Transverse section
througha
transition apex of
Fig.
lid.
Transverse section
through
an
inverted
of
apex
helix
Bryophyllum lubiflorum.
of Bryophyllum
lubiflorum.
Bryophyllum lubiflorum.
apex
of
Bryophyllum lubiflorum.
482
Fig.
P.
12.
Cortex
stripe
measurements
Fig.
13a.
Normal
Fig.
13b.
Plantlet after 7
Fig.
14.
3.2.
In
plantlet
Elongation of
Elongation
all
cases
elongated
the
on
of angles between
of Bryophyilum
days
pair
experiments
basal
plates
leaf
plate
on
on
of the
that
for
A.
LINDENMAYER
Bryophyllum tubiflorum.
Bryophyilum tubiflorum.
of Bryophyilum
very
tubiflorum.
young plantlets (fig.
plantlets
which
the side of the smaller leaf after 7
that the tissue of the
pairs
AND
tubiflorum.
of darkness of
the second leaf
HELLENDOORN
H.
days
bear
the
first
of darkness.
13)
two
This
leaves
means
side in younger. Indeed, the plate is growing
PKYLLOTAXISIN
in size
the
during
but
leaves of the second leaf
3.3.
of the
of the first leaf
of different ages;
are
483
TUBIFLORUM
development
the leaves
treatment
plate,
BRYOPHYLLUM
pair
Measurements of
plantlet.
pair
it is
furthermore,
exactly
are
angles
the
on
without the
Therefore,
located
are
the
on
dark
on
the
from fig. 14 that the
apparent
level.
same
between the
level
same
first
leaf
two
pairs
{table 1)
The
values of the
mean
“one-sided” and
one
opposite,
180°,
and
the
to
the observed values
fact that
of the
position
each
exact
of
centre
of Kalanchoe
between
only
the
the
hybrid,
two
leaves
in
contrast
6°,
tion
helix,
apical
=
1.580
=
1.242
tricussate/bijugate
=
1.966
helix/tricussate
were
to
draw
values
in
180°
deviates
about
primordia by
1.
The
/rd
r
=
t
chapter
apex
been
apex
diffusion
areas
are
as
of the
from
180°
and in
14°. The latter
angle
bijugate phase,
transi-
follows:
2
y/
areas
(rt /r d
1.966
this research
printed.
be
This
,
r
=
radius)
agrees with the
1.4.
was
carried
by
the
primordia,
through
for
place
on
that
out
a
was
in connection with the
based
particular
inhibits the
15
fig.
the
one
of
theory
of
substance,
development
of
which
new
is
leaf
the apex.
developing
approximated by
leafinitiation take
simulation
theory; namely,
assumptions
can
two
angle
1.257
=
since
diffusion
Some of the
first
the
tubifiorum
on
has
the
Bryophyl-
indicate that
In
in
plantlets
conclusions.
leaves in
their outputs
produced
Since the
between the
Bryophyllum
by
exactly.
valid
two
angle
large,
and that the
and also the
computer simulations, obtained by Veen and Lindenmayer.
the
calculated
simulations
Computer
As stated in the first
Schoute,
values
(table 2)
tricussate/bijugate apical
computer prediction,
3.5.
same,
daigremontiana
ones
and inverted helix
phase
tricussate/transition
The ratio of
mean
apical
mean
transition/bijugate
inverted
The
areas
ratios between the
tricussate
be found
between the first
deviate from
to
the
differs from the
with the
values
nearly impossible
possible
Kalanchoe
in
mean
a
180°
exactly
not
the value of the standard deviation.
Estimation of
pairwise
plants
angle
lies
“One-sided”
mean
cannot
was
seemed
hybrid
which appear
The
mean
is
cutting
daigremontiana.
considerably
exceeds
3.4.
first
about
by
it
size,
and in the
tubifiorum
leaves
same
calculated both in
were
pair
The standard deviation is rather
vascular bundle
it is clear that the
Namely,
lum
of the
nearly
sides.
“Two-sided”
transverse
a
leaf
under 180° into its mirror value above
directly.
method used for the three kinds of
were
the
two
on
angle
all values.
over
If
manner.
asymmetry
by translating
averaging
by averaging
due
have
can
obtained
were
in the table,
angles, given
“two-sided”
a
the theoretical model
cylinder
the surface of the
and the
were:
processes
cylindrical section,
leading
to
484
Table
P.
1.
Angles
between the
first
Bryophyllum
2
AND
A.
LINDENMAYER
leafpairs.
tub.
Kalanchoe
daigr.
daigr.
Hybrid
Hybrid
193
293
84
O
0
180
265
86
0
191
191
273
93
93
0
182
182
269
82
0
180
180
269
89
0
156
233
53
53
0
172
265
85
0
180
278
96
0
0
208
305
86
0
195
287
96
0
188
279
90
0
185
242
0
172
274
108
0
186
186
256
103
0
195
273
273
111
0
160
278
89
0
184
184
202
202
91
0
200
259
105
0
143
250
91
0
188
270
270
91
0
153
237
88
0
180
180
277
76
0
191
191
207
106
0
191
278
73
73
0
200
259
101
101
0
187
297
92
0
205
205
290
106
106
0
232
311
130
0
180
255
75
75
0
196
286
86
0
204
284
90
0
188
273
79
79
0
0
195
266
96
0
150
268
88
0
203
270
100
100
0
155
266
91
0
0
182
287
77
0
175
175
263
90
90
0
175
274
94
0
200
286
86
0
196
285
92
0
192
284
112
0
176
175
260
97
0
180
180
270
89
89
0
172
294
90
0
177
177
292
94
0
160
257
75
75
0
165
165
264
74
74
0
190
190
259
112
0
190
190
284
88
88
0
176
274
86
0
194
194
265
85
85
0
184
261
261
106
106
0
184
184
296
91
0
191
191
266
111
179
267
87
0
186
266
100
0
203
291
87
0
191
279
99
0
171
248
248
68
0
0
175
260
260
100
100
0
190
293
90
90
0
197
197
289
109
0
176
268
77
0
180
180
268
80
80
0
182
271
271
69
0
194
194
292
94
0
170
170
284
284
97
0
0
185
259
81
81
0
200
261
77
0
171
233
233
76
0
219
291
291
111
0
158
277
78
78
0
0
179
266
89
0
169
257
88
0
200
275
275
78
78
0
0
180
180
266
266
88
88
0
181
181
262
78
0
179
261
77
0
180
275
95
0
176
270
89
89
0
192
192
272
90
0
182
182
269
89
0
185
258
94
94
0
178
240
81
0
168
233
233
90
0
180
180
280
90
0
190
190
285
113
0
171
290
290
84
0
180
180
269
91
0
172
172
278
89
89
0
180
180
255
101
101
0
174
273
91
0
180
180
274
274
94
0
193
283
101
101
0
181
261
93
0
172
265
85
85
0
186
279
95
95
0
158
253
253
72
0
183
265
90
0
172
172
264
95
95
0
200
200
284
103
122
2
2
1
first
first
second
leafpair
leafpair
leafpair
HEIXHNDOORN
O
0
1
first
H.
mean
value
1
122
1
2
2
first
first
second
leafpair
mean
■sided
one-sided
one-
value
one-sided
194.125° ±
194.125°
±
11.428°
two-sided
two-sided
184.041°
0
± 18.383°
±
54
54
122
2
2
1
1
first
first
second
leafpair
mean
value
one-sided
186.366°
186.366° ±
6.903°
± 6.903°
193.846°
193.846° ±
± 9.396°
two-sided
two-sided
two-sided
182.122° ± 9.147°
184.000° ±
184.000“
±
16.248°
PHYLLOTAXIS IN
Table
2.
The
485
BRYOPHYLLUM TUBIFLORUM
areas
ofapices (in microns) of Bryophyilum tubiflorum.
decussate
decussate
transition
transition
65
helix
helix
tricussate
inverted
helix
121
220
220
51.3
93
105
171
58.8
98
125
165
90
90
118
132
121
121
56.3
121
135
111
91
86
145
117
50
36.3
36.3
120
120
111
89
89
130
97
113
68.8
68.8
86
91
170
71.3
71.3
81
143
176
77
96
150
150
192
192
67.5
67.5
97
102
143
65
97
98
130
130
51.3
87
128
158
73.8
85
85
115
53.8
78
78
143
143
71.3
116
116
138
41.3
76
125
53.8
96
121
78.8
112
143
63.8
86
132
132
60
158
158
56.3
132
132
152
101
115
118
128
128
116
136
133
135
value
mean value
62.62
2.
mean value
12.96
12.95
The cells
lar
3.
±
are
99.20 ±
99.20
approximately
Every
cell has 8
substance
Q1
representing
=
C
0
+
Q
=
a
used
AC„
center
=
X
are
version
value
value
154.79 ±
arranged
on a
D
for
29.71
rectangu-
=
=
following
the diffusion of the inhibitor
of two-dimensional Pick’s
S C, + (1-9X-D)
equation,
C
0
o
cell, C„
diffusion-coefficient,
In this model the
a.
size and
equation
an
digitized
inhibitor concentration
concentration in
=
same
17.28
neighbours.
Lindenmayer
i
X
of the
mean
123.16
123.16 ±
±
grid.
Veen and
where
mean value
16.01
16.01
=
in
the
i-th
cell,
C
=
0
previous
inhibitor concentration in
next
decay-coefficient.
values
can
be
The width of the apex: number of cells
varied;
along
inhibitor
center
circumference of apex,
cell,
486
P.
b. The leaf value:
the
black
by
d.
The diffusion
The
e.
in
A.
LINDENMAYER
the
primordia
primordium arises,
a new
coefficient,
coefficient: the
decay
AND
squares),
The threshold value: the value below which
c.
HELLENDOORN
concentration of inhibitor substance
15 represented
(in figure
H.
inhibitor
decays
at
rate
a
proportional
its
to
concentration in every cell except the leaf primordia,
f.
The
the
In
growth
takes
growth
rate:
place by
adding
new
at
rows
the
top rim of
cylinder.
15 each number
figure
letter represents
or
centration of the inhibitorin
a
rounded-off value for the
con-
cell.
a
The program Blokco 28 for the computer simulations was written in ALGOL
A.
by
H. Veen,
Philips
the program
In
a
Bryophyllum
computations
the
were
rather
bijugate pattern
has been obtained: the
and
width
==0.15(15%
X
=1/18
bijugate
to
should
hypothetical
ly,
remain
printing
more
50
every
realistic
Although
symbol
(since
does
not
did
In the
improve.
not
was
original
values in fig.
Although
decrease
to
a
to
and
T
More
changed
a
are
change
by
relative
ratio for
order
D/X.
In
make
to
printed symbol
improved.
continuous
continuously
with the
simulation this
helix.
was
cell
con-
a
new
a
factor
is
7/12
done
at
of
the
Similar-
15 the width of the
merely
of its
the result
presentation
height).
phyllotaxis
of
Bry-
The transition zone, for instance,
(see figs.
manner
means
factor
same
L/T.
graphical
of the
the
if the leaf
3/2,
concentrations
figure
the
122 for
has been calculated
values for
provides only
observations
a more
a
of the
from
gradually
good approximation
a
it could still be
exactly correspond
the leaf value
in
the width of
the leaf values in
vary
results
twice
via
inhibitor in each
rather than 25. This
array size is
this simulation is
ophyllum tubiflorum,
to
L
only provide
symbols,
of
almost
two
step)
computations
70
pattern. This tenfold
constant.
with
added.
was
inhibitor and the simulation
actually
time
per
after
1/70,
=
programmed
trijugate
starts
trijugate pattern
a
increase of the width of the apex
the simulation can
arrays is
of
was
to an
correspond
value
(L)
changing phyllotaxis
0.4
=
row
12 for the
the
pattern
centrations of the
to
Electrologica
25 cells
=
G
rate
in
over
D
threshold concentration T
leaf value
the
on
were:
W
coefficient
diffusion coefficient
The
out
(inhibitor producing factor), proceeds
then goes
The parameters for this simulation
growth
of
approximation
accurate
tubiflorum
decay
carried
of Utrecht. Further information about
University
leaves of different leaf values
opposite
a
at
be found in Veen & Lindenmayer (1973).
can
15
fig.
with
while the
X-8 computer
(2
3 and
4). By attempting
than in this simulation the
that after each
units per
irregular
new
intervals
growth step
line
(see
added).
list of leaf
15).
it
is
possible
to
improve
the
original
simulation
by
varying
the
PHYLLOTAXIS IN
Fig.
15.
BRYOPHYLLUM
Simulation
of
phyllotaxis
Each
symbol standing for
W
25,
=
D
=
.15.
X
=
487
TUBIFLORUM
of
Bryophyllum tubiflorum by
inhibitor concentration
1/18,
T
=
.4.
G
=
1/70,
L
was
x
=
printed
200,
L
2
Veen
and
Lindenmayer.
double.
=
122, L
decreases to 12.
488
Fig.
16. First column.
489
Fig.
16. Simulation of phyllotaxis of
Bryophyllum tubiflorum
51
W
G
and
=
=
with
width
asymmetry of 7 °.
51,
D
=
1/70,
.019,
=
Ldecreases
X
=
300,
L
T
1/36,
2
=
=
.4,
185,
to 12.
Computation(corresponding
growth) begins
at the
first column and ends
bottom
at the
to
of the
top
of
the second column.
N, B. Due
to
an error
out, in
line 66
shown
in
a
the
in the
faulty position. It
appear two spaces to the
Second
print-
righthand
column.
leaf is
should
right.
490
P.
other
in
that
be reached is
can
for
instance
the
experimental
more
useful
Therefore,
a
4.
is
more
to
to
If
360°).
X
a
more
a
A.
for
possible angle
disappeared.
pattern
less variable asymmetry
or
been tried.
zone
between
than in the
regular
asymmetry
change
to
needed for such
not
are
simulation,
line 77 and
159. The
it
angle,
smaller
to
the results
Although
taken,
was
According
they
This latter simulation is in better agreement with
the
LINDENMAYER
bigger asymmetry angle
and
vary the asymmetry
has
long computer time
{fig. 16).
far
with respect
case
be able
The smallest
pseudo-bijugate
results with
width of 51
because of the
so
14° (1/25
then the
28°,
to
25 cells.
namely
AND
HELLENDOORN
of the simulation is the small width used
shortcoming
another
variables,
the simulation,
H.
to
seems
angles.
final
yet,
positive
are
stripes
cortex
transition helix in this
simulation.
previous
CONCLUSIONS
We
conclude from the
can
results that
sumed in the computer simulation
is realised in the
asymmetry
of the apex size
crease
conditions which
fulfilled. In the first
and in the second
plants,
actually
several of the
are
place
the
place,
occurs
during development
plants
of
in these
we as-
the
mirror
in-
predicted
plants.
In summary,
1.
the first
same
two
leaves
age, thus
of the
they might
exert
Bryophyllum
tubiflorum
different influences
on
the
are
not
of the
following primor-
dia;
2.
the
first
two
leaves
of the
3.
the
apical
a
points
during development
tubiflorum
1 and
with
a
are
180°
not
2);
factor
1.966 from
bijuby
factor of about 1.4.
It is
in
a
increases
area
Bryophyllum
follows from
tricussate and the radius and circumference of the apex increase
to
gate
of
plants
opposite (mirror asymmetry
a
important
plant
is
not
jugal multiple
first
starting
If the
to
the
into
the
plant
a
different type, which is
rate
are
due
to
by necessity
to
phyllotaxis
not
simply
be mirror asymmetry in the
are
of
the
tetracussate
could be
assumed
mostly
the
can
plants
then under the
of the
not
to
the
in
of the
rate
to
starting
leaf,
the
not
cotyledons
starting
single
cotyledon,
dicotyledonous
multiples
must
leaves for
a
phyllotaxis
phyllotaxis patterns
and
leaves of
low
mainly
For
inhibitor pro-
increase.
be the
have
symmetric.
higher indices,
are
the
single starting
have often
position
cotyledons
or
of inhibitordecay has
mirror
plants
grown
helix,
phyllotaxis
increase,
to
irregular phyllotaxis patterns
plantlet
a
higher phyllotaxis system,
a
Monocotyledonous
decussate,
a
or
leaves),
is
fully
obtain
helix and the
cussate
into
there have
one,
the apex has
of
pattern.
tric. Otherwise the
The
previous
decrease,
(alternate
ferent from
order
to
position
however,
changes
leaves.
cotyledons
phyllotaxis
whose
of the
hypothesis
duction has
indices
realize the consequences of these results. If the
phyllotaxis changes
diffusion
The
to
constant, but
which
plants,
are
dif-
of decussate. In
be mirror asymme-
phyllotaxis.
Bryophyllum tubiflorum
like the inverted
pattern represent further interesting problems. Tetra-
be considered
as
completely symmetric
a
at
jugal doubling
the
beginning,
of
decussate,
so
the
and from this pattern
PHYLLOTAXIS
the apex
during development
servations
491
BRYOPHYLLUM TUBIFLORUM
IN
fit
this
idea,
circumference could increase
because
plants
two
pattern, in which the decussate pattern changed
Plants with
have
imperfect
it is obvious
inverted helix
an
trios
quickly
and
tricussate
for
hypothesis
a
a
in
hypothesis
passed
over.
be tested
to
here
becoming
phyllotaxis,
but
constant
Sometimes
phase
plants
was
in
so
too
that the
found with this
too.
apex increased after
than the
increases
size,
were
present
bigger
case
According
being
to
constant
by computer simulation.
be said
can
one.
plants. Furthermore,
that the apex in this
that the
mean
tetracussate
a
a tetracussate
of the inverted helix is
possible
size before
this would
presented
for
area
transitional tricussate
while. This has still
The results
into
times. The ob-
considered in fig. 4 and table 2. These plants
apical
It is
plants.
would be
phase
pattern in which
our
that the
much
too
directly
different from those of the tricussate
(table 2)
of the tricussate
area
are
two
found with
were
they
do
to
be in accordance with the diffusion
not
disprove
the
possibility
of other
of this work with
advice and
hypotheses.
ACKNOWLEDGEMENTS
We
acknowledge gratefully
technical
and
by
suggestions by
Prof.
Dir.
Madrid, Spain.
C.
the
friendly
Dr. R.
Gomez-Campo,
A travel
grant
and enthusiastic support
Soekarjo, Limnologisch Instituut, Nieuwersluis,
Institute
Nacional
de
Netherlands
Investigaciones Agronomicas,
from Utrechts Universiteitsfonds
to P. H. Hellendoom
is also
gratefully acknowledged.
The
following photographs
Utrecht:
The
nal
de
figs. 5, 7, 8,13
photograph
for
were
made
by
Miek
made
by
Maria
van
Rheenen,
Botanisch
Laboratorium,
and 14.
figure lib
was
Investigaciones Agronomicas, Madrid, and
Astrella
the rest
by
Gomez-Campo,
Institute Nacio-
P. H. Hellendoom.
492
P.
H.
HELLENDOORN
AND
A. LINDENMAYER
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in
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