Simulations for pattern formation and
regeneration in hydra
From “The Algorithmic Beauty of Sea Shells”
© Hans Meinhardt and Springer Company
Essential elements of hydra patterning can be modeled by
assuming three pattern-forming systems
Three activator-inhibitor systems are assumed;
one for the head (green), one for the tentacles
(dark red) and one for the foot (pink). These are
coupled via the graded competence of the cells
to perform the head-forming reaction (blue;
source density in our original model, head
activation gradient in the literature). The head
signal increases this competence, the foot
reduces it.
As the simulations shows, the model accounts
for the activation of these signaling systems at
correct positions, e.g., in a fragment of a body
column during regeneration. After growth, newly
separated fragments will regenerate again with
the original polarity. In other words, the intrinsic
polarity is part of the dynamic system and will
persist.
A non-trivial aspect: maintenance of polarity during growth
This simulation shows a highly non-trivial feature of the model: the adaptation to
changing sizes. After the correct initiation, substantial growth is possible without
the formation of additional maxima. Due to the graded competence, the apical
dominance extends over a large region. Without this feature…
(As mentioned, the signals for hypostome (green), tentacle (red) and foot formation (pink) are activator-inhibitor systems.
These systems are liked via source density (blue). The hypostome system elevates the source and appears at the
highest source density, the foot system does the opposite. Tentacles emerge at the highest source density not occupied
by the hypostome signal).
….without a graded competence, multiple maxima
would form
Maintenance of polarity during regeneration
In addition to the enhanced apical dominance, the graded source density
(graded competence, blue) has another important consequence:
regeneration occurs always with the correct polarity. Any fragment has
an intrinsic asymmetry that orients the emerging pattern.
As a sideline and as shown in the next frame, this is not a general
feature of developing systems….
Sea urchin regeneration occurs - in contrast to hydra - with
polarity reversal in one fragment
Left: after separation of a sea urchin embryo, the two fragments regenerate. In
one embryo the polarity is reversed (Hörstadius 1939; the original orientation is
documented by vital staining; stippled).
Model: after separation, the remaining inhibitor gradient in the non-activated
fragment imposes an asymmetry. Since there is no intrinsic asymmetry, the side
most distant to the originally activated region will win the competition (blue arrow).
In sea urchins the predicted activator-inhibitor interaction is realized by Nodal/Lefty2. Nodal has a non-linear feedback on
its own production. The inhibitor lefty2/antivin is under the same control as Nodal but blocks the Nodal-self-enhancement .
In agreement with the model, it is the fragment without Nodal expression (V) that reverses polarity.
Organization by the canonical Wnt pathway
Hyß-cat / Hy-Tcf
Wnt
Hobmayer et al. Nature 407, 186-189 (2000)
Curr.Op.Genet.Dev. 14, 446-454 (2004).
The predicted model accounts for de-novo
pattern formation. The formation of
complete animals from disaggregated
cells is a most convincing example (Gierer
et al., 1972).
The dynamics observed by Hobmayer et
al. (2000) for the appearance of ß-catenin
and TCF in aggregates is in full
agreement with the expectation of the
model: first a more overall activation
appears that sharpens in the course of
time. The head emerges subsequently at
these signalling centers.
Most remarkable, the Wnt maximum
appears with a different dynamics. It
appears somewhat delayed at the
smoother ß-catenin maxima as sharp
peaks, suggesting that Wnt and ß-catenin
are coupled but are not part of the same
positive feedback loop.
Model for the formation of structures next to each other:
tentacle formation as example
Dev. Biol. (1993) 157, 321]
Tentacles appear at the correct position if the head signal generates on long
range the precondition to form the tentacle signal. In contrast, on short range,
the head signal inhibits tentacle formation. Thus, tentacle formation can
appear only next to the head.
In the simulation above, the head signal (green) maintains the source density
(blue; competence, head formation gradient). The tentacles appear at the
highest blue level that is not occupied by the head signal (green)
Dynamics of tentacle regeneration in hydra: in near-head
fragments the tentacle signal appears first at the very tip
Bode et al (1988). Development 102, 223-235
For near-head fragments, Bode et al. 1988 showed
with antibody staining that the tentacle signal appears
first at the very tip and becomes subsequently shifted
to its final position. An intermediate ring decays into
the separate patches.
In the model, the source density (blue) has a long
time constant. After head removal, the source density
is still high enough to trigger tentacle formation
directly. This occurs at the oral end of the fragment.
Only after the trigger of the head signal, the tentacle
signal becomes displaced to the final position….
Dynamics of tentacle regeneration in hydra: in near-head
fragments the tentacle signal appears first at the very tip
This works also in a two-dimensional
simulation.
Bode et al (1988). Development 102, 223-235
Tentacle regeneration in hydra: in more aboral fragments and
in buds tentacle signal appears directly at the correct position
The prediction* was that the sequence of events is other way round in
fragments obtained from a more distal position. There, the competence
is too low to allow a direct trigger of the tentacle signal. First the head
(hypostome-) signal triggers that leads to an elevation of the
competence (blue) After reaching a certain level, the tentacles are
triggered. Since the head signal is already there, the tentacle signal
appear directly at the correct position.
(*) Dev. Biol. (1993) 157, 321; Evidence for this prediction has been found first by Technau and Holstein (1995): Head formation in
hydra is different at apical and basal levels. Development 121,1273-1282. Further evidence came from Smith, Gee and Bode
(2000). Hyalx, an aristaless-related gene... Development 127, 4743-4752:
Hyalx in tentacle formation
Hyalx, an aristaless-related gene...Smith, Gee and Bode (2000). Development 127, 4743-4752
The same dynamics was convincingly demonstrated by Smith et al.(2000) with the
Hyalx gene that is expressed at the tentacle base. In near-head fragments (upper
row) its appears first at the tip that become subsequently shifted to the correct
position. In contrast, during budding, the tentacle signal appears as a ring at the
final position. Later it breaks up into individual small rings that forms the base of
the tentacles. The model accounts for these different modes …
Hyalx in tentacle formation
Hyalx, an aristaless-related gene...Smith, Gee and Bode (2000). Development 127, 4743-4752
Dev. Biol. (1993) 157, 321
In this simulation, the regenerating apical end is approximated by a sheet; the
center corresponds to the tip of the hypostome. The source density (blue) is high
enough to trigger a single tentacle signal (red) directly. This occurs, therefore in
the center. Only after the subsequent trigger of the hypostome signal (green,
top), the tentacle signal is displaced to the final position and splits up.
Hyalx in tentacle formation
Hyalx, an aristaless-related gene...Smith, Gee and Bode (2000). Development 127, 4743-4752
H.M., Dev. Biol. (1993) 157, 321
In more aboral fragments and during budding, the source density (blue, bottom)
is too low to trigger the tentacle signal directly. After elevation due to the newly
formed head signal, the tentacle signal appears in a ring surround the head
signal. The ring disintegrates into discrete spots.
The signal for the first bud has initially a belt-shaped
distribution that sharpens subsequently
ß-catenin
Wnt
ß-catenin
Tcf
Wnt
From: Hobmayer et al.. (2000), WNT signalling act in axis formation in
the diploblastic metazoan Hydra. Nature 407, 186-189 (2000)
Due to the absence of a strong circumferential asymmetry, the first ß-cat / TCF
signal that leads to bud formation has
initially nearly a ring-shaped extension
that sharpens subsequently to a lateral
spot. This is reproduced by the model.
The localization of the tentacles depends on the
graded source density (graded competence)
Broun et al., (2005).
Development
132,2907-2916.
Treatment with the drug Alsterpaullone
stabilizes ß-catenin. This leads to tentacle
formation everywhere. At early stages
supernumerary tentacles form at some distance
from the head, later they are found allover (Top
left: control, top right: after treatment).
In the simulation, first normal tentacles are formed during a regeneration. Later, the
drug Alsterpaullone is added, assumed to elevate the source density everywhere
(blue). Supernumerary tentacles form first at a distance from the existing tentacles.
In this simulation also shifts of the tentacle signals play a role.
Foot formation or how to generate a second
organizer to the opposite position of a field
In many systems different organizers form at antipodal positions. The head and the
foot of hydra is an example. A simple cross-inhibition of the two pattern-forming
systems is insufficient since then, for instance, foot-regeneration would be
impossible in a small fragment containing the head. The cross-inhibition of the head
would suppress the trigger of the foot signal.
The problem disappears if no direct inhibition
exists but the formation of the foot signal
occurs preferentially at a low source density.
The lowest source density occurs at the largest
distance from the head. Therefore, the foot
appears at the maximum distance. The foot, in
turn, is assumed to lower the source density,
stabilizing in this way its own precondition.
Since no direct inhibition of the head onto the
foot exists, foot regeneration can also occur
close to an existing head….
Although forced to be formed at an antipodal position:
a foot can regenerate close to a head
Note that in this simulation the trigger of a new foot signal close to the existing head
has almost no feedback on the head signal. The same would be true if a new head
signal regenerates close to an existing foot.
A candidate for an autocatalytic loop for foot formation is the positive feedback
between Nkx2.5 and pedibin [Thomsen et al., Mech Dev 121,195-204 (2004)]
The reappearance of the Cn-nk-2 signal during foot regeneration
Grens, Gee, Fisher and Bode (1996) Dev. Biol. 180,473-488.
The regeneration of a second head instead of a
foot after DAG treatment.
After treatment with Diacylglycerol (DAG)
supernumerary tentacles are formed (as in the
Alsterpaullone experiment mentioned above and
presumably for the same reason). After removal of
the foot, a head regenerates instead.
Model: with the elevation of the source density,
the tendency for foot formation is reduced in favor of
head formation.
Experiments: Müller, W.A. (1989) Diacylglycerol – induced multihead formation in Hydra
Development 105, 309-316
Model: H.M., Dev. Biol. (1993) 157, 321
Not yet understood: unexpectedly after washing out
DAG, first buds appear in the center;
a new foot comes later.
In the model, after termination of DAG treatment,
first a foot signal is expected to reappear. However,
first buds are formed in the centre. Explanation by
Werner Müller (who did the experiments): with the
lowering of the head activation gradient, first a
threshold for bud formation is reached (dashed
line). Only after further lowering, the level required
for foot formation is obtained (left). This leaves the
question open of how foot regeneration can take
place in a region close to the head, i.e., in a region
of high head activation gradient
Experiments: Müller, W.A. (1989) Diacylglycerol – induced multihead formation in Hydra. Development 105, 309-316
Also unexpected: new heads and feet appear close to each
other when body columns are grafted together;
After Tardent (1954). Roux Arch. 146, 593
Head and foot are expected to emerge at maximum distance from each other.
However, after grafting several body columns to each other (Tardent, 1954), the
terminal organizers appear close to each other. In the model, due to an increase
in the source density by the head signal, the foot signal is expected to shift away
from the head signaling center…
Also unexpected: Tardent grafted body columns together;
new heads and feet appear close to each other.
After Tardent (1954). Roux Arch. 146, 593
Conclusion:
Many basic hydra experiments are explicable by the assumption that the
formation of the head, the tentacles and the foot are under control of activatorinhibitor mechanisms. These three systems are coupled by the source density
(head activation gradient, competence to form the head signal)
The model accounts for the generation and regeneration of all signals at the
correct place, for the reformation of the signals during re-aggregation, for the
maintenance of polarity during regeneration, for the different dynamics of
tentacle formation during head regeneration and budding and for the formation
of tentacles all over the body after Alsterpaullone treatment.
The canonical Wnt pathway with ß-catenin und Wnt plays certainly a central
role in the formation of the primary hydra organizer. The molecular basis of the
long-range inhibition is not yet clear.
The equations used for the hydra-simulations
Head activator a
a s c (a 2  ba )
2a

 ra a  Da 2  ba
t
b
x
b
 2b
2
 s c (a  ba )  rb b  Db 2  bb
t
x
Source density c
c
 2c
 rc a  rc c  Dc 2  bc
t
x
Tentacle activator d
c cd (d 2  bd )
d
2d


r
d

D
d
d
t e (1  sd d 2 ) (1  ce a)
x 2
c cd (d 2  bd )
e
 2e

 re e  De 2  be
t
(1  sd d 2 ) (1  ce a)
x
Foot activator f
2
f rf ( f  b f )
2 f

 rf f  D f
t
cg
x 2
2
g rf ( f  b f )
2 g

 rg g  Dg 2  bg c
t
c
x
H.M., Dev. Biol. (1993) 157, 321