Journal of Theoretical Biology 222 (2003) 37–52
A model for budding in hydra: pattern formation in concentric rings
Stefan Berking*
Zoological Institute, University of Cologne, Weyertal 119, Koln
. 50923, Germany
Received 27 June 2002; received in revised form 11 November 2002; accepted 25 November 2002
Abstract
Current models of pattern formation in Hydra propose head-and foot-specific morphogens to control the development of the
body ends and along the body length axis. In addition, these morphogens are proposed to control a cellular parameter (positional
value, source density) which changes gradually along the axis. This gradient determines the tissue polarity and the regional capacity
to form a head and a foot, respectively, in transplantation experiments. The current models are very successful in explaining
regeneration and transplantation experiments. However, some results obtained render problems, in particular budding, the asexual
way of reproduction is not understood. Here an alternative model is presented to overcome these problems. A primary system of
interactions controls the positional values. At certain positional values secondary systems become active which initiate the local
formation of e.g. mouth, tentacles, and basal disc. (i) A system of autocatalysis and lateral inhibition is suggested to exist as
proposed by Gierer and Meinhardt (Kybernetik 12 (1972) 30). (ii) The activator is neither a head nor a foot activator but rather
causes an increase of the positional value. (iii) On the other hand, a generation of the activator leads to its loss from cells and
therewith to a (local) decrease of the positional value. (iv) An inhibitor is proposed to exist which antagonizes an increase of the
positional value. External conditions like the gradient of positional values in the surroundings and interactions with other sites of
morphogen production decide whether at a certain site of activator generation the positional value will increase (head formation),
decrease (foot formation) or increase in the centre and decrease in the periphery thereby forming concentric rings (bud formation).
Computer-simulation experiments show basic features of budding, regeneration and transplantation.
r 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Hydra; Pattern formation; Mathematical model; Budding; Regeneration
1. Introduction
Hydra has a tube-shaped body. One end comprises
the head with the mouth/anus opening surrounded by
tentacles. The other end is termed foot and includes the
basal disc which closes the tube. The middle part is the
gastric region (Fig. 1A). The body wall consists of two
layers, the ectoderm and the endoderm, separated by an
extracellular matrix, the mesogloea. The body wall
between head and foot has a polar organization: body
pieces obtained by transverse sectioning, regenerate the
head from the apical end and the foot from the basal
end. Tissue pieces obtained from different body levels
display different capacities to transform into a head or a
foot, respectively, when transplanted laterally to a host
animal. The tissue obtained from a more apical position
combines a higher capacity to form a head with a lower
*Tel.: +49-221-470-2248; fax: +49-221-470-5171.
E-mail address: [email protected] (S. Berking).
capacity to form a foot in such transplantation
experiments. The polarity of the tissue and the graded
distribution of the noted capacity is determined by a
scalar tissue property (Gierer et al., 1972) which has
been termed positional value (Wolpert, 1969; Wolpert
et al., 1974) or source density (Gierer and Meinhardt,
1972). By definition, the positional value or the source
density has its highest value at the apical end of a Hydra
and its lowest value in the basal disc (cf. Fig. 2A).
Budding is the way Hydra reproduces asexually.
Budding visibly starts with the formation of a small
protrusion of the parent’s body wall (Fig. 1A). About
one day earlier preparatory steps of bud formation are
initiated (Berking, 1977, 1980). The bud’s tip will
become the head of the new animal. The bud grows by
recruiting tissue of the parent animal and by cell
multiplication within this tissue. At the bud’s base a
foot develops. Then the animal detaches from the
parent. The model presented here is in particular
designed to explain budding. Some regeneration and
0022-5193/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0022-5193(03)00012-2
38
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
Fig. 1. The process of budding in Hydra. (A) Scheme of bud
formation. (B) According to Sanyal (1966) the development of a bud
is determined in a young bud anlage (redrawn from Tardent (1978) and
Sanyal).
transplantation experiments will be discussed as well.
The need to design a new model derived from an
observation made by Rand (1899), Sanyal (1966), and
Tardent (1972): initially within the parent animal the
bud is organized in concentric rings. That means the
head starts as a small patch and the foot as a concentric
ring surrounding the future gastric region and the future
head. This observation strongly disfavours models
which assume head and foot formation to be organized
by symmetrical systems.
2. Results
The evolution of models strongly depends on
observations which do not fit well into the current
models. One possible consequence is to design a model
which starts with that observation which is in variance
with the existing models.
Generally, models for pattern formation in Hydra are
designed to explain head and foot regeneration. The best
evolved model is that of Meinhardt (1993). This model
assumes the existence of a head activator which
stimulates its own release by autocatalysis and causes at
the same time the generation of its antagonist, the head
inhibitor (heterocatalysis). This feature restricts the area
of autocatalysis to a small patch of cells and at the
same time prevents an autocatalysis in the surroundings.
Foot formation is assumed to be similarly organized. In
addition, the activators increase and decrease the source
density/positional value, respectively. Therewith, a
gradient of source density/positional values forms. The
model is also applied to budding. It correctly describes
that a bud forms some distance away from head and
foot and that bud formation starts in a small patch of
cells, the future bud’s head. However, the model is not
able to explain foot formation at the bud’s base and the
separation of the bud from the parent.
In Hydra initially a bud develops by recruiting tissue
of the parent animal (Fig. 1). The bud’s tip will develop
into the head of the bud. When the tip visibly forms, all
body parts including the basal disc are already
determined in the parent body tissue in the form of
concentric rings (Rand, 1899; Sanyal, 1966; Tardent,
1972). Thus, head, gastric region, and foot formation
are organized from one point, the prospective bud’s tip.
This result shows that the assumption of symmetrical
systems for head and foot formation is not appropriate.
A system of autocatalysis and lateral inhibition is
excellently suited to activate a small patch of cells but
it will not cause the activation of a ring of cells. The
future bud’s head, i.e. the region where the head
activator is generated should also cause the generation
of the foot activator, which, however, functions only at
a certain distance from the centre of its generation. An
obviously unattractive conclusion. And in case a ring is
organized by a different system the assumption of a foot
activator and a foot inhibitor with properties similar to
the respective morphogens in head organization (autocatalysis and lateral inhibition) would cause foot patches
to form within this ring. Thus, budding displays a
selective power on models for pattern control in Hydra.
An alternative is to assume that the morphogens do
not control head and foot formation directly but rather
control the source density/positional value. In order to
explain budding the respective system has to be designed
in such a way that it is able to cause an increase of the
positional value in the centre and a decrease in the
periphery. Therewith concentric rings form. According
to a definite positional value secondary systems then
cause head and foot formation (Berking, 1979, 1998).
These thoughts have led to the model described in the
following.
2.1. The model
Pattern formation is proposed to be hierarchically
controlled. A primary system of interactions controls a
rather stable scalar tissue property which is termed
positional value (Wolpert, 1969; Wolpert et al., 1974) or
source density (Gierer and Meinhardt, 1972). The term
source density is well defined in computer simulations.
Thus, to prevent confusion the term positional value is
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
used here. In a normally shaped budless animal
the positional value decreases from the mouth to the
basal disc in the form of a gradient. According to
the local positional value, secondary systems with
secondary morphogens become active. The head is no
longer a unit of structure formation (e.g. Burnett, 1961;
Wolpert, 1969; Gierer and Meinhardt, 1972) but rather
a compound structure (Berking, 1979; Meinhardt,
1993): the maximal positional value causes mouth
formation, a lower value causes tentacle formation.
The lowest possible value causes basal disc formation.
The model proposed here concerns the primary system
only. It appears that the minimal number of morphogens which is able to control the positional value is
three.
da
a2
d2 a
¼ sa ra a þ dba þ Da 2
dt
dx
b
2
db
d b
¼ s b a2 r b b þ bb þ D b 2
dt
dx
or foot) and in addition the positional value/source
density. In the model presented here the respective
structures form when under certain conditions the
positional value increases (head formation), decreases
(foot formation), and increases in the centre of an
activated region but decreases in its periphery (bud
formation).
2.2. Application to budding
2.2.1. In budding, head, gastric region, and foot
formation are organized from one point, the prospective
bud’s tip
A computer simulation of budding is performed in a
row of cells which all are able to generate and to respond
A ¼ activator
B ¼ inhibitor B
2
dc
dc
¼ sc a2 rc c þ dbc þ Dc 2
dt
dx
2
2
dd
a
a
¼ sd
rd d þ bd se sa þ dba
dt
c
b
39
C ¼ inhibitor C
ri ¼ removal rate
bi ¼ basic production
Di ¼ diffusion constant
si ¼ constant
:
D ¼ positional value
Following the rules in physical chemistry and
models proposed for pattern formation, the
capitals A, B, C and D refer to the morphogens and
the positional value, respectively, the small letters a, b, c,
d denote the respective concentrations (used mainly in
the equations). The morphogens are able to diffuse (the
diffusion constant is denoted by Di ), the positional
values are not.
The equations may be interpreted as follows: the
positional value (D) of a cell is determined by its content
in compound A (activator). The activator is produced
within the cells. Therewith, the positional value increases. The activator can be released from the cells
(represented by a subtraction term in the equation for
the positional value), therewith the positional value
decreases. The released activator stimulates both, its
release out of cells and its production within cells. Thus,
two loops of autocatalysis exist. Further, the released
activator stimulates the release of two inhibitors
(heterocatalysis). In the released form one of them (B)
antagonizes the release of the activator out of the cells,
the other (C) antagonizes the production of the activator
within the cells. The equations are almost identical to
those proposed to describe the control of monopodial
growth in the hydrozoon Dynamena pumila (Berking
et al., 2002).
Current models suggest morphogens which are
structure-specific. They directly control a quality (head
to the morphogens A, B, C (Fig. 2). The simulation
results in a local increase of the positional value to a
stable maximal value while in the periphery it decreases
down to (almost) zero (Fig. 2D). The maximal value
causes mouth formation. The minimal value causes
basal disc formation. Thus, a bud is formed with all
tissues and body structures in the normal alignment and
proportion along the body axis. The positional value
adjacent to the bud represents the gastric tissue of the
parent animal.
The simulation shown may represent budding in a
mirror-image transplant obtained by sectioning animals
in the budding region (Fig. 2A). Such a transplant
allows to study the influence of the distance from the
head on bud formation and excludes at the same time a
possible specific influence of the foot. Transplants of this
type were found to result in bud formation (Fig. 2B,
Tripp, 1928). Young and small animals do not form
buds. A short distance to the head and a steep gradient
of positional values should result in an increase of the
concentrations of the inhibitors in the tissue of the
budding region (and the surroundings). When the
concentration of inhibitor B is slightly increased in the
simulations autocatalysis and therewith budding is
prevented (not shown). The same result was found by
Meinhardt (1993) for the initiation of head formation in
the budding region of small animals when applying his
model.
40
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
Fig. 2. Positioning of the bud some distance from the apical end. (A) Large and small (young) animals were sectioned in the budding region
(indicated by the horizontal line) and rejoined to get mirror-image symmetric transplants. In large animals budding is initiated while in small ones it is
not. The scheme also shows the proposed gradient of positional values along the body axis of large and small animals. (B) Development of a mirrorimage transplant made by Tripp (1928). The left part shows the transplant having formed its first bud (on the right side) and two younger buds
opposite the first one. The later formed buds developed so close to each other that they were only separated at their distal end (figure in the middle).
The next buds formed closer to the respective head than the foregoing ones (right side). A foot was not formed in-between the budding regions. (C)
Simulation of the fate of a row of cells in longitudinal orientation along the body wall of such a transplant (indicated by the bracket in (A)). Initially,
all cells are identical with respect to the morphogens they contain and generate (not shown). Lightest grey: activator A; medium grey: inhibitor of
autocatalysis, B, and inhibitor C (broken line) which is involved in the control of the positional value, D (black). In a small region (12 out of 28 cells)
in the centre of the figure the removal rate of inhibitor B, rb, is set higher than in the adjacent region (ratio: 1–0.67). Therewith, in the centre a shallow
depression of the inhibitor concentration is formed. Due to that, autocatalysis starts in the centre of the figure and not at a random position. An
apical basal concentration gradient of all morphogens including inhibitor B is present in both parts of the transplant (see the outcome of this
simulation). Here, only the middle part of these two gradients (restricted to inhibitor B) is shown. For this simulation the gradients are assumed to be
very shallow. When the simulation is carried on the release of the morphogens becomes enhanced by auto- and heterocatalysis in the centre. (D)
When the simulation is further carried on, a stable situation is reached: in the centre the positional value is highest which is suggested to cause mouth
formation (arrow). In the periphery of the peak the positional value has attained the lowest possible value, which is suggested to cause basal disc
formation (arrow). Ordinate: concentration of morphogens/positional value; abscissa: position. Letters a, b, c, d, indicate the morphogen
concentrations and the positional value, respectively. Diffusion rates: Da ¼ 0:015; Db ¼ 0:1; Dc ¼ 0:1; Dd ¼ 0: Removal rates: ra ¼ 0:015; rb ¼ 0:01;
rc ¼ 0:004; rd ¼ 0:000001: Basic production: ba ¼ 0:005; bb ¼ 0:0007; bc ¼ 0:005; bd ¼ 0:000001: Constants: sa ¼ 0:015; sb ¼ 0:017; sc ¼ 0:022; sd ¼
0:00025; se ¼ 0:0075: The parameters rd ; bd ; sd ; and se have been made very small to cause a slow change of parameter D (positional value) compared
to the others (diffusible morphogens). Initial conditions: aa ¼ 0:01; ba ¼ 0:1; ca ¼ 0:1; da ¼ 0:05: The technical basis of this simulation has been
developed by Meinhardt (1995).
2.2.2. Budding takes place some distance away from the
foot in animals of a certain length
In a mirror-image transplant produced by Tripp
(1928) several buds formed. When the first bud had
detached two new buds developed (Fig. 2B). Both of
them formed at a position closer to the respective head.
This type of development continued. While new buds
formed apical to the existing ones, the tissue between the
buds, which looks like so-called peduncle tissue,
elongated. A basal disc did not form for weeks. This
observation indicates that the basal disc is not involved
in the positioning of a bud. Thus, there is no indication
for signals generated by a (hypothetical) foot system
which prevents bud formation in the peduncle.
In the model proposed, budding is prevented by the
low positional value in the peduncle and by a sufficient
high level of inhibitor B. The respective simulation may
represent a mirror-image transplant made out of adult
animals sectioned in the tissue between the budding
region and the basal disc (cf. Fig. 2A). That means, in
the centre of the transplant the positional value is lower
than in the simulation shown above. All other parameters are kept constant (cf. Fig. 2). The simulation
does not result in budding (not shown). Meinhardt
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
41
Fig. 3. The formation of a frustule. Polyp of Haleremita cumulans forming two frustules (fr). Lower part left: a young frustule moving over the
surface. Right: tentacle (t) formation has started indicating the transformation into a complete polyp (after Kuhn,
.
1914). In the simulation the
positional value was reduced from 0.05 to 0.002 and the basic production of B, bb ; was reduced from 0.0007 to 0.00035. The letters indicate
the morphogens and the positional value, respectively. Other parameters as in Fig. 2.
(1993) proposed the same set of conditions to prevent
head formation as an initial step of budding in the
peduncle.
When both, the positional value and the level of
inhibitor B are reduced, a bud-like structure forms
(Fig. 3). However, the maximal positional value attained
is almost as low as the positional value in the
surroundings. According to the model, this is not a
normal bud because the maximal positional value
attained is too low to cause head structures to form.
The result of this pattern forming process is a piece of
gastric tissue with a basal disc at one end which
separates from the parent animal.
Such a process was never described for Hydra. But it
is well known to occur in relatives of Hydra, e.g. in the
fresh water polyp Craspedacusta and in Haleremita
(Fig. 3). Such a piece of tissue is termed frustule. A
frustule can move around for several days and then
transform into a complete polyp. In terms of the model,
following separation from the parent the positional
value slowly increases in the tip of the frustule and
finally reaches the value sufficient for tentacle and
mouth formation. In order to explain that such a
process does not happen in Hydra one has to assume
that in the peduncle the concentration of B does not
decrease strongly below the concentration in the
budding region, i.e. below the value applied in the
‘‘normal’’ simulation.
2.2.3. Certain manipulations cause a bud to develop into a
branch without foot formation
Two experiments will be discussed. (1) When animals
bearing a young bud are sectioned just apical to the bud,
the bud transforms into a branch without signs of foot
formation (Fig. 4A). At the same time head regeneration
is antagonized (Weimer, 1928; Rulon and Child, 1937;
Tardent, 1954, 1972; Sanyal, 1966). (2) Secondary axis
formation is induced by transplantation of small pieces
of hypostomal tissue or isolated heads with the
hypostomal tip in front to various positions into a
Hydra (Fig. 4B): when transplanted between the head
and the budding region, a branch develops. When
transplanted between the budding region and the foot, a
bud develops, which detaches from the parent (Berking,
1979).
A computer simulation of these two experiments was
performed in a two-step process. The first step was to
start budding in a mirror-image transplant, as described
in Fig. 2. Then the concentration of inhibitor B was
slightly increased in the bud as well as in the
surroundings of the bud and the simulation was carried
on. This second step of the simulation should reflect a
short distance of a young bud to a head regenerating
surface (first experiment) or to a still existing head
(second experiment). A head and a regenerating surface
release the morphogens including inhibitor B (see
below). Inhibitor B has a long range. When the second
step of the simulation was performed, a secondary axis
without a basal disc developed (Fig. 4C). Note that here
inhibitor B antagonizes foot formation. In the experiments described in the foregoing paragraph inhibitor B
antagonizes head formation as the initial step of
budding. (The influence of inhibitor C will be discussed
below).
2.2.4. An increase of the positional value in a belt or a
patch may precede budding and additional head formation
In a simulated mirror-image transplant with two
heads the concentration of inhibitor B is adjusted to a
value between the ‘‘normal’’ one which allows budding
and a value which prevents budding. This concentration
of B allows secondary axis formation but the process
takes a very long time and has an interesting feature
(Fig. 5). In a rather broad area there is an autocatalytically enhanced production of the activator and thus an
increase of the positional value. But there is no
autocatalytically enhanced release of the activator.
Finally, the release of the activator becomes autocatalytically enhanced as well, and the positional value
increases strongly which happens in a small area of this
42
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
Fig. 4. The formation of a branch. The simulation should represent two types of experiments: (A) Sectioning just above a young bud causes the
transformation of a bud into a branch and (B) transplantation of a hypostomal tip to a position between the head and the budding region causes
branch formation. The control is shown as well: transplantation of a hypostomal tip to a position between the budding region and the foot causes
bud formation. (C) The simulation was performed in mirror image transplants, as shown above. First step: the simulation was started by using the
parameters shown in Fig. 2. This leads to the release of all morphogens as shown in Fig. 2C. Therewith, budding is initiated (first experiment) or
(second experiment) secondary axis formation is initiated by the transplanted hypostomal tip-tissue. Second step: the concentration of B is raised
(increase of the basic production of B, bb ; from 0.0007 to 0.002). The second step should represent the development of an induced secondary axis at a
position closer to the head than a potential bud (second experiment) or (first experiment) should represent sectioning apical to the bud anlage and the
subsequent development of the bud anlage close to the head-regenerating surface where morphogens including inhibitor B are generated. The
simulation results in a branch and not in a bud. (A bud has a depression of the positional value down to zero at the periphery of the peak of
positional values.) Other parameters as in Fig. 2.
patch. In the periphery of this locally strong increase,
the positional value may decrease below the value
present in the surroundings but will not reach the lowest
value possible. Therewith, a branch forms instead of a
bud. If this took place in reality in an up to now budless
animal the initial slow and slight increase of the
positional value would be expected to form a belt just
above the region where later on a bud visibly forms.
There are two points of interest: (1) in Hydra the
tissue is ‘‘moving’’ down the body column (Tripp, 1928;
Burnett, 1966; Campbell, 1967) due to an almost
random cell multiplication in the gastric tissue (David
and Campbell, 1972). This causes a spreading of the
tissue. Further, there is a loss of cells, e.g. at the basal
disc. Thus, bud development starts closer to the head
than bud separation occurs. It may thus be possible that
budding starts with the noted slow increase of the
positional value in the form of a belt. Experimental
results appear to support that view (Berking, 1977). Due
to the continuous spreading of the tissue between the
head and the bud anlage, the bud may finally reach the
region in which the concentration of inhibitor B is so
low that basal disc formation can take place. (2) There
exists a mutant of Hydra magnipapillata, termed multiheaded one (mh-1) which produces additional heads
along the body column which do not detach (Sugiyama,
1982). It was rather surprising that these extra heads
do not keep their maximal distance from each other.
Rather, the extra heads often form at the same axial
level but opposite each other (Zeretzke and Berking,
2001). This may indicate that the initial event in extra
head formation is an increase of the positional value
within the epithelial cells (Zeretzke and Berking, 2002)
in the form of a belt.
2.2.5. Foot formation at the bud’s base and bud
separation
Certain manipulations prevent foot formation at the
bud’s base. The manipulations include transplantation
(Tardent, 1972; Berking, 1979) and treatments with
chemicals (Hassel and Berking, 1990; Pe! rez and Berking,
1994; Pe! rez, 1996; Zeretzke et al., 2002). The basal disc
may not form at all or may become restricted to a small
part of the original ring. The resulting lateral patch of
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
basal disc cells generally faces the parent animal’s foot.
In both cases the bud does not separate from the parent.
Thus, basal disc formation in the form of a belt appears
to be a prerequisite for bud separation. On the basis of
the observation that in the basal disc, cells are sloughed
off the body column (Tripp, 1928; Burnett, 1966;
Campbell, 1967) one can conclude that a loss of cells
Fig. 5. An increase of the positional value in a large patch or in a belt.
The simulation represents a mirror-image transplant as shown in
Fig. 2. However, in this case the respective animals were sectioned and
rejoined in the gastric region just above the budding region. In this
region the concentration of inhibitor B is higher than in the budding
region. This is simulated by an enhanced basic production of B, bb ;
from 0.0007 to 0.0014. All other parameters remained unchanged.
Even after a large number of iterations an autocatalytically enhanced
release of the activator does not take place. Rather, the positional
value increases slowly. Finally, an autocatalytically enhanced release of
the morphogens starts and a branch forms (not shown; the result is
very similar to that obtained in Fig. 4C, i.e. ‘‘transplantation’’ of tissue
from bb ¼ 0:0007 to 0:0014). Other parameters as in Fig. 2.
43
in the form of a ring is the cause of bud separation.
However, finally the bud only bears the basal disc. The
parent animal’s tissue from which the bud has separated
does not. This asymmetry has to be explained.
The model proposes that basal disc cells develop in a
belt-shaped manner at the junction between the
parent and the bud. In the central part of this belt the
cells may become apoptotic. This splits the belt into two
parts (cf. Fig. 2). A result of this atrophy is that the
communication by morphogens between parent and bud
decreases. At the bud’s basal end the conditions for a
decline of the positional value still exist. Thus, the most
basal cells adjacent to the existing basal disc cells will
reach the threshold for basal disc formation in
particular when the bud grows in length. At the parent
animal’s side of the junction such conditions do not
exist. Therewith, this patch of basal disc tissue will
disappear by apoptosis. Based on these arguments the
basal disc cells do not generate a signal which causes
adjacent cells to reduce their positional value and thus
to develop into basal disc cells (as it was proposed by
Meinhardt, 1993).
There are indications for a low positional value at the
parent’s former junction to the bud: at that site some
few peduncle specific nerve cells have been detected
repeatedly (Pe! rez, pers. comm.). In the course of growth
these cells disappear. Further, at the buds base and in
the adjacent parent animal, the FGFR-like receptor
tyrosine kinase kringelchen is expressed. In the parent
animal’s tissue the expression of kringelchen persists
transiently in the form of a ring after the bud has
separated. Gradually, the diameter of the ring declines
(Hassel, pers. comm.). This may indicate an increase of
the positional value from an initially low to a higher
value normally present in the budding region.
Fig. 6. Bud formation in a growing Hydra: (A) The simulation is made for a row of 5 cells with the equations given. The simulation results in a
graded distribution of the positional value, D, which should represent a small polyp. (B) Then growth is simulated. Initially a new cell has the same
positional value the neighbouring cell has. New cells are added at the apical end. (This is not very realistic but compared to a random insertion this
causes a rather smooth gradient of D.) The insertion of new cells causes the distance between the apical end with the highest positional value and the
budding region to increase gradually as it can be observed in reality. The result of growth is the formation of a bud anlage. (C) When the simulation
including growth is carried on, a new bud anlage forms apical to the old one, which then has developed into a bud. Diffusion rates: Da ¼ 0:015;
Db ¼ 0:1; Dc ¼ 0:1; Dd ¼ 0: Removal rates: ra ¼ 0:015; rb ¼ 0:01; rc ¼ 0:005; rd ¼ 0:0000003: Basic production: ba ¼ 0:005; bb ¼ 0:0003; bc ¼ 0:005;
bd ¼ 0:0000003: Constants: sa ¼ 0:015; sb ¼ 0:017; sc ¼ 0:022; sd ¼ 0:00008; se ¼ 0:0025:
44
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
2.2.6. Budding in a growing Hydra
The simulation starts with a short row of cells in
which the positional values are identical and ends with a
gradient of the positional values. This should represent a
small polyp (Fig. 6) and this small polyp does not form
a bud. Growth is simulated by adding cells. When a
certain body length is reached, the animal develops
a bud anlage. When the simulation is carried on, a new
bud anlage forms apical to the old one which at that
time has developed into a bud. Note that in a budless
Hydra the positional value and the concentration profile
of the morphogens form gradients which all start at the
apical end of the animal. Morphogens of this primary
system are not generated at the basal end.
In order to simulate growth, some of the parameters
are slightly altered to reach a stable endpoint. Of course,
in the long term the same parameters must be used for
all simulations. At present, however, it is difficult to
meet the natural conditions in the simulations. The main
problem appears to be that the concentration of
morphogens changes from the maximal to the minimal
concentration within seconds (real time for simulations)
while a respective change of the positional value needs at
least one day (too long for simulations).
2.3. Regeneration and transplantation
2.3.1. Transplanted tissue can transform into a head
or a foot
When a small piece of body tissue is excised a head
and a foot form at opposite ends. The respective
simulation (Fig. 6A) results in a gradient of positional
values even when the simulation starts with a row of
cells in which the positional values are identical at the
outset.
Several authors found that mirror-image transplants
made out of body pieces not sectioned in the budding
region will give rise either to a head or to a foot at the
junction: a head is formed in double foot animals, a foot
is formed in double headed animals. This takes place
even if, at the outset, the positional value at the junction
was identical in both transplants. Thus, the cause of the
observed different development is either the slope of the
gradient of the positional values (positive or negative)
and/or the differential influence from the ends.
In the model presented, one compound is particularly
responsible for the different developmental fate: it is
inhibitor C. In the double headed transplant the
positional value increases at both sides of the junction.
This causes a net import of C into the tissue at the
junction for two reasons: the basic production of C is
coupled to the positional value (d bc ) and C is
generated at the apical ends. In contrast, in the double
foot transplant there is a net export of C from the tissue
at the junction. This difference causes the different
developmental fate at the junction, though initially the
positional values were identical.
In the model presented, autocatalysis starts only,
when there is a loss of morphogens caused by sectioning.
The differential range of A and B is suggested to be the
cause of the differential loss from the tissue. When the
loss is not strong enough autocatalysis and thus pattern
formation will not start. This may explain why in some
cases a structure does not form at the junction. In the
model proposed by Gierer and Meinhardt (1972), and
Meinhardt (1993) the start of pattern formation does
not depend on a loss of morphogens from the wound.
Autocatalysis starts because the removed structure does
not any longer supply the tissue at the wound with the
inhibitor, specific for the removed structure. In the
model discussed here, structure-specific morphogens do
not exist (in the primary system).
A simulation of mirror-image transplants shows the
role of the level of inhibitor C in pattern formation. The
centre of the figure is comprised by the junction of
the transplant. Here, the concentration of inhibitor B is
slightly reduced as it was in the foregoing experiments.
This should represent the conditions following sectioning which cause autocatalysis to start. To simulate the
development in double headed animals (Fig. 7A) the
concentration of inhibitor C is increased at the outset. In
double foot animals (Fig. 7B) it is decreased. In both
cases the positional value is identical but increased
above the value present in the budding region. The
simulations result in foot formation in double-headed
animals (Fig. 7C) and head formation in double foot
animals (Fig. 7D). Similar results were obtained by
Meinhardt (1993) when applying his model.
The result obtained may help to understand head and
foot regeneration from body sections. In both cases all
three morphogens are generated. Whether the positional
value increases or decreases at the wound depends on
the tissue adjacent to the wound: an increase of the
gradient of positional values favours import into, over
export out of the wound of the decisive compound, the
inhibitor C. This ratio of import to export determines
the developmental fate. Foot regeneration in a growing
hydra is shown below.
2.3.2. Transplantation of regenerating tissue
Tissue excised from a certain body level and
transplanted into an incision made laterally at the same
level in a host will integrate without forming a structure
at all. When transplanted closer to the head, it forms a
foot. When transplanted closer to the foot, it forms a
head. When regenerating tissue is transplanted at the
respective sites, the fate is different. For example, when
the tissue is allowed to start head regeneration some
time before transplanting it to the site, where the
original, not regenerating tissue will transform into a
foot, this tissue will integrate or transform into a head.
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
45
Fig. 7. Head and foot formation at the graft junction of mirror-image transplants. (A) Double head and (B) double foot transplants obtained by
rejoining parts of adult animals sectioned at about the middle of their body column. Double head transplants form a foot, double foot transplants
form a head. In both simulations (C, D) the positional value is elevated from that present in the budding region to a higher one (from 0.05 to 0.1). (C)
The simulation represents the double head transplant: In the centre the level of inhibitor C is increased due to import from the surroundings. In the
simulation this is achieved by decreasing the removal rate of inhibitor C from 0.004 to 0.0035. (An import of inhibitor B from both apical ends is
ignored. It is obvious that in case the concentration of B is too high, autocatalysis will not start.) (D) In order to simulate the double foot transplants,
the concentration of inhibitor C has to be lowered in the centre. This is achieved by increasing the removal rate of C, rc ; from 0.004 to 0.005. Other
parameters as in Fig. 2.
Fig. 8. Reversion of the fate in regenerating tissue following transplantation. (A) Identical to the experiment shown in Fig. 7B a double foot
transplant was obtained by sectioning and rejoining adult animals. The simulation (conditions as in Fig. 7C) was run for some time (about 5000
iterations) which causes the onset of an autocatalytically stimulated release of the activator at the junction and an increase of the positional value.
This represents an early step in the formation of a head. (B) Then the concentration of inhibitor C was increased (conditions as in Fig. 7C). That
causes the positional value to decrease which finally causes basal disc formation. This second step of the simulation should represent the
transplantation of the regenerating tissue (the region with autocatalysis) to a position close to the head where the concentration of C is high. (A
synchronous increase of B below a certain threshold does not change the result. An increase above the threshold prevents autocatalysis and therewith
structure formation in general.) Other parameters as in Fig. 2.
However, when transplanted closer to the head it forms
a foot. With increasing time allowed for regeneration
this tissue has to be transplanted increasingly closer to
the head to still evoke foot formation. Prospective foot
tissue behaves vice versa. Obviously, within the first
hours of regeneration the fate head or foot formation,
respectively, is not fixed. Thus, it was argued that
regeneration does not start with structure specific
processes rather it was proposed that in regenerating
tissue processes start which cause the positional value to
change (Berking, 1979).
This regeneration—transplantation experiment is
simulated by starting with the conditions used for
Fig. 7. Initially, an increase of the positional value was
46
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
Fig. 9. Tandem transplantation of body sections. Head and foot were removed from budless Hydra and transplanted in a line with all pieces having
the same orientation (arrows). The first piece (a) kept its head and the last piece (e) kept its foot. Second and third line show the pieces to regenerate
head and foot. Obtained from Tardent (1954).
allowed to start, representing an early step in head
regeneration (Fig. 8A). Then the overall concentration
of C was increased, representing a transplantation close
to the head. This resulted in foot formation (Fig. 8B).
(At the new position within the transplant the concentration of inhibitor B will increase as well. However,
inhibitor B does not decide whether an increase or a
decrease of the positional value takes place, rather it
decides whether the autocatalysis will proceed or not. In
the present simulation this decision is not of interest.
Thus, at the outset of this simulation the concentration
of inhibitor B is not increased.) The alternative:
transplantation of foot regenerating tissue close to the
foot results in head formation (not shown). It should be
emphasized that during this change of fate the three
morphogens are continuously released.
which finally results in head formation. In the periphery
of the area of enhanced activator release the positional
value either decreases sharply down to a very low value
or—at the opposite site—smoothly in the form of a
gradient. This asymmetry causes an asymmetric concentration profile of morphogens. Of particular interest
is the concentration of inhibitor B. At the periphery of
the activated area where there is a sharp decrease of the
positional values, the concentration of B is low. At the
opposite site it is comparatively high. When the
experiences with bud formation (cf. Figs. 2 and 4) are
applied to the experimental conditions here, it is obvious
that at the site of the sharp decrease of the positional
value a foot is formed but not at the opposite site as
well. The small piece of tissue between head and foot has
an inverted polarity.
2.3.3. In hydra an inversion of the polarity can be
obtained by an experimentally provoked strong
discontinuity of positional values
Tardent (1954) removed the head and the foot from
budless Hydra and rejoined the obtained pieces in a line
with all pieces having the same orientation. The first
piece kept its head and the last piece kept its foot
(Fig. 9). He observed head and foot regeneration and
finally a separation into complete animals. The interesting point is here that at the junction always a head
forms but never a foot. A new foot forms at a short
distance from the new head. The tissue between both
structures is expected to have an inverted polarity. (An
inversion of the polarity has been observed in similar
experiments by Ando et al., 1989; Muller,
.
1996.)
According to the model, an autocatalytically enhanced
release of the activator starts at the junction. Due to the
lower positional value right and left of the junction, the
tissue here can get rid of some of the generated inhibitor
C (cf. the net export of inhibitor C in Fig. 7B and D).
Therewith, at the junction the positional value increases
2.3.4. Foot regeneration and head and foot formation
in aggregates
A normally shaped budless Hydra was simulated by
growth of a small animal. Then the (simulated) body
was sectioned and the apical part was allowed to
regenerate. Autocatalysis resulted in foot formation as
is observed in reality (Fig. 10). One has to keep in mind
that the same activator causes head formation in
budding and head regeneration from homogenous
starting conditions (Fig. 6a). It is the concentration of
the inhibitor C which decides the developmental fate.
In a further simulation, initially the morphogens and
the positional values were almost uniformly distributed
except for inhibitor B which displayed a small depression in the middle part of the row of cells. The
simulation should represent the fate of an aggregate of
single cells as it has been obtained from cells isolated by
dissociation of gastric regions of adult animals
(Gierer et al., 1972). In such aggregates heads and feet
develop. In the simulation this can be observed as well
(Fig. 11). When at a certain site activator is released
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
Fig. 10. Foot regeneration. The simulation was started in 4 cells. In
the first cell the positional value was set to zero. When a gradient of the
positional value had formed, growth was simulated by adding cells at
the apical end. (A) After some time of growth the basal end of the
‘‘animal’’ was removed by sectioning (arrow). For a short period of
time inhibitor B was allowed to leak from the cut surface. (B) This
causes autocatalysis to start. The result of sectioning is foot formation.
Other parameters as in Fig. 6.
autocatalytically all three morphogens are exported
from that site. The local positional value also affects
the local level of these morphogens. The range of
inhibitor B controls the distance at which a new centre
of activation can form. The range of inhibitor C
influences the fate of the activated area, i.e. whether a
head or foot eventually forms.
3. Discussion
3.1. Budding and growth
The model proposed describes head and foot regeneration which has been successfully achieved by other
models as well, and even in more detail (cf. Meinhardt,
1993). For the first time budding is described as
observed by Sanyal (1966): budding starts as a circular
field in the parent animal’s gastric region. The centre of
the field will become the bud’s mouth tip, the very
periphery, the bud’s basal disc. These structures are
47
Fig. 11. Head and foot formation in aggregates. Initially, the
positional value was increased above the value used in the simulation
shown in Fig. 2 and the morphogens and the positional values were
uniformly distributed (d ¼ 0:2). In the middle part the removal rate of
inhibitor B was reduced (ratio 1–0.67). (A) and (B) show two
consecutive stages of the simulation. Note that the autocatalytically
stimulated release of the activator becomes extinct where and when the
basal disc has formed. This has been obtained by assuming the
positional value to reach zero when a certain low threshold (d ¼ 0:01)
is reached. Other parameters as in Fig. 2.
determined when the field is still part of the parent
animal’s body.
According to the model, the size of the area in which
the activator is generated determines the initial size of
the whole prospective bud. Necessarily, the inhibitors
have a much longer range than the activator has.
Therewith, the model ‘‘explains’’ the observation that a
developing bud influences the spacing of new buds and
also the regeneration of head and foot in the parent
when these structures are removed. A suggested head
inhibitor in a model which assumes the existence of head
and foot specific morphogens may be expected to have a
shorter range.
Obviously, the bud field is much smaller than an adult
Hydra. When separated from the parent, the former bud
grows by multiplication of epithelial cells, which occurs
almost randomly but excludes the very ends (David and
Campbell, 1972). Therewith, in an adult animal the
gradient of positional values is much smoother than it is
in a young bud. But certainly, the range of morphogens
is not stretched accordingly. Morphogens generated at
the apical end will hardly reach the basal disc, in
particular the activator will not. Thus, within a large
48
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
part of the body of an adult Hydra the positional value
of a certain cell is no more in accordance with the signals
which control it. This discordance causes the positional
value of these cells to change. Unlike in other models
generally proposed for pattern formation of Hydra, this
model assumes the body pattern to be unstable for adult
animals. A further reason for this discordance is the
removal of cells at four sites: at the tip of the mouth (loss
of cell), at the base of the tentacles (tentacle formation),
in the budding region (if there is budding) and in the
basal disc (loss of cells). All these influences, the almost
uniform gain of cells, the localized loss of cells and the
influence of the morphogens to change the positional
value cause the tissue to ‘‘flow’’. In the model of
Meinhardt (1993) a smooth gradient of the source
density/positional value is obtained by assuming a small
value for the destruction rate and by introducing a
diffusion term for the sources with a diffusion constant
which is larger than that of the head activator. This term
keeps a graded distribution of the source density even in
case the production and the destruction of sources
occurs locally under the control of the head and the foot
activator, respectively. These assumptions represent a
‘‘source’’ and ‘‘sink’’ (Crick, 1970) combined with a
uniform breakdown along the length axis in mathematical terms. In the model presented here the diffusion
constant of the positional value is zero.
3.2. Regeneration
With respect to regeneration there are three features
of particular interest: (1) in regenerating tissue the
pattern forming process can be reverted in case the
tissue is transplanted. Models which assume head and
foot specific morphogens to control the formation of the
respective structures have to explain, for instance, how
the head system is switched off and the foot system is
switched on following transplantation. Such a switch is
not easy to explain because a system property of
autocatalysis is that it maintains itself even under
unfavourable conditions once it is established. In the
model proposed, the autocatalytically stimulated release
of the activator persists while the positional value can
change from increase to decrease or vice versa due to
slight changes of the morphogen concentration in the
environment.
(2) Under certain conditions the development into a
head and a foot, respectively, ends before the terminal
structure is formed, e.g. some tentacles may form, but a
mouth opening is missing. Such developments were
observed in Hydra as a result of transplantation
(Browne, 1909; Rulon and Child, 1937; Webster and
Wolpert, 1966; Webster, 1971; Berking, 1979). Natural
examples are tentaculozooids of Hydractinia which are
polyps that possess neither a mouth opening nor a ring
of tentacles but bear one tentacle instead. Buds of this
type have been described for Hydra by Trembley (1775).
A further example are frustules of various Cnidarians.
When it is assumed that the formation of a structure
directly depends on the generation of the structure
specific activator, the terminal structures will almost
inevitably form because autocatalysis is largely an all-ornone-decision. In the model presented, the autocatalytic
stimulation of the release of the activator is also
assumed to be decision. However, structure formation
does not directly depend on that an all-or-none decision.
A slight supply or an enhanced removal of morphogens
caused by the tissue surrounding the region of autocatalysis is able to influence the positional value reached
in the centre of the area of autocatalysis. Therewith, the
structure formed in the centre does not have to be the
usual terminal one, mouth tissue or basal disc tissue.
(3) In marine hydrozoa, generally, a head either
regenerates at both ends of a polyp’s body section or the
aboral end fails to regenerate at all (for review see
Berking, 1998). However, under certain conditions some
were found to regenerate aboral structures. In Hydractinia this happens only when the hydranths (polyps) are
young and small (Muller
.
et al., 1986), i.e. when the
gradient of positional values is steep. The model
presented explains these results by the differential
concentration ratio of A/C at the site of autocatalysis.
When there is no gradient of positional values, a head
will regenerate (cf. Fig. 4). A foot forms when there is
sufficient import of C into the tissue of the wound. This
takes place only when the gradient of positional values is
steep (cf. Figs. 7 and 10). Models which assume head
and foot systems to control the formation of the
respective structures may have problems to explain
how animals can exist without a foot (cf. Tripp’s
transplant, Fig. 2).
Muller
.
(1990) found foot formation to be favoured
when a head or a bud is located or is formed in close
vicinity. The head (bud tip) supports foot formation. In
particular, supernumerary heads evoke the formation of
additional feet, while additional feet do not evoke the
formation of additional heads. Muller
.
(1995) suggests a
competition of factors necessary for head formation.
Based on the model presented here, inhibitor C
generated by the apical end (head, bud tip) is suggested
to reach a site in the neighbourhood at which the release
of the activator is autocatalytically enhanced but at
which the positional value will only reach zero with the
help of C, supplied from outside. Therewith, a head
favours foot formation. An existing basal disc generates
none of the three morphogens (see below). It cannot
support head formation.
3.3. Foot formation
In budding and regeneration, respectively, the model
predicts foot formation to occur in different ways. In
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
regeneration, the basal disc forms in the centre of the
area where the activator is released (autocatalytically
enhanced). In budding, the basal disc forms in a ring at
the periphery of the activator generating area. Thus, in
the latter, there is no ‘‘own’’ autocatalysis for basal disc
formation. Because a basal disc, formed in the course of
budding, was not found to be structurally and functionally different from a basal disc formed in regeneration,
the model predicts that in foot formation by regeneration, autocatalysis eventually ends. The cause for this
may be that in basal disc tissue the positional value
reaches zero. Due to that, the activator cannot be
generated any more.
Basal disc regeneration is a fast process which in
several Hydra species takes only one day. In the
transplant made by Tripp (1928) (Fig. 2) a basal disc
may have formed but this would have taken several days
or weeks during which the peduncle tissue elongated. It
seems that in the central part of the peduncle tissue the
positional value slowly decreases and finally reaches the
threshold for basal disc formation. It appears difficult to
see a reason for that decrease other than the gradient of
positional values: due to the considerable difference in
the range of A and C (determined by both the diffusion
constant and the removal rate), the concentration ratio
of A/C is high where the positional value is high and low
where the positional value is low. This causes the
positional value to decrease in the centre of the
transplant. Inhibitor C released at the apical end may
contribute to the concentration reached in the centre.
Similar to budding, an autocatalytically enhanced
release of the activator does not appear to be involved
in this type of basal disc formation. These observations
and interpretation shed light on the process of how a
basal disc is maintained though it constantly looses cells
(Tripp, 1928; Burnett, 1966; Campbell, 1967): in the
basal part of the body column the gradient of positional
values is sufficient to cause the positional value to
decrease. This decrease supplies the basal disc with new
basal disc cells. There appears to be no necessity to
assume a signal generated by the basal disc which causes
cells in the basal body part to reduce their positional
value and thus to develop into basal disc cells. The
explanation presented also fits the different fate of the
basal disc at the former junction between the bud and
the parent, as discussed above: the basal disc is
maintained at the bud’s base but not at the parent’s
gastric body wall. The latter cells are obviously unable
to recruit new basal disc cells from their surroundings.
The transplantation experiment made by Tripp further
indicates that the distance between the site of bud
initiation and the basal body end is not controlled by
signals generated from the basal disc.
A pulse treatment with certain chemicals was found to
provoke the development of a bud into a branch without
a foot at the bud’s base (Hassel and Berking, 1990; Pe! rez
49
and Berking, 1994; Pe! rez, 1996; Zeretzke et al., 2002).
Limited concentrations allow the formation of basal disc
tissue in a small part of the ring. This basal disc patch
points to the parent animal’s basal disc. Based on the
model, the chemicals may increase the level of inhibitor
B in the tissue (cf. Fig. 4). Of particular interest is the
existence and the positioning of the basal disc patch:
obviously the basal disc of the parent animal is unable to
prevent basal disc formation in the budding region.
An influence on pattern forming processes by the
foot has been proposed repeatedly (Hyman, 1928;
MacWilliams et al., 1970; Hicklin and Wolpert, 1973;
MacWilliams and Kafatos, 1974; Shostak, 1974; Sinha
et al., 1984; Meinhardt, 1993). For example, MacWilliams et al. (1970) truncated animals by a single cut
at mid-peduncle. After varying periods of time, a lowerhalf peduncle was isolated and grafted to the position
originally occupied by the discarded segment. The graft
either bore a basal disc or not. When a basal disc was
absent, several specimens regenerated a basal disc not
only at the terminal end but also at the graft junction.
When a basal disc was present, a further basal disc was
rarely regenerated at the graft junction. The authors
conclude that a basal disc inhibits basal disc formation.
However, the alternative cannot be ruled out, namely,
the processes occurring at the wound stimulate foot
formation at the graft junction. Based on the model
inhibitor C generated at the cut surface can reach the
tissue at the junction and there C is able to support a
decrease of the positional value down to the lowest value
possible. Without that import, the decrease may be
insufficient. A similar explanation may hold for the
following observation: isolated polyp tissue of the
thecate hydrozoon Eirene viridula regenerates a head
at the apical and a stolon (not identical with but
comparable to a foot (Berking, 1998)) at the basal end,
respectively. When, however, the basal cut, which
isolates the polyp from the colony, is made a few hours
ahead of the apical cut, the apical cut surface
regenerates a stolo, as well, and not a head (Plickert,
1987). Thus a regenerating stolo tip stimulates stolo tip
formation in the surroundings and antagonizes head
formation which is proposed to be caused by inhibitor C
exported from the regenerating tissue. This explanation
also fits an observation made by Muller
.
(1990, 1995); a
(regenerating) head stimulates foot formation in the
vicinity (cf. also head and foot formation in aggregates,
Fig. 11). Thus, it appears necessary to distinguish
between the process foot regeneration and the existence
of a foot. On the basis of the model proposed and the
experiments noted above, a regenerating foot has an
influence on the surroundings, an existing one has not.
In summary, it is proposed that the basal disc (1) is
formed in the process of regeneration in the centre of the
area of an autocatalytically stimulated release of the
activator; (2) is formed in the process of budding in
50
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
the periphery of the area of autocatalytically stimulated
release of the activator (no ‘‘own’’ autocatalysis); (3) can
form without autocatalysis under certain conditions
(Tripp’s transplant); (4) will end autocatalysis, if there
was one, when it differentiates; (5) does not generate
signals which reduce the positional value in the
surroundings; (6) does not generate signals which
control the positioning of buds; (7) does not generate
signals which prevent basal disc formation in the
budding region; (8) does not generate signals which
enhance head formation. It appears that in Hydra the
basal end does not generate morphogens with a long
range.
These propositions may help to understand lateral
branch formation in thecate hydrozoa which form a
stem. Thecate hydrozoa with a sympodial type of
growth produce a stem which grows and finally
differentiates into a hydranth, which is a polyp without
a basal disc (Fig. 12). Then, some distance proximal to
the very end a bud forms which elongates and finally
forms a hydranth at its end, and so on. The apical end is
involved in the positioning of the bud (Kossevitch,
1996). The basal disc is argued to not be involved,
neither in the positioning of the bud nor in the control of
the differentiation of the stem, because a basal disc is far
away and the distance increases when the stem elongates
while the pattern of branching remains unchanged. A
similar explanation was put forward for thecate hydrozoa with a monopodial type of growth. These are
animals which grow by a meristem-like organ at the
distal end of their stem. The most important difference
between monopodial and sympodial type of growth was
Fig. 12. Sympodial growth in thecate Hydrozoa. On top of a stolo (a
hollow tube which connects the different parts of the colony) a stem
grows out whose apical tip develops into a hydranth (polyp)
surrounded by a hydrotheca, a polyp’s housing. Proximal to this
hydranth a new tip (bud) develops which takes over the elongation of
the stem. (Laomedea flexuosa, partly redrawn, after Kossevitch et al.,
2001.)
proposed to be the range of inhibitor C (Berking et al.,
2002).
3.4. Molecular approaches to pattern control
The knowledge of the molecular basis of pattern
formation in Hydra is still rudimentary. The reason is
that functional tests are almost missing. Thus, one can
only speculate on the basis of in situ hybridization
patterns. The Wnt-pathway may serve as an example.
This pathway was found to play a leading role in a large
number of processes including the organization of axis
formation of vertebrates and insects (for references see
Hobmayer et al., 2000). In Hydra Wnt-expression is
found in the very tip of the mouth. The expression starts
early in head regeneration and budding. Two other
members of the WNT-pathway, b-catenin and Tcf are
expressed at the same sites with almost the same kinetics
but within a larger area. The authors propose that
HyWnt signalling is involved in local self-activation in
the Hydra head organizer (Hobmayer et al, 2000).
However, in most foot regenerating animals Hyb-cat
was found to be expressed at the former wound for
hours. (HyTcf has not yet been tested; a faint HyWntexpression was observed in 10% of the foot regenerates
between the 9th and the 12th hour after sectioning) (B.
Hobmayer, pers. comm.). Within the period of Hyb-catexpression (the expression decreases down to zero after
the 18th hour) the foot regenerating tissue has considerably increased its property to cause the formation
of a foot under conditions of transplantation (cf.
experiments to Fig. 8). A few hours later the basal disc
visibly forms. Thus, at least Hyb-cat-expression is not
head-specific. (For a transitory expression of putative
head-specific genes and transitory changes of the
positional value in foot regenerating tissue see also
Muller,
.
1996.) The alternative model proposed in this
paper appears to fit better: the genes (at least Hyb-cat)
are expressed where the hypothetical morphogens
according to this model are generated and/or produced:
both, during head and during foot regeneration, but, as
proposed in the model, transiently during foot regeneration and constantly during head regeneration. The
kinetics of the expression of the three genes in the
course of budding also fit into the model: Hyb-cat and
HyTcf are expressed in a belt like area in the budding
region (cf. Fig. 5) and subsequently not only in the bud’s
head but rather in the whole bud field, including the
prospective basal disc region. Then, all three genes are
expressed at the apical end while Hyb-cat and HyTcf are
not sharply restricted to the head structures. All head
regenerating specimens were found to express the noted
genes. In contrast, some of the foot regenerating
specimens did not. This may indicate that in the course
of foot regeneration the expression oscillates as it was
observed in intact Hydra for the cnidarian homologue of
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
the Serum Response Factor Hv-SRF (Hoffmann and
Kroiher, 2001). In summary, the expression patterns of
the WNT signalling molecules do not support the
proposition of an existing head organizer in Hydra.
Rather, the expression patterns appear to support the
alternative model presented here: they correlate with the
generation of those morphogens which control the
change of the positional value.
Acknowledgements
The technical basis of the presented simulation is
excellent program Meinhardt (1995) developed for
calculation of patterns on sea shells. I thank
Herrmann for helpful discussions and B. Schreiner
critical reading of the manuscript.
the
the
K.
for
51
indicates the autocatalytic feedback of the released
activator on its own release in the form of a dimer. Only
one part of the release is antagonized by inhibitor B. The
second part of the equation
ðsb þ sc Þa2
indicates the loss of the dimeric activator by cleavage
into inhibitors B and C, respectively. The basis of this
interpretation is that this term appears with a positive
sign in the two equations which describe the generation
of the respective inhibitors. The inhibitors B and C may
represent a part of the molecule which includes one of
the two binding sites of the activator by which it is able
to stimulate its production and its release, respectively.
By cleavage of the dimer the competitive inhibitors B
and C emerge.
References
Appendix A
The equations proposed are designed as simple as
possible to describe a set of critical observations and
experiments. They certainly have to be evolved to meet
more results. In particular the influence of D (the
positional value) on the basic production, the degradation and the auto- and heterocatalysis of the three
morphogens is expected to display interesting modulating influences. (Other than here, with respect to
monopodial growth in the thecate hydrozoon Dynamena
pumila, the generation of the inhibitor C was assumed to
be not influenced by the positional value (Berking et al.,
2002).) Such a detailed analysis was not within the scope
of this article.
The activator was proposed to stimulate its own
release out of cells into the intercellular space, its
production within cells and the release of the two
inhibitors. It is possible to interpret the equations in a
different way: the activator does not control the release
of the two inhibitors, rather, the released activator is
cleaved in such a way that two different inhibitors
emerge which compete with the activator at the
respective target.
The equation which describes the activator release
da
a2
d2 a
¼ sa ra a þ dba þ Da 2
dt
dx
b
can be transformed into
s
da
a
¼ a2
þ sb þ sc
dt
b
ðsb þ sc Þa2 ra a þ dba þ Da
The first part of this equation
s
a
þ sb þ sc
a2
b
d2 a
:
dx2
Ando, H., Sawada, Y., Shimizu, H., Sugiyama, T., 1989. Pattern
formation in Hydra without developmental gradients. Dev. Biol.
133, 405–414.
Berking, S., 1977. Analysis of early stages of budding in Hydra by
means of an endogenous inhibitor. W. Roux’s Arch. 182, 117–129.
Berking, S., 1979. Analysis of head and foot formation in Hydra by
means of an endogenous inhibitor. W. Roux’s Arch. 186, 189–210.
Berking, S., 1980. Commitment of stem cells to nerve cells and
migration of nerve cell precursors in preparatory bud development
in Hydra. J. Embryol. Exp. Morphol. 60, 373–387.
Berking, S., 1998. Hydrozoa metamorphosis and pattern formation.
Curr. Top. Dev. Biol. 38, 81–131.
Berking, S., Hesse, M., Herrmann, K., 2002. A shoot meristem-like
organ in animals. Monopodial and sympodial growth in Hydrozoa.
Int. J. Dev. Biol. 46, 301–308.
Browne, E.N., 1909. The production of new Hydranths in Hydra by
insertion of small grafts. J. Exp. Zool. 7, 1–23.
Burnett, A.L., 1961. The growth process in Hydra. J. Exp. Zool. 146,
21–83.
Burnett, A.L., 1966. A model for growth and cell differentiation in
Hydra. Am. Nat. 100, 165–189.
Campbell, R.D., 1967. Tissue dynamics of steady-state growth in Hydra
littoralis. II. Patterns of tissue movements. J. Morphol. 121, 19–28.
Crick, F., 1970. Diffusion in embryogenesis. Nature 225, 420–422.
David, C.N., Campbell, R.D., 1972. Cell cycle kinetics and development of Hydra attenuate. 1. Epithelial cells. J. Cell Sci. 11, 557–568.
Gierer, A., Meinhardt, H., 1972. A theory of biological pattern
formation. Kybernetik 12, 30–39.
Gierer, A., Berking, S., Bode, H., David, C.N., Flick, K., Hansmann,
G., Schaller, H., Treckner, E., 1972. Regeneration of Hydra from
reaggregated cells. Nature, New Biol. 239, 98–101.
Hassel, M., Berking, S., 1990. Lithium ions interfere with pattern
control in Hydra vulgaris. Roux’s Arch. Dev. Biol. 198, 382–388.
Hicklin, J., Wolpert, L., 1973. Positional information and pattern
regulation in Hydra: formation of the foot end. J. Embryol. Exp.
Morph. 30, 727–740.
Hobmayer, B., Rentzsch, F., Kuhn, K., Happel, C.M., Cramer von
Laue, C., Snyder, P., Rothb.acher, U., Holstein, T.W., 2000. WNT
signalling molecules act in axis formation in the diploblastic
metazoan Hydra. Nature 407, 186–189.
Hoffmann, U., Kroiher, M., 2001. A possible role for the cnidarian
homologue of serum response factor on decision making by
undifferentiated cells. Dev. Biol. 236, 304–315.
52
S. Berking / Journal of Theoretical Biology 222 (2003) 37–52
Hyman, L., 1928. Miscellaneous observations on Hydra, with special
reference to reproduction. Biol. Bull. 54, 65–108.
Kossevitch, I.A., 1996. Regulation of formation of the elements of the
hydroid polyps colony. Ontogenez. 27, 95–101 (in Russian).
Kossevitch, I.A., Herrmann, K., Berking, S., 2001. Shaping of colony
elements in Laomedea flexuosa Hinks (Hydrozoa, Thecaphora)
includes a temporal and spatial control of skeleton hardening. Biol.
Bull. 201, 417–423.
Kuhn,
.
A., 1914. Entwicklungsgeschichte und Verwandschaftsbeziehungen der Hydrozoen. 1. Teil: Die Hydroiden. Erg. Fortschr.
Zool. 4, 1–284.
MacWilliams, H.K., Kafatos, F.C., 1974. The basal inhibition in
Hydra may be mediated by a diffusing substance. Am. Zool. 14,
633–645.
MacWilliams, H.K., Kafatos, F.C., Bossert, W.H., 1970. The feedback
inhibition of basal disk regeneration in Hydra has a continuously
variable intensity. Dev. Biol. 23, 380–398.
Meinhardt, H., 1993. A model of biological pattern formation of
hypostome, tentacles and foot in Hydra: how to form structures
close to each other, how to form them at a distance. Dev. Biol. 157,
321–333.
Meinhardt, H., 1995. The Algorithmic Beauty of Sea Shells. Springer,
Berlin, Heidelberg, New York.
Muller,
.
W.A., 1990. Ectopic head and foot formation in Hydra:
diacylglycerol-induced increase in positional value and assistance
of the head in foot formation. Differentiation 42, 131–143.
Muller,
.
W.A., 1995. Competition for factors and cellular resources as a
principle of pattern formation in Hydra II. Assistance of foot
formation by heads and buds and a new model of pattern control.
Dev. Biol. 167, 175–189.
Muller,
.
W.A., 1996. Head formation at the basal end and mirrorimage pattern duplication in Hydra vulgaris. Int. J. Dev. Biol. 40,
1119–1131.
Muller,
.
W.A., Plickert, G., Berking, S., 1986. Regeneration in
Hydrozoa: distal versus proximal transformation in Hydractinia.
Roux’s Arch. Dev. Biol. 195, 513–518.
P!erez, F., 1996. Effects of cantharidin and phorbol ester on bud
formation in Hydra vulgaris. Int. J. Dev. Biol. Suppl. 1, 273.
P!erez, F., Berking, S., 1994. Protein kinase modulators interfere with bud
formation in Hydra vulgaris. Roux’s Arch. Dev. Biol. 203, 284–289.
Plickert, G., 1987. Low-molecular-weight factors from colonial
hydroids affect pattern formation. Roux’s Arch. Dev. Biol. 196,
248–256.
Rand, H.W., 1899. The regulation of graft abnormalities in Hydra.
Archiv EntwMech. 9, 161–214.
Rulon, O., Child, C.M., 1937. Observations and experiments on
developmental pattern in Pelmatohydra oligactis. Physiol. Zool. 10,
1–13.
Sanyal, S., 1966. Bud determination in Hydra. Indian J. Exp. Biol. 4,
88–92.
Shostak, S., 1974. Bipolar inhibitory gradients’ influence on the
budding region of Hydra viridis. Am. Zool. 14, 619–632.
Sinha, S.N.V., Joshi, J., Rao, S., Mookerjee, S., 1984. A 4 variable
model for the pattern forming mechanism in Hydra. Biosystems 17,
15–22.
Sugiyama, T., 1982. Roles of head-activation and head-inhibition
potentials in pattern formation of Hydra: analysis of a multiheaded mutant strain. Am. Zool. 22, 27–34.
Tardent, P., 1954. Axiale Verteilungs-Gradienten der interstitiellen
Zellen bei Hydra und Tubularia und ihre Bedeutung fur
. die
Regeneration. Roux’ Arch. EntwMech. 146, 593–649.
Tardent, P., 1972. Experimente zum Knospungsprozess von Hydra
attenuate Pall. Rev. Suisse. Zool. 79, 355–375.
Tardent, P., 1978. Coelenterata, Cnidaria. In: Seidel, F. (Ed.),
Morphogenese der Tiere. Gustav Fischer Verlag, Stuttgart,
New York.
Trembley, A., 1775. Des Herren Trembley Abhandlungen zur
.
.
Geschichte einer Polypenart des suXen
.
Wassers mit hornerf
ormigen
Armen. Translated from French by Goetze, J.A.E. Quedlinburg:
1775. Tab. X, Fig. 6.
Tripp, K., 1928. Die Regenerationsf.ahigkeit von Hydren in den
.
verschiedenen Korperregionen
Nach Regenerations- und Transplantationsversuchen. Z. Wiss. Zool. 132, 476–525.
Webster, G., 1971. Morphogenesis and pattern formation in hydroids.
Biol. Rev. 46, 1–46.
Webster, G., Wolpert, L., 1966. Studies on pattern regulation in
Hydra. 1. Regional differences in time required for hypostome
determination. J. Embryol. Exp. Morphol. 16, 91–104.
Weimer, B.R., 1928. The physiological gradients in Hydra I.
Reconstitution and budding in relation to length of piece and
. 1, 183–230.
body level in Pelmatohydra oligactis. Phys. Zool.
Wolpert, L., 1969. Positional information and the spatial pattern of
cellular differentiation. J. theor. Biol. 25, 1–47.
Wolpert, L., Hornbruch, A., Clarke, M.R.B., 1974. Positional
information and positional signalling in Hydra. Am. Zool. 14,
647–663.
Zeretzke, S., Berking, S., 2001. Pattern regulation properties of a
Hydra strain which produces additional heads along the body axis.
Int. J. Dev. Biol. 45, 431–439.
Zeretzke, S., Berking, S., 2002. In the multiheaded strain mh-1 of
Hydra magnipapillata the ectodermal epithelial cells are responsible
for the formation of additional heads and the endodermal epithelial
cells for the reduced ability to regenerate a foot. Dev. Growth
Differ. 44, 85–93.
Zeretzke, S., P!erez, F., Velden, K., Berking, S., 2002. Ca2+-ions and
pattern control in Hydra. Int. J. Dev. Biol. 46, 705–710.