A model for budding in hydra: pattern formation in ... - CiteSeerX

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

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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

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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

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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.

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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

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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

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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

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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.

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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:

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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