Is Hydra Axis Definition a Fluctuation-Based Process Picking Up External Cues?

1. Introduction

Hydra (Figure 1a) is a simple freshwater organism known for its unique regenerative abilities. It is able to reorganize from a disordered cell aggregate into a fully formed structure. As Hydra regenerates, it first forms an isotropic, hollow cell spheroid that evolves through a symmetry-breaking event defining the future axis. Research has shown that Wnt signaling is responsible for maintaining Hydra’s body axis [1]. Disruptions in this pathway lead to abnormalities in axis formation [2].

ks1 is a Hydra specific gene known as a marker for the ‘head forming potential’ of a cell [3]. To understand the underlying principles of the observed patterns of ks1 expression on a Hydra cell spheroid, we proposed a mathematical model based on the production and exchange of a hypothetical ks1-expression promoting factor, “X”, which transfers between neighboring cells [4]. The transfer mechanism was taken from the Olami-Feder-Christensen (OFC) model [5], originally developed to simulate earthquakes. The excellent agreement between experiment and model, extending even to the behavior in a temperature gradient, suggests that local cellular interactions drive the observed self-organizing patterns of ks1 expression.

This paper reviews a few facts about Hydra biology before presenting our model, published in a specialized physics journal. Here we present it in a way that is accessible to readers with a low mathematical background. We also discuss the model in the light of recent findings exploring the influence of mechanical stimulation and the actin cytoskeleton on the position of the future axis.

2. Hydra: An Overview

Cnidarians, which include corals, sea anemones, sea pens, sea fans, jellyfish and hydroids, are among the simplest living metazoans [6,7,8,9,10,11,12,13]. The molecular evidence suggests that cnidarians appeared around 741 million years ago, before the Cambrian radiation, a period marked by the rapid emergence of most major animal phyla [14]. Most Cnidarians alternate between two forms: the sessile polyp stage and the free-swimming medusa stage. Their defining characteristic is the stinging cell, the nematocyte (also known as cnidocyte), which contains a unique toxin-producing organelle called nematocyst (also known as cnidocyst), used for predation, defense, and locomotion [12,15,16,17,18,19,20]. The phylum Cnidaria includes more than 9000 animal species, grouped into approximately 7 classes and 25 orders.

Hydra (see Figure 1), a member of phylum Cnidaria, class Hydrozoa, and order Hydroida, is a fresh-water polyp. It was first described in 1702 by Antonie van Leeuwenhoek and studied in more detail by Abraham Trembley [21]. At present, Hydra has become a major model organism studied, among others, for its regeneration property [22,23,24,25,26,27,28,29]. A comprehensive timeline of 300 years of Hydra research can be found in Galliot [30]. The genus Hydra consists of a few dozen species, recent studies suggesting their phylogenetic placement [31]. Hydra is estimated to have originated around 60 million years ago [32], and the most common diploid chromosome number is 30 [33]. The genome of Hydra magnipapillata was published (Chapman, et al. [22]), and the complete mitochondrial genomes of Hydra vulgaris and Hydra sinensis are known (Pan, et al. [34,35]). The Hydra genome contains approximately 20,000 protein-coding genes, excluding transposable elements [22,36], and includes orthologues of genes known to play key developmental roles in bilaterian developmental models [36].

An adult Hydra is usually a few milimeters long. It consists of approximately 100,000 cells. Its body wall, like that of other cnidarians, is composed of two germ layers: the ectoderm (outer cell layer) and endoderm (inner cell layer) separated by mesoglea, a non-cellular, jelly-like substance [37]. Each of these two layers is a single cell thick. Hydra has a mainly cylindrical shape with a head, a body column serving as gastric cavity, and a sticky foot for attachment. The head features a mouth opening, surrounded by typically three to six tentacles used for catching small aquatic invertebrates such as Daphnia and Cyclops, which Hydra feeds on. All Hydra body parts are aligned along a single oral-aboral axis [38].

Hydra can reproduce both, sexually or asexually [39]. Under normal conditions, Hydra reproduces asexually by budding, however, it tends to reproduce sexually if the environment becomes unfavorable [40]. Budding occurs about two thirds of the way down the body column. This is where the body wall evaginates to form a new column that will develop a head at the end. This column eventually detaches as a new, small Hydra [41].

In the adult, new cells constantly emerge within the center column of Hydra, becoming terminally differentiated only when they migrate to either end, where cells are lost. It has been suggested that Hydra escapes senescence and may be potentially immortal [42,43,44,45,46,47]. The genome of Ctenophora Mnemiopsis leidyi was sequenced [48], revealing that ctenophores are the earliest diverging animal lineage. This finding strongly supports a sister relationship between Cnidaria and Bilateria [48]. Consequently, the axial organization of Hydra provides important insights into the transition from radial to bilateral body plans and the evolution of the bilaterian body plan [44,45,46]. The Hydra foot, which pumps gastrovascular fluid, and the blood-pumping vertebrate heart were proposed to have a common ancestry [49], with the foot-to-head extension in Hydra corresponding to the anterior-posterior axis in higher organisms [50].

3. Hydra Regeneration

Due to its simple body plan and its remarkable regenerative ability, Hydra is a frequently considered model organism in developmental biology [7,51,52,53,54,55,56,57]. A small fragment of tissue excised from an adult Hydra is sufficient to regenerate an entire organism within a few days (Figure 1b). Hydra can even reform from a heap of disordered cells [58]. In both situations Hydra will first form a hollow cell ball. Only cells from the body column contribute to regeneration, highlighting their multipotent properties [58,59]. Since the body axis is lost in the disordered cell ball, approximately 10,000 multipotent cells must break the initial isotropy, define a single oral-aboral axis and organize as a function of it.

The study of regeneration is of significant biomedical interest, since it may ultimately lead to the development of therapies for replacing old and damaged tissues with new ones [60]. Regeneration - the regrowth or repair of cells, tissues or body parts - is widespread but highly variable among animal phyla [60,61,62,63,64,65,66,67]. Regeneration may well be a fundamental trait of metazoans, rather than mechanisms that evolved independently for that purpose in various contexts [68]. Similarities between regeneration and post-embryonic development suggest that regeneration may well have originated as a by-product, or epiphenomenon, of development [64,65]. In highly regenerative animals like Hydra and planarians, there is no evidence of frequent amputations occurring in the wild. This implies that their extraordinary regenerative capabilities may be maintained predominantly by mechanisms other than direct selection [64]. According to the phylogenetic inertia hypothesis, the ability to regenerate is an ancestral trait retained for historical reasons; it is not selectively advantageous but persists because it has not been eliminated from the species’ developmental repertoire [64].

4. Hydra Genes Related to Axis Definition

We present genes and pathways involved in defining and maintaining the oral-aboral axis. Among the critical genes involved in early axis definition are β-catenin [69,70,71] and Wnt [1,2,70]. Wnt genes play axis-related roles in many animal species, including sponges. However, unicellular organisms do not contain these genes, suggesting that Wnt signaling played a crucial role in the origin of multicellular animals [72,73,74,75,76,77,78,79,80]. The absence of Wnt signaling mechanism in thale cress (Arabidopsis thaliana), a model organism in plant biology, and in fungi [78] underscores its role specific to multicellular animals. The multicellular organization of plants indeed evolved independently of animals [81,82]. The Wnt gene family in Hydra is comparable in size to those in bilaterian animals [83], suggesting that the diversification of this gene family occurred before the cnidarian-bilaterian split [36].

Hydra expresses Wnt genes only in the organizer region. The organizer is part of the hypostome, located at the tip of the head (see Figure 1). The organizer is the first structure to be restored during regeneration [84]. It guides the fate of all other cells [85,86]. Vogg, et al. [87] demonstrated that a conserved Wnt 3/β-catenin/Sp5 feedback loop restricts head organizer activity in Hydra, highlighting the role of Wnt 3 in axis formation. Studies have shown that disruptions in Wnt signaling pathways can lead to abnormalities in axis formation, underscoring their importance in Hydra development [1,2,69].

The complexity of Wnt intracellular signaling pathways in Hydra is reflected by the diversity of Wnt receptors [84]. These receptors can be categorized into 15 receptors and coreceptors from seven protein families [76], the Fizzled family the most prominent one [88].

The canonical Wnt pathway is characterized by the preservation of β-catenin from degradation in the cytoplasm. β-catenin eventually translocates into the nucleus, where it coactivates the TCF/LEF transcription factor family [89]. The non-canonical pathways include the planar cell polarity (PCP) pathway and the Wnt/calcium pathway, where alterations in gene expression are not mediated via β-catenin [89]. All three pathways, the canonical one, PCP, as well as Wnt/calcium, have roles in Hydra regeneration. For instance, treatment of Hydra cells with a GSK-3β inhibitor (alsterpaullone) leads to β-catenin translocation into the nucleus, causing head formation [90]. This translocation of β-catenin into the nucleus as a downstream step of Wnt pathway has been extensively studied [91,92,93]. Two members of the Dickkopf (Dkk) family—Dkk1/2/4-A and Dkk1/2/4-C—have been identified and shown to be expressed in specific patterns consistent with a role in axial patterning. These Dkks are considered potential Wnt antagonists due to their homology with known Dkk proteins in other animals. However, their precise functional role during regeneration remains only partially understood, with limited direct experimental validation to date [94]. Notably, the Dickkopf (DKK) family is recognized as one of the key inhibitors of the canonical Wnt signaling pathway. DKKs exert their inhibitory function by binding to the Wnt co-receptor LRP5/6, thereby blocking the formation of the Wnt–Frizzled–LRP complex. This prevents β-catenin stabilization and its subsequent translocation to the nucleus, effectively downregulating Wnt-dependent gene transcription. [95].

The study of molecular pathways in Hydra has been undertaken by differential gene expression and single cell expression data analysis [96,97]. This led, among others, to the discovery of ks1, a Hydra specific gene. ks1 is permanently expressed in the adult, just below tentacles and hypostome, that is, before these cells differentiate irreversibly as part of the head. Because ks1 is upregulated in response to early signals of head formation, it has been termed a marker of “cell head forming potential” [3,98,99]. Its specific function remains unknown.

5. Hydra Regeneration Requires Mechanical Stimulation

As a first step of Hydra regeneration, a hollow sphere made of a cell bilayer forms (Figure 1b). This is regardless, whether starting with a cell aggregate or a small piece of tissue. Subsequently, due to osmosis [100,101,102], cycles of steady inflation terminated by bursts are observed (Phase I, Figure 2a). The frequency and amplitude of these cycles abruptly change after about 18 hours (Figure 2a). At this point the release of pressure occurs at a decreased threshold without rupture and increased frequency (Phase II) as if a weak spot in the Hydra cell bilayer had emerged [100,103]. Futterer, et al. [100] showed that this point in time goes along with a transition of the cell spheroid from elastic isotropy to anisotropy (Figure 2b). Moreover, the transition occurs around the time point where Wnt is supposed to be expressed, suggesting a potential connection between Wnt expression, cell adhesion and tissue elasticity [101]. Chiou and Collins [104] showed that indeed the emergence of an early mouth was responsible for the diminished pressure release threshold that induces the sudden change in the inflation burst cycles [101,105]. Soriano, et al. [101] showed that axis definition is delayed by the same time interval as the inflation-burst cycles are prolonged due to changes in osmotic pressure, suggesting the osmotical inflation to be the driving force in the process. Duclut, et al. [106] studied mechanical stimulation from osmotic pressure in cell spheroids in detail. Ferenc, et al. [107] showed that mechanical oscillations orchestrate axial patterning through Wnt activation in Hydra. HyWnt3 is more affected than other Wnt genes by the lack of spheroid inflation in an isotonic environment, indicating its sensitivity to mechanical cues. Wnt 3 overexpression in Hydra spheroids under isotonic conditions was able to rescue the regeneration process, highlighting its pivotal role in response to mechanical stimulation [107]. These observations point towards the role of the cell cytoskeleton as the force sensitive structure of the cell.

The role of mechanical stimulations in conjunction with the cytoskeleton during development has also been observed in other model organisms. For instance, in Drosophila embryogenesis, actomyosin contractions have been shown to generate mechanical stress that regulates apical constriction and cell shape changes during the process of gastrulation [108,109]. Similarly, in Xenopus laevis, mechanical forces play a significant role in convergence and extension movements essential for gastrulation. Here, the cytoskeletal dynamics and cell-cell adhesion are regulated by the non-canonical Wnt/PCP signaling pathway, which is crucial for tissue elongation and axis formation [110]. In mammalian systems, mechanical forces transmitted via the cytoskeleton have been shown to influence stem cell lineage specification. The YAP/TAZ signaling pathway responds to substrate stiffness and cytoskeletal tension, directing mesodermal differentiation [111,112]. These examples across various organisms illustrate the conserved role of mechanical forces and cytoskeletal interactions in regulating development and signaling pathways.

6. Symmetry Breaking in Physics and Biology

In physics, symmetry breaking in non-equilibrium has been studied in physics at length [113,114,115,116]. To make a transition to asymmetry these systems tend to generate transient asymmetries by fluctuations. This is still a symmetric situation. Only if the asymmetric fluctuation reaches critical size so that it becomes locked, the symmetry is considered broken. As an example, consider a vortex as it may appear in a sink. Initially, when the liquid starts to drain, there will be no vortex, but increasingly large fluctuations due to the large flow of liquid create transient asymmetry. Only when these fluctuations reach a sufficient (critical) amplitude to become self-amplifying, the symmetry is broken and the vortex will increase in strength. At that point the vortex will not reverse course if left to itself. It will resist increasingly large perturbations in the opposite direction, even if these would have been large enough to impose a direction if they had been applied from start. In other words, these perturbations can be larger than the initial fluctuation spectrum and still have no effect at this point. At the same time, by very violently spinning the entire mass of fluid in the opposite direction the direction of any vortex can be changed – but this does not mean that the vortex did not undergo symmetry breaking earlier, or that there was no clearly defined direction. We understand that for a study, the magnitude of the applied perturbations matters a lot.

If we ask, how we can show that the vortex broke symmetry in the beginning, it becomes clear that two features are important: initially the direction of rotation will be imposed by perturbations that create rotational flow at the critical fluctuation level – this is still very small compared to an established vortex. Once a direction is chosen, there must be an amplification and stabilization of the movement (Chapter 10.2 “Critical Precursory Fluctuations”).

The same idea is found in Biology. Alberts, et al. [82] (Chapter 22 “Development of Multicellular Organisms”) regarding external influences on developmental decisions says: “The choice between the alternative outcomes can be dictated by an external signal that gives one of the cells a small initial advantage. But once the positive feedback has done its work, this external signal becomes irrelevant. Broken symmetry, once established, is very hard to reverse: positive feedback makes the chosen asymmetric state self-sustaining, even after the biasing signal has disappeared. In this way, positive feedback provides the system with a memory of past signals”.

Transferred to Hydra, this means that we have to show that once a direction is chosen, it amplifies (and as discussed in chapter 11, this seems to be the case). However, the property that small perturbations or advantages are sufficient to direct the axis remains largely unexplored except for the following.

7. A Weak Temperature Gradient Can Direct the Future Axis of a Regenerating Hydra Spheroid

The above can be applied to developing Hydra spheroids that are exposed to a temperature gradient. If the temperature gradient of typically less than a degree was applied early, during phase I (high amplitude, low frequency oscillations), it would set the axis. However, if applied during phase II (low amplitude, high frequency oscillations), the gradient could no longer change the direction of the axis. Soriano, et al. [103] concluded that the axis direction of the regenerating Hydra spheroid becomes irreversibly locked during the transition from phase I to phase II (Figure 2). Larger fragments that retain a certain degree of asymmetry were found to switch to phase II much faster [101].

Naively, one might expect the head to appear on the warmer side, given that the organizer is the first structure to emerge during regeneration. This is because the increased temperature accelerates the development, and this creates the ‘small advantage’ (see above) that can be expected to propagate to axis positioning. However, in cases where the temperature difference between the cold and hot sides was ΔT=0.6°C, the head formed on the cold or the hot side with about equal probability (Figure 3). However, if the gradient was increased to ΔT=0.9°C, a tendency for the head to form on the cold side became apparent [4]. Not only the direction of a temperature gradient imposed the position of the axis, but the magnitude of the gradient had an effect on the orientation – not an intuitive result.

8. Ks1 Expression Patterns at the Axis Defining Moment Are Devoid of a Characteristic Size

Usually, we define our surroundings using specific scales, such as the typical size of organisms or their typical lifespans. However, some phenomena do not fit into this kind of description. Since they lack a characteristic size or scale — they are called scale-free. Mathematically, the absence of scale corresponds to a power law, which is surprisingly common in nature [118]. A power law can be expressed as:

f(x)=b⋅x^a

Here x is the input, for instance the scale of a map, and f(x) the output, for instance the length of the coastline of Britain taking a map at that scale. If the exponent a is one, the two quantities are just proportional. An exponent different from one means that one grows faster or slower than the other. Doubling the scale of the map does not necessarily double the length of the coastline because of its rough, intricate shape. “a” is the exponent of the power law that defines the quantitative connection between the input, the scale of the map, and the output, here the length of the coast. a is called the fractal dimension. More precisely, a is called the “perimeter fractal dimension” of the coastline of Britain. Its value is estimated to be around 1.25 according to Mandelbrot's original analysis. [117]

One key feature of a power law is that if you scale the input x₀ by a factor of k, the output of the function f(k⋅x₀) will be increased by k^a with respect to f(x₀):

f(k⋅x₀) = b (k⋅x₀)^a= b⋅k^a⋅x₀^a= k^a⋅f(x₀).

Since this applies to any x₀, regardless of its precise value, the function f(x) is scale-free: at whatever scale you look at the function, its shape remains the same. Scale-free can be shown to indeed be equivalent to a power-law description [118].

Non-trivial scale-free objects are known as fractals. The term “fractal” was first used by Benoît Mandelbrot in 1975. He derived it from the Latin word "frāctus", meaning broken or irregular. Fractal analysis has been shown to be useful in the biological sciences, helping to describe complex patterns and structures in nature [119,120,121,122,123,124,125,126,127], including investigation of cnidarians [128,129]. Fractal properties of gene expression were reported by Manoel, et al. [130], Ghorbani, et al. [131], and Waliszewski [132].

Fractals are fragmented geometric shapes that can be broken down into smaller parts, each of which is a reduced copy of the whole [126,127,128]. If you take photos with different magnifications of the same fractal, these photos will, again, look strictly the same. An example of this in nature is Romanesco broccoli (Figure 4), where each floret looks like a miniature version of the whole vegetable.

If you cover a straight line with a grid of square boxes (imagine pixels of a camera), the number of boxes N(δ) required is inversely proportional to the box size δ. Mathematically, this is expressed as N(δ)=constant/ δ, leading to a slope of 1 when plotting ln(N(δ)) against (- ln(δ)). The perimeter fractal dimension of a straight line equals 1. Indeed, a line is one-dimensional object. A more complex line shape such as coast of Britain, however, will most likely lead to a different slope, when plotting the logarithms. The slope represents the fractal dimension (Figure 5).

For a simple, two-dimensional shape like a square, the area fractal dimension would be 2, indicating that it fully occupies a flat geometric plane, corresponding to the geometric dimension of flat shapes. This is because the number of boxes needed, N(δ), is inversely proportional to the area of the box δ². This leads to a slope of 2 in the plot of ln(N(δ)) versus -ln(δ). For a more complex shape, again, the fractal dimension is likely to be different from 2.

In the study by Soriano, et al. [103], the spatial distribution of ks1 gene expression on the surface of the Hydra was examined using in situ hybridization. There were domains (or clusters) of cells expressing the ks1 gene. These domains were analyzed to check if they were fractal objects. This was done using the grid (or box-counting) technique as outlined above (see Figure 5).

The perimeter fractal dimension D_per​ of the ks1-expressing domains was determined for three culture times [101]. The results are given in Figure 6. The fractal dimensions reached a maximum above 1.3 after about 25 h, which suggests that the spatial distribution of ks1 expression exhibits fractal properties throughout the critical period of symmetry-breaking and axis definition. Similarly, the area fractal dimension D_area​ was determined as 1.7, using the grid method [133].

Besides studying the shape of the ks1-expressing domains, Soriano, et al. [103] also studied their size distribution. A domain size s means that it contains s cells. To diminish the noise in the analysis, for each value of s, the probability P(s) of observing a domain with a size equal to or larger than s was extracted from the data and plotted as a function of s (Figure 8). Around the transition to phase II, point from which the axis appears locked in a temperature gradient (at a culture time of 25 hours), the relationship between P(s) and s appears linear on a log-log scale. The slope of this relationship corresponds to the scaling exponent -τ, which is close to -0.8. Therefore, the probability P(s) as a function of domain size s can be described by a power law P(s)∼s^−τ. Since the domains scale in size as well as in their shape, which is fractal, one can say that the spatial gene expression pattern of ks1 on the spheroid becomes scale free at the axis locking moment. For the other culture times (16 h and 48 h), the distribution does not fit a power law: here characteristic scales are present in P(s). The observation of scale-free properties often allows one to isolate possible underlying mechanisms because they are unusual. We present a hypothetical model that considers that in the following.

9. Model Based on Production and Exchange of ks1-Promoting Factor

Cells communicate with each other through different types of signaling, such as endocrine, paracrine, and juxtacrine communication [82]. In the case of Hydra, there is currently no evidence for endocrine signaling during axis definition, which raises the question of how the Wnt-expressing organizer communicates with other regenerating cells within the organism. Paracrine signaling is well-documented in other animal developmental processes [134,135,136,137,138,139,140] as well as plant developmental processes [141,142]. Paracrine factors, signaling molecules that diffuse over short distances, can regulate gene expression in neighboring cells [143,144].

Epithelial cells are often connected by gap junctions, microscopic channels that enable transport and communications of signaling molecules between adjacent cells [145,146,147]. In Hydra, the cells in the body column are extensively connected by these gap junctions, facilitating intercellular communication. Their role in intercellular signaling suggests that they could contribute to nearest neighbor signaling during axis definition [148].

Cellular functions often arise from the interactions among neighboring cells, leading to synchronized responses [149,150]. This aligns well with the formation of ks1 expression domains on the Hydra cell spheroids. ks1 expression is either fully turned on or off, resulting in synchronized ks1 expression among neighboring cells that lead to the observed domains.

Nearest neighbor signaling in only one dimension will always lead to exponential decay in that direction. In two dimensions, however, since the signaling may find new directions so that it spreads exponentially, there is a chance that the result obeys to a power law, becomes scale-free. Nevertheless, the probability of signaling to the nearest neighbor needs to be tuned to a precise value for that to happen. Self-organized critical models tune to that parameter by themselves. This often occurs in a robust way, meaning that the result is only slightly changed by reasonable changes in the interaction between neighbors [118]. We therefore searched for a self-critical model that could reflect the situation in Hydra.

The model relies on the production of a hypothetical ks1-promoting factor, “X,” which is transferred between directly neighboring cells [4]. The exact contributors to X remain undefined, however, the concentration of X in a cell integrates various signaling inputs, including paracrine factors or other elements in the surrounding environment, affecting the likelihood of ks1 expression of a given cell. β-catenin, which must be released from the cell junctions into the cell nuclei under the mechanical stretch from the osmotic inflation, may well be the most important contributor to X. Initially, each cell has a concentration of X that is attributed at random between zero and just below [X]_c. Since the osmotic inflation slowly drives the cells to increasingly express ks1, we impose that X increases slowly but steadily at a rate ν = d[X]/dt, which means that in the simulation the [X]_i are increased one by one, each by adding a small amount in random order. As [X]_i in the i-th cell reaches [X]_c​, a determined fraction of [X]_i is distributed evenly to its six neighboring cells. [X]_i​ is set to zero and ks1 is expressed by the i-th cell. As X is shared among neighboring cells, the concentration in these cells may also reach the threshold, in turn initiating expression. This process creates a chain reaction, or avalanche, of ks1 expression that spreads across cells, forming a ks1 expression domain (Figure 7a).

The larger the avalanche, the larger the domain of ks1-expressing cells formed by this process. Once an avalanche concludes, the ks1 expression domain disappears. Eventually, a new avalanche begins, usually at a different location of the sphere. Throughout the simulation, ks1 expression in each cell starts and stops multiple times. For the model to function correctly, the avalanche spreading rate must be much faster than the rate of production of the ks1-promoting factor X [4]. This makes the ks1 expression pattern evolve slowly, leading to progressively larger avalanches. The result is stable if a certain degree of randomness in X production across cells introduces variability in avalanche initiation, reflecting “natural” fluctuations to ks1 expression.

10. Analysis of the Model: Results of Simulations

In the study by Gamba, et al. [4], numerical simulations were carried out to test the proposed model for ks1 expression. Although the body wall of Hydra consists of two cell layers (ectoderm and endoderm), ks1 is expressed exclusively in the ectodermal (outer) layer [3]. Therefore, the simulations modeled the Hydra ball as a single-layered structure representing only the ectoderm. This simplification was justified by additional analyses showing that introducing a second layer did not significantly alter the overall model dynamics [4]. A Hydra spheroid was modeled as a spherical lattice of approximately 10,000 hexagons, each representing a cell (see Figure 7a). Simulations showed the best match to experimental data occurred if 95% of X was conserved during transfer to neighboring cells.

Following the rules outlined above, only small avalanches of X-releasing cells were observed initially. On the sphere, they appeared at random locations. Over time, the domains increased in size, and after approximately 2,000 avalanches, the system reached almost a steady state where domains of all sizes were observed.

Examples of typical domains of cells expressing ks1, emerging from the avalanche mechanism of the model, are shown in Figure 7b. The three domains in Figure 7b represent independent events: each domain emerged and disappeared before the next appeared. To make them appear at the same time, one would have to keep the black color for longer. This would not affect the results in any way.

The simulated expression domains were described by examining the distribution of avalanche sizes, P(s), which represents the probability of observing an avalanche (or domain) of ks1-expressing cells larger than size s. Initially, P(s) displayed a more or less exponential decay. Over time, the distribution evolved to a power-law, indicating that the size distribution of the ks1 expression domains became scale-free. Moreover, at this point (time T), the expression domains formed fractal-like patterns. The model yielded a perimeter fractal dimension D_per≈1.28 and an area fractal dimension D_area≈1.75. The power law of P(s) disappeared for longer times (2T).

11. Comparison of Numerical Simulations with Experimental Data

Figure 7b compares typical shapes of main sizes. Figure 8 shows the size distribution of avalanches (corresponding to the domains of ks1-expressing cells) that appeared at simulation times T/2, T and 2T, where T denotes the time until emergence of a power law P(s)~s^-τ). These results were compared to the equivalent experimentally observed distribution, P(s), of ks1-expressing domains (as described above) for the three culture times of 16 h, 25 h, and 45 h. A strong correspondence appeared between the numerical simulations and experimental data: for both, the linearity of ln(P(s)) vs. ln(s) extends over two orders of magnitude (decades), with an identical exponent of τ≈0.8.

The values of perimeter fractal dimension and area fractal dimension (D_per=1.28±0.05 and D_area=1.75±0.05) obtained from numerical simulations for the conservation level C~0.95 at simulation time T closely match the experimental data, where the perimeter fractal dimension is approximately D_per~1.3 at 25 h and the area fractal dimension is D_area~1.7. This similarity indicates that the time T associated with the emergence of the power law P(s)~s^-τ in the simulations is likely close to the experimental time t≈25 h of the Hydra axis formation.

As described above, experiments demonstrated that a temperature gradient could direct the position of the emerging axis, if applied sufficiently early. Increasing temperature corresponds to accelerating the rates, which is simple to do in the simulation. Usually, the acceleration in rates as a function of temperature in biology follows a Boltzmann description [151,152]. which was realized in the simulation. Supposing that an overcritical, large avalanche of ks1 expression, above a certain threshold, is locked to define the axis, the head emerged at the cold and hot side of the spheroid with about equal probability [4]. However, the head had an increased tendency to appear at the cold side for larger temperature gradients. This corresponds to what was observed experimentally.

12. Mechanical Stimulation and the Actin Cytoskeleton

Since our initial work regarding osmotic inflation and burst cycles of a regenerating Hydra spheroid [100], quite an amount of work has focused on the connections between mechanical stimulation, Hydra axis definition and Wnt expression.

Adult Hydra exhibits filamentous actin, oriented in concentric rings in the endoderm and in direction of the axis in the ectoderm [153]. Braun and Keren [57] found that asymmetry and defects in nematic actin fiber orientation of the cell spheroid correlate with the position of the future axis. An interesting study on regeneration under constrained conditions in narrow tubes of Hydra spheroids with an inherited axis [154]. If the inherited axis did not follow the axis of the tube, multiaxial Hydra emerged that the authors link to multiple defects that occur in actin fiber orientation. Accordingly, the initial direction of the axis can be reversed, however, with difficulty. In our example of vortex formation (section 5), this corresponds to an almost fully developed vortex that is reversed. Ravichandran, et al. [155] showed that toroidal Hydra tissue does not regenerate unless a suitable defect in the actin cytoskeleton rescued the situation.

Through which pathway exactly actin is linked to directional signaling in Hydra remains elusive. However, actin is connected to the cell junctions, where β-catenin is located. Moreover, actin is the force generating structure of cells. Active contraction of cells occurs in the direction of actin [156]. It may well be that the increased stretch of the tissue occurs predominantly at the locations where a defect is located, leading to the result above [154,155,157]. At the same time, the situation may be more complex. Microtubules are connected to actin via the Rho signaling pathway [158]. Both elements are known to act in concert [159]. Often microtubules generate directional information in cells during development [160,161,162,163,164]. We showed that Hydra regeneration is prevented by nocodazole, unless it is rescued by exchanging nocodazole for paclitaxel. The delay in regeneration corresponds to the moment of rescue [165].

Asymmetric mechanical constraints can set the axis [159,160]. The feedback between stretch induced Wnt expression, β-catenin nuclear presence and decreased tissue stiffness has been clearly evidenced [107,166]. During osmotic inflation, this is likely to produce instabilities where elastic weakness entails faster stretching, meaning increased probability of Wnt expression in turn translating to increased elastic weakness. This mechanism successfully explained the experimental data in in Weevers, et al. [166] without the need to consider defects in the actin filament network. Bailles, et al. [167] studied regenerates from single cells where supracellular actin organization was absent at the beginning. The actin fibers gradually self-organized, however, it was not clear whether this structural emergence was functionally relevant in terms of axis definition. Actin organization could not be linked to the emergence of Wnt expression.

One may ask how all the data on mechanical signaling can be linked to the model that we presented earlier. If mechanical stretching increases the probability of Wnt expression and head formation (and by extension ks1), this simply means increased production of the X factor in our model. Obviously, large, avalanche-like, synchronized expression patterns would preferentially appear at the location of increased production. Following this interpretation, stimulation through mechanics means that the imposed stimulus is substantially above the natural fluctuation level, so that it will eventually direct the symmetry breaking process. Compared to the analogy of vortex formation, this means that the liquid is actively stirred in a given direction, above the natural fluctuation level, and before vortex formation so that the emerging vortex just follows this direction.

In the shape analysis of a Hydra spheroid during regeneration [100,157], we did not detect elastic tissue anisotropy before the appearance of the early mouth, in excellent agreement with Bailles, et al. [167]. However, this does not mean that externally applied anisotropic stretch will not translate to asymmetry before that time point. Following our interpretation, it just means that the system does not have to generate by itself a large fluctuation of the X factor that will break the symmetry.

13. Conclusions

We discussed results on the expression of the ‘marker of head forming potential’ ks1 [103] and a proposed model [4] that will produce the results quantitatively. A temperature gradient as a non-molecular parameter changes the biochemical rates on both sides of a regenerating spheroid. Supposing that an overcritical (sufficiently strong to break the symmetry) fluctuation is locked to break symmetry, the intricate behavior of Hydra axis positioning as the animal is exposed to a (decidedly weak!) temperature gradient is predicted by the model. Quoting von Neumann’s saying “with four parameters I can fit an elephant, and with five I can make him wiggle his trunk” [168], one cannot stress enough that this model quantitatively produces all the results with only a single adjustable parameter. We suggest that the gene expression patterns of the cells that form the regenerating Hydra sphere increasingly synchronize until a supercritical fluctuation, that is, a fluctuation that eventually leads to stable Wnt expression emerges. This interpretation is supported by observations that, during axis formation, Wnt-expressing cells can emerge in very limited numbers—too few to give rise to a stable organizer. We interpret this phenomenon as the result of subcritical fluctuations, of insufficient magnitude to break symmetry [169,170].

We show how mechanical perturbations that increase the probability of Wnt expression naturally become part of the model, however, we argue that mechanics is likely to be a strong perturbation overriding the natural fluctuations in an unperturbed setting. A competition between at least two different stimuli that can be varied in magnitude, may offer a better view on the contributions to the symmetry breaking process in the future.

The OFC model is part of so-called self-critical models. They evolve towards a critical state on an infinite lattice. By definition, these states not only scale, but they provide infinite sensitivity towards perturbation [118]. This is because of the absence of characteristic scales. Although Hydra is finite, some of the sensitivity can be expected to remain present. This may offer a lever to test the proposed model.