Animals and whole mount immunocytochemistry
Hydra vulgaris was used for all experiments. For siRNA experiments, a transgenic Hydra vulgaris strain expressing endodermal GFP and ectodermal RFP (reverse water melon) provided by Robert Steele’s laboratory was used as described before [26]. The transgenic reporter line expressing eGFP under control of the Hydra Wnt3 promotor was described by Nakamura et al. [9]. The transgenic vector used to generate the strain expressing HyWnt3 under control of the actin promotor was produced by cloning the full-length HyWnt3 sequence between the actin promotor and actin 3’ flanking sequences of the pBSSA-AR vector using NcoI and XbaI sites [9]. Generation of the transgenic actin::HyWnt3 line was performed by microinjection as described before [9]. All animals were maintained in artificial Hydra medium (HM, 1 mM CaCl2, 0.1 mM MgCl2, 0.1 mM KCl,1 mM NaH2CO3, pH 6.8) at 18 °C in polystyrene dishes (Carl Roth) and fed two to three times per week with freshly hatched Artemia salina nauplii, unless indicated otherwise. Media was renewed 3–4 h after feeding and again the following day. Animals were starved for 24 h prior to experiments, unless indicated otherwise. Whole mount immunocytochemistry using a CPP-1-specific antibody was performed as described previously [53].
Wnt proteolysis assays using tissue lysates
Hydra Wnt3 cDNA lacking the native leader peptide was subcloned into the pCEP-Pu mammalian expression vector, which introduces a BM-40 signal sequence and a C-terminal histidine tag as described previously [12]. Recombinant HyWnt3 was expressed using transiently or stably transfected HEK293T cells. For this, cells were seeded in full DMEM Medium (Gibco) in 6-well plates and transfected at 70% confluency using 1 μg of the respective vector construct together with 5 μl transfection reagent (TransIT®-LT1 Reagent, Mirus Bio) in 200 μl serum-free DMEM medium after 20 min incubation time. The cells were maintained in a humidified 5% CO2 atmosphere at 37 °C. After 48 h the medium was harvested, followed by Ni-NTA sepharose (Qiagen) batch purification. For 1 ml conditioned medium, 40 μl Ni-NTA beads were added and incubated for 1 h at 4 °C under constant rotation. After a 2-min centrifugation step at 500 rpm the beads were washed two times with PBS. Elution was performed with 250 mM Imidazol for 15 min at RT. The beads were centrifuged at 500 rpm and the supernatant containing the histidine-tagged Wnt protein was instantly frozen in liquid nitrogen and stored at -80 °C. The eluted Hydra Wnt3 was verified by Western blotting using 10% SDS-PAGE gels. The proteins were transferred to PVDF membranes by wet blotting. Membranes were blocked for 1 hr at RT in PBS containing 5% BSA and 0.2% Tween-20 (PBST), incubated with mouse Penta His antibody (Qiagen, # 34660) at 1:1000 in 1% BSA at 4 °C overnight, washed 3 × 5 min with PBST, and then incubated with anti-mouse horseradish peroxidase-conjugated antibody (Jackson ImmunoResearch, #115-035-044) at 1:10,000 in PBST containing 5% BSA for 1 hr at RT. The membrane was washed 2 × 5 min with PBST and 2 × 5 min with PBS and blots were developed using a peroxidase substrate for enhanced chemiluminescence.
To generate Hydra body part lysates, 100 hydras were cut into four parts using a scalpel. Tentacles, head without tentacles, upper and lower gastric region were immediately transferred on ice, sonicated for 1 min in 250 μl ice cold PBS (Branson sonifier 250, pulsed mode, duty cycle 10%, output 1.5), centrifugated at 10,000 rpm for 1 min at 4 °C, snap frozen and kept in aliquots at -80 °C. Body part lysate concentrations were adjusted to 4 mg/ml protein concentration using a nanodrop photometer and verified by Western blot using alpha-tubulin antibody (Sigma-Aldrich, clone DM1A, T9026) at 1:1000. For the proteolysis assay, ~ 10 ng of purified HyWnt3-His was incubated with 15 μg of the respective tissue lysates in a total volume of 12 μl in PBS for 0, 1, 2, 4, 6, 8, and 24 h at RT. Peak fractions from the ion exchange chromatography were incubated for 6 h. For inhibitor treatments, 1, 10-Phenanthroline (Sigma-Aldrich), Batimastat (Sigma-Aldrich) or EDTA (AppliChem) were added at 200 μM, respectively. Fetuin-B was diluted in PBS and incubated for the given time periods as indicated in Fig. 1f) The incubation was stopped by adding SDS-PAGE sample buffer (10% glycerol, 50 mM Tris-HCl pH 6.8, 0.02% bromophenol blue, 2% SDS, 1% 2-mercaptoethanol) and heating at 96 °C for 5 min. HyWnt3-His band intensities were documented by immunoblotting as described above. Unspecific proteolysis in HL was monitored by adding 15 μg of the lysate to 1 μg of BSA (Roth) in a total volume of 12 μl in PBS and incubation at RT. The reaction was stopped by adding sample buffer and heat denaturation at 96 °C for 5 min after the same time periods as for HyWnt3-His. Samples were analyzed using 10% SDS-PAGE gels and Coomassie staining.
Full-length recombinant HyDkk1/2/4-His protein was expressed in E. coli BL21 (DE3) cells from a pET15b vector (Novagen) and purified under native conditions from the supernatant of cell pellets lysed by several freeze/thaw steps in PBS. The cleared supernatant was filtered and purified using Ni-NTA agarose beads (Qiagen). The eluted protein solution was dialyzed against PBS and purity was checked by SDS-PAGE. For the proteolysis assay, 10 ng of purified HyDkk/1/2/4-His were incubated with 15 μg of the respective tissue lysates in a total volume of 12 μl in PBS for 0, 1, 2, 4, 6, 8, and 24 h at RT. HyDkk1/2/4-His band intensities were documented by immunoblotting as described above.
The sequence for the Hydra Cadherin CAD1-2 domain (15-239) was amplified by PCR using a large 5′ cDNA fragment of Hydra Cadherin as template. The PCR product was cloned into the pET19B vector (Novagen) using NdeI and BamHI sites and the recombinant protein was expressed in E. coli BL21 (DE3) cells. Inclusion bodies were solubilized in PBS, 8 M urea, and bound to Ni-NTA agarose beads. Refolding was performed by changing to PBS buffer prior to elution with PBS, 250 mM imidazole, 0.4 M L-arginine (AppliChem). The eluted protein solution was dialyzed against PBS and its purity checked by SDS-PAGE. Ten nanograms of the purified protein were used as substrate with 5 μl of either the HyWnt3(+) and HyWnt3(-) HL fraction. Detection of the Hydra cadherin fragment by immunoblotting was performed as described above.
The polyclonal HAS-7 specific antibody was raised against a peptide corresponding to a sequence between the catalytic and ShKT domains (CKGGGNPPTGPPTAPP). For Western blotting, a PVDF membrane was blocked in 5% skimmed milk powder in PBS, 0.2% Tween-20, for 1 h at RT. The primary antibody was applied at 1:1000 in 1% skimmed milk powder in PBST and membranes were incubated overnight at 4 °C. A peroxidase-conjugated anti-rabbit antibody (Jackson ImmunoResearch, #111-035-003) was applied in blocking solution at 1:10,000 for 1 hr at RT. Quantification of Western blot band intensities was performed using the ImageStudioLite software. Uncropped Western blot and gel images are shown in Additional File 11: Fig. S9.
Ion exchange chromatography and mass spectrometry analysis
A pool of head lysate was generated by decapitating 400 animals closely beneath the head region and removing tentacles using a scalpel. Head tissue pieces were resuspended in ice cold 20 mM Tris-HCl, pH 7.5, and lysed on ice by passing at least 10 times through a 20-gauge needle. The lysate was centrifuged for 15 min (14,000 rpm) at 4 °C and the supernatant was applied to anion exchange chromatography using a MonoQ HR5/5 column (GE healthcare). The eluent buffer contained 1 M NaCl in 20 mM Tris-HCl, pH 7.5. Peak fractions indicated in Additional file 1: Fig. S1 were screened for HyWnt3-His proteolytic activity as described above and frozen at − 20 °C until further use. Pooled HyWnt3(+) and HyWnt3(-) fractions were subjected to mass spectrometry analysis after in solution tryptic digestion. Peptide separation was achieved using a nano Acquity UPLC system (Waters). Protein mass spectrometry analyses were performed as previously described [34]. In short: The nano UPLC system was coupled online to an LTQ OrbitrapXL mass spectrometer (Thermo Fisher Scientific). Data dependent acquisition with Xcalibur 2.0.6 (Thermo Fisher Scientific) was performed by one FTMS scan with a resolution of 60,000 and a range from 370 to 2000 m/z in parallel with six MS/MS scans of the most intense precursor ions in the ion trap. The mgf files processed from the MS raw files were used for protein database searches with the MASCOT search engine (Matrix Science, version 2.2) against NCBI GenBank Proteins (version of October 28, 2015). Domain composition of GenBank protein accessions was analyzed against CDD (NCBI) and InterProScan 5 (EMBL-EBI).
For fractionation of full hydra lysates (Additional file 6: Fig. S4d), 1000 animals were dissolved by sonification on ice in 1 ml HEPES(-) buffer (50 mM HEPES in ddH2O pH 7.4). The hydra lysate was cleared at 14,000 rpm for 15 min at 4 °C. Five hundred microliters of the supernatant was applied to anion exchange chromatography as described above. All collected fractions were snap-frozen in liquid nitrogen and stored at − 70 °C.
Expression and purification of recombinant HAS-7
Expression of HAS-7 occurred in insect cells using the Bac-to-Bac system (Invitrogen, Thermo Fisher Scientific). A synthetic fragment of the HAS-7 full-length cDNA (Biomatik) was subcloned into the pCEP-Pu mammalian expression vector, which introduces a C-terminal histidine tag and then the fragment containing the C-terminal His(6) -tag was inserted into the donor plasmid pFASTBac1 to generate bacmids in E. coli DH10Bac cells for transfection of insect cells according to the manufacturer’s manual. Amplification of recombinant baculoviruses occurred in Spodoptera frugiperda 9 cells (Sf9 CCLV-RIE 203, Friedrich–Löffler Institute, Greifswald, Germany) growing as adherent monolayer in Grace’s insect medium supplemented with 100 U/ml penicillin and 100 μg/ml streptomycin. For expression of HAS-7-His(6) Trichoplusia ni cells (BTI-TN-5B1-4/High Five-cells CCLV-RIE 350, Friedrich–Löffler Institute, Greifswald, Germany) were cultured as suspension in Express Five SFM containing 100 U/ml penicillin, 100 μg/ml streptomycin and 16.4 mM L-Glutamine. Infected High Five cells were incubated for 72 h at 27 °C in Fernbach flasks (shaking incubator Multitron, INFORS). Following centrifugation, proteins were precipitated at 10 °C by step wise addition of solid ammonium sulfate to the supernatant resulting in a 60% ammonium sulfate saturation. After further stirring over night at 10 °C and centrifugation (9000×g, 90 min, 10 °C) the harvested pellet was resuspended in 1/10 of the expression volume in 50 mM Tris-HCl pH 7.6, 300 mM NaCl, 20 mM imidazole and dialyzed against the same buffer. The cleared supernatant gained after centrifugation (8000×g, 10 min, 4 °C) was applied onto a Ni-NTA column (Qiagen). Following several washing steps with 50 mM Tris-HCl pH 7.6, 300 mM NaCl containing increasing imidazole concentrations (20 mM, 50 mM and 100 mM), the protein was finally eluted in a buffer containing 250 mM imidazole. The elution fractions were pooled, dialyzed against PBS, and concentrated (Millipore Amicon Ultra, 3 K). SDS-PAGE and transfer to PVDF were performed as described previously [54]. The membrane was blocked in 3% BSA in TBS for 2 h, incubated for 1 h with Penta-His antibody at 1:2000 and further incubated for 1 hr with secondary antibody (goat anti-mouse POX 1:7500 in 7.5% skimmed milk powder in TBS). After each antibody treatment, three washing steps were inserted (2x TBST, 1x TBS). The Clarity Western ECL Substrate (Bio-Rad) was used for detection.
In situ hybridization
Customized LNA digoxygenin-labeled RNA probes were designed and produced by Qiagen corresponding to the antisense strands of the respective astacin cDNAs. The whole-mount ISH procedure was performed as described previously [55]. For hybridization, the LNA probe was added to a final concentration of 1 μM in fresh hybridization solution (1:1 mixture of deionized formamide and buffer containing 5x SSC (750 mM NaCl, 75 mM sodium citrate), 0.2 mg/ml yeast tRNA, 2% of 50x Denhardt’s solution, 0.1 mg/ml heparin, 0.1% Tween-20 and 0.1% CHAPS), and hybridized for ~ 60 h at 55 °C. Digoxygenin-labeled RNA probes for HAS-1, HAS-7, and HMP1 corresponding to the sense and antisense strands were prepared using an RNA labeling in vitro transcription kit (Roche). ISH probe sequences covered the predicted full-length mRNA sequence for each gene. The further procedure was carried out as described previously [12]. Samples were finally mounted in PBS containing 90% glycerol, or in Mowiol 4–88 (Carl Roth), and images were acquired with a Nikon Digital Sight DS-U1 camera mounted on Nikon Eclipse 80i and imaging software NIS Elements (3.10, SP3, Hotfix, Build645). Further image processing was performed with Adobe Photoshop CS6 and Fiji. ISH probe and LNA sequences are summarized in Additional file 12: Table S3. t-SNE plots for the respective genes were designed using online tools provided at https://singlecell.broadinstitute.org/single_cell/study/SCP260/stem-cell-differentiation-trajectories-in-hydra-resolved-at-single-cell-resolution#study-visualize [25].
Alsterpaullone treatment for ISH analysis
Eighty budless hydras, which were fed the day before, were incubated in 5 μM ALP (Sigma-Aldrich) in DMSO in 100 ml HM. The animals were kept in the dark at 18 °C for 24 h. After incubation, the HM was changed every day. Samples were taken for ISH at 24 h, 48 h, and 72 h after ALP treatment. Hydras incubated for 24 h in DMSO and fixed after 72 h served as control. For continuous ALP treatment (Additional file 10: Fig. S8e), the medium was exchanged daily and animals were fed once per week.
Electroporation with siRNAs
siRNAs (HPLC grade) (see Additional file 13: Table S4 for sequences) were purchased from Qiagen. Electroporation of siRNA was performed as described recently [12] using 3 μM of siGFP (1 μM siGFP and 2 μM scrambled siGFP) or a combination of siGFP, scrambled siGFP and target siRNAs. Scrambled siGFP was omitted when two target siRNAs were used. Immediately after the pulse, 500 μl restoration medium consisting of 80% HM and 20% hyperosmotic dissociation medium (6 mM CaCl2, 1.2 mM MgSO4, 3.6 mM KCl, 12.5 mM TES, 6 mM sodium pyruvate, 6 mM sodium citrate, 6 mM glucose and 50 mg/l rifampicin, 100 mg/l streptomycin, 50 mg/l kanamycin, pH 6.9) was added to the cuvette, the animals were transferred to Petri dishes containing restoration medium and allowed to recover for one day. Viable polyps were transferred to new dishes containing HM and maintained under standard culture conditions. For AZK treatment, animals were incubated at 6 days post-electroporation either with 0.1% DMSO or 50 nM AZK in HM for 16 h. Thereafter, they were rinsed in HM several times and cultured under standard conditions. Five days after AZK treatment, the animals were anesthetized in 1 mM Linalool (Sigma-Aldrich) [56] using a Nikon SMZ25 stereomicroscope equipped with Nikon DS-Ri2 high-definition color camera. Note that AZK instead of ALP was used in siRNA experiments allowing a reduction of the total inhibitor concentration applied on the electroporated animals. Fifty nanomolar AZK induces the same grade of ectopic tentacles as 5 μM ALP when applied in the standard assay published previously [36]. Double axes counted in these experiments were defined morphologically to have a duplication of the head structure with fully developed and functional tentacles and to show a clearly detectable split of the body column along two independent axes. Mostly, these animals were Y- or L-shaped sharing the same lower gastric region and foot region. In contrast to buds, secondary axes do not separate after reaching a certain growth limit.
Chromatin immunoprecipitation
Chromatin immunoprecipitation analysis was carried out as described recently by using sheared extracts from formaldehyde-treated Hydra animals, which have been treated without or with 5 μM ALP (48 h) and an antiserum directed against a recombinant Hydra TCF protein [12, 38]. PCR of precipitated DNA was done using specific primers flanking the potential TCF binding sites in the 5′-regulatory regions of the Hydra HAS-7 or the HmTSP gene (Fig. 5g, h). PCR primer sequences: TCF binding motif in HAS-7 (5′-GCTGTTATCTGTCCGCTTTC-3′/5′-CCATATAGAGGCCACACACC-3′), and the proximal TCF binding motif in HmTSP (5′-TTGAAGGCATTTAACAACTTGC-3′/5′-TGCCCAAATGTAAAGTTCTGTG-3′).
Real-time quantitative PCR
The RNA isolation was performed as described previously [34]. Sixty hydra heads per condition were isolated either from steady state AEP polyps or siRNA treated transgenic reverse water melon animals by removing the tentacles and gastric region using a scalpel. The heads were immediately transferred into 200 μl TRIzol and stored at − 20 °C. For siRNA (siGFP or siHAS-7) treated hydras, head samples were isolated 6 days after electroporation. Complete hydras were used for beta-Catenin RNAi treated and transgenic actin::HyWnt3 animals including the respective controls. cDNA synthesis was performed using the SensiFAST cDNA Synthesis Kit (Bioline) according to the manufacturer’s instructions. RT-qPCR was carried out with a StepOnePlus™ instrument (Applied Biosystems, Thermo Fisher Scientific) using the SensiFAST SYBR-Hi-ROX Kit (Bioline) according to the manufacturer’s instructions. The transcript level analysis was done by the ΔΔC(T)-Method with Elongation Factor 1-α (EF-1α) as a house keeping gene for normalization. Three biological replicates were performed for each experiment and triplicate measurements were made for each sample in each experiment. The no-template conditions served as negative controls. The data are presented as relative quantity (RQ) by 2^(-ΔΔC(T)) calculation. qPCR primer sequences are given in Additional file 13: Table S4.
Xenopus experiments
In vitro fertilization, embryo culture, and culture of explants were carried out as described [57]. Staging was done according to Nieuwkopp [58]. mRNA was produced with the mMessage mMachine SP6 trancription Kit (Ambion) from the HyWnt3, XWnt8, HAS-7, and flag-tagged GFP (control mRNA) ORFs in the respective linearized pCS2+ vectors. mRNA was purified with a phenol/chloroform extraction and a subsequent isopropanol precipitation. Injections were done into the marginal zone of both ventral cells of the 4-cell stage. Total amounts of each injected mRNA were XWnt8 (10 pg), HyWnt3 (10 or 100 pg), HyWnt3/HAS-7 (100 pg each), and scrambled RNA (100 pg).
Structural modeling
Protein-Protein docking experiments were performed in ClusPro2 [59] using the crystal structures of Xenopus Wnt8 (pdb-code:4F0A) and promeprin ß (pdb-code: 4GWM). The propetide E25-G66 blocking access to the active site cleft and disordered loop segments in Wnt8 were removed to allow for proper structural analysis. Targeted search matrices were chosen by selecting the zinc-binding site of promeprin ß (H132-H136-H142) and the putative cleavage site of HyWnt3 (K186-D187-P188) as attractive search targets. Based on Lennart-Jones potentials, distance metrices, and electrostatic evaluations the best 100 docking hits were subject to gradient energy minimization in the Crystallography and NMR system. 14 The lowest energy structures were further subject to 500 cycles of unrestrained Powell minimization. Harmonic restraints were imposed on the target molecule (2 kcal/mol Å2) with increased weight (25 kcal/mol Å2). Protein structure and model assessment tools were used to verify the quality of the modeled structure. Additionally, HAS-7 modeling was refined using Modeller [60] implemented in Chimera [61] with astacin complexed to a transition state analog inhibitor (pdb-code: 1QJI) and zebrafish hatching enzyme (3LQB), the latter being the most closely HAS-7-related astacin with known structure to date.
Phylogenetic analysis
Amino acid sequence alignments and phylogenetic trees were computed using the SEAVIEW package (http://doua.prabi.fr/software/seaview) [62, 63]. Alignments were performed by CLUSTAL omega embedded in SEAVIEW [64] using the default adjustments. Refinements of the S1’ regions were arranged based on overlays of the X-ray crystal structures of crayfish astacin (pdb-codes 1ast, 1qji), and zebrafish hatching enzyme 1 (ZHE1, pdb-code 3lqb). Phylogenetic tree calculations were performed using the maximum likelihood approach provided by the PhyML tool implemented in SEAVIEW [65].
Mathematical model
To obtain mechanistic insights into the regulatory function of HAS-7, we propose a mathematical model given in terms of reaction-diffusion equations that model signaling processes. To account for a realistic geometry of the Hydra tissue bilayer, we adopt a mechano-chemical modeling approach coupling the morphogen dynamics with the evolution of infinitely thin deforming tissue [66, 67]. The mechanical part is given by a 4th order partial differential equation (PDE) model based on the minimization of the Helfrich-type energy. It allows describing small tissue deformations induced by patterns of gene expression, such as the initial stage of tentacle development.
The point of departure for the molecular interaction model is the HyWnt3/beta-Catenin model recently proposed in [68], describing three separate (but interacting) pattern formation systems for the body axis (including beta-Catenin/TCF), the head organizer (including HyWnt3), and the tentacle system. In this paper, we extend the HyWnt3/beta-Catenin model to account for experimentally investigated HyWnt3-HAS-7 interactions.
The core of the model accounts for the dynamics of Wnt3 and beta-Catenin/TCF signaling that are coupled through the canonical Wnt signaling pathway [6, 69]. Although it is frequently assumed that beta-Catenin/TCF and Wnt3 molecules act in the confines of the same pattern formation system to coordinate body axis and head formation, e.g., [10, 31, 70, 71], we distinguish between them in the model and describe the dynamics of Wnt3 and beta-Catenin using two model variables, respectively, each controlled by Turing-type activator-inhibitor loops [29, 30]. Although accounting explicitly for HyWnt3 and beta-Catenin/TCF, the proposed model involves hypothetical inhibitors that might be linked to HyDkk1/2/4 [68, 72], the Sp5 transcription factor [10], or Thrombospondin (HmTSP) [12]. Since Sp5 does not diffuse, it is not a HyWnt3 inhibitor in the sense required by the classical activator-inhibitor model. An additional model of the molecular mechanism, possibly including a hypothetical long-range factor locally activating Sp5 is beyond the scope of this work. Therefore, we phenomenologically summarize HyWnt3 inhibition by a diffusive HyWnt3 inhibitor possibly including Sp5 and HmTSP. Similar dynamics may result from multistability in the intracellular signaling [73,74,75], or a negative feedback loop stemming from mechano-chemical interactions [66, 76]. The reduced mathematical representation of the underlying mechanisms is sufficient for the purpose of this paper, since we focus on the role of HAS-7 in the pattern formation process. Specifically, we clarify its function in pattern selection (localization of the Wnt3 spot) that does not depend on a particular molecular agent of the Wnt3/beta-Catenin pattern formation mechanism. The modular structure of the model suggests robustness of the underlying mechanisms that is controlled in a stepwise process.
The HAS-7 function is modeled by coupling the HyWnt3/beta-Catenin model to the dynamics of HAS-7 that is positively regulated by beta-Catenin. The indirect regulation is modeled as a transcriptional HAS-7 activation by head-specific molecules downstream of the organizer (some candidates of the latter are presented in Ref. [7]). The assumption is motivated by (1) our HAS-7 promoter analysis, (2) the HAS-7 expression patterns after AZK treatment presented in this study, and (3) the natural mechanistic assumption that HAS-7 should be activated as soon as a head is established in order to suppress the formation of additional heads. Furthermore, we assume that HAS-7 degrades Wnt3 ligands and that Wnt3 negatively regulates HAS-7. The latter is motivated by (1) our observation that HAS-7 transcript is absent from the upper hypostome, and (2) the natural mechanistic assumption that HAS-7 suppresses organizer formation in the surroundings but not of the existing organizer itself. It is not known if the local negative regulation of HAS-7 by the hypostome is governed directly by Wnt3 or by other molecules downstream of Wnt3. Nevertheless, accounting for such modification of the regulatory mechanism does not influence the results of the model. Hence, for the sake of simplicity, we assume a local negative regulation of HAS-7 by Wnt3.
Model variables are biologically defined as follows: β_cat: Nuclear beta-Catenin/TCF; β_catant: β_cat antagonist, probably involving Dickkopf1/2/4-C [32, 72]; Wnt3: HyWnt3; Wnt3ant: HyWnt3 antagonist (probably Sp5 [10] and HmTSP [12] involved); Head: Head-related factors downstream of HyWnt3 (e.g., multiple Wnts [7]); HAS: HAS-7; SD: Source density (long-term storage of the head forming potential); Tent: Tentacle activator (probably HyAlx [77], Wnt8 [36], BMB5-8b [78] involved); Tentant Tentacle activator antagonist (unknown). Model equations are presented below.
$$ {\partial}_t\beta \_ ca t={a}_1{\Delta}^{\Gamma}\beta \_ ca t+{b}_1\cdot \left(1.0+{c}_1\cdot Wnt3\right)\cdot SD\cdot \frac{0.05+\beta \_{cat}^2}{\beta \_ ca{t}_{ant}}-{d}_1\cdot \beta \_ ca t $$
(1)
$$ {\partial}_t\beta \_{cat}_{ant}={a}_2{\Delta}^{\Gamma}\beta \_{cat}_{ant}+{b}_2\cdot \left(1.0+{c}_2\cdot Wnt3\right)\cdot SD\cdot \beta \_{cat}^2-{d}_2\cdot \beta \_{cat}_{ant} $$
(2)
$$ {\partial}_t Wnt3={a}_3{\Delta}^{\Gamma} Wnt3+{b}_3\cdot \beta \_{cat}_{ant}\cdot \frac{0.05+ Wnt{3}^2}{Wnt{3}_{ant}\cdot \left(1.0+{c}_3\cdot Wnt{3}^2\right)}-{d}_3\cdot \left(1.0+{e}_3\cdot HAS\right)\cdot Wnt3 $$
(3)
$$ {\partial}_t Wnt{3}_{ant}={a}_4{\Delta}^{\Gamma} Wnt{3}_{ant}+{b}_4\cdot \beta \_{cat}_{ant}\cdot \frac{0.005+ Wnt{3}^2}{1.0+{c}_4\cdot Wnt{3}^2}+0.035-{d}_4\cdot Wnt{3}_{ant} $$
(4)
$$ {\partial}_t Head={a}_5{\Delta}^{\Gamma} Head+{b}_5 Wnt3-{d}_5 Head $$
(5)
$$ {\partial}_t HAS={a}_6{\Delta}^{\Gamma} HAS+\frac{b_6 Head}{1.0+{c}_6\cdot Wnt3}-{d}_6 HAS $$
(6)
$$ {\partial}_t Tent={a}_7{\Delta}^{\Gamma} Tent+\frac{b_7\cdot SD\cdot \left(0.005+{ Ten t}^2\right)}{Ten{t}_{ant}\cdot \left(1+{c}_7\cdot { Ten t}^2\right)\cdot \left(1+{e}_7\cdot {Head}_{ant}\right)}-{d}_7\cdot Tent $$
(7)
$$ {\partial}_t{Tent}_{ant}={a}_8{\Delta}^{\Gamma}{Tent}_{ant}+\frac{b_8\cdot SD\cdot \left(0.005+{Tent}^2\right)}{\left(1+{c}_8\cdot {Tent}^2\right)\cdot \left(1+{e}_8\cdot {Head}_{ant}\right)}+0.014-{d}_8\cdot {Tent}_{ant} $$
(8)
$$ {\partial}_t SD={a}_9{\Delta}^{\Gamma} SD+b9\cdot \beta \_ cat+0.00003-{d}_9 SD $$
(9)
We performed numerical simulations of the model to obtain insights into the dependence of the model results on specific choices of parameters. Our study suggests that most of the parameters involved in Eq. (1)–(9) do not influence critically the qualitative HAS-related simulation results (Fig. 7a). In particular, most of them control specific properties of one of the three interplaying de novo pattern formation systems, such as spatial scaling of the pattern (size of expression domain), spacing between the maxima of the pattern, or a condition for de novo patterning. Following these simulation results, we fixed most of the parameters to values taken from Ref. [31, 32]. This allowed focusing on parameters accounting for the novel aspects of the model such as (i) interactions between beta-Catenin and Wnt3 that govern pattern formation on different spatial scales, and (ii) a feedback loop between HAS-7 and Wnt3. In general, changing the corresponding parameters, we observed robust model dynamics (qualitatively the same pattern formation). The only discrepancy was observed in simulations of siHAS-7/AZK animals, where the number of ectopic axes in model simulations depends on the strength of HAS-7 dependent degradation of Wnt3. This observation suggests that there might be an additional mechanism (possibly involving other members of the HAS family) ensuring an experimentally observed robustness with respect to the number of organizers.
For simulations of the unperturbed system, we applied the following parameters following Ref. model [31, 32, 68]):
$$ {a}_1=9\times 1{0}^{-5},{b}_1=3\times 1{0}^{-3},{c}_1=3\times 1{0}^{-2},{d}_1=3\times 1{0}^{-3}, $$
$$ {a}_2=11\times 1{0}^{-3},{b}_2=3\times 1{0}^{-3},{c}_2=3\times 1{0}^{-2},{d}_2=4\times 1{0}^{-3}, $$
$$ {a}_3=6\times 1{0}^{-5},{b}_3=7\times 1{0}^{-3},{c}_3=3\times 1{0}^{-3},{d}_3=12\times 1{0}^{-2},{e}_3=1\times 1{0}^2, $$
$$ {a}_4=24\times 1{0}^{-3},{b}_4=1\times 1{0}^{-2},{c}_4=3\times 1{0}^{-3},{d}_4=18\times 1{0}^{-2}, $$
$$ {a}_5=25\times 1{0}^{-3},{b}_5=1\times 1{0}^0,{d}_5=1\times 1{0}^{-2}, $$
$$ {a}_6=25\times 1{0}^{-3},{b}_6=1\times 1{0}^{-1},{c}_6=1\times 1{0}^1,{d}_6=5\times 1{0}^{-2}, $$
$$ {a}_7=25\times 1{0}^{-5},{b}_7=2\times 1{0}^{-3},{c}_7=12\times 1{0}^{-2},{e}_7=3\times 1{0}^{-2},{d}_7=2\times 1{0}^{-2}, $$
$$ {a}_8=27\times 1{0}^{-3},{b}_8=3\times 1{0}^{-3},{c}_8=12\times 1{0}^{-2},{e}_8=3\times 1{0}^{-2},{d}_8=3\times 1{0}^{-2}, $$
$$ {a}_9=11\times {1}^{-5},{b}_9=3\times 1{0}^{-5},{d}_9=3\times 1{0}^{-5}. $$
To approximate the geometry of the Hydra tissue, initial conditions for X1, X2, and X3 are parametrized over a closed 2D unit-sphere S2 embedded in 3D space with X1(t = 0) ≡ X2(t = 0) ≡ 0 and X3(t = 0) = 4 · s3, thus, leading to a stretch into the direction of s3 (given that s1, s2, s3 are Eulerian coordinates of the S2-surface). For biological molecules, we used a stochastic initial distribution based on the standard random generator provided by C++. The source density is modeled using an initial gradient given by SD(t = 0) = 4.0 · (exp(s3)/exp(1)). Thus, in all simulations, only the geometric and chemical body axis gradient are initially prescribed.
In simulation of the AZK treatment, we modified the initial conditions for the source density adding an offset by SD(t = 0) = 2.0 + 4.0 · (exp(s3)/exp(1)). To simulate HAS knockdown, we set b6 = 0. For Dkk knockdown, we modeled a reduction of Dkk activity by increasing d2 by the two-fold. In the model, a complete deactivation of beta_catant prevents creation of any pattern, since the body-scale system is described by just two components that are a minimal set required for pattern formation. Finally, Wnt3 overexpression was simulated by adding the constant c = 0.1 to the production.
