G protein-coupled receptors (GPCRs) are important signal transducers that are phosphorylated upon activation at intracellular serine and threonine residues. Although antibodies that specifically recognize the phosphorylation state of GPCRs have been available for many years, efficient immunolocalization of phosphorylated receptors in their tissues of origin has not been possible. Here, we show that phosphorylation of receptors is highly unstable during routine immunohistochemical procedures, requiring the use of appropriate phosphatase inhibitors particular during tissue perfusion, post-fixation, and cryoprotection but not during immunostaining of tissue sections. We provide proof of concept using phosphorylation state-specific μ-opioid receptor (MOP) and cannabinoid receptor 1 (CB1) antibodies. Indeed, three of four well-characterized phosphosite-specific MOP antibodies, including pS375-MOP, pT376-MOP, and pT379-MOP, showed robust neuronal immunostaining in brain and spinal cord sections of opioid-treated mice only after inclusion of phosphatase inhibitors. We then extended this approach to the CB1 receptor and demonstrated that one of three newly-generated phosphosite-specific CB1 antibodies, namely pS425-CB1, showed striking staining of fibers and varicosities in brain slices from cannabinoid-treated mice. Although subsequent experiments showed that phospho-CB1 immunostaining was less sensitive to phosphatases, we conclude that the use of phosphatase inhibitors should always be considered in the development of immunohistochemical procedures for new phosphosite-specific GPCR antibodies. In summary, we anticipate that this improved protocol will facilitate the widespread use of phosphorylation state-specific antibodies to monitor the activation of endogenous GPCRs under physiological and pharmacological conditions. Our approach may also prove useful to confirm target engagement of GPCR drug candidates in native tissues. Phosphorylation of receptors is shown to be unstable during routine immunohistochemical procedures in mice thus phosphatase inhibitors should be used alongside phosphosite-specific GPCR antibodies. Phosphorylation also increases the interaction of GPCRs with intracellular adapter proteins such as β-arrestins, which can trigger a second wave of signaling5. Thus, analysis of agonist-driven phosphorylation can provide valuable insights into the receptor activation state and ligand pharmacology. A widely used approach to analyze GPCR phosphorylation is the use of phosphosite-specific antibodies9. When available, such antibodies are valuable tools to elucidate the temporal dynamics of receptor phosphorylation, identify relevant kinases and phosphatases, and detect receptor activation using immunoblotting techniques15. However, previous attempts using routine immunohistochemical approaches to unequivocally reveal agonist-induced phosphorylation of endogenous GPCRs in native tissues largely failed.Agonist-driven serine/threonine phosphorylation is a biologically and pharmacologically important process that primarily initiates desensitization and internalization of G protein-coupled receptors (GPCRs)16. Indeed, phosphosite-specific antibodies in combination with siRNAs have led to the identification of distinct protein phosphatase 1 (PP1) and PP2 catalytic subunits as bona fide GPCR phosphatases22. For many receptors, dephosphorylation is complete within 10 to 30 minutes after agonist washout21. A notable exception is the agonist-induced phosphorylation of S341/S343 at the SST2 somatostatin receptor, which persists for considerably longer periods21. Consequently, activated SST2 receptors could be successfully localized using pS341/pS343-SST2 antibodies under routine immunohistochemical conditions23. However, this approach could not easily be reproduced for other SST2 sites or other GPCRs.In intact cells, GPCR dephosphorylation is regulated in time and space, beginning immediately after receptor activation at the plasma membraneThe ability to visualize activated and phosphorylated GPCRs in their tissues of origin would provide important clues to the physiological and pharmacological regulation of receptor activation. In particular, it would allow for distinction between currently-activated and resting GPCR populations in the context of particular physiological or behavioral conditions. We were therefore motivated to develop and validate immunohistochemical fixation and staining procedures that can be universally applied to prototypical GPCRs. Our improved protocol will facilitate the widespread application of phosphosite-specific antibodies as GPCR activation sensors in academic and pharmaceutical research.16. Serine/threonine phosphatases are known to be activated during cell lysis and tissue fixation procedures. We therefore tested the inclusion of appropriate protein phosphatase inhibitors (PPIs) during tissue fixation and immunohistochemical staining. For this purpose, mice were treated with either methadone or saline, perfusion fixed, and slices of brain and spinal cord were stained with phosphosite-specific µ-opioid receptor (MOP) or phosphorylation-independent np-MOP antibodies. As expected, staining using the np-MOP antibody was present in the superficial layers of the spinal cord, regardless of drug treatment or the presence of phosphatase inhibitors 5. The unique ability of GRKs to recognize activated receptors results in agonist-dependent phosphorylation at intracellular serine and threonine residues5. Thus, analysis of agonist-driven phosphorylation of GPCRs can provide valuable insights into receptor activation. Antibodies that specifically recognize the GPCR phosphorylation state have been available in some cases over the last 20 years. Indeed, phosphosite-specific GPCR antibodies have proven useful to detect receptor activation and to profile the pharmacological properties of new ligands15. To date, however, their utility has been largely limited to immunoblotting approaches using heterologous receptor expression systems, while detection of phosphorylated GPCRs from tissue lysates has been mostly unsuccessful, due to their low abundance in vivo.GPCRs are privileged targets for small molecule drugs in almost all therapeutic areasHere, we have systematically developed an improved immunohistochemical staining procedure that allows for visualization of opioid drug effects in the mouse brain in vivo. We also show that this approach can be easily extended to other prototypical GPCRs. Key to this new protocol was the observation that receptor phosphorylation is highly unstable during routine immunohistochemistry and that for many GPCRs inclusion of protein phosphatase inhibitors may be an absolute requirement during perfusion, post-fixation and cryoprotection of tissues. Several lines of evidence suggest that our approach facilitates the localization of bona fide phosphorylated GPCRs. First, phospho-GPCR immunostaining strongly increased when animals had been treated with agonist. Conversely, immunostaining was not detected when agonist action was blocked by antagonist. Second, robust phospho-GPCR immunostaining was only apparent when appropriate protein phosphatase inhibitors were included in all steps of fixation and staining procedures. Third, phospho-GPCR immunostaining was completely neutralized by preincubation with cognate phosphorylated peptides but not with the corresponding non-phosphorylated peptides.In the present study, we had access to four different phospho-MOP and three different phospho-CB1 antibodies which had previously been validated by immunoblotting techniques. We observed that three out of four antibodies for MOP and one out of three antibodies for CB1 produced specific phospho-immunostaining in tissues suggesting that each phosphosite-specific GPCR antibody has to be validated individually for immunohistochemical applications. Moreover, we observed that phospho-MOP staining was critically dependent on the presence of PPIs, whereas CB1 phosphorylation appeared to be more stable in tissues. Nevertheless, we conclude that use of PPIs should always be considered in the development of immunohistochemical procedures for new phosphosite-specific GPCR antibodies.https://atlas.brain-map.org). Notably, lack of MOP phosphorylation in some brain regions relative to others could have profound consequences on MOP desensitization and may explain well-known differences in tolerance development to different opioid effects. Another example is the different subcellular localization of phosphorylated and non-phosphorylated CB1 receptors. While total CB1 appears to be evenly distributed along fibers and varicosities, phosphorylated CB1 is particular enriched in terminals indicating compartmentalization of GRK-mediated phosphorylation.It is well known that endogenous GPCRs are notoriously difficult to detect in tissues by immunohistochemical staining. Here, we not only establish a protocol for in situ visualization of activated and phosphorylated GPCRs. We also demonstrate that phospho- immunostaining can disclose previously unappreciated functional information not seen with phosphorylation-independent GPCR antibodies. One example is the striking mismatch between total and phospho-immunostaining for MOP in caudate putamen. This mismatch cannot be explained be low detection level of phospho-MOP staining as many other brain regions with comparable total MOP revealed prominent phospho-MOP. The most likely explanation for the relative lack of phospho-MOP staining in caudate putamen is the low or absent expression of relevant GRK family members . In addition, we used rabbit np-MOP from abcam (ab134054), rabbit np-CB1 from Cayman (1000659) and goat np-CB1 from Frontier Institute (MFSR100610). Additional information in Supplemental Table The phosphorylation state-specific rabbit MOP antibodies pT370-MOP 7TM0319B), pS375-MOP (7TM0319C), pT376-MOP (7TM0319D), pT379-MOP (7TM0319E), the phosphorylation state-specific rabbit CB1 antibody pS425-CB1 (7TM0056A) as well as the phosphorylation-independent rabbit antibody np-MOP (7TM0319N) and phosphorylation-independent guinea pig antibody np-MOP (7TM0319N-GP) were provided by 7TM Antibodies  and  for the amplified staining procedure. For the direct stainings we used CyTM3  and CyTM3 , both obtained from Jackson ImmunoResearch.We used a biotinylated AB from Jackson Immuno Research  and the respective streptavidine conjugates from ThermoFisher  were housed 2–5 per cage under a 12-hr light-dark cycle with ad libitum access to food and water. All animal experiments were approved by Thuringian state authorities and performed in accordance with European Commission regulations for the care and use of laboratory animals. Our study is reported in accordance with ARRIVE guidelines28. In all experiments, male and female mice aged 8–30 weeks between 25 and 35 g body weight were used.Mice (C57BL/6 J obtained from JAXn = 4–6 per treatment condition) were used.All drugs were freshly prepared prior to use and were injected subcutaneously (s.c.) or intraperitoneally (i.p.) in unanaesthetized mice at a volume of 10 μl/g bodyweight. Opioid drugs were diluted in 0.9% (w/v) saline and cannabinoid drugs in vehicle solution, containing 13.2% ethanol and 0.25% Tween80 in 0.9% saline, for injections. Drugs were obtained and used as follows: fentanyl citrate  (B. Braun 06900650), levomethadone hydrochloride  (Sanofi-Aventis 07480196), naloxone  (Ratiopharm 04788930), CP-55940  (Sigma-Aldrich C1112) and AM251  (MedChemExpress 183232-66-8). A total of 35 mice , the sections were blocked in PBS containing 0.3% Triton X-100, +/- PPIs and 10% NGS for 2 h. Subsequently, the sections were incubated with primary antibody in PBS containing 0.3% Triton X-100, 2% NGS + /- PPI overnight at 4 °C. Where indicated, primary antibodies were preincubated with 1 µg/ml of their cognate phospho-peptide or the corresponding non-phospho-peptide for 1 h at room temperature. When necessary, staining of primary antibody was detected using the biotin amplification procedure as described30. Briefly, tissue sections were transferred to biotinylated donkey anti-rabbit IgG  for 2 h, washed in PBS-T + /-s PPIs and then incubated in AB solution  for 60 min, washed again in PBS-T + /- PPIs and transferred to biotinylated tyramine (BT) solution  for 20 min, followed by a final incubation step in streptavidin-AlexaFluor555 conjugate 1:400 in PBS + 0.3% Triton X-100 containing 10% NDS +/− PPI) overnight at 4 °C. For double-staining procedures, the primary np-AB was incubated overnight as indicated and after several washing steps on the following day conjugated to the secondary AB, again overnight. Sections were then mounted onto SuperFrost Plus glass slides (ThermoFisher 15438060) and cover slipped with Eukitt (ORSAtec). Specimens were examined using a Zeiss LSM 900 laser scanning confocal microscope equipped with ZEN software for image analysis.Mice were deeply anaesthetized with isoflurane (CP-Pharma 4001404) and then subjected to a transcardial perfusion with calcium-free Tyrode’s solution containing protein phosphatase-inhibitors (+PPIs) (1 tablet PhosSTOP per 10 ml) (Roche 04906845001) followed by Zamboni’s fixative containing 4% paraformaldehyde and 0.2% picric acid in 0.1 M phosphate buffer pH 7.4 +PPIs. Brains and spinal cords were rapidly dissected and postfixed in the same fixative for 4 h at room temperature. The tissue was cryoprotected by immersion in 10% sucrose +PPIs followed by 30% sucrose +PPIs for 48 h at 4 °C before sectioning using a freezing microtome. Tissue was cut into 40 μm sections. Free-floating sections were washed in PBS with or without (+/-) PPIs (depending on the experiment and investigated GPCR) and incubated in methanol containing 0.3% HWildtyp mice were treated with agonist or vehicle. Mice were anesthetized with isoflurane, killed by cervical dislocation, and brains were quickly dissected, excluding the cerebellum. Brain samples were immediately frozen in liquid nitrogen. Brains were transferred to ice-cold detergent buffer , containing protease and phosphatase inhibitors), homogenized, and centrifuged at 14,000 × g for 30 min at 4 °C. The supernatant was then immunoprecipitated with the polyclonal goat np-CB1 antibody bound to protein A-agarose beads  for 90 min at 4 °C. Proteins were eluted from the beads with SDS sample buffer for 25 min at 43 °C and then resolved on 8% SDS-polyacrylamide gels. After electroblotting, membranes were incubated with pS425-MOP antibody, followed by detection using a chemiluminescence detection system. Blots were subsequently stripped and incubated again with the rabbit np-CB1 antibody to confirm equal loading of the gels.Further information on research design is available in the Peer Review FileSupplementary InformationDescription of Additional Supplementary FilesSupplementary Data 1Reporting Summary


Characterization of the zebrafish paralogous genes mutant lines generated by CRISPR/Cas9 approach showed that ambra1b knockout leads to an all-male population.AMBRA1 is an intrinsically disordered protein, working as a scaffold molecule to coordinate, by protein-protein interaction, many cellular processes, including autophagy, mitophagy, apoptosis and cell cycle progression. The zebrafish genome contains two ambra1b gene determines a reduction of primordial germ cells (PGCs), a condition that, in the zebrafish, leads to the development of all-male progeny. PGC reduction was confirmed by knockdown experiments and rescued by injection of ambra1b and human AMBRA1 mRNAs, but not ambra1a mRNA. Moreover, PGC loss was not rescued by injection with human AMBRA1 mRNA mutated in the CUL4-DDB1 binding region, thus suggesting that interaction with this complex is involved in PGC protection from loss. Results from zebrafish embryos injected with murine Stat3 mRNA and stat3 morpholino suggest that Ambra1b could indirectly regulate this protein through CUL4-DDB1 interaction. According to this, Ambra1+/− mice showed a reduced Stat3 expression in the ovary together with a low number of antral follicles and an increase of atretic follicles, indicating a function of Ambra1 in the ovary of mammals as well. Moreover, in agreement with the high expression of these genes in the testis and ovary, we found significant impairment of the reproductive process and pathological alterations, including tumors, mainly limited to the gonads.We demonstrated that the silencing of the ambra1a and ambra1b knockout zebrafish lines, we prove the sub-functionalization between the two paralogous zebrafish genes and uncover a novel function of Ambra1 in the protection from excessive PGC loss, which seems to require binding with the CUL4-DDB1 complex. Both genes seem to play a role in the regulation of reproductive physiology.By exploiting The online version contains supplementary material available at 10.1186/s40659-023-00430-9. Activating Molecule in Beclin-1-Regulated Autophagy Protein 1) is a multifunctional scaffold protein whose intrinsically disordered structure allows high protein-protein interaction plasticity, resulting in its involvement in a plethora of different and critical cellular pathways including autophagy, mitophagy, apoptosis and cell cycle progression [Ambra1+/− mice are characterized by pre-diabetic conditions [AMBRA1  knockdown [ambra1aia35  and ambra1bia36  mutants do not display overt developmental defects, due to the activation of genetic compensation mechanisms with up-regulation of the paralogous gene remaining active [ambra1a−/−/ambra1b−/− mutant line was not possible as double mutants cannot survive after larval stages [ambra1 genes is incompatible with life. Despite the lack of morphological differences between ambra1a−/− and ambra1b−/− fish lines, ambra1b homozygous mutants develop exclusively as males following the achievement of sexual maturity, whereas heterozygous adult ambra1b+/− and ambra1a mutants do not display sex ratio alterations [The zebrafish genome contains two nockdown –17. Sincl stages , 11, thiambra1a and ambra1b on sexual differentiation, we decided to investigate the role of Ambra1 in this process. Sex determination in Danio rerio domesticated lines is controlled by a polygenic system and environmental factors, such as temperature or social cues [Considering the different effects of ial cues . Zebrafiial cues .In addition, experimental reduction of primordial germ cell (PGC) number during the first day of development produces all-male progeny with normal testis, while the total ablation of PGCs generates males with somatic gonad cells organized as testis but devoid of sperm . Hence, ambra1b mutants relies on PGC reduction during early developmental stages. Remarkably, the PGC loss could be rescued with human AMBRA1 (hAMBRA1) mRNA, but not with the hAMBRA1 mRNA mutated in the CUL4-DDB1 complex binding domain, suggesting a possible interaction with this complex in regulating PGC development. Altogether, the high expression of Ambra1 proteins in zebrafish and mouse ovaries and the impairment of zebrafish reproductive capabilities unveil a new role of Ambra1 in reproductive physiology.In this study, we focused our attention on the possible role of Ambra1b on zebrafish sex determination and found that the all-male phenotype of ambra1a and ambra1b paralogous genes in zebrafish and the absence of female individuals in the ambra1b knockout (KO) line suggested a sub-functionalization of these paralogs in zebrafish after the ancestral genome duplication [Ambra1 in adult mouse brain [ambra1a in liver and of ambra1b in muscle. Furthermore, both paralogs displayed high expression levels in ovary and, although in smaller amounts, in testis, suggesting that both genes may play a role in zebrafish reproductive processes  and ambra1b−/−  individuals sacrificed at different ages, from 3 to 20 mpf (months post fertilization). WT  were also analysed for comparative purposes.We then assessed the occurrence of morphological alteration due to the genetic ablation of the two The main pathological signs displayed by the different groups of mutants were found in individuals older than 12 mpf and, in agreement with expression data, morphological changes were mainly located in the gonads in both males and females. Pathological findings include a plethora of different changes, ranging from testicular and ovarian degeneration to inflammation and testicular neoplasia. Other phenotypic features, not related to gonads, included mild hepatic degeneration with intracytoplasmic hepatocellular vacuolar changes and skeletal muscle defects, which were constantly present, albeit at different extents, in all the examined groups except for WT subjects. The main phenotypic features of the ovarian tissue were represented by degeneration of follicles, often associated with mineralization and with an occasional inflammatory chronic granulomatous reaction towards follicular components (mainly yolk) released in the coelomic cavity  and can thus be considered a “mutant escaper”. This individual  displayed the classic female phenotypic features and was able to reproduce with ambra1b−/− males in 2 out of 7 reproductive trials. The two reproductive events resulted in 53 adults, of which only 10 (19%) developed as females (generation 2). One individual of generation 2 was sacrificed at 16 mpf to perform histological analysis, which revealed the presence of all oocytes stages .Although the main feature of the ambra1a and ambra1b in ovaries and testes, the morphological alteration in gonads and the availability of ambra1b−/− females , we assessed the reproductive capabilities of ambra1a−/− and ambra1b−/− males and females. Towards this aim, we used four 10-mpf zebrafish from both mutant lines  and compared them to WT zebrafish of the same age. Every animal was bred with WT individuals. The analysis of male reproductive capabilities showed a reduction, although not significant, of ambra1b−/− male reproductive success compared to WT and ambra1a−/− males .Considering the high expression level of both les Fig. C. This cles Fig. D or offsles Fig. F. Moreovambra1b−/− females of generation 3 obtained as described above were used to assess the reproductive abilities of ambra1b−/− females in parallel with four ambra1a−/− females. Reproductive capabilities were reduced in both mutant lines, although not significantly .The three tly Fig. G. WT maltly Fig. H,J. Agaiambra1b−/− individuals, obtained from heterozygous ambra1b crosses, appear to be all males [ambra1b+/− and male ambra1b−/−. The progeny was raised to adulthood and their sex determined according to zebrafish sexually dimorphic phenotypic traits: body shape, dimorphic colour of the anal fin and abdomen, and appearance of the genital pore with the prominent genital papilla in females [2 test with a p-value < 0.001) and excluded the possibility of female lethality, since the ratio predicted in this case should be 1  ∶ 1  ∶ 0  ∶ 1 , leading to 66% heterozygotes and 33% homozygous mutants.As previously shown, ll males . To excl females . Among tambra1 lines. In zebrafish, juveniles develop a “juvenile or presumptive ovary”, a bipotential gonad that histologically resembles an immature ovary. Depending on multiple signals, this structure becomes a functional ovary in about 50% of individuals. In the remaining half of individuals, the juvenile ovary degenerates, and immature oocytes undergo apoptosis while gonads start developing as testis at around 30 dpf. Then, at the 35-dpf stage, the fate decision toward the male or female phenotype has normally already been taken [Knowing that zebrafish has a window of sex determination in which the sex is decided by a system that integrates the expression of different specific genes with environmental factors , we perfen taken .ambra1a−/−, ambra1b+/− and ambra1b−/−, as well as 6 WT at the same age and classified the gonads of these juveniles as undifferentiated (without any sign of differentiation towards one or the other sex), juvenile ovary, juvenile ovary to testis transition, and ovary  and ambra1b+/+  siblings , indicating that the control of PGC number cannot be only ascribed to the contribution of maternal ambra1b transcripts  or CUL4AMBRA1 reduces MB stem cell growth and migratory potential.A recent study has shown a relationship among AMBRA1, CUL4–DDB1 complex, and STAT3 in medulloblastoma (MB) stem cells, demonstrating that AMBRA1, through its direct interaction with the CUL4–DDB1 complex, is involved in the degradation of SOCS3, an inhibitor of STAT3 activity . In thisStat3 mRNA to ambra1b KO embryos, as well as by injecting a zebrafish stat3 ATG-MO to block the translation of maternal and zygotic stat3 transcripts in WT embryos. Remarkably, injection of murine Stat3 mRNA was able to completely recover the PGC loss of 10 hpf ambra1b KO embryos  at three and ten months of life, using ovaries from Ambra1+/− and Ambra1+/+ sibling mice  and ubiquitination of specific target proteins [Moreover, rescue experiments with the proteins . AMBRA1,proteins .C. elegans where it is involved in the replication of DNA [D. melanogaster [Interestingly, Cul4b, a member of the CUL4-DDB1 complex, initially studied in n of DNA , was founogaster . Moreovenogaster .The functions of this complex in mouse ovary physiology and oocyte survival have been analysed by means of oocyte-specific KO of DDB1, the linker between CUL4 and the DCAF proteins. The silencing of this protein, as well as deletion of DCAF1, determines oocyte loss and ovarian insufficiency . HoweverAMBRA1 binds CUL4-DDB1 complex and targets D-type cyclins for ubiquitin-mediated degradation, thus acting as a tumor suppressor and assuring normal cell cycle progression , 39. Whestat3 ATG-morpholino resulted in a significant reduction in PGC number at 10 hpf, suggesting the involvement of Stat3 in the regulation of PGC number in zebrafish. This hypothesis is also supported by the rescue of PGC number after injection with the murine Stat3 mRNA. To validate this interaction, we analysed Ambra1, Stat3 and Myc expression in the ovary of Ambra1+/+ and Ambra1+/− mice. Ambra1 was found highly expressed in WT ovaries and reduced in heterozygous samples. Interestingly, in heterozygous mouse ovaries the expression of Stat3 and Myc, was also reduced, although significantly only at three months of age for Stat3. Moreover, analysis of the different types of follicles showed a significant increase in atretic follicles and a reduction of the antral ones in Ambra1+/− mice at 10 months of age. Sex determination is driven by different mechanisms in mammals and in zebrafish. Indeed, murine sexual differentiation is genetically determined, whereas in zebrafish it is settled by a combination of genetic and environmental conditions. However, an excessive loss of germ cells is associated with reproductive problems in mammals [Knockdown experiments with a  mammals  and therambra1b−/− females derived from the single mutant escaper we found up to now, and found a significant impairment of the reproductive process in all cases. Notably, the administration of the probiotic L. rhamnosus to zebrafish females increases the expression of genes involved in the autophagic process, including ambra1b, and downregulates transcripts related to apoptosis, thus modulating the balance between the two processes and regulating ovarian functions [The high level of expression of both paralogs in the gonads suggests the involvement of these proteins in other aspects of reproductive physiology. In anticipation of more in-depth studies in this field, we analysed the reproductive performance of both sexes, taking advantage of the unctions .Finally, in agreement with the AMBRA1 role as a tumor suppressor, among the mutant males we analysed we found two cystic degenerations of the seminiferous tubules, a pre-cancer condition, two seminomas and an undifferentiated germ cell tumor. Although testicular lesions such as seminomas and cystic degeneration of seminiferous tubules with or without hyperplasia of spermatogones are common changes in adult zebrafish , seminomambra1a and ambra1b KO zebrafish lines, we were able to prove the sub-functionalization between these two genes and uncover a novel function of Ambra1 in the protection from excessive PGC loss, which seems to require binding with the CUL4-DDB1 complex. Since both Ambra1a and Ambra1b proteins contain the binding domain for CUL4-DDB1, but only Ambra1b is involved in PGC maintenance, it can be speculated that this interaction requires an additional functional domain that is present in the human AMBRA1 but not in the zebrafish Ambra1a isoform. Further work in the near future may allow to dissect the molecular basis of this interaction together with a more precise identification of all actors involved, as well as of the roles played by the two Ambra1 isoforms in ovarian and testicular physiology.In conclusion, by exploiting 4 and 5 g NaHCO3 for 1 L). All husbandry and experimental procedures complied with the Italian and European Legislation for the Protection of Animals used for Scientific Purposes (Directive 2010/63/EU) and were approved by the Animal Ethics Committee of the University of Padua and by the Italian Ministry of Health (Authorization Number 568/2016-PR). The experiments described in this paper were performed with wild-type (WT) zebrafish or with the ambra1a and ambra1b mutant lines whose generation has been previously described [ambra1a+/+/ambra1b+/+ correspond to animals deriving from different or the same batches of the knockouts, respectively. However, the genetic background of the WT fish was the same used for the generation of the ambra1a and ambra1b mutant lines.Zebrafish embryos, larvae, and adults were maintained according to standard procedures . Embryosescribed . Non-mutAmbra1 knockout (Ambra1+/–) mice [Ambra1+/– mice and the age-matched littermate controls Ambra1+/+ were sacrificed by cervical dislocation. Ovaries were dissected, isolated and immediately frozen in liquid nitrogen vapours or fixed in Bouin’s solution for further experiments.Heterozygous whole-body /–) mice  were houBiopsies from the caudal fin of larvae and adult fish were used for fin clips genomic DNA extraction performed with the HotSHOT protocol . For theAmbra1 knockout allele and Ambra1 wild-type allele [Mice genotyping was determined by PCR from digested ear biopsies with specific primers for e allele  . Samples were serially cut into 7–8 μm sections on an LKB microtome. After rehydration, sections were stained with hematoxylin and eosin and mounted with Eukitt  for microscopy examination.ambra1a or ambra1b transcripts (MO-ambra1a-ATG and MO-ambra1b-ATG) and with splice-blocking MOs designed at the exon 3/intron 3 junction sequence of both genes (MO-ambra1a-splice and MO-ambra1b-splice). All MOs were previously described and validated [Stat3 (MO-stat3-ATG) was also used, as previously described and validated [ambra1 MO, 8.2 ng were injected in the yolk of one-cell stage embryos whereas 10 ng were injected for the MO-stat3-ATG. Injections were performed under a dissecting microscope using a microinjector (Leica Microsystems).Morpholino (MO) (Gene Tools) treatment was performed with MOs against the ATG translation initiation sites of either alidated –17. A MOalidated . For eacThe number of PGCs was analysed in WT, knockout (KO), and knockdown (KD) 10-hpf zebrafish embryos by whole-mount immunohistochemistry with an anti-Vasa antibody   kindly pnos1-3’UTR [NotI restriction enzyme (Promega).PGCs were visualized with fluorescence microscopy by injection of mRNA produced using the construct GFP-s1-3’UTR , kindly AMBRA1 (hAMBRA1), as well as hAMBRA1 mutated in the PP2A, LC3, TRAF6 and CULLIN4 binding sites  were inserted in pCS2 + plasmids as stated in a previous paper [HindIII restriction enzyme (Promega). Zebrafish ambra1a1 and ambra1b were inserted in pCS2 + plasmids and transcribed using the SP6 promoter and the mMessage Machine kit, after plasmid linearization with NotI restriction enzyme (Promega). All mRNAs were polyadenylated at the 3’-termini using a Poly(A) Tailing Kit . The mRNAs were injected in one-cell stage embryos at the concentration of 80 pg per embryo for the GFP-nos1-3’UTR mRNA and 40 pg per embryo for the various hAMBRA1 mRNAs. AMBRA1 mutants in LC3, TRAF6, CUL4-DDB1 and PP2A binding sites were generated by using the site-directed mutagenesis kit (Agilent Technologies) respectively. The sequences used are as reported in [mStat3) was also prepared as described [For rescue experiments, cDNAs of human us paper  or produorted in . The murescribed  and injehAMBRA1 mRNA and hAMBRA1-RFP-sspB mRNA [Pools of 50 WT embryos (24 hpf) were injected with spB mRNA , 17, deyGapdh (mice samples) and actb2 (zebrafish samples) mRNAs were used as normalizers. The primer sequences are reported in the Additional file Total RNA was extracted using Quick-RNA Miniprep kit (Zymo Research R1054) from single ovaries of 3- and 10-months-old WT and heterozygote mice, and from organs of 6 mpf WT zebrafish . RNA samples were kept at -80 °C until use. 5 µg of the total RNA obtained from mice ovaries were used for mRNA isolation with Dynabeads mRNA DIRECT Kit . All the mRNA obtained with this procedure as well as 1 µg of the total RNA obtained from zebrafish samples was used for cDNA synthesis, employing FIREScript Reverse Transcriptase Kit  and following the manufacturer’s protocol. Quantitative polymerase chain reaction (qPCR) was performed with SYBR green method on Biorad cfx384 (Biorad). The reaction conditions were as follows: enzyme activation at 95 °C for 15 min followed by 45 cycles of denaturation (30 s at 95 °C), annealing (30 s at 60 °C), and extension (20 s at 72 °C). Fluorescence monitoring occurred at the end of each cycle. in situ hybridization, samples were rehydrated through a graded series of methanol in PBT (PBS and 0.1% Tween-20), permeabilized with proteinase K for 30 min at room temperature and fixed again with PFA 4% in PBT for 30 min at room temperature. Fixation was stopped with several brief washes in PBT; then organs were prepared for in situ hybridization as previously described [ambra1a and ambra1b transcripts [Ovaries and testicles extracted from 6-mpf WT zebrafish were fixed overnight at 4 °C with PFA 4% in PBS added with 0.1% Tween-20 and 0.1% DMSO and then stored in 100% methanol at -20 °C until used. For escribed . Brieflynscripts  were detambra1a−/− and ambra1b−/− males and females was assessed by natural reproductions in spawning tanks under standard aquarium conditions (Westerfield 1995). Four 8-mpf animals were analysed for each sex and genotype. WT partners were randomly chosen each time between the batch of WT males and females (housed separately from the opposite sex). Animals underwent 4 consecutive reproductive rounds, once every 10 days, to ensure we had a realistic mean of the reproductive performances of each individual. We excluded a specific male-female interaction effect by performing repeated reproductive rounds for each animal coupled with different WT partners. At the end of each trial, spawned eggs were collected in a Petri dish with fish water, following the standard husbandry rules [The reproductive performance of ry rules . Reprodury rules .AMBRA1+/− heterozygous mice were scored for each age point (3 and 10 months). The scoring system was set up following the methods proposed by previous literature [An equal number of ovaries from WT and terature , 55.Briefly, the total follicle number was estimated by counting and classifying all the follicles in every 9th section of the ovary. Only follicles in which the nucleus was visible were included in the count. The total follicle number was then calculated by multiplying the raw counts per 9 , 55. CouFor imaging of PGCs at 10 and 24 hpf, embryos were anaesthetized with 0.04% tricaine and mounted in 2% methylcellulose on a depression slide. Images of PGCs at these embryonic stages were recorded with a Leica DMR using a Nikon DS- Fi2 digital camera. Three dpf embryos were instead embedded in 0.8% low-melting agarose and analysed with the Nikon C2 confocal system provided with the software NIS ELEMENTS.WMISH-stained organs, as well as immunohistochemistry embryos, were mounted in 80% glycerol in phosphate-buffered saline plus 0.1% Tween 20 , observed under a Leica M165 FC microscope, and photographed with a Nikon DS-Fi2 digital camera. Histological samples were photographed on a Leica DMR using a Nikon DS- Fi2 digital camera.Statistical analysis was performed with Graph Pad Prism V9.0.1. Data are presented as the means ± SEM. Comparisons between WT and KO/KD zebrafish/mice were performed with the tests reported on the figure captions. The p-values are indicated with the following symbols: *, p < 0.05; **, p < 0.01; ***, p < 0.001; ****, p < 0.0001.Below is the link to the electronic supplementary material.Supplementary Material 1Supplementary Material 2


In the realm of solid water hydrostructures, helium atoms have a tendency to occupy the interstitial spaces formed within the crystal lattice of ice structures. The primary objective of this study is to examine the stability of various ice crystals when influenced by the presence of He atoms. Presenting a first attempt at a detailed computational description of the whole energy components  in the complete crystal unit cells contributes to enhancing the knowledge available about these relatively unexplored helium–water systems, which could potentially benefit future experiments. For this purpose, two different ice structures were considered: the previously established He@ice II system, and the predicted (but currently nonexistent) He@ice XVII system. One of the main features of these He-filled structures is the stability conferred by the weak van der Waals dispersion forces that occur between the host lattice and the guest atoms, in addition to the hydrogen bonds established among the water molecules. Hence, it is crucial to accurately describe these interactions. Therefore, the first part of this research is devoted examining the performance and accuracy of various semi-local and non-local DFT/DFT-D functionals, in comparison with previous experimental and/or high-level computational data. Once the best-performing DFT functional has been identified, the stability of these empty and He-filled structures, including different number of He atoms within the lattices, is analysed in terms of their structural (lattice deformation), mechanical (pressure compression effects) and energetic properties (binding and saturation energies). In this manner, the potential formation of these structures under zero temperature and pressure conditions can be evaluated, while their maximum storage capacity is also determined. The obtained results reveal that, despite the weak underlying interactions, the He encapsulation has a rather notable effect on both lattice parameters and energetics, and therefore, the guest–host interactions are far from being negligible. Besides, both ice crystals are predicted to remain stable when filled with He atoms, with ice XVII exhibiting a higher capacity for accommodating a larger number of guest atoms within its interstitial spaces. Due to their small size and weak interaction with water molecules, helium atoms tend to occupy interstitial spaces within the crystal lattice of ice-like structures, such as hexagonal ice  and D4 schemes ,67,68,69Full geometry optimizations were performed through DFT/DFT-D computations on the empty and He-filled ice structures at various lattice parameters, Therefore, hereafter, we focus the benchmark analysis on r = 0.4816/0.4838  for In turn, with the aim of obtaining the equilibrium lattice constants, e (MEOS)   , respect12.92 Å) . In factMoving forward with the assessment, ice XVII . CuriousTo complete this analysis, we also discuss the Bab initio PW91 calculations [In turn, Lobban et al.  reportedulations , where B237.65 K , B0 = 14nd 200 K , and B0 nd 225 K . AccordiIn conclusion, based on all the aforementioned information, we consider that the exchange-correlation revPBE-D3(0), which is constrained by exact functional conditions with a posteriori incorporation of the D3 dispersion corrections, is the functional that provides a better description of the interactions in both empty and singly/multiply-filled ice structures, in line with previous high-level theoretical and experimental reports on ice polymorphs  and inteTo investigate the capacity of ice II and ice XVII structures and understand how the lattice deforms as the number of He atoms increases, different ice crystals with varying He-fillings are studied by carrying out revPBE-D3(0) calculations. In particular, in addition to the already discussed HeFollowing a similar procedure to that explored in the above-mentioned benchmark study, full geometry optimisation calculations were first carried out at revPBE-D3(0) level of theory for fixed values of r , within the same range of values. Subsequently, the computed optimized total energies were fitted to the MEOS, and The calculated lattice parameter aa et al. ), it seea et al. . Thus, aa et al.  at 2.8 kc-axis has a negligible effect on the crystal, the incorporation of additional He atoms seems to be less favourable, with a difference of ∼1 and 2 kcal/mol for HeFurther analysis of our equilibrium cohesive energies (per water molecule) also reveals the same conclusions. While the inclusion of one He between two hexagons along the pressure . This phIn contrast to ice II, we distinguish a less notorious He-filling effect in ice XVII, with the Heice XVII . Catti eice XVII  have comice XVII .Regarding comparisons of the equilibrium cohesive energies (per water molecule) of HeIn stem see . This suThe ice XVII structure presents wider channels, whatever the empty or He-filled system considered see . The aveat 50 K) .In turn, by using the calculated volume and bulk moduli, the pressure exerted on the ice lattices can be calculated according to the MEOS expression in terms of pressure as a function of the volume, such that The stability of these systems can be evaluated in terms of energy. To this end, the total binding energies of all the He-filled ices with respect to the guest-free systems are calculated, at the respective equilibrium lattice constants, aFurther analysis of the stability entails the computation of the saturation energies for all He-filled structures, as illustrated in In general, considering the energetic and structural evaluation of both empty and He-filled ice II and ice XVII crystals, it seems that the ice II only may allow the inclusion of one He atom per interstitial space, while the ice XVII could accommodate up to two He atoms per void. It is important to consider that the present findings are applicable only under zero temperature and pressure conditions. According to the phase diagram of water, the ice XVII structure is situated in close proximity to these conditions. Hence, it is understandable that our calculations predict that the formation of He-filled ice II, which occurs under extremely high-pressure conditions, is less favourable.This research focused on examining the behavior of crystalline ice structures confining an increasing number of He atoms. We have obtained valuable first-principles insights into the stability of two specific systems: the previously synthesized He@ice II, and a recent hypothetical He@ice XVII structure. First of all, we have conducted a benchmark study, including various semi-local dispersion-corrected GGA and non-local van der Waals-inclusive DFT-D functional approaches, in order to identify the most appropriate one that adequately describes the hydrogen bonds and van der Waals interactions inherent in these systems. Our findings indicate that the revPBE-D3(0) approach yields remarkably accurate results in comparison with available theoretical and/or experimental data. Benchmark studies on the performance of different DFT approaches can assist in resolving controversial issues and/or guide the development of novel/improved functionals.In turn, the stability of different configurations of ice II and ice XVII structures with varying numbers of trapped He atoms within their channels was evaluated. The analysis primarily focused on examining structural, mechanical, and energetic properties, such as equilibrium lattice constants, resistance to compression, and binding/saturation energies. The presence of He atoms within the interstitial spaces of these systems has been observed to have a noteworthy impact. On the one hand, the ice II structure only accepts the inclusion of only one He atom per void, leading to minimal stabilization of the structure. The introduction of additional He atoms results in highly unstable configurations, causing even deformation of the lattice structure. On the other hand, the larger diameter of ice XVII allows for inclusion of up to two He atoms per interstitial space. However, the confinement of extra guest atoms produces a destabilizing effect. Moreover, in contrast to ice II, the addition of just one He atom exerts a significant stabilizing effect on the lattice.Overall, while both ice II and ice XVII structures are predicted to exhibit stability in the presence of He atoms, ice XVII demonstrates more stable configurations under zero temperature and pressure conditions. This observation aligns with the expected behavior in the phase diagram of solid water.To further advance this line of investigation, additional studies can focus on several aspects. Vibrational properties can be explored by conducting phonon calculations to analyze the behavior of lattice vibrations within the He-confined ice structures. Furthermore, examining the response of these systems to variations in temperature and pressure would provide valuable insights into their thermodynamic stability and phase transitions. Molecular dynamics simulations can be employed to investigate the diffusion behavior of helium atoms within the channels, shedding light on their mobility and migration pathways. Understanding the stabilising role of He atoms in these solid hydrostructures holds great potential for the discovery of new low-density ices and materials.


To the Editor,n = 13) or without cross‐reactive food allergy to apple (group 2: n = 12)  were enrolled  where it is hypothesized that IgE is produced only locally but does not reach the systemic circulation.ed Table  and inveed Table . Two patle group : n = 12 The two patients were young females who suffered from birch pollen‐related allergic rhinoconjunctivitis and one (patient 1) suffered also from birch pollen‐induced asthma Table . ImportaAn increase of Bet v 1‐specific IgE levels was observed in both patients 1 and 2 at time point 2 shortly after the birch pollen season Figures , which wWe could not detect at any of the three time points Mal d 1‐specific IgE in the sera of patients 1 and 2 Table  but we fThe increase of Mal d 1‐specific basophil sensitivity was associated with marked increased apple‐induced skin responses at time points 2 and 3 after the birch pollen season as compared to time point 1 before the birch pollen season Figures . By contUsing purified allergen molecules, our study has demonstrated that basophils and mast cells of allergic patients contain already allergen‐specific IgE before it can be detected in serum in the form of free‐allergen‐specific antibodies. This finding is different from LAR regarding the underlying mechanism because it is not a result of local IgE production without spill‐over into the systemic circulation. Furthermore, it is important because it demonstrates that allergen‐specific IgE sensitization may already occur before allergen‐specific IgE can be detected as free IgE in serum because newly produced IgE may be sequentially adsorbed by IgE receptors on cells Figure . The lacWe have studied IgE levels and allergen‐specific and IgE‐mediated effector cell activation using Mal d 1 that cross‐reacts with the major birch pollen allergen Bet v 1. This model is representative for the most common allergens, which cross‐react to a variable extent with homologous proteins from other allergen sources and isoallergens from the same allergen source.Our results are supported by an earlier study, which has demonstrated that after administration of omalizumab, a monoclonal anti‐IgE antibody, which prevents allergen‐specific IgE from binding to the high and low affinity IgE receptor on cells, IgE sensitizations to allergen sources became detectable in serum, which were undetectable before administration of omalizumab.Our results are important for diagnosis of IgE‐mediated allergic sensitization in general because they reveal that a certain threshold of allergen‐specific IgE levels must be reached, which exceeds the capacity of cellular IgE receptors in the body to bind IgE before it can become detectable as free IgE in serum. In this context, it has to be borne in mind that the expression of cellular IgE receptors is regulated by the IgE levelsOur findings are also important when it comes to the investigation of the inception  of allergic sensitization in children because they indicate that early allergen‐specific IgE sensitization may not be accurately detected by serology. This may apply for subjects with low allergen‐specific IgE levels who are sensitized only to few allergen molecules. In fact, monitoring of allergen‐specific IgE to multiple allergen molecules in birth cohorts indeed showed that children are often sensitized only to few dominant allergen molecules in the beginning of sensitization.The strength of our study is that it was conducted with highly purified allergen molecules and hence delivered unambiguous results because they demonstrate the presence of allergen‐specific IgE on basophils and in the tissues of allergic patients without detectable allergen‐specific IgE in serum.RV, AL, AN, MK, DK, OE designed the research studies. AL, ES, OE, EF, NI performed clinical work. AL, MB, ES, AN, OA, RC performed experiments. RV, AL, MB, ES analyzed and interpreted the data. AL wrote the manuscript with contributions from RV and ES.Rudolf Valenta has received research grants from Viravaxx AG, Vienna, Austria, HVD Biotech, Vienna, Austria and Worg Pharmaceuticals, Hangzhou, China. He serves as a consultant for Viravaxx and Worg. The other authors have not conflicts of interest to declare.https://publons.com/publon/10.1111/pai.13891.The peer review history for this article is available at Appendix S1Click here for additional data file.FigureS1Click here for additional data file.


Acer truncatum resources. However, research on one of its by-products, namely seed oil residue (ASR), from seed oil extraction is seriously insufficient, resulting in a waste of these precious resources. We aimed to optimize the conditions of ultrasound-assisted extraction (UAE) using a response surface methodology to obtain high acetylcholinesterase (AChE) inhibitory components from ASR and to tentatively identify the active metabolites in ASR using non-targeted metabolomics. Based on the results of the independent variables test, the interaction effects of three key extracting variables, including methanol concentration, ultrasonic time, and material-to-liquid ratio, were further investigated using the Box–Behnken design (BBD) to obtain prior active components with high AChE inhibitory activity. UPLC-QTOF-MS combined with a multivariate method was used to analyze the metabolites in ASR and investigate the causes of activity differences. Based on the current study, the optimal conditions for UAE were as follows: methanol concentration of 85.06%, ultrasonic time of 39.1 min, and material-to-liquid ratio of 1.06:10 (g/mL). Under these optimal conditions, the obtained extracts show strong inhibitions against AChE with half maximal inhibitory concentration (IC50) values ranging from 0.375 to 0.459 µg/mL according to an Ellman’s method evaluation. Furthermore, 55 metabolites were identified from the ASR extracted using methanol in different concentrations, and 9 biomarkers were subsequently identified as potential compounds responsible for the observed AChE inhibition. The active extracts have potential to be used for the development of functional foods with positive effects on Alzheimer’s disease owing to their high AChE inhibition activity. Altogether, this study provides insights into promoting the comprehensive utilization of A. truncatum resources.The State Council of China has called for the comprehensive development and utilization of  Acer truncatum, a perennial deciduous arbor from the genus Acer (Sapindaceae), is native to China and has been widely cultivated around the world for its ornamental, ecological, edible, and nutritional values oxyphenyl]-2,3-dihydrochromen-4-one and D-Tryptophan, the other seven discriminative metabolites have been reported to display inhibitory effects on AChE or anti-AD. Studies have shown that ferulic acid was the most common active component for the inhibition of Aβ aggregation, and the protection of neurons from oxidative damage, but its interaction with AChE was not significant (<20% inhibition of AChE at 20 mM) oxyphenyl]-2,3-dihydrochromen-4-one are potential biomarkers. The extracts from ASR obtained through UAE could be used for the development of functional foods with positive effects on Alzheimer’s disease.The experimental results indicated that the optimized conditions for UAE were effective in extracting AChE inhibitory components from ASR. The extracts with the IC


Charge regulation is fundamental in most chemical, geochemical,and biochemical systems. Various mineral surfaces and proteins arewell-known to change their charge state as a function of the activityof the hydronium ions, that is, the pH. Besides being modulated bythe pH, the charge state is sensitive to salt concentration and compositiondue to screening and ion correlations. Given the importance of electrostaticinteractions, a reliable and straightforward theory of charge regulationwould be of utmost importance. This Article presents a theory thataccounts for salt screening, site, and ion correlations. Our approachshows an impeccable agreement as compared to Monte Carlo simulationsand experiments of 1:1 and 2:1 salts. We furthermore disentangle therelative importance of site–site, ion–ion, and ion–sitecorrelations. Contrary to previous claims, we find that ion–sitecorrelations for the studied cases are subdominant to the two othercorrelation terms. The electrostatic charge is one key factor forstabilizing colloidal suspensions, as they would otherwise aggregatedue to attractive van der Waals forces unless other protective interactionsare at play . The praisedDLVO-theory,4 including attractive van der Waalsforces and repulsive electrostatic forces, helps us to predict colloidalstability between equal and simple colloidal particles. It successfullypredicts a lowered stability at higher salt concentrations as stabilizingelectrostatic forces are screened out.Surface dissociation or association ofhydronium ions in aqueoussolutions is very important for many chemical and biological systems.Immersed nanoparticles  simulations,10 both of which account for bothpH and salt concentration dependence.10 Unlike MC simulations, the Poisson–Boltzmann theory is well-knownto be insufficient as soon as the counterions of the charged surfaceare multivalent  due to a lack of ion–ioncorrelations in the mean-field theory. Moreover, Poisson–Boltzmanncalculations generally do not account for the discreteness of thesurface sites, even though Bakhshandeh et al.11 have recently accounted for the latter by showing an excellent agreementbetween theory and MC simulations of charge regulating nanoparticlesin the presence of 1:1 salts.The most straightforwardapproach to predict a surface’scharge, as needed for the DLVO theory, uses the Henderson–Hasselbalchformula,2 and previous MC simulations,10 within the primitive model, both in 1:1 and in 2:1 saltsolutions and with excellent agreement. Moreover, we disentangle thevarious contributions to the overall titration behavior and find thatsite–site correlations are important for all of the studiedcases. Ion–ion correlations are, as expected, only relevantat high electrostatic coupling parameters, that is, when the excesscounterion concentration is high close to the surface due to a highsurface charge density and when these counterions are multivalent.In contrast to previous claims,10 we findthat the ion–site correlations are generally subdominant.Here, we extend this work, accountingfor ion correlations, witha straightforward yet highly accurate theory for planar surfaces.We compare our theoretical predictions to available experimental data15 specifies thatlB is theBjerrum length, σ is the surface charge density (in inversearea units), e is the elementary charge, ρibulk and ρis are the bulk reservoir and surface densitiesof species i, zi is the valency of species i, β ≡1/kBT is the inversetemperature, and ϕion is the electrostatic potentialof an ion sitting on the surface. In 2 ≥ 1 parameter,which is not normally included in the contact theorem. This parameteraccounts for ion–site correlations, which give rise to additionalforces over the interface as compared to when surface sites are uniformlysmeared-out. These extra correlations occur when surfaces bear discretecharges and when the electrostatic couplings start to be important16 but also appear within the Poisson–Boltzmanntheory (low-coupling regime) for charge-modulated surfaces.17 For uniformly smeared-out surface charges  or wheneverion–site correlations are negligible, we have ξ2 = 1. Measuring the contact density, as on the right side of lBσ2 is hencean excellent way of quantifying the ion–site correlations.In general, ξ2 is small whenever the electrostaticcoupling parameter is low and/or when the ion–site separationis large as compared to the neighboring site–site distanceof the charged sites.Instead of solving the full nonlinearPoisson–Boltzmannequation, we rely on just solving the boundary condition, assuminga flat impenetrable titratable surface. The (surface charge modulated)contact theorema3O+H is traditionally given in logarithmicunits pH = −log10a3O+H andthe acid equilibrium constant pKa = −log10Ka of the reaction −OH+ H2O(l) ⇌ −O– + H3O+(aq). In most cases, one assumes thatthe activity of the hydronium ion is close to its ideal/entropic part,that is, its concentration, by a3O+H = [H3O+], where [H3O+]  isthe concentration in units of mol/L. Ka is then given in the same units. Our pKa is assumed to be an intrinsic value, independent of salt types andtheir concentrations, as it is based on the individual species’activities, which naturally include electrostatic contributions. Thisdiffers from the sometimes used “apparent” pKaapp values, which assume that activities are only given by the individualspecies concentrations, neglecting any energetic contribution to theactivities. Such a quantity would naturally be heavily salt-dependent,and the use of such should be discouraged.The hydronium activity –) and ϕsite is the electrostatic potential on a surface group. Thesurface charge density is given by σ = −ασsite, where σsite is the surface site density. The ionization degreeof the surface charge density α isgiven by the simple equilibrium relationship:ion = ϕmf + ϕii + ϕex, where ϕmf is the mean-fieldplane-averaged electrostatic potential at the surface . This gives rise to a potential contribution asϕcap = eσ/CS, where CS = ε0εr/dis, dis is the closest separation, perpendicularto the surface, between the charges of an ion and a site, and ε0εr is the absolute dielectric permittivity.We approximate the site–site correlations asMsq ≈1.949 is the Madelung constant for a square lattice.18 The choice is motivated by the site structure used in theMC simulations. This term corresponds to a 2D square lattice crystalenergy where the charges are at a maximum distance away from eachother on the lattice. The 0 < ζ < 1 parameter accountsfor the fact that sites do not correlate perfectly for nonfully occupiedlattices at room temperature, essentially incorporating the entropyeffects of lattice gas. This parameter is one out of two unknownsin the current theory. However, as we will see later, it can be fine-tunedfor one salt case and then kept constant for all other cases.We decompose the electrostatic potentialon a salt ion at the surfaceas ϕface see 1, ϕii is ϕmf see 1 . The c19 as βeϕii = −c0Ξ/zs, where c0 = (π/8 – 0.3104) ≈ 0.08230, and whereΞ is an average coupling parameter, and zs is the average charge of the ions residing at the chargedsurface. However, the ion–ion correlation energy, assumingperfect neutralization, cannot exceed a corresponding Madelung energy,where all counterions are bound to the surface and form a square orhexagonal lattice. Assuming that the ions form a 2D hexagonal layer,being the lowest energy state of the two-dimensional single-speciesionic crystal, the energy per charged site is given by Mhex ≈1.961.18 Note that the Madelung constantsof a square, Msq = 1.949, and hexagonallattice, Mhex = 1.961, differ only byless than 1%; hence, an exact knowledge of which lattice is preferredis not essential for the current theory. To incorporate this upperenergy limit, we use a modified ion–ion correlation:The ion–ion correlations are approximatedvia a first loopcorrection to the mean-fieldThe average valency of the ions residing atthe interface is givenbylB2σzs3.For highly charged surfaces, one obtainsthe valency of the counterion,but it is considerably lower for lower surface charges. In the limitof zero surface charge, the average valency equals zero, as it should.The motivation behind using the contact valencies is that the ion–ioncorrelations are the most important there . Following refs  and 21)21), we ddii isthe closest separation between two ions and is similar in spirit tothe hole corrected Debye–Hückel theory.23 This term, linear in σ, resembles a parallel capacitor correction.While there might be a coupling/overlap between the ion–ioncorrelation and excluded volume, in this theory, we consider themindependent of each other (see the Supporting Information for the effect of excluding this term).The excluded volume correction to the electrostatic potentialofthe ions can be approximated by mf, butthis quantity is redundant as it appears in both 2  and then guessingan initial value for σ. This guess provides us with values forϕex and ϕcap. From ion, which can be used to determine zs andΞ, and hence ϕii. By eliminating ϕmf, we find ϕsite = (ϕion – ϕii – ϕex) + ϕcap + ϕss. For a given pH value, we use site is obtained. This procedure is then iterated until σdoes not vary more between single iterations. Note that, in this procedure,it is very easy to “turn off” one or several of theterms by simply putting the corresponding potential(s) to zero.Theonly undefined quantity is now the ϕKa values,etc.) as the corresponding Monte Carlo simulations.10 These parameters have been shown to successfully reproducethe experimental data of ref (We study the exact same systemsas in refs  and 10)10). To ma of ref  when usilB = 0.714 nm corresponding topure water at room temperature, σsite = 4.8 sites/nm2 (corresponding to a maximum surface charge density of −770mC/m2), pKa = 7.7, dis = 0.35 nm, and dii = 0.4 nm. These values give CS = 1.93F/m2, which is lower than the value CS = 2.6 F/m2 used in ref (Ceff = (CS–1 + Cex–1)–1 = 2.67 F/m2, a value very close to the empirical fittedvalue of ref , the only freeparameter has been ζ. This parameter does, however, only mildlyaffect the ionization curves, as the case ζ = 1 shows . Overall, the theory with site–siteand ion–ion correlations incorporated and MC simulations agreeswell with the experimental data. Combining a) useduniformly smeared-out site charges and (d) accounted for site discreteness.Comparing our theory (solid lines) to their numerical data (dashedor dotted lines) shows an impeccable agreement, with lines being almostindistinguishable, using the same level of approximation (uniformvs discrete). In our theory, assuming that ions–site correlationsare negligible  wouldpredict the same titration curve for the two salt cases. The differencebetween the two corresponding theory lines in 2 for the 2:1 salt. Including also ion–ion correlation effectsinto the theory improves the accuracy for higher surface charge densities,as seen from zs3 is roughly a factor of 8 bigger than the 1:1 salt case at a comparablesurface charge density, explaining the larger importance of ion–ioncorrelations in the former but not the latter. Their combined effectgives an increase of up to 75% at the highest pH values, that is,larger than the sum of the individual contributions. Hence, the twocorrelations work in a synergetic manner.We continue testing ourtheory for 2:1 salts. Again, we neglectany ion–site correlation, setting ξAgain, we can compare to MC simulations with thesame level ofapproximation the (c) uniformly charged surfaces and (d) discretesurface sites for 2:1 salts and find excellent agreement. This indicatesthat each correlation term is well approximated.2 values. ξ2 depends nonlinearly on both Ξand the relative site–ion separation 2 = 1.1), whichshould indicate that we have ξ2 = 1 in our case basedon both our lower coupling parameter and our higher relative ion–siteseparation. Lowering Ξ to ≃1 in our case gives us 2 =1 as Moreira and Netz found ξ2 = 1 for Ξ =1 and 2 can be considered to be equalto 1, which was also assumed in the above analysis, even though thereis a small possibility that an open system behaves slightly differently. Hence,tly for the 1000 mM 2:1 salt case, we can reproducethe MC simulations’ data by using ξ2 = 3 . This is, however, a value much higherthan ref , wedeta of ref ), ξ2 redthan ref  wouldsuNote that, in the above arguments, we used our averagecouplingparameter. However, whenever the average coupling parameter is high,this is near-equal to the standard counterion-only coupling parameter.This is also the regime where ion–site correlations could beimportant. In fact, the average coupling parameter essentially onlyregularizes the low surface charge densities at low pH values, beingthe same as the standard coupling parameter at higher surface chargedensities, corresponding to high pH values.dis, but it would simultaneouslyalso affect the capacitance term. Hence, the outcome is not straightforwardto predict. A lowered dis value wouldcorrespond to neglecting the ions’ and sites’ hydrationlayers, something that is not standardly done within the primitivemodel.Are then ion–sitecorrelations never important? No, theybecome important as one increases the valency of the counterions  or reduces the minimum separation between thesite charges and counterions. This can be easily done in MC simulationsby lowering 10 also exploredthe effectof surface charge density as two surfaces approach each other andfound that the curve was nonmonotonic for 2:1 salts and high pH values.For finite distances, one has a finite pressure βP. According to the contact theorem,15 this nonzero pressure leads toa correction term on the right side of Labbez et al.site, and for the CR condition neither are constant, and σand ϕsite instead related to each other via 24 By definition, the CC condition yields constantsurface charge densities for all separations. However, the surfacecharge densities for the CP condition would depend on the pressure.For a 1:1 salt, the pressure increases monotonically as the separationdecreases; hence, the surface charge density decreases monotonicallyas a function of decreasing separation. However, for a 2:1 salt, thereexist regimes where the pressure is attractive due to ion–ioncorrelations.25 At these separations, thesurface charge density would instead increase, according to This equation is valid for all threestandard boundary conditions(acting on the surfaces): constant charge (CC), constant potential(CP), and charge regulating (CR). CC corresponds to an insulator,CC to a conducting surface (electrode), and CR to a surface with ionizablegroups. The difference between them is what is kept constant and whatis allowed to vary as a function of surface-to-surface separation.For the CC condition, σ is constant, for the CP condition ϕWe have presented a new and simple theoryfor the titration behaviorof charged planar surfaces, including site–site and ion–ioncorrelations. Our approach is in excellent agreement as compared bothto experiments and to Monte Carlo simulations. The results show thatsite–site correlations always are at play, being one of thedominant correction terms. In contrast, the ion–ion correlationsare first significant for divalent salts and high surface charge densities,for example, high effective coupling strengths. Including both site–siteand ion–ion correlations increases the ionization degree by30% for 1:1 salts and by 75% for 2:1 salts. We find that ion–sitecorrelations can largely be neglected, at least for the correspondingstudied MC cases.29 Further work will include excludedvolume effects, ion-specific effects, cases where ion–sitecorrelations are important, and detailed investigations of the densityprofiles close to the surfaces.30This work generalizes the ion–ion correlationcorrectionsapproach for salts, relying on an effective coupling parameter, differentand complementary to the dressed ion approach.


Dimensionality reduction techniques are crucial for enabling deep learning driven quantitative structure-activity relationship (QSAR) models to navigate higher dimensional toxicological spaces, however the use of specific techniques is often arbitrary and poorly explored. Six dimensionality techniques (both linear and non-linear) were hence applied to a higher dimensionality mutagenicity dataset and compared in their ability to power a simple deep learning driven QSAR model, following grid searches for optimal hyperparameter values. It was found that comparatively simpler linear techniques, such as principal component analysis (PCA), were sufficient for enabling optimal QSAR model performances, which indicated that the original dataset was at least approximately linearly separable (in accordance with Cover’s theorem). However certain non-linear techniques such as kernel PCA and autoencoders performed at closely comparable levels, while  being more widely applicable to potentially non-linearly separable datasets. Analysis of the chemical space, in terms of XLogP and molecular weight, uncovered that the vast majority of testing data occurred within the defined applicability domain, as well as that certain regions were measurably more problematic and antagonised performances. It was however indicated that certain dimensionality reduction techniques were able to facilitate uniquely beneficial navigations of the chemical space. The pos utilise .QSAR modelling relies on the premise that molecular structures are correlated with corresponding biological activities ; hence, in-vitro Ames mutagenicity tests [Mutagenicity concerns the ability of molecules to induce genetic mutations ; this enty tests . QSAR moty tests . Deep nety tests , while aty tests . 4 [2 order of magnitude, which enabled overall accuracy scores of ~70%  and ~78%  [Although DNNs have demonstrable advantages for QSAR modelling, naturally high dimensionality may arise in feature spaces built from chemical descriptors, hence deep learning based QSAR model performances may be impaired by “the curse of dimensionality”, where computational cost for a sufficiently complex model scales unfeasibly with increased dimensionality . This ma4 . Princip vector) ,10,11.The aim of this study was therefore to explore, optimise and directly compare the performance of a diverse variety of dimensionality reduction techniques (both linear and non-linear) for enabling deep learning QSAR models of mutagenicity to navigate higher dimensional toxicological/chemical space as effectively as possible. N binary labelled datapoints situated in a D-dimensional space, there is a high likelihood for the data to be linearly separable if N ≤ D + 1  [D increases [approximately linearly separable, with only a comparatively small number of datapoints mishandled by a linear decision boundary.Cover’s theorem statistically shows that for any as DN→∞) . While cncreases . It is sncreases . It shouN of datapoints to be used in this study will be equal to the dataset dimensionality D, however the number of training datapoints in each case will be lower . It is hence hypothesised that, in accordance to Cover’s theorem, linear dimensionality reduction techniques will be sufficient for enabling optimal performance of the deep learning driven QSAR models of mutagenicity trained and tested within this study, but that nonetheless certain non-linear dimensionality reduction techniques will perform equivalently well, while posing as more generally applicable techniques to other datasets where N > D + 1 or otherwise where non-linearly separable distributions may be anticipated to occur.The total number k-fold cross validation. k-fold cross validation was deemed to be a suitable technique for use in this study, given that it is widely applied in similar QSAR modelling studies, as well as that it may enable insight into the entirety of the sampled dataset, avoiding universal exclusion of chosen testing data from training the model [The 2014 AQICP dataset was used, following curation measures such as cross-referencing canonical SMILES (simplified molecular-input line-entry system) and CAS Registry Number descriptors via the online chemical database PubChem , as wellhe model . 5 folds2) TCs, with each row as a feature vector  for each molecule as a sample; this may be visualised below in Feature engineering took place, for the purpose of adequately quantifying molecular structures, via a SC-based approach closely comparable to that used in our previous research . A matriThe dimensionality reduction techniques explored in this study were PCA, kernel PCA (kPCA), independent component analysis (ICA), autoencoders, LLE and isomap. PCA and the version of ICA used in this study are linear dimensionality reduction techniques, whereas kPCA, autoencoders, LLE and isomap are non-linear methods ,19. PCA is a statistical method that attempts to transform higher dimensional data into a lower dimensional space, according to explained variance along different orthogonal axes . kPCA foAll algorithms, with the exception of autoencoders, were implemented via available functions of the open-source Python ML library scikit-learn . AutoencBefore drawing a direct comparison between these dimensionality reduction techniques, it was first imperative to optimise their hyperparameters, to enable a fair and valid comparison . While sPCA, as a comparatively simpler technique and with already demonstrated effectiveness via default values in our previous research (which used the same dataset and closely comparable feature engineering), was neglected from grid search efforts; default values for more trivial hyperparameters were simply used ,20,24.kPCA was similarly not considered for a grid search, although 2 different kernels were used for comparison purposes: an RBF function and a sigmoid function. Aside from this, default values for more trivial hyperparameters were used .The FastICA algorithm was used for ICA, with whitening strategy varied between use of arbitrary variance (default) and use of unit variance , as well as maximum number of allowed iterations of the algorithm linearly varied between 200 (default) and 1000, with steps of size 100. Aside from this, default values were used for other hyperparameters .Autoencoders were configured using the sigmoid activation function in encoder neurons, whereas the ReLU activation function was used in the decoder neurons. The Adam optimisation algorithm was used for stochastic optimisation, along with mean squared error to compute losses. Shuffling of data between epochs was enabled, whereas the number of epochs was varied geometrically between 10 and 1280 (doubling with each step). The number of layers in the encoder and decoder were kept equal and varied via “number of steps to bottleneck” which referred to the number of layers before and after the bottleneck layer, for the encoder and decoder respectively; this number was varied linearly between 1  and 5 . For each given value for number of steps to bottleneck, the size of the first and last autoencoder layer would always be 11,268 , whereas the size of the bottleneck layer would be that of the chosen latent space, however the sizes of the layers between was configured as a geometric series—e.g., for the decoder layers to start from the latent space size and reach 11,268 within the chosen number of steps, maintaining the same common ratio each time. This common ratio naturally changed for different values for number of steps to bottleneck layer and hence was recalculated for these different values. Each common ratio was equivalently used for configuring the sizes of encoder layers, except via division rather than multiplication (as the geometric series would operate in reverse). Any decimal values for layer sizes obtained via these geometric series were simply rounded to the nearest whole number.−6 and 10−2 , with a common ratio of 10. The number of neighbours considered was varied linearly between 5 (default) and 115, with step sizes of 10. Aside from this, default values were used for other hyperparameters [LLE was varied in terms of strength of regularisation constant and number of neighbours considered for each point. The strength of the regularisation constant was varied geometrically between 10rameters .Isomap was varied in terms of number of neighbours considered for each point, as well as eigenvalue decomposition algorithm. The number of neighbours considered was varied in an identical manner to that of LLE, whereas eigenvalue decomposition algorithm was varied between Arnoldi decomposition and the LAPACK  solver . It shouIn all cases, the algorithms (at each state of chosen hyperparameters) were fitted to training data  and then used to transform testing data (a matrix feature vectors from the remaining fold and the permanent testing data pool). This was repeated, using a different fold for the testing data each time, giving a total of 5 iterations. The reduced dimensional space was controlled at 100 dimensions in all cases, for the purposes of the grid search stage, to enable a suitably simple and valid search for hyperparameter value combinations. It is further noted that a reduced dimensionality of 100 (achieved via PCA) was found to be sufficient in our previous research . FollowiAdopting the optimal hyperparameter values from the grid search stage, dimensionality reduction techniques were compared. Final dimensionality was varied over the following values: 2, 4, 8, 10, 20, 50, 100, 200, 300. These values were chosen to cover a sufficiently wide search space that would be likely to demonstrate convergence in performance, to a sufficiently large reduced dimensionality for conserving information. Upon obtainment of transformed feature spaces over all algorithms, iterations and dimensionalities, comparison was enabled via training and testing of identical MLP classifiers as described in k-fold iteration. Any molecules from the testing dataset with  coordinates outside of this region, were deemed as existing outside the AD.There are numerous means for defining the applicability domain (AD), to characterise the applicability of QSAR models . In thisFrom the results of the grid search, it was deemed that for ICA, arbitrary variance performed marginally better than unit variance in terms of overall accuracy scores, with the most optimal scores occurring between 700 and 900 maximum iterations, although this was possibly a mere statistical fluctuation in otherwise considerably uniform data a. UncertFor autoencoders, a tendency for improved model performance was observed as hyperparameter states progressed to the lower-right corner of the heat map, i.e., minimal autoencoder complexity and larger numbers of epochs were optimal a. Affirm−6, whereas the number of neighbours would be controlled at 115 .For LLE, a general trend was uncovered, of more optimal performances for increased numbers of neighbours and decreased regularisation constants a. This wThe results for the grid search for isomap shows negligible variation across varied numbers of neighbours considered, albeit with a slight improvement in average performance as numbers were increased beyond 20 a,b. No vOverall, the grid search stage of this study was of mixed success. Clear and coherent optimisations were reached for autoencoders  and LLE a priori deduction, for such a complex dataset containing 11,268 dimensions; instead a level of starting assumption was necessary. Nonetheless, a potential future research direction could be to expand considered hyperparameters, their respective search spaces and respective levels of resolution, based on the outcomes of an initial grid search . The uneven and arbitrary nature of the resolutions between different grid searches could also be more coherently standardised via a clear methodology, for a future study. Additionally, the inherent limitations of grid searches as a hyperparameter optimisation method  could be addressed via choosing another optimisation approach such as random searches and Bayesian optimisation, perhaps to be ran in parallel to grid searches, followed by a final comparison of optimised hyperparameters found [Overall, grid search offers an appropriate approach for this study, as it would not have been feasible to choose hyperparameters, search spaces and required levels of resolution through any exact theoretical The grid search stage of the study provides an early level of insight into the feasibility of the explored dimensionality reduction algorithms and their respective chosen hyperparameters, as overall accuracy scores approximately between 60–70% were obtained, with comparatively insignificant levels of uncertainty. This indicates that all configured dimensionality reduction techniques were able to reduce the dimensionality of the feature space, while conserving sufficient information to empower simple MLP classifiers to perform measurably more accurately than a random guess and comparable to performance metrics obtained through our previous research . While tDirect comparison of dimensionality reduction technique performances, using optimised hyperparameters, is illustrated in A convergence of overall accuracy scores was observed, for the majority of models, at 100 dimensions and beyond a. HoweveThe overall convergence displayed in Overall, the results indicate that accuracy scores peaked within range of ~70%, even for the best performing dimensionality reduction techniques, which is also in line with the results of our previous study . It is ha priori knowledge, as they in any case can perform in a comparable manner to linear techniques for data that are linearly separable. It is hence a more robust strategy to use generally applicable non-linear dimensionality reduction techniques, such as autoencoders, not just for datasets where N > D + 1 (and hence with increased likelihood of non-linear separability), but also when a priori knowledge of the dataset’s separability is lacking and/or difficult to characterize.The consistently high performance of linear dimensionality reduction techniques, often greater than the performance of non-linear techniques, strongly suggests that the original 11,268-dimensional dataset was linearly separable, or at least approximately so. This is in line with Cover’s theorem and the underlying hypothesis of this study . AlthougIsomap significantly underperformed throughout the different graphs of It was deemed that the chosen selection of PCA, kPCA (using two different non-linear kernels), ICA, autoencoders, LLE and isomap, was sufficiently diverse in providing a valid basis for drawing a limited comparison between different dimensionality reduction techniques, especially in terms of linear techniques vs non-linear, for their suitability in building deep learning based QSAR models of mutagenicity and other toxicological endpoints. However 6 dimensionality reduction techniques compared in this study is not an absolute representation of all dimensionality reduction techniques, especially with regards to the diverse space of non-linear methods; wider comparison could be explored in future research.While it was intended that while all findings of this study would naturally be directly applicable to the specific dataset used, these results were also aimed to provide insight into the general suitability of the algorithms explored, for similar QSAR modelling applications in future. Nonetheless it may be appropriate to extend future studies to include additional datasets.The AD was calculated across all iterations and is irrespective of dimensionality reduction technique used. It was found that an average of 0.15%  of molecules were located outside the AD . This coIt is generally expected that QSAR models are at risk of underperforming when used to test molecular data that exists outside of their AD, hence the performances of the QSAR models of this study, for classifying testing molecules outside of the AD, were investigated :Considering the above findings, it was deemed beneficial to obtain further insight into the classes of molecules which were outside the AD, as well as generally how the classes of molecules were distributed according to the chemical space used to the define the AD.de-facto only being compared by their ability to classify non-mutagenic outliers; their overall improved performances may suggest that the MLP classifier simply learned to classify the majority of statistical outliers as non-mutagenic  means that higher average values were maintained from a denser pool of higher-performance molecules at closer Euclidean distances, with negligible impact from any lower performances over less densely populated farther regions. Isomap generally performed with local correct classification rates of 70–80%  [Aside from the above observations, theorem) , but thaThe trends of This further exploration of QSAR model performance, over all chemical space, uncovered compatible trends with those of N ≤ 11,268 datapoints, behaved as at least approximately linearly separable, as it was sufficiently navigated via linear dimensionality reduction techniques such as PCA and ICA, while non-linear techniques failed to outperform in any significant way . The hypothesis however was further affirmed in its prediction that certain non-linear techniques would demonstrate comparably optimal performance to linear techniques, while naturally holding the advantage of being capable of wider applicability to non-linearly separable datasets; this was indeed the case for kPCA, autoencoders and (to a somewhat lesser extent) LLE. Although kPCA frequently outperformed autoencoders in terms of final QSAR model performance, autoencoders displayed some indication of potential to outperform for higher reduced dimensionalities, while also carrying the advantage of not being limited through a pre-specified kernel function that kPCA requires a priori [Our extensive results and analyses presented in this study concluded that the original hypothesis was largely affirmed; although higher dimensional toxicological space may contain complex relationships that require conservation via dimensionality reduction, the statistical argument of Cover’s theorem appears to hold for this application . The 11,a priori . The grid search for hyperparameter optimisation was of mixed success, especially performing poorly for isomap, while none of the optimised dimensionality reduction techniques were able to consistently or significantly outperform PCA or kPCA (of which both did not undergo any optimisation process). The arbitrary assumptions used for performing the grid search, over limited sets of hyperparameters, as well as limited search spaces and resolutions, were identified as key weaknesses to be addressed in future studies, although the grid search used in this study may be regarded as having successfully enabled some extent of fair and valid comparison of advanced dimensionality reduction techniques, despite its key limitations. The vast majority of molecules tested occurred within the defined AD, although further analysis cast mixed certainty over the suitability of the possibly oversimplistic AD definition used. The XLogP/MW chemical space used to define the AD in this study did however demonstrate relevance in characterising and further investigating the performance of the QSAR models over chemical space, despite using physicochemical descriptors which were considerably different from the SMILES-based descriptors originally used to build the feature space of the QSAR models. From this, it was found that particular regions of the chemical space were significantly more problematic for the QSAR models than others, despite in one case being a data-dense region with an approximately equal statistical distribution of both classes; this particular region significantly contributed to lowering the most optimal QSAR model overall accuracy scores from within range of ~80% to within range of ~70%. It was also further uncovered that, despite underperforming in terms of the majority of summary graphs and metrics, isomap outperformed over certain regions of chemical space that other techniques were negatively impacted by; indeed each technique carried a unique insight into navigating toxicological space, although the exponential distribution of the dataset in chemical space meant that higher performance over the most central and densely populated regions carried the largest weight for determining overall performance.N ≤ D + 1 condition of Cover’s theorem [Future expansions to this study could entail a wider use of other advanced dimensionality reduction techniques, as well as different  datasets for other toxicological endpoints for comparison, especially with application to datasets which do not satisfy the  theorem . Other h theorem . The AD  theorem , while a