Claire Thornton
email: claire.thornton@imperial.ac.uk

Nicola J Bright

Magdalena Sastre

Phillip J Muckett

David Carling

AMP-activated protein kinase (AMPK) is a tau kinase, activated in response to β-amyloid exposure

AMP-activated protein kinase (AMPK) is a tau kinase,

activated in response to β-amyloid exposure

INTRODUCTION

The physiological role of the microtubule-associated protein tau is to bind to tubulin heterodimers and promote microtubule assembly. Phosphorylation of tau is required for efficient neurite outgrowth and the regulation of axonal transport [START_REF] Spittaels | Glycogen Synthase Kinase-3beta Phosphorylates Protein Tau and Rescues the Axonopathy in the Central Nervous System of Human Four-repeat Tau Transgenic Mice[END_REF][START_REF] Stoothoff | Tau phosphorylation: physiological and pathological consequences[END_REF]. However, hyperphosphorylation of tau is believed to underlie the formation of neurofibrillary tangles (NFTs), a neuropathological characteristic of Alzheimer's disease and other "tauopathies" resulting in neurodegeneration [START_REF] Ballatore | Tau-mediated neurodegeneration in Alzheimer's disease and related disorders[END_REF]. This abnormal phosphorylation allows the release of tau from microtubules, its aggregation into paired helical filaments, a major component of NFTs and promotes microtubule disruption [START_REF] Alonso | Alzheimer's disease hyperphosphorylated tau sequesters normal tau into tangles of filaments and disassembles microtubules[END_REF]. Events surrounding the initiation of tau hyperphosphorylation in vivo are not clearly understood. At least 30 protein kinases have been reported to phosphorylate tau in vitro, and of these a number, including glycogen synthase kinase (GSK) 3β, cyclin-dependent kinase 5, casein kinase 1, cAMP-dependent protein kinase A and MAP/microtubule Affinity Regulating Kinases (MARKs) have been suggested to play a role in tau phosphorylation in vivo [START_REF] Hanger | Tau phosphorylation: the therapeutic challenge for neurodegenerative disease[END_REF]. Despite the complexity of the system, only a limited number of the potential tau kinases have been shown to phosphorylate tau at S262 within the KXGS motif of the microtubule-binding domain. Phosphorylation of S262 is thought to influence the ability of tau to bind to microtubules [START_REF] Biernat | Phosphorylation of Ser262 strongly reduces binding of tau to microtubules: distinction between PHF-like immunoreactivity and microtubule binding[END_REF]. MARKs are members of the AMP-activated protein kinase (AMPK)-related kinase family which also includes two recently characterised brain-specific kinases, BRSK1 and BRSK2 [START_REF] Lizcano | LKB1 is a master kinase that activates 13 kinases of the AMPK subfamily, including MARK/PAR-1[END_REF]. All kinases within the group share amino acid sequence homology within their catalytic domains and require phosphorylation of a conserved threonine residue within the activationloop region. LKB1, which was identified as an upstream kinase in the AMPK cascade, phosphorylates and activates all of the AMPK-related kinases, apart from MELK [START_REF] Lizcano | LKB1 is a master kinase that activates 13 kinases of the AMPK subfamily, including MARK/PAR-1[END_REF]. In addition to LKB1, AMPK is phosphorylated and activated by Ca 2+ /calmodulin-dependent protein kinase kinase (CaMKK) β, and this has been shown to mediate activation of AMPK in response to increased intracellular calcium concentrations. [START_REF] Woods | Ca2+/calmodulin-dependent protein kinase kinase-beta acts upstream of AMP-activated protein kinase in mammalian cells[END_REF]. AMPK is also activated in response to an increase in AMP, following a fall in ATP levels within the cell. AMP allosterically activates AMPK as well as protecting the enzyme against dephosphorylation [9]. AMPK is a heterotrimer comprising a catalytic α subunit and regulatory β and γ subunits. Isoforms of all three subunits exist (α1,α2,β1,β2,γ1,γ2,γ3) and all twelve heterotrimeric complex combinations have been identified in vivo. The physiological significance of the different AMPK isoform complexes is only partly understood, although it seems likely that they allow tissue specific regulation of AMPK as well as tissue specific function (reviewed in [10]). AMPK is expressed ubiquitously and is a key regulator of metabolic pathways such as fatty acid and cholesterol synthesis [10]. However, delineating the function of neuronal AMPK is still in its infancy. The most detailed studies have taken place in the arcuate nucleus of the hypothalamus where AMPK is involved in co-ordinating the regulation of appetite and satiety [START_REF] Minokoshi | AMP-kinase regulates food intake by responding to hormonal and nutrient signals in the hypothalamus[END_REF]. AMPK is active in other brain regions but as yet, it is unclear whether activation of AMPK plays a neuroprotective or neurodegenerative role with data supporting either position [START_REF] Garcia-Gil | 5'-aminoimidazole-4-carboxamide riboside induces apoptosis in human neuroblastoma cells[END_REF][START_REF] Kuramoto | Phospho-dependent functional modulation of GABA(B) receptors by the metabolic sensor AMP-dependent protein kinase[END_REF][START_REF] Li | Neuroprotective effects of adenosine monophosphate-activated protein kinase inhibition and gene deletion in stroke[END_REF][START_REF] Spasic | Drosophila alicorn is a neuronal maintenance factor protecting against activity-induced retinal degeneration[END_REF]. More recently, AMPK has been implicated in both beneficial and deleterious modulations of amyloid precursor protein (APP), proteolytic cleavage of which generates Aβ 1- 42 peptide, a key component of disease-characteristic amyloid plaques [START_REF] Hardy | The amyloid hypothesis of Alzheimer's disease: progress and problems on the road to therapeutics[END_REF]. In response to metformin treatment, activation of AMPK was reported to promote the formation of both intracellular and extracellular amyloid β peptide from APP [START_REF] Chen | Antidiabetic drug metformin (GlucophageR) increases biogenesis of Alzheimer's amyloid peptides via upregulating BACE1 transcription[END_REF]. Conversely, resveratrol, a plant polyphenol with potential neuroprotective properties, is reported to negatively regulate the production of Aβ 1-42 peptide through an AMPK-mediated pathway [START_REF] Vingtdeux | AMP-activated Protein Kinase Signaling Activation by Resveratrol Modulates Amyloid-{beta} Peptide Metabolism[END_REF]. Recent research in both drosophila and mice suggest that the MARKs and to a lesser extent, the related BRSKs, may play a role in early priming phosphorylation of tau. However, to date the physiological regulation of these kinases and the signals that modulate their activity remain unknown. [START_REF] Bright | Investigating the regulation of brainspecific kinases 1 and 2 by phosphorylation[END_REF]. Currently, there has been little examination of the role of AMPK in the phosphorylation of tau. Here we present data implicating AMPK as a physiological kinase in a pathway phosphorylating tau. We have identified a number of sites in tau phosphorylated by AMPK, many of which are clustered around the microtubule binding domain of tau. Intriguingly, phosphorylation of tau by AMPK regulates its ability to bind to microtubules. In addition, we show that exposure of primary neurons to Aβ 1-42 increases AMPK activity via CaMKKβ-dependent pathway. Aβ 1-42 -mediated activation of AMPK is blocked by memantine, an NMDA receptor antagonist licensed for treating Alzheimer's disease. These findings place the NMDA receptor upstream of AMPK in a CaMKKβ-mediated activation pathway resulting in the phosphorylation of tau.

EXPERIMENTAL

Preparation of primary neurons Animal use was in accordance with local rules and with the regulations and guidance issued under the Animals (SP) Act (1986). Primary cortical neurons were prepared from WT and CaMKKβ -/-E15 animals as previously described [START_REF] Lesuisse | Long-term culture of mouse cortical neurons as a model for neuronal development, aging, and death[END_REF]. Briefly, cortices from embryos in a single litter were dissected, meninges removed and tissue pooled. Cortices were roughly chopped before incubation in 0.25% trypsin-EDTA followed by trituration. Cells were pelleted by centrifugation, resuspended and plated in Neurobasal media supplemented with B27, 100U/ml penicillin, 100µg/ml streptomycin, 0.25µg/ml amphotericin B, 300 µM glutamine and 25 µM β-mercaptoethanol. Treatments were carried out on neurons grown for a minimum of 7 days in vitro and all media were obtained from Invitrogen. Neuronal cell treatment and lysis Primary cortical neurons were treated with the following compounds for the times indicated: hydrogen peroxide (0.5mM, 15 min, Sigma), ionomycin (2µM, 5 min, Tocris), Aβ 1-42 (20µM, 30 min, Invitrogen), NMDA (50µM, 15 min, Sigma), STO609 (10µg/ml, 3 hr, Tocris), memantine (10µM, 3 hr, Sigma). In cases in which a 3 hr pretreatment was required (memantine, STO609), activators were subsequently added in the presence of the pretreating compound. Aggregated Aβ 1-42 peptide was prepared as per manufacturer's instructions (Invitrogen); lyophilised peptide was first resuspended to 6mg/ml in sterile dH 2 O, vortexed thoroughly then diluted to a final concentration of 1mg/ml in PBS. Aggregation was achieved by incubation at 37ºC for 24 hr. Treated cells were harvested in HEPES buffer A containing Triton (50 mM HEPES pH 7.5, 50 mM NaF, 5 mM sodium pyrophosphate, 1 mM EDTA, 1 mM dithiothreitol, 0.1 mM phenylmethylsulphonyl fluoride, 157 µg/ml benzamidine, 10% [v/v] glycerol, 1% [v/v] Triton X-100) and protein concentration determined by Bradford assay (Biorad). Immunoprecipitation (IP) 50µg total protein lysate was incubated for 2 hr at 4ºC in the presence of protein A-or protein G-sepharose (for rabbit and mouse antibodies respectively) and antibody as follows: AMPK pan β [21], BRSK2 [START_REF] Bright | Investigating the regulation of brainspecific kinases 1 and 2 by phosphorylation[END_REF], 6His and MARK3 (Abcam), AMPKα1 and AMPKα2 (Cell Signalling). Immobilised immune complexes were washed thoroughly in PBS containing 1% triton then used for kinase assay or western blot analysis. In vitro phosphorylation Recombinant bacterially expressed wild-type (α1β1γ1; 1µg) or kinase inactive (α1D157Aβ1γ1; 1µg) AMPK was purified as described previously [START_REF] Neumann | Mammalian AMP-activated protein kinase: functional, heterotrimeric complexes by coexpression of subunits in Escherichia coli[END_REF]. GSK3β was from New England Biolabs (1µg;), MARK4 was from Cell Signaling (1µg;) and recombinant tau (2N4R, longest human tau isoform, 0.5 µg) was from Sigma. Purified protein kinases and recombinant tau (0.5µg) were incubated phosphorylation buffer (50mM Tris-HCl pH7.5, 10mM MgCl 2 , 5mM dithiothreitol, 3mM 32 P-ATP) for 1hr at 37°C. Phosphorylation reactions were resolved by SDS-PAGE, transferred to PVDF membrane and detected by fluorography overnight at -80°C. For western blots of AMPK-phosphorylated recombinant tau, 32 P-ATP was replaced with 3mM unlabelled ATP and 20ng of phosphorylated tau per lane was analysed by phosphotau-specific antibodies.. Kinase assays AMPK, BRSK2 and MARK3 activities was measured using the AMARA substrate peptide as described previously [START_REF] Woods | LKB1 Is the Upstream Kinase in the AMP-Activated Protein Kinase Cascade[END_REF]. Assays were carried out for 20 min at 37ºC. Western blot (WB) 25-50µg of total protein lysate or immobilised immune complexes were boiled in SDS sample buffer and resolved on 10% NuPage BisTris gels (Invitrogen). Proteins were transferred to PVDF and analysed by western blotting with the following antibodies: phospho-tau pS214 (Abcam), 12E8 [24], PHF1 [START_REF] Greenberg | Hydrofluoric acidtreated tau PHF proteins display the same biochemical properties as normal tau[END_REF] pS199/pT231/pS262/pS356/pS396/ pS400/pT404/pS422 (Biosource), Tau5 (Biosource), phosphoAMPK (pT172) and phosphoACC (Cell Signalling), Actin (Sigma). Membranes were washed and incubated with secondary antibodies conjugated to Alexa-Fluor 680 (Invitrogen) or IRDye800 (Li-Cor) and scanned on the Li-Cor Odyssey Infrared Imaging System using Odyssey software 2.0 for band quantification. Typically, primary antibody incubations took place overnight at 4ºC and secondary antibody incubations were for 2 hr at room temperature. Tau over-expression cDNA encoding 2N4R Tau was subcloned into pcDNA3. Mutation of serine 262 to alanine was carried out by site-directed mutagenesis (Invitrogen). pcDNA3-TauWT and -TauS262A were expressed in CCL13 cells by calcium phosphate transfection of 10µg purified plasmid DNA (BD Clontech). Precipitates were removed after overnight incubation and cells harvested 48 hr later. Expressed protein was isolated by immunoprecipitation with anti-His antibodies. Mass Spectrometry Recombinant AMPK-phosphorylated tau (0.5µg) was reesolved by SDS-PAGE, excised and sent for analysis to Harvard Medical School, Taplin Facility. Microtubule binding Taxol-stabilised microtubules were generated according to the manufacturer's instructions (Cytoskeleton, Cambridge). Briefly, a stable population of microtubules was generated by polymerising bovine tubulin (100µg) at 37ºC for 20 min in the presence of GTP (1mM) and taxol (20µM). AMPK-phosphorylated or non-phosphorylated tau was incubated in the presence of taxol-stabilised microtubules at room temperature for 30 min. The reaction was fractionated by ultracentrifugation through taxol-stabilised cushion buffer (50% glycerol [v/v], 20µM taxol) at 100,000 g for 40 min at room temperature. The resulting fractions were resolved by SDS-PAGE and analysed by western blot using a phosphorylationindependent tau antibody. Statistical analyses Data are expressed as mean ± standard deviation from three or more independent experiments. Statistical analysis was performed using Prism (Graphpad). Statistical significance between two conditions was established with a two-tailed Student's ttest whereas significance among multiple datasets was determined by ANOVA followed by post hoc Tukey multiple comparison test. Data was considered significant at *p<0.

RESULTS

AMPK is a tau kinase

In transient transfection studies, we previously observed that BRSK1 and BRSK2 were insensitive to activation by a range of pathophysiological stimuli, including those that might occur early in Alzheimer's disease [START_REF] Bright | Investigating the regulation of brainspecific kinases 1 and 2 by phosphorylation[END_REF]. Here we have extended this study to primary cortical neurons and find that the activities of endogenous MARK3 and BRSK2 are not significantly altered in response to ionomycin (increasing intracellular calcium concentrations) or hydrogen peroxide (to mimic oxidative stress; Figure 1A). Therefore we examined the hypothesis that AMPK itself was a physiological tau kinase as AMPK is activated in response to these and other cellular stresses. We carried out in vitro phosphorylation experiments using active and inactive (mutation of D157A in the ATP-binding pocket of AMPKα) AMPK, GSK3β, MARK4 and recombinant tau (Figure 1B). In addition to GSK3β and MARK4, which have been shown previously to phosphorylate tau, AMPK caused a robust phosphorylation of tau (Figure 1B). An identical experiment analysed by western blot determined that AMPK increased the phosphorylation of S262 (upper panel, Figure 1C), a site previously implicated as a major phosphorylation site for BRSK and MARK kinases [26,[START_REF] Drewes | MARK, a novel family of protein kinases that phosphorylate microtubule-associated proteins and trigger microtubule disruption[END_REF]. Only weak phosphorylation of S262 was observed for GSK3β, although the migration of tau on SDS-PAGE was retarded suggesting phosphorylation at other sites (lower panel, Figure 1C). To determine whether Ser262 was the predominant phosphorylation site for AMPK on tau, we generated His-tagged wild-type tau, or a construct harbouring mutation of S262 to alanine (S262A) and expressed the proteins in CCL13 cells, a mammalian cell line lacking LKB1 [START_REF] Woods | LKB1 Is the Upstream Kinase in the AMP-Activated Protein Kinase Cascade[END_REF]. These cells therefore have the advantage that the basal activity of AMPK is low and overexpression of tau in this line is likely to remain unphosphorylated by AMPK. Tau was isolated from cell lysates by immunoprecipitation and subjected to phosphorylation by AMPK or GSK3β. As expected, mutation of S262A had no detectable effect on the ability of GSK3β to phosphorylate tau. However, tau phosphorylation by AMPK also remained relatively unchanged in the S262A mutant compared to wild-type, indicating the presence of additional AMPK phosphorylation sites within tau (Figure 1D).

AMPK phosphorylates multiple tau epitopes

In order to study further the phosphorylation of tau by AMPK, we compared phosphorylation of tau by MARK4 or AMPK using a panel of phospho-tau antibodies (Figure 2A). Five phosphorylation sites common to both kinases were identified (T231, S262, S356, S396 and S422). In addition, AMPK, but not MARK4, was capable of phosphorylating S214 (Figure 2A). In blots in which there was no phosphorylated signal detected (S199, S400, S404), a positive control phosphorylation was carried out using GSK3β to confirm antibody recognition of the phospho-specific epitope (Supplementary Figure S1). As western blotting revealed that AMPK phosphorylates a number of sites in tau, we used mass spectrometry to comprehensively identify as many AMPK sites as possible. Briefly, AMPK-phosphorylated tau was subjected to mass spectrometric analysis, achieving 81% coverage of 2N4R tau (Supplementary Figure S2). Using this approach, we were able to assign with confidence over 20 AMPK phosphorylation sites within tau (Figure 2B). Interestingly, 9 were identified within the microtubule binding domain and 7 out of 8 conserved serine residues within the four repeated tubulin binding motifs were identified as AMPK sites (Figure 2B, top panel).

AMPK phosphorylation inhibits tau binding of microtubules

As the bulk of AMPK phosphorylation sites are clustered around the microtubule binding domain, we investigated the effect of AMPK phosphorylation on binding of tau to microtubules in vitro. Tau, before and after phosphorylation by AMPK, was incubated with microtubules and subjected to ultracentrifugation to pellet microtubule-binding proteins. The majority (75%) of non-phosphorylated tau was detected in the high-speed pellet, indicating efficient binding to the microtubules (Figure 2C). In contrast, only 20% of the AMPKphosphorylated tau was detected in the microtubule pellet, with 80% remaining in the supernatant fraction.

AMPK is activated in response to Aβ 1-42 exposure

The deposition of extracellular plaques, of which a short amyloid β peptide (Aβ 1-42 ) is a key component, is a pathological characteristic of Alzheimer's disease [START_REF] Hardy | The amyloid hypothesis of Alzheimer's disease: progress and problems on the road to therapeutics[END_REF]. Exposure of primary hippocampal neurons to aggregated Aβ 1-42 peptide has been reported to induce tau phosphorylation [START_REF] Zheng | Amyloid beta peptide induces tau phosphorylation and loss of cholinergic neurons in rat primary septal cultures[END_REF]. To determine whether there is a role for AMPK in the Aβ 1-42 pathway, a mixed cortical/hippocampal culture of primary neurons was prepared and treated with aggregated Aβ 1-42 peptide (20µM) and AMPK activity in immune complexes determined. As shown in Figure 3A there was a significant increase in activity from AMPK isolated from Aβ 1- 42 -treated cell lysates compared with control cells. Phosphorylation of AMPK at T172, another direct measure of AMPK activation, was also increased and this correlated with an increase in phosphorylated acetyl Co-A carboxylase (pACC), a well characterised downstream target of AMPK (Figure 3B). As both BRSK and MARK have been implicated in tau phosphorylation [START_REF] Drewes | MARK, a novel family of protein kinases that phosphorylate microtubule-associated proteins and trigger microtubule disruption[END_REF][START_REF] Kishi | Mammalian SAD kinases are required for neuronal polarization[END_REF] the activity of these kinases following Aβ 1-42 -treatment was also determined. In contrast with AMPK, no change in either BRSK2 or MARK3 activity after Aβ 1-42 treatment was detected (activities for control vs Aβ 1-42 treatment, BRSK2: 0.0537 ± 0.0103 vs 0.0597 ± 0.0083 nmol/min/mg; MARK3: 0.0464 ± 0.0064 vs 0.0463 ± 0.0089 nmol/min/mg, N=4). Acute exposure of primary cortical neurons to aggregated Aβ 1-42 peptide induces a rapid increase in intracellular calcium concentrations [START_REF] Mattson | beta-Amyloid peptides destabilize calcium homeostasis and render human cortical neurons vulnerable to excitotoxicity[END_REF]. Therefore we tested the hypothesis that the regulation of AMPK by Aβ 1-42 was mediated through a CaMKKβ-signalling pathway. We isolated primary neuronal cultures from CaMKKβ-knockout mice, and treated these cells with Aβ 1-42 . In marked contrast to wild-type neurons, Aβ 1-42 -treatment had no effect on AMPK activity in the cells derived from the CaMKKβ-knockout mice (Figure 3C) establishing that CaMKKβ is required for AMPK activation by Aβ 1-42 . Previous studies by us and others have identified a substrate preference in CaMKKβ for activation of AMPKα1-rather than AMPKα2-containing complexes [START_REF] Stahmann | Thrombin Activates AMP-Activated Protein Kinase in Endothelial Cells via a Pathway Involving Ca2+/Calmodulin-Dependent Protein Kinase Kinase {beta}[END_REF][START_REF] Thornton | Muscarinic Receptor Activation of AMP-activated Protein Kinase Inhibits Orexigenic Neuropeptide mRNA Expression[END_REF]. To determine which isoform mediates the response to Aβ 1-42 , isoform-specific depletion of AMPK was repeatedly performed on the same lysate until all isoform complexes were removed. AMPKα1 complexes were significantly activated in response to Aβ 1-42 , whilst there was no significant increase in AMPKα2 activity (Figure 3D). These results are similar to a previous study, where we only observed significant activation of AMPKα1 complexes following activation of CaMKKβ [START_REF] Thornton | Muscarinic Receptor Activation of AMP-activated Protein Kinase Inhibits Orexigenic Neuropeptide mRNA Expression[END_REF].

Activation of AMPK by Aβ 1-42 increases tau phosphorylation at disease-relevant epitopes

Our initial data suggested that AMPK phosphorylates a number of residues on recombinant tau. As Aβ 1-42 treatment activates AMPK in primary neurons, we determined the downstream effect on the phosphorylation of endogenous tau. We obtained 12E8 and PHF-1 antibodies, commonly used to detect pathologically phosphorylated tau [24,[START_REF] Greenberg | Hydrofluoric acidtreated tau PHF proteins display the same biochemical properties as normal tau[END_REF], and examined the phosphorylation state of tau in Aβ 1-42 -treated WT and CaMKKβ -/-neurons. We observed a marked increase in signal for both antibodies in the WT neurons indicating that phosphorylation had occurred at S262/S356 (12E8) and S396/S404 (PHF-1; Figure 4A), consistent with our earlier observations identifying S262, S356 and S396 as candidate AMPK phosphorylation sites in vitro (Figure 2). Conversely, there was little difference in tau phosphorylation between control and Aβ-treated CaMKKβ -/-neurons after western blot analysis with 12E8. More surprisingly perhaps, was that phosphorylation of S396/S404, as detected by the PHF-1 antibody, was reduced following treatment with Aβ 1-42 in CaMKKβ -/- cells (Figure 4B). To determine whether this difference was due to reduced total tau or reduced phosphorylation after Aβ treatment, we compared the levels of total tau in WT and CaMKKβ -/- lysates. Although there was a difference in total tau protein expression in the CaMKKβ -/-cells compared with WT (32% less tau in CaMKKβ -/-neurons), there was no change in tau expression between control and Aβ 1-42 treated neurons (Figure 4C). These data suggest that tau phosphorylation at S396/S404 in response to Aβ 1-42 is significantly influenced by active CaMKKβ.

AMPK activation by Aβ 1-42 is inhibited by memantine

One mechanism by which Aβ 1-42 may exert its intracellular effects is through potentiating glutamate-mediated excitotoxicity [START_REF] Alberdi | Amyloid beta oligomers induce Ca2+ dysregulation and neuronal death through activation of ionotropic glutamate receptors[END_REF]. Neurodegeneration of the glutamatergic pyramidal neurons of the hippocampus and cortex is an early event in Alzheimer's disease and cultured neurons can be protected from Aβ 1-42 toxicity by treatment with glutamatergic N-methyl-Daspartate (NMDA) receptor antagonists [START_REF] Mattson | Glutamate and neurotrophic factors in neuronal plasticity and disease[END_REF]. Interestingly, we found that AMPK was robustly activated in primary neurons after NMDA treatment, and that this activation was abolished by pre-treatment with STO609, a pharmacological inhibitor of CaMKKα and CaMKKβ (Figure 5A). Memantine is a non-competitive NMDA receptor antagonist and recently the use of memantine for treatment in moderate to severe Alzheimer's disease has been approved, highlighting the role of this pathway in the pathology of cognitive decline. We found that treatment of primary neurons with memantine prior to exposure to NMDA prevented the activation of AMPK (Figure 5A). Memantine is reported to reduce both the excitotoxic effects of Aβ 1-42 in rat cortical neurons and the phosphorylation of tau [START_REF] Song | Memantine protects rat cortical cultured neurons against beta-amyloid-induced toxicity by attenuating tau phosphorylation[END_REF]. Therefore we repeated the Aβ 1-42 treatment on primary neurons with or without memantine pre-treatment. As with NMDA, memantine successfully prevented the activation of AMPK by Aβ 1-42 (Figure 5B) implying that Aβ 1-42 acts on AMPK through activation of the NMDA receptor increasing intracellular Ca 2+ and activating CaMKKβ.

DISCUSSION

The hyperphosphorylation of tau is a key event in the progression of Alzheimer's disease yet the mechanisms underlying this event remain unclear. Our data provide the first evidence that AMPK activation may act as a link between exposure of neurons to β-amyloid and subsequent tau phosphorylation.

Mass spectrometry analysis identified a cluster of candidate AMPK phosphorylation sites around the microtubule binding domain of tau including all four conserved serine residues (S262, S293, S324 and S356) that lie within the four KXGS microtubule binding motifs (Fig 2B ,expanded panel). This suggested a role for AMPK phosphorylation in the regulation of microtubule binding. In addition, phosphorylation at S214 and T231 was also detected by western blotting and mass spectrometry. In line with previous research suggesting that phosphorylation of these epitopes induces maximal disruption of binding [START_REF] Drewes | MARK, a novel family of protein kinases that phosphorylate microtubule-associated proteins and trigger microtubule disruption[END_REF][START_REF] Cho | Glycogen synthase kinase 3beta phosphorylates tau at both primed and unprimed sites. Differential impact on microtubule binding[END_REF], the interaction of tau with microtubules was greatly reduced following phosphorylation by AMPK in vitro. Neither mass spectrometry nor western blot analysis identified S404 as a potential AMPK site, suggesting that S396 provides the epitope for recognition by the PHF-1 antibody. Mass spectrometry also highlighted threonine 71 (T71) and threonine 403 as AMPK phosphorylation sites. An online database of tau phospho-epitopes lists both sites as phosphorylated in brain tissue isolated from patients with Alzheimer's disease, but as yet no kinase has been identified (http://cnr.iop.kcl.ac.uk/hangerlab/tautable). The N-terminal sequence of tau containing T71 is referred to as the projection domain as it projects away from the microtubule to interact with cytoskeletal proteins linking it to the plasma membrane and may also regulate tau aggregation [START_REF] Brandt | Interaction of tau with the neural plasma membrane mediated by tau's amino-terminal projection domain[END_REF][START_REF] Horowitz | N-terminal fragments of tau inhibit full-length tau polymerization in vitro[END_REF]. Further work is currently underway to determine whether AMPK phosphorylation at T71 regulates these properties of tau.

Exposure of primary neurons to Aβ 1-42 rapidly increases AMPK activity but is also reported to alter the activities of other kinases. Exposure of primary rat neuronal cultures to Aβ for 6-8hr increased cdk5 and p38 activities [START_REF] Chang | Cdk5 is a major regulator of p38 cascade: relevance to neurotoxicity in Alzheimer's disease[END_REF] and in a similar study, Song and colleagues demonstrated increased ERK1/2 activity after a 1 hr Aβ exposure [START_REF] Song | Memantine protects rat cortical cultured neurons against beta-amyloid-induced toxicity by attenuating tau phosphorylation[END_REF]. In animals, a study describing lentiviral injection of Aβ into rat motor cortex revealed prolonged activity of both cdk5 and GSK3β [START_REF] Rebeck | Beta-amyloid1-42 gene transfer model exhibits intraneuronal amyloid, gliosis, tau phosphorylation, and neuronal loss[END_REF]. In all cases, tau phosphorylation was reported to increase. In contrast, a very recent study found that Aβ exposure of primary rat hippocampal neurons for 3hr resulted in a synaptically localised increase of MARK, BRSK, p70S6K and cdk5 activities whereas GSK3β and MAPK activities did not change [START_REF] Zempel | Abeta oligomers cause localized Ca(2+) elevation, missorting of endogenous Tau into dendrites, Tau phosphorylation, and destruction of microtubules and spines[END_REF]. An increase in tau phosphorylation at 12E8 sites (pS262, pS356) was also observed, and ascribed to the activities of AMPK-related kinases although the activity of AMPK itself was not determined. These data suggest that further work is required to distinguish the effects of short and long term Aβ exposure and intracellular localisation on kinase activities.

In our study, we were unable to detect changes in the activities of BRSK2 or MARK3 in response to 30 min Aβ treatment. The MARK kinase family were originally identified through their ability to regulate the interaction of tau with microtubules [START_REF] Drewes | MARK, a novel family of protein kinases that phosphorylate microtubule-associated proteins and trigger microtubule disruption[END_REF]. Studies of the MARK orthologue in drosophila, PAR-1, show that overexpression of APP upregulated PAR-1 activation and phosphorylation of tau [START_REF] Wang | Activation of PAR-1 kinase and stimulation of tau phosphorylation by diverse signals require the tumor suppressor protein LKB1[END_REF]. In the current study, we show that short term Aβ 1-42 exposure exerts its effects through CaMKKβ activation, consistent with the hypothesis that Ab renders the neuron more susceptible to glutamate receptor neurotoxocity by destabilising calcium homeostasis [START_REF] Mattson | beta-Amyloid peptides destabilize calcium homeostasis and render human cortical neurons vulnerable to excitotoxicity[END_REF]. These findings provide an explanation for the lack of effect of Aβ 1-42 treatment on the activity of the AMPK-RKs tested, since a previous study reported that these kinases were not regulated by calcium signalling pathways, including CaMKKβ [START_REF] Bright | Investigating the regulation of brainspecific kinases 1 and 2 by phosphorylation[END_REF]. AMPK-RKs are activated by LKB1 phosphorylation [START_REF] Lizcano | LKB1 is a master kinase that activates 13 kinases of the AMPK subfamily, including MARK/PAR-1[END_REF] and although the upstream stimulus for this pathway remains unclear, this may be the mechanism by which they play a role in tau phosphorylation.

Antibodies recognising both early (12E8) and late (PHF-1) stage tau phosphorylation were used to determine the effect of Aβ 1-42 treatment on primary neurons. Treatment conditions which activated AMPK also caused a substantial increase in tau phosphorylation at a number of disease-relevant epitopes in wild type neurons. Conversely in CaMKKβ -/-neurons, we did not detect any change in S262/S356 phosphorylation following treatment. We did, however, observe a reduction in phosphorylation of S396/S404 following Aβ 1-42 treatment in CaMKKβ -/- neurons relative to untreated cells. This is not due to reduced levels of tau as the 12E8 epitopes are unaffected and there is no change in the amount of total tau protein expression in CaMKKβ -/-control versus CaMKKβ -/-Aβ 1-42 lysates (Fig 4C ; compare lanes 5 and 6 with lanes 7 and 8). As the actions of kinases and phosphatases are tightly balanced, one possible explanation is that removing CaMKKβ disturbs this balance. Following influx of calcium after Aβ 1-42 treatment, it is possible that calcium-dependent phosphatases are activated, and in the absence of competing activation of CaMKKβ the equilibrium is pushed in favour of dephosphorylation. Interestingly, S396 and S404 have been identified as substrates for calcium-dependent protein phosphatase (PP) 2B (calcineurin; [43]). Although the activity of a number of major tau phosphatases has been reported to decrease in Alzheimer's disease hence promoting tau hyperphosphorylation [START_REF] Tian | Role of serine/threonine protein phosphatase in Alzheimer's disease[END_REF], it has been suggested that the activity of PP2B may increase [43].

Our report demonstrates robust AMPK activity in response to pharmacological activation of the NMDA receptor, an activation that can be inhibited by the NMDA-R antagonist memantine. A recent study identified another NMDA receptor-mediated pathway for AMPK activation. AMPK is involved in the development of excitotoxic apoptosis through a mechanism requiring BH3-only protein, Bim. Prolonged activation of AMPK leads to increased JNK phosphorylation, increased bim mRNA expression and neuronal death [START_REF] Concannon | AMP kinase-mediated activation of the BH3-only protein Bim couples energy depletion to stress-induced apoptosis[END_REF]. Other groups report both JNK activity and Bim are upregulated on exposure to Aβ 1-42 peptide [46,[START_REF] Yao | Beta-amyloid-induced neuronal apoptosis involves c-Jun N-terminal kinase-dependent downregulation of Bcl-w[END_REF]. Taking these studies together, it is interesting to speculate that these effects may be exerted in part by Aβ 1-42 activation of AMPK.

AMPK activity is reported to be altered in the APP/PS2 animal model of Alzheimer's disease which exhibits amyloidosis, cognitive deficits, impaired synaptic plasticity and perturbed cerebrovascular development [START_REF] Richards | PS2APP transgenic mice, coexpressing hPS2mut and hAPPswe, show age-related cognitive deficits associated with discrete brain amyloid deposition and inflammation[END_REF]. In a study carried out in young APP/PS2 animals (less than 8 months old), where the phenotype is mild, AMPK is abnormally activated compared with control mice [START_REF] Lopez-Lopez | Disturbed cross talk between insulin-like growth factor I and AMP-activated protein kinase as a possible cause of vascular dysfunction in the amyloid precursor protein/presenilin 2 mouse model of Alzheimer's disease[END_REF]. This activation, in association with insulin growth factor (IGF)-1 imbalance contributes to increased vascular endothelial growth factor (VEGF) expression and decreased formation of blood vessels. Over-expression of Aβ 1-42 in these animals is apparent as early as 5 months even though cognitive impairment is not detectable until 8 months. These data fit with our results from primary neurons in which there is a rapid activation of AMPK in response to pathological upregulation of Aβ 1-42 .

Currently there is little information regarding the alteration of AMPK activity in human neurological disease. Indeed, a practical limitation of assessing AMPK activity in neural tissue is the rapidity needed for dissection and homogenisation to avoid spurious increases in AMPK activity caused by anoxia and such samples are rarely available. However, a study published while this manuscript was in review carried out a comprehensive analysis of post-mortem human tauopathy samples (including Alzheimer's disease) with phospho-AMPK antibodies.

They observed an increase in phospho-AMPK in both tangle-and pretangle-bearing neurons and confirmed our identification of S396 as a tau epitope phosphorylated by AMPK, using the same PHF-1 antibody [50]. Taken together with our data, it suggests that AMPK may play a key role in governing the early phosphorylation of tau.

In summary, we present data suggesting that AMPK is rapidly activated in primary neurons on exposure to Aβ resulting in an increase in tau phosphorylation at residues critical for microtubule binding. Based on the previously published properties of Aβ, we have identified a key role for the NMDA receptor and the calcium-activated upstream AMPK kinase CaMKKβ in this pathway. Our work suggests that AMPK deserves close examination as a candidate in vivo kinase in the regulation of tau hyperphosphorylation and further studies are required to determine whether AMPK has a clinically relevant role in neurodegeneration.

FIGURE LEGENDS

Figure 1 AMPK is a tau kinase. A) Primary neurons were treated with vehicle control (DMSO), hydrogen peroxide or ionomycin, lysed and immune complexes of BRSK2, MARK3 or AMPK assayed for kinase activity. Average fold activity compared with control is shown for three independent experiments (**p<0.01). B) Recombinant tau was phosphorylated by GSK3β, MARK4 and active or inactive (D157A) AMPK and incorporation of 32 P-ATP into tau determined by fluorography (upper panel). Equivalent amounts of tau was confirmed by commassie blue (CBB) gel staining (lower panel). Additional bands present in lane 1 are from recombinant GSK3β. C) Phosphorylated recombinant tau was analysed by western blot for phospho-tau S262 (pS262 antibody; upper panel) or total tau (lower panel). D) His-tagged cDNA constructs of wild type (WT) or mutated (S262A) tau were transfected into a mammalian cell line. Expressed tau was purified by immunoprecipitation with anti-His antibodies and subjected to phosphorylation by GSK3β (left panels) or AMPK (right panels). Phosphorylation was determined by fluorography as above (upper panels) and equivalent expression of tau confirmed by western blotting (lower panels). Blots and autoradiographs are representative of three independent experiments. A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

4 Biochemical

 4 05 or **p<0.01. Journal Immediate Publication. Published on 05 Jan 2011 as manuscript BJ20101485 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20101485 A c c e p t e d M a n u s c r i p t

Biochemical

  Journal Immediate Publication. Published on 05 Jan 2011 as manuscript BJ20101485 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20101485 A c c e p t e d M a n u s c r i p t

Figure 2

 2 Figure 2 AMPK phosphorylates multiple tau sites and interferes with microtubule binding. A) MARK4 (M)-or AMPK (A)-phosphorylated recombinant tau was analysed by western blotting using phospho-specific tau antibodies (epitope indicated above each panel) and total tau (lower panels). B) A schematic of the longest isoform of tau (2N4R; 441 amino acids) showing domains and known phosphorylation sites of serine / threonine kinases or sites that are phosphorylated in vitro or in post-mortem Alzheimer's disease brain [5, 51]. AMPK sites identified from mass spectrometry analysis are shown in bold. C) Unphosphorylated (tau) or AMPK-phosphorylated (ptau) tau was incubated in the presence or absence of microtubules (MT). Microtubule bound (pellet) or microtubule unbound (supernatant) fractions were generated by ultracentrifugation. Fractions were analysed by western blot for total tau (upper panel). The lower panel shows sedimented microtubules by coomassie staining. The histogram represents the bound versus unbound tau (**p < 0.01) or ptau ( ## p < 0.01) as a percentage of total tau. For A and C, images are representative of four individual experiments.Figure 3 AMPK is activated by Aβ 1-42 . Primary neurons from wild-type (WT) (A,B,D) and CaMKKβ -/-(C) mice were treated with 20µM Aβ 1-42 . Treated and untreated neurons were harvested after 30 min and kinase activity determined in immune complexes of AMPK (A, C). Box plots show the minimum, median and maximum activities obtained for neurons from between 3 and 6 individual experiments (**p < 0.01). B) Lysates from wild-type neurons treated with Aβ 1-42 were analysed by western blot with antibodies recognising phosphorylated (active) AMPK and a known AMPK substrate, acetyl Co-A carboxylase (pACC). Western blot using a panβ-AMPK antibody for total AMPK and anti-actin antibodies demonstrate there is no change in the amount of total protein. Representative images shown are neuronal lysates from two of four individual experiments. D) Lysates from treated and untreated neurons were subjected to sequential immunoprecipitation with either AMPKα1-or AMPKα2-specific antibodies to isolate all available endogenous isoform-specific AMPK complexes. Results shown represent the total isoform-specific activity from neuronal lysates of three individual experiments (*p < 0.05).Figure 4 Increases in tau phosphorylation in response to Aβ 1-42 are muted in cells lacking CaMKKβ. Primary cortical neurons from wild-type (WT) (A, C) and CaMKKβ -/-(B, C) mice were treated with Aβ 1-42 as before and analysed by western blot for phosphorylated tau using 12E8 (pS262/pS356) and PHF-1 (pS396/pS404) antibodies (A,B) or for phosphorylationindependent total tau (C). Quantification of phosphorylation (A,B) and total tau (C) is shown in the histograms normalised relative to actin (*p < 0.05). Images shown are primary cell lysates from two of four (WT) or two of three (CaMKKβ -/-) individual experiments.Figure 5 Activation of AMPK is prevented by the NMDA receptor antagonist memantine. A) Primary neurons were left untreated, pre-treated with either STO609 or memantine alone, or treated with NMDA. Neurons were harvested and AMPK activity determined from immune complexes. B) Primary neurons either untreated or pre-treated with memantine were treated with Aβ 1-42 . Neurons were harvested and assayed for AMPK activity as above. Box plots show the minimum, median and maximum activities obtained for neurons from four (A, **p < 0.01) or three (B, *p < 0.05) individual experiments.

Figure 3

 3 Figure 2 AMPK phosphorylates multiple tau sites and interferes with microtubule binding. A) MARK4 (M)-or AMPK (A)-phosphorylated recombinant tau was analysed by western blotting using phospho-specific tau antibodies (epitope indicated above each panel) and total tau (lower panels). B) A schematic of the longest isoform of tau (2N4R; 441 amino acids) showing domains and known phosphorylation sites of serine / threonine kinases or sites that are phosphorylated in vitro or in post-mortem Alzheimer's disease brain [5, 51]. AMPK sites identified from mass spectrometry analysis are shown in bold. C) Unphosphorylated (tau) or AMPK-phosphorylated (ptau) tau was incubated in the presence or absence of microtubules (MT). Microtubule bound (pellet) or microtubule unbound (supernatant) fractions were generated by ultracentrifugation. Fractions were analysed by western blot for total tau (upper panel). The lower panel shows sedimented microtubules by coomassie staining. The histogram represents the bound versus unbound tau (**p < 0.01) or ptau ( ## p < 0.01) as a percentage of total tau. For A and C, images are representative of four individual experiments.Figure 3 AMPK is activated by Aβ 1-42 . Primary neurons from wild-type (WT) (A,B,D) and CaMKKβ -/-(C) mice were treated with 20µM Aβ 1-42 . Treated and untreated neurons were harvested after 30 min and kinase activity determined in immune complexes of AMPK (A, C). Box plots show the minimum, median and maximum activities obtained for neurons from between 3 and 6 individual experiments (**p < 0.01). B) Lysates from wild-type neurons treated with Aβ 1-42 were analysed by western blot with antibodies recognising phosphorylated (active) AMPK and a known AMPK substrate, acetyl Co-A carboxylase (pACC). Western blot using a panβ-AMPK antibody for total AMPK and anti-actin antibodies demonstrate there is no change in the amount of total protein. Representative images shown are neuronal lysates from two of four individual experiments. D) Lysates from treated and untreated neurons were subjected to sequential immunoprecipitation with either AMPKα1-or AMPKα2-specific antibodies to isolate all available endogenous isoform-specific AMPK complexes. Results shown represent the total isoform-specific activity from neuronal lysates of three individual experiments (*p < 0.05).Figure 4 Increases in tau phosphorylation in response to Aβ 1-42 are muted in cells lacking CaMKKβ. Primary cortical neurons from wild-type (WT) (A, C) and CaMKKβ -/-(B, C) mice were treated with Aβ 1-42 as before and analysed by western blot for phosphorylated tau using 12E8 (pS262/pS356) and PHF-1 (pS396/pS404) antibodies (A,B) or for phosphorylationindependent total tau (C). Quantification of phosphorylation (A,B) and total tau (C) is shown in the histograms normalised relative to actin (*p < 0.05). Images shown are primary cell lysates from two of four (WT) or two of three (CaMKKβ -/-) individual experiments.Figure 5 Activation of AMPK is prevented by the NMDA receptor antagonist memantine. A) Primary neurons were left untreated, pre-treated with either STO609 or memantine alone, or treated with NMDA. Neurons were harvested and AMPK activity determined from immune complexes. B) Primary neurons either untreated or pre-treated with memantine were treated with Aβ 1-42 . Neurons were harvested and assayed for AMPK activity as above. Box plots show the minimum, median and maximum activities obtained for neurons from four (A, **p < 0.01) or three (B, *p < 0.05) individual experiments.

Figure 4

 4 Figure 2 AMPK phosphorylates multiple tau sites and interferes with microtubule binding. A) MARK4 (M)-or AMPK (A)-phosphorylated recombinant tau was analysed by western blotting using phospho-specific tau antibodies (epitope indicated above each panel) and total tau (lower panels). B) A schematic of the longest isoform of tau (2N4R; 441 amino acids) showing domains and known phosphorylation sites of serine / threonine kinases or sites that are phosphorylated in vitro or in post-mortem Alzheimer's disease brain [5, 51]. AMPK sites identified from mass spectrometry analysis are shown in bold. C) Unphosphorylated (tau) or AMPK-phosphorylated (ptau) tau was incubated in the presence or absence of microtubules (MT). Microtubule bound (pellet) or microtubule unbound (supernatant) fractions were generated by ultracentrifugation. Fractions were analysed by western blot for total tau (upper panel). The lower panel shows sedimented microtubules by coomassie staining. The histogram represents the bound versus unbound tau (**p < 0.01) or ptau ( ## p < 0.01) as a percentage of total tau. For A and C, images are representative of four individual experiments.Figure 3 AMPK is activated by Aβ 1-42 . Primary neurons from wild-type (WT) (A,B,D) and CaMKKβ -/-(C) mice were treated with 20µM Aβ 1-42 . Treated and untreated neurons were harvested after 30 min and kinase activity determined in immune complexes of AMPK (A, C). Box plots show the minimum, median and maximum activities obtained for neurons from between 3 and 6 individual experiments (**p < 0.01). B) Lysates from wild-type neurons treated with Aβ 1-42 were analysed by western blot with antibodies recognising phosphorylated (active) AMPK and a known AMPK substrate, acetyl Co-A carboxylase (pACC). Western blot using a panβ-AMPK antibody for total AMPK and anti-actin antibodies demonstrate there is no change in the amount of total protein. Representative images shown are neuronal lysates from two of four individual experiments. D) Lysates from treated and untreated neurons were subjected to sequential immunoprecipitation with either AMPKα1-or AMPKα2-specific antibodies to isolate all available endogenous isoform-specific AMPK complexes. Results shown represent the total isoform-specific activity from neuronal lysates of three individual experiments (*p < 0.05).Figure 4 Increases in tau phosphorylation in response to Aβ 1-42 are muted in cells lacking CaMKKβ. Primary cortical neurons from wild-type (WT) (A, C) and CaMKKβ -/-(B, C) mice were treated with Aβ 1-42 as before and analysed by western blot for phosphorylated tau using 12E8 (pS262/pS356) and PHF-1 (pS396/pS404) antibodies (A,B) or for phosphorylationindependent total tau (C). Quantification of phosphorylation (A,B) and total tau (C) is shown in the histograms normalised relative to actin (*p < 0.05). Images shown are primary cell lysates from two of four (WT) or two of three (CaMKKβ -/-) individual experiments.Figure 5 Activation of AMPK is prevented by the NMDA receptor antagonist memantine. A) Primary neurons were left untreated, pre-treated with either STO609 or memantine alone, or treated with NMDA. Neurons were harvested and AMPK activity determined from immune complexes. B) Primary neurons either untreated or pre-treated with memantine were treated with Aβ 1-42 . Neurons were harvested and assayed for AMPK activity as above. Box plots show the minimum, median and maximum activities obtained for neurons from four (A, **p < 0.01) or three (B, *p < 0.05) individual experiments.

Figure 5

 5 Figure 2 AMPK phosphorylates multiple tau sites and interferes with microtubule binding. A) MARK4 (M)-or AMPK (A)-phosphorylated recombinant tau was analysed by western blotting using phospho-specific tau antibodies (epitope indicated above each panel) and total tau (lower panels). B) A schematic of the longest isoform of tau (2N4R; 441 amino acids) showing domains and known phosphorylation sites of serine / threonine kinases or sites that are phosphorylated in vitro or in post-mortem Alzheimer's disease brain [5, 51]. AMPK sites identified from mass spectrometry analysis are shown in bold. C) Unphosphorylated (tau) or AMPK-phosphorylated (ptau) tau was incubated in the presence or absence of microtubules (MT). Microtubule bound (pellet) or microtubule unbound (supernatant) fractions were generated by ultracentrifugation. Fractions were analysed by western blot for total tau (upper panel). The lower panel shows sedimented microtubules by coomassie staining. The histogram represents the bound versus unbound tau (**p < 0.01) or ptau ( ## p < 0.01) as a percentage of total tau. For A and C, images are representative of four individual experiments.Figure 3 AMPK is activated by Aβ 1-42 . Primary neurons from wild-type (WT) (A,B,D) and CaMKKβ -/-(C) mice were treated with 20µM Aβ 1-42 . Treated and untreated neurons were harvested after 30 min and kinase activity determined in immune complexes of AMPK (A, C). Box plots show the minimum, median and maximum activities obtained for neurons from between 3 and 6 individual experiments (**p < 0.01). B) Lysates from wild-type neurons treated with Aβ 1-42 were analysed by western blot with antibodies recognising phosphorylated (active) AMPK and a known AMPK substrate, acetyl Co-A carboxylase (pACC). Western blot using a panβ-AMPK antibody for total AMPK and anti-actin antibodies demonstrate there is no change in the amount of total protein. Representative images shown are neuronal lysates from two of four individual experiments. D) Lysates from treated and untreated neurons were subjected to sequential immunoprecipitation with either AMPKα1-or AMPKα2-specific antibodies to isolate all available endogenous isoform-specific AMPK complexes. Results shown represent the total isoform-specific activity from neuronal lysates of three individual experiments (*p < 0.05).Figure 4 Increases in tau phosphorylation in response to Aβ 1-42 are muted in cells lacking CaMKKβ. Primary cortical neurons from wild-type (WT) (A, C) and CaMKKβ -/-(B, C) mice were treated with Aβ 1-42 as before and analysed by western blot for phosphorylated tau using 12E8 (pS262/pS356) and PHF-1 (pS396/pS404) antibodies (A,B) or for phosphorylationindependent total tau (C). Quantification of phosphorylation (A,B) and total tau (C) is shown in the histograms normalised relative to actin (*p < 0.05). Images shown are primary cell lysates from two of four (WT) or two of three (CaMKKβ -/-) individual experiments.Figure 5 Activation of AMPK is prevented by the NMDA receptor antagonist memantine. A) Primary neurons were left untreated, pre-treated with either STO609 or memantine alone, or treated with NMDA. Neurons were harvested and AMPK activity determined from immune complexes. B) Primary neurons either untreated or pre-treated with memantine were treated with Aβ 1-42 . Neurons were harvested and assayed for AMPK activity as above. Box plots show the minimum, median and maximum activities obtained for neurons from four (A, **p < 0.01) or three (B, *p < 0.05) individual experiments.

Figure 1

 1 Figure 1 Thornton et al

Figure 3 Figure 5 Biochemical

 35 Figure 3 Thornton et al

Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.© 2011 The Authors Journal compilation © 2011 Portland Press Limited

Biochemical Journal Immediate Publication. Published on 05 Jan 2011 as manuscript BJ20101485 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20101485 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2011 The Authors Journal compilation © 2011 Portland Press Limited

© 2011 The Authors Journal compilation © 2011 Portland Press Limited

ACKNOWLEDGMENTS

We are very grateful to Prof Michel Goedert (LMB, Cambridge) for the pRK172-htau46 plasmid, Peter Davies (Albert Einstein College of Medicine) for the PHF-1 antibody and Peter Seubert (Elan Pharmaceutical) for the 12E8 antibody. This work was supported by the Medical Research Council (N.J.B., P.J.M., D.C), Imperial College (M.S.) and a 30 th anniversary Research into Ageing fellowship award (C.T.)


H E T Holgersson
email: thomas.holgersson@ihh.hj.se

A modified skewness measure for testing asymmetry By

Keywords: Skewnes, Median, Bootstrap, graphics Skewness, median, bootstrap, graphics. 1

Statistical practitioners frequently wish to know whether a variable is symmetrically distributed. There are a number of different tests available but the most commonly used one is perhaps that based on the standardised third central moment, as defined by Pearson and Fisher in the early 1900:s. While this traditional skewness measure uniquely determines the symmetry of a variable within the Pearson family, it does not uniquely determine symmetry for a general distribution. In this paper we propose a modified version of the classical skewness test which is easy to conduct and consistent against a wide family of asymmetric distributions.

Statistical analysis frequently involves assessing the shape of the distribution. In particular, it is often of great importance to know if a variable is symmetric about its centre or not. Examples include studies conducted of income distributions, stochastic variation in quality control and pharmaceutical analyses, which requires appropriate statistical methods. According to the literature it appears that in the early 19th century, skewness and asymmetry were almost considered as identical concepts (for example, [START_REF] Yule | An introduction to the theory of statistics[END_REF][START_REF] Pearson | Das fehlergesetz und seine verallgemeinerungen durch Fechner und Pearson[END_REF]. Later on the standardised third order central moment came to dominate the literature, and it is sometimes considered to be identical to the term "skewness statistic", even though it is formally only one of several possible measures. [START_REF] Fisher | Moments and product moments of sampling distributions[END_REF] derived sampling distributions of the third central moment and therefore, in a sense, put this statistic into a context of hypothesis testing. The third central moment is well known to uniquely determine the (a)symmetry of distributions within the Pearson family [START_REF] Ord | Families of frequency distributions[END_REF] and is nowadays a standard option in most computer packages. In the 1940s and later, it was recognised that tests based on this statistic cannot be expected to have power against asymmetric distributions with zero third order central moment [START_REF] Churchill | Information given by odd moments[END_REF]. Hence a number of alternative tests have been developed which all have their own pros and cons. To mention some important contributions, [START_REF] Antille | Testing symmetry[END_REF] considered a family of asymmetry tests for unimodal distributions based on symmetric differences of gaps, while [START_REF] Doksum | Plots and tests for symmetry[END_REF] proposed plots with confidence bounds to assess asymmetry. [START_REF] Csörgö | Testing for symmetry[END_REF] proposed an asymmetry test based on the fact that a random variable is symmetrically distributed about its mean if, and only if, its characteristic function is real. In general, no test can be said to be uniformly superior; issues of robustness to irrelevant distributional properties, levels of complication and size/power properties vary strongly between families of tests.

In this paper we introduce a modification of the standardized third central moment to test whether a random variable is symmetrically distributed. More specifically, the test statistic proposed is shown to be a function of the standardized third central moment plus a function of the difference between the mean and median. Hence it has the potential to detect asymmetric distributions that tests based only on the third central moment cannot.

Moreover, bootstrap consistency of the test statistic is proved and hence an operational It is also demonstrated that the bootstrap consistency of the proposed statistic allows for graphical investigation of the (a)symmetry of the investigated variable. In the following sections we will pay more attention to these matters.

Testing symmetry versus asymmetry

A random variable X is said to be symmetric if and only if

( ) ( ) 1 X X F a x F a x -+ + = for
all x in . The definition itself does not lead to a feasible test of symmetry and a number of approaches have been proposed in the literature. In this section we will present a modified skewness coefficient for testing asymmetry, restricting ourselves to properties of purely continuous variables. One of the most commonly applied measures is perhaps the classical skewness coefficient defined by (  )

1 1 j n j i i m n X X = = - ∑ . If X is distributed independently with kurtosis 2 4 2 3 β µ µ = = and skewness 0 γ = , then
we have the well-known result ( )

6 0,1 n N γ → l , (2.2)
(e.g. [START_REF] Bowman | Omnibus test contours for departures from normality based on 1 2 and b b[END_REF]. Hence the null hypothesis 0 H : 0 γ = may be rejected when the squared left hand side of (2.2) exceeds a pre-determined percentile of the ( )

2 1 χ distribution.
In the following we will refer to that test as the Fisher skewness test. An alternative test was proposed by [START_REF] Holgersson | Robust testing for skewness[END_REF] who showed that the distribution of m may be consistently estimated by bootstrapping, without the assumptions of homoscedasticity and a kurtosis of the value 3. This paper is not so much concerned with robustness aspects but rather the family of asymmetric distributions that the skewness coefficient can detect. For distributions within the Pearson family the γ coefficient is well known to uniquely determining whether a variable is symmetric or not [START_REF] Ord | Families of frequency distributions[END_REF].

On the other hand, there are asymmetric distributions outside the Pearson family where the third central moment vanishes and the test based on (2.2) thus cannot have power against these distributions (Churchill, 1946, Stuart and[START_REF] Stuart | Kendall´s Advanced Theory of Statistics[END_REF]. We will therefore consider a modified version of γ . Let the population mean and variance be defined by

[ ] E X µ = and [ ] 2 2 E X σ µ = -
respectively and let ψ be an arbitrary measure of centrality of X that coincides with µ for symmetric distributions. We may then obtain a family of skewness measures by [ ]

3 3 E X θ σ ψ - = - .
In order to investigate the relation between this statistic and the regular skewness measure we expand it about µ :

[ ] [
] ( )

3 3 3 3 2 3 E X E X θ σ µ µ ψ σ µ λ λ - - = -+ - = - + + (2.3)
where ( )

λ µ ψ σ = -
. The first term of (2.3) is clearly the γ coefficient. Thus, even though θ cannot be expected to uniquely determine symmetry for all variables, it is evidently more general than the traditional γ skewness measure and has therefore great potential for asymmetry testing. The crucial question is then which ψ to employ. Two obvious candidates are the mode ( m ) and median ( M ) respectively, but other possibilities include linear functions of order statistics (L-statistics). Ideally, one would like to have ψ far from µ for asymmetric distributions but there is also a simplicity aspect. [START_REF] Stuart | Kendall´s Advanced Theory of Statistics[END_REF] gives a definition of unimodality and argue that ( ) ( )

3 m M µ µ
-≈ for unimodal distributions with only moderate asymmetry, while [START_REF] Hotelling | The limits of a measure of skewness[END_REF] argue that ( )

M µ σ - is bounded whereas ( ) m µ σ - is
not. This speaks in favour of the mode over the median. The mode will not, however, be very practical for multimodal distributions as the modes may then be far from the centre of the distribution, even for symmetric distributions. Moreover, using the mode will lead to difficulties in finding an operational test statistic. As simplicity is an important issue in this paper we will therefore disregard the mode and restrict ourselves to the median, M .

Hence our proposed measure of asymmetry takes the form [ ]

3 3 E X M θ σ - = - , i.e.,
[ ] ( )

3 3 2 3 + E X θ σ µ λ λ γ ν - = - + + = (2.4) ( )
where

M λ µ σ = - , [ ] 3 3 E X γ σ µ - = - and
( )

2 3 ν λ λ = + . A sample version of θ is obtained by ( ) ( ) ( ) ( ) 3 3 1 2 1 ˆ1 n i n i n X X θ σ - + = = - ∑ (2.5)
where

( ) ( ) 1 2 n X +
is the sample median defined by the ( )

1 2 : n th + ordered value of { } 1 n i i X = , ( ) ( ) 2 2 1 ˆ1 n i i n X X σ = = - ∑
and the sample size n is taken to be odd, i.e.

( )

2 1 n + ∈ + .
Even sample sizes require a small modification of the sample median (e.g. [START_REF] David | Order statistics[END_REF]. Note that (2.5) may be expanded in two equivalent forms:

ˆˆθ γ ν = + (2.6) and ( ) ( ) ( ) ( ) ( ) ( )
( )

2 3 3 3 2 1 1 2 1 2 1 2 ˆ3 3 n n n m m X m X X θ σ + + + ′ ′ ′ = - + - (2.7)
where

( ) 1 1 n r r i i m n X = ′ = ∑ , ( )
3 3 1 1 ˆˆn i i n X X γ σ -- = = - ∑ , ( ) ( ) ( ) 1 1 2 ˆn m X λ σ + ′ = - and
( ) 2 ˆ3 ν λ λ = + . Equation (2.6
) is convenient in examining the difference between θ and the γ while (2.7) is more convenient when establishing asymptotical properties. In particular, if [START_REF] Serfling | Approximation theorems of mathematical statistics[END_REF]). Hence it follows that Theorem 1: Let i X be a strictly continuous iid random variable such that ( ) ( )

( ) r r i E X µ = < ∞ and i X is iid distributed with 9 i E X δ +   < ∞   then ( ) ( ) 2 1 2 r r p E m n δ µ ο -   ′ - =   1, 2,3 r = , for all 0 δ > . Furthermore, if i X is continuous
0 f M f M < -= + < ∞ (the
( ) { } ( ) ( ) ( ) 1 2 2 1 2 0, 1 4 n n X M N f M + - → l (e.g.
( ) ( ) ( ) ( ) 2 1 2 1 2 p n E X M n δ ο - +   - =     and so . .
0 f M f M < -= + < ∞ and [ ] 9 i E X δ + < ∞ , 0 δ > . Also, let [ ] 3 E X M φ = - , ( ) ( ) ( ) ( ) ( )
( )

3 1 2 1 ˆ1 n i n i n X M M X φ + = = - + - ∑
and * φ be the bootstrap version of φ .

Then

* * ˆˆˆ0 p P n x P n x ρ φ φ φ φ ∞ -≤ - -≤ → ( ) ( ) ( ) ( ) ( )
, where ρ ∞ is the sup-norm distance and * P denotes bootstrap probability.

Proof:

Note that

( ) ( ) ( ) ( ) ( ) ( ) 3 1 2 1 ˆ1 n i n i n X M M X φ + = = - + - ∑ 2 3 3 2 1 3 3 µ δµ δ µ δ = - + - % % % % % % where ( ) ( ) 1 1 n k k i i n X M µ = = - ∑ % and ( ) ( ) ( ) 1 2 n X M δ + = - %
. The finite ninth order moment is sufficient for

( ) ( ) ( ) ( ) ( ) . . * * 0 a s k k k k P n x P n x ρ µ µ µ µ ∞ - ≤ - - ≤ → % % % , 1, 2,3 k =
, (see [START_REF] Holgersson | Robust testing for skewness[END_REF] while the smoothness of f is sufficient for [START_REF] Shao | The Jacknife and Bootstrap[END_REF] and θ is a smooth, measurable function of φ and 2 σ .

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . . * * 1 2 1 2 1 2 0 a s n n n P n X X x P n X M x ρ ∞ + + + - ≤ - - ≤ → (Shao
( ) ( ) ( ) ( ) . . 2* 2 2 2 * ˆˆˆ0 a s P n x P n x ρ σ σ σ σ ∞ - ≤ - - ≤ → (
Corollary 2:

Let { } 2 : 3 ν λ λ = + , { } 2 ˆ: 3 ν λ λ = + , ( ) ( ) 2
* * * ˆˆˆˆˆ, , 0 p P n x n y P n x n y ρ γ γ ν ν γ γ ν ν ∞ -≤ -≤ - -≤ -≤ → , ( ) ( ) ( ) ( ) ( ) ( ) ( )
that is, * γ and * ν converge jointly in 2 . This follows since * ν converges according to Theorem 1 and Corollary 1. The convergence of * γ is established in [START_REF] Holgersson | Robust testing for skewness[END_REF].

Finally, when ( )

* * ˆ,
γ ν are obtained jointly from the same resample, the joint distribution follows.

Hence, according to Corollary 1 the proposed test statistic θ may be consistently Before the properties of the proposed test is examined further some things should be noted. Firstly, the proposed statistic θ is here mainly used to provide a simple and (in some sense) useful test of symmetry versus asymmetry. In case the analysis primarily

( ) ( )

vs. G x F x , then other measures may be more adequate. For example, just because ( ) G x is more skewed than ( ) F x with respect to γ , it does not imply that the same skewness ordering is preserved with respect to θ . More specifically, the λ statistic (and hence the θ statistic) does not preserve the skewness ordering in the sense of van Zwet (1964). An in-depth discussion of orderings of various skewness/asymmetry measures, including γ and λ , is given in [START_REF] Macgillivray | Skewness and assymmetry: Measures and orderings[END_REF]. Secondly, apart from the power to detect "unexpected" asymmetric distributions (e.g. with zero third central moment) there is also an issue of the power to detect "expected" deviations from asymmetry. In particular, since the γ coefficient does uniquely determine (a)symmetry of Pearson distributions, and such distributions arise naturally within many sciences, it is tractable to have a test with high power to detect these distributions. Since the γ coefficient is an ingredient within the proposed measure it, the test based on it may indeed be expected to have relatively high power against such distributions. Thirdly, as there are two terms of the θ statistic there is a possibility that they are of opposite sign and hence cancel out each other. Specifically, since ( ) and0 σ > , the sign of λ will play an important role in the comparison of θ and γ . Sufficient conditions for determining the sign of ( )

2 3 0 λ + >
M µ -
are given in van Zweet (1979) and Groenveld and Meeden (1984). A particularly important contribution is that if x is unimodal and ( ) ( )

f x f x µ µ
+ -change sign once in x, then λ and γ will bear the same sign (MacGillivray, 1981) so that the proposed test should have relatively high power against these distributions.

However, there does exist distributions such that γ and ν will have opposite signs, even though the set of such distributions seem rather small (empirically). Fortunately this problem need not be that serious. According to Corollary 2, there is an option of analysing the distribution of γ and ν separately or even jointly. For example, bootstrap histograms of * γ and * ν will be helpful to empirically assess their individual signs.

Cases when one of them is significantly smaller than zero and the other significantly larger than zero would indicate a peculiar distribution of X, and further investigation such Doksum, 1977, Wilk andGnanadesikan, 1968) should be conducted. In the next section we will investigate some basic properties of the proposed test, including the option of separate analysis of γ and ν .

Empirical investigation

In this section we will investigate some of the properties of the proposed test and also compare it with the bootstrapped γ coefficient and the traditional Fisher skewness test based on the asymptotic ( )

2 1
χ distribution. We will investigate the size and power properties by calculating the rejection frequencies of r = 10 000 replications of different symmetric and asymmetric distributions respectively, using 99 B = bootstrap resamples in each replicate. The distributions used in the simulation are specified in Table 1 below.

Details of the distributions may be found in e.g. [START_REF] Johnson | .ca Communications in Statistics -Simulation and Computation class of densities[END_REF] and [START_REF] Ramberg | A probability distribution and its uses in fitting data[END_REF]. Moreover, to demonstrate the possibility of investigating the signs of γ vs. bootstrap resamples (Fig. 123).

Insert Table 1 about here

Power simulations are usually conducted by letting the parameter of relevance increase away from the null hypothesis while keeping the sample size fixed. This is not quite appropriate when investigating tests for distributional properties as there is no unique way to quantify asymmetry. We will instead investigate the proposed test for a fixed distribution and let the sample size increase from 30 pseudo observations up to 10 000.

All simulations involve testing at the nominal level of 5%. In Table 2 below we display the rejection frequencies of the three tests when operating on the normal distribution and t (4) distribution respectively. All tests are seen to behave well for the normal distribution.

For the t (4) distribution the tests of the bootstrapped γ and θ statistic over reject with up to 14%, which may seem high. It should, however, be remembered that the null distribution of these tests assumes all first nine moments to be finite, while the t (4) distribution lacks moments above order four. Thus it is curious to see that the bootstrap tests actually do limit their nominal levels asymptotically even though higher order moments do not exist. The null distribution of the Fisher test requires the kurtosis coefficient to be 3, and it is clearly seen that it diverges for the t (4) distribution. We therefore present it in brackets in cases when it is not valid (i.e. for all distributions but the normal distribution). As regards the other two symmetric distributions in Table 3, the proposed test is seen to behave well and rapidly limits its nominal size, as does the test based on the bootstrapped γ skewness coefficient. The Fisher test is again seen to behave peculiarly. In Table 4 some multimodal distributions are investigated. The trimodal distribution is symmetric and the rejection frequencies stays fairly close to the nominal size for this distribution though both the bootstrapped γ test and the proposed test over rejects, but it is revealing that multimodality does not impose any serious distortions.

Also, both bootstrapped tests detect the bimodal asymmetric t-distribution but the rejection frequencies does not quite limit 100% within the investigated sample sizes. The bootstrap histograms of the separate components of the proposed statistic ( )

ˆˆ and γ ν is presented in Figure 1 where it is seen that they have the same sign although the ν statistic has a higher absolute value and also a smaller sampling variation. In Table 5 some power properties of unimodal distributions are investigated. The gamma distribution with shape parameter equal to 500 yields a distribution that is very close to the normal distribution though not exactly symmetric. The proposed test is clearly consistent against this distribution though slightly inferior to that of the bootstrapped γ coefficient. The power properties against the exponential distribution are clearly higher than those of the near-normal distribution, and again the bootstrapped skewness coefficient is slightly better than the proposed test. The rejection frequencies for the asymmetric distributions with zero third order central moment are shown in Table 6.

Indeed, the power of the bootstrapped γ coefficient limits the nominal size (0.05) for the centred gamma distribution while the proposed test is consistent as expected, in the sense

Conclusive summary

In this paper we propose a modification of the traditional skewness coefficient (defined by the standardized third central moment) to test for symmetry versus (a)symmetry.

Instead of the third moment being centred about its mean value it is centred about the median. This adjusted measure is shown to be a function of the traditional skewness coefficient plus a function of the difference between the mean and median. While the proposed measure might not be optimal for the problem of ordering the amount of skewness between two distributions, it is argued that it is more useful than the traditional skewness test for the problem of discriminating between symmetric and asymmetric distributions, specifically for distributions with zero third order central moment.

Bootstrap consistency is proved for the sample version of the proposed measure and so an operational test procedure is available. Moreover, a Monte Carlo simulation was conducted to investigate some size and power properties of the proposed test versus two tests based on the third central moment, and it is demonstrated that the new test is indeed consistent against asymmetric distributions which tests based on the classical skewness measure cannot detect. The test only relies on some moment restrictions and that the distribution being analysed is smooth at the median. In particular, it is demonstrated that multimodality imposes no distortions to the test. It is also shown that the two terms within the proposed test statistic may be investigated separately by bootstrap histograms.

This graphical option will provide some additional insight into the (a)symmetry of the variable being investigated. Generally, the proposed test is not meant to replace other methods, such as graphical tools, but rather it is seen as a supplementary test. In particular, the proposed test along with separate analysis of the two components of the proposed measure and plots of ordered observations should provide a powerful tool set for investigating whether a variable is symmetrically distributed.

  . A Monte Carlo simulation is included to investigate some size and power properties of the proposed test relative to tests based on only the third central moment.

  r:th central moment. The skewness coefficient plays an important role as a characteristic of a random variable and is frequently used as a measure of shape. The null hypothesis 0

  other words, the point estimates of θ and λ are consistent and have limiting distributions. We will now establish bootstrap consistencies of the test statistic.

  [START_REF] Shao | The Jacknife and Bootstrap[END_REF]. Finally, since φ is a measurable function of k µ % and k δ % ( 1

  version of γ . Then

  of comparing skewness between two distributions, e.g.

  that the rejection frequency increases monotonically with the sample size. Moreover, Fisher test simultaneously tests against the kurtosis and skewness, the power of the proposed test is nevertheless of the same magnitude as that of the Fisher test. The same pattern is seen from the Tukey´s lambda distribution where the rejection frequencies of the bootstrapped γ coefficient again limits 0.05 while the proposed test fairly quickly limits a rejection frequency of 100%. Moreover, the sampling distributions of the separate components of the proposed statistic are displayed in Figures2 and 3, and clearly reveal that γ is centered about zero while ν has no overlap at zero. These figures hence gives some insight to the very different behavior of the rejection frequencies of

Figure 1 .

 1 Figure 1. Bootstrap histogram of the bimodal t (4) distribution

  left and right derivatives of the densities at M are

	equal	and	finite)	we	have	the	well	known	result
	( )

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Table 6 ,

 6 and also demonstrates the use of separate analyses of the two components of the proposed statistic. Finally, it should be stressed that even though the power of the proposed test may seem rather low against some of the distributions, the level of asymmetry (whatever the criteria) vary largely among the distributions which must be taken into consideration. To sum up the discussion, the size properties of the proposed test are more or less the same as those of the test of the bootstrapped γ skewness coefficient but it does have additional power against asymmetric distributions which tests based on γ cannot detect. Hence the proposed test should be a useful supplement to other tests for asymmetry versus symmetry.

	Insert

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	Wilk, M. B. and Gnanadesikan R. (1968). Probability plotting methods for analysis of
	data. Biometrika. 55:1-17.
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Table 1 .

 1 Variates used in the simulation

	Table 2. Standard normal distribution Nobs γ θ Fisher	γ	t(4) θ	Fisher
	30	0.048	0.039	0.033	0.113	0.084	(0.297)
	50	0.049	0.049	0.043	0.132	0.103	(0.385)
	100	0.049	0.047	0.043	0.137	0.112	(0.501)
	300	0.051	0.050	0.053	0.119	0.108	(0.644)
	500	0.050	0.051	0.048	0.108	0.100	(0.682)
	2000	0.049	0.046	0.051	0.075	0.072	(0.790)
	5000	0.052	0.052	0.052	0.061	0.062	(0.846)
	10000	0.049	0.051	0.050	0.054	0.054	(0.871)
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	Nobs	γ	Beta	θ	Fisher	γ	Uniform θ	Fisher
	30	0.038	0.053	(0.005)	0.034	0.039	(0.002)
	50	0.047	0.055	(0.004)	0.046	0.040	(0.001)
	100	0.044	0.050	(0.003)	0.051	0.044	(0.001)
	300	0.050	0.054	(0.002)	0.048	0.047	(0.001)
	500	0.049	0.051	(0.003)	0.048	0.048	(0.001)
	2000	0.049	0.051	(0.003)	0.050	0.052	(0.001)
	5000	0.050	0.049	(0.004)	0.053	0.054	(0.001)
	10000	0.051	0.050	(0.003)	0.051	0.049	(0.001)

Table 4 .

 4

	Nobs	γ	Trimodal normal θ	Fisher	γ	Bimodal t(4) θ	Fisher
	30	0.089	0.046	(0.047)	0.495	0.496	(0.209)
	50	0.085	0.038	(0.043)	0.626	0.741	(0.354)
	100	0.073	0.031	(0.035)	0.778	0.930	(0.642)
	300	0.065	0.045	(0.044)	0.883	0.974	(0.938)
	500	0.065	0.051	(0.052)	0.901	0.976	(0.966)
	2000	0.061	0.056	(0.064)	0.946	0.990	(0.994)
	5000	0.066	0.059	(0.065)	0.960	0.992	1
	10000	0.064	0.059	(0.071)	0.970	0.993	1

Table 5 .

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	Nobs	γ	Gamma	θ	Fisher	γ	Exponential θ	Fisher
	30	0.063	0.022	(.039)	0.779	0.437	(0.867)
	50	0.073	0.036	(.053)	0.832	0.673	(0.987)
	100	0.083	0.054	(.068)	0.884	0.873	(1)
	300	0.137	0.101	(.126)	0.968	0.980	(1)
	500	0.189	0.145	(.178)	0.991	0.996	(1)
	2000	0.574	0.473	(.559)	1	1	(1)
	5000	0.915	0.840	(.911)	1	1	(1)
	10000	0.997	0.986	(.997)	1	1	(1)

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Table 6 .

 6 Centred Gamma

	Tukey´s lambda

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Distribution

Parameter values Shape

Skewness coefficient

Standard normal ( )

) . **The addition here refers to "mixing", with the respective probabilities adding up to one.