Mi Deng

Pei-Chao Chen

Sisi Xie

Junqiong Zhao

Lili Gong

Jinping Liu

Lan Zhang

Shuming Sun

Jiao Liu

Haili Ma

Surinder Batra

Dr David Wan-Cheng Li

The Small Heal Shock Protein AlphaA-Crystallin Is Expressed In Pancreas and Acts as A Negative Regulator of Carcinogenesis (Short Title: αA-Crystallin Is Down-Regulated in Pancreatic carcinoma)

Keywords: Small heat shock protein, αA, Pancreas, AP-1, Smad2/3/5, TGFβ Pancreatic Cancer, cell migration

The small heat shock protein αA-crystallin is a structural protein in the ocular lens. In addition, recent studies have also revealed that it is a molecular chaperone, an autokinase and a strong anti-apoptotic regulator. Besides its lenticular distribution, a previous study demonstrates that a detectable level of αA-crystallin is found in other tissues including thymus and spleen. In the present study, we have re-examined the distribution of αA-crystallin in various normal human and mouse tissues and found that the normal pancreas expresses a moderate level of αA-crystallin. Moreover, αA-crystallin is found significantly downregulated in 60 cases of pancreatic carcinoma of different types than it is in 11 normal human pancreas samples. In addition, we demonstrate that αA-crystallin can enhance activity of the activating protein-1 (AP-1) through modulating the function of the MAP kinase, and also upregulates components of TGFβ pathway. Finally, expression of αA-crystallin in a pancreatic cancer cell line, miapaca, results in retarded cell migration.

Together, these results suggest that αA-crystallin seems to negatively regulate pancreatic carcinogenesis.

Introduction

Heat-shock proteins (HSPs) are implicated in multiple cellular functions including signaling transduction, protein degradation, survival promotion, exocytosis and endocytosis [START_REF] Benndorf | Shocking degeneration[END_REF][START_REF] Sun | The small heat shock proteins and their role in human disease[END_REF][START_REF] Bennardini | Alpha B-crystallin in cardiac tissue. Association with actin and desmin filaments[END_REF][START_REF] Li | The two lens structural proteins, αAand αB-crystallins, prevent stress-induced apoptosis through regulation of multiple signaling transduction pathways[END_REF][START_REF] Benjamin | CRYAB and HSPB2 deficiency alters cardiac metabolism and paradoxically confers protection against myocardial ischemia in aging mice[END_REF]. Recent studies have revealed that HSPs also play an important role in tumorigenesis [START_REF] Sherman | Heat shock proteins in cancer[END_REF][START_REF] Didelot | Anti-cancer therapeutic approaches based on intracellular and extracellular heat shock proteins[END_REF][START_REF] Ciocca | Heat shock proteins in cancer: diagnostic, prognostic, predictive, and treatment implications[END_REF]. The lens structural proteins, αAand αB-crystallins are characterized as HSPs [START_REF] Reddy | Chaperone-like activity and hydrophobicity of alpha-crystallin[END_REF]. Besides their robust expression in mammalian lens [START_REF] Robinson | Differential expression of alpha A-and alpha B-crystallin during murine ocular development[END_REF][START_REF] Piatigorsky | Lens crystallins and their genes: diversity and tissue-specific expression[END_REF], the two α-crystallins are also expressed in non-lenticular tissues. αB-crystallin is significantly expressed in retina, muscle, heart and brain, and acts to prevent neural degradation, heart fail, but promote carcinogenesis [START_REF] Li | The two lens structural proteins, αAand αB-crystallins, prevent stress-induced apoptosis through regulation of multiple signaling transduction pathways[END_REF][START_REF] Benjamin | CRYAB and HSPB2 deficiency alters cardiac metabolism and paradoxically confers protection against myocardial ischemia in aging mice[END_REF][START_REF] Bhat | alpha B subunit of lens-specific protein alphacrystallin is present in other ocular and non-ocular tissues[END_REF][START_REF] Dubin | Expression of the murine alpha B-crystallin gene is not restricted to the lens[END_REF][START_REF] Kase | Expression of alpha-crystallin in retinoblastoma[END_REF][START_REF] Moyano | AlphaBcrystallin is a novel oncoprotein that predicts poor clinical outcome in breast cancer[END_REF]. αA-crystallin is also expressed in some non-lenticular tissues, such as retina, spleen and thymus [START_REF] Kato | Immunoreactive alpha A crystallin in rat non-lenticular tissues detected with a sensitive immunoassay method[END_REF].

However, the function of non-lenticular αA-crystallin remains unknown.

Pancreatic cancer in human poses clinical challenges in diagnosis and treatment [START_REF] Ujiki | Surgical management of pancreatic cancer[END_REF][START_REF] Niederhuber | The National Cancer Data Base report on pancreatic cancer[END_REF]. Among the various types of pancreatic malignancies, pancreatic duct adenocarcinoma is dominant and in addition, the mucinous adenocarcinoma, islet adenocarcinoma and acinic cell carcinoma are also observed with less frequency of occurrence [START_REF] Pezzilli | Pancreatic ductal carcinoma: from the bench to the bedside[END_REF][START_REF] Hezel | Genetics and biology of pancreatic ductal adenocarcinoma[END_REF]. At the molecular level, development of pancreatic malignancies is associated with canonical oncogenes and tumor suppressor genes, such as Ki-Ras, p16, p53, Smad4 and BRCA2, which are involved in several cross-talking cellular signaling pathways including TGFβ/SMAD, PI3K/AKT, and MAPK pathways [START_REF] Hezel | Genetics and biology of pancreatic ductal adenocarcinoma[END_REF][START_REF] Bardeesy | Pancreatic cancer biology and genetics[END_REF][START_REF] Rustgi | The molecular pathogenesis of pancreatic cancer: clarifying a complex circuitry[END_REF].

In the present study, we present evidence to show that αA-crystallin is expressed in the normal human and mouse pancreas at a moderate level. Moreover, analysis of αA-crystallin in the tissue samples from normal human pancreas and 60 cases of pancreatic carcinoma reveals significant difference. αA-crystallin is downregulated more than 10-fold in the pancreatic carcinoma of various types than

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that in normal pancreas. To explore the possible role of αA-crystallin in pancreatic carcinoma, we demonstrate here for the first time that αA-crystallin positively regulates the DNA binding and transactivity of the activating protein-1 (AP-1). This upregulation of AP-1 activity is derived from changed MAPK activity, which can positively regulate c-Jun and c-Fos. Moreover, expression of αA-crystallin in pancreatic cells enhances TGFβ signaling pathway. Finally, expression of αAcrystallin in pancreatic cancer cells leads to retarded cell migration. Together, these results suggest that αA-crystallin appears to suppress pancreatic carcinogenesis.

Materials and Methods

Reagents and antibodies

Anti-αA/B-crystallin antibodies were described before [START_REF] Mao | Human alphaA-and alphaBcrystallins bind to Bax and Bcl-X(S) to sequester their translocation during staurosporine-induced apoptosis[END_REF]. ERK2 kinase assay kit, and anti-phosphorylated Elk antibody were purchased from Cell Signaling, Inc.

Anti-c-Jun antibody was a kind gift from Dr. Tom Curran's Laboratory. Anti-c-Fos, anti-Lamin B, and anti-β-actin antibodies were purchased from Santa Cruz Biotechnology. Human normal pancreas and pancreatic carcinoma tissue samples were obtained from US Biomax, Inc.

Immunohistochemistry analysis of αA-Crystallin, measure of fluorescence intensity and HE staining

Immunohistochemistry analysis was conducted as recently described [START_REF] Xiao | Protein Phosphatase-1 regulates Akt1 signaling pathway to control gene expression, cell survival and differentiation[END_REF].

Briefly, the sections were rehydrated in graded alcohols, heated up to boiling for 5 min in citrate buffer at pH 6. The sections were washed three times with PBS. For non-specific blocking, each section was incubated in 500 µl of 5% normal goat serum (Sigma-Aldrich, CA) in PBS for 1 h at room temperature and then incubated

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overnight in 400 µl of diluted anti-αA-crystallin antibody (1:100) in a humidified chamber at 4 °C. The sections were then washed with PBS three times (5 min each) followed by incubation with 400 µl of FITC-linked secondary antibody and DAPI (Vector Laboratories, CA) in blocking solution for 1 h in the absence of visible light.

After incubation, sections were washed with PBS at room temperature 6 times (5 min each) and then observed under a Carl Zeiss Laser Scanning Confocal microscope.

For negative controls, the sections were treated in the same way except that the primary antibody was replaced with normal serum IgG.

A quantitative measure of fluorescence intensity was conducted using confocal microscopy for assessment of the level of αA-crystallin expression. During analysis of the obtained results, the designated values of 0 to 10, 10 to 100, 100 to 1000 and more than 1000 represent negative background, weak, moderate and strong expression of αA-crystallin, respectively. The same immunohistochemical quantitative system was used for tissue microarray analysis by others [START_REF] Marinov | AKT/mTOR pathway activation and BCL-2 family proteins modulate the sensitivity of human small cell lung cancer cells to RAD001[END_REF]. Each core was measured individually. If more than one core was evaluated from each tissue the mean intensity and standard deviation were calculated and shown as Mean ± SD.

For histology analysis, hematoxylin and eosin (HE) staining was conducted as recently described [START_REF] Xiao | JNK1 downregulation is associated with sex reversal of the ricefield eel[END_REF].

Cell culture and cell fraction extraction

The preparation of the pEGFP-C3 and pEGFP-C3-αA/αB-crystallin expression constructs, and establishment of stable transfected cell lines were conducted as previously described [START_REF] Mao | Human alphaA-and alphaBcrystallins bind to Bax and Bcl-X(S) to sequester their translocation during staurosporine-induced apoptosis[END_REF][START_REF] Xiao | Protein Phosphatase-1 regulates Akt1 signaling pathway to control gene expression, cell survival and differentiation[END_REF]. The pEGFP-αTN4-1, pEGFP-αA-αTN4- and 400 µg/ml neomycin as described before [START_REF] Mao | Human alphaA-and alphaBcrystallins bind to Bax and Bcl-X(S) to sequester their translocation during staurosporine-induced apoptosis[END_REF][START_REF] Xiao | Protein Phosphatase-1 regulates Akt1 signaling pathway to control gene expression, cell survival and differentiation[END_REF]. All cells were kept at 37 °C and 5% CO2 gas phase. Cell fractions were extracted by differential sucrose gradient as previously described [START_REF] Mao | Human alphaA-and alphaBcrystallins bind to Bax and Bcl-X(S) to sequester their translocation during staurosporine-induced apoptosis[END_REF].

The cytoplasmic and nuclear extracts of vector (pEGFP)-, αA-crystallin (pEGFP-αA)-or αB-crystallin (pEGFP-αB)-transfected cells were used for western blot analysis. The nuclear extracts from parent, vector (pEGFP)-, αA-crystallin (pEGFP-αA)-or αB-crystallin (pEGFP-αB)-transfected cells were used for gel mobility shifting assays.

Reverse transcription-linked polymerase chain reaction (RT-PCR)

Total RNA samples were extracted from αA-crystallin-transfected cells (pEGFP-αA-Miapaca), αB-crystallin-transfected cells (pEGFP-αB-Miapaca), vector transfected cells (pEGFP-Miapaca) and control parent pancreatic cancer cells (Miapaca) using TRIZOL reagent. Five µg of total RNA from each sample was used for cDNA synthesis. Several components of TGFβ signaling pathway were examined by RT-PCR using specific primer pairs listed in

Gel mobility shifting assay

Gel mobility shifting assays were conducted as previously described [START_REF] Xiao | Protein Phosphatase-1 regulates Akt1 signaling pathway to control gene expression, cell survival and differentiation[END_REF][START_REF] Feng | Human Bcl-2 activates RAF/MEK/ERK pathway and regulates AP-1 activity and downstream genes in rabbit Lens epithelial cells[END_REF][START_REF] Qin | Protein phosphatase-2A is a target of epigallocatechin-3-gallate and modulates the p53-Bak apoptotic pathway[END_REF][START_REF] Chen | Transcriptional regulation of PP2A-A alpha is mediated by multiple factors including AP-2alpha, CREB, ETS-1, and SP-1[END_REF]. The following oligos were used: 5'-GTCCTTCATTACGTCACGCATAG-3' for conserved AP-1 binding site, 5'-GTCCTTCATTCATTCACGCATAG-3' for mutated

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AP-1 binding site; Twenty µg of nuclear extracts or cytoplasmic extracts prepared from the pEGFP-αTN4-1, pEGFP-αA-αTN4-1, pEGFP-Miapaca, and pEGFP-αA-Miapaca cells were incubated with 1x10 5 cpm of 32 P-labeled double-stranded synthetic oligonucleotides for 30 minutes at 37 °C in binding shifting buffer [START_REF] Feng | Human Bcl-2 activates RAF/MEK/ERK pathway and regulates AP-1 activity and downstream genes in rabbit Lens epithelial cells[END_REF][START_REF] Qin | Protein phosphatase-2A is a target of epigallocatechin-3-gallate and modulates the p53-Bak apoptotic pathway[END_REF][START_REF] Chen | Transcriptional regulation of PP2A-A alpha is mediated by multiple factors including AP-2alpha, CREB, ETS-1, and SP-1[END_REF].

For competition experiments, 50-fold of the non-labeled wild type or mutant doublestranded synthetic oligonucleotides were pre-incubated with the nuclear extracts for 30 minutes before the labeled probe was added into the reaction. For the pre-cleared experiments, 20 µg of nuclear extract prepared from pEGFP-αTN4-1, pEGFP-αA-αTN4-1, pEGFP-Miapaca, and pEGFP-αA-Miapaca cells was pre-incubated with 10 µg antibody against c-Jun or/and c-Fos for 1 hour on ice, then mixed with protein A/G agarose and rotated for additional 1 hour at 4 °C, and then the supernatant was recovered for incubation with 1x10 5 cpm of 32 P-labeled double-stranded synthetic AP-1 oligonucleotides for 30 minutes at 37 °C in a binding shifting buffer. After the binding reactions, the mixtures were loaded onto 6% native PAGE and detected by autoradiography.

Kinase activity assay

Substrate Elk protein was incubated with different doses of ERK2 kinase, 0.1, 1, 5 or 10 ng or mock kinase, GSK3β in the kinase buffer for 30 min at 37 °C in the presence of γ-32 P-ATP. For the effect of αA-crystallin on ERK kinase, differential concentrations (0, 1 and 10 µg) of purified bovine αA-crystallin were added to the kinase assay reactions. After reaction, the mixtures were loaded to SDS-PAGE and detected by western blot analysis using anti-phosphorylated-Elk antibody.

Western blot analysis

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Preparation of total proteins from mouse tissues and various transfected cells and Western blot analysis of different protein samples were conducted as previously described [START_REF] Li | Lens epithelial cell apoptosis appears to be a common cellular basis for non-congenital cataract formation in humans and animals[END_REF][START_REF] Li | Protein expression patterns of the signaling molecules for MAPK pathways in human, bovine and rat lenses[END_REF][START_REF] Li | Calcium-Activated RAF/MEK/ERK pathway mediates p53-dependent apoptosis and is abrogated by αB-crystallin through Inhibition of Ras Activation[END_REF].

Wound healing assay.

The three types of stable clones of pEGFP-Miapaca, pEGFP-αA-Miapaca and pEGFP-αB-Miapaca cells were seed in 6-well plates and cultured until 100% confluent. A straight scratch was made by using a 1 ml blue pipette tip to simulate the wound in each well. After PBS washing for 2 times, new DMEM medium was added for a continuous growth of another 48 hours. The wound healing process was recorded daily using the Leica Inverted Phase Contrast Fluorescence Microscopy with a 10x objective.

Quantitation and Statistical analysis

The mRNA gels were analyzed using the UN-SCAN-IT software from Silk Scientific Corporation (Orem, UT, USA). Total pixel data were averaged from three different groups of samples of each species after normalization against the background. The total pixels from the target band were divided by the total pixels from the corresponding β-actin. The student t-test was used to compare the medians of two unpaired groups. P < 0.05 was considered significant [START_REF] Li | Protein expression patterns of the signaling molecules for MAPK pathways in human, bovine and rat lenses[END_REF].

Results

Expression of αA-crystallin in human non-lenticular tissues

Recent studies have shown that αB-crystallin is implicated in promotion of carcinogenesis [START_REF] Sherman | Heat shock proteins in cancer[END_REF][START_REF] Moyano | AlphaBcrystallin is a novel oncoprotein that predicts poor clinical outcome in breast cancer[END_REF][START_REF] Chelouche-Lev | alphaBcrystallin as a marker of lymph node in breast carcinoma[END_REF]. Whether αA-crystallin has similar functions remains to be

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explored. For this reason, we have re-examined the tissue-specific expression of αAcrystallin in human non-lenticular tissues using immunohistochemistry analysis of human multi-organ tissue array samples and confocal microscope measurement of the relative fluorescence density derived from immunohistochemistry staining. As shown in Fig. 1A, a strong lenticular expression of αA-crystallin in the developing mouse eye was observed, which is consistent with previously reported result [START_REF] Robinson | Differential expression of alpha A-and alpha B-crystallin during murine ocular development[END_REF]. In addition, we observed a moderate level of αA-crystallin expression in human pancreas and to a less degree, in human liver (Fig. 1A). Similar pancreatic level of αA-crystallin was observed in 7 out of 11 human samples examined (Fig. 1B). In pancreas, αA-crystallin was specifically expressed in islets, acinus and interlobular tubes (Fig. 1A). In other non-lenticular ocular tissues, αA-crystallin was also expressed at moderate level in retina and ciliary processes (data not shown). The relative mean values of fluorescence intensities in various non-lenticular tissues were measured with confocal microscope coupled with related software and were presented in Fig. 1B.

Expression of αA-crystallin in mouse tissues

To validate the observed results of αA-crystallin expression in human samples, we next dissected 10 different tissues from adult mouse: lens, heart, thymus, kidney, brain, spleen, pancreas, lung, liver and testis, and analyzed the expression level of αA-crystallin in these tissues. As shown in the top panel of Fig. 2, Western blot analysis revealed that besides its robust expression in the ocular lens (lane 1 contains only 0.5 µg protein, rest of the lanes 100 µg proteins), αA-crystallin was detected in thymus and pancreas at a relatively strong level, then to a less degree in brain. The αA-crystallin level in heart, kidney, spleen and liver was much lower than that in

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10 thymus and pancreas. Finally, a trace of αA-crystallin was also observed in testis.

No αA-crystallin was detected in lung. As comparison, we also examined the tissue distribution of αB-crystallin in mouse. Besides the strongest expression in lens, αBcrystallin was highly expressed in heart, thymus and brain, moderately expressed in kidney and lung. However, different from the tissue specific expression patterns of αA-crystallin, αB-crystallin was barely detectable in pancreas and spleen, and absent in liver and testis examined. Thus, our results confirm that the pancreas expresses a moderate level of αA-crystallin. In addition, we also show that several other nonlenticular tissues also express low level of αA-crystallin. These results are slightly different from the pattern of αA-crystallin detected in different tissues from rat [START_REF] Kato | Immunoreactive alpha A crystallin in rat non-lenticular tissues detected with a sensitive immunoassay method[END_REF][START_REF] Srinivasan | αA-Crystallin is expressed in nonocular tissues[END_REF].

Down-regulation of αA-crystallin in pancreatic tumors

Since both Hsp27 and αB-crystallin are implicated in carcinogenesis [6-8, 15, 33, 36-37], we reason that αA-crystallin may be also involved in regulation of carcinogenesis. To test this possibility, we compared the expression pattern of αAcrystallin in human normal pancreatic tissues and various types of pancreatic carcinoma using the tissue array samples. As shown in Fig. 3A, expression of αAcrystallin in human pancreatic cancer tissues was lower than that in human normal pancreas. Among the 11 normal pancreas samples examined, 7 samples displayed moderate expression level of αA-crystallin (normal pancreas A in Fig. 3A) and 4

samples relatively weak expression (normal pancreas B in Fig. 3A). In contrast, To further characterize the expression of αA-crystallin in different human pancreatic cancer, we divided the pancreatic cancer samples into 4 different types:

among
duct adenocarcinoma (n=48), mucinous adenocarcinoma (n=3), islet adenocarcinoma (n=6) and acinic cell carcinoma (n=3). The fluorescence signal in each individual sample and the statistic data with the mean value of fluorescence intensity of αAcrystallin expression in each type of pancreatic carcinoma were presented in Fig. 3B.

The expression level of each normal pancreas and the statistical data derived from the normal pancreas were also included for comparison (Fig. 3B). Together, our results reveal that αA-crystallin is significantly down-regulated in human pancreatic carcinoma comparing with normal pancreas.

Expression of αA-crystallin in miapaca cells upregulates expression of c-Jun and c-Fos.

It is well established that the transcriptional factor complex, activating protein-1 (AP-1) plays a significant role in tumorigenesis [START_REF] Curran | Fos and Jun: the AP-1 connection[END_REF][START_REF] Imler | AP1, a composite transcription factor implicated in abnormal growth control[END_REF][START_REF] Varmus | Oncogenes and transcriptional control[END_REF][START_REF] Durchdewald | The transcription factor Fos: a Janus-type regulator in health and disease[END_REF][START_REF] Karin | From JNK to pay dirt: jun kinases, their biochemistry, physiology and clinical importance[END_REF]. AP-1 is either a homodimer containing two Jun family members (c-Jun, Jun-B or Jun D) or a heterodimer containing one member from the Jun family and another from the Fos family (C-Fos,

Fra-1 and Fra-2) [START_REF] Curran | Fos and Jun: the AP-1 connection[END_REF]. To explore the possible function of αA-crystallin in human pancreatic cancer, we overexpressed αA-crystallin in both pancreatic cancer cells, miapaca and non-cancer cells, αTN4-1, a SV40 large-T-transformed mouse lens epithelial cell line expressing very little endogenous αA-crystallin. Then, we examined the AP-1 expression and activity in these transfected cells. As shown in Figure 4, expression and phosphorylation of c-Jun were significantly upregulated in

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12 the nucleus of the miapaca cells expressing αA-crystallin than in the nucleus of the vector-transfected cells, expression of c-Fos was also upregulated, though to a much less degree, in both nucleus and cytoplasm of the miapaca cells expressing αAcrystallin than in the same fractions of the vector-transfected cells. Similar results were also obtained in the nucleus of the αTN4-1 cells expressing αA-crystallin (Supplementary Fig. 1).

Expression of αA-crystallin in miapaca cells upregulates AP-1 activity.

To determine whether the upregulated c-Jun and c-Fos actually led to increased AP-1 activity, we first conducted DNA binding assays using the oligos contained either the conserved AP-1 binding site (TGAGTCA) or the mutated site (ATCGGTA). As shown in Fig. 5A, the nuclear extract prepared from either parent miapaca cells (lane 13) or vector pEGFP transfected Miapaca cells can bind to the wild type AP-1 oligo (lane 2) which was competed off by cold wild type oligo (lane 3) but much less by the mutated oligo (Lane 4). The nuclear extracts pre-cleared by antibodies against c-Jun (lane 5), c-Fos (Lane 6) or both antibodies (lane 7) lost most of the AP-1 activity so that the formed AP-1 complex was either lost or appeared very weak. In contrast, the nuclear extracts pre-cleared by normal IgG only slightly attenuated the AP-1 complex (Lane 8). These results suggest that the specific band we detected between the nuclear extract from pEGFP-Miapaca cells and the wild type oligo is the AP-1/DNA complex. Next, we compared the AP-1 DNA binding activity in vector-and αAcrystallin-transfected cells. As shown in lane 9 and 11 of Fig. 5A, miapaca cells expressing αA-crystallin display significant upregulation of AP-1 activity. Similar results were obtained with the lens epithelial cells, αTN4-1 expressing either only EGFP or EGFP-αA-crystallin (Supplementary Fig. 2). As a control for the specific

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binding to the AP-1 site, we conducted the reaction using purified p53 protein and AP-1 oligo, lane 12 of Fig. 5A clearly showed that p53 did not bind to AP-1 oligo.

To further confirm that the transactivity of AP-1 was indeed upregulated in miapaca cells expressing αA-crystallin, we introduced two reporter genes into the two types of stable transfectants: pEGFP-Miapaca and pEGFP-αA-Miapaca. As shown in Fig. 5B, assayed with either pJun-Luc (the luciferase gene was driven by the Jun promoter), or pProlactin-1xAP-1-Luc (the luciferase gene was driven by a mini-prolactin gene promoter containing a copy of AP-1 binding site), the miapaca cells expressing αAcrystallin showed statistically significant upregulation of luciferase activity, indicating the upregulation of AP-1 activity in αA-crystallin expression miapaca cells (Fig. 5B). Thus, expression of αA-crystallin in pancreatic cancer cells positively regulates AP-1 activity. As shown in Fig. 6C, both expression and activity of ERK2 were clearly upregulated in miapaca cells expressing αA-crystallin than vector-transfected cells.

αA-crystallin reglates TGF-β Signaling Pathway

To determine whether αA-crystallin actively regulates TGFβ signaling pathway which plays key role in pancreatic carcinogenesis [START_REF] Hezel | Genetics and biology of pancreatic ductal adenocarcinoma[END_REF], we first examined expression of several genes in the TGFβ signaling pathway at the mRNA level. As shown in Fig. 7, TGFβ was increased about 2-fold when αA-crystallin was overexpressed in Miapaca (Fig. 7A and 7B, similar result was observed in αA-crystallin-transfected

Capan-1 cells, data not shown). For the receptor, TGFβ receptor 1 and 2 are not detected and TGFβ receptor 3 displayed little change in the three types of cells (Fig.

7A and 7C

). To further confirm the modulation of TGFβ signaling pathway by αAcrystallin, we examined the protein expression levels of TGFβ and the downstream smads, as well as the phosphorylation status of the smads. As shown in Fig. 8, Western blot analysis revealed that TGFβ was also upregulated at the protein level.

In addition, expression and phosphorylation of Smad 3 and Smad 5 were significantly upregulated. Thus, αA-crystallin actively enhances TGFβ signaling pathway.

Retardation of cell migration in αA-crystallin-overexpressed cells

To explore how αA-crystallin may regulate carcinogenesis, we conducted wound healing assay using the established stable cell lines: pEGFP-miapaca, pEGFP-αA-Miapaca and pEGFP-αB-Miapaca. When these cells with similar expression levels

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ACCEPTED MANUSCRIPT 15 were selected to conduct wound-healing assays (Fig. 9), it was found that miapaca cells expressing αA-Crystallin displayed statistically significant retardation in cell migration. In contrast, the same cells expressing either the vector or αB-Crystallin did not show such effect. Thus, αA-Crystallin retards migration of pancreatic cancer cells, which is consistent with its downregulation in various types of pancreatic carcinoma (Fig. 3).

Discussion

In the present study, we have demonstrated the following: 1) αA-crystallin is expressed at moderate level in pancreas from human and mouse, a non-ocular tissue previously not documented; 2) αA-crystallin is significantly downregulated in the pancreatic tumors than in the normal human pancreas; 3) Expression of αA-crystallin in pancreatic carcinoma cells induces significant upregulation of the AP-1 components and activity, 4) αA-crystallin can enhance activity of ERK2 MAP kinase, through which it can regulate AP-1 expression and activity; 5) Expression of αAcrystallin in pancreatic carcinoma cells significantly enhanced the TGFβ signaling pathways; and 6) expression of αA-crystallin in pancreatic cells led to obvious retardation of the cell migration. Together, our results suggest that αA-crystallin appear to negatively regulate pancreatic carcinogenesis, which is contrast to the function of αB-crystallin in breast cancer.

αA-crystallin Is Expressed in Normal Pancreas Besides Other non-Lenticular

Tissues

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αA-crystallin is first known as a lens structural protein [START_REF] Robinson | Differential expression of alpha A-and alpha B-crystallin during murine ocular development[END_REF][START_REF] Piatigorsky | Lens crystallins and their genes: diversity and tissue-specific expression[END_REF]. It shares about 60% identity in amino acid sequence with αB-crystallin, another lens structure protein [START_REF] Robinson | Differential expression of alpha A-and alpha B-crystallin during murine ocular development[END_REF][START_REF] Piatigorsky | Lens crystallins and their genes: diversity and tissue-specific expression[END_REF]. Several lines of evidence have shown that although the two genes encoding αAand αB-crystallins may arise from gene duplication, they have diverged significantly. First, during murine development, the initial turning on of the two genes occurs at different time. While αB mRNA is first becoming detectable at E9.5, expression of αA mRNA appears at E10.5 [START_REF] Robinson | Differential expression of alpha A-and alpha B-crystallin during murine ocular development[END_REF]. Such differential turning on time reflects the different control mechanisms in the promoters of the two α-crystallin genes. Second, the two genes display distinct tissue-specific expression patterns.

While αA is highly restricted to lens during mouse embryonic developmental process, αB-crystallin is expressed in the developing heart, nasal epithelium, and retinal pigment epithelium [START_REF] Robinson | Differential expression of alpha A-and alpha B-crystallin during murine ocular development[END_REF]. In the adult vertebrates, although both αA and αB are abundantly expressed in the lens, they display significantly difference in nonlenticular tissue expressions. αB is strongly expressed in heart, skeletal muscle, kidney and brain [START_REF] Piatigorsky | Lens crystallins and their genes: diversity and tissue-specific expression[END_REF][START_REF] Bhat | alpha B subunit of lens-specific protein alphacrystallin is present in other ocular and non-ocular tissues[END_REF][START_REF] Dubin | Expression of the murine alpha B-crystallin gene is not restricted to the lens[END_REF]. In contrast, αA is reported to be expressed at very low level in some non-lenticular tissues including spleen, thymus, heart, brain and liver [START_REF] Kato | Immunoreactive alpha A crystallin in rat non-lenticular tissues detected with a sensitive immunoassay method[END_REF][START_REF] Ujiki | Surgical management of pancreatic cancer[END_REF]. In the present study, our results not only confirm the presence of a low level of αA-crystallin in kidney and liver, but also reveal a moderate level of αA-crystallin expression in a novel non-lenticular tissue, the normal human pancreas. Among the 11 normal human pancreas samples examined, 7 samples display consistently moderate level of αA-crystallin expression and the remaining 4 samples also showed a low level of αA-crystallin expression (Fig. 1). The pancreatic distribution of αAcrystallin is further confirmed from analysis of mouse tissue samples (Fig. 2).

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In a previous study, Kato et al. [START_REF] Kato | Immunoreactive alpha A crystallin in rat non-lenticular tissues detected with a sensitive immunoassay method[END_REF] compared the relative abundance αAcrystallin in both lens and non-lenticular tissues (ng αA-crystallin/per mg total protein in organ). These authors found that in the adult rat (8-16 weeks), the lens contains 244 µg αA-crystallin/per mg total lens protein, and the spleen contains 16.6

ng αA-crystallin/per mg total spleen protein, so the relative abundance of lens αAcrystallin is about 14,698-fold. Using confocal fluorescence microscopy, we quantitated the relative abundance of αA-crystallin in the developing mouse lens and the pancreatic tissue, and found the corresponding values are 807+/-48 and 283+/-67, respectively. Therefore, the human adult pancreatic signal of αA-crystallin is about 1/3 of the developing mouse eye lens (Fig. 1). In mouse tissue, we estimate that αAcrystallin in lens is about 500-fold higher than that in pancreas (see Fig. 2, 200-fold difference in sample load x 2.5-fold difference in αA-crystallin band in lens verse in pancreas). Together, our results demonstrate that the pancreas expresses a moderate level of αA-crystallin.

αΑ-Crystallin Is Implicated in Carcinogenesis

It is well established that small heat shock proteins are implicated in carcinogenesis [START_REF] Ciocca | Heat shock proteins in cancer: diagnostic, prognostic, predictive, and treatment implications[END_REF]. One of the most studied members is Hsp27, which is involved in carcinogenesis of many different types of tissues [START_REF] Arrigo | Hsp27 (HspB1) and alphaB-crystallin (HspB5) as therapeutic targets[END_REF][START_REF] Kostenko | Heat shock protein 27 phosphorylation: kinases, phosphatases, functions and pathology[END_REF][START_REF] Storm | Distribution of Hsp-27 and HER-2/neu in in situ and invasive ductal breast carcinomas[END_REF][START_REF] Porter | Transcriptional activation of heat shock protein 27 gene expression by 17betaestradiol and modulation by antiestrogens and aryl hydrocarbon receptor agonists[END_REF][START_REF] Ciocca | Colocalization of estrogen and progesterone receptors with an estrogen-regulated heat shock protein in paraffin sections of human breast and endometrial cancer tissue[END_REF][START_REF] Ciocca | The presence of an estrogen-regulated protein detected by monoclonal antibody in abnormal human endometrium[END_REF][START_REF] Navarro | Endometrial stromal sarcoma expression of estrogen receptors, progesterone receptors and estrogen-induced srp27 (24K) suggests hormone responsiveness[END_REF][START_REF] Langdon | Expression of the heat shock protein HSP27 in human ovarian cancer[END_REF][START_REF] Foster | Hsp-27 expression at diagnosis predicts poor clinical outcome in prostate cancer independent of ETS-gene rearrangement[END_REF]. Another member of the HSP family implicated in carcinogenesis is αB-crystallin. Iwaki and Tateishi [START_REF] Iwaki | Immunohistochemical demonstration od alphaB-crystallin in hamartomas of tuberous sclerosis[END_REF] first demonstrated the existence of αB-crystallin in hamartomas of tuberous sclerosis.

Then, it was found that concentrations of αB-crystallin in prostatic carcinoma tissues were significantly higher than in benign prostatic hyperplasia [START_REF] Takashi | Different concentrations of two small stress proteins, alphaB crystallin and HSP27 in human urological tumor tissues[END_REF]. In breast cancer cells, αB-crystallin was found expressing constitutively in certain breast carcinoma

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 18
cell lines, and increased intensity of αB-crystallin expression was correlated with shorter survival [START_REF] Chelouche-Lev | alphaBcrystallin as a marker of lymph node in breast carcinoma[END_REF]. Moreover, expression of αB-crystallin results in transformation of immortalized human mammary epithelial cells, induction of EGF-and anchorageindependent growth, and enhancement of cell migration and invasion [START_REF] Moyano | AlphaBcrystallin is a novel oncoprotein that predicts poor clinical outcome in breast cancer[END_REF]. Thus, αBcrystallin is considered as a novel oncoprotein [START_REF] Moyano | AlphaBcrystallin is a novel oncoprotein that predicts poor clinical outcome in breast cancer[END_REF].

Both αAand αB-Crystallins belong to the heat shock protein (Hsp) family [START_REF] Sun | The small heat shock proteins and their role in human disease[END_REF].

Whether αA-crystallin is also in implicated in carcinogenesis remains to be explored.

Compared with αB-Crystallin, the limited distribution of αA-crystallin in nonlenticular tissues may restrict its function in carcinogenesis [START_REF] Robinson | Differential expression of alpha A-and alpha B-crystallin during murine ocular development[END_REF][START_REF] Piatigorsky | Lens crystallins and their genes: diversity and tissue-specific expression[END_REF][START_REF] Bhat | alpha B subunit of lens-specific protein alphacrystallin is present in other ocular and non-ocular tissues[END_REF][START_REF] Dubin | Expression of the murine alpha B-crystallin gene is not restricted to the lens[END_REF][START_REF] Kase | Expression of alpha-crystallin in retinoblastoma[END_REF]. Nevertheless, several recent studies suggest that αA-crystallin may be also implicated in tumor development. First, in 4 cases of human sebaceous carcinoma of the eyelid examined, both αA-Crystallin and αB-Crystallin were found highly expressed [START_REF] Rigas | Expression of alpha-crystallins in human sebaceous carcinoma of the eyelid[END_REF]. Second, in the retinocytoma, αA-crystallin was expressed in the cytoplasm of all tumor cells and the apoptotic index was significantly higher in those cases with weak signal of αAcrystallin than in those that were strongly positive [START_REF] Kase | Expression of alpha-crystallin in retinoblastoma[END_REF]. These results suggest that αA-crystallin, acting like αB-crystallin, seems to promote carcinogenesis. On the other hand, a recent study revealed that in 6 cases of retinoblastoma, preoperative chemotherapy induced strong expression of only Hsp27 and αB-Crystallin but not αA-Crystallin [START_REF] Kase | Expression of heat shock protein 27 and alphacrystallins in human retinoblastoma after chemoreduction[END_REF]. Moreover, the viable tumor cells survived contained high levels of Hsp27 and αB-Crystallin but not αA-Crystallin. Therefore, these results indicate that αA-crystallin does not seem to act the same way as Hsp27 and αB-Crystallin do in promoting carcinogenesis. Our present finding that in 60 different cases of pancreatic carcinoma, the expression level of αA-crystallin was consistently downregulated than that in 11 normal human pancreas samples also support the negative

A C C E P T E D M A N U S C R I P T

regulation of carcinogenesis by αA-crystallin. In addition, our results also suggest that αA-crystallin may be used as one of the biomarkers for pancreatic cancer diagnosis. Quantitative results of the Elk phosphorylation under different conditions. The relative pixel of each band was quantitated using the software from Silk Scientific Inc.

αA-crystallin

Note that at the concentration of 0.1 and 1ng ERK2 kinase conditions, addition of 1 or 10 µg of αA-crystallin significantly enhances the ERK2 activity as reflected by the differential intensity of Elk phosphorylation. C. Western blot analysis of expression and activities of ERK1/2 in vector-or αA-transfected miapaca cells. Western blot analysis was conducted as described in Fig. 2. Note that both expression and activity of ERK2 was upregulated in miapaca cells expressing αA-crystallin than vectortransfected cells.

  the 60 cases of pancreatic cancer samples, 4 had weak expression of αAcrystallin signal (shown as pancreatic cancer #A of Fig.3A), 17 displayed barely A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 11 detectable αA-crystallin signal and the remaining 39 samples displayed no detectable signals (shown as pancreatic cancer #B of Fig.3A).

3. 6 .

 6 αA-crystallin positively reglates ERK2 MAPK.It has been shown that AP-1 expression and activity can be enhanced by MAP kinases[START_REF] Gutierrez-Venegas | Characterization of the transduction pathway involved in c-fos and c-jun expression induced by Aggregatibacter actinomycetemcomitans lipopolysaccharides in human gingival fibroblasts[END_REF][START_REF] Chen | Epidermal growth factor induces c-fos and c-jun mRNA via Raf-1/MEK1/ERK-dependent and -independent pathways in bovine luteal cells[END_REF][START_REF] Cavigelli | Induction of c-fos expression through JNK-mediated TCF/Elk-1 phosphorylation[END_REF][START_REF] Monje | Phosphorylation of the carboxylterminal transactivation domain of c-Fos by extracellular signal-regulated kinase mediates the transcriptional activation of AP-1 and cellular transformation induced by platelet-derived growth factor[END_REF][START_REF] Babu | Phosphorylation of elk-1 by MEK/ERK pathway is necessary for c-fos gene activation during cardiac myocyte hypertrophy[END_REF]. To explore how might αA-crystallin regulate AP-1, we examined the possible modulation of ERK kinase activity by αA-crystallin since ERK is a major MAP Kinase regulating AP-1 components in both expression and activation[START_REF] Gutierrez-Venegas | Characterization of the transduction pathway involved in c-fos and c-jun expression induced by Aggregatibacter actinomycetemcomitans lipopolysaccharides in human gingival fibroblasts[END_REF][START_REF] Chen | Epidermal growth factor induces c-fos and c-jun mRNA via Raf-1/MEK1/ERK-dependent and -independent pathways in bovine luteal cells[END_REF][START_REF] Cavigelli | Induction of c-fos expression through JNK-mediated TCF/Elk-1 phosphorylation[END_REF][START_REF] Monje | Phosphorylation of the carboxylterminal transactivation domain of c-Fos by extracellular signal-regulated kinase mediates the transcriptional activation of AP-1 and cellular transformation induced by platelet-derived growth factor[END_REF][START_REF] Babu | Phosphorylation of elk-1 by MEK/ERK pathway is necessary for c-fos gene activation during cardiac myocyte hypertrophy[END_REF]. As shown in Fig.6A, we found that Elk was dose-dependently phosphorylated by ERK2 in vitro. Moreover, in the presence of αA-crystallin, phosphorylation of Elk by 0.1 or 1 ng ERK2 was significantly enhanced. Addition of 1 µg of αA-crystallin into the ERK2 kinase reaction with low level of ERK2 (1 ng) enhanced more than 10-fold Elk phosphorylation, reflecting the corresponding enhancement of ERK2 activity. Together, these results reveal that αA-crystallin can positively regulate AP-1 activity through modulation of ERK activity. To further A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 14 confirm that αA-crystallin enhances ERK2 kinase activity, we analyzed the ERK1/2 expression and activity in vector-and αA-crystallin transfected stable lines of cells.

Figure 1 .

 1 Figure Legends:Figure 1. Expression of αA-crystallin in human normal tissues. (A) Immunohistochemistry of αA-crystallin in human lens, pancreas, liver, kidney, brain,

Figure 2 .

 2 Figure 2. Western blot analysis of αAand αB-crystallins in 11 tissues from normal

Figure 3 .

 3 Figure 3. Expression of αA-crystallin in human pancreatic cancer tissues. (A) Immunohistochemistry results of αA-crystallin in human pancreatic cancer. As comparison, immunohistochemistry results of αA-crystallin in two human normal

Figure 4 .

 4 Figure 4. Expression of αA-crystallin in pancreatic cancer cell line, Miapaca,

Figure 5 .

 5 Figure 5. Expression of αA-crystallin in pancreatic cancer cell line, Miapaca,

Figure 6 .

 6 Figure 6. αA-crystallin enhances MAPK (ERK2) activity in the in vitro kinase assay.

Figure 7 .

 7 Figure 7. αA-crystallin modulates expression of TGFβ as demonstrated by RT-PCR analysis. A. RT-PCR. Total cellular RNAs were extracted from vector-, αAor αB-

table 1

 1

	. The PCR reaction is: 94 o C

Acknowledgements

We thank Drs. J. Horvitz for the αA/αB-Crystallin antibodies and Dr. Tom Curran for the C-Jun antibody. This work is supported in part by the NIH grants 1R01 EY018380, and 1R01EY015765, and the Lotus Scholar Program Funds from Hunan Province Government and Hunan Normal University, and the Changjiang Scholar Team Award from the National Education Ministry of China to DWL. A C C E P T E D M A N U S C R I P T

A C C E P T E D

M A N U S C R I P T level in vector-, αAor αB-crystallin-transfected miapaca cells. Note that αAcrystallin enhances expression of TGFβ mRNA but has little effect on TGF receptors.


S Longhi
email: longhi@fisi.polimi.it

Optical realization of the two-site Bose-Hubbard model in waveguide lattices

Keywords: numbers: 42.82.Et, 03.75.-b, 03.75.Lm

A classical realization of the two-site Bose-Hubbard Hamiltonian, based on light transport in engineered optical waveguide lattices, is theoretically proposed. The optical lattice enables a direct visualization of the Bose-Hubbard dynamics in Fock space.

The Bose-Hubbard Hamiltonian provides a paradigmatic theoretical framework to investigate the physics of strongly interacting bosonic systems [1]. In particular, the two-site Bose-Hubbard model has received a great deal of attention in the last decade as a simple model to investigate the dynamics of ultracold bosonic atoms confined in a double-well potential [2,3,4,5,6,7,8], the so-called bosonic junction (for a recent review see [9]). A remarkable property of the bosonic junction is the existence of Josephson-like oscillations of the atomic populations and of macroscopic self trapping due to atom-atom interaction, which were originally predicted within a semiclassical (mean-field) limit of the Bose-Hubbard model [2,3]. However, various quantum features of the Bose-Hubbard dynamics, such as quantum fluctuations, fragmentation [10], the cat-states formation in the supercritical attractive case [11], or the many-body coherent control of tunneling [12,13], are not accessible within the mean-field limit. This has lead to intensive studies on the relation between the full quantum and mean-field dynamics (see, e.g., [14,15,16,17] and references therein). A comprehensive description of the entire variety of phenomena rooted in the Bose-Hubbard Hamiltonian would benefit from a direct access to quantum dynamical evolution in the Fock space. However, in experiments with ultracold gases the measurable quantities are usually atomic population imbalance and relative phases [3], whereas full information on Fock state occupation evolution is generally not accessible.

In another physical context, light transport in waveguide lattices has provided a test bench to visualize in photonic systems the classical analogues of a wide variety of coherent single-particle quantum phenomena generally encountered in condensed matter or matter wave systems [18,19,20], such as Bloch oscillations and Zener tunneling [18,21], dynamic localization [22,23], and Anderson localization [24,25] to mention a few. As compared to quantum systems, photonic lattices enable a direct and complete visualization of the dynamics by mapping the temporal evolution of the quantum system into spatial propagation of light waves in the engineered lattice [19]. Most of such previous quantum-optical analogue studies, however, focused on single-particle effects, whereas the possibility to visualize in a classical optical setting the analogues of many-particle phenomena, such as those rooted in the Bose-Hubbard model, remains unexplored. Recently, photonic lattices with engineered coupling constants have been proposed as classical analogs to quantum coherent and displaced Fock states [26].

In this Letter it is shown that photonic lattices with suitably engineered coupling and propagation constants provide a simple realization in the Fock space of two-site Bose-Hubbard Hamiltonian.

The starting point of the analysis is provided by a standard two-site Bose-Hubbard Hamiltonian that describes a system of N interacting bosons occupying two weaklycoupled lowest states of a symmetric double-well potential, which reads (see, for instance, [9])

Ĥ = -J(â † 1 â2 + â † 2 â1 ) + U 2 (â † 2 1 â2 1 + â † 2 2 â2 2 ) ( 1
)
where â1,2 (â † 1,2 ) are the bosonic particle annihilation (creation) operators for the modes in the two wells, J > 0 accounts for the coupling constant between the two modes and U is the strength of the on-site interaction (U > 0 for a repulsive interaction). If we expand the vector state of the system |ψ(t) on the basis of Fock states with constant particle number N , i.e. after setting

|ψ(t) = N l=0 c l (t) l!(N -l)! â † l 1 â † N -l 2 |0 (2)
the evolution equations for the amplitude probabilities c l (t) to find l particles in the left well and the other (N -l) particles in the right well, as obtained from the Schrödinger equation i ∂ t |ψ(t) = Ĥ|ψ(t) , read explicitly

i dc l dt = -(κ l c l+1 + κ l-1 c l-1 ) + V l c l (3)
(l = 0, 1, 2, ..., N), where we have set

κ l = J (l + 1)(N -l) , V l = U 2 l 2 + (N -l) 2 -N . (4)
The normalization condition N l=0 |c l (t)| 2 = 1 is assumed. The evolution equations for the Fock state amplitudes c l can be viewed as formally analogous to the coupled-mode equations describing the transport of light waves in a tight-binding array composed by (N + 1) waveguides with engineered propagation constant shift V l and coupling rates κ l between adjacent waveguides, in which the temporal evolution of the Fock-state amplitudes of Bose-Hubbard Hamiltonian is mapped into the spatial evolution of the modal field amplitudes of light waves in the various waveguides along the axial direction z (see, for instance, [19,20]). The fractional light power distribution |c l | 2 in the various waveguides of the array at the propagation distance z thus reproduces the occupation probabilities of the bosons between the two sites of the double-well.

In the optical context, the tight-binding model (3) can be derived using a variational procedure starting from the paraxial and scalar wave equation for the electric field amplitude φ(x, z) describing the propagation of monochromatic light waves at wavelength λ in an array of (N + 1) waveguides with refractive index profile n(x) and substrate index n s

iλ∂ z φ = - λ 2 2n s ∂ 2 x φ + [n s -n(x)]φ, (5)
where λ = λ/(2π) is the reduced wavelength of photons (for details see, for instance, [29]). In particular, to independently engineer the coupling constants κ l and propagation constant shifts V l of the waveguides, one can assume a chain of waveguides with equal normalized refractive index profile g(x), but with different index contrasts ∆n l (l = 0, 1, 2, ..., N) and spacing As discussed in the example below, to realize the Bose-Hubbard lattice (3) the values of d l and ∆n l slightly vary around some mean values d r and ∆n that define a uniform array. The waveguide separation d l mainly determines the value of the coupling rate κ l-1 , with a characteristic exponential dependence of κ l-1 from d l , whereas the index change ∆n l mainly defines the propagation constant mismatch V l . It is worth noticing that, in the absence of particle interaction, i.e. for U = 0, the lattice model (3) associated to the Bose-Hubbard Hamiltonian was previously introduced in the photonic context to realize exact spatial beam self-imaging in finite waveguide arrays [27] and shown to belong to a rather general class of exactly-solvable self-imaging tight-binding lattices with equallyspaced energy levels [28]. A nonvanishing value of the interaction strength U breaks the periodic self-imaging property of the array and enables to visualize with light waves the rich dynamical features of the two-site Bose-Hubbard Hamiltonian directly in the Fock space (see, for instance, [2,3,4,5,6,7,8,9,14,15,16,17] and references therein). Here two important dynamical features will be considered for the sake of example, namely (i) the transition from Josephson-like oscillations to self-trapping as the interaction U is increased [2,3,9], and (ii) the transition from single-atom to correlated pair tunneling of two bosons in a double well potential [8,30]. To visualize the first effect, we specifically design three arrays comprising N + 1 = 10 waveguides, which mimic the evolution of N = 9 atoms in a double well, for three increasing values of the interaction U . The arrays were designed to operate at λ = 633 nm in a transparent glass with a bulk refractive index n s = 1.45 and with a normalized waveguide channel profile g

d l = x l -x l-1 (l = 1, 2, ..., N), i.e. n(x) -n s = - N l=0 ∆n l g(x -x l ). ( 6
)
(x) = {erf[(x + w)/D x ] -erf[(x -w)/D x ]}/[2erf(w/D x )]
with channel width 2w = 4 µm and diffusion length D x = 0.3 µm -1 . The distances between waveguides d l were designed to realize the coupling rates κ l given Eq.( 4) with J = 0.0781 mm -1 . To determine the distribution of distances d l , a reference value ∆n = 2 × 10 -3 of refractive index contrast was assumed, and correspondingly the coupling rate κ between two adjacent waveguides versus distance d was computed, yielding to a good approximation the exponential dependence κ(d) = κ 0 exp[-γ(d -d r )] for distances not too far from the reference distance d r = 8 µm, where κ 0 0.3907 mm -1 and γ 0.6 µm -1 . The resulting distance distribution, depicted in Fig. 1(a), shows that the waveguide distances slightly vary around the reference value d r indeed. By modulating the index contrasts ∆n l of the waveguides around the reference value ∆n, three different arrays were then designed to realize the three different interaction regimes U = 0, U = 0.0174 mm -1 , and U = 0.1043 mm -1 , as shown in Figs. 1(b-d) [31]. Such arrays could be fabricated in fused silica by the recently developed femtosecond laser writing technique [20], in which the different refractive index contrasts are obtained by varying the speed of the writing laser beam. Figures 2 show the evolution of light intensity |φ(x, z)| 2 along the three arrays, as obtained by numerical simulations of Eq.( 5) using a standard pseudospectral split-step method, when the left boundary waveguide is excited at the input plane. Such an excitation corresponds to the initial conditions c l (0) = δ l,0 , i.e. to the entire N bosons in the right well at the initial time. Note that in case U = 0 [Fig. 2(a)] periodic self-imaging of the light pattern is observed because of the equispacing of the energy levels of the Bose-Hubbard Hamiltonian. Such periodic oscillations of the light intensity pattern, previously predicted in Ref. [27] and referred to as harmonic oscillations, are thus the optical analogue of the bosonic Josephson oscillations in the non-interacting regime [2]. A quantitative measure of the Josephson oscillations, generally used in the atom optics context, is provided by the normalized population imbalance P (z), which is defined as the normalized difference between the average numbers of atoms in the two wells, i.e.

P (z) = N l=0 [(N -2l)/N ]|c l | 2 .
Note that in an optical experiment P (z) can be retrieved from a measurement of the beam center of mass position x(z) = dxx|φ(x, z)| 2 / dx|φ(x, z)| 2 using the simple relation P (z)

1 -2 x(z) /(Nd r ). For the U = 0 case, the evolution of P (z) along the array is depicted in Fig. 3(a) and compared to the exact curve obtained from the tightbinding lattice model (3). Note that P (z) oscillates between -1 and 1 with spatial period z R = π/J 4 cm, which is precisely the period of Josephson oscillations of the bosonic junction in the absence of interaction. As the interaction strength is increased, the self-imaging property of the array is clearly broken [see Figs. 2(b) and (c)], with a clear transition to damped Josephson oscillations [Fig. 3(b)] and to the self-trapping regime [Fig. 3(c)]. It should be noted that the mean-field limit of the two-site Bose-Hubbard model in the large N limit, which is described by two nonlinear coupled mode equations [2,9], could be realized in an optical setting by a simple nonlinear optical directional coupler [32], and self-trapping phenomena in such nonlinear couplers have been previously investigated. However, our waveguide lattice system provides an exact realization of the Bose-Hubbard model even for a low number of particles (as in the example previously discussed) where the mean field approximation fails.

As a second example, we discuss the phenomenon of correlated pair tunneling of two bosons in a double well potential induced by interaction, which was investigated theoretically and experimentally in previous works (see, for instance, [8,30]). Even if the two-mode Bose-Hubbard model is capable of describing the dynamics of such a system solely for weak interactions and far from the fragmentation regime [30], it well explains the transition from Rabi oscillations of single particles to correlated pair tunneling in the presence of a weak interaction. Such a kind of correlated dynamics of a pair of interacting atoms was experimentally observed in Ref. [8] by recording both the atom position and phase coherence over time, and was explained on the basis of the simplified Bose-Hubbard Hamiltonian (2). In the N = 2 particle case, the coupled-mode equations (3) reduce to the following ones

i dc 0 dt = - √ 2Jc 1 + Uc 0 i dc 1 dt = - √ 2J(c 0 + c 2 ) ( 7 ) i dc 2 dt = - √ 2Jc 1 + Uc 2 .
The tunneling dynamics can be analyzed by the introduction of the percentage of bosons in the right well, p R (t) = |c 0 (t)| 7) with the initial condition c 0 (0) = 1 and c 1 (0) = c 2 (0) = 0. One obtains

|c 1 (t)| 2 = 2J 2 M 2 sin 2 (Mt) |c 2 (t)| 2 = 1 4 1 + cos 2 (Mt) -2 cos Ut 2 cos(Mt) - U M sin Ut 2 sin(Mt) + U 2 4M 2 sin 2 (Mt) ( 8
)
|c 0 (t)| 2 = 1 -|c 1 (t)| 2 -|c 2 (t)| 2
where we have set M = 4J 2 + U 2 /4. Correspondingly, the behavior of p R (t) and p 2 (t) reads

p R (t) = 1 - J 2 M 2 sin 2 (Mt) - 1 4 1 + cos 2 (Mt) -2 cos Ut 2 cos(Mt) - U M sin Ut 2 sin(Mt) + U 2 4M 2 sin 2 (Mt) (9) p 2 (t) = 1 - 2J 2 M 2 sin 2 (Mt) ( 1 0 )
A typical behavior of both p R (t) and p 2 (t) for increasing values of the ratio U/J is shown in Figs. 4(a) and (b), respectively. Note that, for the non-interacting case (U = 0, curve 1 in Fig. 4) the atoms simply Rabi oscillate back and forth between both wells, and they tunnel independently. As a correlation is introduced, the tunneling becomes a two-mode process and the tunneling period increases [curves 2 and 3 in Fig. 4(a)]. More interestingly, from the behavior of p 2 (t) one can see that as the interaction is increased both atoms remain essentially in the same well in the course of tunneling, i.e they tunnel as pairs and the amplitude c 1 (t) gets negligible [30]. In our optical setting, such a dynamical behavior simply describes light tunneling among three coupled waveguides, with the same coupling rate but with the propagation constant of the central waveguide detuned from that of the outer waveguides [see Eq.( 7)], the detuning being proportional to the interaction strength U in the quantum mechanical analogue. Such an optical structure was recently proposed and realized for the observation of an optical analogue of two-photon Rabi oscillations in Ref. [33]. Indeed, the large interaction regime |U/J| 1 of the Bose-Hubbard model, corresponding to |c 1 (t)| 1 and to pair tunneling (also referred to as second-order tunneling [8]), leads to Rabi oscillations of light power between the outer waveguides of the triplet system, with small excitation of the central waveguide.

In conclusion, a photonic realization of the two-site Bose-Hubbard Hamiltonian in engineered waveguide lattices has been proposed. Such an optical setting enables to visualize in the Fock space the main dynamical aspects of interacting bosons. In particular, waveguide lattices have been designed to visualize the transition from Josephson-like oscillations to self-trapping, as well as the transition from single-atom to correlated pair tunneling in a simple two-boson system. It is envisaged that the idea proposed in this work to use engineered waveguide lattices to simulate in a purely classical setting the quantum physics of interacting particles should motivate further theoretical and experimental studies. For example, longitudinal modulation of the refractive index in the lattice or the introduction of balanced loss and gain in the waveguides might be used to mimic the recently proposed kicked Bose-Hubbard [12] and non-Hermitian PT-symmetric Bose-Hubbard models [34,35].

This work was supported by the Italian MIUR (Grant No. PRIN-20082YCAAK).

Figure 1 .

 1 Figure 1. (color online) (a) Distribution of waveguide distance (d l is the distance between waveguides (l -1) and l, l = 1, 2, ..., 9) in a waveguide array composed by N + 1 = 10 waveguides that realizes the distribution of the coupling constants defined by Eq.(4) for J = 0.0781 mm -1 . The panels (b-d) show the refractive index profile n(x) -n s of the waveguide arrays that realize the Bose-Hubbard Hamiltonian for increasing values of the interaction strength U : in (b) U = 0, in (c) U = 0.0174 mm -1 , whereas in (d) U = 0.1043 mm -1 .

Figure 2 .

 2 Figure 2. (color online) Light intensity evolution (snapshot of φ(x, z)| 2 ) in the three waveguide arrays of Figs.1(b-d) corresponding to excitation of the left boundary waveguide, as obtained by numerical simulations of the paraxial wave equation (5). The images in (a), (b) and (c) correspond to the Bose-Hubbard Hamiltonian for increasing values of the interaction strength U (from left to right).

Figure 3 .

 3 Figure 3. (color online) Behavior of the numerically-computed population imbalance P (z) for the three waveguide arrays of Fig.2 (solid curves), and corresponding behavior predicted by the tight-binding Bose-Hubbard Hamiltonian (3). From (a) to (c), the interaction strength takes the values U = 0, U = 0.0174 mm -1 , and U = 0.1043 mm -1 .

Figure 4 .

 4 Figure 4. (color online) Behavior of (a) the percentage of bosons in the right well p R (t), and (b) of the pair (same-site) boson probability p 2 (t) in the Bose-Hubbard model with two atoms for for increasing values of the intercation strength U , normalized to the hopping rate J. Curve 1: U/J = 0; curve 2: U/J = 4; curve 3: U/J = 8.

  2 + (1/2)|c 1 (t)| 2 , and the pair (or same-site) boson probability p 2(t) = |c 0 (t)| 2 + |c 2 (t)| 2[30], i.e. the probability to find the two bosons in the same well (either the left or the right well). The evolution of the particle occupation probabilities |c l (t)| 2 can be readily obtained by solving Eq.(


E M Staicu Casagrande

C Li

A Lahmam-Bennani

Dal Cappello

M Schulz

M Ciappina

Experimental and theoretical confirmation of the role of higher order mechanisms in the electron impact double ionisation of Helium

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The study of double and multiple ionisation processes by charged particle impact has received a considerable interest in the last decade or so, with a strong focus being put on getting a close insight of the role and of the relative importance of the various mechanisms responsible for double ionisation (DI) processes. The progress in computational methods, together with the spectacular development of multi-parameter detection techniques, opened up the way to performing complete experiments in which all kinematical parameters (vector momenta and energies) of all involved particles are determined. Electron impact (e,3e)

experiments for DI are one of the most fundamental of such processes, though they remain difficult to perform because they necessitate the triple coincidence detection of three particles in the final state (three electrons [1,2] or alternatively the residual ion plus two electrons [3]).

To circumvent this difficulty, Lahmam-Bennani and coworkers developed, in a series of papers [4][5][6][7][8][9][10][11], (e,3-1e) experiments where, with respect to the (e,3e) case, one electron is not detected, hence only necessitating a double coincidence experiment similar to an (e,2e) single ionisation (SI) but with energetics corresponding to DI. In all these works as well as in further works by other groups [12][13][14][15], it was clearly demonstrated that (e,3-1e) experiments provide a very sensitive mean to identify the role of various mechanisms responsible for electron impact DI and to gauge their relative importance.

In a recent study [16] we reported the measurement of the (e,3-1e) four-fold differential cross sections (4DCS) for DI of helium in coplanar asymmetric geometry for a large range of ejected electron energies and at an incident energy of about 600 eV. The experimental angular distributions of the 4DCS displayed large angular shifts of the forward and backward lobes with respect to the momentum transfer direction or it's opposite, respectively, as well as a breaking of symmetry with respect to these directions. These observations are clear signatures of the presence of non-first order mechanisms in the DI process. [We briefly recall (see e.g. [16][17][18]) that first order or first Born mechanisms (in the Born series) involve only one single interaction of the projectile with the target. This is the case for the so-called 'shake-off' (SO) and 'two-step 1' (TS1) mechanisms. Whereas second order or second Born mechanisms such as 'two-step 2' (TS2) involve two successive projectile -target interactions]. A qualitative, kinematical analysis was given in [16] which allowed relating these shifts and the observed structures in the intensity distributions to the second order, 'two-step 2' DI mechanism, which was shown to predominate over the first-order 'shake-off' and 'two-step 1' mechanisms under the investigated kinematics. Such conclusion strongly supports those previously drawn from (e,3-1e) experiments by Lahmam-Bennani et al [11] or from full (e,3e) experiments by Lahmam-Bennani et al [19] and by Dorn et al [20] and also from the theoretical analyses given in [19] and more recently in [21,22].

The aim of the present work is to extend the study in [16] in two directions: (i) first, by producing more experimental data so as to extend the investigated kinematics to larger ejected electron energies and hence larger energy loss suffered by the projectile, in order to validate or invalidate the observations made in [16] and the kinematical analysis therein given; (ii) and second by comparing the experimental results with two different, newly developed non-first order theoretical predictions, based on a second Born treatment on the one side [21] and on a Monte Carlo approach to TS2, on the other side [22]. (In the absence at the time of elaborate non-first order calculations, the experimental data in [16] were only compared to first order theoretical results.) Our objective is to add further evidence from both experiment and theory that at the impact energy of the present work (~600-700 eV) DI of Helium atom is dominated by non-first order mechanisms such as TS2.

Experiment

The experimental set-up and experimental procedure used in the present work are described in detail in [23]. The main characteristic of the spectrometer is the unique combination of three high-efficiency, multi angle toroidal electrostatic energy analysers.

Briefly, a monochromatic electron beam with energy E 0 ~600 -700 eV (see below) impinges on the gas jet formed at the collision centre. A coplanar geometry is used, where all electrons are observed in the collision plane defined by the incident and scattered momentum vectors, k 0 and k a , respectively. The fast, forward-scattered electron (indexed 'a') is detected at the scattered energy E a = 500 eV and at two symmetrical scattering angles, θ a = + (6°± 3°) and - 1 where for completeness we also include the data sets published in [16] as they are also used in the discussion given in the present paper.

E a = 500 eV Experimental data for the cases indicated with the superscript * have been published in [16]. The last column indicates the directions of ejection of the 'b'-electron in the forward and backward directions (θ F-TS2 and θ B-TS2 , respectively) as predicted by the given kinematical analysis, see text.

θ a = -6 deg Case E 0 (eV) E b (eV) detected E c (eV) undetected K (au) θ K / θ -K (deg) θ F-TS2 / θ Β-TS2 (deg)
Given the above experimental parameters, the momentum transfer to the target, defined by K = k 0 -k a , varies in magnitude from K = 0.88 au at E b + E c = 22 eV (case (a) of Table 1)

to K = 1.46 au at E b + E c = 156 eV (case (e)), while its direction θ K (shown in the 6 th column of Table 1) varies from ~46°to ~26°for these two extreme cases. Simultaneously, due to the quite large acceptance in θ a -angle, ∆θ a = ± 3°, the momentum transfer resolution amounts to ∆Κ ~± 0.2 au and the uncertainty in the momentum transfer direction is ∆ K ~±10 .

Results and discussion

The angular distributions of the (e,3-1e) four-fold differential cross sections (4DCS), pronounced disagreement is found between this first-order theory and experiments. For all energy sharings considered in figure 1, the most remarkable differences are:

(i) the breaking of symmetry about ±K directions in the measured distributions whereas the calculated ones do show such symmetry, (ii) the large shift in the angular position of the experimental lobes, ~30°to 70°with respect to θ ±K . The uncertainty in the momentum transfer direction due to the angular resolution in θ a is ~10°, that is significantly smaller than the observed shift, and hence it only marginally affects this shift.

(iii) the existence of structures in these lobes.

These features are clear evidence that strong non-first Born effects are present in the (e,3-1e)

4DCS distributions, that is, the contribution of the TS2 mechanism to the DI process is sufficiently important with respect to that of SO and/or TS1 to impose its finger print on the angular distributions. We thus confirm the observations made in [16] and earlier in [11].

As a second facet of this discussion, we now apply the kinematical analysis proposed in [16] which is meant to give a qualitative interpretation of the observed structures as well as an estimate of the angular positions of the lobes in terms of the physical effects due to the TS2 mechanism. This analysis is described in details in [16], hence it will only briefly be summarized here. TS2 is a two-step process: in the first step, the slowest 'c'-electron is ejected in an (e,2e) ionising process of the He atom, where the corresponding scattered electron (called 'a*') appears with the highest probability at the Compton scattering angle, ±θ a* , corresponding to the Bethe ridge [26,27]. The ± sign stands for the fact that the intermediate 'a*'-electron may appear on both sides of the incident beam direction. In the second step of TS2, the 'a*'-electron plays the role of the incident projectile in a new (e,2e) ionisation of the intermediate He + ion, resulting in the pair ('a':'b') of electrons effectively detected, where the 'a'-scattered one is observed with E a = 500 eV under θ a = -6°, whereas the 'b'-ejected one appears mostly along the corresponding momentum transfer direction of this new collision, noted here θ TS2 . To define this direction, one has to consider two scenarios, depending whether the intermediate 'a*'-electron is scattered at +θ a* or -θ a* . Each of these scenarios yields an ejected 'b'-electron either in the forward direction or in the backward direction, noted θ F-TS2 and θ B-TS2 , respectively. The so predicted θ F-TS2 and θ B-TS2 values are listed in the last column of Table 1 for each of the kinematics considered here. They are also indicated in Fig. 1 by the vertical dashed lines. Remarkably, all the experimental data shown in Fig. 1 do display prominent structures (for both the forward and the backward distributions). The angular positions of the structures located at the largest angles very nicely agree with the predicted θ F-TS2 and θ B-TS2 values. This holds for all energy sharings considered in this work, from the equal (case (f)) to the highly unequal (case (e)) sharing. We thus conclude that the forward and backward lobes positions in Fig. 1 are essentially determined by the TS2 contribution, according to the above qualitative kinematical analysis where the TS2 process is considered as two successive, independent (e,2e) single ionisation of the target. First-order SO and TS1 contributions are also present as seen by the significant intensity near the momentum transfer direction, θ K , and its opposite, θ -K . In some cases, even separate maxima at θ -K are observed. Nevertheless, there the intensity appears to be appreciably smaller than that of the TS2, more so for the backward lobe.

In the third facet of this discussion we compare our measured 4DCS distributions with the calculated results obtained using two newly developed theoretical models. The first model is based on the first Born approximation (FBA), but higher order effects are incorporated using the Monte Carlo Event Generator (MCEG) technique. This approach was successfully applied first to DI by ion impact [28] and more recently extended to electron impact [22]. The method makes use of the MCEG technique to simulate the TS2 mechanism by convoluting two SI events which are both calculated in the FBA. These are successively the SI of the He atom by the incident electron and that of the resulting He + ion by the intermediate scattered electron. In the following, this model is termed TS2-MCEG. Note that the basic idea of this two steps 'decomposition' is the same here as that of the kinematical model considered above.

The second theoretical model is more elaborate as it makes use of the second Born approximation and the closure approximation in the way described in [21]. In this model (termed here B2) the final state is described by the approximate BBK wave function, since the use of the full 3C wave-function with the second Born approximation needs much computer time especially for (e,3-1e). The initial state is again described by a wave function which only includes a part of the radial correlation [21] and the final state is orthogonalized to the initial one. The well-known closure approximation [29] is used with a parameter corresponding to the average excitation energy fixed here to 79 eV. This value corresponds to the energy of the initial state.

The results obtained within these two models, TS2-MCEG and B2, are also displayed in Fig. 1. The main general observation is the overall improved agreement with experiments, as compared to FBA-3C results discussed above. Indeed, although both theories do yield some differences in their results, in particular for electron emission in the backward direction, they both succeed in producing forward and backward lobes which now are shifted with respect to ± K directions and also have additional structures, especially in the backward region, similar to the behaviour found in the experimental data. In particular, the measured positions of the structures located at angles larger than θ +K and θ -K , respectively, are at least qualitatively (if not quantitatively) predicted by both non-first order models, may be save the equal energy sharing case in Fig. 1(f) where the agreement between second-order theories and experiments is somewhat less satisfactory. Remarkably, these predicted positions are in quite good agreement with the θ F-TS2 and θ B-TS2 values listed in Table 1 and derived from our very simple kinematical model discussed above. These observations, together with the construction of the TS2 description (based on two successive SI (e,2e) events) as done both in the kinematical model and in TS2-MCEG model, leads us to the conclusion that the lobes located at the largest angles can be attributed to the TS2 contribution, whereas the ones located at smallest angles are grossly peaking at +K or -K direction, and hence may be attributed to the firstorder SO and/or TS1 contributions. This conclusion appears to be valid for all cases considered here, independently of the energy sharing among the two ejected electrons.

We also note two further observations: • at least for the backward lobes, the relative contribution to the measured as well as the calculated 4DCS of the second-order contribution (located at about θ B-TS2 ) is generally larger than that of the first-order contribution at about θ -K meaning that "non-first order mechanisms such as TS2" dominates over the first-order ones, SO and TS1.

• Apart from the structures we discussed up to now, additional structures in the measured intensity distributions of Fig. 1 seem to be present. Two causes for these structures are conceivable: other non-first order mechanisms, and/or interference effects. (i) A possible candidate for non-first order mechanisms other than TS2 which have not been considered here was recently invoked by Schulz et al [30] in the context of fast p + He collisions, where it is the dominant double ionisation process. The authors labelled it Two-Step 1-Elastic mechanism (TS1-El). It can be viewed as follows: first, a TS1 process leads to the ejection of both target electrons, and then the projectile is elastically scattered from one of the two continuum electrons (hence the alternative label '3steps -2 interactions' 3S2). As in TS2, the symmetry about K direction is broken because of this double interaction of the projectile with the target. However, the kinematical analysis presented in this work as well as the model calculations within TS2-MCEG and B2 frameworks cannot be straightforwardly applied to this mechanism. (ii) the observed additional structures might also be due to interferences between all these various competing processes which all lead to the same final state. With a detailed theoretical description missing, the discussion of these interference effects is difficult.

Conclusion

We have extended our previous (e,3-1e) experiments for double ionisation of He at ~600 -700 eV impact energy to larger ejected energies and more asymmetric energy-sharing. Our new data also display large shifts as well as marked structures in the forward and backward lobes for the ejected electron angular distributions, similar to the observations made in [16].

For all data sets, our previous qualitative analysis based on kinematical arguments shows that under the present kinematics the two-step 2 (TS2) mechanism dominates over the SO and the TS1 and is mostly responsible for the structures and angular positions of the measured lobes.

The first Born-3C prediction does not reproduce the observed shifts and structures of the cross section distribution, whereas the theoretical results from two newly developed models, namely the TS2-MCEG and B2 treatments, very clearly constitute a considerable improvement, both for the position of the forward lobe which is correctly reproduced and for the structure of the backward lobe which is qualitatively well predicted. This confirms again that under the present kinematics the non-first order mechanisms such as TS2 are mostly responsible for the structures observed in the measured distributions. Other mechanisms such as TS1-El may also contribute, but the present models cannot be straightforwardly applied to these mechanisms.

Caption to figure

  Experimental and theoretical confirmation of the role of higher order mechanisms in the electron impact double ionization of helium E M Staicu Casagrande, C Li, A Lahmam-Bennani, C Dal Cappello, M Schulz, M Ciappina To cite this version: E M Staicu Casagrande, C Li, A Lahmam-Bennani, C Dal Cappello, M Schulz, et al.. Experimental and theoretical confirmation of the role of higher order mechanisms in the electron impact double ionization of helium. Journal of Physics B: Atomic, Molecular and Optical Physics, 2011, 44 (5), pp.55201. 10.1088/0953-4075/44/5/055201. hal-00600136

(

  6°± 3°) as set by input slits at the entrance to the electrostatic lenses associated with the 'a'toroidal analyser. Throughout this work, positive angles are counted clockwise starting from the incident electron beam direction. Among the two ejected electrons resulting from DI of the target, (labelled 'b' for the faster and 'c' for the slower), we choose to detect only the faster one, with energy E b , in coincidence with the 'a'-scattered electron, hence an (e,3-1e) experiment. Of course, such distinction does not hold for the equal energy sharing condition, but the same labelling 'b' is kept for the detected electron. These 'b'-electrons are multi-angle analysed in a double toroidal analyser over the angular ranges θ b = 20°-160°and 200°-340°, where 0°is defined by the incident beam direction. In the off-line analysis, the total θ bangular range is divided into sectors of width ∆θ b = 5°. Though the emission direction of the third, 'c'-electron is unknown, its kinetic energy E c is known from the energy conservation E 0 -IP 2+ = E a +E b +E c , where IP 2+ is the He DI potential. [Here, the translational energy of the target atom and the recoil energy transferred to the ion after the emission of the two electrons are neglected, due to the small electron to ion mass ratio]. The experiments were performed at a variety of ejected electron energies ranging from 5 to 144 eV and corresponding either to an equal (E b = E c ) or an unequal (E b > E c ) energy sharing among electrons 'b' and 'c'. We also note the very large range of energy loss (E 0 -E a ) suffered by the projectile, from 101 to 235 eV. The incident energy (E 0 ) is consequently adjusted to fulfil the energy conservation requirement for the Helium target under study, with IP 2+ = 79 eV. The investigated kinematical conditions are summarized in Table

d 4 σ

 4 /dE a dE b d a d b , for DI of He are shown in figure1(a) to 1(f) at the various energy sharings listed in Table1. The experimental results in Fig.1(a) to (c) and in 1(f) have been previously published in[16]. They are reproduced here for the completeness of the discussion. The other cases are unpublished, new results.The data are first compared with the calculated results obtained using the first Born BBK or 3C model where the final state, orthogonalized to the initial state, is described by the product of three Coulomb waves[24,25]. Two of these Coulomb functions describe each electron in the field of the target nucleus and the third function describes one electron in the field of the second electron, i.e. takes into account the electron-electron correlation in the continuum. The initial state is described by a wave function which only includes a part of the radial correlation[21]. These calculations, which are shown as dotted curves in Fig.1, only include first-order DI mechanisms, namely the SO and TS1. Note that the experiments are obtained on a relative scale and have been normalized to the calculations at the maximum of the forward lobe. Here, the labels forward lobe and backward lobe designate the lobe pointing roughly in the momentum transfer direction (+K) and in the opposite direction (-K), respectively. These two directions are indicated by the vertical dotted lines in the figures. A

Figure 1 (

 1 Figure 1 (Color online): Four-fold differential cross sections (4DCS) for double ionisation of He. The scattered electron with energy E a = 500 eV is detected at an angle θ a = -6°in coincidence with the fast-emitted electron with energy E b , whereas the slow-emitted electron with energy E c remains undetected. Panel (a): (E b : E c )= (17 : 5) eV, (b): (E b : E c )= (37 : 5) eV, (c) (E b : E c )= (74 : 5) eV, (d): (E b : E c )= (72 : 12) eV, (e): (E b : E c )= (144 : 12) eV and (f): (E b : E c )= (17 : 17) eV. Full squares are the experimental data, with one standard deviation statistical error bar. Theoretical models' predictions are from: first-order FBA-3C (dotted blue curves), TS2-MCEG (dashed green curves) and second-order B2 (full black curves). The 4DCS scale shown is arbitrary, where all experimental and theoretical results are internormalised for best visual fit at the maximum of the forward lobe. The thin dotted vertical lines indicate the direction of the momentum transfer (θ K ) and its opposite (θ -K ). The heavy dashed vertical lines indicate the directions of ejection of the 'b'-electron, (θ F-TS2 and θ B-TS2 ) as predicted by the given kinematical analysis, see text.

Table 1 .

 1 Kinematical parameters (energies and momenta) used in this study.

	(a) *	601	17	5	0.88	46 / 226	82 / 290
	(b) *	621	37	5	0.96	41 / 221	79 / 297
	(c) *	658	74	5	1.12	34 / 214	74 / 307
	(d)	663	72	12	1.14	34 / 214	76 / 300
	(e)	735	144	12	1.46	26 / 206	67 / 315
	(f) *	613	17	17	0.93	43 / 223	84 / 282

Acknowledgment

We wish to acknowledge Tom Kirchner for his significant involvement in the development of the MCEG based calculations. C Li acknowledges a PhD grant from the China Scholarship Council.

Staicu Casagrande E M et al