Edouard Alphandéry
email: edouardalphandery@hotmail.com

Darine Abi

+++ , Olivier Haidar ++

Seksek ++

François Guyot

Imène Chebbi

Fluorescent magnetosomes for controlled and repetitive drug release under the application of an alternating magnetic field under conditions of limited temperature increase (< 2.5 C)

Keywords: magnetosomes, magnetotactic bacteria, probe, thermometer, fluorescence, luminescence, nano-thermometry, fluorescent nanoparticles, nanotechnology

ABREVIATIONS: MCR400: magnetosomes produced by magnetotactic bacteria cultivated in the presence of 400 µM of rhodamine B introduced in the bacterial growth medium followed by magnetosome extraction from these bacteria; MC: magnetosomes produced by magnetotactic bacteria cultivated without rhodamine B introduced in the bacterial growth medium followed by magnetosome extraction from these bacteria; FACS: Fluorescence activated cell sorting; AMF: alternating magnetic field; Ca: calcein acetoxymethyl ester; RhB: rhodamine B DEFINITION: Magnetosome mineral core: Magnetosomes studied here are composed of an iron oxide mineral core essentially composed of crystallized iron oxide surrounded by a layer containing biological material (LPS, proteins, lipids) and rhodamine B. For a schematic picture of these magnetosomes, see scheme 2.

ABSTRACT: Therapeutic substances bound to nanoparticles have been shown to dissociate following excitation by various external sources of energies or chemical disturbance, resulting in controllable and efficient antitumor activity. Bioconjugation is used to produce magnetosomes associated with Rhodamine B (RhB), whose fluorescence is partially quenched by the presence of the iron oxide and becomes strongly enhanced when RhB dissociates from the magnetosomes under the application of an alternating magnetic field. This novel approach enables to simultaneously release a RhB model molecule and to monitor such mechanism by fluorescence. The dissociating mechanism of RhB is highlighted by exposing a suspension of fluorescent magnetosomes to an alternating magnetic field, by magnetically isolating the supernate of this suspension, and by showing fluorescence enhancement of the supernate. Furthermore, to approach in vivo conditions, fluorescent magnetosomes are mixed with tissue or introduced in mouse brain and exposed to the alternating magnetic field. Most interestingly, the percentages of RhB dissociation measured at the beginning of magnetic excitation (R/t) or 600 seconds afterwards (R 600s ) are R/t ~ 0.13% and R 600s ~ 50% under conditions of limited temperature increases (<2.5 C), larger values than those of R/t ~ 0.02-0.11% and R 600s ~ 13%, estimated at temperatures above 2.5 C. Furthermore, when magnetic excitations are repeated two to five times, temperature increase becomes undetectable, but RhB dissociation continues to occur up to the fifth magnetic excitation. Since large heating temperatures may be damaging for tissues, this study paves the way towards the development of a safe theranostic dissociating nano-probe operating at low temperature.

INTRODUCTION

Antitumor activity can be reached with nanoparticles introduced to tumors and exposed to an external source of energy, either producing a temperature increase, 1-5, or the release and activation of a therapeutic substance, 6. The first method of treatment is usually designated as magnetic hyperthermia. It led to preclinical efficacy on various cancer types, including prostate, 7, glioblastoma, 8-10, head and neck, 11, melanoma, 12, and to a 6 months increase in survival among patients bearing glioblastoma compared with the standard of care, 13, 14. It also appears to be safer than conventional thermotherapies such as ultrasounds since it yields antitumor activity at more moderate heating temperatures, i.e. 41-46 C, 15-17, compared with temperatures larger than 60 C for high intensity ultrasound treatments, 18. To strengthen safety even more, drugs could be released from nanoparticles under magnetic excitation in the absence of a significant temperature increase. Chemical synthesis methods have been used to fabricate dissociating nano-probes, 19-21], but it appears that they often require high temperatures to reach efficient dissociation, possibly due to the relatively strong bonds between the nanoparticles and the therapeutic substance that can't easily be broken at low temperatures.

In this manuscript, we describe the synthesis method of a dissociating nano-probe operating in conditions of limited temperature increase (< 2.5 C). This nano-probe is further made fluorescent to monitor the dissociation. For that, we introduce in the growth medium of magnetotactic bacteria a fluorescent substance mimicking a therapeutic agent (RhB), 22, we wait for these bacteria to incorporate RhB in the magnetosomes, and we finally extract fluorescent magnetosomes designated as MCR400 from these bacteria. We show that RhB is sufficiently strongly bound to the magnetosomes to enable MCR400 to maintain their fluorescence following their isolation and re-suspension in water. To examine if MCR400 can be used as intracellular probe, fluorescent magnetosomes are then brought into contact with cancer cells and their internalization, cytotoxicity, and fluorescent properties are studied.

The operating mechanism of the MCR400 probe is deciphered by exposing a MCR400 suspension to an alternating magnetic field (AMF), by magnetically separating from bacterial debris the supernate of this suspension in which RhB has potentially been released, and by measuring its fluorescence intensity. To approach real magnetic hyperthermia treatment conditions, MCR400 are either mixed with brain tissue or introduced in a mouse brain and exposed one to five times to an AMF. MCR400 concentration and environment are varied to modulate the magnitude of temperature increases. The initial rate and quantity of RhB released are deduced from the variation of the MCR400 fluorescence intensity, measured at the beginning of magnetic excitation and 600 seconds afterwards, as a function of various heating temperatures.

RESULTS AND DISCUSSION

Properties of whole fluorescent magnetotactic bacteria. In the absence of a compound that can permeabilize the membrane of gram-negative bacteria such as PEI, it is difficult to introduce fluorescent substances inside such bacteria during their growth, 23. For this reason, it was usually attempted to stain bacteria with RhB following their amplification. It either yielded very weak fluorescence, 24, or more significant fluorescence, 25, depending on the use (or not) of a specific protocol to bind RhB to bacterial membranes. By using magnetotactic bacteria, we were able to trigger a truly unusual behavior, i.e. incorporate RhB in these bacteria by simply adding RhB to their growth medium, which in turn seems to have changed their metabolism since both whole bacteria and magnetosomes became larger in the presence of RhB as explained below. The fluorescence of magnetotactic bacteria cultivated in the presence of 400 µM of RhB was observed by optical microscopy. Whereas whole magnetotactic bacteria cultivated without RhB appeared non-fluorescent under optical microscopy observation, 26, those grown in the presence of 400 µM of RhB displayed a strong intracellular fluorescent signal (Fig. 1). This behavior was further supported by flow cytometer measurements. Indeed, the FL3-H signal, which is proportional to RhB fluorescence, is lower than 10 a.u. for bacteria grown without RhB indicating that the bacteria are non-fluorescent according to a criterion of the flow cytometer. It increases to a value larger than 10 a.u. for magnetotactic bacteria cultivated in the presence of RhB, yielding 54% of fluorescent bacteria according to this criterion (Fig. 2(a) and table S1). To examine possible changes in bacterial morphologies induced by RhB, we analyzed the cloud of points representing the side scatter height (SSC-H) signal as a function of the forward scatter height (FSC-H) signal. Between bacteria grown in the absence and presence of RhB, Figs. 2(b) and 2(c) show that the FSC-H and SSC-H signals increase from 20-200 a.u. up to 200-10000 a.u. and from 30-300 a.u. up to 200-10000 a.u., respectively. Such behavior may be associated with an increase in size of magnetotactic bacteria and magnetosome chains, 27, when RhB is added to the growth medium. It may arise from the chelation of iron by RhB that could simultaneously increase the amount of iron and RhB introduced in magnetotactic bacteria. On the one hand, chelation could result in enhanced stress applied on these bacteria or to change in their metabolism, which would modify their size, 28. On the other hand, it could favor the formation of larger magnetosomes or longer magnetosome chains due to more iron introduced in these bacteria, 28, 29. This behavior is further confirmed by comparing the histograms in size of MC and MCR400 presented in Figs. S2( small magnetosomes (between 0 and 30 nm) and 40 nm for large magnetosomes (between 30 and 60 nm). In the presence of RhB, these two sizes increase to 30 and 52 nm, respectively (Fig. S2(b))., [START_REF] Alphandéry | The effect of iron-chelating agents on Magnetospirillum magneticum strain AMB-1: stimulated growth and magnetosome production and improved magnetosome heating properties[END_REF].

Properties of MCR400 extracted from magnetotactic bacteria. For the magnetic hyperthermia treatment of tumor, it is difficult to use whole magnetotactic bacteria, due to the too low magnetosome concentration in a bacterial suspension that does not enable sufficient heat production under AMF application, [START_REF] Alphandéry | Heat Production by Bacterial Magnetosomes Exposed to an Oscillating Magnetic Field[END_REF]. To enable their potential further use in magnetic hyperthermia, magnetosomes were therefore extracted from fluorescent magnetotactic bacteria. We first analyzed whether the fluorescence of RhB which was observed in the bacterial cells as underlined in the previous section, was maintained following extraction. The FT-IR spectrum of magnetosomes extracted from fluorescent magnetotactic bacteria is presented in Fig. 3(a). It shows the superposition of two series of peaks. The first series of peaks is also present in RhB free magnetosomes MC (in blue), i.e. amide I at 1650 cm -1 , amide II at 1530 cm -1 , lipopolysaccharides (LPS) or phospholipids at 1050 cm -1 and at 1250 cm -1 , maghemite iron oxide at 580 cm -1 , Fig. 3(a), [START_REF] Li | Investigation of adsorbtion and photocatalytic activities of in situ cetyltrimethylammonium bromide-modified Bi/BiOCl heterojunction photocatalyst for organic contaminants removal[END_REF]. The second series of peaks is due to RhB (in red), i.e. peaks at 1700 cm -1 , 1600 cm -1 , 1475 cm -1 , 1250 cm -1 , 1175 cm -1 , 1125 cm -1 , 675 cm -1 , Fig. 3(c), 32. These results suggest that in MCR400, RhB is either adsorbed at magnetosome surface or complexed with magnetosomes, possibly due to the carboxylic acid function of RhB that can interact with the FeOH groups at magnetosome surface, [START_REF] Farag | absorption and optical dispersion characteristics of rhodamine B thin films prepared by drop casting technique[END_REF]. Further proofs of the association of RhB to the magnetosomes in MCR400 are provided by the absorption spectrum of a MCR400 suspension that displays a shoulder at 550 nm (Fig. 3(d)), a wavelength corresponding to maximum absorption by free RhB (Fig. 3(f)) and to the absence of an absorption signal from MC at this wavelength (Fig. 3(e)). Moreover, the fluorescence spectrum of the MCR400 suspension excited at 405 nm, displays a peak at 569 nm (Fig. 3(g)), which is absent in the spectrum of the MC suspension excited in the same conditions as MCR400 (Fig. 3(h)) and is positioned at a slightly lower wavelength than that of 576 nm observed for free RhB (Fig. 3(i)). This wavelength shift is possibly due to interactions between RhB molecules and the magnetosome surface in MCR400, as previously reported for fluorescent molecules associated to other types of nanoparticles, [START_REF] Boyer | The design and utility of polymerstabilized iron-oxide nanoparticles for nanomedicine applications[END_REF].

Cytotoxicity, fluorescence, and internalization properties of MCR400 in the presence of cancer cells. In a previous study, we showed that MCR400 incubation with Zymozan-activated murine peritoneal macrophages leads to enhanced fluorescence intensity, a behavior which was attributed to the capture of MCR400 by lysosome following their cellular internalization leading to RhB dissociation from the magnetosomes at acidic pH, 35. To be able to use MCR400 as an intracellular fluorescent probe, MCR400 shall not only maintain a high level of fluorescence but also yield limited cytotoxicity.

In this study, MCR400 were therefore incubated with MDA-MB-231 cells during 6 hours in the presence (or not) of Calcein AM. The Calcein AM fluorescence emission intensity (FL1-H), which is proportional to the number of viable cells, 36, was observed to increase from 2 for cells alone to ~ 1000 for cells in the presence of Calcein AM both with and without MCR400 (Figure 4(a)), indicating that ~97% of cells are viable following MCR400 incubation (table S1). MCR400 therefore do not induce any significant cytotoxicity towards these cancer cells. This result is consistent with the absence of RhB cytotoxicity below a certain concentration threshold, 37. Furthermore, between cells alone and cells incubated with MCR400, the fluorescence signal of RhB (FL3-H) and its associated percentage of fluorescent cells increase from 1 to 55 (Fig. 4(b)) and from 0% to 99% (table S1), respectively.

Therefore, almost all cancer cells become fluorescent following their incubation with MCR400. Finally, to determine if MCR400 internalize in these cancer cells, two types of epi-fluorescence microscopy images of MCR400 incubated with cancer cells were taken, either in the absence (Fig. 4(c and macrophages, 38, making this behavior likely to occur for different cell types. We can conclude that MCR400 may be used as intracellular fluorescent probe, which could have potential diagnostic and therapeutic applications both in vitro and in vivo.

Operating mechanism of the MCR400 probe studied in suspension. We further examined the operating mechanism of the MCR400 probe by suspending MCR400 in water at an iron oxide concentration of 1 mg Fe /mL and by applying an AMF of 25 mT and 198 kHz during 1200 seconds that produced a temperature increase of 7 °C (Fig. 5(a)). This yielded conditions in terms of AMF parameters and temperature that are typical to those reached during magnetic hyperthermia, 15, 16.

During magnetic excitation, Fig. 5(b) shows that the fluorescence intensity of this suspension remains unchanged, possibly due to fluorescence quenching by the iron oxide of the magnetosome mineral core, which could limit RhB emission, [START_REF] Yu | Bare magnetic nanoparticles as fluorescence quenchers for detection of thrombin[END_REF]. By contrast, the fluorescence intensity of the supernate of this suspension, which was magnetically separated from the magnetosome mineral cores, increased during the first 100 seconds of the magnetic excitation from 1 to 19 a.u. and then stabilized at 18.5-19.5 a.u. between 100 and 1200 seconds (Fig. 5(b)). Such behavior may not come from the variation of the intrinsic RhB fluorescence, which decreases with increasing temperature, [START_REF] Löw | High-Spatial-Resolution Surface-Temperature Mapping Using Fluorescent Thermometry[END_REF]. Instead, it may be due to the dissociation of RhB from the magnetosomes under magnetic excitation, which could lead to fluorescence de-quenching. Furthermore, the emission wavelength, corresponding to the maximum fluorescence intensity for an excitation at 405 nm, is observed to increase from 569 nm for MCR400 to 576 nm for the supernate of MCR400. In fact, RhB molecules may either interact with the magnetosomes in the MCR400 suspension leading to an emission wavelength below that of free RhB or may produce such interactions in the supernate of this suspension due to the diffusion of RhB molecules away from the magnetosomes, yielding for the supernate the same emission wavelength as that of free RhB, (Fig. S4(b)). We have highlighted a mechanism by which RhB molecules initially interact with the iron oxide of the magnetosome mineral, resulting in fluorescence quenching. Following AMF application, RhB molecules are released from the magnetosomes, leading to fluorescence intensity increase, which could also be designated as fluorescence de-quenching. Such mechanism could potentially be used to control the release of a drug under the application of an AMF provided RhB is replaced by a therapeutic substance.

Ex vivo release of RhB from magnetosome minerals in the presence of a temperature increase comprised between 2.5 °C and 10 °C.

To complement studies in solution and come closer to the real conditions of treatment, we have introduced 2 µl of a suspension containing 40 or 400 µg of MCR400 at a depth of 1 to 3 mm in a mouse brain extracted from a dead mouse (Scheme 3) that we have exposed to an alternating magnetic field of frequency 200 kHz and strength 25 mT for 30 minutes. Figs. 6(a) and 6(b) show that these conditions yield a temperature increase lying within the range of 2.5-10 °C, a similar temperature increase than that reached in a magnetic hyperthermia treatment, which was previously determined as ~ 3-10 C from preclinical studies carried out with magnetosomes, [START_REF] Alphandéry | Chains of magnetosomes with controlled endotoxin release and partial tumor occupation induce full destruction of intracranial U87-Luc glioma in mice under the application of an alternating magnetic field[END_REF], [START_REF] Alphandéry | Development of non-pyrogenic magnetosome minerals coated with poly-l-lysine leading to full disappearance of intracranial U87-Luc glioblastoma in 100% of treated mice using magnetic hyperthermia[END_REF].

Furthermore, the influence of the position of the excitation/detection fiber on MCR400 fluorescence properties was studied in sample 1 by mixing homogenously 40 µg of MCR400 with brain tissue and by positioning this fiber at different heights of 1 to 3 mm above the tissue. The various fiber heights led to similar values of MCR400 fluorescence intensity (Fig. 7(a)), indicating that the distance between the fiber end and the magnetosome region (Scheme 3) is not an important parameter in this distance range.

Between samples 1 and 2, we increased the amount of MCR400 from 40 µg (sample 1) to 400 µg (sample 2), resulting in enhanced heat production and diffusion outside of the injection volume, two behaviors that compensated each other to yield in both cases a similar value of fluorescence intensity, F 600s ~ 125 a.u., and percentage of RhB released from the magnetosomes, R 600sec ~ 12%, measured 600 seconds following magnetic excitation (Figs. 6(c) to 6(f)).

Given that after a certain time of magnetic excitation (600 seconds), the fluorescence intensity and percentage of released RhB appear to reach saturation and to become independent of temperature and conditions of excitation, we have examined another parameter, i.e. the initial rate of released RhB, R/t, which takes place just after activation of the magnetic excitation. It appears to be larger and less influenced by MCR400 interactions with its surrounding environment than rates of release estimated at later stages, hence possibly being a controllable parameter to trigger a therapeutic activity. Most interestingly, between samples 1 and 2, R/t increases from 0.02 %/sec. to 0.11 %/sec (Figs. 7(c) and 7(d)). On the one hand, such sharp transition could be due to T/t increase between samples 1 and 2, reaching a certain threshold value of ~ 0.008-0.12 C/sec. (Figs. 7(c) and7(d)). However, this hypothesis seems unlikely since Fig. 7 shows that temperature variations influence rather weekly fluorescence intensity and RhB release. On the other hand, it could come from more brain tissue material surrounding the nanoparticles in sample 1 than in sample 2, which could more strongly prevent efficient RhB diffusion. In fact, in sample 1 brain tissue material could have adsorbed at MCR400 surface or fill MCR400 environment when MCR400 were mixed with tissue, while in sample 2 MCR400 could mainly be surrounded by brain liquid and much less by brain tissue materials as was observed during MCR400 injection in mouse brain. The optimal conditions for efficient release of RhB from magnetosomes and for the monitoring of such mechanism therefore seem to be an environment that does not prevent RhB diffusion away from the magnetosomes, a sufficiently large initial temperature gradient, and the measurement of fluorescence variations carried out at the beginning of magnetic excitation. For in vivo or clinical applications, these results imply that magnetosomes could be used to release a drug provided RhB is replaced by a therapeutic substance. Heat and pharmacological treatments could hence potentially be combined during magnetic hyperthermia by using a drug, which would diffuse away from the magnetosomes and thus be activated following magnetic excitation.

Repetitive ex vivo release of RhB from magnetosomes under magnetic excitation in the absence of significant temperature increase (< 2.5 °C). Previous studies have reported antitumor activity when magnetic nanoparticles were mixed with tumor cells or introduced to tumors and exposed to an AMF without inducing any measurable temperature increase, 15. Such behavior was either explained by mechanical disruption of tumor cells induced by AMF application, 41, or to an indirect mechanism of tumor destruction, possibly involving immune cells, such as natural killer, T cells or polynuclear neutrophils, 15, which could eradicate unheated tumor cells. In order to control such mechanism, an immunogenic substance could be bound to nanoparticles and be progressively released from them under AMF application, as recently suggested for lipopolysaccharide (LPS) dissociating from magnetosomes under magnetic excitation, possibly leading to the activation of the immune system against the tumor, 15. Most interestingly, such activation could be repeated by re-applying the AMF, 15, 16. Here, by using RhB as a model molecule that mimics the behavior of a therapeutic substance, we examined if such a substance could repetitively dissociate from the magnetosomes in the absence of any significant temperature increase over several AMF applications. For that, we administered 40 µg of MCR400 at a depth of 3 mm in a mouse brain (Scheme 3) that we exposed to five successive magnetic sessions (MS1 to MS5) during which an AMF of 25 mT and 200 kHz was applied for 30 minutes. Whereas MS1 produced a limited temperature increase of 2.4 °C, MS2 to MS5 did not induce any detectable change in temperature (Fig. 8(a), Fig. 9(c)), presumably due to magnetosome diffusion away from the exposed area. Most interestingly, in the absence of any detectable temperature increase (MS2 to MS5), although the four different parameters F 600s , F/t, R 600s , R/t, decreased during increasing number of magnetic sessions (Figs. 9(a) and 9(b)), they remained significant, indicating that RhB could be repetitively released from the magnetosomes and that such release could be monitored by measuring the parameters F 600s and F/t. This result differs from previous studies which suggested that significant temperature increase was necessary to trigger drug release from nanoparticles, 42-44. However, to our knowledge such studies did not consider the change in nanoparticle environment or nanoparticle-nanoparticle interactions with increasing temperature, which could strongly influence and possibly hinder efficient drug release. Our finding is interesting for medical applications since it indicates that efficient drug release could be induced by AMF application in the absence of a detectable temperature increase, hence using safer conditions than the high temperatures usually reported to be required for triggering an efficient release mechanism, which could be damaging towards healthy tissues and dangerous for the patient.

CONCLUSION:

In this study, we describe an original method of fabrication of a theranostic fluorescent nano-probe, in which magnetotactic bacteria are cultivated in the presence of RhB followed by magnetosome extraction from these bacteria, yielding a suspension of MCR400. Magnetosomes thus made are shown to maintain their fluorescence properties following internalization in cancer cells, showing their potential use as intracellular theranostic probes. When a MCR400 suspension is exposed to an AMF, it produces enhanced fluorescence intensity of its supernate, indicating the release of RhB from the magnetosomes. The operation of this probe is further highlighted in conditions that are close to those reached in a magnetic hyperthermia treatment, i.e. for 40 µg or 400 µg of MCR400 mixed with tissue or introduced in a mouse brain and exposed to the AMF. Indeed, with increasing duration of the magnetic excitation, they lead to an enhancement of MCR400 fluorescence intensity and hence also of RhB release. Moreover, we demonstrate that we can control such release mechanism by monitoring it through fluorescent measurements, and by adjusting the parameters of the magnetic excitation, the temperature, as well as MCR400 environment and concentration.

To summarize, we have shown that:

i)

It is possible to release a substance from a nano-probe and to monitor the dissociation by fluorescence.

ii)

The fluorescence, which is initially quenched by the iron oxide, becomes enhanced following dissociation, making this nano-probe more sensitive than other fluorescent nanoparticles operating under conditions of fluorescence intensity decrease, 45.

iii) Provided the fluorescent substance (Rhodamine B) is replaced by a therapeutic drug, this nanoprobe could also be used as a therapeutic drug operating in conditions of limited temperature increase (< 2.5 C), hence providing safer treatment conditions than conventional thermotherapies working at higher temperatures.

iv)

The release of a therapeutic substance could be repeated by re-applying the alternating, hence possibly enabling a therapeutic activity to be triggered repetitively.

The field of drug released from nanoparticles, which is rapidly expanding, 46, 47, should benefit from the new approach presented in this study.

MATERIALS AND METHODS

Culture of magnetotactic bacteria in the presence (or not) of RhB. Magnetospirillum magneticum

AMB-1 magnetotactic bacteria were obtained from ATCC (ATCC 700274) and grown under microaerophile conditions at 30 °C in a growth medium slightly different from the MSGM medium (ATCC medium 1653). In one liter, this medium contained 0.68 g of monobasic potassium phosphate, 0.85 g of sodium succinate, 0.57 g of sodium tartrate, 0.083 g of sodium acetate, 225 μl of 0.2% resazurin, 0.17 g of sodium nitrate, 0.04 g of L-ascorbic acid, 2 ml of a 10 mM iron quinate solution, 10 ml of a solution of Woolf vitamins and 5 ml of a solution of Woolf minerals. While RhB was not introduced in the bacterial growth medium to synthesize non-fluorescent magnetosomes (MC), 400 µM of RhB were introduced in this medium to synthetize fluorescent magnetosomes (MCR400). The pH of the culture medium was adjusted to 6.85 using a 1M sodium hydroxide solution. The bacteria were collected during the stationary phase and concentrated using a tangential flow filtration column (mPES, 500 kDa) with a flow rate of 950 mL/min and then washed 5 times for 30 minutes with a solution of saline phosphate buffer at pH 7.4 (137 mM NaCl, 2.7 mM KCl, 10 mM Na2HPO 4 , 1.76 mM KH2PO 4 ). Bacteria were collected by centrifugation at 4000 rpm for 1 hour, the supernatant was removed and the bacteria were re-suspended in 50 mM Tris-HCl buffer solution at pH 7.4 and diluted to yield an optical density of 5 at 600 nm.

Preparation of MC and MCR400 in suspension. Suspensions of magnetotactic bacteria cultivated in the presence (or not) of 400 µM RhB were lysed, magnetosomes were extracted from whole magnetotactic bacteria, bacterial suspensions were sonicated at 30 W at 5 °C during 60 minutes with pulses of 2 sec. and an interval between pulses of 1 sec. Following sonication, magnetosome chains were magnetically isolated from cellular debris using a neodymium magnet. The supernate containing the cellular debris was removed and fluorescent (MCR400) or non-fluorescent (MC) magnetosome chains were washed five times magnetically with a 50 mM Tris-HCl buffer solution at pH 7.4 and then fifteen times with Millipore® water. They were finally re-suspended in Millipore® sterile water to obtain suspensions of MC and MCR400.

Preparation of MC or MCR400 brought into contact with MDA-MB-231 cells. MDA-MB-231 cells

were purchased from the American Type Culture Collections (ATCC). They were cultivated in Dulbecco's modified Eagle's medium (DMEM) supplement, which contained 10% fetal calf serum (FCS), 2 mM l-glutamine, 1 mM sodium pyruvate, and 50 U/ml streptomycin, which were purchased from Life Technologies Inc. The cells were then incubated during 6 hours with suspensions of MC and MCR400 at an amount of 63 µg in iron oxide and then studied by flow cytometry or epifluorescence.

Preparation of MC or MCR400 mixed with brain tissue or introduced in a mouse brain.

To prepare MC and MCR400 mixed in brain tissue, 2 l of a MC or MCR400 suspension at a concentration of 20 mg/mL in iron oxide were mixed homogenously with mouse brain tissue contained in a 2 mm 3 volume. To obtain MC and MCR400 introduced in mouse brains, 2 or 20 l of a MC or MCR400 suspension at a concentration of 20 mg/mL in iron oxide were introduced at a depth of 1, 2 and 3 mm of a mouse brain extracted from a dead mouse.

Magnetic treatment of MC and MCR400 in suspension, mixed with brain tissue, or introduced in a mouse brain. 300 µl of MC and MCR400 dispersed in suspensions at 3.6 mg/mL in iron oxide were exposed to one magnetic session (MS). MC and MCR400 mixed in brain tissue or introduced in a mouse brain as described above were also exposed to 1 MS. Each MS consisted in the application of an alternating magnetic field (AMF) of 25 mT and 200 kHz for 1200 to 20000 seconds.

Optical microscopy imaging of whole magnetotactic bacteria. 10 µl of a suspension of whole magnetotactic bacteria were deposited on a microscope slide and observed with an inverted phasecontrast microscope (Zeiss Primovert) using an oil immersion objective (100 X, Ph1/0.4) with a LD condenser 0.4 (working distance: 55 mm).

Epifluorescence microscopy of cells brought into contact with MCR400.

We imaged MDA-MB-231 cells incubated in the presence of MCR400 to determine if MCR400 were internalized inside these cells.

We used the same conditions of incubation as for flow cytometer measurements except that we fixed cells using 5% PFA. We used an epi-fluorescence microscope with an oil immersion objective x64 for optical imaging with/without RhB detection.

Flow cytometer measurements on whole magnetotactic bacteria. We used a flow cytometer to compare the properties of magnetotactic bacteria cultivated in the absence and presence of 400 µM RhB. Before flow cytometer measurements, the bacteria were washed 5 times with deionized water. for each wash, the suspension was placed against a magnet, the supernate was removed, and replaced by deionized water. 500 μl of suspensions containing magnetotactic bacteria cultivated with/without RhB were then mixed with Water Millipore® and introduced into a flow cytometer. The study was carried out using 50 000 bacteria for each condition. The flow cytometer, a FACS-calibur, was equipped with a laser of excitation wavelength 488 nm and enabled to measure various parameters such as size, FT-IR characterization of MC, MCR400, and RhB. Suspensions at pH 7 containing MC, RhB and MCR400 with a concentration of 1 mg/mL in iron oxide were lyophilized and integrated in a KBr matrix. Their Fourier Transform infra-red (FT-IR) absorption spectra were recorded using a Nicolet FT-IR model 380 with resolution of 0.5 cm -1 and a number of scans of 30.

Transmission electron microscopy of whole bacteria. Electron transmission microscopy images of whole bacteria, M and M R400 were obtained with a EM-2100 from JEOL. For that, 5 µl of a suspension of whole bacteria, MC, and MCR400, were deposited on top of a carbon grid and dried. Size distributions of magnetosomes MC and MCR400, were measured using 300 magnetosomes and plotted in histograms.

Absorption and fluorescence measurements of MC, MCR400, and RhB suspended or mixed in water. The absorption spectra of MC and MCR400 (suspensions containing 40 µg in iron oxide of nanoparticles contained in 1 mL of water) or of RhB (solution containing 4 µg of RhB in 1 mL of water)

were recorded between 350 nm and 700 nm using a Varian Cary 3E UV-Vis spectrophotometer. The fluorescence spectra of MC and MCR400 (170 µg in iron oxide of nanoparticles contained in 800µL of water) or of RhB (0.4µg of RhB in 800 µL of water) were recorder on a Aminco-Bowman 2 spectrofluorimeter (Edison, NJ). Fluorescence intensity was recorded between 550 and 650 nm under excitation at 405 nm.

Fluorescence measurements of MCR400 suspended in water in the presence of a magnetic treatment. 1 mg/mL in iron oxide of MCR400 were mixed in 100 µL of water and exposed to one MS, we measured the fluorescence intensity of this suspension and of its supernate, which was magnetically isolated from the magnetosomes using a Neodynium magnet of 0.6 T. We measured the fluorescence intensity of the MCR400 suspension and of its supernate at different times following magnetic excitation (0, 100, 200, 300, 600, 900 or 1200 seconds). Fluorescence intensity was measured at 576 nm for an excitation at 555 nm using a Fluoroskan Ascent (Thermo Scientific) microplate spectrometer.

Fluorescence measurements of MCR400 mixed with brain tissue or introduced in a mouse brain, using a bi-fiber set-up, in the presence of a magnetic treatment. For MCR400 and MC mixed in brain tissue or introduced in mouse brain as described above, samples were excited by a pulsed diode laser emitting at 405 nm with a mean power of 1 mW and a repetition rates of 40 MHz from PicoQuant (GmbH, Berlin, Germany). A bi-fibered configuration was employed for excitation and collection positioned 3 mm above the surface of the brain or mixed issue. The fibers used for excitation and collection had a core diameter of 200 µm and 365 µm, respectively, with a numerical aperture of 0.22.

The spatial resolution was 500 µm. Collected fluorescence signal was sent toward a computer controlled cooled spectrometer (Ocean optics QP600-1-UV-VIS) for spectroscopic analysis. For fluorescence detection, the fiber was positioned 1, 2, or 3 mm, above the lamella for the sample containing MCR400 mixed with brain tissue while it was fixed at a height of 3 mm above the lamella for the sample containing MCR400 introduced in mouse brain at a depth of 1, 2, or 3 mm. In this way, the distance between the fiber end and the center of the samples was the same for a height of 1, 2, or 3 mm in mixed tissue than for a depth of 1, 2, or 3 mm in mouse brain.

Temperature measurements of MCR400 in suspensions mixed with brain tissue or introduced in a mouse brain during a magnetic treatment. For studies in brain tissues and ex vivo, spatial temperature distribution in the region containing MCR400 was measured using an Easir 2 thermographic infrared camera positioned 20 cm above the tissues or brain. The plotted temperatures correspond to the highest temperature measured at each time point within the heated surface of ~ 3 mm 2 . A microprobe thermocouple (IT-18, Physitemp, Clifton, USA) was inserted inside the magnetosme suspensions and employed for measurements in solution using a Thermes USB (Physitemp) thermometer. The spatial resolutions of the infrared camera and of the thermocouple were 660 µm and 600 µm, respectively.

Temperature parameters T 600sec. , T/t, and deduction of magnetosome biodistribution properties from the values of T/t. T 600sec. and T/t correspond to the temperature increase measured 600 seconds following the beginning of magnetic excitation and to the initial slope of the temperature variation with time, measured at the beginning of magnetic excitation, respectively. For MCR400 mixed in tissue or introduced in mouse brain, we could deduce MCR400 biodistribution properties from the different values T/t by using the relation between the magnetosome specific absorption rate (SAR), T/t, and the magnetosome concentration, C mag , which is given by: SAR=C v (T/t)/C mag , where C v is the specific heat of water. Assuming that C v and SAR remain unchanged between the different samples, we deduced that the percentage of MCR400 in the injection volume is proportional to (T/t) 2 /(T/t) 1 , where (T/t) 2 and (T/t) 1 are the initial slopes of the temperature variation, measured when MCR400 occupy part or 100% of the injection volume, respectively.

Fluorescence measurement of RhB introduced in mouse brain or mixed with brain tissue, using a bi-fiber set-up, in the absence of magnetic treatment. We followed the same protocols of preparation as for MCR400 mixed with brain tissue or introduced in mouse brain. On the one hand, different

  a) and S2(b) deduced from transmission electron microscopy images of bacteria cultivated in the absence (Fig. S3(a)) or presence (Fig. S3(b)) of RhB.. In the absence of RhB, Fig. S2(a) shows that magnetosome sizes with maximum frequency are 22 nm for

  )) or presence (Fig. 4 (d)) of RhB emission. The superposition of Figs. 4 (c) and 4 (d), presented in Fig. 4(e), shows that RhB molecules fluoresce where MDA-MB231 cells are located, which suggests that MCR400 have penetrated inside MDA-MB-231 cells without losing their fluorescence properties. The other possible location of MCR400 at cell surface, which may also be considered by analyzing Fig. 4(d), appears unlikely since Fig. 4(c) doesn't show any magnetosome aggregates at this location. MCR400 enhanced fluorescence intensity following cellular internalization was observed with cancer cells (MDA-MB-231)

  granularity and fluorescence of magnetotactic bacteria (or cancer cells see below). These parameters were analyzed simultaneously by means of an optical filter set, which decomposes the light emitted by the fluorochromes and directs each light signal to a different photomultiplier to record it. The detected optical signals were then converted into electrical signals and then stored and analyzed in digital values. This light-matter interaction also led to the diffusion of light by bacteria. These scattering signals were collected along an axial axis (FSC-H: Forward Scattering or axial diffusion) and a lateral axis (SSC-H: Side scattering). The FSC-H signal is proportional to bacterial sizes whereas the SSC-H signal gives information on the intracellular aspect or content of magnetotactic bacteria, i.e. on their granularity. The luminescence signal of RhB was also collected through a FL3-H channel at emission wavelengths between 560 nm and 627 nm. According to the flow cytometer specifications, a FL3-H signal lower and larger than the threshold of 10 a.u. corresponds to the absence and presence of RhB fluorescence, respectively. The flow cytometer also estimated the percentage of fluorescent bacteria.Flow cytometer measurements on MDA-MB231 cells. We used the same flow cytometer as described above to study the viability and fluorescent properties of MDA-MB-231 cancer cells incubated in the presence of MC and MCR400. For that, 500 000 MDA-MB-231 cells contained in a petri dish of 2 mL were incubated during 6 hours with 31.5 µg/mL in iron oxide of MC or MCR400. Then, a cell viability probe, calcein acetoxymethyl ester (Ca), which becomes luminescent in the presence of living cells, was added to the petri dish and the resulting assemblies were collected after trypsinization and centrifugation in tube for flow cytometer measurements. The flow cytometer detected either the fluorescence of Ca (FL1-H) at 530 nm or that of RhB at 585 nm (FL3-H). According to the flow cytometer specifications, FL1-H and FL3-H signals are either lower or larger than a threshold of 10, corresponding to the threshold value above which cells are viable for FL1-H or fluorescent for FL3-H. The flow cytometer also estimated the percentages of viable and fluorescent cells.

Scheme 1 :Scheme 2 :Figure 1 :

 121 SCHEME:

Figure 2 :

 2 Figure 2: (a), Number of bacteria counted with the flow cytometer as a function of the FL3-H

Figure 3 :

 3 Figure 3: FT-IR spectra of chains of magnetosomes extracted from magnetotactic bacteria cultivated in

Figure 4 :

 4 Figure 4: (a) Number of cells counted with a flow cytometer (FACS-Calibur) as a function of FL1-H

Figure 5 :

 5 Figure 5: (a), for a 100 µL of suspension of MCR400 at an iron oxide concentration of 1 mg/mL

Figure 6 :

 6 Figure 6: Fluorescence analysis of the MCR400 nano-probe in conditions where the application of an

Figure 7 :

 7 Figure 7: Fluorescence analysis of the MCR400 nano-probe in conditions where the application of an

Figure 8 :

 8 Figure 8: Fluorescence analysis of the MCR400 nano-probe in conditions where the application of an

Figure 9 :

 9 Figure 9: MCR400 fluorescence properties for 20 µl of a MCR400 suspension at 20 mg/mL

718 35


Marc P Bellon

Alain Connes

Michel Dubois

Lectures on the three-dimensional non-commutative spheres

Lectures on the three-dimensional non-commutative

spheres.

Introduction

The description of the moduli space of non-commutative three-dimensional spheres is a beautiful piece of mathematics, first presented by Alain Connes and Michel Dubois-Violette in [START_REF] Connes | Noncommutative Finite-Dimensional Manifolds. I. Spherical Manifolds and Related Examples[END_REF][START_REF] Connes | Moduli space and structure of noncommutative 3-spheres[END_REF][START_REF] Connes | Noncommutative finite-dimensional manifolds. II. Moduli space and structure of non commutative 3-spheres[END_REF]. In non-commutative geometry, it is the first exhaustive study of a category of non-commutative spaces.

Non-commutative spaces have been studied both from the mathematical and physical perspectives. In the mathematical realm, a strong motivation has been the study of "bad" quotients of spaces, like foliations with dense leaves and quotients by ergodic group actions.

Physical motivations have come from the realm of quantum gravity, since absolute limitations on the measurements of positions of events might translate in non-commuting coordinates. In string theories, background gauge field may induce non-commutation of position fields.

The most complete presentation of non-commutative geometry can be found in the book of the creator of the field, Alain Connes [START_REF] Connes | Noncommutative geometry[END_REF], but the shorter presentation of [START_REF] Khalkhali | Very basic noncommutative geometry[END_REF] provides an interesting introduction. Finally, a panorama of the domain and its applications can be found in [START_REF] Connes | A walk in the noncommutative garden[END_REF].

I will first briefly motivate the definition of non-commutative spheres, showing in particular that ordinary spheres as well as some non-commutative deformations are included. In one and two dimensions, the only objects which satisfy the conditions are the commutative spheres, so that we have to go to three dimensions to have non-trivial solutions. Then the moduli space of three-dimensional non-commutative spheres is introduced, the corresponding quadratic algebras are described. A large place is devoted to the geometric data, which allows to show that differing parameters really describe differing algebras, with some lights about the special cases. The last sections introduce the covering of the generic quantum spheres by a cross-product algebra of functions with Z.

These lecture notes use two principles to reach a fairly complete treatment in a reasonable space. Avoid any use of the transcendental functions and only present one of the different possible presentations. In particular, I always work with the variables which are used only for the complexified moduli space in [START_REF] Connes | Noncommutative finite-dimensional manifolds. II. Moduli space and structure of non commutative 3-spheres[END_REF]. This means that the generator of the algebra are not self-adjoint and it requires some care to express the reality conditions, but it greatly simplifies other aspects.

Basic facts in non-commutative geometry

In ordinary differential geometry, a large use is made of the differential forms which allow to define the De Rham complex. Forms can be considered as elements of an algebra generated by the functions and symbols df for any function f with the relations d1 = 0, d(f g) = (df )g + f (dg), (Leibniz rule).

The differential d can be extended to these forms by making it obey a graded Leibniz rule and imposing that d 2 = 0. The classical De Rham complex is obtained adding the commutation rules f (dg) = (dg)f, df dg = -dg df However these rules are inconsistent if the product of f and g is not commutative. An important step has therefore been to realize that cyclic symmetry is sufficient to establish many properties, giving rise to cyclic (co-)homology.

A fundamental object for any non-commutative space A is the K-homology, which takes different forms in even and odd dimensions. The even dimensional case is linked to the classification of finitely generated projective modules. Projective modules over the algebra of functions can be realized as the vector space of sections of vector bundles and therefore allow to define direct analogs of Kgroups. The trivial modules corresponding to trivial bundles are simply powers A n . The projective modules are defined by projection operators e in M n (A) singling out a submodule of A n . Instead of the projector e, which satisfy e * = e and e 2 = e, it can be easier to deal with the involution s = 2e -1 which is also hermitian and satisfy s 2 = 1.

In the odd dimensional case, the generators of K-theory are unitary operators U in M n (A). They are characterized by U U * = U * U = 1. In both case there is a coupling between the K-groups and the cyclic homology, the Chern characters. To describe them, it is convenient to introduce the operation ⊙ in

M n (A) ≃ A ⊗ M n (C): ⊙ : M n (A) × M n (A) → M n (A ⊗ A) (a ⊗ m, b ⊗ n) → (a ⊗ m) ⊙ (b ⊗ n) = (a ⊗ b) ⊗ (m • n) (1)
With a trace acting only on the matrix part of the expressions, we have the following formulas for the Chern characters:

Ch 2n (s) = Tr( 2n+1 s ⊙ s ⊙ • • • ⊙ s) (2) Ch 2n+1 (U, U * ) = Tr( n+1 U ⊙ U * ⊙ • • • ⊙ U ⊙ U * - n+1 U * ⊙ U ⊙ • • • ⊙ U * ⊙ U ) (3)
3 Definition of non-commutative spheres.

The sphere of dimension n has the property that the cohomology and the homology is supported only in the dimensions 0 and n. Equipped with the notion of Chern character in cyclic cohomology, the following definition has been proposed [START_REF] Connes | Noncommutative Finite-Dimensional Manifolds. I. Spherical Manifolds and Related Examples[END_REF] for the non-commutative spheres.

Definition 1 An even non-commutative sphere of dimension 2n is an algebra generated by the matrix elements of a hermitian involution s such that Ch 2p (s) = 0 for all p smaller than n and Ch 2n (s) is the volume form of the sphere.

An odd non-commutative sphere of dimension 2n + 1 is an algebra generated by the matrix elements of a unitary U together with it hermitian conjugate U * such that Ch 2p+1 (U, U * ) = 0 for all p smaller than n and Ch 2n+1 (U, U * ) is the volume form of the sphere.

In many cases, it will be convenient to separate the conditions that U is unitary or s is an involution in a first one that s 2 or U U * = U * U be proportional to the identity

s 2 = CI, U U * = U * U = CI (4)
and a second one that this proportionality constant is 1. The advantage is that the first part is purely quadratic in the matrix elements of s or U and U * . The proportionality constant C can be seen to be central in the algebra by considering the products s 3 , U U * U or U * U U * . It is therefore always possible to consider the quantum sphere as a quotient of a quantum affine space which is a quadratic algebra by the ideal generated by C -1, where C is a central element quadratic in the generators.

With this formula, it is easy to see that a quantum sphere allows to define a quantum sphere with one dimension more, by an operation called suspension. If we add a central hermitian element x to the algebra, U = xI+is and U * = xI-is satisfy U U * = U * U = (x 2 + C)I, with the new constant C ′ = x 2 + C. The odd Chern forms are cup products of the even Chern forms with the homology of C[x], so that if s defines a sphere of dimension 2n, U and U * define a sphere of dimension 2n + 1. If we start from an odd sphere, we have to double the matrix dimension to define s through:

s = xI U U * -xI (5)
The quadratic relation is satisfied again with C ′ = C + x 2 , Tr s is evidently 0 and the Chern forms of higher rank are again cup products of the odd Chern forms for U with the ones of the line.

In [START_REF] Connes | Noncommutative Finite-Dimensional Manifolds. I. Spherical Manifolds and Related Examples[END_REF], it was shown that the commutative spheres, as well as some noncommutative deformations, satisfy this definition. Indeed, the matrix s or U can be defined in terms of the Clifford algebra of the space of dimension 2n + 1:

s = 2n j=0 x j γ j , (6)
U = x 0 I + i 2n+1 i=1 x j γ j . (7)
The properties of the Clifford algebra ensure that

s 2 = 2n j=0 x 2 j I, (8)
U U * = U * U = 2n+1 j=0 x 2 j I, (9)
so that the defining properties of the involutions or the unitary operators are satisfied on the sphere. The properties of the Clifford algebra allow to show that the Chern characters of low order are. In the even case, the Chern character involves the trace of an odd number of γ matrices. This is zero unless the number of matrices is at least equal to 2n + 1, in which case it is proportional to the totally antisymmetric symbol. The low order Chern characters are therefore zero and we get for the character Ch 2n that it is proportional to the volume form. In the odd case, it can be seen that the two terms annihilate themselves if there are not an odd number of terms proportional to x 0 . Again we remain with the trace of an odd number of γ matrices, and the lowest order non-zero Chern character is Ch 2n+1 with a single term proportional to x 0 . Again it can be shown that this non-zero Chern character is proportional to the volume form. One could also obtain the commutative spheres by iterated suspension of the circle, the sphere of dimension 1.

In low dimensions, the only possible spheres are the commutative ones. In dimension one, there are no conditions and we have the algebra generated by a single unitary element u which is the algebra of function on the circle. In dimension two, the generator s must be taken as a two by two matrix and it satisfies the condition Ch 0 (s) = 0 which is simply that it has zero trace. We can therefore write it, allowing for the hermitian constraint:

s = x y y * -x . ( 10
)
The condition s 2 = I then implies that x, y and y * mutually commute and satisfy x 2 + yy * = 1. The first non-trivial case is the three-dimensional one.

4 Moduli space of non-commutative three-spheres

For three-dimensional spheres, the generator of K-theory is a unitary operator U in M 2 (A). In terms of the Pauli matrices σ, U and its adjoint are written:

U = z 0 I + i 3 j=1 z j σ j . (11)
U * = z * 0 I -i 3 j=1 z * j σ j . (12)
The defining equations of the three-sphere are invariant by left and right multiplication of the matrix U by unitary matrices A and B. This corresponds to an arbitrary SO(4) rotation mixing the algebra elements z µ and their multiplication by a common phase. With these parameters, the Chern character Ch 1 (U, U * ) reads:

3 µ=0 z µ ⊗ z * µ - 3 µ=0 z * µ ⊗ z µ . (13)
For this Chern character to be zero, the z * µ must be linear combinations of the z µ . If the z µ are linearly independent, there exists a unique matrix Λ such that

z * µ = Λ µν z ν (14)
The case where the z µ are linearly dependent is not really important since it does not allow to form spaces with non-trivial three-volumes. We can use the redefinition by a four dimensional rotation to put some generators to zero and remain with independent generators. A formula similar to eq. ( 14) can be written for the non-zero generators. Inserting in eq. ( 13) shows that Λ is symmetric and taking the adjoint of eq. ( 14) shows that it is unitary. The real and imaginary part of Λ are real symmetric matrices, the unitarity of Λ implies that they commute and they can therefore be simultaneously diagonalized by a four dimensional rotation:

z * µ = λ µ z µ , (15)
without summation over µ. The four eigenvalues of Λ are unit norm complex numbers and they can all be multiplied by a common phase by a redefinition of U . The moduli space is therefore of dimension three.

After the diagonalization, we keep the freedom of changing the order of the eigenvalues. There is a natural cyclic order on the circle, but the determination of the starting point, which can be set to one, remains. This can be settled if we impose that the imaginary parts of λ 2 /λ 0 and λ 3 /λ 1 are positive. In terms of the parameters φ i such that:

λ 0 = 1, λ i = e iφ i , (16)
this gives two possible set of constraints:

0 ≤ φ 1 ≤ φ 2 ≤ φ 3 ≤ π, (17)
φ 1 ≤ φ 2 ≤ π ≤ φ 3 ≤ π + φ 1 . (18)
The boundaries of this fundamental domain correspond to either the equality of two λ or to a relation λ i = -λ j .

It remains to show that different parameters really correspond to distinct algebras and that an algebra can be realized with any choice of these parameters.

5 The quadratic algebras.

The unitarity condition on U can be expanded on the basis of the matrices I and σ i . The I term is identical for the U U * and U * U products and gives the quadratic relation:

3 µ=0 λ µ z 2 µ = 1. ( 19
)
This is the inhomogeneous relation in the algebra and its left hand side is automatically central.

There remains six equations which can be deduced from the following two by a circular permutation of the indices 1, 2 and 3.

λ 0 z 1 z 0 -λ 1 z 0 z 1 + λ 3 z 2 z 3 -λ 2 z 3 z 2 = 0, -λ 1 z 1 z 0 + λ 0 z 0 z 1 + λ 2 z 2 z 3 -λ 3 z 3 z 2 = 0. ( 20
)
These two equations are related if we take new generators z ′ 0 = -λ 0 z 0 , z ′ i = λ i z i , and define new parameters λ ′ µ = λ -1 µ . This is linked to the symmetry of the roles of U and U * in the definition of the algebra. The primed parameter determine U * in the same manner that the other ones determine U . This only changes the sign of the Chern character, that is the orientation of the sphere. We have therefore an isomorphism between the algebra of a sphere and the algebra for the sphere with all of the parameters λ replaced by their inverses. Since the λ are unit norm complex numbers, taking the inverses or the complex conjugates are equivalent.

The sum and the difference of the equations (20) express them in terms of commutators and anticommutators:

(λ 1 -λ 0 )[z 0 , z 1 ] + = (λ 2 + λ 3 )[z 2 , z 3 ] -, (λ 1 + λ 0 )[z 0 , z 1 ] -= (λ 3 -λ 2 )[z 2 , z 3 ] + . (21)
In this form, it is clear that when the λ's are equal, all these relations reduce to the mutual commutation of all the generators z µ , so that we obtain the commutative sphere. We can also remark that the algebra and its opposite defined by a • b • = (ba) • are isomorphic. Indeed, the commutators change sign and the anticommutators are unchanged in the change to the opposite algebra and this can be compensated in eqs. ( 21) by changing the sign of any of the generators, since each of them appears once in each of the equations (21). Changing the sign of two generators yields an automorphism.

In the form (21), changing the scale of the generators can allow to put one of the equations to a standard form, if we are not in one of the special cases where some of the sums or differences of the λ are zero. According to the one put to standard form, we can obtain a Sklyanin algebra or a similar one given by the equations:

[Z 0 , Z 1 ] -= [Z 2 , Z 3 ] + , (λ 1 -λ 0 )(λ 2 -λ 3 )[Z 0 , Z 1 ] + = (λ 1 + λ 0 )(λ 3 + λ 2 )[Z 2 , Z 3 ] -. (22)
Again, the equations with circular permutations of the indices are understood.

From now on, we will adopt the convention that the indices k, l, m will stand for 1, 2, 3 or any of its circular permutations. In any expression with these three indices, a sum over k or a product over k will stand for the sum or the product on the three possible permutations. If we introduce the quantities

a k = (λ k + λ 0 )(λ l + λ m ), the second set of equations takes the form (a m -a l )[Z 0 , Z k ] + = a k [Z l , Z m ] -. (23)
This shows that the quadratic part of the algebra only depends on two parameters, since these equations are invariant by a rescaling of the a i . This corresponds to the fact that there is always a second central element quadratic in the generators, so that the sphere can be deformed by modifying the inhomogeneous condition by adding this other central element. This second central element gives a foliation of the sphere in tori.

The explicit relation between z µ and Z µ is given by

Z µ = ρ µ z µ , (24)
with ρ satisfying the following equations:

ρ 0 ρ k (λ m -λ l ) = ρ m ρ l (λ 0 + λ k ) (25)
We can choose

ρ 2 0 = (λ 0 + λ k ), ρ 2 k = (λ 0 + λ k )(λ l -λ k )(λ k -λ m ) ρ 0 ρ 1 ρ 2 ρ 3 = k [(λ 0 + λ k )(λ m -λ l )] (26)
It is interesting to note that with this choice for ρ µ and the normalization λ µ = 1, the new generator Z µ are hermitian or anti-hermitian.

Z * µ = ρ * µ z * µ = λ µ ρ * µ ρ µ Z µ (27)
The relation does not depend on the choice of square roots in the definition of ρ µ . One has for the unit norm λ:

λ µ + λ ν λ * µ + λ * ν = λ µ + λ ν 1/λ µ + 1/λ ν = λ µ λ ν ( 28
)
The square of the factor between Z µ and Z * µ is one. The product of these factors however is -1 using the constraint on the product of the ρ. This result could be anticipated by considering the adjoint of eqs. (22). In the case of eq. ( 17), one can further show that all the new generators are hermitian except Z 2 .

With this presentation of the algebra, the center of the algebra contains three elements Q k , whose sum is zero:

Q k = (a m -a l )(Z 2 0 + Z 2 k ) + a k (Z 2 m -Z 2 l ) (29)
The proof uses the identities:

[Z µ , Z 2 ν ] -= [[Z µ , Z ν ] -, Z ν ] + = [[Z µ , Z ν ] + , Z ν ] -. (30)
By choosing the appropriate one of these two possibilities for each term of the commutator of Z µ and Q k , the identities (22,23) allow to obtain a k or a l -a m as common factor of double (anti-)commutators which sum up to zero.

6 Geometric data.

In order to show that the algebras associated to distinct parameters a i are really inequivalent, we introduce the geometric data. The quadratic relations R defining the algebra are elements of the tensor product V ⊗ V of the vector space of the generators of the algebra. They define quadratic equations in the product of projective spaces P V * × P V * . The resulting space Γ define a correspondence σ between the projection of Γ on the first and the second factor.

The association of the projection E of Γ and the correspondence σ defined by Γ is the geometric data. Isomorphic algebras will give isomorphic geometric data, so that we can show that some quadratic algebras are inequivalent. These geometric data, plus the line bundle stemming from the canonical one on P V * , will furthermore allow to represent the algebra on a space of sections of line bundles.

It appears that the first form of the relations (20) is the most interesting for the determination of the geometric data. If we denote by (y µ ) and (y ′ µ ) coordinates in the first and second P V * factor in a basis dual to the z µ , the equations of Γ read:

λ 0 y 1 y ′ 0 -λ 1 y 0 y ′ 1 + λ 3 y 2 y ′ 3 -λ 2 y 3 y ′ 2 = 0, -λ 1 y 1 y ′ 0 + λ 0 y 0 y ′ 1 + λ 2 y 2 y ′ 3 -λ 3 y 3 y ′ 2 = 0, (31)
plus the one deduced by circular permutation of the indices (1,2,3). The point of E are such that there exist a non zero solution in (y ′ 0 , y ′ 1 , y ′ 2 , y ′ 3 ). This means that the following matrix is of rank less than four:

        λ 0 y 1 -λ 1 y 0 -λ 2 y 3 λ 3 y 2 λ 0 y 2 λ 1 y 3 -λ 2 y 0 -λ 3 y 1 λ 0 y 3 -λ 1 y 2 λ 2 y 1 -λ 3 y 0 -λ 1 y 1 λ 0 y 0 -λ 3 y 3 λ 2 y 2 -λ 2 y 2 λ 3 y 3 λ 0 y 0 -λ 1 y 1 -λ 3 y 3 -λ 2 y 2 λ 1 y 1 λ 0 y 0         (32)
cases to consider are therefore the four projective points with only one non-zero coordinate. It is easy to verify that each of these points is indeed in the variety E and that they are associated to themselves by the correspondence σ.

The case where the equation ( 34) is not verified is completely similar. In fact, a change of variable allows to exchange the upper and lower block of the matrix, so that no new calculations are necessary to ensure that in the generic case, the variety E is the union of the elliptic curve defined by the equations (34,35) and the four points P µ . The correspondence σ is the identity on the points P µ . The geometric data for the opposite algebra involve the same variety E but the inverse correspondence σ -1 . The isomorphism of the algebras of the three-spheres with their opposites implies that σ -1 can be obtained from σ simply by conjugating with the change of sign of y 0 . This identifies σ as a translation of the elliptic curve. We therefore have two parameters to describe the geometric data, the modulus of the elliptic curve and the parameter of the translation.

When different λ 2 µ become equal, the situation is very similar. The equations (34,35) define a curve which now split in different rational components. As in the generic case, if one of these equations is not satisfied, the equations (36) or a similar set are necessarily satisfied. The solutions are either included in the generic variety or have some zero components. Substituting in the matrix allows to show that these points with some zero components really are in the characteristic variety and that the correspondence is either the identity or a symmetry with some of the coordinates changing sign. The full description of these non-generic cases is better done after the study of symmetries which allow to relate some cases.

7 Reality and symmetries.

Before completing the analysis of the geometric data, we have to consider the reality conditions and the symmetries for the algebras and the relations between different algebras that they allow.

The generators z µ allow for a simpler description of the characteristic variety, but their conjugation properties make the reality conditions for this variety subtler. The restriction to the space of generators V of the adjunction in the algebra maps to V , so that it defines an antilinear map j. In V * , the complex conjugation is defined by j(L)(Z) = L(j(Z)), which gives in coordinates:

(y 0 , y 1 , y 2 , y 3 ) -→ (λ * 0 y * 0 , λ * 1 y * 1 , λ * 2 y * 2 , λ * 3 y * 3 ) (38)
Since the λ are complex numbers of unit norm, they do not obey the reality condition x * = x but the one x * = x -1 . Combined with the complex conjugation given by eq. ( 38), this allows to show that the generic variety is indeed invariant by complex conjugation, since the equations (34,35) on the complex conjugates are the complex conjugates of these same equations for the initial point.

From the formula (ab) * = b * a * , we obtain that the set of relation R is invariant by the action of j ⊗ j followed by the exchange of the factors. This shows that the characteristic variety is invariant by the action of j and that we have:

j • σ = σ -1 • j (39)
The algebra automorphisms introduced in section 5 allow to define twisting of the non-commutative spheres which are again non-commutative spheres with related parameters. These automorphisms come from the diagonal transformations in SO( 4) and act on the generators by changing the sign of two of them. Now, with any algebra *-automorphism τ one can build a cross-product algebra A × τ Z by adjoining to the algebra A a unitary symbol W satisfying the relation:

W a = τ (a)W (40)
In the cases where the automorphism act by linear transformation on the generators x i , we can recover an algebra with the same set of generators by considering the subalgebra of the cross-product with generators x i W . The geometric data for this new algebra can easily be deduced from those of the original algebra.

The relations for the new algebra are deduced from the old ones by the action of I ⊗ τ -1 so that the space Γ is changed by the action of the identity on the first factor and the transpose of τ -1 on the second. The variety E is unchanged and the correspondence becomes t τ -1 • σ.

In the case of the non-commutative three sphere, τ 2 is the identity so that we can add the condition W 2 = 1. We change the matrix generators U to the generator U W . The quadratic relations are deduced from the relation U U * = U * U = I and keep the same form. What will change is the relation between U and U * .

(U W ) * = W * U * = W U * = τ (U )W. ( 41
)
The eigenvalues λ µ corresponding to the generators on which τ has a non-trivial action will change sign.

In particular, changing the sign of λ 0 and λ 3 and exchanging them allows to relate the algebras corresponding to the case of (18) with those described by (17).

Special cases.

In the study of the geometric data, these symmetries allow to reduce the number of different case to study. The non-generic cases can be first classified by the number of different values of the λ 2 , which can go from 1 to 4. In the cases of two different values of λ 2 , the eigenvalues can be split in two groups of two, or one group of one and one of three.

In the case with three different values of λ 2 , we can have the two relations λ 1 = λ 2 or λ 1 = -λ 2 but the two can be related by a symmetry. The generic variety stemming from equations (34,35) split in two conics. To the four special points we must add the line through the points P 1 and P 2 .

In the case with two different values of λ 2 and unequal groups, again the different possibilities for the sign relation between the λ can be related by the symmetry. The special variety now contain the plane through the three points P 1 , P 2 and P 3 . The generic variety is now embedded in this plane, where σ is the identity in the simplest case where three λ are equal.

The case of say λ 1 = λ 2 and λ 0 = λ 3 divide in three cases. The two cases where either the λ are pairwise equal or opposite can be related by symmetry, but the third possibility is distinct. With λ 0 = -λ 3 and λ 1 = λ 2 , two of the equations becomes equal, so that the algebra grows exponentially. On the geometric data, this translates in the apparition of coarse correspondences, where some points map to whole lines.

Finally, when all the λ 2 are equal and can therefore be taken to be 1, there are three cases according to the number of With all 1, we have the commutative case, the entire projective space is the characteristic variety and the correspondence is the identity. The case with two -1 is related to the first one by a symmetry. The case with one -1 is very special. All the a i are zero and we only have three relations in the algebra. The matrix (32) has only three independent lines and is therefore of maximum rank 3. However, on the surface with equation (34), its rank becomes 2 and the correspondence σ associates with such points whole lines. 9 The fundamental elliptic curve.

In section 5, we saw that the quadratic algebra R 4 λ only depends on two parameters, with all parameters lying on a curve describing a unique quadratic algebra. The identification is linked to a rescaling of the generators such that the algebra is defined by eqs. (22,23), which only depend on three homogeneous parameters. What is most remarkable is that the curve in the space of parameters λ which maps to a given set of the a parameters is identical with the generic part of the characteristic variety. This was first an observation on the identity of the elliptic parameters, but the two curves can be explicitly identified.

The scaled generators of the algebra are linked to a new coordinate system for V * such that Y µ = ρ µ y µ . The equations (34,35) become, after eliminating the denominators:

k (λ m -λ l )Y 2 0 + k (λ 0 + λ m )(λ 0 + λ l )(λ m -λ l )Y 2 k = 0, λ 2 0 k (λ m -λ l )Y 2 0 + k λ 2 k (λ 0 + λ m )(λ 0 + λ l )(λ m -λ l )Y 2 k = 0. ( 42
)
The important property of these equations is that they are linear combinations of

B k = Y 2 0 -Y 2 k .
Indeed the sums of the coefficients are polynomials of degree less than 2 in λ 0 which are zero at the three points -λ k . The linear combination of the equations (42) such that the coefficient of say Y 2 3 is zero allows to obtain the ratio of B 1 and B 2 on the curve. It turns out to be the same as the one of a 1 and a 2 .

The B k can be converted to the a k through a linear change of variables with an involutive matrix M :

M = 1 2     1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1     . (43)
We therefore have identified the two elliptic curves, in the parameter space of the non-commutative spheres and in the characteristic variety. Another interest of this presentation of the quadratic algebra is that the correspondence σ has a simple form, since the first three relations of the algebra, which are independent on the parameters, are sufficient to determine the image of a point. The image σ(Z) of a point is determined by the condition that its image by the matrix N (Z) is zero:

N (Z) =         Y 1 -Y 0 Y 3 Y 2 Y 2 Y 3 -Y 0 Y 1 Y 3 Y 2 Y 1 -Y 0 (a 2 -a 3 )Y 1 (a 2 -a 3 )Y 0 -a 1 Y 3 a 1 Y 2 (a 3 -a 1 )Y 2 a 2 Y 3 (a 3 -a 1 )Y 0 -a 2 Y 1 (a 1 -a 2 )Y 3 -a 3 Y 2 a 3 Y 1 (a 1 -a 2 )Y 0         . (44)
The first three lines form a system of maximal rank, so that they are sufficient to determine σ(Z). Furthermore, if we replace Y 0 by its opposite, the first three lines of N (Z)M read:

  λ 0 λ 1 -λ 2 -λ 3 λ 0 -λ 1 λ 2 -λ 3 λ 0 -λ 1 -λ 2 λ 3   . (45)
This shows that σ is the product I • I 0 of two involutions: I 0 changes the sign of Y 0 and I corresponds to taking the inverse of all coordinates in the λ coordinates. These two operations are symmetries for the elliptic curves since they have four fixed points. On these curves there are three different notions of reality. There is the simple one corresponding to taking real coordinates. The one associated to the λ, which are of unit norm, so that we have v = I(v). As I corresponds to taking the opposite in the elliptic curve, they are imaginary points. Finally, the j operation on V * , linked to the adjunction properties of the algebra, changes the sign of an odd number of the complex conjugated coordinates: the j-real points are again purely imaginary points of the curve, but starting from a different origin. The transformation σ has real, even integer, coefficients, so that it is a translation by a real element of the curve.

10 Uses of the geometric data.

The geometric data allow for the definition of a representation of the algebra. Indeed [START_REF] Artin | Some algebras associated to automorphisms of elliptic curves. the Grothendieck festschrift[END_REF], the algebra maps to a cross-product algebra of sections of line bundles by the map σ. Since the variety E is given as a subvariety of the projective space P V * , it is equipped with a naturally defined line bundle L, the restriction of the canonical bundle of P V * . To each element of V , we can associate a section s V of this bundle L. The generator of the algebra are mapped to products s V W , with the symbol W satisfying the commutation relation W s V = s V •σW . The product of n generators is therefore the product of a holomorphic section of a bundle L n by W n . The bundle L n is defined in terms of pullbacks of L by the maps σ n :

L n := L ⊗ L σ ⊗ • • • ⊗ L σ n-1 (46)
The defining relations of the algebra are satisfied from the definition of σ. However, this cannot provide a faithful representation for the non-commutative spheres, since all central elements of the algebra are represented by 0 in the generic case. Indeed, if a central element would have a non-zero representation, we would get inconsistencies for the divisor of its product with any generator. The trouble can be spotted also from the dimensions of the spaces of products of n factors. The Riemann-Roch theorem shows that, in the generic case, this dimension is at most 4n when the dimension of the corresponding space is n + 3 3 for the quadratic algebra and (n+1) 2 for the non-commutative sphere.

An other limitation of this representation is that it is not a * -representation. Alain Connes and Michel Dubois-Violette introduced an extension of this construction based on the notion of "central quadratic form". This allows for a representation where the inhomogeneous equation of the algebra (19) is satisfied. The main point is to turn the algebra of the previous representation in a *-algebra. Taking the complex conjugate of holomorphic functions gives antiholomorphic ones. We therefore introduce a second variable to represent the complex conjugate. In a first time, the two variables are better kept independent, so that the space we consider is the product E × E with the complex conjugation j: j(Z, Z ′ ) = (j(Z ′ ), j(Z)).

In the end, we will restrict to the invariant subspace of j, formed of points (Z, j(Z)). With c denoting complex conjugation in the image space, the adjunction for the function F is defined by:

F * = c • F • j. ( 48
)
With the following representation for the generators,

ρ(Y ) = Y (Z)W + W * Y (Z ′ ), (49)
it is easy to see that ρ(Y

) * = ρ(j(Y )). (50)
The transformation σ must be extended to E × E. In view of the relation between σ and j (39), we define σ so that it commutes with j:

σ(Z, Z ′ ) = (σ(Z), σ -1 (Z ′ ). (51)
It is natural to suppose that W is a unitary, so that the commutation with W * gives a translation by σ-1 .

Finally, we want the map (49) to define a representation of the algebra of the non-commutative sphere. Terms W * Y i (Z ′ )Y j (Z)W and Y i (Z)W W * Y j (Z ′ ) will correspond to fibers in differing points of E, so that a trivialization is necessary for their comparison. Such a trivialization will also be necessary to allow for the inhomogeneous equation ( 19).

The product of the line bundle L on the variable Z and Z ′ is trivialized by dividing by a quadratic form Q(Z, Z ′ ). Then, the quadratic relations read:

ω ij ρ(Y i )ρ(Y j ) = ω(Z, σ(Z))W 2 + (W * ) 2 ω(σ -1 (Z ′ ), Z ′ ) + ω(Z, Z ′ ) Q(Z, Z ′ ) + ω(σ(Z ′ ), σ -1 (Z)) Q(σ -1 (Z), σ(Z ′ )) (52)
The parts proportional to W 2 or (W * ) 2 are zero from the defining property of σ. The remaining part is zero if Q is symmetric and satisfies the defining property of a central quadratic form:

ω(Z, Z ′ )Q(σ(Z ′ ), σ -1 (Z)) + ω(σ(Z ′ ), σ -1 (Z))Q(Z, Z ′ ) = 0 (53)
In the generic case, this condition can be verified if both Z and Z ′ belong to the generic curve, for any quadratic form deduced from a central quadratic elements of the algebra. The calculation is done in the coordinates introduced in section 9, for any of the central forms Q k : the transformation σ has a simple explicit expression and the fact that Z and Z ′ are on the curve can be expressed simply by substituting y 2 0 + Ca k to y 2 k . The equation ( 53) is not verified if Z is a special point and Z ′ is on the elliptic curve, but this does not preclude the possibility to obtain a representation of the algebra. By specializing eq. ( 53) to the case Z ′ = σ(Z), we can further prove that Q(Z, σ(Z)) is zero is σ 4 is not the identity. The calculation of (52) applied to ω = 1 2 Q now shows that

1 2 Q ij Y i Y j is represented by 1. This representation is a *-representation if Q verifies Q( j(Z, Z ′ )) = Q(Z, Z ′ ).
If we now specialize to a j invariant subset, i.e., to points of the form (Z, j(Z)), we can further demand that Q is strictly positive, so that it defines an hermitian metric on L. This is certainly the case for the central element C of the algebra, which can also be written z µ z * µ . Division by Q(Z, j(Z)) does not introduce any singularity and the algebra of the sphere is faithfully represented in the space of regular sections. We have therefore obtained a representation of the non-commutative spheres for generic parameters. The fact that an elliptic curve is topologically a torus can be used to further understand the structure of the algebra. Essentially, we have a family of non-commutative torus with a parameter determined from the transformation σ.

However, this does not provide for a full representation in every cases. In the case where three of the λ are equal, we have a central element z 0 which play the role of the Planck constant for the su(2) algebra formed by the three others generators. The correspondence σ is the identity, so that the algebra obtained from the geometric data is commutative. This is compatible with the algebra structure because the characteristic variety is the union of a point and the plane y 0 = 0: on the plane, the geometric data represent z 0 by 0 and all other generators are zero on the isolated point.

The full structure of this algebra is much more interesting. We get a discrete structure, because for a n dimensional representation of the su(2) algebra, z 2 1 + z 2 2 + z 2 3 = (n 2 -1)z 2 0 . We therefore have a discrete family of fuzzy spheres which converge to the ordinary sphere obtained for z 0 = 0.

Conclusion

We arrive at the end of this tour of non-commutative three-spheres. The definition of the non-commutative three-spheres has been introduced, the moduli space discovered. A large place has been given to the geometric data, first to provide for intrinsic parameters to single out the quadratic algebras, then to show how they give rise to representations for the algebra of the noncommutative spheres.

In a sequel to these lectures, it would be interesting to evaluate the volume form provided by the Chern form of order 3 and the Jacobian of the transformation to the cross-product algebra F u × σ,L Z. In keeping to the spirit of these lectures, we could make a purely algebraic version of the computation of section 12 of [START_REF] Connes | Noncommutative finite-dimensional manifolds. II. Moduli space and structure of non commutative 3-spheres[END_REF].

We also would like to build spectral triples for these algebras, in order to complete the description of their geometry. However, one expects that this would be rather difficult. In the cases where the characteristic variety is rational, which have quantum group structures of SU q (2), the construction has met with some difficulties: in the generic case, we do not have any symmetry to assist in the process.

Each of the fifteen four by four determinants that can be extracted from this matrix must be simultaneously zero. This calculation is made manageable by the factorization of the four three by three minors extracted from the first three lines of the matrix: they all have the common quadratic factor y 2 0 + y 2 1 + y 2 2 + y 2 3 . Similarly, the three by three minors extracted from the last three lines have the common factor λ 2 µ y 2 µ . Therefore on the curve defined by the intersection of these two quadrics, six of the four by four submatrices have determinant 0. Furthermore, since the upper and the lower subblock are of rank 2, it will not be necessary to verify that the remaining nine submatrices are of determinant 0. Indeed, apart from isolated points on the curve, the third line of each group is a linear combination of the two others. If the determinant of the matrix formed with the lines 1,2,4,5 is 0, we therefore have that they generate a space of dimension at most 3 and all other determinants are also zero. Continuity allows to ensure that these other determinants remain zero on the points where the lines 1 and 2 for example are proportional. That the determinant of the matrix formed with the lines 1,2,4,5 is zero can be seen by writing it:

The two quadratic equations define a curve which is generically an elliptic one:

If different λ 2 become equal, this curve decompose in a product of rational curves. Suppose that equation ( 35) is not verified. We then have three necessary conditions stemming from the determinants including the lines 4,5,6.

Again, these equations only depend on the squares of the λ µ . In the case where all the squares are distinct, these homogeneous equations have exactly eight projective solutions. The four trivial solutions P µ such that only y µ is non zero and four other ones which can be deduced from one of them by changing the sign of a pair of y µ . However these additional solutions belong to the generic variety and we do not have to consider them independently. Indeed, in the case where all the y µ are not zero, the product of the first two equations for example gives the ratio y 2 0 /y 2 3 so that we have:

The expressions y 2 µ and λ 2 µ y 2 µ are polynomials of degree two in say λ 2 0 which vanish for the three values λ 2 i , therefore they are zero. The only particular