Iris Dadole

Kevin Huguet

Didier Blaha

Nicolas Personnic

Protocol for high-throughput isolation of bacterial intracellular nonreplicating persisters

Keywords: Persisters, Single-cell, Timer, Growth rate, Infection, Pathogens, Host, Antibiotic, FACS

Failure of antibiotics to clear a bacterial infection is arguably one of the most severe threat to global health. Bacteria ability to withstand drug treatment partly relies on the formation of a subpopulation of persisters. Persisters are rare nonreplicating individuals, transiently refractory to antibiotic treatment and responsible for nonresolving bacterial infection and relapse. Studying persisters is challenging as it is a reversible physiological state occurring in a small fraction of the pathogen population hidden in a complex environment. Persisters have been primarily analyzed in broth, hardly reflecting the conditions prevailing within a host. In addition, indirect read-outs used to detect the persisters, a posteriori, limited a comprehensive understanding of the persisters peculiar physiology during the infection. Here, we describe a high-throughput method to track, and collect the bacterial persisters generated during the infection of phagocytic cells, with singlecell resolution. Persisters are detected based on the bacterial division rate measured for each individual bacterium using the Timer fluorescent growth rate reporter and subsequently recover using fluorescence-activated cell sorting (FACS). We will illustrate this method with Legionella pneumophila, a facultative intracellular pathogen and the agent of the Legionnaires' disease for which persistence to antibiotics and relapsing infections are clinically documented. Isolated persisters can be analyzed biochemically and functionally, yielding fundamental new insights into the molecular and cellular processes underlying persistence, and into the peculiar physiology of the persisters within the host. This method is adaptable to other microbial pathogens and alternative infection models.

termed phenotypic heterogeneity is of major clinical importance as it implies the formation of persisters [START_REF] Balaban | Bacterial persistence as a phenotypic switch[END_REF]. Persisters are preestablished or stress induced individuals that transiently evade the bactericidal activity of antibiotics by entering a rare and peculiar physiological state at the cost of abrogation of bacterial division, and finally resume growth on the onset of the treatment termination [START_REF] Harms | Mechanisms of bacterial persistence during stress and antibiotic exposure[END_REF]. The emergence of persisters has been documented for major bacterial pathogens including Staphylococcus aureus, Mycobacterium tuberculosis, Escherichia coli, Salmonella enterica, Pseudomonas spp., Listeria monocytogenes and Legionella pneumophila [4,[START_REF] Balaban | Bacterial persistence as a phenotypic switch[END_REF][START_REF] Josephw | TREATMENT OF STAPHYLOCOCCAL INFECTIONS WITH PENICILLIN BY INTERMITTENT STERILISATION[END_REF][START_REF] Conlon | Persister formation in Staphylococcus aureus is associated with ATP depletion[END_REF][START_REF] Helaine | Internalization of Salmonella by Macrophages Induces Formation of Nonreplicating Persisters[END_REF][START_REF] Kortebi | Listeria monocytogenes switches from dissemination to persistence by adopting a vacuolar lifestyle in epithelial cells[END_REF][START_REF] Manina | Stress and Host Immunity Amplify Mycobacterium tuberculosis Phenotypic Heterogeneity and Induce Nongrowing Metabolically Active Forms[END_REF][START_REF] Moyed | hipA, a newly recognized gene of Escherichia coli K-12 that affects frequency of persistence after inhibition of murein synthesis[END_REF][START_REF] Mulcahy | Emergence of Pseudomonas aeruginosa Strains Producing High Levels of Persister Cells in Patients with Cystic Fibrosis[END_REF][START_REF] Personnic | Quorum sensing modulates the formation of virulent Legionella persisters within infected cells[END_REF][START_REF] Personnic | Quorum sensing controls persistence, resuscitation, and virulence of Legionella subpopulations in biofilms[END_REF]. Traditionally, the presence of bacterial persisters has been revealed by the "biphasic killing" phenomenon of bacterial cultures exposed to bactericidal antibiotics [START_REF] Balaban | Definitions and guidelines for research on antibiotic persistence[END_REF]. However, such population-averaging methods, performed a posteriori, are poorly suited to comprehensively and dynamically analyze a phenomenon where reversible cell-to-cell variations plays a key role. The development of high-throughput fluorescence-based single-cell technologies has rendered important breakthroughs reachable. Notably, the microfluidic device called mother machine has been incredibly useful to directly monitor the formation of persisters and the underlying regulatory mechanisms using a combination of genetic and environmental manipulations [START_REF] Balaban | Bacterial persistence as a phenotypic switch[END_REF]. Persisters have been mostly studied using non-infectious model organisms cultivated in broth, hardly reflecting the conditions prevailing within a host. Deciphering the biology of the persisters during the infection has remained technically challenging as the persisters represent a rare and unstable subpopulation of the pathogen, that is generated within complex environments.

The ubiquitous environmental Gram-negative bacterium Legionella pneumophila is a waterborne human pathogen responsible for a life-threatening pneumonia termed Legionnaires' disease (LD) [START_REF] Newton | Molecular pathogenesis of infections caused by Legionella pneumophila[END_REF], the inventoried cases of which have doubled since 2013 [START_REF]Legionnaires' disease[END_REF]. LD is fatal for 5-15% of the patients (up to 50% among the elderly), even when promptly diagnosed and treated [START_REF] Brady | Legionnaires' Disease. StatPearls. Treasure Island[END_REF]. A nation-wide retrospective study describing those cases of slowly or non-resolving LD proposed that recalcitrance to antibiotic treatments or relapsing infections were the expression of bacterial persistence [START_REF] Pouderoux | Slowly or Nonresolving Legionnaires' Disease: Case Series and Literature Review[END_REF]. L. pneumophila survives its ingestion by a diverse array of protozoan predators, as well as lung alveolar macrophages, and establishes a replicative membrane-bound compartment termed the Legionella-containing vacuole (LCV) [START_REF] Boamah | From Many Hosts, One Accidental Pathogen: The Diverse Protozoan Hosts of Legionella[END_REF][START_REF] Isberg | The Legionella pneumophila replication vacuole: making a cosy niche inside host cells[END_REF]. LCV formation requires the Icm/Dot type IV secretion system (T4SS) and involves approximately 300 different so-called « effector » proteins, which are translocated into host cells [START_REF] Burstein | Genomic analysis of 38 Legionella species identifies large and diverse effector repertoires[END_REF][START_REF] Personnic | Subversion of Retrograde Trafficking by Translocated Pathogen Effectors[END_REF] to hijack numerous host cell functions. The amoebozoa Acanthamoeba castellanii has proved a reliable, easy-to-use and cost effective model organism to investigate the molecular interactions of L. pneumophila with its host cell, and at large, the pathogenic potential and process of many other human opportunistic pathogens [START_REF] Moliner | Genome analysis of microorganisms living in amoebae reveals a melting pot of evolution[END_REF]. The plasmid based Timer growth rate reporter [4] is a powerful technology to track the nonreplicating persisters produced during an infection. Timer is a stable fluorescent protein that slowly maturates from a green to a red fluorescent protein. As a consequence, in growing bacteria, constitutively producing Timer, green protocols.io | https://dx.doi.org/10.17504/protocols.io.3byl4j5xolo5/v1

Oct 25 2023

fluorescent Timer dominates over red fluorescent Timer, which is diluted by cell division before maturation, and the individual bacteria show a high [500 nm (green)/600 nm (red)] fluorescence (color) ratio (i.e. "green" fluorescence). By contrast, growth-arrested bacteria accumulate both green and red fluorescent Timer, and the individual bacteria show a low green/red color ratio (i.e., "red" fluorescence).

Here, coupling the protozoan model of infection, the innovative Timer fluorescent growth rate reporter and high-throughput single-cell technologies, we are defining a pipeline to track and collect the persisters produced by L. pneumophila with an unprecedented resolution. Isolated persisters can be analyzed biochemically and functionally. To date, using this methodology, we demonstrated that L. pneumophila persisters are a preestablished subpopulation of nongrowers and the expression of a regulated pathogen strategy which is applied to evolutionarily distant host cells (i.e., protists and macrophages) [START_REF] Personnic | Quorum sensing modulates the formation of virulent Legionella persisters within infected cells[END_REF][START_REF] Personnic | Quorum sensing controls persistence, resuscitation, and virulence of Legionella subpopulations in biofilms[END_REF]. Near in situ biochemical characterization of the intracellular L. pneumophila persisters unveiled a unique physiology and, remarkably, a specific virulence program in order to survive the combined action of the host defenses and the antibiotics, hence, breaking the dogma that persisters are dormant bacteria [START_REF] Personnic | Quorum sensing modulates the formation of virulent Legionella persisters within infected cells[END_REF]. Of note, this approach can be applied to the study nonreplicating persisters produced by other pathogens during the infection or within complex microbial communities.

ACKNOWLEDGMENTS: ACKNOWLEDGMENTS:

Research at the CIRI is supported by the Inserm, CNRS, University Lyon 1, and ENS Lyon. N.P. was the recipient of an SNF Ambizione fellowship (PZ00P3_161492, PZ00P3_185529). The research in the laboratory of N.P. is supported by a CIRI seed fund and the ANR. The funders had no role in study design, decision to publish, or preparation of the manuscript.

DISCLOSURES: DISCLOSURES:

The authors have nothing to disclose. REFERENCES: REFERENCES:

1. Bumann D,

Note

For long-term storage, the cell preparation shall be deposited in either the liquid or the vapor phase of a nitrogen freezer. Warning. L. pneumophila JR32 is used. This lab strains originally derived from a Philadelphia-1 clinical strain isolated from an outbreak in Philadelphia in 1976 [START_REF] Rao | Phylogenetic Reconstruction of the Legionella pneumophila Philadelphia-1 Laboratory Strains through Comparative Genomics[END_REF][START_REF] Samrakandi | Genetic and Phenotypic Differences between Legionella pneumophila Strains[END_REF]. L. pneumophila is the causing agent of the Legionnaires' disease and must therefore be manipulated in a biosafety containment level 2 in a microbial safety cabinet in compliance with local rules. Note 6: L. pneumophila D icmT lacks a functional T4SS and therefore cannot grow within infected cells. It is a convenient control to define the Timer spectral properties corresponding to the lack of bacterial division during the infection.

Generation of a

2

  A. castellanii strain Douglas (ATCC 30234) is used. A. castellanii is an ubiquitous amoeba with a 2-stage lifecycle: an active trophozoite stage that exhibits vegetative growth and a stress induced dormant cyst stage with minimal metabolic activity[START_REF] Siddiqui | Biology and pathogenesis of Acanthamoeba[END_REF]. Only A. castellanii trophozoites are suited for infection. Note 2: Warning.Warning. A. castellanii is an opportunistic pathogen producing rare and severe human infections. Therefore, it must be manipulated in a biosafety containment level 2 in a microbial safety cabinet in compliance with local rules.Cultivation ofAcanthamoeba castellanii: Axenic cultivation of… protocols.io | https://dx.doi.org/10.17504/protocols.io.3byl4j5xolo5/v1 Oct 25 2023 D(+)glucose monohydrate (50 mL of a 2 the components (except the glucose) to 950 mL H 2 O, adjust the pH with 1 Molarity (M) HCl to 6.5±0.1 and autoclave. Solubilize 11 g of D(+)glucose in 50 mL of warmed H 2 O ( 37 °C ) and add to the autoclaved medium (see Note 4 Note 4). Sterilize the medium using a 0.2 µm filter cartouche and store at 4 °C . Note Note 3: For a high reproducibility of growth characteristics, the components should be tested beforehand, and the same suppliers and batches should be used for all experiments. Note 4: Glucose caramelizes upon autoclaving in combination with the medium. Suspend the D(+)glucose in warmed 50 mL of H 2 O, filter sterilize (0.2 µm) and add to the autoclaved medium. Thaw an ampule of frozen amoeba at Room temperature and aseptically transfer the contents into a 13 mL sterile Falcon round-bottom polypropylene tubes containing 10 mL of PYG. Centrifuge 250 x g, 00:10:00 and discard the supernatant. Resuspend the pellet in 10 mL of PYG medium to inoculate a tissue culture treated flask (75 cm 2 : 10-15mL medium) for initial amplification.

35 6

 6 Cultivate the amoebae at 23 °C in PYG medium optionally supplemented with antibiotics (e.g. Pen/Strep or Fungizone) to avoid bacterial and fungal contaminations. Use either 100 mm tissue culture treated Petri dishes (minimum of 10 mL medium) or tissue culture treated flasks (25 cm 2 : 5-7 mL medium, 75 cm 2 : 10-15mL medium). Change the medium every 2-3 days. Split the culture when cell reach 80% confluence by repeated pipetting of the media over the plate. 4 Remove the PYG medium and harvest the amoeba, when in logarithmic growth phase, by repeated pipetting of fresh refrigerated PYG medium over the plate. Determine the cell concentration by counting the amoeba with a haemocytometer. Adjust the concentration to about 10 7 amoeba/mL. 10m Cultivation of Acanthamoeba castellanii: Cryopreservation of A… protocols.io | https://dx.doi.org/10.17504/protocols.io.3byl4j5xolo5/v1 Oct 25 2023 Prepare freshly a 15% solution of sterile dimethyl sulfoxide (DMSO) in refrigerated PYG medium and mix to the cell suspension in equal portions to reach a final concentration between 10 6 and 10 7 cells/mL and 7.5 % DMSO. Dispense in 0.5 mL aliquots into 2.0 mL sterile plastic screw-capped CryoTube vials. Place the vials in a Nalgene Cryo 1 °C freezing container and store at -80 °C , Overnight .

7 NotePrepare

 7 CYE (charcoal yeast extract) agar plates Prepare CYE (charcoal yeast extract) agar plates: yeast extract in 950 mL of H 2 O and adjust the pH to 6.9 with 10 Molarity (M) KOH. Transfer the solution to a 1 L Schott bottle containing 2 g of activated charcoal powder, 15 g of agar and a stir bar. Autoclave and let the agar solution cool down to 50 °C . Add filter sterilized 0.4 g / 10 mL L-cysteine and 0.25 g / 10 mL Fe(NO 3 ) 3 solutions (see Note 8 Note 8). If required add antibiotics. Mix the solution on a magnetic stirrer and Note 5: Warning.

  Timer producing Legionella pneumophila protocols.io | https://dx.doi.org/10.17504/protocols.io.3byl4j5xolo5/v1 Oct 25 2023

Adjust the drop delay. protocols.io | https://dx.doi.org/10.17504/protocols.io.3byl4j5xolo5/v1 Oct 25 2023

protocols.io | https://dx.doi.org/10.17504/protocols.io.3byl4j5xolo5/v1 Oct 25 2023

pour plates (approximately 40 plates per L of medium). Let the plates dry for 1 day at Room temperature and store at 4 °C .

Note

Note 7: The source and quality of yeast extract and peptone affect the physiology of L. pneumophila. For a high reproducibility of virulence traits and growth characteristics, the components should be tested beforehand, and the same suppliers and batches should be used for all experiments. Note 8: Dissolve L-cysteine and Fe(NO 3 ) 3 each separately in 10 mL of H 2 O in a 15 mL tube. Stir the medium and slowly add the L-cysteine solution first, followed by the iron solution to prevent precipitation.

Note

Note 11: Electrocompetence of L. pneumophila is increased when using freshly prepare bacteria grown at 30°C.

10

Generously harvest the bacterial lawn with a disposable inoculation loop and inoculate 1 mL of PBS in a 1.5 mL sterile Eppendorf tube.

11

Centrifuge at 5000 rpm, 00:10:00 , discard the supernatant and resuspend the bacterial pellet with a 10 % glycerol solution. Repeat the operation 3-4 times.

12

Finally, resuspend the bacterial pellet with 100 mL of a 10% glycerol solution.

13

Directly proceed with the electroporation.

14

Add 100 ng of the plasmid pNP107 to the bacterial suspension and transfer into a 2 mm electroporation cuvette.

15

Electroporate using the following settings: U=2.5 kV; C= 25 µF; 200 Ω=∞; t.c.≈2.5.

16

Immediately add 500 µL of pre-warmed AYE, transfer into a 13 mL Falcon round-bottom polypropylene tubes and incubate on a shaker set at 200 rpm, 37°C, 05:00:00 . Note 16: Under our conditions an OD 600 nm of 5.0 corresponds to approximately 2 × 10 9 bacteria/mL. At 21 h, the exact correlation of the OD 600 nm with the bacterial concentration depends on the spectrophotometer used and must be determined experimentally.

21

Proceed with infection of A. castellanii.

22

Split A. castellanii cultures one day before the experiment to obtain cells in logarithmic growth phase, which have been fed the day before (see Note 17 Note 17).

Infection of Acanthamobea castellanii

23

Wash exponentially growing A. castellanii with PYG and detach the cells by repeated pipetting of the media over the plate.

24

Determine cell concentration by counting the cells with a haemocytometer.

25

Seed cells at a density of 9 x 10 5 amoeba per well of a tissue culture treated 6-well plate. Let the phagocytes settle and adhere to the plastic surface for 24:00:00 under normal culture conditions (see Note 18 Note 18).

Note

Note 18: Use 2 mL of PYG medium per well. The next day, cell number will have doubled.

Alternatively, cell can be seeded the day of the infection as cell adhesion to the plastic surface takes 00:30:00 .

26

Prior to infection, remove the PYG medium and wash out antibiotics if necessary (see Note 19 Note 19). Determine the cell numbers per well using a haemocytometer. Calculate the appropriate number of bacteria to infect host cells with a multiplicity of infection (MOI) of 2.

Note

Note 19: Cell culture media used in assays must not contain antibiotics.

Note

Note 20: Subpopulations of proliferative L. pneumophila and nonreplicating persisters of are distinctly observed and were characterized at 24 h post-infection (p.i.), in infected A. castellanii. Note 21: Infected A. castellanii loses motility and develops tomette-like shape.

28

Proceed with the sample preparation for FACS. Filter sterilize and store at 4 °C .

30

On ice , detach the infected amoeba by repeated pipetting of the media over the wells and transfer into 2 mL Eppendorf tubes.

31

Pellet by centrifugation 250 x g, 4°C, 00:10:00 , remove the supernatant and wash once with 1 mL ice-cold PBS.

32

On ice , lyse the cell pellet with 1 mL 0.1 % Triton TX-100 (Sigma) in HS buffer for 00:10:00 .

35

Rinse well the ball homogenizer.

35.1

Remove the plunger of one syringe and fill the volume with ddH 2 O. Install the plunger and flush. (1/3)

35.2

Remove the plunger of one syringe and fill the volume with ddH 2 O. Install the plunger and flush. (2/3)

35.3

Remove the plunger of one syringe and fill the volume with ddH 2 O. Install the plunger and flush. (3/3)

35.4

Finally flush the homogenizer once with ice-cold HS buffer. Make sure there is no leakage or clogging.

38

Proceed with the isolation of the nonreplicating persisters. Note 26: The procedure detailed bellow was established using a high speed multicolor FACSAria III 4L sorter equipped with 4 spatially separated laserlines, for excitation, and the associated detector arrays: 405 nm (violet), 488 nm (blue), 561 nm (yellow/green) and 633 nm (red). Data acquisition using the FACSDiva software. Note 27: Warning. Warning. FACS generates aerosol that poses a threat to the operator when working with respiratory pathogens. Local rules may impose a biosafety containment level 3 biosafety containment level 3 when working on aerosolized L pneumophila. Ensure the sorter is equipped with an aerosol management system to evacuate the sort collection chamber and to trap aerosolized particles during sorting. Wear mask and goggles. Notify your colleagues.

39.1

Decontaminate the sort block, the sort chamber, the sample injection port and the sort collection device (tube holder and lid) with 70% ethanol. Clean the deflecting plates with ddH 2 O. Wipe to dry carefully all surfaces that will be charged.

39.2

39.3

Turn on the aerosol management system.

39.4

Turn on the sorter (see Note 30 Note 30).

Note

Note 30: Ensure the excitation laser 488 nm and 561 nm are set to "ON".

39.5

In the FACSDiva menu, turn on the fluidic system by selecting "Fluidic startup"(see Note 31 Note 31) and follow the instructions.

Note

Note 31: Ensure that the lid of the sheath tank is properly closed and the tank pressurized.

39.6

In the sort block, remove the close-loop nozzle and insert the nozzle with an orifice of 70 mm at the dedicated position at the lower end of the cuvette flow cell. Set the correct configuration for the 70 mm nozzle in the FACSDiva menu "view configurations".

39.7

Turn on the external temperature control system for the collection tube holder and set the temperature to 4 °C . In the FACSDiva menu "Cytometer", set the agitation of the sample in the injection chamber to 300 rpm and temperature to 4 °C .

40

Establish a stable stream and set up the drop formation.

Note

Note 32: During sorting, drop drive energy is applied to the stream to break it into highly uniform droplets containing particle of interest. Droplets detach from the stream a few millimeters downstream from the nozzle Note 33: In the FACSDiva software, use the breakoff window to control the stream and the drop formation.

40.1

Start the stream that should appear in the breakoff window.

40.2

Within the sort block, ensure the stream hit the center of the waste aspirator (see Note 34 Note 34).

Note

Note 34: The entire sort block assembly can be rotated on a fixed pivot point to adjust the position of the stream in the waste aspirator.

40.3

Stabilize the position of the drop breakoff point by adjusting the amplitude (see Note 35 Note 35). Drops before the breakoff point will appear pear-shaped. Ensure satellite drop merge with leading drop.

Note

Note 35: Each nozzle size configuration is optimized to a preset sheath pressure and also change the related values for drop drive frequency, drop charge levels, laser delay and area scaling. Keep the preset values unless you cannot establish a stable stream (i.e. pressure: 70 undetermined ; gap: 6 and frequency: 87 KHz).

40.4

Copy the generated Drop 1 value (i.e., the distance between the top of the image and the center of the first broken-off drop) in the target field.

40.5

Turn on the Sweet Spot to maintain the drop breakoff point during the sorting.

40.6

Wait 15-20 min and control the stability of the stream and drop formation. For the Drop 1, a difference of 10 units is acceptable. For the GAP a difference of 1 unit is acceptable.

Note

Note 36: The drop delay calibrates the time between the particle analysis and the droplet formation in order to charge the drop the sort. Note 37: Side streams, deflection, and drop delay are defined using the side stream window in the FACSDiva software. Note 38: Sorting is controlled using the sort layout window in the FACSDiva software.

41.1

Ensure the maximal illumination of the central stream using the micrometer dial.

41.2

Turn on the deflection plates (voltage) in the side stream window (see Note 39 Note 39).

Note

Note 39: Warning. High voltage potential. Do not touch the deflection plates when the voltage is ON (red warning light). Keep the sort block door closed.

41.3

In the FACSDiva menu, create an AccuDrop experiment.

41.4

Place a tube filled with a dilute suspension of AccuDrop beads on the sample injection port and load into the sample injection chamber (see Note 40 Note 40). Adjust the flow rate to reach a threshold rate of 2'000-3'000 events per second (see Note 41 Note 41).

Note

Note 40: Tube loading, data acquisition and recording as well as flow rate are controlled using the FACSDiva acquisition dash board. Note 41: Avoid applying high flow rate.

41.5

Activate the test sort in the side stream window.

41.6

Activate the optical Filter in the side stream window.

41.7

Adjust the voltage slider to place the left side stream in the center of the box on the left. Ensure that the central stream is visible in the right box (see Note 42 Note 42).

Note

Note 42: Adjust the 2nd, 3rd, and 4th Drop settings to tighten the center stream and fine-tune the side streams, if needed.

41.8

Stop the test sort in the side stream window.

41.9

In the FACSDiva sort layout window select "2 tube" as device and "Initial" as a precision mode.

41.10

Start sorting the beads by selecting "sort" in the sort layout window (see Note 43 Note 43).

Note

Note 43: Keep the aspirator drawer closed.

41.11

In the side stream window, adjust the drop delay value in 1-drop increments to achieve close to 100% intensity in the left side stream (see Note 44 Note 44).

Note

Note 44: Alternatively, the drop delay can be set automatically using the Auto Delay button in the side stream window.

41.12

In the sort layout window, change the precision mode to "Fine Tune" and continue optimizing the drop delay by small increments until the left side stream intensity is > 95 % (see Note 45 Note 45).

Note

Note 45: Wait a few seconds after each small increment for a complete response to the delay change.

41.13

In the side stream window, inactivate the Optical Filter and set the voltage slider to "0". In the sort layout, stop the sort. Close the AccuDrop experiment.

41.14

Unload the tube filled with AccuDrop beads using the FACSDiva acquisition dash board. Discard the tube or keep it in a fridge for future sort.

42

Set up the side streams for the 2-way sort.

42.1

Insert one 15 mL tube in the 2-way sort tube holder. Close with the universal top. Insert the sort collection device into the sort collection chamber and connect to the temperature control system in order to refrigerate the collection tube.

42.2

In the side stream window, turn on the deflection plates (voltage) and activate "test sort". Optimize the position of the first either right or left side stream using the corresponding voltage sliders (see

Note 46

Note 46).

Note

Note 46: Keep the aspirator drawer closed.

42.3

In the side stream window, open the aspirator drawer to confirm the side stream are properly aligned with the collection 15 mL tube.

42.4

Close the aspirator drawer, inactivate "test sort", turn off the voltage, remove the sort collection device and trash the 15 mL collection tube.

43

Create the gates to sort nonreplicating persisters.

Note

Note 47: To ensure that your cytometer is performing consistently over time, you should regularly run performance check. Note 48: Acquisition setup and gating strategy are carried out on the FACSDiva software workplace.

43.1

In FACSDiva browser, open a new experiment and click on "Cytometer settings".

43.2

In the Cytometer window, in the parameter tab, delete all unneeded optical configuration. Keep forward scatter (FSC); side scatter(SSC); [Laser 488 nm -502LP -BP 530/30] (FITC) and [Laser 561 nm -600LP -BP 610/20] (mCherry).

43.3

For each parameter, only record the height (H) of the pulse signal detected.

43.4

In the cytometer window, in the threshold tab, set both the FSC and SSC values to the minimum (i.e., 200).

43.5

On the global worksheet, create (i) a SSC-H vs FSC-H plot; (ii) a FITC-H vs FSC-H plot and (iii) a mCherry-H vs FITC-H plot.

43.6

In the inspector window, select the checkbox for X Axis and Y Axis under biexponential display.

43.7

Place the sample tube filled with homogenized lysates on the sample injection port and load into the sample injection chamber using the FACSDiva acquisition dash board and start acquisition (see Note Note 49 49). Detected signals appear in the plots created on the global worksheet.

Note

Note 49: In the FACSDiva browser / experiment / specimen, make sure that the tube is activated.

43.8

In the cytometer window, parameter tab, set the photomultiplier (PMT) voltage for FSC and SSC in order to center the detected signal on the SSC-H vs FSC-H plot.

43.9

In the cytometer window, parameter tab, set the PMT voltage for FITC-H in order to separate the green Timer signal from the host debris autofluorescence and the electronic noise on the FITC-H vs

43.10

In the mCherry-H vs FITC-H plot, apply Gate 1. In the cytometer window, parameter tab, set the PMT voltage for mCherry-H in order to center the red Timer signal (see Note 51 Note 51).

Note

Note 51: The use of homogenized lysate from cell infected with mcherry or GFP producing L. pneumophila shall confirm the absence of fluorescence overlap in themCherry-H vs FITC-H plot. When using a sorter without spatially separated laserlines and detector arrays possible spectral overlap must be corrected via a process of fluorescence compensation.

43.11

In the mCherry-H vs FITC-H plot, two well-separated subpopulations with similar FITC-H signal intensity appear. The population with the higher mCherry intensity corresponds to the nonreplicating persisters. Using the gating tools, draw the Gate 2 Gate 2 around this subpopulation (see Note 52 and 53 Note 52 and 53).

Note

Note 52: L. pneumophilaDicmT cannot replicate intracellularly. To control the gating strategy, use homogenized lysate from cell infected with Timer producing L. pneumophilaDicmT mutant. All bacteria shall be contained in Gate 2 Gate 2. Note 53: On the global worksheet verify the gate are properly hierarchized by showing the "population hierarchy view".

43.12

In the FACSDiva acquisition dash board, select Gate 2 as stopping gate and 10'000 as events to record. Start recording by clicking "record" (see Note 54 Note 54).

Note

Note 54: Adjust the flow rate to reach a threshold rate < 12'000 events per second.

43.13

In the mCherry-H vs FITC-H plot, using the population hierarchy view, determine the population size in Gate 2 (% Parent).

44

Sort the nonreplicating persisters.

Note

Note 55: Make sure the aerosol management system is ON and both the sort block and sort chamber tighly closed in order to avoid aerosols diffusion.

44.1

Insert one 15 mL tube prefilled with 3 mL ice-cold PBS in the 2-way sort tube holder. In the sort layout window, define the sort location field to align the collection tube with the side stream as setup in 5.3. In the selected sort location field, add the subpopulation to be sorted (i.e., Gate 2 Gate 2). Close with the universal top. Insert the sort collection device into the sort collection chamber (as in 42.1.).

44.2

In the FACSDiva sort layout window select "2 tube" as device and "purity" as a precision mode (see Note 56 Note 56).

Note

Note 56: Precision modes define the sorting stringency. Sorting in Purity mode results in a sorted sample that is highly pure, at the expense of recovery and yield.

44.3

In the side stream window, turn on the deflection plates (voltage), open the aspirator drawer and initiate the sample flow ("acquire").

44.4

In the FACSDiva sort layout window, lunch the sort by clicking "sort".

44.5

In the sort layout window, monitor the sorting by observing the sorting rate, the conflicts rate and the sorting efficiency (ideally above 95 %). Inspect regularly the central and side stream in the side stream window.

44.6

Sort until the required number of nonreplicating persisters has been sorted (see Note 57-58 Note 57-58).

Note

Note 57: For continuous sorting, select "Continuous" from the Target Events menu in the sort layout window. If sample reloading is requested, pause the sorting, by clicking the "Pause" button in the Sort Layout. The counting of sorted cells is retained when you restart sorting by clicking the "Resume" button.

Note 58: Open the sort chamber to visually inspect the 15 mL collection tube filling to avoid spill over. When necessary pause the sort and replace the 15 mL collection tube. Write on the tube the number of nonreplicating persisters sorted and store on ice.

44.7

Stop the sort by inactivating "sort" in the sort layout window and save the "sort report". In the side stream window, control the deflection plates (voltage) are turned off and the aspirator drawer closed.

44.8

Unload the sample tube using the FACSDiva acquisition dash board and eliminate into the dedicated biological wastes disposal in compliance with local rules.

44.9

Open the sort chamber, remove and disassemble the sort collection device to collect and the store On ice the 15 mL collection tube.

45

Re-analysis of the sorted nonreplicating persisters.

45.1

In the FACSDiva browser, under experiment and specimen create a new tube (see Note 59 Note 59).

Note

Note 59: A new tube can be created directly from the FACSDiva acquisition dash board.

45.2

Harvest only 100 mL of sorted nonreplicating persisters from the 15 mL collection tube and transfer into a sample tube.

45.3

Place the sample tube on the sample injection port and load into the sample injection chamber using the FACSDiva acquisition dash board and activate acquisition. Detected signals appear in the plots created on the global worksheet.

45.4

In the FACSDiva acquisition dash board, adjust the flow rate to reach a threshold rate of 2'000-3'000 events per second.

45.5

In the FACSDiva acquisition dash board, select Gate 2 as stopping. Start recording by clicking "record". Stop recording and acquisition after 500-1000 events are recorded (see Note 60 Note 60).

Note

Note 60: In the mCherry-H vs FITC-H plot, sorted Timer producing nonreplicating persisters shall be contained in the Gate 2 Gate 2.

45.6

In the mCherry-H vs FITC-H plot, using the population hierarchy view, determinethe population size in Gate 2 (% Parent) to infer the sort purity.

45.7

Unload the sample tube using the FACSDiva acquisition dash board and eliminate into the dedicated biological wastes disposal in compliance with local rules.

45.8

Save your date and proceed with the shut-down.

46

Shut-down.

46.1

Clean by sequentially loading and running sample tubes filled with FACSClean FACSRinse, 70% ethanol and finally sterile ddH 2 O.

46.2

In the side stream window, turn off the stream, depressurized and disassemble the PBS tank (see Note 61 Note 61).

Note

Note 61: Autoclaved the tank, decontaminate the sheath level sensor with 70 % ethanol.

46.3

Decontaminate the sort block, the sort chamber, the sample injection port, the sort collection device (tube holder and lid) with 70% ethanol. Clean the deflecting plates with ethanol and ddH 2 O. Wipe to dry carefully all surfaces.

46.4

Assemble and filled the ethanol tank with 70% ethanol and connect to the sorter fluidics system (see Note 62 Note 62).

Note

Note 62: Ensure that the lid of the ethanol tank is properly closed and the tank pressurized.

46.5

In the FACSDiva menu, turn off the fluidic system by selecting "Fluidic shutdown" and follow the instructions.

46.6

In the sort block, remove the 70 mm nozzle. Decontaminate the nozzle and its slot at the lower end of the cuvette flow cell with 70 % ethanol. Insert the close-loop nozzle.

46.7

Turn off the external temperature control system and the aerosol management system.

46.8

Turn off the sorter.


B Sareni

L Kra ¨henbu ¨hl

A Beroual
email: beroual@trotek.ec-lyon.fr

C Brosseau

Complex effective permittivity of a lossy composite material

In recent work, boundary integral equations and finite elements were used to study the ͑real͒ effective permittivity for two-component dense composite materials in the quasistatic limit. In the present work, this approach is extended to investigate in detail the role of losses. We consider the special but important case of the axisymmetric configuration consisting of infinite circular cylinders ͑assumed to be parallel to the z axis and of permittivity ⑀ 1 ͒ organized into a crystalline arrangement ͑simple square lattice͒ within a homogeneous background medium of permittivity ⑀ 2 ϭ1. The intersections of the cylinders with the x -y plane form a periodic two-dimensional structure. We carried out simulations taking ⑀ 1 ϭ3Ϫ0.03i or ⑀ 1 ϭ30Ϫ0.3i and ⑀ 2 ϭ1. The concentration dependence of the loss tangent of the composite material can be fitted very well, at low and intermediate concentrations of inhomogeneities, with a power law. In the case at hand, it is found that the exponent parameter depends significantly on the ratio of the real part of the permittivity of the components. We argue, moreover, that the numerical results discussed here are consistent with the Bergman and Milton theory ͓D.

I. INTRODUCTION

Major new applications for aeronautics, space, and telecommunications technology using composite materials are springing up at an ever-increasing pace. The continued innovations in materials such as fiber-reinforced resins, polymer blends, and multilayered media include the recent impressive performance levels of microwave-absorbing composite devices, i.e., photonic band structures. 1 At the same time the understanding of more fundamental problems, i.e., propagation of electromagnetic waves in periodic and random dielectric structures, is of great importance to optimize these materials for a given application and has been a long standing issue. [2][START_REF]Progress in Electromagnetics Research: Dielectric Properties of Heterogenous Materials[END_REF][START_REF]For recent reviews on electrical and optical properties[END_REF][START_REF] Landauer | Electric Transport and Optical Properties of Inhomogeneous Media[END_REF] One of the more important, yet less frequently discussed, problems in the study of the dielectric properties of dense composite materials, composed of two or more components, is the role of losses. By comparison, a great deal more attention has been given to evaluate the ͑real͒ effective dielectric constant, in the quasistatic limit, of two-component composites made of a constituent of real permittivity ⑀ 1 embedded in crystalline fashion in a homogeneous threedimensional matrix of real permittivity ⑀ 2 . [2][START_REF]Progress in Electromagnetics Research: Dielectric Properties of Heterogenous Materials[END_REF][START_REF]For recent reviews on electrical and optical properties[END_REF][START_REF] Landauer | Electric Transport and Optical Properties of Inhomogeneous Media[END_REF] Most reported calculations for effective dielectric constant considered simple geometries such as spheres and ellipsoids. Some phenomenological mixture equations have been presented because of their potentially useful applications, the fit differing from material to material. [2][START_REF]Progress in Electromagnetics Research: Dielectric Properties of Heterogenous Materials[END_REF][START_REF]For recent reviews on electrical and optical properties[END_REF][START_REF] Landauer | Electric Transport and Optical Properties of Inhomogeneous Media[END_REF][START_REF] Tinga | [END_REF][START_REF] Dukhin | Surface and Colloid Science[END_REF][START_REF] Van Beek | [END_REF] The characterization of lossy composite materials is challenging, both experimentally and theoretically, and is of considerable interest for technological applications. To obtain reliable results for the complex permittivity, one must have theoretically and computationally demanding methods that accurately account for the details of the geometric microstructure of the material. Our own interest in this problem arose from our studies, based on the resolution of boundary integral equations, of the effective dielectric constant of composite nonlossy materials in the quasistatic limit. 2 We showed that the interplay between the geometrical shape and the arrangement in space of the inclusions gives rise to a large variety of behaviors. In this article we add to these works by investigating the behavior of losses in detail. For our simulations we use the method of finite elements ͑FE͒ which deals with the details of the geometry of the composite in a particularly efficient manner. A secondary purpose is to test whether the prediction of analytical theories can be reproduced in these simulations. More specifically, the Bergman-Milton analysis is discussed in order to establish a perspective from which to assess our simulation results and thus help to decide whether our numerical method is able to correctly describe the effective permittivity of lossy composite materials.

The remainder of the article is organized in the following way. Section II provides a brief summary of the overall theoretical framework of the problem at hand. We also introduce some of the details of our model and of our simulation. Section III presents numerical evaluations of the effective complex permittivity of a simple lossy two-component composite material using the FE scheme and compares our findings with the literature. Section IV summarizes the content of the article and gives conclusions.

II. BACKGROUND

In this section we briefly summarize the context of the problem at hand. To establish notation and terminology, we present a number of relevant definitions and the principle of our numerical approach.

A considerable amount of reliable work has already been achieved in the area of electric properties of heterogeneous media. The reader can find good introductions to the theory in some review articles [START_REF] Landauer | Electric Transport and Optical Properties of Inhomogeneous Media[END_REF][START_REF] Tinga | [END_REF][START_REF] Dukhin | Surface and Colloid Science[END_REF][START_REF] Van Beek | [END_REF] and a list of most relevant references in Ref. 3. Dielectric mixture theory dates back the pioneering researches in the late nineteenth century by such famous physicists as Maxwell and Lorentz, the ͑mean-field͒ results of which could prove to be a useful benchmark to assist those who wish to test computer simulations under the assumption that one of the constituents had much smaller volume than the other ͑dilute limit͒. However, depending on the geometric shape of the components and the topological arrangement in the two-component material, these results have varying degrees of success at high inclusion concentration. Of course, any valid model of the effective dielectric constant of mixtures must correctly describe the properties not only at low concentrations, but at high concentrations as well. In fact it turns out that an exact solution of the problem is possible only for a few specific and geometrically welldefined systems. One can also notice that the problem of the theoretical determination of the effective dielectric constant of composite materials was also approached on the basis of rigorous variational techniques which lead to upper and lower bounds for this quantity and also by an analyticsimulation method based on multipolar expansion of the fields around each inclusion arising from the presence of the other inclusions. [9][10][11] The complex effective dielectric constant, relative to free space, ⑀ϭ⑀ЈϪi⑀Љ carries information about the average polarization in the heterogeneous medium and is defined as the ratio of the average displacement field vector and the applied electric field. We may write this quantity as E ¯2⑀ϭ(1/⍀)͐ ⍀ ⑀ ˆ͉"V͉ 2 d⍀, where ⑀ ˆis the local permittivity value, E ¯ϭ(1/⍀)͐ ⍀ Ed⍀ denotes the average field which depends on the applied potentials, E is the local electric field in the cell, and ⍀ is the volume of the medium. The absolute complex dielectric constant is obtained by multiplying ⑀ by the permittivity of free space ⑀ 0 . Here the imaginary part ⑀Љ is due to the absorption since scattering attenuation is negligible. The ratio of ⑀Љ to ⑀Ј is referred to as the loss tangent of the material, i.e., tan ␦ where ␦ is the loss angle of the material. The complex effective permittivity will depend on the permittivity of each constituent in the mixture, their volume fractions, and eventually on the spatial arrangement in the mixture.

For simplicity we consider a composite medium made up of two nonmagnetic ͑we take 1 ϭ 2 ϭ1͒ materials with dielectric constants ⑀ 1 and ⑀ 2 . We denote by f the concentration occupied by the inclusions of permittivity ⑀ 1 periodically placed within the host material with permittivity ⑀ 2 . When considering the propagation of an electromagnetic wave in heterogeneous media, two length scales are of importance. The first scale is the wavelength of the wave probing the medium. The second one is the typical size of the inhomogeneities. When the two conditions k 1 Ӷ1 and k 2 Ӷ1 are met, where we have set k i ϭ(⑀ i i ) 1/2 ͑2/͒, iϭ1,2, the wave cannot resolve the individual scatterers and thus the material appears homogeneous to the probing wave. In this quasistatic limit, the system can be described by an effective ͑average͒ dielectric constant ⑀ which is a linear homogeneous function of ⑀ 1 and ⑀ 2 . In the numerical experiments of Sec. III the component 1 is a lossy material, i.e.,

⑀ 1 ϭ ⑀ 1 Ј Ϫ i⑀ 1 Љ , with ⑀ 1 Ј ӷ ⑀ 1 Љ and the component 2 is a pure dielectric, i.e., ⑀ 2 ϭ ⑀ 2 Ј .
Bergman and Milton presented independently an analytical treatment for computing the effective dielectric constant ͑or conductivity͒ of a two-component mixture as a function of the ratio of the dielectric constants ͑or conductivities͒ of those components. 12,13 For our problem, the results of Bergman and Milton suggest that if the geometry of the composite is known, the effective permittivity ⑀ of the mixture can be written as a function of a simple complex variable ⑀/⑀ 1 ϭ1ϩ͚ jϭ1 N ͓A j /͑s j Ϫs͔͒, where sϭ1/͓1Ϫ͑⑀ 2 /⑀ 1 ͔͒, N is the number of simple poles determined by the geometry of the material, A j and s j denoting respectively the residue and pole in the real interval ͓0,1͔ of the jth term in the summation. Since the above expression of ⑀ is an analytic function of ͓0,1͔ in the s plane, ⑀ can be implicitely determined by the positions of its zeros and poles in the complex plane. As discussed in Refs. 10-12, an important feature of the Bergman-Milton analysis is that the location of the poles is under the dependence of the geometry of the structure only and is invariant for all possible values of ⑀ 1 and ⑀ 2 . Liu and Shen have compared the Bergman-Milton simple pole analysis and the Fourier series expansion technique to compute the effective dielectric constant of two-component twodimensional composite materials. 14 They found that both methods give comparable results in predicting ⑀ even when the dielectric constants of the components are complex ͑this can be seen in Table IV of Ref. 14͒. However, it is important to observe that most of the numerical experiments of these authors concern the case of weak contrast ratios between the dielectric constants of the background medium and the inclusions. Now we describe the simulation procedure as follows. The calculation of the effective dielectric constant of composite materials from the known properties of the pure, homogeneous components is an electrostatics problem which involve the resolution of partial differential equations and taking into account boundary conditions defined on domains with given geometries. Here we give only some of the most important details of the model investigated and the essential features of the simulation. More details can be found in Ref.

2. We assume a system as displayed in Fig. 1, where an arbitrarily shaped homogeneous inclusion with permittivity ⑀ 1 is embedded in a homogeneous matrix with permittivity ⑀ 2 . The solution of Laplace's equation is computed with the field calculation package FLUX2D, [START_REF]FLUX2D Users Manual[END_REF] using the FE method. Its implementation consists in dividing the domain S into triangular finite elements and for each element, the calculation is carried out by interpolation of the potential V and its normal derivative ץV/ץn with the corresponding nodal values Vϭ ͚ j j V j and ץV/ץnϭ͚ j j (ץV/ץn) j , where j denotes interpolating functions. [START_REF] Zienkiewicz | The Finite Element Method ͑McGraw-Hill[END_REF] The field and potential distributions are obtained from the boundary conditions using the Galerkin method [START_REF] Zienkiewicz | The Finite Element Method ͑McGraw-Hill[END_REF][START_REF] Finlaysson | The Method of Weighted Residuals and Variational Principles ͑Academic[END_REF] and solving the matrix equations resulting from the discretization procedure by standard numerical techniques, e.g., the Gauss method. [START_REF] Press | Numerical Recipes ͑Cambridge[END_REF] Having computed the potential and its normal derivative on each node of the mesh, the electrostatic energy

␦W e ͑ k ͒ϭ 1 2 ͵ S k k Ј ͫͩ ץV ץx ͪ 2 ϩ ͩ ץV ץy ͪ 2 ͬ dx dy
for each triangular element is evaluated, where ⑀ k Ј and S k denote the real part of the permittivity and the surface of the kth triangular element, respectively. Thus, the total energy of the entire composite can be written by summation over the n k elements such as W e ϭ ͚ kϭ1 n k ␦W e (k). In the problem at hand, we consider a portion of the composite material which is the filler of a parallel-plate capacitor. In this manner we obtain the real part of the complex effective permittivity from the electrostatic energy stored in such a capacitor, i.e., W e ϭ 1 2 ⑀Ј(S d /e)(V2ϪV1) 2 when a given potential difference is applied across the plates ͑see Fig. 1͒, S d ϭLd stands for the surface of the plates with side of length L ͑for the twodimensional structures considered below we take dϭ1 unit of length͒. Now, we must relate the dielectric losses to the imaginary part of ⑀. To do so, the dielectric losses are evaluated on each element of the mesh as

␦͑k͒ϭ 1 2 ͵ S k ⑀ k Ј tan ␦ kͫ ͩ ץV ץx ͪ 2 ϩ ͩ ץV ץy ͪ 2 ͬ dx dy,
where tan ␦ k is the loss tangent of the kth element and is the angular frequency of the electric field. The total losses of the entire composite are then obtained by summation over the n k elements such as ϭ ͚ kϭ1 n k ␦(k) and finally its connection with the imaginary part of the complex effective per-mittivity is ϭ 1 2 ⑀Љ(S d /e)(V2ϪV1) 2 . With these considerations in mind, we now turn to the presentation of the results of our numerical experiments.

III. NUMERICAL RESULTS AND DISCUSSION

Let us consider the details of Fig. 2 with Cartesian spatial coordinates x, y, and z. This figure represents the axisymmetric unit cell of the two-component structure under study. It has been chosen for purpose of comparison with the results of Liu and Shen. 14 The structure consists of an infinite circular cylinder ͑⑀ 1 , 1 ϭ1͒ of radius r and with generator parallel to the z axis embedded in a host matrix ͑⑀ 2 , 2 ϭ1͒, i.e., the intersections of the cylinders with the x -y plane form a periodic two-dimensional structure ͑simple square lattice of side eϭ1͒. This symmetrical structure in both the x and y directions renders the two-component composite material to be both translationally and rotationally invariant. We denote by f the fractional occupancy of constituent 1: f ϭr 2 below the percolation threshold ͑0рrр 1 2 , nonoverlapping cylinders͒ and

f ϭr 2 Ϫ4r 2 arcos ͩ 1 2r ͪ ϩͱ4r 2 Ϫ1
beyond the percolation threshold ͑ 1 2 рrр1/&, overlapping cylinders͒. The rу 1 2 case is special because the symmetry of the structure provides that we can exclude in our numerical calculations the four sections of the component of permittivity ⑀ 1 exceeding from the unit square cell.

To facilitate the comparison with the results of Ref. 14, we first use the following set of parameters: ⑀ 1 ϭ1Ϫ3i and ⑀ 2 ϭ5Ϫ8i. In Fig. 3 we show a comparison of the effective permittivity computed for the two-pole approximation ͑Bergman-Milton theory͒ and the FE method versus the radius of the circular cylinder. Figures 3͑a͒ and3͑b͒ are for the real part and imaginary part of ⑀, respectively. As can be recognized from these figures, in the range of r investigated, the effective complex permittivity obtained from the FIG. 1. Schematic drawing of the two-dimensional composite medium made of arbitrarily shaped homogeneous inclusions with permittivity ⑀ 1 embedded in a homogeneous matrix with permittivity ⑀ 2 . On the boundary of each cell ͑with sides of length L and e͒ the potential is kept constant ͑V 1 or V 2 ͒ on the top and bottom faces and its normal derivative ץV/ץn is equal to zero along the vertical walls. FIG. 2. Notation relating to the configuration of the two-component periodic composite material ͑simple square lattice͒ investigated in this study. The geometry is symmetrical in both the x and y directions. The unit cell consists of a square of side eϭ1 containing an infinite circular cylinder, of radius r, along the z direction and of permittivity ⑀ 1 . The dielectric constant of the remaining space is ⑀ 2 .

Bergman-Milton theory are quantitatively very similar to the ones presented in this work. The only thing that changes is that the values of ⑀Ј and ⑀Љ are slightly higher than those given by the two-pole approximation. These results confirm the usefulness of the FE algorithm as an efficient tool for computing the complex effective permittivity of lossy composite media. Liu and Shen's method is based on a Fourier expansion technique. 14 As discussed in Ref. 14 this computation technique is costly in computer time even in the twodimensional case considered here. By contrast, our method is not time consuming. The CPU time for calculating the permittivity of a typical two-dimensional configuration is of the order of a few seconds. As an aside, we note that ⑀Х⑀ 1 when r is close to 0.65, for both cases.

Having checked the FE method for the computation of the complex effective permittivity of a lossy composite medium, we now investigate what the relative importance of ⑀ 1 and ⑀ 2 is. Under many physically interesting conditions ͑e.g., polymer carbon-black composites͒, the dielectric constant of the matrix material is much smaller than the dielectric constant of the inhomogeneities. In our simulations we take ⑀ 1 ϭ3Ϫ0.03i or ⑀ 1 ϭ30Ϫ0.3i and ⑀ 2 ϭ1, i.e., tan ␦ 1 ϭ10 Ϫ2 and tan ␦ 2 ϭ0. The differences in physical behavior can only arise from a difference in the value of the ratio ⑀ 1 Ј/⑀ 2 Ј since the physical structure and losses of both components are invariant in this study. In Figs. 4 and5 we present the results for the real and imaginary parts of ⑀ versus the concentration f for the two values of ⑀ 1 investigated. Naturally, for the area fraction value f ϭ1, the complex effective permittivity ⑀ equals the inclusion permittivity ⑀ 1 . From these figures it is of interest to observe that for ⑀ 1 ϭ30Ϫ0.3i the real and imaginary parts of the permittivity increase smoothly but display a sharp increase at a concentration close to 0.7. Note, however, that this phenomenon is absent for ⑀ 1 ϭ3Ϫ0.03i. An important insight can be obtained by examining the concentration dependence of the tangent of losses of the composite samples in the logarithmic plot shown in Fig. 6. Most interestingly we find that the electric loss factor increases, at low and intermediate concentrations of inhomogeneities, with increasing concentration of inhomogeneities as a power law, tan ␦ϳf a , with an exponent parameter aϷ1.07 for ⑀ 1 ϭ3Ϫ0.03i and aϷ1.31 for ⑀ 1 ϭ30Ϫ0.3i. Thus, we find that the exponent parameter a is not constant but depends on the actual values of ⑀Ј 1 ; in fact it will depend, here, on the ratio of the real part of the permittivity of the two components. This behavior changes strongly when we go to a higher concentration of inhomogeneities. We make the additional observation that the dielectric losses are higher ͑by a factor of 8͒ for ⑀ 1 ϭ3Ϫ0.03i compared to the case ⑀ 1 ϭ30Ϫ0.3i, while at high concentration they tend to be close for both cases. It is also seen in Fig. 6 that the two plots of tan ␦ vs f intersect at a concentration close to f *ϭ/4

where the cylinders touch.

IV. CONCLUSIONS

In summary, we have continued our comprehensive analysis of the dielectric properties of composite materials. In contrast to our earlier article in which we mainly concentrated on the investigation of the real effective permittivity of composite materials, we have described here a numerical scheme to compute the complex effective permittivity of a lossy composite material in the quasistatic limit. Our computational tool for studying this problem, i.e., the FE method, provides an accurate approach to evaluate this quantity. The present analysis has been carried out in a circular cylinder geometry. Similar consideration can be readily applied to other axisymmetric geometries. One goal of our investigation was to test if the Bergman-Milton theory is able to give a correct description of the effective complex dielectric constant of a lossy two-component composite medium. In this respect, we would like to point out that numerical experiments discussed here are consistent with the Bergman and Milton theory. The conclusions reached in this article, being strictly valid for the case of periodic two-dimensional configurations, are also guidelines in general context, i.e., periodic composites with deliberate introduction of imperfections. It is hoped that this work will stimulate experimental interest in these problems. For reasons of mathematical analogy, these results are also valid for the magnetic permeability of two-component composite materials. We conclude this study reminding the reader that all the simulation results presented here must be regarded as simply preliminary attempts to identify the major features of the modeling of the effective permittivity of real composite materials. In particular the present work should be extended in several directions. First, and in the light of this work and results presented in Refs. 3, 19-22 it would be of great interest to study the effect of frequency of the applied field on the effective permittivity in order to check explicitly theories aiming at elucidating the mechanism of dielectric relaxation, e.g., the scaling behaviors proposed by Jonscher. [START_REF] Jonscher | Dielectric Relaxation in Solids ͑Chelsea Dielectrics[END_REF] An even more formidable chal- lenge will be to investigate the random spatial arrangement of the aggregate topology. Work is in progress to meet these next challenges and full details of the above calculations will be given elsewhere.

FIG. 3 .

 3 FIG. 3. ͑a͒ Comparison of the real part of the complex effective dielectric constant evaluated by the finite-element method ͑᭹͒ and the two-pole Bergman-Milton approximation ͑ϫ, Ref. 14͒ as a function of the radius of the circular cylinder r. The dielectric constants of the two-component composite material are ⑀ 1 ϭ1Ϫ3i and ⑀ 2 ϭ5Ϫ8i. ͑b͒ Comparison of the imaginary part of the complex effective dielectric constant evaluated by the finiteelement method ͑᭹͒ and the two-pole Bergman-Milton approximation ͑ϫ, Ref. 14͒ as a function of the radius of the circular cylinder r. The dielectric constants of the two-component composite material are ⑀ 1 ϭ1Ϫ3i and ⑀ 2 ϭ5Ϫ8i.

FIG. 5 .

 5 FIG. 5. ͑a͒ The real part of the effective permittivity is shown as a function of the area fraction f of the inclusion phase. Inclusions ͑permittivity ⑀ 1 ϭ30Ϫ0.3i͒ are placed in a host matrix material of permittivity ⑀ 2 ϭ1. ͑b͒ The imaginary part of the effective permittivity is shown as a function of the area fraction f of the inclusion phase. Inclusions ͑permittivity ⑀ 1 ϭ30Ϫ0.3i͒ are placed in a host matrix material of permittivity ⑀ 2 ϭ1.

ACKNOWLEDGMENTS

The authors would like to thank the CEDRAT and MAGSOFT corporations for their help in using the field calculation package FLUX2D which allowed us to perform these numerical experiments. CEGELY is Unite ´Associe ´e 829 au CNRS.