Matti ; Karp

Mariel G Pikkemaat
email: mariel.pikkemaat@wur.nl

Michel Lba

Matti T Karp

J W Alexander Elferink

Application of a luminescent bacterial biosensor for the detection of 1 tetracyclines in routine analysis of poultry muscle samples 2 3 4

Keywords: Bioassay, Method validation, Screening -microbial screening Additives/Contaminants: Veterinary drug residues -antibiotics, Veterinary drug residuesoxytetracyclin, Veterinary drug residues -tetracycline cell-biosensor, bioassay, veterinary drug residues, tetracyclines 33

Application of a luminescent bacterial biosensor for

Veterinary drug use is under major public debate. There is great concern about resistance 36 development against antibiotics, and subsequently the possibility of transfer from animal 37 pathogens or commensals ultimately to human pathogens. This has elicited a example led to the withdrawal of enrofloxacin for use in poultry in the United States (US 40 Food and Drug Administration. 2005). The WHO has defined a list of antibiotics 41 considered critically important for human treatment (World Health Organization. 2005), 42 that should preferably not be used for animal purposes. In Europe concerns about 43 antibiotic resistance resulted in a ban on the use of antibiotics for growth promoting 44 purposes (European Commission. 2003.) 45 46 Compared to the resistance threat, the toxicological risks of veterinary drug residues in 47 food products of animal origin, can be considered relatively minor. Nevertheless, to 48 protect the consumer from being exposed to harmful levels of residues, the EU has set 49 procedures for determining Maximum Residue Limits (MRLs) for antibiotics in food 50 products of animal origin (European Commission. 2009). To enforce residue legislation, 51 national monitoring programs are statutorily required under EU directive 96/23 52 (European Commission. 1996). Apart from basic surveillance to safeguard consumers 53 from unwanted levels of residues in their food, effective residue monitoring also serves as 54 an instrument to establish prudent use of antibiotics. Fast and inexpensive high-55 throughput methodology is a prerequisite for effective enforcement. 56 57 Tetracyclines are in many countries the most used veterinary drug group. They are 58 inexpensive broad spectrum drugs, widely available, and their residues are frequently 59 found in food commodities of animal origin (Okerman et al.1998;Myllyniemi et al.1999;60 Schneider et al. 2008;Pikkemaat et al. 2009a). Detection of tetracycline residues in 61 animal matrices can be achieved using a variety of methods. On one end of the spectrum 62 are the physical chemical LC-MS based methods [START_REF] Oka | Chromatographic analysis of tetracycline antibiotics 336 in foods[END_REF]. These require 63 sophisticated equipment and specialized technicians to operate it, and are generally 64 considered too expensive to be applied for initial screening. On the other end of the 65 spectrum are the microbiological assays (Pikkemaat. 2009b). These tests rely on growth 66 inhibition of a sensitive test bacterium. They are very cheap and can be applied with 67 minimal training and equipment. For detection of tetracyclines B. cereus ATCC 11778 is 68 the most commonly used bacterium [START_REF] Bovee | Bioactivity-based screening of antibiotics and 295 hormones[END_REF]. The inhibition zones 69 can be considered a semi-quantitative result. A major drawback of this type of test is the 70 lengthy incubation (over night) before the result is obtained. 71 72 Recently we reported the development of an assay for the detection of tetracyclines in 73 poultry based on a bioluminescent bacterial sensor as an alternative for "classical" 74 microbial screening methods [START_REF] Virolainen | Rapid detection of 365 tetracyclines and their 4-epimer derivatives from poultry meat with bioluminescent 366 biosensor bacteria[END_REF]. The sensor bacterium is an E. coli 75 strain harboring a plasmid containing a bacterial luciferase operon under the control of a 76 tetracycline sensitive repressor (Korpela et al.1998). Induction of the lux operon, and 77 subsequently the production of a bioluminescent signal, only occurs in the presence of 78 tetracycline. The test is much faster than microbial inhibition tests, as results can be 79 obtained after 3 hours, and very inexpensive since it does not require any additional 80 reagents. So far the sensor had been tested on several animal food products, but it had 81 only been tested using spiked materials [START_REF] Kurittu | Qualitative detection of tetracycline 323 residues in milk with a luminescence-based microbial method: the effect of milk 324 composition and assay performance in relation to an immunoassay and a microbial 325 inhibition assay[END_REF][START_REF] Pellinen | Detection of traces of 343 tetracyclines from fish with a bioluminescent sensor strain incorporating bacterial 344 luciferase reporter genes[END_REF]82 Virolainen et al. 2008). For determining practical applicability in routine monitoring 83 conditions, a large scale evaluation using field samples was performed, running the test in 84 parallel with the conventional microbial inhibition test. 85 86 87

Materials & Methods 88

Sample material 89

A total of 337 poultry samples (chicken, turkey and guinea fowl) originating from a 90 national monitoring program and other surveillance programs were analyzed. For sample 91 preparation, muscle tissue was chopped in a kitchen blender and transferred to a 15 ml 92 centrifuge tube. After incubating the tube for 20 min at 65 °C, it was centrifuged for 10 93 min at 27000 x g. The supernatant (muscle fluid/extract) was transferred to a fresh tube 94 and kept at -20 °C until further use. 95

Microbial inhibition test 97

The microbial inhibition test for tetracycline is based on growth inhibition of Bacillus 98 cereus ATCC 11778. The test plate contained 31,4 g L -1 Iso-sensitest agar (Oxoid) 99 adjusted to pH 6.0. After sterilization the agar was cooled to 48 ºC, supplemented with 100 500 ug L -1 chloramphenicol and inoculated with approximately 10 5 CFU ml -1 of the test 101 organism. The inoculated agar was poured in square petri dishes as a 2.2 mm layer. After 102 solidifying, holes with a diameter of 14 mm were punched in the agar and samples of 250 103 µl muscle extract were applied. The samples were supplemented with 40 µl of a 1M 104 phosphate buffer pH 6.0, containing 12.5 µg ml -1 chloramphenicol. After 16-18 hours 105 incubation at 30 ºC the presence of tetracyclines became visible as a zone of growth 106 inhibition around the punch hole. Muscle extract from a sample spiked with 100 µg kg -1 107 oxytetracycline was included in each test plate as a reference. 108 109

Cell-biosensor assay 110

The tetracycline cell biosensor assay was performed as described in (Virolainen et al. 111 2008). In short, lyophilized sensor cells were reconstituted in Luria Broth (LB) 112 containing 0.5 µg ml -1 polymyxin B for 1 hr at room temperature. The reconstituted cells 113 were diluted four-fold in LB containing a 100 mM pH 6 phosphate buffer and 0.5 µg ml -1 114 polymyxin B. Aliquots of 50 µl were dispensed in white 96-well flat bottom microtiter 115 plates (Greiner Bio-One GmbH, Frickenhausen, Germany). Additionally, the assay 116 contained 150 µl of poultry meat fluid and 25 µl of 225 mM EDTA solution to obtain a 117 final EDTA concentration 25 mM. Each sample was analyzed in triplicate. The microtiter 118 plates were incubated at 30 °C and 300 rpm. At timepoints 0 and 3 hours 119 bioluminescence was read using a Synergy™ HT Multi-detection Microplate Reader 120 (BioTek Instruments Inc., Winooski, VT, USA). Because the absolute bioluminescence 121 signal strongly depends on the type of instrument used, results are reported as the ratio 122 between the signal of induced and non-induced cells. This induction coefficient (IC) was 123 calculated by dividing the average signal after three hours of incubation by a blank 124 control, a poultry meat fluid sample which previously was found to be free of 125 step with water and buffer the analytes were extracted from the column using methanol. 135

The methanol was evaporated under a steam of nitrogen. The residues were reconstituted 136 in acetonitrile/water solution (2/98;v/v). The analytes were separated from the matrix 137 components and detected by using liquid chromatography in combination with triple 138 quadrupole mass spectrometric detection, using electrospray ionization in the positive 139 mode (LC-ESI(+)QqQ-MS). The LC separation was on a Symmetry ® C18 column 5 µm 140 3.0*150 mm (Waters) using a gradient of (solvent A, water 1 mM ammonium acetate pH 141 (the MRL). Oxytetracycline is used because it is the least detectable tetracycline residue, 166 so if a sample shows an inhibition zone smaller than the control, it is assumed to be 167 compliant. The positive samples in this study provide an opportunity to evaluate how 168 well spiked samples represent incurred material. Figure 1 shows the concentration of the 169 samples plotted against the inhibition zone (corrected for the QC inhibition), and the 170 result of a concentration series of doxycycline spiked samples. From the graph it is clear 171 that differences between spiked and incurred materials are almost negligible. Since the 172 trend line crosses the x-axis approximately at 50 µg kg -1 , it can also be concluded from 173 the graph that doxycycline can be detected approximately two-fold more sensitive 174 compared to oxytetracycline, the reference compound. 175

[Insert figure 1 about here] 177 178

Cell-biosensor assay 179

Without knowledge on the outcome of the initial microbial screening, all 337 samples 180 were additionally tested using the bacterial biosensor. The assay is based on the induction 181 of a luminescence signal upon the presence of tetracyclines. In previous studies (Kurittu 182 et al. 2000;[START_REF] Pellinen | Detection of traces of 343 tetracyclines from fish with a bioluminescent sensor strain incorporating bacterial 344 luciferase reporter genes[END_REF][START_REF] Virolainen | Rapid detection of 365 tetracyclines and their 4-epimer derivatives from poultry meat with bioluminescent 366 biosensor bacteria[END_REF] the assay proved to be sensitive to 183 all veterinary relevant tetracyclines (tetracycline, doxycycline, chlortetracycline and 184 oxytetracycline). Typical dose-response (induction coefficient, IC) curves of the assay 185 show a parabolic behavior, since high concentrations will inhibit protein production (and 186 ultimately kill the test organism), decreasing the signal upon increasing concentrations. 187 These samples included the 7 samples that were also identified with the microbial 195 inhibition assays. The additionally identified samples were analyzed by LC-MS/MS 196 (except for three samples for which no material was remaining). In all positive samples 197 the presence tetracyclines was confirmed, while two negative samples, which were 198 included as a (blind) control, tested negative. The concentrations appeared to be very low, 199 2 samples contained concentrations between 10 and 50 µg kg -1 , but most samples were 200 even below the limit of quantification (10 µg kg -1 ). 201

[Insert table 1 about here] 203 204

The majority of the samples contained doxycycline. In one sample oxytetracycline was 205 found, while three samples contained chlortetracycline, of which one additionally 206 contained a similar amount of tetracycline. Remarkably in almost all of the doxycycline 207 positive samples also tetracycline was found. Since tetracycline itself in the Netherlands 208 is not registered for use in poultry, this could indicate that the applied doxycycline 209 product contained a substantial percentage of tetracycline. 210

211

The absolute as well as the relative (compared to the oxytetracycline calibration line) IC 212 value appeared to have very limited predictive value, which becomes evident from the 213 oxytetracycline calibration data (figure 2), showing a large standard deviation. The peak 214 of the curve on the other hand, always occurred at 125 µg kg -1 , which is similar to former 215 results [START_REF] Virolainen | Rapid detection of 365 tetracyclines and their 4-epimer derivatives from poultry meat with bioluminescent 366 biosensor bacteria[END_REF]. Although the lowest oxytetracycline spike (15 µg kg -1 ) in 216 one experiment failed to produce a positive result, the single incurred sample obtained in 217 this study showed that the detection limit for oxytetracycline probably lies around the 218 limit of quantification of the chemical confirmation (10 µg kg -1 ). 219 220[insert figure 2 about here] 221 222 One explanation for the variations in the IC is the presence of varying levels of 223 hemoglobin. The bioluminescence spectrum overlaps with the absorption spectrum of 224 hemoglobin, attenuating the bioluminescence intensity [START_REF] Luker | Applications of bioluminescence imaging to antiviral 328 research and therapy: multiple luciferase enzymes and quantitation[END_REF]. In 225 practice, color of the extracted fluid from chicken muscle varied between almost 226 colorless to pink. This phenomenon became very obvious when the assay was subjected 227 to guinea fowl which has a very dark red color. The samples initially showed an unusual 228 low luminescence signal, and when they were fortified with doxycycline, no measurable 229 signal could be produced. Another source of variation in the induction factors may be 230 introduced by the use of reconsituted lyophilized cells. Batches of cells may not be 231 identical with respect to the number of cells and thus may yield batch dependend ICs. 232 This variation could probably be minimized by standardizing (by OD measurement) the 233 cell suspension to be used in the assay. 234 235 As mentioned before, to prevent false-negative results, the experimental setup comprised 236 three subsequent dilutions of the same sample. In contrast to the IC value itself, the 237 pattern of the IC values generated by such a series of samples did appear to be a suitable 238 predictor for the concentration. The 7 samples showing a response in the microbial 239 inhibition assay all showed an increased signal upon twofold dilution. Six additional 240 samples that did not show microbial growth inhibition, also showed this pattern. Of these 241 13 samples only the MRL violation and one sample confirmed at 47 µg kg -1 doxycycline, 242 showed increasing signal in the four-fold dilution as well. Using this pattern as a cut-off 243 leads to a reduction of 96% samples requiring additional chemical confirmation. 244 245

Effect of other antibiotics (specificity) 246

Since the assay requires active bacterial metabolism, other antibiotics could potentially 247 affect the assay. Only one of the 337 samples appeared to contain antibiotic residues 248 2). It appeared to be able to detect 281 tetracycline levels far below the MRL, and by applying a test format using a series of 282 three sample dilutions, an appropriate cut-off for further confirmatory analysis can be 283 made. The assay appeared to be very specific and robust, with the limitation that it is not 284 suitable for the analysis of guinea fowl and most likely also not for other ´red´ poultry 285 species, though this probably could be solved by some additional sample preparation. 286

287

The result of this study furthermore shows that application of tetracyclines in poultry is 288 widely spread, but withdrawal times are generally obliged, since the concentrations that 289

  samples typically yielded IC values between 0.5 and 1.5, so a 126 sample was considered positive if the IC > 2. 127 128Chemical confirmation 129Quantification and identification of the suspect samples from the microbial inhibition test 130 were performed using an in-house optimized and according to Commission Decision 131 657/2002/EC validated method. Samples were extracted with an EDTA-McLLvain buffer 132 (0.1 M; pH 4.0). The primary extract was transferred to an OASIS TM HLB solid-phase 133 column (Waters, Milford, MA) preconditioned with methanol and buffer. After a washing 134

  2.6; solvent B, 10 mM ammonium acetate pH 2.6/acetontiril (10/90;v/v): 0-1 min, 0% B; 142 1-10 min, linear increase to 50%B; 10-14 min, linear increase to 100% B with a final 143 hold of 4 min. For identification the selective reaction monitoring (SRM) mode was used 144 for the specific detection of each analyte of interest. The ions monitored for tetracyclines 145 were m/z 445> 410 and m/z 445>154; for oxytetracycline m/z 461> 337 and m/z 146 461>201; for doxycycline m/z 445> 428 and m/z 445>154; for chlorotetracycline m/z 147 479> 153 and m/z 479>444 and for the internal standard demeclocycline m/z 465>154. 148For quantification, a 'detector response' -peak area ratios IS/standard tetracycline -149 versus 'tetracycline concentration' plot was constructed. To this end, blank samples were 150 fortified with different concentrations of the tetracyclines (0 -250 µg/kg). The collected 151 samples were analysed together with the calibration samples (Matrix Match Standard or 152 MMS); concentrations were calculated using the least squares linear regression method. Out of the 337 samples, 7 (2%) tested positive on the B. cereus test 158 plate. After LC-MS/MS analysis, in all samples the presence of doxycycline was 159 confirmed. One non-compliant result was found (195 µg kg -1 ), the remainder of the 160 suspect samples contained concentrations between 26 and 50 µg kg -1 doxycycline. 161 162 In practice, the result of the microbial inhibition test is compared to a quality control 163 (QC) reference sample, to visualize day to day variation in sensitivity of the test. This 164 control sample is obtained from poultry muscle spiked with 100 µg kg -1 oxytetracycline 165

  , the assay was performed using three subsequent dilutions of the 189 same sample, so if a sample contains high concentrations of tetracycline, the diluted 190 samples will yield an increased IC. The sample analysis was performed on 9 individual 191 days, and each series included an oxytetracycline calibration line (figure2). 192 193 Out of the 337 samples 36 (10.7%) showed a positive cell-biosensor result (IC > 2). 194

  traditional microbial inhibition tests (a comparison of the performance 280 and requirements of the methods given Table

Figure 1 .Figure 2 .

 12 Figure captions 373 374 Figure 1. Inhibition zones obtained from the microbial inhibition assay plotted against the 375 concentration of doxycycline. Squares indicate a series of spiked poultry samples, the 376 diamonds represent the incurred samples obtained in this study. The inhibition zone was 377 compared to a reference obtained from a sample spiked with 100 µg kg-1 378 oxytetracycline. 379 380 381 Figure 2. Induction coefficient (IC) plotted against the concentration of oxytetracycline in 382 spiked samples, results obtained from 9 individual experiments. 383 384

Table 1 .

 1 Overview of the results

		Food Additives and Contaminants
	Total nr of	Suspect inhibition	Positive biosensor	MRL violation
	samples	assay	assay
	337	7 (2%)	36 (10.7%)	1 (0.3%)

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Table 2 .

 2 Comparison of the performance and requirements of the methods

	Page 17 of 19
		Microbiological assay Cell-biosensor
	Results	Qualitative	Qualitative
	Sensitivity	50 -100 µg kg -1	10 µg kg -1
	Specificity	Tetracyclines	Tetracyclines
	Sample preparation	Isolation of muscle fluid Isolation of muscle fluid
	Measurement time	12-16 h	3 h
	Instruments required	Incubator	Incubator
			Bioluminescence reader
	Applicability	Laboratory and on-site	Restricted to laboratory
	Market price per sample 15 €	7.5 €

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uk Food Additives and Contaminants

reconsideration of the application of certain antibiotics for non-human use, which has for

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants

et al. 2007). This sample did not show any aberrant behavior. Sensitivity to other

active substances in foodstuffs of animal origin, repealing Council Regulation (EEC) No

2377/90 and amending Directive 2001/82/EC of the European Parliament and of the

Council and Regulation (EC) No 726/2004 of the European Parliament and of the

medicine for risk management strategies of non-human use. Available from:


Olivier ; Fumière
email: fumiere@cra.wallonie.be

Keywords: PAP, animal proteins, BSE, feed ban, real-time PCR, NIRM

At present, European legislation prohibits totally the use of processed animal 24 proteins (PAPs) in feed for all farmed animals [Commission Regulation (EC) 25 n°1234/2003 -extended feed ban]. A softening of the feed ban for non-ruminants 26 would nevertheless be considered if alternative methods could be used to gain more 27 information concerning the species origin of PAPs than that which can be provided by 28 classical optical microscopy. This would allow control provisions such as the ban of 29 feeding animals with proteins from the same species or intra-species recycling 30 [Regulation (EC) n°1774/2002]. Two promising alternative methods, Near Infrared 31 Microscopy (NIRM) and real-time Polymerase Chain Reaction (PCR) were combined 32 to authenticate, at the species level, the presence of animal particles. The present 33 paper describes the improvements of the real-time PCR method which were made to 34 the DNA extraction protocol, allowing 5 PCR analyses to be performed with the DNA 35 extracted from a single particle. 36

(EC) 152/2009] but the method offers very limited species discrimination perspectives. The Regulation (EC) 152/2009, however, stated that alternative methods can be used to gain more information about the origin of the PAPs. The capabilities of the different analytical methods for the detection, the species identification and the quantification of PAPs in feedingstuffs were recently reviewed by [START_REF] Fumière | Methods of detection, species identification and quantification of processed animal proteins in feedingstuffs[END_REF]. Among the developed alternative methods, NIRM (Near Infrared Microscopy) and PCR (Polymerase Chain reaction) are in-house validated and routinely used under accreditation (ISO 17025 standard) at the Walloon Agricultural Research Centre. Moreover, they have been successfully assessed in international inter-laboratory studies describing their performance profile and their specific pros and cons (e.g. [START_REF] Prado | Detection of ruminant meat and bone meals in animal feed by real time Polymerase Chain Reaction: result of an interlaboratory study[END_REF]. Other validation studies were conducted under the framework of the European SAFEED-PAP project. The results of these studies will be published soon.

Since 1998, the NIRM method that combines a NIR spectrometer and a microscope is in development at CRA-W for the detection of animal by-products in feed [START_REF] Piraux | Microscopie-NIR appliqué aux aliments du bétail[END_REF]. Through different Belgian and European projects e.g. STRATFEED and SAFEED-PAP, the method was continuously improved and the discrimination between terrestrial PAPs and fish by-products on particles from the whole sample (raw fraction) and the particles having a density > 1.62 obtained through a sedimentation process with tetrachloroethylene (sediment fraction) has been demonstrated [START_REF] Baeten | Strategies and methods to detect and quantify mammalian tissues in feedingstuffs. Brussels: European Commission. The near infrared microscopic (NIRM) method: combination of the advantages of optical microscopy and near infrared spectroscopy[END_REF][START_REF] Baeten | Detection of banned meat and bone meal in feedingstuffs by near-infrared analysis of the sediment fraction[END_REF][START_REF] De | Discrimination of fish bones from other animal bones in the sedimented fraction of compound feeds by near infrared microscopy (NIRM)[END_REF]von Holst et al. 2008). Looking at the discrimination between PAPs particles from terrestrial animal species, the results obtained in previous studies tend to indicate that it might be possible. Nevertheless, such information is only considered as an indication and could not be used as a forensic result. An interesting aspect of the NIRM is the non-destructive property of the method towards the analysed samples thereby allowing to isolate the suspicious particles and to perform additional analyses.

PCR is currently the only technique able to determine the origin of animal by-products present in a feed at the species level. Thanks to the good stability of the DNA to high temperatures and rendering processes, different real-time PCR methods using small sized multi-copy targets already proved their efficiency for the detection of PAPs in animal feed at low level [START_REF] Aarts | Detection of bovine meat and bone meal in animal feed at level of 0.1%[END_REF][START_REF] Fumière | Effective PCR detection of animal species in highly processed animal byproducts and compound feeds[END_REF][START_REF] Prado | Detection of ruminant meat and bone meals in animal feed by real time Polymerase Chain Reaction: result of an interlaboratory study[END_REF][START_REF] Cawthraw | Real-Time PCR Detection and Identification of prohibited Mammalian and Avian Material in Animal Feeds[END_REF]). Nevertheless, PCR alone is not sufficient to evidence non compliance with the feed ban as it cannot distinguish authorised animal DNA sources (e.g. blood or milk products) from forbidden ones. Due to the high sensitivity of some PCR methods, the species origin of by-products containing only traces of DNA such as fat (e.g. tallow) can be determined [START_REF] Bellorini | Discriminating animal fats and their origins: assessing the potentials of Fourier Transform infrared microscopy, gas chromatography, immunoassay and polymerase chain reaction techniques[END_REF] and could even lead to false positive results from a legal point of view [START_REF] Prado | Detection of ruminant meat and bone meals in animal feed by real time Polymerase Chain Reaction: result of an interlaboratory study[END_REF]).

The combination of both techniques allowing the detection of individual PAPs particles by NIRM and their authentication at a species level by real-time PCR was therefore investigated. This paper describes the last developments achieved in the PCR protocol to determine the origin of a PAPs particle at a species level with 5 PCR. NIRM analysis. The raw materials were ground with a Retsch ZM200 grinding mill (Retsch GmbH & Co., Haan, Germany) to obtain particles with a diameter < 2 mm. Prior to the NIRM analysis, the samples were sieved to select the fraction of the particles with a size with a diameter > 250 µm. Sixty particles were then manually spread on a spectralon plate. Using the microscope pointer of an auto image microscope connected to a Fourier transform near infrared spectrometer (FT-NIR; Perkin Elmer, Waltham, MA, USA), the infrared beam was focused on each particle and their NIR-spectrum (1100-2500 nm) was collected. The particles were arranged to be able to track the link between the spectrum and the corresponding particle. After the NIRM analysis, each particle was individually and carefully transferred in a 200 µl vial for PCR analysis and its origin (animal or vegetal) was predicted by existing discrimination models [START_REF] Baeten | Strategies and methods to detect and quantify mammalian tissues in feedingstuffs. Brussels: European Commission. The near infrared microscopic (NIRM) method: combination of the advantages of optical microscopy and near infrared spectroscopy[END_REF]).

Materials and methods

Samples

Cleaning of the particles. After NIRM analysis, the particles detected as animal are isolated and put in a 200 µl vial. A cleaning procedure is then required to remove unspecific traces of DNA stuck to the particle before the DNA extraction (see the result section). Different reagents degrading the DNA were tested alone or in combination to clean the particles: DNA Erase (MP Biomedicals, Solon, Ohio, USA), RBS (Chemical Products R. Borghgraef s.a., Brussels, Belgium), hydrogen peroxide, acetone and isopropanol. Different incubation times and/or concentrations were tested. The protocols described here are the final ones giving the best results.

Cleaning with DNA Erase. The DNA Erase is classically used in laboratories to clean and decontaminate from DNA traces small equipments (e.g. pipets) or working surfaces. The cleaning is realised by adding 100 µl of DNA Erase to the particle. The vial is gently shaken during 4 min and briefly centrifuged in a centrifuge 2K15 (SIGMA Laborzentrifugen GmbH, Osterode, Germany) before to remove the DNA Erase. The particle is then rinsed with 100 µl of PCR-grade water (ICN Biomedicals Inc., Aurora, OH, USA) and dried in a concentrator 5301 (Eppendorf AG, Hamburg, Germany).

Cleaning with RBS. The RBS is a detergent used in our lab for the cleaning of the mill used to grind the samples. The particle is incubated in 100 µl of RBS diluted 1 vol:19 vol in water for 2 min and briefly centrifuged in a centrifuge 2K15 (SIGMA Laborzentrifugen GmbH, Osterode, Germany) before to remove the RBS. The particle is then rinsed with 100 µl of PCR-grade water (ICN Biomedicals Inc., Aurora, OH, USA) and dried in a concentrator 5301 (Eppendorf AG, Hamburg, Germany).

Cleaning with hydrogen peroxide. The particle incubates in 100 µl of a solution of hydrogen peroxide at 30 % during 2 min. The vial is then briefly centrifuged in a centrifuge 2K15 (SIGMA Laborzentrifugen GmbH, Osterode, Germany) and the hydrogen peroxide is removed. The particle is then rinsed with 100 µl of PCR-grade water (ICN Biomedicals Inc., Aurora, OH, USA) and dried in a concentrator 5301 (Eppendorf AG, Hamburg, Germany).

Cleaning with acetone and isopropanol. A first cleaning is realised by adding 100 µl of acetone (ICN Biomedicals Inc., Aurora, OH, USA) to the particle. The vial is gently shaken by inverting four times and briefly centrifuged in a centrifuge 2K15 (SIGMA Laborzentrifugen GmbH, Osterode, Germany) before to remove the acetone. The cleaning process is managed so that the particle is in contact with the reagent during 2 min. A second cleaning with 100 µl of isopropanol (Sigma Aldrich Chemie GmbH, Steinheim, Germany) is immediately performed using the same way and timing. The particle is then rinsed with 100 µl of PCR-grade water (ICN Biomedicals Inc., Aurora, OH, USA) and dried in a concentrator 5301 (Eppendorf AG, Hamburg, Germany).

DNA extraction from a single particle. Several kits dedicated to DNA extraction from small amount of starting material were identified and tested: Invisorb® Spin Tissue Mini Kit and Invisorb® Spin Forensic Kit (Invitek GmbH, Berlin, Germany) are two kits in which a lysis step with a buffer is followed by a purification step on a spin column. Invisorb® Forensic Kit I (Invitek GmbH, Berlin, Germany) is a kit using the nanoparticles technology to recover the DNA extracted after the lysis step. Two other DNA extraction kits, PicoPure TM DNA Extraction Kit (Arcturus Bioscience, Mountain View, CA, USA) and Direct PCR® (tail) Lysis Reagent for Genotyping using Crude Lysates (Viagen Biotech Inc., Los Angeles, CA, USA), were also tested. These latter 2 kits allow to get a ready-to-use DNA extract directly after the lysis in a buffer.

The protocols presented here are the ones adapted to our specific application according to the constraints developed in the Results and discussion part of the paper from the original guidelines provided by the suppliers.

Invisorb® Spin Tissue Mini Kit is used to extract and purify DNA from small amounts (0.5 -40 mg) of animal tissue. The lysis step of the particle is performed with 400 µl of Lysis Buffer G containing 40 µl of a Proteinase K solution provided with the kit. The lysate is centrifuged at 12000 rpm during 2 min. Fourty µl of RNAse A (10 mg/ml) are added to the supernatant recovered and incubated at room temperature during 5 min. After the addition of 200 µl of Binding Buffer T, the lysate is dropped on a spin column. After 1 minute, the spin column is centrifuged at 12000 rpm for 2 min. The spin column is then cleaned twice by the addition of 550 µl of Wash Buffer followed by a centrifugation at 12000 rpm during 1 min. The remaining ethanol is removed by a centrifugation at 12000 rpm during 2 min.

Finally, the DNA is eluated by a centrifugation at 10000 rpm during 2 min after an incubation during 3 min in 60 µl of Elution buffer warmed at 52 °C.

Invisorb® Spin Forensic Kit is optimised to recover small amounts of DNA from forensic samples (e.g. blood, sperm or saliva stains, stamps, cigarette butts,...). The lysis step of the particle is performed at 56 °C for 2 hours in 500 µl of Lysis Buffer G containing 25 µl of a Proteinase K solution provided with the kit. The lysate is centrifuged at 10000 rpm during 2 min. After the addition of 250 µl of Binding Buffer T to the supernatant recovered, the lysate is dropped on a spin column. After 1 min, the spin column is centrifuged at 10000 rpm for 1 min. The spin column is then cleaned twice by the addition of 800 µl of Wash Buffer followed by a centrifugation at 10000 rpm during 1 min. The remaining ethanol is removed by a centrifugation at 12000 rpm during 3 min. Finally, the DNA is eluated by a centrifugation at 8000 rpm during 1 min after an incubation during 1 min in 50 µl of Elution buffer warmed at 52 °C.

Invisorb® Forensic Kit I is optimised to extract DNA from the same forensic samples as the Invisorb® Forensic Kit but can also be used with bones (the following protocol is directly derived from the one describing DNA isolation from bone or ancient bone material in the manual of the kit). The lysis step of the particle is performed at temperature between 50 °C and 60 °C for 15-20 hours in 20 µl of Lysis Buffer G containing 4 µl of Bone Lysis Enhancer solution. The lysate is centrifuged at 12000 rpm during 2 min and the supernatant is recovered. Two µl of Carrier Suspension buffer made of nanoparticles are then added for an incubation at room temperature for 5 min. The mix is centrifuged at 10000 rpm for 1 second to pellet down the carrier and discard the supernatant. The nanoparticles are then cleaned twice with 20 µl of Wash Buffer. After a centrifugation, the pellet of nanoparticles is dried into a vacuum dessicator at 60 °C to completely evaporate the residual ethanol. The pellet is resuspended in 20 µl of Elution Buffer D and incubated at 60 °C for 5 min to release the DNA from the nanoparticles. A last centrifugation allows the removing of the nanoparticles and the recovering of the DNA.

PicoPure TM DNA Extraction Kit is developed for processing small samples of few as ten cells or several milligrams of tissue. It enables to recover high amounts of DNA ready for use in PCR assays. For optimal enzymatic activity, the lyophilised Proteinase K is freshly reconstituted in 155 µl of Chicken specific primers and probe were designed using the Primer Express version 2.0.0 (Applied Biosystems, Foster City, CA, USA) from the chicken mitochondrial sequence of the Genebank database. The sequences are: for the forward primer 5'-AGC ACT AGC CTT TTA AG-3', for the reverse primer 5'-GGT TTA ATT GGG GCA T-3' and for double dye probe 5'-ACA CCC TCC CCC TTA ATG AC-3'. The thermal protocol is the same as for the cattle and pig probes. The PCR is performed in a total volume of 35 µl. The reaction mix contains 5 µl of DNA extract, 17.5 µl of Real Time PCR Mastermix (Diagenode, Seraing, Belgium), 1.25 µl of each primer at 10 µM (Eurogentec, Seraing, Belgium), 1.25 µl of double dye probe at 10 µM (Eurogentec, Seraing, Belgium) and 8.75 µl of PCR-grade water (ICN Biomedicals Inc., Aurora, OH, USA).

Maize and soja DNA detection were performed according to the protocols described by Kuribara et al. (2002) and [START_REF] Shindo | Validation of real time PCR analyses for line-specific quantitation of genetically modified maize and soybean using new references molecules[END_REF].

Results and discussion

The first tests aiming to evaluate the feasibility to amplify the DNA from a single particle were performed using a very basic protocol: the particle was soaked in a solution of proteinase K at the concentration of 1 µg/µl in the well of the PCR plate and submitted to an incubation at 60 °C (nearby the optimum t° of proteinase K) during 20 min before the inactivation of the enzyme at 95 °C during 10 min. Only after that, the PCR mix is added to perform the PCR thermal program. The results obtained were very encouraging but it rapidly appeared that a separate DNA extraction step would be needed to work with a purified DNA template and avoid PCR troubles such as inhibition or probe degradation leading to unspecific noisy signals (Figure 1).

The origin of these unspecific signals is not totally explained but the action of nucleases degrading the probe is a credible hypothesis: particles were analysed with a mix containing all the PCR reagents with the exception of the primers so that no amplification was possible. Similar signals were observed that substantiated the presence of molecules in the mix able to hydrolyse the double dye probe as measured by the emitted fluorescence. As presented in the Figure 2, the pre-PCR analysis of particles is composed of a cleaning step followed by a DNA extraction. Experimentally, this last step was however developed before the cleaning tests. Indeed it was mandatory to first obtain the best PCR usable DNA before to evaluate the effect of a cleaning process on the PCR results. That is why the developments of the protocols are described in this order.

Development of a DNA extraction protocol from a single particle. Starting from the general guidelines indicated in the manuals of the kits tested, the protocols were adapted to this particular application. As our aim was to obtain enough DNA to perform more than one PCR, the modifications took mainly into account the very limited amount of starting material (1 animal particle between 250 and 500 µm weighting around 0.03 mg - Abbas et al., submitted in 2010) and consequently, the limited amounts of the DNA extracted by reducing the volumes of the different buffers used to extract and purify the DNA. The influence of a second parameter, the incubation lysis time, was also studied. Nevertheless, the modifications of the protocols were sometimes limited by the design of the kits: the Invisorb® Spin Tissue Mini Kit and the Invisorb® Forensic Kit involve the use of a spin column for the purification of the DNA with the requirement to use minimum volumes of buffers (e.g. for the elution, a minimum volume of 50 µl is recommended by the manufacturer of the Invisorb® Spin Tissue Mini Kit).

Ten particles coming from the same three pure MBMs heat treated at 133 °C were analysed with optimised protocols. The performances obtained with the five kits are compared in Table 1.

Beside the cost of the DNA extraction per particle analysed and the time needed for extraction process, different criteria having a direct impact on the reliability of the method were examined: the risks of contamination due to manipulation, the number of available PCR replicates, the number of The numerous centrifugations to perform with the spin column after the lysis step (Invisorb® Spin Tissue Mini Kit and the Invisorb® Forensic Kit) to purify the DNA appeared to us not convenient and a possible source of contamination. Moreover, it forced us to use minimum volumes of buffers probably inducing dilutions and loss of the DNA extracts. The rates of successful PCR amplification were closely dependent of the sample and the species origin of a significant part of the particles cannot be determined. In the case of the Invisorb® Forensic Kit, additional tests were performed: a longer lysis time (e.g. overnight at 42 °C) and the use of a Carrier RNA can be also performed in the case of extremely low amounts of DNA. Longer lysis time did not improve the successful PCR rates neither the number of particles successfully detected (data not shown). Concerning the use of a Carrier RNA, we tested a Poly(A) RNA from Roche Diagnostics as recommended by the manufacturer.

Unfortunately, traces of bovine DNA were detected in the product making its use impossible for our purpose. The use of a home-made vegetal DNA coming from sojabean was also unsuccessfully tested. With the Invisorb® Forensic Kit I, the absence of spin column in the protocol helps us to adapt the volumes of the different buffers to use. Nevertheless, it was not possible to obtain any DNA extract usable for the PCR and the investigations were not carried on with that kit. With the PicoPure TM DNA Extraction Kit (Arcturus Bioscience, Mountain View, CA, USA) and the Direct PCR® (tail) Lysis Reagent for Genotyping using Crude Lysates, manipulations are limited to an incubation of the particle in the lysis buffer to obtain a PCR ready to use DNA extract. Therefore contaminations risks are reduced. As with the Invisorb® Forensic Kit I, it was easier to modify the initial provider's protocol of these two latter kits and to adapt it to our very particular application. The successful PCR rates and the number of particles successfully detected were significantly higher than with the three Invitek kits. Nevertheless the variability of the results with respect to the type of particles (muscle or bone) submitted to the analysis is clearly higher with the PicoPure TM DNA Extraction Kit than with the Direct PCR® Lysis Reagent. Even an increase of the lysis incubation time from 3 hours to 18 hours (overnight) had a limited impact on the DNA extraction efficiency. Finally, the Direct PCR® Lysis Reagent was selected to continue the development of the method as it allowed to perform more successful PCR replicates than with the PicoPure TM DNA Extraction Kit, 5 and 2 PCR respectively, to authenticate the species origin of nearly all the particles. It must be stressed that with the DNA extraction protocol using the Direct PCR® Lysis Reagent no more unspecific signal as described above (Figure 1) was observed.

Development of a particle cleaning protocol. The PCR analysis of a single PAPs particle is only interesting if the DNA comes from the core of this particle and not from any external source especially from possible authorised ingredients present in a mix such as a compound feed. The possible presence of DNA molecules stuck to the surface of the particle was checked by preparing 8 mixes containing ingredients from two different animal species. The DNAs of the 8 mixes extracted from a test portion of 100 mg according to the Promega protocol described by [START_REF] Fumière | Effective PCR detection of animal species in highly processed animal byproducts and compound feeds[END_REF] were firstly analysed by PCR. The composition of the 8 mixes and the Ct values obtained with the different PCR tests are presented in the Table 2.

Twenty MBM particles detected as from animal origin by NIRM analysis (MBM particles) and coming from 5 mixes containing one MBM and one authorised ingredient were analysed by PCR without any cleaning protocol before the extraction with the Direct PCR Lysis Reagent.

The results presented in the Table 3 show that in most of the cases a large majority of the MBM particles analysed gives a positive signal only with target corresponding to their species. These results show that the impact of the authorised ingredients can be limited when the particles are analysed according the protocol described in this paper. Nevertheless, the third mix (92 % of soja + 5 % of chicken MBM + 3 % of milk powder) shows, that in some cases, the percentage of particles giving ambiguous results (positive result with more than one target) can increase and confirm that a cleaning would be useful.

The first cleaning protocol tested used a solution of bleach at 0.5 % in volume as recommended by Toyoda et al. (2004) for the cleaning of sedimented bone particles. Despite modified incubation times and bleach dilutions, the DNA denaturing action of bleach was too efficient and reduced drastically the number of successful PCR. Various other degrading reagents were also tested alone or in combination as described in the material and method section: DNA Erase, RBS, hydrogen peroxide (H 2 O 2 ), acetone and isopropanol. Even if the NIRM analysis is able to distinguish an animal particle from a vegetal one, 20 particles of each ingredient from a mix containing 92 % of Maize, 5 % of a chicken MBM and 3 % of cattle blood were analysed by PCR after being submitted to the different cleaning protocols. The results obtained are presented in the Table 4.

The DNA Erase cleaning protocol gives clearly the more reliable results for the analysis of MBM particles: all the chicken MBM particles remains detectable with the chicken probe whereas no trace of DNA external to the particles was detected. For the other types of particles (blood and maize), the cleaning with the DNA Erase allows high levels of detection, 100 % and 90 %, for the blood particles and the maize particles respectively. With these two last types of particles, the frequencies of detection with the other probes are however higher. A maximum of 30 % of cattle blood particles gave a signal with the chicken probe. It must be stress that MBM particles can be easily distinguished from blood and vegetal ones by a NIRM analysis. So in a global analysis process combining NIRM and PCR analysis, these types of particles would not be analysed. Compared to the other cleaning protocols, the performances obtained with the cleaning with DNA Erase were considered most promising to test it with 3 other mixes. The results of the PCR analysis of 20 particles of MBM and of the vegetal ingredient are presented in the Table 5. With the MBM particles, the DNA Erase cleaning protocol allows an important decreasing of the simultaneous detection with the other animal probes. A notable exception must nevertheless be pointed out in the case of the mix containing blood powder. Seventy-five percent of the chicken MBM particles still give a positive detection with the pig probe. This result must however be attenuated by the figures presented in the Table 6. Looking at the Ct values obtained with chicken MBM particles analysed without and with the DNA Erase cleaning protocol, it clearly appears that the signals obtained with the chicken probe are significantly earlier than with the cattle probe. These results indicate that, considering similar PCR efficiencies with both probes, the number of targets coming from the core of the particle is higher than the number of targets coming from the environment of the particle. This data could be considered in itself as a first evidence of the origin of the MBM particle. Moreover, the cleaning protocol does not have the same influence on both signals: whereas the signals with the chicken probe (the probe for the species the MBM particle comes from) is slightly affected (no change or a delay of less than 2 cycles), the signals obtained with the other probes are strongly delayed (+/-6 cycles) or completely eliminated (second mix tested in the Table 6).

Conclusion

The full application of the European legislation on the animal by-products requires reliable specific methods. Up to now, PCR is the only method able to detect the presence of animal ingredients up to the species level. Unfortunately, the impact of the PCR is limited by the use of authorised animal ingredients that are also sources of animal DNA and can interfere with the results from a legal point of view. In this paper, we demonstrate that the combination of NIRM with real-time PCR is feasible and could be integrated in a global analytical strategy as the one proposed by [START_REF] Fumière | Methods of detection, species identification and quantification of processed animal proteins in feedingstuffs[END_REF] to determine if the presence of animal constituents in feed is conform or not to the European legislation. Using a special buffer, the Direct PCR® Lysis Reagent, a rapid extraction protocol allowing the analysis of the DNA coming from a single PAPs particle with up to five different PCR targets has been developed. A protocol for the cleaning of the particles was also developed to eliminate possible DNA molecules stuck to the MBM particles surface. The protocol using the DNA Erase is the most efficient but its efficiency remains largely dependent on the sample and on the particle tested. No universal conditions could be found to eliminate totally the presence of DNA coming from authorised ingredients keeping in the same time enough PCR usable DNA from the core of the MBM particles.

Nevertheless, due to their costs, the use of authorised ingredients such as egg and milk powder in feed is limited. The results obtained with real world samples containing limited amounts (< 5%) of authorised ingredients are quite encouraging and allow to authenticate the origin of a majority of the particles. Moreover when a signal is observed with more than one probe, the Ct values obtained reflecting the copy numbers of each species target can be used to determine the species origin of the MBM particle with a high level of confidence.

When possible, the results of previous analyses (presence of PAPs particles by classical microscopy, presence of animal species in the sample by PCR and NIRM) could even be used to focus the particle PCR analysis on a limited number of species and to use rationally the available PCR replicates for a reliable species authentication. Nevertheless, its use for routine analysis would be very tedious. The limitation of the method is essentially the number of particles that are manually analysed by NIRM and isolated previously to the analysis of their DNA. The overall process will be assessed soon with more complex samples.

It must be stressed that the combination of the real-time PCR could be also done with the classical microscopy following a similar protocol. Here again the analysis of a representative amount of particles is a critical point. An attractive solution to automate the identification of the suspicious particles and their isolation for the subsequent PCR analysis could be the use of a microscope coupled with a laser microdissection and pressure catapulting system. obtained for the detection of chicken, cattle, pig, sojabean and maize DNA.

Table 3. PCR analysis of 20 MBM particles detected as from animal origin after NIRM analysis.

Percentages of detection with the Chicken probe, the Cattle probe and the Pig probe.

Table 4. PCR analysis of 20 particles of each ingredient from a mix containing 92 % of Maize, 5 % of a chicken MBM and 3 % of cattle blood. Percentages of detection with the Chicken probe, the Cattle probe and the Maize probe.

Table 5. PCR analysis of 20 particles of MBM and of vegetal ingredient from three mixes. Percentages of detection with the Chicken probe, the Cattle probe, the Pig probe, the Maize probe and the Sojabean probe. Table 6. PCR analysis of MBM and blood particles of 2 mixes. Comparison of mean Ct values in number of cycles with the Chicken probe, the Cattle probe and the Pig probe with and without a DNA Erase cleaning protocol of the particles.

  of bovine spongiform encephalopathy (BSE), the use of processed animal proteins (PAPs) including meat and bone meal as feed ingredients for farmed animals is drastically controlled within the European Union through several regulations (Regulation (EC) 999/2001, Regulation (EC) 1774/2002 and Regulation (EC) 1234/2003). Presently, classical optical microscopy is the only official method for the detection of PAPs in compound feed in the European Union [Commission Regulation

.

  The raw materials used in this study are all representative of the real world feed ingredients: the meat and bone meals were obtained from plants members of the European Fat Processors and Renderers Association (EFPRA ) and were produced as required by European treatment at a temperature of ≥ 133 °C and 3 bars for 20 min). Concerning the authorised ingredients, the blood powders were also provided by the EFPRA. The egg powder came from a Belgian feed producer. The milk powders were obtained from the dairy industry.

  Reconstitution Buffer. Thirty µl of the prepared buffer are added to the particle in a 200 µl vial and are incubated at 65 °C for 3 hours. Proteinase K is then inactivated at 95 °C for 10 min.Direct PCR® (tail) Lysis Reagent for Genotyping using Crude Lysates is dedicated to mouse tail cells lysis and gives a DNA extract ready-to-use for the PCR. The extraction buffer is freshly prepared by adding Proteinase K (Invitrogen, Carlsbad, CA, USA) at a concentration of 0.4 µg/µl to the Direct PCR® Lysis Reagent. Seven µl of the prepared buffer are added to the particle in a 200 µl vial and are incubated at 85 °C for 45 min. The extract with the particle is then centrifuged. Six µl of supernatant are recovered and are diluted with 24 µl of PCR-grade water (ICN Biomedicals Inc., Aurora, OH, USA). Real-time PCR analysis. Cattle and pig primers and probes as well as real-time PCR conditions are the same as the ones presented by Fumière et al. (2006). The PCR amplifications are performed on a LightCycler 480 (Roche Diagnostics GmbH, Mannheim, Germany) without any modification of the thermal program optimised on a GeneAmp 5700 real-time PCR device (Applied Biosystems, Foster City, CA, USA) composed of an initial decontamination step by uracyl DNA glycosylase at 50 °C for 2 min, a DNA denaturation combined to the activation of the Taq DNA polymerase performed at 95 °C for 10 min and 50 cycles including a melting step at 95 °C for 15 s, an annealing/elongation step at 50 °C for 60 s. The PCR is performed in a total volume of 35 µl. The reaction mix contains 5 µl of DNA extract, 17.5 µl of Real Time PCR Mastermix (Diagenode, Seraing, Belgium), 1.25 µl of each primer at 3 µM (Eurogentec, Seraing, Belgium), 1.25 µl of double dye probe at 10 µM (Eurogentec, Seraing, Belgium) and 8.75 µl of PCR-grade water (ICN Biomedicals Inc., Aurora, OH, USA).

  successful PCR and the number of particles successfully detected by at least one of the PCR replicates.

  Nakajo M, Kawachi H, Matsui T, Yano A. 2004. PCR Detection of Bovine Mitochondrial DNA Derived from Meat and Bone Meal in Feed. Journal of Food Protection 67(12):2829-2832. von Holst C, Baeten V, Boix A, Slowikowski B, Fernández Pierna JA, Tirendi S, Dardenne P. 2008.Transferability study of a near-infrared microscopic method for the detection of banned and bone meal in feedingstuffs.

Figure 1 .

 1 Figure 1. Real time PCR signals obtained with a pig MBM particle using the cattle and pig probes. The

Figure 2 .Figure 1 .Figure 2 .Figure 2 .

 2122 Figure 2. Overall analytical process of PAPs particles combining NIRM and Real Time PCR analysis.

  %) + Chicken MBM #2 (5 %) + Milk powder #2 (3 %) Maize (92 %) + Cattle MBM (5 %) + Egg powder (3 %) Maize (92 %) + Pig MBM (5 %) + Egg powder (3 %) Maize (92 %) + Chicken MBM #1 (5 %) + Cattle blood (3 %) Maize (92 %) + Chicken MBM #3 (5 %) + Pig blood (3 %) Chicken MBM #1 (50 %) + Milk powder #2

Table 1 .

 1 Comparative performances of Invisorb® Spin Tissue Mini Kit, Invisorb® Forensic Kit, Invisorb® Forensic Kit I, PicoPure TM DNA Extraction Kit and Direct PCR® (tail) Lysis Reagent for Genotyping using

	Crude Lysates.

Table 2 .

 2 PCR analysis of DNAs from 8 mixes containing 2 animal ingredients: Ct values (n = 2 extracts)

Table 1 .

 1 Comparative performances of Invisorb® Spin Tissue Mini Kit, Invisorb® Forensic Kit, Invisorb® Forensic Kit I, PicoPure TM DNA Extraction Kit and Direct PCR® (tail) Lysis Reagent for Genotyping using Crude Lysates.

		Invisorb® Spin	Invisorb® Forensic	Invisorb® Forensic	PicoPure	TM DNA	Direct PCR®
		Tissue Mini Kit	Kit	Kit I	Extraction kit	Lysis Reagent
	Cost /DNA extraction	2.58 €	3.17 €	0.80 €	2.56 €	0.01 €
	Extraction time	1.5 hour	2.5 hours	15 -20 hours	3 hours	45 minutes
	Contamination risk	High	High	Medium	Low	Low
	Number of PCR	5	4	2	2		5
	Successful PCR rates	10-65 %	70-85 %	0 %	75-100 %	96-100 %
	Particles successfully detected	10-70 %	70-90 %	0 %	80-100 %	100 %

Table 2 .

 2 PCR analysis of DNAs from 8 mixes containing 2 animal ingredients: Ct values (n = 2 extracts) obtained for the detection of chicken, cattle, pig, sojabean and maize DNA.

	Mixes (% in weight of ingredients)	Ct Chicken	Ct Cattle	Ct pig	Ct Sojabean	Ct Maize
	Sojabean (92 %) + Chicken MBM #1 (5 %) + Milk powder #1 (3 %) Sojabean (92 %) + Chicken MBM #1 (5 %) + Milk powder #2 (3 %) Sojabean (92 %) + Chicken MBM #2 (5 %) + Milk powder #2 (3 %) Maize (92 %) + Cattle MBM (5 %) + Egg powder (3 %) Maize (92 %) + Pig MBM (5 %) + Egg powder (3 %) Maize (92 %) + Chicken MBM #1 (5 %) + Cattle blood (3 %) Maize (92 %) + Chicken MBM #3 (5 %) + Pig blood (3 %) Chicken MBM #1 (50 %) + Milk powder #2 (50 %)	25.90 24.90 27.52 29.74 29.73 23.09 23.03 19.08	38.47 32.09 32.49 25.41 NT a 34.79 NT a 25.98	NT a NT a NT a a NT 32.56 NT a 25.79 NT a	28.90 28.05 28.50 a NT NT a NT a NT a a NT	NT a NT a NT a 29.16 28.81 30.37 29.08 NT a
	a NT : not tested

Table 3 .

 3 PCR analysis of 20 MBM particles detected as from animal origin after NIRM analysis. Percentages of detection with the Chicken probe, the Cattle probe and the Pig probe.

Table 4 .

 4 PCR analysis of 20 particles of each ingredient from a mix containing 92 % of Maize, 5 % of a chicken MBM and 3 % of cattle blood. Percentages of detection with the Chicken probe, the Cattle probe and the Maize probe.

	Type of	Without cleaning protocol	With RBS cleaning protocol	With DNA Erase cleaning	With H2O2 cleaning protocol	With acetone-isopropanol
	particle								protocol					cleaning protocol
		Chicken	Cattle	Maize	Chicken	Cattle	Maize	Chicken	Cattle	Maize	Chicken	Cattle	Maize	Chicken	Cattle	Maize
	Chicken MBM	100 %	60 %	90 %	100 %	5 %	5 %	100 %	0 %	0 %	100 %	5 %	40 %	100 %	40 %	60 %
	Cattle blood	100 %	100 %	100 %	90 %	100 %	45 %	30 %	100 %	15 %	35 %	95 %	60 %	100 %	100 %	95 %
	Maize	100 %	45 %	95 %	15 %	0 %	95 %	5 %	10 %	90 %	100 %	20 %	95 %	95 %	10 %	90 %

Table 5 .

 5 PCR analysis of 20 particles of MBM and of vegetal ingredient from three mixes. Percentages of detection with the Chicken probe, the Cattle probe, the Pig probe, the Maize probe and the Sojabean probe.

				Without cleaning protocol			With DNA Erase cleaning protocol
	Mixes (% in weight of ingredients)	Particles	Chicken	Cattle	Pig	Maize	Sojabean	Chicken	Cattle	Pig	Maize	Sojabean
		analysed
	Chicken MBM #1 (50 %) + Milk powder #2 (50 %)	Chicken MBM	100 %	70 %	NT a	NT a	NT a	95 %	5 %	NT a	NT a	NT a
	Soja (92 %) + Chicken MBM #1 (5 %)	Chicken MBM	95 %	20 %	NT a	NT a	85 %	100 %	0 %	NT a	NT a	0 %
	+ Milk powder #2 (3 %)	Sojabean	80 %	5 %	NT a	NT a	95 %	5 %	0 %	NT a	NT a	95 %
	Maize (92 %) + Chicken MBM #3 (5 %)	Chicken MBM	100 %	NT a	100 %	95 %	NT a	100 %	NT a	75 %	5 %	NT a
	+ Pig blood (3 %)	Maize	100 %	NT a	100 %	95 %	NT a	10 %	NT a	15 %	85 %	NT a
	F o r
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Table 6 .

 6 PCR analysis of MBM and blood particles of 2 mixes. Comparison of mean Ct values in number of cycles with the Chicken probe, the Cattle probe and the Pig probe with and without a DNA Erase cleaning protocol of the particles.

	Without cleaning protocol	With DNA Erase cleaning protocol

b ND : not detected

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants

Acknowledgements

The research presented in this paper was supported by the Belgian Federal Public Service for Health, Food Chain Safety and Environment (Contract Nr S-6168) and the European Community within the framework of the SAFEED-PAP project (FOOD-CT-2006-036221) of the 6 th EC Framework Program.


A Shevchenko
email: andriy.shevchenko@tkk.fi

M Kaivola

Electromagnetic force density in dissipative isotropic media

Keywords: numbers: 42.25.Bs, 03.50.De, 75.80.+q, 77.65.-j

We derive an expression for the macroscopic force density that a narrowband electromagnetic field imposes on a dissipative isotropic medium. The result is obtained by averaging the microscopic form for Lorentz force density. The derived expression allows us to calculate realistic electromagnetic forces in a wide range of materials that are described by complex-valued electric permittivity and magnetic permeability. The three-dimensional energy-momentum tensor in our expression reduces for lossless media to the so-called Helmholtz tensor that has not been contradicted in any experiment so far. The momentum density of the field does not coincide with any well-known expression, but for non-magnetic materials it matches the Abraham expression.

After the first well-known theoretical model for static electromagnetic forces in ponderable media, introduced by H. von Helmholtz [1], several other descriptions have been proposed to account for the action of both static and oscillating electromagnetic fields on a medium. The two most famous of them were proposed by H. Minkowski [2] and M. Abraham [3]. Their models give different predictions for the value of the force density in certain particular cases, which has created a century-long scientific debate on the topic [START_REF] De Groot | Foundations of Electrodynamics[END_REF]- [15]. It is well known that both these models ignore the existing electroand magnetostrictive forces [7,16]. A model by Einstein and Laub [17] was intended to include these forces, but it turned out to be in disagreement with experiments (see, e.g., Ref. [7] and references therein). A reliable way to also include electro-and magnetostrictive forces can be based on the so-called Helmholtz energy-momentum tensor. Previously we have shown that the Helmholtz tensor can be derived in a relatively simple way starting with the microscopic field-matter interaction picture [18]. However, this tensor and the other models discussed so far are not applicable to dissipative media and, therefore, they do not allow calculating electromagnetic forces in many realistic materials.

The aim of this work is to derive a general expression for the macroscopic force density imposed on a dissipative and electrically conducting inhomogeneous isotropic medium by a narrow-band electromagnetic field. We do this by spatially averaging the microscopic Lorentz force density (that assumes a unique and commonly accepted form, in contrast to the macroscopic electromagnetic force, energy, and momentum densities). The result is written in terms of the three-dimensional electromagnetic energy-momentum tensor and the momentum density of the field, which explicitly depend on the complex relative permittivity and permeability of the material. In the limit of lossless medium, the obtained energy-momentum tensor becomes the Helmholtz tensor. We note that the force density expressed in terms of complex-valued quantities has been introduced previously (see, e.g., [19] and [20]), but not in connection with the Helmholtz tensor. The expression obtained by us for the field momentum density does not coincide with any of the well-known expressions, but for lossless dielectrics it converges to an expression obtained by us previously [18], and if the material is not magnetic, it matches the Abraham expression.

The derivation of the force density in dissipative media is similar to that introduced by us in Ref. [18] for lossless materials. Since essential new details appear along the derivation, we present it here in its entirety. Within classical electrodynamics, the interaction of an electromagnetic field with matter is unambiguously described by the microscopic Maxwell's equations,

0 ∇ • e = ξ , (1)
∇ • b = 0 , (2)
-∇ × e = ḃ , (3)
1 µ 0 ∇ × b = 0 ė + j , (4)
and the microscopic Lorentz force density

f mic = ξe + j × b. (5)
Here e and b are the microscopic electric and magnetic fields in the medium and ė and ḃ are their time derivatives. The electric charge and current densities ξ and j, respectively, can be written in terms of point electric charges q i as

ξ = i q i δ(r -r i ) , (6)
j = i q i ṙi δ(r -r i ) , (7)
where δ(rr i ) is the Dirac delta function centered at the coordinate r i of charge q i . The bound electric charges in a medium can be combined into localized groups that belong to individual atoms (or molecules). For each such group one can expand the charge and current densities into Taylor series around the group's center r l and then truncate the series to include contributions from electric and magnetic dipole moments d only [21]. Within this approximation the atoms are treated as point dipoles and the bound charge and current densities in the medium are

ξ b = - l d (b) l • ∇δ(r -r l ) , (8)
j b = l ḋ(b) l δ(r -r l ) + l ∇ × m (b) l δ(r -r l ) , (9)
The moments d contain the contributions of all bound electric charges of atom l. The second term in the expression for j b originates from the electric current loops due to rotational motion of the charges in the atoms [21].

If the medium contains conduction charges, an oscillating external field will put these charges into oscillating motion. For an electrically neutral medium, the oscillation of the conduction electrons is accompanied by an out-of-phase oscillation of the residual positive charges of the atoms, which can as well be treated as conduction charges. The oscillation of such positive and negative conduction charges produces their own dipole moments d (c) l and m (c) l . If the oscillating electron does not move far from a certain positively charged atom l (which is the case for high-frequency fields), the conduction charge and current densities can be written in the same form as ξ b and j b ,

i.e., ξ c = -l d (c) l • ∇δ(r -r l ) and j c = l ḋ(c) l δ(r -r l ) + l ∇ × m (c)
l δ(rr l ). In order to take into account possible dissipation of electromagnetic energy in the medium, the moments d

(b) l , m (b) l , d (c)
l , and m (c) l are considered to be complex vectors. For example, when calculating an expression for the frequency-dependent dielectric constant , the contribution of each bound charge k to an atomic dipole moment turns out to be proportional to e -iωt /(ω 2 k -ω 2 -iωγ k ), while the dipole moment originating from each conduction charge is proportional to e -iωt /(-ω 2 -iωγ j ) [START_REF] Jackson | Classical Electrodynamics[END_REF]. Here ω k and γ k are the resonance frequency and damping constant of the electron, respectively. For conduction electrons, ω j are equal to zero but γ j are not because of ohmic losses. From this point of view the division of electric charges into bound and conduction charges is conditional. The total charge and current densities, ξ = ξ b + ξ c and j = j b + j c , are now given by the complex quantities

ξ = - l d l • ∇δ(r -r l ) , (10)
j = l ḋl δ(r -r l ) + l ∇ × m l δ(r -r l ), (11)
where

d l = d (b) l + d (c) l and m l = m (b) l + m (c) l
are, respectively, the total electric and magnetic dipole moments per atom.

When in addition to ξ and j the fields e and b are replaced with complex oscillating functions, the form of the microscopic Maxwell's equations and Eqs. ( 10) and ( 11) are preserved, but the Lorentz force density must be rewritten. In terms of complex quantities, one can write the microscopic Lorentz force density averaged over one oscillation period as

fmic = 1 2 Re{f mic } = 1 2 Re{ξ * e + j * × b}, (12)
where Re denotes the real part, and the asterisk stands for complex conjugation. Here the functions e and b are assumed to oscillate harmonically with slowly varying complex amplitudes, in which case ∆ω ω, with ∆ω and ω denoting the field's bandwidth and carrier frequency, respectively. In this case, the changes in the field amplitudes during a single oscillation period of the field are negligibly small. In Eq. ( 12) the fast oscillation at the carrier frequency is averaged out and fmic remains a slowly varying function of time.

Using Eqs. (10) and (11), the function f mic is written as

f mic = l -d * l • ∇δ(r -r l )e(r) + ḋ * l δ(r -r l ) × b(r) ( 1 3 )
+ (∇ × m * l δ(r -r l )) × b(r)
. By integrating this function over a representative elementary volume δV and dividing the result by this volume, one can find the macroscopic force density function f ≡ f mic , where the angle brackets denote spatial averaging. The function f is given by [18]

f = 1 δV l in δV ∇(d * l • e(r l )) + ∇(m * l • b(r l )) + d dt (d * l × b(r l )) , (14)
where the summation is performed over atoms belonging to δV . Equation (12) implies that the macroscopic force density averaged over one oscillation period of the field is

f = fmic = 1 2 Re{f }. (15)
Note that the order of temporal and spatial averaging does not affect the result. For charges within δV and for an arbitrary coordinate r in δV , one can write e(r) = e ext (r) + e int (r) and b(r) = b ext (r) + b int (r), where e int (r) and b int (r) are the fields produced by the charges in δV and the fields e ext (r) and b ext (r) are created by all sources that are external to δV . The external fields are independent of the charges and their coordinates in δV , while the "internal" fields are strongly inhomogeneous around the internal charges. Substituting these expansions into Eq. ( 14) and using Eq. ( 15), one obtains

f = 1 2 Re 1 δV l in δV ∇(d * l • e ext (r l )) + ∇(m * l • b ext (r l )) (16)
+ d dt (d * l × b ext (r l )) + fint ,
where the second term, fint , has the same form as the first one, but with the external fields replaced with the internal ones [18]. The term fint is equal to 1/δV times the total force imposed on all electric charges in δV by the spiking fields e int (r) and b int (r) produced by the charges themselves. According to Newton's third law, this force is equal to zero and thus fint = 0. Removing fint from Eq. ( 16) and assuming that in the small volume δV the total electric and magnetic dipole moments of each atom are equal to constant vectors d and m, one obtains an expression for f = Re{f }/2, where f is given by

f = 1 δV k=x,y,z d * k ∇ l in δV e ext,k (r l ) + m * k ∇ l in δV b ext,k (r l ) (17)
+ d dt d * × 1 δV l in δV b ext (r l ) .
Here the quantities with subindex k are the Cartesian components of the corresponding vector quantities.

The atoms can to the first order be considered to be distributed uniformly within the small δV . Since in δV the fields e ext (r) and b ext (r) are smooth charge-independent functions, one can replace the averaging of the fields over the coordinates of the atoms in Eq. ( 17) with volume averaging, which results in f = 1 2 Re k=x,y,z

P * k ∇E ext,k + M * k ∇B ext,k + d dt (P * × B ext ) , (18)
where E ext ≡ e ext (r) and B ext ≡ b ext (r) . The electric polarization P and magnetization M are given by

P ≡ N δV d/δV = P b + P c , (19)
M ≡ N δV m/δV = M b + M c , (20)
with N δV denoting the number of atoms in δV . In the above equations, P b and M b originate from bound charges and P c and M c from conduction charges. We note that Eq. ( 18) holds also for anisotropic and nonlinear materials. In what follows, however, it is assumed that the medium is isotropic and linear, so that the polarization components are described by P b = 0 ( b -1)E and P c = iEσ/( 0 ω), where b is the complex dielectric constant due to bound charges and σ the complex electric conductivity; i is the imaginary unity. Usually the polarization is written as P = 0 ( -1)E, where = b + iσ/( 0 ω) is the overall dielectric constant [START_REF] Jackson | Classical Electrodynamics[END_REF]. Similarly, the magnetization can be written as M = (µ -1)H, with µ being the overall complex relative permeability of the medium. In this case, the medium is treated as an "absorptive medium without dispersion" discussed, e.g., in [START_REF] Yu | [END_REF]. The above equations are valid, if the size δV 1/3 is much larger than the distance l e over which a conduction electron moves during one oscillation period of the field. This condition limits the theory to high-frequency fields. However, even for copper in an electric field with an amplitude of 1 kV/cm, the distance l e is on the order of 1 µm or less already at a frequency of 1 GHz.

The averaged external electric field can be found from E ext = Ee int [18]. For an arbitrary charge distribution in δV , the field e int is equal to -D δV /(3 0 δV ), where D δV = PδV is the total dipole moment of the medium within δV (see also sect. 2.13 in [START_REF] Lorrain | Electromagnetic Fields and Waves[END_REF]). Therefore, the external field is

E ext = E + P/(3 0 ) = E( + 2)/3, (21)
which has the same form as the traditional local field with the Lorentz correction. Note, however, that the polarization P also contains the contribution from the electric dipole moments of the conduction charges. Similarly, the external magnetic field is calculated as B ext = Bb int . The field b int created by microscopic electric current loops in δV is given by b int = 2µ 0 M/3 (see Eq. (9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)[START_REF] Jackson | Classical Electrodynamics[END_REF] in [START_REF] Lorrain | Electromagnetic Fields and Waves[END_REF]), which yields

B ext = B -2µ 0 M/3 = µ 0 H(µ + 2)/3, (22)
where the relation B = µ 0 µH has been used. Substituting the calculated E ext and B ext into Eq. ( 18) and expressing P and M through E and H, one obtains the force density

f = 1 2 Re 0 ( * -1) 3 [|E| 2 ∇ + ( + 2) k=x,y,z E * k ∇E k ] ( 2 3 )
+ µ 0 (µ * -1) 3 [|H| 2 ∇µ + (µ + 2) k=x,y,z H * k ∇H k ] + ∂ ∂t 0 µ 0 ( * -1)(µ + 2) 3 E * × H
that depends on E and H and on the complex parameters and µ.

The macroscopic fields E and H satisfy the macroscopic Maxwell's equations

∇ • D = 0 , (24)
∇ • B = 0 , (25)
-∇ × E = Ḃ , (26)
∇ × H = Ḋ , ( 27
)
where the electric charge and current densities due to conduction charges are absorbed in D = 0 E. With the help of these equations and using standard differential identities involving vectors and dyads, the obtained equation for f can be rewritten in a general form as

f = -∇ • T - d Ḡ dt , (28)
where the energy-momentum tensor T and the momentum density Ḡ are given by

T = 0 2 (-Re{ * E * E} + 2 + 2Re{ } -| | 2 6 |E| 2 Î) ( 2 9 ) + µ 0 2 (-Re{µ * H * H} + 2 + 2Re{µ} -|µ| 2 6 |H| 2 Î) , Ḡ = 1 6c 2 Re{(2 + 2 * µ -2 * + µ)E * × H}. ( 30
)
Here, E * E and H * H denote the outer products of the vectors, E and H are the complex amplitudes of the fields, Î denotes the unit tensor, and c is the speed of light in vacuum. form of Eq. ( 28) is not only physically insightful, but also very convenient in view of calculations, since for stationary fields the electromagnetic force on an object in a medium can be calculated simply by integrating n • T over the surface inclosing the object instead of integrating f over the volume of the object; n is the unit vector normal to the surface. We emphasize here that not only the field, but also the surrounding medium can impose a force on the object, which has to be taken into account in each particular case. It can be seen that if the medium is lossless, so that both and µ are real, Eqs. ( 29) and (30) converge to Eqs. ( 35) and (36) in [18]. The tensor in Eq. (35) of Ref. [18] is the Helmholtz tensor (see, e.g., Ref. [7]), and the field momentum density of Ref. [18] still has to be experimentally verified. At high frequencies, µ is equal to 1 for most materials, and the momentum density Ḡ becomes

Ḡµ=1 = 1 2 Re 1 c 2 E * × H (31)
independently of . Equation ( 31) is seen to match the Abraham's expression for G averaged over one oscillation period of the field within the slowly varying envelope approximation. Since Eqs. ( 28)-(30) are valid for dissipative media, they can be applied also to metamaterials that can have µ = 1 even at optical frequencies. The quantity d Ḡ/dt is equal to zero for constant-amplitude harmonic fields, and in the case of a slowly varying amplitude the contribution of this quantity to f is small compared to the contribution of ∇ • T. We still believe that in the future it will be possible to verify Eq. (30) by using pulsed or intensity-modulated light in a magnetic metamaterial liquid.

It is worth mentioning that if one deals with dielectric liquids at hydrodynamic equilibrium with the field, the negative of the gradient of the hydrostatic excess pressure can be found to compensate for the electromagnetic strictive force density at each point in the liquid. Inclusion of this hydrostatic force density often leads to a result that can as well be obtained by using Minkowksi's or Abraham's tensors [7]. However, in a non-equilibrium case, e.g., immediately after introducing the field in the medium, the pressure evolves and does not compensate for the electromagnetic strictive force.

As a simple example of new phenomena that can be revealed by applying Eqs. ( 28)-(30), let us consider a monochromatic electromagnetic plane wave (d Ḡ/dt = 0) that propagates along z-axis in a dissipative homogeneous medium and has an attenuation length of z a . At z = 0, the complex amplitudes of the field are E 0 and H 0 . The timeaveraged force imposed by the field on the part of the medium that is confined between the planes z = 0 and z = z is

F = - z=0 n 1 • Tdxdy - z=z n 2 • Tdxdy, ( 32
)
where the surface integrals are obtained by applying Gauss' integration law to the original volume integral; n 1 and n 2 are the normals to the integration surfaces directed outward from the section of the medium in question. If E and H are everywhere perpendicular to z and z z a , Eq. ( 32) becomes

F = ẑ 2 + 2Re{ } -| | 2 6 z=0 |E 0 | 2 dxdy (33) + 2 + 2Re{µ} -|µ| 2 6 z=0 |H 0 | 2 dxdy ,
where Eq. (29) has been used; ẑ is the unit vector along z. The integrals in Eq. ( 33) are proportional to the intensity I 0 of the plane wave at z = 0, since

|E 0 | 2 = 2I 0 µ 0 / 0 |µ|/Re{ √ µ} and |H 0 | 2 = 2I 0 0 /µ 0 | |/Re{ √ µ}.
In terms of I 0 , the force F per unit cross-sectional area of the wave, electromagnetic pressure p, reads

p = ẑI 0 cRe{ √ } 2 + 2Re{ } -| | 2 + 3| | 6 , (34)
where the medium is assumed to be non-magnetic so that µ ≡ 1.

It can be seen that for sufficiently large | |, the pressure p becomes directed opposite to the propagation direction of the beam. Thus, if one would consider the matter-field interaction purely in terms of momentum exchange between photons in the wave and the medium, the obtained average momentum per photon would be negative. Obviously, photons can not only share their momenta with the medium, but also impose gradient forces on it. The pressure p in Eq. (34) contains both a positive component due to the momentum exchange (the usual radiation-pressure component) and a negative component due to the intensity gradient originating from the attenuation of the wave. For large | |, the second contribution can be stronger than the first one. Single-crystal silicon can be considered as an example. For a wavelength of, say, 532 nm (frequencydoubled Nd:YAG laser), we have ≈ 20.3 + i1.0, Re{ √ } ≈ 4.5, and | | ≈ 20.3 [START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF]. Substituting these values into Eq. (34) yields p = -11.4ẑI 0 /c. This pressure points against ẑ and is 11.4 times higher than a pressure that would result from full absorption of the same wave by an object in vacuum. Another example is salt water at 1-THz field frequency. For a mass fraction of 0.25 of NaCl in the solution, the solution is characterized by ≈ 4.9 + i3.1, Re{ √ } ≈ 2.3, and | | ≈ 5.8 [START_REF] Jepsen | [END_REF]. The resulting pressure is p = -0.3ẑI 0 /c that is also opposite to z.

Suppose now that an electromagnetic beam instead of a plane wave is interacting with the previous medium of salt water. If the beam diameter is large compared to z a , then switching on the beam will lead to a motion of the liquid toward the beam source. In time, the hydrostatic pressure will be redistributed to compensate for the gradient force density due to the field. However, the positive radiation-pressure force component will remain uncompensated and, hence, the liquid will eventually flow along the beam axis in the positive direction of z. In principle, the same motion of the hydrodynamically equilibrated liquid can be obtained also within Minkowski's and Abraham's theories written in the domain of complex functions, but what cannot be obtained using these theories is the intensity dependent hydrostatic pressure. This is explained by the fact that Minkowski's and Abraham's force densities implicitly contain the action of the medium on itself in the form of an Archimedes-like force density that at equilibrium compensates for the compressive action of the field.

In conclusion, we have derived the equation for the force density imposed by a narrow-band electromagnetic field on a medium that is characterized by complex-valued electric permittivity and magnetic permeability. Equation (28) expresses the force density in the form that allows one to conveniently calculate the overall force imposed by the field on an arbitrary part of the medium by evaluating a surface instead of a volume integral, if the field is stationary. For narrow-band fields, Eq. ( 30) can be used to evaluate the momentum density of the field averaged over its single oscillation pediod. This quantity depends on and µ, but it takes on the Abraham's form if the material is not magnetic, which is the case for essentially all materials in high-frequency fields. However, Eqs. ( 28)-( 30) can be applied also to high-frequency magnetic metamaterials, which can reveal unexpected phenomena associated with electromagnetic forces. We finally note that, formally, the theory should be applicable also to optical gain materials, since and µ can have arbitrary complex values.

Acknowledgments

We acknowledge financial support from the Academy of Finland and thank Prof. B. J. Hoenders for useful discussions on the subject.


Fabien Flori
email: fabien.flori@ufe.edu.eg

Catherine Giacomoni

Espaces de Hardy et compacité par compensation Un r ésultat en dimension 2

Nous donnons un résultat faisant apparaître un phénomène de léger gain en régularité dans des quantités non linéaires par rapport à leur régularité apparente.

On définit l'espace de Hardy généralisé introduit par E. Stein et G. Weiss [3] que nous noterons H 1

H 1 (R 2 ) = f ∈ L 1 (R 2 )/ sup η≥0 |h η f | ∈ L 1 (R 2 ) où h η (x) = 1 η 2 h x η ≥ 0 appartient à C ∞ 0 (R 2 ) et vérifie : supph η (x) ⊂ B x η , R 2 h η (x)dx = 1.
On travaille dans ce qui suit sur des espaces de Hardy définis sur des domaines bornés et en particulier on pose :

H 1 z (Ω) = f ∈ L 1 (Ω)/f z ∈ H 1 (R 2 )
où f z est le prolongement de f par 0 dans R 2 . Notons que toute fonction f dans

H 1 z (Ω) vérifie Ω f dx = 0.
Une norme sur cet espace est donnée par

||f || H 1 z (Ω) = ||f z || H 1 z (R 2 ) . On définit l'espace de Hardy-Sobolev W 1 H 1 z (Ω) comme l'ensemble des fonctions φ de H 1 z (Ω) dont les dérivées ∂Φ ∂x i appartiennent aussi à H 1 z (Ω). On note W 1 0 H 1 z (Ω) la fermeture de C ∞ 0 (Ω) dans W 1 H 1 z (Ω). On munit W 1 0 H 1 z (Ω) de la norme : ||φ|| W 1 0 H 1 z (Ω) = ||div φ|| H 1 z (Ω) + ||curl φ|| H 1 z (Ω) . On montre le résultat suivant : Théorème 1 -Soit Ω un ouvert borné de R 2 et v = (v 1 , v 2 ) ∈ L ∞ (0, T ; L 2 (Ω))∩ L 2 (0, T ; H 1 0 (Ω)) alors v i v j ∈ L 2 (0, T ; W 1 H 1 z (Ω)) ∀i, j ∈ {1, 2}.
Remarque 1 -On retrouve ainsi un résultat classique en mécanique des fluides lorsque le fluide est incompressible et que div v = 0. En effet, si

v i v j ∈ L 2 (0, T ; W 1 H 1 z (Ω)) alors le terme d'advection (v.∇)v appartient à L 2 (0, T ; H 1 z (Ω)) avec H 1 z (Ω) ⊂ H -1 (Ω) dans le cas de la dimension 2 [2]. Lemme 1 -Si v ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)) alors : curl (v j v) ∈ L 2 (0, T ; H 1 z (Ω)) ∀j ∈ {1, 2}. Preuve du lemme 1 -Dans ce qui suit, comme v ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)), on prolonge v et ∂v j ∂x i par 0 sur R 2 -Ω. Pour tout x ∈ Ω, on note B x
η la boule de R 2 de centre x et de rayon η. Sur chaque boule B x η , on peut toujours décomposer v de façon unique sous la forme :

v = ∇p x η + Curl q x η avec Curlq = ( ∂q ∂x 2 , -∂q ∂x 1 ), p x η ∈ H 1 (B x η ) et q x η ∈ H 1 0 (B x η ). Bien sûr, si B x η et B x η sont deux boules telles que B ∩ = B x η ∩B x η = ∅,
sur chacune de ces boules, v se décompose de façon unique et il est clair que l'on n'a pas nécessairement Curl q x η = Curl q x η sur B ∩ . Toutefois, on a curl u = curl Curl q x η = curl Curl q x η sur B ∩ . Cette propriété sera utile par la suite.

Sur chaque boule B x

η on a la décomposition :

curl (v j v) = curl (v j Curl q x η ) -∇p x η Curl v j .
On estime séparément chacun des termes du second membre en utilisant les propriétés de la décomposition v = ∇p x η + Curl q x η sur chaque boule. On obtient une borne de curl (v j v) dans L 2 (0, T ; H 1 z (Ω)) indépendante de η. 1ère étape -Estimation de ∇p x η Curl v j . On adapte le théorème div -curl [1] au terme ∇p x η Curl v j .

∇p x η Curl v j * h η (x) = B x η (∇p x η Curl v j )(y) 1 η 2 h x -y η dy
Comme h η (x) est à support compact sur B x η , il vient :

∇p x η Curl v j * h η (x) = B x η ∇p x η (y) v j -v j η (y) 1 η 2 Curl h x -y η dy où v j = B x η -v j dy et ∇p x η = v -Curl q x η . Si on pose C 0 = π||Curl h|| ∞ et mes(B x η ) = πη 2 on obtient : ∇p x η Curl v j * h η (x) ≤ C 0 B x η - v j (y) -v j η |v(y)|dy+C 0 B x η -|v j (y) -v j | Curl q x η (y) η dy
Avec l'inégalité de Holder, en posant 1 β + 1 β = 1, on a :

∇p x η Curl v j * h η (x) ≤ C 1 B x η - v j (y) -v j η β dy 1 β B x η -|v| β dy 1 β +C 1 B x η -|v j (y) -v j | β dy 1 β B x η - Curl q x η η β dy 1 β
On applique ensuite l'inégalité de Sobolev-Poincaré avec

1 α -1 2 = 1 -1 β = 1 β . Il vient d'une part : B x η - v j (y) -v j η β dy 1 β ≤ C 2 B x η -|∇v j | α dy 1 α et d'autre part, comme B x η -Curl q x η dy = 0, Curl q x η .n = 0 sur ∂B x η et curl v = curl Curl q x
η sur B x η , il est clair avec l'inégalité de Sobolev-Poincaré que :

B x η - Curl q x η η β dy 1 β ≤ C 3 B x η -|curl v| α dy 1 α Ainsi : sup η>0 ∇p x η Curl v j * h η (x) ≤ C 4 sup η>0 B x η -|∇v j | α dy 1 α sup η>0 B x η -|v| β dy 1 β +C 5 B x η -|v j (y) -v j | β dy 1 β sup η>0 B x η -|curl v| α dy 1 α avec B x η -|v j (y) -v j | β dy 1 β ≤ C 5 B x η -|v j (y)| β dy 1 β + C 5 B x η -|v j (y)| dy En posant sup η>0 B x η -|f |dy = M (f ), on parvient à : Ω sup η>0 ∇p x η Curl v j * h η (x) dx ≤ C 4 Ω (M (|∇v j | α )) 1 α M |v| β 1 β dx +C 5 Ω M |v j | β 1 β (M (|curl v| α )) 1 α dx + C 5 Ω (M (|v j |)) (M (|curl v| α )) 1 α dx
On utilise à nouveau l'inégalité de Holder 1 p + 1 p = 1 :

Ω sup η>0 ∇p x η Curl v j * h η (x) dx ≤ C 6 ||M (|∇v j | α ) || 1 α L p α (Ω) M |v| β 1 β L p β (Ω) +C 7 M |v j | β 1 β L p β (Ω) ||M (|curl v| α ) || 1 α L p α (Ω) +C 8 ||M (|v j |)|| L p (Ω) ||M (|curl v| α ) || 1 α L p α (Ω)
.

Finalement en choisissant β < p et α < p on peut appliquer le théorème maximal de Hardy-Littlewood :

Ω sup η>0 ∇p x η Curl v j * h η (x) dx ≤ C 9 ||∇v j || L p (Ω) ||v|| L p (Ω) +C 10 ||v j || L p (Ω) ||curl v|| L p (Ω) Si on fixe p = p = 2, comme v ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 [Ω)), alors : Ω sup η>0 ∇p x η Curl v j * h η (x) dx ≤ C(t) ∈ L 2 (0, T ) Finalement, comme d'une part Ω ∇p x η Curl v j dx = 0 et que d'autre part ∇p x η Curl v j = 0 sur R 2 -Ω (car on a prolongé v j et ∂v j ∂x i par 0 sur R 2 -Ω), on obtient : ∇p x η Curl v j ∈ L 2 (0, T ; H 1 z (Ω)) (0.1)
2ème étape -Estimation de curl (v j Curl q x η ). On a :

curl (v j Curl q x η ) * h η (x) = B x η 1 η 2 Curl h x -y η v j (y) Curl q x η (y) η dy et avec l'inégalité de Holder si 1 β + 1 β = 1 : curl (v j Curl q x η ) * h η (x) ≤ C 1 - B x η |v j | β 1 β   - B x η Curl q x η η β   1 β . Comme B x η -Curl q x η dy = 0, Curl q x η .n = 0 sur ∂B x η et curl v = curl Curl q x η sur B x η
, il est clair avec l'inégalité de Sobolev-Poincaré que :

B x η - Curl q x η η β dy 1 β ≤ C 2 B x η -|curl v| α dy 1 α avec 1 α -1 2 = 1 -1 β = 1 β . Ainsi, en posant sup η>0 - B x η |f | = M (f ), on parvient à : Ω sup η>0 curl (v j Curl q x η ) * h η (x) dx ≤ C 4 Ω M |v j | β 1 β (M (|curl v| α )) 1 α dx
et toujours avec l'inégalité de Holder si 1 p + 1 p = 1 :

Ω sup η>0 curl (v j Curl q x η ) * h η (x) dx ≤ C 5 M |v j | β 1 β L p β (Ω) ||M (|curl v| α )|| 1 α L p α (Ω)
.

Finalement, si on choisit β < p = 2 et α < p = 2, le théorème du maximum de Hardy-Littlewood montre que :

Ω sup η>0 curl (v j Curl q x η ) * h η (x) dx ≤ C 6 ||v j || L 2 (Ω) ||curl v|| L 2 (Ω) ∈ L 2 (0, T ).
Finalement, comme d'une part :

Ω curl (v j Curl q x η )dx = 0 et que d'autre part curl (v j Curl q x η ) = 0 sur R 2 -Ω (car on a prolongé v j et ∂v j ∂x i par 0 sur R 2 -Ω), on obtient : curl (v j Curl q x η ) ∈ L 2 (0, T ; H 1 z (Ω)). (0.2)
De (0.1) et (0.2) on tire le résultat annoncé dans le lemme.

Lemme 2 -Si v ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 0 (Ω)) alors : div (v j v) ∈ L 2 (0, T ; H 1 z (Ω)). Preuve du lemme 2 -En dimension 2 on remarque que : div (v j v) = -curl (v j α(v)), α(v) = (-v 2 , v 1 ).
Sur chaque boule B x η , on peut toujours décomposer α(v) de façon unique sous la forme : α(v) = ∇r x η + Curl s x η avec r x η ∈ H 1 (B x η ) et s x η ∈ H 1 0 (B x η ). On remarque que : -div (v j v) = curl (v j α(v)) = curl (v j Curl s x η ) -Curl v j ∇r x η . On procède alors comme au lemme 1 et on prouve que : Preuve du théorème 1 -La preuve du théorème est une simple conséquence des lemmes 1 et 2.

Références

η>0

  Curl v j ∇r x η * h η (x) dx ≤ C 1 ||∇v j || L 2 (Ω) ||α(v)|| L 2 (Ω) +C 2 ||v j || L 2 (Ω) ||curl α(v)|| L 2 (Ω) ∈ L 2 (0, T ), et Ω sup η>0 curl (v j Curl s x η ) * h η (x) dx ≤ C 3 ||u j || L 2 (Ω) ||curl α(u)|| L 2 (Ω) ∈ L 2 (0, T )d'où l'on déduit le résultat annoncé.