Maha Diekan Abbas

Rachel E Marschang

Volker Schmidt

Astrid Kasper

Tibor Papp

A unique novel reptilian paramyxovirus, four atadenovirus types and a reovirus identified in a concurrent infection of a corn snake (Pantherophis guttatus) collection in Germany

In 2009, 26 clinical samples (organs and oral/cloacal swabs) from a total of 24 corn snakes (Pantherophis guttatus) from a single owner were sent to our laboratory to be tested for the presence of viruses. Paramyxoviruses (PMV), adenoviruses (AdV) and reoviruses were detected by RT-PCR, PCR and virus isolation methods. Three snakes were infected with all three viruses at the same time, while two other snakes had a double infection (PMV and reo, AdV and reo) and nine other snakes had a single infection with any of the three viruses. No viruses were detected in 10 animals.

All isolated reoviruses were identical to one another and to the reptilian orthoreovirus isolate 55-02 in the partial RNA dependent RNA polymerase (RDRP) gene sequence. AdV partial polymerase sequences represented four different types, one *Manuscript of which was first described here: most similar to SnAdV-1, while the other three were identical to known types: SnAV-1, -2 and -3. However, the detected single PMV differed distinctly from described reptile PMV and was a new type. According to partial L gene, HN gene and U gene sequences it may be the first described representative of a third squamatid PMV cluster: "group C" within the proposed reptilian PMV genus "Ferlavirus". Nucleotide identity values for the L gene of the new PMV compared to group A viruses range between 76.5-80.3%, and between 80.5-81.2% compared to group B viruses. For the HN gene, these values were similar: 78.2-80% (A) and 79.9-80.5% (B) and somewhat lower for the U gene: 72.7-75.4% (A) and 69.7-70% (B). No reports on the prevalence of concurrent viral infection in captive snake populations have been published so far. The possibility of concurrent infection with several different viruses and subsequent consequences for animal health should be kept in mind when testing reptile samples for viruses.

A c c e p t e d M a n u s c r i p t Introduction Paramyxoviruses (PMV) have been isolated from several reptiles, mainly snakes, and have been characterized morphologically, immunologically and/or molecular biologically [START_REF] Blahak | Isolation and characterization of paramyxoviruses from snakes and their relationship to avian paramyxoviruses[END_REF][START_REF] Richter | Characterization of paramyxoviruses isolated from three snakes[END_REF][START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF][START_REF] Franke | Identification and molecular characterization of 18 paramyxoviruses isolated from snakes[END_REF][START_REF] Kindermann | Phylogenetic analysis of the L and HN gene of ophidian paramyxoviruses[END_REF][START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF].

The first recorded outbreak of PMV occurred in 1972 in a snake farm in Switzerland in common lancehead vipers (Bothrops atrox) [START_REF] Fölsch | Fatale endemische Infektion in einem Serpentarium[END_REF]. The causative agent was named Fer de Lance virus (FDLV) and is regarded as the type species for reptilian PMV [START_REF] Clark | Fer de Lance virus (FDLV): a probable paramyxovirus isolated from a reptile[END_REF]. Recently, the complete genome of FDLV has been fully sequenced [START_REF] Kurath | Complete genome sequence of Fer-de-Lance virus reveals a novel gene in reptilian paramyxoviruses[END_REF]. This genome has a unique gene (U gene) that is absent in other PMVs, which encodes a short protein of Viperidae. These viruses were first named ophidian (snake) paramyxoviruses (OPMV) [START_REF] Essbauer | Viruses of lower vertebrates[END_REF], but have since also been described in lizards and tortoises [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF]Papp et al., 2010b). Reptilian PMV are considered one of the most important pathogens of snakes and have been isolated from both private and zoological collections [START_REF] Jacobson | Viruses and viral diseases of reptiles[END_REF]. They have also been described as emerging infectious diseases which may endanger wildlife [START_REF] Daszak | Emerging infectious diseases of wildlife threats to biodiversity and human health[END_REF][START_REF] Jacobson | Implications of infectious diseases for captive propagation and introduction programs of threatened/endangered reptiles[END_REF][START_REF] Azevedo | Characterization of a paramyxovirus from a Fer de Lance Viper (Bothrops jararaca): partial nucleotide sequence of the putative fusion protein[END_REF]. Clinical signs observed in PMV infected snakes most commonly involve the respiratory tract, but central nervous system (CNS) disease is also regularly observed [START_REF] Jacobson | Viruses and viral diseases of reptiles[END_REF]. Phylogenetic analysis of partial gene sequences of PMV isolated from snakes but also from lizards and a tortoise revealed that there are at least two squamatid lineages (groups) both containing snake and lizard isolates, whereas the sole tortoise PMV isolate from a Hermann´s tortoise (Testudo hermanni) clusters as an ancient, separate lineage of the proposed new reptilian PMV genus "Ferlavirus" [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF]). An additional study detected a squamatid PMV in a diseased leopard tortoise (Papp et al., 2010b), demonstrating the wide host range of some of these viruses.

Reovirus infections have been reported in several species of reptiles including snakes [START_REF] Ahne | Isolation of a reovirus from the snake, Python regius[END_REF][START_REF] Blahak | A case reported of a reovirus infection in an emerald tree boa (Corallus caninus)[END_REF][START_REF] Blahak | Isolation and characterization of paramyxoviruses from snakes and their relationship to avian paramyxoviruses[END_REF][START_REF] Jacobson | Viruses and viral associated diseases of reptiles[END_REF], lizards [START_REF] Ahne | Isolation of a reovirus from the snake, Python regius[END_REF][START_REF] Drury | Isolation and identification of a reovirus from a lizard, Uromastyx hardwickii, in the United Kingdom[END_REF][START_REF] Marschang | Paramyxovirus and reovirus infections in wild-caught Mexican lizards (Xenosaurus and Abronia spp.)[END_REF][START_REF] Marschang | Isolation and partial characterization of three reoviruses from lizards[END_REF], and tortoises [START_REF] Drury | Isolation and identification of a reovirus from a lizard, Uromastyx hardwickii, in the United Kingdom[END_REF][START_REF] Marschang | Infectious diseases[END_REF]. Clinical signs associated with reovirus infections have been pneumonia and A c c e p t e d M a n u s c r i p t 4 neurologic problems that may appear similar to PMV infections [START_REF] Wellehan | Consensus nested PCR amplification and sequencing of diverse reptilian, avian, and mammalian Orthoreoviruses[END_REF].

The number of reports of adenovirus (AdV) infections in reptiles is increasing, apparently due to the growing popularity of reptiles as pets [START_REF] Essbauer | Viruses of lower vertebrates[END_REF] and to sensitive diagnostic methods [START_REF] Wellehan | Detection and analysis of six lizard adenoviruses by consensus primer PCR provides further evidence of a reptilian origin for the atadenoviruses[END_REF]. AdV infections have been reported in several reptile species including crocodiles, snakes, and lizards [START_REF] Frye | Adenovirus-like hepatitis in a group of related Rankin's dragon lizards (Pogona henrylawsoni)[END_REF][START_REF] Jacobson | Adeno-like virus in esophageal and tracheal mucosa of a Jackson's chameleon (Chamaeleo jacksoni)[END_REF][START_REF] Kinsel | Small intestinal adeno-like virus in a mountain chameleon (Chameleo montium)[END_REF][START_REF] Ramis | Adenovirus hepatitis in a boa constrictor (Boa constrictor)[END_REF]. In these species, adenovirus infection was considered the cause of gastroenteritis, hepatitis, nephritis, pneumonia and encephalitis [START_REF] Frye | Adenovirus-like hepatitis in a group of related Rankin's dragon lizards (Pogona henrylawsoni)[END_REF]Helstab and Bestetti, 1984;Schumacher et al., 1994). The single fully sequenced reptilian isolate is from a corn snake (Pantherophis guttatus) [START_REF] Farkas | Genomic and phylogenetic analyses of an adenovirus isolated from a corn snake (Elaphe guttata) imply a common origin with members of the proposed new genus Atadenovirus[END_REF]2008). This snake adenovirus type 1 (SnAdV-1) represents the supposed reptilian lineage of the Atadenovirus genus. Further squamatid AdVs (including new snake types) have been partially sequenced and found to belong to the same genus [START_REF] Wellehan | Detection and analysis of six lizard adenoviruses by consensus primer PCR provides further evidence of a reptilian origin for the atadenoviruses[END_REF][START_REF] Garner | Characterization of enteric infections associated with two novel atadenoviruses in colubrid snakes[END_REF]Papp et al., 2009a;[START_REF] Benkő | How to find "new" adenoviruses[END_REF][START_REF] Pénzes | Further proofs for the reptilian origin of atadenoviruses[END_REF]. Interestingly, unlike PMVs, the recently described chelonian AdVs cluster separate from the squamatid AdVs, either to the supposed amphibian lineage Siadenovirus genus [START_REF] Rivera | Systemic adenovirus infection in Sulawesi tortoises (Indotestudo forsteni) caused by a novel siadenovirus[END_REF] or outside of the five accepted genera of the family Adenoviridae [START_REF] Farkas | Adenovirus and mycoplasma infection in an ornate box turtle (Terrapene ornata ornata) in Hungary[END_REF]Doszpoly and Harrach, personal communication;Wellehan et al., personal communication).

During routine diagnostic testing for reptilian viruses we found a concurrent ongoing infection with all three above described viruses in a collection of corn snakes including a snake that died suddenly showing dispnoe and vomitus prior to exitus.

The survivors showed no specific symptoms. A follow-up study was performed in the animals from two consecutive collections for over 5 months, before they were finally dispersed. Detected viruses were partially sequence characterized, and most of the A c c e p t e d M a n u s c r i p t viruses were also isolated on permanent cell lines. The PMV detected in these corn snakes is new to science, and seems to be the first described representative of a third major squamatid PMV group.

Materials and Methods:

Samples

In January 2009, a dead juvenile female corn snake (Pantherophis guttatus) (Lab.

No. 6/3/09) was dissected at the Bird and Reptile Clinic, University of Leipzig, Leipzig, Germany. Its body weight was 4 g and the head-cloaca length 29 cm. Prior to death, the snake had dyspnoea and vomitus. Gross necropsy showed cachexia.

The stomach was moderately filled with brown pasty content, the gut was empty and no macroscopic lesions were observed. Histopathologically the only finding was hyperplasia of melanophagocytes in the liver, while other organs showed advanced autolysis. According to the clinical symptoms, the hyperplasia of melanophagocytes, the juvenile age and the history of ananother death in the snake collection, a viral disease was suspected. Samples of lung, kidney, and intestine were sent to the University of Hohenheim, Stuttgart, Germany for virus detection. One month later, 13 young corn snakes (body weight: 7-30 g) remaining in the possession of the same owners were swabbed (oral and cloacal) and tested (Lab. No. 14/1/09 to 14/13/09).

Later, without further notice the owner dispersed the stock (some were reported to have died afterwards) and established a new snake collection. At this point -five months after the first tests -these 10 animals (Lab. No. 54/1/09 to 54/10/09) were also swabbed and tested. All together 26 clinical diagnostic samples including 3 organs and 23 oral and cloacal swabs from a total of 24 corn snakes of two consecutive stocks kept at the same enclosure of one owner were sent to our laboratory and tested for the presence of PMV, AdV, and reoviruses (Table 1).

Virus isolation

Isolation of viruses was attempted from all samples on Russell's viper heart cells (VH-2, ATCC: CCL-140) and iguana heart cells (IgH2, ATCC, CCL-108). Samples in DMEM were sonicated for destruction of cells and suspension of viruses. The samples were then centrifuged at 3000 x g for 10 min. for the removal of cell-debris and bacteria, then 200 µl of the homogenate was inoculated onto medium-free, approximately 70% confluent cell monolayers in 30 mm diameter Cellstar ® tissue culture dishes (Greiner Bio-One GmbH, Frickenhausen, Germany). After incubating for 2 h (at 28 °C, 5% CO 2 ) 2 ml nutrient medium (DMEM supplemented with 2% foetal calf serum, 1% non-essential amino acids and antibiotics (200 U/ml penicillin-G, 380 U/ml streptomycin-sulfate, 6.4 U/ml gentamycinsulfate and 0.5 µg/ml amphotericin-B) were added to each dish. Cells were examined for the presence of cytopathic effects (CPE) approximately every 3 days with an inverted light microscope (Leitz, Weitzlar, Germany), dishes were frozen when extended CPE was seen. Dishes showing no CPE were frozen after 2 weeks of incubation for blind passaging. Two additional passages were performed from each dish after a freeze and thaw cycle and low speed centrifugation. RNA and DNA were prepared from the cell culture supernatant as described below for further identification and characterization of isolates.

In the case of mixed virus isolates, a 10 fold serial dilution of the supernatant was made. From each dilution step at least four wells of a 96-well FALCON MICROTEST A c c e p t e d M a n u s c r i p t plate (Becton Dickinson Labware, Franklin Lakes, USA) were inoculated in parallel in order to separate the different viruses (end-point dilution, Reed and Muench, 1938).

To determine whether virus isolates were enveloped, chloroform treatment was performed by mixing chloroform and virus suspension 1:10, followed by shaking on a vortex for 1h and centrifugation at 10.000 x g for 10 min. The supernatant was then diluted 1:10 in maintenance medium and inoculated into tissue culture dishes as described. Infectious cell culture supernatant without chloroform treatment was inoculated parallel into tissue culture dishes. Plague purification assay was also performed, as described by [START_REF] Bijlenga | Isolation and plaque purification of a syncytial forming component of Myxoma virus[END_REF] with some modifications.

Nutrient medium (see above) was prepared with an additional 1% of agar-agar (Merk, Darmstadt, Germany). Tissue culture dishes were infected with the mixed virus isolate, and after two hours of incubation 2 ml of this semi-solid medium was added. Dishes were incubated at 28 ºC, 5% CO 2 for 3 days, microscopic agar-plaque samples were collected randomly using pipette tips and placed separately in 500 µl DMEM supplemented with a 2x concentration of antibiotics. Samples were then sonicated in an ice bath (Bandelin electronics, Berlin, Germany, 40 KHz) for 5 minutes and re-inoculated onto one day-old tissue culture dishes and incubated for ten days.

RNA and DNA extraction.

RNA was prepared from 300 µl of the sonicated clinical samples and virus isolate supernatants using the guanidinium isothiocyanate method described by [START_REF] Boom | Rapid and simple Method for Purification of Nucleic Acid[END_REF]

PCR/RT-PCR and sequencing

A nested RT-PCR targeting a portion of the L gene of ophidian PMV was carried out using the primers described by [START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF] in 25 µl reaction mixtures. The RT-PCR targeting portion of HN gene was performed as for L gene using primers F1 and R2 [START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF] in the first round, and HN-cons Fwd-in and HN-cons Rev-in [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF] in the second round. The RT PCR targeting the complete ORFs of the U gene was carried out similarly using U-cons Fwd-out and U-cons Revout in the first round, and U-cons Fwd-in and U-cons Rev-in the second round [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF]. The chemistry used for all PCRs is decribed in the latter paper.

For reovirus detection, a nested RT-PCR targeting a portion of the RNA-dependent RNA polymerase (RDRP) gene was carried out as described by [START_REF] Garner | Characterization of enteric infections associated with two novel atadenoviruses in colubrid snakes[END_REF]. The primers 1607F and 2608R were used for the RT and first round PCR reactions, the second round of PCR amplification was carried out using the primers 2090F and 2200R.

Nested AdV PCRs were carried out as described by [START_REF] Papp | PCRsequence characterization of new adenovirus found in reptiles and the first successful isolation of a lizards adenovirus[END_REF] using the primers originally described by [START_REF] Wellehan | Detection and analysis of six lizard adenoviruses by consensus primer PCR provides further evidence of a reptilian origin for the atadenoviruses[END_REF]. The PCR protocol was optimized for some samples to remove unspecific bands by raising the annealing temperature to 50 °C and adding a total of 2 mM MgCl 2 in the second round.

Products of all PCR protocols were separated on 1.5% agarose gels (Bioenzym, Oldendorf, Germany) in TAE puffer containing 0.5 mg/ml ethidium-bromide and visualised under 320 nm UV light. Gel purified PCR amplicons (Invisorb Spin DNA Extraction Kit; Ivitek GmbH, Berlin, Germany) were sequenced directly using a Big-

Analysis of sequences

Raw sequences were processed by the ABI Sequence Analysis Programme 5.1.1 (Applied Biosystems, Foster City, USA) then edited, assembled and compared using the STADEN Package version 2003.0 Pregap4 and Gap4 programmes [START_REF] Bonfield | A new DNA sequence assembly program[END_REF]. The sequences were compared to data in GenBank (National Center for Biotechnology Information, Bethesda, USA) online (www.ncbi.nih.gov) using BLASTN and BLASTX options. Multiple alignments of nucleotide sequences were performed with ClustalW algorithm of the BioEdit Sequence Alignment Editor programme [START_REF] Hall | BioEdit: a user-friendly biological sequence alignment editor and analysis program for Windows 95/98/NT[END_REF]. This alignment was further used for phylogenetic calculations in the PHYLIP program Package version 3.6. [START_REF] Felsenstein | PHYLIP -Phylogeny Inference Package[END_REF] comparing parsimony, maximum-likelihood, and distance based methods to obtain an optimal tree. Bootstrap analysis of 100 replicates was carried out.

Results

In this study we have investigated 26 organs and oral/cloacal swabs originating from 24 corn snakes of two successive stock of a single owner and were able to detect PMV by PCR in the 7 snakes of the fist stock, and AdVs from 3 snakes both in the first and second stocks. Reoviruses were successfully isolated from 7 & 2, PMV from 2 & 0 and AdV from 2 & 0 snakes of the first & second stock respectively (Table 1).

(RT-)PCR results

PMV L gene nested RT-PCR gave a positive signal (566 bp product) with the kidney and intestine of the dead animal and six swabs of the surviving snakes from the 1 st A c c e p t e d M a n u s c r i p t 10 collection (50%) (Table 1), but none from the 2 nd collection. From the L-gene positive samples each (50% of group one) were positive in the HN gene nested RT-PCR (399 bp), and all but one (sample "14/8/09", 42.8% positive of group one) using the U gene nested RT-PCR (640 bp). The homologous PMV partial gene sequences found in different samples were identical to one another, and represent a new reptilian PMV (assigned name: PanGut-GER09) (Fig. 1 andFig S-1). The PanGut-GER09 partial gene sequences have been submitted to GenBank and assigned the accession numbers: HQ148084 to HQ148087.

AdVs were detected from the diagnostic samples using consensus nested PCR (product size: 320 bp) in 6 animals altogether (25%), 3 from the 1 st group (21.4 %) (Table 1). The AdV from the kidney and intestine of the dead snake (6/3/09) had a partial polymerase sequence identical to the GenBank (FJ012163) snake adenovirus type 2 (SnAdV-2) sequence. Two surviving snakes of this 1 st collection (14/2/09 & 14/3/09) that were also identified with AdV infection by PCR revealed two different viruses based on their partial polymerase sequences with 100% and 93% identity respectively to the SnAdV-1 (DQ106414). This latter, new sequence is identical to that of a recently reported tiger python (Python molurus) AdV isolate [START_REF] Papp | New squamatid adenoviruses: partial PCR-sequence characterisation of the first lizard isolates together with other snake and lizard field samples[END_REF]Romanova et al., 2010), a supposed new serotype of the SnAdV-A virus species, and has been submitted to GenBank with an accession number: HQ148088. Three animals of the 2 nd collection (30%) were AdV positive. The sequencing revealed three different viruses: beside the above mentioned SnAdV-1 (54/4/09) and SnAV-2 (54/6/09), a third type, identical in its partial polymerase sequence to the SnAV-3 (FJ012164) was also detected in a snake (54/1/09).

Reoviruses were not detected directly from the diagnostic samples using the nested RT-PCR targeting the RNA dependent RNA polymerase (RDRP).

Isolation of viruses

Adenoviruses were isolated in two cases (14.3%), once from the intestine of the dead snake (6/3/09) and from a swab taken from a survivor (14/2/09) in the first collection (Table 1). Observed cytopathic effects (CPE) with rounding and detachment of some cells (Fig S -1b) were typical of AdV. Paramyxoviruses were isolated from the kidney and intestine of the dead animal and from the swab of the same survivor as above.

Typical CPE of giant fusion cells (syncitia) was observed. Reoviruses were isolated from a total of 9 snakes (37.5%), including all three organs of the dead snake and six survivors of the first collection, as well as two swabs from the second collection (Table 1). The cytopathic effect (CPE) caused by the PMV and reoviruses were both characterized by cytolysis and giant syncytial cell formation. Verification of the presence of reoviruses in cell culture supernatant by consensus nested RT-PCR (110 bp) was positive in all cases. All isolated reoviruses had identical partial RDRP genes sequences to one another and to the reptilian GenBank carpet python (Morelia spilota) orthoreovirus isolate 55-02 (GenBank accession No. E4309703).

In both cases of AdV isolation, PMV and reoviruses were also present in the isolates.

The PMV isolates are easily overgrown with reovirus, yet the presence of PMV was verified by L, U, and HN gene RT-PCR amplification of PMV RNA from the cell culture supernatant. Using the end-point dilution method followed by chloroform treatment pure AdV was obtained from the 14/2/09 sample. Using an end-point dilution method followed by a plaque purification assay, a pure PMV isolate was obtained from the same sample, as verified by verified via (RT-)PCR. The PMV grew slowly in cell culture, with gradual development of a specific CPE with small syncytial cell formation (on day 10 post inoculation). In the case of 6/3/09, following chloroform treatment and end-point dilution of the isolate, the AdV specific CPE was still obscured by the syncitia of the fusogenic reovirus and it could not be isolated in a

Infection rate

Summarizing the PCR and virus isolation results, the following infection rates were observed. Three snakes (21.4%) of the 1 st collection (6/3/09, 14/2/09 & 14/3/09) were identified with a triple viral infection with reoviruses, PMV, and AdV. One snake (7.1%) in this collection (14/6/09) had a double infection with a reovirus and PMV, three snakes (14/5/09, 14/10/09 & 14/13/09) had single infections with a reovirus and three further ones (14/8/09, 14/11/09 & 14/12/09) had a single infection with PMV.

The remaining four (28.6%) of the 14 tested animals of the 1 st collection were negative in all virus detection tests (Table 1). In the case of the 2 nd collection one snake (54/4/09) (10%) had a double infection with a reovirus and AdV, two snakes (54/1/09 & 54/6/09) (20%) were infected with only AdV and one (54/3/09) with only reovirus. An additional six (60%) animals from this group were tested and found negative in the virus diagnostic tests.

Sequence analysis of new reptilian PMV

L, HN, and U gene partial sequence data revealed a novel PMV type (PanGut-GER09) in seven animals of the first snake collection. This virus is nearly equally distant from the previously described "A" and "B" squamatid (snake and lizard) rPMV groups (Marshang et al., 2009). Nucleotide identity values for the new partial L gene sequence compared to group "A" member sequences are 76.5-80.3% and 80.5-A c c e p t e d M a n u s c r i p t 13 81.2% compared to group "B" ones. The identity value compared to the single tortoise isolate is 73.8%.

For the HN gene, these values are similar: 78.2-80% ("A") and 79.9-80.5% ("B") and somewhat lower for the U gene (including non-coding regions): 72.7-75.4% ("A") and 69.7-70% ("B"). Phylogenetic calculations (Fig. 1) based on the three partial gene sequences show that all rPMV form a single monophyletic cluster, representing the proposed new "Ferlavirus" genus. Within this genus the earlier described tortoise isolate and the "A" and "B" squamatid PMVs form separate groups. The novel PanGut-GER09 virus clusters with moderate bootstrap values either to the "B" group (L gene, Fig. 1a), or to group "A" (HN & U gene, Fig. 1bc), as a sister taxon.

Alignments are shown in Figures S1-2.

In the transcriptional analysis of the squamatid (or reptile) PMV specific gene of unknown function (U gene) both earlier described consensus reading frames (ORF-1 & -2) [START_REF] Kurath | Complete genome sequence of Fer-de-Lance virus reveals a novel gene in reptilian paramyxoviruses[END_REF][START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF] were identified in the new isolate (Fig S1). The putative protein coded by ORF-1 has a predicted length of 82 aa (suppl.: Fig. S2c), whereas ORF-2 encodes a 144 aa protein (suppl: Fig. 2d), 1 or 2 aa longer than those found in rPMV of group B or A respectively. Nucleotide identity values of the overlapping presumed coding regions of the U gene were comparable to those seen in the L and HN gene partial CDS. Yet unlike the latter two genes, the U gene putative ORFs of the new isolate had a distinctly higher similarity to the group "A" viruses than to group "B" ones: 85.9-86.7% ("A") / 77.1-77.9% ("B") for ORF1 or 81.9-83.5% ("A") / 78.4-78.6% ("B") for ORF-2. Additionally, repeated sequencing verified that there is a single nucleotide variation at position 383 (Fig. S1/c) of the U gene sequence, resembling four (6/3/09, 14/2/09, 14/3/09, 14/11/09) and two (14/6/09, 14/12/09) samples respectively of the detected six cases, and coding a silent mutation of ORF-2.

A c c e p t e d M a n u s c r i p t

Discussion

The purchase of new reptiles is associated with a particularly high risk of introducing disease into a collection. Viral infections such as inclusion body disease (IBD) and PMV often lead to devastating disease when they are transmitted to healthy snakes [START_REF] Pasmans | Introducing reptiles into a captive collection: the role of the veterinarian[END_REF]. Some studies demonstrate that there is a significant risk of hidden infectious diseases including PMV, inclusion body disease (IBD) and some bacterial infections in apparently healthy snake collections [START_REF] Pees | Prevalence of viral infections in captive collections of boid snakes in Germany[END_REF]. Basic veterinary health checks and quarantine measures are of great importance to minimize the risk of infection among newly established snake groups.

In the current study, multiple viral infections were detected in captive corn snakes.

The pathogenic role of the concurrent viral infection in the present case could not be fully clarified as only limited data were made available, and the owner dispersed the population without further notice soon after our second positive tests. We presume that mixing of previously untested animals of uncertain origins to form a new breeding population might be the source of infection in the 1 st collection of this study, while in the case of the 2 nd collection after dispersing the 1 st snake group, inappropriate disinfection of the terrariums could also be a factor for transmitting viral infection into the newly introduced animals, as the same types of non-enveloped viruses (reo-, AdV) were detected in these animals as before, whereas the environmentally less stable PMV was no longer present. Detecting a further type of virus (SnAdV-3) in the 2 nd collection might be due to previous infection in one of the new animals, leading to the introduction of this type into the newly formed collection as a result of a lack of screening and entry examinations. In addition to quarantine and disinfection procedures, screening and entry examinations are also of great A c c e p t e d M a n u s c r i p t importance for detecting the prevalence of such infectious diseases [START_REF] Pees | Prevalence of viral infections in captive collections of boid snakes in Germany[END_REF].

Viral polymerases are often highly conserved, as they are essential for viral function.

They are therefore often good targets for viral diagnostic assays. In the present paper we have used earlier described nested (RT-) PCRs targeting the polymerase genes to detect infection with reoviruses, PMV, and AdV [START_REF] Garner | Characterization of enteric infections associated with two novel atadenoviruses in colubrid snakes[END_REF][START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF][START_REF] Wellehan | Detection and analysis of six lizard adenoviruses by consensus primer PCR provides further evidence of a reptilian origin for the atadenoviruses[END_REF] from snake tissue samples and swabs. We have also used classical virus isolation method on permanent reptilian cell lines, and two further PCRs for the characterization of PMV [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF].

Considering the results and in consensus with earlier papers [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF]Papp et al., 2010a) we can state that the nested RT-PCR targeting the L-gene [START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF] is the most sensitive method currently available and the recommended procedure for basic diagnostic testing for PMV detection. A nested PCR targeting the polymerase gene [START_REF] Wellehan | Detection and analysis of six lizard adenoviruses by consensus primer PCR provides further evidence of a reptilian origin for the atadenoviruses[END_REF]) is also the most sensitive assay to date for the detection of AdV. However, in the case of reoviruses, the consensus nested RT-PCR targeting the RDRP gene [START_REF] Garner | Characterization of enteric infections associated with two novel atadenoviruses in colubrid snakes[END_REF] is less sensitive than the virus isolation method (Table 1, see also [START_REF] Marschang | Isolation and partial characterization of three reoviruses from lizards[END_REF], although it provides a useful tool for the initial characterization of these viruses. All reoviruses detected in the current study using the RT-PCR were from RNA extracts of cell culture isolates. No reoviruses were detected in original samples prior to isolation on cell culture. Optimization trials along with designing new primers might be necessary to increase the efficiency of this RT-PCR protocol.

Reoviruses are commonly described in reptiles, mostly in snakes [START_REF] Jacobson | Viruses and viral diseases of reptiles[END_REF].

Previous studies have shown that reoviruses isolated from reptiles can differ from one another both serologically and molecularly [START_REF] Blahak | Isolation and characterization of paramyxoviruses from snakes and their relationship to avian paramyxoviruses[END_REF]others 1995, Wellehan et al., 2009;[START_REF] Pees | Prevalence of viral infections in captive collections of boid snakes in Germany[END_REF], however, in the present study all isolated reoviruses A c c e p t e d M a n u s c r i p t 16 from nine snakes (including tissues of a dead one, seven survivors of the 1 st collection and two snakes in the 2 nd collection, Table 1) were identical to one another and 100% identical to the reptilian orthoreovirus isolate 55-02 (E4309703), suggesting a single source for the infection.

The AdV sequences detected in six snakes from the two collections, on the other hand, could be divided into four distinct types, representing the previously described SnAdV-1, -2 and -3 types [START_REF] Farkas | Genomic and phylogenetic analyses of an adenovirus isolated from a corn snake (Elaphe guttata) imply a common origin with members of the proposed new genus Atadenovirus[END_REF]2008;[START_REF] Garner | Characterization of enteric infections associated with two novel atadenoviruses in colubrid snakes[END_REF] and an additional SnAdV-1-like (93% id.) virus of the same species: SnAdV-A [START_REF] Papp | New squamatid adenoviruses: partial PCR-sequence characterisation of the first lizard isolates together with other snake and lizard field samples[END_REF]. This novel virus type is to be better characterized in a separate publication together with similar novel types [START_REF] Romanova | Three novel snake atadenoviruses, proposed members of SnAdV-A species[END_REF]. This finding is of great interest as it contributes new aspects to the postulated coevolution theory of adenoviruses [START_REF] Harrach | Reptile adenoviruses in cattle?[END_REF][START_REF] Davison | Genetic content and evolution of adenoviruses[END_REF]. The present paper shows that the same corn snake species (Pantherophis guttatus, Colubridae) can be infected with 4 different atadenovirus types (3 AdV species) that were earlier detected in other colubrid, boid, pythonid or viperid snake species.

Despite the numerous detections of reptilian AdV by electron microscopy and PCR, there are very few reported cases in which the virus was successfully isolated (Juhász and Ahne, 1992;[START_REF] Ogawa | Reptilian viruses: adenovirus like agent isolated from royal python (Python regius)[END_REF]Papp et al., 2009a). Partial sequence data of the different snake isolates suggest that these are the same type: SnAdV-1, which was also isolated in this study. However, this study is the first report on the isolation of the proposed type 2 snake AdV in cell culture (6/3/09 intestine, mixed with reo-& PMV). This snake AdV type 2 has been identified previously in the USA [START_REF] Garner | Characterization of enteric infections associated with two novel atadenoviruses in colubrid snakes[END_REF], Germany (Papp et al., 2009a) and Hungary (Farkas, unpublished data) PMV infections have been involved in high mortality diseases in snake collections [START_REF] Fölsch | Fatale endemische Infektion in einem Serpentarium[END_REF][START_REF] Jacobson | Viruses and viral associated diseases of reptiles[END_REF]. A recent publication reports concurrent infection with several distinct PMV in a single snake or group of snakes (Papp et al., 2010a) during one outbreak, yet all earlier papers [START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF][START_REF] Franke | Identification and molecular characterization of 18 paramyxoviruses isolated from snakes[END_REF] identified the same PMV in animals of the same population or outbreak. In the present study, sequencing of three gene portions revealed a single novel unique rPMV in seven (one dead and six living) snakes from the 1 st collection, which could be isolated first in a mixed, then in pure culture.

The first molecular based (L and HN gene) classification of reptilian PMV by Ahne and co-authors (1999) described two distinct "subgroups" with very low inner variance, called "a" and "b" and a set of more variable intermediate isolates. Later, Franke and co-authors (2001) described three clusters (A, B and C) on their L gene phylogenetic tree. Unfortunately, this latter publication did not include sequences from the earlier paper in the comparison, thus the denominations of their clusters do not refer to those of the subgroups ("cluster A" corresponded to the earlier "subgroup b", "cluster B" to the "intermediate isolates", and "cluster C" to the earlier "subgroup a"). This discrepancy in the names was resolved in recent publications by our research group [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF]. Considering the seniority rule in nomenclature, and the high bootstrap monophyletic clusters of the L, HN and U gene trees, a revised grouping for the squamatid rPMV was suggested: A "sensu lato" group "A" contained the previous "subgroup a" and the old and new "intermediate isolates", while the "sensu lato" group "B" those from "subgroup b" and similar squamatid PMVs. The use of "C" as a denomination for any group was omitted. A single tortoise isolate was suggested as the first representative of a separate chelonid PMV group, clustering with the two squamatid PMV groups into the proposed "Ferlavirus" genus [START_REF] Kurath | Complete genome sequence of Fer-de-Lance virus reveals a novel gene in reptilian paramyxoviruses[END_REF][START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF]. The results of a later publication (Papp et al., 2010a) characterizing 38 samples from 25 snakes, foreshadowed that all rPMV found in captive snake collections in Germany belong to one of the two squamatid PMV groups. The current study based on partial L, HN and U gene sequences first describes a representative of a novel third squamatid PMV cluster: "group C", equally distant from the other two squamatid groups, within the proposed genus "Ferlavirus" genus (Fig. 1).

These interesting new findings underline the significance of sequencing in routine reptilian virus diagnostics. It also shows the importance of testing for several different pathogens. Although each of the viruses detected in these snakes has been reported in connection with disease, the effect of combined infections and the prevalence of such combined infections is unknown. Concentrating on a single pathogen (e.g. PMV in snakes) can lead to underestimation of the importance of other infectious agents.

The case report also emphasizes the importance of disinfection together with entry assessment prior to establishing new snake collections or introducing new animals into formerly established groups. # For reoviruses, a 110 nt long portion of the RNA-dependent RNA polymerase (RDRP) gene was amplified and used for the sequence comparison [START_REF] Garner | Characterization of enteric infections associated with two novel atadenoviruses in colubrid snakes[END_REF]. For AdV, a 318 nt long portion of the DNA dependent DNA polymerase gene was amplified and compared [START_REF] Wellehan | Detection and analysis of six lizard adenoviruses by consensus primer PCR provides further evidence of a reptilian origin for the atadenoviruses[END_REF]. For PMV, a 566 nt long portion of the the RDRP gene (L-gene) was amplified in the diagnostic PCR [START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF]. That sequence, together with a 537 nt & a 648 nt long portion of two further genes (the hemagglutinin-neuraminidase (HN-gene) & the unique (U-gene)) were used for the seuquence comparison [START_REF] Marschang | Comparison of paramyxovirus isolates from snakes, lizards and a tortoise[END_REF]. (All sizes are for the unedited PCR products.) GenBank accession numbers: orthoreo isolate 55-02 (E4309703), snake AdV type 1 (DQ106414), SnAdV-1 like new type from 14/3/09 swab (HQ148088), snake AdV type 2 (FJ 012163),snake AdV type 3 (FJ 012164). Paramyxovirus PanGut-GER09 portions of L-, HN-& U-genes (HQ148084 to HQ148087). Orth-GER05 (GQ277616), Pyth-GER01 (GQ277612), TORTOISE-GER99 ++ (GQ277615), Xeno-USA99 + (GQ277614). For diagnostic non-isolated rPMV samples (Papp et al., 2010a): Pyt-2 kidney, Col-2c kidney, Pyt-6 intestine (GU393344 to GU393348). The following sequences [START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF] were kindly provided by Dr.

Gael Kurath and Dr. William N. Batts: Boa-CA98, Biti-CA98, Biti-GER 87, Call- GenBank accession numbers: Bush viper virus /ATCC-VR-1409/ (AF286044), Dasy-GER00 (GQ277618), Fer de Lance virus (NC_005084), Igu-GER00 + (GQ277619), NTV /ATCC-VR-1408/ (AF286046), Orth-GER05 (GQ277620), Xeno-USA99 + (GQ277621). The other reptilian sequences [START_REF] Ahne | Comparative sequence analyses of sixteen reptilian paramyxoviruses[END_REF] were kindly provided by Dr. Gael Kurath and Dr. William N. Batts.

(c) Phylogenetic distance tree of the reptilian PMV based on 568 bp of the unique (U) gene. Please note that no homologous sequence from non-reptilian PMV could be applied as an outgroup. The maximum likelihood analysis resulted in the same topology tree as the FITCH analysis, bootstrap values are both indicated on the branches. "Variant 1 and 2" represent the single nucleotide sequence variation of "Pangut Ger09" U gene sequence found in samples: 6/3/09, 14/2/09, 14/3/09, 14/11/09 and samples 14/6/09, 14/12/09 respectively.

GenBank accession numbers: Biti-CA98 (AY534645), Crot-GER03 (GQ277625), Dasy-GER00 (GQ277627), Fer de Lance virus (NC_005084), Gono-GER85 (AY534644), Igu-GER00 + (GQ277623), Orth-GER05 (GQ277622), Pyth-GER01 (GQ277624), Xeno-USA99 + (GQ277626)

  This feature together with the results of phylogenetic calculations supports the classification of FDLV in a new proposed genus "Ferlavirus" of the Paramyxovirinae subfamily. PMV have been detected in many different snake species from all major families including Boidae, Elapidae, Colubridae, and

6

  Tissue samples or swabs were immersed in 3 ml Dulbecco's modified Eagle's medium (DMEM) (Biochrom AG, Berlin, Germany) supplemented with antibiotics for further processing.

  and resuspended in 75 µl of RNase free water. DNA extraction of the sonicated clinical samples and virus isolate supernatants (both from 200 µl) was carried out using the DNeasy ® Blood and Tissue Kit (Quiagen A c c e p t e d M a n u s c r i p t GmbH, Hilden, Germany) following the protocol of the manufacturer in a final elution volume of 100 µl.

  A c c e p t e d M a n u s c r i p t Dye terminator kit v.1.1 and analysed on an ABI prism 310 automated DNA sequencer (both Applied Biosystems, Foster City, USA).

  the methods above. Verification of the presence of AdV in cell culture was carried out by PCR amplification of AdV specific DNA in cell culture supernatant. Sequencing of the amplicons verified that there are two different AdV types (SnAdV-1 and -2) in the two isolates. All of the above listed isolates could be further passaged on homologous cells.

  by PCR. Continuous efforts are being made to obtain a pure culture of this AdV.

  Abbreviations: AdV = adenovirus, o/cl = oral & cloacal, PMV = paramyxovirus, Sn =

  Figure legends

( a )

 a Phylogenetic distance tree, based on 443 bp of the large RNA-dependent RNA polymerase (L) gene. GenBank accession numbers for the reptilian PMV: Bush viper virus /ATCC-VR-1409/ (AF286043), CeraCe-98 (AF351137), Crot-GER03 (GQ277611), CrotX1-96 (AF349405), Dasy-GER00 (GQ277613), ElaGut-91 (AF349408), Fer de Lance virus (NC_005084), GonoGER-85 (AF349404), Igu-GER00 + (GQ277617), Neotropical rattlesnake virus /ATCC-VR-1408/ (AF286045),

  Crot1-VA95, Crot2-OH90, Crot3-CA98, Ela1-GER94, Ela2-GER93, Ela-FL93, Lamp-MD96, More-GER86, Pyth-GER88, Trim-MD97. (b) Phylogenetic distance tree based on 351 bp of the hemagglutinin-neuraminidase (HN) gene. The maximum likelihood analysis resulted in the same topology tree as the FITCH analysis, bootstrap values from both are indicated on the branches.

Fig

  Fig. 1/b

Table legend Table 1 . Virus positive corn snake samples.

 legend1

A c c e p t e d M a n u s c r i p t


Matiasovic Jan

Stepanova Hana

Volf Jiri

Kubala Lukas

Ovesna Petra

Rychlik Ivan

Faldyna Martin

Jan Matiasovic
email: matiasovic@vri.cz

Influence of the lipopolysaccharide structure of Salmonella enterica serovar Enteritidis on interactions with pig neutrophils

Keywords: Salmonella, pig, pathogen-associated molecular patterns, lipopolysaccharide, neutrophil

The key process for immune response development is the recognition of bacteria by the immune system of the host based on the sensing of pathogen-associated molecular patterns (PAMP). One of the most important PAMP is the lipopolysaccharide (LPS) molecule, a complex molecule present in the outer membrane of Gram negative bacteria. In this study we were interested in how different parts of the LPS of Salmonella enterica serovar Enteritidis are recognized by porcine neutrophils. To this aim, we constructed S. Enteritidis mutants with *Manuscript rfaL and rfaC genes disabled in the attachment of the O-antigen and in the synthesis of the inner oligosaccharide core of LPS, respectively. We found that in the absence of serum, both the rfa mutants associated with neutrophils and stimulated them for reactive oxygen species (ROS) production significantly more than the wild-type strain. Addition of polymyxin B, which neutralized lipid A, the endotoxic moiety of LPS, effectively decreased the association of the wild-type strain and the rfaC mutant with neutrophils, but not the rfaL mutant. This indicates that the oligosaccharide core newly exposed on the surface in the rfaL mutant, protected from interaction in the wild-type strain by the O-antigen but completely absent in the rfaC mutant, may represent a new ligand for porcine neutrophils that cannot be neutralized by polymyxin B.

A c c e p t e d M a n u s c r i p t

Introduction

After ingestion by susceptible host, Salmonella enterica serovar Enteritidis (S. Enteritidis) multiplies in gut lumen and invades intestinal epithelial cell which start to produce cytokines and chemokines (e.g. IL-8 or TNFα). The cytokine signaling than serve as a signal for recruitment of innate immune system cells, neutrophils in particular, to the site of infection [START_REF] Mccormick | Salmonella typhimurium attachment to human intestinal epithelial monolayers: transcellular signaling to subepithelial neutrophils[END_REF][START_REF] Scharek | The porcine immune system--differences compared to man and mouse and possible consequences for infections by Salmonella serovars[END_REF]. During primary infection, i.e. in the absence of specific antibodies, immune system cells interact with S. Enteritidis by two mutually complementary ways. The first one is based on binding of complement opsonized bacteria by complement receptors present on a surface of professional phagocytes and is aimed primarily to pathogen uptake and inactivation [START_REF] Underhill | Phagocytosis of microbes: complexity in action[END_REF]. The second one is focused on a more detailed perception and identification of a pathogen and is A c c e p t e d M a n u s c r i p t 3 based on the recognition of structural motifs common to many pathogens, i.e. pathogenassociated molecular patterns (PAMP). In the case of S. Enteritidis, the most important PAMPs include flagellin recognized by Toll-like receptor (TLR) 5 and lipopolysaccharide (LPS) which is recognized by TLR4 receptor [START_REF] Gerold | What is the role of Toll-like receptors in bacterial infections?[END_REF].

LPS of Gram-bacteria consists of lipid A, KDO domain, oligosaccharide core and O-antigen [START_REF] Rickles | Structural features of Salmonella Typhimurium lipopolysacharide required for activation of tissue factor in human mononuclear cells[END_REF]. Full sized LPS is synthesized in a step by step manner, addition of each new part of growing LPS being dependent on a particular enzyme, and it is therefore possible to generate single gene mutants which produce LPS of different complexity. Despite the complex nature of LPS, till now, out of the whole LPS molecule only the lipid A has been identified as a TLR4 ligand (Bäckhead et al. 2003[START_REF] Jerala | Structural biology of the LPS recognition[END_REF]. The interaction of LPS with TLR4 can be neutralized by antimicrobial peptides present in blood serum, or by polymyxin B, a cationic antimicrobial peptide structurally and functionally similar to those in blood serum [START_REF] Shimazu | MD-2, a molecule that confers lipopolysaccharide responsiveness on Toll-like receptor 4[END_REF][START_REF] Cruz | Effectiveness of polymyxin Bimmobilized fiber column in sepsis: a systematic review[END_REF].

Rough mutants, i.e. the mutants with incomplete LPS can grow under optimal conditions but are affected in the survival under the non-favorable conditions indicating a passive barrier function of distal parts of LPS. When we previously tested rfaC and rfaL mutants of S.

Enteritidis in vivo [START_REF] Karasova | Comparative analysis of Salmonella enterica serovar Enteritidis mutants with a vaccine potential[END_REF], we considered a hypothesis that the unusual heptoses and octoses present in the oligosaccharide core of LPS may serve as a ligand for either the association or recognition by receptors present in the cells of host immune system.

If so, these interactions could be detected by experiments with mutants defective in different steps of LPS biosynthesis performed in the presence and absence of polymyxin B. Polymyxin B should neutralize lipid A-TLR4 interaction and the residual interactions would be a function of the remaining parts of LPS present in the mutants. In this study we therefore A c c e p t e d M a n u s c r i p t 4 tested a mutant with inactivated rfaL gene which was unable to attach O-antigen to the core LPS and a mutant with inactivated rfaC gene which expressed LPS which consisting of only the lipid A and KDO domain, together with the wild strain for their ability to associate with porcine neutrophils in vitro and stimulate them for ROS production. We found that both the rfa mutants associated with porcine neutrophils more and stimulated them for a higher ROS production than did the wild type S. Enteritidis. Interestingly, polymyxin B treatment decreased the interaction of the rfaC mutant with porcine neutrophils to a greater extent than the interaction of the rfaL mutant. This indicates, that the oligosaccharide core newly exposed on the surface of the rfaL mutant after the removal of O-antigen, but protected from any interaction by the O-antigen in the wild type strain or completely absent in the rfaC mutant, may represent a new ligand for porcine neutrophils which cannot be neutralized by polymyxin B.

Material and methods

S.

Enteritidis 147 (wild-type strain) was used in this study [START_REF] Methner | Intestinal colonisation-inhibition and virulence of Salmonella phoP, rpoS and ompC deletion mutants in chickens[END_REF]. The mutations in the rfaL and rfaC genes were generated by a one-step PCR inactivation method as described previously [START_REF] Karasova | Comparative analysis of Salmonella enterica serovar Enteritidis mutants with a vaccine potential[END_REF]. All of the strains were transformed with the pFPV25.1 plasmid constitutively expressing green fluorescent protein (GFP). The strains were grown for 16 h in Luria Bertani (LB) media, then washed twice with Dulbecco's phosphate buffered saline (DPBS, Cambrex, IA) and stored in PBS supplemented with 10% glycerol at -70 °C until use. The bacterial concentration was determined both spectrophotometrically and by serial dilutions and plating on LB agar followed by colony counting. The integrity of bacterial cell membranes before and during the experiments was measured by propidium iodide staining followed by flow cytometry.

A c c e p t e d M a n u s c r i p t 5 Peripheral blood was taken from the vena jugularis of seven three-month-old pigs. White blood cells (WBC) were isolated independently by hypotonic lysis of the erythrocytes as follows: 5 ml of heparinized blood was mixed with 40 ml of apyrogenic distilled water for 20 s and then the osmotic pressure was normalized with 5 ml of ten times concentrated DPBS.

The WBC were washed twice with 1× DPBS and finally resuspended in Hank's balanced salt solution (HBSS, Cambrex, IA) at a concentration of 10 7 per ml.

One million WBC from each pig were separately incubated at 37 °C in 100 µl of HBSS with 10 7 (multiplicity of infection of 10) GFP-expressing wild-type strain or rfaC and rfaL mutants under serum-free conditions [START_REF] Underhill | Phagocytosis of microbes: complexity in action[END_REF][START_REF] Blander | Regulation of phagosome maturation by signals from Toll-like receptors[END_REF]. In parallel, for blocking lipid A signalling, the WBC were treated for 30 min before and during the infection with all strains [START_REF] Tohno | Molecular cloning of porcine RP105/MD-1 involved in recognition of extracellular phosphopolysaccharides from Lactococcus lactis ssp[END_REF]) by polymyxin B (Sigma Aldrich, Germany) at a concentration of 20 µg/ml.

For the flow cytometry analysis, the cells were resuspended in PBS containing 0.2% gelatine from cold water fish skin (Sigma Aldrich, Germany), 0.1% NaN 3 and 0.63 mM EDTA (Sigma Aldrich, Germany). Neutrophils were gated based on forward and side scatter characteristics (R1, Fig. 1) and GFP positivity was used for the detection of S. Enteritidis-positive neutrophils (R2, Fig. 1). The associations were measured at 30 min intervals during the 3 h incubation. Flow cytometry was performed with a FACSCalibur operated by CELLQuest software (BD Biosciences, MD).

The release of reactive oxygen species (ROS) was measured by 8-amino-5-chloro-7phenylpyridopyridazine (L-012, Wako Pure Chemical Industries, Ltd., Osaka, Japan)enhanced luminometry. Samples were run in triplicate with a 20 µM concentration of L-012 Correlations between neutrophil-Salmonella associations and ROS production were assessed using Pearson's correlation. Differences with a value of P < 0.05 were considered statistically significant. Statistica software (StatSoft, Inc., OK) was used for data analyses.

Results

First, we were interested in the time-dependent associations of all S. Enteritidis strains used with the neutrophils (Fig. 2). The wild-type strain association gradually increased over the 3 h period. When compared with time 0, significantly more wild-type strain associated neutrophils were recorded after 60 minutes of incubation (Fig. 2 Tab. A). At the end of the experiment, 22% of the neutrophils were associated with the wild-type strain. Mutant rfaC exhibited significantly higher associations as early as 30 minutes after the beginning of the A c c e p t e d M a n u s c r i p t 7 experiment, but rfaL, similarly to wild-type strain, at 60 minutes after the beginning of the experiment. In the case of the rfaC mutant, the peak of the association was at 60 minutes, when it reached 49% and then remained at the same level (there were no significant differences between time points 30-180 minutes). In the case of the rfaL mutant, the associations continuously increased over the 3 hours period, as confirmed by significantly different associations at time points 30 and 180 minutes. At the end of the experiment, 58% of the neutrophils were associated with rfaL mutant. The addition of polymyxin B significantly decreased the associations of wild-type S. Enteritidis and the rfaC mutant with porcine neutrophils when compared to the conditions without polymyxin B, but it had no effect on the association of the rfaL mutant with the neutrophils (Fig. 2 Tab. B).

In order to investigate not only the ability of the strains used to bind to neutrophils but also their competence in stimulating them, in the next experiment we determined ROS production by luminometry (Fig. 3). The ROS production of neutrophils stimulated by the wild-type strain gradually increased during the first 150 minutes, although when compared with time 0, significantly higher ROS production was recorded after 90 minutes of incubation (Fig. 3 Tab.

A). The rfaL and rfaC mutants stimulated ROS production more rapidly and to a greater extent than the wild-type strain, both stimulating significantly higher ROS production as early as 30 minutes after the beginning of the experiment when compared with time 0. The maximum ROS production stimulated by both rfa mutants was observed 60 minutes after infection, followed by a slow decrease in time points 120-180 minutes for the rfaL mutant. In the case of the rfaC mutant, no change after time point 60 mintes was observed. The addition of polymyxin B decreased the global ROS production for all strains (Fig. 3 Tab. B), although this decrease was more significant for the wild-type strain and the rfaC mutant than for the rfaL mutant. Moreover, when the effect of polymyxin B was determined in individual A c c e p t e d M a n u s c r i p t 8 measurement time points, out of six measurements (not considering time 0), polymyxin B significantly decreased the ROS production stimulated by the wild-type and the rfaC mutant four and five times, respectively. However, in the case of the rfaL-stimulated ROS production, this was significantly reduced only once, i.e. at 60 minutes after the beginning of the experiment.

When analysing the correlations between the neutrophil-Salmonella association and ROS production, a significantly positive correlation was observed for the wild-type strain and the rfaC mutant, both with and without polymyxin B (Tab. 1). On the other hand, the same correlation was insignificant for the rfaL mutant, regardless of the presence or absence of polymyxin B (Tab. 1). This was most likely caused by maximal ROS production stimulated by the rfaL mutant 60 minutes after the beginning of the experiment, whereas the association continuously increased throughout the whole experiment.

Discussion

In this study, we were interested in the interactions between porcine neutrophils and S.

Enteritidis mutants defective in the synthesis of full sized LPS. Since we were interested in the recognition of S. Enteritidis antigens independent of Fc or complement receptors present in the plasma membrane of neutrophils, the study was performed in the absence of serum.

Such conditions have been already used for the identification of neutrophil receptors that recognize the surface structures of bacteria [START_REF] Underhill | Phagocytosis of microbes: complexity in action[END_REF][START_REF] Blander | Regulation of phagosome maturation by signals from Toll-like receptors[END_REF].

The complete structure of LPS present in the wild-type strain acted as an inhibitor of the neutrophil-Salmonella association since removal of the O-antigen in the rfaL mutant and the A c c e p t e d M a n u s c r i p t oligosaccharide core in the rfaC mutant led to a more intensive association. This masking effect of the O-antigen has also been documented in Escherichia coli (Bäckhead et al., 2003), and the ability of neutrophils to bind S. Enteritidis was negatively correlated with the complexity of LPS. The dynamic of ROS production was very slow for the wild-type strain but much faster for both mutants, similar to human neutrophils infected with the S.

Typhimurium 395 MR10 strain [START_REF] Lock | Characteristics of the granulocyte chemiluminescence reaction following an interaction between human neutrophils and Salmonella typhimurium bacteria[END_REF]. Maximal ROS production was observed at 60 minutes after the interaction of both rfa mutants with the neutrophils.

Although we do not have any explanation for this observation, we can exclude that maximal ROS production was achieved by the neutrophils since when the opsonized zymosan particles were used for neutrophil stimulation, ROS production ten times higher than that induced by the rfa mutants was recorded (data not shown).

Although significantly different associations and ROS induction profiles were observed for the wild-type strain and the rfaC mutant, when we analysed the correlation between these two factors, a positive and significant correlation was observed for these two strains. On the other hand, in the case of the rfaL mutant, the correlation between neutrophil association and ROS production appeared to be insignificant, indicating a difference in the interactions between the two rfa mutants and porcine neutrophils. This difference became even more apparent when the experiments were performed in the presence of polymyxin B. Polymyxin B had a lower effect on both the association and the ROS production by porcine neutrophils induced by the rfaL mutant than by the rfaC mutant. This indicated that the interaction of the rfaL mutant with porcine neutrophils was less dependent on lipid A, and therefore a new ligand absent in the wild-type strain and the rfaC mutant had appeared in the rfaL mutant. The likely candidates are the core oligosaccharides containing unusual heptose residues. These sugar residues were protected by the O-antigen in the wild-type strain and were absent in the rfaC A c c e p t e d M a n u s c r i p t 10 mutant. We also considered that completely new ligands not related to the LPS structure could appear in the rfaL mutant, but similar interactions of human monocytes were described with purified LPS isolated from wild-type S. Typhimurium and mutants similar to rfaC and rfaL in the presence of polymyxin B [START_REF] Rickles | Structural features of Salmonella Typhimurium lipopolysacharide required for activation of tissue factor in human mononuclear cells[END_REF]. Based on our and [START_REF] Rickles | Structural features of Salmonella Typhimurium lipopolysacharide required for activation of tissue factor in human mononuclear cells[END_REF] observations, it is probable that new PAMP was really represented by the core oligosaccharides of the LPS, rather than by the unmasking of new structures independent of LPS.

Conclusion

When compared with the wild-type S. Enteritidis, the rfaC and rfaL mutants exhibited increased associations with porcine neutrophils and stimulated them for a higher production of reactive oxygen species. Polymyxin B treatment decreased the interactions of the wildtype strain and the rfaC mutant with neutrophils more effectively than in the case of rfaL mutant, thus showing a lower influence of lipid A recognition on rfaL mutant interactions with porcine neutrophils. A: Table of significance levels of the differences between sampling time points tested by oneway ANOVA; significant levels (P < 0.05) are in bold.

1 Global difference between sampling time points 2 Test of differences between sampling times (Tukey's post hoc test): significantly different groups (P < 0.05) are marked by different letters ("a" differ from "bc", but not "ab", etc.). B: Table of significance levels of the F tests of repeated measures two-way ANOVA incorporating time and Salmonella strains and treatments as the main factors, significant levels (P < 0.05) are in bold.

1 Significance level (P) of F tests for Salmonella strains and treatment factors; all significance levels (P) of the F tests for the time factor and interactions were lower than 0.001 and are not shown in the table.

2 Tests of differences between Salmonella strains and treatments performed separately within each sampling time point (Tukey's post hoc tests). A: Table of significance levels of the differences between sampling time points tested by oneway ANOVA, significant levels (P < 0.05) are in bold. 1 Global difference between sampling time points.

2 Test of differences between sampling times (Tukey's post hoc test): significantly different groups (P < 0.05) are marked by different letters ("a" differ from "bc", but not "ab", etc.). B: Table of significance

  A c c e p t e d M a n u s c r i p t 6 using a multi-detection microplate reader Synergy TM 2 (BioTek Instruments, VT). Neutrophils were stimulated with opsonized zymosan particles (InvivoGen, CA) as a positive control.In all of the figures, the mean ±standard errors are shown. The data obtained were tested for normality by the Kolmogorov-Smirnov test and the homogeneity of the variance among the Salmonella strains and treatments was compared by the F ratio test. The significance (P) of the differences between sampling time points for a given strain or treatment was tested by one-way ANOVA followed by Tukey's post hoc tests for comparisons among given time points. The repeated measures model of the two-way analysis of variance (rmANOVA) was used to analyse the influence of Salmonella strains, treatments and time on the neutrophil-Salmonella association and ROS production followed by Tukey's post hoc tests for the mutual comparison of Salmonella types within the given single time point of the follow-up.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Association of S. Enteritidis and its rfaC and rfaL mutants with the neutrophils and

Figure 3 .

 3 Figure 3. Kinetics of ROS production and the influence of polymyxin B treatment during 3 h

  levels (P) of F tests of repeated measures two-way ANOVA incorporating time, Salmonella strains and treatments as the main factors; significant levels (P < 0.05) are in bold 1 Significance level (P) of F tests for Salmonella strains and treatment factors; all significance levels of the F tests for time factor and interactions were lower than 0.001 and are not shown in the table.

2

  Tests of differences between Salmonella strains and treatments performed separately within each sampling time point (Tukey's post hoc tests).

  Figure1Click here to download high resolution image

TableTable 1 .

 1 Correlations between the neutrophil-Salmonella associations and ROS production of the wild-type S. Enteritidis (wt), rfaC and rfaL mutant strains in the absence and presence of polymyxin B (PmxB). Significant correlations and corresponding P values are in bold.

Acknowledgements

This study was supported by the Ministry of Agriculture of the Czech Republic (projects QH81062 and MZe 0002716202) and the Ministry of Education, Youth and Sports of the Czech Republic ( AdmireVet project: CZ.1.05/2.1.00/01.0006, ED0006/01/01).

Conflict of interest statement

The authors have no financial or personal relationships with other people or organizations that could have inappropriately influenced our work.


D Ieşan

Torsion of chiral Cosserat elastic rods

Keywords: Torsion of chiral Cosserat elastic rods Chiral materials, Cosserat cylinder, Torsion, Bending

This paper is concerned with the torsion of isotropic chiral Cosserat elastic cylinders. First, the generalized plane strain problem is defined and an existence result is presented. Then, the three-dimensional problem is reduced to the study of some generalized plane strain problems. In general, the torsion of the cylinder is accompanied by bending and extension. The method is applied to study the torsion of a circular cylinder.

Introduction

In recent years the behaviour of chiral materials has been the object of intensive research. A body which is not isotropic with respect to inversion is called chiral. Chirality can be observed in some carbon nanotubes, biological molecules (DNA), bones, honeycomb structures as well as in composites with inclusions.

It is well-known that chiral effects cannot be described within classical elasticity (see [START_REF] Lakes | Elastic and viscoelastic behaviour of chiral materials[END_REF]. The theory of Cosserat elasticity is adequate to describe the mechanical behaviour of chiral elastic bodies [START_REF] Lakes | Elastic and viscoelastic behaviour of chiral materials[END_REF][START_REF] Donescu | On the Young's modulus of an auxetic composite structure[END_REF]. The Cosserat theory studies continua with oriented particles which have the six degrees of freedom of a rigid body. The theory of Cosserat elastic solids was extensively studied in the last decades. For the historical development and the analysis of various results on the subject see [START_REF] Nowacki | Theory of Asymmetric Elasticity[END_REF], [START_REF] Ciarletta | Non-Classical Elastic Solids[END_REF], [START_REF] Eringen | Microcontinuum Field Theories, I: Foundations and Solids[END_REF], [START_REF] Rubin | Cosserat Theories: Shells, Rods and Points[END_REF], [START_REF] Dyszlewicz | Micropolar Theory of Elasticity[END_REF], [START_REF] Jasiuk | From lattices and composites to micropolar continua. Analysis of materials with complex microstructure[END_REF]. A large number of papers have been devoted to the study of chiral Cosserat elastic bodies [START_REF] Nowacki | Green functions for a hemitropic micropolar continuum[END_REF]Lakes and Email address: iesan@uaic.ro (D. Ieşan).

Preprint submitted to Elsevier

19 July 2010 [START_REF] Lakes | Noncentrosymmetry in micropolar elasticity[END_REF][START_REF] Dyszlewicz | Selected one-dimensional problems of elastostatics and thermoelastostatics of the hemitropic micropolar medium[END_REF]. This class of materials continues to attract attention both from the theoretical and the technical point of view [START_REF] Healey | Material symmetry and chirality in nonlinearly elastic rods[END_REF][START_REF] Dyszlewicz | Micropolar Theory of Elasticity[END_REF][START_REF] Teodorescu | On the wave propagation in chiral media[END_REF][START_REF] Natroshvili | Representation formulae of general solutions in the theory of hemitropic elasticity[END_REF][START_REF] Khurana | Longitudinal wave response of a chiral slab interposed between micropolar solid half-spaces[END_REF]. A Cosserat material is called isotropic chiral if its symmetry group equals the proper orthogonal group. In the linear theory an isotropic chiral Cosserat elastic material is characterized by nine constitutive coefficients.

In this paper we study the deformation of homogeneous and isotropic chiral Cosserat elastic cylinders subjected to torsion, extension and bending. This work is motivated by the recent interest in the using of the isotropic chiral Cosserat elastic solid as model for cellular solids and composite materials [START_REF] Prall | Properties of a chiral honeycomb with a Poisson's ratio-1[END_REF]Lakes, 1998[START_REF] Lakes | Elastic and viscoelastic behaviour of chiral materials[END_REF][START_REF] Donescu | On the Young's modulus of an auxetic composite structure[END_REF][START_REF] Chandraseker | An atomisticcontinuum Cosserat rod model of carbon nontubes[END_REF] and for bones [START_REF] Lakes | Slow compressional wave propagation in wet human and bovine cortical bone[END_REF]. We note that [START_REF] Lakes | Noncentrosymmetry in micropolar elasticity[END_REF] studied the problem of a chiral Cosserat elastic rod of circular cross section, stretched axially and free to twist. In this paper we consider the problem of extension, bending and torsion of a right cylinder with general cross section. First, we introduce the generalized plane strain problem and present an existence result. In contrast with the case of centrosymmetric solids, in the generalized plane strain of a chiral Cosserat elastic cylinder, all components of displacement vector and microrotation vector are different from zero. We introduce four special generalized plane strain problems characterized by external data which depend only on the constitutive coefficients. The solutions of these problems are used to solve the torsion problem. We show that, in general, the torsion of an isotropic chiral Cosserat elastic cylinder is accompanied by extension and bending. The results are applied to study the torsion of a circular cylinder. We determine the solutions of the auxiliary generalized plane strain problems for a circular domain. Then, the displacement and microrotation vector fields in the torsion problem are determined.

Basic equations

Throughout this paper B denotes the interior of a right cylinder of length h with open cross-section Σ and lateral boundary Π (Fig. 1). We call ∂B the boundary of B, and designate by n i the components of the outward unit normal of ∂B. Throughout this paper a rectangular cartesian coordinate system Ox k , (k = 1, 2, 3), is used. The rectangular cartesian coordinate frame is chosen such that the x 3 -axis is parallel to the generators of B and the x 1 Ox 2 plane contains one of terminal cross-sections. We denote by Σ 1 and Σ 2 , respectively, the cross-section located at x 3 = 0 and x 3 = h.

Fig. 1. A general prismatic rod

We assume that the generic cross-section Σ is a simply connected regular region. Let L be the boundary of the region Σ. We shall employ the usual summation and differentiation conventions: Greek subscripts are understood to range over the integers (1, 2), whereas Latin subscripts (unless otherwise specified) to the range (1, 2, 3); summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. Letters in boldface stand for tensors of an order p ≥ 1, and if v has the order p, we write v ij...s (p subscripts) for the components of v in the cartesian coordinate frame.

Throughout this paper we consider the linear theory of homogeneous and isotropic chiral Cosserat elastic bodies. Let u be the displacement vector field on B. We denote by ϕ the microrotation vector field. The strain measures are given by

e ij = u j,i + ε jik ϕ k , κ ij = ϕ j,i , (1)
where ε ijk is the alternating symbol. Let t ij be the stress tensor and let m ij be the couple stress tensor. The equilibrium equations, in the absence of body loads, are t ji,j = 0, m ji,j + ε irs t rs = 0.

(2)

The constitutive equations can be presented in the form [START_REF] Lakes | Noncentrosymmetry in micropolar elasticity[END_REF][START_REF] Dyszlewicz | Micropolar Theory of Elasticity[END_REF])

t ij = λe rr δ ij + (µ + κ)e ij + µe ji + C 1 κ ss δ ij + C 2 κ ji + C 3 κ ij , m ij = ακ rr δ ij + βκ ji + γκ ij + C 1 e rr δ ij + C 2 e ji + C 3 e ij , (3)
where δ ij is Kronecker's delta, and γ, µ, κ, α, β, γ, C 1 , C 2 and C 3 are constitutive constants. In the case of a centrosymmetric material the coefficients C 1 , C 2 and C 3 are equal to zero. The surface force and the surface moment acting at a regular point of ∂B are given by

t i = t ji n j , m i = m ji n j .
We assume that the considered cylinder is free of lateral loads. Thus, we have the following conditions

t αi n α = 0, m αi n α = 0 on Π. ( 4
)
We suppose that the cylinder B is subjected to extension, bending and torsion. Let R = (0, 0, R 3 ) and M = (M 1 , M 2 , M 3 ) be prescribed vectors representing the resultant force and the resultant moment about O of the tractions acting on Σ 1 . On Σ 2 there are tractions applied so as to satisfy the equilibrium conditions of the body. Consequently, for x 3 = 0 we have the conditions

Σ 1 t 3α da = 0, (5)
Σ 1 t 33 da = -R 3 , (6) Σ 1 (x α t 33 -ε 3αβ m 3β )da = ε αβ3 M β , (7)
Σ 1 (ε αβ3 x α t 3β + m 33 )da = -M 3 . (8)
The problem consists in finding of the functions u i and ϕ i of class C 2 (B) ∩ C 1 (B) which satisfy the equations ( 1)-(3) on B, the conditions (4) on the lateral surface, and the conditions ( 5)-( 8) on the end Σ 1 . We have introduced the mechanical loads R 3 and M α in order to emphasize the torsional effects on the deformation of the cylinder.

We assume that Σ is C ∞ -smooth. We have chosen this hypothesis in order to best describe the method for the solution of the problem. We consider only a "C ∞ -theory" but it is possible to get a classical solution for more general assumptions of regularity. In what follows we suppose that the elastic potential is a positive definite quadratic form. The restrictions imposed by this hypothesis on the constitutive coefficients have been presented in various works [START_REF] Lakes | Noncentrosymmetry in micropolar elasticity[END_REF][START_REF] Dyszlewicz | Micropolar Theory of Elasticity[END_REF]. We note only the following inequalities

κ > 0, 2µ + κ > 0, γ + β > 0, γ -β > 0, λ + 2µ + κ > 0, (λ + 2µ + κ)(α + β + γ) -(C 1 + C 2 + C 3 ) 2 > 0. ( 9
)
3 Generalized plane strain

Let us assume now that a body force f and a body couple g are prescribed on B. Moreover, we consider that on the lateral boundary of the cylinder there are prescribed the surface force t and the surface moment m. We suppose that f , g, t and m are all independent of the axial coordinate. We define the state of generalized plane strain of the cylinder to be that state in which the displacement vector and microrotation vector are independent of the axial coordinate,

u i = u i (x 1 , x 2 ), ϕ i = ϕ i (x 1 , x 2 ), (x 1 , x 2 ) ∈ Σ 1 . ( 10
)
The above restrictions, in conjunction with the geometrical equations (1) and the constitutive equations ( 2), imply that e ij , κ ij , t ij and m ij are all independent of the axial coordinate. The equations (1) reduce to

e αi = u i,α + ε iαj ϕ j , e 3i = ε i3β ϕ β , κ αi = ϕ i,α , κ 3i = 0. ( 11
)
The constitutive equations for a generalized plane strain are given by

t αβ = λe ρρ δ αβ + (µ + κ)e αβ + µe βα + C 1 κ ρρ δ αβ + C 2 κ βα + C 3 κ αβ , t α3 = (µ + κ)e α3 + µe 3α + C 3 κ α3 , t 3α = (µ + κ)e 3α + µe α3 + C 2 κ α3 , t 33 = λe ρρ + C 1 κ ρρ , (12)
m νη = ακ ρρ δ νη + βκ ην + γκ νη + C 1 e ρρ δ νη + C 2 e ην + C 3 e νη , m α3 = γκ α3 + C 2 e 3α + C 3 e α3 , m 3α = βκ α3 + C 2 e α3 + C 3 e 3α , m 33 = ακ ρρ + C 1 e ρρ .
The equations of equilibrium can be written as

t βi,β + f i = 0, m βi,β + ε irs t rs + g i = 0, (13)
on Σ 1 . The conditions on the lateral surface become

t βi n β = t i , m βi n β = m i on L. ( 14
)
We assume that f i , g i , t i and m i are functions of class C ∞ .

The generalized plane strain problem consists in finding of the displacement u i and microrotations ϕ i which satisfy the equations ( 11)-( 13) on Σ 1 , and the boundary conditions ( 14) on L. In view of ( 11) and ( 12), the equations of equilibrium can be expressed in terms of the functions u i and ϕ i in the form

(µ + κ)∆u α + (λ + µ)u β,βα + κε αβ3 ϕ 3,β + C 3 ∆ϕ α + (C 1 + C 2 )ϕ β,βα + f α = 0, (µ + κ)∆u 3 + κε 3βα ϕ α,β + C 3 ∆ϕ 3 + f 3 = 0, C 3 ∆u ν + (C 1 + C 2 )u ρ,ρν + κε νβ3 u 3,β + γ∆ϕ ν + (15) + (α + β)ϕ ρ,ρν + 2(C 3 -C 2 )ε νη3 ϕ 3,η -2κϕ ν + g ν = 0, C 3 ∆u 3 + κε 3νη u η,ν + γ∆ϕ 3 + 2(C 3 -C 2 )ε 3νη ϕ η,ν -2κϕ 3 + g 3 = 0,
on Σ 1 , where ∆ is the two-dimensional Laplacian.

In the case of the generalized plane strain the elastic potential W is given by 2W = λe ρρ e νν + (µ + κ)e αβ e αβ + µe βα e αβ + (µ + κ)e α3 e α3 + + 2µe α3 e 3α + (µ + κ)e 3α e 3α + ακ ρρ κ νν + βκ ην κ νη + (16)

+ γκ νη κ νη + γκ α3 κ α3 + 2C 1 κ ρρ e νν + 2C 2 e αβ κ βα + + 2C 3 e αβ κ αβ + 2C 2 e 3α κ α3 + 2C 3 e α3 κ α3 . Let u * 1 = c 1 -c 4 x 2 , u * 2 = c 2 + c 4 x 1 , u * 3 = c 3 , ϕ * 1 = 0, ϕ * 2 = 0, ϕ * 3 = c 4 , (17)
where c s , (s = 1, 2, 3, 4), are arbitrary constants. Then

U * = (u * 1 , u * 2 , u * 3 , ϕ * 1 , ϕ * 2 , ϕ * 3
) is called a plane rigid vector field.

Theorem 1. If the elastic potential W is a positive definite quadratic form, then any two solutions of the generalized plane strain problem (11) -( 14) are equal modulo a plane rigid vector field.

Proof. In view of ( 11)-( 13) and ( 16), we get

2W = t βi e βi + t 3i e 3i + m βi κ βi = = (t βi u i + m βi ϕ i ) ,β + f i u i + g i ϕ i .
With the help of the divergence theorem we obtain 2

Σ 1 W da = L (t βi u i + m βi ϕ i )n β ds + Σ 1 (f i u i + g i ϕ i )da. ( 18
)
Let (u ′ i , ϕ ′ i ) and (u ′′ i , ϕ ′′ i ) be two solutions of the problem ( 11)-( 14). We denote

u 0 i = u ′ i -u ′′ i and ϕ 0 i = ϕ ′ i -ϕ ′′ i .
The functions u 0 i and ϕ 0 i satisfy the generalized plane strain problem corresponding to null data. Thus, from (18) we find that

Σ 1 W 0 da = 0, ( 19
)
where W 0 is the elastic potential corresponding to u 0 i and ϕ 0 i . Since W 0 is positive definite, we find e 0 ji = 0, and κ 0 αi = 0 on Σ 1 . These equations imply that (u 0 1 , u 0 2 , u 0 3 , ϕ 0 1 , ϕ 0 2 , ϕ 0 3 ) is a plane rigid vector field.

The six-dimensional vector

U * = (u * 1 , u * 2 , u * 3 , ϕ * 1 , ϕ * 2 , ϕ *
3 ) defined by ( 17) can be expressed in the form

U * = 4 k=1 c k U (k) ,
where U (1) = (1, 0, 0, 0, 0, 0), U (2) = (0, 1, 0, 0, 0, 0),

U (3) = (0, 0, 1, 0, 0, 0), U (4) = (-x 2 , x 1 , 0, 0, 0, 1). ( 20
)
Let us consider the equations ( 11)-( 13) on Σ 1 with the boundary conditions

t βi n β = 0, m βi n β = 0 on L. ( 21
)
Following [START_REF] Fichera | Existence Theorems in Elasticity[END_REF], a C ∞ solution of the boundary valued problem ( 11)-( 13), ( 21) exists if and only if

Σ 1 4 j=1 H j U (k) j da = 0, (k = 1, 2, 3, 4), (22)
where

H = (H 1 , H 2 , . . . , H 6 ) with H i = f i , H 3+i = g i , and
U (k) = (U (k) 1 , U (k) 2 , . . . , U (k) 6 ), U (1) 1 = 1, U (1) 1+s = 0, (s = 1, 2, . . . , 5), U (2) 2 = 1, U (2) 1 = U (2) 2+m = 0, 6 (m = 1, 2, 3, 4), U (3) 3 = 1, U (3) α = U (3) 3+p = 0, U (4) 1 = -x 2 , U (4) 2 = x 1 , U (4) 6 = 1, U (4) 2+k = 0. The conditions (22) reduce to Σ 1 f i a = 0, Σ 1 (x 1 f 2 -x 2 f 1 + g 3 )da = 0.
It is a simple matter to see that in the case of the boundary value problem ( 11)-( 14), the above conditions are replaced by

Σ 1 f i da + L t i ds = 0, Σ (ε αβ3 x α f β + g 3 )da + L (ε αβ3 x α t β + m 3 )ds = 0. (23)
Thus, we have the following existence result.

Theorem 2. The boundary value problem (11) -( 14) has solutions if and only if the relations (23) hold.

In the next section we will have occasion to use four generalized plane strain problems P (k) , (k = 1, 2, 3, 4). The problem P (1) corresponds to the body loads

f (1) i = λδ 1i , g (1) i = (C 1 + C 3 -C 2 )δ 1i , (24)
and to the boundary data

t (1) α = -λx 1 n α , t (1)
3 = C 2 n 2 , m (1) ρ = -C 1 x 1 n ρ , m (1)
3 = βn 2 . ( 25
)
In the problem P (2) the body loads and the boundary data are

f (2) i = λδ 2i , g (2)
i = (C 1 + C 3 -C 2 )δ 2i , t (2) α = -λx 2 n α , t (2)
3 = -C 2 n 1 , m (2) α = -C 1 x 2 n α , m (2)
3 = -βn 1 . (26)
The problem P (3) corresponds to the following loading

f (3) i = 0, g (3)
i = 0, t (3) α = -λn α , t (3)
3 = 0, m (3)
β = -C 1 n β , m (3)
3 = 0. ( 27
)
We denote by P (4) the problem characterized by the following body loads and boundary tractions

f (4) i = 0, g (4) α = -κx α , g (4) 3 = 0, t (4) α = -C 1 n α , t (4) 3 = -µε 3ρα x ρ n α , m (4) ρ = -αn ρ , m (4) 3 = C 2 ε 3ρν x ν n ρ . ( 28
)
We note that the necessary and sufficient conditions (23) for the existence of the solution are satisfied for each boundary value problem P (k) , (k = 1, 2, 3, 4). We denote by u

(k) i , ϕ (k)
i , e

(k) ij , κ (k) ij , t (k) ij and m (k)
ij the displacement, microrotation, strain measures, stress tensor and couple stress tensor in the problem P (k) , respectively. Clearly, the problems P (k) are characterized by the geometrical equations

e (k) αi = u (k) i,α + ε iαj ϕ (k) j , e (k) 3i = ε i3β ϕ (k) β , κ (k) αi = ϕ (k) i,α , (29)
the constitutive equations

t (k) αβ = λe (k) ρρ δ αβ + (µ + κ)e (k) αβ + µe (k) βα + C 1 κ (k) ρρ δ αβ + C 2 κ (k) βα + C 3 κ (k) αβ , t (k) α3 = (µ + κ)e (k) α3 + µe (k) 3α + C 3 κ (k) α3 , t (k) 3α = (µ + κ)e (k) 3α + µe (k) α3 + C 2 κ (k) α3 , t (k) 33 = λe (k) ρρ + C 1 κ (k) ρρ , (30)
m (k) νη = ακ (k) ρρ δ νη + βκ (k) ην + γκ (k) νη + C 1 e (k) ρρ δ νη + C 2 e (k) ην + C 3 e (k) νη , m (k) α3 = γκ (k) α3 + C 2 e (k) 3α + C 3 e (k) α3 , m (k) 3α = βκ (k) α3 + C 2 e (k) α3 + C 3 e (k) 3α , m (k) 33 = ακ (k) ρρ + C 1 e (k) ρρ ,
and the equations of equilibrium

t (k) βi,β + f (k) i = 0, m (k) βi,β + ε irs t (k) rs + g (k) i = 0, (31)
on Σ 1 , and the following boundary conditions

t (k) βi n β = t (k) i , m (k) βi n β = m (k) i on L, (32)
(k = 1, 2, 3, 4). Here, f

(k) i , g (k)
i , t

(k) i and m

(k) i are defined by ( 24)-( 28).

Solution of the problem

In this section we reduce the three-dimensional problem to the study of four generalized plane strain problems. Following [START_REF] Ieşan | On Saint-Venant's problem[END_REF] we seek the solution of the problem formulated in Section 2 in the form

u α = - 1 2 a α x 2 3 + ε 3βα a 4 x β x 3 + 4 k=1 a k u (k) α , u 3 = (a 1 x 1 + a 2 x 2 + a 3 )x 3 + 4 k=1 a k u (k) 3 , (33)
ϕ α = ε 3αβ a β x 3 + 4 k=1 a k ϕ (k) α , ϕ 3 = a 4 x 3 + 4 k=1 a k ϕ (k) 3 , (x 1 , x 2 , x 3 ) ∈ B,
where u (k) i and ϕ

(k) i are the displacements and microrotations from the problems P (k) introduced in Section 3, and a k , (k = 1, 2, 3, 4), are unknown constants. It follows from ( 1) and ( 30) that

e αi = 4 k=1 a k e (k) αi , e 3α = ε 3βα x β a 4 + 4 k=1 a k e (k) 3α , e 33 = a 1 x 1 + a 2 x 2 + a 3 , κ αi = 4 k=1 a k κ (k) αi , κ 3α = ε 3αβ a β , κ 33 = a 4 , (34)
where e

(k) ij and κ (k)
ij are given by ( 29). By ( 34) and the constitutive equations (3) we obtain

t αβ = λ(a 1 x 1 + a 2 x 2 + a 3 )δ αβ + C 1 δ αβ a 4 + 4 k=1 a k t (k) αβ , t α3 = µε 3βα x β a 4 + C 2 ε 3αβ a β + 4 k=1 a k t (k) α3 , t 3α = (µ + κ)ε 3βα x β a 4 + C 3 ε 3αβ a β + 4 k=1 a k t (k) 3α , t 33 = (λ + µ + κ)(a 1 x 1 + a 2 x 2 + a 3 ) + (C 1 + C 2 + C 3 )a 4 + 4 k=1 a k t (k) 33 , (35) m νη = [αa 4 + C 1 (a 1 x 1 + a 2 x 2 + a 3 )]δ νη + 4 k=1 a k m (k) νη , m ν3 = βε 3νρ a ρ + C 2 ε 3ρν x ρ a 4 + 4 k=1 a k m (k) ν3 , m 3ν = γε 3νρ a ρ + C 3 ε 3ρν x ρ a 4 + 4 k=1 a k m (k) 3ν , m 33 = (α + β + γ)a 4 + (C 1 + C 2 + C 3 )(a 1 x 1 + a 2 x 2 + a 3 ) + 4 k=1 a k m (k) 33 ,
where t

(k) ij and m (k)
ij are the stress tensor and the couple stress tensor from the problem P (k) , (k = 1, 2, 3, 4).

It is a simple matter to see that the equations of equilibrium (2) and the boundary conditions (4) are satisfied on the basis of equations ( 31) and the boundary conditions (32). Let us study the end conditions (5). With the help of equilibrium equations (2) and boundary conditions (4) we can write

Σ 1 t 31 da = Σ 1 (t 13 -m ρ2,ρ )da = Σ 1 [(x 1 t ρ3 ) ,ρ -m α2,α ]da = = L (x 1 t α3 n α -m α2 n α )ds = 0.
Similarly we can prove that the second condition from ( 5) is satisfied.

The functions t ij and m ij from ( 35) can be expressed in the form

t ij = 4 k=1 a k s (k) ij , m ij = 4 k=1 a k π (k) ij , (36)
where

s (ν) αβ = λx ν δ αβ + t (ν)
αβ , s

(3)

αβ = λδ αβ + t (3) αβ , s (4)
αβ = C 1 δ αβ + t (4) αβ , s (ν) α3 = C 2 ε 3αν + t (ν) α3 , s (3) α3 = t (3) α3 , s (4) α3 = µε 3ρα x ρ + t (4) α3 , s (ν) 3α = C 3 ε 3αν + t (ν) 3α , s (3) 3α = t (3) 3α , s (4) 3α = (µ + κ)ε 3ρα x ρ + t (4) 3α , (37)
s (ν) 33 = (λ + 2µ + κ)x ν + t (ν)
33 , s

(3) 33 = λ + 2µ + κ + t (3) 33 , s (4) 33 = C 1 + C 2 + C 3 + t (4) 33 , π (ν) αβ = C 1 x ν δ αβ + m (ν) αβ , π (3) αβ = C 1 δ αβ + m (3) αβ , π (4) νη = αδ νη + m (4) νη , π (ρ) 3ν = γε 3νρ + m (ρ) 3ν , π (3) 3ν = m (3) 3ν , π (4) 3ν = C 3 ε 3ρν x ρ + m (4) 3ν , π (ρ) 33 = (C 1 + C 2 + C 3 )x ρ + m (ρ) 33 , π (3) 33 = C 1 + C 2 + C 3 + m (3) 33 , π (4) 33 = α + β + γ + m (4) 33 .
In view of (36), the conditions ( 6)-( 8) reduce to the following system for the constants a 1 , a 2 , a 3 and a 4 ,

4 k=1 D αk a k = ε αβ3 M β , 4 k=1 D 3k a k = -R 3 , 4 k=1 D 4k a k = -M 3 , (38)
where we have introduced the notations

D αk = Σ 1 (x α s (k) 33 -ε 3αβ π (k) 3β )da, D 3k = Σ 1 s (k) 33 da, D 4k = Σ 1 [ε 3αβ x α s (k) 3β + π (k) 33 ]da. ( 39
)
We note that the constants D rs , (r, s = 1, 2, 3, 4), depend only on the cross section and the constitutive coefficients. As in classical theory [START_REF] Ieşan | Classical and Generalized Models of Elastic Rods[END_REF] we can prove that det(D rs ) > 0, D rs = D sr .

(40) The inequality from (40) follows from (36) and the positive definiteness of the elastic potential. The symmetry of D rs is a direct consequence of the reciprocal theorem. The system (38) uniquely determines the constants a k , (k = 1, 2, 3, 4).

We conclude that the solution of the problem has the form (33), where the where

l 1 = (C 1 -σλ)I 1 (qa) + a[C 1 + C 2 + C 3 -σ(λ + 2µ + κ)]I ′ 1 (qa), l 2 = (α -σC 1 )I 1 (qa) + a[α + β + γ -σ(C 1 + C 2 + C 3 )]I ′ 1 (qa). (61)
We introduce the notation

Λ = [(2λ + 2µ + κ)l 2 -(2C 1 + C 2 + C 3 )l 1 ]a. (62)
From (60) we get

A 1 = (C 1 l 1 -λl 2 )aΛ -1 , B 1 = [(C 2 + C 3 )λ -(2µ + κ)C 1 ]a 2 Λ -1 . (63)
Let us study now the problem P (4) . We seek the solution of this problem in the form

u (4) α = x α F (r), u (4)
3 = 0, ϕ (4) α = x α [Ψ(r) - 1 2 ], ϕ (4)
3 = 0, (64)
where F and Ψ are unknown functions. We note that in this case we have e

βα = u (4) α,β = F δ αβ + x α x β r -1 F ′ , e (4) αβ = e (4)
= ε 3αβ x β (Ψ - 1 2 ), κ (4) 33 = 0, e (4) α3 = -e (4) 3α
α3 = κ (4) 3α = 0, (4)
αβ = κ (4) βα = ϕ (4) α,β = (Ψ - 1 2 )δ αβ + x α x β r -1 Ψ ′ . (65) κ (4)
The equations of equilibrium (15) with the body loads f i = 0, g α = -κx α , g 3 = 0, reduce to

(λ + 2µ + κ)(F ′′ + 3r -1 F ) + (C 1 + C 2 + C 3 )(Ψ ′′ + 3r -1 Ψ ′ ) = 0, ( 66
) (C 1 + C 2 + C 3 )(F ′′ + 3r -1 F ) + (α + β + γ)(Ψ ′′ + 3r -1 Ψ ′ -s 2 Ψ) = 0,
where s 2 is given by (45). From these equations we find that F and Ψ satisfy the equations

d 2 dr 2 + 3 r d dr d 2 dr 2 + 3 r d dr -q 2 F = 0, Ψ ′′ + 3r -1 Ψ ′ -q 2 Ψ = 0,
where q 2 is defined in (47). The functions F and Ψ which are bounded at the origin are given by

F = E 1 + N 1 r -1 I 1 (qr), Ψ = H 1 r -1 I 1 (qr),
where E 1 , N 1 and H 1 are arbitrary constants. It is easy to see that F and Ψ satisfy the equations (66) only if N 1 = -σH 1 , where σ is given by (55). Thus, the displacements u (4) α and the microrotations ϕ (4) α have the form

u (4) α = x α [E 1 -σH 1 r -1 I 1 (qr)], ϕ (4) α = x α [H 1 r -1 I 1 (qr) - 1 2 ]. (67)
It follows from ( 65), ( 29) and (30) that t (4)

αβ = λ(2F + rF ′ )δ αβ + (2µ + κ)(F δ αβ + x α x β r -1 F ′ )+ + C 1 (2Ψ + rΨ ′ -1)δ αβ + (C 2 + C 3 )(Ψδ αβ - 1 2 δ αβ + x α x β r -1 Ψ ′ ), t (4) α3 = -t (4) 3α = κε 3αβ x β (Ψ - 1 2 ), t (4) 33 = λ(2F + rF ′ ) + C 1 (2Ψ + rΨ ′ -1), m (4) νη = α(2Ψ + rΨ ′ -1)δ νη + (β + γ)(Ψδ νη - 1 2 δ νη + x ν x η r -1 Ψ ′ )+ (68) + C 1 (2F + rF ′ )δ νη + (C 3 + C 4 )(F δ αβ + x α x β r -1 F ′ ), m (4) ν3 = -m (4) 3ν = (C 3 -C 2 )ε 3νβ x β (Ψ - 1 2 ), m (4) 33 = α(2Ψ + rΨ ′ -1) + C 1 (2F + rF ′ ).
Taking into account (68) and the form of F and Ψ, we find that on the boundary L the following relations hold t (4)

αβ n α = a -2 x β [(2λ + 2µ + κ)aE 1 + lH 1 - 1 2 a(2C 1 + C 2 + C 3 )], t (4)
α3 n α = 0, m

α3 n α = 0, (69)

m (4) αρ n α = a -2 x ρ [(2C 1 + C 2 + C 3 )aE 1 + l 2 H 1 - 1 2 a(2α + β + γ)],
where l 1 and l 2 are defined in (61). Thus, the boundary conditions (32), for k = 4, reduce to the following system for the constants E 1 and H 1 ,

(2λ + 2µ + κ)aE 1 + l 1 H 1 = 1 2 (C 2 + C 3 )a, (2C 1 + C 2 + C 3 )aE 1 + l 2 H 1 = 1 2 (β + γ)a. ( 70
)
We obtain

E 1 = 1 2 aΛ -1 [(C 2 + C 3 )l 2 -(β + γ)l 1 ], (71)
H 1 = - 1 2 Λ -1 a 2 [(2C 1 + C 2 + C 3 )(C 2 + C 3 ) -(2λ + 2µ + κ)(β + γ)],
where the constant Λ is given by (62).

As the functions u

(3) i , ϕ (3)
i , u

i and ϕ

(4)

i are known, from (39) we can determine the constants D k3 and D k4 , (k = 1, 2, 3, 4). It follows from ( 37), ( 39) and ( 57) that

D α3 = Σ 1 x α [λ + 2µ + κ + t (3) 33 + (C 2 -C 3 )Φ]da, D 33 = Σ 1 [λ + 2µ + κ + t (3) 33 ]da, (72)
D α4 = Σ 1 [x α (C 1 + C 2 + C 3 + t (4) 33 ) -ε 3αβ (C 3 ε 3ρβ x ρ + m (4) 3β )da], D 34 = Σ 1 (C 1 + C 2 + C 3 + t (4) 33 )da, D 44 = Σ 1 [(µ + κ)x α x α + ε 3αβ x α t (4) 3β + α + β + γ + m (4) 33 ]da.
With the help of ( 56), ( 57), ( 67) and ( 68), the relations (72) reduce to

D α3 = 0, D α4 = 0, D 33 = π(λ + 2µ + κ)a 2 + 2π a 0 {2λA 1 r+ + (C 1 -λσ)[I 1 (qr) + rI ′ 1 (qr)]B 1 }dr, (73)
D 34 = π(C 2 + C 3 )a 2 + 2πλa 2 E 1 + 2π(C 1 -λσ)H 1 a 0 [I 1 (qr) + rI ′ 1 (qr)]dr, D 44 = π(β + γ)a 2 + 1 4 (2µ + κ)πa 4 + 2π a 0 r[κr 2 Ψ+ + α(2Ψ + rΨ ′ ) + C 1 (2F + rF ′ )]dr.
We note that

a 0 r 3 Ψdr = H 1 a 0 r 2 I 1 (qr)dr, α(2Ψ + rΨ ′ ) + C 1 (2F + rF ′ ) = 2C 1 E 1 + +(α -C 1 σ)H 1 [r -1 I 1 (qr) + I ′ 1 (qr)], (74)
[x 2 I 2 (x)] ′ = x 2 I 1 (x), I 1 (0) = 0.
By using the relations ( 73) and (74) we find

D 33 = π(λ + 2µ + κ)a 2 + 2πλa 2 A 1 + 2π(C 1 -λσ)aB 1 I 1 (qa), D 34 = π(C 2 + C 3 )a 2 + 2πλa 2 E 1 + 2π(C 1 -λσ)aH 1 I 1 (qa), (75)
D 44 = π(β + γ)a 2 + 1 4 (2µ + κ)πa 4 + 2πa 2 C 1 E 1 + + 2πκq -1 a 2 H 1 I 2 (qa) + 2π(α -C 1 σ)aH 1 I 1 (qa).
Sice D rs = D sr , from (73) we obtain D 43 = D 34 , and

D 3α = D α3 = 0, D 4α = D α4 = 0. ( 76
)
The system (38) becomes

D αβ a β = 0, D 33 a 3 + D 34 a 4 = -R 3 , D 34 a 3 + D 44 a 4 = -M 3 . (77)
Thus, for a right circular cylinder the system (38) reduces to two uncoupled systems: one for the unknowns a 1 and a 2 and the other for the constants a 3 and a 4 . It follows from ( 76) that in this case the problem of bending can be treated independently. From (77) we obtain

a 1 = a 2 = 0, a 3 = d -1 (D 34 M 3 -D 44 R 3 ), a 4 = d -1 (D 34 R 3 -D 33 M 3 ), ( 78
)
where d = D 33 D 44 -D 2 34 . It follows from ( 33), ( 41), ( 56), ( 64) and ( 78) that the solution of the problem of torsion and extension is

u α = ε 3βα a 4 x β x 3 + x α [a 3 A 1 + a 4 E 1 -σr -1 (a 3 B 1 + a 4 H 1 )I 1 (qr)], u 3 = a 3 x 3 , ϕ 3 = a 4 x 3 , (79)
ϕ α = x α [r -1 (a 3 B 1 + a 4 H 1 )I 1 (qr) - 1 2 a 4 ],
where A 1 , E 1 , B 1 and H 1 are defined in ( 63), ( 71), and the constants a 3 and a 4 are given by ( 78). We conclude that the torsion of a right circular cylinder made of an isotropic chiral Cosserat elastic material is accompanied by extension.

Clearly, if M 3 = 0 then we obtain the solution of the extension problem.

The torsion problem for an isotropic centrosymmetric Cosserat elastic cylinder has been studied by [START_REF] Ieşan | Torsion of micropolar elastic beams[END_REF]. In the case of centrosymmetric materials we have C j = 0, and the relations ( 53), ( 61) and ( 70) imply that σ = 0, l 1 = 0, E 1 = 0. (80)

It follows from ( 75) and ( 80) that for an isotropic and centrosymmetric circular cylinder we have D 34 = 0, so that in this case the torsion is not accompanied by extension. Moreover, it is a simple matter to see from ( 75) and ( 80) that the constant D 44 reduces to the torsional rigidity calculated by [START_REF] Usidus | Torsion of cylindrical shafts made of micropolar materials[END_REF]. The extension, bending and flexure of a circular cylinder made of an isotropic centrosymmetric Cosserat elastic material have been investigated by [START_REF] Reddy | Saint-Venant's problems for a micropolar elastic circular cylinder[END_REF].

Conclusions

The original results established in this paper can be summarized as follows:

(a) We present a solution of the torsion, bending and extension problem for a right cylinder made of an isotropic chiral Cosserat elastic material. The threedimensional problem is reduced to the study of two-dimensional problems.

(b) We study the generalized plane strain problem in the case of chiral materials and present an existence result. We introduce four generalized plane strain problems characterized by external data which depend only on the constitutive coefficients. The solution of the torsion problem is expressed in terms of solutions of these problems.

(c) We prove that, in general, the torsion of an isotropic chiral Cosserat elastic cylinder is accompanied by bending and extension.

(d) We apply the method to study the torsion of a circular cylinder. We establish the solutions of the auxiliary generalized plane strain problems for a circular domain. The displacement and microrotation vector fields in the torsion problem are determined. We show that in this case the torsion is accompanied by extension and a plane deformation. The solution established in the case of a circular cylinder could be of interest for experimental investigations.

Figure

  Figure

Acknowledgements. This work was supported by CNCSIS-UEFISCU, project number PN II-IDEI 89/2008. I express my gratitude to the referees for their helpful suggestions.

constants a k are given by ( 38) and (u (k) j , ϕ (k) j ) are the solutions of the twodimensional problems P (k) , (k = 1, 2, 3, 4). Generally, the torsion of an isotropic chiral Cosserat elastic cylinder is accompanied by bending and extension.

Torsion of a circular cylinder

In this section we use the solution given in Section 4 in order to study the torsion of a circular cylinder made of an isotropic chiral Cosserat elastic material. We assume that the cylinder B is defined by B = {x : x 2 1 + x 2 2 < a 2 , 0 < x 3 < h}, (a > 0), and suppose that M 1 = M 2 = 0. The solution has the form (33) where the functions u

are the displacements and microrotations in the problem P (k) , (k = 1, 2, 3, 4) defined by ( 29)-( 32), and the constants a k are given by the system (38).

First, we study the generalized plane strain problem P (3) characterized by the external loading (27). We seek the solution of this problem in the form

where r = (x 2 1 + x 2 2 ) 1/2 , and G and Φ are unknown functions. Clearly,

where f ′ = df /dr. It follows from ( 29) and ( 42) that e

(3)

3ρ , e

(3)

In view of ( 42), the equations of equilibrium ( 15) for the problem P (3) , reduce to

If we eliminate the function G, from (44) we find that Φ satisfies the equation

where

Clearly, from ( 9) we find that s 2 > 0, Q > 0 and q 2 > 0. We introduce the function Ω by Ω = rΦ.

(48) Then, the equation ( 46) becomes

In what follows we denote by I n and K n the modified Bessel functions of order n. The general solution of the equation ( 49) is

where B 1 and B 2 are arbitrary constants. Since the functions ϕ (3) α must be bounded for r = 0, we have B 2 = 0. Thus, we find that

It follows from (44) that the function G satisfies the equation

We can write G = G 1 +G 2 where the functions G 1 and G 2 satisfy the equations

Clearly,

where A α and D α are arbitrary constants. Since the functions u (3) α are bounded for r = 0 we must take A 2 = 0 and D 2 = 0. Thus the function G has the form

We note that the functions G and Φ, given by ( 50) and ( 52), must satisfy the system of second-order (44). Substituting G and Φ into the equations ( 44) we obtain the relations

12

The first equation from ( 53) implies that

where

(55) It is a simple matter to see that the second equation from ( 53) is identically satisfied on the basis of the relations ( 47) and ( 55). We conclude that

Let us impose the boundary conditions (32) for k = 3. It follows from ( 43) and the constitutive equations (30) that t

(3)

On the boundary L we have r = a and n α = x α /a. From ( 57) we obtain t

(3)

(3)

In view of (58), the boundary conditions (32), for k = 3, reduce to By using the relations (56) and ( 59) we find that the constants A 1 and B 1 must satisfy the equations (2λ + 2µ + κ)aA 1 + l 1 B 1 = -λa,