Sophie Thomas

Ferechte Encha-Razavi

Louise Devisme

Heather C Etchevers

Bettina Bessieres-Grattagliano

Geraldine Goudefroye

Nadia Elkhartoufi

Emilie Pateau

Amale Ichkou

Maryse Bonniere

Keywords: Fowler syndrome, cerebral proliferative vasculopathy, <i>FLVCR2</i>, Hydranencephaly, Fetal lethality, Arthrogryposis Fowler syndrome, cerebral proliferative vasculopathy, FLVCR2, hydranencephaly, fetal lethality, arthrogryposis

come

Sophie Thomas 1,2 , Ferechté Encha-Razavi 1,2,3 , Louise Devisme 4 , Heather Etchevers 1,2 , Bettina Bessieres-Grattagliano 5 , Géraldine Goudefroye 3 , Nadia Elkhartoufi 3 , Emilie Pateau 6 , Amale Ichkou 3 , Maryse Bonnière with limb deformities [START_REF] Fowler | Congenital hydrocephalus-hydrencephaly in five siblings, with autopsy studies: a new disease[END_REF]. PGV disrupts the developing central nervous system (CNS) but the reason for which abnormal angiogenesis is restricted to the CNS parenchyme remains unknown. Arthryogryposis, when present, appears to be a secondary result of CNS motoneuron degeneration, itself one potential outcome of perfusion failure.

Since its earliest description, 42 PGV cases from 26 families have been reported on the basis of histological criteria [START_REF] Bessieres-Grattagliano | Refining the clinicopathological pattern of cerebral proliferative glomeruloid vasculopathy (Fowler syndrome): report of 16 fetal cases[END_REF][START_REF] Williams | Fowler syndrome-a clinical, radiological, and pathological study of 14 cases[END_REF].

Identification of a causative gene for a very rare lethal syndrome is a challenge at many levels. The first issue is to find a family that allows the identification of a linkage interval. Such an interval may contain too many genes to make the classical subsequent strategy practical, consisting in designing primers that will permit sequencing of each exon of all the genes of the region. The second difficulty is that sequencing of all the exons is sometimes vain in light of the growing number of non-coding regions identified as pathogenic alleles [START_REF] Benko | Highly conserved non-coding elements on either side of SOX9 associated with Pierre Robin sequence[END_REF][START_REF] Kleinjan | Long-range control of gene expression: emerging mechanisms and disruption in disease[END_REF][START_REF] Lettice | A long-range Shh enhancer regulates expression in the developing limb and fin and is associated with preaxial polydactyly[END_REF].

MATERIALS AND METHODS

Patients : The 7 families analysed have been previously reported (Families I to VII [START_REF] Bessieres-Grattagliano | Refining the clinicopathological pattern of cerebral proliferative glomeruloid vasculopathy (Fowler syndrome): report of 16 fetal cases[END_REF]). Genomic DNA was extracted from frozen tissue or cultured amniocyte cells in fetal cases and from peripheral blood samples for parents and unaffected siblings.

Genome linkage screening and linkage analysis: Genome-wide homozygosity mapping was performed using 250K Affymetrix SNP arrays in 5 affected and 3 unaffected individuals of two Turkish and one French multiplex, consanguineous families. Data were evaluated by calculating multipoint lod scores across the whole genome using MERLIN software, assuming recessive inheritance with complete penetrance. Immunohistochemistry : Immunohistochemistry was carried out on six micrometer selected sections using antisera directed against smooth muscle actin (diluted 1:800).

Immunohistochemical procedures included a classical microwave pre-treatment protocol in citrate buffer to aid antigen retrieval. Incubations were performed for one hour at room temperature, using the TECHMATE system (DAKOPATTS-Trappes-France). After incubation, histological slides were processed using the LSAB detection kit (DAKOPPATTS-Trappes-France). Peroxidase was visualized by means of either 3-3' diaminobenzidine or amino-ethyl carbazole.

RESULTS

We have collected DNA from fetuses of 7 families reported earlier (Families I to VII [START_REF] Bessieres-Grattagliano | Refining the clinicopathological pattern of cerebral proliferative glomeruloid vasculopathy (Fowler syndrome): report of 16 fetal cases[END_REF]). All 14 fetal cases bore the brain-specific angiogenic anomalies characteristic of PGV, resulting in thickened and aberrant perforating vessels and glomeruloids, as exemplified in Figure 1A. Endothelial cells (ECs) were positive for CD34 in both control fetal brains (Figure 1B) and in the tortuous glomerular capillaries (Figure 1C). VEGF-A, while not normally expressed by small brain capillaries (Figure 1D), was strikingly found in the glomerular ECs of PGV fetuses (Figure 1E arrowhead). Like normal Ecs though, PGV ECs expressed VEGFR2 and, weakly, Glut-1 (not shown). CD68, characteristic of macrophages, was completely absent (data not shown). Numerous GFAP-positive astrocytes were observed throughout the cerebral mantle, with immunoreactive endfeet justaposed to glomeruloids (Figure 1F). An antibody to alpha-smooth muscle actin (aSMA) stained vessels within the outer leptomeninges and the walls of perforating vessels in normal fetal brains (Figure 1G). In contrast, although PGV meningeal vessels had similar aSMA expression, the dysplastic intraparenchymous vessels were irregularly stained, if at all (Figure 1H), while most glomeruloid vessels were negative for aSMA (Figure 1I).

To find the molecular basis for this phenotype, we first undertook a genome-wide SNP analysis using an Affymetrix 250K SNP chip with 5 affected and 3 unaffected members of two Turkish and one French multiplex, consanguineous families. Informed consent was obtained from all patients and their relatives; clinical data of all families have previously been reported [START_REF] Bessieres-Grattagliano | Refining the clinicopathological pattern of cerebral proliferative glomeruloid vasculopathy (Fowler syndrome): report of 16 fetal cases[END_REF]. Genome-wide linkage To identify the causative gene, we applied array-based sequence capture of the complete 4.1 MB region followed by high-throughput sequencing. DNA from one proband of 6 families, the heterozygous mother from family I, and a healthy brother not carrying the at-risk allele were selected (Figure 2). Coverage varied from 2X to 12X in individuals depending on the integrity of their DNA (Table 1), with an average coverage depth of 7X; 60% (851,147) of the enriched reads were located on the targeted regions. Only 25% of the targeted regions reached 10X coverage depth.

The number of the detected variations was too large to handle them manually. To facilitate the analysis of these variations a specific genome browser was set up to visualise the locations of variations on the genome, and at the same time an analysis tool has been developed. This analysis tool applied a series of filters to the identified variations. These filters were based on the following criteria: 1) the quality of the sequence variant measured as the number of reads that detected the variant 2) the presence or absence of variants in public databases such as dbSNP and HapMap. 3) the Thus, initial analysis of the sequence data that met an arbitrary threshold of at least three reads, of which at least one was required to be in the opposing orientation, detected a total of 23,262 variations, 17,031 of which were on chromosome 14 (73%, Table S1). Of these, 3,457 variants were found to not correspond to known SNPs, and were absent from the normal control individual (E). After initial exclusion of non-exonic and synonymous variants, 42 variants in 29 candidate genes remained. In 20 of these genes, a single variation was found in one individual, while two and three variations were found in six and two genes, respectively (Figure 3). 4B).

Reasons for false-negative results using HTS approaches are summarized in Table 1, and emphasize the need for complementary confirmation. In particular, in family IV, a second heterozygous mutation was found by direct resequencing, although it had an apparently homozygous mutation as indicated by the HTS analysis. In family III, the homozygous mutation found with Sanger sequencing had only been read 2 times in the HTS and had thus been excluded by the stringency of the filter. As a third example, the second heterozygous mutation in family VII had been read 4 times but was excluded for unidirectionality. Interestingly, in family VI, not known to be consanguineous, the identical nonsense mutation was found in the 3 affected sibs (homozygous in fetuses and heterozygous in parents), suggesting more distant consanguinity or a founder effect.

FLVCR2 is a member of the major facilitator superfamily (MFS) of transporter proteins, that shuttle small molecules in response to ion gradients [START_REF] Pao | Evolution of the MIP family of integral membrane transport proteins[END_REF]. Like other MFS members, FLVCR2 is predicted to contain 12 membrane-spanning segments and six extracellular loops. As shown in figure 5A, the 3 homozygous mutations are predicted to alter an amino-acid localized to one transmembrane domain (TM) : TM2 in family VI, TM8 in family III and TM10 in family V. In family IV, one of the 2 mutations alters an amino-acid predicted to be localized in TM8 and the other in the intracellular loop 5.

Amino acid sequence alignment for FLVCR2 from 10 different species showed that T430 and G412 have been conserved since our common ancestor with C. elegans, whereas R84 has been conserved in common with D. melanogaster (Figure 5B). T352R and L398V mutations identified in this study were predicted to be damaging to protein function.

Thus, the pathogenicity of these two last mutations is likely but not totally proven. In total, eight different mutations including one nonsense mutation (homozygous in family VI), six missense mutations, and one homozygous deletion in two families (I and II) have been found in FLVCR2.

DISCUSSION

PGV is a very rare and lethal genetic condition. Since its first description, 42 cases from 26 families have been reported on the basis of histological criteria of PGV [START_REF] Bessieres-Grattagliano | Refining the clinicopathological pattern of cerebral proliferative glomeruloid vasculopathy (Fowler syndrome): report of 16 fetal cases[END_REF][START_REF] Williams | Fowler syndrome-a clinical, radiological, and pathological study of 14 cases[END_REF] might be linked to the embryonic lineage of the telencephalic pericytes, of a distinct neural crest cell origin from those of the spinal cord [START_REF] Etchevers | The cephalic neural crest provides pericytes and smooth muscle cells to all blood vessels of the face and forebrain[END_REF].

Interestingly, immunostaining for alpha smooth muscle actin (aSMA, a marker for mature pericytes) in fetal PGV brains was drastically reduced in the PGV within the CNS while normal aSMA expression was found in the leptomeninges (Figure 1I). Further studies should elucidate whether this observed effect on pericytes is the primary cause or an effect of this disease.

Recently, FLVCR2 mutations were also reported in 5 families with Fowler syndrome (Meyer, et al., 2010), with the same homozygous Thr430Arg mutation in three families, and 2 compound heterozygous cases. Interestingely, Thr430Arg is associated with both forms of the disease, namely with or without spinal cord involvement, suggesting no genotype phenotype correlations. It is noteworthy that the mutation concerned the same codon (Thr430) as in our family IV, the only one of our series without spinal cord involvement. More recently, Lalonde et al. also reported four FLVCR2 compound mutations in 2 FS families with spinal cord involvement ( (Lalonde, et al., 2010).

Interestingly, the only missense mutation predicted to be "benign" in our study (L398V) was identified by two distinct approaches in a common case reported by both Lalonde et The FLVCR2 gene encodes a transmembrane protein that belongs to the major facilitator superfamily (MFS) of secondary carriers that transport small solutes such as calcium [START_REF] Pao | Evolution of the MIP family of integral membrane transport proteins[END_REF]. It is closely related in both sequence and topology to the better-known FLVCR1, sharing 60% amino acid identity [START_REF] Lipovich | Genomic structure and evolutionary context of the human feline leukemia virus subgroup C receptor (hFLVCR) gene: evidence for block duplications and de novo gene formation within duplicons of the hFLVCR locus[END_REF]. FLVCR1 has been identified as the receptor for a feline leukemia virus (FeLV-C), and like FLVCR2 and other MFS members, is predicted to contain 12 membrane-spanning segments and six extracellular loops. A single mutation in the sixth extracellular loop is sufficient to confer

FeLV-C receptor activity on FLVCR2, which does not otherwise bind the native virus [START_REF] Brown | Comprehensive mapping of receptor-functioning domains in feline leukemia virus subgroup C receptor FLVCR1[END_REF]. However, FLVCR2 functions as a receptor for the FeLV-C variant FY981 [START_REF] Shalev | Identification of a feline leukemia virus variant that can use THTR1, FLVCR1, and FLVCR2 for infection[END_REF]. FLVCR1 is found only in hematopoietic tissues, the pancreas and kidney [START_REF] Tailor | A putative cell surface receptor for anemiainducing feline leukemia virus subgroup C is a member of a transporter superfamily[END_REF], but rodent Flvcr2 is widely expressed during embryonic development, in particular within the CNS and in the vessels of the maturing retina, and human FLVCR2, within the fetal pituitary [START_REF] Brasier | Novel hexad repeats conserved in a putative transporter with restricted expression in cell types associated with growth, calcium exchange and homeostasis[END_REF]. FLVCR1 has been shown to function as a heme exporter, essential for erythropoiesis [START_REF] Quigley | Identification of a human heme exporter that is essential for erythropoiesis[END_REF]. Interestingly, the five glutamate residues in the C-terminal putative coiled-coiled domain of FLVCR2, not present in FLVCR1, may serve an analogous function to the same ferric ion-binding glutamate sequence in glycine-extended gastrin, by stimulating cell proliferation [START_REF] He | Biological activity and ferric ion binding of fragments of glycine-extended gastrin[END_REF]. Based on the cell types in which it is expressed and MFS transport of chelated complexes of divalent metal ions, the FLVCR2 transporter was postulated to be a gatekeeper for the controlled entry of calcium into target cell types [START_REF] Brasier | Novel hexad repeats conserved in a putative transporter with restricted expression in cell types associated with growth, calcium exchange and homeostasis[END_REF]. Calcium signalling is involved in virtually all cellular processes and its homeostasis is tightly regulated. Angiogenic factors such as VEGF-A and FGF2

induce a transient increase of endothelial cell intracellular calcium concentrations , which 14 acts as a second messenger to induce proliferation, among other effects [START_REF] Tomatis | Cytosolic calcium microdomains by arachidonic acid and nitric oxide in endothelial cells[END_REF]. Blood vessels are susceptible to responding to angiogenic signals and undergoing calcification when their pericytic coverage has been disrupted [START_REF] Collett | Angiogenesis and pericytes in the initiation of ectopic calcification[END_REF], both of which signs we have observed in PGV patient brain sections.

HTS of the entire exome has been used so far to identify disease-causing genes in the rare Miller and Bartter syndromes, respectively [START_REF] Choi | Genetic diagnosis by whole exome capture and massively parallel DNA sequencing[END_REF][START_REF] Ng | Exome sequencing identifies the cause of a mendelian disorder[END_REF]. Recently, targeted exon-specific sequencing within a restricted 40 MB linkage interval allowed the identification of an additional gene for Familial Exudative

Vitreoretinopathy [START_REF] Nikopoulos | Next-generation sequencing of a 40 Mb linkage interval reveals TSPAN12 mutations in patients with familial exudative vitreoretinopathy[END_REF]. Our study underlines the use of HTS for the coverage of an entire linkage interval with no compelling candidate genes and no justification for the exclusion of non-coding regions. Our nested analysis approach led rapidly to the identification of a disease-causing gene. While it further demonstrates the power of this new technology, it also highlights other potential risks of missing mutations during data analyses. The number of patients, diagnostic accuracy and genetic homogeneity allowed us to compensate for low capture efficiency due to suboptimal DNA quality, and in the future, as the technology develops, furthering the depth of coverage should ensure a better distinction of background from true mutations. Finally, identification of the gene for Fowler syndrome will permit accurate genetic counselling for PGV and prenatal diagnosis, in particular for the late-onset forms of the disease without spinal cord involvement. [START_REF] Corpet | Multiple sequence alignment with hierarchical clustering[END_REF]. Highly conserved amino acids are represented in red, moderately conserved amino acids are in blue and non-conserved ones are in black. Mutated amino acids are boxed. Mutation numbering is based on cDNA sequence with a 'c.' symbol before the number, where +1 corresponds to the A of ATG translation codon (codon 1) of the cDNA reference sequences (NM_017791). Mutation names were checked by the Mutalyzer program [START_REF] Wildeman | Improving sequence variant descriptions in mutation databases and literature using the Mutalyzer sequence variation nomenclature checker[END_REF].

COMPETING INTERESTS STATEMENT

F

  Sequencing of a 4.1 Mb Linkage Interval Reveals FLVCR2 Deletions and Mutations in Lethal Cerebral Vasculopathy

Foliguet 20 ,

 20 Michel Vekemans 1,2,4 , Tania Attié-Bitach 1,2,4 1-INSERM U-781, Hôpital Necker-Enfants Malades, Paris, France 2-Université René Descartes, Paris 5, France 3-Département de Génétique, Hôpital Necker-Enfants Malades, AP-HP, Paris, France 4-Pôle de Pathologie, Centre de Biologie Pathologie, CHRU Lille, France 5-Laboratoire d'Anatomo-Foeto-Pathologie, Institut de Puériculture et de Périnatalogie, Paris, France 6-CEA, DSV, Institut de Génomique, Genoscope, Evry, France 7-Laboratoire Nord Pathologie, Lille, France 8 -Laboratoire d'anatomopathologie, CHRU Hôpital Morvan, Brest, France 9 -Département de pédiatrie et génétique médicale, CHRU Hôpital Morvan, Brest, France 10-Service de Génétique Clinique et Université Lille 2, CHRU de Lille, Hôpital Jeanne de Flandre, Lille, France 11-Laboratoire d'Anatomie Pathologique, Hôpital de Rouen, Rouen, France 12-Service d'Anatomie et de Cytologie Pathologiques,CHI Poissy, Saint Germain en Laye, France 13 -Génétique Médicale, CHI Poissy, Saint Germain en Laye, France 14 -Service de Cytogénétique, Génétique moléculaire et Biologie de la reproduction, Hôpital Farhat Hached, Sousse, Tunisie de Génétique et d'Embryologie Médicales, Hôpital Armand Trousseau, AP-HP, et Université Pierre et Marie Curie, Paris 6, France 16-Plateforme de génomique, Fondation IMAGINE 17-Service de Bioinformatique, Université Paris -Descartes 18-Service de Gynécologie Obstétrique, Hôpital Necker Enfants Malades, Paris, France 19-Department of Medicine, University of Melbourne, Australia 20-Laboratoire de Biologie de la Reproduction et du Développement, .attie@inserm.fr INTRODUCTION Cerebral proliferative glomeruloid vasculopathy (PGV) is a severe autosomal recessive disorder of brain angiogenesis, resulting in abnormally thickened and aberrant perforating vessels, forming glomeruloids with inclusion-bearing endothelial cells. This peculiar vascular malformation was delineated by Fowler in 1972 in relation to a stereotyped, lethal fetal phenotype (OMIM 225790), associating hydranencephaly and hydrocephaly

  Finally, for prenatally lethal syndromes such as PGV, technical constraints such as poor quality genomic DNA samples are added. Recent advances in biotechnology permit the sequencing of all the DNA, including the non-coding regions, in most genomic homozygosity mapping of a 4.1 Mb region, we applied targeted genome capture by using a NimbleGen array and high-throughput Roche 454 GS FLX sequencing to the genomic DNA of the proband of 6 families. Bioinformatic analysis of the data allowed us to identify FLVCR2 (MIM 610865) as the gene responsible for Fowler syndrome (FS). High-throughput sequencing (HTS) generated false positive and false negative results, in part due to insufficient sequencing coverage, and unless care is taken, these can engender the risk of missing mutations during the analysis.

  the MERLIN program revealed a 13 Mb genomic region on chromosome 14 from rs10151019 to rs12897284, with a lodscore of 5.4. Moreover, four affected sibs from the two Turkish families shared the same alleles for 280 consecutive SNPs, suggesting a founder effect and reducing the interval to 4.1 MB, from rs2803958 to rs11159220. These two families originated from villages 12 km apart in Khramanmaraps (central Turkey). Microsatellite marker analysis further confirmed the same disease allele in both families, and showed linkage in 3 additional families (Figure2).

  of the variants among the studied samples 4) annotation of the sequence variants based on their location (intron, exon, etc ...) and the characteristics of the resulting change such as synonymous, non-synonymous or stop mutation. Filtered results were visualised in an interactive table permitting us to sort and analyse the results.

FLVCR2

  was the only gene with variations identified in four out of seven individuals. In addition, careful examination of the FLVCR2 locus in the proband of family II revealed a homozygous deletion of exons 2 to 10, as the absence of both nucleotide variations and reads over a 46.8 KB genomic region (Figure4A). The deletion was confirmed to segregate in families I and II, and cloning of the breakpoints revealed the inclusion of the last two exons of the neighbouring C14orf1 gene, with no repeated DNA sequences at the boundaries. It is noteworthy that this deletion was not detected by Affymetrix 250K SNP chip. Indeed, only one SNP was located in the non-deleted portion of intron 1. Direct sequencing of the 10 exons of FLVCR2 (Supp.

  evolutionary conserved, especially L398V. However, those mutations are absent from both the dbSNP and the 1000 Genome database not yet integrated in dbSNP. While the L398V mutation was predicted to be benign by the Polyphen algorithm (http://genetics.bwh.harvard.edu/pph/), the T352R mutation as well as the other missense

  al.and Meyer et al., adding to the likely pathogenicity of this variation. To sum up, 15 different FLVCR2 mutations (including those described in our study) have now been reported in 13 cases: one large deletion, two nonsense mutations, one splice site mutation, one insertion/deletion change and 10 missense variations.

Figure 1 :

 1 Figure 1: Marker analysis in Fowler syndrome fetal brain

Figure 2 :

 2 Figure 2 : Pedigree and linkage analysis results

Figure 3 :

 3 Figure 3 : Summary of HTS data analysis

Figure 4 :

 4 Figure 4 : FLVCR2 deletion and mutations

Figure 5 :

 5 Figure 5: Localisation of mutations in FLVCR2 and conservation of mutated FLVCR2 amino

  Figure 1 115x86mm (300 x 300 DPI)

  Figure 5 199x149mm (300 x 300 DPI)

  3,7 , Pascale Marcorelle 8 , Philippe Parent 9 , Sylvie Manouvrier 10 , Muriel Holder 10 , Annie Laquerrière 11 , Laurence Loeuillet 12 , Joelle Roume 13 , Jelena Martinovic 3 , Soumaya Mougou-Zerelli 1,2,14 , Marie Gonzales 15 , Vincent Meyer 6 , Marc Wessner 6 , Christine Bole Feysot 16 , Patrick Nitschke 17 , Nadia Leticee 18 , Arnold Munnich 1,2,4 , Stanislas Lyonnet 1,2,4 , Peter Wookey 19 , Gabor Gyapay 6 , Bernard

NimbleGen Sequence capture and high-throughput sequencing:

	with the manufacturer's instructions (Roche NimbleGen) and a final amount of 3
	micrograms of amplified enriched DNA was used as input for generating a ssDNA
	library for HTS; 25% lane of a Roche 454 GS FLX sequencer with Titanium reagents)
	yielding 135 Mb of sequence data per sample.
	Capillary sequencing of FLVCR2 : Primers were designed in introns flanking the 10
	exons using the "Primer 3" program (http://fokker.wi.mit.edu/primer3/input.htm) and are
	listed in Supp. Table S2. PCR were all performed in the same conditions, with a F o touchdown protocol consisting of denaturation for 30s at 96°C, annealing for 30s at a
	r temperature ranging from 64°C to 50°C (decreasing 1° during 14 cycles, then 20 cycles at
	P 50°) and extension at 72°C for 30s. PCR products were treated with Exo-SAP IT (AP
	e Biotech), and both strands were sequenced with the appropriate primer and the "BigDye" e r terminator cycle sequencing kit (Applied Biosystems Inc.) and analyzed on ABI3130
	automated sequencers. Mutation numbering is based on cDNA reference sequence R w e i v e NM_017791.
	A custom sequence
	capture array was designed and manufactured by Roche NimbleGen (Madison, WI,
	USA). 21 micrograms of genomic DNA was used for sequence-capture in accordance
	6

Table S1

 S1

	the 7 families studied (5 homozygotes and 2 compound heterozygotes; Table 1 and
	Figure
	F o
	r
	P
	e e r
	R e v i e w
	10

), identified mutations in two additional families (Table

1

), such that mutant FLVCR2 alleles were identified in each of

Table 1 :

 1 Analysis of variations by individual and FLVCR2 variations identified by highthroughput or capillary sequencing.E is a healthy brother in family V not carrying the disease allele by haplotyping, and taken as healthy control. Mutation numbering is based on cDNA sequence with a 'c.' symbol before the number, where +1 corresponds to the A of ATG translation codon (codon 1) of the cDNA reference sequences (NM_017791). Mutation names were checked by the Mutalyzer program[START_REF] Wildeman | Improving sequence variant descriptions in mutation databases and literature using the Mutalyzer sequence variation nomenclature checker[END_REF].

	Supplementary

Table S1 :

 S1 Analysis of total number of variations detected by high-throughput sequencing Supplementary

Table S2 :

 S2 Primers and PCR conditions

Table 1 :

 1 Analysis of variations by individual and FLVCR2 variations identified by next generation or capillary sequencing.

		Individual	A (F am II)	B (F am VI)	C (F am VII)	D (F am IV)	F (F am V)	G (F am I)	H (F am III)	T otal
		O rigin	T urkis h	F rench	F rench	F rench	Maroccan	T urkis h	F rench
		C overage	8,8X	4X	8,6X	2,3X	11,8X	11,6X	6,6X	7X
		All	2852	1804	2639	823	3067	3841	2005	17 031
	Number of Variations	Variations in E removed	1379	790	1154	282	1527	2075	1182
	(total)	S NP removed	565	380	608	112	695	872	821
		Variations in E and S NP s removed	546	300	465	80	569	750	747
	Number of variations	T otal	100	74	105	41	87	139	58
	on mR NA	Variations in E and S NP removed	23	14	17	6	20	26	29
		T otal	41	22	44	13	42	60	25
	Number of	Variations in E and S NP removed	8	2	4	2	11	12	15
	variations on C DS	non s ynonymous	22	8	23	9	23	28	13
		non s ynonymous and S NP removed	8	2	4	2	9	8	9	42
		Next generation s equencing	D el E x 2-10 hmz	c.402C >G , p.T yr134S top hmz	c.251G >A, p.Arg84His htz	c.1056C >G , p.T hr352Arg hmz	c.1234G >C , hmz p.G ly412Arg	(mother)	-
			D el E x 2-10	c.402C >G ,	c.251G >A, p.Arg84His	c.1056C >G ,	c.1234G >C ,	D el E x 2-10 htz	c.1076T >C ,
		C apillary	hmz	p.T yr134S top	htz c.1192C >G ,	p.T hr352Arg htz	p.G ly412Arg	in mother	p.L eu359P ro hmz
	F L VC R 2	s equencing		hmz	p.L eu398Val htz	c.1289C >T ,	hmz
	variations					p.T hr430Met htz
		C omparis on and	confirmation	confirmation	Arg84H is : confirmation	T 352R : 4 reads of	confirmation	D eletion	2 reads of only
		reas on for			L eu398Val: 7 reads , 4	only the mutated		confirmed in	the mutated allele
		dis crepancy			with the mutation, but	allele, T 430M: no		foetus es (hmz),
					excluded for	reads		htz in parents
					unidirectionality

John Wiley & Sons, Inc.Human Mutation

ACKNOWLEDGEMENTS

We are grateful to families and to the French Society of Fetal Pathology (SOFFOET) for participating in the study. We thank Chantal Esculpavit for technical help. This work was funded by the GIS-Maladies Rares. ST is supported by grant NS039818 from the US National Institute of Health (NIH).


Karen Gaudon

Isabelle Pénisson-Besnier

Brigitte Chabrol

Françoise Bouhour

Laurence Demay

Asma Ben Ammar

Stéphanie Bauché

Christophe Vial

Guillaume Nicolas

Bruno Eymard

Dr Daniel Hantaï
email: daniel.hantai@upmc.fr

Pascale Richard

Multiexon deletions account for 15% of Congenital Myasthenic Syndrome with RAPSN mutations after negative DNA Sequencing

Post-synaptic congenital myasthenic syndromes (CMSs) (OMIM_ #608931) is a group of genetic disorders affecting neuromuscular transmission and due to acetylcholine receptor (AChR) deficiency in 80% of cases. [START_REF] Engel | Current understanding of congenital myasthenic syndromes[END_REF] These autosomal recessive CMSs may be caused by mutations in genes encoding the AChR or one of the AChR-clustering or anchoring proteins, rapsyn, Dok-7 or MuSK. [START_REF] Engel | Current understanding of congenital myasthenic syndromes[END_REF][START_REF] Ohno | Rapsyn mutations in humans cause endplate acetylcholinereceptor deficiency and myasthenic syndrome[END_REF][START_REF] Chevessier | MUSK, a new target for mutations causing congenital myasthenic syndrome[END_REF][START_REF] Beeson | Dok-7 mutations underlie a neuromuscular junction synaptopathy[END_REF] Spectra of rapsyn mutations show allelic heterogeneity and suggest that the common substitution p.Asn88Lys (N88K) (variant_021217 in Q13702) results in less stable AChR clusters. [START_REF] Cossins | Diverse molecular mechanisms involved in AChR deficiency due to rapsyn mutations[END_REF] Until recently, all patients harbouring mutations in RAPSN are either homozygous for the p.Asn88Lys substitution or heteroallelic for p.Asn88Lys and a mutation which is in most of cases an amino acid substitution but can be also a null allele. [START_REF] Banwell | Novel truncating RAPSN mutations causing congenital myasthenic syndrome responsive to 3,4-diaminopyridine[END_REF] Analysis of disease severity in patients suggested that the second mutant allele may largely determine severity of the phenotype. [START_REF] Ioos | Congenital myasthenic syndrome due to rapsyn deficiency: three cases with arthrogryposis and bulbar symptoms[END_REF] Recently, a patient with two non p.Asn88Lys in RAPSN has been described and the first chromosomal deletion event was described by Müller and colleagues. [START_REF] Maselli | Congenital myasthenic syndrome caused by two non-N88K rapsyn mutations[END_REF][START_REF] Müller | A newly identified chromosomal microdeletion of the rapsyn gene causes a congenital myasthenic syndrome[END_REF] When analysing 20 patients presenting recessive CMS for RAPSN mutations three out of them were carriers of different large (multi-exonic) but partial deletions of RAPSN that could not be identified by gene sequencing. Patient 1 is a 22-year-old girl presenting a severe generalized hypotonia at birth with arthrogryposis, retrognathism and amimic face with no sucking reflex. There was no respiratory problem or dysphagia. She achieved independent walking on tiptoes at 16 months and presented several episodes during which she was unable to walk. At age 17, she showed a mild proximal muscle weakness in the four limbs and in neck flexors, a reduced mouth opening and a slight limitation in left eye abduction. EMG demonstrated a clear decrement on 3-Hz repetitive stimulation in trapezius and quadriceps muscles.

Patient 2 is a 27-year-old woman presenting a severe generalized hypotonia at birth with arthrogryposis of the ankles, knees, elbows, and fingers. She had facial diplegia, respiratory failure with weak cries, no suction, no swallowing, and a permanent defect of orbicular muscles of lips and eyes. The diagnosis of CMS was confirmed after a negative test for AChR antibodies, a 15% decrement and a positive prostigmine test.

Patient 3 is a 7-years-old girl with no family history. At birth, a major hypotonia and arthrogryposis of the hands and feet were noted. She was not able to swallow and had frequent respiratory failure episodes leading to mechanical ventilation. The prostigmine test was positive and the EMG showed a decrement confirming the diagnosis of CMS.

Molecular analyses revealed in Patient 1, an "apparently" homozygous RAPSN substitution (c.264c>a), p.Asn88Lys transmitted by the unaffected mother but not carried by the father suggesting a "missed" variant not detected by usual sequencing approach.

In both Patients 2 and 3, the missense p.Asn88Lys substitution was identified on a single allele with no identification of the second allelic mutation. In both families, heterozygous parent for p.Asn88Lys was the mother. We determined the RAPSN copy number in each patient, using three SNPs known to cosegregate (rs7111873, rs34729771 and rs7126210; fig 1). In these three patients, the SNPs do not cosegregate correctly. In patient 1, rs34729771 and rs7126210 SNPs were heterozygous while rs7111873 was homozygous. In patient 2, only the SNPs rs7111873 was heterozygous. In patient 3, only rs7111873 and rs34729771 were heterozygous. A relative quantification for each exon of RAPSN was performed by a qPCR approach. In patient 1, a deletion involving 5'UTR, promoter, exon 1 and exon 2 was detected (fig 2). Partial delimitation of breakpoints showed that the deletion corresponding to a ~30kb occurred between rs2242081 and intron 2. In patient 2, allelic quantification confirms the loss of exon 3 to 7 on one allele and the long range PCR/sequencing showed that the recombination occurred between a short sequence of 7 nucleotides (CCTGCAG) in intron 2 at the junction with exon 3 (c.532-7; c.532-1) and the same sequence at the 5' end of exon 8 (c.1170_1176) resulting in a deletion spanning on 4.785 kb (g.47,416,165_47,420,949del). In patient 3, the deletion involves exon 7 to 8 and part of 3'UTR and the recombination occurred between two sequences of 25 nucleotides (GCTAATTTTTGTATTTTTAGTAGAG), the first located in intron 6 (c.967-398_967-374) and the other one located in 3'UTR region resulting in a deletion of 10.334 kb (g. 47,417,456_47,407,123del). The allele quantification was performed in the parents and showed that in the 3 families, the chromosomal micro-deletion was transmitted by the father (for experimental details see Supplemental data files 1 and 2).

In our series of CMS patients recruited via the French CMS National Network, twenty patients were found with disease-causing mutations in RAPSN. Among these patients, three (15%) had the recurrent p.Asn88Lys substitution but the sequencing approach failed to identify the second allelic mutation. We hypothesized that genomic deletions may account in these patients and developed a simple molecular assay based on qPCR analysis. This led us to identify three different chromosomal micro-deletions due to recombinations. All these multi-exon deletions corresponded to the missing disease-causing allelic mutation in patients.

The description of Müller and colleagues and ours demonstrates that RAPSN, containing numerous repeated sequences, is particularly subject to multiple genomic recombinations. The precise deletions breakpoints determination showed that the recombination involves multiple and different sequences in all the described cases.

Altogether these findings lead to the following comments: (i) a negative result obtained by direct sequencing of genomic DNA raises the question of an incomplete detection of some mutations; (ii) in recessive transmissions, the analysis of patient parents is crucial to confirm the inheritance; (iii) phenotype-genotype correlations should also be considered:while patient 1 did not exhibit a very severe phenotype during childhood, patients 2 and 3 presented a CMS with all criteria of severity corresponding to the phenotype observed in patients carrying the p.Asn88Lys substitution associated with another allelic mutation. [START_REF] Burke | Rapsyn mutations in hereditary myasthenia: distinct earlyand late-onset phenotypes[END_REF] In conclusion, this mutational mechanism represents 15% of patients with RAPSN mutations referred to our laboratory. For diagnostic purposes, carrier detection, genetic counselling, and prenatal diagnosis, it is critical to know the exact functional gene copy number that an individual carries.

Key Words: Congenital myasthenic syndrome, rapsyn, chromosomic microdeletion, loss of heterozygosity, allele copy number.

Figure legends : Figure 1 :

 :1 Figure legends:

Figure 2 :

 2 Figure 2: Graphic representation of the relative quantification of RAPSN exons. Results were expressed in N-fold changes in RAPSN exon copies, normalized to β-globin relative to the copy

Acknowledgements

The authors would like to thank the patients and their families for their helpful collaboration. This work was supported by Assistance Publique-Hôpitaux de Paris (PHRC AOM 1036), Réseaux Inserm, ANR-Maladies Rares (ANR-07-MRAR-001), Association Française contre les Myopathies and a Contrat d'Interface AP-HP Inserm (to DH).

Competing Interests: None to declare.

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J Bičák

M Scholtz
email: scholtzzz@gmail.com

P Tod

On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions

By an argument similar to that of Gibbons and Stewart [10], but in a different coordinate system and less restrictive gauge, we show that any weaklyasymptotically-simple, analytic vacuum or electrovacuum solutions of the Einstein equations which are periodic in time are necessarily stationary.

Introduction

The inspiral and coalescence of binary black holes or neutron stars appears to be the most promising source for the detectors of gravitational waves, so that there has been much effort going into the development of numerical codes and analytic approximation methods to find the corresponding solutions of Einstein's equations. One of the recent approaches assumes the existence of a helical Killing vector k (see e.g. [START_REF]Max Planck Institute for Gravitational Physics[END_REF]). The field is assumed stationary in a rotating frame where k generates time translations but k becomes null at the light cylinder and is spacelike outside. k has the form k = ∂ t +ω∂ φ where ∂ t is timelike and ∂ φ is spacelike with circular orbits with parameter length 2π (except where ∂ φ = 0); ω =constant. The space-time is not stationary but it is still periodic where k is spacelike. Requiring the helical symmetry for a binary system implies equal amounts of outgoing and incoming radiation so that the space-time, containing energy radiated all times is not expected to be asymptotically flat. A corresponding solution in Maxwell's theory for two opposite point charges moving on circular orbits was considered a long time ago by Schild [START_REF] Schild | Electromagnetic two-body problem[END_REF]. The properties of the field were analyzed recently in the Newman-Penrose formalism in [START_REF] Bičák | Helical symmetry in linear systems[END_REF]. The rather complicated periodicity properties of the solution became apparent as well as Confidential: not for distribution. Submitted to IOP Publishing for peer review 11 January 2010 its asymptotic behaviour: at I -the advanced fields exhibit the standard Bondi-type expansion and peeling, whereas the retarded fields do decay with r → ∞ but in an oscillatory manner like (sin r)/r. Hence for the retarded plus advanced solution no radiation field is asymptotically defined. Naturally, one would like to go beyond the linearized theory. There are special exact, time-dependent, solutions known, for example, Szekeres's dust solution, which has in general no Killing vector, which can be matched to an exterior Schwarzschild metric [START_REF] Bonnor | Non-radiative solutions of Einstein's equations for dust[END_REF]. One can construct oscillating spherical shells of dust particles moving with the same angular momentum, but in every tangential direction, or oscillating Einstein clusters which are matched to the Schwarzschild spacetime outside [START_REF] Gair | Spherical universes with anisotropic pressure Class[END_REF]. Can there be periodic solutions representing "bound states" of gravitational or electromagnetic waves so that the radiation field at infinity vanishes and the Bondi mass remains constant?

There have been various attempts to prove that, while solutions of the vacuum Einstein equations can be genuinely periodic in a suitable time-coordinate (so not time-independent), these solutions cannot be asymptotically flat. These started with [START_REF] Papapetrou | Über periodische nichtsinguläre Lösungen in der allgemeinen Relativitätstheorie[END_REF] and [START_REF] Papapetrou | Über periodische Gravitations-und elektromagnetische Felder in der allgemeinen Relativitätstheorie[END_REF], with a summary in English in [START_REF] Papapetrou | Non-existence of periodically varying non-singular gravitational fields in Les théories relativistes de la gravitation[END_REF], and [START_REF] Trautman | Proof of the non-existence of periodic gravitational fields representing radiation Bull[END_REF] and more recently was considered in [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF]. The method in [START_REF] Papapetrou | Über periodische nichtsinguläre Lösungen in der allgemeinen Relativitätstheorie[END_REF] considers vacuum metrics which are everywhere nonsingular, weak and asymptotically-flat and which can be expanded in a series in some parameter, with the flat metric as the first term in the series. Each term in the series is assumed to be periodic in a fixed Minkowski time-coordinate and to satisfy the de Donder gauge condition. The second and third terms, call them v ab and w ab respectively, are expanded as Fourier series in the background time-coordinate and the Einstein equations then imply that v ab satisfies the source-free wave equation, and w ab satisfies a wave equation whose source is a quadratic expression in v ab . Assuming that the solution for v ab is everywhere regular, the author shows that there cannot be an asymptotically-flat solution for w ab unless v ab vanishes. Therefore the spacetime is flat. In [START_REF] Papapetrou | Über periodische Gravitations-und elektromagnetische Felder in der allgemeinen Relativitätstheorie[END_REF], a similar calculation when v ab is regular only outside a certain radius leads to the conclusion that v ab must be time-independent in order to have asymptotically-flat w ab , and the space-time is stationary. In [START_REF] Trautman | Proof of the non-existence of periodic gravitational fields representing radiation Bull[END_REF] it was observed by integrating the Einstein pseudotensor and matter energy-momentum tensor over a 4-dimensional volume that "the mean value of power radiated by a periodic, asymptotically Minkowskian gravitational field is equal to zero". The question of existence of periodic fields was left open. In [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] the authors used the spin-coefficient formalism (see e.g. [START_REF] Newman | An approach to gravitational radiation by a method of spin coefficients[END_REF], [START_REF] Stewart | Advanced general relativity[END_REF]) to study the system of conformal Einstein equations of Friedrich [START_REF] Friedrich | On the regular and the asymptotic characteristic initial value problem for Einstein's vacuum field equations[END_REF]. A coordinate system is based on two families of null hypersurfaces, incoming from past null infinity I -and labelled by constant v and outgoing near I -and labeled by constant u. The authors make a definition of periodicity which enables them to prove that, at I -, the u-derivatives of all orders of all components of the metric are independent of v. They conclude that if the metric is analytic in these coordinates, then it necessarily has a Killing vector, which in these coordinates is ∂ v , at least in a neighbourhood of I -. Thus any analytic metric, periodic in their sense, has such a Killing vector. While certainly correct, there is a problem with this conclusion in that, by construction, the Killing vector is null wherever it is defined, and reduces at I -to a constant translation along the generators. These are strong conditions and in fact no Killing vector in flat space has these properties (any null Killing vector is necessarily a null translation, and a null translation is zero along one generator of I) ‡. Thus flat space is not periodic according to the definition of [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] ‡ For example the null translation ∂t + ∂z becomes 2 cos 2 (θ/2) ∂v on I -, which vanishes at θ = π. and nor is any of the familiar stationary, asymptotically flat solutions, for example the Schwarzschild solution.

For convenience, we follow [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] in working at I -rather than I + , though this is trivial to switch, but we shall make a weaker definition of 'periodic in time' which will permit metrics stationary near I -and indeed will allow only these for analytic, asymptotically-flat vacuum or electrovac metrics. We follow the method of [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] for both the vacuum and electrovac field equations, deferring other cases to a second article, but in a different coordinate and tetrad system. Our coordinate and tetrad system is similar to the one used at I in [START_REF] Newman | An approach to gravitational radiation by a method of spin coefficients[END_REF], and to prove the existence of a symmetry at the event horizon in [START_REF] Hawking | The large scale structure of space-time[END_REF] and at a compact Cauchy horizon in [START_REF] Moncrief | Symmetries of cosmological Cauchy horizons[END_REF]. We also differ from [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] in the choice of conformal gauge. In [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] the unphysical Ricci scalar is set to zero by a choice of conformal factor obtained by solving a wave equation. However, the solution of the characteristic IVP for this wave equation as posed in [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] will not in general be periodic, so that the rescaled, unphysical metric would not in general share the periodicity of the physical metric -in fact, in the particular case of the Reissner-Nordström solution this gauge choice is compatible with periodicity only for zero mass, as we show in Appendix C. Thus we assume that there is at least one conformal factor which is periodic and then modify this choice in the course of the calculation in order to simplify the spin coefficients. From this point on, our method is then essentially the same as in [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF], though a little more complicated, and we arrive at the same conclusion, but now with a Killing vector which is time-like in the interior, at least near to I -. The condition of time-like periodicity which we impose is as follows: a space-time is time-like periodic if there is a discrete isometry taking any point of the physical space-time to a point in its chronological future. To define timelike periodicity at I -for an asymptotically-flat space-time, we require this isometry to extend to an isometry of a neighbourhood of I -which preserves the generators of I -. In particular, we require the existence of at least one Ω which conformally compactifies the spacetime and preserves the periodicity. The isometry has to be a supertranslation [START_REF] Stewart | Advanced general relativity[END_REF],

v → v + a(θ, φ), ( 1
)
on I -, in terms of the usual coordinates (v, θ, φ) on I -and we shall assume that a = 0. (We could imagine allowing a to vanish on some generators of I -, since as noted above periodicity along a null translation in flat space would appear like this at I -, but this would be null-periodicity rather than time-like periodicity.) We could assume further that a is actually a positive constant but this turns out not to be necessary, as we shall find that, for analytic space-times, this assumption of periodicity necessarily leads to a space-time metric with a Killing vector which, in coordinates to be defined, is ∂ v and is time-like near I -. Our result is Theorem 1.1. A weakly-asymptotically simple, vacuum or electrovac, time-periodic space-time which is analytic in a neighbourhood of I -in the coordinates introduced below necessarily has a Killing vector which is time-like in the interior and extends to a translation on I -.

Thus there are no non-trivial time-periodic solutions satisfying these conditions, in the sense that they would necessarily be actually time-independent if time-periodic. In a later article, we shall prove the corresponding result for the Einstein equations coupled to either a massless scalar field with the usual energy-momentum tensor, or a solution of the conformally-invariant wave equation with the energy-momentum tensor from p125 of [START_REF] Penrose | Spinors and space-time vol II Cambridge Monographs on Mathematical Physics[END_REF] (sometimes called the 'new improved energy momentum tensor').

The method of proof requires the assumption of analyticity. It was shown in [START_REF] Friedrich | On the existence of analytic null asymptotically flat solutions of Einstein's vacuum field equations[END_REF] that there are vacuum solutions analytic near I -. However, one would like either to drop the assumption of analyticity, for example following the lead of [START_REF] Friedrich | On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon[END_REF] or [START_REF] Galloway | Splitting theorems for spatially closed space-times[END_REF] with a similar problem, or to prove that it follows from the assumptions of periodicity and asymptotic-flatness. It remains to be seen in what circumstances this can be done since, as noted above, there are non-analytic solutions with matter in periodic motion and matched to a (static) Schwarzschild exterior.

While this work is primarily motivated by an interest in the possibility or impossibility of helical motions, it is worth noting the connection with the question of the inheritance of symmetry. Recall that, for a solution of Einstein's field equations with matter, the matter is said to inherit the symmetry of the metric if any isometry of the metric is necessarily a symmetry of the matter. There are explicit solutions of the Einstein-Maxwell equations known for which an isometry of the metric is not a symmetry of the Maxwell field [START_REF] Maccallum | Non-inheritance of static symmetry by Maxwell fields, in Galaxies, axisymmetric systems and relativity[END_REF] but these solutions are not asymptotically-flat. In [START_REF] Stephani | Exact solutions to Einstein's field equations[END_REF] some other references may be found for explicit solutions with Maxwell fields which do not share the symmetry of the metric. The same will be true for some Robinson-Trautman solutions with null electromagnetic field which may depend on time though the metric is static (see [START_REF] Stephani | Exact solutions to Einstein's field equations[END_REF], §28.2) These solutions will very likely have wire singularities extending to infinity. From Theorem 1.1 noninheritance cannot happen with asymptotically-flat, analytic solutions: Corollary 1.2. In any weakly-asymptotically simple, stationary electrovac space-time which is analytic in a neighbourhood of I -in the coordinates introduced below, the Maxwell field is also stationary.

One can raise the question of inheritance also for Einstein-scalar field solutions but the answer is rather different: for a massive (complex) Klein-Gordon field there do exist solutions, the so-called 'boson stars', for which the metric is spherically-symmetric, asymptotically-flat and static but the scalar field has a phase linear in time (see e.g. [START_REF] Bizoń | On existence of mini-boson stars[END_REF]); however these solutions are not analytic at infinity and, by a scaling argument, such solutions do not exist with massless scalar fields. In a later article, we shall obtain this result as a corollary of the result corresponding to Theorem 1.1. In that subsequent work we start from the conformal Einstein field equations with a general energy-momentum tensor as a source and specialize them to scalar field cases.

In Section 2 we analyze the conformal Einstein-Maxwell equations. We first rewrite Maxwell's equations in the unphysical space-time, then translate the physical Bianchi identities and obtain differential equations for the unphysical Weyl spinor and Ricci spinor. In Appendix A we summarize a number of quantities, their relations and behaviour under conformal transformations in the Newman-Penrose formalism ( [START_REF] Newman | An approach to gravitational radiation by a method of spin coefficients[END_REF]); these are extensively used in the main text and in Appendices B and C. In particular, all conformal equations for the gravitational and electromagnetic field analyzed in terms of spinors in Section 2 are projected on the spin basis (i.e. the null tetrad) and written down in the Newman-Penrose formalism in Appendix B. In Section 3 a suitable coordinate system and a convenient Newman-Penrose null tetrad which gives special values to some of the Newman-Penrose spin-coefficients are introduced in the neighbourhood of I -. As noted above, these differ from those used by the authors of [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF]. At he end of Section 3 we explain in detail in what our choice of the coordinate system and the null tetrad differs from that of [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF]. In Appendix C we demonstrate that in contrast to [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] our choice of gauge admits simple static (i.e. "periodic") spacetimes like flat space and the Reissner-Nordström metric. In Section 4 we follow [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] (although in a different conformal gauge) and study the problem in the NP formalism in the unphysical space-time, with data on I -. Assuming periodicity along I -we first discover that the only possibility is the independence of all geometric quantities of an affine parameter v along I -. By induction we then prove that all derivatives of all geometric quantities, including the physical metric components, in the direction into the physical spacetime must also be v-independent. The proof of Theorem 1.1 and Corollary 1.2 then follows from the assumed analyticity.

This paper arose from a collaboration after P. T. posted his work [START_REF] Tod | On analytic asymptotically-flat vacuum and electrovac metrics[END_REF] on the gr-qc arXive and J. B. informed him that he and his PhD. student M. S. were already engaged in tackling the same problem [START_REF] Scholtz | On the non-existence of asymptotically flat periodic solutions of Einstein's equations in Week of doctoral students[END_REF].

The conformal Einstein-Maxwell equations

We first introduce conformal equations for the gravitational and electromagnetic field in the formalism of 2-component spinors. In Appendix B these equations are written down explicitly after the projection on a spin basis, in the form employed in the Newman-Penrose formalism. In the physical space-time, Maxwell's equations without sources are simply § (see e.g. [START_REF] Stewart | Advanced general relativity[END_REF])

∇AA φAB = 0. ( 2
)
They are conformally invariant if under conformal rescaling the Maxwell spinor φ AB tranforms with conformal weight 1, φAB = Ω φ AB , (

when the convention used in this article for conformal rescaling is ˜ AB = Ω -1 AB . From the transformation of the derivative operator (see (A12)), in the unphysical space-time equations (2) become

∇ AA φ AB = 0. ( 4
)
The situation is more complicated in the case of the gravitational field. The physical Bianchi identities read

∇D C ΨABCD = ∇D (C ΦAB)C D , (5)
where ΨABCD and ΦABC D are the Weyl and the Ricci spinor, respectively. Using the rules for the conformal transformation of these spinors (eq. ( A15) and (A17)) we find

Ω 2 ∇ D C ψ ABCD = Ω ∇ D (C Φ AB)C D + ∇ D (A Ω Φ BC)C D + ∇ D (C ∇ A(C ∇ D )B) Ω, (6)
where ψ ABCD = Ω -1 Ψ ABCD . These equations are the physical Bianchi identities written in terms of the quantities in the unphysical space-time. We simplify them by employing Einstein's equations in the physical space-time, Here we used the fact that the physical scalar curvature vanishes for the electromagnetic field; we put the constant factor k on the r.h.s. of ( 7) equal to 1 following the convention of [START_REF] Newman | An approach to gravitational radiation by a method of spin coefficients[END_REF], unlike, e.g. [START_REF] Penrose | Spinors and space-time vol II Cambridge Monographs on Mathematical Physics[END_REF]. From eqs. (A15), ( 7) and ( 3) we obtain

ΦABA B = k φAB φA B . ( 7
∇ A(A ∇ B )B Ω = Ω 3 φ AB φA B -Ω Φ ABA B . ( 8
)
Applying ∇ D C , symmetrizing and using Maxwell's equations (4), we can express the term containing the third derivative of Ω appearing in [START_REF] Friedrich | On the existence of analytic null asymptotically flat solutions of Einstein's vacuum field equations[END_REF] as follows:

∇ D (C ∇ A(C ∇ D )B) Ω = 3Ω 2 φC D φ (AB ∇ D C) Ω + Ω 3 φC D ∇ D (C φ AB) -Ω∇ D (C Φ AB)C D -(∇ D (C Ω)Φ AB)C D .
Inserting this result into (6) we arrive at the conformal Bianchi identities for the Einstein-Maxwell field expressed in terms of the quantities in the unphysical spacetime:

∇ D C ψ ABCD = 3 φA B φ (AB ∇ B C) Ω + Ω φA B ∇ B (C φ AB) . ( 9
)
Projecting these equations onto the spin basis we obtain the set of the equations which are explicitly written down (using the NP formalism) in Appendix B, see (B5a)-(B5h). Equations ( 9) are differential equations for the unphysical Weyl spinor. To obtain the equations for the Ricci spinor we use the Bianchi identities valid for quantities in the unphysical spacetime:

∇ D C Ψ ABCD = ∇ D (C Φ AB)C D . ( 10
)
Combining the last two equations, we get

∇ B (C Φ AB)A B = ψ ABCD ∇ D A Ω + 3 Ω φA B φ (AB ∇ B C) Ω + Ω 2 φA B ∇ B (C φ AB) . (11)
In the following we shall also need the expression for quantities ∇ AA ∇ BB Ω. Let us decompose ∇ AA ∇ BB Ω into its symmetric and antisymmetric parts,

∇ AA ∇ BB Ω = ∇ A(A ∇ B )B Ω + 1 2 A B ∇ AC ∇ C B Ω. ( 12
)
The first term on the r.h.s. is given in [START_REF] Gair | Spherical universes with anisotropic pressure Class[END_REF], the second term can be decomposed again:

∇ AC ∇ C B Ω = ∇ C (A ∇ C B) Ω + 1 2 AB Ω. ( 13
)
Since the operator

∇ C (A ∇ C B) is just the commutator ∇ [a ∇ b] contracted by A B
, it annihilates scalar quantities. Using equations ( 8), ( 12) and ( 13) we obtain

∇ AA ∇ BB Ω = Ω 3 φ AB φA B -Ω Φ ABA B + 1 4 A B AB Ω. ( 14
)
It will be convenient to introduce the quantity

F = 1 2 Ω -1 (∇ AA Ω) (∇ AA Ω), (15)
which can be seen to be smooth in the unphysical space-time from the rule for the conformal transformation of the scalar curvature (A15) in the form

Ω = 4 Ω Λ -4 Ω -1 Λ + 4 F, ( 16
)
since the physical scalar curvature Λ = 0 for the electromagnetic field. From equation ( 14) we now obtain the following expressions for the second derivatives of Ω:

∇ AA ∇ BB Ω = Ω 3 φ AB φA B -Ω Φ ABA B + A B AB (F + Ω Λ). ( 17
)
Finally we wish to derive expressions for ∇ AA F . Directly from the definition of the unphysical Riemann tensor and from the decomposition (A3), we have

(∇ AA ∇ BB -∇ BB ∇ AA ) ∇ BB Ω = -2Φ ABA B ∇ BB Ω + 6 Λ ∇ AA Ω. ( 18
)
Employing Maxwell's equations ( 4) and the contracted Bianchi identities (A21), we find that equations ( 14) and ( 18) imply

∇ AA F = Ω 2 φ B A φB A ∇ BB Ω -Φ ABA B ∇ BB Ω + Λ ∇ AA Ω. ( 19
)

Coordinates, tetrad and conformal gauge

We assume that we have an analytic, time-periodic solution of the Einstein-Maxwell equations and an analytic, time-periodic conformal factor so that the unphysical metric with I -also has these properties. We construct a convenient coordinate system and a Newman-Penrose null tetrad in the neighborhood of I -(see Figs. 1,2). We stay in the unphysical space-time in order to include I -. Let S ⊂ I -be a particular spacelike 2-sphere. We can introduce arbitrary coordinates x I , I = 2, 3 on S and propagate them along I -by the condition

∇ γ x I = 0, (20)
where γ = γ(v) is an affinely-parametrized null generator of I -. We may set v = 0 on S. The triple (v, x 2 , x 3 ) represents suitable coordinates on I -. In order to go into the interior of space-time we introduce the family of null hypersurfaces N v orthogonal to I -and intersecting I -in the space-like cuts S v of constant v. Let γ = γ (r) be the null generators of the surface N v labeled by x I . Here, r is the affine parameter which can be chosen so that r = 0 on I -and g(dv, dr) = 1 at I -. We propagate the coordinates v and x I onto N v by conditions

∇ γ x I = 0, ∇ γ v = 0. ( 21
)
We thus have established a coordinate chart

x µ = (v, r, x 2 , x 3 ) , µ = 0, 1, 2, 3 , (22)
in the neighbourhood of past null infinity.

Next we construct a suitable Newman-Penrose null tetrad. N v are null hypersurfaces v = constant, therefore the gradient of v is both tangent and normal to N v ; we denote it by

n a = ∇ a v . ( 23
)
Since n a is tangent to γ along which only r varies,

n = ∂ ∂r . ( 24
)
Components of tensors with respect to the basis induced by these coordinates will be labelled by Greek letters µ, ν, .... Components with respect to an arbitrary tetrad will be labelled by Latin letters a, b, ... from the beginning of the alphabet. Indices labelled by capital letters I, J, .. have values 2, 3. On each cut S vr : v, r = constant there exists exactly one null direction normal to S vr not proportional to n a . We choose the vector field l a to be parallel to this direction and normalize it by n a l a = 1. It can be written in the form

I - S g ( ) v r v S v g ( r ) ' x I x = c o n s t . I N v N 0 x = c o n s t . I S vr
l = ∂ ∂v -H ∂ ∂r + C I ∂ ∂x I . ( 25
)
On I -l is tangent to the generators γ(v), so functions H and C I vanish on I -. The conformal gauge can be chosen so that ∂Ω ∂r = 1 on I -. (26) Let us now turn to the 2-spheres S vr on which ∂ I are basis vectors. Since S vr is a space-like sphere, we choose, following standard procedure, a complex vector m and its complex conjugate m, such that m a m a = 0 , m a ma = -1 ;

I - S g ( ) v
(27) m has the form

m = P I ∂ ∂x I , ( 28
)
where P 2 , P 3 are complex functions. The coordinates x I can be chosen to be the standard spherical coordinates, x I = (θ, φ). Then the appropriate choice of the null vector m at I -is (see e.g. [START_REF] Stewart | Advanced general relativity[END_REF])

m = 1 √ 2 ∂ θ + i sin θ ∂ φ , P I = 1 √ 2 1, i sin θ . ( 29
)
The vectors m, m are orthogonal to l and n. The contravariant components of the tetrad read l µ = (1, -H, C 2 , C 3 ), n µ = (0, 1, 0, 0), m µ = (0, 0, P 2 , P 3 ).

(30)

The contravariant components of the metric tensor are given, regarding the relation g µ ν = 2l (µ n ν) -2m (µ mν) , by the matrix

g µ ν =     0 1 0 0 1 -2H C 2 C 3 0 C 2 -2P 2 P 2 -P 2 P 3 -P 3 P 2 0 C 3 -P 2 P 3 -P 3 P 2 -2P 3 P 3     . ( 31
)
Using (30) and the inverse of (31) we find the covariant components of the tetrad vectors:

l µ = (H, 1, 0, 0), n µ = (1, 0, 0, 0), m µ = (ω, 0, R 2 , R 3 ), (32)
where

R 2 = P 3 P 2 P 3 -P 3 P 2 ,R 3 = P 2 P 3 P 2 -P 2 P 3 , ω = -C I R I . (33)
The covariant components of the metric are

g µ ν =     2H -2ω ω 1 -ω R2 -ωR 2 -ω R3 -ωR 3 1 0 0 0 -ω R2 -ωR 2 0 -2R 2 R2 -R 3 R2 -R 2 R3 -ω R3 -ωR 3 0 -R 3 R2 -R 2 R3 -2R 3 R3     . ( 34
)
The vectors l, n, m and m constitute the NP tetrad. However, it is not unique since there is a rotation gauge freedom m → e iχ m which will be used later. Following the standard notation of the NP formalism (e.g. [START_REF] Newman | An approach to gravitational radiation by a method of spin coefficients[END_REF], [START_REF] Stewart | Advanced general relativity[END_REF]), we define the operators:

D = l a ∇ a , ∆ = n a ∇ a , δ = m a ∇ a . ( 35
)
We shall also employ the spin basis (o A , ι A ) associated with the null tetrad,

l a = o A ōA , n a = ι A ῑA , m a = o A ῑA , ( 36
)
normalized by o A ι A = 1. Note that this coordinate and tetrad system has some more gauge freedom associated with it. In particular we may make another choice Ω with Ω = ΘΩ where Θ is also periodic and takes the value one at I -. Thus gab = Ω-2 ĝab = Ω -2 g ab , and so ĝab = Θ 2 g ab . We assume that Θ = 1 + f (v, r, x I ) with f = O(r). This will change the definition of the affine parameter r , to r say, and then we must accompany the change of conformal factor with a null-rotation of the tetrad so that δ is tangent to the sphere S vr , thus

na = n a , ma = Θ(m a + Zn a ), la = Θ 2 (l a + Z ma + Zm a + Z Zn a ), (37)
where Z, which parametrises the null rotation, is fixed by requiring δr = 0; the associated operators change according to

∆ = Θ -2 ∆, δ = Θ -1 (δ + Z∆), D = D + Z δ + Zδ + Z Z∆. ( 38
)
With the coordinate v common to both systems, we define r as the affine parameter with ∆r = Θ -2 ∆r = 1.

This can be integrated to give

r = r 0 Θ 2 dr = r + O(r 2 ), (39)
and we need 0 = δr = Θ -1 (δr + Z∆r), so that

Z = -Θ -2 δr,
which can be calculated from (39). Note that Z = O(r 2 ). We shall need to exploit this gauge freedom below. Next we examine what special values some of the spin coefficients take due to the above choice of the null tetrad (we calculate for the unhatted system, but the same relations hold in the hatted systems). Acting by the commutators (A2) on the coordinate v, we find

γ + γ = ᾱ + β -π = ν = µ -μ = 0. (40)
Furthermore, commutators [δ, ∆]r and [ δ, δ]r give

τ -ᾱ -β = ρ -ρ = 0. ( 41
)
Applying the remaining commutators on the variables v, r and x I leads to the "frame equations", i.e. the equations for the metric functions H, C I and P I :

∆ H = -(ε + ε), (42a) δ H = -κ, (42b)
∆ C I = -2 π P I -2 π P I , (42c) δ P I -δ P I = (α -β) P I -(ᾱ -β) P I , (42d)

∆ P I = -(µ -γ + γ) P I -λ P I , (42e) δ C I -D P I = -(ρ + ε -ε) P I -σ P I . ( 42f
)
Since the generators γ(v) of I -are affinely parametrized null geodesics, Dl a = 0 on I -. Comparing this with the general relation

Dl a = (ε + ε) l a -κ m a -κ ma , ( 43
)
we see that

ε + ε = κ = 0 on I -. ( 44
)
Next we wish to show that the freedom in choosing the basis (m, m) of the space tangential to S vr allows us to set γ = 0. From the definition of γ (eq. A1) we have γγ = m a ∆ ma . Under the rotation through χ, m a → e iχ m a , (45) the quantity γγ transforms according to

γ -γ → γ -γ + i∆χ, ( 46
)
so by solving the equation

∆χ = i(γ -γ), ( 47
)
and regarding (40) we can set

γ = 0. ( 48
)
Because the ∆-operator is the derivative with respect to the coordinate r, further rotation (45) with an r-independent function χ does not violate the equality (48).

The quantity εε under the rotation (45) transforms according to

ε -ε → ε -ε + iDχ. ( 49
)
Solving the equation

Dχ = i(ε -ε) ( 50
)
on I -, where r = 0, we set ε = ε which, together with (44), implies

ε = 0 on I -. ( 51
)
To end this section, we exploit the gauge freedom ( 37) and ( 38 Having done this, we omit the hats. In order to elucidate the differences between our choice of the coordinate system and the null tetrad and those used by Gibbons and Stewart, we conclude this section by giving the details of their construction. Instead of affine parameter r they use coordinate u, defined as follows. Let S be a spacelike cut on I -and N , S ⊂ N , a null hypersurface such that the null generators of N are orthogonal to S. Now, the real function u on N is defined in such a way, that u = 0 on S, and the (spacelike, two-dimensional) hypersurface S u : u =constant is orthogonal to the null generators of N . The cut S u defines another null hypersurface N u with null generators orthogonal to S u . Coordinate u is obtained by setting u =constant on N u . Similarly, the family of null hypersurfaces N v orthogonal to spacelike cuts S v on I -, with v being the affine parameter along the null generators of I -, is constructed. Coordinates x I are chosen freely on S and propagated into the spacetime along N and N u . Functions x µ = (u, v, x 2 , x 3 ) constitute a coordinate system in the neighbourhood of I -.

The NP tetrad used in [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] consists of vectors l, tangent to N u , n, tangent to N v , and m, m spanning the tangent space of S and propagated into the spacetime.

Coordinate expressions of their tetrad read (this should be compared with our expressions ( 24), ( 25) and ( 28))

l = Q ∂ v , n = ∂ u + C I ∂ I , m = P I ∂ I , ( 52
)
where Q, C I and P I are metric functions. In this tetrad, following equation holds:

∆n a = -(γ + γ)n a .
Therefore, the null generators of N v are geodesics, but u is not an affine parameter. The periodicity of the spacetime is defined as the periodicity of all geometrical quantities in variable v. It is shown in [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] that K = ∂ v is the Killing vector of the metric and concluded that the spacetime is stationary. However, K is null everywhere by construction as it is tangent to the null generators of N v , while the stationarity requires timelike Killing vector. Thus, it is impossible to conclude that the spacetime is stationary from the fact that K is the Killing vector. As was mentioned in the introduction, even the Minkowski spacetime does not posses Killing vector which is everywhere null and tangent to I -.

In the following, we use the coordinates and the tetrad introduced in the beginning of this section. We show that K = ∂ v is the Killing vector null on I -but timelike in its neighbourhood.

Proof of the theorem

Having chosen coordinates and tetrad and fixed special values of some of the NP coefficients we now analyse all geometric quantities assuming analyticity in the chosen coordinates and periodicity on I -in v. Following [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] we introduce the notation

S 0 = DΩ , S 1 = δΩ , S 2 = ∆Ω , F = 1 Ω S 0 S 2 -S 1 S1 , ψ n = Ψ n Ω , n = 0, 1 , 2, 3, 4, (53)
where Ψ n are the NP components of the Weyl spinor (see eq. (A10)). In the case of asymptotically-flat space-time they vanish on I -, so assuming smoothness, the ψ n are regular there. Tangential derivatives of the conformal factor vanish on I -, i.e. S 0 = S 1 = 0, and so, again by smoothness, the quantity F is regular on I -. The remaining component of ∇Ω is S 2 which is 1 on I -(cf. (26)), so that its tangential derivatives also vanish on I -. Equations ( 17) and ( 19) are explicitly written down in the NP formalism in Appendix B as (B2a) -(B4c). Equations (B2d)-(B2j) show that on

I - σ = 0, (54a) F = 0, (54b) ρ = 0 (54c) π = 0 = β + ᾱ = τ, ( 54d
) ∆S 0 = 0, (54e) ∆S 2 = 0, ( 54f
) ∆S 1 = 0. (54g)
Since F = 0 on I -, also the tangential derivatives DF and δF vanish there. From equations (B4a) and (B4b) we thus obtain

Φ 00 = Φ 01 = 0 on I -. ( 55
)
The metric functions P I on I -are given by (29). Inserting this expression into the frame equation (42d) and using relation (54d) we find

α = -β = - 1 2 √ 2 cot θ on I -. ( 56
)
The Ricci identity (A20q) now shows that

Λ + Φ 11 = 1 2 on I -. ( 57
)
In order to discover the behaviour of the other relevant quantities we shall take into account the properties of the Bondi mass. In a general asymptotically flat electrovacuum spacetime the total mass-energy at I + is defined by the formula (see e.g. [START_REF] Exton | Conserved quantities in the Einstein-Maxwell theory[END_REF])

M B = - 1 2 √ π dS Ψ0 2 + σ0 σ0 . ( 58
)
By the superscript 0 we denote the leading term in the asymptotic expansion of a quantity, superscripts 1, 2, ... then denote higher-order terms, for example, σ = σ0 Ω 2 + σ1 Ω 3 + O(Ω 4 ). The rate of decrease of the Bondi mass is given by

ṀB = - 1 2 √ π dS σ0 σ0 + φ0 2 φ0 2 . ( 59
)
The quantities σ and φ i , i = 0, 1, 2, are defined in (A1) and (A26). Following the "conversion table" between I + and I -(see (A14)), we analogously define the Bondi mass at I -by

M B = - 1 2 √ π dS Ψ0 2 + λ0 λ0 . ( 60
)
Since radiation comes into the physical spacetime through I -but can't exit through it, the total mass energy at I -cannot decrease. Its rate of change in (advanced) time v along I -is given by

ṀB = 1 2 √ π dS λ0 λ0 + φ0 0 φ0 0 . ( 61
)
Now we assume periodicity. But a non-decreasing periodic function must be a constant. Hence, our assumption of periodicity of the mass-energy at

I -requires λ0 = 0, φ0 0 = 0 . ( 62
)
The leading term in the asymptotic expansion of Ψ0 is then Ψ0 0 = λ0 = 0. Regarding equations (A16) and (A18) and putting Ψ0 0 = 0, we can write the asymptotic expansion of Ψ 0 near I -as

Ψ 0 = Ψ 1 0 Ω 2 + O(Ω 3 ), (63)
or (cf. eq. ( 53))

ψ 0 = O(Ω). (64)
Equation ( 63) implies

∆Ψ 0 = 0 on I -. (65)
Similarly, eq. (A27), where we put φ0 0 = 0, implies φ 0 ∈ O(Ω) and ∆φ

0 = φ 1 0 S 2 on I -. ( 66
)
The geometrical quantities consist of the tetrad components, which give the metric functions, the spin coefficients and the components of the Weyl and the Ricci tensor on I -. Because of our assumption of the periodicity of gravitational field, the geometrical quantities are all assumed to be periodic in the variable v on I -. We do not assume the periodicity of the electromagnetic field since this field may not have the same symmetries as the gravitational field (this is the issue of inheritance which we shall return to). We have shown that the following spin coefficients vanish on I -(and thus do not depend on v): µ,ρ,σ,κ,ε,ν,γ,π,τ. (

The spin coefficients α and β are v-independent because of (56). Now we wish to show that also the last spin coefficient λ is independent of v. The Bianchi identity (A23a) together with (65) and (55) shows that

DΦ 02 = 0 on I -. ( 68
)
If we now apply D to the Ricci identity (A20g), we get

D 2 λ = 0 on I -. ( 69
)
The general solution of this equation on I -is λ = λ (0) + v λ (1) , (

where λ (0) and λ (1) are functions independent of v. Since λ is assumed to be periodic and a polynomial in v can be periodic only if it is constant, we get λ = λ (0) and

Dλ = 0 on I -. ( 71
)
(we borrow this style of argument from [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] where it is used extensively). The Ricci identity (A20g) then implies

Φ 02 = 0 on I -. ( 72
)
The Ricci identity (A20h) on

I -becomes Λ = 0, (73)
and then by (57) Φ 11 = 1/2 there. Now from (A22c) and D on (A20k), DΦ 12 and DΦ 22 vanish at I -. We collect these results and some similar ones as a lemma: Proof. The first line is done already, as is the second line up to Dψ 1 , which comes from (B5a). From D applied to (B5b)-(B5d) we obtain D 2 ψ i = 0 whence by periodicity Dψ i = 0 at I -, in order for i = 2, 3, 4 . The same procedure applied to (A29a), (A29b) takes care of Dφ 1 , Dφ 2 . Then the third line follows from D applied to (B2h)-(B2j). Now we turn to the proof of the Theorem. We set up an induction with the following inductive hypothesis: Suppose inductively that

∂ v ∆ j Q = 0 at I -for 0 ≤ j ≤ k with Q one of H, C I , P I , , π, λ, β, α, ρ, σ, κ, F, ψ i , Φ ij , φ i , Λ (74)
and for 0 ≤ j ≤ k + 1 with Q = S i . This is easily seen by the Lemma to hold for k = 0, so we need to deduce it for j = k+1 from its truth for j ≤ k. In this calculation, we use the fact that ∂ v = D at I -, and make extensive use of the commutators (A2). Under the inductive hypothesis, the inductive step follows

• for H, C I , P I from (42a), (42c) and (42e);

• for , π, λ, β, α, ρ, σ, κ, respectively, from (A20f), (A20i), (A20j), (A20l), (A20o), (A20n), (A20m) and (A20c); • for F from (B4c); • for φ 0 and φ 1 directly from (A29c) and (A29d) respectively; for φ 2 , from (A29b) we deduce at I - D 2 ∆ k+1 φ 2 = 0, and then periodicity implies

D∆ k+1 φ 2 = 0;
• for ψ i , i = 0, 1, 2, 3 from (B5e)-(B5h); for ψ 4 , under the inductive hypothesis, we deduce at I - D 2 ∆ k+1 ψ 4 = 0 from (B5d) and then periodicity implies D∆ k+1 ψ 4 = 0;

• for Φ 00 , Φ 01 , Φ 02 , Φ 12 from (A22b), (A23b), (A22d) and (A23d) respectively, all with Ψ n = Ωψ n ; then for Λ, Φ 11 and Φ 22 we use (A20h), (A24c) and (A20k).

This completes the inductive step for the first set of quantities Q. For Q = S i we use D∆ k+1 applied to (B2h)-(B2j). Thus r-derivatives of all orders of the quantities in (74), which includes the metric functions H, C I and P I , are independent of v. Now analyticity in r forces these functions to be independent of v. Therefore, by (30), the metric components are all independent of v and so K := ∂/∂v is a Killing vector of the unphysical metric. However, for any j,

∂ v ∆ j Ω = ∂ v ∆ j-1 S 2 ,
at I -and the r.h.s. vanishes for all j. Thus, by analyticity in r, Ω is also independent of v and so K is a Killing vector of the physical metric too. The norm-squared of the Killing vector is

g(K, K) = 2(H -ω ω).
This is O(r 2 ) at I -but there

∆ 2 g(K, K) = 2∆ 2 H = -2∆(ε + ε) = 2
so that K is null at I -but time-like just inside: the metric is stationary. This completes the proof of the theorem. Note that we have shown that, under the assumption of periodicity of both fields are necessarily time-independent. A slightly different question is whether a stationary asymptotically-flat gravitational field might be produced by an electromagnetic field which is not itself stationary. The content of the Corollary 1.2 is that the answer is no.

Proof of Corollary 1.2. Starting from the assumption that the metric admits ∂ v as a Killing vector, we want to show that this is also a symmetry of the Maxwell field. We have

Φij = Ω 2 φ i φj , ( 75
)
and ∂ v Φij = 0 so that, for some χ possibly depending on v, we have

φ i = e i χ ϕ i , ( 76
)
where ϕ i is v-independent. From the Maxwell equation (A29a), with φ 0 = 0 on I -, we find φ 1 Dχ = 0 on I -so that Dχ = 0 unless φ 1 = 0 there. If φ 1 = 0 there, (A29b) gives Dχ = 0 unless φ 2 = 0, so we can conclude that Dφ i = 0 on I -. Now we set up an induction to show that D∆ n φ i = 0 on I -for all n ∈ N and i = 0, 1, 2. The inductive hypothesis will be

(∀k ≤ n)(∀i ∈ {0, 1, 2})(D∆ k φ i = 0 on I -). (77)
Then by D∆ n on (A29c) and (A29d) we obtain this for k = n + 1 and i = 0, 1. For i = 2, D∆ n+1 on (A29b) gives

D 2 ∆ n+1 φ 2 = 0 on I -, (78)
which integrates to give ∆ n+1 φ 2 = av + b. This would contribute a v-dependent term to Φ22 at O(Ω 2n+4 ), a contradiction unless a = 0. Then D∆ n+1 φ 2 = 0 on I -, which completes the induction.

By assumption, the Maxwell field is analytic and so has a convergent power series in r near to I -and we have shown that all coefficients are v-independent. Since the spinor dyad is Lie-dragged by the Killing vector, this proves that the Maxwell field is too: in this situation the Maxwell field inherits the symmetry.

The spinor equivalent of the Einstein tensor is

G ab = -2 Φ ABA B -6 Λ AB A B , (A7)
and the spinor equivalent of Einstein's equations is

Φ ABA B = -3 Λ AB A B + 4 π T ABA B . (A8)
Taking the symmetric part or contracting them with AB A B , respectively, we obtain two equations, equivalent to (A8):

Φ ABA B = 4 π T (AB)(A B ) , 3 Λ = π T A A A A . (A9)
The five complex components of the Weyl spinor are

Ψ 0 = C abcd l a m b l c m d = Ψ ABCD o A o B o C o D , Ψ 1 = C abcd l a n b l c m d = Ψ ABCD o A o B o C ι D , Ψ 2 = C abcd l a m b mc n d = Ψ ABCD o A o B ι C ι D , Ψ 3 = C abcd l a n b mc n d = Ψ ABCD o A ι B ι C ι D , Ψ 4 = C abcd ma n b mc n d = Ψ ABCD ι A ι B ι C ι D . (A10)
The traceless Ricci tensor has the following components (3 real and 3 complex):

Φ 00 = - 1 2 R ab l a l b = Φ ABA B o A o B ōA ōB , Φ 01 = - 1 2 R ab l a m b = Φ ABA B o A o B ōA ῑB , Φ 02 = - 1 2 R ab m a m b = Φ ABA B o A o B ῑA ῑB , Φ 11 = - 1 4 R ab l a n b + m a mb = Φ ABA B o A ι B ōA ῑB , Φ 12 = - 1 2 R ab n a m b = Φ ABA B o A ι B ῑA ῑB , Φ 22 = - 1 2 R ab n a n b = Φ ABA B ι A ι B ῑA ῑB . (A11)
The three remaining components can be obtained via the condition Φ ij = Φji . Under the conformal rescaling g ab = Ω 2 gab the covariant derivative acting on a 2-component spinor transforms as

∇AA ξ B = ∇ AA ξ B + Ω -1 ξ A ∇ BA Ω. (A12)
The NP quantities also transform. To find relations between the physical and unphysical quantities we have to transform the null tetrad. We wish to keep n a = ña = ∂ a v so the correct choice is

o A = õA , ι A = Ω -1 ιA , o A = Ω õA , ι A = ιA , l a = la , n a = Ω -2 ña , m a = Ω -1 ma , ma = Ω -1 ma l a = Ω 2l a , n a = ña , m a = Ω ma , ma = Ω ma , (A13)
from which the transformation of the spin-coefficients can be found. The geometrical meaning of the spin coefficients depends on the choice of the null tetrad. With our choices, the vector l is pointing into I + , while n is tangent to I + . On I -the role of these vectors is interchanged, n is pointing from I -and l is tangent to it. To convert quantities from I + to I -we have only to switch the spinors o A and The projections of the equation [START_REF] Schild | Electromagnetic two-body problem[END_REF], The electromagnetic 4-potential and corresponding electromagnetic tensor in these coordinates are A µ = (Q r, 0, 0, 0) ,

∇ AA F = Ω 2 φ B A φB A ∇ BB Ω -Φ ABA B ∇ BB Ω + Λ ∇ AA Ω, ( B3
F µ ν = -Q µ ν 2 3 . (C10)
The only non-vanishing NP component of F µν is

φ 1 = Q, (C11)
as one would expect. All these results are in accordance with results obtained in the text. On the other hand, the gauge condition Λ = 0 everywhere, imposed in [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF], leads to a periodic unphysical metric only if m = 0, i.e. flat spacetime. This can be seen as follows: we need to rescale the metric (C6) say to ĝab = Θ -2 g ab so that, by (A15),

Λ = Θ -2 (ΘΛ + 1 4 Θ) = 0,
where the boundary conditions on Θ are that Θ = 1 on r = 0 and, say, v = 0 (in order to preserve the conditions that ρ = 0 on r = 0, µ = 0 on v = 0 and Θ = 1 on v = r = 0). With the metric (C6) this wave equation on Θ becomes

2∂ v ∂ r Θ = ∂ r (A∂ r Θ) -L 2 Θ -2(mr -Qr 2 )Θ, ( C12
)
with A = r 2 (1 -2mr + Q 2 r 2 ) and

L 2 Θ = 1 sin θ ∂ ∂θ ∂Θ ∂θ + 1 sin 2 θ ∂ 2 Θ ∂θ 2 .
Now from (C12) evaluated at r = 0 we calculate ∂ v ∂ r Θ = 0 so that ∂ r Θ is constant on I -, but it vanishes at v = 0 so it is zero for all v. Then from (C12) again at I -, ∂ v ∂ 2 r Θ = -m. Thus Θ cannot be periodic in v unless m = 0, in which case the physical metric is flat.

  ) § Spinor indices are labelled by A, A , B, B , ... and have values 0, 1. The metric has signature -2.

Figure 1 .

 1 Figure 1. Construction of coordinate system.

Figure 2 .

 2 Figure 2. NP null tetrad.

  ) to achieve a further simplification. From the commutator [ δ, ∆] (see (A2) with the values of the spin coefficients fixed above) we calculate μ = Θ -2 (µ + Θ -1 ∆Θ), so that we can set μ = 0 by choosing Θ = exp -

Lemma 4 . 1 .

 41 The following are zero on I -:H, C A , ρ, σ, π, κ, , S 0 , S 1 , F, ψ 0 , Φ 00 , Φ 01 , Φ 02 , φ 0 , Λ, DP A ,Dα, Dβ, DS 2 , Dλ, DΦ 11 , DΦ 12 , DΦ 22 , Dψ 1 , Dψ 2 , Dψ 3 , Dψ 4 , Dφ 1 , Dφ 2 , D∆S 0 , D∆S 1 , D∆S 2 .

3 Ω [ φ0 δφ 1 -δφ 2 - 3 Ωδφ 2 -

 31232 )give DF = -S 0 Φ 11 + S 1 Φ 10 + S1 Φ 01 -S 2 Φ 00 + Ω 2 S 0 φ 1 φ1 -S 1 φ 1 φ0 -S1 φ 0 φ1 + S 2 φ 0 φ0 + Λ S 0 , (B4a)δF = -S 0 Φ 12 + S 1 Φ 11 + S1 Φ 02 -S 2 Φ 01 + Ω 2 S 0 φ 1 φ2 -S 1 φ 1 φ1 -S1 φ 0 φ2 + S 2 φ 0 φ1 + Λ S 1 , (B4b) ∆F = -S 0 Φ 22 + S 1 Φ 21 + S1 Φ 12 -S 2 Φ 11 + Ω 2 S 0 φ 2 φ2 -S 1 φ 2 φ1 -S1 φ 1 φ2 + S 2 φ 1 φ1 + Λ S 2 . (B4c)The conformal Bianchi identities[START_REF] Galloway | Splitting theorems for spatially closed space-times[END_REF] for the Einstein-Maxwell field projected onto the spin basis imply the following system:Dψ 1 -δψ 0 = (π -4α)ψ 0 + 2(ε + 2ρ)ψ 1 -3κψ 2 -3S 1 φ 0 φ0 + 3S 0 φ 0 φ1 +Ω [2σφ 1 φ0 -2βφ 0 φ0 + 2εφ 0 φ1 -2κφ 1 φ1 + φ0 δφ 0 -φ1 Dφ 0 ], (B5a) Dψ 2 -δψ 1 = -λψ 0 + 2(π -α)ψ 1 + 2ρψ 2 -2κψ 3 -S 2 φ 0 φ0 -2S 1 φ 1 φ0 +2S 0 φ 1 φ1 + S1 φ 0 φ1 + 2 φ1 Dφ 1 -(γ + µ)φ 0 φ0 +τφ 1 φ0 + (α + π)φ 0 φ1 + σφ 2 φ0 -ρφ 1 φ1 -κφ 2 φ1 ] + 1 3 Ω φ0 ∆φ 0 -φ1 δφ 0 , (B5b)Dψ 3 -δψ 2 = -2λψ 1 + 3πψ 2 + 2(ρ -ε)ψ 3 -κψ 4 -2S 2 φ 1 φ0 -S 1 φ 2 φ0 + S 0 φ 2 φ1 + 2 S1 φ 1 νφ 0 φ0 -µφ 1 φ0 + λφ 0 φ1 + (β + τ )φ 2 φ0 + πφ 1 φ1 -(ε + ρ)φ 2 φ1 + φ0 ∆φ 1 -φ1 δφ 1 ] φ1 Dφ 2 , (B5c) Dψ 4 -δψ 3 = -3λψ 2 + 2(α + 2π)ψ 3 + (ρ -4ε)ψ 4 -3S 2 φ 2 φ0 + 3 S1 φ 2 φ1 +Ω φ0 ∆φ 2 -φ1 δφ 2 -2νφ 1 φ0 + 2γφ 2 φ0 + 2λφ 1 φ1 -2αφ 2 φ1 , (B5d) δψ 1 -∆ψ 0 = (µ -4γ)ψ 0 + 2(β + 2τ )ψ 1 -3σψ 2 -3S 1 φ 0 φ1 + 3S 0 φ 0 φ2 +Ω [-2βφ 0 φ1 + 2σφ 1 φ1 + 2εφ 0 φ2 -2κφ 1 φ2 -φ2 Dφ 0 + φ1 δφ 0 ], (B5e) δψ 2 -∆ψ 1 = -νψ 0 + 2(µ -γ)ψ 1 + 3τψ 2 -2σψ 3 -S 2 φ 0 φ1 -2S 1 φ 1 φ1 + 2S 0 φ 1 φ1 + S1 φ 0 φ2 + 2 3 Ω [-(γ + µ)φ 0 φ1 + τφ 1 φ1 + σφ 2 φ1 + (π + α)φ 0 φ2 -ρφ 1 φ2 -κφ 2 φ2 + φ1 δφ 1 -φ2 Dφ 1 ] + 1 φ1 ∆φ 0 -φ2 δφ 0 , (B5f)δψ 3 -∆ψ 2 = -2νψ 1 + 3µψ 2 + 2(τ -β)ψ 3 -σψ 4 -2S 2 φ 1 φ1 -S 1 φ 2 φ1 + S 0 φ 2 φ2 + 2 S1 φ νφ 0 φ1 -µφ 1 φ1 + (β + τ )φ 2 φ1 + λφ 0 φ2 + πφ 1 φ2 -(ε + ρ)φ 2 φ2 + φ1 ∆φ 1 -φ2 δφ 1 ] φ2 Dφ 2 , (B5g) δψ 4 -∆ψ 3 = -3νψ 2 + 2(2γ + 2µ)ψ 3 + (τ -4β)ψ 4 -3S 2 φ 2 φ1 + 3 S1 φ 1 φ2 +Ω -2νφ 1 φ1 + 2γφ 2 φ1 + 2λφ 1 φ2 -2αφ 2 φ2 + φ1 ∆φ 2 -φ2 δφ 2 . (B5h)

Acknowledgments P. T. gratefully acknowledges hospitality and financial support from the Mittag-Leffler Institute, Djursholm, Sweden and the Charles University, Prague, and useful discussions with Gary Gibbons and John Stewart. J.B. also acknowledges the discussions with Gary Gibbons and the partial support from the Grant GA CR 202/09/00772 of the Czech Republic, of Grant No LC06014 and MSM0021620860 of the Ministry of Education. The work of M. S. was supported by the Grant GAUK no. 22708 of the Charles University, Czech Republic.

Appendix A: The Newman-Penrose formalism and conformal transformations in Einstein-Maxwell spacetimes

A1. Gravitational field

In the NP formalism, the spin coefficients are the Ricci rotation coefficients with respect to a null tetrad {l, n, m} with the corresponding spin basis o A , ι A ; they encode the connection. The twelve independent complex coefficients are defined by (see e.g. [START_REF] Stewart | Advanced general relativity[END_REF], [START_REF] Newman | An approach to gravitational radiation by a method of spin coefficients[END_REF] for details)

where

Acting on a scalar, the operators D, ∆, δ obey the commutation relations:

The Riemann tensor can be decomposed as follows:

The first part is the Weyl tensor whose spinor equivalent is the totally symmetric Weyl spinor Ψ ABCD :

The scalar Λ is related to the scalar curvature R by

The symmetric Ricci spinor Φ ABC D is equivalent to the trace-free part of the Ricci tensor:

ι A (and adjust some signs). The correspondence between the quantities on I + and I -is given in the following table:

The scalar curvature and the Ricci spinor transform according to the formulas

the NP components of the Weyl spinor as Ψn = Ω n Ψ n .

(A16) The Weyl spinor is conformally invariant with weight zero:

Ψ ABCD = ΨABCD . (A17) Because the physical Weyl spinor vanishes on I -, so does the unphysical one, and assuming smoothness is therefore O(Ω). Then we get Ψn ∈ O(Ω n+1 ) .

(A18) The Ricci identities can be written in the spinor form as follows:

Substituting the basis spinors o A and ι A for ξ A and projecting the last equations onto the spin basis we obtain the Ricci identities in the NP-formalism:

The spinor form of the Bianchi identities is

Projecting these equations onto the spin basis leads to the Bianchi identities in the NP formalism:

)

A2. Electromagnetic field

For the description of an electromagnetic field we use the electromagnetic field tensor F ab and its spinor equivalent φ AB :

The NP components of the Maxwell spinor are defined by

The conformal transformation of these quantities is given by φAB

Maxwell's equations without sources are equivalent to the (conformally-invariant) spin-1 zero-rest-mass equation

Projecting this onto the spin basis we obtain Maxwell's equations in the NP formalism:

(A29d)

Appendix B: Conformal field equations

B1. Einstein-Maxwell fields

The projections of the equation [START_REF] Papapetrou | Non-existence of periodically varying non-singular gravitational fields in Les théories relativistes de la gravitation[END_REF],

onto the null tetrad imply the following system of equations:

Appendix C: Reissner-Nordström spacetime

To justify our choice of gauge and show that the choice made by [START_REF] Gibbons | Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity in Classical general relativity[END_REF] is too restrictive we shall show here how a simple space-time, namely, the Reissner-Nordström solution, appears in our gauge. The physical metric is

where Q is the charge and dΣ 2 = dθ 2 + sin 2 θdφ 2 . In the standard conformal compactification of the Reissner-Nordström spacetime one introduces the "tortoise coordinate" r * and the advanced time v by

In these coordinates the physical metric acquires the form

(C7)

From the metric the other geometrical quantities follow. The spin coefficients are all zero, except for

The non-zero components of the Weyl and Ricci tensor read

(C9)