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Journal Pre-proof Lessons already learnt from the Covid-19 pandemic Letter to the Editor of The Journal of Thoracic Oncology Lessons already learnt from the Covid-19 pandemic
2020
MDJosé Sanz-Santos jsanzsantos@mutuaterrassa.cat
PhDRamón Rami-Porta
PhD, FETCSSergi Call
MD, PhDJosé Sanz-Santos
Department of Pulmonology
Hospital Universitari Mútua Terrassa
University of Barcelona
Terrassa (Barcelona)Spain
Department of Medicine
Medical School
University of Barcelona
BarcelonaSpain
Network of Centers for Biomedical Research in Respiratory Diseases (CIBERES) Lung Cancer Group
Terrassa (Barcelona)Spain
MDRamón Rami-Porta
Network of Centers for Biomedical Research in Respiratory Diseases (CIBERES) Lung Cancer Group
Terrassa (Barcelona)Spain
MDSergi Call
Department of Thoracic Surgery
Hospital Universitari Mútua Terrassa
University of Barcelona
Terrassa, BarcelonaSpain
MDJosé Sanz-Santos
Department of Thoracic Surgery
Hospital Universitari Mútua Terrassa
University of Barcelona
Terrassa, BarcelonaSpain
Department of Morphological Sciences
Medical School
Department of Pulmonology. Hospital
Unit of Human Anatomy and Embryology
Autonomous University of Barcelona
Bellaterra, BarcelonaSpain
Universitari Mútua Terrassa
Plaça Dr. Robert 508221Terrassa (Barcelona)Spain
Journal Pre-proof Lessons already learnt from the Covid-19 pandemic Letter to the Editor of The Journal of Thoracic Oncology Lessons already learnt from the Covid-19 pandemic
Journal of Thoracic Oncology
202010.1016/j.jtho.2020.04.012Received Date: 10 April 2020 Accepted Date: 10 April 2020Since January 2020 Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre -including this research content -immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active. Please cite this article as: Sanz-Santos J, Rami-Porta R, Call S, Lessons already learnt from the This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. 1 Corresponding author:coronavirusCovid-19lung cancerthoracic malignancies Disclosure: Nothing to disclose
The first Covid-19 patient in Spain was registered on 31 st January 2020. Since then, the escalating growth of the disease has affected more than 150,000 patients, has caused over 15,000 deaths, and a similar number of health professionals has been infected. As of 10 th April 2020, Spain is the European country with the highest number of patients and the third in the world regarding deaths 1 .
The thoracic surgery service of our 400-bed hospital serves a population of 1,200,000 inhabitants, performs about 120 lung resections for lung cancer and over 90 surgical of delays are difficult to quantify, but most likely will jeopardize the fate of many patients. The workload, once we can resume normal activities, will be enormous.
The Covid-19 pandemic may not be the only one we will have to face in our professional lives. If there were another one, all measures should be taken to keep a section of the hospital clean, so that the regular activities could continue for as long as possible. Sooner or later patients have to be transferred to makeshift hospitals. If this is done in the early phase of the disease 2 , normal activities could be continued for much longer and priority cancer patients would benefit of timely treatment. The early testing of health personnel and patients and the use of adequate protective gear would prevent the dissemination of disease in the hospital, which has been catastrophic in our case. These measures would reduce the number of infections among the health care personnel, would maintain an area of the hospital clean, and would increase the capacity to continue the regular activities with non-Covid-19 patients.
explorations of the mediastinum per year. In early March, the commission in charge of the hospital organization during the pandemic restricted the outpatient clinic to the day-hospital for onco-hematologic treatments and the surgical activity to priority oncologic operations. However, with the exponential increase of Covid-19 patients, the outpatient clinics and the postsurgical recovery rooms had to be transformed into hospitalization wards and intensive care units, respectively, and the respirators of the operating rooms had to be used to ventilate Covid-19 patients. The result is that no surgery can be performed other than emergency cases and no new patients can be seen in the outpatient clinics. Our hospital is like a casualty hospital. All wards are filled with Covid-19 patients. Nearly 300 health professionals are infected and more are in quarantine for having been in close contact with infected patients or colleagues. Those who can still work are devoted exclusively to Covid-19 patients. Some patients have been externalized in a nearby hotel because there was no room in the hospital. Pulmonologists, thoracic surgeons and oncologists, who usually meet in tumor boards, are now working in improvised medical teams attending infected patients. There is only one thoracic 3 surgeon who controls the chest tubes and performs tracheostomies. Tumor boards are now conducted virtually, if there are patients to discuss. The numbers in surgical waiting lists and of patients waiting to be diagnosed are increasing. The consequences
Thoracic Surgery Outcomes Research Network, Inc. COVID-19 giuidance for triage of operations for thoracic malignancies: a consensus statement from Thoracic Surgery Research Network, the Society of Thoracic Surgeons and the American Association for Thoracic Surgery. 10.1016/j.jtcvs.2020.03.061J Thorac Cardiovasc Surg. Thoracic Surgery Outcomes Research Network, Inc. COVID-19 giuidance for triage of operations for thoracic malignancies: a consensus statement from Thoracic Surgery Research Network, the Society of Thoracic Surgeons and the American Association for Thoracic Surgery. J Thorac Cardiovasc Surg 2020. Doi: https://doi.org/10.1016/j.jtcvs.2020.03.061.
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Supplementary Table-1: SARS-CoV-2 variant spike proteins. Variants Pango Mutations Stabilization Length Furin Wuhan-Hu-1 A -2P, 6P, 6P-Mut7 1-1208 GSAS Alpha
GSAS Beta
B.1.1.7 del69-70, del144, B.1.351 L18FN501Y, A570D, D614G, P681H, T716I, S982A, D1118H 6P-Mut7 1-1208, D80A, D215G, del242-244, K417N, E484K, N501Y, D614G, A701V 6P-Mut7 1-1208
GSAS Gamma
P.1 L18FT20N, P26S, D138Y, R190S, K417T, E484K, N501Y, D614G, H655Y, T1027I, V1176F
Supplementary Table-1: SARS-CoV-2 variant spike proteins. Variants Pango Mutations Stabilization Length Furin Wuhan-Hu-1 A -2P, 6P, 6P-Mut7 1-1208 GSAS Alpha
71C9CB3974FC78C276433CB30FF83E34Delta B.1.617.2 T19RG142Ddel156-157R158GL452RT478KD614GP681RD950N 6P6P-Mut7 1-1208 GSAS Omicron B.1.1.529 A67Vdel69-70T95IG142Ddel143-145del211ins214EPEG339DS371LS373PS375FK417NN440KG446SS477NT478KE484AQ493RG496SQ498RN501YY505HT547KD614GH655YN679KP681HN764KD796YN856KQ954HN969KL981F
Figure S1
: Previous quantitation of site-specific N-glycosylation on SARS-CoV-2 spike-protein trimer vary.Shown are the results from site-specific N-glycosylation analysis by standard MS-based glycoproteomic method on Wuhan-Hu-1 as reported by Watanabe et. al., 2020 [S1] (Report 1) ;Chawla et. al., 2022 [S2] (Report 2); Newby et. al., 2023 [S3] (Report 3); Shajahan et. al., 2020 [S4] (Report 4); Shajahan et. al., 2023 [S5] (Report 5); and Wang et. al., 2021 [S6] (Report 6), compared to results from DeGlyPHER (top 2 rows, 2P followed by 6P).A visual estimate from these reports is plotted here and compared at each N-glycosylation site.Variation in glycosylation pattern observed in different reports are apparent at many sites.Color-coding of "Report #", groups the reports based on research laboratories authoring them.Cited studies used either 2P stabilized mutant of SARS-CoV-2 spike-protein [S7, S8] or 6P stabilized mutant [S9], as indicated, except report 4 that used independently expressed S1 and S2 subunits of S-protein.Error bars are absent here because all the cited reports here except this study, have not estimated error in their calculations, presumably owing to low sampling.*Report 4 does not claim quantitation; thus, estimated values are from the types of glycoforms reported.Only this study and Report 1 have validated that S-proteins examined are well-folded trimers (using negative-stain electron microscopy).n.d.: glycosylation not determined.
Key:
Foldon, T4 fibritin trimerization domain [S10] HRV3C protease cleavage sequence (modified from [S11]) TwinStrep tag for affinity purification [S12] 8x His-tag Possible conflict * stop codon
Figure S2 :
S2
Figure S2: Evaluating SARS-CoV-2 spike-protein trimer purity and integrity.(A) Size exclusion chromatography of SARS-CoV-2 spike-protein trimers on Superose 6 columns.Trimer elution peaks at 68-and 13-mL post-injection with Superose 6 Hiload 16/600 pg and Superose 6 Increase 10/300 GL columns, respectively.(B) Representative negative-stain TEM micrographs and 2D classes of SARS-CoV-2 spike-protein trimers show highly homogenous particles in various orientations.
Figure S3 :
S3
Figure S3: Peptide sampling at each NGS across all spike-protein trimers analyzed.(A) Number of unique peptides mapping to each NGS shows that N709, N717 and N1134 are inconsistently sampled, but all other NGS are sufficiently sampled.(B) Bar graph demonstrating N-glycan heterogeneity determined at N1134.The N-glycans present at N1134 are mostly complex as seen in previous studies, though the sampling at this NGS is inconsistent in our study.N-glycosylation states are color-coded.Error bars represent mean-SEM.
Supplementary Table-2: SARS-CoV-2 spike-protein stabilizing mutations.
SARS-CoV-2 spike-protein sequences.Wuhan-Hu-1 2P, 1-1208 (2P -K986P, V987P; Furin CS -R682G, R683S, R685S)MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIG6P, 6P-Mut7 VTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDI 1-1208 GSASLSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHSAWSHPQFEKGGGSGGGGSGGSAWSHPQFEK*MuB1.1.621 T95I, Y144T, Y145S,6P-Mut71-1208 GSASins146N, R346K, E484K,N501Y, D614G, P681H,D950NLambdaC.37G75V, T76I, del246-252,6P-Mut71-1208 GSASL452Q, F490S, D614G,T859NNVX-CoV2373--2P1-1273 QQAQVersions Mutations2PK986P, V987P6PF817P, A892P, A899P,A942P, K986P, V987P6P-Mut7 F817P, A892P, A899P,A942P, K986P, V987P;Mut7 -V705C, T883C
Wuhan-Hu-1 6P, 1-1208 (6P -F817P, A892P, A899P, A942P, K986P, V987P; Furin CS -R682G, R683S, R685S)
YAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIAD QDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYF IAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIGVTQPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFL NVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLT LDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLG QSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSAENSCAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGI GNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDC NESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGGSLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIG GGGGSGGSAWSHPQFEK*VTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLM Gamma (P.1) Variant 6P-Mut7, 1-1208 (L18F, T20N, P26S, D138Y, R190S, K417T, E484K, N501Y,SFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNT D614G, H655Y, T1027I, V1176F; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C,FVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVA T883C; Furin CS -R682G, R683S, R685S)KNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKG MFVFLVLLPLVSSQCVNFTNRTQLPSAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVGGSGGGGSGGSAWSHPQFEK* SGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNYPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLSEFVFKNIDGYFKIYSKHTPIAlpha (B.1.1.7) Variant 6P-Mut7, 1-1208 (del69-70, del144, N501Y, A570D, D614G, P681H, T716I, NLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNS982A, D1118H; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C, T883C; Furin CS ENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASV-R682G, R683S, R685S) YAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGTIADMFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAISG YNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVKGFNCYFTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLG PLQSYGFQPTYGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLVYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLV PFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENG PTWRVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAW AENSCAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGINRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNY AVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQ LGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIGMFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHV SYGFQPTYGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQ VTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDISGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPF QFGRDIDDTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTW LSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAAIKMSECVLGQSKRVDFCGKGYHLMLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPI RVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSHGSASSVASQSIIAYTMSLGAEN SFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTNLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYN SCAYSNNSIAIPINFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVE FVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASFVNIQKEIDRLNEVAENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASV QDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGD KNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIAD IAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIGVTQ GGSGGGGSGGSAWSHPQFEK*YNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYF NVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILARPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFL QSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTHNTFVS PFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLT LDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFP Delta (BPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLG GNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGI NESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGGSAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDC GGGGSGGSAWSHPQFEK*LGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDI Beta (B.1.351) Variant 6P-Mut7, 1-1208 (L18F, D80A, D215G, del242-244, K417N, E484K, N501Y, LSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLM D614G, A701V; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C, T883C; Furin CSSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNT -R682G, R683S, R685S)FVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVA MFVFLVLLPLVSSQCVNFTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKG SGTNGTKRFANPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFGGSGGGGSGGSAWSHPQFEK* LGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRGLPQGFSALEPLVDLPIGINITRFQTLHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGWuhan-Hu-1 6P-Mut7, 1-1208 (6P -F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C, TITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWT883C; Furin CS -R682G, R683S, R685S) NRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGNIADYNYMFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHV KLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVKGFNCYFPLQSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPF SYGFQPTYGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPI QFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWNLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYN RVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGVENENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASV SCAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVE
.1.617.2) Variant 6P, 1-1208 (T19R, G142D, del156-157, R158G, L452R, T478K, D614G, P681R, D950N; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Furin CS -R682G, R683S, R685S)
LNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGG Omicron (B.1.1.529) Variant 6P-Mut7, 1-1208 (A67V, del69-70, T95I, G142D, del143-145, del211, MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGGGGSGGSAWSHPQFEK* L212I, ins214EPE, G339D, S371L, S373P, S375F, K417N, N440K, G446S, S477N, T478K, E484A, SGTNVIKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFQ493R, G496S, Q498R, N501Y, Y505H, T547K, D614G, H655Y, N679K, P681H, N764K, D796Y, LGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPIMFVFLVLLPLVSSQCVNLRTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHV SGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPF LDVYYHKNNKSWMESGVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINL VRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNEN GTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYA WNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYN EQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLG QSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVS GGGSGGGGSGGSAWSHPQFEK* NSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAV LDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFP AKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEK WRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSRGSASSVASQSIIAYTMSLGAE NVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDIFSR TFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEV QQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPT IAARDLICAQKFKGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQ MSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDN QSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPF QDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGD ILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHL YKLPDDFTGCVIAWNSNNLDSKVGGNYNYRYRLFRKSNLKPFERDISTEIYQAGSKPCNGVEGFNCYFPL Delta (B.1.617.2) Variant 6P-Mut7, 1-1208 (T19R, G142D, del156-157, R158G, L452R, T478K, D614G, P681R, D950N; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C, T883C; Furin CS -R682G, R683S, R685S) MFVFLVLLPLVSSQCVNLRTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHV SGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPF LDVYYHKNNKSWMESGVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINL VRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNEN GTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYA WNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYN YKLPDDFTGCVIAWNSNNLDSKVGGNYNYRYRLFRKSNLKPFERDISTEIYQAGSKPCNGVEGFNCYFPL QSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPF QQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPT WRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSRGSASSVASQSIIAYTMSLGAE NSCAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAV EQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLG DIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIGVT N856K, Q954H, N969K, L981F; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C, NLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITD T883C; Furin CS -R682G, R683S, R685S) AVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKR MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHVISG ISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPD TNGTKRFDNPVLPFNDGVYFASIEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLD DFTGCVIAWNSNNLDSKVGGNYNYQYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYSPLQSYGF HKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPILVREP QPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGR EDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENG DIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYS TITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFDEVFNATRFASVYAW TGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSCAY NRKRISNCVADYSVLYNLAPFFTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGNIADYNY SNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKN KLPDDFTGCVIAWNSNKLDSKVSGNYNYLYRLFRKSNLKPFERDISTEIYQAGNKPCNGVAGFNCYFPLR TQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAAR SYSFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLKGTGVLTESNKKFLPFQ DLICAQKFNGLNVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLY QFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTW ENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPP RVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSHGSASSVASQSIIAYTMSLGAEN EAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAP SCAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVE HGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCD QDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGD VVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESL IAARDLICAQKFKGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGIGVTQ IDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGGSGGGG NVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDIFSR SGGSAWSHPQFEK* LDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFP QNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQNVVNQNAQALNTLVKQLSSNFGAISSVLNDILS RLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSF PQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFV SGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKN LNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGG QSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVS Reference Sequence (Wuhan-Hu-1, 1-1273); UniProt# P0DCT2 GNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNL >QHD43416.1 surface glycoprotein [severe acute respiratory syndrome coronavirus 2] NESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGGS MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHV GGGGSGGSAWSHPQFEK* SGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPF SGGGGSGGSAWSHPQFEK* Omicron (B.1.1.529) Variant 6P, 1-1208 (A67V, del69-70, T95I, G142D, del143-145, del211, L212I, ins214EPE, G339D, S371L, S373P, S375F, K417N, N440K, G446S, S477N, T478K, E484A, Q493R, LGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPI Mu (B.1.621) Variant 6P-Mut7, 1-1208 (T95I, Y144T, Y145S, ins146N, R346K, E484K, N501Y, D614G, NLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYN P681H, D950N; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C, T883C; Furin CS ENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASV -R682G, R683S, R685S) MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHV YAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIAD G496S, Q498R, N501Y, Y505H, T547K, D614G, H655Y, N679K, P681H, N764K, D796Y, N856K, SGTNGTKRFDNPVLPFNDGVYFASIEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPF YNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYF Q954H, N969K, L981F; 6P -F817P, A892P, A899P, A942P, K986P, V987P; Furin CS -R682G, LGVTSNHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTP PLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFL R683S, R685S) INLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKY PFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLT MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHVISG NENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATKFAS PTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPRRARSVASQSIIAYTMSLG TNGTKRFDNPVLPFNDGVYFASIEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLD VYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIA AENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGI HKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPILVREP DYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVKGFNCY AVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDC EDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENG FPLQSYGFQPTYGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKF LGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIG TITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFDEVFNATRFASVYAW LPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQL VTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDI NRKRISNCVADYSVLYNLAPFFTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGNIADYNY TPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSHGSASSVASQSIIAYTMSL LSRLDKVEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLM KLPDDFTGCVIAWNSNKLDSKVSGNYNYLYRLFRKSNLKPFERDISTEIYQAGNKPCNGVAGFNCYFPLR GAENSCAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTG SFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNT SYSFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLKGTGVLTESNKKFLPFQ IAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGD FVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVA QFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTW CLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTICSGWTFGAGPALQIPFPMQMAYRFNGI KNLNESLIDLQELGKYEQYIKWPWYIWLGFIAGLIAIVMVTIMLCCMTSCCSCLKGCCSCGSCCKFDEDD RVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSHGSASSVASQSIIAYTMSLGAEN SVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVE GVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQNVVNQNAQALNTLVKQLSSNFGAISSVLND SEPVLKGVKLHYT*DIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVT GNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQNVVNQNAQALNTLVKQLSSNFGAISSVLNDILS NESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGSAWSHPQFEKGGGS Lambda (C.37) Variant 6P Mut7, 1-1208 (G75V, T76I, del246-252, L452Q, F490S, D614G, T859N; 6P -GGGGSGGSAWSHPQFEK* F817P, A892P, A899P, A942P, K986P, V987P; Mut7 -V705C, T883C; Furin CS -R682G, R683S, RLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSF PQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFV R685S)SGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKN
Site-specific glycan analysis of the SARS-CoV-2 spike. Y Watanabe, J D Allen, D Wrapp, J S Mclellan, M Crispin, 10.1126/science.abb998332366695PMC7199903Science. 36965012020 Jul 17. 2020 May 4
Glycosylation and Serological Reactivity of an Expression-enhanced SARS-CoV-2 Viral Spike Mimetic. H Chawla, S E Jossi, S E Faustini, F Samsudin, J D Allen, Y Watanabe, M L Newby, E Marcial-Juárez, R E Lamerton, J S Mclellan, P J Bond, A G Richter, A F Cunningham, M Crispin, 10.1016/j.jmb.2021.16733234717971PMC8550889J Mol Biol. 43421673322022 Jan 30. 2021 Oct 27
Variations within the Glycan Shield of SARS-CoV-2 Impact Viral Spike Dynamics. M L Newby, C A Fogarty, J D Allen, J Butler, E Fadda, M Crispin, 10.1016/j.jmb.2022.16792836565991PMC9769069J Mol Biol. 43541679282023 Feb 28. 2022 Dec 21
Site specific N-and O-glycosylation mapping of the spike proteins of SARS-CoV-2 variants of concern. Sci Rep. A Shajahan, L E Pepi, B Kumar, N B Murray, P Azadi, 10.1038/s41598-023-33088-037344512PMC102849062023 Jun 211310053
Deducing the N-and O-glycosylation profile of the spike protein of novel coronavirus SARS-CoV-2. Glycobiology. A Shajahan, N T Supekar, A S Gleinich, P Azadi, 10.1093/glycob/cwaa04232363391PMC72391832020 Dec 930
N-glycosylation profiles of the SARS-CoV-2 spike D614G mutant and its ancestral protein characterized by advanced mass spectrometry. Sci Rep. D Wang, B Zhou, T R Keppel, M Solano, J Baudys, J Goldstein, M G Finn, X Fan, A P Chapman, J L Bundy, A R Woolfitt, S H Osman, J L Pirkle, D E Wentworth, J R Barr, 10.1038/s41598-021-02904-w34876606PMC86516362021 Dec 71123561
Immunogenicity and structures of a rationally designed prefusion MERS-CoV spike antigen. J Pallesen, N Wang, K S Corbett, D Wrapp, R N Kirchdoerfer, H L Turner, C A Cottrell, M M Becker, L Wang, W Shi, W P Kong, Andres El Kettenbach, A N Denison, M R Chappell, J D Graham, B S Ward, A B Mclellan, J S , 10.1073/pnas.170730411428807998PMC5584442Proc Natl Acad Sci. 114352017 Aug 29. 2017 Aug 14
Structural analysis of full-length SARS-CoV-2 spike protein from an advanced vaccine candidate. S Bangaru, G Ozorowski, H L Turner, A Antanasijevic, D Huang, X Wang, J L Torres, J K Diedrich, J H Tian, A D Portnoff, N Patel, M J Massare, Yates Jr 3rd, D Nemazee, J C Paulson, G Glenn, G Smith, A B Ward, 10.1126/science.abe150233082295PMC7857404Science. 37065202020 Nov 27. 2020 Oct 20
Structure-based design of prefusion-stabilized SARS-CoV-2 spikes. C L Hsieh, J A Goldsmith, J M Schaub, A M Divenere, H C Kuo, K Javanmardi, K C Le, D Wrapp, A G Lee, Y Liu, C W Chou, P O Byrne, C K Hjorth, N V Johnson, J Ludes-Meyers, A W Nguyen, J Park, N Wang, D Amengor, J J Lavinder, G C Ippolito, J A Maynard, I J Finkelstein, J S Mclellan, 10.1126/science.abd082632703906PMC7402631Science. 36965102020 Sep 18. 2020 Jul 23
the natural trimerization domain of T4 fibritin, dissociates into a monomeric A-state form containing a stable beta-hairpin: atomic details of trimer dissociation and local beta-hairpin stability from residual dipolar couplings. S Meier, S Güthe, T Kiefhaber, Grzesiek S Foldon, 10.1016/j.jmb.2004.09.07915544812J Mol Biol. 34442004 Dec 3
Substrate requirements of human rhinovirus 3C protease for peptide cleavage in vitro. M G Cordingley, P L Callahan, V V Sardana, V M Garsky, R J Colonno, 2160953J Biol Chem. 265161990 Jun 5
Development of the Twin-Strep-tag® and its application for purification of recombinant proteins from cell culture supernatants. T G Schmidt, L Batz, L Bonet, U Carl, G Holzapfel, K Kiem, K Matulewicz, D Niermeier, I Schuchardt, K Stanar, 10.1016/j.pep.2013.08.02124012791Protein Expr Purif. 9212013 Nov. 2013 Sep 6
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Postprandial sleep in short-sleeping Mexican cavefish
Kathryn Gallman
Department of Biology
Texas A&M University
77840College StationTX
Aakriti Rastogi
Department of Biology
Texas A&M University
77840College StationTX
Owen North
Department of Biology
Texas A&M University
77840College StationTX
Morgan O'gorman
Department of Biology
Texas A&M University
77840College StationTX
Pierce Hutton
Department of Biology
Texas A&M University
77840College StationTX
Evan Lloyd
Department of Biology
Texas A&M University
77840College StationTX
Wes Warren
Department of Genomics
University of Missouri
65201ColumbiaMO
Johanna E Kowalko
Department of Biological Sciences
Lehigh University
18015BethlehemPA
Erik R Duboue
Florida Atlantic University
33458JupiterFL
Nicolas Rohner
Stowers Institute for Medical Research
64110Kansas CityMO
Alex C Keene
Department of Biology
Texas A&M University
77840College StationTX
Postprandial sleep in short-sleeping Mexican cavefish
0BC8F28AE58DC0D644771A585783494410.1101/2024.07.03.602003
Interaction between sleep and feeding behaviors are critical for adaptive fitness.Diverse species suppress sleep when food is scarce to increase the time spent foraging.Post-prandial sleep, an increase in sleep time following a feeding event, has been documented in vertebrate and invertebrate animals.While interactions between sleep and feeding appear to be highly conserved, the evolution of postprandial sleep in response to changes in food availability remains poorly understood.Multiple populations of the Mexican cavefish, Astyanax mexicanus, have independently evolved sleep loss and increased food consumption compared to surface-dwelling fish of the same species, providing the opportunity to investigate the evolution of interactions between sleep and feeding.Here, we investigate effects of feeding on sleep in larval and adult surface fish, and two parallelly evolved cave populations of A. mexicanus.Larval surface and cave populations of A. mexicanus increase sleep immediately following a meal, providing the first evidence of postprandial sleep in a fish model.The amount of sleep was not correlated to meal size and occurred independently of feeding time.In contrast to larvae, postprandial sleep was not detected in adult surface or cavefish, that can survive for months without food.Together, these findings reveal that postprandial sleep is present in multiple short-sleeping populations of cavefish, suggesting sleep-feeding interactions are retained despite the evolution of sleep loss.These findings raise the possibility that postprandial sleep is critical for energy conservation and survival in larvae that are highly sensitive to food deprivation.
Introduction
Sleep and metabolic regulation are highly variable throughout the animal kingdom (Lesku et al. 2006;Joiner 2016;Keene and Duboue 2018;Seebacher 2018).This variability is reflected by the diversity of food availability and foraging strategy, which potently impact the duration and timing of sleep.There is an interaction between sleep and feeding, regardless of life history strategy, that is critical for organismal survival, and therefore, under selection (Capellini et al. 2008;Yurgel et al. 2014;Slocumb et al. 2015;Aulsebrook et al. 2016;Brown et al. 2019).While both of these behavioral processes have been studied in detail, much less is known about interactions between sleep and feeding, particularly in the context of evolution.
In many species, sleep deprivation results in increased food intake, while prolonged periods of food deprivation lead to a reduction in metabolic rate and suppression of sleep (Keene et al. 2010;Arble et al. 2015;Stahl et al. 2017;Regalado et al. 2017;Goldstein et al. 2018).Conversely, animals ranging from the nematode, C. elegans, to humans, increase sleep immediately following a meal, revealing an acute effect of dietary nutrients on sleep regulation (Stahl et al. 1983;Murphy et al. 2016;Makino et al. 2021).Defining how evolution has shaped interactions between sleep, metabolic regulation, and feeding is critical to determine the functions of these traits.
The rapidly increasing number of organisms used to study sleep provides new opportunities to study interactions between sleep and metabolism (McNamara et al. 2009;Anafi et al. 2019).Fish have become a model to study the biological basis of sleep regulation (Chiu and Prober 2013; Levitas-Djerbi and Appelbaum 2017; Keene and Appelbaum 2019).Growing evidence suggests the genetic and functional basis of sleep is conserved across multiple fish species (Chiu and Prober 2013;Levitas-Djerbi and Appelbaum 2017;Keene and Appelbaum 2019).Further, the small size and amenability to genetic manipulation of these fish allows for high-throughput genetic and pharmacological screens to identify novel regulators of sleep (Rihel et al. 2010;Chiu et al. 2016;Kroll et al. 2021).Furthermore, at larval stages, many fish models are transparent, allowing for mapping of sleep and feeding circuits across the entire brain (Semmelhack et al. 2014;Leung et al. 2019;Wee et al. 2019;Förster et al. 2020).Therefore, zebrafish and other fish models are exceptionally well positioned to examine interactions between sleep and feeding.
The Mexican tetra, A. mexicanus exist as river-dwelling surface fish and at least 30 blind populations of cavefish, which have evolved in nutrient-limited environments, providing the opportunity to examine sleep after fasting and postprandial sleep in an evolutionary context (Jeffery 2009;Gross 2012;McGaugh et al. 2020).Multiple cavefish populations have evolved behavioral and physiological differences relative to surface fish including sleep loss, reduced metabolic rate, and increased feeding (Duboué et al. 2011;Moran et al. 2014;Aspiras et al. 2015;Yoshizawa 2015;Volkoff 2016).Long-term starvation has opposing effects on sleep between the surface and cave populations.Starved surface fish suppress sleep, while starved cavefish increase sleep, suggesting that the evolutionary factors shaping the sleep-feeding interaction differ between populations (Jaggard et al. 2018).However, sleep-feeding interactions are poorly understood, and postprandial sleep has to our knowledge not been identified in any fish model to date.Examining the effects of feeding state on sleep in surface and cave populations of A. mexicanus has the potential to identify whether these behaviors evolved through shared genetic mechanisms and to provide insight into how sleep-feeding interactions are influenced by adaptation to a nutrient-poor cave environment.
Larval A. mexicanus provide a particularly tractable model for examining the effects of feeding on sleep regulation.Multiple populations of cavefish larvae have converged on sleep loss similar to adults (Duboué et al. 2011;Yoshizawa et al. 2015).However, while adult fish can live for months without food, larval fish live for only a matter of days (Salin et al. 2010;Medley et al. 2022;Pozo-Morales et al. 2024).Therefore, interactions between feeding and other behaviors may be particularly important for the survival of larvae and young juvenile fish.Feeding larval fish Artemia is readily quantifiable and large numbers of larval fish can be tested without the need to grow fish to adulthood (Espinasa et al. 2014(Espinasa et al. , 2017;;Lloyd et al. 2018).The experimental amenability of larval fish allows for efficient characterization of sleep-feeding interactions across different behavioral and genetic contexts, providing a model to investigate the evolutionary relationship between these processes.
Here, we characterize the effects of starvation and acute feeding on sleep in surface fish and multiple A. mexicanus cavefish populations.We identify multiple sleep-feeding interactions in A. mexicanus, including the presence of post-prandial sleep in multiple, parallelly evolved cavefish populations.Feeding promotes sleep, independent of time-of-day, revealing the presence of postprandial sleep in both surface and cavefish.Together, these findings reveal interactions between feeding and sleep and provide a model system to examine how these interactions evolved.
Results
To investigate the effects of feeding on sleep, we compared sleep in different populations of cavefish immediately following a meal.Briefly, fish were fed a meal, and baseline sleep and activity were measured for 24 hours prior to sleep and feeding measurements.At Zeitgeber Time (ZT) 0 on the second day, fish were fed 70 Artemia over two hours, followed by a four-hour recording of sleep (Fig 1A).In agreement with previous findings, baseline sleep was lower in both Pachón and Tinaja cavefish compared to surface fish (Fig 1B;Duboué et al. 2011a;Jaggard et al. 2020;O'Gorman et al. 2021a).When sleep was measured following a two-hour feeding period, surface fish slept significantly more than cavefish from both populations (Fig 1C).Consistent with previous findings, quantification of Artemia consumed during the two-hour feeding window revealed significantly greater consumption in Tinaja fish, but not Pachón cavefish, compared to surface fish (Aspiras et al. 2015;Alié et al. 2018)(Fig 1D).Taken together, these findings reveal difference in sleep and feeding behavior of larval A. mexicanus populations.
It is possible that sleep is elevated across A. mexicanus populations from ZT2-ZT6 due to postprandial sleep or light-regulated rest-activity rhythms.To differentiate between these possibilities, we compared sleep following meals prior to ZT2, ZT6, and ZT10.Feeding time was limited to half an hour to provide additional resolution for postprandial sleep (Fig 2A -C).Across feeding time courses, surface fish slept more than cavefish populations (Fig 2D-F), supporting the notion that surface fish sleep more than cavefish independent of feeding treatment.To measure for postprandial sleep, we compared sleep duration during the four hours following feeding to the remaining hours of daytime (excluding the time for the feeding assay) to determine the percent change in sleep post feeding.Sleep was increased following the meal across all three timepoints, for surface fish and both cavefish populations (Fig 2G-I).Strikingly, for all timepoints tested, there was a significant increase in the amount of postprandial sleep, measured by the increase over the baseline sleep (Fig 2G -I).Variation in the degree of postprandial sleep increase across populations were dependent of feeding time.There were no differences in the percent increase in postprandial sleep between populations fed prior to ZT2, but Surface fish had a significantly greater increase in postprandial sleep than Tinaja cavefish fed prior to ZT6, and Pachón fish had a significantly greater increase in postprandial sleep than either surface and Tinaja cavefish fed prior to ZT10.Similarly, both surface and Pachón cavefish, but not Tinaja cavefish, experienced a significantly greater increase in postprandial sleep prior to ZT10 than for the timepoints earlier in the day.Therefore, while postprandial sleep occurs across A. mexicanus populations, the degree to which sleep is increased in each population is dependent on the time of day that feeding occurs.Taken together, these findings reveal the presence of postprandial sleep in surface and cave populations of A. mexicanus.
It is possible that meal size, or its caloric value, contributes to the duration of postprandial sleep.
To determine whether the amount of postprandial sleep is related to meal size, we examined the correlation between the number of Artemia consumed and the duration of sleep in the four hours following the meal.For surface fish fed prior to ZT2, there was a significant positive correlation between meal size and post prandial sleep, however there was no significant correlation for surface fish fed prior to ZT6 and ZT10 (Fig 3A-C).For both Pachón (Fig 3D -F) and Tinaja (Fig 3G-H) cavefish, there was no correlation between Artemia consumed and postprandial sleep.
Therefore, postprandial sleep is largely driven by the presence of a meal and does not appear to be directly linked to meal size.
Postprandial sleep may provide a mechanism for conserving energy immediately following successful foraging.Conversely, many animals suppress sleep under food-deprived conditions, presumably to forage for food (Macfadyen et al. 1973;Danguir and Nicolaidis 1979;Keene et al. 2010;Goldstein et al. 2018).Larval A. mexicanus survive for only a few days without food, raising the possibility that sleep will be acutely impacted by feeding state.To directly examine the effects of feeding state on sleep, we compared sleep in 20 days post fertilization (dpf) fish that were fed from ZT0-ZT2 to unfed fish that had been starved for the previous 24 hours (Fig 4A -C).Surface fish and both populations of cavefish slept significantly more during the four hours following feeding than unfed controls (Fig 4D-F).To further examine the effects of feeding on sleep, we analyzed the activity patterns of fed and unfed fish using a Markov model that predicts the sleep and wake propensity, both indicators of sleep drive (Wiggin et al. 2020).Across all three populations, fed fish had a significantly greater sleep propensity P(Doze) and a significantly lower waking propensity P(Wake) than unfed fish, suggesting that sleep drive is increased following feeding (Fig 4G-I).Together, these findings reveal that both surface and cavefish suppress sleep when starved, and that starvation-induced sleep suppression is intact in short-sleeping cavefish.
Adult A. mexicanus live months without food and are thought to be highly adapted to survive periods of starvation (Cobham and Rohner 2024).Previously, we have shown that surface fish suppress sleep during periods of prolonged starvation, while cavefish increase sleep (Jaggard et al. 2018).To determine whether differences in sleep response extend to acute behavior following meals, we examined postprandial sleep in adult surface and cavefish.Fish were starved for five days prior to recording to synchronize meal patterns and then fed a blood-worm meal at ZT6.In agreement with previous findings (Jaggard et al. 2018), control surface fish that were not fed slept significantly more than Pachón and Tinaja cavefish (Fig 5 A, I).Similarly, in fish fed at ZT6, surface fish slept significantly more than Tinaja and Pachòn cavefish (Fig 5B , J).To examine whether postprandial sleep is present in adult A. mexicanus, we compared sleep during the four hours following feeding to unfed counterparts (Fig 5C -E).Within this four-hour duration, there were no significant differences in sleep duration (Fig 5F -H) or sleep propensity (Fig 5K -M) between fed and unfed fish across the three A. mexicanus populations.Therefore, there is no evident postprandial sleep for adults under the conditions tested, supporting the notion that post prandial sleep is less robust at a life stage when fish are more starvation resistant.
Discussion
To date, five populations of A. mexicanus cavefish have been studied under laboratory conditions, all of which have significantly reduced sleep compared to surface fish populations (Yoshizawa et al. 2015).These findings have led to the speculation that reduced sleep is adaptive in the foodpoor cave environment because it provides more time to forage (Keene et al. 2015;Keene and Duboue 2018).However, nearly all studies to date have examined sleep in fed animals, using daily averages.Therefore, little is known about how sleep differs between populations under natural conditions and in response to feeding.Here, we describe interactions between sleep and feeding behavior in surface fish and two different populations of cavefish.All three populations sleep more following feeding than under food-deprived conditions, revealing that feeding is required for baseline sleep.Furthermore, all three populations sleep more in the period following a meal as larvae, but not as adults.These findings suggest that despite robust sleep loss across cavefish populations, sleep-feeding interactions have remained intact.
Numerous neural mechanisms associated with sleep loss in cavefish have been identified including elevated levels of the wake-promoting neuropeptide Hypocretin (HCRT), changes in wake-promoting catecholamine systems (Duboué et al. 2012;Bilandzija et al. 2013;Gallman et al. 2019) providing candidate regulators of postprandial sleep.Similarly, feeding is increased in multiple populations of adult A. mexicanus (Aspiras et al. 2015).In agreement with previous findings, we find that feeding is elevated in 20 days post fertilization juvenile cavefish from the Tinaja, but not Pachón population (O'Gorman et al. 2021).In adults, differences in feeding are at least partially attributable to polymorphisms in the GPCR Melanocortin 4 receptor (Mc4r) which is associated with obesity in humans and animal models (Aspiras et al. 2015).While there is little evidence that MC4R directly regulates sleep, it is thought to contribute to obesity-induced sleep apnea that in turn regulates sleep (Larkin et al. 2010;Pillai et al. 2014).Our findings that postprandial sleep is intact in Tinaja cavefish suggests that Mc4r, and other genes involved in feeding, are likely dispensable for sleep feeding interactions.There are also numerous genes that have been identified to regulate sleep or feeding in fish models that are potential regulators of sleepmetabolism interactions.For example, the orexigenic neuropeptides Neuropetide Y (Npy) and Hcrt both induce wakefulness, providing a potential molecular mechanism for feeding-dependent modulation of sleep (Appelbaum et al. 2009;Penney and Volkoff 2014;Singh et al. 2015Singh et al. , 2017;;Jaggard et al. 2018).Future functional analysis is required to define whether these candidate genes regulate interactions between sleep and feeding.
In A. mexicanus, rhythmic transcription is significantly diminished under dark-dark conditions, and cavefish have elevated levels of light-inducible genes (Beale et al. 2013).The circadian clock plays a critical role in the timing of both sleep and feeding, raising the possibility that the circadian clock may be critical for sleep-feeding interactions.Transcriptome-wide analysis in larvae, reveals a loss of rhythmic gene expression across all cave populations tested (Mack et al. 2021) Therefore, because identified postprandial sleep in all of the populations tested across three different timepoints during the day, postprandial sleep may be independent of time-of-day and may not require a functioning circadian clock.
A. mexicanus larvae, like zebrafish, can subsist on a variety of foods including paramecium, rotifers, and fish feed that differ in micronutrients.In this study, A. mexicanus larvae were fed a standard diet of Artemia.Artemia is comprised of macronutrients that include diverse fatty acids, proteins, and carbohydrates.Analysis suggests that Artemia is ~40-60% protein, raising the possibility that consumption of dietary protein may impact sleep (de Clercq et al. 2005).In Drosophila, dietary protein promotes post-prandial sleep, while a loss of dietary protein disrupts sleep depth (Murphy et al. 2016;Brown et al. 2020;Titos et al., 2023).Therefore, it is possible that changes in protein detection, or its downstream targets, regulate the physiology of sleep circuits that are responsible for the different effects of feeding on sleep between Pachón and Tinaja cavefish.Understanding the effects of different diets on sleep, and how individual macronutrients regulate sleep across populations could reveal evolved differences in sleepfeeding interactions across different A. mexicanus populations.
The identification of postprandial sleep in cavefish provides an avenue for future studies examining the genetic basis of this behavior.Mapping genetic loci associated with trait variation has been used to identify candidate regulators of many morphological and behavioral traits, including regulators of sleep, activity, feeding posture, and metabolism (Kowalko et al. 2013;Yoshizawa et al. 2015;Carlson et al. 2018;Riddle et al. 2021).Further, population genetic approaches have identified genome-wide markers of selection across multiple cave populations, and this genetic variation may provide insight into genes impacting sleep-feeding interactions (Herman et al. 2018;Warren et al. 2021;Moran et al. 2022).Genes with signatures of selection that have previously been implicated in sleep or feeding could provide candidate regulators of postprandial sleep.In A. mexicanus, like zebrafish, CRISPR-based gene editing has been used to functionally validate genes identified through genomics approaches and could be applied to the investigation of postprandial sleep (Klaassen et al. 2018;Kroll et al. 2021).Genetic studies will require the use of CRISPR for forward genetic screens, or the identification of A. mexicanus with diminished or highly variable post-prandial sleep that can be used for genetic mapping studies.
In conclusion, these studies identify postprandial sleep in A. mexicanus and suggest it is under independent genetic regulation from total sleep duration and meal size in surface fish and two parallely evolved populations of cavefish.These studies lay the groundwork for future analysis that apply currently available population genetics, neural anatomical, and genetic screening toolsets in A. mexicanus to examine the integration of feeding and sleep regulation
Materials and Methods
Methods
Husbandry
Throughout this study, we followed previously described standard animal husbandry and breeding for A. mexicanus (Borowsky 2008a).All fish were housed under standard temperature (23°C for adults, 25°C for embryos and larvae) and lighting conditions (14:10 hr light:dark cycle).Adult fish were bred by increasing water temperature to 27±1°C and feeding a high-calorie diet that includes thawed frozen bloodworms three times per day (Elipot et al. 2014) .Larvae were fed brine shrimp (Artemia nauplii) ad libitum from 6 -20 days post-fertilization (dpf; Borowsky 2008b).Embryos and larvae were held in small glass bowls until behavioral testing.All procedures in this study were approved under the Florida Atlantic University and Texas A&M University IACUC.
Sleep behavior
These experiments focused on three distinct A. mexicanus morphotypes: the sighted, surfacedwelling Río Choy, and two blind, cave-dwelling populations, Pachón and Tinaja.We quantified sleep behavior in these fish using previously described methods (Jaggard et al. 2019a) and baseline sleep data (O'Gorman et al. 2021).Briefly, we used Ethovision XT 17.0 software (Noldus Information Technology, Wageningen, the Netherlands) to track locomotor behavior.Raw locomotor behavior was used to calculate sleep behavior parameters using a custom Perl script (Jaggard et al. 2019b).We operationally define sleep as 60 seconds or more of immobility given that previous studies show both surface and Pachón cavefish exhibit increased arousal thresholds after this period (Jaggard et al. 2019b).We defined immobility as a velocity below 6 mm/sec for larval fish and a velocity below 4 cm/sec for adult fish.All recordings were performed at 23 °C under a 14:10 hour light/dark cycle.
Larval behavior recordings
All larval used to quantify sleep behavior were 20 dpf.Fish were fed and then acclimated individually in 24-well plates for at least 15 hours prior to behavior recordings.Recordings began at ZT0 and lasted for 24 hours, with interruptions for feeding at specific time points.The 24-well plates were placed on light boxes made from white acrylic housing infrared (IR) lights (Figure 1A).
Basler ace acA1300-200um Monochrome USB 3.0 Cameras with mounted IR filters were mounted above the well plates and recordings were taken using Pylon Viewer software.
The effects of feeding on sleep were tested throughout the light cycle at time points prior to ZT0, ZT2, ZT6, and ZT10.Each 24-well plate was either not fed as a control or fed at a single time point.We conducted two separate feeding experiments.In the first experiment, larvae were fed for 10 mins immediately before a 24-hour recording beginning at ZT0.This 24-hour recording was followed by a 2-hour feeding behavior assay (described below) and then another behavior recording for 4 hours from ZT2-ZT6 (Fig 1).In the second experiment, we recorded behavior for 24 hours around a 45-minute window for feeding prior to either ZT2, ZT6, or ZT10.
Larval feeding behavior assay
To quantify the relationship between the amount of food consumption and post-prandial sleep duration, we performed feeding assays that allowed us to count the number of Artemia over a given time.The duration of the feeding assay was 2 hours for the first experiment, starting at ZT0 following 24 hours of recording.The duration of the feeding assay was 30 minutes for the second experiment, starting prior to ZT2, ZT6, or ZT10.For the 2-hour feeding assay, fish were given exactly 70 Artemia, for the 30 minute feeding assay, Artemia were provided ad libitum.We filled a new 24-well plate with Artemia hatched within 24 hours and recorded for at least one minute prior to transferring the larval fish from the recording well plate to this new feeding well plate.At the end of the recording duration, fish were removed from the feeding assay, placed back into the original 24-well recording plate with clean water and returned to the behavior recording.We used FIJI (Schindelin et al. 2012) to count the number of Artemia both before the fish were added to the wells and at the end of the feeding assay.Subtraction of the former from the latter allowed us to determine the amount of Artemia eaten over the duration of the feeding assay.
Adult behavior recordings
Adult fish used for behavior recordings were approximately 1 year old with an equal number of males and females per treatment.Food was withheld for 5 days prior to recording.Fish were placed in individual glass tanks of approximately 30 x 17 cm in a 2 x 2 grid in front of an IR light board and left to acclimate for at least 24 hours.Recordings began at ZT0 and lasted 24 hours.
In the top two tanks, 4 oz of thawed, frozen blood worms were added at ZT5.5 and any uneaten worms were removed after 30 minutes at ZT6.The fish in the bottom two tanks were not fed as a control.
Analysis
Warren, W. C., T. E. Boggs, R. Borowsky, B. M. Carlson, E. Ferrufino et al., A) 20 dpf fish were briefly fed prior to 24 h behavioral sleep recordings.At ZT0 the following day, fish were assayed for feeding behavior until ZT2, immediately after which we recorded sleep behaviors between ZT2 and 6.B) Sleep profiles of wild type surface, Pachón, and Tinaja fish taken over the experiment.Lines and error bars represent the mean ± SD.C) Cross-population comparison of total sleep duration immediately following the feeding experiment.Cavefish slept significantly less than surface fish (ANOVA: F2, 34 = 8.123, p = 0.0013; Tukey's HSD for surface-Pachón, p = 0.0202, p = 0.0024; Tukey's HSD for surface-Tinaja, p = 0.0024).D) Cross-population comparison of the number of Artemia eaten during the two-h feeding experiment.Tinaja ate significantly more than surface fish (ANOVA: F2, 76 = 3.91, p = 0.0242; Tukey's HSD for surface-Tinaja, p = 0.0178).
no significant difference across populations in the percentage of increase in postprandial sleep (Anova: F2, 104 = 3.36, p = 0.0417).H) Percent change of postprandial sleep after ZT6 feeding window.Surface: t = 13.65,df = 47, p < 0.0001; Pachón: t = 2.67, df = 23, p = 0.0137; Tinaja: t = 2.480, df = 26, p = 0.0200.There was no significant different in the percentage of increase in postprandial sleep between surface and Pachón cavefish, but surface fish had a significantly greater increase in sleep than Tinaja cavefish (ANOVA: F2, 96 = 5.758, p = 0.0072; Tukey's HSD for surface-Tinaja, p = 0.0101).I) Percent change of postprandial sleep after ZT10 feeding window.Surface: t = 8.619, df = 52, p < 0.0001; Pachón: t = 10.27,df = 43, p < 0.0001; Tinaja: t = 3.636, df = 16, p = 0.0022.Pachón cavefish had a significantly greater percent increase in postprandial sleep than both surface and Tinaja cavefish (ANOVA: F2, 111 = 4.727, p = 0.0107; Tukey's HSD for surface-Pachón, p = 0.0298; Tukey's HSD for Pachón-Tinaja, p = 0.0275).For surface fish and Pachón cavefish, the percentage of increase in postprandial sleep was significantly greater after a ZT10 feeding window than at any other timepoint (Surface Anova: F2, 144 = 13.84,p < 0.0001; Pachón Anova: F2, 197 = 19.56,p < 0.0001).There were no other significant differences in the percent increase for postprandial sleep between timepoints or for Tinaja cavefish (Tinaja Anova: F2, 70 = 3.978, p = 0.0231).
Figure 3 :
3
Figure 3: Postprandial sleep in larval Astyanax is not dependent on the amount of food consumed, regardless of the time of day that feeding occurs.Correlation of amount of Artemia nauplii consumed with sleep duration in the four hours following feeding with a simple linear regression for surface (A-C), Pachón (D-F), and Tinaja (G-I).A, D, G) Larvae were fed prior to ZT2.B, E, H) Larvae were fed prior to ZT6.C, F, I) Larvae were fed prior to ZT10.
Figure 5 :
5
Figure 5: Adult Astyanax do not display post prandial sleep behavior.A, B) Sleep profiles of adult Surface, Pachón, and Tinaja, in minutes per hour.Lines and error bars represent the mean ± SD.A, I) Fish were not fed over the course of the day.B, J) Fish were provided food from ZT5.5 (indicated by the arrow and dotted black line in B) to ZT6.I, J) Cross-population comparison of total sleep duration in hours over the 24-hour day.Letters represent significant differences.I) Total sleep duration in 24 hours was significantly different between unfed surface and cave populations of A. mexicanus ((ANOVA: F2, 28 = 15.5, p < 0.0001; Tukey's HSD for Surface-Pachón, p < 0.0001 and Surface-Tinaja, p = 0.0015).J) Total sleep duration in was significantly different between fed surface and cave populations of A. mexicanus ((ANOVA: F2, 25 = 15.04,p < 0.0001; Tukey's HSD for Surface-Pachón, p < 0.0001 and Surface-Tinaja, p = 0.0008).C-E) Four-hour sleep profiles comparing the sleep of fed (orange) and unfed (black) individuals in each population.Lines and error bars represent the mean ± SEM.F-H) There are no significant differences in sleep during the four hours following feeding, regardless of the population.F) Surface: Mann-Whitney U = 88, nfed = 12, nunfed = 15, p = 0.9317.G) Pachon: Mann-Whitney U = 31.5,nfed = 8, nunfed = 8, p > 0.9999.H) Tinaja: Mann-Whitney U = 22.5, nfed = 8, nunfed = 8, p > 0.2.K-M) There are no significant differences in activity state transitions between fed and unfed fish.K) Surface: P(Wake) t = 0.271, df = 22, p = 0.7888; P(Doze) t = 2.041, df = 22, p = 0.054.L) Pachon: Mann-Whitney U = 24, nfed = 8, nunfed = 8; P(Wake) p = 0.4667; P(Doze) p = 0.4667.M) Tinaja: Mann-Whitney U = 23, nfed = 8, nunfed = 8; P(Wake) p = 0.5714; P(Doze) p = 0.1319).Horizontal lines represent quartiles.
Sleep, feeding, and post-prandial sleep behaviors across three populations of wild- type Astyanax mexicanus.
2021 A chromosomelevel genome of Astyanax mexicanus surface fish for comparing population-specific genetic differences contributing to trait evolution.Nat Commun 12: 1447.Wee, C. L., E. Y. Song, R. E. Johnson, D. Ailani, O. Randlett et al., 2019 A bidirectional network for appetite control in larval zebrafish.Elife 8:.Wiggin, T. D., P. R. Goodwin, N. C. Donelson, C. Liu, K. Trinh et al., 2020 Covert sleep-related biological processes are revealed by probabilistic analysis in Drosophila.Proc Natl Acad Sci U S A 117: 10024-10034.Yoshizawa, M., 2015 Behaviors of cavefish offer insight into developmental evolution.Mol Reprod Dev 82: 268-280.Yoshizawa, M., B. G. Robinson, E. R. Duboué, P. Masek, J. B. J. Jaggard et al., 2015 Distinct genetic architecture underlies the emergence of sleep loss and prey-seeking behavior in the Mexican cavefish.BMC Biol 20: 15.Yurgel, M., P. Masek, J. R. DiAngelo, and A. Keene, 2014 Genetic dissection of sleep-metabolism interactions in the fruit fly.J Comp Physiol A Neuroethol Sens Neural Behav Physiol.epub ahead:
Figures Figure 1.
Acknowledgements: This work was supported by NIH Grants NIH 1R01GM127872 to SEM, NR, and ACK; R24 OD030214 to WW, NR and ACK; R21 NS122166 to ACK and JEK; 1DP2AG071466-01 to NR; NSF Grant NSF grant IOS 2202359 to JEK and SEM.The authors are grateful for technical assistance from Kaya Harper and Lawaal Agboola.Statistical analyses were performed in GraphPad Prism (version # 9.5.0) and R (version 4.0.4).When assumptions of normality and equal variances were met, we used parametric t-tests, ANOVA, and Pearson's r tests, otherwise we used non-parametric Mann-Whitney U, Kruskal-Wallis, and Spearman's ρ tests.Following a significant ANOVA or Kruskal-Wallis test, pairwise comparisons were made using Tukey's HSD or Dunn's test, respectively.To quantify the percent change in sleep duration during the 4 hours following feeding, we determined the proportion of total daylight sleep to total daylight recording time as well as the proportion of sleep to the 4 hour post prandial recording period.We then calculated percent change as the proportion of post prandial sleep minus the proportion of total daylight sleep divided by the proportion of total daylight sleep.Finally, to test whether the amount of Artemia consumed was related to post-prandial sleep duration, we analyzed the goodness of fit from a linear regression.
Developmental evolution of the forebrain in cavefish: from natural variations in neuropeptides to behavior. A Alié, L Devos, J Torres-Paz, L Prunier, F Boulet, 2018Elife
Exploring phylogeny to find the function of sleep. R C Anafi, M S Kayser, D M Raizen, Nat Rev Neurosci. 202019
Sleep-wake regulation and hypocretin-melatonin interaction in zebrafish. L Appelbaum, G X Wang, G S Maro, R Mori, A Tovin, Proc Natl Acad Sci U S A. 1062009
D M D M Arble, J Bass, C D C D Behn, M P M P Butler, E Challet, Impact of Sleep and Circadian Disruption on Energy Balance and Diabetes : A Summary of Workshop Discussions. 201538
Melanocortin 4 receptor mutations contribute to the adaptation of cavefish to nutrient-poor conditions. A Aspiras, N Rohner, B Marineau, R Borowsky, J Tabin, Proceedings of the National Academy of Sciences. 1122015
A E Aulsebrook, T M Jones, N C Rattenborg, T C Roth, J A Lesku, Sleep Ecophysiology: Integrating Neuroscience and Ecology. 201631
Circadian rhythms in Mexican blind cavefish Astyanax mexicanus in the lab and in the field. A Beale, C Guibal, T K Tamai, L Klotz, S Cowen, Nat Commun. 427692013
A potential benefit of albinism in Astyanax cavefish: downregulation of the oca2 gene increases tyrosine and catecholamine levels as an alternative to melanin synthesis. H Bilandzija, L Ma, A Parkhurst, W Jeffery, PLoS One. 8e808232013
Breeding Astyanax mexicanus through natural spawning. R Borowsky, Cold Spring Harb Protoc. 32008a
Drosophila insulin-like peptide 2 mediates dietary regulation of sleep intensity. R Borowsky, E B Brown, K D Shah, R Faville, B Kottler, A C Keene, Cold Spring Harb Protoc. 32008b. 2020PLoS Genet
Starvation resistance is associated with developmentally specified changes in sleep, feeding and metabolic rate. E B Brown, M E Slocumb, M Szuperak, A Kerbs, A G Gibbs, Journal of Experimental Biology. 2222019
Phylogenetic analysis of the ecology and evolution of mammalian sleep. I Capellini, R A Barton, P Mcnamara, B T Preston, C L Nunn, Evolution (N Y). 622008
Genetic analysis reveals candidate genes for activity QTL in the blind Mexican tetra, Astyanax mexicanus. B M Carlson, I B Klingler, B J Meyer, J B Gross, PeerJ. 6e51892018
Regulation of zebrafish sleep and arousal states: current and prospective approaches. C N Chiu, D Prober, Front Neural Circuits. 7582013
A zebrafish genetic screen identifies neuromedin U as a regulator of sleep/wake states. C N Chiu, J Rihel, D A Lee, C Singh, E A Mosser, Neuron. 892016
Nutritional value of brine shrimp cysts as a factitious food for Orius laevigatus (Heteroptera: Anthocoridae). P De Clercq, Y Arijs, T Van Meir, G Van Stappen, P Sorgeloos, Biocontrol Sci Technol. 152005
Unraveling stress resilience: Insights from adaptations to extreme environments by Astyanax mexicanus cavefish. A E Cobham, N Rohner, J Exp Zool B Mol Dev Evol. 2024
Dependence of sleep on nutrient's availability. J Danguir, S Nicolaidis, Physiol Behav. 221979
2012 β-adrenergic signaling regulates evolutionarily derived sleep loss in the mexican cavefish. E R E R Duboué, R L R L Borowsky, A C A C Keene, Brain Behav Evol. 80
Evolutionary convergence on sleep loss in cavefish populations. E R Duboué, A C Keene, R L Borowsky, Current Biology. 212011
Astyanax transgenesis and husbandry: how cavefish enters the laboratory. Y Elipot, L Legendre, S Père, F Sohm, S Rétaux, Zebrafish. 112014
Enhanced prey capture skills in Astyanax cavefish larvae are independent from eye loss. L Espinasa, J Bibliowicz, W R Jeffery, S Rétaux, Evodevo. 52014
Contrasting feeding habits of post-larval and adult Astyanax cavefish. L Espinasa, N Bonaroti, J Wong, K Pottin, E Queinnec, Subterr Biol. 212017
Retinotectal circuitry of larval zebrafish is adapted to detection and pursuit of prey. D Förster, T O Helmbrecht, D S Mearns, L Jordan, N Mokayes, 20209
Evolutionary increases in catecholamine signaling may underlie the emergence of adaptive traits and behaviors in the blind cavefish, Astyanax mexicanus. K Gallman, D Rivera, D Soares, 10.1101/72:2019
Hypothalamic Neurons that Regulate Feeding Can Influence Sleep/Wake States Based on Homeostatic Need. N Goldstein, B J Levine, K A Loy, W L Duke, O S Meyerson, Curr Biol. 282018e3
The complex origin of Astyanax cavefish. J B Gross, 10.1186/1471-2148-12-105BMC Evol. Biol. 121052012
The role of gene flow in rapid and repeated evolution of cave-related traits in Mexican tetra. A Herman, Y Brandvain, J Weagley, W R Jeffery, A C Keene, Astyanax mexicanus. Mol Ecol. 222018
Automated Measurements of Sleep and Locomotor Activity in Mexican Cavefish. J B Jaggard, E Lloyd, A Lopatto, E R Duboue, A C Keene, J Vis Exp. 2019a
Automated Measurements of Sleep and Locomotor Activity in Mexican Cavefish. J B Jaggard, E Lloyd, A Lopatto, E R Duboue, A C Keene, J Vis Exp. 2019b
Cavefish brain atlases reveal functional and anatomical convergence across independently evolved populations. J B Jaggard, E Lloyd, A Yuiska, A Patch, Y Fily, Sci Adv. 62020. 2018Hypocretin underlies the evolution of sleep loss in the Mexican cavefish
Regressive evolution in Astyanax cavefish. W R Jeffery, Annu Rev Genet. 432009
Unraveling the Evolutionary Determinants of Sleep. W J Joiner, Current Biology. 262016
A C Keene, L Appelbaum, Sleep in Fish Models. 2019
Clock and cycle limit starvation-induced sleep loss in drosophila. A C Keene, E R Duboue ; Keene, A C A C , E R E R Duboué, D M D M Mcdonald, M Dus, G S B G S B Suh, Current Biology. 202018. 2010J Exp Biol
. A C Keene, M Yoshizawa, S E Mcgaugh, Biology and Evolution of the Mexican Cavefish. 2015
CRISPR mutagenesis confirms the role of oca2 in melanin pigmentation in Astyanax mexicanus. H Klaassen, Y Wang, K Adamski, N Rohner, J E Kowalko, Dev Biol. 4412018
Convergence in feeding posture occurs through different genetic loci in independently evolved cave populations of Astyanax mexicanus. J E Kowalko, N Rohner, T A Linden, S B Rompani, W C Warren, Proceedings of the National Academy of Sciences. 1102013
A simple and effective F0 knockout method for rapid screening of behaviour and other complex phenotypes. F Kroll, G T Powell, M Ghosh, G Gestri, P Antinucci, Am J Respir Crit Care Med. 102021. 2010A Candidate Gene Study of Obstructive Sleep Apnea in European Americans and African Americans
A phylogenetic analysis of sleep architecture in mammals: the integration of anatomy, physiology, and ecology. J A Lesku, T C Roth, C J Amlaner, S L Lima, Am Nat. 1682006
Neural signatures of sleep in zebrafish. L C Leung, G X Wang, R Madelaine, G Skariah, K Kawakami, Nature. 5712019
Modeling sleep and neuropsychiatric disorders in zebrafish. T Levitas-Djerbi, L Appelbaum, Curr Opin Neurobiol. 442017
Evolutionary shift towards lateral line dependent prey capture behavior in the blind Mexican cavefish. E Lloyd, C Olive, B A Stahl, J B Jaggard, P , Dev Biol. 2018
Starvation and human slow-wave sleep. U Macfadyen, I Oswald, S Lewis, J Appl Physiol. 351973
Repeated evolution of circadian clock dysregulation in cavefish populations. K L Mack, J B Jaggard, J L Persons, E Y Roback, C N Passow, PLoS Genet. 172021
Regulation of Satiety Quiescence by Neuropeptide Signaling in Caenorhabditis elegans. M Makino, E Ulzii, R Shirasaki, J Kim, Y.-J You, Front Neurosci. 156785902021
Dark world rises: The emergence of cavefish as a model for the study of evolution, development, behavior, and disease. S E Mcgaugh, J E Kowalko, E Duboué, P Lewis, T A Franz-Odendaal, J Exp Zool B Mol Dev Evol. 3342020
Evolution of sleep: phylogenetic and functional perspectives. P Mcnamara, R Barton, C Nunn, 2009
The metabolome of Mexican cavefish shows a convergent signature highlighting sugar, antioxidant, and Ageing-Related metabolites. J K Medley, J Persons, T Biswas, L Olsen, R Peuß, 202211
Selection-driven trait loss in independently evolved cavefish populations. R Moran, E Richards, C Ornelas-Garcia, J Gross, A Gross, BioRxiv. 5181852022
Eyeless Mexican Cavefish Save Energy by Eliminating the Circadian Rhythm in Metabolism. D Moran, R Softley, E J Warrant, PLoS One. 9e1078772014
. K R K R Murphy, S A S A Deshpande, M E M E Yurgel, J P J P Quinn, J L J L Weissbach, Postprandial sleep mechanics in Drosophila. Elife. 52016
Pleiotropic function of the oca2 gene underlies the evolution of sleep loss and albinism in cavefish. M O'gorman, S Thakur, G Imrie, R L Moran, S Choy, Curr Biol. 312021e4
Peripheral injections of cholecystokinin, apelin, ghrelin and orexin in cavefish (Astyanax fasciatus mexicanus): effects on feeding and on the brain expression levels of tyrosine hydroxylase, mechanistic target of rapamycin and appetiterelated hormones. C Penney, H Volkoff, Gen Comp Endocrinol. 1962014
Severe obstructive sleep apnea in a child with melanocortin-4 receptor deficiency. S Pillai, K Nandalike, Y Kogelman, R Muzumdar, S J Balk, J Clin Sleep Med. 102014
Starvationresistant cavefish reveal conserved mechanisms of starvation-induced hepatic lipotoxicity. M Pozo-Morales, A E Cobham, C Centola, M C Mckinney, P Liu, Life Sci Alliance. 72024
Increased food intake after starvation enhances sleep in Drosophila melanogaster. J M Regalado, M B Cortez, J Grubbs, J A Link, A Van Der Linden, J Genet Genomics. 442017
Genetic mapping of metabolic traits in the blind Mexican cavefish reveals sex-dependent quantitative trait loci associated with cave adaptation. M R Riddle, A Aspiras, F Damen, S Mcgaugh, J A Tabin, BMC Ecol Evol. 21942021
Zebrafish behavioral profiling links drugs to biological targets and rest/wake regulation. J Rihel, D A Prober, A Arvanites, K Lam, S Zimmerman, Science. 3272010
Cave colonization without fasting capacities: An example with the fish Astyanax fasciatus mexicanus. K Salin, Y Voituron, J Mourin, F Hervant, Comp Biochem Physiol A Mol Integr Physiol. 1562010
Fiji: an open-source platform for biological-image analysis. J Schindelin, I Arganda-Carreras, E Frise, V Kaynig, M Longair, Nat Methods. 92012
The evolution of metabolic regulation in animals. F Seebacher, Comp Biochem Physiol B Biochem Mol Biol. 2242018
A dedicated visual pathway for prey detection in larval zebrafish. J L Semmelhack, J C Donovan, T R Thiele, E Kuehn, E Laurell, 20143
Norepinephrine is required to promote wakefulness and for hypocretin-induced arousal in zebrafish. C Singh, G Oikonomou, D A Prober, 20154e070000
. C Singh, J Rihel, D A Prober, 2017
Y Neuropeptide, Regulates, Sleep by Modulating Noradrenergic Signaling. 27e5
Enhanced Sleep Is an Evolutionarily Adaptive Response to Starvation Stress in Drosophila. M E Slocumb, J M Regalado, M Yoshizawa, G G Neely, P Masek, PLoS One. 10e01312752015
Postprandial sleepiness: objective documentation via polysomnography. M L Stahl, W C Orr, C Bollinger, Sleep. 61983
Sleep-Dependent Modulation of Metabolic Rate in Drosophila. B A Stahl, M E Slocumb, H Chaitin, J R Diangelo, A C Keene, Sleep. 402017
Feeding Behavior, Starvation Response, and Endocrine Regulation of Feeding in Mexican Blind Cavefish (Astyanax fasciatus mexicanus). H Volkoff, Biology and Evolution of the Mexican Cavefish. 2016Elsevier
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THE LOWER BOUND OF THE PCM QUANTIZATION ERROR IN HIGH DIMENSION
18 Mar 2014
Heng Zhou
Zhiqiang Xu
THE LOWER BOUND OF THE PCM QUANTIZATION ERROR IN HIGH DIMENSION
18 Mar 2014D1E03A7F3DAAD18AC06BB2BDE1C7ADF3arXiv:1403.4311v1[math.NA]AMS Subject Classification 2000 42C1533C1042A05
In this note, we investigate the performance of the PCM scheme with linear quantization rule for quantizing unit-norm tight frame expansions for R d without the White Noise Hypothesis.In [4], Wang and Xu showed that for asymptotically equidistributed unit-norm tight frame the PCM quantization error has an upper bound O(δ (d+1)/2 ) and they conjecture the upper bound is sharp.In this note, we confirm the conjecture with employing the asymptotic estimate of the Bessel functions.
Introduction
In signal processing, one of the primary goals is to find a digital representation for a given signal that is suitable for storage, transmission, and recovery.We assume that the signal x is an element of a finite-dimensional Hilbert space H = R d .One often begins to expand x over a dictionary F = {e j } N j=1 , i.e.,
x = N j=1 c j e j ,
where c j are real numbers.We say F is a tight frame of
R d if x = d N N j=1
x, e j e j holds for all x ∈ R d .The tight frame is called unit-norm if e j 2 = 1 holds for all 1 ≤ j ≤ N .
In the digital domain the coefficients x j = x, e j must be mapped to a discrete set of values A which is called the quantization alphabet.The simplest way for such a mapping is the Pulse Code Modulation (PCM) quantization scheme, which has A = δZ with δ > 0 and the mapping is done by the function
Q δ (t) := argmin r∈A |t − r| = δ t δ + 1 2 .
Z. Xu is supported by the National Natural Science Foundation of China (11171336 and 11331012) and by the Funds for Creative Research Groups of China (Grant No. 11021101).
Thus in practical applications we in fact have only a quantized representation q j := Q δ ( x, e j ), j = 1, . . ., N
for each x ∈ R d .The linear reconstruction is
xF = d N N j=1
q j e j .
Naturally we are interested in the error for this reconstruction, i.e.
E δ (x, F ) := x − xF ,
where • is ℓ 2 norm.To simplify the investigation of E δ (x, F ), one employs the White Noise Hypothesis (WNH) in this area (see [6,8,9,10,11,5]), which asserts that the quantization error sequence {x j − q j } N j=1 can be modeled as an independent sequence of i.i.d.random variables that are uniformly distributed on the interval (−δ/2, δ/2).Under the WNH, one can obtain the mean square error
M SE = E( x − xF 2 ) = d 2 δ 2 12N .
The result implies that the MES of E δ (x, F ) tends to 0 with N tending to infinity.However, as pointed out in [11,5], the WNH only asymptotically holds for fine quantization (i.e. as δ tends to 0) under rather general conditions.So, for a fixed x, one is interested in whether E δ (x, F ) really tending to 0 without WNH.The result in [4] gives a solution for the case where d = 2 which shows that for some x ∈ R 2 the quantization error E δ (x, F ) does not diminish to 0 with N tending to infinity.Naturally, one would like to know whether it is possible to extend the result to higher dimension.In [4], Wang and Xu investigate the case where F is the asymptotically equidistributed unit-norm tight frame in R d .A sequence of finite sets A m ⊂ S d−1 with cardinality N m = #A m is said to be asymptotically equidistributed on S d−1 if for any piecewise continuous function f on S d−1 we have
lim m→∞ 1 N m v∈Am f (v) = z∈S d−1 f (z)dν,
where f are piecewise continuous functions on S d−1 and dν denotes the normalized Lebesgue measure on S d−1 .Then the following theorem presents an upper bound for lim m→∞ E δ (x, F m ).
Theorem 1.1.[4] Assume that F m are asymptotically equidistributed unit-norm tight frames in R d .Then for any x ∈ R d we have
lim m→∞ E δ (x, F m ) ≤ C d δ (d+1)/2 r (d−1)/2 ,
where r = x and C d is a constant depending on d.
A main tool for obtaining Theorem 1.1 is Euler-Maclaurin formula.However, it seems that it is difficult to extend the method to obtain the lower bound.In [4], Wang and Xu conjecture the bound O δ (d+1)/2 r (d−1)/2 is sharp.In this note, we employ the tools of Bessel function and hence confirm the conjecture.In particular, we have: Theorem 1.2.Suppose that d > 2 is an integer.Assume that x ∈ R d and that F m are asymptotically equidistributed unit-norm tight frames in R d .Set r := x , R := r/δ, ǫ := R − ⌊R⌋, and
I := d 2π 0 |(sin(θ 2 )) d−3 • • • sin θ d−2 |dθ 2 • • • dθ d−2 > 0. (i) If d = 2n and 1/4 ≤ ǫ ≤ 1/2, then lim m→∞ E δ (x, F m ) ≥ C 1,d δ d+1 2 r d−1 2
, provided that R = x /δ is big enough, where
C 1,d = (n − 1)! • 2n−2 n−1 • M 1 2 2n−2 π n • I, M 1 = 4 5 | cos(2πǫ − 3 4 π)| − 5 4 +∞ k=2 1 k 2n+1 2 > 0. (ii) If d = 2n + 1 and 1/6 ≤ ǫ ≤ 1/3, then lim m→∞ E δ (x, F m ) ≥ C 2,d δ d+1 2 r d−1 2
, provided that R = x /δ is big enough, where
C 2,d = (n − 1)! • M 2 π n+1 • I, M 2 = 7 8 | cos(2πǫ − 1 2 π)| − 8 7 +∞ k=2 1 k n+1 > 0.
After introducing some necessary concepts and results to be used in our investigation in Section 2, we present the proof of Theorem 1.2 in Section 3.
Preliminaries
Bessel function.(see [12]) For α > 0, the Bessel function J α is defined by the series representation (1)
J α (x) = +∞ k=0 (−1) k k! • Γ(k + α + 1) x 2 2k+α .
Particularly, when α ∈ N, we have
J α (x) = 1 π π 0 cos(ατ − x sin(τ ))dτ.
We also need an asymptotic estimate for the Bessel function J α which is presented in [2].
Theorem 2.1.( [2]) For the Bessel function J α , we have
J α (x) = 2 πx cos(x − ω α ) + θcµx − 3 2 ,
where
ω α = πα 2 + 1 4 π, µ = |α 2 − 1 4 |, |θ| ≤ 1, and c = (2/π) 3/2 , x ≥ 0, |α| ≤ 1 2 √ 2/2, x ≥ √ µ, α > 1 2 5/4, 0 < x < √ µ, α > 1 2 .
Combinatorics identity.(see (7.7) of Table 4 in [1])
(2)
h m=0 (−1) m n + h h − m n + h h + m = 1 2 n + h h + 1 2 n + h h 2 3. Proof of Theorem 1.2
To this end, we first introduce several lemmas:
Lemma 3.1. For all n ∈ N + and h ∈ N we have h m=0 (−1) m (2m + 1) n + h h − m n + h h + m + 1 = n n + h n ,(3)and h m=l (−1) m (2m + 1) 2h + 1 h − m m + l 2l = 0, l = 0, 1, . . . , h − 1. (4)
Proof.We prove (3) by induction.To state conveniently, set
A h n := h m=0 (−1) m (2m + 1) n + h h − m n + h h + m + 1 .
A simple observation is that (3) holds when n ∈ N + , h = 0 and when n = 0, h ∈ N + .Assume that n 0 , h 0 ∈ N + .For the induction step, we assume that ( 3) is true both for n ≤ n 0 ∈ N + , h ∈ N + and for n = n 0 + 1, h ≤ h 0 − 1 ∈ N + .To this end, we just need prove that the result holds for n = n 0 + 1, h = h 0 .We have
A h 0 n 0 +1 = h 0 m=0 (−1) m (2m + 1) n0 + h0 + 1 h0 − m n0 + h0 + 1 h0 + m + 1 = h 0 m=0 (−1) m (2m + 1) n0 + h0 h0 − m + n0 + h0 h0 − m − 1 n0 + h0 h0 + m + 1 + n0 + h0 h0 + m = A h 0 n 0 + A h 0 −1 n 0 +1 + h 0 m=0 (−1) m (2m + 1) n0 + h0 h0 − m n0 + h0 h0 + m + n0 + h0 h0 − m − 1 n0 + h0 h0 + m + 1 = A h 0 n 0 + A h 0 −1 n 0 +1 + 2 h 0 m=0 (−1) m n0 + h0 h0 − m n0 + h0 h0 + m − n0 + h0 h0 2 = A h 0 n 0 + A h 0 −1 n 0 +1 + n0 + h0 n0 = (n0 + 1) n0 + h0 + 1 n0 + 1 ,
where the last equality uses the identity (2) and the induction assumption.We now turn to (4).Set
(5)
g m := (−1) m+1 (h + m + 1)(m − l) 2h+1 h−m m+l 2l h − l .
A simple calculation shows that (6)
g m+1 − g m =(−1) m 2h+1 h−m m+l 2l h − l ((h − m)(l + m + 1) + (h + m + 1)(m − l)) =(−1) m (2m + 1) 2h + 1 h − m m + l 2l . Then h m=l (−1) m (2m + 1) 2h + 1 h − m m + l 2l = m≥l (−1) m (2m + 1) 2h + 1 h − m m + l 2l = m≥l g m+1 − g m = g l = 0.
Here, the first equality holds since n k = 0 provided k < 0.
Remark 1.A key step to prove ( 4) is to construct the sequence g m which satisfies (6).In the proof of Lemma 3.1, we obtain g m using Gosper algorithm [7].However, it is also simple to verify (6) by hand.
We introduce the following results for Bessel functions
Lemma 3.2. Set L m := π −π cos(2m + 1)θ cos θ(sin θ) 2n−2 dθ D m := π 0 cos(2m + 1)θ cos θ(sin θ) 2n−1 dθ.
Then we have
n−1 m=0 (−1) m L m J 2m+1 (x) = L 0 2 n−1 n! 1 x n−1 J n (x), (7) +∞ m=0 (−1) m D m J 2m+1 (x) = √ π2 n− 3 2 (n − 1)! 1 x n− 1 2 J n+ 1 2 (x).(8)
Proof.To this end, we first calculate the value of L m .Using the expansion
sin 2n−2 θ = 1 2 2n−2 2n − 2 n − 1 + 2 2 2n−2 n−2 k=0 (−1) n−1−k 2n − 2 k cos((2n − 2 − 2k)θ),
we can obtain that (9)
L m = (−1) m π 2 2n−2 2n − 2 n + m − 1 2m + 1 n + m .
Recall that the series representation of the Bessel function J α (10)
J α (x) = +∞ k=0 (−1) k k!Γ(k + α + 1) x 2 2k+α .
Substituting (10) into (7) we obtain that (11
) +∞ k=0 (−1) k k! n−1 m=0 (−1) m L m 1 Γ(2m + k + 2) x 2 2k+2m+1 = L 0 •n!• +∞ k=0 (−1) k k! • Γ(k + n + 1) x 2 2k+1 .
To this end, we just need prove (11).Comparing the coefficients of the powers of x on the both sides of (11), we only need prove
h m=0 L m 1 (h − m)! • Γ(h + m + 2) = L 0 • n! • 1 h! • Γ(h + n + 1) ,(12)
which is equivalent to
(13) h m=0 (−1) m (2m + 1) n + h h − m n + h h + m + 1 = n n + h n .
Here, we use (9).According to Lemma 3.1, (13) holds which in turn implies (7).
We next turn to (8).Substitute ( 10) into ( 8) and compare the coefficients of the powers of x on the two sides of this equation, we only need to prove
h m=0 D m (h − m)! • (h + m + 1)! = (n − 1)! 4 • 2 2h+2n+3 • (h + n + 1)! h! • (2h + 2n + 2)! , h = 0, 1, . . . . (14) Using cos nx = n 2 ⌊ n 2 ⌋ k=0 (−1) k n − k k (2 cos x) n−2k n − k ,
and
π 2 0 (sin t) x (cos t) y dt = π 2 x+y+1 x! • y! ( x 2 )! • ( y 2 )! • ( x+y 2 )!
, where x! = Γ(x + 1) for x > 0, we have
(15) D m = (2m + 1) m k=0 (−1) k 2m + 1 − k k √ π(n − 1)! 4(2m + 1 − k) (2m + 2 − 2k)! (m + 1 − k)! • ( 2m+2n−2k+1 2 )! .
Substituting (15) into (14), we can rewrite (14) as ( 16)
h m=0 2m + 1 (h − m)! • (h + m + 1)! m k=0 (−1) k 2m + 1 − k k 1 2m + 1 − k (2m + 2 − 2k)! (m + 1 − k)! • ( 2m+2n−2k+1 2 )! = 1 √ π 2 2h+2n+3 (h + n + 1)! h! • (2h + 2n + 2)! .
On the other hand, we can rewrite the left side of ( 16) as (17)
h m=0 2m + 1 (h − m)! • (h + m + 1)! m k=0 (−1) k 2m + 1 − k k 1 2m + 1 − k (2m + 2 − 2k)! (m + 1 − k)! • ( 2m+2n−2k+1 2 )! = h m=0 2m + 1 (h − m)! • (h + m + 1)! m l=0 (−1) m−l m + l + 1 m − l 1 m + l + 1 (2l + 2)! (l + 1)! • ( 2l+2n+1 2 )! = h l=0 1 ( 2l+2n+1 2 )! h m=l 2m + 1 (h − m)! • (h + m + 1)! (−1) m−l m + l + 1 m − l 1 m + l + 1 (2l + 2)! (l + 1)! .
Here, in the first equality, we set a new variable l := m − k.To this end, we consider the second term on the right side of the last equality in (17).Note that, for l = 0, . . ., h − 1,
h m=l 2m + 1 (h − m)! • (h + m + 1)! (−1) m−l m + l + 1 m − l 1 m + l + 1 (2l + 2)! (l + 1)! = (−1) l (2l + 2)! (2l + 1)(2h + 1)! • (l + 1)! h m=l (2m + 1) 2h + 1 h − m (−1) m m + l 2l = 0.
Here, the last equality follows from (4) in Lemma 3.1.Hence the last summation in (17
) is reduced to 1 ( 2h+2n+1 2 )! 2h + 1 (h + h + 1)! (−1) h−h h + h + 1 h − h 1 h + h + 1 (2h + 2)! (h + 1)! = 1 ( 2h+2n+1 2 )! 2 h! = 1 √ π 2 2h+2n+3 (h + n + 1)! h! • (2h + 2n + 2)! .
Here, the last equality uses
2h + 2n + 1 2 ! = Γ h + n + 1 + 1 2 = (2h + 2n + 2)! 2 2h+2n+2 (h + n + 1)! √ π.
We arrive at the conclusion.
Now we can give an estimation for the integrals
(n − 1)! • 2n−2 n−1 M 1 2 2n−2 • π n δ 2n+1 2 r 2n−1 2 ≤ π 0 ∆ δ (r cos θ) cos θ(sin θ) 2n−2 dθ ≤ 5 4 (n − 1)! • 2n−2 n−1 +∞ k=1 1 k 2n+1 2 2 2n−2 • π n δ 2n+1 2 r 2n−1 2 , provided that R = r δ is big enough, where M 1 = 4 5 | cos(2πǫ − 3 4 π)| − 5 4 +∞ k=2 1 k 2n+1 2 > 0 and n ≥ 2. When 1/6 ≤ ǫ ≤ 1/3, we have (19) M 2 (n − 1)! π n+1 δ n+1 r n ≤ π 0 ∆ δ (r cos θ) cos θ(sin θ) 2n−1 dθ ≤ 8 7 (n − 1)! +∞ k=1 1 k n+1 π n+1 δ n+1 r n , provided that R = r δ is big enough, where M 2 = 7 8 | cos(2πǫ − 1 2 π)| − 8 7 +∞ k=21k n+1 > 0 and n ≥ 1.
Proof.Firstly we consider π 0 ∆ δ (r cos θ) cos θ(sin θ) 2n−2 dθ.Using the Fourier expansion for ⌊x⌋ with x ∈ R \ Z,
⌊x⌋ = x − 1 2 + 1 π +∞ k=1 sin(2kπx) k ,
we have (20
) π 0 ∆ δ (r cos θ) cos θ(sin θ) 2n−2 dθ = 1 2 π −π ∆ δ (r cos θ) cos θ(sin θ) 2n−2 dθ = − δ 2π π −π +∞ k=1 sin(2kπ r cos θ δ + kπ) k cos θ(sin θ) 2n−2 dθ = − δ 2π +∞ k=1 (−1) k k π −π sin(2kπ r δ cos θ) cos θ(sin θ) 2n−2 dθ = − δ π +∞ k=1 (−1) k k n−1 m=0 (−1) m L m J 2m+1 (2kπ r δ ) = − δ π • L 0 • 2 n−1 n! +∞ k=1 (−1) k k 1 (2kπ r δ ) n−1 • J n (2kπ r δ ) = − 1 π n • δ n r n−1 • L 0 • n! • ∞ k=1 (−1) k k n J n (2kπ r δ ) = − 1 π n • δ n r n−1 • π 2 2n−2 • 2n − 2 n − 1 • (n − 1)! • ∞ k=1 (−1) k k n J n (2kπ r δ ).
In the fourth equality, we use the formula
sin(x cos θ) = 2 +∞ m=0 (−1) m cos((2m + 1)θ)J 2m+1 (x)
and the orthogonality of the systems {cos kx} +∞ k=0 on the interval [−π, π].We use (7) in the fifth equality.To this end, according to (20), we only need to estimate
+∞ k=1 (−1) k k n J n (2kπ r δ ) .
In fact, note that
(21) |J n (2π r δ )| − +∞ k=2 1 k n |J n (2kπ r δ )| ≤ +∞ k=1 (−1) k k n J n (2kπ r δ ) ≤ +∞ k=1 1 k n |J n (2kπ r δ )|. We first consider |J n (2π r δ )| − +∞ k=21k n |J n (2kπ r δ )|.
Using the asymptotic estimate for J n (x) in Theorem 2.1, we have
(22) J n (2π r δ ) = 1 π δ r cos(2π r δ − ω n ) + θcµ(2π r δ ) −3/2 = 1 π δ r cos(2πǫ − (2n + 1)π 4 ) + θcµ( 1 2π δ r ) 3/2
which implies that
J n (2π r δ ) ≥ 4 5 1 π δ r cos(2πǫ −3π 4 )
provided that R = r δ is big enough.On the other hand,
+∞ k=2 1 k n |J n (2kπ r δ )| = +∞ k=2 1 k n 1 π δ kr cos(2πk r δ − ω n ) + θcµ(2πk r δ ) −3/2 = +∞ k=2 1 k n 1 π δ kr cos(2πkǫ − (2n + 1)π 4 ) + θcµ(2πk r δ ) −3/2 .(23)
Therefore, according to (22),
+∞ k=2 1 k n |J n (2kπ r δ )| ≤ 5 4π δ r +∞ k=2 1 k 2n+1 2
provided that R = r δ is big enough.Combining above results, we obtain that
(24) |J n (2π r δ )| − +∞ k=2 1 k n |J n (2kπ r δ )| ≥ 1 π δ r 4 5 cos(2πǫ − 3π 4 ) − 5 4 +∞ k=2 1 k 2n+1 2
, provided that R = r δ is big enough.When 1/4 ≤ ǫ ≤ 1/2 and n ≥ 2,
M 1 := 4 5 cos(2πǫ − 3π 4 ) − 5 4 +∞ k=2 1 k 2n+1 2 ≥ 4 5 • √ 2 2 − 5 4 +∞ k=2 1 k 5 2 ≈ 0.138 > 0.
Combining (20), ( 21) and ( 24), we obtain the left side of (18).Similarly, based on (23), we have
+∞ k=1 (−1) k k n J n (2kπ r δ ) ≤ +∞ k=1 1 k n J n (2kπ r δ ) ≤ 5 4 1 π δ r +∞ k=1 1 k 2n+1 2
provided that R = r δ is big enough, which implies the right side of (18).Now let us turn to
= − δ π +∞ k=1 (−1) k k π 0 sin(2kπ r δ cos θ) cos θ(sin θ) 2n−1 dθ = − 2δ π +∞ k=1 (−1) k k +∞ m=0 (−1) m D m J 2m+1 (2kπ r δ ) = − 2δ π √ π2 n− 3 2 (n − 1)! +∞ k=1 (−1) k k 1 (2kπ r δ ) n− 1 2 J n+ 1 2 (2kπ r δ ) = − (n − 1)! π n δ n+ 1 2 r n− 1 2 +∞ k=1 (−1) k k n+ 1 2 J n+ 1 2 (2kπ r δ ),
where we use (8) in Lemma 3.2 for the fourth equality.Using the asymptotic estimate for J n+ 1 2 (x) in Theorem 2.1, similarly with the above, we can show that (26) 1 π δ r
7 8 | cos(2πǫ − 1 2 π)| − 8 7 +∞ k=2 1 k n+1 ≤ | +∞ k=1 (−1) k k n+ 1 2 J n+ 1 2 (2kπ r δ )| ≤ 8 7 1 π δ r +∞ k=1 1 k n+1 provided that R = r δ is big enough. When 1/6 ≤ ǫ ≤ 1/3 and n ≥ 1, M 2 := 7 8 | cos(2πǫ − 1 2 π)| − 8 7 +∞ k=2 1 k n+1 ≥ 7 8 • √ 3 2 − 8 7 +∞ k=2 1 k 2 ≈ 0.02 > 0.
Combing (25) and (26), we arrive at (19).
We now can state the proof of the main theorem.
Proof of Theorem 1.2.The idea to prove Theorem 1.2 is similar to one of proving Theorem 1.1 in [4] with using Lemma 3.3 to estimate lim m→∞ E δ (x, F m ).We state the proof of (i) for the completeness.In fact, (ii) can be proved using a similar method.
We denote the number of the non-zero entries in x by x 0 , i.e.,
x 0 := #{j : x j = 0}.
The proof is by induction on x 0 .Note that
lim m→∞ E δ (x, F m ) = lim m→∞ d N m Nm j=1 ∆ δ (x • e j )e j = d z∈S d ∆ δ (x • z)zdω .
We begin with x 0 = 1.Without loss of generality, we suppose x = [x 1 , 0, . . ., 0] T ∈ R d and consider lim m→∞ E δ (x, F m ).By the sphere coordinate system, each z = [z 1 , . . ., z d ] ∈ S d−1 can be written in the form of
[cos θ 1 , sin θ 1 cos θ 2 , sin θ 1 sin θ 2 cos θ 3 , . . . , sin θ 1 • • • sin θ d−1 ] ⊤ ,
where θ 1 ∈ [0, π) and θ j ∈ [−π, π), 2 ≤ j ≤ d − 1.To state conveniently, we set where the last inequality follows from Lemma 3.3.
Θ := [0, π) × [−π, π) × • • • × [−π, π) d−2 , S t(
For the induction step, we suppose that the conclusion holds for the case where x 0 ≤ k.We now consider x 0 ≤ k + 1.Without loss of generality, we suppose x is in the form of [0, . . ., 0, x d−k , . . ., x d ] ∈ R d .We can write [x d−1 , x d ] in the form of (r 0 cos ϕ 0 , r 0 sin ϕ 0 ), where r 0 ∈ R + and ϕ 0 ∈ [0, 2π).Then where the last inequality follows from the fact x 0 ≤ k provided ϕ 0 = 0.
π 0 ∆
0
δ (r cos θ) cos θ(sin θ) 2n−2 dθ and π 0 ∆ δ (r cos θ) cos θ(sin θ) 2n−1 dθ.Lemma 3.3.Set R := r δ and ǫ := R − ⌊R⌋.Then when 1/4 ≤ ǫ ≤ 1/2 (18)
π 0 ∆
0
δ (r cos θ) cos θ(sin θ) 2n−1 dθ.Similar with the above,we have (25) π 0 ∆ δ (r cos θ) cos θ(sin θ) 2n−1 dθ sin θ) 2n−1 dθ
− 1 ∆− 1 ∆ δ (x • z)zdω = d z∈S d− 1 ∆ 1 ≥
1111
θ) := t j=1 sin θ j and J d (θ) := (sin θ 1 ) d−2 (sin θ 2 ) d−3 • • • (sin θ d−2 ) .Noting that dω = J d (θ)dθ 1 • • • dθ d−1 and z∈S dδ (x 1 z 1 )z j dω = 0, 2 ≤ j ≤ d − 1, we have lim m→∞ E δ (x, F m ) = d z∈S dδ (x 1 z 1 )z 1 dω = d θ∈Θ ∆ δ (x 1 cos θ 1 ) cos θ 1 (sin θ 1 ) d−2 |(sin θ 2 ) d−3 • • • (sin θ d−2 )|dθ 1 • • • dθ d−C 1,d • δ (d+1)/2 /|x 1 | (d−1)/2
2 .
2
x j S j (θ) cos θ j + r 0 sin θ1 • • • sin θ d−2 cos(θ d−1 − ϕ 0 ) =: T (ϕ 0 ).A simple observation is θ∈Θ ∆ δ (T (ϕ 0 ))S d−2 (θ)J d (θ) cos θ d−1 dθ 2 + θ∈Θ ∆ δ (T (ϕ 0 ))S d−2 (θ)J d (θ) sin θ d−1 dθ 2 = θ∈Θ ∆ δ (T (0))S d−2 (θ)J d (θ) cos θ d−1 dθ 2 + θ∈Θ ∆ δ (T (0))S d−2 (θ)J d (θ) sin θ d−1 dθ Then we have lim m→∞ E δ (x, F m ) = d z∈S d−1 ∆ δ (x • z)zdω
(
θ∈Θ ∆ δ (T (ϕ 0 ))S t−1 (θ)J d (θ) cos θ t dθ) 2 + ( θ∈Θ ∆ δ (T (ϕ 0 ))S d−1 (θ)J d (θ)dθ)
( 2 1/ 2 ≥
22
θ∈Θ ∆ δ (T (0))S t−1 (θ)J d (θ) cos θ t dθ) 2 + ( θ∈Θ ∆ δ (T (0))S d−1 (θ)J d (θ)dθ) C 1,d • δ (d+1)/2 /r (d−1)/2
Combinatorial Identities -eight tables based on seven unpublished manuscript notebooks. H W Gould, Prof.Jocelyn Quaintance1945-1990. May 2010
Ilia Krasikov, arXiv:1107.2007arxiv.orgSome asymptotics for the Bessel functions with an explicit error term. 2011arXiv preprint
Optimal frames for erasures. R Holmes, V Paulsen, Linear Algebra Appl. 3772004
The performance of PCM quantization under tight frames representations. Yang Wang, Zhiqiang Xu, SIAM J. MATH. ANAL. 442012
White noise hypothesis for uniform quantization errors. D Jimenez, L Wang, Y Wang, SIAM J. Math. Anal. 382007
Quantized overcomplete expansions in R n : Analysis, synthesis, and algorithms. V Goyal, M Vetterli, N Thao, IEEE Trans. Inform. Theory. 441998
Decision Procedure for Indefinite Hypergeometric Summation. Rw Gosper, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USAJanuary 197875
W Bennett, Spectra of quantized signals. 194827
Sigma-delta quantization and finite frames. J Benedetto, A M Powell, Ö Yılmaz, IEEE Trans. Inform. Theory. 522006
Second order sigmal-delta quantization of finite frame expansions. J Benedetto, A M Powell, Ö Yılmaz, Appl. Comput. Harmon. Anal. 202006
On the distribution of uniform quantization errors. S Borodachov, Y Wang, Appl. Comput. Harmon. Anal. 272009
NIST Handbook of Mathematical Functions. F W J Olver, D W Lozier, R F Boisvert, C W Clark, Cambridge University Press2010
Email: zhouheng7598@sina.com.cn LSEC, Institute of Computational Mathematics. Beijing; China100190Chinese Academy of Sciences
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