A Weierstrass-type theorem for homogeneous polynomials

ORIGINAL ARTICLERegulation of MEIS1by distal enhancer elements in acute leukemiaQ-f Wang 1,2,Y-j Li 2,3,J-f Dong 1,B Li 2,JJ Kaberlein 1,L Zhang 2,FE Arimura 4,RT Luo 1,J Ni 5,F He 2,J Wu 2,R Mattison 1,J Zhou 2,C-z Wang 6,S Prabhakar 7,MA Nobrega 4and MJ Thirman 1Aberrant activation of the three-amino-acid-loop extension homeobox gene MEIS1shortens the latency and accelerates the onset and progression of acute leukemia,yet the molecular mechanism underlying persistent activation of the MEIS1gene in leukemia remains poorly understood.Here we used a combined comparative genomics analysis and an in vivo transgenic zebrafish assay to identify six regulatory DNA elements that are able to direct green fluorescent protein expression in a spatiotemporal manner during zebrafish embryonic hematopoiesis.Analysis of chromatin characteristics and regulatory signatures suggests that many of these predicted elements are potential enhancers in mammalian hematopoiesis.Strikingly,one of the enhancer elements (E9)is a frequent integration site in retroviral-induced mouse acute leukemia.The genomic region corresponding to enhancer E9isdifferentially marked by H3K4monomethylation and H3K27acetylation,hallmarks of active enhancers,in multiple leukemia cell lines.Decreased enrichment of these histone marks is associated with downregulation of MEIS1expression during hematopoietic differentiation.Further,MEIS1/HOXA9transactivate this enhancer via a conserved binding motif in vitro ,and participate in an autoregulatory loop that modulates MEIS1expression in vivo .Our results suggest that an intronic enhancer regulates the expression of MEIS1in hematopoiesis and contributes to its aberrant expression in acute leukemia.Leukemia (2014)28,138–146;doi:10.1038/leu.2013.260Keywords:MEIS1;comparative genomic analysis;enhancer elements;hematopoiesisINTRODUCTIONThe myeloid ectropic viral integration site 1(MEIS1)gene was first identified as a common retroviral integration site in BXH-2leukemic mice.1It encodes a homeodomain-containing transcription factor belonging to the three-amino-acid loop extension superfamily.As a cofactor of HOXA-family members and PBX1,MEIS1is critical for the development of hematopoietic cells and many other tissues including the central nervous system,vasculature,lung and eye.2–4During normal hematopoiesis,the expression of the MEIS1gene is tightly controlled.MEIS1shows lineage and developmental stage-specific expression,with high levels observed in hematopoietic stem cells and early progenitor cells.MEIS1expression downregulates in later stages of hematopoietic development.5Meis1knockout mice die by embryonic day 12.5–14.5because of the lack of megakaryocytes.2In addition to its role in normal hematopoiesis,MEIS1is critical to leukemogenesis.In acute leukemia patients,persistent over-expression of MEIS1has been consistently observed.6–10The level of MEIS1expression is inversely correlated with prognosis in human acute myeloid leukemia.11,12Further,it has been reported that overexpression of Meis1correlated with shorter latency and accelerated progression in various leukemogenic models,such as mouse MLL-associated leukemia models,13,14leukemogenic cells with NUP98-HOX translocations 15and CD34þNPM1-mutated acute myeloid leukemia cells.16Downregulation of MEIS1inMLL-rearranged leukemia cell lines resulted in decreased proliferation as well as transcriptional repression of cell cycle entry-related genes.8,17,18Despite the essential role of MEIS1overexpression in acute leukemia,the molecular mechanism underlying persistent activation of the MEIS1gene in leukemia remains poorly understood.Cellular gene expression is critically determined by DNA regulatory elements,sequence-specific transcription factors,as well as chromatin modifications.The human MEIS1gene is located on chromosome 2p13–2p14and spans B 1300kb in length.19The strict temporal and spatial pattern of MEIS1expression suggests that it is under the tight control of cis -regulatory sequences.The Meis1promoter is regulated by ELF1and CREB.20,21In hematopoietic stem cells,the expression of MEIS1gene is under the combinatorial control of multiple hematopoietic transcription factors.The binding of those factors is not limited to the promoter region and is distributed along the MEIS1locus.22These data suggest that hematopoietic-specific regulatory elements may exist in the MEIS1locus.We therefore sought to determine the genetic and epigenetic mechanisms involved in the persistent expression of MEIS1in leukemogenesis.In this study,we report the systematic identification of distal enhancer sequences in the 1300-kb genomic region of the MEIS1locus.Traditionally,labor-intensive and time-consuming techniques have been employed to hunt for distal regulatory1Department of Medicine,Section of Hematology/Oncology,University of Chicago,Chicago,IL,USA;2CAS Key Laboratory of Genome Sciences and Information,Beijing Institute of Genomics,Chinese Academy of Sciences,Beijing,China;3Beijing Institute of Traditional Chinese Medicine,Beijing Hospital of Traditional Chinese Medicine Affiliated to Capital Medical University,Beijing,China;4Department of Human Genetics,University of Chicago,Chicago,IL,USA;5Tongji Medical College of Huazhong,University of Science &Technology,Wuhan,China;6Department of Anesthesia &Critical Care,University of Chicago,Chicago,IL,USA and 7Genome Institute of Singapore,Singapore.Correspondence:Dr Q-f Wang,CAS Key Laboratory of Genome Sciences and Information,Beijing Institute of Genomics,Chinese Academy of Sciences,Beijing,China.E-mail:Wangqf@ or Dr MJ Thirman,Department of Medicine,Section of Hematology/Oncology,University of Chicago,5841S.Maryland Avenue,MC21115,Chicago 60637,IL,USA.E-mail:mthirman@Received 1July 2013;revised 22August 2013;accepted 29August 2013;accepted article preview online 11September 2013;advance online publication,15October 2013Leukemia (2014)28,138–146&2014Macmillan Publishers Limited All rights reserved 0887-6924/14/leuparative genomic strategies are based on the concept that sequences important for gene regulation are conserved throughout evolution.23When coupled with appropriate functional tests,this genomic strategy has proved to be a powerful approach for systematic discovery of enhancer sequences.24Using a combined comparative genomic and molecular characterization strategy,we identified six regulatory elements in the MEIS1locus.These elements contributed to tissue-specific gene expression of normal hematopoiesis/vasculogenesis in a zebrafish reporter assay. One of those six elements,designated‘E9’,corresponds to a common retroviral integration site in retrovirus-induced mouse leukemia models.We demonstrate that increased levels of histone H3K4monomethylation(H3K4me1)and H3K27 acetylation(H3K27ac)at this intronic E9region are associated with active MEIS1expression in multiple human leukemic cell lines.In an inducible MLL-ENL leukemia system,the levels of those histone marks diminish when the expression of MEIS1is downregulated during cellular differentiation.Finally,we show that HOXA9and MEIS1directly bind to a conserved binding motif in the E9region.Knockdown of HOXA9in THP-1cells results in downregulation of the MEIS1gene.These studies suggest that expression of MEIS1can be driven by an autoregulatory loop mediated through a distal intronic enhancer.MATERIALS AND METHODSAnalysis of sequence conservationHuman,mouse,rat,fugu,zebrafish and tetraodon sequences were downloaded from the UCSC Genome Bioinformatics(http://www. ).On the basis of the human May2004assembly hg17, the coordinates for the analyzed MEIS1human locus are:chr2:66,223, 424-67,536,101.This region spans1312678bp and includes all the MEIS1 exons and introns,as well as the entire intergenic region on each side of the gene.We aligned the human MEIS1locus to its orthologs in mouse,rat,fugu, zebrafish and tetraodon using MLAGAN.25Aligned sequences were scanned for statistically significant evolutionarily conserved regions using Gumby.26We performed human-fish(human-fugu-zebrafish-tetraodon) and human-mouse-rat comparisons,and selected top-ranked elements from each analysis.Conserved noncoding sequences were also identified based on significant conservation in an alignment of17vertebrate genomes.27Selection of14potential regulatory sequences tested in the transgenic zebrafish assay was based on prediction by more than one analysis.Reporter gene assay in transgenic zebrafishThe transgenic zebrafish system were based on the Tol2transposon.28 Corresponding human sequences of predicted enhancer elements(Figure1 and Supplementary Table S1)were individually cloned.Each PCR product was recombinedfirst into the pDONR221vector,and then into pXIG_c-fos_GW,using Gateway reagents(Invitrogen,Carlsbad,CA,USA). Approximately100fertilized eggs were injected for each reporter gene construct.The transparent embryos were then examined byfluorescence microscopy for greenfluorescent protein(GFP)expression at the hematopoiesis sites of both posterior lateral mesoderm(PLM)and anterior lateral mesoderm(ALM)over a time period of3days.A functional enhancer should exhibit an expression pattern consistent across435%of embryos.Cell isolation and antibody stainingHematopoietic stem and progenitor cells(HSPCs)were isolated as described.29In brief,bone marrow cells from femurs of4-week-old C57/BL6mice were incubated with the mouse Lineage Cell Depletion Kit(Miltenyi Biotec,Bergisch Gladbach,Germany;130-090-858)to obtain LinÀcells.To isolate HSPC cells that are LinÀ,c-Kitþand CD34þ,LinÀcells were stained with an allophycocyanin-conjugated anti-c-Kit antibody(clone2B8)and afluorescein isothiocyanate-conjugated anti-CD34antibody.Cell cultureLeukemic cell lines were maintained in Roswell Park Memorial Institute 1640(Gibco,Carlsbad,CA,USA)with10%fetal bovine serum(Hyclone, Logan,UT,USA).Two clones of MLL-ENL-inducible cell line(csh2and csh3)were cultured in the presence of interleukin-3(10ng/ml),interleukin-6(10ng/ml),granulocyte-macrophage colony-stimulating factor(10ng/ml)and stem cell factor(100ng/ml).30Chromatin immunoprecipitationThe chromatin immunoprecipitation(ChIP)was performed as described.31 Antibodies used were as follows:anti HOXA9(Upstate,Billerica,MA,USA;07-178),anti-MEIS1(Santa Cruz,Dallas,TX,USA;sc-10599x),anti-H3K4me1 (Abcam,Cambridge,UK;ab8895),anti H3K4me3(Upstate,07-473),H3K27ac(Abcam,ab4729)and rabbit IgG(Santa Cruz,sc2027).The amount of purified DNA was measured using the PicoGreen system (Molecular Probes,Carlsbad,CA,USA),and was subjected to PCR using SYBR Green Master mix(Applied Biosystems,Carlsbad,CA,USA).The results are shown as fold enrichment of ChIP DNA over input DNA.Primersare listed in Supplementary Table S3.Transient transfection and luciferase reporter assayThe human promoter was cloned in the proper orientation upstream of the luciferase complementary DNA in the pGL3Basic construct(Promega, Madison,WI,USA).The enhancer elements were PCR cloned into polylinkersites upstream of the promoter.Site-specific point mutations and deletionswere introduced into the element E9using QuickChangeII site-directed mutagenesis kit(Stratagene,La Jolla,CA,USA).The sequences wereconfirmed by sequencing(Supplementary Table S2).The cells were grown in12-well plates and transfected using Fugene (Roche Molecular Biochemicals,Basel,Switzerland)following the manu-facturer’s protocol.The activity of luciferase and b-galactosidase was measured using the Luciferase Assay System(Promega)and galacto-LightPlus(Applied Biosystems),respectively.Luciferase activity for each samplewas normalized to the b-galactosidase assay control.Transfections were carried out in duplicates.All experiments are representative of at least three independent transfections.Short interfering RNA knockdown experimentsHOXA9short interfering(Sigma,St Louis,MO,USA)was transfected intoTHP-1cells with Fugene HD transfection kit(Roche Molecular Biochem-icals).At24h after transfection,the total RNA was extracted by TRIZOL,and treated with DNaseI Turbo kit(Ambion,Carlsbad,CA,USA;AM1907).The complementary DNA was then obtained by reverse transcription(PromegaA3500)and real-time PCR was performed.The gene expression level was normalized with the expression of the PGK1gene.RESULTSMultiple hematopoietic regulatory elements in the MEIS1genelocus identified by combined comparative genomic analysis and functional assayOn the basis of the temporal and spatial expression pattern of MEIS1during embryonic and adult hematopoiesis,we hypothe-sized that distal enhancer sequences are needed to control the expression of parative sequence analysis of the human MEIS1locus showed that the evolutionary conservation of MEIS1is limited not only to exons but also to introns and adjacent downstream and upstream regions.To predict putative enhancer elements,we employed bioinformatic analytical tools including VISTA,32GUMBY,26PReMod33and ECR34(Supplementary Table S1),to scan the1312678-bp MEIS1locus.This scanned region includesall the MEIS1exons and introns,as well as the entire intergenic region on each side of the gene.We identified14noncoding DNA elements that are conserved in vertebrate species at varying evolutionary distance,from75million years in rodents to450 million years in zebrafish(See Materials and Methods and Supplementary Table S1,Figure1a,element1–14).These sequences also contain clusters of binding sites for multiple transcription factors(data not shown),a sequence characteristic of regulatory modules.23,33Althoughfive of these DNA sequencesare located in the intronic regions of the gene,the remaining nine Regulation of MEIS1by distal enhancer elementsQ-f Wang et al139&2014Macmillan Publishers Limited Leukemia(2014)138–146elements were found at the intergenic regions between MEIS1and its neighboring genes (Figure 1a).Among these conserved regions,element E6has nervous system-associated enhancer activity in a transgenic zebrafish system;35and element E13has been reported to exhibit hindbrain-specific enhancer activity in a transgenic mouse model (element 831of the VISTA enhancer browser at /cgi-bin/gateway2).Using a transgenic zebrafish system,28we tested the ability of the 14predicted regulatory sequence elements to drive GFP reporter gene expression in vivo .The corresponding human sequence of predicted enhancer elements (Supplementary Table S1)were individually cloned upstream of the minimal c-fos promoter of the reporter gene construct,and then injected into fertilized zebrafish eggs.With a minimal c-fos promoter,the GFP reporter gene showed no expression in developing embryos of transgenic zebrafish (Figure 1b).When cloned upstream of the minimal promoter,6out of 14predicted regulatory elements showed hematopoietic-specific gene-activating activity in this in vivo transgenic zebrafish assay (Supplementary Table S1).The PLM and ALM are major sites of zebrafish embryonic blood development (Figure 1b).In our transgenic zebrafish system,the corresponding human sequence of E6and E14were able to drive GFP expression at ALM,elements E5,E7,E9were able to drive GFP expression at PLM and E8element was able to drive GFP expression at both PLM and ALM at 24h post fertilization (Figure 1b,Supplementary Table S1).Thirty-eight out of 95(39%)fertilized zebrafish eggs injected with the E9reporter gene,and 41,38,39,66and 40%for E5,E6,E7,E8and E14,respectively,consistently exhibited the same pattern of restricted GFP expression (Figure 1b).In addition,elements E6and E9were able to drive GFP expression in the heart.The transgenic zebrafishresults suggest that these distal regulatory elements may contribute to hematopoietic expression of MEIS1.5We further explored whether the enhancer elements identified in zebrafish reporter assay are active during mammalian hematopoiesis.Regulatory DNA elements such as enhancers and promoters are marked by distinct histone modification patterns,and show characteristic of open chromatin structure in the mammalian genome.H3K4me1is commonly identified at both active and poised enhancer regions,36,37,38and H3K27ac is frequently enriched at both active enhancers and promoters.39,40We employed ChIP assay to determine the levels of acetylated H3K27in primary mouse HSPCs in which MEIS1is actively expressed.We isolated Lin-CD34þc-kit þHSPCs,a combination of hematopoietic stem cells,common myeloid progenitors and granulocyte/monocyte lineage progenitor cells,from C57/BL6mouse bone marrows by fluorescence-activated cell sorting (Figure 2a).The sorted population had high c-kit mRNA expression,lacked expression of Gr1,a marker for differentiated mature myeloid cells (Figure 2b).The ChIP results showed that the active histone mark H3K27ac was significantly enriched at genomic regions corresponding to enhancer elements E6and E9,as well as the MEIS1promoter (Figure 2c),suggesting that those regulatory sequences may have a role in regulating MEIS1expression in primary murine HSPCs.The enhancer E9region is a frequent retroviral integration site in mouse leukemia models and is specifically marked by active histone modifications in MEIS1-expressing leukemic cellsTo better understand the molecular mechanism in disregulated MEIS1expression,we examined whether any of the enhancers identified may be involved in leukemeogenesis.Inmouse100%50%ALMPLMPromoter only E7(39%)E6(38%)E5(41%)E14(40%)E9(39%)E8(66%)14567891014Sequence Identity (Human/Mouse)010030050070090011001300KbElements MEIS1a13231112Figure 1.Multiple regulatory elements within the MEIS1gene.(a )Schematic map of enhancer elements in the MEIS1Locus.Orthologous sequences in the MEIS1locus are compared between human and mouse (Vista),with human sequence as the base.Numbered green bars (1–14)correspond to predicted enhancer elements conserved in multiple vertebrate species.The elements showing positive expression in the transgenic zebrafish assay are marked in red.Hg17:chr2:66,223,424-67,536,101(corresponding to Hg19:66311773-66680696).(b )Identification of enhancer activity by transgenic Zebrafish assay.The GFP reporter gene is under the control of a minimal promoter and the human sequence of putative enhancers.The elements predicted by sequence conservation and the cloned elements used in functional testing are indicated in Supplementary Table S1.GFP reporter constructs were injected into fertilized zebrafish eggs.Representative transgenic zebrafish embryos (24hpf)are shown,with the PLM area framed in blue and ALM area circled in red.The percentage of positive embryos is marked in parentheses.Regulation of MEIS1by distal enhancer elementsQ-f Wang et al140Leukemia (2014)138–146&2014Macmillan Publishers Limitedleukemia models,the Meis1locus harbors frequent retrovirus intergration sites,which result in overexpression of Meis1leading to the development of AML.The genomic location of an active enhancer possesses an open chromatin structure that often renders the region susceptible to retroviral integration.Upregula-tion of the affected gene may result from strong regulatory element carried by the retrovirus.41A search in the Retrovirus Tagged Cancer Gene Database (/rtcgd/)42revealed 12mouse clones that carry proviral insertions interrupting introns or exons of Meis1.Strikingly,these viral integration events formed two distinct clusters along the MEIS1locus.In addition to an integration hot spot at the promoter region,3out 12of the viral integrations occurred at the genomic sequence corresponding to enhancer E9region (Figure 3a).Among them,the retroviral integration sites of clone NUP53LD PCR and 403_4_9are located within the enhancer E9,whereas clone 73_4_4is only 52bp away from enhancer E9.In contrast,none of the remaining 13predicted putative enhancers harbored retroviral integration sites.A survey of the UCSC genomic browser indicates a high-level enrichment of H3K4me1,H3K27ac and presence of DNaseI hypersensitive site at the E9region in H1-hESC,K562and GM12878cells (Figure 3a).Altogether,these data suggest that enhancer E9may be functional in mammalian hematopoiesis and is likely involved in retrovirus-induced leukemogenesis.We therefore focused subsequent analysis on this regulatory element.To explore whether enhancer element E9is active in human leukemias,we determined the level of monomethylated H3K4,a hallmark of distal enhancers,37in patient-derived leukemia cell lines.In MEIS1-expressing THP-1cells,the genomic region of enhancer E9was marked by extensive monomethylation of H3K4(Figure 3b).In contrast,no enrichment of H3K4me1was observed in HL-60cells,which have a low level of MEIS1mRNA.We did not detect significant level of H3K4me1in the genomic regions corresponding to the other five identified enhancers (data not shown).MEIS1is expressed at high levels in certain subtypes of AML patients.We further selected a panel of leukemic cell linesthat exhibit varied levels of MEIS1expression and examined the enhancer E9region for the presence of H3K4me1.We found that the presence of H3K4me1is associated with the active expression of MEIS1(Figure 3c).Comparing with leukemia cell line HL-60,the level of H3K4me1enrichment is significantly high in cells harboring MLL fusion proteins (THP-1,RS4;11and ML-2).U937cells,which carry the CALM-AF10fusion gene,also exhibit a high level of H3K4me1(Figure 3c).Considering the developmental stage-specific expression of MEIS1,we sought to determine the chromatin characteristics of enhancer E9during induced differentiation of leukemic cells.We employed a murine Meis1-expressing myeloblast cell line,which harbors an oncogenic MLL-ENL gene that can be induced to terminal differentiation upon inactivation of the MLL-ENL fusion protein.43,44Consistent with reports that the level of Meis1is downregulated when myeloid progenitor cells undergo terminal differentiation to neutrophils,5Meis1expression diminishes with induced differentiation in this MLL-ENL cell line (Figure 4a).Using this inducible MLL-ENL cell line,we determined the levels of H3K4me1,H3K4me3and H3K27ac enrichment at both the promoter and enhancer E9regions.When endogenous Meis1expression was downregulated after inactivation of MLL-ENL,active histone marks H3K4me1and H3K27ac were significantly decreased across the E9region (Figures 4b and c).In contrast,a high level of histone H3K4me3modification,a marker at the promoter of actively expressed genes,remained largely unchanged at the promoter region upon inactivation of the MLL-ENL fusion protein (Figure 4d).As the presence of both H3K4me1and H3K27ac have been associated with active enhancers,these data suggest that enhancer E9is a functional regulatory sequence in controlling expression of Meis1in MLL-rearranged leukemic cells.MEIS1/HOXA9regulates MEIS1expression through enhancer E9In an effort to explore the molecular basis underlying the observed regulatory function of enhancer E9,wediscoveredR e l a t i v e E n ri c h m e n t t o i n p u tR e l a t i v e E x p r e s s i o nFigure 2.Chromatin characteristics of MEIS1regulatory elements in mammalian hematopoietic cells.(a )HSPC cells (Lin-CD34þc-Kit þ)were isolated from mouse bone marrow by fluorescence-activated cell sorting.(b )mRNA expression of c-Kit ,Gr1and Meis1were detected in HSPC population.(c )H3K27ac enhancer marks of the Meis1gene during hematopoiesis in mice.ChIP assay was performed with the antibody against H3K27ac in HSPC cells.The amounts of input DNA and ChIP DNA were normalized,and the data are shown as the relative enrichment ratio of precipitated DNA to input DNA.P1is in the promoter of Meis1gene,and D130is located 140kb downstream of the 30region of Meis1gene.Regulation of MEIS1by distal enhancer elements Q-f Wang et al141&2014Macmillan Publishers LimitedLeukemia (2014)138–146several evolutionary conserved MEIS1/HOXA9binding sites at the enhancer E9region using the Genomatix tool (http://www.genomatix.de,Germany;Figures 5a and b).To test possible recruitment of these two proteins to enhancer E9in vivo ,ChIP-quantitative PCR primers were designed to amplify eight locations across the cloned 2.1kb E9enhancer (see Supplementary Table S1),with an average spacing of 200–300bp.The binding of MEIS1and HOXA9proteins was enriched both at the MEIS1promoter (data not shown)and the primer hE9-5region corresponding to two overlapping conserved sites (M2/M3in Figures 5a and b)in MEIS1-expressing THP-1cells.The MEIS1and HOXA9-binding signals were diminished at flanking regions B 300bp in both directions (Figure 5c).To examine whether the predicted MEIS1/HOXA9consensus sites are critical for the function of enhancer E9,we mutated the MEIS1/HOXA9-binding sites and determined their contributions to the enhancer activity in REH cells (Figure 6a).A 14-bp deletion/mutations were introduced into the E9element to destroy the two overlapping MEIS1/HOXA9-binding sites (M2/M3).Compared with the construct with a cytomegalovirus promoter only,addition of enhancer E9upstream of the promoter increased the luciferase activity in a transient transfection assay.Mutation of the MEIS1/HOXA9-binding sites diminished the enhancer activation activity(Figure 6a and Supplementary Table S2).These results suggest that the binding of MEIS1/HOXA9to enhancer E9might regulate the expression level of MEIS1itself.Further,short interfering knockdown of HOXA9resulted in a downregulation of MEIS1expression in the THP-1cell line (Figure 6b).Altogether,these results demonstrated that MEIS1/HOXA9may transactivate the MEIS1gene through a conserved binding site in the enhancer E9.DISCUSSIONAs its level of expression is inversely correlated with prognosis,MEIS1expression is an important predictive marker in human acute leukemia.Further,knockdown of Meis1delays the develop-ment of overt leukemia in mouse models.8Despite the essential role of MEIS1overexpression in acute leukemia,the underlying genetic and epigenetic mechanisms regulating its expression are largely ing combined comparative genomics analysis and functional assays,we identified multiple intronic regulatory DNA elements that may have a role in regulating MEIS1expression in zebrafish and mammalian hematopoiesis.One of the identified enhancer elements,designated E9,may mediate the autoregulation of MEIS1through an evolutionarily conserved binding motif.-MEIS1 expressionh E9h E9h E9h E9h E9h E9h E9h E9R e l a t i v e E n r i c h m e n t t o I n p u tH3K4me1R e l a t i v e E n r i c h m e n t t o I n p u thE9-4Meis1Retroviral Integration SitesMeis1Retroviral Integration SitesMammal ConsH3K4me1H3K27ac DNaseImm950kb1kbmm9E9 (964bp)+++++++++++Figure 3.Retroviral integration sites and enriched histone H3K4me1at enhancer E9;correlation with active MEIS1expression in acute myeloidleukemia cells.(a )Retroviral integration sites in the mouse Meis 1gene.The 30region of the mouse Meis1gene (UCSC genomic browser,NCBI36/mm9,chr11:18,780,325-18,786,183)is enlarged.The genomic location of the predicted enhancer E9(964bp)is marked in the gene structure as a black solid box.The open boxes below the gene structure represent the three genomic sequences recovered from the integration sites in individual leukemia mouse clones (data obtained from Retrovirus Tagged Cancer Gene Database).Retroviral integration site of clone NUP53LD PCR and 403_4_9are located within the enhancer E9,whereas clone 73_4_4is only 52bp away from enhancer E9.The DNA sequence of the enhancer E9is aligned with the human MEIS1genomic region in UCSC genomic browser.Histone modifications of H3K4me1,H3K4me3and H3K27ac are the ChIP sequence results from the H1-hESC,K562and GM12878cells in the UCSC genomic browser,and the DNaseI hypersensitive site (ENCODE project)and the mammalian conservation (Mammal Cons)are presented for the enhancer element E9region.(b ,c )Histone H3K4me1at enhancer E9is associated with MEIS1activation.Crosslinked chromatin from THP-1(in light grey)and HL-60(in dark grey)were immunoprecipitated with antibodies against H3K4me1,a marker for potential enhancers.The precipitated DNA was amplified using eight primer pairs (hE9-1–hE9-8)spanning the cloned 2.1kb E9enhancer (see Supplementary Table S1).Data are shown as fold change versus input DNA.(c )Crosslinked chromatin was immunoprecipitated with antibodies against H3K4me1in multiple leukemic cell lines.The precipitated DNA was amplified using primer hE9-4.Data are shown as fold change versus input DNA.The mRNA expression level of each cell line is marked under the each column,and are shown as the ‘À,þ,þþand þþþ’,representing negative or a graded relative expression level.Regulation of MEIS1by distal enhancer elementsQ-f Wang et al142Leukemia (2014)138–146&2014Macmillan Publishers Limited。

a rXiv:g r-qc/94338v 118Mar1994gr-qc/9403038,CGPG-94/1-1Overview and OutlookAbhay AshtekarCenter for Gravitational Physics and Geometry Pennsylvania State University,University Park,PA 16802-6300When you do some mountaineering...you sometimes...want to climb to some peak but there is fog everywhere...you have your map or some other indication where you probably have to go and still you are com-pletely lost in the fog.Then...all of a sudden you see quite vaguely in the fog,just a few minute things from which you say,“Oh,this is the rock I want.”In the very moment that you have seen that,then the whole picture changes completely,because although you still don’t know whether you will make the rock,nevertheless for a moment you say,“...Now I know where I am;I have to go closer to that and then I will certainly find the way to go ...”(Werner Heisenberg,1963)1.Introduction The problem of quantum gravity is an old one and over the course of time several distinct lines of thought have evolved.However,for several decades,there was very little communication between the two main communities in this area:particle physicists and gravitation theorists.Indeed,there was a lack of agreement on even what the key problems are.By and large,particle physics approaches focused on perturbative techniques.The space-time metric was split into two parts:g µν=ηµν+Gh µν,ηµνbeing regarded as a flat kinematic piece,h µνbeing assigned the role of the dynamical variable and Newton’s constant G playing the role of the coupling constant.The field h µνwas then quantized on the ηµν-background and perturbative techniques that had been so successful in quantum electrodynamics were applied to the Einstein-Hilbert action.The key problems then were those of handling the infinities.The gravity community,on the other hand,felt that acentral lesson of general relativity is that the space-time metric plays a dual role:it is important that one and the same mathematical object determine geometry and encode the physical gravitational field.From this perspective,an ad-hoc split of the metric goes against the very spirit of the theory and must be avoided.If one does not carry out the split,however,a theory of quantum gravity would be simultaneously a theory of quantum geometry and the notion of quantum geometry raises a variety of conceptual difficulties.If there is no background space-time geometry –but only a probability amplitude for various possibilities–how does one do physics?What does causality mean?What is time?What does dynamics mean?Gravity theorists focused on such conceptual issues.To simplify mathematics,they often truncated the theory by imposing various symmetry conditionsand thus avoided thefield theoretic difficulties.Technically,the emphasis was on geometry rather than functional analysis.It is not that each community was completely unaware of the work of the other(although,by and large,neither had fully absorbed what the other side was saying).Rather,each side had its list of central problems and believed that once these issues were resolved,the remaining ones could be handled without much difficulty. To high energy theorists,the conceptual problems of relativists were perhaps analogous to the issues in foundations of quantum mechanics which they considered to be“unimportant for real physical predictions.”To relativists,thefield theoretic difficulties of high energy physicists were technicalities which could be sorted out after the conceptual issues had been resolved.Even on the rare occasions when they got together,the two sides seemed to“talk past each other.”Over the last decade,however,there has been a certain rapprochement of ideas on quantum gravity.Each side has become increasingly aware of the difficulties that were emphasized by the other.There are some agreements and increasing clarity.In the poetic imagery of Heisenberg,we have seen faint outlines of some of the rocks we want to climb.Perhaps the most important among these developments are the following:1.Recognition that non-perturbative methods are essential.There is a growing senti-ment that,at a fundamental level,the theory should not involvefluctuations arounda classical geometry.Indeed,there should be no background geometry or any back-groundfields for that matter.The theory should be diffeomorphism invariant.2.Acceptance that conceptual issues such as the problem of time will have to be handledsatisfactorily.The unease with the standard measurement theory of quantum mechan-ics is brought to the forefront by the absence of a background,classical space-time.The interface of classical and quantum domains and the decoherence processes that make the world seem classical are receiving greater attention.3.Agreement that thefield theoretic divergences have to be faced squarely.There isgrowing awareness that although the mini-superspaces that feature in quantum cos-mology are obviously interesting,from the perspective of full quantum gravity,they are essentially toy models.One has to learn to deal with the infinite number of degrees of freedom“honestly.”Time has come to give proper mathematical meaning to the formal equations such as the Wheeler-DeWitt equation.New ideas and mathematical techniques are needed since the issue of regulating quantum operators in absence of a background geometry has rarely been faced.4.Reconciliation with the possibility that the“quantum geometry”governing the Planckscale may not at all resemble the Riemannian picture.It is likely that differential geometry,in the standard sense of the term,will itself be inadequate to capture physics.Some discrete structures are likely to emerge and new mathematical tools will be needed to handle them.Thus,the goals of the two communities have moved closer.These recognitions do not imply,however,that there is a general consensus on how all these problems are to be resolved.Thus,there are again many approaches.But I strongly support belief that this diversity in the lines of attack is very healthy.In a problem like quantum gravity,where directly relevant experimental data is scarce,it would be a grave error if everyone followed the same path.Indeed,the most promising way of enhancingour chances at success is to increase the amount of variety.What is striking is that,in spite of the diversity of methods,some of the results are often qualitatively similar.For example,in the last few years,the fourth point I mentioned above has come up again and again in quite different contexts.The idea that the continuum picture itself is likely to break down is not new.For thefirst time,however,it is arising as a concrete result of calculations within well-defined,coherent schemes which were not explicitly constructed to obtain such a result.It arose from certain computer simulations of4-dimensional Euclidean gravity,(see e.g.Agishtein&Migdal(1992)and Br¨u gmann&Marinari(1993)),from string theory(see e.g.Gross&Mende(1988),Amati et al(1990),Aspinwall(1993)),and,from canonical quantization of4-dimensional general relativity(Ashtekar et al(1992)).The detailed pictures of the micro-structure of space-time that arise in these approaches are quite different at least atfirst sight.Nonetheless,there are certain similarities in the results;most of them are obtained by using genuinely non-perturbative techniques.The overall situation makes me believe that we have,so to say,entered a new era in thefield of quantum gravity.Through the dense fog,we have caught afleeting glimpse of the rocks that,we think,will bring us to the distant peak and,in the spirit of Heisenberg,various groups are charting their paths even though they still do not know if they will make it.In this article,I will attempt to provide a broad overview of the canonical approach based on connections and loops and suggest some directions for future work.I should emphasize that this is not a systematic survey.I will not even attempt to cover all areas in which significant developments have occurred.Rather,I will concentrate on issues which are related to the topics covered by other speakers but which were not discussed in their talks in any detail.I should also emphasize that most of what I will say are personal perspectives which are not necessarily shared by others in thefield.(For comprehensive reviews of the canonical approach based on connections and loops see other articles in this volume and,e.g.,Ashtekar(1991,1992)and Pullin(1993).)I begin in section2with an assessment of the canonical approach and,using2+1-dimensional general relativity,illustrate the vision that underlies the program.In section 3,I will summarize some recent mathematical developments which were mentioned by Bernd Br¨u gmann(1994)and Renate Loll(1994)in their talks.These results have made it possible to define the loop transform rigorously for non-Abelian connections with an infinite number of degrees of freedom and they provide a basis for regulating and solving the quantum constraints.Section4illustrates how low energy(boratory scale) physics can arise from non-perturbative quantum gravity and section5presents some promising directions for future work.2.The canonical approachThis workshop as a whole was devoted to canonical methods in general relativity,with a particular emphasis on quantization.Therefore,in thefirst part of this section,I will present an assessment of the strengths and weaknesses of these methods in general terms, without restricting myself to the choice of a specific type of dynamical variables or to the details of how the canonical program is carried out.In the second part,I will present my own view of the general framework one can hope to arrive at in quantum gravity through the use of canonical methods.2.1An assessmentAt the mathematical as well as conceptual level,the quantum theory we best un-derstand is the non-relativistic quantum mechanics of point particles.In this case,thecanonical approach provides the“royal road”to quantization.One can even formulate the fundamental kinematic problem as that of obtaining the appropriate representation of the basic canonical commutation relations.The problem of dynamics is then that ofdefining the Hamiltonian operator and of fully understanding its action on states.Note that the path integral method is not a substitute for canonical quantization.It is ratheran alternative–and,in the case of scattering problems,a powerful–method of tackling the issue of quantum dynamics.What it provides is the transition amplitudes x′,t′| x,t for the particle to go from the position x at time t to the position x′at time t′.The full quantum mechanical results,however,involve calculating the amplitude for a transitionfrom one physically realizable quantum state to another—say,from a stateΨ0( x)with angular momentum0to a stateΨm( x)with angular momentum m.And,to obtain these, we need to know the Hilbert space of states,the action of physically interesting operators and the eigenfunctions and eigenvalues of these operators.These have to be supplied ex-ternally—typically through canonical quantization.It is for this reason that most courses on non-relativistic quantum mechanics focus on the canonical method.Coming tofield theory,the canonical approach is again the oldest coherent method to tackle the problem of quantization.It is deeply rooted in the Heisenberg uncertainty relation and hence in the fundamental physical principles of quantum mechanics.Why is it then that no modern course infield theory develops the subject within the canonical method?The main reasons,I believe,are the following.First,the approach lacks manifest covariance.Second,it is not well-suited to the use of Feynman diagrams and other pertur-bative techniques.Thefirst drawback arises from obvious reasons:The basic canonically conjugatefields satisfy equal time commutation relations and are therefore defined on a Cauchy slice,whence the4-dimensional covariance is broken at the outset.The origin of the second drawback lies in the fact that the quantum states in the canonical approach arise as suitable functionals offields and this representation is quite inconvenient if one wants to work with particle states.In perturbativefield theory,on the other hand,the emphasis is on particles–real and virtual–and the powerful machinery of Feynman dia-grams is tuned to the particle picture.Furthermore,each diagram represents a space-time process.The exchange of virtual particles,particles moving forward in time and the anti-particles moving backwards,real particles scattering offeach other–each of these processes involves the passage of time and is a representation of a4-dimensional,space-time integral. It is in principle possible but in practice cumbersome to capture all this in the canonical framework.It is hard to prove renormalizability.It is much easier to forego the descrip-tions in terms of functionals of3-dimensionalfields and consider instead the Fock spaces of particle states.Indeed,this is hardly surprising.It is always the case that calculations simplify when one tailors the framework to the dynamics in question.In the perturbative treatment of scattering theory,then,simplifications should occur when we use eigenstates of the asymptotic,free Hamiltonian.And these are precisely the particle states in the Fock space.One might adopt the viewpoint that from the standpoint of what is true“in principle,”both the drawbacks stem from aesthetic considerations.Indeed,although the procedure is not manifestly covariant,it is nonetheless true that,in thefinal picture,the quantum field theory one obtains,say in the case of a free Maxwellfield,is completely equivalent to the more familiar one featuring the Fock space of photons.In the canonical approach,the states are represented by(gauge invariant)functionalsΨ(A)of the vector potential A a( x) on a3-dimensional plane(or,more generally,a Cauchy surface)in Minkowski space.This description breaks the manifest covariance.However,the Poincar´e group is unitarily imple-mented.The vacuum state is Poincar´e invariant.The spectrum of the(free)4-momentum operator is causal and future-directed,and so on.Similarly,as we noted already,using the equivalence with the Fock representation,one can translate,step by step,any pertur-bative calculation in,say,quantum electrodynamics to the canonical language.This is all true.But the price one would pay would be substantial.It would be analogous to–and much more complicated than–insisting on using Cartesian coordinates in a problem where the Hamiltonian has manifest spherical symmetry.Not only would it make calculations enormously more difficult but one would also lose all sorts of physical insights.This is not to say that canonical quantization has no place infield theory.It does.Most text books begin the discussion with canonical quantization perhaps because the underlying basic structure is clearest in this framework.Similarly,at the odd place when one is not sure of the measure in the path integral,one returns to the canonical picture and“derives the correct measure”starting with the Liouville form on the phase space.However,for the actual calculations of scattering cross sections or probing general properties such as renormalizability,the language of particles and the4-dimensional pictures are obviously better-suited.What about general relativity?The canonical approach has the great advantage that it does not require us to introduce a background metric.It is well suited for a non-perturbative treatment.Thefinal Hamiltonian framework does have a number of features not encountered in Minkowskianfield theories.However,in view of the profound conceptual differences between these theories and general relativity,the emergence of such features is but to be expected.In particular,much of the dynamical information of the theory is now contained in constraints.Indeed,as discussed by Robert Beig(1994),in the spatially compact case,the Hamiltonian vanishes identically on the constraint surface, i.e.,on the physical states of the classical theory.In quantum theory then,to begin with, there is no Hamiltonian,no time and no evolution.One only has physical states–the solutions to quantum constraints.Yet,it may be possible to“extract”dynamics from these solutions by identifying a suitable physical variable as an internal clock.Indeed, this has already been done in2+1-dimensional gravity(see e.g.Carlip(1993)).Thus,at least in principle,the approach provides us with a well-defined,precise strategy that is sufficiently sophisticated to extract physical information at the quantum level in spite of the absence of a background geometry.This is its greatest strength.How does the approach fare with respect to the two key difficulties it faces in Minkow-skian quantumfield theories?In quantum gravity,the scattering cross sections are not our prime concerns.Since there is no background space-time metric,notions such as particle states–and gravitons in particular–which are tied to the Poincar´e group would presumably be only approximate concepts.The fact that quantum general relativity(andvarious modifications thereof)fails to provide us with a consistent,local quantumfield theory perturbatively also indicates that the fundamental theory should not be formulated in terms of these particle states.The whole imagery of processes mediated by particles moving forward and backward in time is probably inappropriate.So,the second reason for abandoning the canonical method now loses its force.Indeed,the key question now is if the quantum dynamics of the gravitationalfield can be made simple in an appropriate representation within the canonical scheme.And,as we have seen in this workshop,there are indeed strong indications that the loop representation is well-suited for this purpose (see e.g.Br¨u gmann(1994)).The lack of manifest covariance of the canonical scheme,however,is still with us. And indeed it is now a much more serious issue.In Minkowskianfield theories the issue was an aesthetic one.While the canonical procedure violates manifest covariance,it is covariant;the Poincar´e group is unitarily implemented.We can,if we wish,describe dynamics in the space-time picture.In quantum gravity this is no longer the case.In classical general relativity,it is the space-time geometry that is the dynamical variable.A space-time represents a possible history–analogous to a trajectory in particle mechanics. Just as in non-relativistic quantum mechanics the particle trajectories have no basic role to play in thefinal quantum description,one would expect that in full canonical quantum gravity,space-times would have no distinguished place.There will presumably exist some special(“semi-classical”)states which can be approximated by4-dimensional space-times. Given any such state,one would be able to speak of approximate4-dimensional covariance in an effective theory which ignores large quantumflucutations away from that state. At a fundamental level,however,there would not be a4-dimensional geometric entity to replace the classical space-time.And if there is no such entity,how can one even speak of4-dimensional covariance?It does seem that there is a branching of ways here.Furthermore, it is the qualitative features of the canonical approach that force this branching;it is not tied to the use of specific variables.To preserve the space-time covariance,it is tempting to choose the path integral ap-proach.Even though in thefinished picture there is no preferred space-time,one does have an underlying4-manifold and the basic objects that one calculates with are4-metrics.And the diffeomorphism group does act on the space of these metrics.Since the Einstein-Hilbert action is diffeomorphism invariant,one would expect to get diffeomorphism invariant an-swers for(the correctly formulated)physical questions.Thus,the strategy seems attractive. There are,however,three problems.First,since path integrals in full quantum gravity are only formal expressions,ev-erything one does from the very beginning involves formal manipulations;the level of mathematical precision leaves much to be desired.In practice,one generally proceeds by making some“approximations”which mimic the successful strategies in otherfield theo-ries.However,it is not at all obvious that these strategies can be taken over to gravity in a meaningful fashion.Indeed,in Minkowskianfield theories such as quantum electrody-namics,the only way we know how to make sense of path integrals is through perturbation theory.In fact,the power of the method lies in the fact that it captures the enormous information in the perturbation expansions very succinctly.In a theory that does not exist perturbatively,we are at a loss.It would be surprising indeed if all of the approximationmethods that are useful in the perturbative context will continue to be useful in quantum gravity.A much more sophisticated approach is needed.One might imagine using the techniques developed in rigorous quantumfield theory.However,in constructive quan-tumfield theories,path integrals are defined in the Euclidean regime and the physical, Lorentzian Green’s functions are obtained through a wick rotation.We all know that,in gravity,this simple route is not available.The second problem is that while the method is manifestly covariant in the sense of classical general relativity,it lacks quantum covariance even in the case of particle dynamics.Let me explain this point in some detail since many of the readers may be unfamiliar with the issue.In the sum over histories approach,onefixes,at the outset, a preferred configuration space—all histories are to be trajectories in this space.In quantum mechanics,this corresponds tofixing a representation(such as the position or the momentum)of the observable algebra.Now,one of the technically powerful and,I feel,conceptually deep features of quantum physics is Dirac’s transformation theory†which establishes the covariance of the quantum theory under the change of representation.This covariance is manifest in the canonical approach.In the path integral approach,not only it is not manifest but it is often cumbersome to incorporate.Consequently,just as it is often hard to phrase“space-time questions”in the canonical approach,it is hard to phrase questions which involve different representations in the path integral approach (unless of course one passes through the canonical approach and makes use of the Dirac transformation theory).Consider for example an harmonic oscillator.If one wants to ask for the probability for the particle which passes through the interval∆0of the position space at time t0and the interval∆1at time t1to end up in an interval∆2at time t2, we immediately know we should integrate over the set of paths which pass through the given intervals at the given times.Now suppose we change the question somewhat by replacing the condition at the intermediate time t1and ask instead that,at that time,the particle have an energy¯hω(n+1†about which Dirac said in1977:“I think that is the piece of work which has most pleased me of all the works that I have done in my life...The transformation theory became my darling.”(Pais(1987))ultimately the two will be used together so that the framework has both classical and quantum covariance.The dynamics coded in the quantum constraints in the canonical approach could perhaps be re-interpreted in terms of mathematically well-defined path integrals to demonstrate that there is a well-defined sense in which the theory enjoys space-time covariance.Such an interpretation would be especially helpful in the analy-sis of whether topology changes occur in quantum gravity and,if so,whether they have physically significant ramifications in the low energy regime.The full quantum covariance could be established by switching to the canonical framework.It may also turn out,however,that in the gravitational case,canonical quantization can not be reconciled with path integrals in the full quantum theory.As the discussion in the next sub-section indicates,even the“spatial”3-manifold could be a secondary con-struct.At the fundamental level,there may only be discrete structures and combinatorial operations.In such a scenario,the continuum picture and objects such as manifolds may arise as useful mathematical constructs only in semi-classical physics.And the theory would have the desired covariance only in these regimes.It may be that the predictions of the two methods agree in such regimes but in the fully Planckian domain,the two theories are quite different.To summarize then,the canonical methods are well adapted to non-perturbative treat-ments of quantum gravity.However,there appears to be a fundamental tension between the quantum gravity scenarios that are natural to canonical approaches and space-time pictures that we are so accustomed to in the classical regime.Whether this tension is real or only apparent is not yet clear.Technical progress,particularly in the path integral approach,would be of great help to settle this issue.2.2An underlying visionIn this workshop,Thomas Thiemann discussed2+1-dimensional gravity in some detail and Hermann Nicolai presented an extension(due to de Wit,Matschull and himself(1993)) of those results to give an elegant clarification of the issue of the“size”of the space of physical states in supergravity.I would now like to use2+1gravity for a different purpose: to illustrate my own expectations of non-perturbative quantum gravity in3+1dimensions. (For details on2+1-dimensional gravity,in addition to these proceedings,see,e.g.,Carlip (1990,1993),and Ashtekar(1991),chapter17.)Let me begin with a brief historical detour.Conceptually–and,in certain respects, also technically–2+1-dimensional general relativity is very similar to the3+1-dimensional theory.There is no background structure;the theory is diffeomorphism invariant.One and the same object–the space-time metric gµν–determines the geometry and encodes the gravitationalfield.In the canonical description,the Hamiltonian again vanishes in the spatially compact case and the dynamics is driven byfirst class constraints.Furthermore, in geometrodynamics,the general structure of the constraint algebra is the same as in the 3+1theory(see,e.g.,Beig1994);it is not a true Lie algebra.Consequently,for a number of years,it was believed that a non-perturbative,canonical quantization of this theory is as difficult as that of the3+1-dimensional theory.In particular,not a single solution to the Wheeler-DeWitt equation was known.In the perturbative treatment,onefinds that Newton’s constant is again dimensionful(with dimension[L],thus a positive power of length in the¯h=c=1units)and simple power counting arguments suggested that thetheory is not renormalizable.Therefore,there appeared several papers in the literature saying that2+1-dimensional general relativity is as intractable as the3+1-dimensional theory.This would have been somewhat puzzling since in2+1-dimensions there are no local degrees of freedom;all solutions to the Einsteinfield equations areflat.Indeed,the conclusions of these papers turned out to be wrong.There are no ing connections as basic variables rather than metrics,the quantum theory can in fact be solved exactly in the spatially compact case for any genus.(Achucarro&Townsend(1986), Witten(1988).See also Ashtekar et al(1989)and Nelson&Regge(1989)).Since then, the theory has also been solved in the metric representation in the case when the spatial topology is that of a2-torus(Moncrief1989,Hosoya&Nakao(1990))and work is in progress on the higher genus case.One of the lessons that one can draw from this work is that connections and holonomies are better suited to the mathematics of the quantum theory than metrics and light cones.It is not that the theory can not be solved in the metric picture;as I just remarked,this has already been achieved in the2-torus topology. Rather,connections seem to be better adapted to deal with the quantum constraints and to write down Dirac observables.Variables which are most useful to the macro-physics are not the ones which make the micro-physics most transparent.There are several equivalent ways of quantizing2+1gravity on a torus(see,e.g., Carlip1993).The variety of representations that result provide us with a rich example of the Dirac transformation theory in action.Here,I will consider three and argue that each clarifies and emphasizes a different aspect of quantum gravity.Atfirst sight,it appears that each representation provides its own,distinct picture of reality.And yet,thanks to the Dirac transformation theory,they are all equivalent.First,there is the loop representation.Here,quantum states are functions of homotopy classes of loops on the spatial2-manifold;the domain space of quantum states is thus discrete.In the case of a torus,each homotopy class is represented by a pair of integers (n1,n2)which tell us how many times the loop winds around the two generators of the homotopy group.Hence,in this case,the quantum statesΨ(n1,n2)are functions just of two integers.The basic observables of the theory,T0(n1,n2)and T1(n1,n2),are also labelled by two integers and their action on states just shifts the arguments of the wave functions.Thus,the whole mathematical structure is combinatorial.There is no space,no time,no continuum.Hence there is no diffeomorphism group to implement and no issue of space-time covariance to worry about.I would like to regard this as the“fundamental description.”Now,we learnt from Thomas Thiemann’s talk(1994)that this description is completely equivalent to that in terms of connections on a2-torus†.Put differently,suppose we knew nothing about general relativity but were provided with just the combinatorial description given above.Staring at this description,a clever young student could have realized that the description would become“nice and geometric”if one were to introduce a。

圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of two sheets柱面cylindrical surface ,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general form equations of a space curve空间曲线的参数方程parametric equations of a space curve螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角anglebetween two lines垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Severa l Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point,boundary point聚点point of accumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative梯度gradient数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程solve a dirrerential equation常微分方程ordinary differential equation偏微分方程partial differential equation,PDE微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation初值问题initial value problem微分方程的积分曲线integral curve of a differential equation可分离变量的微分方程variable separable differential equation隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant暂态电流transient stata current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linera differential equation of higher order自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linera differential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程second order homogeneour lineardifferential equation withconstant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping固有频率natural frequency简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linera differential equations with constant coefficient。

SIAM J.M ATH.A NAL.c 2006Society for Industrial and Applied Mathematics Vol.37,No.5,pp.1417–1434A BLOW-UP CRITERION FOR THE NONHOMOGENEOUSINCOMPRESSIBLE NA VIER–STOKES EQUATIONS∗HYUNSEOK KIM†Abstract.Let(ρ,u)be a strong or smooth solution of the nonhomogeneous incompressible Navier–Stokes equations in(0,T∗)×Ω,where T∗is afinite positive time andΩis a bounded domain in R3with smooth boundary or the whole space R3.We show that if(ρ,u)blows up at T∗,thenT∗0|u(t)|sL r w(Ω)dt=∞for any(r,s)with2s+3r=1and3<r≤∞.As immediate applications,we obtain a regularity theorem and a global existence theorem for strong solutions.Key words.blow-up criterion,nonhomogeneous incompressible Navier–Stokes equations AMS subject classifications.35Q30,76D05DOI.10.1137/S00361410044421971.Introduction.The motion of a nonhomogeneous incompressible viscousfluid in a domainΩof R3is governed by the Navier–Stokes equations(ρu)t+div(ρu⊗u)−Δu+∇p=ρf,(1.1)ρt+div(ρu)=0in(0,∞)×Ω,(1.2)div u=0(1.3)and the initial and boundary conditions(ρ,ρu)|t=0=(ρ0,ρ0u0)inΩ,u=0on(0,T)×∂Ω,(1.4)ρ(t,x)→0,u(t,x)→0as|x|→∞,(t,x)∈(0,T)×Ω.Here we denote byρ,u,and p the unknown density,velocity,and pressurefields of thefluid,respectively.f is a given external force driving the motion.Ωis either a bounded domain in R3with smooth boundary or the whole space R3.Throughout this paper,we adopt the following simplified notation for standard homogeneous and inhomogeneous Sobolev spaces:L r=L r(Ω),D k,r={v∈L1loc(Ω):|v|D k,r<∞};H k,r=L r∩D k,r,D k=D k,2,H k=H k,2;D10={v∈L6:|v|D1<∞and v=0on∂Ω};H10=L2∩D10,,D10,σ={v∈D10:div v=0inΩ};|v|D k,r=|∇k v|L r and|v|D10=|v|D10,σ=|∇v|L2.Note that the space D10is the completion of C∞c(Ω)in D1,and thus there holds the following Sobolev inequality:|v|L6≤2√3|v|D1for all v∈D10.(1.5)∗Received by the editors March17,2004;accepted for publication(in revised form)May9,2005; published electronically January10,2006.This work was supported by the Japan Society for the Promotion of Science under the JSPS Postdoctoral Fellowship for Foreign Researchers./journals/sima/37-5/44219.html†School of Mathematics,Korea Institute for Advanced Study207-43Cheongnyangni2-dong, Dongdaemun-gu,Seoul,130-722,Korea(khs319@postech.ac.kr,khs319@kias.re.kr).14171418HYUNSEOK KIMFor a proof of (1.5),see sections II.5and II.6in the book by Galdi [11].The global existence of weak solutions has been established by Antontsev and Kazhikov [1],Fernandez-Cara and Guillen [10],Kazhikov [13],Lions [21],and Simon [26,27].From these results (see [21]in particular),it follows that for any data (ρ0,u 0,f )with the regularity0≤ρ0∈L 32∩L ∞,u 0∈L 6,and f ∈L 2(0,∞;L 2),there exists at least one weak solution (ρ,u )to the initial boundary value problem (1.1)–(1.4)satisfying the regularity ρ∈L ∞(0,∞;L 32∩L ∞),√ρu ∈L ∞(0,∞;L 2),and u ∈L 2(0,∞;D 10,σ)(1.6)as well as the natural energy inequality.Then an associated pressure p is determinedas a distribution in (0,∞)×Ω.But the global existence of strong or smooth solutions is still an open problem and only local existence results have been obtained for sufficiently regular data satisfying some compatibility conditions.For details,we refer to the papers by Choe and the author [6],Kim [15],Ladyzhenskaya and Solonnikov [19],Okamoto [22],Padula [23],and Salvi [24].In particular,it is shown in [6](see also [7])that if the data ρ0,u 0,and f satisfy 0≤ρ0∈L 32∩H 2,u 0∈D 10,σ∩D 2,−Δu 0+∇p 0=ρ120g,(1.7)f ∈L 2(0,∞;H 1),and f t ∈L 2(0,∞;L 2)for some (p 0,g )∈D 1×L 2,then there exist a positive time T and a unique strongsolution (ρ,u )to the problem (1.1)–(1.4)such that ρ∈C ([0,T ];L 32∩H 2),u ∈C ([0,T ];D 10,σ∩D 2)∩L 2(0,T ;D 3),(1.8)u t ∈L 2(0,T ;D 10,σ),and √ρu t ∈L ∞(0,T ;L 2).Moreover,the existence of a pressure p in C ([0,T ];D 1)∩L 2(0,T ;D 2)can be deducedfrom (1.1)–(1.3).See [5]for a detailed argument.Let (ρ,u )be a global weak solution to the problem (1.1)–(1.4)with the data (ρ0,u 0,f )satisfying condition (1.7).Then from the above local existence result and weak-strong uniqueness results in [6]and [21],we conclude that the solution (ρ,u )satisfies the regularity (1.8)for some positive time T .One fundamental problem in mathematical fluid mechanics is to determine whether or not (ρ,u )satisfies (1.8)for all time T .As an equivalent formulation,we may ask the following.Fundamental question 1.1.Does the solution (ρ,u )blow up at some finite time T ∗?Such a time T ∗,if it exists,is called the finite blow-up time of the solution (ρ,u )in the class H 2.In spite of great efforts since the pioneering works by Leray [20]in 1930s,there have been no definite answers to the fundamental question even for the case of the homogenous Navier–Stokes equations with only some blow-up criteria available.The first criterion is due to Leray [20]who proved,among other things,that if T ∗is the finite blow-up time of a strong solution u to the Cauchy problem for the homogeneous Navier–Stokes equations,then for each r with 3<r ≤∞,there exists a constant C =C (r )>0such that |u (t )|L r ≥C (T ∗−t )−12(1−3r )for all near t <T ∗.(1.9)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1419This estimate near the blow-up time was extended by Giga [12]to the case of bounded domains.An immediate consequence of (1.9)is the following well-known blow-up criterion in terms of the so-called Serrin class (see [9,25,29]):T ∗0|u (t )|s L r dt =∞for any (r,s )with 2s +3r =1,3<r ≤∞.(1.10)By virtue of Sobolev inequality (1.5),we deduce from (1.10)thatT ∗|∇u (t )|4L 2dt =∞.(1.11)Further generalizations of (1.10)and (1.11)have been obtained by Beirao da Veiga [2],Berselli [3],Chae and Choe [4],and Kozono and Taniuchi [17].The major purpose of this paper is to prove the blow-up criterion (1.10)for strong solutions of the nonhomogeneous Navier–Stokes equations (1.1)–(1.3).In fact,we establish a more general result.To state our main result precisely,we first introduce the notion of the blow-up time of solutions in the class H 2m with m ≥1.Let (ρ,u )be a strong solution to the initial boundary value problem (1.1)–(1.4)with the regularityρ∈C ([0,T ];L 32∩H 2m ),u ∈C ([0,T ];D 10,σ∩D2m)∩L 2(0,T ;D 2m +1),∂j t u ∈C ([0,T ];D 10,σ∩D2m −2j)∩L 2(0,T ;D 2m −2j +1)for 1≤j <m,(1.12)∂m t u ∈L 2(0,T ;D 10,σ),and √ρ∂m t u ∈L ∞(0,T ;L 2)for any T <T ∗,where T ∗is a finite positive time.Then we can defineΦm (T )=1+sup 0≤t ≤T|∇ρ(t )|H 2m −1+|u (t )|D 10∩D 2m + T|u (t )|2D 2m +1dt+sup1≤j<msup 0≤t ≤T|∂jt u (t )|D 10∩D 2m −2j +T|∂jt u (t )|2D 2m −2j +1dt(1.13)+ess sup 0≤t ≤T|√ρ∂mtu (t )|L 2+T|∂mt u (t )|2D 10dtfor any T <T ∗.Hereafter we use the obvious notation|·|X ∩Y =|·|X +|·|Yfor (semi-)normed spaces X,Y.Definition 1.2.A finite positive number T ∗is called the finite blow-up time ofthe solution (ρ,u )in the class H 2m ,provided thatΦm (T )<∞for0<T <T ∗andlim T →T ∗Φm (T )=∞.We are now ready to state the main result of this paper.Theorem 1.3.For a given integer m ≥1,we assume that∂mtf ∈L 2(0,∞;L 2)and∂jt f ∈L 2(0,∞;H 2m −2j −1)for 0≤j <m.Let (ρ,u )be a strong solution of the nonhomogeneous Navier–Stokes equations (1.1)–(1.3)satisfying the regularity (1.12)for any T <T ∗.If T ∗is the finite blow-up time of (ρ,u )in the class H 2m ,then we haveT ∗|u (t )|s L r wdt =∞for any (r,s )with 2s +3r =1,3<r ≤∞.(1.14)1420HYUNSEOK KIMHere L r w denotes the weak L r-space,that is,the space consisting of all vector fields v ∈L 1loc (Ω)such that |v |L r w=sup α>0α|{x ∈Ω:|v (x )|>α}|1r <∞for 3<r <∞and |v |L ∞w=|v |L ∞<∞.In the case when 3<r <∞,L ris a proper subspace of L r w (|x |−3/r ∈L r w (R 3)for instance)and so Theorem 1.3is in fact a generalizationof the blow-up criterion (1.10)due to Leray and Giga even for the homogeneous Navier–Stokes equations.Theorem 1.3is proved in the next two sections.In section 2,we provide a proof of the theorem for the very special case m =1.Our method of the proof is quite well known in the case of the homogeneous Navier–Stokes equations and was also applied in an earlier paper [6]by Choe and the author to the nonhomogeneous case:combining classical regularity results on the Stokes equations with H¨o lder and Sobolev inequal-ities,we show that Φ1(T )is bounded in a double exponential way by T0|u (t )|s L r wdt for any T less than the blow-up time T ∗.But the use of weak Lebesgue spaces in space variables makes it more difficult to estimate the nonlinear convection term.To overcome this technical difficulty,we utilize some basic theory of the Lorenz spaces developed in [18]and [30].See the derivations of (2.6)and (2.7)for details.Concern-ing the proof for the general case m ≥2,the basic idea is also to show that Φm (T )isbounded in some specific way by T0|u (t )|s L r wdt for any T <T ∗.Such an approach is more or less standard in the case of the homogeneous Navier–Stokes equations,but its extension to the nonhomogeneous case is not straightforward and indeed much complicated due to the evolution of the density.A detailed argument is provided in section 3.Some corollaries of Theorem 1.3can be deduced from a local existence result on strong solutions in the class H 2m .For instance,as an immediate consequence of Theorem 1.3,the local existence result in the class H 2,and the weak-strong uniqueness result,we obtain the following regularity result whose obvious proof may be omitted.Corollary 1.4.Let (ρ,u )be a global weak solution to the initial boundary value problem (1.1)–(1.4)with the data satisfying condition (1.7).If there exists a finite positive time T ∗such thatu ∈L s (0,T ∗;L r w )for some (r,s )with2s +3r=1,3<r ≤∞,(1.15)then the solution (ρ,u )satisfies regularity (1.8)for some T >T ∗.A similar result was obtained by Choe and the author [6]assuming,however,astronger condition on u ,that is,u ∈L 4(0,T ∗;D 10).By virtue of Corollary 1.4,we may conclude that the class (which we call a weak Serrin class )in (1.15)is a regularity class for weak solutions of the nonhomogeneous Navier–Stokes equations,which was already proved by Sohr [28]for the homogeneous case.See also a local version of Sohr’s result in [14].Moreover,thanks to a recent result by Dubois [8],the weak Serrin class is a uniqueness class for the homogeneous Navier–Stokes equations.It is also noticed that the same results can be easily derived from regularity and uniqueness results due to Kozono by adapting the arguments in the remarks of Theorem 3in [16].Theorem 1.3and its proof can be used to obtain a global existence result on solutions in the class H 2under some smallness condition on u 0and f (but not on ρ0).Theorem 1.5.For each K >1,there exists a small constant ε>0,depending only on K and the domain Ω,with the following property:if the data ρ0,u 0,and f satisfy|ρ0|L 32∩L∞≤K,|u 0|D 10≤ε,and ∞|f (t )|2L 2dt ≤ε2(1.16)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1421in addition to condition (1.7),then there exists a unique global strong solution (ρ,u )to problem (1.1)–(1.4)satisfying regularity (1.8)for any T <∞.A rather simple proof of Theorem 1.5is provided in section 4.Finally,we recall that in the case when ρ0is bounded away from zero and Ωis a bounded domain in R 3with smooth boundary,Salvi [24]proved the local existence of strong solutions in the class H 2m for every m ≥1.Hence adapting the proofs of Corollary 1.4and Theorem 1.5,we can also prove analogous regularity and global existence results on strong solutions in every class H 2m provided that Ω⊂⊂R 3and ρ0>0on Ω.2.Proof of Theorem 1.3with m =1.In this section,we prove Theorem 1.3for the special case m =1.Let t 0be a fixed time with 0<t 0<T ∗and let us denoteΦ0(T )= T|u (t )|s L r wdt for t 0≤T <T ∗,where (r,s )is any pair satisfying condition (1.14).To prove Theorem 1.3,we haveonly to show that Φ1(T )≤C exp (C exp (C Φ0(T )))for t 0≤T <T ∗.(2.1)Throughout this paper,we denote by C a generic positive constant depending onlyon r ,m ,Φm (t 0),T ∗,Ω,|ρ(0)|L 32∩L ∞,|u (0)|L 6,and the norm of f ,but independent of T in particular.To begin with,we recall from (1.6)that sup0≤t ≤T|ρ(t )|L 32∩L ∞+|√ρu (t )|L 2 +T|u (t )|2D 10dt ≤C(2.2)for t 0≤T <T ∗.2.1.Estimates for T 0|√ρu t (t )|2L 2dt and sup 0≤t ≤T |u (t )|D 10.Next,we willshow thatT 0|√ρu t (t )|2L 2+|u (t )|2D 10∩D 2 dt +sup 0≤t ≤T|u (t )|2D 10≤C exp (C Φ0(T ))(2.3)for t 0≤T <T ∗.To show this,we multiply the momentum equation (1.1)by u t andintegrate over Ω.Then using (1.2)and (1.3),we easily deriveρ|u t |2dx +12d dt |∇u |2dx = ρ(f −u ·∇u )·u t dx andρ|u t |2dx +ddt|∇u |2dx ≤2ρ|f |2dx +2ρ|u ·∇u |2dx.(2.4)On the other hand,since (u,p )is a solution of the stationary Stokes equations−Δu +∇p =Fanddiv u =0inΩ,where F =ρ(f −u ·∇u −u t ),it follows from the classical regularity theory that |∇u |H 1≤C (|F |L 2+|∇u |L 2)(2.5)≤C (|f |L 2+|√ρu t |L 2+|u ·∇u |L 2+|∇u |L 2).1422HYUNSEOK KIMTo estimate the right-hand sides of(2.4)and(2.5),wefirst observe that|u·∇u|L2=|u·∇u|L2,2≤C|u|L rw |∇u|L2rr−2,2,(2.6)which follows from H¨o lder inequality in the Lorenz spaces.See Proposition2.1in[18]. Next,we will show that|∇u|L2rr−2,2≤C|∇u|1−3rL2|∇u|3rH1.(2.7)If r=∞,then(2.7)is obvious.Assuming that3<r<∞,we choose r1and r2suchthat3<r1<r<r2<∞and2r =1r1+1r2.Then in view of H¨o lder and Sobolevinequalities,we have|∇u|L2r ir i−2≤|∇u|1−3r iL2|∇u|3r iL6≤|∇u|1−3r iL2(C|∇u|H1)3r ifor each i=1,2.Since L2r r−2,2is a real interpolation space of L2r2r2−2and L2r1r1−2,moreprecisely,L2r r−2,2=(L2r2r2−2,L2r1r1−2)12,2,it thus follows that|∇u|L2rr−2,2≤C|∇u|12L2r2r2−2|∇u|12L2r1r1−2≤C|∇u|1−3r2L2(C|∇u|H1)3r212|∇u|1−3r1L2(C|∇u|H1)3r112,which proves(2.7).For some facts on the real interpolation theory and Lorenz spaces used above,we refer to sections1.3.3and1.18.6in Triebel’s book[30].The estimates(2.6)and(2.7)yield|u·∇u|L2≤C|u|L rw |∇u|2sL2|∇u|3rH1≤η−3s2r C|u|s2L rw|∇u|L2+η|∇u|H1for any small numberη∈(0,1).Substituting this into(2.5),we obtain|∇u|H1≤C|f|L2+|√ρu t|L2+|u|s2L rw|∇u|L2+|∇u|L2,(2.8)and thus|u·∇u|L2≤η−3s2r C|u|s2L rw+1|∇u|L2+C|f|L2+η|√ρu t|L2.Therefore,substituting this estimate into(2.4)and choosing a sufficiently smallη>0, we conclude that1 2|√ρu t(t)|2L2+ddt|∇u(t)|2L2≤C|f(t)|2L2+C|u(t)|s L rw+1|∇u(t)|2L2(2.9)for t0≤t<T∗.In view of Gronwall’s inequality,we haveT0|√ρu t(t)|2L2dt+sup0≤t≤T|∇u(t)|2L2≤C exp(CΦ0(T))for any T with t0≤T<T∗.Combining this and(2.8),we obtain the desired estimate (2.3).NONHOMOGENEOUS NAVIER–STOKES EQUATIONS14232.2.Estimates for ess sup 0≤t ≤T |√ρu t (t )|2L 2and T 0|u t (t )|2D 10dt .To de-rive these estimates,we differentiate the momentum equation (1.1)with respect to time t and obtainρu tt +ρu ·∇u t −Δu t +∇p t =ρt (f −u t −u ·∇u )+ρ(f t −u t ·∇u ).Then multiplying this by u t ,integrating over Ω,and using (1.2)and (1.3),we have12d dtρ|u t |2dx + |∇u t |2dx(2.10)=(ρt (f −u t −u ·∇u )+ρ(f t −u t ·∇u ))·u t dx.Note that since ρ∈C ([0,T ];L 32∩L ∞),ρt ∈C ([0,T ];L 32),and u t ∈L 2(0,T ;D 10)forany T <T ∗,the right-hand side of (2.10)is well defined for almost all t ∈(0,T ∗).Hence using finite differences in time,we can easily show that the identity (2.10)holds for almost all t ∈(0,T ∗).In view of the continuity equation (1.2)again,we deduce from (2.10)that12ddtρ|u t |2dx + |∇u t |2dx≤2ρ|u ||u t ||∇u t |+ρ|u ||u t ||∇u |2+ρ|u |2|u t ||∇2u |(2.11)+ρ|u |2|∇u ||∇u t |+ρ|u t |2|∇u |+ρ|u ||u t ||∇f |+ρ|u ||f ||∇u t |+ρ|f t ||u t |dx ≡8 j =1I j .Following the arguments in [6],we can estimate each term I j :I 1,I 5≤C |ρ|12L ∞|∇u |L 2|√ρu t |L 3|∇u t |L 2≤C |ρ|34L ∞|∇u |L 2|√ρu t |12L 2|∇u t |32L 2≤C |∇u |4L 2|√ρu t |2L 2+116|∇u t |2L 2,I 2,I 3,I 4≤C |ρ|L ∞|∇u |2L 2|∇u t |L 2|∇u |H 1≤C |∇u |4L 2|∇u |2H 1+116|∇u t |2L 2,I 6,I 7≤C |ρ|L 6|∇u |L 2|f |H 1|∇u t |L 2≤C |∇u |2L 2|f |2H 1+116|∇u t |2L 2,and finallyI 8≤C |ρ|L 3|f t |L 2|∇u t |L 2≤C |f t |2L 2+116|∇u t |2L 2.Substitution of these estimates into (2.11)yieldsd dt |√ρu t |2L 2+|∇u t |2L 2≤C |∇u |4L 2 |√ρu t |2L 2+|∇u |2H 1+|f |2H 1 +C |f |2H 1+|f t |2L 2 .1424HYUNSEOK KIMTherefore,by virtue of estimate (2.3),we conclude that ess sup 0≤t ≤T|√ρu t (t )|2L 2+T|∇u t (t )|2L 2dt ≤C exp (C Φ0(T ))(2.12)for t 0≤T <T ∗.On the other hand,using the regularity theory of the Stokesequations again,we have|∇u |H 1≤C (|f |L 2+|√ρu t |L 2+|u ·∇u |L 2+|∇u |L 2)≤C |f |L 2+|√ρu t |L 2+|∇u |32L 2|∇u |12H 1+|∇u |L 2and|∇u |H 1,6≤C (|u t |L 6+|u ·∇u |L 6+|f |L 6+|∇u |L 6)≤C|∇u t |L 2+|∇u |2H1+|f |H 1+|∇u |H 1 .Hence it follows immediately from (2.3)and (2.12)that sup0≤t ≤T|u (t )|2D 10∩D 2+T|∇u (t )|2H 1,6dt ≤C exp (C Φ0(T ))(2.13)for t 0≤T <T ∗.2.3.Estimates for sup 0≤t ≤T |∇ρ(t )|H 1and T0|u (t )|2D 3dt .To derive these,we first observe that each ρx j (j =1,2,3)satisfiesρx jt +u ·∇ρx j =−u x j ·∇ρ.Then multiplying this by ρx j ,integrating over Ω,and summing up,we obtaind dt|∇ρ|2dx ≤C|∇u ||∇ρ|2dx ≤C |∇u |L ∞|∇ρ|2L 2.A similar argument shows thatddt|∇2ρ|2dx ≤C|∇u ||∇2ρ|2+|∇2u ||∇ρ||∇2ρ| dx ≤C |∇u |L ∞|∇2ρ|2L 2+|∇2u |L 6|∇ρ|L 3|∇2ρ|L 2.Hence using Sobolev embedding results and then Gronwall’s inequality,we derive thewell-known estimate|∇ρ(t )|H 1≤C exp C t 0|∇u (τ)|H 1,6dτ≤C exp tC |∇u (τ)|2H 1,6+1 dτ .Therefore by virtue of (2.13),we conclude that sup 0≤t ≤T|∇ρ(t )|H 1≤C exp (C exp (C Φ0(T )))(2.14)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1425for t 0≤T <T ∗.Finally,observing from the regularity theory on the Stokes equations that|u |D 3≤C (|ρu t |H 1+|ρu ·∇u |H 1+|ρf |H 1)≤C (|∇ρ|L 3+1) |∇u t |L 2+|∇u |2H 1+|f |H 1 ,we easily deduce from (2.3),(2.12),(2.13),and (2.14)thatT|u (t )|2D 3dt ≤C exp (C exp (C Φ0(T )))(2.15)for t 0≤T <T ∗.This completes the proof of (2.1)and thus the proof of Theorem 1.3with m =1.3.Proof of Theorem 1.3with m ≥ 2.Assume that m ≥ 2.Then to prove Theorem 1.3,it suffices to show that the following estimate holds for each k ,0≤k <m :Φk +1(T )≤C exp C expC Φk (T )10m for t 0≤T <T ∗.(3.1)The case k =0was already proved in section 2and so it remains to prove (3.1)forthe case 1≤k <m .Let k be a fixed integer with 1≤k <m .From (1.13),we recall thatΦk (T )=1+sup0≤j<ksup0≤t ≤T|∂jt u (t )|D 10∩D 2k −2j +T|∂jt u (t )|2D 2k −2j +1dt(3.2)+sup 0≤t ≤T|∇ρ(t )|H 2k −1+ess sup 0≤t ≤T|√ρ∂kt u (t )|L 2+T|∂kt u (t )|2D 10dtfor any T <T ∗.3.1.Estimates for ∂j t(u ·∇u ),∂j +1t ρ,and ∂jt (ρu )with 0≤j ≤k .To estimate nonlinear terms,we will make repeated use of the following simple lemma whose proof is omitted.Lemma 3.1.If g ∈D 10∩D j ,h ∈H i ,0≤i ≤j ,and j ≥2,thengh ∈H iand|gh |H i ≤C |g |D 10∩D j |h |H i for some constant C >0depending only on j and Ω.Using this lemma together with the fact that∂j t (u·∇u )=j i =0j !i !(j −i )!∂itu ·∇∂j −i t u,we can estimate ∂jt(u ·∇u )as follows:for 0≤j <k ,|∂jt(u ·∇u )|H 2k −2j −1≤Cj i =0|∂it u ·∇∂j −i t u |H 2k −2j −1≤C j i =0|∂it u |D 10∩D 2k −2j |∇∂j −i tu |H 2k −2j −1≤Cj i =0|∂it u |D 10∩D 2k −2i |∂j −i tu |D 10∩D 2k −2(j −i )1426HYUNSEOK KIM and|∂k t(u·∇u)|L2≤C k−1i=0|∂i t u·∇∂k−itu|L2+|∂k t u·∇u|L2≤C k−1i=0|∂i t u|D1∩D2|∇∂k−itu|L2+|∂k t u|D1|∇u|H1≤C k−1i=0|∂i t u|D1∩D2k−2i|∂k−itu|D1+|∂k t u|D1|u|D1∩D2.Hence it follows from(3.2)thatsup 0≤j<ksup0≤t≤T|∂j t(u·∇u)(t)|H2k−2j−1+T|∂k t(u·∇u)(t)|2L2dt≤CΦk(T)4(3.3)for t0≤T<T∗.Applying Lemma3.1to the continuity equationρt=−div(ρu)=−u·∇ρ,(3.4)we also deduce thatsup 0≤t≤T |ρt(t)|H2k−1≤CΦk(T)2for t0≤T<T∗.(3.5)Using(3.4)and(3.5),we can show thatsup 1≤j<ksup0≤t≤T|∂j+1tρ(t)|H2k−2j+T|∂k+1tρ(t)|2L2dt≤CΦk(T)2k+4(3.6)for t0≤T<T∗.A simple inductive proof of(3.6)may be based on the observation that for1≤j<k,|∂j+1tρ|H2k−2j=|−∂j t(u·∇ρ)|H2k−2j≤Cji=0|∂j−itu·∇∂i tρ|H2k−2j≤Cji=0|∂j−itu|D1∩D2k−2j|∇∂itρ|H2k−2j≤Cji=0|∂j−itu|D1∩D2k−2(j−i)|∂itρ|H2k−2(i−1)and|∂k+1t ρ|L2≤Cki=0|∂k−itu·∇∂i tρ|L2≤Cki=0|∂k−itu|D1|∂i tρ|H2.Moreover,it follows easily from(3.6)thatsup 0≤j<ksup0≤t≤T|∂j t(ρu)(t)|H2k−2j+T|∂k t(ρu)(t)|2H1dt≤CΦk(T)4k+10(3.7)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1427for t 0≤T <T ∗.Finally,recalling that∂k +1t f ∈L 2(0,∞;L 2)and ∂jt f ∈L 2(0,∞;H 2k −2j +1)for 0≤j ≤k,we deduce from standard embedding results that∂jt f ∈C ([0,∞);H 2k −2j )for0≤j ≤k.3.2.Estimates for T 0|√ρ∂k +1t u (t )|2L 2dt and sup 0≤t ≤T |∂k t u (t )|D 10.Fromthe momentum equation (1.1),we deriveρ ∂k t u t −Δ∂k t u +∇∂k t p =∂k t (ρf −ρu ·∇u )+ ρ∂k t u t −∂k t (ρu t ) .Hence multiplying this by ∂k +1t u and integrating over Ω,we haveρ|∂k +1t u |2dx +12d dt|∇∂k t u |2dx=∂k t (ρf −ρu ·∇u )+ ρ∂k t u t −∂kt (ρu t ) ·∂k +1tu dx (3.8)=I 0,1+k j =1k !j !(k −j )!(I j,1+I j,2),whereI j,1=∂j t ρ∂k −j t (f −u ·∇u )·∂k +1t u dx,I j,2=−∂j t ρ∂k −j t u t ·∂k +1t u dx.We easily estimate I 0,1as follows:I 0,1≤|ρ|12L ∞ |∂k t f |L 2+|∂k t (u ·∇u )|L 2 |√ρ∂k +1t u |L 2≤C |∂k t f |2L 2+|∂kt (u ·∇u )|2L2 +12|√ρ∂k +1t u |2L 2.To estimate I j,1for 1≤j ≤k ,we rewrite it asI j,1=d dt∂j t ρ∂k −j t (f −u ·∇u )·∂k t u dx − ∂j +1t ρ∂k −j t (f −u ·∇u )·∂kt u dx −∂j t ρ∂k −j +1t (f −u ·∇u )·∂kt u dx and observe that −∂j +1tρ∂k −j t (f −u ·∇u )·∂kt u dx ≤C |∂k −j t f |2H 1+|∂k −j t (u ·∇u )|2H 1 |∂j +1t ρ|2L 2+|∇∂k t u |2L 2and−∂j tρ∂k −j +1t (f −u ·∇u )·∂k t u dx ≤C |∂j t ρ|2H 1 |∂k −j +1t f |2L 2+|∂k −j +1t (u ·∇u )|2L 2 +|∇∂k t u |2L 2.1428HYUNSEOK KIMUsing the continuity equation (1.2),we can also estimate I j,2as follows:I 1,2=− ρt 12|∂k t u |2 t dx =−d dt ρt 12|∂k t u |2dx + ∂2t ρ12|∂k t u |2dx =−d dt ρu ·∇ 12|∂k t u |2 dx + ∂t (ρu )·∇ 12|∂k tu |2dx ≤−d dt (ρu ·∇∂k t u )·∂k t u dx +C |∂t (ρu )|H 1|∇∂k t u |2L 2and similarlyI j,2=−d dt ∂j t ρ∂k −j t u t ·∂k t u dx + ∂j +1t ρ∂k −j t u t +∂j t ρ∂k −j +1t u t ·∂kt u dx ≤−d dt ∂j −1t (ρu )·∇ ∂k −j +1t u ·∂kt u dx +C |∂j t (ρu )|H 1|∇∂k −j +1t u |L 2+|∂j −1t (ρu )|H 1|∇∂k −j +2t u |L 2 |∇∂kt u |L 2for 2≤j ≤k .Substituting all the estimates into (3.8),we have12 ρ|∂k +1t u |2dx +12d dt|∇∂k t u |2dx ≤d dt ⎛⎝k j =1k !j !(k −j )!∂j t ρ∂k −j t (f −u ·∇u )·∂k t u −(ρu ·∇∂k t u )·∂k t u ⎞⎠dx −d dt kj =2k !j !(k −j )!∂j −1t(ρu )·∇ ∂k −j +1t u ·∂kt u dx+Ck −1 j =1|∂jt (ρu )|2H 1+|∂j +1t ρ|2H 1|∂k −j t f |2H 1+|∂k −jt (u·∇u )|2H 1+|∂k −j +1t u |2D 1+C |∂k +1t ρ|2L 2|f |2H 1+|u |2D 10∩D 2+C 1+|∂t ρ|2H 1 |∂k t f |2L 2+|∂k t (u ·∇u )|2L2 +C |∂k t (ρu )|2H 1+C 1+|∂t (ρu )|2H 1+|∇∂t u |2L 2 |∇∂k t u |2L 2.Hence,integrating this in time over (t 0,T )and using (3.3),(3.5),(3.6),and (3.7)together with the estimates|∂j t ρ||∂k −j t (f −u ·∇u )||∂k t u |dx≤η−1|∂j t ρ|2H 1|∂k −j t (f −u ·∇u )|2L 2+η|∇∂k t u |L 2,ρ|u ||∇∂k t u ||∂k t u |dx ≤η−3C |ρ|3L ∞|∇u |4L 2|√ρ∂k t u |2L 2+η|∇∂k t u |2L 2and|∂j −1t (ρu )||∇∂k −j +1t u ||∂kt u |+|∂k −j +1t u ||∇∂kt u |dx≤η−1C |∂j −1t (ρu )|2H 1|∇∂k −j +1tu |2L 2+η|∇∂kt u |L 2,NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1429 whereηis any small positive number,we deduce thatTt0|√ρ∂k+1tu(t)|2L2dt+|∇∂k t u(T)|2L2≤CΦk(T)20m+CTt01+|∂t(ρu)(t)|2H1+|u t(t)|2D1|∇∂k t u(t)|2L2dtfor t0≤T<T∗.Note thatT t01+|∂t(ρu)(t)|2H1+|u t(t)|2D1dt≤CΦk(T)10m.Therefore,in view of Gronwall’s inequality,we conclude that T0|√ρ∂k+1tu(t)|2L2dt+sup0≤t≤T|∂k t u(t)|2D1≤C expCΦk(T)10m(3.9)for any T with t0≤T<T∗.3.3.Estimates for ess sup0≤t≤T|√ρ∂k+1tu(t)|L2andT|∂k+1tu(t)|2D1dt.From the momentum equation(1.1),it follows thatρ∂k+1tut+ρu·∇∂k+1tu−Δ∂k+1tu+∇∂k+1tp=∂k+1t(ρf)+ρ∂k+1tu t−∂k+1t(ρu t)+ρu·∇∂k+1tu−∂k+1t(ρu·∇u).Multiplying this by∂k+1tu and integrating overΩ,we have1 2ddtρ|∂k+1tu|2dx+|∇∂k+1tu|2dx=∂k+1t(ρf)·∂k+1tu dx+ρ∂k+1tu t−∂k+1t(ρu t)·∂k+1tu dx(3.10)+ρu·∇∂k+1tu−∂k+1t(ρu·∇u)·∂k+1tu dx.This identity can be derived rigorously by using a standardfinite difference method because if0<T<T∗,then∂m tρ∈L2(0,T;L32∩L2)and∂j tρ∈C([0,T];L32∩L∞) for0≤j<m.Thefirst term of the right-hand side in(3.10)is bounded byCkj=0|∂j tρ||∂k−j+1tf||∂k+1tu|dx+|∂k+1tρ||f||∂k+1tu|dx≤Ckj=0|∂j tρ|2H1|∂k−j+1tf|2L2+C|∂k+1tρ|2L2|f|2H1+16|∇∂k+1tu|2L2.In view of the continuity equation(1.2),we can rewrite the second term as−k+1j=1(k+1)!j!(k−j+1)!∂j tρ∂k−j+1tu t·∂k+1tu dx=−k+1j=1(k+1)!j!(k−j+1)!∂j−1t(ρu)·∇∂k−j+2tu·∂k+1tudx,1430HYUNSEOK KIM which is bounded byC|ρ|L∞|u|2D10∩D2|√ρ∂k+1tu|2L2+Ckj=1|∂j t(ρu)|2H1|∂k−j+1tu|2D1+16|∇∂k+1tu|2L2.Finally,the last term is bounded byCkj=1|∂j t(ρu)||∇∂k−j+1tu||∂k+1tu|dx+|∂k+1t(ρu)||∇u||∂k+1tu|dx≤Ckj=1|∂j t(ρu)|2H1+|∂j tρ|2H1|u|2D1∩D2|∂k−j+1tu|2D1+C|∂k+1tρ|2L2|u|4D1∩D2+C|ρ|L∞|u|2D1∩D2|√ρ∂k+1tu|2L2+1|∇∂k+1tu|2L2.Hence substituting these estimates into(3.10),we haved dtρ|∂k+1tu|2dx+|∇∂k+1tu|2dx≤C1+|u|2D1∩D2|√ρ∂k+1tu|2L2+Ckj=0|∂j tρ|2H1|∂k−j+1tf|2L2+C|∂k+1tρ|2L2|f|2H1 +Ckj=1|∂j t(ρu)|2H1+|∂j tρ|2H1|u|2D1∩D2|∂k−j+1tu|2D1+C|∂k+1tρ|2L2|u|4D1∩D2.Therefore,by virtue of(3.5),(3.6),(3.7),and(3.9),we conclude thatess sup0≤t≤T |√ρ∂k+1tu(t)|L2+T|∂k+1tu(t)|2D1dt≤C expCΦk(T)10m(3.11)for any T with t0≤T<T∗.3.4.Estimates for sup0≤t≤T|∂j t u(t)|D10∩D2k−2j+2with0≤j≤k.Toderive these estimates,we observe that∂j t u∈C([0,T∗);D10,σ)and−Δ∂j t u+∇∂j t p=∂j t(ρf−ρu·∇u−ρu t) (3.12)for each j≤k.From(3.5),(3.6),(3.9),and(3.11),it follows easily thatess sup0≤t≤T|∂k t(ρu·∇u)(t)|L2+|∂k t(ρu t)(t)|L2≤C expCΦk(T)10mfor t0≤T<T∗.Hence applying the regularity theory of the Stokes equations to (3.12)with j=k,we obtainsup 0≤t≤T |∂k t u(t)|D1∩D2≤C expCΦk(T)10mfor t0≤T<T∗.It also follows from the Stokes regularity theory that for0≤j<k,|∂j t u|D10∩D2k−2j+2≤C|∂jt(ρf−ρu·∇u−ρu t)|H2k−2j+C|∂j t u|D1.(3.13)。

BMBreports338BMB reports*Corresponding author. T el: 82-31-201-2688; Fax: 82-31-202-2687;E-mail: dcyang@khu.ac.krReceived 17 October 2007, Accepted 26 December 2007K eywords: Abiotic stress, Codonopsis lanceolata , Dehydrin (DHN), Semi-quantitative RT-PCRIsolation of a novel dehydrin gene from Codonopsis lanceolata and analysis of its response to abiotic stressesRama Krishna Pulla 1,2, Yu-Jin Kim 1, Myung Kyum Kim 1, Kalai Selvi Senthil 3, Jun-Gyo In 4 & Deok-Chun Yang 1,*1Korean Ginseng Center and Ginseng Genetic Resource Bank, Kyung Hee University, Seocheon-dong, Kiheung-gu Yongin, Kyunggi-do, South Korea, 2Kongunadu Arts and Science College, Coimbatore, Tamil Nadu, 641029, India. 3Avinashilingam University for Women, Coimbatore, 641043, India. 4Biopia Co., Ltd., Yongin, KoreaDehydrins (DHNs) compose a family of intrinsically unstructured proteins that have high water solubility and accumulate during late seed development at low temperature or in water-deficit conditions. They are believed to play a protective role in freez-ing and drought-tolerance in plants. A full-length cDNA encod-ing DHN (designated as ClDhn ) was isolated from an oriental medicinal plant Codonopsis lanceolata , which has been used widely in Asia for its anticancer and anti-inflammatory properties. The full-length cDNA of ClDhn was 813 bp and contained a 477 bp open reading frame (ORF) encoding a polypeptide of 159 amino acids. Deduced ClDhn protein had high similarities with other plant DHNs. RT-PCR analysis showed that different abiotic stresses such as salt, wounding, chilling and light, trig-gered a significant induction of ClDhn at different time points within 4-48 hrs post-treatment. This study revealed that ClDhn assisted C. lanceolata in becoming resistant to dehydration. [BMB reports 2008; 41(4): 338-343]INTRODUCTIONPlants have developed defensive strategies against various stresses that arise from frequent environmental fluctuations to which they are exposed. Drought and low temperatures are the most severe factors limiting plant growth and yield. More than 100 genes have been shown to be responsive to such conditions and they are believed to function either during the physiological protection of cells from water-deficiencies or temperature-changes or in the regulation of gene expression (1-3).DHNs are proteins that are known to accumulate in vegetative plant tissues under stress conditions, such as low temperature, drought, or salt-stress (2, 4-6). These proteins have been catego-rized as late embryogenesis abundant (LEA) proteins (7, 8).DHNs have been subdivided into five classes according to thepresence of highly conservative segments: YnSK 2, Kn, KnS, SKn and Y 2Kn. The K-segment (EKKIGIMDKIKEKLPG) is a conserved 15-mer lysine-rich sequence characteristic of DHNs, which may be present in one or several copies (5). The K-segment can form an amphiphathic α-helix structure that may interact with lipid components of bio-membranes and partially denatured proteins like chaperones (6, 9). The S-segment consists of contiguous ser-ine residues in the centre of the protein, which may be phosphorylated. They are involved in nuclear transport through their binding to nuclear localization signal peptides (6). The Y-segment with the consensus sequence DEYGNP, shares some similarities to the nucleotide-binding site of chaperones in plants and bacteria (5, 10). Another conserved domain contained in many DHNs is ϕ-segment (repeated Gly and polar amino acids), which interacts with and stabilizes membranes and macro-molecules, preventing structural damage and maintaining the activity of essential enzymes (11).DHNs have been found in the cytoplasm (12), nucleus (12, 13), mitochondria (14), vacuole (15), and chloroplasts (16). They are known to associate with membranes (17, 18), pro-teins (19) and excess salt ions (15, 20). Several DHN genes have been isolated and characterized from different species, including cor47, erd10 and erd14 from Arabidopsis thaliana ; Hsp90, BN59, BN115 and Bnerd10 from Brassica napus ; cor39 and wcs19 from Triticum aestivum (bread wheat); and cor25 from Brassica rapa subsp. Pekinensis (21). Many studies have reported a positive correlation between the accumulation of DHN transcripts or proteins and tolerance to freezing, drought, and salinity (12, 17, 22-24). Moreover, mod-ulation of transcripts by light has been reported for many DHN-encoding genes in drought- or cold-stressed plants (25-28). Although the biochemical functions and physiological roles of DHNs are still unclear, their sequence character-izations and expression patterns suggest that they may play a positive role in plant-response and adaptation to abiotic stress that leads to cellular dehydration. Indeed, many studies have indicated that transgenic plants with DHNs have a better stress-tolerance, recovery or re-growth after drought and freez-ing stress than that of the control (8, 29, 30).Thus far, there are no reports on isolation of the DHN gene from the oriental medicinal plant Codonopsis lanceolata . ThisCodonopsis lanceolata dehydrin geneRama Krishna Pulla, et al.339BMBreportsFig. 1. Nucleotide sequence and de-duced amino acid sequence of a ClDhn cDNA isolated from C. lanceolata . Num-bers on the left represent nucleotide positions. The deduced amino acid se-quence is shown in a single-letter code below the nucleotide sequence. The as-terisk denotes the translation stop signal.Amino acids in two double boxes repre-sent the Y-segment and amino acids in a single box the S-segment, respectively.The two underlined sequences represent the K-segments.plant belongs to the family of Campanulaceae (bellflower fam-ily), which contains many famous oriental medicinal plants such as Platycodon grandiflorum (Chinese bellflower or balloon flow-er), Codonopsis pilosula and Adenophora triphylla (nan sha shen). The roots of these plants have been used as herbal drugs to treat bronchitis, cough, spasm, macrophage-mediated immune responses and inflammation, and has also been administered as a tonic (31). C. lanceolata grows in North-eastern china, Korea, and far eastern Siberia. Despite their medicinal importance, little genomic study of this plant has been carried out. In this study, we characterized an Y 2SK 2 type DHN gene from C. lanceolata and analyzed its expression in response to various abiotic stresses.RESULTS AND DISCUSSIONIsolation and characterization of the full length cDNA of the ClDhn geneAs part of a genomic project to identify genes in the medicinal plant C. lanceolata , a cDNA library consisting of about 1,000 cDNAs was previously constructed. A cDNA encoding a dehy-drin (DHN), designated ClDhn was isolated and sequenced. The sequence data of ClDhn has been deposited in GenBank under accession number AB126059. As shown in Fig. 1, ClDhn is 813 bp in length and it has an open reading frame (ORF) of 477 bp nucleotide with an 87-nucleotide upstream sequence and a 248-nucleotide downstream sequence. The ORF of ClDhn starts at nucleotide position 88 and ends at position 565. ClDhn encodes a precursor protein of 159 amino acids resi-dues with no predicted signal peptide at the N-terminal. The calculated molecular mass of the protein is approximately 16.7kDa with a predicated isoelectric point of 6.87. In the deduced amino acid sequence of ClDhn protein, the total number of neg-atively charged residues (Asp +Glu) amounted to 21 while the total number of positively charged residues (Arg +Lys) was 20. In addition, transmembrane helix prediction (TMHMMv2.0) did not identify any transmembrane helices in the deduced protein, implying that the protein did not function in the membrane but might function within the cytosolic or nuclear compartment.Homology analysisA GenBank Blastp search revealed that ClDhn had the highest sequence homology to the carrot (Daucus carota ) DHN (BAD86644) with 51% identity and 61% similarity. ClDhn also shared homology with ginseng (Panax ginseng ) DHN5 (ABF48478, 50% identity and 60% similarity), wild potato (Solanum commersonii ) DHN (CAA75798, 50% identity and 58% similarity), robusta coffee (Coffea canephora ) DHN1α (ABC55670, 47% identity and 55% similarity), grape (Vitis vin-ifera ) DHN (ABN79618, 47% identity and 57% similarity), American beech (Fagus sylvatica ) DHN (CAE54590, 46% iden-tity and 56% similarity), tobacco (Nicotiana tabacum ) DHN (BAD13498, 45% identity and 56% similarity), sunflower (Helianthus annuus ) DHN (CAC80719, 45% identity and 52% similarity), and soybean (Glycine max ) DHN (AAB71225, 44% identity and 52% similarity). The DHNs showing the highest similarities were Y 2SK 2 type DHNs except grape (Vitis vinifera ) DHN (YSK 2 type) (32). Thus ClDhn might belong to Y 2SK 2 type DHNs based on the two Y-segments, one S-segment, and two K-segments present in its amino acid sequence. Phylogenetic analysis of ten of the plant DHNs were carried out using theCodonopsis lanceolata dehydrin gene Rama Krishna Pulla, et al.340BMB reportsFig. 2. A phylogenetic tree based on DHN amino acid sequence, showing the phylogenetic relationship between ClDhn and other plant DHNs . The tree was constructed using the Clustal X method (Neighbor-joining method) and a bar represents 0.1 substitutions peramino acid position.Fig. 3. Alignment of ClDhn with the most closely related DHNs from carrot (Daucus carota , BAD86644), ginseng (Panax ginseng DHN5, ABF48478), com-merson’s wild potato (Solanum commer-sonii , CAA75798), robusta coffee (Coffea canephora , ABC55670), grape (Vitis vin-ifera , ABN79618), American beech (Fagus sylvatica , CAE54590), tobacco (Nicotiana tabacum , BAD13498), sunflower (Helian-thus annuus , CAC80719) and soybean (Glycine max , AAB71225). Gaps are marked with dashes. The conserved ami-no acid residues are shaded and Y-, S-, and K-segments are shown.Clustal X program (Fig. 2). Fig. 3 is a sequence alignment result of ClDhn and other closely related DHNs .The differential expression of ClDhn in different organs of C . lanceolataThe expression patterns of ClDhn in different C . lanceolata or-gans were examined using RT-PCR analysis. Almost similar levels of ClDhn -mRNA expression were observed in leaves and roots, whereas ClDhn was expressed in slightly higher lev-els in the stems. (Data was not shown).Expression of ClDhn in response to various stressesExpression patterns of ClDhn under various conditions were ex-amined using RT-PCR analysis. Fig. 4A showed the accumu-lation of ClDhn -mRNA in response to 100 mM ABA in MS agar. ABA is a hormone secreted when environmental conditions be-come dry. Expression of ClDhn was induced and reached a maximum level after 12 hrs, and then gradually decreased. When plants are submitted to dehydration the endogenous con-tent of ABA increases, with ABA mediating the closure of the stomata. Several studies have identified ABA as a key hormone in the induction pathway of many inducible genes including DHN , in response to drought (33-36). 100 μM of ABA in sprayCodonopsis lanceolata dehydrin geneRama Krishna Pulla, et al.341BMBreportsFig. 4. RT-PCR analyses of the expressions of ClDhn gene in the leaves of C. lanceolata at various time points (h) post-treatment with various stresses: A, 100 mM ABA; B, 100 mM NaCl; C, wounding; D, chilling and E, light treatment. Actin was used as an internal control.induced DHN-levels in Brassica napus and increased its ex-pression up to 48 hrs after treatment with ABA (37). 100 μM of ABA in MS agar induced DHN -levels in rice and cause a max-imum expression level at 1 hr post-treatment (10).Fig. 4B shows the accumulation of ClDhn mRNA in re-sponse to salt stress (100 mM NaCl). ClDhn expression was in-duced at 4 hrs post-treatment and gradually increased until 48 hrs. In Brassica napus , 250 mM NaCl added in the nutrient medium induced DHN-expression and reached a maximum at 48 hrs post-treatment (37). The application of NaCl to soil brought on a progressive decrease of the pre-dawn leaf water potential, a decrease of stomatal-conductance and a growth- reduction. Osmotic potential increase during salt treatmentcould result from Na ＋ or Cl −absorption and from the synthesis of compatible compounds (38).Under wounding stress, ClDhn gene transcription was in-duced at 4 hrs post-treatment and gradually increased until 48 hrs (Fig. 4C). Richard et al . (39) discussed that the cumulative effect of wounding on transcript accumulation could also be associated with greater water-loss through more open surfaces arising from the wounding treatment.Under cold treatment, increase of ClDhn transcripts was ob-served at 4 hrs post-treatment and gradually increased until 48 hrs (Fig. 4D). Induction of DHN by low temperatures has been observed in numerous plants (17, 38). Overexpression of citrus DHN improved the cold tolerance in tobacco (18). Overexpre-ssion of multiple DHN genes in Arabidopsis resulted in accu-mulation of the corresponding DHNs to levels similar or higher than in cold-acclimated wild-type plants (24). Another example showed that overexpression of the acidic DHN WCOR410 could improve freezing tolerance in transgenic strawberry leaves (29). Fig. 4E shows that ClDhn gene expression was induced bylight stress and increased continuously until 48 hrs post-treat-ment. Natali et al . (40) showed that the G-box (CACGTGGC), a motif found in the promoter region of many light regulated genes, was found in the DHN gene promoter of helianthus and that DHN was responsive to light stress (41).In conclusion, we isolated a new dehydrin gene (ClDhn ) from C. lanceolata and characterized its expression in response to various stresses. ClDhn was induced by various stresses related to wa-ter-deficiency (ABA, salt, wounding and cold) and was induced by light, similar to other DHN genes isolated from other plants.MATERIALS AND METHODSPlant materialsCodonopsis lanceolata were grown in vitro on MS medium supplemented with 3% sucrose and 0.8% agar under the 16 hrs light and 8 hrs dark period. Its growth was maintained by regular subculture every 4 weeks. Abiotic stress studies were carried out on plants that were subcultured for one month. To analyze gene expression in different organs, samples were col-lected from leaves, roots and stem of C. lanceolata plants.Sequence analysesThe full-length ClDhn gene was analyzed using the softwares BioEdit, Clustal X, Mega 3 and other databases listed below; NCBI (http://www.ncbi.nlm.nih), SOPMA (http://npsa-pbil.ibcp /npsaautomat.pl?page=npsopma.html).Stress assaysTo investigate the response of the ClDhn gene to various stress-es, the third leaves with petioles from C. lanceolata were used. For treatment with ABA (100 mM) and NaCl (100 mM), leaf samples were incubated in media containing each compound at 25o C for 48 hrs. For mechanical wounding stress, excised leaves were wounded with a needle puncher (42). Chilling stress was applied by exposing the leaves to a temperature of 4o C (43). To investigate the ClDhn gene-expressions in light, leaves were incubated under an electrical lamp with a light in-tensity of 24 mol m-2 s-1 for 48 hrs. All treatments were carried out on MS media with or without the treatment solution (ABA, NaCl). All treated plant materials were immediately frozen in liquid nitrogen and stored at -70o C until further analysis.Semi-quantitative RT-PCR analysisTotal RNA was extracted from various whole plant tissues (leaves, stem, roots) of C. lancolata using the Rneasy mini kit (Qiagen, Valencia, CA, USA). For RT-PCR (reverse tran-scriptase-PCR), 800 ng of total RNA was used as a template for reverse transcription using oligo (dT) primer (0.2 mM)(INTRON Biotechnology, Inc., South Korea) for 5 mins at 75oC. The reaction mixture was then incubated with AMV Reverse Transcriptase (10 U/μl) (INTRON Biotechology, Inc., SouthKorea) for 60 mins at 42oC. The reaction was inactivated byheating the mixture at 94oC for 5 mins. PCR was then per-Codonopsis lanceolata dehydrin gene Rama Krishna Pulla, et al.342BMB reportsformed using a 1 μl aliquot of the first stand cDNA in a final volume of 25 μl containing 5 pmol of specific primers for cod-ing of ClDhn gene (forward, 5'-AAA GAG AGA GAA AAT GGC AGG TTA C-3'; reverse, 5'-GGA GTA GTT GTT GAA GTT CTC TGC T-3') were used. As a control, the primers spe-cific to the C. lanceolata actin gene were used (forward, 5'-CAA GAA GAG CTA CGA GCT ACC CGA TGG-3'; reverse, 5'-CTC GGT GCT AGG GCA GTG ATC TCT TTG CT-3'). PCR was carried out using 1 μl of taq DNA polymerase (Solgent Co., South Korea) in a thermal cycler programmed as follows:an initial denaturation for 5 mins at 95oC, 30 amplification cy-cles [30 s at 95o C (denaturation), 30 s at 53o C (annealing), and90 s at 72oC (polymerization)], followed by a final elongation for 10 mins at 72o C. 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高二英语科学家名称单选题20题1. Who discovered the theory of relativity?A. Isaac NewtonB. Albert EinsteinC. Galileo GalileiD. Marie Curie答案：B。

本题考查著名物理学家的成就。

选项A 艾萨克·牛顿，主要贡献是万有引力定律等；选项C 伽利略·伽利雷，在天文学和力学方面有重要成就；选项D 玛丽·居里，是著名的化学家。

而阿尔伯特·爱因斯坦发现了相对论，所以选B。

2. Which physicist is known for his work on quantum mechanics?A. Max PlanckB. Nikola TeslaC. Thomas EdisonD. Werner Heisenberg答案：D。

本题考查对量子力学有贡献的物理学家。

选项A 马克斯·普朗克，是量子力学的奠基人之一；选项B 尼古拉·特斯拉，在电学领域有突出成就；选项C 托马斯·爱迪生，是发明大王。

维尔纳·海森堡在量子力学方面有重要贡献，故选D。

3. The law of universal gravitation was proposed by _____.A. ArchimedesB. CopernicusC. KeplerD. Isaac Newton答案：D。

本题考查万有引力定律的提出者。

选项A 阿基米德，在浮力方面有重要发现；选项B 哥白尼，提出日心说；选项C 开普勒，发现了行星运动定律。

艾萨克·牛顿提出了万有引力定律，所以答案是D。

4. Who is famous for his experiments with electricity?A. Benjamin FranklinB. Michael FaradayC. James Clerk MaxwellD. Alessandro V olta答案：B。

Bolzano-Weierstrass Theorem1. IntroductionThe Bolzano-Weierstrass Theorem is a fundamental result in real analysis, providing essential insights into the behavior of sequences of real numbers. Named after Bernard Bolzano and Karl Weierstrass, this theorem has important applications in various branches of mathematics, including calculus, functional analysis, and mathematical physics.2. Statement of the TheoremThe Bolzano-Weierstrass Theorem states that every bounded sequence of real numbers has a convergent subsequence. In other words, if a sequence (an) is bounded, then there exists a subsequence (an_k) that converges to a real number.3. Historical BackgroundThe theorem's origins can be traced back to the 19th century, with both Bolzano and Weierstrass making significant contributions to its development. Bernard Bolzano, a Czech mathematician and philosopher, laid the groundwork for the theorem in the early 1800s by investigating the properties of continuous functions and the intermediate value theorem. KarlWeierstrass, a German mathematician known for his rigorous approach to analysis, further refined the theorem in the mid-1800s, emphasizing the importance of boundedness in the context of convergent sequences.4. Proof of the TheoremThe proof of the Bolzano-Weierstrass Theorem typically involves constructing a convergent subsequence from a given bounded sequence. By repeatedly dividing the interval on which the sequence is defined and selecting subsequences from the resulting subintervals, one can establish the existence of a convergent subsequence. The continuity of the real number line and the properties of convergent sequences are crucial elements in the proof.5. Applications and SignificanceThe Bolzano-Weierstrass Theorem has far-reaching implications in mathematics and its various applications. In calculus, the theorem is used to establish the existence of convergent subsequences of real-valued functions, supporting the study of limits and continuity. In functional analysis, the theorem plays a central role in the study ofpact sets and the properties of topological spaces. Furthermore, in mathematical physics, thetheorem provides essential tools for analyzing the behavior of physical systems through the study of convergent sequences.6. Related Results and ExtensionsExtensions and variants of the Bolzano-Weierstrass Theorem have been developed to address specific settings and generalizations. For instance, the theorem has been extended to sequences in metric spaces and to sequences of vectors in infinite-dimensional spaces. Additionally, the concept of sequentialpactness, closely related to the theorem, has been explored in the context of topological spaces and generalizations ofpactness.7. ConclusionIn conclusion, the Bolzano-Weierstrass Theorem stands as a cornerstone of real analysis, offering deep insights into the behavior of sequences of real numbers. Its historical significance, rigorous proof, and broad applications underscore its importance in mathematics and its foundational role in various fields of study. As mathematicians continue to explore the implications of the theorem and its extensions, its profound impact on the understanding of convergent sequences andboundedness will persist as a guiding principle in mathematical theory and practice.。