Intact Element Research Articles

Pdx1 and BETA2/NeuroD1 Participate in a Transcriptional Complex That Mediates Short-range DNA Looping at the Insulin Gene

The activity of the insulin gene, Ins, in islet beta cells is thought to arise in part from the synergistic action of the transcription factors Pdx1 and BETA2/NeuroD1. We asked how the binding of these factors to A and E elements many tens or hundreds of base pairs upstream of the start site could influence activity of transcriptional machinery. We therefore tested the hypothesis that the complex of Pdx1 and BETA2/NeuroD1 maintains a DNA conformation such that distal regions of the gene are brought into proximity of the promoter and coding region. We show by coimmunoprecipitation that Pdx1 and BETA2/NeuroD1 exist within a complex and that the two physically interact with one another in this complex as assessed by fluorescence resonance energy transfer. Consistent with this interaction, we found that the two factors simultaneously occupy the same fragment of the Ins gene in beta cell lines using the chromatin immunoprecipitation/re-chromatin immunoprecipitation assay. Using a modification of the chromosome conformation capture assay in vitro and in beta cells, we observed that Pdx1 and BETA2/NeuroD1 mediate looping of a segment of Ins that brings EcoRI sites located at -623 and +761 bp (relative to the transcriptional start site) in proximity to one another. This looping appears to be dependent in vitro upon an intact A3 binding element, but not upon the E2 element. Based on our findings, we propose a model whereby Pdx1 and BETA2/NeuroD1 physically interact to form a nucleoprotein complex on the Ins gene that mediates formation of a short DNA loop. Our results suggest that such short loop conformations may be a general mechanism to permit interactions between transcription factors and basal transcriptional machinery.

Stabilized binding of TBP to the TATA box of herpes simplex virus type 1 early (tk) and late (gC) promoters by TFIIA and ICP4.

We have recently shown that ICP4 has a differential requirement for the general transcription factor TFIIA in vitro (S. Zabierowski and N. DeLuca, J. Virol. 78:6162-6170, 2004). TFIIA was dispensable for ICP4 activation of a late promoter (gC) but was required for the efficient activation of an early promoter (tk). An intact INR element was required for proficient ICP4 activation of the late promoter in the absence of TFIIA. Because TFIIA is known to stabilize the binding of both TATA binding protein (TBP) and TFIID to the TATA box of core promoters and ICP4 has been shown to interact with TFIID, we tested the ability of ICP4 to stabilize the binding of either TBP or TFIID to the TATA box of representative early, late, and INR-mutated late promoters (tk, gC, and gC8, respectively). Utilizing DNase I footprinting analysis, we found that ICP4 was able to facilitate TFIIA stabilized binding of TBP to the TATA box of the early tk promoter. Using mutant ICP4 proteins, the ability to stabilize the binding of TBP to both the wild-type and the INR-mutated gC promoters was located in the amino-terminal region of ICP4. When TFIID was substituted for TBP, ICP4 could stabilize the binding of TFIID to the TATA box of the wild-type gC promoter. ICP4, however, could not effectively stabilize TFIID binding to the TATA box of the INR-mutated late promoter. The additional activities of TFIIA were required to stabilize the binding of TFIID to the INR-mutated late promoter. Collectively, these data suggest that TFIIA may be dispensable for ICP4 activation of the wild-type late promoter because ICP4 can substitute for TFIIA's ability to stabilize the binding of TFIID to the TATA box. In the absence of a functional INR, ICP4 can no longer stabilize TFIID binding to the TATA box of the late promoter and requires the additional activities of TFIIA. The stabilized binding of TFIID by TFIIA may in turn allow ICP4 to more efficiently activate transcription from non-INR containing promoters.

Three Dimensional Frame Structures with Edge-Cracks of Uncertain Depth and Location

In this paper, a numerical study on the structural behaviour of three-dimensional cracked structures have been presented. The stiffness matrix of the cracked element is found as the inverse of the compliance matrix. This matrix is given by the sum of the compliance matrix of the intact element and an additional compliance matrix which contains all the flexibilities given by the presence of the crack. The flexibilities are related to the stress intensity factors. A simple method for obtaining approximate stress intensity factors is applied. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. Moreover, crack depth and location are modelled as random variables in order to take into account the unavoidable uncertainty that always affects damaged structures. A simple and accurate method for the probabilistic characterization of the linear elastic response of cracked structures with uncertain damage is employed. According to this method, the uncertainties are transformed into superimposed deformations depending on the distribution of internal forces and an iterative procedure is established to solve the resultant equations. Numerical tests evidence excellent accuracy for multicracked structures with large fluctuation of damage. Present manuscript also discusses relevant patents. Keywords: Stress intensity factors, three-dimensional frame structure, edge crack, probabilistic analysis

Molecular characterization of the first intact Transib transposon from Helicoverpa zea

Transib is a superfamily of DNA transposons recently reconstructed in silico from degenerate elements in the genomes of Drosophila melanogaster and Anopheles gambiae. Here we report characterization of the first intact Transib transposon designated Hztransib from Helicoverpa zea. At least two Hztransib copies (one full-length, one internally-deleted) exist in the genomes of a midgut cell line and a laboratory strain of H. zea. The full-length Hztransib has 3518 bp including a 5′ terminal inverted repeat (TIR) of 552 bp, a promoter sequence of 381 bp, an intact open reading frame encoding 507 amino acids, and a 3′ TIR of 502 bp, and is flanked by 5-bp (CGTCG) target site duplications. The full-length Hztransib is transcriptionally active, producing an 3′-truncated mRNA lacking a termination codon (known as nonstop mRNA) due to alternative polyadenylation in two somatic tissues (midgut and fat body) and one germline tissue (ovary). A BLAST search with the deduced Hztransib transposase identified 51 novel Transib elements from 11 insect genomes. While the full-length Hztransib inserts into the 5′-flanking region of a xenobiotic-metabolizing P450 gene CYP6B8 in the midgut cell line, it does not insert into the 5′-flanking region of CYP6B8 in the laboratory strain. Such an insertion dimorphism, together with its complete structural features and mRNA transcripts, demonstrates that the full-length Hztransib not only represents the first known intact Transib element in any organism, but has been recently transposed in H. zea. The fact that the intact Hztransib is transcribed into a 3′-truncated nonstop mRNA which may encode a non-functional transposase or be blocked from further translation via nonstop mRNA decay suggests that it is silenced at the translational or transpositional level.

Continuum mathematical modeling of slip weakening in geological systems

We describe a framework for mathematical modeling of slip weakening in an initially intact rock mass due to shear strain localization along any arbitrary slip plane. The modeling technique considered is based on continuum mechanics and may be cast directly into a standard nonlinear finite element algorithm for the analysis of prefailure and postfailure responses of geological systems in a boundary value problem. The prefailure behavior is represented by a continuum constitutive model; the postfailure behavior is characterized by frictional yielding on a slip surface with state‐ and velocity‐dependent coefficient of friction. In the context of finite element analysis, slip planes are represented by an embedded strong discontinuity introduced into an initially intact finite element to signal the beginning of postfailure behavior. This paper focuses on the narrow time interval of slip weakening, from the moment the strong discontinuity has been embedded into a finite element until the relative slip has grown to a large enough value for the coefficient of friction to reach steady state. To this end, we formulate a linear slip weakening constitutive law in which the weakening component decays to zero at the same time that the frictional component increases to its value at residual state.

A Sequence Motif in the Simian Virus 40 (SV40) Early Core Promoter Affects Alternative Splicing of Transcribed mRNA

To identify new sequence elements in the promoter that affect splicing patterns of pre-mRNAs, we analyzed effects of different promoters on alternative splicing of model reporter genes. We compared the E1a alternative splicing pattern in transcripts expressed from the full-length cytomegalovirus, SV40 early, or a hybrid cytomegalovirus/SV40 early promoter and found that the hybrid promoter improved selection of the suboptimal E1a 5'SS-1. Expressing RNA from the hybrid promoter also enhanced selection of suboptimal splice sites in other alternatively spliced reporter genes, demonstrating the generality of this effect. Unlike previously defined promoter elements shown to affect alternative splicing, which were located in the enhancer/upstream activating sequences, the motif identified in this work is positioned within the core promoter; it is comprised of eight T-residues directly upstream of the SV40 early TATA box. This motif was previously implicated in DNA bending and negative regulation of transcription. Together, these results suggest that the identity of transcription complex assembled in the core promoter-dependent fashion can affect splice site selection during pre-mRNA splicing, perhaps by influencing the processivity of transcription elongation.

Meiotic interference among MLH1 foci requires neither an intact axial element structure nor full synapsis

During meiosis, homologous chromosomes (homologs) perform reciprocal exchanges (crossovers) at a high frequency. Crossovers display interference, i.e. their spacing is more even than would be expected if they were placed randomly along the chromosomes. Concomitantly with crossover formation, synaptonemal complexes (SCs) appear between homologs: each chromosome forms an axial structure, the axial element (AE); the AEs of homologs align, and numerous transverse filaments connect the AEs to form an SC. Both the AE and the SC have been implicated in the imposition of interference. We investigated whether intact AEs or SCs are required for crossover interference in the mouse, using a mutant lacking AE protein SYCP3, which displays structurally abnormal AEs and incomplete synapsis. We estimated the level of interference from the spacing of immunofluorescent MLH1 foci, which mark almost all crossover sites in the mouse, along the SCs. The levels of interference among MLH1 foci in wild-type and Sycp3(-/-) mice were comparable, implying that neither an intact AE structure nor full synapsis is required for wild-type levels of interference.

Intact mariner‐like element in tobacco budworm, Heliothis virescens (Lepidoptera: Noctuidae)

An intact mariner-like element was isolated from the genome of tobacco budworm, Heliothis virescens. This is the first report of an intact mariner element after the initial identification of Mos1 from Drosophila mauritiana. The full-length Hvmar1 has 30 bp inverted terminal repeats and a complete 1065 bp open reading frame (ORF) encoding the transposase with a 'D,D(34)D' motif in the catalytic domain. Polymerase chain reaction results show that at least one insertion of the Hvmar1 element is conserved in this Heliothis strain. Phylogenetic analysis demonstrated that Hvmar1 belongs to the irritans subfamily.

A histone H3 Ser10 phosphorylation‐independent function of Snf1 and Reg1 proteins rescues a gcn5¬C mutant in HIS3 expression

Gcn5 serves as the catalytic subunit of the SAGA histone acetyltransferase (HAT) complex and is critical for HIS3 activation in Saccharomyces cerevisiae. A truncated Reg1 protein, Reg1(1–740), was found to be a dominant suppressor rescuing the HIS3 transcriptional defects of a gcn5 mutant (E173H). The function of Reg1(1–740) protein requires an intact cis-acting element for the Gcn4 transcriptional activator, and another SAGA component, Spt3, suggesting that the observed suppression was achieved via the normal SAGA regulatory pathway. Reg1 protein functionally and physically interacts with a histone H3 kinase, Snf1. Indeed, Snf1 plays an important role for normal and Reg1(1–740)-dependent HIS3 activation. However, substituting the phosphorylation residue in H3, i.e. Ser10, with alanine or glutamate neither attenuates nor augments the suppression phenotypes. These results argue against an essential role of H3 phosphorylation in HIS3 expression. Both Reg1(1–740) and overexpressed Snf1 rescue the E173H allele of gcn5 selectively, suggesting physical interactions between Gcn5 and these two proteins. In vivo co-purification experiments confirmed this notion, and importantly, in vitro data showed that Gcn5 can be phosphorylated by Snf1 protein. Together, these data suggest that Reg1 and Snf1 proteins activate HIS3 in a histone H3 phosphorylation-independent manner that may also involve a non-catalytic function or a novel HAT activity of the Gcn5 protein. This work was supported by NIH grant GM 62282

Nuclease sensitive element binding protein 1 associates with the selenocysteine insertion sequence and functions in mammalian selenoprotein translation

Biosynthesis of selenium-containing proteins requires insertion of the unusual amino acid selenocysteine by alternative translation of a UGA codon, which ordinarily serves as a stop codon. In eukaryotes, selenoprotein translation depends upon one or more selenocysteine insertion sequence (SECIS) elements located in the 3'-untranslated region of the mRNA, as well as several SECIS-binding proteins. Our laboratory has previously identified nuclease sensitive element binding protein 1 (NSEP1) as another SECIS-binding protein, but evidence has been presented both for and against its role in SECIS binding in vivo and in selenoprotein translation. Our current studies sought to resolve this controversy, first by investigating whether NSEP1 interacts closely with SECIS elements within intact cells. After reversible in vivo cross-linking and ribonucleoprotein immunoprecipitation, mRNAs encoding two glutathione peroxidase family members co-precipitated with NSEP1 in both human and rat cell lines. Co-immunoprecipitation of an epitope-tagged GPX1 construct depended upon an intact SECIS element in its 3'-untranslated region. To test the functional importance of this interaction on selenoprotein translation, we used small inhibitory RNAs to reduce the NSEP1 content of tissue culture cells and then examined the effect of that reduction on the activity of a SECIS-dependent luciferase reporter gene for which expression depends upon readthrough of a UGA codon. Co-transfection of small inhibitory RNAs directed against NSEP1 decreased its expression by approximately 50% and significantly reduced luciferase activity. These studies demonstrate that NSEP1 is an authentic SECIS binding protein that is structurally associated with the selenoprotein translation complex and functionally involved in the translation of selenoproteins in mammalian cells.

Expression of SRG3, a core component of mouse SWI/SNF chromatin-remodeling complex, is regulated by cooperative interactions between Sp1/Sp3 and Ets transcription factors

SRG3, a mouse homolog of yeast SWI3 and human BAF155, is known to be a core component of SWI/SNF chromatin-remodeling complex. We have previously shown that SRG3 plays essential roles in early mouse embryogenesis, brain development, and T-cell development. SRG3 gene expression was differentially regulated depending on the developmental stages and exhibited tissue-specific pattern. In this study, we showed that the functional interactions between Sp and Ets family transcription factors are crucial for the SRG3 expression. Sp1 and Sp3 specifically bound to the two canonical Sp-binding sites (GC boxes) at −152 and −114, and a non-canonical Sp-binding site (CCTCCT motif) at −108 in the SRG3 promoter. Using Drosophila SL2 cells, we found that various Sp or Ets family members activate the SRG3 promoter through these Sp- or Ets-binding sites, respectively, in a dose-dependent manner. Intriguingly, different combinatorial expression of Ets and Sp factors in SL2 cells resulted in either strong synergistic activation or repression of the SRG3 promoter activity. Moreover, the Sp-mediated activation of SRG3 promoter required the intact Ets-binding element. Taken together, these results suggest that diverse interactions between Sp1/Sp3 and Ets factors are crucial for the SRG3 gene expression.

A Segmental Deletion Series Generated by Sister-Chromatid Transposition of Ac Transposable Elements in Maize

Certain configurations of maize Ac/Ds transposon termini can undergo alternative transposition reactions leading to chromosome breakage and various types of stable chromosome rearrangements. Here, we show that a particular allele of the maize p1 gene containing an intact Ac element and a nearby terminally deleted Ac element (fAc) can undergo sister-chromatid transposition (SCT) reactions that generate large flanking deletions. Among 35 deletions characterized, all begin at the Ac termini in the p1 gene and extend to various flanking sites proximal to p1. The deletions range in size from the smallest of 12,567 bp to the largest of >4.6 cM; >80% of the deletions removed the p2 gene, a paralog of p1 located approximately 60 kb from p1 in the p1-vv allele and its derivatives. Sequencing of representative cases shows that the deletions have precise junctions between the transposon termini and the flanking genomic sequences. These results show that SCT events can efficiently generate interstitial deletions that are useful for in vivo dissection of local genome regions and for the rapid correlation of genetic and physical maps. Finally, we discuss evidence suggesting that deletions induced by alternative transposition reactions can occur at other genomic loci, indicating that this mechanism may have had a significant impact on genome evolution.

A site‐specific insertion sequence in flax genotrophs induced by environment

A single-copy 5.7 kilobase (kb) DNA fragment, termed Linum Insertion Sequence 1 (LIS-1), has been identified and characterized. This is one of the DNA changes associated with the environmentally induced heritable changes resulting in stable lines termed genotrophs in flax (Linum usitatissimum). The insertion sequence and its insertion site have been cloned from genomic libraries and sequenced. PCR products across the insertion and surrounding regions have also been cloned and sequenced. The 5.7 kb DNA fragment is inserted into a 3.7 kb EcoRI fragment in the plastic line (Pl) with the generation of a 3 base pair duplication at the insertion site, as well as additional sequence changes. The identical insertion was also found in other genotrophs and flax varieties. The intact element was not present in Pl but appeared to be generated by a reproducible series of complex rearrangements or insertion events. LIS-1 is the result of a targeted, highly specific, complex insertion event that occurs during the formation of some of the genotrophs, and occurs naturally in many flax and linseed varieties.

Ligand-dependent Activation of the Farnesoid X-receptor Directs Arginine Methylation of Histone H3 by CARM1

In this study we demonstrate that the class II nuclear hormone receptor, farnesoid X-receptor (FXR), incorporates histone methyltransferase activity within the gene locus for bile salt export pump (BSEP), a well established FXR target gene that functions as an ATP-dependent canalicular bile acid transporter. This methyltransferase activity is directed specifically to arginine 17 of histone H3. We demonstrate that FXR is directly associated with co-activator-associated arginine methyltransferase 1 (CARM1) activity. Furthermore, we show by chromatin immunoprecipitation that the ligand-dependent activation of the human BSEP locus is associated with a simultaneous increase of FXR and CARM1 occupation. The increased occupation of the BSEP locus by CARM1 also corresponds with the increased deposition of Arg-17 methylation and Lys-9 acetylation of histone H3 within the FXR DNA-binding element of BSEP. Consistent with these findings, CARM1 led to increased BSEP promoter activity with an intact FXR regulatory element, whereas CARM1 failed to transactivate the BSEP promoter with a mutated FXRE. Induction of endogenous BSEP mRNA and Arg-17 methylation by FXR regulatory element ligand, CDCA, requires CARM1 activity. Therefore, histone methylation at Arg-17 by CARM1 is a downstream target of signaling through ligand-mediated activation of FXR. Our studies provide evidence that FXR directly recruits specific chromatin modifying activity of CARM1 necessary for full potentiation of the BSEP locus in vivo.

Removal of biological and non-biological viral surrogates by spiral-wound reverse osmosis membrane elements with intact and compromised integrity.

The removal of bacteriophage MS2 and fluorescent-dyed polystyrene microspheres with intact and purposely compromised spiral-wound RO membrane elements was investigated. MS2 rejection with intact membrane elements was >99.9995%. A model developed for data evaluation revealed that the advective passage of MS2 through imperfections of intact membrane elements was <2 x 10(-5)% of the overall product water flow produced. The advective passage of MS2 and microspheres through a pinhole induced in one of the elements was 0.05-0.1% of the overall product water flow. Prolonged testing of both intact and compromised elements resulted in increased MS2 rejection corresponding to advective MS2 passage through membrane imperfections of <3 x 10(-7)% of the overall product water flow. The permeate flow rate obtained with an element with a larger pinhole was 5-13% greater than that of the intact element, and the corresponding rejection of MS2 and microspheres was similar to that observed for sodium chloride. The use of a cracked o-ring in the connection of the permeate tube to the element vessel end-cup resulted in advective passage of MS2 through the crack of <0.0001% of the overall permeate flow.

Transposition of reversed Ac element ends generates chromosome rearrangements in maize.

In classical "cut-and-paste" transposition, transposons are excised from donor sites and inserted at new locations. We have identified an alternative pathway in which transposition involves the 5' end of an intact Ac element and the 3' end of a nearby terminally deleted fAc (fractured Ac). The Ac and fAc elements are inserted at the maize p1 locus on chromosome 1s in the same orientation; the adjacent ends of the separate elements are thus in reversed orientation with respect to each other and are separated by a distance of approximately 13 kb. Transposition involving the two ends in reversed orientation generates inversions, deletions, and a novel type of local rearrangement. The rearrangement breakpoints are bounded by the characteristic footprint or target site duplications typical of Ac transposition reactions. These results demonstrate a new intramolecular transposition mechanism by which transposons can greatly impact genome evolution.

Negative and positive effects of an IAP-LTR on nearby Pcdaα gene expression in the central nervous system and neuroblastoma cell lines

Intracisternal A-particles (IAPs) are defective retrovirions encoded by members of a large family of endogenous proviral elements in the murine genome. An intact IAP element was found in the protocadherin alpha ( Pcdhα) gene cluster of five laboratory mouse strains. However, IAP insertion was not detected in three wild mouse strains we investigated. This IAP insertion caused the disruption of one variable exon of laboratory mouse and down-regulated expression of the Pcdhα v8 exon, which is located just downstream of the IAP in the brain following the methylation of 5′ regulatory region of Pcdhα v8. In contrast, the Pcdhα v8 exon was highly expressed in mouse neuroblastoma cell lines. This suggested that the IAP insertion activates the expression of the nearby Pcdhα v8 exon in these cell lines. In fact, the Pcdhα v8 exon expression was driven by the IAP-long terminal repeat (LTR) following the de-methylation of 5′ regulatory region of Pcdhα v8. To investigate the promoter activity of the IAP, we constructed an IAP-LTR-ECFP reporter gene and introduced it into neuroblastoma, melanoma, lymphoma, and plasmacytoma cell lines. Interestingly, ECFP-positive cells were observed only in the neuroblastoma cell lines. Moreover, there were no differences in the promoter activities of the IAP-LTR whether it was in the sense or complimentary orientation. Thus, this IAP-LTR has negative and positive regulation on near by gene expression in the brain and neuroblastoma cell lines.

Differential cellular requirements for activation of herpes simplex virus type 1 early (tk) and late (gC) promoters by ICP4.

The herpes simplex virus type 1 immediate-early protein, ICP4, activates the transcription of viral early and late genes and is essential for viral growth. It has been shown to bind DNA and interact with components of the general transcription machinery to activate or repress viral transcription, depending upon promoter context. Since early and late gene promoters have different architectures and cellular metabolism may be very different at early and late times after infection, the cellular requirements for ICP4-mediated activation of early and late genes may differ. This hypothesis was tested using tk and gC as representative early and late promoters, respectively. Nuclear extracts and phosphocellulose column fractions derived from nuclear extracts were able to reconstitute basal and ICP4-activated transcription of both promoters in vitro. When examining the contribution of the general transcription factors on the ability of ICP4 to activate transcription, the fraction containing the general transcription factor TFIIA was not essential for ICP4 activation of the gC promoter, but it was required for efficient activation of the tk promoter. The addition of recombinant TFIIA restored the ability of ICP4 to efficiently activate the tk promoter, but it had no net effect on activation of the gC promoter. The dispensability of TFIIA for ICP4 activation of the gC promoter required an intact INR element. In addition, microarray and Northern blot analysis indicated that TFIIA abundance may be reduced at late times of infection. This decrease in TFIIA expression during infection and its dispensability for activation of late but not early genes suggest one of possibly many mechanisms for the transition from viral early to late gene expression.

ABI3 mediates expression of the peroxiredoxin antioxidant AtPER1 gene and induction by oxidative stress.

The peroxiredoxin antioxidant gene AtPER1 has been considered to be specifically expressed in the embryo and aleurone layer during maturation and desiccation stages of development, and in the mature seed, typically for late embryogenesis-abundant (lea) transcripts. In the abscisic acid-insensitive abi3-1 mutant, the AtPER1 transcript level is strongly reduced, suggesting ABI3 to be a prime regulator of AtPER1. We have studied the expression pattern and regulation of AtPER1 with a series of nine promoter::GUS constructs with deletions and/or mutations in putative regulatory elements. Arabidopsis lines harbouring these constructs revealed AtPER1 promoter activity in the endosperm, especially the chalazal cyst, already when the embryo is in the late globular stage, in the embryo from the late torpedo stage, and in distinct cells of unfertilized and fertilized ovules. Early expression seems to be dependent on a putative antioxidant-responsive promoter element (ARE), while from the bent cotyledon stage endosperm and embryo expression is dependent on an ABA-responsive element (ABRE) likely to bind ABI5. The shortest promoter fragment (113 bp), devoid of ARE, ABRE and without an intact RY/Sph element thought to bind ABI3 did not drive GUS expression. The AtPER1::GUS construct also revealed expression in cotyledons, meristems and stem branching points. In general, seed and vegetative expression coincided with the expression pattern of ABI3. In plants ectopically expressing ABI3, AtPER1::GUS expression was found in true leaves, and AtPER1 could be induced by exogenous ABA and oxidative stress (H2O2 and hydroquinone). ABI3-mediated oxidative stress induction was dependent on the presence of an intact ARE element.

Free vibration analysis of a cracked beam by finite element method

In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.