Genetic organization and diversity of the hepatitis C virus

The nucleotide sequence of the RNA genome of the human hepatitis C virus (HCV) has been determined from overlapping cDNA clones. The sequence (9379 nucleotides) has a single large open reading frame that could encode a viral polyprotein precursor of 3011 amino acids. While there as little overall amino acid and nucleotide sequence homology with other viruses, the 5' HCV nucleotide sequence upstream of this large open reading frame has substantial similarity to the 5' termini of pestiviral genomes. The polyprotein also has significant sequence similarity to helicases encoded by animal pestiviruses, plant potyviruses, and human flaviviruses, and it contains sequence motifs widely conserved among viral replicases and trypsin-like proteases. A basic, presumed nucleocapsid domain is located at the N terminus upstream of a region containing numerous potential N-linked glycosylation sites. These HCV domains are located in the same relative position as observed in the pestiviruses and flaviviruses and the hydrophobic profiles of all three viral polyproteins are similar. These combined data indicate that HCV is an unusual virus that is most related to the pestiviruses. Significant genome diversity is apparent within the putative 5' structural gene region of different HCV isolates, suggesting the presence of closely related but distinct viral genotypes.

ICOS co-stimulatory receptor is essential for T-cell activation and function

T-lymphocyte activation and immune function are regulated by co-stimulatory molecules. CD28, a receptor for B7 gene products, has a chief role in initiating T-cell immune responses. CTLA4, which binds B7 with a higher affinity, is induced after T-cell activation and is involved in downregulating T-cell responses. The inducible co-stimulatory molecule (ICOS), a third member of the CD28/CTLA4 family, is expressed on activated T cells. Its ligand B7H/B7RP-1 is expressed on B cells and in non-immune tissues after injection of lipopolysaccharide into animals. To understand the role of ICOS in T-cell activation and function, we generated and analysed ICOS-deficient mice. Here we show that T-cell activation and proliferation are defective in the absence of ICOS. In addition, ICOS -/- T cells fail to produce interleukin-4 when differentiated in vitro or when primed in vivo. ICOS is required for humoral immune responses after immunization with several antigens. ICOS-/- mice showed greatly enhanced susceptibility to experimental autoimmune encephalomyelitis, indicating that ICOS has a protective role in inflammatory autoimmune diseases.

Defective T cell differentiation in the absence of Jnk1

The c-Jun NH2-terminal kinase (JNK) signaling pathway has been implicated in the immune response that is mediated by the activation and differentiation of CD4 helper T (TH) cells into TH1 and TH2 effector cells. JNK activity observed in wild-type activated TH cells was severely reduced in TH cells from Jnk1-/- mice. The Jnk1-/- T cells hyperproliferated, exhibited decreased activation-induced cell death, and preferentially differentiated to TH2 cells. The enhanced production of TH2 cytokines by Jnk1-/- cells was associated with increased nuclear accumulation of the transcription factor NFATc. Thus, the JNK1 signaling pathway plays a key role in T cell receptor-initiated TH cell proliferation, apoptosis, and differentiation.

MKK7 is an essential component of the JNK signal transduction pathway activated by proinflammatory cytokines

Mitogen-activated protein kinases (MAPK) are activated by phosphorylation on Thr and Tyr by MAPK kinases. Two MAPK kinases (MKK4 and MKK7) can activate the c-Jun NH(2)-terminal kinase (JNK) group of MAPK in vitro. JNK is phosphorylated preferentially on Tyr by MKK4 and on Thr by MKK7. Targeted gene-disruption studies in mice were performed to examine the role of MKK4 and MKK7 in vivo. Simultaneous disruption of the Mkk4 and Mkk7 genes was required to block JNK activation caused by exposure of cells to environmental stress. In contrast, disruption of the Mkk7 gene alone was sufficient to prevent JNK activation caused by proinflammatory cytokines. These data demonstrate that MKK4 and MKK7 serve different functions in the JNK signal transduction pathway.

A quantitative test to estimate neutralizing antibodies to the hepatitis C virus: cytofluorimetric assessment of envelope glycoprotein 2 binding to target cells

Hepatitis C virus (HCV) is a major cause of chronic hepatitis. The virus does not replicate efficiently in cell cultures, and it is therefore difficult to assess infection-neutralizing antibodies and to evaluate protective immunity in vitro. To study the binding of the HCV envelope to cell-surface receptors, we developed an assay to assess specific binding of recombinant envelope proteins to human cells and neutralization thereof. HCV recombinant envelope proteins expressed in various systems were incubated with human cells, and binding was assessed by flow cytometry using anti-envelope antibodies. Envelope glycoprotein 2 (E2) expressed in mammalian cells, but not in yeast or insect cells, binds human cells with high affinity (Kd approximately 10(-8) M). We then assessed antibodies able to neutralize E2 binding in the sera of both vaccinated and carrier chimpanzees, as well as in the sera of humans infected with various HCV genotypes. Vaccination with recombinant envelope proteins expressed in mammalian cells elicited high titers of neutralizing antibodies that correlated with protection from HCV challenge. HCV infection does not elicit neutralizing antibodies in most chimpanzees and humans, although low titers of neutralizing antibodies were detectable in a minority of infections. The ability to neutralize binding of E2 derived from the HCV-1 genotype was equally distributed among sera from patients infected with HCV genotypes 1, 2, and 3, demonstrating that binding of E2 is partly independent of E2 hypervariable regions. However, a mouse monoclonal antibody raised against the E2 hypervariable region 1 can partially neutralize binding of E2, indicating that at least two neutralizing epitopes, one of which is hypervariable, should exist on the E2 protein. The neutralization-of-binding assay described will be useful to study protective immunity to HCV infection and for vaccine development.

JNK is required for effector T-cell function but not for T-cell activation

The hallmark of T-cell activation is the production of interleukin 2 (IL-2). c-Jun amino-terminal kinase (JNK), a MAP kinase that phosphorylates c-Jun and other components of the AP-1 group of transcription factors, has been implicated in the activation of IL-2 expression. Previously, we found that T cells from mice deficient in the Jnk1 or Jnk2 gene can be activated and produce IL-2 normally, but are deficient in functional differentiation into Th1 or Th2 subsets. However, studies of mice with compound mutations indicate that JNK1 and JNK2 are redundant during mouse development. Here we use three new mouse models in which peripheral T cells completely lack JNK proteins or signalling, to test whether the JNK signalling pathway is crucial for IL-2 expression and T-cell activation. Unexpectedly, these T cells made more IL-2 and proliferated better than wild-type cells. However, production of effector T-cell cytokines did require JNK. Thus, JNK is necessary for T-cell differentiation but not for naive T-cell activation.

Microtubule binding to Smads may regulate TGF beta activity

Smad proteins are intracellular signaling effectors of the TGF beta superfamily. We show that endogenous Smad2, 3, and 4 bind microtubules (MTs) in several cell lines. Binding of Smads to MTs does not require TGF beta stimulation. TGF beta triggers dissociation from MTs, phosphorylation, and nuclear translocation of Smad2 and 3, with consequent activation of transcription in CCL64 cells. Destabilization of the MT network by nocodazole, colchicine, or a tubulin mutant disrupts the complex between Smads and MTs and increases TGF beta-induced Smad2 phosphorylation and transcriptional response in CCL64 cells. These data demonstrate that MTs may serve as a cytoplasmic sequestering network for Smads, controlling Smad2 association with and phosphorylation by activated TGF beta receptor I, and suggest a novel mechanism for the MT network to negatively regulate TGF beta function.

Defective antigen processing in GILT-free mice

Processing of proteins for major histocompatibility complex (MHC) class II-restricted presentation to CD4-positive T lymphocytes occurs after they are internalized by antigen-presenting cells (APCs). Antigenic proteins frequently contain disulfide bonds, and their reduction in the endocytic pathway facilitates processing. In humans, a gamma interferon-inducible lysosomal thiol reductase (GILT) is constitutively present in late endocytic compartments of APCs. Here, we identified the mouse homolog of GILT and generated a GILT knockout mouse. GILT facilitated the processing and presentation to antigen-specific T cells of protein antigens containing disulfide bonds. The response to hen egg lysozyme, a model antigen with a compact structure containing four disulfide bonds, was examined in detail.

Polymorphisms in inflammation-related genes are associated with susceptibility to major depression and antidepressant response

There are clinical parallels between the nature and course of depressive symptoms in major depressive disorder (MDD) and those of inflammatory disorders. However, the characterization of a possible immune system dysregulation in MDD has been challenging. Emerging data support the role of T-cell dysfunction. Here we report the association of MDD and antidepressant response to genes important in the modulation of the hypothalamic-pituitary-adrenal axis and immune functions in Mexican Americans with major depression. Specifically, single nucleotide polymorphisms (SNPs) in two genes critical for T-cell function are associated with susceptibility to MDD: PSMB4 (proteasome beta4 subunit), important for antigen processing, and TBX21 (T bet), critical for differentiation. Our analyses revealed a significant combined allele dose-effect: individuals who had one, two and three risk alleles were 2.3, 3.2 and 9.8 times more likely to have the diagnosis of MDD, respectively. We found associations of several SNPs and antidepressant response; those genes support the role of T cell (CD3E, PRKCH, PSMD9 and STAT3) and hypothalamic-pituitary-adrenal axis (UCN3) functions in treatment response. We also describe in MDD increased levels of CXCL10/IP-10, which decreased in response to antidepressants. This further suggests predominance of type 1 T-cell activity in MDD. T-cell function variations that we describe here may account for 47.8% of the attributable risk in Mexican Americans with moderate MDD. Immune function genes are highly variable; therefore, different genes might be implicated in distinct population groups.

Cutting edge: critical role of inducible costimulator in germinal center reactions

Inducible costimulator (ICOS) is a new member of the CD28/CTLA-4 family that is expressed on activated and germinal center (GC) T cells. Recently, we reported that ICOS-deficient mice exhibited profound defects in T cell activation and effector function. Ab responses in a T-dependent primary reaction and in a murine asthma model were also diminished. In the current study, we investigate the mechanism by which ICOS regulates humoral immunity and examine B cell GC reactions in the absence of ICOS. We found that ICOS(-/-) mice, when immunized with SRBC, had smaller GCs. Furthermore, IgG1 class switching in the GCs was impaired. Remarkably, GC formation in response to a secondary recall challenge was completely absent in ICOS knockout mice. These data establish a critical role of ICOS in regulation of humoral immunity.

Interaction with Suv39H1 is critical for Snail-mediated E-cadherin repression in breast cancer

Expression of E-cadherin, a hallmark of epithelial-mesenchymal transition (EMT), is often lost due to promoter DNA methylation in basal-like breast cancer (BLBC), which contributes to the metastatic advantage of this disease; however, the underlying mechanism remains unclear. Here, we identified that Snail interacted with Suv39H1 (suppressor of variegation 3-9 homolog 1), a major methyltransferase responsible for H3K9me3 that intimately links to DNA methylation. We demonstrated that the SNAG domain of Snail and the SET domain of Suv39H1 were required for their mutual interactions. We found that H3K9me3 and DNA methylation on the E-cadherin promoter were higher in BLBC cell lines. We showed that Snail interacted with Suv39H1 and recruited it to the E-cadherin promoter for transcriptional repression. Knockdown of Suv39H1 restored E-cadherin expression by blocking H3K9me3 and DNA methylation and resulted in the inhibition of cell migration, invasion and metastasis of BLBC. Our study not only reveals a critical mechanism underlying the epigenetic regulation of EMT, but also paves a way for the development of new treatment strategies against this disease.

Second primary neoplasms among 53 159 haematolymphoproliferative malignancy patients in Sweden, 1958-1996: a search for common mechanisms

The Swedish Family-Cancer Database was used to analyse site-specific risk of second primary malignancies following 53 159 haematolymphoproliferative disorders (HLPD) diagnosed between 1958 and 1996. Standardized incidence ratio (SIR) of a second malignancy was calculated as the ratio of observed to expected numbers of second malignancies by applying site-, sex-, age-, period-, residence- and occupation-specific rates in the corresponding population in the Database to the appropriate person-years at risk. Among 18 960 patients with non-Hodgkin's lymphoma (NHL), there was over a 3-fold significant increase in cancer of the tongue, small intestine, nose, kidney and nervous system, squamous cell carcinoma (SCC) of the skin, NHL, Hodgkin's disease (HD) and lymphoid and myeloid leukaemia. Among 5353 patients with HD, there was over a 4-fold significant increase in cancer of the salivary glands, nasopharynx and thyroid, NHL and myeloid leukaemia, and over a 1.6-fold increase in cancer of the stomach, colon, lung, breast, skin (melanoma and SCC), nervous system and soft tissues and lymphoid leukaemia. Among 28 846 patients with myeloma and leukaemia, there was a significant increase in cancer of the skin, nervous system and non-thyroid endocrine glands and all HLPD except for myeloma. Our findings showed some clustering between first and second primaries among Epstein-Barr virus-, ultraviolet radiation- and immunosuppression-related cancers.

Regulatory mechanisms of mitogen-activated kinase signaling

MAP kinases (MAPKs) are evolutionarily conserved regulators that mediate signal transduction and play essential roles in various physiological processes. There are three main families of MAPKs in mammals, whose functions are regulated by activators, inactivators, substrates and scaffolds, which together form delicate signaling cascades in response to different extracellular or intracellular stimulation. MAPK signaling is tightly regulated so that optimal biological activities are achieved and health is maintained. However, how the specificity of the signaling flow along each cascade is achieved is still relatively unclear. In this review, we summarize recent advances in understanding the regulation of MAPK cascades and the roles of MAP kinases and their regulators in development and in immune responses.

Sibling risks in cancer: clues to recessive or X-linked genes?

A systematic analysis of cancer risks to offspring and to siblings of cancer cases was carried out based on the nation-wide Swedish Family-Cancer Database. For all 13 cancer sites examined, risks to both offspring and siblings of cases of cancer at the same site were significantly elevated. The relative risk to siblings was approximately 2 fold more than the offspring risk for cancers of the prostate, testis, kidney and bladder, suggesting that recessive or X-linked susceptibility genes may be important for these cancers. Risks to siblings of cases where a parent was also affected were increased >20 fold over population rates for colorectal, ovarian, prostate and renal cancer, and for leukaemia, consistent with the effects of rare high-risk susceptibility alleles.

Impairment of T and B cell development by treatment with a type I interferon

Type I interferons alpha and beta, naturally produced regulators of cell growth and differentiation, have been shown to inhibit IL-7-induced growth and survival of B cell precursors in vitro. After confirming an inhibitory effect on B lymphopoiesis in an ex vivo assay, we treated newborn mice with an active IFN-alpha2/alpha1 hybrid molecule to assess its potential for regulating B and T cell development in vivo. Bone marrow and splenic cellularity was greatly reduced in the IFN-alpha2/alpha1-treated mice, and B lineage cells were reduced by >80%. The bone marrow progenitor population of CD43+B220+HSA- cells was unaffected, but development of the CD19+ pro-B cells and their B lineage progeny was severely impaired. Correspondingly, IL-7-responsive cells in the bone marrow were virtually eliminated by the interferon treatment. Thymus cellularity was also reduced by >80% in the treated mice. Phenotypic analysis of the residual thymocytes indicated that the inhibitory effect was exerted during the pro-T cell stage in differentiation. In IFN-alpha/beta receptor-/- mice, T and B cell development were unaffected by the IFN-alpha2/alpha1 treatment. The data suggest that type I interferons can reversibly inhibit early T and B cell development by opposing the essential IL-7 response.

Relationship of obesity to depression: a family-based study

OBJECTIVE

To examine the relationship between obesity and depression in a sample of extremely obese individuals and their siblings and parents.

SUBJECTS

A total of 1730 European Americans (558 men, 1172 women, aged 49.29+/-15.42 y, body mass index (BMI) of 35.57+/-11.53 kg/m(2)) and 373 African Americans (103 men, 270 women, aged 44.85+/-15.08 years, BMI of 36.83+/-11.31 kg/m(2)) in a sample of 482 nuclear families segregating extreme obesity and normal weight.

MEASUREMENTS

Individual BMI, history of depression treatment and covariates (age, sex, race, education, marital status, socioeconomic status, chronic medical conditions and exercise program).

RESULTS

Greater odds for depression were found for the obese, European American, women, the unmarried, the more educated, those with chronic physical disorder(s) and the offspring of depressed parents. A trend test found that the odds ratios for depression increased with BMI and number of chronic medical conditions (P<0.0001). Multivariate logistic regression analyses indicated that BMI, race, marital status, chronic medical conditions and family history were the predicators of depression for both the genders. Hierarchical analyses revealed that BMI significantly increased the risk above that predicated by the combined effects of all other variables.

CONCLUSIONS

Extreme obesity was associated with the increased risk for depression across gender and racial groups, even after controlling for chronic physical disease, familial depression and demographic risk factors. More detailed research is needed to determine the underlying mechanisms.

Biomechanics of cell rolling: shear flow, cell-surface adhesion, and cell deformability

The mechanics of leukocyte (white blood cell; WBC) deformation and adhesion to endothelial cells (EC) has been investigated using a novel in vitro side-view flow assay. HL-60 cell rolling adhesion to surface-immobilized P-selectin was used to model the WBC-EC adhesion process. Changes in flow shear stress, cell deformability, or substrate ligand strength resulted in significant changes in the characteristic adhesion binding time, cell-surface contact and cell rolling velocity. A 2-D model indicated that cell-substrate contact area under a high wall shear stress (20 dyn/cm2) could be nearly twice of that under a low stress (0.5 dyn/cm2) due to shear flow-induced cell deformation. An increase in contact area resulted in more energy dissipation to both adhesion bonds and viscous cytoplasm, whereas the fluid energy that inputs to a cell decreased due to a flattened cell shape. The model also predicted a plateau of WBC rolling velocity as flow shear stresses further increased. Both experimental and computational studies have described how WBC deformation influences the WBC-EC adhesion process in shear flow.

Second primary neoplasms in 633,964 cancer patients in Sweden, 1958-1996

The Swedish Family-Cancer Database was used to analyze concordant (same site) and discordant (different site) second primary neoplasms in 633,964 cancer patients diagnosed from 1958 to 1996. Cases of second malignant neoplasms were extracted from the Database if the diagnosis date of the first and second cancer differed by at least 1 month. The expected numbers of cancers were obtained by applying site-, sex-, age-, period-, residence- and socioeconomic level-specific rates in the corresponding population in the Database to the appropriate person-years at risk. The standardized incidence ratio (SIRs) of a second cancer was taken to be the ratio of observed to expected numbers of second cancers. Of all cancers, 8.5% were subsequent neoplasms (8.4% for males and 8.7% for females). SIRs for both concordant and discordant subsequent cancer were elevated in patients with cancer of the upper aerodigestive tract, colon, nose, breast, other female genitals, testis, kidney, urinary, bladder, skin, nervous system, endocrine, bone, connective tissue, melanoma, lymphoma and leukemia. The risks at some concordant sites, such as nose, squamous cell skin, bone and connective tissue in both sexes, breast in males and upper aerodigestive tract and leukemia in females, were very high (>10). At discordant sites, SIRs were less than 2 but significantly increased after all but gastric and prostatic cancer. Compared with the general population, cancer patients were at a modestly increased risk for new primary cancer after cancers at many sites, calling for attention in treatment, management and prevention.

Sox genes and cancer

Sox genes encode transcription factors belonging to the HMG (High Mobility Group) superfamily. They are conserved across species and involved in a number of developmental processes. In vitro studies have shown at least one Sox gene to be capable of inducing oncogenic transformation of fibroblast cells. In addition, overexpression and/or amplification of Sox genes are associated with a large number of tumour types in vivo. We review here the available evidence linking Sox gene expression and cancer, and show that this link is supported by extensive EST database analysis. This work provides a basis for further studies aimed at investigating the possible role of Sox genes in the oncogenic process.

c-Jun NH(2)-terminal kinase inhibits targeting of the protein phosphatase calcineurin to NFATc1

The protein phosphatase calcineurin is a critical mediator of calcium signals during T-cell activation. One substrate of calcineurin is the transcription factor NFATc1, which is retained in the cytoplasm of quiescent cells. NFATc1 activation requires the translocation of the transcription factor into the nucleus, a process that is mediated by calcineurin. This interaction with calcineurin requires a targeting domain (PxIxIT motif) located in the NH(2)-terminal region of NFATc1. Here we demonstrate that the calcineurin targeting domain of NFATc1 is phosphorylated and inactivated by the c-Jun NH(2)-terminal kinase (JNK). This disruption of calcineurin targeting inhibits the nuclear accumulation and transcription activity of NFATc1 and accounts for the observation that Jnk1(-/-) T cells exhibit greatly increased NFATc1-dependent nuclear responses.

Th1 and Th2 cells

After activation, CD4 helper T (Th) cells differentiate into Th1or Th2 effector subsets. These two types of cells produce distinct profiles of cytokines and regulate different immune responses. This review summarizes recent progress on different features of co-stimulatory regulation and chemokine-mediated homing of Th1 and Th2 cells. Transcription factors and signaling pathways that are selectively expressed or activated in Th1 and Th2 cells to regulate cytokine gene expression are discussed.

Involvement of the mt1 melatonin receptor in human breast cancer

Two putative melatonin receptors have been described including the cell surface G-protein-linked receptors, mt1 and MT2, and the nuclear retinoic orphan receptor alpha (RORalpha). The mt1 receptor, but not the MT2 receptor, is expressed in human breast tumor cell lines, and melatonin-induced growth suppression can be mimicked by the mt1 and MT2 agonist, AMMTC, and blocked by the antagonist, CBPT. RORalpha receptors are also expressed in MCF-7 breast cancer cells and the putative RORalpha agonist CPG-52608 inhibits MCF-7 cell growth but with a very different dose-response than melatonin. Finally, melatonin and AMMTC, but not CPG-52608, can repress RORalpha transcriptional activity in MCF-7 cells.

Tonsillar and other upper aerodigestive tract cancers among cervical cancer patients and their husbands

The study aimed at probing the possible role of human papillomavirus (HPV) infection in squamous cell carcinomas of the upper aerodigestive tract, with a special reference to tonsillar cancer. We used the Swedish Family Cancer Database to analyse second cancers in the upper aerodigestive tract of women first diagnosed with in-situ or invasive cervical cancer. First cancers of their husbands were also analysed. Standardized incidence ratios (SIRs) were calculated for female and male cancers, adjusted for age at diagnosis, period, sex, socio-economic status and residential area. Among women, increases were observed at many sites, but tonsillar cancers were increased only among women aged 50 years or more at diagnosis of in-situ cervical cancer (SIR 2.58). The increases at these sites are probably ascribed to the effects HPV, smoking, alcohol or their interaction. Husbands of cervical cancer patients developed an excess (SIR over 2.00) of both tonsillar cancer (SIR 2.39 when wife with in-situ cancer and SIR 2.72 when wife with invasive cervical cancer) and cancer of the tongue. The excess of tonsillar cancer among husbands of women with HPV-associated neoplastic lesions of the cervix supports the a priori hypothesis that HPV may be involved in tonsillar carcinogenesis.

Mechanics of leukocyte deformation and adhesion to endothelium in shear flow

The mechanics of leukocyte [white blood cell (WBC)] deformation and adhesion to endothelial cells (EC) in shear flow has been investigated. Experimental data on transient WBC-EC adhesion were obtained from in vivo measurements. Microscopic images of WBC-EC contact during incipient WBC rolling revealed that for a given wall shear stress, the contact area increases with time as new bonds are formed at the leading edge, and then decreases with time as the trailing edge of the WBC membrane peels away from the EC. A two-dimensional model (2D) was developed consisting of an elastic ring adhered to a surface under fluid stresses. This ring represents an actin-rich WBC cortical layer and contains an incompressible fluid as the cell interior. All molecular bonds are modeled as elastic springs distributed in the WBC-EC contact region. Variations of the proportionality between wall shear stress (tau(w)) in the vicinity of the WBC and the resulting drag force (F(s)), i.e., F(s)/tau(w), reveal its decrease with WBC deformation and increasing vessel channel height (2D). The computations also find that the peeling zone between adherent WBC and EC may account for less than 5% of the total contact interface. Computational studies describe the WBC-EC adhesion and the extent of WBC deformation during the adhesive process.

Passive deformation analysis of human leukocytes

The following analysis presents an experimental and theoretical study of the passive viscoelastic behavior of human leukocytes. Individual neutrophils in EDTA were observed both during their partial aspiration into a small micropipette and after expulsion from a large micropipette where the cell had been totally aspirated and deformed into a sausage shape. To analyze the data, a passive model of leukocyte rheology has been developed consisting of a cortical shell containing a Maxwell fluid which describes the average properties of the cell cytoplasm. The cortical shell represents a crosslinked actin layer near the surface of the cell and is assumed to be under pre-stressed tension. This model can reproduce the results of experiments using micropipette for both short-time small deformation and slow recovery data after large deformation. In addition, a finite element scheme has been established for the same model which shows close agreement with the analytical solution.