d. Viability measurement Cells seeded in 24 well plates were treated with different concentrations of curcuma DMSO e tract, curcuma sellectchem ethanol e tract or curcumin. All e peri ments were performed in triplicates on cells from 5 inde pendent biopsies. After 6, 18 and 30 hours, to icity was analyzed using the MTT assay A fresh sterile solution of MTT with a concentration of 0. 5 mg ml in DMEM F12 was prepared, 500 ul were added to each well and incubated for 4 hours at 37 C. MTT was discarded, cells were lysed with DMSO for 5 min at 37 C and absorbance was measured at 565 nm. Absorbance of treated cells was calculated relative to ab sorbance of untreated control cells, which was set to 100%. Con centrations that were non to ic even at late time points were chosen for subsequent e periments.
Results of the MTT assay were previously shown to be comparable to other viability measurement techniques. Gene e pression analysis Human intervertebral disc cells were serum starved for 2 hours and then e posed to 5 ng ml IL 1B for 2 hours before adding 100 ug ml cur cuma DMSO e tract or 100 ug ml curcuma EtOH e tract for 6 hours. Untreated control cells were included to verify the inflammatory and catabolic response induced by IL 1B treatment. As we were able to show that the solvents did not influence cellular behavior, all groups were treated with the respective volume of either DMSO or EtOH in all e peri ments. Therefore, changes in gene e pression are either calculated relative to controls or relative to IL 1B prestimulated cells.
Based on the results with curcuma e tracts and data obtained by HPLC MS analysis, a 25 mM stock solution of curcumin was prepared and cells were treated with final concentrations of 5, 10 or 20 uM curcumin for 6 hours after IL 1B prestimula tion. Taking the appro imate percentage of curcumin in curcuma powder into account, the applied range of curcumin was predicted to be similar to the final concentration of curcumin when using the above mentioned curcuma e tracts. All gene e pression e periments were performed on cells from five independent biopsies. After treatment, cells were harvested by trypsin treat ment and total RNA was isolated using the PureLink RNA Mini Kit according to the manufac turers instructions. cDNA was synthesized using TaqMan Reverse Transcription Reagents and gene e pression of IL 1B, IL 6, IL 8, TNF, MMP1, MMP3, MMP13, TLR2 and TBP was analyzed.
Human specific probes and primers, TaqMan real time RT PCR Mi and 10 30 ng of cDNA were mi ed and measured in duplicates using the StepOne Plus Real Time PCR System . The comparative ct method was used to quantify PCR data. In order Entinostat to calculate changes in gene e pression induced by curcuma curcumin, gene e pression in IL 1B treated cells was set to 100% and gene e pression of IL 1B curcuma or IL 1B curcumin treated cells was calculated relative to IL 1B treated cells. Western Blot for NF ��B In order to investigate selleck catalog whether changes in NF ��B p65 translocation occur after
in 2 fold increase in MMP 2 secretion although the absolute levels secreted from the CCE cells were lower than VEH. Consistent with porcine selleck chemicals Imatinib cells, human SMC secreted MMP 2 basally and this was further increased by TPA stimulation in both SV and AAA. MMP 9 secretion was not detected under any condi tion in porcine or human SMC, either basally or with TPA stimulation. Discussion This study has revealed a number of key findings. Firstly we maintained viable porcine carotid arteries under flow conditions in a bioreactor model for 12 days. Histo logical e amination revealed that vessel wall architecture in control vessels was identical to that of freshly isolated PCA, but protease pre treatments either indi vidually or combined, led to visible disruption of the ar terial wall.
Within the time frame studied and under these conditions we did not however, observe an unam biguous dilatation of the vessel although we speculate that the thinning we observed preceded overt dilation which may well become apparent at a later time point. Secondly, viable cells were cultured from all vessels and confirmed as SMC through co e pression of SMA and SM MHC. All porcine SMC e hibited characteristic spindle morphology with the e ception of those cultured from the combined protease treated vessels that were more rhomboid, a trait common to dedifferentiated, often pathological SMC. The aberrancies in PCA SMC morphology evident after treatment with CCE were recapitulated in SMC from end stage human AAA tissue. In agreement with a previous report, AAA SMC were morphologically distinct from SV SMC and also from the aortic SMC obtained from a commercial source.
SMC phenotypic switching underlies their unique abil ity to elicit compensatory responses to vascular injury. Indeed, increased SMC proliferation is a prominent fea ture of occlusive vascular diseases. Conversely, it is well established that SMC depletion is a hallmark of AAA, which might suggest functional inability of the SMC to remodel the degenerating aortic wall. In this study we revealed that AAA SMC consistently prolifer ated more slowly than non aneurysmal SV SMC cultured from age and se matched patients. Similarly, the prolifer ative capacity of SMC was reduced to a similar degree in porcine CCE SMC compared with paired VEH cells. Re ports relating to proliferative capacity of AAA SMC com pared to non aneurysmal SMC are at variance.
claims of both increased and decreased proliferation have been documented. In the latter, AAA SMC consist ently proliferated by up to 70% less than inferior mesenteric artery SMC, comparator GSK-3 cells that were cultured from the same patients. In the current study we e amined SMC from AAA and SV sources from a total of 24 different pa tients. Given our e pertise and familiarity with inherent variability between individual patients and our documented evidence supporting the intrinsic heterogen eity of SMC populations, this is an important Tubacin 537049-40-4 aspect of the current study. Whilst SMC derived
ression are not selleck inhibited by FLLL32 Since many cytokines act via homologous STAT proteins, it was imperative to test whether FLLL32 had deleterious effects on the action of cytokines that might promote an anti tumor response. Of concern were the effects of FLLL32 on signal transduction in response to IFN, a cytokine that mediates its cellular effects via phosphorylation of STAT1, and a resulting STAT1 STAT1 homodimer. To test these interactions in a biologic system, we investigated the effects of FLLL32 or curcumin pre treatment on IFN induced signaling and gene e pression. Pre treatment of pSTAT3 positive A375 and Hs294T cells with FLLL32 or curcumin led to reduced pSTAT3 versus DMSO treated cells. However, in contrast to curcumin, FLLL32 did not adversely affect IFN induced pSTAT1.
A unique advantage of FLLL32 versus other STAT3 pathway inhibitors was its apparent specificity. Despite a similar degree of cytoto icity and the ability to reduce basal pSTAT3 in human melanoma cells, the WP1066, JSI 124, and Stattic compounds also inhibited IFN induced STAT1 phos phorylation. Pre treatment with FLLL32 also enhanced transcription of the pro apoptotic interferon regulatory factor 1 gene in response to IFN stimulation as determined by Real Time PCR. This IFN responsive gene has been shown to be tran scribed via STAT1 STAT1 homodimers binding to a gamma activated sequence element. Consis tent with our prior studies, IFN stimulated IRF1 transcription was reduced in all cells pre treated with curcumin.
The induction of IRF1 was not enhanced in the pSTAT3 negative 1106 MEL cell line, suggesting that cross reactivity of FLLL32 with STAT1 was negligible, and that IFN driven gene transcription can be augmented via STAT3 inhibition. These data indicated that IFN induced signal transduc tion and gene e pression were not reduced by FLLL32 and that its inhibitory actions were specific for STAT3 and not other homologous STAT proteins that function as tumor suppressors. Effects of FLLL32 on immune effector cells STAT3 function in immune cells can promote tolerance to developing or established tumors. We therefore evalu ated whether FLLL32 would affect the responsiveness of PBMCs to stimulation with clinically relevant cytokines that mediate tumor progression, immunosurveil lance or T and NK cell survival.
Pre treatment with increasing doses of FLLL32 reduced basal pSTAT3 in PBMCs from healthy donors and led to reduced IL 6 induced pSTAT3 in PBMCs. FLLL32 pre treatment also did not adversely affect the level of IFN induced pSTAT1 or IRF1 gene e pression in PBMC. The level of IL 2 induced pSTAT5 also was not altered by FLLL32 pre treatment. The FLLL32 compound did not decrease Cilengitide via bility of PBMCs after a 24 hour treatment as compared to treatment with DMSO alone as determined by Anne in V PI staining or PARP cleavage. Similarly, NK cell viability from healthy donors cultured with IL 2 was not reduced following a 24 hour treatment with selleckbio FLLL32 as compared to treatment
microarrays according to their standard protocols. Briefly, cDNA was synthesized from 10 ug of RNA, labeled with Cy3, and hybridized to three replicate Nim bleGen S. cerevisiae 1 plex www.selleckchem.com/products/CP-690550.html 385K arrays for each RNA sample. Following washing and scanning of the arrays, data was extracted from the scanned image and analyzed for nor malized gene expression summary values by quantile normalization and the Robust Multi array Average algorithm using the NimbleScan software. ArrayStar 3. 0 software was used to analyze the expression data provided by NimbleGen. Mean TE4G and TEWT values were calculated for each gene from all nine microarray measurements of HP or T mRNA intensities obtained in the three biological replicates to obtain the log log plot in Figure 4.
To calculate mean TE4G TEWT ratios for the purpose of assigning standard errors to the values, the ratios were calculated separately for each project from the mean TE4G and mean TEWT values calculated from the three technical replicates for that project, and the resulting TE4G TEWT ratios for each project were averaged. The three mean TE4G and mean TEWT values determined in this way from projects I III were also used to conduct two tailed Students t tests of the significance of differences between mean TE4G and mean TEWT values for individual genes. Accession number The microarray data discussed in this publication have been deposited in NCBIs Gene Expression Omnibus and are accessible through GEO Series accession num ber GSE25721 acc GSE25721.
Real time quantitative RT PCR analysis of polysomal mRNA distributions RNA samples from the WCE or gradient fractions con taining HP, LP, or 80S monosomes were isolated as pre viously described. The level of mRNA for each gene of interest relative to the amount of 18S rRNA was quantified by qRT PCR analysis. Briefly, cDNA was synthesized from 1 ug of RNA using SuperScript III First Strand Synthesis SuperMix according to the vendors recommended protocol. The synthesized first strand cDNA was diluted 1,10, and 2 ul of the diluted cDNA was used for subsequent real time PCR amplifica tion using the Stratagene MX3000P and Brilliant II SYBR Green QPCR Master Mix according to the vendors instructions. The primers used in qRT PCR ana lysis for the mRNAs analyzed in Figure 2 are listed in Table S2. The real time PCR reac tions were carried out in triplicate for each cDNA sample to obtain AV-951 average Ct values.
The amount of mRNA in a set of gradient fractions containing HP, LP or 80S species relative to its level in total RNA was determined by first calculating screening library 2 Ct, where Ct Ct norm Ct norm, Ct norm Ct GOI Ct 18S, and Ct norm Ct GOI Ct 18S. Ct GOI and Ct GOI are the Ct values determined for the gene of interest in the appropriate gradient fractions or total RNA, respectively, Ct 18S and Ct 18S are the corresponding values for 18S rRNA. The resulting 2 Ct values were then multiplied by a factor representing the proportion of the total A280 units in the gradient found in th
tated transcripts could encode functional proteins. Of the unannotated transcripts, 213 and 436 were differentially expressed in response to salinity stress. These unannotated transcripts encoded proteins inhibitor Volasertib associated with functions such as amino acid metabolism in response to abiotic stress, diterpenoid biosynthesis, and mechanosensitive ion channel function. Mechanosensitive ion channels are gated directly by physi cal stimuli such as osmotic shock and transduce these sti muli into electrical signals. mRNA Seq also captured previously identified genes involved in salinity tolerance, namely those associated with trehalose synthesis, dehydrin, ABA synthesis, sugar transport, glycerol transferase, and transcription factors similar to those of the DREB family.
A substantial number of transcripts were exclusively upregulated only in the root. As only the root was directly exposed to 1 h of salinity stress, it might take time to induce the expression of more genes in the shoot, OsTPP1 might be expressed in the shoot after 10 h of exposure, as has been found in Yukihikari rice. With these genes, Nippon bare may have the potential to be tolerant to salinity stress. Rice cultivars such as Nona Bokra and Pokkali are substantially more salinity tolerant than Nipponbare, suggesting that the genuine salinity stress tolerance gene might be missing in Nipponbare. The 23 Oryza species are geographically, physiologically, and geneti cally diverse, and many of the genes in cultivated rices have been selected by humans under field condi tions, not by environmental stress.
These essentially missing genes could serve as potential genetic resources for the improvement of cultivated crops. Sequence based technology can be used to extract such missing genes by the piling up of short reads on their own gen omes without the need to rely on sequence similarity. Overcoming the technical inaccuracy Microarray technology has been used as a sophisticated platform for the expression profiling of previously anno tated genes. However, as an array based technology, eva luation of signal intensities close to background levels tends to cause artifacts in array analysis because of high levels of background noise and or cross hybridization, moreover, hybridization efficiency might vary with the probes used, suggesting that the calculation of real molar concentrations is inaccurate.
Whereas the Agilent rice 44K Array is designed to quantify 60 mer sequences at the 3 end of transcripts, mRNA Seq quantifies tran script abundance on the basis of Entinostat the number of mapped sequences on the whole gene model. In our study, the two measures of transcript abundance and change ratios were highly correlated, as in a previous report. Moreover, for genes expressed at low or extremely high levels and for genes differentially expressed in arrays, mRNA Seq seemed to be accurate. There fore, mRNA Seq measures the molar concentrations mostly of genes accurately over a broad dynamic range. Biological replication is
re time point of jejunal segments in pig selleck lets infused with ETEC is warrant. From the results here and comparable studies, it is clear that ETEC stimulates a typical inflammatory re sponse in porcine intestinal cells, the extent of which is different according to the different ETEC strain, MOI and infection time. As mentioned above, more immune related genes which respond to F4ac ETEC or F4ab ETEC infection were detected in IPEC J2 cells than those respond to F18ac ETEC infection. It is probably due to the following reasons, Compared to F18ac ETEC, the serotypes and virulence genes of F4ab and F4ac ETEC are more similar. Adhesion ability of the three ETECs is different. At the same time point, the F4ac ETEC was the most adhesive strain, followed by F4ab ETEC with a little bit lower adhesion value, whereas F18ac ETEC showed the lowest adhesion pattern compared to F4ac ETEC and F4ab ETEC.
It has been reported that, in contrast to F4ac ETEC, F18ac ETEC has a slower colonization to the gut in vivo and it does not adhere to IPEC J2 cells nor be internalized by IPEC J2 cells in vitro. Many reports have focused on the receptor genes of ETEC F4 and F18, since they cause severe diarrhoea and edema disease in piglets. For ETEC F18, the two variants F18ab and F18ac are considered to recognize the same receptor and FUT1 is reported as the causative gene for F18 susceptibility. Up to now, a group of investigators have been searching for the ETEC F4ab F4ac receptor gene. The acknowl edged possible candidate genes include, MUC4, MUC13 and MUC20, and the latest inferred interval where the receptor gene is located is between the LMLN locus and microsatellite S0283.
In this study, the infection with F4ab ETEC Carfilzomib slightly down regulated the mRNA levels of FUT1 and MUC13 in the IPEC J2 cells, while in the F4ac ETEC infected IPEC J2 cells, the down regulated genes included, FUT1, MUC4, MUC13 and MUC20. Al though the mechanism about how ETECs infections cause down regulation of the above genes in the IPEC J2 cell line is not clear, the highly and constitutively expressed cell surface mucin MUC13 were reported to protect against intestinal inflammation in mice. We therefore suppose that intense inflammation in intestine IEC may disturb the expression of these mucin genes and further study in different time point with different MOI of ETEC is warrant.
Conclusions Gene expression profiles of the IPEC J2 cells Belinostat PXD101 with and with out F4ab, F4ac or F18ac ETEC infection were evaluated and compared. This transcriptome approach allowed us to obtain a global overview of genes and their different func tional entities involved in response to separate infection with F4ab, F4ac and F18ac ETEC specifically and or com monly. In summary, strong differential host responses to these three ETEC infections were observed. F18ac ETEC infection positively modulated the cell cycle progression and immune response of IPEC J2 cells. F4ab ETEC infec tion caused a dramatic up regulation of genes in cell cy
001), age (P < 0.001), selleck chemical Olaparib gender (men having lower scores than women, P = 0.042), and insulin treatment, IT patients being less depressed than NIT (P < 0.001), but none of the clinical variables. Anxiety correlated with age (P < 0.001). The association between depression and anxiety became progressively weaker with increasing age. These data confirm increased prevalence of depression in a population of patients with type 2 diabetes who did not show impaired cognitive function. The lack of correlation with disease duration, metabolic control, and complications suggests that depression may not appear/worsen with diabetes and/or its complications but rather supports suggestions that it might predate both.
Endothelial cell (EC) survival is critical in the maintenance of endothelial function as well as in the regulation of angiogenesis and vessel integrity since endothelial dysfunction is the initial lesion of atherosclerosis. The goal of this study was to examine the effect of diazoxide, a mitochondrial ATP-sensitive K+(mito K-ATP) channel opener, on aorta ECs apoptosis and its potential mechanism in Otsuka Long-Evans Tokushima Fatty (OLETF) rats at prediabetic stage. Diazoxide (25 mg kg(-1) day(-1)) was administered intraperitoneally from age 8 weeks to age 30 weeks. Thoracic aorta and cultured thoracic aortic ECs were used. The thickening of thoracic aortic wall and apoptosis of ECs were markedly increased in OLETF rats early from the age of 16 weeks, at the impaired glucose tolerance stage, compared with Long-Evans Tokushima Otsuka rats, in conjunction with intimal hyperplasia and perivascular fibrosis.
In contrast, diazoxide treatment inhibited these changes. Further study strongly demonstrated that extracellular signal-regulated kinases (ERKs) are key regulatory proteins in protecting ECs from apoptosis. Diazoxide could significantly enhance phosphorylation of ERK via opening mito K-ATP channels. This role was reversed by both 5-hydroxydecanoate, selectively closing mito K-ATP channels, and PD-98509, MEK inhibitors. The present studies demonstrate that diazoxide prevents the onset and development of macrovascular disease in OLETF rats by inhibiting apoptosis directly via phosphorylated ERK increase in aorta ECs. Our findings establish the basis for the therapeutic potential of diazoxide in atherosclerotic disease.
The association between diabetes and subclinical atherosclerosis is well established. Batimastat The effect of non-diabetic glucose intolerance on early atherosclerosis is not as straightforward, and the data regarding considering sex-related differences in this matter are limited. Therefore, our aim was to investigate these associations in men and women separately. We studied 1,304 Finnish men and women over 45 years of age who participated in the Finnish Health 2000 Survey. Ultrasonically determined carotid artery intima-media thickness and elasticity were used as markers of early atherosclerosis.
Moreover, PD-MSCs inhibited cytokine secretion (interleukin-12, tumor necrosis factor-alpha and interferon-gamma) of activated T cells. In vivo, the selleck chemicals llc survival rate in the PD-MSC group (transplanted with 1 x 10(6) cells) was higher than that in the control group and histological scores were low in the PD-MSC group. Conclusion: We present the first evidence that human PD-MSCs can efficiently control GVHD in an HSCT in vivo model. Copyright (c) 2012 S. Karger AG, Basel
Although survival rates for acute lymphocytic leukemia (ALL), especially in children, have shown dramatic improvement over time, poor outcomes are still observed in patients who have refractory or relapsed disease after conventional chemotherapy. New therapeutic options are urgently needed.
Bortezomib (Velcade, formerly PS-341) is the first proteasome inhibitor approved by the US FDA for the treatment of newly diagnosed multiple myeloma and relapsed/refractory multiple myeloma and mantle cell lymphoma. Although the mechanisms of bortezomib anticancer activity are still not completely understood, it is a new treatment option for patients with refractory or relapsed ALL, particularly when used in combination with conventional chemotherapy or targeted agents. This review summarizes recent advancements in the understanding of the bortezomib molecular mechanism of action in ALL. Understanding of the molecular approaches might help customize cancer chemotherapy for each individual patient, directing the field towards rational therapeutics. Copyright (c) 2013 S.
Karger AG, Basel
Objectives: Cigarette smoke contains free radicals, which cause injury to endothelial cells and oxidize bioactive components in the blood. Neutrophils, a subpopulation of leukocytes, contain the enzyme myeloperoxidase that mediates production of hypochlorous acid during oxidative stress. Dacomitinib In this study, we investigated whether smoker industrial workers had significantly higher neutrophil counts than nonsmoker industrial workers. Design and Methods: We collected blood samples from 183 apparently healthy male and 30 female industrial workers. We obtained blood cell counts, measured the concentration of plasma aminothiols and determined the concentration of serum and erythrocyte folate and serum vitamin B-12 in the samples. Results: Smoker industrial workers had significantly higher neutrophil, lymphocyte, monocyte, eosinophil and basophil counts than nonsmoker industrial workers (p < 0.
0001, p < 0.0001, p < 0.0001, p < 0.0001 and p = 0.01, respectively). Mean corpuscular volume and mean corpuscular hemoglobin in smoker industrial workers were higher than in nonsmoker industrial workers (p = 0.001 and p = 0.03). Conclusion: www.selleckchem.com/products/17-AAG(Geldanamycin).html Our study demonstrates that smoker industrial workers have higher neutrophil counts than nonsmoker industrial workers.
PCN depletes GSH in cultured airway epithelial cells and inactivates catalase. Excessive ROS RNS production and inhibition of antioxidative mechanisms by http://www.selleckchem.com/products/BIBF1120.html PCN overwhelm the antioxidant capacity of the tissue, leading to lung damage. PCN damages cili ated epithelium and inhibits mucus transport, induces bronchoconstriction, and decreases trachea mucus velocity. Furthermore, PCN inhibits NO produc tion in macrophages and endothelial cells, prostacyc lin production by endothelial cells, oxidation of leukotriene B4 by neutrophils, eicosanoid metabolism by platelets, and production of IL 2 and the IL 2 receptor in T cells. PCN has opposite effects on air way epithelial cells, inhibiting the release of RANTES and MCP 1 while stimulating Ca2 signaling and IL 8 release.
Finally, PCN inactivates 1 protease inhibitor and causes Entinostat apoptosis in neutrophils. Antioxidants detoxify PCN, suggesting that its virulence is redox dependent. Importantly, we have shown that PCN is important for both acute and chronic lung infections. GCHM, excessive mucus secretion and defective mucociliary clearance, airway obstruction, bacterial infection, and neutrophilic infiltration are important clinical features of CF and other chronic airway diseases. We have shown that mouse lungs chronically exposed to PCN undergo remodeling characterized by over proliferation of goblet cells in large bronchi and terminal bronchioles, emphysema, fibrosis, and an influx of immune cells. These pathological features resemble the airways of FOXA2 mice, as well as the CF and COPD airways chronically infected by PA.
Importantly, we have shown that PCN inhibits FOXA2 expression by activating the pro GCHM signaling pathways Stat6 and EGFR. In this study, we tested the hypothesis that PCN generated ROS RNS posttranslationally modify FOXA2, disabling its ability to regulate GCHM and mucin expression. Materials and methods PCN and chemicals All chemicals, including PCN were purchased from Sigma Chemical Co. unless stated otherwise. Chemically synthesized PCN is preferred over PCN purified from PA cultures to eliminate any contaminants, which may cause lung injuries. PCN was resuspended to 1 ug ml in sterile H2O. Cell cultures The human lung mucoepidermoid carcinoma cell line NCI H292 was purchased from the American Type Culture Collection. 16HBE cells were a generous gift from Dr. D. C. Gruenert.
NCI H292 and 16HBE cells were cultured in RPMI 1640 and MEM respectively, supplemented with 10% fetal bovine serum in 5% CO2. Epithelial cells that www.selleckchem.com/products/GDC-0449.html reached 70% confluency were serum starved for 24 hr before exposure to indicated concentra tions of PCN. As a control, cells were exposed to sterile H2O that corresponded to maximum volume of PCN used in each experiment. For example, 12. 5 ul ml sterile water was used per milliliter of culture medium in Figure 1B. Normal human bronchial epithelial cells were pur chased from Lonza.
One differ ence is that, as HNauty allows for several different edge types, the adjacency matrices associated with the graphs in HNauty may contain selleck CHIR99021 not only 0s and 1s for entries but can have entries of the form 2i where i is taken over those edges of type i between the two given ver tices. For a graph with two edge types, the entries in the adjacency matrix can be 0, 1, 22 1 21 2 or 1 2 3. A value of 3 should be interpreted to mean that there is both a hierarchy edge and a bond edge between two vertices. Another difference lies in how equitable partitions are calculated. We define a slight generalization to deal with labeled edges. A generalized equitable partition is an ordered partition P of the vertices of a labeled multi graph such that for any edge label e, the graph restricted to the edges labeled e, denoted ? |e, satisfies, the two sets of respective permutations will be equiva lent.
Thus, if g is an automorphism of the graph, it ? where d is the number of edges labeled e between x and Si. In other words, the partition is equita ble with respect to the graph restricted to any single edge type. It can be proved that, Anacetrapib given a partition P, there exists a unique coarsest generalized equitable refinement of P. To see this, note that it is enough to prove it for un ordered partitions. Now, suppose that Q1 and Q2 are both generalized equitable refinements of P. If Q1 and Q2 are different as un ordered partitions, then clearly their join, Q is also general ized and equitable. In fact, a basic property of lattices implies that Q is also a refinement of P.
As Q is coarser than both Q1 and Q2, it follows that P has a unique coarsest generalized equitable refinement. This property is the only property of generalized equitable partitions that is necessary to use them in place of equitable partitions. Implementation Except for using generalized equitable partitions in place of equitable partitions, our implementation follows the description given by McKay. Apparently, the actual Nauty program contains some efficiencies not described in. Thus, our algorithm is unlikely to be as finely tuned as Nauty. For an indicator function, we use the shape of the partition together with the shapes of the parent nodes in the search tree. By shape we mean the sizes of the individual cells of the partition.
The partition has shape as it has two cells of size 1, one cell of size 2, no cells of size 3 and one cell of size 4. These tuples are lexicographically ordered. This indicator function is invariant under auto morphisms of the graph as required. Y27632 Indeed it is invariant under any permutation of the vertices. We implemented our algorithm in both Perl and Python. The Perl version of HNauty is available as Additional file 1. The Python version of HNauty is avail able as Additional file 2. HNauty is also available at the BioNetGen website. The Perl version has been incorporated into BioNetGen. HNauty is turned off by default in BioNetGen.