Approximate Profile Maximum Likelihood

We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML has appealing theoretical properties, but is difficult to compute exactly. Inspired by observations gleaned from exactly solvable cases, we look for an approximate PML solution, which, intuitively, clumps comparably frequent symbols into one symbol. This amounts to lower-bounding a certain matrix permanent by summing over a subgroup of the symmetric group rather than the whole group during the computation. We extensively experiment with the approximate solution, and find the empirical performance of our approach is competitive and sometimes significantly better than state-of-the-art performance for various estimation problems

Additional file 13 of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

: Multiple alignment of the 42 sncRNAs over represented in CAF-EXO samples. (FASTA 8 kb

Additional file 2: of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

Absolute and average relative counts of reads mapped to each ncRNAs biotype per sample and fraction. (XLSX 15 kb

Additional file 11: of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

Table S1. sncRNAs distributed differently in CAF-EXO samples from in NF-EXO ones. Highly significant lncRNAs and sncRNAs (FDR < 1E-04) are highlighted in bold. (DOCX 79 kb

Additional file 4: of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

Figure S1. logFC-based MDS-plots (one for each type of lncRNA), where the first dimension corresponds to differences due to the type of sample (i.e. whether it is a normal cell, a tumor cell or an exosomal sample) and the second dimension corresponds to the differences between the samples themselves as biological replicates. The inferred dispersion and the Biological Coefficient of Variation (BCV) of all assayed samples for lncRNAs are 0.21201 and 0.4604, respectively, while the coefficients for sncRNA content are 0.25164 and 0.5016. These two coefficients reveal some interesting variation among samples. In this respect the multidimensional scaling (MDS) plots where the differences in ncRNA content between cellular and exosomal fractions and the heterogeneity of CAF samples, and especially their exosomes, are indicated as the main cause of this variation. Moreover, since each sequenced sample was prepared as a pool of three others and CAF samples and their exosomes are, as expected, more heterogenous because of their tumoral condition, then their variability should also be greater than that of NFs and their exosomes. Figure S1 also gives some differences that exist between NF and CAF exosomes. (PNG 104 kb

Additional file 5: of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

Excel file with two documents summarizing the results obtained from the differential expression NF-CELL versus NF-EXO analyses for differentially distributed lncRNAs and sncRNAs. (XLSX 73 kb

Additional file 7: of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

Mini web site presenting a dynamic venn diagram intersecting the relationships of significance from the assayed ncRNAs in the differential expression analyses performed between NF- and CAF- exosomes versus their respective cellular environments (i.e. NF-CELL versus NF-EXO and CAF-CELL versus CAF-EXO). Clicking on any intersected number, the web site opens a dialog summarizing the ncRNAs species that correspond to the intersection. (HTML 120 kb

Additional file 3: of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

Excel document with the two count files used as input to EdgeR for differential expression analysis; one with the counts of reads of all samples mapped on the lncRNA references and another with the read counts of reads mapped on scnRNAs. (XLSX 221 kb

Additional file 8: of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

Mini web site presenting a dynamic venn diagram showing the relationships between the results cellular or exosomal over represented in the two analyses performed between NF- and CAF- exosomes. Clicking on any intersected number, the web site opens a dialog summarizing the ncRNAs species that correspond to the intersection. (HTML 47 kb

Additional file 14 of Differential distribution and enrichment of non-coding RNAs in exosomes from normal and Cancer-associated fibroblasts in colorectal cancer

: Table S2. Target genes for CAF-EXO over-distributed sncRNAs supported by FDR < 1E-04. (DOCX 14 kb