The Probability Mass Function of the Kaplan-Meier Product-Limit Estimator

Kaplan andMeier’s 1958 article developed a nonparametric estimator of the survivor function from a right censored dataset. Determining the size of the support of the estimator as a function of the sample size provides a challenging exercise for students in an advanced course in mathematical statistics. We devise two algorithms for calculating the support size and calculate the associated probability mass function for small sample sizes and particular probability distributions for the failure and censoring times

Proteotranscriptomic Analysis Reveals Stage Specific Changes in the Molecular Landscape of Clear-Cell Renal Cell Carcinoma

<div><p>Renal cell carcinoma comprises 2 to 3% of malignancies in adults with the most prevalent subtype being clear-cell RCC (ccRCC). This type of cancer is well characterized at the genomic and transcriptomic level and is associated with a loss of <i>VHL</i> that results in stabilization of HIF1. The current study focused on evaluating ccRCC stage dependent changes at the proteome level to provide insight into the molecular pathogenesis of ccRCC progression. To accomplish this, label-free proteomics was used to characterize matched tumor and normal-adjacent tissues from 84 patients with stage I to IV ccRCC. Using pooled samples 1551 proteins were identified, of which 290 were differentially abundant, while 783 proteins were identified using individual samples, with 344 being differentially abundant. These 344 differentially abundant proteins were enriched in metabolic pathways and further examination revealed metabolic dysfunction consistent with the Warburg effect. Additionally, the protein data indicated activation of ESRRA and ESRRG, and HIF1A, as well as inhibition of FOXA1, MAPK1 and WISP2. A subset analysis of complementary gene expression array data on 47 pairs of these same tissues indicated similar upstream changes, such as increased HIF1A activation with stage, though ESRRA and ESRRG activation and FOXA1 inhibition were not predicted from the transcriptomic data. The activation of ESRRA and ESRRG implied that HIF2A may also be activated during later stages of ccRCC, which was confirmed in the transcriptional analysis. This combined analysis highlights the importance of HIF1A and HIF2A in developing the ccRCC molecular phenotype as well as the potential involvement of ESRRA and ESRRG in driving these changes. In addition, cofilin-1, profilin-1, nicotinamide N-methyltransferase, and fructose-bisphosphate aldolase A were identified as candidate markers of late stage ccRCC. Utilization of data collected from heterogeneous biological domains strengthened the findings from each domain, demonstrating the complementary nature of such an analysis. Together these results highlight the importance of the VHL/HIF1A/HIF2A axis and provide a foundation and therapeutic targets for future studies. (Data are available via ProteomeXchange with identifier PXD003271 and MassIVE with identifier MSV000079511.)</p></div

Differential protein abundance and gene expression between tumor and normal-adjacent ccRCC tissues.

<p>(A) Heatmap of 344 proteins with differential abundance between tumor and normal-adjacent samples (moderated <i>t</i>-test, Benjamini-Hockberg adjusted <i>p</i>-value < 0.05). (B) Heatmap of 1003 genes with differential expression between 94 tumor and normal-adjacent samples (moderated <i>t</i>-test BH adjusted <i>p</i> < 0.001 and absolute fold-change ≥ 4). Scale bar is standard deviation units around the mean of each protein abundance or gene expression level.</p

Correlation of differentially abundant proteins and respective gene expression levels in matched samples.

<p>(A) There were 725 proteins identified which had 1764 corresponding probes in the corresponding transcriptomic data. Pearson's linear correlation coefficient was used to correlate normalized spectral count levels and RMA normalized microarray data in matched samples (94 samples). The average Pearson's linear correlation coefficient (<i>r</i>) was 0.157 (dotted line). (B) Distribution of <i>r</i> for just the 344 differentially abundant proteins, of which 318 were also measured in the corresponding transcriptomic study. The average <i>r</i> was 0.347 (dotted line).</p

Candidate markers of advanced stage ccRCC.

<p>Heatmap of log₂ fold-change in protein abundance of four candidate markers of late stage ccRCC, cofilin-1 (CFL1), profilin-1 (PFN1), nicotinamide N-methyltransferase (NNMT), and fructose-bisphosphate aldolase A (ALDOA), in paired tumor and normal-adjacent tissues from 84 individuals, as well as 9 pairs that also included metastasis tissue.</p

Upstream regulation with stage.

<p>(A) Heatmap of six upstream targets predicted to be activated/inhibited in tumor versus normal at each ccRCC stage and using all stage data together. Scale is activation z-score, with ≥ 2 being likely activation and ≤ -2 being likely inhibition. Using all stage data together, activated or inhibited upstream targets (ESRRA, ESRRG, HIF1A, FOXA1, MAPK1, and WISP2) are shown. (B) Heatmap of six upstream targets predicted to be activated/inhibited in tumor versus normal-adjacent tissues using the proteomic data set. Corresponding activation z-scores from transcriptomic data analysis are included to demonstrate conserved trends at each ccRCC stage and using all stage data together.</p

Protein abundance changes with stage in the glycolysis pathway.

<p>Using only proteins that were differentially abundant at each stage, the glycolysis pathway was interrogated. Directionality of change is indicated by red (increased in tumor) or green (decreased in tumor) and the small bar graph next to each protein symbol in the pathway is protein fold-change (tumor/normal) at stage I, II, III, IV and then all stages (from left to right). Levels of two proteins, PFKP and PKM, were confirmed using IHC staining of stage I ccRCC tissue (other stages were not evaluated by IHC). A representative IHC image is shown for PFKP and PKM along with average staining intensity (H value) ± standard error of a stage I ccRCC TMA. For both PFKP and PKM the average log₂ fold-change (FC) levels ± standard error for protein abundance (stage I-IV and metastasis tissue) and gene expression (stage I-IV) are displayed as bar graphs (‘*’ indicates significance, BH adjusted <i>p</i> < 0.05). Below each pair of bar graphs is a scatter plot of log₂ normalized spectral counts (protein) versus RMA normalized array gene expression data from the 94 tissues with proteomic and transcriptomic data.</p