Computing persistent homology in biological datasets

National Institute Of Diabetes And Digestive And Kidney Diseases

Investigators

Linked publications, trials & patents

Abstract

Persistent homology (PH) is a tool from Topological Data Analysis that can determine robust topological features in a data set. We work on computing PH for a point-cloud data set, specifically in our case, a set of points in a two- or three-dimensional space. The topological features that we compute can be interpreted as holes in the point-cloud in two-dimensional space and as hollow polyhedra in three-dimensional space. After determining topological features that are significant, we aim to explain possible functional relevance of these features in the physical system underlying the data. Such an analysis might reveal mechanistic features of the system that are related to or driven by its spatial structure. A data set of interest to us is the genome-wide Hi-C interaction map at 1 kb resolution, that is comprised of around 3 million points when considered as a point-cloud. Significant topological features in this data set are indicative of genes that are far apart along the linear chromosome to be spatially close to each other in the folded genome. This might elucidate long-range genomic interaction and regulation, that arises as a feature of the folding of the chromosome. However, when computing the PH, the size of this data set turned out to be beyond the computational ability of pre-existing software packagesthey either ran out of memory or were running for hours before we manually interrupted them. To surpass this hurdle, we developed a novel algorithm that was able to process the same data set in under four minutes, using only 4 GB of memory. Further, we computed PH of human genome under two different experimental conditions, with and without auxin. Auxin is a molecule that impairs function of cohesin, which is a protein complex that has been observed to localize at anchors of chromatin loops in the DNA. The results showed a decrease in the number of significant topological features upon addition of auxin. This provides supporting evidence for the prevalent hypothesis that cohesin is integral for loop formation in the human genome. In general, we have shown that our algorithm to compute PH outperforms others in most cases and has an efficient balance between memory consumption and computation time. We call it Dory and make it available as a user-friendly Python package. We have submitted this work to a journal for review. After computing PH, we explore possible functional significance of the computed topological features. This requires determining a representative location or boundary of significant topological features in the point-cloud data set. However, this computation is not well-defined and the resulting locations are not geometrically precise. As a result, most analyses that use PH are limited to studying significance of topological features. To surmount this hurdle, we developed new strategies to compute representative boundaries with improved geometric precision. We used our tool to analyze arrangement of cells (alpha, delta, beta) in human pancreatic islets. It has been hypothesized that these cells are arranged such that there are clusters of beta cells that are surrounded by alpha-delta mantle to confer efficient signaling in the pancreatic endocrine system. We compared the possibility of such an arrangement between control and diabetic subjects. This was done by using our algorithm to compute locations of holes in the alpha-delta structure (experimental data from 2D slices of human pancreatic islets), and then counting beta-cells in them. Our results showed that, compared to control subjects, a higher percentage of diabetic subjects have a low percentage of beta-cells that are surrounded by an alpha-delta mantle. This can possibly support the alpha-delta mantle hypothesis by suggesting that disruption in this particular arrangement of cells might be a contributing factor in impairment of proper function of the endocrine system in diabetic subjects.