Computing persistent homology in biological datasets

National Institute Of Diabetes And Digestive And Kidney Diseases

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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. 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. The islets of Langerhans are critical endocrine micro-organs that secrete hormones regulating energy metabolism in animals. Insulin and glucagon, secreted by beta and alpha cells, respectively, are responsible for metabolic switching between fat and glucose utilization. Dysfunction in their secretion and/or counter-regulatory influence leads to diabetes. Debate in the field centers on the cytoarchitecture of islets, as the signaling that governs hormonal secretion depends on structural and functional factors, including electrical connectivity, innervation, vascularization, and physical proximity. Much effort has therefore been devoted to elucidating which architectural features are significant for function and how derangements in these features are correlated or causative for dysfunction, especially using quantitative network science or graph theory characterizations. Here, we ask if there are non-local features in islet cytoarchitecture, going beyond standard network statistics, that are relevant to islet function. An example is ring structures, or cycles, of α and δ cells surrounding β cell clusters or the opposite, β cells surrounding α and δ cells. These could appear in two-dimensional islet section images if a sphere consisting of one cell type surrounds a cluster of another cell type. To address these issues, we developed two independent computational approaches, geometric and topological, for such characterizations. For the latter, we introduce an application of topological data analysis to determine locations of topological features that are biologically significant. We show that both approaches, applied to a large collection of islet sections, are in complete agreement in the context both of developmental and diabetes-related changes in islet characteristics. The topological approach can be applied to three-dimensional imaging data for islets as well. Recent research has shown that modern deep learning (DL) models can predict gene expression just from histopathology slides stained with hematoxylin (H) and eosin (E). This is of clinical importance since gene expression in turn can predict response to cancer pretreatment. This also indicates that tissue morphology by itself contains crucial and actionable biological information. However, it is not clear what morphological features the DL models are using to make these predictions, information that is crucial for biological validation and hypothesis development for efficient precision cancer treatment therapies. Additionally, there is no singular quantification of morphology that can comprehensively characterize complex spatial patterns and their various geometric properties. Our approach is to first separate the nuclear and the nonnuclear morphologies by separating concentrations of H and E stains. However, we found that pre-existing computational methods to separate stains either did not account for staining variability or they did not accurately label the separated stains (H stain labeled as E and vice-versa), which is crucial for our downstream analysis. Hence, we developed a novel stain separation method that is robust to staining variation, extracts spatially resolved concentrations of the two stains, and accurately labels them. After stain separation, we computed embeddings of each of the two spatially resolved stains using a DL model that was pretrained on billions of images in the Imagenet-21k data set. The underlying assumption is that the DL models pretrained on large image data sets learn to embed important distinguishing morphological features. We used the computed nuclear and nonnuclear embeddings to train another DL model to predict gene expression. To determine what this model is using to predict gene expression, we used our model agnostic XAI tool, SensX. SensX showed that the gene expression predictions are more sensitive to the nuclear morphology than nonnuclear morphology. Importantly, it found localized regions in the tissue that the model is using to make predictions. We are studying whether the topology of the arrangement of nuclei is one of the distinguishing characteristics of these localized regions. This is ongoing work.