Quantifying Cellular Pluripotency and Pathway Robustness through Forman-Ricci Curvature

Abstract

In stem cell biology, cellular pluripotency describes the capacity of a given cell to differentiate into multiple cell types. From a statistical physics perspective, entropy provides a statistical measure of randomness and has been demonstrated as a way to quantitate pluripotency when considering biological gene networks. Furthermore, recent theoretical work has established a relationship between Ricci curvature (a geometric measure of "flatness") and entropy (also related to robustness), which one can exploit to link the geometric quantity of curvature to the statistical quantity of entropy. Therefore, this study seeks to explore Ricci curvature in biological gene networks as a descriptor of pluripotency and robustness among gene pathways. Here, we investigate Forman-Ricci curvature, a combinatorial discretization of Ricci curvature, along with network entropy, to explore the relationship of the two quantities as they occur in gene networks. First, we demonstrate our approach on an experiment of stem cell gene expression data. As expected, we find Ricci curvature directly correlates with network entropy, suggesting Ricci curvature could serve as an indicator for cellular pluripotency much like entropy. Second, we measure Forman-Ricci curvature in a dataset of cancer and non-cancer cells from melanoma patients. We again find Ricci curvature is increased in the cancer state, reflecting increased pluripotency or "stemness". Further, we locally examine curvature on the gene level to identify several genes and gene pathways with known relevance to melanoma. In turn, we conclude Forman-Ricci curvature provides valuable biological information related to pluripotency and pathway functionality. In particular, the advantages of this geometric approach are promising for extension to higher-order topological structures in order to represent more complex features of biological systems.