The Alan Turing Institute, The British Library, London, NW1 2DB, UK.
School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK.
Nat Commun. 2024 Mar 13;15(1):2258. doi: 10.1038/s41467-024-45889-6.
Complex biological processes, such as cellular differentiation, require intricate rewiring of intra-cellular signalling networks. Previous characterisations revealed a raised network entropy underlies less differentiated and malignant cell states. A connection between entropy and Ricci curvature led to applications of discrete curvatures to biological networks. However, predicting dynamic biological network rewiring remains an open problem. Here we apply Ricci curvature and Ricci flow to biological network rewiring. By investigating the relationship between network entropy and Forman-Ricci curvature, theoretically and empirically on single-cell RNA-sequencing data, we demonstrate that the two measures do not always positively correlate, as previously suggested, and provide complementary rather than interchangeable information. We next employ Ricci flow to derive network rewiring trajectories from stem cells to differentiated cells, accurately predicting true intermediate time points in gene expression time courses. In summary, we present a differential geometry toolkit for understanding dynamic network rewiring during cellular differentiation and cancer.
复杂的生物过程，如细胞分化，需要细胞内信号网络的复杂重连。先前的特征描述揭示了较低分化和恶性细胞状态下存在较高的网络熵。熵和 Ricci 曲率之间的联系导致了离散曲率在生物网络中的应用。然而，预测动态生物网络重连仍然是一个悬而未决的问题。在这里，我们将 Ricci 曲率和 Ricci 流应用于生物网络重连。通过在单细胞 RNA 测序数据上进行理论和实证研究，研究网络熵和 Forman-Ricci 曲率之间的关系，我们证明了这两个度量并不总是如先前所述呈正相关，而是提供互补而非可互换的信息。接下来，我们采用 Ricci 流从干细胞到分化细胞推导出网络重连轨迹，准确地预测了基因表达时间序列中的真实中间时间点。总之，我们提出了一个微分几何工具包，用于理解细胞分化和癌症过程中的动态网络重连。
