ncts

We describe recent developments in understanding the topological structures of complex biological systems of 1) molecular reactions of activated processes and 2) probability landscape of reaction networks. For molecular reactions of activated processes, we describe an exact approach based on persistent homology to quantify the topology of dynamic probability surface of barrier-crossing. For the simplest complex activated process of alanine-dipeptide isomerization, we find that instead of a saddle-point in the free energy surface, the transition state ensemble (TSE) at the barrier top forms the most prominent probability H₀ peak in the high-dimensional configuration-time space after reactants/products. Furthermore, this reactive region is vorticle and has strong rotational fluxes, which cannot be described by Kramers' diffusion-based theory. For stochastic reaction networks, we show the persistent homology of the high-dimensional probability landscapes obtained using the finite-buffer ACME methods can be computed. Furthermore, defining topological properties of the landscapes can be quantified, including the degree of multistability by H₀, the oscillatory cycles by H₁, and higher-order oscillatory membranes by H₂. We also give examples of phase diagrams over the parameter space of multistability of a class of stochastic reaction network. These results show that topological data analysis and persistent homology can provide powerful means towards understanding naturally occurring physical dynamic processes and stochastic interaction networks. (Joint work with Farid Manuchehrfar, Wei Tian, Huiyu Li, Ao Ma, Hubert Wagner, and Herbert Edelsbrunner).