Program Director/Principal Investigator (Liang, Jie): PROJECT SUMMARY/ABSTRACT We will continue our study of biopolymers and their interactions at two levels. At the molecular network level, we will study the A) probability landscape of stochastic networks of molecular reactions. We will develop efficient computational tools to construct exact probability landscapes in high-dimension, quantify probability discrete fluxes, and characterize their exact topology. These powerful tools will be applied to gain mechanistic understanding of stochastic control of network phenotypes in a number of important biological problems. At the (sub)cellular level, we will study B) biophysics of 3D chromatin folding. We will develop algorithms to identify driver interactiomes that can generate large ensembles of accurate models of single-cell 3D chromatin conformations consistent with Hi-C and single-cell experimental data. Our methods will be applied to study foundational problems of 3D genome to gain understanding of principles of genome organization. In A), we will study stochastic reaction networks of molecules to gain mechanistic understanding of their behavior. Many important cellular processes involve a small copy number of molecules of transcription factors, enzymes, and signaling molecules. Stochasticity and rare events arising from such low copy number reactions are important for processes such as embryonic development, stem cell differentiation, and nongenetic heterogeneity. Our approach will be based on the fundamental framework of the stochastic kinetic processes and the discrete chemical master equation (dCME). The central tasks are: 1) constructing the probability landscape of the network, and from which to 2) gain analytical insight into mechanism of network behavior. For 1), we have developed the ACME method that can construct the exact probability landscapes of a large class of complex stochastic reaction networks and will make further improvement. For 2), we will develop landscape analysis tools using persistent homology that can compute the exact topology of the high-dimensional probability landscape. We have also developed the concept of discrete fluxes and methods for its computation. We will further formulate and generalize the concept of discrete rotational flux to higher dimension. These developments will enable global and mechanistic understanding of the behavior of stochastic networks through accurately constructed probability landscape and exactly computed topological structures. Our work will open up new frontiers for investigations, many of which are currently not computationally feasible. Specifically, we will construct probability landscapes of networks, study how global flux maps evolve and how phenotype switching occur. We will generalize the discovery of stochastic oscillation and investigate higher-order oscillatory behavior of networks, where probability mass may be transferred through higher-dimensional k-channels. In addition, ...