Probability Landscape and Its Rotational Flux in Complex Systems: Construction and Topological Structure

Biomolecules such as proteins, DNAs, and RNA interact in cells to regulate cellular functions. These interactions can be modeled as reaction networks. At small copy numbers, there is strong stochasticity in the network due to thermal fluctuations. Such stochasticity plays important roles in cellular functions. Stochasticity can be investigated using models of stochastic chemical kinetics (SCK), which is governed by the discrete chemical master equation (dCME). The probability landscapes over the discrete states of copy numbers of molecules are dictated by the dCME of networks and provide detailed information on these stochastic processes. However, there are two difficulties. First difficulty is that there are often many molecular species. Thus, stochastic networks are usually of high dimensionality. The exact computation of the probability landscape is challenging because of the high dimensionality. Second, even if one can construct a landscape accurately, it is difficult to navigate and understand the high dimensional probability landscape, to gain insight into the stochastic behavior of the system. This work addresses these two difficulties. First, we expand the theoretical framework for state space truncation and provide the error bounds for truncating the state space 1) below a specific total mass of the system, 2) below a minimum copy number of each molecular specie in the network, and 3) over a maximum copy number of each molecular specie in the reaction network, and provide error for such truncations. Second, we develop algorithms to characterize topological structure of high dimensional probability surfaces. Third, we formulate the concept of the rotation of probability flux governed by reaction stoichiometry in discrete space using 1-forms. Our analysis reveals important findings in a number of dynamic processes, including phenotypic switching in feedback loops, high dimensional repressilator networks, and activated process of alanine dipeptide isomerization in vacuum. Our findings were not possible without the concepts and algorithms developed in this work.