  Conventional studies of biomolecular behaviors rely largely on the
construction of kinetic schemes. Since the selection of these networks is not
unique, a concern is raised whether and under which conditions hierarchical
schemes can reveal the same experimentally measured fluctuating behaviors and
unique fluctuation related physical properties. To clarify these questions, we
introduce stochasticity into the traditional lumping analysis, generalize it
from rate equations to chemical master equations and stochastic differential
equations, and extract the fluctuation relations between kinetically and
thermodynamically equivalent networks under intrinsic and extrinsic noises. The
results provide a theoretical basis for the legitimate use of low-dimensional
models in the studies of macromolecular fluctuations and, more generally, for
exploring stochastic features in different levels of contracted networks in
chemical and biological kinetic systems.
