  Quantum Fisher information (QFI) is an important feature for the precision of
quantum parameter estimation based on the quantum Cram\'er-Rao inequality. When
the quantum state satisfies the von Neumann-Landau equation, the local quantum
uncertainty (LQU), as a kind of quantum correlation, present in a bipartite
mixed state guarantees a lower bound on QFI in the optimal phase estimation
protocol [Phys. Rev. Lett. 110 (2013) 240402]. However, in the open quantum
systems, there is not an explicit relation between LQU and QFI generally. In
this paper, we study the relation between LQU and QFI in open systems which is
composed of two interacting two-level systems coupled to independent
non-Markovian environments with the entangled initial state embedded by a phase
parameter $\theta$. The analytical calculations show that the QFI does't depend
on the phase parameter $\theta$, and its decay can be restrained through
enhancing the coupling strength or non-Markovianity. Meanwhile, the LQU is
related to the phase parameter $\theta$ and shows plentiful phenomena. In
particular, we find that the LQU can well bound the QFI when the coupling
between the two systems is switched off or the initial state is Bell state.
