  Pre-stellar cores are the basic unit for the formation of stars and stellar
systems. The anatomy of the physical and chemical structures of pre-stellar
cores is critical for understanding the star formation process. L1544 is a
prototypical pre-stellar core, which shows significant chemical differentiation
surrounding the dust peak. We aim to constrain the physical conditions at the
different molecular emission peaks. This study allows us to compare the
abundance profiles predicted from chemical models together with the classical
density structure of Bonnor-Ebert (BE) sphere. We conducted multi-transition
pointed observations of CH$_{3}$OH, c-C$_{3}$H$_{2}$ and HNCO with the IRAM 30m
telescope, towards the dust peak and the respective molecular peaks of L1544.
With non-LTE radiative transfer calculations and a 1-dimensional model, we
revisit the physical structure of L1544, and benchmark with the abundance
profiles from current chemical models. We find that the HNCO, c-C$_{3}$H$_{2}$
and CH$_{3}$OH lines in L1544 are tracing progressively higher density gas,
from $\sim$10$^{4}$ to several times 10$^{5}$ cm$^{-3}$. Particularly, we find
that to produce the observed intensities and ratios of the CH$_{3}$OH lines, a
local gas density enhancement upon the BE sphere is required. This suggests
that the physical structure of an early-stage core may not necessarily follow a
smooth decrease of gas density profile locally, but can be intercepted by
clumpy substructures surrounding the gravitational center. Multiple transitions
of molecular lines from different molecular species can provide a tomographic
view of the density structure of pre-stellar cores. The local gas density
enhancement deviating from the BE sphere may reflect the impact of accretion
flows that appear asymmetric and are enhanced at the meeting point of
large-scale cloud structures.
