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Pre-stellar cores are the basic unit for the formation of stars and stellarsystems. The anatomy of the physical and chemical structures of pre-stellarcores is critical for understanding the star formation process. L1544 is aprototypical pre-stellar core, which shows significant chemical differentiationsurrounding the dust peak. We aim to constrain the physical conditions at thedifferent molecular emission peaks. This study allows us to compare theabundance profiles predicted from chemical models together with the classicaldensity structure of Bonnor-Ebert (BE) sphere. We conducted multi-transitionpointed observations of CH$_{3}$OH, c-C$_{3}$H$_{2}$ and HNCO with the IRAM 30mtelescope, towards the dust peak and the respective molecular peaks of L1544.With non-LTE radiative transfer calculations and a 1-dimensional model, werevisit the physical structure of L1544, and benchmark with the abundanceprofiles 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 findthat to produce the observed intensities and ratios of the CH$_{3}$OH lines, alocal gas density enhancement upon the BE sphere is required. This suggeststhat the physical structure of an early-stage core may not necessarily follow asmooth decrease of gas density profile locally, but can be intercepted byclumpy substructures surrounding the gravitational center. Multiple transitionsof molecular lines from different molecular species can provide a tomographicview of the density structure of pre-stellar cores. The local gas densityenhancement deviating from the BE sphere may reflect the impact of accretionflows that appear asymmetric and are enhanced at the meeting point oflarge-scale cloud structures.

Multi-line observations of CH$_{3}$OH, c-C$_{3}$H$_{2}$ and HNCO towards L1544: Dissecting the core structure with chemical differentiation