  Pre-stellar cores represent the earliest stage of the star- and
planet-formation process. By characterizing the physical and chemical structure
of these cores we can establish the initial conditions for star and planet
formation and determine to what degree the chemical composition of pre-stellar
cores is inherited to the later stages. A 3D MHD model of a pre-stellar core
embedded in a dynamic star-forming cloud is post-processed using sequentially
continuum radiative transfer, a gas-grain chemical model, and a line-radiative
transfer model. Results are analyzed and compared to observations of CH$_3$OH
and $c$-C$_3$H$_2$ in L1544. Nine different chemical models are compared to the
observations to determine which initial conditions are compatible with the
observed chemical segregation in the prototypical pre-stellar core L1544. The
model is able to reproduce several aspects of the observed chemical
differentiation in L1544. Extended methanol emission is shifted towards colder
and more shielded regions of the core envelope while $c$-C$_3$H$_2$ emission
overlaps with the dust continuum, consistent with the observed chemical
structure. Increasing the strength of the interstellar radiation field or the
cosmic-ray ionization rate with respect to the typical values assumed in nearby
star-forming regions leads to synthetic maps that are inconsistent with the
observed chemical structure. Our model shows that the observed chemical
dichotomy in L1544 can arise as a result of uneven illumination due to the
asymmetrical structure of the 3D core and the environment within which the core
has formed. This highlights the importance of the 3D structure at the
core-cloud transition on the chemistry of pre-stellar cores.
