Context. Pre-stellar cores represent the earliest stage of the formation process of stars and planets. 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 by the later stages. Aims. We aim to determine the underlying causes of spatial chemical segregation observed in pre-stellar cores and study the effects of the core structure and external environment on the chemical structure of pre-stellar cores. Methods. A three-dimensional (3D) magnetohydrodynamic model of a pre-stellar core embedded in a dynamic star-forming cloud was post-processed with a sequentially continuum radiative transfer, a gas-grain chemical model, and a line-radiative transfer model. The results were analyzed and compared to observations of CH3OH and c-C3H2 in L1544. We compared nine different chemical models to the observations to determine which initial conditions are compatible with the observed chemical segregation in the prototypical pre-stellar core L1544. Results. 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-C3H2 emission overlaps with the dust continuum, which is consistent with the observed chemical structure. Furthermore, these results are consistent across a broad spectrum of chemical models. 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. Conclusions. 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. The reported effect is likely to affect later stages of the formation process of stars and planets through chemical inheritance.