Fine-tuning the complex organic molecule formation: sulfur and CO ice as regulators of surface chemistry

Grain-surface chemistry plays a crucial role in the formation of molecules of astrobiological interest, including H$_{2}$S and complex organic molecules (COMs). They are commonly observed in the gas phase toward star-forming regions, but their detection in ices remains limited. Combining gas-phase observations with chemical modeling is therefore essential for advancing our understanding of their chemistry. In this paper we investigate the factors that promote or hinder molecular complexity combining gas-phase observations of CH$_{3}$OH, H$_{2}$S, OCS, N$_{2}$H$^{+}$, and C$^{18}$O with chemical modeling in two dense cores: Barnard-1b and IC348. We observed millimeter emission lines of CH$_{3}$OH, H$_{2}$S, OCS, N$_{2}$H$^{+}$, and C$^{18}$O along strips using the IRAM 30m and Yebes 40m telescopes. We used the gas-grain chemical model \texttt{Nautilus} to reproduce the observed abundance profiles adjusting parameters such as initial sulfur abundances and binding energies. H$_{2}$S, N$_{2}$H$^{+}$ and C$^{18}$O gas-phase abundances vary up to one order of magnitude towards the extinction peak. CH$_{3}$OH abundance remains quite uniform. These abundances can only be reproduced assuming a decreasing sulfur budget, which lowers H$_{2}$S and enhances CH$_{3}$OH abundances. Decreasing binding energies, which are expected in CO-rich apolar ices, are also required. The sulfur depletion required by H$_2$S is generally higher than that required by CH$_3$OH, suggesting unknown sulfur sinks. These findings highlight the intricate relationship between sulfur chemistry and COM formation, driven by the competition between sulfur and CO for hydrogen atoms. Our study emphasizes that the growth of CO ice and the progressive sequestration of hydrogen atoms by sulfur are critical in determining whether chemical complexity can develop, providing key insights into the early stages of star and planet formation.