Recent observations of the HDO/H$_2$O ratio toward protostars in isolated and clustered environments show an apparent dichotomy, where isolated sources show higher D/H ratios than clustered counterparts. Establishing which physical and chemical processes create this differentiation can provide insights into the chemical evolution of water during star formation and the chemical diversity during the star formation process and in young planetary systems. Methods: The evolution of water is modeled using 3D physicochemical models of a dynamic star-forming environment. The physical evolution during the protostellar collapse is described by tracer particles from a 3D MHD simulation of a molecular cloud region. Each particle trajectory is post-processed using RADMC-3D to calculate the temperature and radiation field. The chemical evolution is simulated using a three-phase grain-surface chemistry model and the results are compared with interferometric observations of H$_2$O, HDO, and D$_2$O in hot corinos toward low-mass protostars. Results: The physicochemical model reproduces the observed HDO/H$_2$O and D$_2$O/HDO ratios in hot corinos, but shows no correlation with cloud environment for similar identical conditions. The observed dichotomy in water D/H ratios requires variation in the initial conditions (e.g., the duration and temperature of the prestellar phase). Reproducing the observed D/H ratios in hot corinos requires a prestellar phase duration $tsim$1-3 Myr and temperatures in the range $T sim$ 10-20 K prior to collapse. This work demonstrates that the observed differentiation between clustered and isolated protostars stems from differences in the molecular cloud or prestellar core conditions and does not arise during the protostellar collapse itself.
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