  Extremely large deuteration of several molecules has been observed towards
prestellar cores and low-mass protostars for a decade. New observations
performed towards low-mass protostars suggest that water presents a lower
deuteration in the warm inner gas than in the cold external envelope. We
coupled a gas-grain astrochemical model with a one-dimension model of
collapsing core to properly follow the formation and the deuteration of
interstellar ices as well as their subsequent evaporation in the low-mass
protostellar envelopes with the aim of interpreting the spatial and temporal
evolutions of their deuteration. The astrochemical model follows the formation
and the evaporation of ices with a multilayer approach and also includes a
state-of-the-art deuterated chemical network by taking the spin states of H$_2$
and light ions into account. Because of their slow formation, interstellar ices
are chemically heterogeneous and show an increase of their deuterium
fractionation towards the surface. The differentiation of the deuteration in
ices induces an evolution of the deuteration within protostellar envelopes. The
warm inner region is poorly deuterated because it includes the whole molecular
content of ices while the deuteration predicted in the cold external envelope
scales with the highly deuterated surface of ices. We are able to reproduce the
observed evolution of water deuteration within protostellar envelopes but we
are still unable to predict the super-high deuteration observed for
formaldehyde and methanol. Finally, the extension of this study to the
deuteration of complex organics (COMs), important for the prebiotic chemistry,
shows a good agreement with the observations, suggesting that we can use the
deuteration to retrace their mechanisms and their moments of formation.
