Isogenic cells sensing identical external signals can take markedly different decisions. Such decisions often correlate with pre-existing cell-to-cell differences in protein levels. When not neglected, these differences are accounted for in a static manner by assuming randomly-distributed initial protein levels. Capturing the actual temporal fluctuations of protein levels resulting from protein stochastic turnover would be more appropriate. Adopting this dynamical view on noise amounts to recast extrinsic variability into intrinsic noise. Here, we propose a generic approach to merge in a systematic and principled manner signal transduction models with stochastic protein turnover models. When applied to an established kinetic model of TRAIL-induced apoptosis, our approach markedly increased model prediction capabilities. One obtains a mechanistic explanation of yet-unexplained observations on fractional killing and non-trivial robust predictions of the temporal evolution of cell resistance to TRAIL. Our results challenge the role of surviving pathways in cell survival since no TRAIL-induced regulations are needed and strongly suggest that short-lived anti-apoptotic proteins Flip/Mcl1 exhibit large and rare fluctuations. More generally, our results highlight the importance of accounting for stochastic protein turnover to quantitatively understand signal transduction over extended durations, and imply that fluctuations of short-lived proteins deserve particular attention.