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Measurements of the nitrogen isotopic ratio in Solar System comets show a constant value, ~140, which is three times lower than the protosolar ratio, a highly significant difference that remains unexplained. Observations of static starless cores at early stages of collapse confirm the theoretical expectation that nitrogen fractionation in interstellar conditions is marginal for most species. Yet, observed isotopic ratios in N2H+ are at variance with model predictions. These gaps in our understanding of how the isotopic reservoirs of nitrogen evolve, from interstellar clouds to comets, and, more generally, to protosolar nebulae, may have their origin in missing processes or misconceptions in the chemistry of interstellar nitrogen. So far, theoretical studies of nitrogen fractionation in starless cores have addressed the quasi-static phase of their evolution such that the effect of dynamical collapse on the isotopic ratio is not known. In this paper, we investigate the fractionation of 14N and 15N during the gravitational collapse of a pre-stellar core through gas-phase and grain adsorption and desorption reactions. The initial chemical conditions, which are obtained in steady state after typically a few Myr, show low degrees of fractionation in the gas phase, in agreement with earlier studies. However, during collapse, the differential rate of adsorption of 14N- and 15N-containing species onto grains results in enhanced 15N:14N ratios, in better agreement with the observations. Furthermore, we find differences in the behavior, with increasing density, of the isotopic ratio in different species. We find that the collapse must take place on approximately one free-fall timescale, based on the CO abundance profile in L183 [see the end in the PDF file]