We develop a simple, generic global model of carbon cycling by terrestrial vegetation driven by climate data, observed greenness from global vegetation index data, and a drought-stress indicator calculated with a one-layer bucket model. Modelled CO2 fluxes are then fed into a three-dimensional atmospheric tracer transport model, so that results can be checked against observed concentrations of CO2 at various monitoring sites. By exploring a wide range of model formulations, we find an optimal fit of the model to atmospheric CO2 data. To check these results, we use a second model with a coupled photosynthesis-hydrology scheme, using the Penman-Monteith equation to calculate evapotranspiration. This model simulates feedbacks between drought-stress and photosynthesis through the soil-water balance. We show that CO2 measurements at tropical and southern-hemisphere stations can be used to constrain the seasonal atmosphere-biosphere carbon exchange in the wet-dry tropics. In both models, this seasonality is strongly suppressed, more strongly in fact than predicted by some complex terrestrial-vegetation models. We also find some evidence of a considerable CO2 release from soils during the northern-hemisphere winter. An exponential air-temperature dependence of soil release with a Q10 of 1.5 is found to be most appropriate, with no cutoff at low freezing temperatures. The results of this study should indicate in how far measurements of atmospheric CO2 concentration can provide a constraint on global models of terrestrial-vegetation activity.