This paper is about neural mechanisms of direction selectivity (DS) in Macaque primary visual cortex, V1. DS arises in V1 layer 4Ca, which receives afferent input from the Magnocellular division of the Lateral Geniculate Nucleus (LGN). LGN itself, however, is not direction-selective. To understand the mechanisms of DS, we built a new computational model (DSV1) of 4Ca. DSV1 is a realistic, large-scale mechanistic model that simulates many V1 properties: orientation selectivity, spatial and temporal tuning, contrast response, and DS. In the model, DS is initiated by the dynamic difference of OFF and ON Magnocellular cell activity that excites the model's layer 4Ca; the recurrent network has no intra-cortical direction-specific connections. In experiments, and in DSV1, most 4Ca Simple cells were highly direction-selective but few 4Ca Complex cells had high DS. Furthermore, the preferred directions of the model's direction- selective Simple cells were invariant with spatial and temporal frequency, in this way emulating the experimental data. The distribution of DS across the model's population of cells was very close to that found in experiments. Analyzing DSV1, we found that the dynamic interaction of feedforward and intra-cortical synaptic currents led to cortical enhancement of DS for a majority of cells. In view of the strong quantitative agreement between DS in data and in model simulations, the neural mechanisms of DS in DSV1 may be indicative of those in the real visual cortex.