NGC 4438 is a highly perturbed spiral with a stellar tidal tail and extraplanar molecular gas, now very HI deficient, crossing the center of the Virgo cluster at high speed. Various authors have attributed the perturbed appearance to the ram pressure of the intracluster medium, the tidal interaction with NGC 4435, and an ISM-ISM collision between the ISM of NGC 4438 and NGC 4435. We present new CO observations covering virtually all of NGC 4438 and the center of NGC 4435 and detailed simulations including all of the above effects. For the first time CO is detected in NGC 4435. In NGC 4438 we find double line profiles at distances up to 40'' to the west and south-west and redshifted lines with respect to galactic rotation in the south of the center. The lack of gas to the North and East coupled with the large gaseous extent to the West and the redshifted and double line profiles can only be reproduced with a ram pressure wind. NGC 4438 is most probably on its first passage through the cluster center and has been stripped of its HI only over the past 100 Myr. While an ISM-ISM collision between NGC 4435 and NGC 4438 may occur, the effect is not significant compared to ram pressure and tidal forces, not surprising for the passage of an S0 galaxy 5-10 kpc from the center of NGC 4438. We also detect narrow CO lines, in the absence of detected HI, in the northern tidal arm some 15 kpc from the center of NGC 4438. This can be understood from the simulations assuming a few percent of the gas is too dense to experience the ram pressure wind. NGC 4438 has changed greatly over the past 100 Myr due to its plunge through the center of the Virgo cluster and the interaction with the S0 NGC 4435. The ram pressure wind has a strong effect which is increased by the interaction with NGC 4435, bringing gas further from NGC 4438 where the ram pressure strips it away.< Réduire