We have developed ab initio calculations to determine the interaction between carbon atoms and screw dislocations in BCC tungsten. A strong attraction is found, leading to segregation of carbon atoms into the core of dislocation, modifying its structure and mobility. We have then calculated the energy landscape experienced by a screw dislocation saturated by carbon when it glides by nucleation and propagation of kink-pairs. Dislocation velocity is obtained, first with kinetic Monte-Carlo simulations, then analytically with classical nucleation theory. Results obtained with both approaches are in good agreement. The obtained mobility law allows us to estimate the local stress existing in TEM in-situ straining experiments, leading to values comparable with the ones known from macroscopic tensile tests at high temperature when carbon is mobile.