In this paper we present a molecular dynamics calculation of the Frank elastic constants of a nematic liquid crystal. We study two well-known variants of the Gay-Berne potential, and determine the elastic constants by measuring orientational fluctuations as a function of wave vector, using reasonably large system sizes in the range 1000-8000 molecules. For some of the simulations, a set of Lagrangian constraints was applied in order to keep the director fixed along one of the box axes, facilitating the measurement of fluctuations in components of the reciprocal-space order tensor (Q) over cap(k) in the director frame.