Diatoms are unicellular microalgae with a rigid cell wall, able to glide on surfaces by releasing nanopolymeric fibers through central slits known as raphes. Here we consider the model Nitszchia communis to perform quantitative studies on two complementary aspects involved in diatom gliding. Using video microscopy and automated image analysis, we measure the motion of test beads as they are pulled by extracellular polymeric substances (EPS) fibers at the diatom raphe (particle streaming). A multimodal distribution of particle speed is found, evidencing the appearance of short-time events of high speed and acceleration (known as jerky motion) and suggesting that different mechanisms contribute to set diatom velocity during gliding. Furthermore, we use optical tweezers to obtain force-extension records for extracellular diatom nanofibers; records are well described by the worm-like chain model of polymer elasticity. In contrast to previous studies based on application of denaturing force (in the nN regime), application of low force (up to 6 pN) and using enable us to obtain the persistence length of intact fibers. From these measurements, mechanical parameters of EPS fibers such as radius and elastic constant are estimated. Furthermore, by modeling particle streaming as a spring in parallel with a dashpot, we show that the time involved in the release of mechanical energy after fiber detachment from beads (elastic snapping) agrees with our observations of jerky motion. We conclude that the smooth and jerky motions displayed by gliding diatoms correspond to molecular motors and elastic snapping, respectively, thus providing quantitative elements that incorporate to current models of the mechanics behind diatom locomotion.