Automatic tracking of particles in time-lapse fluorescence microscopy images is essential for quantifying the dynamic behavior of subcellular structures and virus structures. We introduce a novel particle tracking approach based on a deep recurrent neural network architecture that exploits past and future information in both forward and backward direction. Assignment probabilities are determined jointly across multiple detections, and the probability of missing detections is computed. In addition, existence probabilities are determined by the network to handle track initiation and termination. For correspondence finding, track hypotheses are propagated to future time points so that information at later time points can be used to resolve ambiguities. A handcrafted similarity measure and handcrafted motion features are not necessary. Manually labeled data is not required for network training. We evaluated the performance of our approach using image data of the Particle Tracking Challenge as well as real fluorescence microscopy image sequences of virus structures. It turned out that the proposed approach outperforms previous methods.