We performed replica exchange molecular dynamics and forward flux sampling simulations of hexapeptide VQIINK and VQIVYK systems, also known as, respectively, fragments PHF6* and PHF6 from the tau :protein. Being a part of the microtubule binding region, these fragments are known to be aggregation prone, and at least one of them is a prerequisite for fibril formation of the tau protein. Using a coarse-grained force field, we establish the phase behavior of both fragments, and investigate the nucleation kinetics for the conversion into a beta-sheet fibril. As the conversion is, in principle, a reversible process, we predict the rate constants for both the fibril formation and melting, and examine the corresponding mechanisms. Our simulations indicate that, while both fragments form disordered aggregates, only PHF6 is able to form beta-sheet fibrils. This observation provides a possible explanation for the lack of available steric zipper crystal structures for PHF6*.