We studied the stability and dynamics of a model of a nucleosome, the fundamental unit for the packing of long DNA in eukaryotes, using a Brownian dynamics simulation. For the proper folding of a stiff polymer on a core particle, moderate attractive interaction is shown to be essentially important, which explains the empirical experimental protocol for the reconstitution of nucleosomes. The effect of the chain end on the positioning of the core particle is examined and compared with the experimental data by atomic force microscopy measurement. It is also suggested that the core particle exhibits sliding motion along the chain as a manifestation of Brownian motion.