We simulated the docking of human immunodeficiency virus (HIV) with a cell membrane using Brownian adhesive dynamics. The main advance in the current version of Brownian adhesive dynamics is that we use a simple bead-spring model to coarsely approximate the role of gp120 trimerization on HIV docking. We used our simulations to elucidate the effect of env spike density on the rate and probability of HIV binding, as well as the probability that each individual gp120 trimer is fully engaged. We found that for typical CD4 surface densities, viruses expressing as few as 8 env spikes will dock with binding rate constants comparable to viruses expressing 72 spikes. We investigated the role of cellular receptor diffusion on the degree of binding achieved by the virus on both short timescales (where binding has reached steady state but before substantial receptor accumulation in the viral-cell contact zone has occurred) and long timescales (where the system has reached steady state). On short timescales, viruses with 10-23 env trimers most efficiently form fully engaged trimers. On long timescales, all gp120 in the contact area will become bound to CD4. We found that it takes seconds for engaged trimers to cluster CD4 molecules in the contact zone, which partially explains the deleay in viral entry.