We present a combined experimental-simulation study of self-assembly into one-dimensional filaments. Experimentally, we study amphiphilic Au-1-metallopeptides in neutral aqueous media. Our model focuses on two aspects of the monomers, their geometry and their hydrophobic core. We demonstrate numerically that a coarse-grained, generic model is sufficient to obtain spontaneous formation of nearly defect-free, long supramolecular polymers at experimentally relevant monomer concentrations. In the simulations, we find both nonchiral and helical polymer morphologies depending on the effective hydrophobic strength. We analyze the assembly kinetics, comparing simulations to experimental data obtained from circular dichroism spectroscopy, and relate both to kinetic models of nucleated polymerization. While the simulations proceed via nucleation and elongation, experimental kinetics seems to involve fragmentation.