We present ab initio calculations of the electronic and transport properties of one-dimensional π-conjugated stacks of guanine- and melanin-based molecules. The electron and hole bandwidths are found to be very large for these organic materials (up to 2 eV), and the effective masses of the charge carriers are rather small. Since these material parameters basically determine the intermolecular charge-transport properties, we predict highly ordered self-assembled one-dimensional stacks of such biological molecules to be promising candidates for applications in organic electronics.