We study by small angle neutron scattering and UV titration how the ratio of negative to positive charges, [-]/[+](intro), acts on the structure of complexes formed by short negatively charged polyelectrolyte chains (PSS) and globular positively charged proteins (lysozyme). The range of [-]/[+](intro) lies between 0.65 and 3.33. In all ratios, dense primary complexes are formed with radii around 10 nm. The species composition and the water content of the primary complexes are precisely obtained by the systematic use of the contrast matching of (deuterated) polymer or protein in SANS, yielding the compactness and the inner charge ratio [-]/[+](inner). The primary complexes have (i) an inner charge ratio [-]/[+](inner) close to 1 whatever [-]/[+](intro), (ii) a high total volume fraction (0.25-0.4), (iii) a constant radius (75 A) for [-]/[+](intro) <or= 1 that increases up to 150 A for [-]/[+](introduced) > 1, and (iv) a shell of PSS chains when [-]/[+](intro) > 1. Moreover, UV titration shows that there are free proteins if [-]/[+](introduced) < 1 and free PSS chains if [-]/[+](intro) is largely superior to 1. Hence, we observe that the primary complexes reach a finite size, controlled by electrostatic repulsion, and then aggregate at a higher scale with a fractal dimension of 2.1 characteristic of reaction-limited colloidal aggregation.