Self-assembly of extended structures via cross-linking of individual biomolecules often occurs in solutions at concentrations well below the estimated threshold for random cross-link percolation. This requires solute-solute correlations. Here we study bovine serum albumin. Its unfolding causes the appearance of an instability region of the sol, not observed for native bovine serum albumin. As a consequence, spinodal demixing of the sol is observed. The thermodynamic phase transition corresponding to this demixing is the determinative symmetry-breaking step allowing the subsequent occurrence of (correlated) cross-linking and its progress up to the topological phase transition of gelation. The occurrence of this sequence is of marked interest to theories of spontaneous symmetry-breaking leading to morphogenesis, as well as to percolation theories. The present results extend the validity of conclusions drawn from our previous studies of other systems, by showing in one single case, system features that we have hitherto observed separately in different systems. Time-resolved experimental observations of the present type also bring kinetic and diffusional processes and solute-solvent interactions into the picture of cross-link percolation.