This study presents the theory for liquid-liquid phase separation for systems of molecules modeling monoclonal antibodies. Individual molecule is depicted as an assembly of seven hard spheres, organized to mimic the Y-shaped antibody. We consider the antibody- antibody interactions either through Fab, Fab' (two Fab fragments may be different), or Fc domain. Interaction between these three domains of the molecule (hereafter denoted as A, B, and C, respectively) is modeled by a short-range square-well attraction. To obtain numerical results for the model under study, we adapt Wertheim's thermodynamic perturbation theory. We use this model to calculate the liquid-liquid phase separation curve and the second virial coefficient B-2. Various interaction scenarios are examined to see how the strength of the site-site interactions and their range shape the coexistence curve. In the asymmetric case, where an attraction between two sites is favored and the interaction energies for the other sites kept constant, critical temperature first increases and than strongly decreases. Some more microscopic information, for example, the probability for the particular two sites to be connected, has been calculated. Analysis of the experimental liquid- liquid phase diagrams, obtained from literature, is presented. In addition, we calculate the second virial coefficient under conditions leading to the liquid-liquid phase separation and present this quantity on the graph B-2 versus protein concentration.