  Cellular functions are established through biological evolution, but are
constrained by the laws of physics. For instance, the physics of protein
folding limits the lengths of cellular polypeptide chains. Consequently, many
cellular functions are carried out not by long, isolated proteins, but rather
by multi-protein complexes. Protein complexes themselves do not escape physical
constraints, one of the most important being the difficulty to assemble
reliably in the presence of cellular noise. In order to lay the foundation for
a theory of reliable protein complex assembly, we study here an equilibrium
thermodynamic model of self-assembly that exhibits four distinct assembly
behaviors: diluted protein solution, liquid mixture, "chimeric assembly" and
"multifarious assembly". In the latter regime, different protein complexes can
coexist without forming erroneous chimeric structures. We show that two
conditions have to be fulfilled to attain this regime: (i) the composition of
the complexes needs to be sufficiently heterogeneous, and (ii) the use of the
set of components by the complexes has to be sparse. Our analysis of publicly
available databases of protein complexes indicates that cellular protein
systems might have indeed evolved so to satisfy both of these conditions.
