  Inside a cell, heterotypic proteins assemble in inhomogeneous, crowded
systems where the abundance of these proteins vary with cell types. While some
protein complexes form putative structures that can be visualized with imaging,
there are far more protein complexes that are yet to be solved because of their
dynamic associations with one another. Yet, it is possible to infer these
protein complexes through a physical model. However, it is often not clear to
physicists what kind of data from biology is necessary for such a modeling
endeavor. Here, we aim to model these clusters of coarse-grained protein
assemblies from multiple subunits through the constraints of interactions among
the subunits and the chemical potential of each subunit. We obtained the
constraints on the interactions among subunits from the known protein
structures. We inferred the chemical potential, that dictates the particle
number distribution of each protein subunit, from the knowledge of protein
abundance from experimental data. Guided by the maximum entropy principle, we
formulate an inverse statistical mechanical method to infer the distribution of
particle numbers from the data of protein abundance as chemical potentials for
a grand canonical multi-component mixture. Using grand canonical Monte Carlo
simulations, we captured a distribution of high-order clusters in a protein
complex of Succinate Dehydrogenase (SDH) with four known subunits. The
complexity of hierarchical clusters varies with the relative protein abundance
of each subunit in distinctive cell types such as lung, heart, and brain. When
the crowding content increases, we observed that crowding stabilizes emergent
clusters that do not exist in dilute conditions. We, therefore, proposed a
testable hypothesis that the hierarchical complexity of protein clusters on a
molecular scale is a plausible biomarker of predicting the phenotypes of a
cell.
