  The two classes of membrane transport, namely, secondary active and passive
transport, are understood as different reaction pathways in the same protein
structure, described by the 16-state model in this paper. To quantify the
thermodynamic difference between secondary active transport and passive
transport, we extend the second law of information thermodynamics of the
autonomous demon in the four-state model composed of two subsystems to the
16-state model composed of four subsystems representing the membrane transport.
We reduce the 16 states to 4 states and derive the coarse-grained second law of
information thermodynamics, which provides an upper bound of the free energy
transport by the coarse-grained information flow. We also derive an upper bound
on the free energy transport by the multi-body information flow representing
the two-body or four-body correlations in the 16-state model by exploiting the
cycle decomposition. The coarse-grained information flow and the multi-body
information flows express the quantitative difference between secondary active
and passive transport. The numerical analysis shows that the coarse-grained
information flow is positive for secondary active transport and negative for
passive transport. The four-body correlation is dominant in the multi-body
information flows for secondary active transport. In contrast, the two-body
correlation is dominant for passive transport. This result shows that both the
sign of the coarse-grained information flow and the difference of the four-body
correlation and the two-body correlation explain the difference of the free
energy transport in secondary active and passive transport.
