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The two classes of membrane transport, namely, secondary active and passivetransport, are understood as different reaction pathways in the same proteinstructure, described by the 16-state model in this paper. To quantify thethermodynamic difference between secondary active transport and passivetransport, we extend the second law of information thermodynamics of theautonomous demon in the four-state model composed of two subsystems to the16-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 ofinformation thermodynamics, which provides an upper bound of the free energytransport by the coarse-grained information flow. We also derive an upper boundon the free energy transport by the multi-body information flow representingthe two-body or four-body correlations in the 16-state model by exploiting thecycle decomposition. The coarse-grained information flow and the multi-bodyinformation flows express the quantitative difference between secondary activeand passive transport. The numerical analysis shows that the coarse-grainedinformation flow is positive for secondary active transport and negative forpassive transport. The four-body correlation is dominant in the multi-bodyinformation flows for secondary active transport. In contrast, the two-bodycorrelation is dominant for passive transport. This result shows that both thesign of the coarse-grained information flow and the difference of the four-bodycorrelation and the two-body correlation explain the difference of the freeenergy transport in secondary active and passive transport.

Thermodynamic role of main reaction pathway and multi-body information flow in membrane transport