Equilibrium Submanifold for a Biological System

The complexity in a biological system may be caused by both the number of variables involved and the number of system constants that can vary. A biological system in the subcellular level often stabilizes after a certain period of time. Its asymptote can then be described as an equilibrium under certain continuity assumptions. The biological quantities at the equilibrium can be detected by experiments and they observe some mathematical equations. The purpose of this paper is to study the equilibrium submanifold of vesicle trafficking in a two-compartment system. We compute the equilibrium submanifold under some fairly general assumption on the system constants. The disconnectedness of the equilibrium submanifold may have biological implications. We show that, unlike many other systems, the equilibrium is determined largely by system constants rather than the initial state. In particular, the equilibrium submanifold is locally a real algebraic variety, with small generic dimension and large degenerate dimension. Our result suggests that some biological system may be studied by algebraic or geometric methods.