Efficient Switches in Biology and Computer Science
Fig 6
Condensed and extended wiring diagrams of AM (above) and MI (below) and their deterministic behaviour in time-course diagrams. A morphism m:(S,R)→(S’,R’) between two reaction networks (S,R) and (S’,R’) is a mapping of species S (e.g., y0, y1, y2, z0, z1, z2 of MI) to species S’ (e.g., x0, x1, x2 of AM, by corresponding colours) and of reactions R to reactions R’. Structural properties: A morphism that preserves the reactants and products of each reaction under the mapping is called a homomorphism. One that preserves stoichiometry under the mapping (by appropriately summing multiplicities and rates) is called a stoichiomorphism. These properties can be calculated directly on the network representation. Dynamical properties: A morphism m is an emulation if it preserves all trajectories of species concentrations over time under the mapping (e.g., the trajectories on the right are preserved). That is, m is an emulation if for any choice of initial conditions I’ for S’ there exist initial conditions I for S such that the trajectory of each species s in S overlaps exactly the trajectory of m(s) in S’. Theorem [44]: A morphism that is a homomorphism and a stoichiomorphism is also an emulation.