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Table 2 Hierarchical transformation that realizes a Kalman decomposition for the example system depicted in Figure 2A.

From: Exact model reduction of combinatorial reaction networks

[R(*, *)] = [R(0, 0)] + [R(0, E1)] + [R(0, E2)] + [R(0, E3)] + [R(0, E4)] + [R(L, 0)] + [R(L, E1)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[E1(*)] = [E1(0)] + [E1(E2)] + [E1(E3)] + [E1(E4)] + [R(0, E1)] + [R(0, E2)] + [R(0, E3)] + [R(0, E4)] + [R(L, E1)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[E2(*)] = [E1(E2)] + [E1(E3)] + [E1(E4)] + [E2(0)] + [E2(E3)] + [E2(E4)] + [R(0, E2)] + [R(0, E3)] + [R(0, E4)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[E3(*)] = [E1(E3)] + [E1(E4)] + [E2(E3)] + [E2(E4)] + [E3(0)] + [E3(E4)] + [R(0, E3)] + [R(0, E4)] + [R(L, E3)] + [R(L, E4)]
[E4(*)] = [E1(E4)] + [E2(E4)] + [E3(E4)] + [E4(0)] + [R(0, E4)] + [R(L, E4)]
[R(L, *)] = [R(L, 0)] + [R(L, E1)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[R(*, E1(*))] = [R(0, E1)] + [R(0, E2)] + [R(0, E3)] + [R(0, E4)] + [R(L, E1)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[E1(E2(*)] = [E1(E2)] + [E1(E3)] + [E1(E4)] + [R(0, E2)] + [R(0, E3)] + [R(0, E4)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[E2(E3(*))] = [E1(E3)] + [E1(E4)] + [E2(E3)] + [E2(E4)] + [R(0, E3)] + [R(0, E4)] + [R(L, E3)] + [R(L, E4)]
[E3(E4(*))] = [E1(E4)] + [E2(E4)] + [E3(E4)] + [R(0, E4)] + [R(L, E4)]
[R(L, E1(*))] = [R(L, E1)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[R(*, E2(*))] = [R(0, E2)] + [R(0, E3)] + [R(0, E4)] + [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[E1(E3(*)] = [E1(E3)] + [E1(E4)] + [R(0, E3)] + [R(0, E4)] + [R(L, E3)] + [R(L, E4)]
[E2(E4(*))] = [E1(E4)] + [E2(E4)] + [R(0, E4)] + [R(L, E4)]
[R(L, E2(*))] = [R(L, E2)] + [R(L, E3)] + [R(L, E4)]
[R(*, E3(*))] = [R(0, E3)] + [R(0, E4)] + [R(L, E3)] + [R(L, E4)]
[E1(E4(*)] = [E1(E4)] + [R(0, E4)] + [R(L, E4)]
[R(L, E3(*))] = [R(L, E3)] + [R(L, E4)]
[R(*, E4(*))] = [R(0, E4)] + [R(L, E4)]
[R(L, E4(*))] = [R(L, E4)]
  1. The new states correspond to the occurrence levels of different subcomplexes. The transformation can be structured in different tiers. The previously discussed case of single protein ligand systems can be considered as border case of the underlying transformation pattern. The transformation is independent of the chosen output variables as well as the kinetic properties of the reaction network. However, another choice of output variables may lead to a higher or lower number of observable states. The same holds true for varying kinetic parameters. For given input and output signals the kinetic properties determine whether states are observable and/or controllable. Furthermore, the kinetic parameters also define whether the model equations can be modularized or not. In the considered example the system does not comprise unobservable states and can be divided into five modules if k8k9 and k-8k-9. If k8 = k9 and k-8 = k-9 the system can be reduced to ten ODEs.