## Excluding circular self regulation

In this part I will introduce a new global consistency rule to ensure that every
change is justified by a chain of influences that can be traced back to an input
node. This natural constraint is especially useful to exclude self-justification of
changes via positive feedback loops.

### Circular regulation

In most biological systems we find circular regulation via feedback loops. For
the consistency rules that we’ve covered so far circular regulations pose the
problem that they allow to represent state transistions for which we can not
identify the reason that trigger the state change. For example in Figure 1 we
see an IG with labelings that explain an increase in d. Both total labelings
satisfy the propagation rules (Rule 1 and 3). For μ_{1} we see a circular up
regulation in b and c, which is used as explanation for the increase in d, but we
don’t know why b and c have increase in the first place. Only the labeling
μ_{2} allows us to identify a trigger for the state change, i.e. the increase in
a.

To filter labelings which represent circular explanations we introduce the following
consistency rule.

Rule 4 (a change must be founded in an input). Let (V,E,σ) be an IG and
I ⊆ V the input nodes. Then a labeling μ : V →{+,–,0} satisfies Rule 4 for i ∈ V
if

- i ∈ I, or
- μ(i) = 0, or
- there exist a path (v
_{0},…,v_{k}) in E with v_{0} ∈ I, v_{k} = i and
μ(v_{n-1})σ(v_{n-1},v_{n}) = μ(v_{n}) for all n = 1…k.

Using Rule 4 we can avoid manual removal of positive feedback loops as done in
other approaches, and identify state transitions which can be explained by external
perturbations. Only the labeling μ_{2} satisfies Rule 4. Consistency rule 4 is especially
useful if we want apply sign consistency methods in the context of perturbation
experiments. When we are actually interested in the response to perturbations,
or if we want to identify possible perturbations that trigger desired state
transitions.

### Conclusion

This part introduced a consistency rule that allows us to exclude unfounded self
regulations. In the next part I will show how we can relate our model to actual
measurement data via sign constraints.