## Minimum and maximum level constraints

Previously, we covered the sign consistency rule for backward propagation. In this
part I will introduce a new rules that discards solutions that violate minimum and
maximum constraints on system variables.

### Constrained values for system variables

A lot of variables in biological systems have minimum (resp. maximum)
constraints. Concentration level cannot go below 0 or above 100%, and signals
which are below the detection threshold cannot drop any further. Figure 1
shows an IG with 4 variables a, b, c and d. Lets say the variable c represent a
concentration and only values in the range of 0 to 100 are valid. Further, we know
that in our reference state S_{R} the variable c is at its minimum. Let’s try to
find all labelings μ_{i} that represent transisitions from S_{R} to a state S_{i} where
the level of d has increased, μ_{i}(d) = +. Figure 2 shows all labelings μ_{i} that
satisfy consistency Rule 1. The labelings μ_{2} and μ_{3} violate the the minimum
constraint on variable c, because there exists no value for c in [0,100] such that
sign(c - 0) = –.

To deal with minimum (resp. maximum) constraints we introduce a new
consistency rule.

Rule 2 (obey minima/maxima) A variable that is on its minimum cannot
decrease and an variable at its maximum cannot increase.

Let (V,E,σ) be an IG, MIN ⊆ V variables that are at their minimum, and
MAX ⊆ V variables that are at their maximum. Then a labeling μ : V →{+,–,0}
satisfies Rule 2 for node i ∈ V iff

- μ(i) = 0, or
- μ(i) = –, and iMIN or
- μ(i) = +, and iMAX.

Rule 2 allows us to exclude solutions that violate the constraints on
minimal/maximal values. In Figure 2 only labeling μ_{1} satisfies both consistency
rules 1 and 2.

### Conclusion

In this part I introduced an consistency rule that allows us to filter solutions that
violate constraints restricting the minimum or maximum level of system variables.
I have shown how it can be combined with other consistency rules to get
better explainations for state transistions. In the next part I will introduce a
new consistency rule that increases the predictive power of sign consistency
methods.