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 SR the variable c is at its minimum. Let’s try to find all labelings μi that represent transisitions from SR to a state Si where the level of d has increased, μi(d) = +. Figure 2 shows all labelings μi that satisfy consistency Rule 1. The labelings μ2 and μ3 violate the the minimum constraint on variable c, because there exists no value for c in [0,100] such that sign(c - 0) = .



Fig. 1: An IG and with the reference state SR where variable c on its minimum.


Fig. 2: All labelings μi with μi(d) = + that satisfy Rule 1. Only μ1 is consistent with the value restrictions of variable c.


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

Rule 2 allows us to exclude solutions that violate the constraints on minimal/maximal values. In Figure 2 only labeling μ1 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.