In this part I want to give an introduction to interaction graph models and how they are used by sign consistency methods to model the changes in a dynamic system.

Interaction or influence graphs are a widely used representation for complex systems. Nodes represent the components or players in the system and edges denote how these components interact with each other. A lot of biological systems have a representation as interaction graph: hunter prey models, gene regulatory networks, signaling networks, etc. Here is a more formal definition.

Definition 1. An interaction graph is a signed directed graph (V,E,σ), where V is a set of nodes, E a set of edges, and σ : E →{+,–} a labeling of the edges. Every node in V represents a state variable in the modeled system and an edge i → j means that the change of i in time influences the value of j. Every edge i → j of an interaction graph can be labeled with a sign, either + or –, denoted by σ(i,j), where + (–) indicates that i tends to increase (decrease) j.

An example of an interaction graph is given in Figure 1. There exist many variants of interaction graphs some have weighted edges and some have other kind of edges or different types of nodes. But with this definition we will come pretty far.

Interaction graphs are an abstraction of dynamic quantitative systems where a
quantitative state of the system is a mapping S_{i} : V → ℝ^{+}. Sign consistency
methods use signs to denote changes in the variables of the modeled system.
Examples for such changes could be increased or decreased in metabolite
concentrations or expression levels of genes. The signs + and – are used to denote
increase and decrease and 0 signifies no-change. Sign consistency methods relate
the IG model of the system and the variations in between system states by
representing the variations as labels on the nodes in the graph. For example,
the changes between two states of the system can be represented as a sign
labeling of the IG. Given two system states S_{R} and S_{O} the differences between
these states can be represented as the labeling μ_{RO} : V →{+,–,0} with
μ_{RO}(x) = sign(x_{SO} - x_{SR}). See Figure 2 for an example of two states and the
corresponding sign labeling. We use the colors
,
,
to represent the signs
+,–,0.

Further, sign consistency methods define rules that determine which labelings of the graph are considered consistent and which are considered inconsistent. There exist several different consistency rules which are useful to model different properties of a biological system. For now we only consider the following.

Rule 1 (backwards propagation) Every change in a node must be explained by a change in one of its predecessors.

Let (V,E,σ) be an IG. Then a labeling μ : V →{+,–,0} satisfies Rule 1 for node i ∈ V iff

- μ(i) = 0, or
- there is some edge j → i in E such that μ(i) = μ(j)σ(j,i).

Rule 1
implements backward reasoning. Given an effect we look backwards to
verify its cause. Labelings that are consistent with this rule represent the differences
between steady states. In a steady state the values of all state variables are
balanced. Hence, the change in one variable must be sustained by the change
in one of its predecessors. In other words, if S_{R} and S_{O} are steady states
of the system then the labeling μ_{RO} is consistent with Rule 1. The trivial
example is when both state are the same S_{R} = S_{O} then nothing changes
∀x ∈ V : μ_{RO}(x) = 0.

Let’s see what else we can do with that. Figure 3 shows an interaction graph and
all labelings μ_{i} : V →{+,–,0} with μ_{i}(a) = + that satisfy Rule 1.

Often it is useful to represent the labelings in a table as shown in Table 1. As you
can see, there exist only four labelings that satisfy these constraints. In every of
these labeling it holds μ_{i}(e) = + and μ_{i}(f) = –. We can use this table to
predict the behavior of the system. We see that in every steady state, with
an increase in a we also have an increase in e and a decrease in f. For b
and c we can predict that they will not decrease, and for d that it will not
increase.

In this part I introduced interaction graphs and explained how they can be used to model biological systems. We have seen how sign consistency methods represent variations in the system as sign labelings, and how sign consistency constraints can be used to derive predictions over the steady states of a system. So far we have modeled a closed system. In Part 2 I will introduce external inputs and perturbations.