Approximating Attractors of Boolean Networks by Iterative CTL Model Checking

This paper introduces the notion of approximating asynchronous attractors of Boolean networks by minimal trap spaces. We define three criteria for determining the quality of an approximation: “faithfulness” which requires that the oscillating variables of all attractors in a trap space correspond to their dimensions, “univocality” which requires that there is a unique attractor in each trap space, and “completeness” which requires that there are no attractors outside of a given set of trap spaces. Each is a reachability property for which we give equivalent model checking queries. Whereas faithfulness and univocality can be decided by model checking the corresponding subnetworks, the naive query for completeness must be evaluated on the full state space. Our main result is an alternative approach which is based on the iterative refinement of an initially poor approximation. The algorithm detects so-called autonomous sets in the interaction graph, variables that contain all their regulators, and considers their intersection and extension in order to perform model checking on the smallest possible state spaces. A benchmark, in which we apply the algorithm to 18 published Boolean networks, is given. In each case, the minimal trap spaces are faithful, univocal, and complete, which suggests that they are in general good approximations for the asymptotics of Boolean networks.

Proposition 1 (Fig. 1). If p is a trap space and A ⊆ S [p] an attractor of (S , →) Proof. The percolation p of a trap space p is defined by iterative substitution (see Sec. 3.1 in main text), i.e., by a sequence of trap spaces p = p 0 , p 1 , p 2 . . . , p K = p where each pair p k , p k+1 is a single percolation step and K the first index that satisfies p K = p K+1 . Witout loss of generality we can assume that K = 1 because the statement is trivially true for K = 0 and will follow for K > 1 by induction. Hence, let p be a trap space whose percolation is achieved by a single step.
Synchronous update: Any state in the subspace p will reach the subspace p by a single transition because f v (x) = p(v) holds for any v in the domain of p (by definition of p). Since p is a trap space this implies that there can not be a SCC in between p and p, i.e., intersecting S [p] \ S [ p].
Asynchronous update: For any state x in the subspace p and any variable v that is fixed in p there is a transition to some state y such that p(v) = y(v). Since this argument can be repeated for y there is a path from x to the subspace p (of at most |D p \ D p | transitions). As before, since p is a trap space there can be no attractor in between p and p. Proposition 2 (Attractor State, Fig. 2). Let p be a trap space and x ∈ S [p].

The state x belongs to an attractor
Proof. Let p be a trap space and x ∈ S [p] with x ∈ A for some attractor A ⊆ [p]. Since A is an attractor it is an inclusion-wise minimal trap set (by definition) and must therefore be strongly connected because otherwise it would contain a smaller trap set. Hence any state in A, and therefore any state reachable from x, has a path back to x. With respect to the reduced system (S Vp , →) this means that any state z ∈ S Vp that is reachable from the projection y satisfies EF(ϕ y ). Since the states reachable from y are referenced by AG it follows that AG(EF(ϕ y )) is true for y. So the transition system (S Vp , →) with initial states {y} satisfies |= AG(EF(ϕ y )).
Let p be a trap space such that TS = (S Vp , →, {y}) |= AG(EF(ϕ y )) where y ∈ S Vp is the projection of some state x ∈ S V onto V p . Then all states reachable from y (AG) have a path back to y (EF(ϕ y ) and hence y belongs to a strongly connected component A (all states of A are connected via y). A must also be a trap set because the connectedness holds for every state reachable from y. Hence A is an attractor. Note that A ⊆ S Vp so far, but given p we can position A in S V , call it A, by assigning values to the variables D p according to p such that A ⊆ S V is an attractor of (S , →) and x ∈ A.
Proof. If p is univocal in (S , →) then A is the only attractor of (S , →) and x ∈ A can be reached from every state in S Vp . Hence the transition system (S Vp , →) with initial states S Vp satisfies EF(ϕ y ). If the transition system TS = (S Vp , →, S Vp ) satisfies EF(ϕ y ) then y belongs to the unique attractor A ⊆ S Vp of (S Vp , →). As in the previous proof we can use p to position A in the original transition system (S , →) and this set A will be the unique attractor A ⊆ S [p] and x ∈ A holds.

y S[p]
full system (S, →) Figure 4: The attractors of a trap space p are faithful iff for every y ∈ S Vp and v ∈ V p there is a path to a state z that satisfies z |= δ v = 0.
Proposition 4 (Faithfulness, Fig. 4) Proof. Let p be faithful and x ∈ S Vp arbitrary. We want to prove that path between x 1 and x 2 and hence a transition in which the activity of v ∈ V p changes. Let Since v ∈ V p was chosen arbitrarily, Eq. 1 holds.
For the other direction let the transition system (S Vp , →) with initial states S Vp be such that Eq. 1 holds for every x ∈ S Vp . The equation therefore holds in particular for every x ∈ A where A is an attractor of S [p]. Hence, for every v ∈ V p and attractor A there is y ∈ A such that δ v (y) = 0 and hence a transition y → y such that y(v) = y (v). Hence Sub(A) = p and so p is faithful. Proof. Let P be a complete set or trap spaces of (S , →) and x ∈ S arbitrary. We want to show that x |= p∈P EF(ϕ p ).
Let A be an arbitrary attractor that is reachable from x. Since P is complete there is p ∈ P such that A ⊆ S [p]. Since there is a path from x to A it follows that x |= EF(ϕ p ) and therefore Eq. 2 holds. For the other direction note that if Eq. 2 holds for all x ∈ S that it holds in particular for all states of every attractor. But if for every attractor A there is a p ∈ P such that there is a path from A to S [p] then A ⊆ S [p] and P is complete.
Proposition 6 (Refinement of Complete Sets, Fig. 6). Let P ⊆ S F be complete in (S , →) and p ∈ P some trap space. If Q ⊆ S Fp is complete in (S Vp , →) then P := (P \ {p}) ∪ {q p | q ∈ Q} is complete in (S , →). Proof. Let P be a complete set of trap spaces of (S , →) and p ∈ P arbitrary. Consider the reduced system (F p , V p ) and its trap spaces S Fp and let Q ⊆ S Fp be complete in (S Vp , →). Note that we defined subspaces as mappings p : D p → B. Hence, although a trap space q of (V p , F p ) is well-defined when considered as a subspace of (V, F ), we need to intersect it with p to assign values to the variables that are implicitly fixed in q when considered as a subspace of (V p , F p ). The completeness of P then follows from the completeness of Q in (S Vp , →) because the dynamics inside p is identical with the dynamics of the reduced system (V p , F p ).
full system (S, →) Proposition 7 (Failure Criterion, Fig. 7). If there is a trap space p such that min(S Fp ) is not complete in (S Vp , →) then min(S F ) is not complete in (S , →).
Proof. Suppose p is such that Q := min(S Fp ) is not complete in (S Vp , →). The main observation is that P := {p q | q ∈ min(S Fp )} ⊆ min(S F ). That is, if the subspace Q are positioned correctly within (S , →), i.e., intersected with p, then they are also minimal trap spaces of (V p , F p ). The statement then follows because if Q is not complete in (S Vp , →) then there is a state x ∈ S [p] that can not reach any trap space in P . But, since p is a trap space x must reach some attractor A which is therefore outside of P and hence outside of min(S F ) which implies that min(S F ) is not complete in (S , →).
U Figure 8: A schematic drawing of the interaction graph, enclosed are SCCs, and an autonomous set U .
Proposition 8 (Fig. 8). Let U be autonomous and Q := min(S F |U ) the minimal trap spaces of the restriction (U, F |U ).
Proof. Observations: The dynamics in the restricted and full transition systems can be related to each other. For any path (y 0 , y 1 , . . . , y k ) of (S U , →) and any x 0 ∈ S [y 0 ] there is a path (x 0 , x 1 , . . . , x k ) of (S , →) such that x i (u) = y i (u) for all u ∈ U and 1 ≤ i ≤ k. Also, for any path (x 0 , x 1 , . . . , x k ) of (S , →) there is a unique path (y 0 , y 1 , . . . , y r ) in (S U , →) with r ≤ k, x 0 ∈ S [y 0 ] and x k ∈ S [y r ] that describes the projected dynamics. It follows that a trap space q of (U, F |U ) is also a trap space of (V, F ) because otherwise we could consider the projection of the path that proves that q is not a trap space in (V, F ) and deduce that q is not a trap space in (U, F |U ), a contradiction. Hence Q is a set of trap spaces of (V, F ). Proof of (a): Let Q be complete in (S U , →) and x ∈ S an arbitrary state. We want to show that there is a path from x to some q ∈ Q. Let y be the projection of x onto U . Since Q is complete there is a path (y 0 , y 1 , . . . , y k ) such that y 0 = y and y k ∈ S [q] for some q ∈ Q. By the observations above there is therefore a path (x 0 , x 1 , . . . , x k ) with x 0 = x and x k ∈ S [q]. Hence Q is complete in (S , →).
Proof of (b): The main observation is that since U is autonomous and since Q = min(S F |U ) it follows that for any p ∈ min(S F ) there is q ∈ Q such that p ≤ q. If Q is not complete in (S U , →) then there is y ∈ S U that can not reach any q ∈ Q. Any x whose projection on U is equal to y can therefore not reach any q ∈ Q in (S , →). Hence it can not reach any p ∈ min(S F ) (because for any p there is a q ∈ Q with q ≤ p). Hence min(S F ) is not complete in (S , →).
Proof. (a) ⇒ (b): Let U be minimal and autonomous in (V, →). We need to show that U is strongly connected. Let u, v ∈ U be arbitrary. If there is no path from u to v then u is not above v and so Above(v) is a proper autonomous subset U , a contradiction to minimality. Hence, there is a path from u to v and so U is stronlgy connected.
(b) ⇒ (a): Let U be autonomous and strongly connected. We need to show that U does not contain a smaller autonomous set. Assume there is U ⊂ U with U = U and U is autonomous. Let u ∈ U \ U . Since U is strongly connected there is a path from u to any u ∈ U . Hence u is above u and so u ∈ U which contradicts u ∈ U \ U . Hence such U does not exist and U is minimal. Proposition 10 (Fig. 9). If P, Q ⊆ S F are complete in (S , →) then P Q := {p q | p ∈ P, q ∈ Q : p and q are consistent} is also complete in (S , →).
Proof. Let A be an attractor of (S , →). Since P and Q are complete there are p ∈ P and q ∈ Q such that A ⊆ S [p] and A ⊆ S [q]. Hence, p and q are consistent and (p q) ∈ P Q. Hence P Q is complete in (S , →).
Proposition 11. Let (Z, ) be the condensation graph of a constant-free network (V, F ). A set U ⊆ V is minimal and autonomous iff U ∈ Z and Lay(U ) = 1.
Proof. Let U be minimal and autonomous. It follows from Prop. 9 that U ∈ SCCs(V, →). We need to show that Lay(U ) = 1. If Lay(U ) > 1 then Above(U ) ⊇ U with Above(U ) = U which contradicts U being autonomous.
For the other direction assume that U ∈ Z and Lay(U ) = 1. We will again use Prop. 9. Note that Z = SCCs(V, →). Also, U is autonomous because if Above(U ) ⊃ U with U = Above(U ) then Lay(U ) > 1, i.e., there would have to be an SCC above U . Note that the last deduction uses the fact that (V, F ) is constant-free.

Update Functions
The update functions for the three Boolean networks are given in Fig. 10.