Setting negative regulations (red edges) at the same rate a_ij = a_r = −1, positive regulations (green edges) at the same rate a_ij = a_g = 1 and protein lifetime t_d = 1, the main result of the cell-cycle TBN, reported in Li et al. (2004), is the set of attractors in Table 1A. The largest basin of attraction shown is 1764. Of 2¹¹ = 2048 possible network states, 1764 states flow toward the fixed point (0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0), in which inhibitor proteins Cdh1 and Sic1 stay active indefinitely even when the rest of the network is off. Although the cell-cycle network is very complex, the attractor set has only seven attractors, which are all fixed points.

Thomas (1981) explored the effects of different regulatory circuits or feedback loops on the composition of the attractor set. Regulatory circuits are classified as positive or negative depending on whether the number of negative regulations (red edges) in the circuit is odd or even. Thomas proposed that positive circuits are necessary to generate multiple attractors and negative circuits are necessary to generate fixed points and periodic attractors. These ideas were later formalized in theorems by Remy et al. (2008); Richard (2010), and various conditions for a unique fixed point attractor set have been developed by Robert (1980); Shih and Dong (2005); Richard (2013). The theorems and results in this manuscript build upon these works by examining the effect of self-degradation and regulatory circuits on a network's long term behavior.

Theorem 1. Let = (, ) be a TBN of the form in Equation (2) with N nodes, = {1, …, N} and edges . Suppose each threshold parameter satisfies α_i ≥ 0 for each i. If every node has a self-degradation loop and network cycles must have at least 1 negative regulation (red edge), then the network's attractor is a unique fixed point, the null state.

The proof requires the following definition. Let = (, ) be a graph. An ordering of nodes 1, …, N is a topological ordering relative to if, whenever we have i → j ∈ , then i < j. A parent node has a lower order than a child node. Most importantly, a graph is directed acyclic or DAG if and only if it has a topological ordering.

Proof. Denote the set of nodes having either an incoming or outgoing positive regulation (green edge) as _n = {1, …, n} ⊂ . Given that cycles with all positive regulation (green edges) do not exist, choose a topological ordering (with respect to green edges only) for _n, say , and add directed null edges, which have no real regulatory effect, to all pairs of nodes in _n not having an edge such that is not violated. Then _n has the unique topological ordering = 1, …, n. The expression of a node in = (, ) at time t is a function of nodes with smaller topological order and other nodes in at the previous time t − 1, i.e., xi(t)=fi({x1(t−1),…,xi − 1(t−1)},{xi(t−1),…,            xn(t−1),…,xN(t−1)}) where f_i is the transition function for node i of the form in Equation (2) in which the parameter a_ij can take any magnitude so long as positive regulation is defined by a positive sign and negative regulation by a negative sign.

The proof proceeds from the observation that, under the stated hypothesis, if for t_d consecutive time points all nodes with topological ordering smaller than i have value 0, at the time point t immediately following we must also have x_i(t) = 0.

By mathematical induction, we will show that (x₁(k), …, x_n(k)) = (0, …, 0) for some time k and remains at 0→ after time k. At some time t < k, x1(t)=f1({∅},{x1(t−1),…,xN(t−1)})       =0, and remains at 0 indefinitely through negative regulation or self-degradation. At some t′ > t, x2(t′)=f2(t′−t)({x1(t)},{x2(t),…,xn(t),…,xN(t)})        =f2(t′−t)({0},{x2(t),…,xn(t),…,xN(t)})        =0 where the composite function f^{(t′ − t)}₂ is the (t′ − t)th iteration of the transition function f₂, and (t′ − t) ≤ t_d, for any t_d. Node 2 remains at 0 indefinitely through negative regulation or self-degradation. Assume that for some l nodes, all with order less than n, satisfies at time t″ > t′, x1(t′′)=…=xl(t′′)=0, and remains at 0 indefinitely through negative regulation or self-degradation.

Then at time k > t″, xn(k)=fn(k−t′′)({x1(t′′),…,xn−1(t′′)},{xn(t′′),…,xN(t′′)})         =fn(k−t′′)({0,…,0},{xn(t′′),…,xN(t′′)})         =0. where f^(k−t″)_n is the (k − t″)th iteration of the transition function f_n, and (k − t″) ≤ t_d, for any t_d. Node n remains at 0 indefinitely. For all nodes not in _n, they remain at state 0 through negative regulation or self-degradation. Therefore, (x_i(k), …, x_N(k)) = 0→ and remains a fixed point after time k.     □

In short, the proof shows that when upstream positive regulations are shut down by self-degradation, the network turns off in a cascading fashion due to the topological order and self-degradation. The theorem applies to an entire class of networks whose member graphs may have any number of genes, any number of cycles with at least one negative regulation (red edge), differing interaction coefficients a_ij and differing protein lifetimes t_d. The theorem is invariant to a_ij and t_d because these parameters only work to speed up or slow down the rate at which the network reaches the null attractor. An example of a network belonging to this class is displayed in Figure 2A.

Consider a more general network class that is still acyclic in the positive regulations (green edges) but has the additional feature of persistence (green self loops). An example of such a network is shown in Figure 2B.

We noted above that the degradation model defined here implies an assignment to each gene of either a yellow loop or a green self loop. Theorem 1 concerns the special case in which all genes are assigned yellow loops. A green self loop is formally a cycle (which does not contain a red edge), and so the hypothesis of Theorem 1 does not hold if any persistent nodes are present.

However, suppose we are given a TBN which does satisfy the hypothesis of Theorem 1, but we then alter the model by designating a set of nodes as persistent, otherwise leaving the model unchanged. We wish to determine how this affects the complexity of the resulting attractor structure. It must have some effect. To take a trivial case, suppose we have n unconnected persistent nodes. Each may be analyzed as an independent TBN, each of which can sustain a fixed point of value 0 or 1. The total number of unique fixed points for the entire network is therefore 2ⁿ. Of course, the complexity of the attractor structure in this case is due entirely to the lack of any exogenous degradation pathways, and not to any connectivity structure of the network (which does not exist in our example).

We next show that this type of reasoning can be extended to TBNs which have the type of acyclicity defined by Theorem 1, but which also have persistent nodes. It is possible to describe mathematically weaker properties of acyclicity within cyclic networks in a way which bounds the complexity of attractor structure. For example, Skodawessely and Klemm (2011) found the maximum number of fixed points in such a network to be 2^|V| where V ⊆ N is a set of nodes whose removal leaves the network acyclic.

Here, we extend our notion of acyclicity in the following way. We say j is an ancestor of i if there is a directed path from j to i. Define the two sets of nodes: (3)SG={ all persistent nodes }SA={ all nonpersistent nodes not possessing         a persistent node as an ancestor}.

Theorem 2. Suppose we are given a TBN in which the subnetwork defined by the nodes S_A of (3) satisfies the hypothesis of Theorem 1, or for which S_A = ∅.

Next, define the following sequence of subsets of nodes: E1=SG∪SA,Ej={all nodes not in ∪i<jEi with all parents in∪i<jEi},j>1, and suppose for some J all nodes are included in ∪_i≤JE_i. Then any two fixed points with identical values for the persistent nodes must be equal, and therefore the maximum number of fixed points is 2^g, where g is the number of persistent nodes.

Proof. Suppose we are given any fixed point. The nodes in S_A (if any) form a TBN satisfying the hypothesis of Theorem 1, so any fixed point must be 0 on these nodes. This implies that the fixed point values of the nodes in E₂ are determined entirely by those of S_G. The argument may be repeated for E₃, E₄, …, until the fixed point values of all nodes are determined.     □

Theorem 2 complements the result of Skodawessely and Klemm (2011). The conclusion implies a similar upper bound of 2^g for the number of distinct fixed points, where g is the number of persistent nodes. However, while the class of BoNs considered by Theorem 2 is more restricted, removal of the persistent nodes does not necessarily leave the network acyclic, so that the result of Skodawessely and Klemm (2011) does not imply Theorem 2.

The hypothesis of Theorem 2 is satisfied by both TBNs of Figure 2. In particular, for (B) we have S_G = {1, 3}, S_A = ∅, E₂ = {4}, E₃ = {2}. However, if a negative regulation from node 2 to node 4 was added, the hypothesis would no longer hold (we would have E_j = ∅ for all j ≥ 2) and a counter-example could be constructed.

Next, consider, the cell-cycle network of Figure 1. This TBN satisfies the hypothesis of Theorem 2 by setting SG={MBF,Clb5&6,Cdh1,SBF,Sic1,Clb1&2}SA={Cln3}E2={Mcm1/SFF,Cln1&2}E3={Cdc20&14}E4={Swi5}.

It is interesting to note that the hypothesis of Theorem 2 is satisfied despite the existence of a cycle of green edges between Mcm1/SFF and Clb1&2 (due the the fact that one of these nodes is persistent).

We can see from the application of Theorem 2 to the cell-cycle network that the relationship between the attractor structure and the configuration of persistent nodes is similar to the previous example of the completely unconnected TBN, in the sense that all fixed points are fully determined by their values on the persistent nodes, so that the complexity of the attractor structure must be understood to be driven by a selective lack of exogenous degradation pathways.

The assignment of self-degradation (yellow loops) to certain proteins in a network is not a trivial task and cannot be completed ad-hoc because self-degradation influences the network's long term behavior. The simplicity of the attractor set associated with the cell-cycle network in Table 1A is attributable to the presence of self-degradation and a lack of active network cycles composed entirely of positive regulations (green edges). We exemplify this claim with protein Swi5, the transcription factor for inhibitor protein Sic1. According to Li et al.'s rule of assigning self degradation only to proteins without negative regulators (incoming red edges), Swi5 should not self-degrade since it has the inhibitor Clb1&2. However, their representation of the network allowed Swi5 to have both attributes. Suppose we don't allow Swi5 to self-degrade since it has an inhibitor. How would this change affect the network's steady state behavior? We computed the attractor set for the cell-cycle TBN (Equation (1)) disallowing Swi5 to have the self-degradation property in Table 2. Compared to the attractor set with Swi5 self degrading (yellow loop) in Table 1A, the attractor set in Table 2 is bigger with 14 fixed points, half of which has Swi5 on. The attractor set in Table 1A is a subset of that in Table 2, meaning that the new attractors are due to Swi5 not degrading to 0. The biggest attractor in this new set is (0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0) which differs from the biggest attractor in Table 1A only by the presence of Swi5. This exercise has shown that slightly altering the degradation assumption dramatically affected the size and complexity of the cell-cycle's long term behavior.

As noted above, the only cycle constructed with all positive regulations in Figure 1 is between Clb1&2 and Mcm1/SFF, and this cycle is not sustained (both proteins are at state 0) in the network's long term behavior. To leave the cycle on indefinitely, that is, to keep Clb1&2 and Mcm1/SFF at state 1 perpetually, the sum of the interaction coefficients a_g associated with the positive regulations (green edges) must exceed the sum of a_r associated with the negative regulations (red edges) acting on Clb1&2. Since −a_r = a_g = 1, the cycle between Clb1&2 and Mcm1/SFF may get turned on, but does not endure. If this cycle is deleted, the network satisfies the hypothesis of Theorem 1. Because the cycle between Clb1&2 and Mcm1/SFF does not stay on, the network therefore yields a null attractor when all proteins are forced to self-degrade. Thus, following Theorem 2, the variety of fixed points in Table 1A is attributable to the 6 proteins with persistence (green self loop) and the cardinality of the attractor set satisfies the upper bound of 2⁶. Note that the fixed points in Table 1A differ at the proteins with persistence (green self loop), as predicted by Theorem 2. In Section 3.3, we present a network in which the cycle remains active in the steady state.

Since self-degradation is not built into the Markovian transition functions of the TBN model in Equation (1), specifying incremental degradation is a cumbersome separate process that requires tracking each gene with the self-degradation property and counting the t_d time steps prior to a state change. More importantly, by not explicitly modeling degradation, the model in Equation (1) does not have the typical Boolean network behavior. In particular, a state can be repeated without the network having reached an attractor. For example, suppose we have a two member network in which the only regulations are: protein 1 positively regulates (green edge) protein 2, protein 1 self degrades (yellow loop), and protein 2 persists (green self loop). The interaction coefficient is a₂₁ = 1. Further, suppose that a protein's lifetime is t_d = 2. Using the TBN of Equation (1), a network path is (1, 1) → (1, 1) → (0, 1). Markovianizing degradation via the following model eliminates this problem by augmenting the state space to express the degradation counter.

Here I(x_j(t) > 0) is an expression indicator for protein j; ϵ_i ∈ [0, 1] is the degradation rate for protein i; all other parameters are as previously defined in Equations (1) and (2). Whether a protein degrades is determined by the degradation parameter ϵ_i. A protein degrades quickly with a large value of ϵ_i and persists at ϵ_i = 0. The TBN model in Equation (1) with the protein lifetime parameter t_d = 1 is equivalent to setting ϵ = 1 for proteins with self-degradation (yellow loop) and ϵ = 0 for proteins with persistence (self green loop). Note that ϵ = 1/t_d. Compared to the TBN model in Equation (1) for which self-degradation must modeled in a side process, Equation (4) explicitly models self-degradation as part of the TBN.

The third line in Equation (4) is meant solely as a device for Markovianizing degradation and persistence. Thus, x_i(t + 1) ∈ [0, 1], but the regulatory relationships remain Boolean via the indicator I(x_j(t) > 0). The state space has simply been augmented to allow self-degradation. A further modification that would bring a TBN model closer to a system of differential equations would be to eliminate I(x_j(t) > 0) and allow node j to take state x_j ∈ [0, 1] in Equation (4).

So far self-degradation has been treated as a triggered event, i.e., decays occurs after the net influence on the protein is equal to the threshold. The model can be extended to have decay in the presence of a net regulatory effect (Hanel et al., 2012) by letting a protein be its own parent. The sums in Equation (4) would then include node i and line 3 could be omitted with < α_i replaced by ≤ α_i. These extensions of Equation (4) need to be further studied to understand their properties and appropriateness for modeling a genetic regulatory network.