Role of self-loop in cell-cycle network of budding yeast

Study of network dynamics is very active area in biological and social sciences. However, the relationship between the network structure and the attractors of the dynamics has not been fully understood yet. In this study, we numerically investigated the role of degenerate self-loops on the attractors and its basin size using the budding yeast cell-cycle network model. In the network, all self-loops negatively surpress the node (self-inhibition loops) and the attractors are only fixed points, i.e. point attractors. It is found that there is a simple division rule of the state space by removing the self-loops when the attractors consist only of point attractors. The point attractor with largest basin size is robust against the change of the self-inhibition loop. Furthermore, some limit cycles of period 2 appear as new attractor when a self-activation loop is added to the original network. It is also shown that even in that case, the point attractor with largest basin size is robust.


S. Kinoshita, H. Yamada
with such network structures have been widely studied on the properties of the attractors that represent cellular activity states. This is because the state space is finite, so it is easy to search the fixed points and the periodic solutions using computer power. For example, Kauffman et al. modeled the early cells before differentiation with the dynamics of the network, and made the type of the attractors correspond to the type of cells after the differentiation [6] [7] [8] [9]. On the other hand, Li et al. discovered that in the model of the gene regulatory network related to the cell-cycle, there is a fixed point with a very large basin size, and the transition process to the fixed point corresponds to the expression pattern of the gene in each process of the cell-cycle [10]. It should be noticed that in the network of the Kauffman et al., there is no self-regulating factor (self-loop), but in the model of Li et al. the existence of the self-loops has influence on the attractors. Very recently, in other systems such as fission yeast cell cycle and mammalian cell cycle, the Boolean network models for the regulation have also been studied [11] [12] [13].
In this study, using the same gene regulatory network as Li et al. for the budding yeast, we clarify the relationship between the fixed points (point attractors) with large basin size and the presence of the self-loops in the network. It is found that there is a simple division rule of the state space by removing the self-loops, and the point attractors with largest basin size (BS) is robust against the changing the self-loops. The similar results are obtained for C. elegans early embryonic cell cycles as well [14].

Model
Here, we give some basic properties of the Boolean network model of the cell-cycle regulation for the budding yeast. Let us take the binary value { } 0,1 as the state i S of each node i corresponding to the numbered genes as given in Table 1. The states 1 and 0 correspond to expressed and unexpressed genes, respectively and the attractors of the dynamics are associated to cell differentiation. The effect on the node i from the other node ( ) where N is the total number of the nodes, and ij a denotes matrix element of the weighted adjacency matrix A representing the interaction between the genes.
We take 1 ij a = + when the node j positively regulates the node i (positive interaction), and 1 ij a = − when the node j negatively suppresses the node i (negative interaction).
The node without the self-loop, i.e. 0 ii a = , follows a threshold dynamics from discrete time t to 1 t + ( t ∈ N ) by using the parallel updating scheme as follows: .) The third line shows that there is a degenerate self-loop when mark ○ is present in the node. In the decimal notation, each attractor is displayed as, ( ) The last column (BS) represents the basin size of the attractors. Note that Cln 1 represents Cln 1, 2, Clb 5 represents Clb 5, 6, and Clb 1 represents Clb 1, 2. are self-degeneration loops on the 5 nodes, Cln3, Cln1-2, Swi5, Cbe/Cdc14, Mcm1/SFF. Note that this rule is the same as that of Refs. [5] and [10], but it differs from that of [15]. In this network, the total state number is and all steady states are seven point attractors by numbering as The state of the point attractor with the largest basin size among these is  A is stable against external perturbation.
In addition, the result for the basin size of the attractors in the similar random networks with same conditions of the structure as the ( ) 0 G is given in Appendix A. We confirmed that the occurrence probability of the point attractors with the large basin size (≥1700) is less than 20 percent. This result is consistent with those in Ref. [4].
These results may be due to all self-loops being degenerate and threshold values being zero, and all the attractors are point attractors only. Generally, the threshold values are related to adding the active self-loops at each node. Note that for fission yeast cell-cycle model with similar network structure some limit cycles of period two appear as the attractor because some of the threshold value are not zero [15] [16]. Further, notice that when an active self-loop is attached to the node the state update rule becomes different from those of Tran et al. due to the existence of rule (3).

Numerical Result
In this section, we investigate the effect of the degenerate self-loops on the at-

Case of Removing Degenerate Self-Loop
In Figure 1 of the original network, degenerate self-loops are included in five control factors of Cln3, Cln1-2, Swi5, Cbc20/Cdc14, Mcm1/SFF, and Table 1 shows the 7 attractors. We show in Table 2  We compare the attractors of the network ( ) That is, all of the attractor sets ( ) Next, we focus on the change of the basin size. It follows that the basin size of the attractor ( )

1
A with the largest basin size is reduced by the elimination of the degenerate self-loop. Also, the basin size of the other attractors are also reduced from those of ( ) 0 A . Figure 2 shows the basin structure of the 2048 initial states Table 2. Eleven attractors in the gene regulatory network ( ) 1 G − which removed the degenerate self-loop of Cln3 (the first node). (All are point attractors.) The last column (BS) represents the basin size of the attractors. In the decimal notation, each attractor is displayed as,   flowing to the fixed points given in Table 2. The red circles are the point attractors of ( ) In Figure 3, we show the coloring basin structure of ( ) 0 G depending on each basin of the attractors of ( ) − . (Figure 4 shows the one that removed the color-coded state other than red from the attractor of the largest basin.) It is found that the newly appearing attractors of ( ) 1 G − are created by connecting the the leaf states to the other leaf states in the original gene state in the transition diagram.
Although above results are for the specific case which the degenerate self-loop of Cln3 has been removed, but also it is found that the similar results are also true for the cases removing the other degenerate self-loops. Furthermore, if we apply this rule repeatedly in the process of removing the self-loops, we can see that in general the above relations of the attractors and the basin size also applies to the relationship before and after removing the self-loops.

Case of Adding Active Self-Loop
It is noting that in the general network which both the self-regression loops and self-activation loops exist, limit cycles can appear as the attractors, as shown in case of the fission yeast. In networks which the self-activation loop is added to S. Kinoshita, H. Yamada   Table 3. It follows that the attractors ( ) ( ) , exist also in the network ( ) 0 G , and the limit cycle attractors of period 2, ( )  Table 3 is shown in Figure 5. It is found Table 3. Five attractors present in gene regulatory network ( ) 8 G + which an active self-loop is added to Clb5 (the 8th node). The three attractors ( ) 8 2 A + , ( ) 8 3 A + , ( ) 8 4 A + are limit cycle. G + . The 2 red circles present the point attractors. The 6 blue circles represent the states that belong to the three limit cycles of period 2 two each, respectively. that the limit cycles are constituted by combining the gene states with the relatively small basin size. In such a case the limit cycles with large basin size do not occur.
These features occur even if the self-activated loop is added to the other nodes without the self-loop. Furthermore, the similar phenomena can also be confirmed by changing any of the degenerate self-loop of the five nodes to the active one.

Summary and Discussion
In this short report, we investigated the influence of the degenerate self-loop on attractor of the gene regulatory network model of the cell-cycle of budding yeast.
In the case of networks with degenerate self-loops removed from the original network ( ) 0 G , only the point attractor appears because all of the self-loops are degenerate. The attractor set of the network without the degenerate self-loops includes all attractors of the original network ( ) 0 G . In addition, when self-regression loops and self-activation loops coexist, limit cycles with the period more than 2 appear other than point attractor, and many attractors of ( ) 0 G are not included in the attractor set, but the attractor with the largest basin size was relatively stable against the deletions and additions of the self-loop. Above result can apply to Boolean genetic network model of C. elegans early embryonic cell-cycle network as it is, because the self-loops of network are only self-inhibitation loops, and the attractors are only fixed points [4] [14].
Note that necessary and sufficient condition that the network attractors does not become limit cycle but only point attractors is not known yet [5] [15] [17]. However, we expect that the result in Subsec. 3.1 holds when at least the attractors are only fixed points in the random network with only degenerate self-loops.
There is a theorem in the graph theory [15]: Consider a Boolean network such that each gene is governed with a threshold function. Then, if the associated incidence graph, without considering the diagonal elements, is a directed acyclic graph (DAG) and the thresholds are non negative, 0 i θ ≥ , then the attractors are only fixed points. The network of the budding yeast satisfies the following sufficient condition for the fixed points. The result of the Subsec. 3.2 seems to contradict above theorem at first glance. However, considering that the update rule (3) is different from one in Ref. [15], we can see that it is not necessarily contradictory to the theorem.