Recently, some networks representing metabolic reactions in the cell and gene regulatory responses through transcription factors have been elucidated along with progress of experimental systems and accumulation technology in the database [1] . In addition, researches on characterizing the state of the cells as a complex network utilizing these databases have been actively investigated [2] [3] [4] [5] .

Moreover, the deterministic discrete-time dynamics for discrete-state model 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] .

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 S i 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 j ( ≠ i ) is defined as

B i = ∑ j ( ≠ i ) N a i j S j , (1)

where N is the total number of the nodes, and a i j denotes matrix element of the weighted adjacency matrix A representing the interaction between the genes. We take a i j = + 1 when the node j positively regulates the node i (positive interaction), and a i j = − 1 when the node j negatively suppresses the node i (negative interaction).

The node without the self-loop, i.e. a i i = 0 , follows a threshold dynamics from discrete time t to t + 1 ( t ∈ N ) by using the parallel updating scheme as follows:

S i ( t + 1 ) = ( 0 ( B i ( t ) < θ i ) 1 ( B i ( t ) > θ i ) S i ( t ) ( B i ( t ) = θ i ) (2)

where θ i denotes the threshold value of the node i. Also, if the self-loop acts inactively when B i ( t ) = θ i , the effect of the protein degradation called “degeneration”, which is distinguished from a simple inhibition effect, is given as follows;

S i ( t + 1 ) = ( 0 ( B i ( t ) = θ i , a i i = − 1 ) 1 ( B i ( t ) = θ i , a i i = + 1 ) (3)

The budding yeast cell-cycle network model (denoted by G ( 0 ) ) by Li et al. is a special one in a sense that all nodes of the existing self-loops are given as a i i = − 1 . The network is shown in Figure 1. We take the values θ i = 0 for all i in this report. Each regulatory factor is represented by each numbered node ( i = 1 , 2 , ⋯ , 11 ), and the activation effect ( a i j = + 1 ) and suppression effect ( a i j = − 1 ) are indicated by solid and dashed arrows between the nodes. There 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 W = 2 11 = 2048 , and all steady states are seven point attractors by numbering as

A ( 0 ) = { A 1 ( 0 ) , A 2 ( 0 ) , A 3 ( 0 ) , A 4 ( 0 ) , A 5 ( 0 ) , A 6 ( 0 ) , A 7 ( 0 ) } .

The state of the point attractor with the largest basin size among these is A 1 ( 0 ) = 00001000100 = 68 , where the last number is in decimal. According to the study of Li et al. the following facts are known. 1) The attractor with the largest basin A 1 ( 0 ) = 68 corresponds to the stationary G 1 state in the cell-cycle of the budding yeast. 2) When creating the random network model of the same system size N = 11 , there is no attractor that corresponds to A 1 ( 0 ) with a very large

basin size. 3) One of the trajectories to reach the attractor A 1 ( 0 ) coincides with the trajectory of the actual biological cell-cycle. 4) The trajectory corresponding to the biological cell-cycle leading to A 1 ( 0 ) 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 G ( 0 ) 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).

In this section, we investigate the effect of the degenerate self-loops on the attractors of the original network G ( 0 ) . Therefore, we write the network from which the degenerate self-loop of the kth node is removed from G ( 0 ) as G ( − k ) , and the network with self-activating loop is added to the mth node of G ( 0 ) as G ( + m ) . Here, k selects from the nodes with the self-loop, and m selects from the nodes without the self-loop. The attractor sets are indicated as

A ( − k ) = { A 1 ( − k ) , A 2 ( − k ) , ⋯ , A n − k ( − k ) } , A ( + m ) = { A 1 ( + m ) , A 2 ( + m ) , ⋯ , A n + m ( + m ) } ,

and so on, respectively, where n − k and n + m means the number of attractors in the networks G ( − k ) and G ( + m ) , respectively. We can numerically decide the all attractors and the basin size because the network has a state space of 2 11 = 2048 states.

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, 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. 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 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, , 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.

This work was supported with funds from research funding at Musashino University.

The authors declare that they have no competing interests.

Kinoshita, S. and Yamada, H. (2019) Role of Self-Loop in Cell-Cycle Network of Budding Yeast. Open Journal of Biophysics, 9, 10-20. https://doi.org/10.4236/ojbiphy.2019.91002

We randomly construct the cell-cycle network with the same number of nodes and links as the budding yeast, and examined the attractors and its basin size x by re-linking in the network. The number of nodes is 11, the number of active links is 14, number of suppressing links is 15, and (suppressing) self loop number is 5. For each sample, attractors with the largest basin size were examined. In the cases, all are point attractors because the networks satisfy the sufficient condition. Figure 6 shows the probability distribution of the random network that the largest basin size is smaller than x. It follows that about 20 percent even on a random network maintaining the same structure as the budding yeast have attractor with the similar or the larger basin size (≥1700) than the budding yeast. The result is similar with those for ES cell network of C. elegans in Ref. [4] .