Boolean gene regulatory network model of centromere function in Saccharomyces cerevisiae

Abstract

Centromeres, a highly conserved locus of eukaryotic chromosomes, have critical function for genome stability and integrity. Because their centromeric DNA sequences are necessary and sufficient for kinetochore recruitment and DNA segregation, point centromeres of Saccharomyces cerevisiae chromosomes provide an attractive system for the study of the regulation of centromere function. Using the mathematical model of Boolean gene regulatory networks, the gene regulatory dynamics of centromere region of S. cerevisiae (budding yeast), which is actively involved in the cell-cycle, has been examined. A gene regulatory network containing the relevant centromere genes of the model organism from biological databases was established and all possible cellular phenotypes subjected to a synchronous gene regulation and attracted to several basins. Gene expression in the largest attractor was compared with the biological data by obtaining changes in the cell-cycle. We show that the model for centromere function recovers a single cyclic attractor. The trajectory flow diagram plotted over all initial conditions of the system also shows good correspondence with the cell-cycle phases. Although other upstream signals are possibly involved in the regulation of centromere genes, proposed interactions with selected cell-cycle genes were sufficient to recover whole cell-cycle process. To truly clarify these proposed regulatory interactions of candidate genes for centromere function, profiling and analyzing their expression levels over time with expanded nodes/edges are required. Moreover, a previously modeled gene knock-down mechanism applied to the network and robustness versus knock-down was interpreted based on the obtained consequences.

Introduction

Dynamic regulation of the genes which are actively involved in a particular biological process and mapping their regulatory patterns are a main issue of interest in current computational biology. Identifying major regulatory elements and their spatiotemporal dynamics is critical to understanding the precise organization of cellular functions.

The concept of gene networks with a focus on levels and patterns of gene expression has become an important part of the current computational genomics studies [1–7]. Regulatory interactions between transcription factors and their target sequences provide a network structure named a gene regulatory network (GRN) [8]. Profiling and analyzing expressions levels and patterns of genes over time allows inferring regulatory networks and the states of the cell where each have a certain phenotype. Mutational dynamics of GRNs also enables to infer the possible regulatory actions of genes playing a key role in a specific process [2, 9–11]. GRN approach has been widely used for describing and modeling regulatory mechanisms involved in various important processes such as, cell-cycle progression, cellular development, and cancer dynamics.

GRNs are graphical representations of parameterized functions, which describe the regulatory interactions between a set of genes, applying correlations, differential equations, Boolean functions, and probability distribution of random variables or more complex functions. Wide ranges of modeling approaches using different mathematical or statistical functions have evolved to infer GRNs. In general, GRN models integrate available data sources and predict the roles of the players as regulators or targets based on their activity levels, directions, and the behavior of the system. These predictions of the GRN model provide a testable hypothesis for comprehensive analysis and can also be used for further experimental design. Therefore, selection and application of an optimal GRN inference method is critical for valuable predictions and depends on available data regarding the regulatory system.

Large-scale genomic, transcriptomic, and proteomic technologies have begun to generate powerful data sets including complete genome sequences, long-term measurements of gene expression profiles, protein levels, and other active components of the system for GRN studies. These resources, together with high-performance computing systems, have greatly enhanced the scope of GRN analysis, providing an abstraction of complex relationships that are not always detectable with conventional techniques. For instance, when the state of the system is expressed in the ordinary differential equations (ODE) models, it is possible to identify the instantaneous changes in the gene expression levels using a set of differential equations [12, 13], while correlating the gene expression profiles with protein content in the environment. Unlike the deterministic models, there are no values for the equilibrium state in stochastic models. Since each reaction or iteration towards the next time constantly changes the number of molecules, the probability distribution of the molecules becomes the output of the system. However, the binary characteristics of regulatory elements can be described as a similarity between digital computation and gene regulatory interactions [14, 15].

In fact, network models based on Boolean states with discrete dynamics are suitable for investigating the dynamical sequences of protein activation patterns. As each node in the network can have a value of 1 or 0, they are also called Boolean networks [16, 17]. The behavior of the system over time is determined by the Boolean functions. Each gene that constitutes a node of the network is active if it has a value of 1 and inactive if it has a value of 0. Activity and inactivity indicate the expression of a gene. Possible states are N element vectors representing the expression of all genes at the corresponding position in the Boolean model [18–20]. Through the dynamical systems approach, if a network of interacting molecules produces a dynamic output signal, then the control sequence is captured. Furthermore, such a control sequence, depending on the interaction network, gives the dynamic trajectory of the system [21, 22].

A GRN may cover an entire genome or may be reduced to sub-cellular tasks. In other words, there are gene assemblies, each of which being responsible for each function of the cell. The genes responsible for the functions of the centromere are also one of these clusters.

The GRN discussed below has been formed by gene interaction data from the literature. There are also many other methods related to Boolean GRN reconstruction from binarized gene expressions. One of them is the REVEAL, which uses a deterministic transition table to infer the Boolean relationships. The other is best-fit, which utilizes partially defined Boolean functions and determine variables with the least error size [23]. Moreover, there are evolutionary computational approaches and neural networks for understanding the behaviors of genes in a cell [24, 25].

Yeast centromere genes and regulation

Centromeres are originally assumed to be primary constrictions of eukaryotic chromosomes providing kinetochore recruitment that is essential for attachment of chromosomes to the spindle microtubules during cell division. Although their primary sequences show complex and diverse patterns in eukaryotes [26–32], they have emerged as central locus for genome stability and integrity. Extensive studies on eukaryotic model organisms have revealed that centromeres are characterized by the presence of specialized nucleosomes containing H3 variant centromeric protein A [33]. Their function and specification is primarily related to epigenetic mechanisms [34].

Because of their centromeric DNA sequences’ necessity and sufficiency for kinetochore recruitment and DNA segregation [26], point centromeres of S. cerevisiae chromosomes provide an attractive system for the study of the regulation of centromere function. Additionally, the cell-cycle of S. cerevisiae, in which mitosis can be observed completely, has entered the literature since 1974 [35, 36] and the gene regulatory dynamics have also been examined using the Boolean model [19, 22]. The point centromeres of S. cerevisiae have certain common features, including a centromeric DNA element (CDEIII), which is specifically recognized by Cbf3 complex (Cbf1–3) and centromeric nucleosome containing histone H3 variant protein Cse4p, nonhistone protein component Scm3p and histone H4 protein encoded by Hhf1-Hhf2. Therefore, in this work, centromere function is considered as regulatory action of these genes, which are related to common features of budding yeast point centromeres and genes whose regulatory actions formerly examined in the budding yeast cell-cycle progression [22]. Additionally, major transcriptional regulator is regarded as Spt10, which is the sequence-specific activator of histone genes.

The regulatory actions of these candidate genes have been analyzed using a mathematical model with a Boolean GRN approach, with the aim of describing the cycle of the centromere function in S. cerevisiae. Additionally, a previously modeled gene knock-down mechanism has been applied to the network and robustness versus knock-down was interpreted based on the obtained results.

The model description

Boolean model

Boolean variables indicate whether a gene or a protein is expressed or whether a kinase or many other molecular entities are activated as an on-off switch [37]. Thus, states of each gene (g_i, i = 1, … , N) assume values 1 for active, or 0 for inactive at any given time. A full state also called a cellular phenotype, which can be written as follows

$$\Phi \left(t\right)=g_1\left(t\right),g_1\left(t\right),\dots ,g_N\left(t\right)$$ 1

where each g_i(t) is in the Boolean domain.

Accordingly, the number of all possible states (phenotypes) of a system is 2^N, in which some specific phenotypes, also called steady states, are the key findings of pathways of biological systems. They are state trajectories updated in discrete timesteps and each gene of the state updated to next timestep at the same time, indicating a synchronous machinery. In more biologically realistic GRNs, time delays between genes are often caused by transcription and translation. When a gene changes its expression, its corresponding product would follow within some unspecified delay, which causes an asynchronous update scheme. However, a resulting synchronous model resembles an asynchronous automation of biological reality [38] and steady states of a discrete system are equivalent to steady states of a continuous system [39] only if continuous models are sufficiently sigmoidal.

Update of each gene is performed via the following rule

$$g_i\left(t+1\right)=\{\begin{array}{c}1,\sum _j{\omega }_{\mathit{ji}}{g}_j\left(t\right)>00,\sum _j{\omega }_{\mathit{ji}}{g}_j\left(t\right)<0g_i\left(t\right),\sum _j{\omega }_{\mathit{ji}}{g}_j\left(t\right)=0\end{array}$$ 2

where ω_ji is the adjacency matrix element of the GRN. Gene states take on the values 1 for activation, −1 for inactivation, and 0 for no interaction at all. Update of all genes in a phenotype shows an increase in the timestep and following timesteps form a trajectory.

Although the study focuses on a deterministic Boolean network with the synchronized updating scheme where each gene passes through the next timestep at the same time, several update schemes have also been applied to our GRN for comparison [40]. In the sequential update scheme, every node is updated at the same time but with a predefined sequence. Regardless of which sequence we determine, the size of the attractor has not changed due to Eq. (2). Block-sequential update schemes separate the nodes into blocks (we have adjusted five blocks with each having two genes) where nodes of the same block update synchronously in a timestep. Here we have identified a sequence from first block to the fifth block. The asynchronous deterministic update scheme provides the update of a random node (chosen with a probability of 1/N) in a timestep, but every node can be updated once. An asynchronous generalized update scheme is similar, but every node can be updated more than once in a timestep.

Inference of the nodes and edges of the network

To generate the yeast centromere GRN, we have collected data from the literature about key genes involved in yeast centromere function and their regulatory interactions. The regulatory interactions between the budding yeast’s centromere and cell-cycle genes have also been gained from the literature and various databases such as NCBI, SGD, Yeastract, BioGRID [41–46]. Table 1 shows the respective literature review for gene interactions.

Table 1.

The regulatory interactions between the budding yeast’s centromere and cell-cycle genes from the literature and various databases

Regulator		Target	Interaction	Reference
Spt10	→	Hhf2	The centromeric histone UAS is specifically recognized by the Spt10 transcription factor, which has the unusual property of requiring two histone UAS elements for high-affinity binding	[47, 48]
Spt10	→	Cse4		[47, 48]
Cse4	→	Cbf1–3	Cse4-depleted cells had severe chromosome segregation defects. It was proposed that yeast kinetochore assembly requires both centromeric DNA specificity and centromeric chromatin. It was shown that centromere binding protein Cbf1 is required for centromeric transcription that plays an important role in the centromere function of budding yeast	[49]
Cbf1–3	X	Sic1	It was shown that centromere binding protein Cbf1 is required for centromeric transcription that plays an important role in the centromere function of budding yeast. Although no direct regulatory association with Sic1 gene was identified, combinatorial effect of transcription factors involved in Sic1 regulation had been evidenced	[50, 51]
Hhf2	→	Clb5&6	Altered histone levels had been shown to cause problems with DNA replication, chromatin structure, chromosome segregation, transcription, and other aspects of cell growth Clb6 was involved in DNA replication during S phase. It was shown that repressing histone H4 gene transcription results in the elevation of replication-stress-induced transcripts	[52–55]
Scm3	X	Cbf1	Negative genetic interaction is described	[56]
	X	Cse4	No direct evidence was reported for negative genetic interaction. But the localization and the number of Cse4p in centromeric nucleosome is regulated by the histone chaperone Scm3p. We propose that Scm3 may have involved in the transcriptional inactivation of Cse4	[57]
	X	Hhf2	Negative genetic interaction is described	[56]
Cdh1	X	Clb1&2	Degradation of Clb1&2	[58]
Sic1	X	Clb1&2	Binding of Sic1 to Clb1&2/Cdc28 and inactivation	[59]
Clb1&2	X	Sic1	Phosphorylation of Sic1 with Clb1&2	[60]
	→	Cdc14&20	Phosphorylation and activation of anaphase promoting complex (APC)	[61]
	X	Cdh1	Phosphorylation and inactivation of Cdh1 with Clb1&2	[62]
Cdc14&20	→	Sic1	Dephosphorylation and activation with Cdc14	[63]
	→	Cdh1	Dephosphorylation and activation with Cdc14	[63]
	X	Clb1&2	Degradation of Clb1&2 with Cdc20/APC Negative genetic interaction is described	[56, 64]
	X	Clb5&6	Cdc20 regulate the ubiquitination of Clb5&6 with APC	[58, 65]
Clb5&6	→	Clb1&2		[22, 61]
	X	Cdh1	Phosphorylation and inactivation of Cdh1 with Clb5&6	[63]
	X	Sic1	Phosphorylation of Sic1 with Clb5&6	[63]

However, now that the number of genes is very large, it is necessary to sort out numerous interactions. A core set of genes has been selected for implementation as a Boolean model by their significance for centromere function and cell division as well. In the consequence of conducted surveys, ten candidates have been selected from those genes, which consist of the histone activator Spt10, the histone proteins Cse4-Hhf2, the non-histone chromatin component Scm3, the sequence specific binding protein Cbf1–3, the cyclins which are Clb5&6-Clb1&2 complexes and the cyclin repressors Cdc14&20-Sic1-Cdh1. We have also obtained the time-dependent expression profile of each gene from Cyclebase 3.0 [66] and a sufficient amount of data available concerning the regulation of their expression (Table 1). To concentrate on the core regulatory effects, genes integrated into the network contain input as well as output relations.

Gene silencing as a knock-down mechanism

Next, theoretical and numerical modeling of gene silencing in Boolean gene regulation has been investigated for the GRN. Gene silencing can be defined as a prevention of expression of a gene and is one of the gene knock-down mechanisms. The product of the silenced gene is insufficient for its own molecular function [67], meaning the removal of the node from GRN [68]. Mathematically speaking, a specific active gene is selected, transformed from 1 to 0 and fixed. According to the following equation [69], a silenced gene may induce a phenotype switch due to the probability to alter the expression for one or more target genes.

$$P_{\varphi }=\frac{{\left(2N-1\right)}^{C_k^{\mathit{\text{out}}}}-{\left(2N-2\right)}^{C_k^{\mathit{\text{out}}}}}{{\left(2N-1\right)}^{C_k^{\mathit{\text{out}}}}}$$ 3

Here N is the number of genes in system and $C_k^{\mathit{\text{out}}}$ is the out-degree of the silenced gene. The robustness of other nodes (or whole phenotype) has been observed by not being affected by the silenced gene.

The simulation of network dynamics, which is the gene regulation related to the proposed GRN, has been applied in the following section. Trajectories have been drawn based on the behavior of the cell during a cell-cycle. In order to find the attractors of GRN dynamics, all phenotypes (2¹⁰) have been subjected to gene regulation. Furthermore, effects of the gene silencing on the super state (largest attractor) of the system have been examined. Trajectories of the altered phenotypes entering the cell-cycle have been compared with the original one. Flow diagrams are drawn in Pajek (v1.18), a program that enables us to easily draw and analyze large networks and dynamic orbits [70].

Results

Proposed GRN and the cell-cycle

The cycle of centromere function is described in terms of regulatory actions of these ten candidate genes (five of centromere and five of cell-cycle genes). Some genes are linked to act as a single gene for the simplicity of the GRN and the cell-cycle. These genes are associated with their activity throughout the cell-cycle and their connections with the same genes. The proposed network is shown in Fig. 1. In order to find the attractors of GRN dynamics, all phenotypes (2¹⁰) have been subjected to gene regulation. Seven attractive phenotypes have been found and thus for 1024 initial phenotypes, how many of them collapsed correlates with the size of the basins of attraction (Table 2).

Fig. 1.

Proposed gene regulatory network model of centromere function of S. cerevisiae. Five centromere (yellow) and five other cell-cycle genes (blue for single and blue-purple for complexes) are blended in the GRN of budding yeast. Green arrows indicate activation whereas red arrows indicate inactivation. Start node is an upstream activator of the network for the cell-cycle

Table 2.

Basin sizes and gene expressions of the attractive phenotypes

Green is used for active, red for inactive states of genes

Our results showed that approximately 84.5% of all the initial phenotypes fall into a basin (size 865) as the largest attractor, which indicates the G1 phase, the most stable one of the cell-cycle. All gene expressions of the cell-cycle obtained from experiments on the budding yeast are available in the literature [22, 57, 71–74] and it has been verified by comparison that the largest attractor is indicative of the G1 phase (Table 2).

Genes in each update scheme have been updated via the rule in Eq. (2) and each update scheme has been averaged over a hundred experiments and the average percentages of the attractor G1 are listed in Table 3.

Table 3.

Sizes and ranking of the attractor G1 for different update schemes

Update scheme	Ranking	Percentage
Synchronous	1	84.5
Sequential	1	84.5
Block-sequential	1	71.8
Asynchronous deterministic	1	58.2
Asynchronous generalized	2	12.4

The interpretation of a stable state of a cell to correspond to a largest attractor phenotype of the network in several update schemes is obvious. The vital stability of the cell is guaranteed and the GRN probably has the dynamical robustness to maintain its activity against some perturbations. Under normal conditions, the cell would be paused in the G1 phase and would wait for the start signal to induce division phases. However, the centromere function genes Spt10, Cse4, Cbf1–3, Hhf2, and Scm3 are inactive at G1 phase because chromosomes’ centromere genes are actively involved in the S, G2, and M phases [44].

As all the accessible states of phenotypes form the phase space of the Boolean gene regulation, each transform between them forms an element of the transfer matrix, where the dynamic trajectories leading to the G1 phase alongside with six other attractors are examined. Transfer matrix is a 2^N × 2^N square matrix as follows;

$$\mathrm{\Phi }=\left(\begin{array}{cccc}{\varphi }_{1,1}& {\varphi }_{1,2}& \dots & {\varphi }_{1,1024}\\ {\varphi }_{2,1}& {\varphi }_{2,2}& \dots & {\varphi }_{2,1024}\\ ⋮& ⋮& \dots & ⋮\\ {\varphi }_{1024,1}& {\varphi }_{1024,2}& \dots & {\varphi }_{1024,1024}\end{array}\right)$$ 4

The result of the gene regulation is the corresponding matrix element 1 when a phenotype turns to the other, others are 0. The transfer matrix is quite large and the inter-phenotypic connections are very infrequent because a phenotype can only turn into another. In the flow diagram formed by the transfer matrix, each point corresponds to a phenotype (Fig. 2).

Fig. 2.

Flow diagram from the regulation of 1024 initial phenotypes. Arrows between the phenotypes indicate the direction of flow to attractors

Temporal dynamics of the largest attractor corresponding to the G1 phase gives rise to a cell-cycle. The cell resting in G1 phase receives the initiation signal with biochemical reactions for the continuation of its viability and starts the cycle. To participate in the cell-cycle of the centromere genes, the cell-cycle emerges from the stimulated G1 phase with the activation of the Spt10 gene, which receives the initial signal outside the network (Table 4).

Table 4.

Expressions of the network genes in the cell-cycle

Each row represent the state of each gene during the cell-cycle and the columns are related cellular phenotypes. The row below shows the corresponding cell-cycle phases

Cell-cycle phenotypes are shown in the flow diagram of the largest attractor of phase space. The excited G1 phase is in the uppermost zone and has reached the G1 phase in nine timesteps (Fig. 3).

Fig. 3.

Trajectory of cell-cycle and corresponding phases in flow diagram. Cell-cycle sequence of budding yeast shown in bold arrows

Comparison with a computer-generated (CG) network

In this section, a network has been generated by the computer that can provide the temporal evolution in Table 3. In the algorithm, it is aimed to create a network with the maximum number of edges different from the original GRN. Thus, although the cell-cycle flow has remained the same, it could be observed how the dynamics of the system established from the real data changes. By comparing the generated adjacency matrix with that of the proposed GRN (Fig. 4), six matrix elements have been found to be different. Considering node connections, where all different edges transform from zero into nonzero elements, it appears that these edges have not been already in the proposed GRN. New activators and inhibitors have been linked to the Spt10, Hhf2, and Scm3 genes and Clb5&6 gene complex. Since the temporal evolution in Table 4 does not change, it can be said that the activities of these four genes with three centromere genes are robust throughout the cell-cycle.

Fig. 4.

Comparison of the adjacency matrices of the proposed GRN (left) and the CG network (right). The red matrix elements are connections that are different from the original GRN

In addition to these, Table 5 shows the attractor comparison of the CG network and the proposed GRN. All of the attractors found in the CG network are same with the proposed GRN which has one more attractor. Alongside that, sizes of the attractors also show a similar ranking.

Table 5.

Size and gene expression profile comparison of the attractors

Size	Attractor (GRN)	Attractor (CG network)	Size
865	0000011000	0000011000	907
61	0000000000	0000000000	40
54	0110000000	–
31	0001011000	0001011000	59
5	0000001000	0000001000	5
5	0000010000	0000010000	4
3	0001001000	0001001000	9

One of the results that can be obtained here is that the main dynamics of the system is not disturbed extremely when a CG network is applied in which the cell-cycle flow is kept constant. In other tested simulations, it was observed that the addition of new edges to other genes or changes of the signs of formerly existing edges led the system to different attractors.

Gene silencing

In the G1 attractor, which has a dynamical trajectory (cell-cycle), there are two genes active. By silencing Cdh1 and Sic1 genes one by one, the results on the fate of the cell-cycle have been investigated. Since the Cdh1 gene has an out-degree of 1, the probability of altering the expression of its target gene is about 0.053 (Eq. 4). With the initiation of cell-cycle by silencing the Cdh1 gene, no phenotypic change has been observed (Table 6).

Table 6.

Attractors recovered by simulations of silencing Cdh1 and Sic1 components

A) In the simulation of silenced Cdh1 no phenotypic change has been observed. B) Silencing Sic1 induces a change in the cell-cycle with 10% probability. Asterisk indicate the network configurations which show different expression patterns (altered phases)

The probability of inducing a change in other genes is approximately 0.102 when Sic1 gene with the out-degree 2 is silenced. A sequencing reaction has been achieved by silencing it (Table 6). First, Sic1 has changed the expression of the Clb5&6 gene, which has in turn influenced Clb5&6 to finally cause a change in Scm3. In the next time steps, Scm3 has changed Cse4, Cbf1–3, Hhf2 and the other three centromere genes which are all extensively affected. The silenced gene induces a change in the cell-cycle with 10% probability. Cell-cycle normalized at the 5th timestep and at the 7th timestep the genes other than Sic1 gene have reached the unchanged G1 phase, which can be interpreted as the completion of the cell-cycle through a different path.

Discussion

The eukaryotic cell-cycle represents an evolutionarily conserved mechanism and requires integration and coordination of different levels of information that are fundamental for successful cell generations. The whole process of one cell-cycle comprises distinct events, which ensure cell growth, DNA replication, and cell division. The eukaryotic cell-cycle consists of four phases: RNA and protein synthesis occur during G1 and G2, DNA is replicated during S-phase and cells undergo mitosis and cytokinesis during M-phase. Eventually, after the M-phase, the new cell generation enters G1 again, thereby completing one cycle [68, 75].

Eukaryotic cells constantly monitor their conditions throughout three checkpoints for progression of normal cell-cycle. Failure of the checkpoints generally activates repair mechanisms that ensure the necessary conditions allowing cells to enter next phases. The primary checkpoint occurs in late G1 and the cells inquire the DNA integrity and cell machinery. In the G1 checkpoint, the cells also decide to enter the S phase and initiate DNA replication. In the second checkpoint, which exists in the S phase, before the G2/M transition, the cells check the accuracy of DNA replication and activate DNA repair mechanisms or apoptosis if necessary. Finally, cells decide to complete cell division throughout the checkpoint control mechanisms within mitosis.

Extensive studies on yeast and several animal models have shown that the cell-cycle process is mainly regulated by cyclins, cyclin-dependent kinases, and Cdk inhibitors, which are regulated by phosphorylation and dephosphorylation events [76]. The cell-cycle gene regulatory dynamics of cyclins and cyclin-dependent kinases have been extensively examined and well documented (Cyclebase and Biogrid).

On the other hand, centromeres that are a highly conserved locus of eukaryotic chromosomes have critical function for chromosome segregation. The genome stability and integrity of eukaryotic cells require the proper organization and behavior of centromeres during the cell-cycle. Recent studies on diverse animal models have shown that centromere function requires some specific features including centromeric nucleosomes, CEN DNA sequences, chromatin marks, and interactions with kinetochore proteins. Although molecular properties and mechanisms underlying centromere function have been extensively examined [77, 78], GRN models for centromere function have not yet been studied.

The budding yeast S. cerevisiae is a eukaryotic model organism for genetics, molecular biology, biochemistry, medicine and mathematical sciences. Although almost all gene regions, proteins, and main regulatory interactions obtained using a wide range of experimental and theoretical approaches, GRN dynamics of most of the cellular processes are an ongoing challenge. The cell-cycle process of the budding and fission yeast are well documented and some of the main genes responsible for the cell-cycle modeled with GRN approach [22, 68].

In this study, we have proposed a GRN model for centromere function of budding yeast. We show that the model for centromere function recovers a single cyclic attractor (Table 2). When the generated network is subjected to synchronous gene regulation, all of the initial conditions (phase space) are reduced to certain attractors. Rather than the synchronous update, four more update schemes have also been subjected to the GRN and three of them indicate the same largest attractor phenotype. When the largest attractor phenotype is exposed to the upstream transcriptional signal of the Spt10 gene, the cell-cycle is obtained by all phases. The trajectory flow diagram plotted over all initial conditions of the system also shows good correspondence with the cell-cycle phases (Fig. 3). Spt10 is the histone gene-specific transcription activator. The activation of histones is mainly mediated by Spt10, incorporation with the major G1/S-phase activators, SBF and MBF [47, 48].

In the largest attractor phenotype, while Sic1 and Cdh1 genes were found to be transcriptionally active, all the centromere genes were inactive (Fig. 2). This super state of the system was considered as G1 phase because of the activity of Sic1 (inhibitor of cyclin-dependent kinase that forms complexes and controls the G1/S phase transition). Previous studies show that Sic1 also plays an important role in ensuring that the genomic integrity and Sic1 deletions lead to increased frequencies of broken and lost chromosomes [79]. The absence of mitotic cyclins (Clb1&2) and the presence of Cdh1 until G1/S transition also considered as indicative of the G1 phase. Cdh1 is the activator of anaphase-promoting complex/cyclosome (APC/C) and required for mitotic exit and passage into G1 of the next cycle [80].

In the network configurations (Fig. 2), centromere genes recovered as inactive until late G1. This result is also consistent with previous findings on G1/S transcriptional activations of the centromeric histone genes [47, 48] and transcriptional profiles of each of the genes gained from Cyclebase. Previous studies have shown that yeast kinetochore assembly requires both centromeric DNA specificity and centromeric chromatin. Centromeric H3 histone variant Cse4 is an essential gene for centromeric nucleosomes and plays a critical role in kinetochore formation [81]. Cse4-depleted cells had severe chromosome segregation defects [49]. Studies on the assembly of centromeric nucleosomes have shown that Scm3 is required for localization of Cse4 to centromeres and G₂/M progression [57]. Centromere binding protein Cbf1 is involved in centromeric transcription, which plays an important role in centromere function in budding yeast [51].

For progression of the whole cell-cycle process, genes that are involved in centromere function should receive feedback from other cell-cycle genes. However, the number of genes which are responsible for the cell-cycle of the budding yeast, is very large, and only five of them are included in the proposed GRN. Although other upstream signals are possibly involved in regulation of centromere genes, proposed interactions with selected cell-cycle genes were sufficient to recover the whole cell-cycle process. Our results show that Clb5–6 is active in S phase and Clb1–2 in G2. These two genes encode B-type cyclins involved in cell-cycle progression. Clb5 is required for the initiation of DNA synthesis during S phase and Clb2 required for G2/M transition [60]. Cdc14–20 is recovered in M phase, required for mitotic exit and cytokinesis [65]. To truly clarify these proposed regulatory interactions of candidate genes for centromere function, profiling and analyzing their expression levels over time with expanded nodes/edges are required.

Upon application of gene silencing to the active genes of the super state, the probability of expression alteration of other genes has been found to be low. Nevertheless, when there is a change, it has been observed that the altered phenotypes entering the cell-cycle try to adjust themselves to the original cycle, from which it follows that the cell-cycle trajectory is a stable trajectory and the other phenotypes are less likely to deviate from the cell-cycle trajectory. Although this phenotypic change can alter the cell-cycle phases in Boolean networks, external factors may also account for these type of alterations [82, 83]. However, in the absence of experimentally supported effects of any perturbative quantity for the organism’s genes or gene groups, it is not possible to make a strong conclusion about the major forces underlying these alterations.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.