
==== Front
PLoS One
PLoS One
plos
PLOS ONE
1932-6203
Public Library of Science San Francisco, CA USA

10.1371/journal.pone.0306523
PONE-D-23-41629
Research Article
Biology and Life Sciences
Cell Biology
Cell Processes
Cell Cycle and Cell Division
Computer and Information Sciences
Network Analysis
Computer and Information Sciences
Network Analysis
Protein Interaction Networks
Biology and Life Sciences
Biochemistry
Proteomics
Protein Interaction Networks
Earth Sciences
Soil Science
Soil Perturbation
Research and Analysis Methods
Animal Studies
Experimental Organism Systems
Model Organisms
Saccharomyces Cerevisiae
Research and Analysis Methods
Model Organisms
Saccharomyces Cerevisiae
Biology and Life Sciences
Organisms
Eukaryota
Fungi
Yeast
Saccharomyces
Saccharomyces Cerevisiae
Research and Analysis Methods
Animal Studies
Experimental Organism Systems
Yeast and Fungal Models
Saccharomyces Cerevisiae
Physical Sciences
Mathematics
Statistics
Statistical Data
Biology and Life Sciences
Biochemistry
Proteins
Protein Interactions
Protein-Protein Interactions
Research and Analysis Methods
Database and Informatics Methods
Database Searching
A perturbation approach for refining Boolean models of cell cycle regulation
A perturbation approach for refining boolean models of cell cycle regulation
https://orcid.org/0000-0002-2727-3290
Banerjee Anand Conceptualization Formal analysis Supervision Writing – original draft Writing – review & editing 1 2 *
Rahaman Asif Iqbal Data curation Formal analysis Investigation Writing – review & editing 3
Mehandale Alok Data curation Investigation Writing – review & editing 3
https://orcid.org/0000-0001-7742-0140
Kraikivski Pavel Conceptualization Formal analysis Supervision Writing – original draft Writing – review & editing 1 2 *
1 Division of Systems Biology, Academy of Integrated Science, Virginia Polytechnic Institute and State University, Blacksburg, VA, United States of America
2 VT-Center for the Mathematics of Biosystems, Virginia Polytechnic Institute and State University, Blacksburg, VA, United States of America
3 Department of Computer Science, Virginia Polytechnic Institute and State University, Blacksburg, VA, United States of America
Ruz Gonzalo A. Editor
Universidad Adolfo Ibanez, CHILE
Competing Interests: The authors have declared that no competing interests exist.

* E-mail: anandban@vt.edu (AB); pavelkr@vt.edu (PK)
6 9 2024
2024
19 9 e030652311 12 2023
19 6 2024
© 2024 Banerjee et al
2024
Banerjee et al
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Considerable effort is required to build mathematical models of large protein regulatory networks. Utilizing computational algorithms that guide model development can significantly streamline the process and enhance the reliability of the resulting models. In this article, we present a perturbation approach for developing data-centric Boolean models of cell cycle regulation. To evaluate networks, we assign a score based on their steady states and the dynamical trajectories corresponding to the initial conditions. Then, perturbation analysis is used to find new networks with lower scores, in which dynamical trajectories traverse through the correct cell cycle path with high frequency. We apply this method to refine Boolean models of cell cycle regulation in budding yeast and mammalian cells.

Academy of Data Science Discovery Fund https://orcid.org/0000-0002-2727-3290
Banerjee Anand Academy of Data Science Discovery Fund https://orcid.org/0000-0001-7742-0140
Kraikivski Pavel The work was supported by the Academy of Data Science Discovery Fund awarded to AB and PK. Data AvailabilityAll relevant data are within the paper and its Supporting Information files. Python codes used for the simulations and for generating figure panels can be accessed on GitHub: https://github.com/asif256000/boolean_cellcycle_analysis/releases/tag/v1.1.
Data Availability

All relevant data are within the paper and its Supporting Information files. Python codes used for the simulations and for generating figure panels can be accessed on GitHub: https://github.com/asif256000/boolean_cellcycle_analysis/releases/tag/v1.1.
==== Body
pmcIntroduction

The cell cycle is the complex process through which a cell grows and divides into two genetically identical daughter cells. The progression of a cell through the cell cycle can be viewed as a trajectory through a multidimensional space of protein activation states which is controlled by an intricate network of biochemical reactions. The reaction network is fundamentally identical in all eukaryotes, and involves the activation/deactivation of different Cyclin-Cdk complexes at different stages of the cell cycle.

Most mathematical models of cell cycle are Ordinary Differential Equation (ODE) based [1–6]. While these models provide complete information about the time evolution of concentrations of different molecular species, they often require a large number of parameters in the form of reaction rate constants, whose values are often not known.

In studies of biological processes where detailed temporal information is not necessary, Boolean modeling can be used to analyze qualitative behavior of the system [7]. This approach eliminates the need for kinetic parameters. The model consists of a network with nodes and edges. The nodes represent genes or proteins, and the edges signify reaction connecting them. Gene/protein activity is often represented by binary values, 1 and 0, denoting active and inactive states, respectively. The activity values are updated using logic-based rules. Boolean modeling has been used to study a wide range of biological phenomena, including gene regulatory networks [8], signaling pathways [9], cell fate decisions [10], and cell cycle [11–14].

The dynamics of a Boolean network depend not only on its update rules but also on the order in which these updates occur, commonly referred to as the update scheme [15]. In general, the update scheme can be defined as synchronous or asynchronous. In the synchronous update scheme, the values of all variables are updated simultaneously in one update step. The synchronous update is computationally easy to implement; however, it can produce spurious cycles in the dynamics. In contrast, in asynchronous update schemes, a randomly chosen variable is updated in each time step, resulting in stochastic representation of the dynamics. In this method, the same reaction can occur multiple times before the next reaction occurs, allowing simulation of biological processes that involve reactions on slow and fast time scales. A drawback of the asynchronous scheme is that the run time for simulations increases rapidly with the number of nodes in the network [16].

Among the Boolean models of cell cycle, Li et al. [11] studied the cell cycle network in budding yeast using synchronous update scheme, and found that the steady state with the largest basin of attraction is stable to perturbations, and the correct sequence of protein activation states during cell cycle, is the globally attracting trajectory of the dynamics. Davidich et al. [12] found similar results in the case of cell cycle networks in fission yeast. Faure et al. [13] used both synchronous and asynchronous update schemes to analyze mammalian cell cycle networks, and found that synchronous update resulted in two attractors: a steady state corresponding to G0 (with the inhibitors of cell cycle active), and a cyclic attractor corresponding to the cell cycle. Asynchronous update preserved the G0 steady state, but the cyclic attractor observed in the case of synchronous update scheme changed to a complex attractor made up of many intertwined cycles.

Although these studies have laid the groundwork, they include only a small subset of the genes/proteins involved in the cell cycle. Furthermore, the models were developed in a user-centric manner as opposed to a data-centric manner. To develop more comprehensive data-centric models, a systematic and automated approach is needed to construct the initial network and then incorporate additional nodes into a growing network.

In this study, our aim was to develop a method for refining Boolean models of cell cycle so that the model dynamics follows the correct cell cycle trajectory with high frequency. To achieve this goal, we analyzed three cell cycle models shown in Fig 1: (a) Model A: the budding yeast cell cycle model [11], (b) Model B: the mammalian cell cycle regulation derived from Novak-Tyson model [1] and, (c) Model C: the mammalian cell cycle control derived from Gerard-Goldbeter model [2]. We defined a network score which quantifies the consistency of the model dynamics with that of the cell cycle. All single-edge and double-edge perturbations of the networks were analyzed to search for new interactions that improve the network score. For each model, we found new interactions that improve the network score. We compared those interactions with the ones listed in the protein-protein interaction database SIGNOR 3.0 [17]. By filtering out the interactions that are inconsistent with the database, we were able to produce refined versions of all the three models under study. Our future plan is to is to enhance the level of automation in our analysis and develop approaches to handle larger number of nodes and edges in the network.

10.1371/journal.pone.0306523.g001 Fig 1 Three cell cycle models.

(a) The budding yeast cell cycle regulation network derived from Li et al., model [11]. (b), (c) Mammalian cell cycle regulation networks derived from Novak-Tyson and Gerard-Goldbeter models [1, 2], respectively. Oval shapes (network nodes) are proteins and lines between proteins (directed edges) are interactions that are of two types: activation represented by green arrow-headed lines and inhibition represented by red bar-headed lines.

Results

Budding yeast: Model A

To test and optimize our perturbation-based approach, we first analyzed the interaction network controlling the cell cycle in budding yeast, shown in Fig 1A. The network was taken from Li et al. [11]. As mentioned previously, Li et al., used synchronous update scheme and analyzed the robustness of the network by calculating the effect of single perturbations on the size of the largest attractor. We furthered that analysis by using the asynchronous update method, and a more comprehensive scoring scheme to quantify perturbations–one that includes the basin sizes of all the steady states, as well as the sequence of protein activation events. (See Methods for the definition of network score)

The network in Fig 1A has 11 nodes, resulting in 211 = 2048 initial conditions. All initial conditions eventually reach one of the 7 steady states listed in Table 1. As expected, these 7 steady states are same as those found in Ref. Li et al. [11], but due to the asynchronous updating rule used in our simulations, the size of the basin of attraction corresponding to each steady state is slightly different in comparison. The steady state with the largest basin of attraction (in bold) corresponds to the G0 phase of the cell cycle, in which the inhibitors of the cell cycle Cdh1 and Sic1 are active, and the cell is in a resting phase (neither dividing nor preparing to divide).

10.1371/journal.pone.0306523.t001 Table 1 List of steady states and the corresponding sizes of basin of attraction.

The basin sizes were obtained using asynchronous update scheme.

Cln3	MBF	SBF	Cln1,2	Cdh1	Swi5	Cdc20,14	Clb5,6	Sic1	Clb1,2	Mcm1,SFF	Basin size	
0	0	0	0	1	0	0	0	1	0	0	1151	
0	0	0	0	0	0	0	0	1	0	0	374	
0	0	1	1	0	0	0	0	0	0	0	257	
0	1	0	0	1	0	0	0	1	0	0	155	
0	1	0	0	0	0	0	0	1	0	0	48	
0	0	0	0	0	0	0	0	0	0	0	37	
0	0	0	0	1	0	0	0	0	0	0	26	

Next, we introduced single-edge and double-edge perturbations (see Methods for the definitions) to the network, aiming to identify perturbations that decrease the network score. Fig 2 shows the histogram of normalized scores after single-edge and double-edge perturbations of the network. Notably, in both instances, the distribution’s peak aligns with the score of the original network. However, intriguingly, a significant proportion of the single-edge (28% of the total) and double-edge perturbations (14% of the total) led to a lower network score, i.e., a normalized score less than one. The complete list of network scores corresponding to single-edge and double-edge perturbations is given in the S1 Table.

10.1371/journal.pone.0306523.g002 Fig 2 Model A: Histogram of normalized network scores after single-edge perturbation (a) and double-edge perturbation (b). The normalized scores were calculated by dividing the scores of the perturbed networks with the score of the original network. 28% of the single-edge and 14% of the double-edge perturbations had a normalized score less than one.

From the perturbations resulting in a score smaller than the original graph, the ones which are consistent with experimental data are listed in Table 2. For example, Skotheim et al., observed that Cln1/2 dependent positive feedback promotes coherent SBF and MBF regulated gene expression [15]. Therefore, the addition of a positive influence of Cln1/2 on MBF can improve the consistency of the modeled cell cycle regulation with data. For the other two interactions in Table 2, it is well known that the transcription factor SBF controls CLN1/2 transcription, and the transcription factor MBF regulates CLB5/6 [18, 19], some data also suggest that in the absence of MBF, SBF is able to regulate expression of Clb5 [20]. Also, deletion analysis of upstream DNA sequences shows that Cln2 transcription can be induced by MBF transcription factor [21]. Furthermore, some known cell cycle models incorporate the positive regulation from SBF on Clb5,6 and MBF on Cln1,2 in addition to SBF→Cln1,2 and MBF→Clb5,6 [4, 22]. These additional interactions allowed the models to correctly characterize a vast amount of gene deletion mutant strains and better explain the regulation of cell cycle START transition. Further, removing a direct regulation of Mcm1 transcription factor by Clb5/6 is another way to improve the network score. This interaction has been introduced by Li et al. [11] to simplify the original cell-cycle regulation from Chen et al. [3] by removing DNA replication event that is controlled by Clb5/6. Thus, the control of DNA replication by Clb5/6 was bypassed by the direct Cln5/6→Mcm1 interaction that drives cell cycle into G2-M phases. Therefore, the direct activation of Mcm1 by Clb5/6 can be removed since this regulation is not supported by experimental data.

10.1371/journal.pone.0306523.t002 Table 2 List of single-edge and double-edge perturbations of the network in Model A that are consistent with experimental data and result in a network score smaller than the original network.

The last three columns show the percentage of trajectories that go through the correct sequence of protein activation states (correct), incorrect sequence of protein activation states (incorrect), and return back to G0 state without ever reaching the S phase of the cell cycle (did-not-start).

Perturbation	Normalized Network Score	steady state count	Largest basin size	Correct	Incorrect	Did-not-start	
Original Graph	1	7	1151	42%	16%	42%	
Single-edge perturbation	
Cln1,2-to-MBF: 0 to 1	0.59	6	1322	55%	20%	25%	
MBF-to-Cln1,2: 0 to 1	0.85	5	1296	54%	21%	25%	
SBF-to-Clb5,6: 0 to 1	0.87	7	1280	46.5%	28%	25.5%	
Clb5,6-to-Mcm1: 1 to 0	0.88	7	1161	58%	0%	42%	
Double-edge perturbation	
Cln1,2-to-MBF: 0 to 1 Clb5,6-to-Mcm1: 1 to 0	0.44	6	1344	74%	0%	26%	
Cln1,2-to-MBF: 0 to 1 Cdh1 to Swi5: 0 to -1	0.61	6	1317	52%	19%	29%	
MBF-to-Cln1,2: 0 to 1 SBF-to-Clb5,6: 0 to 1	0.71	5	1420	59%	34%	7%	

Interestingly, double perturbation Cln1,2-to-MBF: 0 to 1 and Clb5,6-to-Mcm1: 1 to 0 resulted in the largest drop in the network score. The combination of these perturbations increased the basin size of the largest steady state as well as the frequency with which the dynamics follows the correct trajectory. From the network it is clear that positive regulation of transcription factor MBF by Cln1,2 improves the chances of Clb5,6 activation, which marks the point-of-no-return for this model. This results in more trajectories going through the cell cycle. We also found that the non-biological steady state [Cln3 = 0, MBF = 0, SBF = 1, Cln1,2 = 1, Cdh1 = 0, Swi5 = 0, Cdc2014 = 0, Clb5,6 = 0, Sic1 = 0, Clb1,2 = 0, Mcm1,SFF = 0] is absent in the presence of this double perturbation.

Mammalian cell: Model B

Next, we analyzed the ODE-based Tyson-Novak model [1] of cell cycle in mammalian cells. Guided by the ODEs, we deduced the molecular interaction network (Fig 1B) that we analyzed using the Boolean approach. Our model has 10 nodes, resulting in 210 = 1024 initial conditions. We found the largest attractor was the G0 state with a basin of attraction of size 283. The complete list of steady states (98 in total) and the corresponding basin sizes are given in the S2 Table.

Like previously, we used the perturbation analysis to find networks with a lower score. The distributions of scores after single-edge and double-edge perturbations are shown in Fig 3. The distribution again peaks at the original graph score, but in this case, we find more perturbations– 39.5% of single-edge and 35.2% of double-edge—resulting in a lower graph score. The full list of network scores corresponding to single-edge and double-edge perturbations is given in the S2 Table.

10.1371/journal.pone.0306523.g003 Fig 3 Model B: Histogram of normalized network scores after single-edge perturbation (a) and double-edge perturbation (b). The normalized scores were calculated by dividing the scores of perturbed networks with the score of the original network. 39.5% of the single-edge and 35.2% of the double-edge perturbations had a normalized score less than one.

In Table 3 we list the perturbations that are consistent with experimental data and also result in a score lower than the score computed for the original network. In this case, the validity of perturbations were determined by comparing them against known protein-protein interaction listed in the database SIGNOR 3.0 [17] (see S1 Text for details).

10.1371/journal.pone.0306523.t003 Table 3 List of single-edge and double-edge perturbations of the network in Model B that are consistent with experimental data and result in a network score smaller than the original network.

The last three columns show the percentage of trajectories that go through the correct sequence of protein activation states (correct), incorrect sequence of protein activation states (incorrect), and return back to G0 state without ever reaching the S phase of the cell cycle (did-not-start).

Perturbation	Normalized Network Score	Steady state count	Largest basin size	Correct	Incorrect	Did-not-start	
Original Graph	1	98	283	58%	0%	42%	
Single-edge perturbation	
CycE-to-CycE: 0 to -1	0.5	167	640	53%	3%	44%	
CycA-to-CycE: 0 to -1	0.52	219	640	55%	0%	45%	
pRB-to-CycE: 0 to -1	0.65	34	640	50%	2%	48%	
Double-edge perturbation	
CycE-to-CycE: 0 to -1 CycA-to-CycE: 0 to -1	0.58	171	633	59%	4%	37%	

The single-edge perturbations CycE-to-CycE: 0 to -1 and CycA-to-CycE: 0 to -1 cause large drops in the network score. The effect of both perturbations is to turn off the CycE activity after the cell cycle has entered the S phase. The interaction corresponding to the perturbation CycA-to-CycE: 0 to -1, namely, negative autoregulation of CycE, was indeed observed in Ref. [23], where autophosphorylation of CycE-Cdk2 complex results in ubiquitin-mediated degradation of CycE. The interaction corresponding to the perturbation CycA-to-CycE: 0 to -1, namely negative regulation of CycE by CycA does not exist in databases. However, the known sequence of interactions CycA ⊣ Cdh1 ⊣ Skp2 ⊣ CycE results in the indirect negative regulation of CycE by CycA. Since Skp2 is not a part of Model B, we think the perturbation CycA-to-CycE: 0 to -1 tries to capture the indirect negative interaction with a direct one. Interestingly, the perturbation pRB-to-CycE: 0 to -1 is also present in our network for the Model C (Fig 1C). Further, Model B has a lower graph score when the negative regulation of Cdh1 by CycE is removed from the network. This interaction is not found in databases and also not included in Model C (Fig 1C).

When both network edges CycE-to-CycE: 0 to -1 and CycA-to-CycE: 0 to -1 in Model B are simultaneously modified the graph score drops a little bit lower compared with the score results for corresponding single-edge perturbations. The graph score is also better when the activation of p27 by phosphatases PPs is removed and self-activation loop to Cdh1 is added. The last regulation is also present in Model C (Fig 1C). Overall, almost all interaction modifications that give lower graph scores for Model B in our perturbation analysis are already present in Model C that we analyze next.

Mammalian cell: Model C

Finally, we constructed a Boolean model using the ODE-based Goldbeter model of mammalian cell cycle [2]. The network of interaction is shown in Fig 1C. Compared to Model B in Fig 1B, this network has additional nodes Skp2, Wee1, and Cdc25. For this model we find that the dynamics for all initial conditions converge to the attractor corresponding to G0 state listed in Table 4.

10.1371/journal.pone.0306523.t004 Table 4 Steady state and the basin size for the Goldbeter model.

CycD	CycE	CycA	CycB	E2F	Skp2	Cdh1	Cdc25	pRB	p21-p27	Cdc20	Wee1	PPs	Basin size	
0	0	0	0	0	0	1	0	1	0	0	1	0	8192	

The distribution of scores after single-edge and double-edge perturbations are shown in Fig 4. The original network gives excellent results, and as a consequence, relatively few perturbations resulted in a lower score. The complete list of network scores corresponding to single-edge and double-edge perturbations is given in the S3 Table.

10.1371/journal.pone.0306523.g004 Fig 4 Histogram of normalized network scores after single perturbation (a) and double perturbation (b). The normalized scores were calculated by dividing the network scores with the score of the original network. 35.2% of the single-edge and 2% of the double-edge perturbations had a normalized score less than one.

The perturbations that are consistent with experimental data and also result in a score lower than the score for the original network are listed in Table 5. The double-edge perturbation CycA-to-CycE: 0 to -1 and RB-to-RB: 1 to 0, resulted in the lowest graph score. Remarkably, the indirect negative regulation of CycE by CycA (CycA ⊣ E2F1 → CycE) is present in Model B. This double-edge perturbation improved the frequency of starting the cell cycle as well as the frequency of correct trajectories. Interestingly, the single perturbation RB-to-RB: 1 to 0 has a score of 13.8, but when combined with CycA-to-CycE: 0 to -1, results in a score of 0.85. The second lowest graph score was obtained by adding a negative interaction from phosphatases PPs on Skp2 and a negative regulation of p21 and p27 by Skp2. Both perturbations are found in databases. For example, dephosphorylation of Skp2 by the mitotic phosphatase Cdc14B promotes the degradation of Skp2 [24]. Also, Skp2 is required for ubiquitin-mediated degradation of p27 and p21 [25, 26]. The double-edge perturbation Cdc25-to-CycE: 0 to 1 and Cdh1-to-Cdc25: 0 to -1 also improves the graph score. The up-regulation of CycE by Cdc25A has been confirmed in Ref. [27]. It has been also observed that Cdc25A degradation is mediated by the anaphase-promoting complex (APC/C)(Cdh1) [28]. The final suggestion in the list of Table 5 is pRB-to-CycA: -1 to 0 and PPs-to-Wee1: 0 to 1 double-edge perturbation. The positive regulation of Wee1 by phosphatases agrees with studies reporting that phosphatase Cdc14A inhibits Wee1 degradation through dephosphorylation [29]. Also, the direct negative regulation of CycA by pRB does not appear in databases. Further, Model C has already a negative indirect influence of pRB on CycA (pRB ⊣ E2F1 → CycA). Thus, the direct inhibition of CycA by pRB is redundant.

10.1371/journal.pone.0306523.t005 Table 5 List of single-edge and double-edge perturbations of the network in Model C that are consistent with experimental data and result in a network score smaller than the original network.

The last three columns show the percentage of trajectories that go through the correct sequence of protein activation states (correct), incorrect sequence of protein activation states (incorrect), and return back to G0 state without ever reaching the S phase of the cell cycle (did-not-start).

Perturbation	Normalized Network Score	Steady state count	Largest basin size	Correct	Incorrect	Did-not-start	
Original Graph	1	1	8192	54%	2%	44%	
Single Perturbation	
CycA-to-CycE: 0 to -1	1	1	8192	54%	1%	45%	
CycE-to-CycA: 0 to 1	0.98	1	8192	55%	2%	43%	
CycA-to-p21-p27: 0 to -1	0.98	1	8192	55%	2%	43%	
pRB-to-CycA: -1 to 0	0.98	1	8192	55%	2%	43%	
Double Perturbation	
CycA-to-CycE: 0 to -1 pRB-to-pRB: 1 to 0	0.85	1	8192	61%	0%	39%	
PPs-to-Skp2: 0 to -1 Skp2-to-p21-p27: 0 to -1	0.9	1	8192	58%	3%	39%	
Cdc25-to-CycE: 0 to 1 Cdh1-to-Cdc25: 0 to-1	0.94	1	8192	57%	7%	36%	
pRB-to-CycA: -1 to 0 PPs-to-Wee1: 0 to 1	0.96	1	8192	56%	2%	42%	

Discussion

Understanding the complex dynamics of the cell cycle is of both theoretical and practical interest. Considerable effort has been put into developing ODE-based and Boolean dynamical models of cell cycle, that are consistent with experimental data. The main challenge in developing such dynamical models is the lack of a systematic approach in putting together the model, and later incorporating new genes and proteins in the model as new information becomes available. The lack of such a structured approach is apparent among existing cell cycle models, which exhibit differences in their sets of variables and the interaction networks connecting them.

In this manuscript, we developed a semi-automated method to construct and improve Boolean models of cell cycle. The main steps in our approach are as follows: (a) manually construct an initial Boolean model using existing Boolean or ODE models of the cell cycle (b) define a network score to quantify the consistency of network dynamics with that of the cell cycle, (c) using single-edge and double-edge perturbations of the network, find interactions that result in a lower network score, and (d) compare the proposed interactions in (c) with protein-protein interaction databases to find those that are consistent with experimental data. Edge perturbations have been previously used in several studies, including the study different classes of mutations in human inherited disorders [30], genotype-phenotype associations using alleles that lack single interactions in C. elegans [31], removal of unwanted attractors such as the cancerous attractor in T-LGL leukemia [32], and the stability of the largest attractor in budding yeast cell cycle [11]. Our goal, however, was to develop a perturbation technique that allows us to refine cell cycle models, ensuring that the refined models exhibit correct cell cycle dynamics. Importantly, our approach should also facilitate automation of the model refinement process.

We first applied our perturbation method to the well-known model of cell cycle in budding yeast. Through double perturbation analysis, we found interactions that significantly improved the basin size of the largest attractor and the frequency of transitions through the correct cell cycle trajectory. Interestingly, these interactions (Cln1,2-to-MBF: 0 to 1, Clb5,6-to-Mcm1: 1 to 0) are not present in any database, but since they emerged naturally from our analysis, searching for them in the literature became much easier. We then applied the same method to two different models of cell cycle in mammalian cells: the Tyson-Novak and Goldbeter model. In both cases, we found perturbations that improved the network score. Thus, our approach resulted in the refinement of all three models that we studied.

Currently, the main limitation of our approach is that after selecting the interactions that are consistent with the data available in the SIGNOR database, we still need to manually accept or reject them. This is because the SIGNOR database is not complete, and sometimes the format in which the information is stored does not allow us to uniquely identify the interaction nodes. For example, CDK2 in the SIGNOR database corresponds to both Cyclin E and Cyclin A. Nevertheless, as the quality of databases improves, the need for manual intervention will also go away. The best-case scenario will be that all interactions with a lower network score and consistent with the database are incorporated to create a new refined model.

Another aspect of developing data-centric models needing attention is an automated method for adding more cell cycle genes to the existing network. This would result in more detailed models that can be used to study gene knockout experiments. As a preliminary step, we explored adding one new node at a time, selecting the gene with the highest connectivity (number of interactions) to the existing nodes in the network. We searched through the SIGNOR database and identified MYC, CHECK1, and TP53 as the new genes that should be added to the network, one gene at a time. These genes play important roles in the cell cycle and would be reasonable additions. Thus, this simple approach provides an automated method of adding new genes to the existing network. We envision that the iterative process of network growth and refinement will provide the means for developing data-centric Boolean models of cell cycle.

The number of double-edge perturbations increases as N4 with the number of nodes in the network. Combining this with the fact that for each network perturbation multiple simulation runs are needed to compute the statistical properties of the dynamics, the computation time increases significantly with the number of nodes. However, we found that retaining only the feasible single perturbations for the double perturbation analysis, significantly reduced the computational burden. For example, for Model C, about 80% of all single perturbations were classified as ‘false’ after database search, and retaining only the ‘true’ single perturbations for the double perturbation analysis reduced the search to only about 4% of all possible double perturbations.

Methods

Cell-cycle models and update scheme

The Boolean models of cell cycle contain nodes and directed edges. The nodes represent proteins and the directed edges correspond to the interactions between them. The edges are of two different types, inhibitors (-1 weight) and activators (+1 weight). The edges can also be self-loops, i.e. directed edges starting and ending at the same node. The state of the system is described by an N-component vector (X1, X2,…,XN), where N is the number of nodes in the network and Xk describes the activity of the protein corresponding to k-th node in the network; Xk = 1 if the k-th node is active and Xk = 0 if it is inactive.

We use the asynchronous update scheme, in which each node is equally likely to be updated at any time step, to calculate the network dynamics. At each time step, a node is selected randomly and its value is updated using the following rule Xi(t+1)={1if∑jaijXi(t)>00if∑jaijXi(t)<0Xi(t)if∑jaijXi(t)=0 (1)

Here aij are the weights of the edges joining node i to node j. We chose aij = 1 for activation and aij = −1 for inhibition. Since an asynchronous update scheme results in stochastic trajectories, we simulated multiple trajectories for each initial condition to obtain the statistical details of the model dynamics. We follow the same update scheme throughout the publication, except where explicitly mentioned.

Classification of trajectories

For each model, we fix a protein whose activation is considered as the start of the cell cycle. We call the activation of this protein as the point-of-no-return. If in a simulation the protein is not activated and the trajectory reaches the steady state corresponding to G0, we classify the trajectory as ‘did-not-start’. If the dynamics passes the point-of-no-return, we check for a predefined order of protein activation to classify the dynamics as correct or incorrect. If the dynamics progresses via the predefined sequence of states, we classify the trajectory as ‘correct’, and otherwise ‘incorrect’. Examples of each type of trajectory is given in the SI, and the point of no return, G1 states, and the correct sequence for different models are given below.

Model A (Budding Yeast Model)

In the network shown in Fig 1A, the cell size acts as the start signal. In the absence of this signal the cell remains in G0 phase (a steady state of the model), with the inhibitors of the cell cycle Cdh1 and Sic1 being active. In the presence of the signal the cell enters the G1 phase where Cln3 is activated. At this point the cell cycle can move forward to the S phase by activating Clb5,6 or go back to the G0 phase (we refer to this as the did-not-start case). Upon Clb5,6 activation the cell is committed to go through the cell cycle. The cycle is characterized by the sequential activation of Cln3, Cln1,2, Clb5,6, and Clb1,2.

Point of no return: Clb5,6 activation

G1 states: [Cln3, Cdh1, Sic1] = 1 and [Cdc20,14, Clb5,6, Cln1,2, Clb1,2, Mcm1, SFF] = 0.

Correct sequence: [Clb5,6 = 1, Clb1,2 = 0, Cdc20,14 = 0] → [Clb1,2 = 1, Cdc20,14 = 0] → [Cdc20,14 = 1]

Model B (Modified Tyson-Novak Mammal Model)

Point of no return: CycE activation

G1 states: [CycD, RB, P27, Cdh1] = 1 and [CycE, CycA, CycB, Cdc20] = 0.

Correct sequence: [E2F = 1, CycE = 0, CycA = 0, CycB = 0, Cdc20 = 0] → [CycE = 1, CycA = 0, CycB = 0, Cdc20 = 0] → [CycA = 1, CycB = 0, Cdc20 = 0] → [CycB = 1, Cdc20 = 0] → [Cdc20 = 1]

Model C (Modified Goldbeter Mammal Model)

Point of no return: CycE activation

G1 states: [CycD, RB, Wee1] = 1 and [CycE, CycA, CycB, E2F, Cdc20] = 0.

Correct sequence: [E2F = 1, CycE = 0, CycA = 0, CycB = 0, Cdc20 = 0] → [CycE = 1, CycA = 0, CycB = 0, Cdc20 = 0] → [CycA = 1, CycB = 0, Cdc20 = 0] → [CycB = 1, Cdc20 = 0] → [Cdc20 = 1]

Perturbations and scoring

We introduced single-edge perturbations in the network by altering an edge value according to the options outlined in Table 6.

10.1371/journal.pone.0306523.t006 Table 6 Description of single-edge perturbations.

Perturbation	Description	
0 → -1	Addition of an inhibiting edge	
0 → 1	Addition of an activating edge	
1 → 0	Activating edge replaced with no edge	
1 → -1	Activating edge replaced with inhibiting edge	
-1 → 0	Inhibiting edge replaced with no edge	
-1 → 1	Inhibiting edge replaced with activating edge	

We analyzed all possible perturbation to all edges (including self-loops), resulting in a total of 2N2 single-edge perturbations. Double-edge perturbations were introduced by applying single-edge perturbation to two different edges at a time.

The perturbation analysis was designed to search for networks in which the model dynamics follows the correct cell cycle trajectory with high frequency. Since the trajectories follow stochastic dynamics, for each network we observed trajectories that fall into the groups ‘correct’, ‘incorrect’, and ‘did-not-start’. Using the frequency with which each type of trajectory was observed, we defined a network score to quantify the performance of perturbed networks. The network score is defined as networkscore=∑n=12NS(icn)+2NNG1∑n=1NG1Y(icn)+2NNG1∑n=1NG1Z(icn), (2)

where S(icn)=∑k=1N|Xkss(icn)−G0k|, (3)

The first term in the network score is the penalty associated with initial condition not reaching the G0 state. It is determined by summing S(icn): the element-wise difference between the G0 state and the steady state corresponding initial conditions over all 2N initial conditions. In Eq 3, Xkss(icn) is the kth component of the steady state corresponding to the initial condition icn, and G0k is the kth component of the G0 state. In the second term in the network score, Y, is for the sequence-related penalty; it takes the value 1 if the stochastic trajectory is classified as ‘incorrect’ and 0 otherwise. The quantity Z is the penalty associated with the ‘did-not-start’ case. It takes the value 1 if the trajectory did-not-start and 0 otherwise. The ‘incorrect’ and did-not-start related penalties were introduced only for the G1 states because we were interested in biologically meaningful trajectories that start in the G1 phase and end in the G0 phase. Also, the weights associated with ‘incorrect’ and ‘did-not-start’ penalties were chosen to be 2N/NG1 so that these penalties scales with the number of nodes in the network. Note, based on the definition of the network score, in networks with relatively smaller score, the model dynamics follows the correct cell cycle trajectory with high frequency.

Supporting information

S1 Table Statistical data related to perturbation analysis of cell cycle model in budding yeast cells.

This Excel file contains statistical data related to the single-edge (sheet 1) and double-edge (sheet 2) perturbation analysis of Model A (cell cycle in budding yeast based on Ref. [11]). The data was calculated after performing 500 iterations per single-edge perturbations and 20 iterations per double perturbation. Sheet 3 shows the final states (attractors) of the model after and their corresponding sizes.

(XLSX)

S2 Table Statistical data related to perturbation analysis of cell cycle model in mammalian cells.

This Excel file contains statistical data related to the single-edge (sheet 1) and double-edge (sheet 2) perturbation analysis of Model B (cell cycle in mammalian cell based on the Tyson-Novak model [1]). The data was calculated after performing 500 iterations per single-edge perturbations and 20 iterations per double perturbation. Sheet 3 shows the final states (attractors) of the model after and their corresponding sizes.

(XLSX)

S3 Table Statistical data related to perturbation analysis of cell cycle model in mammalian cells.

This Excel file contains statistical data related to the single-edge (sheet 1) and double-edge (sheet 2) perturbation analysis of Model C (cell cycle in mammalian cell based on the Goldbeter model [2]). The data was calculated after performing 300 iterations per single-edge perturbations and 8 iterations per double perturbation. Sheet 3 shows the final states (attractors) of the model after and their corresponding sizes.

(XLSX)

S1 Text Description of details of simulation methods and statistical analysis.

(DOCX)

10.1371/journal.pone.0306523.r001
Decision Letter 0
Ruz Gonzalo A. Academic Editor
© 2024 Gonzalo A. Ruz
2024
Gonzalo A. Ruz
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version0
25 Jan 2024

PONE-D-23-41629A Perturbation Approach for Refining Boolean Models of Cell Cycle RegulationPLOS ONE

Dear Dr. Banerjee,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by Mar 10 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Gonzalo A. Ruz, Ph.D.

Academic Editor

PLOS ONE

Journal requirements:

When submitting your revision, we need you to address these additional requirements.

1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at

https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and

https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2. Please note that PLOS ONE has specific guidelines on code sharing for submissions in which author-generated code underpins the findings in the manuscript. In these cases, all author-generated code must be made available without restrictions upon publication of the work. Please review our guidelines at https://journals.plos.org/plosone/s/materials-and-software-sharing#loc-sharing-code and ensure that your code is shared in a way that follows best practice and facilitates reproducibility and reuse.

3. Thank you for stating the following in the Acknowledgments Section of your manuscript:

“The work was supported by the Academy of Data Science Discovery Fund awarded to AB and PK.”

We note that you have provided funding information that is not currently declared in your Funding Statement. However, funding information should not appear in the Acknowledgments section or other areas of your manuscript. We will only publish funding information present in the Funding Statement section of the online submission form.

Please remove any funding-related text from the manuscript and let us know how you would like to update your Funding Statement. Currently, your Funding Statement reads as follows:

“The author(s) received no specific funding for this work.”

Please include your amended statements within your cover letter; we will change the online submission form on your behalf.

4. Please include captions for your Supporting Information files at the end of your manuscript, and update any in-text citations to match accordingly. Please see our Supporting Information guidelines for more information: http://journals.plos.org/plosone/s/supporting-information.

Additional Editor Comments:

Both reviewers have raised important issues, which the authors must address.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: No

Reviewer #2: Yes

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: N/A

Reviewer #2: Yes

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: TITLE: A Perturbation Approach for Refining Boolean Models of Cell Cycle Regulation

AUTHORS: Anand Banerjee, Asif Iqbal Rahaman, Alok Mehandale, Pavel Kraikivski

----------- Recommendation -----------

Major revision

----------- Overall Evaluation -------------------------

Summary: The paper focuses on the challenge of constructing models for large protein regulatory networks, with a particular emphasis on boolean network models. This model is composed by a directed signed graph, where each protein is represented by a node. Nodes can be active (state 1) or inactive (state 0), with regulatory dependencies depicted by directed edges. A $-1$ label on an edge indicates an inhibition link, while +1 denotes an activation link. Local functions in the paper are threshold functions i.e. the future state of a node depends on the sum of the states of its in-neighbors multiplied by the sign of its corresponding edges. Each node (representing a protein) needs this sum to surpass certain threshold in order to activate. In the paper, activation occurs when the sum of the states of the in-neighbors of a node is positive; non-activation results from a negative sum. If the sum is 0, the state of the node remains unchanged. Dynamics are defined asynchronously by randomly selecting a node for update in each time step. The paper proposes a "semi-automatic approach" for constructing reliable models, using single-edge and double-edge perturbations, involving changes in edge signs. These perturbation protocols are simply defined by a change on the sign of an edge (or two edges for the double edge perturbation). More precisely, the single edge perturbation protocol consists on one of the following operations: i) to change the sign of an edge from -1 to 1 (or symmetrically from 1 to -1) or ii) to remove an edge or adding an edge (wether a positive or a negative edge). Double-edge perturbations are simply single perturbations done on two edges at the same time.

The proposed method can be summarized in four main steps:

(a) Manually construct an initial Boolean model using existing Boolean or ODE models of the cell cycle.

(b) Define a network score to quantify the consistency of network dynamics with the cell cycle.

(c) Use single-edge and double-edge perturbations to find interactions resulting in a lower score.

(d) Compare proposed interactions with databases to identify those consistent with experimental data.

While the main idea is interesting, the paper's presentation is inadequate. Figures representing the original networks are unreadable, and several elements lack clear definitions and explanations. For instance, the mention of "very complicated" trajectories for the asynchronous update schemes (lines 53-56) lacks elaboration (it is not clear what does it mean to be “very complicated”), and the score function's definition is unclear due to the ambiguous concept of the "k-th component" for a steady state. In the context of graph theory, a connected component of an undirected graph (or a strongly connected component of a directed graph) does not correspond to a specific numerical value but to a maximal connected subgraph (maximal strongly connected subgraph in the case of a strongly connected component). I assume that in the context of the paper it may refer to the convergence time starting from a particular initial condition to a specific fixed point but, it is not clear from the text. Additionally, the idea behind the definition of the score as a measure of model reliability is not well-explained, beyond the concept of penalizing "incorrect" and "did-not-start" trajectories.

The automatic aspect of the method requires more emphasis and comparison with other approaches. For example, it could be interesting to explore a potential connection with the concept of the neutral space used in [1].

Considering these weaknesses, the paper is not suitable for publication in its current state. However, with improvements in presentation and addressing the latter issues, it could become a valuable contribution.

Score:

Positive Aspects:

• The main idea of the paper is original and interesting.

• The paper's organization is good.

Negative Aspects:

• Presentation is subpar; figures are poorly formatted and unreadable.

• Critical concepts like "k-th component" and "more complicated" trajectories lack clear definitions.

• The automatic aspect of the approach needs greater emphasis and comparison with other methods.

Typos: 333 - it says $2N^2$ and I think it should be $2^{N^{2}}.$

References

[1] Travisany, D., Goles, E., Latorre, M., Cortés, M. P., & Maass, A. (2020). Generation and robustness of Boolean networks to model Clostridium difficile infection. Natural Computing, 19, 111-134.

Reviewer #2: The manuscript presents an interesting study that applies a perturbation approach to refine Boolean models for cell cycle regulation. The approach is well-defined, and the methods used for building the models and analyzing the data seem robust and appropriate. The conclusions drawn from the study are supported by the data presented.

However, there are several areas where further clarification and additional information would strengthen the manuscript:

- Further information on how the proposed method compares to existing methods of model refinement would be valuable. A discussion on the potential advantages and limitations of the perturbation approach would provide greater context for the significance of the findings.

- The authors should consider providing a clearer explanation of how the proposed interactions are validated against experimental data. The use of the SIGNOR 3.0 database is mentioned, but a more detailed description of this validation process may be beneficial for the readers (I acknowledge that my understanding on the database may not be comprehensive).

- Details of how the method could be fully automated in the future would be a valuable addition. This would offer insight into the potential scalability and applicability of the approach for other networks and systems.

- It would be beneficial to enhance the visual clarity and resolution of the graphs for better comprehension.

Finally, the manuscript would be strengthened by a critical discussion of the limitations and potential future directions of the research.

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: Yes: Luis Gómez Guzmán

**********

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

10.1371/journal.pone.0306523.r002
Author response to Decision Letter 0
Submission Version1
29 Apr 2024

We have submitted a file labeled 'Response to Reviewers' in which we have addressed all the suggestions and questions raised by the reviewers.

Attachment Submitted filename: Response to Reviewers.docx

10.1371/journal.pone.0306523.r003
Decision Letter 1
Ruz Gonzalo A. Academic Editor
© 2024 Gonzalo A. Ruz
2024
Gonzalo A. Ruz
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version1
27 May 2024

PONE-D-23-41629R1A Perturbation Approach for Refining Boolean Models of Cell Cycle RegulationPLOS ONE

Dear Dr. Banerjee,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by Jul 11 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Gonzalo A. Ruz, Ph.D.

Academic Editor

PLOS ONE

Journal Requirements:

Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

Additional Editor Comments:

Reviewer 1 points out that some minor aspects still need to be addressed. The authors should work on these points for the paper to be accepted.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Partly

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: While the paper has significantly improved the aspects highlighted in the previous review, there are still some points to improve in order to consider it for publication:

1) Although the resolution of the images has improved considerably, Figure 1 is still difficult to understand. I think the authors should rework this figure to make it clearer.

2) Although it was added in the introduction that: "A drawback of the asynchronous scheme is that the run time for simulations increases rapidly with the number of nodes in the network," no references are provided to support it. It would be interesting to add references from other papers that theoretically or experimentally justify the statement. In addition, regarding the following statement: “Asynchronous update preserved the G0 steady state, but the cyclic attractor became very complex with many intertwined cycles.” The authors should provide more details in what does it mean “cyclic attractor” in the context of stochastic dynamics.

3) I still think that the paper could be greatly improved if it is better situated in the context of the state of the art. How does it compare with other methods?

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

**********

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

10.1371/journal.pone.0306523.r004
Author response to Decision Letter 1
Submission Version2
17 Jun 2024

Please see the file 'Response to Reviewers' in our submission

Attachment Submitted filename: Response to Reviewers.docx

10.1371/journal.pone.0306523.r005
Decision Letter 2
Ruz Gonzalo A. Academic Editor
© 2024 Gonzalo A. Ruz
2024
Gonzalo A. Ruz
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version2
20 Jun 2024

A Perturbation Approach for Refining Boolean Models of Cell Cycle Regulation

PONE-D-23-41629R2

Dear Dr. Banerjee,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. If you have any questions relating to publication charges, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Gonzalo A. Ruz, Ph.D.

Academic Editor

PLOS ONE

10.1371/journal.pone.0306523.r006
Acceptance letter
Ruz Gonzalo A. Academic Editor
© 2024 Gonzalo A. Ruz
2024
Gonzalo A. Ruz
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
7 Jul 2024

PONE-D-23-41629R2

PLOS ONE

Dear Dr. Banerjee,

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now being handed over to our production team.

At this stage, our production department will prepare your paper for publication. This includes ensuring the following:

* All references, tables, and figures are properly cited

* All relevant supporting information is included in the manuscript submission,

* There are no issues that prevent the paper from being properly typeset

If revisions are needed, the production department will contact you directly to resolve them. If no revisions are needed, you will receive an email when the publication date has been set. At this time, we do not offer pre-publication proofs to authors during production of the accepted work. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few weeks to review your paper and let you know the next and final steps.

Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

If we can help with anything else, please email us at customercare@plos.org.

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Prof. Gonzalo A. Ruz

Academic Editor

PLOS ONE
==== Refs
References

1 Novak B , Tyson JJ . A model for restriction point control of the mammalian cell cycle. J Theor Biol. 2004;230 (4 ):563–79. Epub 2004/09/15. doi: 10.1016/j.jtbi.2004.04.039 .15363676
2 Gerard C , Goldbeter A . Temporal self-organization of the cyclin/Cdk network driving the mammalian cell cycle. Proc Natl Acad Sci U S A. 2009;106 (51 ):21643–8. Epub 2009/12/17. doi: 10.1073/pnas.0903827106 ; PubMed Central PMCID: PMC2799800.20007375
3 Chen KC , Calzone L , Csikasz-Nagy A , Cross FR , Novak B , Tyson JJ . Integrative analysis of cell cycle control in budding yeast. Mol Biol Cell. 2004;15 (8 ):3841–62. Epub 2004/06/01. doi: 10.1091/mbc.e03-11-0794 ; PubMed Central PMCID: PMC491841.15169868
4 Kraikivski P , Chen KC , Laomettachit T , Murali TM , Tyson JJ . From START to FINISH: computational analysis of cell cycle control in budding yeast. NPJ Syst Biol Appl. 2015;1 :15016. Epub 2015/12/10. doi: 10.1038/npjsba.2015.16 ; PubMed Central PMCID: PMC5516803.28725464
5 Jung Y , Kraikivski P , Shafiekhani S , Terhune SS , Dash RK . Crosstalk between Plk1, p53, cell cycle, and G2/M DNA damage checkpoint regulation in cancer: computational modeling and analysis. NPJ Syst Biol Appl. 2021;7 (1 ):46. Epub 2021/12/11. doi: 10.1038/s41540-021-00203-8 ; PubMed Central PMCID: PMC8660825.34887439
6 Csikasz-Nagy A. Computational systems biology of the cell cycle. Brief Bioinform. 2009;10 (4 ):424–34. Epub 2009/03/10. doi: 10.1093/bib/bbp005 .19270018
7 Wang RS , Saadatpour A , Albert R . Boolean modeling in systems biology: an overview of methodology and applications. Phys Biol. 2012;9 (5 ):055001. Epub 2012/09/27. doi: 10.1088/1478-3975/9/5/055001 .23011283
8 Chaves M , Albert R , Sontag ED . Robustness and fragility of Boolean models for genetic regulatory networks. J Theor Biol. 2005;235 (3 ):431–49. Epub 2005/05/11. doi: 10.1016/j.jtbi.2005.01.023 .15882705
9 Fumia HF , Martins ML . Boolean network model for cancer pathways: predicting carcinogenesis and targeted therapy outcomes. PLoS One. 2013;8 (7 ):e69008. Epub 2013/08/08. doi: 10.1371/journal.pone.0069008 ; PubMed Central PMCID: PMC3724878.23922675
10 Zanudo JG , Albert R . Cell fate reprogramming by control of intracellular network dynamics. PLoS Comput Biol. 2015;11 (4 ):e1004193. Epub 2015/04/08. doi: 10.1371/journal.pcbi.1004193 ; PubMed Central PMCID: PMC4388852.25849586
11 Li F , Long T , Lu Y , Ouyang Q , Tang C . The yeast cell-cycle network is robustly designed. Proc Natl Acad Sci U S A. 2004;101 (14 ):4781–6. Epub 2004/03/24. doi: 10.1073/pnas.0305937101 ; PubMed Central PMCID: PMC387325.15037758
12 Davidich MI , Bornholdt S . Boolean network model predicts cell cycle sequence of fission yeast. PLoS One. 2008;3 (2 ):e1672. Epub 2008/02/28. doi: 10.1371/journal.pone.0001672 ; PubMed Central PMCID: PMC2243020.18301750
13 Faure A , Naldi A , Chaouiya C , Thieffry D . Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics. 2006;22 (14 ):e124–31. Epub 2006/07/29. doi: 10.1093/bioinformatics/btl210 .16873462
14 Laomettachit T , Kraikivski P , Tyson JJ . A continuous-time stochastic Boolean model provides a quantitative description of the budding yeast cell cycle. Sci Rep. 2022;12 (1 ):20302. Epub 2022/11/27. doi: 10.1038/s41598-022-24302-6 ; PubMed Central PMCID: PMC9700812.36434030
15 Skotheim JM , Di Talia S , Siggia ED , Cross FR . Positive feedback of G1 cyclins ensures coherent cell cycle entry. Nature. 2008;454 (7202 ):291–6. Epub 2008/07/18. doi: 10.1038/nature07118 ; PubMed Central PMCID: PMC2606905.18633409
16 Garg A , Di Cara A , Xenarios I , Mendoza L , De Micheli G . Synchronous versus asynchronous modeling of gene regulatory networks. Bioinformatics. 2008;24 (17 ):1917–25. Epub 2008/07/11. doi: 10.1093/bioinformatics/btn336 ; PubMed Central PMCID: PMC2519162.18614585
17 Licata L , Lo Surdo P , Iannuccelli M , Palma A , Micarelli E , Perfetto L , et al . SIGNOR 2.0, the SIGnaling Network Open Resource 2.0: 2019 update. Nucleic Acids Res. 2020;48 (D1 ):D504–D10. Epub 2019/10/31. doi: 10.1093/nar/gkz949 ; PubMed Central PMCID: PMC7145695.31665520
18 Nasmyth K , Dirick L . The role of SWI4 and SWI6 in the activity of G1 cyclins in yeast. Cell. 1991;66 (5 ):995–1013. Epub 1991/09/06. doi: 10.1016/0092-8674(91)90444-4 .1832338
19 Schwob E , Nasmyth K . CLB5 and CLB6, a new pair of B cyclins involved in DNA replication in Saccharomyces cerevisiae. Genes Dev. 1993;7 (7A ):1160–75. Epub 1993/07/01. doi: 10.1101/gad.7.7a.1160 .8319908
20 Bean JM , Siggia ED , Cross FR . High functional overlap between MluI cell-cycle box binding factor and Swi4/6 cell-cycle box binding factor in the G1/S transcriptional program in Saccharomyces cerevisiae. Genetics. 2005;171 (1 ):49–61. Epub 2005/06/21. doi: 10.1534/genetics.105.044560 ; PubMed Central PMCID: PMC1456534.15965243
21 Stuart D , Wittenberg C . Cell cycle-dependent transcription of CLN2 is conferred by multiple distinct cis-acting regulatory elements. Mol Cell Biol. 1994;14 (7 ):4788–801. Epub 1994/07/01. doi: 10.1128/mcb.14.7.4788-4801.1994 ; PubMed Central PMCID: PMC358852.8007978
22 Adames NR , Schuck PL , Chen KC , Murali TM , Tyson JJ , Peccoud J . Experimental testing of a new integrated model of the budding yeast Start transition. Mol Biol Cell. 2015;26 (22 ):3966–84. Epub 2015/08/28. doi: 10.1091/mbc.E15-06-0358 ; PubMed Central PMCID: PMC4710230.26310445
23 Clurman BE , Sheaff RJ , Thress K , Groudine M , Roberts JM . Turnover of cyclin E by the ubiquitin-proteasome pathway is regulated by cdk2 binding and cyclin phosphorylation. Genes Dev. 1996;10 (16 ):1979–90. Epub 1996/08/15. doi: 10.1101/gad.10.16.1979 .8769642
24 Rodier G , Coulombe P , Tanguay PL , Boutonnet C , Meloche S . Phosphorylation of Skp2 regulated by CDK2 and Cdc14B protects it from degradation by APC(Cdh1) in G1 phase. EMBO J. 2008;27 (4 ):679–91. Epub 2008/02/02. doi: 10.1038/emboj.2008.6 ; PubMed Central PMCID: PMC2262036.18239684
25 Carrano AC , Eytan E , Hershko A , Pagano M . SKP2 is required for ubiquitin-mediated degradation of the CDK inhibitor p27. Nat Cell Biol. 1999;1 (4 ):193–9. Epub 1999/11/13. doi: 10.1038/12013 .10559916
26 Bornstein G , Bloom J , Sitry-Shevah D , Nakayama K , Pagano M , Hershko A . Role of the SCFSkp2 ubiquitin ligase in the degradation of p21Cip1 in S phase. J Biol Chem. 2003;278 (28 ):25752–7. Epub 2003/05/06. doi: 10.1074/jbc.M301774200 .12730199
27 Foster JS , Henley DC , Bukovsky A , Seth P , Wimalasena J . Multifaceted regulation of cell cycle progression by estrogen: regulation of Cdk inhibitors and Cdc25A independent of cyclin D1-Cdk4 function. Mol Cell Biol. 2001;21 (3 ):794–810. Epub 2001/01/12. doi: 10.1128/MCB.21.3.794-810.2001 ; PubMed Central PMCID: PMC86671.11154267
28 Donzelli M , Squatrito M , Ganoth D , Hershko A , Pagano M , Draetta GF . Dual mode of degradation of Cdc25 A phosphatase. EMBO J. 2002;21 (18 ):4875–84. Epub 2002/09/18. doi: 10.1093/emboj/cdf491 ; PubMed Central PMCID: PMC126287.12234927
29 Ovejero S , Ayala P , Bueno A , Sacristan MP . Human Cdc14A regulates Wee1 stability by counteracting CDK-mediated phosphorylation. Mol Biol Cell. 2012;23 (23 ):4515–25. Epub 2012/10/12. doi: 10.1091/mbc.E12-04-0260 ; PubMed Central PMCID: PMC3510014.23051732
30 Zhong Q , Simonis N , Li QR , Charloteaux B , Heuze F , Klitgord N , et al . Edgetic perturbation models of human inherited disorders. Mol Syst Biol. 2009;5 :321. Epub 2009/11/06. doi: 10.1038/msb.2009.80 ; PubMed Central PMCID: PMC2795474.19888216
31 Dreze M , Charloteaux B , Milstein S , Vidalain PO , Yildirim MA , Zhong Q , et al . ’Edgetic’ perturbation of a C. elegans BCL2 ortholog. Nat Methods. 2009;6 (11 ):843–9. Epub 2009/10/27. doi: 10.1038/nmeth.1394 ; PubMed Central PMCID: PMC2865203.19855391
32 Campbell C , Albert R . Edgetic perturbations to eliminate fixed-point attractors in Boolean regulatory networks. Chaos. 2019;29 (2 ):023130. Epub 2019/03/03. doi: 10.1063/1.5083060 .30823730
