PCSY-D-25-00101

Fixed points and multistability in monotone Boolean network models

PLOS Complex Systems

Dear Dr. Gedeon,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems'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 22 2026 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 complexsystems@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pcsy/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

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We look forward to receiving your revised manuscript.

Kind regards,

Vera Pancaldi

Section Editor

PLOS Complex Systems

Vera Pancaldi

Section Editor

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

Journal Requirements:

If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise.

Additional Editor Comments (if provided):

Dear authors,

thank you for submitting your article to our journal. As you will see, 3 referees recognize the potential value of your paper, while highlighting some potential points for improvement.

We hope that you will be able to address these comments, especially regarding the comparison of your approach to the BoNesis framework, for example. I also agree that changing the title into something more explicit might help the community to identify the contribution you are making to the literature.

We hope the comment provided will help you improve your paper and we look forward to receiving your revised version.

Many thanks,

Best wishes,

Vera

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Reviewers' Comments:

Reviewer's Responses to Questions

Comments to the Author

1. Does this manuscript meet PLOS Complex Systems’s publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.-->?>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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

Reviewer #1: Yes

Reviewer #2: N/A

Reviewer #3: N/A

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3. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)??>

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. 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.requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. 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: No

Reviewer #3: Yes

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

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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Reviewer #1: In this paper, fixed points and multistability for monotone Boolean network models are analyzed based on lattice theory.

Major comments:

Although interesting results are obtained in this paper, the reviewer considers that the structure of this paper should be improved.

Please reconsider the contents of the ``Results'' and ``Methods'' sections.

In this paper, it may not be appropriate to write the proposed method as theorems/lemma and their proofs.

Minor comments:

1) Definition 2.6: In this definition, the notation is stated.

The notation should not be given as Definition.

2) Proof of Lemma 2.11: It is confusing that Theorem 3.11 appeared later is used for the proof of Lemma 2.11.

The structure of this paper should be improved.

Reviewer #2: The authors provide a method to enumerate all monotone Boolean models that are consistent with a given regulatory network and allow certain steady states. The authors show this method on a positive network, and claim that it can be extended to any regulatory network without negative loops. The enumeration is done by finding all the positive functions for each node that can stay consistent with the steady state, and finding the combinations of such functions. Models allowing multistability can be found by taking the intersections of sets of Boolean functions that allow those steady states, and then finding the combinations of them again. The authors apply this method to various regulatory networks and identify the most common steady states or bistability consistent with them.

The identification of possible dynamics that a network allows is an important problem in the study of complex systems. The overall approach is appealing and the manuscript is well organized. I believe that the paper could be strengthened further with some revisions.

Before outlining my comments, I want to mention that I have no strong preferences about how the authors ultimately choose to present the material. I am mainly requesting clarification on a few points that were unclear to me while reviewing the manuscript.

Major comments

1. It is unclear to me how a monotone Boolean model compatible with a negative-loop-free network can be converted into a positive Boolean model with a change of variables. For example, a network with incoherent feed-forward loops may not have any negative loops, but no change of variables would be able to resolve the incoherence. If the authors are focusing only on strongly connected networks, as in all the examples, it should be clarified.

If the authors are indeed only interested in strongly connected networks without negative loops, then the network will be sign-consistent. It may be beneficial to state that the manuscript considers sign-consistent networks instead, as converting them into networks with only positive edges is well-known.

2. The paper can be strengthened if the authors can describe the novelty of their method in comparison to the existing methods for enumerating Boolean models with certain steady states. For example, the paper below describes the enumeration of all Boolean models that are consistent with a set of given constraints, such as having certain steady states.

Stéphanie Chevalier, Christine Froidevaux, Loïc Paulevé, Andrei Zinovyev. Synthesis of Boolean Networks from Biological Dynamical Constraints using Answer-Set Programming. 31st International Conference on Tools with Artificial Intelligence, 2019, Portland, Oregon, United States. ffhal-02276921v1

3. I believe that removing the self-inhibition on SNAI1 for both EMT models requires some justification, especially as it leads to the key dynamics of the system that is biologically interpretable. It may be worth mentioning that both networks are constructed for the study of ODE systems, whereas the authors are working on Boolean models. One key aspect of Boolean modeling is that the self-inhibitions are innate; deactivation of positive inputs immediately leads to deactivation of the node.

4. The authors are allowing the inputs to be redundant (input does not affect the output) when considering all the Boolean functions consistent with the regulatory network. However, allowing redundancies means that the regulations are not actually present in the model. This may be worth some discussion, as this is closely related to the number of Boolean models allowing a certain steady state. For example, the steady state of all 0s is not allowed if and only if one of the functions of the positive Boolean model is a constant 1. The all 1s or the all 0s will always be allowed for models with no redundant inputs, and the bistability will always happen.

5. I am guessing that listing the top 12 most common steady states for Table 2 required calculation of the number of models for every steady state. If this was done using code, providing it could make the results be more reproducible, and help the readers apply the method to other regulatory networks.

Minor comments

1. Line 119: Please check the indices. It may have to be changed to: fj is increasing (decreasing) in xi, etc.

2. Line 120: It is unclear what “if xi=fi(x)” means.

3. Line 574: I wonder if the equation should be written a1(x1)=a1f1(x), or for more clarity a1(x1)=a1f1(a1(x1), x2, ..., xN)

4. Line 575: Please check the indices. It may have to be changed to: g1:=a1f1

5. Line 586: It is unclear how sources of g1 differ from sources of node x1

6. Line 590: It is not clear to me how each node can be considered only once, as the change of variables may turn originally positive edges to negative.

Reviewer #3: This manuscript studies the set of Boolean network models supporting a given steady state. The authors focus on the class of models made up of monotone Boolean functions and with no negative feedback loops. Specifically, they study the problem of which steady states are supported by most models. They provide theoretical results (e.g., formulas) for the number of models supporting two relevant steady states after a mapping into a positive network. They also provide examples of their results in small networks such as toggle switches, and they applied their methods to a model relevant to cancer where they discuss potential biological implications of their findings.

I found this manuscript mostly well written and their results will be useful and important contributions for the analysis of Boolean network models. However, I found few potential issues (detailed below) with the current version and suggest addressing these before recommending it for publication.

Major Revisions:

1. Define positive Boolean model before Theorem 2.8. The concept was loosely described in the introduction, but it was not formally defined before using it in the theorem.

2. The title of the manuscript does not seem to be very informative. It generically indicates that it has to do with fixed points and multistationarity, but it does not indicate that it is about the set of models that support a specific steady state and/or the percentage of models supporting certain fixed points. Besides, it sounds very general that it might be a bit misleading that results of the manuscript apply to all monotone Boolean network models. Since current results are only for NL-free networks, I would suggest adding “with no negative loops” to the end of the current title or modifying it in some other way.

Minor Revisions:

3. Proofread the author summary, especially the last part where it says, “We show that the most common steady states are those correspond to ”. Consider changing “correspond” to “corresponding”.

4. On line 117, replace the period after the function f with a comma.

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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Characterization of monotone Boolean modelssupporting fixed points and multistability in balancednetworks

PCSY-D-25-00101R1

Dear Dr. Gedeon,

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.

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Kind regards,

Vera Pancaldi

Section Editor

PLOS Complex Systems

Additional Editor Comments (optional):

Dear authors,

thanks for addressing the reviewers' comments. I'm glad to tell you that your paper can now be acepted for publication.

Many thanks for choosing our journal, I hope you found the reviewers' comments useful.

Best wishes,

Vera

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

Reviewer #3: All comments have been addressed

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publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.-->?>

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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

Reviewer #1: Yes

Reviewer #2: N/A

Reviewer #3: N/A

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4. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)??>

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. 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.requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. 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

Reviewer #3: Yes

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

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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Reviewer #1: According to my comments, the paper has been improved.

I have no further comments.

Reviewer #2: Line 141: i /∈ E_i may be a misprint.

Reviewer #3: The authors have addressed my comments from the previous round of reviews.

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what does this mean?). If published, this will include your full peer review and any attached files.). If published, this will include your full peer review and any attached files.). If published, this will include your full peer review and any attached files.). If published, this will include your full peer review and any attached files.

Do you want your identity to be public for this peer review? If you choose “no”, your identity will remain anonymous but your review may still be made public.If you choose “no”, your identity will remain anonymous but your review may still be made public.If you choose “no”, your identity will remain anonymous but your review may still be made public.If you choose “no”, your identity will remain anonymous but your review may still be made public.

For information about this choice, including consent withdrawal, please see our Privacy Policy..-->

Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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