PCSY-D-24-00042

Difficult control is related to instability in biologically inspired Boolean networks

PLOS Complex Systems

Dear Dr. Borriello,

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Luis M. Rocha, Ph.D.

Section Editor

PLOS Complex Systems

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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.

Reviewer #1: Partly

Reviewer #2: Partly

Reviewer #3: Partly

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

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: No

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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.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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

PLOS Complex Systems 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

Reviewer #3: Yes

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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: Comments on this paper:

1.In this paper, Boolean networks are proposed, and several results are obtained, could the obtained results be extended to stochastic Boolean networks, or Boolean networks under different update rules.

2.Are there any other control methods for Boolean networks except by fixing the values of certain nodes, like dynamical control methods?

3.The reviewer is wondering whether the method of control kernel can drive a Boolean networks globally into a desired attractor from any initial conditions?

4.The English of the paper should be polished, and the presentation and organization of this paper should be also carefully improved, in addition, the format of the reference parts should be checked and revised.

Reviewer #2: The authors extend a previous study about the relationship between the minimal number of nodes used in pinning control and the number of attractors in Boolean networks. They have incorporated a large number of new models from Kadelka et al., and they have also conducted a deeper analysis of networks that do not follow the trend identified in previous work. This leads to a new correction to the relationship between control kernel size and attractor count that depends on the number of isolated fixed points.

It is not particularly surprising that isolated fixed points lead to difficult pinning control, though it is good that other (independent) factors are effectively ruled out. The results presented here are interesting, though I find myself wishing that the investigation of exceptional networks had been included in Ref 4 instead; indeed much of the manuscript is dedicated to recalling the contents of that earlier work. My biggest criticism, discussed in detail below, relates to how input nodes are handled.

Overall, the manuscript is very well written, though some key details appear to be missing or buried in external references.

Major Issues:

I have a few questions and concerns about how input nodes are handled (Park et al. [PRX Life 1, 023009] recently showed that stability measures in BNs are very sensitive to this and that the Cell Collective is not consistent in how inputs are represented):

- It is not clear from the text whether the inputs are included in the count for a control kernel.

- It is not clear whether inputs are regarded as having self-loops or in-degree zero for the purposes of topological analyses (there is no universal convention, so hopefully the authors took special care to identify inputs in both conventions).

- The authors take special care to handle acyclic networks because the attractors are fully determined by the inputs, but don't appear to address the cases when inputs fully determine the attractor, but the graph is not acyclic.

- Ideally, inputs would be fixed and percolated prior to analysis. Perhaps all or most of the "difficult" networks are actually just networks with trivial dynamics governed entirely or almost entirely by their inputs.

The authors claim that difficulty of control depends on number of attractors, but is independent of network size. However, number of attractors generally scales with network size (on average, at least). It's not clear to me how this interdependence is handled.

It appears that the Kauffman networks and the ER networks with threshold dynamics are grouped together as "Random networks", but presumably these could have a quite different character and should probably be plotted separately. Related to this, the authors say there are 3 ensembles but only seem to list 2. Finally, please include the parameters used to generate the random networks (rather than just referring to Ref 4).

Regarding the Kauffman networks, it is possible to generate nodes with fully redundant regulators. Is network topology reevaluated after removing these regulators? This would have an effect on n_c. In fact, Gates et al. [PNAS 118 (12) e2022598118] showed that there are also fully redundant regulators in the Cell Collective (I don't recall any instances where this affected the SCC sizes, but I'm not certain that this doesn't occur).

Minor Issues:

I am glad to see the adoption of the network database in Ref 14. The authors may be interested in an even larger database (Pastva et al 2024: https://doi.org/10.1101/2023.06.12.544361) for future work.

What features of a Boolean network tend to give rise to isolated fixed points? Have the authors observed any patterns in the network topology or regulatory functions associated with these attractors, for example?

What do the symbol shapes indicate in Figure 3?

It would be good if the authors could indicate a goodness of fit parameter in Figure 3.

Perhaps the authors could briefly comment on how (or whether) the chosen update scheme might affect their results or conclusions.

The authors should add a data availability statement indicating where the code and network files can be found. In the cover-page, it is indicated that this is all available on GitHub, but the actual repository location is not given anywhere.

In the discussion (and also in the results), the authors state that identifying the globally unstable fixed points avoids the NP-hard problem of evaluating control kernels. But they neglect to point out that determining whether the network has fixed points is itself NP-hard (and that finding them all is #P-hard). This is not to undermine the computational savings of their approach--clearly it is much easier to identify isolated fixed points than it is to find control kernels for all attractors. Still, the current presentation gives an incorrect impression about the computational complexity of the approach.

In the intro, regarding the choice of Refs 5-10:

- Refs 6-8 refer to the same method (FVS control); only ref 6 considers application to BNs; 7 & 8 explicitly do *not* apply to discontinuous dynamics, and ref 10 is also about ODE control.

- All Boolean control methods listed are designed to control toward the attractors of the unconstrained dynamics; the only reference to arbitrary state control is a review paper (ref 10) that focuses on ODEs, not BNs.

- Of the 6 reference cited, only two are bona fide BN control methods. Perhaps the authors could cite a couple of tools like ActoNet, AEON.py, CABEAN, CANA, Caspo, pyboolnet, pystablemotifs (successor to Ref 5), or any of a large number of others that have different approaches to control.

Reviewer #3: In this study, the authors investigate exceptions in Boolean dynamical networks where attractors require controlling most nodes, contrary to the typical pattern. These cases are characterized by unstable fixed points. By efficiently identifying these points, the authors claim to shed light on the variance in control kernel sizes across biological models and random dynamics. However, the authors also conclude that these exceptions may likely be artifacts of proposed deterministic models, and therefore suggest that the ease of controllability seen in biological networks remains broadly applicable.

1)The introduction is very brief and lacks sufficient context to understand the authors' contribution compared to previous work, both by the authors themselves and other research groups. While the authors mention that the main result can be summarized in Equation 1, which computes the number of control kernels (CK) using log_2(r), where r is the number of attractors, they also acknowledge that a similar result was previously obtained or reinterpreted in a study referenced as Ref. [4]. However, upon reviewing Ref. [4], it becomes apparent that the result described there was already established in 2019 by Hou et al. in "On the number of driver nodes for controlling a Boolean network when the targets are restricted to attractors" (Journal of Theoretical Biology, 463, 1-11), albeit through different theoretical analyses. Ref. [4] seems to offer only a different theoretical method and an additional network data analysis. Therefore, the authors should expand and clarify the background and novelty of their work in the introduction and discussion sections, including a fair comparison with theoretical results obtained in previous works [4], as well as Hou et al. (2019).

For readers with not enough experience on Boolean networks, a minimum set of definitions of the main concepts such as fixed-point attractor etc should be clearly stated in the introduction.

2)In the Discussion section, the authors state, "This work represents a more in-depth analysis of these exceptions. Similarly to Ref. [4], we compare the results..." suggesting that this work is an extension of Ref. [4] focusing on exceptional cases. However, it is unclear how the theoretical results presented here differ from those by Hou et al. Further elucidation is required.

3)When proposing a new model or approach, it is important to compare it with existing models, which this work fails to do. For example using continuous models, based on differential equations, or stochastic models to analyse some exception cases shown in small graphs.

4) Figures 2-4 compare biological networks and random networks in terms of control kernel sizes but only analyze variance without conducting other statistical analyses. It would be beneficial to examine the distributions of these two groups of networks statistically, including assessing their statistical significance.

5) Upon examining the biological networks analyzed in this work, it is evident that some are related to cancers or diseases while others are associated with biological processes such as development. Are there significant structural differences observed when comparing normal and disease-related networks? Tend to be more into the so-called "exception" group?

6) More details regarding the construction or structure of the analyzed random networks are needed. The topological features of these networks are not specified in the tables. The authors mention that they found only one similar network in the random ensembles they studied and conjecture that such contrived examples are "unlikely" in real biological systems, underscoring the importance of understanding the construction of random networks. So the construction of the random networks seems relevant.

7) On page 14, the authors employ the Louvain Modularity detection algorithm, but modularity in complex networks is a nuanced topic, and using

multiple algorithms such as Leiden could enhance the robustness of the results. The authors should extend the analysis using different algorithms such as Leiden for example. Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. Sci Rep 9, 5233 (2019)

8)Adding the number of links to Tables S1 and S2 would provide additional clarity.

9) Many of the networks analyzed have fewer than 10 nodes, with most having fewer than 20 nodes. While this limitation is inherent in Boolean network approaches, can it be justified that these small subsystems operate independently of the complete interactome network in a cell? Such biological explanation should be added to the text.

10) Figure 4 comprises three subfigures, but the figure captions label them as A, B, C, which is inconsistent with the figures themselves (no labels A,B,C are shown).

11) In the caption for Figure 3, the authors refer to "exceptional networks," but it appears that there are numerous exceptions! It would be helpful to provide p-values for these cases or its ensemble.

12) In the References section, some references are incomplete or have incorrect author names, such as Refs. [5,6].

13) Fig. 1 is too simple. This space/figure should be used to illustrate more clearly the concepts of this work. For example, SI (eqs. 6-7) denote a "exceptional network", could this specific graph topology be represented in a figure?

In conclusion, I do not recommend publication of this manuscript in its current form. Overall, the paper lacks a clear statement of its concrete findings and robust validation of results through comprehensive data, statistical analysis and model comparisons. The use of terms like "likely" undermines the confidence in the validity of the findings. Therefore, it is strongly suggested that the authors enhance the solidity of their presentation by addressing the points outlined above.

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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Difficult control is related to instability in biologically inspired Boolean networks

PCSY-D-24-00042R1

Dear Prof. Borriello,

We are pleased to inform you that your manuscript 'Difficult control is related to instability in biologically inspired Boolean networks' has been provisionally accepted for publication in PLOS Complex Systems.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow-up email from a member of our team. 

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Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Complex Systems.

Best regards,

Luis M. Rocha, Ph.D.

Section Editor

PLOS Complex Systems

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Please pay attention to a few comments from two of the reviewers in preparing your final version. Also, please note that the revised version (PDF) contained several copies of the manuscript. I understand that the final one contains the edited manuscript. However, the second and third versions differ in the ordering/numbering of the references, so please be careful which version is sent to as camera ready.

Reviewer Comments (if any, and for reference):

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

Reviewer #2: (No Response)

Reviewer #3: All comments have been addressed

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2. 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.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

**********

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

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

**********

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.

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?

PLOS Complex Systems 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

Reviewer #3: 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: No more comments.

Reviewer #2: The manuscript is overall much improved and better contextualized. The introduction is especially improved, and now the paper is better able to stand on its own. Crucial gaps in the model generation procedure are now filled. Many of the issues I raised in my previous review are fully addressed. I have only a few minor remaining comments.

1. My point about inputs nodes is adequately addressed, but I want to highlight a minor detail that may be of interest to the authors. Though it is true that only a very few Cell Collective models contain the "explicit" constant nodes the authors have discussed, many of them do contain "implicit" constant nodes: self-referential input nodes of the form x(t+1)=x(t) for which an initial condition is stated and enforced as a modeling assumption (but, crucially, not in the update functions!). Such nodes are effectively constant, but appear self-referential. Identifying these implicit constants is very tedious, as it essentially requires reading the original paper in detail. The Park et al. paper I previously mentioned has done this for the Cell Collective and provided updated models that explicitly encode these "implicit" constant nodes, which may save the authors some effort in future work should they embark on a study for which this distinction is important. However, the authors have made a convincing case section S2 and elsewhere that this will not affect the conclusions of this paper. The authors need not take any action in response to this point--it is for their information only.

2. The addition of figure S1 is a much-appreciated closer look at the models in figure 4B. However, it would be good to also see a similar figure based on figure 4D, which is one of the main results of this paper.

3. The argument made in the "inoperable edges" part of section S2 is sound, but I found it took me a few readings to parse it, particularly the passage beginning with "For a given network . . .". I suggest breaking up the points more for clarity, perhaps something along the lines of:

"A network that lacks output trees cannot have isolated fixed points because any node in an output tree will always return to its original value upon perturbation from a fixed point (and so the perturbed states are in the basin of attraction of the fixed point). Thus, detection of an isolated fixed point precludes the possibility that an output tree was hidden by inoperative edges and included in the core. If we remove inoperable edges, then the core size can only change in cases when no isolated fixed points were detected. We find empirically that difficult control occurs only in networks with isolated fixed points, so removing inoperable edges would not affect our conclusions about difficult control."

This is only a suggested rewording; the authors may do with it what they wish. Ultimately this is a matter of stylistic preference.

Reviewer #3: The reference [10] has a typo: 'nature' should be 'Nature.'

I believe the authors have properly addressed my comments.

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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.

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

Reviewer #1: None

Reviewer #2: No

Reviewer #3: No

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