Dear Prof. Khammash,

Thank you very much for submitting your manuscript "DeepCME: A deep learning framework for solving the Chemical Master Equation" for consideration at PLOS Computational Biology. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations.

Please prepare and submit your revised manuscript within 30 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email.

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[1] A letter containing a detailed list of your responses to all review comments, and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Thank you again for your submission to our journal. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

James R. Faeder

Associate Editor

PLOS Computational Biology

Daniel Beard

Deputy Editor

PLOS Computational Biology

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A link appears below if there are any accompanying review attachments. If you believe any reviews to be missing, please contact ploscompbiol@plos.org immediately:

[LINK]

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: review is uploaded as attachment

Reviewer #2: At a high level, the authors are using machine learning techniques (basic feedforward neural networks) to solve for (time dependent) expectations of stochastically modeled reaction networks. I think that this paper has the potential to be very good. It is the rare paper I review in which I think to myself "I wish I had done this!". Here are my main comments/critiques that I would like the authors to consider.

1. The title claims to solve the chemical master equation (Kolmogorov's forward equation). However, the paper is actually focused on solving for expectations. Now, I agree that a solution to the CME can come from using indicator functions. However, that does not really seem to be a natural application for this method (and there are no examples showing if the method is good for that).

2. A very important part of the paper is Theorem 3.3. More precisely: equation (19), which gives an equality that must hold (almost surely) for every *path*. In my view, the hear of the proof is that m(t) (unnumbered equation between (23) and (24)) is a local martingale. A few more words can be given here (or a reference) proving/saying that \\Delta \\hat V_k(s,X(s)) is adapted to the proper filtration and \\tilde R_k is a local martingale, thus.... Pointing to the proper reference will be helpful to readers not familiar with these things.

3. Page 10. Why was \\phi as given chosen? (this is minor)

4. This is my only major critique. I could not figure out exactly what the input to the NN was. This is explained on pages 10 and 11, but I did not understand it. Of course, I would really like to know how a path gets inputed since this is an absolutely key part of the method. I wonder if a simple example would be helpful here (I would suggest the birth death process of section 5.1).

5. Why were the NN's used so small (L = 2 and N_H = 4)?

6. The method does not seem viable at this stage for computing sensitivities. Perhaps soften your language to something along the lines of "we note that sensitivities can theoretically be computed as well via these methods" you could then give the reasoning as is. However, you could point out that it sometimes works and sometimes doesn't (since the examples are all over the map), and then point to this as future research.

Reviewer #3: The DeepCME is a novel deep learning framework for numerical solution of the chemical master equation (CME). The authors reformulate the problem of obtaining expectation of specific functions of state space and their sensitives using the Kolmogorov's backward equation (theorem 3.3). Using this reformulation they construct appropriate loss functions to train deep neural networks using a reinforcement framework using relatively small number of stochastic simulations (SSA). They demonstrate the power and utility of the their method using a series of examples. This is a timely, well written and important study. I have the following specific comments:

- Could the authors provide some insight the advantage of the specific formulation compared for example to forward equation in solving CME using deep neural networks?

- At the beginning of section 5 the author's describe specific choice of a large number of hyper parameters (number of layer's, nodes, etc). Could the authors explore the significant of at least some of these choices in an example on the performance?

- The author's show the CPU time required for direct SSA vs DeepCME. Could the author's also make comparison in term's of number of simulations used in the training and how does that change with n and the quality if the results?

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Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code 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 and code 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 or code —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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Reviewer #1: No

Reviewer #2: No

Reviewer #3: Yes: Vahid Shahrezaei

Figure 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. 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 us at figures@plos.org.

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Reproducibility:

To enhance the reproducibility of your results, we recommend that you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. Additionally, PLOS ONE offers an option to publish peer-reviewed clinical study protocols. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols

References:

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.

Attachments

Attachment

Submitted filename: Gupta_review.pdf

Dear Prof. Khammash,

Thank you very much for submitting your manuscript "DeepCME: A deep learning framework for computing solution statistics of the Chemical Master Equation" for consideration at PLOS Computational Biology. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations. In particular, Reviewer 2 is asking for clarification on the exact form of input to the neural net, which would be needed to replicate the method.

Please prepare and submit your revised manuscript within 30 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email.

When you are ready to resubmit, please upload the following:

[1] A letter containing a detailed list of your responses to all review comments, and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Thank you again for your submission to our journal. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

James R. Faeder

Associate Editor

PLOS Computational Biology

Daniel Beard

Deputy Editor

PLOS Computational Biology

***********************

A link appears below if there are any accompanying review attachments. If you believe any reviews to be missing, please contact ploscompbiol@plos.org immediately:

[LINK]

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: The authors have sufficiently addressed my comments.

Reviewer #2: The manuscript has improved. However, one of my main worries about the manuscript (and apparently this worry was also shared by Reviewer #1) remains unresolved. In particular, the inputs to the NN are still quite unclear to me. Let me be very explicit: I have no idea what is going on with the lambda's and phi's of what is now section 4.1. The authors describe them as "temporal features", but that doesn't mean anything to me. A trajectory is simply a listing of states and times. The authors need to explain how a trajectory is inputed into the NN. If there is a mapping to the lambda's and phi's somehow, then so be it, but please tell us clearly what that mapping is. At this point I would not be able to implement this method as the whole portion on "temporal features" is mysterious.

Here are some smaller comments:

1. In the abstract it again says “The goal… estimating solutions of high-dimensional CMEs…” this is not really what it is doing. The next line is more accurate. Maybe merge and fix? (Except the next line says "Arbitrarily chosen functions" which seems strange)

2. (Minor). Line 63. “Which for” to “for which”

3. Line 123, this is super small but in the definition of the Q matrix you seem to be assuming that each reaction vector is uniquely associated with a reaction. This is not always the case.

4. Line 143. “On which our deep learning approach depends on” — one too many “on”s

5. Same line. Which is same —> which is the same.

6. Line 250. Why is this line here: “The relation between Φ and its realisation R(Φ) as a map is not one-to-one.” I guess it’s true: there can be lots of choices of rho’s and T’s that give R. But is this important (i.e., I’m concerned I’m missing something)?

Reviewer #3: The authors have addressed all of my concerns in the revised manuscript.

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Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code 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 and code 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 or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: None

Reviewer #2: Yes

Reviewer #3: Yes

**********

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: No

Reviewer #3: No

Figure 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. 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 us at figures@plos.org.

Data Requirements:

Please note that, as a condition of publication, PLOS' data policy requires that you make available all data used to draw the conclusions outlined in your manuscript. Data must be deposited in an appropriate repository, included within the body of the manuscript, or uploaded as supporting information. This includes all numerical values that were used to generate graphs, histograms etc.. For an example in PLOS Biology see here: http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001908#s5.

Reproducibility:

To enhance the reproducibility of your results, we recommend that you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. Additionally, PLOS ONE offers an option to publish peer-reviewed clinical study protocols. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols

References:

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.

Dear Prof. Khammash,

We are pleased to inform you that your manuscript 'DeepCME: A deep learning framework for computing solution statistics of the Chemical Master Equation' has been provisionally accepted for publication in PLOS Computational Biology.

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

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.

IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.

Should you, your institution's press office or the journal office choose to press release your paper, you will automatically be opted out of early publication. We ask that you notify us now if you or your institution is planning to press release the article. All press must be co-ordinated with PLOS.

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

Best regards,

James R. Faeder

Associate Editor

PLOS Computational Biology

Daniel Beard

Deputy Editor

PLOS Computational Biology

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