Size matters: tissue size as a marker for a transition between reaction–diffusion regimes in spatio-temporal distribution of morphogens

The reaction–diffusion model constitutes one of the most influential mathematical models to study distribution of morphogens in tissues. Despite its widespread use, the effect of finite tissue size on model-predicted spatio-temporal morphogen distributions has not been completely elucidated. In this study, we analytically investigated the spatio-temporal distributions of morphogens predicted by a reaction–diffusion model in a finite one-dimensional domain, as a proxy for a biological tissue, and compared it with the solution of the infinite-domain model. We explored the reduced parameter, the tissue length in units of a characteristic reaction–diffusion length, and identified two reaction–diffusion regimes separated by a crossover tissue size estimated in approximately three characteristic reaction–diffusion lengths. While above this crossover the infinite-domain model constitutes a good approximation, it breaks below this crossover, whereas the finite-domain model faithfully describes the entire parameter space. We evaluated whether the infinite-domain model renders accurate estimations of diffusion coefficients when fitted to finite spatial profiles, a procedure typically followed in fluorescence recovery after photobleaching (FRAP) experiments. We found that the infinite-domain model overestimates diffusion coefficients when the domain is smaller than the crossover tissue size. Thus, the crossover tissue size may be instrumental in selecting the suitable reaction–diffusion model to study tissue morphogenesis.


Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
Size matters: an analytical study on the role of tissue size in spatiotemporal distribution of morphogens unveils a transition between different Reaction-Diffusion regimes by Alberto S. Ceccarelli and Augusto Borges and Osvaldo Chara In this paper Ceccarelli et al study the spatio-temporal evolution of a morphogen diffusing and decaying in a 1D domain. Particularly, they compare the predictions in a finite domain with the predictions obtained assuming that the domain is infinite. This is an interesting question worthy of investigation and the results are technically sound and explained clearly. This analysis could have potentially important implications for the interpretation of experimental measurements with FRAP and it could also help to select the appropriate approximation in formulation of biological models involving the spread of morphogens. However, for both purposes there are important (yet solvable) issues that cast doubt into the relevance of this work. These points and other minor observations are elaborated below.
Firstly, the authors limit their analysis, at least in the main text, to the study of the RD equation with zero initial concentration, a constant flow of morphogen at the origin and a vanishing concentration at the opposite end. These type of boundary conditions are not necessarily the most relevant in a realistic biological setting. Arguably, a no-flux boundary condition at the opposite end of the source could describe more realistically the confinement of a morphogen in a tissue or in a cell. Importantly, the boundary conditions in this particular problem have a major effect in the results. For example, statements like "morphogen concentrations predicted by the model assuming an infinite domain are higher than those predicted by the model assuming a finite domain" would not hold true with a no-flux boundary condition at R. Furthermore, many of the differences between the finite and infinite domain explored at length in the text and figures, and even some of formulas derived analytically, are dependent on the boundary conditions. Since the authors mention that that they have in fact derived the analytical solution with alternative boundary conditions, I would suggest that they include at least a discussion of how they compare with the solution already discussed. The crossover length and time to reach steady states with alternative BCs should also be included. In addition, it would be important that the authors discuss if this was not included in the main text due to the overlap with ref.13 (Umulis 2009) or if their formulation of the problem and derivation of a solution are different in any important way.
Secondly, the relevance of the present study for the interpretation of FRAP experiments is not justified by the contents of the main text. Section 2.7 is almost entirely devoted to demonstrate once again that the two alternative assumptions, namely an infinite vs finite domain, produce different steady state predictions in domain lengths below a certain threshold, and that this threshold is the crossover length obtained in previous sections. This result is just a rehashing of the previous results but derived in a more convoluted way, which in fact obscures a point that has already been firmly established in the preceding sections: The steady state concentrations predicted using the two alternative assumptions differ significantly for lengths L<3*\lambda. This derivation is redundant given the contents of the previous sections. Instead, I would suggest that the authors attempt to demonstrate that the theoretical formulation under study is relevant to interpret FRAP essays. For example, it would be helpful to show that the present mathematical formulation is a reasonable description of a FRAP essay. In a typical FRAP essay, a circular or square region containing a fluorescently labeled molecule is bleached and the recovery of the fluorescence levels caused by the diffusion of molecule back into the bleached region allow to estimate the diffusion rates and decay parameters. Again, the boundary conditions discussed in the main text, and in this case also the geometrical setup (with a single source of morphogen in one of the boundaries) are not necessarily a good description of a FRAP essay. Would it be possible to derive the analytical solution for a case that more closely represents it? In addition, finding examples in the literature in which the estimation of morphogen parameters could be improved with the finite domain assumption and even providing the improved estimations would go a long way to attract attention to this work (much like it is done in the Discussion section with the analysis of the validity of the infinite-domain assumption for specific processes involving FGF8 and Dpp).
These two are the main issues that I find would improve this work, other less important points are explained next. Section 2.6 explores the differences in time to reach steady state in the infinite vs finite-domain scenarios. This is done introducing the mean time and the standard deviation of the time to reach the steady state. These variables are typically associated with stochastic processes, but since all the equations studied throughout the manuscript are deterministic, it is not entirely clear what they represent. The reader is referred to the Supplementary Material for the derivation of their analytical expressions. This is fine, but I would suggest to include a brief explanation of how they enter in the deterministic description.
In several instances it is described as remarkable that the the analytical predictions and numerical simulations match. This, rather than a remarkable result, is a reassuring feature that confirms that the analytical derivation are correct.
Related to this, in the Discussion it is said this work is valuable for numerical packages because it is more accurate and efficient. This is not very convincing, since the numerical solution of this type of simple equations can be made as accurate as to be virtually indistinguishable from the analytical solutions, and also because typically these packages are used to simulate complex problems in "D or 3D geometries, for which the simple 1D analytical solution is not useful.

Comments to the Author(s)
How tissue size affects the diffusion dynamics of morphogens is an interesting problem. In this sense I find the analyses performed in this study a useful addition to the literature on mathematical modeling of development. The results on the "crossover tissue size" is particularly interesting. A major problem of the manuscript is that the authors seem to have confused the "reaction-diffusion" model with the "French flag" model. As far as I can tell, this study has dealt with only the properties of morphogen diffusion; there is no "reaction" component. The solution for the real "reaction-diffusion" dynamics would be very different from the solution for a simple diffusion-only mechanism. I assume there are two ways to fix this: either reanalyze everything with a reaction component (which could be much more challenging than the current analysis) or rewrite the manuscript to put the work in the context of the "French flag" model rather than the "reaction-diffusion" model.

Decision letter (RSOS-211112.R0)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Dr Chara
The Editors assigned to your paper RSOS-211112 "Size matters: an analytical study on the role of tissue size in spatiotemporal distribution of morphogens unveils a transition between different react" have now received comments from reviewers and would like you to revise the paper in accordance with the reviewer comments and any comments from the Editors. Please note this decision does not guarantee eventual acceptance.
We invite you to respond to the comments supplied below and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
We do not generally allow multiple rounds of revision so we urge you to make every effort to fully address all of the comments at this stage. If deemed necessary by the Editors, your manuscript will be sent back to one or more of the original reviewers for assessment. If the original reviewers are not available, we may invite new reviewers.
Please submit your revised manuscript and required files (see below) no later than 21 days from today's (ie 13-Sep-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 21 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers).
Thank you for submitting your manuscript to Royal Society Open Science and we look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. In this paper Ceccarelli et al study the spatio-temporal evolution of a morphogen diffusing and decaying in a 1D domain. Particularly, they compare the predictions in a finite domain with the predictions obtained assuming that the domain is infinite. This is an interesting question worthy of investigation and the results are technically sound and explained clearly. This analysis could have potentially important implications for the interpretation of experimental measurements with FRAP and it could also help to select the appropriate approximation in formulation of biological models involving the spread of morphogens. However, for both purposes there are important (yet solvable) issues that cast doubt into the relevance of this work. These points and other minor observations are elaborated below.
Firstly, the authors limit their analysis, at least in the main text, to the study of the RD equation with zero initial concentration, a constant flow of morphogen at the origin and a vanishing concentration at the opposite end. These type of boundary conditions are not necessarily the most relevant in a realistic biological setting. Arguably, a no-flux boundary condition at the opposite end of the source could describe more realistically the confinement of a morphogen in a tissue or in a cell. Importantly, the boundary conditions in this particular problem have a major effect in the results. For example, statements like "morphogen concentrations predicted by the model assuming an infinite domain are higher than those predicted by the model assuming a finite domain" would not hold true with a no-flux boundary condition at R. Furthermore, many of the differences between the finite and infinite domain explored at length in the text and figures, and even some of formulas derived analytically, are dependent on the boundary conditions. Since the authors mention that that they have in fact derived the analytical solution with alternative boundary conditions, I would suggest that they include at least a discussion of how they compare with the solution already discussed. The crossover length and time to reach steady states with alternative BCs should also be included. In addition, it would be important that the authors discuss if this was not included in the main text due to the overlap with ref.13 (Umulis 2009) or if their formulation of the problem and derivation of a solution are different in any important way.
Secondly, the relevance of the present study for the interpretation of FRAP experiments is not justified by the contents of the main text. Section 2.7 is almost entirely devoted to demonstrate once again that the two alternative assumptions, namely an infinite vs finite domain, produce different steady state predictions in domain lengths below a certain threshold, and that this threshold is the crossover length obtained in previous sections. This result is just a rehashing of the previous results but derived in a more convoluted way, which in fact obscures a point that has already been firmly established in the preceding sections: The steady state concentrations predicted using the two alternative assumptions differ significantly for lengths L<3*\lambda. This derivation is redundant given the contents of the previous sections. Instead, I would suggest that the authors attempt to demonstrate that the theoretical formulation under study is relevant to interpret FRAP essays. For example, it would be helpful to show that the present mathematical formulation is a reasonable description of a FRAP essay. In a typical FRAP essay, a circular or square region containing a fluorescently labeled molecule is bleached and the recovery of the fluorescence levels caused by the diffusion of molecule back into the bleached region allow to estimate the diffusion rates and decay parameters. Again, the boundary conditions discussed in the main text, and in this case also the geometrical setup (with a single source of morphogen in one of the boundaries) are not necessarily a good description of a FRAP essay. Would it be possible to derive the analytical solution for a case that more closely represents it? In addition, finding examples in the literature in which the estimation of morphogen parameters could be improved with the finite domain assumption and even providing the improved estimations would go a long way to attract attention to this work (much like it is done in the Discussion section with the analysis of the validity of the infinite-domain assumption for specific processes involving FGF8 and Dpp).
These two are the main issues that I find would improve this work, other less important points are explained next.
Section 2.6 explores the differences in time to reach steady state in the infinite vs finite-domain scenarios. This is done introducing the mean time and the standard deviation of the time to reach the steady state. These variables are typically associated with stochastic processes, but since all the equations studied throughout the manuscript are deterministic, it is not entirely clear what they represent. The reader is referred to the Supplementary Material for the derivation of their analytical expressions. This is fine, but I would suggest to include a brief explanation of how they enter in the deterministic description.
In several instances it is described as remarkable that the the analytical predictions and numerical simulations match. This, rather than a remarkable result, is a reassuring feature that confirms that the analytical derivation are correct.
Related to this, in the Discussion it is said this work is valuable for numerical packages because it is more accurate and efficient. This is not very convincing, since the numerical solution of this type of simple equations can be made as accurate as to be virtually indistinguishable from the analytical solutions, and also because typically these packages are used to simulate complex problems in "D or 3D geometries, for which the simple 1D analytical solution is not useful.
Reviewer: 2 Comments to the Author(s) How tissue size affects the diffusion dynamics of morphogens is an interesting problem. In this sense I find the analyses performed in this study a useful addition to the literature on mathematical modeling of development. The results on the "crossover tissue size" is particularly interesting. A major problem of the manuscript is that the authors seem to have confused the "reaction-diffusion" model with the "French flag" model. As far as I can tell, this study has dealt with only the properties of morphogen diffusion; there is no "reaction" component. The solution for the real "reaction-diffusion" dynamics would be very different from the solution for a simple diffusion-only mechanism. I assume there are two ways to fix this: either reanalyze everything with a reaction component (which could be much more challenging than the current analysis) or rewrite the manuscript to put the work in the context of the "French flag" model rather than the "reaction-diffusion" model.

===PREPARING YOUR MANUSCRIPT===
Your revised paper should include the changes requested by the referees and Editors of your manuscript. You should provide two versions of this manuscript and both versions must be provided in an editable format: one version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting if your manuscript is accepted.
Please ensure that any equations included in the paper are editable text and not embedded images.
Please ensure that you include an acknowledgements' section before your reference list/bibliography. This should acknowledge anyone who assisted with your work, but does not qualify as an author per the guidelines at https://royalsociety.org/journals/ethicspolicies/openness/.
While not essential, it will speed up the preparation of your manuscript proof if accepted if you format your references/bibliography in Vancouver style (please see https://royalsociety.org/journals/authors/author-guidelines/#formatting). You should include DOIs for as many of the references as possible.
If you have been asked to revise the written English in your submission as a condition of publication, you must do so, and you are expected to provide evidence that you have received language editing support. The journal would prefer that you use a professional language editing service and provide a certificate of editing, but a signed letter from a colleague who is a native speaker of English is acceptable. Note the journal has arranged a number of discounts for authors using professional language editing services (https://royalsociety.org/journals/authors/benefits/language-editing/).

===PREPARING YOUR REVISION IN SCHOLARONE===
To revise your manuscript, log into https://mc.manuscriptcentral.com/rsos and enter your Author Centre -this may be accessed by clicking on "Author" in the dark toolbar at the top of the page (just below the journal name). You will find your manuscript listed under "Manuscripts with Decisions". Under "Actions", click on "Create a Revision".
Attach your point-by-point response to referees and Editors at Step 1 'View and respond to decision letter'. This document should be uploaded in an editable file type (.doc or .docx are preferred). This is essential.
Please ensure that you include a summary of your paper at Step 2 'Type, Title, & Abstract'. This should be no more than 100 words to explain to a non-scientific audience the key findings of your research. This will be included in a weekly highlights email circulated by the Royal Society press office to national UK, international, and scientific news outlets to promote your work.

At
Step 3 'File upload' you should include the following files: --Your revised manuscript in editable file format (.doc, .docx, or .tex preferred). You should upload two versions: 1) One version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); 2) A 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them.
--An individual file of each figure (EPS or print-quality PDF preferred [either format should be produced directly from original creation package], or original software format).
--An editable file of each table (.doc, .docx, .xls, .xlsx, or .csv --If you are requesting a discretionary waiver for the article processing charge, the waiver form must be included at this step.
--If you are providing image files for potential cover images, please upload these at this step, and inform the editorial office you have done so. You must hold the copyright to any image provided.
--A copy of your point-by-point response to referees and Editors. This will expedite the preparation of your proof.

At
Step 6 'Details & comments', you should review and respond to the queries on the electronic submission form. In particular, we would ask that you do the following: --Ensure that your data access statement meets the requirements at https://royalsociety.org/journals/authors/author-guidelines/#data. You should ensure that you cite the dataset in your reference list. If you have deposited data etc in the Dryad repository, please include both the 'For publication' link and 'For review' link at this stage.
--If you are requesting an article processing charge waiver, you must select the relevant waiver option (if requesting a discretionary waiver, the form should have been uploaded at Step 3 'File upload' above).
--If you have uploaded ESM files, please ensure you follow the guidance at https://royalsociety.org/journals/authors/author-guidelines/#supplementary-material to include a suitable title and informative caption. An example of appropriate titling and captioning may be found at https://figshare.com/articles/Table_S2_from_Is_there_a_trade-off_between_peak_performance_and_performance_breadth_across_temperatures_for_aerobic_sc ope_in_teleost_fishes_/3843624.

At
Step 7 'Review & submit', you must view the PDF proof of the manuscript before you will be able to submit the revision. Note: if any parts of the electronic submission form have not been completed, these will be noted by red message boxes.
Author's Response to Decision Letter for (RSOS-211112.R0) See Appendix A.

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
The authors have addressed all my major concerns. I would suggest to add a short paragraph in the discussion describing in plain words the major differences in behavior between the alternative assumptions (Dirichlet vs Neumann), like the time to reach steady state, the shape of the steady state, how they depart from the infinite-length solution and so on.
There is a statement in in pag. 64 that I find surprising: "With this boundary condition (no-flux), the total amount of morphogen accumulated in the tissue at the 218 steady state (NSS) is conserved and consequently, independent of R." Is this true, given that there is a decay term?
-The FRAP section is in much better shape. The authors tackled the problem of a bleached gradient, which is probably a much harder problem than what a typical FRAP essay entails. In a typical FRAP essay, a uniform distribution of a molecule is bleached. Since this is just a particular case of the more general scenario that they have solved, perhaps I'd be fitting if they briefly discuss it.
Aside from that, and to finish on a positive note, I'd like to congratulate the authors for this interesting piece of work.

Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
I appreciate the substantial effort that went into the revision of this manuscript, which made it an even stronger paper. I maintain my initial enthusiasm about the study, but I want to encourage the authors to think again about choosing the term "reaction-diffusion" over the more accurate alternatives such as SDD, Diffusion-decay, or Diffusion-degradation. The second term in the equation (-kC) describes the degradation of the same (diffusible) morphogen, not a "reaction" between two morphogens. If one mentions the term "reaction-diffusion model" to a developmental biologist, they will immediately think about two (or more) morphogens interacting with each other like in the "activator-inhibitor" or "activator-substrate depletion" systems. I assume the ultimate audience of this study are developmental biologists who are interested in mechanisms of pattern formation. Using a term that potentially confuses them will likely reduce the impact of the work. In both the "French flag" model and the real "Reaction-Diffusion" model, the properties of morphogen diffusion, be it over finite or infinite domain, are critically important. Therefore, choosing a more accurate but perhaps not as buzzing a term is not going to devalue the work. With that said, I am not demanding any changes. I'll leave this suggestion to the authors to consider.

Decision letter (RSOS-211112.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Dr Chara
On behalf of the Editors, we are pleased to inform you that your Manuscript RSOS-211112.R1 "Size matters: Tissue size as a marker for a transition between Reaction-Diffusion regimes in spatiotemporal distribution of morphogens" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referees' reports. Please find the referees' comments along with any feedback from the Editors below my signature.
We invite you to respond to the comments and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
Please submit your revised manuscript and required files (see below) no later than 7 days from today's (ie 13-Dec-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 7 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers).
Thank you for submitting your manuscript to Royal Society Open Science and we look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch.
Kind regards, Royal Society Open Science Editorial Office Royal Society Open Science openscience@royalsociety.org on behalf of Dr Jose Carrillo (Associate Editor) and Mark Chaplain (Subject Editor) openscience@royalsociety.org Reviewer comments to Author: Reviewer: 2 Comments to the Author(s) I appreciate the substantial effort that went into the revision of this manuscript, which made it an even stronger paper. I maintain my initial enthusiasm about the study, but I want to encourage the authors to think again about choosing the term "reaction-diffusion" over the more accurate alternatives such as SDD, Diffusion-decay, or Diffusion-degradation. The second term in the equation (-kC) describes the degradation of the same (diffusible) morphogen, not a "reaction" between two morphogens. If one mentions the term "reaction-diffusion model" to a developmental biologist, they will immediately think about two (or more) morphogens interacting with each other like in the "activator-inhibitor" or "activator-substrate depletion" systems. I assume the ultimate audience of this study are developmental biologists who are interested in mechanisms of pattern formation. Using a term that potentially confuses them will likely reduce the impact of the work. In both the "French flag" model and the real "Reaction-Diffusion" model, the properties of morphogen diffusion, be it over finite or infinite domain, are critically important. Therefore, choosing a more accurate but perhaps not as buzzing a term is not going to devalue the work. With that said, I am not demanding any changes. I'll leave this suggestion to the authors to consider.
Reviewer: 1 Comments to the Author(s) The authors have addressed all my major concerns. I would suggest to add a short paragraph in the discussion describing in plain words the major differences in behavior between the alternative assumptions (Dirichlet vs Neumann), like the time to reach steady state, the shape of the steady state, how they depart from the infinite-length solution and so on.
There is a statement in in pag. 64 that I find surprising: "With this boundary condition (no-flux), the total amount of morphogen accumulated in the tissue at the 218 steady state (NSS) is conserved and consequently, independent of R." Is this true, given that there is a decay term?
-The FRAP section is in much better shape. The authors tackled the problem of a bleached gradient, which is probably a much harder problem than what a typical FRAP essay entails. In a typical FRAP essay, a uniform distribution of a molecule is bleached. Since this is just a particular case of the more general scenario that they have solved, perhaps I'd be fitting if they briefly discuss it.
Aside from that, and to finish on a positive note, I'd like to congratulate the authors for this interesting piece of work.

===PREPARING YOUR MANUSCRIPT===
Your revised paper should include the changes requested by the referees and Editors of your manuscript.
You should provide two versions of this manuscript and both versions must be provided in an editable format: one version should clearly identify all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting.
Please ensure that any equations included in the paper are editable text and not embedded images.
Please ensure that you include an acknowledgements' section before your reference list/bibliography. This should acknowledge anyone who assisted with your work, but does not qualify as an author per the guidelines at <a href="https://royalsociety.org/journals/ethicspolicies/openness/">https://royalsociety.org/journals/ethics-policies/openness/</a>.
While not essential, it will speed up the preparation of your manuscript proof if you format your references/bibliography in Vancouver style (please see https://royalsociety.org/journals/authors/author-guidelines/#formatting). You should include DOIs for as many of the references as possible.
If you have been asked to revise the written English in your submission as a condition of publication, you must do so, and you are expected to provide evidence that you have received language editing support. The journal would prefer that you use a professional language editing service and provide a certificate of editing, but a signed letter from a colleague who is a proficient user of English is acceptable. Note the journal has arranged a number of discounts for authors using professional language editing services (https://royalsociety.org/journals/authors/benefits/language-editing/).

===PREPARING YOUR REVISION IN SCHOLARONE===
To revise your manuscript, log into https://mc.manuscriptcentral.com/rsos and enter your Author Centre -this may be accessed by clicking on "Author" in the dark toolbar at the top of the page (just below the journal name). You will find your manuscript listed under "Manuscripts with Decisions". Under "Actions", click on "Create a Revision".
Attach your point-by-point response to referees and Editors at the 'View and respond to decision letter' step. This document should be uploaded in an editable file type (.doc or .docx are preferred). This is essential, and your manuscript will be returned to you if you do not provide it.
Please ensure that you include a summary of your paper at the 'Type, Title, & Abstract' step. This should be no more than 100 words to explain to a non-scientific audience the key findings of your research. This will be included in a weekly highlights email circulated by the Royal Society press office to national UK, international, and scientific news outlets to promote your work. An effective summary can substantially increase the readership of your paper.
At the 'File upload' step you should include the following files: --Your revised manuscript in editable file format (.doc, .docx, or .tex preferred). You should upload two versions: 1) One version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); 2) A 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them.
--An individual file of each figure (EPS or print-quality PDF preferred [either format should be produced directly from original creation package], or original software format).
--An editable file of all figure and table captions. Note: you may upload the figure, table, and caption files in a single Zip folder.
--If you are requesting a discretionary waiver for the article processing charge, the waiver form must be included at this step.
--If you are providing image files for potential cover images, please upload these at this step, and inform the editorial office you have done so. You must hold the copyright to any image provided.
--A copy of your point-by-point response to referees and Editors. This will expedite the preparation of your proof.
At the 'Details & comments' step, you should review and respond to the queries on the electronic submission form. In particular, we would ask that you do the following: --Ensure that your data access statement meets the requirements at https://royalsociety.org/journals/authors/author-guidelines/#data. You should ensure that you cite the dataset in your reference list. If you have deposited data etc in the Dryad repository, please only include the 'For publication' link at this stage. You should remove the 'For review' link.
--If you are requesting an article processing charge waiver, you must select the relevant waiver option (if requesting a discretionary waiver, the form should have been uploaded, see 'File upload' above).
--If you have uploaded any electronic supplementary (ESM) files, please ensure you follow the guidance at https://royalsociety.org/journals/authors/author-guidelines/#supplementarymaterial to include a suitable title and informative caption. An example of appropriate titling and captioning may be found at https://figshare.com/articles/Table_S2_from_Is_there_a_trade-off_between_peak_performance_and_performance_breadth_across_temperatures_for_aerobic_sc ope_in_teleost_fishes_/3843624. At the 'Review & submit' step, you must view the PDF proof of the manuscript before you will be able to submit the revision. Note: if any parts of the electronic submission form have not been completed, these will be noted by red message boxes -you will need to resolve these errors before you can submit the revision.

See Appendix B.
Decision letter (RSOS-211112.R2) We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Dr Chara, I am pleased to inform you that your manuscript entitled "Size matters: Tissue size as a marker for a transition between Reaction-Diffusion regimes in spatiotemporal distribution of morphogens" is now accepted for publication in Royal Society Open Science.
Please remember to make any datasets or code libraries 'live' prior to publication, and update any links as needed when you receive a proof to check -for instance, from a private 'for review' URL to a publicly accessible 'for publication' URL. It is good practice to also add data sets, code and other digital materials to your reference list.
The proof of your paper will be available for review using the Royal Society online proofing system and you will receive details of how to access this in the near future from our production office (openscience_proofs@royalsociety.org). We aim to maintain rapid times to publication after acceptance of your manuscript and we would ask you to please contact both the production office and editorial office if you are likely to be away from e-mail contact to minimise delays to publication. If you are going to be away, please nominate a co-author (if available) to manage the proofing process, and ensure they are copied into your email to the journal.
Please see the Royal Society Publishing guidance on how you may share your accepted author manuscript at https://royalsociety.org/journals/ethics-policies/media-embargo/. After publication, some additional ways to effectively promote your article can also be found here https://royalsociety.org/blog/2020/07/promoting-your-latest-paper-and-tracking-yourresults/.
On behalf of the Editors of Royal Society Open Science, thank you for your support of the journal and we look forward to your continued contributions to Royal Society Open Science. we would like to thank you and the reviewers for the prompt and thoughtful review. The very constructive suggestions and comments of the reviewers allowed us to build, in our opinion, a more complete and thus stronger article. Essentially, the main elements of this rebuttal are as follows: We included in our study the analysis of the no-flux boundary condition at the opposite end of the source, as suggested by the first reviewer. We performed the geometrical characterization of the morphogen gradients together with the calculation of the mean time to achieve the steady state predicted by our model with the no-flux boundary condition, as we did with the sink boundary condition presented in our original submission. The new results show differences between the sink and the no-flux boundary conditions. Noteworthy, the main result of the study holds: the existence of a crossover tissue length separating two reaction-diffusion regimes, regardless of the particular characteristics of the boundary conditions studied.
Thanks also to a suggestion of the first reviewer, we reformulated the section devoted to Fluorescence Recovery After Photobleaching (FRAP), where we recreated a simplified FRAP scenario. Our new results show that while a reaction-diffusion model assuming an infinite-domain could be used to study tissues larger than the crossover tissue length, the here proposed finite-domain model should be used for smaller tissues, in agreement with the main body of our results.
We also included a comparison between the model and numerical simulations in 2D in the supplementary information.
Finally, and thanks to the second reviewer, we improved the description of the model, making it, in our eyes, clearer.
Please find below a point-by-point response where we describe the additional results and changes in text. We look forward to your response. In this paper Ceccarelli et al study the spatio-temporal evolution of a morphogen diffusing and decaying in a 1D domain. Particularly, they compare the predictions in a finite domain with the predictions obtained assuming that the domain is infinite. This is an interesting question worthy of investigation and the results are technically sound and explained clearly. This analysis could have potentially important implications for the interpretation of experimental measurements with FRAP and it could also help to select the appropriate approximation in formulation of biological models involving the spread of morphogens. However, for both purposes there are important (yet solvable) issues that cast doubt into the relevance of this work. These points and other minor observations are elaborated below.
Firstly, the authors limit their analysis, at least in the main text, to the study of the RD equation with zero initial concentration, a constant flow of morphogen at the origin and a vanishing concentration at the opposite end. These type of boundary conditions are not necessarily the most relevant in a realistic biological setting. Arguably, a no-flux boundary condition at the opposite end of the source could describe more realistically the confinement of a morphogen in a tissue or in a cell. Importantly, the boundary conditions in this particular problem have a major effect in the results. For example, statements like "morphogen concentrations predicted by the model assuming an infinite domain are higher than those predicted by the model assuming a finite domain" would not hold true with a noflux boundary condition at R. Furthermore, many of the differences between the finite and infinite domain explored at length in the text and figures, and even some of formulas derived analytically, are dependent on the boundary conditions. Since the authors mention that that they have in fact derived the analytical solution with alternative boundary conditions, I would suggest that they include at least a discussion of how they compare with the solution already discussed. The crossover length and time to reach steady states with alternative BCs should also be included. In addition, it would be important that the authors discuss if this was not included in the main text due to the overlap with ref.13 (Umulis 2009) or if their formulation of the problem and derivation of a solution are different in any important way.

Answer:
We thank the reviewer for her/his commentaries and the very constructive criticism. We fully agree with the suggestion of incorporating the no-flux boundary condition of the reactiondiffusion finite-domain model in our study. Thus, we decided to add the solution in the new version of the manuscript. We then extended the geometric characterization of the morphogen gradient together with the calculation of the main time to establish the steady state to this particular boundary condition. We were also able to define a crossover length for the model with the no-flux boundary condition which was similar to the one calculated from the sink boundary condition. As with the sink boundary condition, we observed that although the infinite-domain model cannot be distinguished from the finite-domain model for tissue lengths higher than the crossover tissue length, the morphogen gradients predicted by both models are clearly different for tissues smaller than the crossover length. Interestingly, and more prominently for these small tissues, the choice of the boundary condition at the opposite end of the source determines whether the steady state concentrations are higher or smaller than the concentrations predicted by the infinite-domain model. Because of that, while the sink boundary condition in the finite-domain model leads to values of the mean time to reach the steady state smaller than those predicted by the infinite-domain model, the opposite occurs with the no-flux boundary condition. In our eyes, the reviewer suggestion helped to build a more cohesive study comparing the finite-domain with the infinite-domain reaction-diffusion models.
As for the study of Umulis (2009), we acknowledge in our article the fact that our solution with the no-flux boundary condition coincides with the one previously obtained by him (reference 13). Nevertheless, as we mentioned in the discussion section, his study focused on the very interesting but completely different problem of morphogen scaling (which is reflected on the fact that he normalized space in units of the tissue length = ; in such units the effect of tissue size that we are interested in our study cannot be easily studied).
Secondly, the relevance of the present study for the interpretation of FRAP experiments is not justified by the contents of the main text. Section 2.7 is almost entirely devoted to demonstrate once again that the two alternative assumptions, namely an infinite vs finite domain, produce different steady state predictions in domain lengths below a certain threshold, and that this threshold is the crossover length obtained in previous sections. This result is just a rehashing of the previous results but derived in a more convoluted way, which in fact obscures a point that has already been firmly established in the preceding sections: The steady state concentrations predicted using the two alternative assumptions differ significantly for lengths L<3*\lambda. This derivation is redundant given the contents of the previous sections. Instead, I would suggest that the authors attempt to demonstrate that the theoretical formulation under study is relevant to interpret FRAP essays. For example, it would be helpful to show that the present mathematical formulation is a reasonable description of a FRAP essay. In a typical FRAP essay, a circular or square region containing a fluorescently labeled molecule is bleached and the recovery of the fluorescence levels caused by the diffusion of molecule back into the bleached region allow to estimate the diffusion rates and decay parameters. Again, the boundary conditions discussed in the main text, and in this case also the geometrical setup (with a single source of morphogen in one of the boundaries) are not necessarily a good description of a FRAP essay. Would it be possible to derive the analytical solution for a case that more closely represents it? In addition, finding examples in the literature in which the estimation of morphogen parameters could be improved with the finite domain assumption and even providing the improved estimations would go a long way to attract attention to this work (much like it is done