Revealing the Genetic Code Symmetries through Computations Involving Fibonacci-like Sequences and Their Properties

Round 1

Reviewer 1 Report

Tidjani Négadi looks into symmetries of the genetic code and observes their various mathematical properties. This isn't the first paper on this subject - other works approach it in a similar way, primarily focusing on the mathematical side rather than on biochemical side. Hence, the research design appears appropriate.

My biggest problem is that this manuscript also appears to be mostly a conglomerate of some of Author's previous papers. I use the word "appears" because currently there is no clear distinction of what's new and what's old. For example:

1. Content is shared with Reference 2: NeuroQuantology 2009,7(1),181-187, like the unnamed table in Appendix A, which is a copy of Table 2 from Reference 2 with some extra sums at the bottom.

2. Fibonacci-like sequences are mentioned in the title of Symmetry: Culture and Science 2014,30(2),261-268 (Reference 14) - are they different from what is presented in the manuscript (Section 3)?

3. Fibonacci numbers and the four ribonucleotides UMP, CMP, AMP and GMP appear in the abstract of (uncited or citation error) Symmetry: Culture and Science 2019,30(2),181-185: "In this short paper, we present an interesting and welcome connection between two different approaches to the mathematical structure of the standard genetic code, we have considered in the last years. The first one relies on the use of the unique number 23! and the second is based on the atomic composition of the four ribonucleotides UMP, CMP, AMP and GMP, the building-blocks of RNA, where several Fibonacci numbers are seen to occur"

The Author has published multiple papers on the same subject. It is crucial that they are presented in their own subsection (preferably at the end of Section 2), so that the reader can see how they tie with this research. Computation is an open access journal, which gives the Author an opportunity to explain, highlight and promote his previous works.

There are also other issues that must be resolved. They are listed below.

1. I've spotted multiple English errors or hard to understand sentences in the manuscript. One is in the first line of the abstract: "we present a new way to study the genetic code mathematical and chemical structure". They all have to be corrected.

2. Author should use his official university email address rather than gmail (it looks unprofessional).

3. There are only 14 references. It must be increased to the number required by the Editorial Office. There are already three subsection in Section 2, but there is enough papers regarding the genetic code symmetry to cite. In particular, Rosandić and Paar (not "Parr") have published a couple of new papers on this subject since 2014. Also, none of the references in the manuscript contain DOI, making it harder to find the cited works.

4. The introduction must be split into paragraphs. Add breaks at these lines: 37, 49, 51, 78, 88. Turn the "multiplets" (lines 51-58) into a bullet list - this is crucial information and needs to be presented clearly.

5. Lines 45-41 should be rewritten. Start by explaining that in order to be able to encode all standard 20 amino acids using a combination of 4 nucleotides, each amino acid must be encoded by 3-nucleotide long codons. 4^3 = 64, hence either some codons would not produce a protein residue, or the same protein residue is produced by multiple codons, or both. And both is the right answer, since there are 3 stop codons + 61 protein sequence codons due to redundancy.

6. The style of Table 1 makes it hard to see the data, in particular the nucleic -> protein relationships mentioned in introduction. At this point the reader should be mostly concerned with amino acids and their codon groups, so it's better if the cells were colored like that, e.g. all ALA red. The color spectrum should change from - say - red to blue in alphabetic order of 3-letter sequence: ALA, ARG, ..., VAL. See Figure 2 in 10.1016/j.biosystems.2016.11.004 for comparison. Or, at least, each amino acid's code should be in the same color, e.g., red "A" in "GCC (A)". The gray background is also not explained here (only later in Section 2.1). The current table could be moved to Section 2.1 where it visualizes Rumer's approach.

7. Line 75. Reference 5 does not claim that the number of AMINO ACIDS should be increased to 23, but writes about the increase of CODON FAMILIES to 23. It is a fundamental difference.

8. Line 80. "Here, Nc is shown to be a better predictor". Make sure that it is clear that "here" refers to Reference 5, not this manuscript.

9. Lines 83-84. Again, its not 23 amino acids but ("effective") codons families.

10. Line 86. Reference 2 from 2009 is hardly an "our recent works". "Our previous works" would be fine.

11. Lines 94 and 104. "61 amino acids". Again, codons, not amino acids. The same happens multiple times in various forms in the manuscript.

12. Line 108. There is no visualization in the manuscript - no figures. Author means here presentation of examples, but actual figures would make this paper much more accessible to non-mathematicians (as claimed in the last sentence of the abstract).

13. Line 125. A mapping is also a relation. Clarify what relation and what mapping.

14. Lines 217-613. I find this part is hard to follow. The long paragraphs need to be broken down, even into subsubsections, Section 4.2 in particular (4.2.1, 4.2.2, etc).

15. Line 767. Contrary to this statement: "those shown in bold, and others, are all mathematically derived in this paper", "other" rows and row "Total (20)" were derived in Reference 2.

16. Line 777. Is there a reason to not call the "block" as simply the backbone? Also, the sidechain is bound to C-alpha carbon, not "linked to" it.

17. Line 787. This table has no label nor caption. Again, there is >20 amino acids in the green rows. Should 1s in the M columns be in the same cell, like they are in Reference 2?

18. Line 818. It won't hurt to define the coprime here (i.e., GCD = 1). If the appendix is to help the reader, it should be self-contained, i.e., all information, like the Euler's Theorem's formula, should be included in it. Each component should be described in its own paragraph (i.e., add break it at lines 823, 834).

19. Line 824. Again, 2009 is not the year of a "recent work" (Reference 13).

20. Line 831. Typo: "fctorization".

Author Response

July 05 2023

Thank you for taking time to assess my manuscript entitled “Revealing the genetic code symmetries through computations involving Fibonacci-like sequences and their properties”.

I tried to address the great majority, if not all, of the concerns you raised, including the English issues which were corrected by using, in particular, online grammar checkers.

You are right when you are using the word “appears”, probably because of some writing clumsiness on my part. I fact, the paper is not, at all, a conglomerate of my previous papers. Some few of these, in fact only two, were cited in this manuscript in appropriate places, when necessary. The whole content of the present manuscript, including the mathematical formalism based on Fibonacci-like sequences and the numerous results derived from it, is new and has never been published. I have tried to clarify this point all along the paper and hope that it is now satisfying.

Below, are my replies to the 20 points and what I did.

• Concern 1: The table in Appendix A has now a name and the data it contains is necessary for the reader to check the calculations. It is my own table of the elemental composition of the amino acids and, it’s normal that I use it again, xxxxx
• Concern 2: The table in Appendix A has now a name and the data it contains is necessary for the reader to check the calculations. It is my own table of the elemental composition of the amino acids and, it’s normal that I use it again, xxxxx
• Concern 3: “Fibonacci-like sequences” are not mentioned in the paper to which you are refer. The title of that paper is “The genetic code invariance: when Euler and Fibonacci meet” (please, follow the link to the paper, if needed, here:

https://journal-scs.symmetry.hu/issue-content/?volume=25&issue=3

             This paper is almost entirely based on the use of the 12^th  Fibonacci number, 144, with some

             few applications of the ordinary Fibonacci numbers are made in a last section. No mention, at  

             all, of  the word  “Fibonacci-like sequences”. By the way, the reference to this paper has been

             dropped in the present manuscript.

Issues:          

1. Many English errors have been corrected and some sentences were revised by using online grammar checkers. A large number of these corrections are highlighted in yellow color.

2. The official email address has replaced the old one.

3. Another reference of a recent paper by Rosandić and Paar has been added. Following and interesting comment by reviewer 2, I added in the conclusion (now entitled “conclusion and perspectives”) an additional part which constitutes a beginning answer to one of the questions he/she raised. This added part is also new and connected with symmetry. It has been worked out while revising the manuscript. This gave me the opportunity to add three more appropriate references having something to do with symmetry. There is now a total of 19 references. All available DOIs have been added.

4. The introduction has been split into paragraphs by introducing breaks at the indicated lines. Also, the “multiplets” list has been turned into bullets.

5. The part in lines 49-454 (old lines 41-45) has been rewritten and are is, I hope, comprehensible.

6. Table 1 has been rearranged by introducing colors. I added below the table a short comment about the grey color background which visualizes one of Rumer’s sets by also referring to the corresponding section where it is considered. I would prefer to let the table at its present place because it follows immediately the description of the genetic code.

7. “number of codon families” has been written in line 85 and its reference has moved to #4.

8. I now make sure that N_c refers to Ref.4; see at line 89.

9. “effective number of codon families” has been explicitly written in line 87.

10. The old reference 2 (Négadi, 2009) has been dropped, as I said above, but the words “our previous works”, as suggested, are written two times in other places (line 85 and line 905).

11. Concerning “61 amino acids”: this wording is also used by many genetic code researchers and there is no ambiguity when I say almost each time “61 amino acids corresponding to the 61 coding codons”. This is true when one consider the degeneracy; each one of the 20 amino acids, except the singlets, can be coded several times by several codons. Eventually dropping this from the manuscript would demand many days, if not more, of corrections with many chances of new errors. As, I explained above, there is no ambiguity and, in this respect, I added a clear explanation, with emphasis, in lines 284-285.

12. In this paper, there is no need of figures (only equations and tables). In the last line of the old abstract, I used the word “visualization”. In fact, this was a wrong choice I made. I replaced is by “presentation of some computation examples”. This was made, in particular, in a new appendix (Appendix C), suggested to me by the Editor, in his review.

13. I added some sentences (lines 141-149) to explain what is an “f-mapping”.

14. Section 4.2, the longest one, has been broken into three sub-sections: 4.2.1 (line 288), 4.2.2 (line 330) and 4.2.3 (line 450).

15. When I say “Those shown in bold, and others, are all mathematically derived in this paper using the present new approach”, this true. In the table, they are evaluated from the data of the amino acids given in the table and, in the text, throughout the manuscript, they are all effectively computed by using the mathematics of the defined Fibonacci-like sequences and their properties.

16. The word “block” has been changed to “backbone” everywhere it appears. Also, “side chain is bound to the a-carbon” has been added in line 856.

17. The table has now the label “The elemental composition of the 20 amino acids”. No numbering is necessary.

18. A sentence about Euler’s theorem (with a reference in a footnote) has been included in line 901-902

19. See point 10 above.

20. The typo has been corrected: “factorization”(line 913).

Sincerely,

Dr. Tidjani Négadi

Physics Department

Faculty of Exact and Applied Science

University of Oran, Oran1, Algeria

Email: [email protected]

Reviewer 2 Report

The paper presents a number of numerical relations between various properties of the amino acids and nucleic acids that form the basic building blocks of biological systems. Besides addressing a few minor issues listed below, I suggest that, if at all possible, (a) the authors discuss the possible biological use of the relations discovered and (b) the authors examine the question whether the number of symmetries/relations are specific to the current choice of codon table (i.e., can the codon table be considered optimal in the sense of highest level of symmetry). I understand that these are non-trivial questions and the authors may not have an answer, but addressing them at any extent would greatly enhance the paper.

It is not clear from the paper that the idea of (p,q) Fibonacci sequence is not new. As I understand it, the only thing new is the specific choice of p and q.

L 52-58: List the various multiplets in ascending or descending order

L 179: ‘resourse’ -> ‘resource’

 The number of hydrogens depends on the protonation state for several amino acids – it should be discussed and explained what state the number of hydrogens refers to.

The English is fine.

Author Response

July 05 2023

Thank you for taking time to assess my manuscript entitled “Revealing the genetic code symmetries through computations involving Fibonacci-like sequences and their properties”.

Thank you, also, for the questions you raised. They are certainly difficult ones and, as you suspect, I have right now no answer. However, in the meantime, I have worked out, concretely, a case related to the second part of your questions; it is related to the vertebrate mitochondrial genetic code, which is known to exhibit the highest symmetry. I have obtained, using my Fibonacci-like sequences, as custom, remarkable results, I believe, with simple calculations (a pocket calculator let’s do them quickly). These (new) results have been included in the conclusion (now entitled “conclusion and perspectives”).

Issues:

1. I clarified what is new in my use of the Fibonacci-like sequences (in lines 203-204)

2. The “multiplets” have been arranged in descending order, as suggested, and I introduced “bullets”, as also suggested by reviewer 1 (lines 56-63).

3. The misprint has been corrected (“resource”), line 201.

• A comment about the protonation/deprotonation of some amino acids and also a word concerning some early results, I have already obtained, but waiting for a future consideration, have been also included in the conclusion (lines 792-801).

Sincerely,

Dr. Tidjani Négadi

Physics Department

Faculty of Exact and Applied Science

University of Oran, Oran1, Algeria

Email: [email protected]

Round 2

Reviewer 1 Report

This is round 2 review of „Revealing the genetic code symmetries through computations involving Fibonacci-like sequences and their properties” by Tidjani Négadi.

While the Author has made some changes to the manuscript following round 1 comments, in my opinion, this submission still needs a significant rework before it can be considered for publication.

The two biggest problems I’m observing are that (1) the manuscript is (still) doing a poor job with telling what was previously done by the Author in this research area and what new is brought by this submission (in light of lines 95-99 in particular), and (2) that it (still) feels like too much content is mashed together with narrative that is lost in the wall-of-text paragraphs. This stands in contradiction to the claim from the end of the abstract: “We organize the content of the text in such a way that, besides the presentation of several new research results, it has also an educational dimension. The paper could therefore be read, the computations easily worked out, also by non-experts with mathematical backgrounds.” If those claims are to hold, the paper should have a straightforward, logically and visually divided narrative. I have also identified additional problems with this submission. Some of them are related to changes in v2, the others were carried from v1. They are given below.

1. Line 25: Author has added following line to the abstract: „We emphasize that this work is new, both in terms of the formalism used and in terms of the results obtained which have never been published before.”. But the reader should see it from the text, not be just told. Novelty is expected from scientific papers anyway. Such statements are detrimental to the submission. Same goes for “totally original way” at line 111.

2. Line 71. Author insists on showing Rumer’s set in Table 1, but mixing methods with introduction is not the right way. Table 1 should only visualize the codon families, emphasizing the basics of genetic to protein sequence mapping. It is fine to duplicate this table for use in Section 2.1.  Colors have been added to Table 1, which is good, but some of them make the text barely visible, especially the greens. Try to use darker shades. In hindsight, readability can also be improved by writing “ACC – Thr” instead of “ACC (Thr)”.

3. Line 82: The last two sentences of this paragraph must be removed.

4. Line 85: Codon families = multiplets. Not explained. This and the previous paragraph should be rewritten together since they pretty much talk about the same thing. Also define the effective number of codons and show its relationship to the family boxes from line 76.

5. Lines 95-99 contain “It has been established in our previous works, by other means (see for example [5]). Here, in the present work, it is derived, together with a multitude of other significant quantities, from a completely different and novel approach and also integrated into a coherent set of Fibonacci-like sequences and relationships.”. Portion of Author’s previous works is presented only in Appendix A, which should be moved to somewhere after Section 2.3 together with the table, which contains amino acid data necessary to understand the calculations (like when it is referenced at line 296). I already wrote about this during round 1. The reader should not be just referred to the appendix, especially when the Author has already worked on this subject before (see point #1 above).

6. Line 200: Some rationale behind using Fibonacci numbers in the context of genetic code research should be given. Can you cite any papers? The old reference 14 that had some connection to this subject was dropped from version 2 of the manuscript.

7. Line 208: Where is the justification for p and q of sequences a and a`? Line 343 only mentions sequences b and c. It is important that the reader knows how the Author came up with these equations.

8. Line 210: The p and q values, being crucial coefficients here, should be moved to a new equation or given in Table 4 in two additional columns.

9. Line 219: Where is the proof that the relations in eq 2 hold for any n? Appendix C only shows how to check some of them for given n. If the Author cannot prove it, it should be mentioned and a program should be provided that will check them all for all k between 1 and n given by the user. Table 4 should likewise contain values up to n that can be reached in this research, which seems to be 15 (after eq 2.vii, unverified).

10. Line 231: It should be -b_{5} = 0, not -a_{5} = 0. What’s the significance of the next sentence about n = 8? b_{6} does not appear anywhere else in the text. Remove it if it’s not needed.

11. Line 233: “See below” should say exactly where. Next paragraph or Section 4.2.1?

12. Line 243: For readability, every time the Author presents a numeric relationship he finds interesting, or starts a new thought (like that at line 260), new paragraph should be started.

13. Line 291: I do not agree that “61 amino acids” is the correct nomenclature, even if it is “used by many genetic code researchers” (who?). The number of standard amino acids is fixed at 20. The number of codons that facilitate translation of the genetic sequence into protein residues is 61. But saying that “61 amino acid  = 61 codons” is nonsensical. The number of codon families is 20, or 23 if we split the families of Arg, Ser and Leu into the two “effective” subsets. This must be changed throughout the entire manuscript. Considering other researchers, Rosandić and Paar do not write “61 amino acids” but “61 codons” in their cited works.

14. Line 303: “Equ.(2) which, we recall, is valid for any n”. Again, where is the proof? It can be placed in an Appendix.

15. Line 373: “It could be easily verified that they give the same result and both hold for any n” As above. Remember the educational dimension.

16. Line 522: as above.

17. Line 670: as above.

18. Line 769: Conclusions need additional summary in form of a matching between the Fibonacci sequences and the special numbers - something like “aX_{n} + bY_{n} = 358, which is <one-liner explanation>”. A numbered list would be fine. Everything important Author has noticed throughout sections 3-5 has to be clearly summarized like this for an easy-to-understand takeaway by the readers.

19. Line 809. Conclusions is not the right place for these calculations and the accompanying text. Move them to a separate section or a new manuscript. It doesn’t conclude but add another layer of complexity to this already complex paper.

20. Line 929: The code and equations in Appendix C are not very helpful in their current form. They might be useful for those who have knowledge of the software Author is using, but the code is also in paths and can’t be copied from the PDF. Since this appendix does not help much, it would be better if its contents were transferred to a working source code repository in the supplementary materials, allowing reader’s computer to check the identities from all the equations, ideally in a form of one human- and machine-readable script per equation (with inline comments). Author must already have this code, so it’s only the matter of sharing it.

Author Response

Review

(x) I would not like to sign my review report

( ) I would like to sign my review report

Quality of English Language

(x) I am not qualified to assess the quality of English in this paper

( ) English very difficult to understand/incomprehensible

( ) Extensive editing of English language required

( ) Moderate editing of English language required

( ) Minor editing of English language required

( ) English language fine. No issues detected

YesCan be improvedMust be improvedNot applicable

Does the introduction provide sufficient background and include all relevant references?

( )(x)( )( )

Are all the cited references relevant to the research?

(x)( )( )( )

Is the research design appropriate?

(x)( )( )( )

Are the methods adequately described?

( )( )(x)( )

Are the results clearly presented?

( )( )(x)( )

Are the conclusions supported by the results?

( )( )(x)( )

Comments and Suggestions for Authors

This is round 2 review of „Revealing the genetic code symmetries through computations involving Fibonacci-like sequences and their properties” by Tidjani Négadi.

While the Author has made some changes to the manuscript following round 1 comments, in my opinion, this submission still needs a significant rework before it can be considered for publication.

The two biggest problems I’m observing are that

(1) the manuscript is (still) doing a poor job with telling what was previously done by the Author in this research area and what new is brought by this submission (in light of lines 95-99 in particular), and (2) that it (still) feels like too much content is mashed together with narrative that is lost in the wall-of-text paragraphs. This stands in contradiction to the claim from the end of the abstract: “We organize the content of the text in such a way that, besides the presentation of several new research results, it has also an educational dimension. The paper could therefore be read, the computations easily worked out, also by non-experts with mathematical backgrounds.” If those claims are to hold, the paper should have a straightforward, logically and visually divided narrative. I have also identified additional problems with this submission. Some of them are related to changes in v2, the others were carried from v1. They are given below.

Response to (1) : A new paragraph has been added, explaing in some detail my previous  works and what is new in the present paper, with the addition of new references. (see more in point 5 below).

Response to (2) : The used verb « organize », from the end of the old abstract, was very badly chosen. It was not my intention to organize anything in the text, or structure it, concerning an « educational dimension ».  What was meant, instead, was to give the paper something more than a mere presentation of new results, an educational dimension, that of giving also the non-expert reader the possibility to learn about the GC. That erroneous sentence has been removed and replaced by another one which wouldn’ imply a contradiction later.

1. Line 25: Author has added following line to the abstract: „We emphasize that this work is new, both in terms of the formalism used and in terms of the results obtained which have never been published before.”. But the reader should see it from the text, not be just told. Novelty is expected from scientific papers anyway. Such statements are detrimental to the submission. Same goes for “totally original way” at line 111.

Response : The mentioned sentence in the abstract has been removed. Idem for the three words words at old line 111 ; see lines 109-110.

2. Line 71. Author insists on showing Rumer’s set in Table 1, but mixing methods with introduction is not the right way. Table 1 should only visualize the codon families, emphasizing the basics of genetic to protein sequence mapping. It is fine to duplicate this table for use in Section 2.1. Colors have been added to Table 1, which is good, but some of them make the text barely visible, especially the greens. Try to use darker shades. In hindsight, readability can also be improved by writing “ACC – Thr” instead of “ACC (Thr)”.

Response : Table 1 (now Table 1a) is duplicated and moved to section 2.1 (named there Table 1b). The mentioned colors are now darker and the notation ex. UUU-Phe adopted.

3. Line 82: The last two sentences of this paragraph must be removed.

Response : These two sentences have been removed.

4. Line 85: Codon families = multiplets. Not explained. This and the previous paragraph should be rewritten together since they pretty much talk about the same thing. Also define the effective number of codons and show its relationship to the family boxes from line 76.

Response : The relation between « codons families » and « multiplets » is explained in lines 84-94.

I give only a brief definition of the « effective number of codons », in lines 98-99 (from ref.4), together with some more details above (line 96-98), concerning the work in reference 4. This, I think, is enough as these elements are not the subject of this paper nor used in it. The part concerning the work by Sun et a. (Ref.4) is mentioned in lines 97-101) only to highlight the importance of the role of the three sextets.

As far as I know, there is no visible and direct relationship, to show, between « the effective number of codons » and the 16 « family boxes », which are described in lines 75-79. The first is an « index », used in bioinformatics, and the second is a term used in the genetic code research to explain, among other things, that each one of the three sextets has its 6 codons distributed over two separate « familly boxes ». Each one of the 8 « family boxes » in Rumer’s set M1 (8 quartets or 4-fold) could eventually serve as an « effective number of codons » and they do in Ref.4, as, each, contains four synonymous codons. This is not the case for the remaining 8 « familly boxes » of Rumer’s set M2 because they contain, each, groups of non synonymous codons.). Once again, it would be useless to add all these details which would only make the text heavier and obscure the understanding of the rest of the paper.

5. Lines 95-99 contain “It has been established in our previous works, by other means (see for example [5]). Here, in the present work, it is derived, together with a multitude of other significant quantities, from a completely different and novel approach and also integrated into a coherent set of Fibonacci-like sequences and relationships.”. Portion of Author’s previous works is presented only in Appendix A, which should be moved to somewhere after Section 2.3 together with the table, which contains amino acid data necessary to understand the calculations (like when it is referenced at line 296). I already wrote about this during round 1. The reader should not be just referred to the appendix, especially when the Author has already worked on this subject before (see point #1 above).

Response : The two sentences in lines 95-99 have been removed and, In connection with point #1, a (quite) long paragraph has been added (lines 110-154) to explain in detail what was previously done (with several references added including the one which was removed) and what is new in this paper .

As for the table in Appendix A, it cannot be moved because putting it elsewhere would disrupt completely the entire structure of the paper It is called upon in almost all sections. I recommend to the reader (in lines 174-180) that a prior consultation of this appendix would be beneficial for the understanding of the paper. The calculations in this appendix are necessary, there, where they can be checked with the table next to them. So, I keep it where it is.

6. Line 200: Some rationale behind using Fibonacci numbers in the context of genetic code research should be given. Can you cite any papers? The old reference 14 that had some connection to this subject was dropped from version 2 of the manuscript.

Response : Fibonacci numbers are so well-known so it is not necessary to present them in this paper. In the context of the genetic code research, there very few papers and these latter have no connection with the subject of this paper so why cite them ? Also, when googling « Fibonacci+genetic code », the first displayed results are mine and I only mention one of them (as Ref.9) which is the reference that was removed earlier. The present pape ris about « Fibonacci-like » sequences, not « Fibonacci » sequences and I do not know of works, on the genetic code, using Fibonacci-like sequences.

7. Line 208: Where is the justification for p and q of sequences a and a`? Line 343 only mentions sequences b and c. It is important that the reader knows how the Author came up with these equations.

Response : Following this comment, an entire section (4.2.5) was added, discussing, in detail, this point. (lines 585-625). A first piece, necessary for understanding the content of section 4.2.2 is first given (in lines 411-424). Next, in the new section, 4.2.5, the whole question is discussed.

8. Line 210: The p and q values, being crucial coefficients here, should be moved to a new equation or given in Table 4 in two additional columns.

Response : They have been included near the name of each sequence (adding two additional columns is not easy (because of n) so they are placed just before the names of the sequences ).

9. Line 219: Where is the proof that the relations in eq 2 hold for any n? Appendix C only shows how to check some of them for given n. If the Author cannot prove it, it should be mentioned and a program should be provided that will check them all for all k between 1 and n given by the user. Table 4 should likewise contain values up to n that can be reached in this research, which seems to be 15 (after eq 2.vii, unverified).

Response : The relations in Eq.(2) can only be verified by direct computer calculations, according to the hints given in Appendix C which has been rewritten with examples and explanations for the reader who wants to check by himeslf/herself the verifications and/or the calculations, using either their mathematical software or using the indicated links to some source codes (in footnotes). I have done these verifications several times for various bigger and bigger values of n, for ALL the identities. For one of them, for example (), while always giving the right answer (equality of the two sides) the software (here Maple) hits the limit of what can be done at about n=50000; there, it displays only a part of the 10251 digits. The Fibonacci sequence, from which the functions a, a’, b, c and g are defined, grows fast.

The calculations in the paper itself can be done by anyone using only pocket calculator (they are  all low n cases).

As far as I know, nowhere, in the paper, is the value n=15 refered to. As the table is already large, especially after incuding the (p,q)-values, I keep it as it is (n=13). Of course, if someone would like to know the following values, only elementary additions are involved .

10. Line 231: It should be -b_{5} = 0, not -a_{5} = 0. What’s the significance of the next sentence about n = 8? b_{6} does not appear anywhere else in the text. Remove it if it’s not needed.

Response : True, it should be . Corrected. The case n=8 is needed later : just before Equ.(20) in line 525 and also below Equ.(60) in line 926.

On the other hand, the case n=9 which was mentioned after the case n=8, in a short sentence, is not necessary, although having a possible application (139-137=2 which can be used to transform the recurrence relation 137+221=358, from Table 4,  to the equality 139+219=358). This is another way to derive Equ.(10) but this point was not considered in the paper. This short sentence has therefore been removed.

11. Line 233: “See below” should say exactly where. Next paragraph or Section 4.2.1?

Response. The sentence has been removed (see point 10 above).

12. Line 243: For readability, every time the Author presents a numeric relationship he finds interesting, or starts a new thought (like that at line 260), new paragraph should be started.

Response. A new paragraph is started (at old line 260, after Equ.(8)). Same above Equ.(14) line 431. Same several line below Equ.(18) line 493.

13. Line 291: I do not agree that “61 amino acids” is the correct nomenclature, even if it is “used by many genetic code researchers” (who?). The number of standard amino acids is fixed at 20. The number of codons that facilitate translation of the genetic sequence into protein residues is 61. But saying that “61 amino acid = 61 codons” is nonsensical. The number of codon families is 20, or 23 if we split the families of Arg, Ser and Leu into the two “effective” subsets. This must be changed throughout the entire manuscript. Considering other researchers, Rosandić and Paar do not write “61 amino acids” but “61 codons” in their cited works.

Response : « who ? » : Downes and Richardson in their paper, published in J Mol Evol and cited in the present manuscript, use this wording (see page 477 in their paper). I followed however your recommendation and replaced this short sentence « 61 amino acids », in all places where it appears, by another one or some variant of it, necessarily much longer to convey properly the idea. This wouldn’t induce confusion for the readers.

14. Line 303: “Equ.(2) which, we recall, is valid for any n”. Again, where is the proof? It can be placed in an Appendix.

Response :  Already answered in point 9 above.

15. Line 373: “It could be easily verified that they give the same result and both hold for any n” As above. Remember the educational dimension.

Response : This has also been addressed above in point (9)

16. Line 522: as above.

Response : see above 

17. Line 670: as above.

Response : see above 

18. Line 769: Conclusions need additional summary in form of a matching between the Fibonacci sequences and the special numbers - something like “aX_{n} + bY_{n} = 358, which is <one-liner explanation>”. A numbered list would be fine. Everything important Author has noticed throughout sections 3-5 has to be clearly summarized like this for an easy-to-understand takeaway by the readers.

Response : At first I was reluctant ; how to put all this mass of information in such a small place? Subsequently, after reflection, I found this idea appropriate and I included at the end of the introduction a short « summary » as a list of four numbered points, in a "one-liner" format.

19. Line 809. Conclusions is not the right place for these calculations and the accompanying text. Move them to a separate section or a new manuscript. It doesn’t conclude but add another layer of complexity to this already complex paper.

Response : The text and the calculations were moved to a new 7th section (lines 892-932)

20. Line 929: The code and equations in Appendix C are not very helpful in their current form. They might be useful for those who have knowledge of the software Author is using, but the code is also in paths and can’t be copied from the PDF. Since this appendix does not help much, it would be better if its contents were transferred to a working source code repository in the supplementary materials, allowing reader’s computer to check the identities from all the equations, ideally in a form of one human-and machine-readable script per equation (with inline comments.) Author must already have this code, so it’s only a matter of sharing it.

Response : The Appendix C has been rewritten another way with examples from the text and others. Also, some source code links were included (in two footnotes).

Note: The writing style of the paper has been taken over. Some sentences have been removed (see the points above) and many sentences have been scrutinized, in particular with the help of "grammar checkers". Also, several, if not all, "interesting" qualifiers present in the old version, you seem to not appreciate too much, have been removed, although they are at several occasions. A neutral style has therefore been adopted.

(many characters reappear in bold, even after correcting them, several times; this is something unfortunate with word 2007.)

Round 3

Reviewer 1 Report

This is round 3 review of „Revealing the genetic code symmetries through computations involving Fibonacci-like sequences and their properties” by Tidjani Négadi.

I wasn’t impressed by the original submission and the changes in version 2, but here Author made significant improvements to the manuscript. It seems that the file is now on the right track. Hence, this review round is less critical than the previous two. However, I still believe more changes are required. They are listed below, intermixed with additional suggestions for improvements.

1. Abstract: (a) Drop “some” from line 10 and maybe replace “few” with the actual number of sequences used; (b) see point 3 below for “initial conditions”; (c) change “like” to “such as” at line 16; (d) remove “the detailed number of hydrogen atoms” - it is covered by “the detailed number of atoms (H/CNOS)”; (e) compress “Also, unexpectedly but interestingly, we find, as a by-product” to “We also discover, as a by-product”.

2. Reading the abstract again, I think that the last two yellow paragraphs should be moved to Introduction (as a separate paragraph, either at the beginning or the very end) since they are more of Author's opinion that a summary about what’s going on in the manuscript, which – let’s be honest – is not an easy read for non-experts. The related works (Rumer, etc) could be mentioned somewhere in the abstract instead.

3. “Initial terms” is simpler than “initial conditions” and can be written without quotes since it is related to the beginning of a sequence by definition.

4. Maybe “genetic code symmetry” should be added as a keyword too.

5. The “61 amino acids” is gone, replaced by “61 coding codons”, which is closer to reality, but also sounds redundant. I understand that Author means “codons other than stop codons”, but it can be improved further. For example, at line 52 Author writes “61 sense codons” – why not go with that?

6. Table 1 is better now, although Gln still looks washed out on my monitor, and Thr, Trp and Ser blend with each other due to similar hues. Try to use shades of darker yellow somewhere. You may also want to underline Ser’s and Arg’s text to emphasize the fact that they have two zones on this map. Check if adding thin spaces around “–” improves the readability. Caption (or footer) of the table should explain the meaning of the color boxes to make it self-contained.

7. I’d drop the entire sentence at line 39 and start the paragraph with “The genetic code is the basis of…”. Crick’s central dogma should be mentioned (and cited) somewhere around here (possibly before ref 1) as it is the basis of understanding of information transfer in biological systems.

8. Line 65. Replace “The table, below” with “Table 1”. Do not call it Table 1a, just Table 1. The same goes for Table 1b – name it Table 2.

9. Line 110. This new section is a needed addition to the manuscript, explaining Author’s previous contributions to genetic code research. As such, it could be placed in its own subsection, say “1.1. Previous works” (or something), to logically separate it from the rest of the Introduction. This is because it is necessary that this wall of text is broken into paragraphs any time new thought is started. For example, before “In [5]”, “In [6]”, etc. Line 39 could also start “1.1. Genetic code” section, turning “previous works” into 1.2. Following this logic, section 1.2 (or 1.3), titled “structure of the text” (or something) can be started at line 155. And again, with every section mentioned here (lines 156, 158, 169, etc) new paragraph should be started.

10. Line 147. The sentence starting with “The use of these sequences” must be rewritten and split into smaller pieces. It’s hard to understand.

11. Line 182. Plural of “software” is “software”.

12.  Line 186. Author insists on keeping Appendix A where it is, but also states in the response letter that the table there is referred to in multiple locations (“see the table in Appendix A”) and that moving it would disrupt the narrative. The problem is that appendices are meant for short extras that (1) do not need / should not be placed in supplementary materials and (2) are not necessary for the main portion of the article to be complete. Appendix A is referred in the main text 19 times (!), so the reader would need to jump back and forth quite a lot, which is what actually disrupts the narrative. This table contains necessary data that grounds the theoretical analysis (results) with reality. Appendix A is also short (1.5 page, but it can be surely shortened to 1 page). Given its importance and the number of times it is used, it should be moved to a new section before current Section 2. I’d call it simply “2. Data” or “2. Data and Definitions”. The table then can be referred to as Table 2 with proper internal PDF links. Shorten the text as necessary and optionally include commonly used definitions, such as “61 sense codons” = “61 of 64 genetic code codons (without stop codons)”, “perfect number” = “…”, etc. It can be a bullet list. This way the reader will see the numeric database derived from Table 1 before seeing results working on those numbers.

13. Line 345. There are two unmarked tables here. Same for the tiny table at line 932, which can be converted to text.

14. Line 449: Author admits in the response letter that the relations are only verified computationally rather than proven mathematically and gives n = 50000 as a practical limit. It works, but 50000 is not “for any n”, so all such statements should be removed from the text (there is a few of them) or converted to statements about verification up to a finite n that is (1) doable and (2) meaningful to this research, which seems to be 13. It also means that the examples in Appendix C could use n = 13 rather than n = 150.

15. Line 592. Replace “Guessed” with “chosen arbitrarily”. Also in lines 853 and 943. Hint at the thought process that lead to their choice.

16. Line 611. Avoid jargon (“They also have something to tell”). This applies to the entire manuscript. Another example is “striking” (line 844, etc).

17. Line 935. Conclusions must be split into smaller paragraphs with proper links to sections of the manuscript they summarize. The addition of the takeaway equations is very good. Is 4 enough?

18. Line 1036. “Many areas” appears twice in the same sentence.

19. Line 1053 and others. The links in the footnotes should be converted to proper references with online access dates.

Author Response

Open Review

(x) I would not like to sign my review report
( ) I would like to sign my review report

Quality of English Language

(x) I am not qualified to assess the quality of English in this paper
( ) English very difficult to understand/incomprehensible
( ) Extensive editing of English language required
( ) Moderate editing of English language required
( ) Minor editing of English language required
( ) English language fine. No issues detected

Comments and Suggestions for Authors

This is round 3 review of „Revealing the genetic code symmetries through computations involving Fibonacci-like sequences and their properties” by Tidjani Négadi.

I wasn’t impressed by the original submission and the changes in version 2, but here Author made significant improvements to the manuscript. It seems that the file is now on the right track. Hence, this review round is less critical than the previous two. However, I still believe more changes are required. They are listed below, intermixed with additional suggestions for improvements.

1. Abstract: (a) Drop “some” from line 10 and maybe replace “few” with the actual number of sequences used; (b) see point 3 below for “initial conditions”; (c) change “like” to “such as” at line 16; (d) remove “the detailed number of hydrogen atoms” - it is covered by “the detailed number of atoms (H/CNOS)”; (e) compress “Also, unexpectedly but interestingly, we find, as a by-product” to “We also discover, as a by-product”.

Respone : All done.lines 10, 16, 17, 19.

2. Reading the abstract again, I think that the last two yellow paragraphs should be moved to Introduction (as a separate paragraph, either at the beginning or the very end) since they are more of Author's opinion that a summary about what’s going on in the manuscript, which – let’s be honest – is not an easy read for non-experts. The related works (Rumer, etc) could be mentioned somewhere in the abstract instead.

Response : The old (two yellow) sentences have been moved, as a small separate parapgraph, in the beginning of the introduction (lines 35-38). Also, Rumer is mentioned at line 24.

3. “Initial terms” is simpler than “initial conditions” and can be written without quotes since it is related to the beginning of a sequence by definition.

Response : The term « initial conditions » is THE term used in mathematics; please make a quick google search.  (It is also used in the physics of the dynamical systems.) Obviously, they are the first two terms of the sequence. I added « i.e, their first two terms» for a pedagogical clarification, once, not in the abstract but in the introduction, the first time it appears in lines 153-154. In the rest of the paper, the commas are the removed (lines 172, 178, 278, 281, 423, 952)

4. Maybe “genetic code symmetry” should be added as a keyword too.

Response : I merged the two old keywords « genetic code » and « symmetries » into one keyword « genetic code symmetries »

5. The “61 amino acids” is gone, replaced by “61 coding codons”, which is closer to reality, but also sounds redundant. I understand that Author means “codons other than stop codons”, but it can be improved further. For example, at line 52 Author writes “61 sense codons” – why not go with that?

Response : « 61 sense codons » replaced “61 coding codons” everywhere (lines 53, 142, 167, 365, etc.)

6. Table 1 is better now, although Gln still looks washed out on my monitor, and Thr, Trp and Ser blend with each other due to similar hues. Try to use shades of darker yellow somewhere. You may also want to underline Ser’s and Arg’s text to emphasize the fact that they have two zones on this map. Check if adding thin spaces around “–” improves the readability. Caption (or footer) of the table should explain the meaning of the color boxes to make it self-contained (RVB, TSL, percentages, etc.)

The colors for Gln, Thr, Ser and Trp (and also Phe) have been improved. The readability has been improved.by adding spaces around “–”.  Also, the three amino acids, as suggested, are now underlined. The colors are self-explanatory.There is no need to explain the meaning of the colors, first, because it would require additional place as text and, second, this would take a long long time to find the precise names of the colors (RVB, TSL, percentages for each color, etc.)

7. I’d drop the entire sentence at line 39 and start the paragraph with “The genetic code is the basis of…”. Crick’s central dogma should be mentioned (and cited) somewhere around here (possibly before ref 1) as it is the basis of understanding of information transfer in biological systems.

Response : The mentioned sentence in line 39 was dropped. Now, there is no need to quote Crick’s central dogma and add a reference. My manuscript is only about the genetic code, as it is, and the important thing to mention is its experimental decryption by Nirenberg et al., my reference 1 (if this comment had been important, why not have done it during the two previous revisions?)

8. Line 65. Replace “The table, below” with “Table 1”. Do not call it Table 1a, just Table 1. The same goes for Table 1b – name it Table 2.

Response : All the tables have been renamed (: All the tables have been renamed (1, 2, 3, 4, 5, inclusing Tables 6, 7 and 8 ).

9. Line 110. This new section is a needed addition to the manuscript, explaining Author’s previous contributions to genetic code research. As such, it could be placed in its own subsection, say “1.1. Previous works” (or something), to logically separate it from the rest of the Introduction. This is because it is necessary that this wall of text is broken into paragraphs any time new thought is started. For example, before “In [5]”, “In [6]”, etc. Line 39 could also start “1.1. Genetic code” section, turning “previous works” into 1.2. Following this logic, section 1.2 (or 1.3), titled “structure of the text” (or something) can be started at line 155. And again, with every section mentioned here (lines 156, 158, 169, etc) new paragraph should be started.

Response : I followed the suggestion and splitted that part of the introduction into three subsections 1.1 (line 39),  1.2 (line 111) and 1.3 (line 150).

10. Line 147. The sentence starting with “The use of these sequences” must be rewritten and split into smaller pieces. It’s hard to understand.

Response : The mentioned long sentence has been cut into two easier to understand sentences(lines 156-162).

11. Line 182. Plural of “software” is “software”.

Response : corrected (line 191).

12. Line 186. Author insists on keeping Appendix A where it is, but also states in the response letter that the table there is referred to in multiple locations (“see the table in Appendix A”) and that moving it would disrupt the narrative. The problem is that appendices are meant for short extras that (1) do not need / should not be placed in supplementary materials and (2) are not necessary for the main portion of the article to be complete. Appendix A is referred in the main text 19 times (!), so the reader would need to jump back and forth quite a lot, which is what actually disrupts the narrative. This table contains necessary data that grounds the theoretical analysis (results) with reality. Appendix A is also short (1.5 page, but it can be surely shortened to 1 page). Given its importance and the number of times it is used, it should be moved to a new section before current Section 2. I’d call it simply “2. Data” or “2. Data and Definitions”. The table then can be referred to as Table 2 with proper internal PDF links. Shorten the text as necessary and optionally include commonly used definitions, such as “61 sense codons” = “61 of 64 genetic code codons (without stop codons)”, “perfect number” = “…”, etc. It can be a bullet list. This way the reader will see the numeric database derived from Table 1 before seeing results working on those numbers.

Response : Thank you for your efforts to convince me but I still stick to leave Appendix A where it is and also as it is. Making such suggested changes would disrupt all what I wanted to write in the manuscript, from the start. (In fact, the reader does not need to go back and forth each time to consult Appendix A. He may have consulted it, carefully, once, as I advise him to do at the end of the introductio. Also, the author still expects the reader to show a certain dose of confidence. However, if the reader wishes, or if he/she is somewhat suspicious, he/she can always have the idea of printing copy of the appendix, next to him/her, and keeps on  reading, at ease.)

13. Line 345. There are two unmarked tables here. Same for the tiny table at line 932, which can be converted to text.14. Line 449: Author admits in the response letter that the relations are only verified computationally rather than proven mathematically and gives n = 50000 as a practical limit. It works, but 50000 is not “for any n”, so all such statements should be removed from the text (there is a few of them) or converted to statements about verification up to a finite n that is (1) doable and (2) meaningful to this research, which seems to be 13. It also means that the examples in Appendix C could use n = 13 rather than n = 150.

Response : In line 355, there is now a named table (Table 6,). Same for old tiny table now Table 7 (line 945). Also, the table in Appendix A is now named Table 8 and it is mentioned in several parts in the text.The mention « for any n » has been removed but the reference to Appendix C is still there for explanations about larger values of n. The case n=150 is conserved to show the reader that he/she can verify the relations for n large or even for very large.n’s.

15. Line 592. Replace “Guessed” with “chosen arbitrarily”. Also in lines 853 and 943. Hint at the thought process that lead to their choice.

Response : guessed  has be replaced by « found after a trial and error thought process. (lines 601, 863).

16. Line 611. Avoid jargon (“They also have something to tell”). This applies to the entire manuscript. Another example is “striking” (line 844, etc).

Response : This is not « jargon ». (jargon, in french, means a way of expressing oneself specific to a group, a profession or an activity difficult to understand for the layman). I changed that sentence anyway (in line 621)..(A total dryness, even in a scientific text, is a bit boring.). « strinking », in line 844, has been replaced by « remarkable » which is true and « striking » in line 672 has been replaced by « it is worth noting that », which is also true.

17. Line 935. Conclusions must be split into smaller paragraphs with proper links to sections of the manuscript they summarize. The addition of the takeaway equations is very good. Is 4 enough?

Response : The conclusion was remodeled and split into small paragraphs and some few parts were removed..  4 is enough.

18. Line 1036. “Many areas” appears twice in the same sentence.

Response : I wanted to add "and still many other fields". FinaIly, I shortened the sentence. (line 1050).

19. Line 1053 and others. The links in the footnotes should be converted to proper references with online access dates.

Response : These two links in the two footnotes have been included in the references (25 and 26).

Note : In Microsoft word (2007), several bold characters reappear, frequently all over the text, even after correcting them. To avoid this problem, for later use of the manuscript, I changed the old trully desired charactères in bold in Table 5 and in Table 8 as well as in Equ.(27) into dark red colors.

Thank you.

Tidjani Négadi

Author Response File: Author Response.docx