Stochastic fluctuations in apoptotic threshold of tumour cells can enhance apoptosis and combat fractional killing

Fractional killing, which is a significant impediment to successful chemotherapy, is observed even in a population of genetically identical cancer cells exposed to apoptosis-inducing agents. This phenomenon arises not from genetic mutation but from cell-to-cell variation in the activation timing and level of the proteins that regulates apoptosis. To understand the mechanism behind the phenomenon, we formulate complex fractional killing processes as a first-passage time (FPT) problem with a stochastically fluctuating boundary. Analytical calculations are performed for the FPT distribution in a toy model of stochastic p53 gene expression, where the cancer cell is killed only when the p53 expression level crosses an active apoptotic threshold. Counterintuitively, we find that threshold fluctuations can effectively enhance cellular killing by significantly decreasing the mean time that the p53 protein reaches the threshold level for the first time. Moreover, faster fluctuations lead to the killing of more cells. These qualitative results imply that fluctuations in threshold are a non-negligible stochastic source, and can be taken as a strategy for combating fractional killing of cancer cells.

The results presented here are to my mind novel and interesting. The theoretical framework is elegant and provides a powerful approach for understanding the phenomenon of fractional killing of cells in response to drugs. Therefore this work would potentially be of interest to a wide range of researchers, and a valuable addition to the literature.
However, there are a number of places where I feel the manuscript needs to be improved, before it can be published. Some of the most important (and counterintuitive) results have been provided with no deeper analysis of why the results occur in the first place. Limitations of the model in the context of the underlying biology and recent reports in the literature have not been carefully discussed. Finally, a number of statements in the manuscript are not very clear and are potentially misleading, including in my opinion, the title of the manuscript. I elaborate below: Major points: (1) The single most important (and interesting) result in this work is the observation that a fluctuating apoptotic threshold leads to shorter MFPT as compared to a non-fluctuating threshold. This is fascinating mainly because it is very counterintuitive to me . Fluctuations in the threshold would result in the threshold decreasing sometimes compared to the average threshold. But at other times, these fluctuations would also result in increasing the threshold over the average, right? Therefore, for some p53 trajectories the FPT will become smaller, but for other trajectories the FPT would increase. Then why should the MFPT decrease in the fluctuating barrier situation? This seems quite counter-intuitive to me, and the authors need to provide an explanation for this observation. Since this is the main result, the authors should provide a detailed discussion with additional calculations/simulations explaining why this result is true.
(2) In various places of the manuscript, the authors refer to the previous paper by Paek et al, Cell 2016 and talk about an apoptotic threshold that increases with time. Fig 1A of this manuscript even provides a graphical example of this situation. However, if I understand correctly, this manuscript deals ONLY with the situation of a threshold that on average is fixed at a constant level -is that correct? The comparison in this work is only between a constant threshold (called 'A_threshold') with no fluctuations vs a threshold that is constant on average but with fluctuations, right? In that case, I would strongly urge the authors to show this situation in Fig 1A  instead of showing a threshold that increases with time -since the latter scenario is not being studied in the current manuscript, the current Fig 1A could potentially cause confusion. Related to this point is the use of the terminology "dynamically fluctuating threshold" by the authors in many places. Why not simply say "fluctuating threshold"? The word "dynamic" tends to be associated with the idea of a threshold that changes on average with time as demonstrated in Paek et al, Cell 2016. The authors should therefore be careful in terms of the terminology they use.
(3) A number of earlier papers have shown that cell-fate decisions (whether the cell will eventually divide or go into cell cycle arrest) seem to happen early in the lifetime of the cell. Therefore the eventual fate outcome of a cell is to a large extent already determined by the cellular state inherited by that cell from its mother at the time of the mother's division. For example, in the paper "Hidden heterogeneity and circadian-controlled cell fate inferred from single cell lineages", Nat Comm 2018, the authors looked at cell fates of sister cells and found that they shared the same fate about 80% of the time, regardless of whether they were born before or after cisplatin treatment, indicating early decision making. Conceptually similar results were found using very different techniques in the papers "The proliferation-quiescence decision is controlled by a bifurcation in CDK2 activity at mitotic exit", Cell 2013 and "Competing memories of mitogen and p53 signalling control cell-cycle entry", Nature 2017.
If I understand the theory presented here correctly, this phenomenon of early decision making is not captured by the FPT framework, is it? Given that the 'decision' of death vs survival in the author's model happens as a result of reaching an absorbing barrier for the first time, the 'decision' making in the authors' model seems to be happening late in the cell's lifetime, at the moment the p53 trajectory crosses the level of A. The authors should provide a discussion on this interesting aspect in the context of the papers cited above and their own model. Is this indeed a limitation of their model? If so, can the authors propose some way their FPT model can be improved in future work to account for the early decision-making process in cells?
(4) The title of the paper is in my opinion not a fair representation of the contents of this paper. The authors show that increasing the fluctuations in the absorbing barrier can reduce the MFPT and hence increase the fraction of cells killed. However, in order to achieve this, the production and degradation rates have to be simultaneously increased for the protein A. How can this be achieved realistically, in actual experiments? The authors do not provide any discussion on this important point. Indeed, to my knowledge, this may not be an easy task to achieve. In that case, suggesting that controlling the fluctuations in the threshold level might be a 'strategy' to combat fractional killing seems too strong a conclusion in my opinion. The authors should provide a discussion on the feasibility of this 'strategy' and also potentially change the title of the manuscript to better reflect their interesting, but theoretical findings.
Minor Points: (1) In many places throughout the manuscript, the authors use the terminology "fast fluctuations" and "slow fluctuations". Would it be more accurate to say "large fluctuations" and "small fluctuations"? By increasing the magnitude of the production and degradation rates of p53 or A, the amplitude of the noise increases, not how fast the fluctuations occur.
(2) On page 6 of the manuscript, the line "Note that the more threshold-crossing events are, the fewer cancer cells are killed…." is quite confusing. Did the authors mean "more" cancer cells are killed?
(3) On page 22, the authors say "However, the results in the case of timing variability is almost converse to those…". This statement seems incorrect to me after looking at Fig 5 -it seems to me that BOTH the MFPT and the timing variability are decreasing with increase of the p53 transcription rate.
(4) In the heatmaps shown, for example in Figures 5 and 6, the authors may contemplate adding contour lines to aid the eye. It's a little difficult to see the trends in the heatmaps as shown in the current figures.

02-Oct-2019
Dear Dr Zhang, The editors assigned to your paper ("Dynamic variability in apoptotic threshold as a strategy for combating fractional killing") have now received comments from reviewers. We would like you to revise your paper in accordance with the referee and Associate Editor suggestions which can be found below (not including confidential reports to the Editor). Please note this decision does not guarantee eventual acceptance.
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Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. This manuscript proposed a stochastic model to simulate dynamics and variation in fractional killing of cancer cells, which is a very important topic in the study of drug resistance. The firstpassage time problem was modeled by a stochastic model, and this manuscript derived the analytical distribution of the first-passage time. Interesting results are found for the influence of the fluctuations in nearly every parameter in the model, which may provide insights into the mechanisms of drug resistance. This is an interesting and important work.
The major comment is the writing of this manuscript. Much work is needed to improve the writing. Here I do not give a detailed list since this list may quite long. Please check the manuscript carefully and polish the English writing thoroughly.
The following are some suggestions for the main structure of the manuscript.
(1) The discussion on page 4 for model introduction may be shortened.
(2) The subsection title on page 7 actually is the master equation of the model rather than "stochastic model formulation". The stochastic model has already been introduced in Eq (3).
(3) Subsection from page 10 is too long. (D) "the effectiveness …" may be the first subsection in Results section. The subsection titles (A, B, C) may be removed.
(4) There are certain overlaps between the figure legends and main text. It is suggested to write the legends briefly.
(5) Supplementary Information discussed four cases of different absorbing domains. The main text only discussed case 1 in detail. There are totally 13 supplementary figures that are not mentioned in the manuscript at all. Thus, a subsection (or a paragraph) in Results section may be needed to compare (6) In the proposed model, the expression of p53 is modeled by a bursting process but the expression of protein A is Poisson. It is not clear for the results if the expression of protein A is also bursting. Some simulation results may be enough to explain this.

Reviewer: 2 Comments to the Author(s)
In the manuscript "Dynamic variability in apoptotic threshold as a strategy for combating fractional killing", Qiu et al present a theoretical framework based on first passage time (FPT) distributions to study the phenomenon of fractional killing of cells in response to drugs. Building a model for a simple two-gene network of p53 and a gene 'A' that sets a threshold for the p53 level (at which point the cell dies), the authors develop an elegant approach to compute the mean FPT (MFPT) and the variability in FPT for this model. The authors use the Chemical Master Equation to model the two-dimensional state space of p53 and A, and then use the finite state projection method to convert the infinite-dimensional transition probability matrix M into a finite dimensional matrix that is amenable to numerical solutions. The main finding from this theory is that fluctuations in the apoptotic threshold reduce the MFPT compared to a non-fluctuating fixed threshold, implying that fractional killing of cells will be larger in the presence of a fluctuating threshold.
The results presented here are to my mind novel and interesting. The theoretical framework is elegant and provides a powerful approach for understanding the phenomenon of fractional killing of cells in response to drugs. Therefore this work would potentially be of interest to a wide range of researchers, and a valuable addition to the literature.
However, there are a number of places where I feel the manuscript needs to be improved, before it can be published. Some of the most important (and counterintuitive) results have been provided with no deeper analysis of why the results occur in the first place. Limitations of the model in the context of the underlying biology and recent reports in the literature have not been carefully discussed. Finally, a number of statements in the manuscript are not very clear and are potentially misleading, including in my opinion, the title of the manuscript. I elaborate below: (1) The single most important (and interesting) result in this work is the observation that a fluctuating apoptotic threshold leads to shorter MFPT as compared to a non-fluctuating threshold. This is fascinating mainly because it is very counterintuitive to me . Fluctuations in the threshold would result in the threshold decreasing sometimes compared to the average threshold. But at other times, these fluctuations would also result in increasing the threshold over the average, right? Therefore, for some p53 trajectories the FPT will become smaller, but for other trajectories the FPT would increase. Then why should the MFPT decrease in the fluctuating barrier situation? This seems quite counter-intuitive to me, and the authors need to provide an explanation for this observation. Since this is the main result, the authors should provide a detailed discussion with additional calculations/simulations explaining why this result is true.
(2) In various places of the manuscript, the authors refer to the previous paper by Paek et al, Cell 2016 and talk about an apoptotic threshold that increases with time. Fig 1A of this manuscript even provides a graphical example of this situation. However, if I understand correctly, this manuscript deals ONLY with the situation of a threshold that on average is fixed at a constant level -is that correct? The comparison in this work is only between a constant threshold (called 'A_threshold') with no fluctuations vs a threshold that is constant on average but with fluctuations, right? In that case, I would strongly urge the authors to show this situation in Fig 1A  instead of showing a threshold that increases with time -since the latter scenario is not being studied in the current manuscript, the current Fig 1A could potentially cause confusion. Related to this point is the use of the terminology "dynamically fluctuating threshold" by the authors in many places. Why not simply say "fluctuating threshold"? The word "dynamic" tends to be associated with the idea of a threshold that changes on average with time as demonstrated in Paek et al, Cell 2016. The authors should therefore be careful in terms of the terminology they use.
(3) A number of earlier papers have shown that cell-fate decisions (whether the cell will eventually divide or go into cell cycle arrest) seem to happen early in the lifetime of the cell. Therefore the eventual fate outcome of a cell is to a large extent already determined by the cellular state inherited by that cell from its mother at the time of the mother's division. For example, in the paper "Hidden heterogeneity and circadian-controlled cell fate inferred from single cell lineages", Nat Comm 2018, the authors looked at cell fates of sister cells and found that they shared the same fate about 80% of the time, regardless of whether they were born before or after cisplatin treatment, indicating early decision making. Conceptually similar results were found using very different techniques in the papers "The proliferation-quiescence decision is controlled by a bifurcation in CDK2 activity at mitotic exit", Cell 2013 and "Competing memories of mitogen and p53 signalling control cell-cycle entry", Nature 2017.
If I understand the theory presented here correctly, this phenomenon of early decision making is not captured by the FPT framework, is it? Given that the 'decision' of death vs survival in the author's model happens as a result of reaching an absorbing barrier for the first time, the 'decision' making in the authors' model seems to be happening late in the cell's lifetime, at the moment the p53 trajectory crosses the level of A. The authors should provide a discussion on this interesting aspect in the context of the papers cited above and their own model. Is this indeed a limitation of their model? If so, can the authors propose some way their FPT model can be improved in future work to account for the early decision-making process in cells?
(4) The title of the paper is in my opinion not a fair representation of the contents of this paper. The authors show that increasing the fluctuations in the absorbing barrier can reduce the MFPT and hence increase the fraction of cells killed. However, in order to achieve this, the production and degradation rates have to be simultaneously increased for the protein A. How can this be achieved realistically, in actual experiments? The authors do not provide any discussion on this important point. Indeed, to my knowledge, this may not be an easy task to achieve. In that case, suggesting that controlling the fluctuations in the threshold level might be a 'strategy' to combat fractional killing seems too strong a conclusion in my opinion. The authors should provide a discussion on the feasibility of this 'strategy' and also potentially change the title of the manuscript to better reflect their interesting, but theoretical findings.
Minor Points: (1) In many places throughout the manuscript, the authors use the terminology "fast fluctuations" and "slow fluctuations". Would it be more accurate to say "large fluctuations" and "small fluctuations"? By increasing the magnitude of the production and degradation rates of p53 or A, the amplitude of the noise increases, not how fast the fluctuations occur.
(2) On page 6 of the manuscript, the line "Note that the more threshold-crossing events are, the fewer cancer cells are killed…." is quite confusing. Did the authors mean "more" cancer cells are killed?
(3) On page 22, the authors say "However, the results in the case of timing variability is almost converse to those…". This statement seems incorrect to me after looking at Fig 5 -it seems to me that BOTH the MFPT and the timing variability are decreasing with increase of the p53 transcription rate.
(4) In the heatmaps shown, for example in Figures 5 and 6

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

Comments to the Author(s)
The authors have addressed the comments in the first report very well. The quality of this paper was improved substantially, in both research and writing. There is no further comments in this second report.

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

Recommendation?
Accept as is

Comments to the Author(s)
The authors have now satisfactorily responded to all points and have updated their manuscript accordingly. I thank the authors for changing the title of the manuscript to a more appropriate one. Overall I think this is a nice piece of work, and is now ready for publishing.

20-Jan-2020
Dear Dr Zhang, It is a pleasure to accept your manuscript entitled "Stochastic fluctuations in apoptotic threshold of tumor cells can enhance apoptosis and combat fractional killing" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
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Thank you for your fine contribution. On behalf of the Editors of Royal Society Open Science, we look forward to your continued contributions to the Journal. The authors have addressed the comments in the first report very well. The quality of this paper was improved substantially, in both research and writing. There is no further comments in this second report.
Reviewer: 2 Comments to the Author(s)