Homeostasis and systematic ageing as non-equilibrium phase transitions in computational multicellular organizations

Being a fatal threat to life, the breakdown of homeostasis in tissues is believed to involve multiscale factors ranging from the accumulation of genetic damages to the deregulation of metabolic processes. Here, we present a prototypical multicellular homeostasis model in the form of a two-dimensional stochastic cellular automaton with three cellular states, cell division, cell death and cell cycle arrest, of which the state-updating rules are based on fundamental cell biology. Despite the simplicity, this model illustrates how multicellular organizations can develop into diverse homeostatic patterns with distinct morphologies, turnover rates and lifespans without considering genetic, metabolic or other exogenous variations. Through mean-field analysis and Monte–Carlo simulations, those homeostatic states are found to be classified into extinctive, proliferative and degenerative phases, whereas healthy multicellular organizations evolve from proliferative to degenerative phases over a long time, undergoing a systematic ageing akin to a transition into an absorbing state in non-equilibrium physical systems. It is suggested that the collapse of homeostasis at the multicellular level may originate from the fundamental nature of cell biology regarding the physics of some non-equilibrium processes instead of subcellular details.


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

Comments to the Author(s)
Yuting et al present a model of multicellular homeostasis using a 2D stochastic cellular automaton framework. Although quite technical in places all the methods, including the master equations and mean field approximation are well presented and the assumptions made are appropriate. Overall, the development of the model is well explained and the exploration of the phase planes both statistically (Monte Carlo) and analytically (Mean Fields followed by Jacobian) is well done. The sharp transitions seen in the phase planes are not that surprising. Towards the end of the paper (Figure 9), the authors introduce realism into the model by including a threshold-dependent cell death they although the oscillating value for cell death is a little odd.
One point that I found unclear was the technical details of the correlation length (method behind Figure 7) but it makes sense that the agents will be highly correlated at the start of the simulation and at the end of the simulation if they undergo a quick extinction or a degradation as they seem to suggest.

28-May-2019
Dear Ms Lou On behalf of the Editors, I am pleased to inform you that your Manuscript RSOS-190012 entitled "Homeostasis and systematic aging as nonequilibrium phase transitions in computational multicellular organizations" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referee suggestions. Please find the referees' comments at the end of this email.
The reviewers and handling editors have recommended publication, but also suggest some minor revisions to your manuscript. Therefore, I invite you to respond to the comments and revise your manuscript.
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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. Comments to the Author(s) I enjoyed this paper, and think it is a sound approach to an interesting set of scientific problems.
One thing that caught my eye was that the phase portraits and system trajectories in section 3 were strongly reminiscent of this earlier work, which was not discussed or cited: Warren, P.B., 2009. Cells, cancer, and rare events: Homeostatic metastability in stochastic nonlinear dynamical models of skin cell proliferation. Physical Review E, 80(3), p.030903.
In particular, this work derives similar phase portraits and sketches of the system dynamics, including tumorigenesis as an attractor of biological systems that escape some basin of stability, representing homeostasis. While the current paper by Y. Lou et al. is a more substantial and detailed study of the same problems, I feel this paper would be enriched by a brief (2 sentence?) discussion of P.B. Warren 2009.
Otherwise I think their survey of the literature seems fine. The essential papers by D. Drasdo and B.D. Simons are all there.
The introduction of the model could also be described a bit more clearly. Something like "Cells switch at a rate proportional to the number of neighbours of type X" would help a lot. At present, the simple description is very short and the abstract mathematical description much longer, which has the effect of swamping the simple explanation. It is a simple and elegant model and I think this should be emphasised.

Reviewer: 2
Comments to the Author(s) Yuting et al present a model of multicellular homeostasis using a 2D stochastic cellular automaton framework. Although quite technical in places all the methods, including the master equations and mean field approximation are well presented and the assumptions made are appropriate. Overall, the development of the model is well explained and the exploration of the phase planes both statistically (Monte Carlo) and analytically (Mean Fields followed by Jacobian) is well done. The sharp transitions seen in the phase planes are not that surprising. Towards the end of the paper (Figure 9), the authors introduce realism into the model by including a threshold-dependent cell death they although the oscillating value for cell death is a little odd.
One point that I found unclear was the technical details of the correlation length (method behind Figure 7) but it makes sense that the agents will be highly correlated at the start of the simulation and at the end of the simulation if they undergo a quick extinction or a degradation as they seem to suggest.

04-Jun-2019
Dear Ms Lou, I am pleased to inform you that your manuscript entitled "Homeostasis and systematic aging as nonequilibrium phase transitions in computational multicellular organizations" is now accepted for publication in Royal Society Open Science.
You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience_proofs@royalsociety.org and openscience@royalsociety.org) to let us know if you are likely to be away from e-mail contact. Due to rapid publication and an extremely tight schedule, if comments are not received, your paper may experience a delay in publication.
Royal Society Open Science operates under a continuous publication model (http://bit.ly/cpFAQ). Your article will be published straight into the next open issue and this will be the final version of the paper. As such, it can be cited immediately by other researchers. As the issue version of your paper will be the only version to be published I would advise you to check your proofs thoroughly as changes cannot be made once the paper is published. We thank the editors and reviewer's comments on this manuscript and complete minor revisions on it. We have added some reference papers to the introduction part and revised the description of our model in a more straightforward manner (highlighted in yellow). Besides, we added some technical demonstrations on how to obtain the correlation length in Sec. 4.2 (highlighted in pink). We hope that these revisions improve the quality of this paper. The detailed answers to each point by the reviewers are listed as below.

On the reference of P.B. Warren 2009
One thing that caught my eye was that the phase portraits and system trajectories in section 3 were strongly reminiscent of this earlier work, which was not discussed or cited: Warren, P.B., 2009. Cells, cancer, and rare events: Homeostatic metastability in stochastic nonlinear dynamical models of skin cell proliferation. Physical Review E, 80(3), p.030903.
In particular, this work derives similar phase portraits and sketches of the system dynamics, including tumorigenesis as an attractor of biological systems that escape some basin of stability, representing homeostasis. While the current paper by Y. Lou et al. is a more substantial and detailed study of the same problems, I feel this paper would be enriched by a brief (2 sentence?) discussion of P.B. Warren 2009.

Answer:
We thank the reviewer for the recommendation of this paper. It is indeed very important and relevant to our research and so are the other related papers therein. We have added several sentences in the introduction as suggested (line 37-42) to enrich the background part.

On the description of model in Introduction
The introduction of the model could also be described a bit more clearly. Something like "Cells switch at a rate proportional to the number of neighbours of type X" would help a lot.
At present, the simple description is very short and the abstract mathematical description much longer, which has the effect of swamping the simple explanation. It is a simple and elegant model and I think this should be emphasised.

Appendix A
We take the suggestion here and try to make model description in Introduction easier to read (line 52-54). We thank the reviewer's appreciation on our model. Reviewer 2 (correspondent revisions highlighted in pink in the manuscript)

On the odd types of cell death
The authors introduce realism into the model by including a threshold-dependent cell death they although the oscillating value for cell death is a little odd.

Answer:
We are sorry for unclear demonstration on the motivation of introducing these modifications. These may not have accurate biological meaning and we propose them here only to demonstrate the model behaviour under some non-uniform and constant setting of the control parameters and provide some hints for more realistic extensions.
We add some explanations on our motivation in line 494-495.

On the technical details of the correlation length
One point that I found unclear was the technical details of the correlation length (method behind Figure 7) but it makes sense that the agents will be highly correlated at the start of the simulation and at the end of the simulation if they undergo a quick extinction or a degradation as they seem to suggest.

Answer:
We thank the reviewer for pointing out this problem of unclear demonstration on correlation length. The correlation length is calculated by fitting parameters in the correlation function (eq.10). The average  is taken over a period of homeostatic growth (where the numbers of proliferative and quiescent cells are fluctuating around some values before the system goes to extinct or degenerate) and over 40000 sessions of simulations. A very important thing is that correlation length is only measurable when the system is in the proliferative phase with infinite size (see the phase P in Fig.5(c) ).
The details of fitting procedures is as such: 1) by varying the control parameter d, one can find the critical points where the correlation function C(l) decays in a powerlaw function C(l)  l -α . 2)Measure the powers α of decay at the critical points. 3) Calculate the correlation function C(l) for the range of control parameter near this critical point and fit Eq.10 to obtain the correlation length ξ.
We make the technical demonstration on correlation length clearer in the Sec. (line 421-423 and line 431-435) and hope this revision may improve the readability of this part.