Growth-dependent concentration gradient of the oscillating Min system in Escherichia coli

  1. Claudia Parada
  2. Ching-Cher Sanders Yan
  3. Cheng-Yu Hung
  4. I-Ping Tu
  5. Chao-Ping Hsu
  6. Yu-Ling Shih

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Abstract

The Min system contributes to the spatiotemporal regulation of division sites in Escherichia coli . The MinD and MinE proteins of this system self-organize into oscillatory waves in the form of concentration gradients. How the intracellular Min protein concentration gradients are coordinated with cell growth to achieve spatiotemporal accuracy of cell division is unknown. Here, we report that the MinD concentration gradient becomes progressively steeper as cells elongate, suggesting that the division inhibitory activity at the midcell also decreases with cell growth. Interestingly, the oscillation period appears relatively stable across different cell lengths. Similar features were found in cells under carbon stress conditions, but the gradient was even steeper, likely favoring division at shorter cell lengths. The length-dependent variation of the concentration gradient was further examined in silico using a reaction-diffusion model, which not only supported the above features, but also revealed a decrease in the midcell concentration as the shape of the gradient becomes steeper in growing cells. This growth-dependent regulation of the midcell concentration of MinD may be coupled with the FtsZ ring formation through the MinD-interacting protein MinC. We found that the variable concentration gradients occur by coordinating the reaction rates of the recruitment of MinD and MinE to the membrane and the recharging of MinD with ATP in the cytoplasm. In conclusion, this work uncovers the plasticity of MinD concentration gradients during interpolar oscillations throughout cell growth, an intrinsic property integrated during cell division.

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    Reply to the reviewers

    A. General Statements

    We thank the reviewers for their constructive feedback. We have made significant revisions to the mathematical modelling section of the manuscript to address your concerns. Therefore, some of the specific issues and concerns raised in previous reviews no longer apply. Where that is the case, please see the relevant context in the revision as indicated in the point-by-point description section below. We summarize the key points in the revised manuscript as follows.

    1. The key finding of our study, involving experimental measurements and mathematical modelling, is plasticity in the MinD concentration gradient, which results from spatial differences in molecular interactions and is an intrinsic property of the Min system during cell growth. This study reveals not only the role of the MinD concentration gradient in modulating bacterial cell division site placement but also showcasing an example of cellular components in the form of a concentration gradient in fundamental cellular processes, a concept crucial in cell biology. This work provides conceptual advancement in a quantitative understanding of MinD oscillations in the cellular environment and provides implications for bacterial cell division regulation for further studies in the field.
    1. The reviewer requested clarification on the differences between our study and previous studies involving experimental measurements and mathematical modelling of Min oscillations in cells. We would like to emphasize that although the goal of the previous works was to measure the spatiotemporal distribution of oscillating MinD concentration gradients as a function of cell length, these works conceived the problem differently and therefore used different experimental designs and execution methods, which differentiates our key conclusions from theirs. This is also true for mathematical modelling. Although similar observations can be found in some respects, they are not directly comparable due to the different mathematics and assumptions used in the simulations. For example, our model was built to adequately investigate the biological question of the MinD concentration gradient during cell elongation but not to evaluate the impact of cell shape and confinement or the nucleation effect of MinD. Thus, our model cannot be generalized to other shapes, such as those observed in the study by Wu et al., 2015 (Wu et al, 2015). Therefore, we would like to draw attention to the experimental rigor and to the specific points and views that contribute to our understanding of Min systems. We now provide a comprehensive comparison between them in the Supplemental Information.
    1. We have re-run the simulation to refine and improve the modelling procedures and results, and the corresponding text and illustration are provided in the Results section of the main text (Lines 265-279, 614-653) and Fig. S6. In brief, we fixed the diffusion coefficients D_D and D_E from Meacci et al. (2006) (Meacci et al, 2006); the dissociation rate constant k_de from a previous simulation (Wu et al., 2015); and the experimentally measured MinD and MinE concentrations in this study. Meanwhile, the diffusion coefficients D_d and D_de were assumed values based on bacterial membrane protein diffusion (Schavemaker et al, 2018). This operation allowed us to probe for the general behaviours of the system. As a result, we were able to obtain a few parameter sets, including #2728, that generate features of the oscillation period, λN and I_Ratio, that highly mimic MinD oscillation in the cellular context (Figs. 4C, S7-9). We further tested the impact of different kinetic constants, k_de, k_dD, k_dE, k_D, and k(ADP→ATP), which represent different molecular interactions influencing the oscillation period, λ_N and I_Ratio (Fig 4D-H). Our findings have provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions. Furthermore, the modelling results help us understand the possible mechanisms associated with oscillation cycle maintenance and length-dependent variable concentration gradients.

    1. Regarding the inclusion or removal of results from more culture conditions, we decided to keep only one condition as in the previous version for the following reasons. In order to draw convincing conclusions, we consider it more important to characterize all aspects under the same growth condition and avoid manipulation. Therefore, the main conclusions are drawn from our experiments characterizing several aspects of MinD oscillations in cells growing with 0.4% glucose. In support of these observations, we decided to maintain only one other condition, 0.1% glucose. Further analysis of cells growing under other conditions will not change the main conclusions but will increase the difficulty of determining how the MinD concentration changes with cell growth.

    2. Studying the variable concentration gradient underlying the dynamic oscillations of the Min system may be of broad interest to cell biologists since the concentration gradient plays a fundamental role in various cellular processes, and the concept of concentration gradients is crucial in cell biology. Examples of related processes include passive and active transport, osmosis, cell signalling, and maintenance of cellular homeostasis. These processes allow cells to respond to their environment, regulate their internal conditions, and perform important functions required for survival and normal function. In addition, variable concentration gradients, characterized by the numerical descriptor λ_N and was reproduced in a simple mathematical model, demonstrate a nonlinear dynamics behaviour in physical biology. Therefore, the audience of this work can include the broader general audience of cell biology and physical biology rather than just the immediate specialized audience interested in the Min system. We will also reiterate the importance of specialized research, which often provides the basis for broader application and understanding.

    B. Point-by-point description of the revisions

    Reviewer #1 (Evidence, reproducibility and clarity (Required)):

    Summary: Parada et al. studied both experimentally and theoretically the MinD concentration distribution of Min waves during cell growth. The main finding was that (i) the gradient of MinD is steeper for longer cells and accordingly the MinD concentration at the middle of cell is lower, (ii) period of the oscillation is independent to the cell length, and (iii) those features are shared even under glucose starvation except the MinD gradient is steeper. (iv) Those results are supplemented by the analyses of the reaction-diffusion equations in which parameters that can reproduce the MinD concentration distribution are identified. I think the results are interesting; basically, as the cell grows, the contrast of the wave becomes clearer, such the MinD concentration at the cell centre decreases. The results may clarify the mechanism of FtsZ accumulation at the cell centre more quantitatively. The experiments were performed by measuring the fluorescent intensity of MinD during cell growth and analysing the intensity distribution along the long axis of the cell. The theoretical results were based on the analyses of the reaction-diffusion model. Both approaches are already well established and the results sound. Nevertheless, I do not think the novelty of this work is not well highlighted in the current manuscript; I think most of the results, except (iii) and (iv), have already been shown explicitly or implicitly in the previous studies. Min oscillations in a growing cell have been analysed both theoretically and experimentally in (Meacci 2005) and [1] (Fischer-Friedrich et al, 2010). The concentration distribution and period of the oscillation were measured. The complete results were presented in [2] (Meacci et al., 2006), and I am not aware of those results in scientific journals (the thesis is available online). Nevertheless, I think it is fair to cite those studies and compare the current results with them. In fact, in [2], it was shown that the concentration of MinD near the cell centre decreases as the cell grows, the total MinD concentration is approximately constant during the growth (therefore, the number of the molecules increases), and that the variance of the period becomes smaller as the cell grows. I do not think those previous studies spoil this work, and this work deserves publication somewhere. Still, the authors should highlight the novelty of this study more clearly.

    ANS: We thank the reviewer for recognizing the soundness of our experimental and theoretical approaches and results. The key finding of our study, involving experimental measurements and mathematical modelling, is plasticity in the MinD concentration gradient, which results from spatial differences in molecular interactions and is an intrinsic property of the Min system during cell growth. This study reveals not only the role of the MinD concentration gradient in modulating bacterial cell division site placement but also showcasing an example of cellular components in the form of a concentration gradient in fundamental cellular processes, a concept crucial in cell biology. We believe that the established techniques and methods are integral to a broad range of works and provide confidence in improving them and using them to test hypotheses and obtain results. We also appreciate the reviewer for pointing out that Meacci's PhD thesis entitled "Physical aspects of Min oscillations in Escherichia coli" (Meacci & Kruse, 2005) is available online for public access. This thesis, along with two publications (Meacci & Kruse, 2005) (Meacci et al., 2006), explored Min oscillations in growing cells and used mathematical models. These two published works are cited in the previous version of the manuscript because we agree that these earlier works provide valuable context. As recommended, we went through these works again and the work by Fischer-Friedrich et al. (2010) (Fischer-Friedrich et al., 2010) to compare their wet experiments and mathematical models with ours, which are detailed in the Supplemental Information (Lines 26-147). Here, we emphasize that although the published works and our work set the goal of measuring the spatiotemporal distribution of oscillating MinD concentration gradients as a function of cell length, we conceived the problem differently and therefore used different experimental designs and analysis approaches, which have led to the key conclusions that differentiate our work from theirs.

    Major comments: (i) In (Meacci 2005) and [1,2], it was claimed that the standard deviation of the period is comparable with the mean period, particularly for the shorter cell. Therefore, they did not claim the period is independent to the cell length. As far as I understood, the variance arises from the variance of the total protein concentration in the assemble of cells. I am wondering how the authors are able to conclude the constant period in different cell length. I also point out that in the theoretical part of (Meacci 2005), the period is, in fact, increasing as the cell grows and suddenly decreases at the length in which cell division occurs.

    ANS: In our experiments, we found that the oscillation periods ranged from 36.8 to 65.6 sec, as measured from a population of cells (length of 1.9-4.5 µm; main text, Fig. 1E). Moreover, the standard deviations of the period ranged from 5.4% to 34.8% of the period, with larger standard deviations more common in shorter cells (Fig. 1D), indicating that regular interpolar oscillations are more likely to occur in longer cells. This observation echoes the study by Fischer-Friedrich et al. (2010) (Fischer-Friedrich et al., 2010), who reported stochastic switching MinD oscillation between two cell poles in cells below 2.5 μm. MinD starts to oscillate regularly from pole-to-pole between 2.5-3 μm with an oscillation period of 80 sec. Above 3.5 μm, MinD invariably undergoes regular oscillation with an initial period of 87 sec and then decreases to 70 sec at the end. In their study, they focused on the length-dependent switching from stochastic to regular oscillation states and speculated that the amount of MinE bound to the membrane critically influenced the shift from stochastic to regular interpolar oscillations. In addition, their observation of a longer period at the initial phase and a shorter period after the cells grew beyond 3.5 μm somewhat coincided with our simulation results, as shown in Fig. 4C-H, left. In Meacci's work (Thesis: Figure 2.14; Meacci and Kruse (2005) (Meacci & Kruse, 2005): Figure 5(b)), the temporal oscillation periods were measured from 40 to 120 sec when focusing on cells with lengths similar to those in our measurements (black dots in Meacci's chart). Our measurements of oscillation periods clearly show much smaller fluctuations than those in Meacci's study and are more comparable to Fischer-Friedrich's measurements. Differences can arise across different bacterial strains and culture conditions that may significantly affect the amount and quality of protein expressed in individual studies. In short, all three works differ in terms of experimental design and execution. Although similar observations can be found in some aspects, they are not directly comparable. Therefore, we would like to draw attention to the experimental rigor and specific points and views that contribute to our understanding of the Min system. We have changed the wording from 'constant period' to 'fairly stable period' throughout the manuscript. This description is based on our experimental measurements (Fig. 1D, E) and is also supported by our mathematical modelling (Fig. 4C-H, left). In response to the statement from the theoretical model of (Meacci & Kruse, 2005): "the period is increasing as the cell grows and suddenly decreases at the length in which cell division occurs." First, our simulation results revealed a mild increase in the oscillation period during cell elongation (Fig. 4C). The increase is adjustable by varying the reaction rate constants in the simulation (Fig. 4D-H). Second, although we did not simulate dividing cells, our experimental measurements clearly showed that this period increased in newborn cells (Fig. S4). As mentioned above, although similar observations can be found in different studies, they are not directly comparable because the experiments were performed differently for different purposes. We have added comparison of different models in the Supplemental Information (Lines 26-147).

    (ii) I do not think the explanations of the reaction-diffusion model were well described. The authors mentioned that they studied a one-dimensional model and used the delta function to describe the membrane reaction. Did the authors study 1D cytosol and 0D membrane? Then, why the surface diffusion term exists in (4) and (5)? I believe the authors simply assumed that both the membrane and the cytosol are 1D (with larger diffusion constants for cytosolic Min concentrations). Then, the delta functions in (1)-(5) are not necessary. In (Wu 2015), the delta function was used in order to treat a 2D membrane embedded in 3D space.

    Besides that, there is no description of the initial conditions for the concentration fields to solve the reaction-diffusion equations. I think the description of the no-flux boundary condition is better put in the Methods rather than supplementary materials.

    ANS: Thank you for your suggestions to improve the description of the numerical model. As summarized below, we have rewritten this section of 'Simulating the dynamic MinD concentration gradient in growing cells' in the manuscript (Lines 237-279). We have specified the dimensionality of the rate and diffusion constants of each molecule, where applicable, in our 1D model from Lines 237-264. Their dimensionality can also be conceived from their units, as listed in Tables 2 and S4. We have specified the initial 'no-flux' boundary conditions in Lines 267, 630, and 647. We agree that the delta function is not necessary and have removed it from the equations.

    (iii) As in the previous comment, the current model did not take into account the geometry of the system; namely, cytosol is in 3D and membrane is on 2D. Recent theoretical studies can handle the effect, and also the effect of confinement. I would appreciate it if the authors would make a comment on whether those issues are relevant or not for the conclusion of this work.

    ANS: Thank you for pointing out this interesting aspect of cell geometry as investigated in Wu et al., 2015 (Wu et al., 2015). Our model is built to adequately describe changes in the MinD concentration gradient during cell elongation under the assumption that a 1D description is sufficient. Thus, our model cannot be generalized to other shapes, such as those observed in Wu et al., 2015 (Wu et al., 2015). This point is now commented upon in Supplemental Information, lines 120-123.

    (iv) I would appreciate it if the authors would describe the screening process more clearly. I did understand the first screening is a finite imaginary part and a positive real part at the first mode of spatial inhomogeneity in the eigenvalues. However, I did not understand the other processes clearly. The second screening is based on \lambda_N and I_Ratio, but its criteria is not clear. I think both quantities fluctuated in experimental results and I am not sure what to define numerical results match them. The third process is based on a fitting error using the fitting function of linear increase plus a constant. I am not sure why we need to exclude, for example, the bottom right example in Fig.S6 because it shows no oscillation until the cell length of 3um but then the gradient linearly increases. Please clarify how to justify the criteria. The same argument applies to the fourth screening process. It is not clear why the slope should be smaller than 2.

    ANS: Thank you for your suggestions to improve the description of the screening process. We have re-run the simulation to refine and improve the screening process, and the corresponding text and illustration are provided in the Results section of the main text (Lines 237-279, 614-653) and Fig. S6.

    (v) The authors claimed that the steeper gradient of MinD under glucose starvation results in cell division for shorter cells. I do not think the claim is convincing. It is necessary to measure the correlation between the length at the cell division and the gradient. It would also be nicer to show the correlation under other parameters. I think those studies truly support the authors' claim and the novelty of this work.

    ANS: Thank you for the comments. We would like to draw your attention to the right side of the graph shown in Fig. 3B, E, where measurements were obtained from cells prior to division. Our claim that "the steeper gradient of MinD under glucose starvation results in cell division for shorter cells" is also supported by the wave slope (λ_N range): 0.4% glucose of 1.49-2.66 (cell length range: 1.7-4.5 µm) and glucose starvation of 1.34-3.54 (cell length range: 2.1-3.8 µm). Therefore, under glucose starvation, λ_N increases more significantly with increasing length, allowing us to speculate on the contribution of steeper concentration gradient in stressed shorter cell to division. In the revised manuscript, the statement is kept in the Results section (Lines 217-218), but removed from the abstract. About the correlation between the concentration gradient and cell length at division under different conditions, we consider it more important to characterize all aspects under the same growth condition and avoid manipulation. In this study, the main conclusions are drawn from our experiments characterizing several aspects of MinD oscillations in cells growing with 0.4% glucose. In support of these observations, we decided to maintain only one other condition, 0.1% glucose. Further analysis of cells growing under other conditions will not change the main conclusions but will increase the difficulty of determining how the MinD concentration changes with cell growth.

    (vi) The conclusion at Line 346 "This plasticity arises from spatial differences in molecular interactions between MinD and MinE, as demonstrated..." looks unclear to me. My understanding is that (i) by screening the randomly sampled parameters in the reaction-diffusion model, the authors found the parameters that "match" experimental results, and (ii) the parameters after screening show the correlation between them (k_dD-k_dE and k_D-k_ATP->ADP). The logic heavily relies on the reaction-diffusion model is quantitatively correct. First, I think it is better to explain the logic more explicitly, that is, the claim of the molecular interaction is not based on the experimental facts. Second, I personally think the reaction-diffusion model used in this work does not reproduce quantitatively the experimental results, as discussed in (iii) and also (iv). Please make some discussions on how to justify the comparison between the model and experiments.

    ANS: Thank you for your constructive comments. To address these questions, we have re-run the simulation to refine and improve the results, and the corresponding text and illustration are provided in the Results section of the main text (Lines 237-279, 614-653) and Fig. S6. The kinetic parameters used in this study are described in the main text, lines 258-264: 'To randomly search for combinations of the parameter sets k_dD, k_dE, k_D, and k_(ADP→ATP), the following parameters were fixed in the simulation: the diffusion coefficients D_d and D_de were assumed values based on bacterial membrane proteins (Schavemaker et al., 2018), the diffusion coefficients D_D and D_E were from Meacci et al. (2006) (Meacci et al., 2006), and the dissociation rate constant k_de were from a previous simulation (Wu et al., 2015). This operation allowed us to probe for the general behaviours of the system.' Lines 277-279: 'This screening process reduced the parameter sets to 23, including set #2827, which, judging by the correlation plots for length vs. period, λ_N, and I_Ratio (Figs. S7-S9), showed features similar to those of the experimental data (Figs. 1E, 3B, C).' Based on the parameters of set #2827, we rigorously tested the impact of different kinetic constants that represent different molecular interactions on the oscillation period, λ_N and I_Ratio (Fig 4D-H). The results are described in the section of 'Effect of the kinetic rate constant on the MinD concentration gradient' of the main text, lines 323-349. This effort has provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions. In addition, a comparison between our modelling and experimental results is described in the main text, section 'In silico oscillation resembles oscillation in a cellular context', lines 300-321.

    (vii) I did not capture the point why the authors can claim "... further distinguishing in vivo and in vitro observations. " at Line 350. I did not find the results comparing with vitro studies. I would appreciate a demonstration of vitro results and/or references.

    ANS: To avoid confusion, this sentence has been removed.

    Minor comments: (1) Line 214: It should be "Fange and Elf".

    ANS: Line 238 in the revised manuscript: This has been corrected.

    (2) I think it is better to show sampled points in Fig. 4C and 4D to show how dense the authors sampled in the parameter space.

    ANS: Since we have rewritten this part, the suggested revision is no longer applicable.

    REFERENCES: [1] Fischer-Friedrich, Elisabeth / Meacci, Giovanni / Lutkenhaus, Joe / Chaté, Hugues / Kruse, Karsten, "Intra- and intercellular fluctuations in Min-protein dynamics decrease with cell length", Proceedings of the National Academy of Sciences, 107, 6134-6139 (2010). [2] Meacci, Giovanni, "Physical Aspects of Min Oscillations in Escherichia Coli", PhD thesis (2006) available at

    Reviewer #1 (Significance (Required)):

    General assessment: I think the strength of this study is that it potentially shows the quantitative correlation between the MinD concentration gradient during the oscillation and the cell length when it divides. However, the current data of glucose starvation is not convincing enough. The model parts are interesting but their connection to the experiments is not clear in the current manuscript.

    ANS: Thank you for your comment. The key finding of our study, involving experimental measurements and mathematical modelling, is plasticity in the MinD concentration gradient, which results from spatial differences in molecular interactions and is an intrinsic property of the Min system during cell growth. We hypothesized that if the plasticity of the MinD concentration gradient is an intrinsic property of the system, then this property would be robust and show consistent behaviour under different growth conditions. Therefore, we tested this hypothesis by studying MinD oscillations under a low-glucose condition, and the results strengthened the main conclusion derived from experiments under the regular growth condition containing 0.4 % glucose. We believe that further analysis of cells growing under other conditions will not change the main conclusions but may increase the difficulty of determining how the MinD concentration changes with cell growth. Therefore, we decide to make this section concise, containing only one additional condition, even though we have more data than presented here. As mentioned earlier in this response letter, we have re-run the simulation to refine and improve the results, and the corresponding text and illustration are provided in the Results section of the main text (Lines 237-279, 614-653) and Fig. S6. This operation allowed us to probe for the general behaviours of the system. As a result, we were able to obtain a few parameter sets, including #2728, that generate features of the oscillation period, λN and I_Ratio, that strongly mimic MinD oscillation in the cellular context (Figs. 4C, S7-9). We further tested the impact of different kinetic constants, k_de, k_dD, k_dE, k_D, and k(ADP→ATP), which represent different molecular interactions influencing the oscillation period, λ_N and I_Ratio (Figs. 4D-H). This effort has provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions.

    Advance: The advance of this study is to measure the MinD concentration gradient under glucose starvation, and to compare the experimental results with the (simplified) model under a wide range of parameters. I do not think the advance in the current manuscript looks conceptual level because the conceptual conclusions are not really convincing from the results. In this respect, the advance of this work may be technical.

    ANS: Thank you for this constructive comment and have responded as follows. In combination with both experimental and theoretical efforts in the revised manuscript, this work provides conceptual advancement in a quantitative understanding of MinD oscillations in the cellular environment and provides implications for bacterial cell division regulation for further studies in the field. Specifically, we would like to emphasize that this work revealed the inherent plasticity and adaptability of the MinD concentration gradient that contributes to division site selection. The mathematical modelling provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions.

    Audience: As a theoretician working on biophysics, including the model of the Min system, I think a specialised audience would be interested in this study. People who are studying the mechanism of the Min oscillation and resulting cell division, particularly those who are interested in both experiments and models, would be interested in this work. For the broad audience, I do not think the novelty of this study is well described.

    ANS: Thank you for your comment. We would like to point out that studying the variable concentration gradient underlying the dynamic oscillations of the Min system may be of broad interest to cell biologists since the concentration gradient plays a fundamental role in various cellular processes, and the concept of concentration gradients is crucial in cell biology. Examples include passive and active transport, osmosis, cell signalling, and maintenance of cellular homeostasis. These processes allow cells to respond to their environment, regulate their internal conditions, and perform important functions required for survival and normal function. In addition, the variable concentration gradient, characterized by the numerical descriptor λ_N and reproduced in a simple mathematical model, demonstrates a nonlinear dynamics behaviour in physical biology. Therefore, the audience of this work may include the broader general audience of cell biology and physical biology rather than just the immediate specialized audience interested in the Min system. We will also reiterate the importance of specialized research, which often provides the basis for broader application and understanding.

    Reviewer #2 (Evidence, reproducibility and clarity (Required)):

    Summary: This work by Parada et al showed that in the oscillatory Min System, MinD gradient was steeper in longer e.coli cells, while period was stable. This behavior was recapitulated in a mathematical model and it also revealed coordinated reaction rates in a wide range of parameter space.

    ANS: We thank the reviewer for the concise summary of our work.

    Major comments:

    1. There were some inconsistencies between experimental and modeling data. Wave slope (𝜆𝑁) plateaued at ~3um in the model but not shown in the experiment (Fig.3B). The period was much less in the model (Fig. S8) than in the experiment (Fig. 1B).

    ANS: Thank you for pointing out this problem. We have re-run the simulation to refine and improve the results, and the corresponding text and illustration are provided in the Results section of the main text (Lines 237-279, 614-653) and Fig. S6. This operation allowed us to probe for the general behaviours of the system. As a result, we were able to obtain a few parameter sets, including #2728, that generate features of the oscillation period, λ_N and I_Ratio, that highly mimic MinD oscillation in the cellular context (Figs. 4C, S7-9). Regarding oscillation period, the simulation result was shorter than the experimental measurements. Even though, based on the parameters of set #2827, we rigorously tested the impact of different kinetic constants that represent different molecular interactions on the oscillation period, λN and I_Ratio (Main text, lines 323-349; Fig 4D-H). This effort has provided us with a theoretical view of how oscillation features may be controlled by different molecular interactions. We found that the rate constants k_de, representing detachment of the MinDE complex from the membrane, and k(ADP→ATP), representing recharging of MinD-ADP with ATP, more significantly affected the oscillation period. The results suggested that the oscillation cycle time is tunable. In response to the question of the wave slope (λ_N) plateaued at ~3um in the modelling (Fig. 3B) but not shown in the experiment (Fig. 1D), we think this is due to experimental examination of a heterogenous population of cells versus simulating a growing bacterial cell. We came up with conclusions and hypotheses through wet experiments, which were further strengthened using mathematical modelling, providing insights into kinetic properties of the Min system.

    1. Generally, I found that the data of starved condition added little to the major message. Unless the model can recapitulate the even steeper gradient in such condition by tuning starvation-related parameters, it may be removed.

    ANS: We thank the reviewer for this suggestion. The key finding of our study, involving experimental measurements and mathematical modelling, is plasticity in the MinD concentration gradient, which results from spatial differences in molecular interactions and is an intrinsic property of the Min system during cell growth. We hypothesized that if the plasticity of the MinD concentration gradient is an intrinsic property of the system, then this property would be robust and show consistent behaviour under different growth conditions. Therefore, we tested this hypothesis by studying MinD oscillations under a low-glucose condition, and the results strengthened the main conclusion derived from experiments under the regular growth condition containing 0.4 % glucose. We agree that further analysis of cells growing under other conditions will not change the main conclusions but may increase the difficulty of determining how the MinD concentration changes with cell growth. Therefore, we decide to make this section concise, containing only one additional condition, even though we have more data than presented here.

    1. The authors need to compare what was different/novel between the model in this study and previous models such as Wu, et al 2015 and highlight the uniqueness of this work.

    ANS: Thank you for this suggestion. We now provide a comprehensive comparison between them in the Supplemental Information (Lines 26-147). We would like to emphasize that although the goal of the previous works was to measure the spatiotemporal distribution of oscillating MinD concentration gradients as a function of cell length, these works conceived the problem differently and therefore used different experimental designs and execution methods, which differentiates our key conclusions from theirs. This is also true for mathematical modelling. Although similar observations can be found in some respects, they are not directly comparable due to the different mathematics and assumptions used in the simulations. Therefore, we would like to draw attention to the experimental rigor and to the specific points and views that contribute to our understanding of Min systems.

    1. The model explored parameter space of reaction rates and found 60 sets. The KdE, KD, KdD, KADP-ATP ranged 6 orders of magnitude. It is interesting data in itself, but cells were not likely to vary that much for reaction rates. The relevance should be discussed.

    ANS: Thank you for pointing out this problem. For this revision, we re-ran the simulation to refine and improve the results, allowing us to identify parameter sets that generate features resembling the experimental measurements. Using set #2728 as an example, the variations in the five rate constants k_de, k_dD, k_dE, k_D, and k_(ADP→ATP) fall within a small range (Table 2, S4), eliminating the concern that arose from the previous version of the manuscript. We found that this parameter set allows for maximum utilization of MinD and MinE molecules, which are fixed in number according to experimental measurements, to drive membrane-associated oscillations in the simulation.

    Minor comments:

    1. Fig.1B colors were conflicting. The legend was different than diagram. Fig.1C no scale for x axis.

    ANS: We have resolved the colour conflict in Fig. 1B, and a time range has been added to Fig. 1C.

    1. Fig.S6A How the 638 oscillatory parameter sets were matched with experimental data and screened to 174 sets was not clear. Data of fitting errorANS: Thank you for your suggestions to improve the description of the screening process. In this revision, we have re-run the simulation to refine and improve the results, and the corresponding text and illustration are provided in the Results section of the main text (Lines 237-279, 614-653) and Fig. S6. This operation allowed us to probe for the general behaviours of the system. The mentioned filter no longer applies.

    2. Significant digits were not used properly. For example, the period (table 1) was showed as 46.00 sec, but the imaging interval was 12 sec, the 2 decimal digits were thus meaningless. The same argument goes for length measurement at 2.84 um, while the optical resolution of the microscope used should be no good than 200nm.

    ANS: We have corrected this significant digit throughout the manuscript.

    1. For scatter plot like Fig.1D-G, generally smaller dots would show trend more obvious.

    ANS: We have modified the plots and used smaller dots in Figs. 1D-G, 3B, C, E, F, S3D, and S5B, C.

    1. The molecular mechanism of why MinD gradient increases with length was not the scope of the current study, but better to be discussed.

    ANS: Let me address this comment in another way. The key finding of our study, involving experimental measurements and mathematical modelling, is plasticity in the MinD concentration gradient, which results from spatial differences in molecular interactions and is an intrinsic property of the Min system during cell growth. In the revised manuscript, we have re-run the simulation to refine and improve the modelling procedures and results, and the corresponding text and illustration are provided in the Results section of the main text (Lines 265-279, 614-653) and Fig. S6. In brief, we fixed the diffusion coefficients D_D and D_Efrom Meacci et al. (2006) (Meacci et al., 2006); the dissociation rate constant k_de from a previous simulation (Wu et al., 2015); and the experimentally measured MinD and MinE concentrations in this study. Meanwhile, the diffusion coefficients D_d and D_de were assumed values based on bacterial membrane protein diffusion (Schavemaker et al., 2018). This operation allowed us to probe for the general behaviours of the system. As a result, we were able to obtain a few parameter sets, including #2728, that generate features of the oscillation period, λN and I_Ratio, that highly mimic MinD oscillation in the cellular context (Figs. 4C, S7-9). We further tested the impact of different kinetic constants, k_de, k_dD, k_dE, k_D, and k(ADP→ATP), which represent different molecular interactions influencing the oscillation period, λ_N and I_Ratio (Fig 4D-H). Our findings have provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions. Furthermore, the modelling results help us understand the possible mechanisms associated with oscillation cycle maintenance and length-dependent variable concentration gradients.

    1. Fig. S8, why sudden jump in period in many of the sets of both groups?

    ANS: This supplemental figure is now Fig. S7. A slower oscillation at the initiation of oscillation appears to be a common property in our simulation.

    Reviewer #2 (Significance (Required)):

    Min system was well-studied oscillation mechanism to restrict FtsZ at cell center. Previous work has shown how the system work molecularly, simulated the behavior and reconstituted many different patterns in vitro. The major new information from this work was: 1. the rigorously measured endogenous level of MinD and MinE; 2. gradient increased with length; 3. a model recapitulated this relationship and explored parameter space of reaction rates. The paper was well presented, experiments and analysis were rigorous, and the conclusions were not overstated. It should interest specialized cell biologists studying cell size, oscillation pattern.

    ANS: Many thanks to Reviewer 2 for recognizing the contributions of our work to the understanding of the Min system and its role in cell division. We also thank you for identifying professional cell biologists studying cell size and oscillation patterns as readers of our paper. We would like to emphasize that cellular concentration gradients play a fundamental role in various cellular processes and that the concept of concentration gradients is crucial in cell biology. These concentration gradient-mediated processes allow cells to respond to their environment, regulate their internal conditions and perform important functions required for survival. In addition, the variable concentration gradient, characterized by the numerical descriptor λ_N and reproduced in a simple mathematical model, demonstrates a nonlinear dynamics behaviour in physical biology. Therefore, the audience of this work may include a broader audience in the field of cell biology and physical biology rather than just an immediate specialist audience. We will also reiterate the importance of specialized research, which often provides the basis for broader application and understanding.

    Reviewer #3 (Evidence, reproducibility and clarity (Required)):

    The manuscript shows that the concentration of MinD does not change during the division cycle of E. coli. Due to the oscillation pattern the concentration of MinD decreases at the mid-cell which makes it favorable for the division. The mid-cell decrease in concentration of MinD is majorly length dependent. The oscillation pattern is not due to the change in concentration of MinD, but due to the plasticity arises from the spatial differences in molecular interactions between MinD and MinE. The manuscript is well written, the experiments are performed carefully and the results will be of interest to readers from variety of field. However, there are several concerns need explanation.

    ANS: We greatly appreciate the positive feedback from the reviewer, and we address the specific concerns below.

    Major concerns: One of my major concern is these interactions are not shown experimentally but explained using either the previously published literature or mathematical models. Further, the previous literatures are shown on in vitro models which does not mimic the in vivo system fully.

    ANS: We thank the reviewer for the important point that reaction rates in previous studies and in our model of Min oscillations have not been experimentally tested. We are aware of the lack of experimental measurements, but these reaction rates cannot be measured in batch reactions using classical biochemical methods. To accurately measure these reaction rates, the experiments require advanced techniques and methods to handle spatial and temporal resolution, which is beyond the scope of our current study. However, in the revised manuscript, we have re-run the simulation to refine and improve the results, and the corresponding text and illustration are provided in the Results section of the main text (Lines 237-279, 614-653) and Fig. S6. In our simulation, we fixed the diffusion coefficients D_D and D_E from Meacci et al. (2006) (Meacci et al., 2006); the dissociation rate constant k_de from a previous simulation (Wu et al., 2015); and the experimentally measured MinD and MinE concentrations in this study. Meanwhile, the diffusion coefficients D_d and D_de were assumed values based on bacterial membrane protein diffusion (Schavemaker et al., 2018). This operation allowed us to probe for the general behaviours of the system. As a result, we were able to obtain a few parameter sets, including #2728, that generate features of the oscillation period, λN and I_Ratio, that highly mimic MinD oscillation in the cellular context (Figs. 4C, S7-9). Interestingly, we found that this parameter set allows for maximum utilization of MinD and MinE molecules, which are fixed numbers from experimental measurements, to drive membrane-associated oscillations in the simulation. We further tested the impact of different kinetic constants, k_de, k_dD, k_dE, k_D, and k(ADP→ATP), which represent different molecular interactions influencing the oscillation period, λ_N and I_Ratio (Figs. 4D-H). Our findings have provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions, and help us understand the possible mechanisms associated with oscillation cycle maintenance and length-dependent variable concentration gradients.

    The concentration of MinD does not change with the increasing length of the cell. Is the MinD concentration (or copy numbers) is different in the case of cells growing in low glucose and when compared to the cells growing at high glucose?

    ANS: Thank you for the comments. As shown in Figs. 2B, C, the concentration of MinD changed with cell length, but the number of MinD molecules per unit area did not change significantly with cell length. Although how the number of MinD molecules changes when cells are grown under low-glucose conditions is unclear, this number does not appear to be essential for the following reasons. We focused on studying Min oscillations during the normal growth cycle, minimizing experimental manipulations to analyse oscillation dynamics. Measurements of oscillations in cells grown under low-glucose conditions support the primary measurements. We think that further analysis of MinD concentration changes in growing cells under low-glucose conditions will not change the main conclusion of this manuscript: 'plasticity in the MinD concentration gradient is an intrinsic property of the Min system during cell growth',

    As per the current study a particular I-ratio at the mid-cell is required to initiate the cell division. In the case of cells growing at low glucose, how this required I-ratio is achieved at the mid-cell?

    ANS: Thank you for the excellent question. As described in the main text, lines 199-201, I_Ratio is defined as the ratio of the minimum intensity to the maximum intensity measured from the experimental data, which gradually decreases as the cell length increases (Fig. 3C). Since the minimum and maximum intensities were measured from the concentration gradient, which is characterized by the slope of the concentration gradient (λ_N), there exists a correlation between I_Ratio and λ_N. That is, a larger λ_N will result in a smaller I_Ratio, and vice versa. When comparing measurements made from cells grown with 0.4% and 0.1% glucose (Fig. 3B, C, E, F), the changes in λ_N are more drastic within a shorter length under low-glucose condition, which is accompanied by more drastic changes in I_Ratio. Furthermore, when the I_Ratio value was approximately 0.5, the corresponding cell length was significantly shorter under low-glucose condition. Therefore, we speculate that there may be an effective I_Ratio that is low enough for stable FtsZ ring formation. This effective I_Ratio can occur at any cell length, allowing us to see that bacteria divide at shorter cell lengths under low-glucose conditions. This property necessitates a faster reduction in the concentration gradient to reach the effective I_Ratio for cells dividing at shorter lengths. As a result, by adjusting λ_N as a function of length, the steepness of the I_Ratio reduction can be altered. Please see the main text, lines 389-406.

    There is decrease in the MinD oscillation time observed in low glucose condition. As explained by the authors the MinD oscillation is mainly guided by the FtsE induced removal of MinD from the membrane, how the authors can explain this decrease?

    ANS: Thank you for raising the question of how the MinE-induced detachment of membrane-bound MinD contributes to the oscillation time of MinD under low-glucose conditions. Although this is an interesting question, determining what regulates MinE-induced detachment of membrane-bound MinD under low-glucose conditions is beyond the scope of the current study. This unknown regulatory mechanism that regulates MinD-MinE interactions in growing cells under low glucose conditions is worthy of further investigation. However, our modelling results have provided a theoretical view of how oscillation features may be controlled by different molecular interactions between MinD and MinE and may guide future experiments investigating the underlying mechanism involved. Please refer to the Results section: 'Spatiotemporal distribution of the concentration gradient' in the main text, lines 351-373.

    Further, it is explained that the concentration of cellular ATP is in much higher concentration compared to the required amount for this oscillation. As the Iratio is majorly dependent on the cell length, what could be the reason for the differential N in the case of low and high glucose condition?

    ANS: Please refer to the previous answer to the question: 'As per the current study a particular I-ratio at the mid-cell is required to initiate the cell division. In the case of cells growing at low glucose, how this required I-ratio is achieved at the mid-cell?'. (this letter, Lines 764-779) In addition, our modelling in search of parameter sets that generate characteristics of MinD oscillation resembling oscillation in vivo allowed us to evaluate the impact of different molecular interactions, as represented by different rate constants (Fig. 4), which has provided important information for future mechanistic investigations, although not in the present study. Please see the Results section: 'Effect of the kinetic rate constant on the MinD concentration gradient' in the main text, lines 323-349.

    MinD is a highly insoluble protein. It also has an amphipathic helix and thus most of the time it binds to the membrane. The method used by the author to determine the cellular MinD concentration (mentioned in Fig S1) will only give the concentration of the soluble MinD and not of the total MinD. How the authors justify this as the total concentration. This is also the same in the case of MinE copy number calculation. Authors may need to perform the transcriptome analysis and compare both the data.

    ANS: We thank the reviewer for the comments. Since the attachment of MinD and MinE to the membrane is transient and MinD-membrane interactions require ATP, we expected that most of the protein would be released from the membrane into the cytoplasm after cell disruption, sufficiently representing the total MinD concentration. Furthermore, our measurements of molecule numbers are within the range of previous measurements (Di Ventura & Sourjik, 2011; Juarez & Margolin, 2010; Meacci & Kruse, 2005; Tostevin & Howard, 2006; Touhami et al, 2006). Thus, we believe that our current measurements are reliable and sufficient for subsequent interpretation.

    One of the main question asked by the authors in the abstract is. "How the intracellular Min protein concentration gradients are coordinated with cell growth to achieve spatiotemporal accuracy of cell division is unknown". Although the authors have shown that there is a change in concentration gradient during cell growth, the mechanism for the same is not very well explained. Authors have not provided any specific explanation for the increase in the velocity of the MinD oscillation and the gradient formation. How the velocity of MinD is increasing although there is no increase in the MinD concentration.

    ANS: We have changed 'the mechanism' to 'the exact way' in the abstract (Abstract, line 28). Moreover, in the revised manuscript, we have improved the mathematical model and performed a thorough investigation of the variations in the kinetic constants. This effort has provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions. The results may guide future experiments investigating the underlying mechanism involved. Please refer the answers to previous questions above.

    Figure 2B: shows the overall concentration of MinD in a single cell varies between 1180 - 1160 molecules/um2. In Fig 2C it is mentioned that mid-cell has a MinD concentration of 120-20 molecuels/ um2. Further, Fig3C and 3F shows I-ratio values varies between 0.6-0.4. Considering the values given the I-ratio (I min/ I max) should be between 0.1- 0.01. Authors need to explain the same. Figure 2C: The data in both the Y-axes are not matching and needs more clarification in the legend. Whether the number of molecules were counted only in the marked 200 nm area? If so, why the Y-axis 1 (molecules/um2) is decreasing 7 times, whereas, Y-axis 2 (molecules) is only by 2 times.

    ANS: In this work, we measured sfGFP-MinD intensity through fluorescence microscopy. The fluorescence intensity was converted into molecular numbers based on estimates from Western blot analyses (Fig. S1). This number of molecules for MinD and MinE was assumed to be the mean number, which was fit into the midpoint of the doubling time (Fig. 2B, black dashed line; main text, lines 166-167). Fig. 2C was obtained by further processing the same dataset to restrict the region of analysis to the midcell zone. Please refer to the main text, lines 158-178. However, the λ_N and I_Ratio values were calculated from the processed intensity data (Fig. S2; main text, lines 190-209, 533-559). Because of the conversion from intensity to molecule number in Figs. S2B, C and the image processing procedure applied to the calculation of λ_N and I_Ratio, it is not possible to directly compare the fold change and the upper and lower limits between molecule numbers and the λ_N and I_Ratio values.

    Other comments: Line 84: Requires reference for this statement.

    ANS: A recent review article has been added in the main text, line 84: '(Cameron & Margolin, 2024)'.

    Line 96: Can authors provide other evidence or validation for the determination of the copy numbers such as transcriptome analysis.

    ANS: We thank the reviewer for this suggestion. However, we believe that direct measurement of cellular protein abundance is reliable and sufficient for our purposes. Furthermore, transcriptome-measured RNA abundance does not translate directly to protein abundance in living cells because posttranscriptional processing, translation, posttranslational processing, and protein stability issues complicate the interpretation. Therefore, protein abundance measurement from cell extracts is straightforward for our purpose.

    Fig 1C: what is the units of time in Fig 1C? Is it equal for all the cell lengths?

    ANS: As described in the main text, lines 511-512, 'Time-lapse images of sfGFP-MinD were acquired at 12-sec intervals for 10 min or before the fluorescence diminished'. This condition is applied to all the acquired images in this work.

    Page 6, line 136-138: what could be the possible mechanism for change in velocity at different cell cycle time?

    ANS: To avoid confusion, we have modified the text and tone down the velocity when mentioned. This is because the mentioned velocity is inferred from the measured oscillation period and cell length but not from direct measurements; our emphasis is on understanding how the oscillation period remains fairly stable during cell growth rather than how the velocity changes. In the revised manuscript, we used modelling results to elucidate the possible mechanism related to period maintenance. The corresponding text and illustration are provided in the Results section (Lines 300-373) and the Discussion section of the main text (Lines 407-446) and Figs. 4, 5. In brief, this simulation allowed us to probe for general behaviours of the system, allowing us to obtain a few parameter sets that generate features of the oscillation period, λN and I_Ratio highly mimicking MinD oscillation in the cellular context (Fig 4C, S7-9). We further tested the impact of different kinetic constants, k_de, k_dD, k_dE, k_D, and k(ADP→ATP), which represent different molecular interactions influencing the oscillation period, λ_N and I_Ratio (Fig 4D-H). This effort has provided us with a solid theoretical view of how oscillation features may be controlled by different molecular interactions. Please see the Results section: 'Effect of the kinetic rate constant on the MinD concentration gradient' in the main text, lines 323-349.

    Page 7, line 155: Any evidence for claiming the same?

    ANS: The sentence has been modified as follows: 'Thus, the fairly stable oscillation period and variable velocity did not change the precision of the septum placement.' (Main text, lines 155-156)

    Page 7, line 156: Is there any proof authors can show that burst MinD synthesis occurs during the division? If not in the case of MinD, is it shown in any other protein?

    ANS: The text is now in line 168-171: 'Interestingly, the value after division was not doubled, which could indicate a balanced outcome between de novo synthesis and degradation or a burst of MinD synthesis at cell division followed by constant synthesis.' In previous studies by Männik et al. (2018) (Mannik et al, 2018) and Vischer et al. (2015) (Vischer et al, 2015), the division protein FtsZ increased the cellular concentration throughout the cell cycle under slow growth conditions and degraded rapidly at the end of the cell cycle, a process controlled by the ClpXP protease. Because we do not know the relevance of these observations to our study, which focused on the plasticity of the MinD concentration gradient, we decided not to discuss them in the manuscript.

    Page 9, line 217: The Fig 4A is not explained clearly and all the terms mentioned needs to be explained. This figure is used to explain the differential concentration of MinD at the poles and the mid-cell, thus needs to be explain more clearly.

    ANS: Thank you for your comments. Please refer to the above answer to the question: 'One of my major concern is these interactions are not shown experimentally but explained using either the previously published literature or mathematical models. Further, the previous literatures are shown on in vitro models which does not mimic the in vivo system fully.', in this letter, lines 691-715.

    Page 12, line 285: What is meaning of default speed of MinD oscillation in new-born cells? Do the authors observed any specific velocity in the new-born cells? What is the explanation for length dependent oscillation velocity for MinD?

    ANS: Thank you for the questions. As mentioned earlier, the emphasis of this study is on understanding how the oscillation period remains relatively stable while showing plasticity of the concentration gradient during cell growth. The velocity is inferred from the oscillation period and cell length but is not a direct measurement. To avoid confusion, we have modified the text and placed less emphasis on the velocity when mentioned.

    Reviewer #3 (Significance (Required)):

    General assessment: Major work of the manuscript is relying on the mathematical models, whereas the audience are majorly from the biology fields and thus simplified explanations are required in many places. Many of the legends in the figures require more explanation for better understanding. If possible more experimental data can be added, specifically to explain the model mentioned in figure 4A.

    ANS: We have modified the figure legends to include more explanations. As mentioned above, we have also revised Fig. 4 to include improvements in modelling results to better fit the experimental data and to examine the impacts of the kinetics constants of the reaction steps in the Min system. Please refer to lines 691-715 in this letter.

    Advance: The study is adding to the existing knowledge and will be helpful to fill the conceptual gaps in understanding the mid-cell MinD concentration and what may favor the initiation of bacterial division. Audience: Majorly the microbiology community will be interested in the study. This will also be interest to Physicists and mathematical persons working to understand bacterial division.

    ANS: We thank the reviewer for this positive comment.

    Reviewer #4 (Evidence, reproducibility and clarity (Required)):

    The study by Parada et al. illuminates the intricate interplay between Min proteins, exemplified by MinD, and cell growth in E. coli. Their findings demonstrate that the MinD concentration gradient steepens progressively as cells elongate, potentially influencing FtsZ ring formation via MinC. Moreover, their comprehensive reaction-diffusion model not only corroborates experimental observations of length-dependent concentration gradients but also underscores the critical role of kinetic interactions involving Min proteins, the membrane, and ATP. This elucidation significantly advances our understanding of the oscillatory mechanisms within the Min system. Both the experimental and simulation data are robust, and the manuscript is exceptionally well-written. I express my full support for publication pending the satisfactory resolution of the outlined concerns.

    ANS: We appreciate the reviewer's positive feedback and have addressed most issues to the best of our ability.

    1. Remove the dot in front of "Min" in line 57.

    ANS: This has now been removed.

    1. In lines 82-84, the statement "...The distribution of the division inhibitor MinC may be synchronized with spatiotemporal differences in MinD concentrations, leading to a stable placement of the FtsZ ring at the midcell..." suggests a potential synchronization between MinC and MinD oscillations. It is crucial to investigate if sfGFP-MinC exhibits similar concentration gradient oscillatory behavior in vivo as observed with MinD.

    ANS: Thank you for bringing up this question. The key finding of our study, involving experimental measurements and mathematical modelling, is plasticity in the MinD concentration gradient, which results from spatial differences in molecular interactions and is an intrinsic property of the Min system during cell growth. With many investigations already covered in this manuscript, we prefer to investigate sfGFP-MinC in future studies, which will have different focuses on how MinC dynamics are coupled with the variable MinD concentration gradient to directly impact FtsZ ring formation.

    1. Ensure consistent significant digits throughout the text. For instance, 1.95{plus minus}0.16 μM in line 97, 1.4{plus minus}0.13 μM in line 98, and 1.9 {plus minus} 0.2 μM in line 100 have varying precision. Consider using integers for molecules.

    ANS: We have corrected the significant digits in the main text and supplemental information.

    1. Address the discrepancy in expression levels of MinD and MinE between strain FW1541 and its parental strain W3110. Given the labeling effect, it is possible that MinD expression levels differ. However, MinC's expression level should be approximately the same. Conduct whole-genome sequencing of both strains to identify any additional mutations.

    ANS: Thank you for the comments. As described in the main text (Lines 67-70), the most important aspect is the concentration ratio between MinD and MinE. Although the numbers are not the same, they are comparable to those in previous studies (Hale et al, 2001; Li et al, 2014; Schmidt et al, 2016; Shih et al, 2002) (Main text, lines 113-115). Furthermore, we performed whole-genome sequencing of the W3110 and FW1541 strains. We confirmed that sfGFP was correctly inserted. The sequence alignment of the minCDE locus is provided for your reference but not for publication. Although there are some sporatic point mutations, there is no obvious reason to believe that the mutations would impact Min protein expression. We will organize the deposition data as soon as I can.

    1. Clarify the apparent discrepancy between lines 112 and 127. Line 112 suggests that the periodic regularity of interpolar oscillations increases with cell length, as demonstrated in Fig 1B-C, 1E, Fig S5. However, in the subsequent section (starting from line 127), the authors state that oscillation periods remain relatively stable across cells of different lengths. Provide clarification on this apparent discrepancy.

    ANS: Thank you for pointing out this confusion caused by misuse of the term. In Lines 122-123, the statement has been modified as follows: '...the uniformity of the oscillation intervals appears to increase with length...' In line 139, 'The oscillation period' refers to the time required for the oscillation cycle. Since the correction in line 123 should suffice to clarify, we did not modify the statement in line 139.

    1. Specify if the analysis was limited to non-constricted cells. If so, state this explicitly in the text, as it could impact the interpretation of results, especially in relation to the linear dependence of cell length on time before constriction, as shown in Fig S3C.

    ANS: We did not specifically remove those constricted cells, but cells before splitting were considered one cell. We have added a statement to clarify in Lines 144-145.

    1. Improve clarity in Fig 2A by using distinct colors (e.g., green and red) for differentiation on the Y-axis.

    ANS: The Y axes of Fig. 2A have been modified.

    1. Correct "of" to "from" in line 223 for improved clarity and accuracy.

    ANS: Corrected.

    1. Include the missing "A" in Fig S6A for completeness and accuracy.

    ANS: This figure has been updated.

    1. Ensure consistency in referencing style (full names versus short names) throughout the manuscript.

    ANS: This has now been done.

    Reviewer #4 (Significance (Required)):

    While numerous commendable in vitro studies have explored the oscillatory behavior of the Min system, this work uniquely delves into the oscillation of MinD within live cells. It unveils the remarkable coordination between intracellular Min protein concentration gradients and cell growth, shedding light on the precise spatiotemporal regulation of cell division.

    ANS: We thank the reviewer for this positive comment.

    References Di Ventura B, Sourjik V (2011) Self-organized partitioning of dynamically localized proteins in bacterial cell division. Molecular systems biology 7: 457 Fischer-Friedrich E, Meacci G, Lutkenhaus J, Chate H, Kruse K (2010) Intra- and intercellular fluctuations in Min-protein dynamics decrease with cell length. Proceedings of the National Academy of Sciences of the United States of America 107: 6134-6139 Hale CA, Meinhardt H, de Boer PA (2001) Dynamic localization cycle of the cell division regulator MinE in Escherichia coli. The EMBO journal 20: 1563-1572 Juarez JR, Margolin W (2010) Changes in the Min oscillation pattern before and after cell birth. Journal of bacteriology 192: 4134-4142 Li GW, Burkhardt D, Gross C, Weissman JS (2014) Quantifying absolute protein synthesis rates reveals principles underlying allocation of cellular resources. Cell 157: 624-635 Mannik J, Walker BE, Mannik J (2018) Cell cycle-dependent regulation of FtsZ in Escherichia coli in slow growth conditions. Molecular microbiology 110: 1030-1044 Meacci G, Kruse K (2005) Min-oscillations in Escherichia coli induced by interactions of membrane-bound proteins. Phys Biol 2: 89-97 Meacci G, Ries J, Fischer-Friedrich E, Kahya N, Schwille P, Kruse K (2006) Mobility of Min-proteins in Escherichia coli measured by fluorescence correlation spectroscopy. Phys Biol 3: 255-263 Schavemaker PE, Boersma AJ, Poolman B (2018) How Important Is Protein Diffusion in Prokaryotes? Front Mol Biosci 5: 93 Schmidt A, Kochanowski K, Vedelaar S, Ahrne E, Volkmer B, Callipo L, Knoops K, Bauer M, Aebersold R, Heinemann M (2016) The quantitative and condition-dependent Escherichia coli proteome. Nature biotechnology 34: 104-110 Shih YL, Fu X, King GF, Le T, Rothfield L (2002) Division site placement in E. coli: mutations that prevent formation of the MinE ring lead to loss of the normal midcell arrest of growth of polar MinD membrane domains. The EMBO journal 21: 3347-3357 Tostevin F, Howard M (2006) A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division. Phys Biol 3: 1-12 Touhami A, Jericho M, Rutenberg AD (2006) Temperature dependence of MinD oscillation in Escherichia coli: running hot and fast. Journal of bacteriology 188: 7661-7667 Vischer NO, Verheul J, Postma M, van den Berg van Saparoea B, Galli E, Natale P, Gerdes K, Luirink J, Vollmer W, Vicente M, den Blaauwen T (2015) Cell age dependent concentration of Escherichia coli divisome proteins analyzed with ImageJ and ObjectJ. Front Microbiol 6: 586 Wu F, van Schie BG, Keymer JE, Dekker C (2015) Symmetry and scale orient Min protein patterns in shaped bacterial sculptures. Nature nanotechnology 10: 719-726

  2. Review Commons

    Note: This preprint has been reviewed by subject experts for Review Commons. Content has not been altered except for formatting.

    Learn more at Review Commons

    Referee #4

    Evidence, reproducibility and clarity

    The study by Parada et al. illuminates the intricate interplay between Min proteins, exemplified by MinD, and cell growth in E. coli. Their findings demonstrate that the MinD concentration gradient steepens progressively as cells elongate, potentially influencing FtsZ ring formation via MinC. Moreover, their comprehensive reaction-diffusion model not only corroborates experimental observations of length-dependent concentration gradients but also underscores the critical role of kinetic interactions involving Min proteins, the membrane, and ATP. This elucidation significantly advances our understanding of the oscillatory mechanisms within the Min system. Both the experimental and simulation data are robust, and the manuscript is exceptionally well-written. I express my full support for publication pending the satisfactory resolution of the outlined concerns.

    1. Remove the dot in front of "Min" in line 57.
    2. In lines 82-84, the statement "...The distribution of the division inhibitor MinC may be synchronized with spatiotemporal differences in MinD concentrations, leading to a stable placement of the FtsZ ring at the midcell..." suggests a potential synchronization between MinC and MinD oscillations. It is crucial to investigate if sfGFP-MinC exhibits similar concentration gradient oscillatory behavior in vivo as observed with MinD.
    3. Ensure consistent significant digits throughout the text. For instance, 1.95{plus minus}0.16 μM in line 97, 1.4{plus minus}0.13 μM in line 98, and 1.9 {plus minus} 0.2 μM in line 100 have varying precision. Consider using integers for molecules.
    4. Address the discrepancy in expression levels of MinD and MinE between strain FW1541 and its parental strain W3110. Given the labeling effect, it is possible that MinD expression levels differ. However, MinC's expression level should be approximately the same. Conduct whole-genome sequencing of both strains to identify any additional mutations.
    5. Clarify the apparent discrepancy between lines 112 and 127. Line 112 suggests that the periodic regularity of interpolar oscillations increases with cell length, as demonstrated in Fig 1B-C, 1E, Fig S5. However, in the subsequent section (starting from line 127), the authors state that oscillation periods remain relatively stable across cells of different lengths. Provide clarification on this apparent discrepancy.
    6. Specify if the analysis was limited to non-constricted cells. If so, state this explicitly in the text, as it could impact the interpretation of results, especially in relation to the linear dependence of cell length on time before constriction, as shown in Fig S3C.
    7. Improve clarity in Fig 2A by using distinct colors (e.g., green and red) for differentiation on the Y-axis.
    8. Correct "of" to "from" in line 223 for improved clarity and accuracy.
    9. Include the missing "A" in Fig S6A for completeness and accuracy.
    10. Ensure consistency in referencing style (full names versus short names) throughout the manuscript.

    Significance

    While numerous commendable in vitro studies have explored the oscillatory behavior of the Min system, this work uniquely delves into the oscillation of MinD within live cells. It unveils the remarkable coordination between intracellular Min protein concentration gradients and cell growth, shedding light on the precise spatiotemporal regulation of cell division.

  3. Review Commons

    Note: This preprint has been reviewed by subject experts for Review Commons. Content has not been altered except for formatting.

    Learn more at Review Commons

    Referee #3

    Evidence, reproducibility and clarity

    The manuscript shows that the concentration of MinD does not change during the division cycle of E. coli. Due to the oscillation pattern the concentration of MinD decreases at the mid-cell which makes it favorable for the division. The mid-cell decrease in concentration of MinD is majorly length dependent. The oscillation pattern is not due to the change in concentration of MinD, but due to the plasticity arises from the spatial differences in molecular interactions between MinD and MinE. The manuscript is well written, the experiments are performed carefully and the results will be of interest to readers from variety of field. However, there are several concerns need explanation.

    Major concerns:

    One of my major concern is these interactions are not shown experimentally but explained using either the previously published literature or mathematical models. Further, the previous literatures are shown on in vitro models which does not mimic the in vivo system fully.

    The concentration of MinD does not change with the increasing length of the cell. Is the MinD concentration (or copy numbers) is different in the case of cells growing in low glucose and when compared to the cells growing at high glucose? As per the current study a particular I-ratio at the mid-cell is required to initiate the cell division. In the case of cells growing at low glucose, how this required I-ratio is achieved at the mid-cell? There is decrease in the MinD oscillation time observed in low glucose condition. As explained by the authors the MinD oscillation is mainly guided by the FtsE induced removal of MinD from the membrane, how the authors can explain this decrease? Further, it is explained that the concentration of cellular ATP is in much higher concentration compared to the required amount for this oscillation. As the Iratio is majorly dependent on the cell length, what could be the reason for the differential N in the case of low and high glucose condition? MinD is a highly insoluble protein. It also has an amphipathic helix and thus most of the time it binds to the membrane. The method used by the author to determine the cellular MinD concentration (mentioned in Fig S1) will only give the concentration of the soluble MinD and not of the total MinD. How the authors justify this as the total concentration. This is also the same in the case of MinE copy number calculation. Authors may need to perform the transcriptome analysis and compare both the data.

    One of the main question asked by the authors in the abstract is. "How the intracellular Min protein concentration gradients are coordinated with cell growth to achieve spatiotemporal accuracy of cell division is unknown". Although the authors have shown that there is a change in concentration gradient during cell growth, the mechanism for the same is not very well explained. Authors have not provided any specific explanation for the increase in the velocity of the MinD oscillation and the gradient formation. How the velocity of MinD is increasing although there is no increase in the MinD concentration. Figure 2B: shows the overall concentration of MinD in a single cell varies between 1180 - 1160 molecules/um2. In Fig 2C it is mentioned that mid-cell has a MinD concentration of 120-20 molecuels/ um2. Further, Fig3C and 3F shows I-ratio values varies between 0.6-0.4. Considering the values given the I-ratio (I min/ I max) should be between 0.1- 0.01. Authors need to explain the same. Figure 2C: The data in both the Y-axes are not matching and needs more clarification in the legend. Whether the number of molecules were counted only in the marked 200 nm area? If so, why the Y-axis 1 (molecules/um2) is decreasing 7 times, whereas, Y-axis 2 (molecules) is only by 2 times.

    Other comments:

    Line 84: Requires reference for this statement.

    Line 96: Can authors provide other evidence or validation for the determination of the copy numbers such as transcriptome analysis.

    Fig 1C: what is the units of time in Fig 1C? Is it equal for all the cell lengths?

    Page 6, line 136-138: what could be the possible mechanism for change in velocity at different cell cycle time?

    Page 7, line 155: Any evidence for claiming the same?

    Page 7, line 156: Is there any proof authors can show that burst MinD synthesis occurs during the division? If not in the case of MinD, is it shown in any other protein?

    Page 9, line 217: The Fig 4A is not explained clearly and all the terms mentioned needs to be explained. This figure is used to explain the differential concentration of MinD at the poles and the mid-cell, thus needs to be explain more clearly.

    Page 12, line 285: What is meaning of default speed of MinD oscillation in new-born cells? Do the authors observed any specific velocity in the new-born cells? What is the explanation for length dependent oscillation velocity for MinD?

    Significance

    General assessment: Major work of the manuscript is relying on the mathematical models, whereas the audience are majorly from the biology fields and thus simplified explanations are required in many places. Many of the legends in the figures require more explanation for better understanding. If possible more experimental data can be added, specifically to explain the model mentioned in figure 4A.

    Advance: The study is adding to the existing knowledge and will be helpful to fill the conceptual gaps in understanding the mid-cell MinD concentration and what may favor the initiation of bacterial division.

    Audience: Majorly the microbiology community will be interested in the study. This will also be interest to Physicists and mathematical persons working to understand bacterial division.

  4. Review Commons

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    Referee #2

    Evidence, reproducibility and clarity

    Summary:

    This work by Parada et al showed that in the oscillatory Min System, MinD gradient was steeper in longer e.coli cells, while period was stable. This behavior was recapitulated in a mathematical model and it also revealed coordinated reaction rates in a wide range of parameter space.

    Major comments:

    1. There were some inconsistencies between experimental and modeling data. Wave slope (𝜆𝑁) plateaued at ~3um in the model but not shown in the experiment (Fig.3B). The period was much less in the model (Fig. S8) than in the experiment (Fig. 1B).
    2. Generally, I found that the data of starved condition added little to the major message. Unless the model can recapitulate the even steeper gradient in such condition by tuning starvation-related parameters, it may be removed.
    3. The authors need to compare what was different/novel between the model in this study and previous models such as Wu, et al 2015 and highlight the uniqueness of this work.
    4. The model explored parameter space of reaction rates and found 60 sets. The KdE, KD, KdD, KADP-ATP ranged 6 orders of magnitude. It is interesting data in itself, but cells were not likely to vary that much for reaction rates. The relevance should be discussed.

    Minor comments:

    1. Fig.1B colors were conflicting. The legend was different than diagram. Fig.1C no scale for x axis.
    2. Fig.S6A How the 638 oscillatory parameter sets were matched with experimental data and screened to 174 sets was not clear. Data of fitting error<0.12 and slope<2 were filtered. Authors should explain the criterion for data filtering.
    3. Significant digits were not used properly. For example, the period (table 1) was showed as 46.00 sec, but the imaging interval was 12 sec, the 2 decimal digits were thus meaningless. The same argument goes for length measurement at 2.84 um, while the optical resolution of the microscope used should be no good than 200nm.
    4. For scatter plot like Fig.1D-G, generally smaller dots would show trend more obvious.
    5. The molecular mechanism of why MinD gradient increases with length was not the scope of the current study, but better to be discussed.
    6. Fig.S8, why sudden jump in period in many of the sets of both groups?

    Significance

    Min system was well-studied oscillation mechanism to restrict FtsZ at cell center. Previous work has shown how the system work molecularly, simulated the behavior and reconstituted many different patterns in vitro. The major new information from this work was: 1. the rigorously measured endogenous level of MinD and MinE; 2. gradient increased with length; 3. a model recapitulated this relationship and explored parameter space of reaction rates.

    The paper was well presented, experiments and analysis were rigorous, and the conclusions were not overstated. It should interest specialized cell biologists studying cell size, oscillation pattern.

  5. Review Commons

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    Referee #1

    Evidence, reproducibility and clarity

    Summary:

    Parada et al. studied both experimentally and theoretically the MinD concentration distribution of Min waves during cell growth. The main finding was that (i) the gradient of MinD is steeper for longer cells and accordingly the MinD concentration at the middle of cell is lower, (ii) period of the oscillation is independent to the cell length, and (iii) those features are shared even under glucose starvation except the MinD gradient is steeper. (iv) Those results are supplemented by the analyses of the reaction-diffusion equations in which parameters that can reproduce the MinD concentration distribution are identified.

    I think the results are interesting; basically, as the cell grows, the contrast of the wave becomes clearer, such the MinD concentration at the cell centre decreases. The results may clarify the mechanism of FtsZ accumulation at the cell centre more quantitatively. The experiments were performed by measuring the fluorescent intensity of MinD during cell growth and analysing the intensity distribution along the long axis of the cell. The theoretical results were based on the analyses of the reaction-diffusion model. Both approaches are already well established and the results sound. Nevertheless, I do not think the novelty of this work is not well highlighted in the current manuscript; I think most of the results, except (iii) and (iv), have already been shown explicitly or implicitly in the previous studies. Min oscillations in a growing cell have been analysed both theoretically and experimentally in (Meacci 2005) and [1]. The concentration distribution and period of the oscillation were measured. The complete results were presented in [2], and I am not aware of those results in scientific journals (the thesis is available online). Nevertheless, I think it is fair to cite those studies and compare the current results with them. In fact, in [2], it was shown that the concentration of MinD near the cell centre decreases as the cell grows, the total MinD concentration is approximately constant during the growth (therefore, the number of the molecules increases), and that the variance of the period becomes smaller as the cell grows. I do not think those previous studies spoil this work, and this work deserves publication somewhere. Still, the authors should highlight the novelty of this study more clearly.

    Major comments:

    (i) In (Meacci 2005) and [1,2], it was claimed that the standard deviation of the period is comparable with the mean period, particularly for the shorter cell. Therefore, they did not claim the period is independent to the cell length. As far as I understood, the variance arises from the variance of the total protein concentration in the assemble of cells. I am wondering how the authors are able to conclude the constant period in different cell length. I also point out that in the theoretical part of (Meacci 2005), the period is, in fact, increasing as the cell grows and suddenly decreases at the length in which cell division occurs.

    (ii) I do not think the explanations of the reaction-diffusion model were well described. The authors mentioned that they studied a one-dimensional model and used the delta function to describe the membrane reaction. Did the authors study 1D cytosol and 0D membrane? Then, why the surface diffusion term exists in (4) and (5)? I believe the authors simply assumed that both the membrane and the cytosol are 1D (with larger diffusion constants for cytosolic Min concentrations). Then, the delta functions in (1)-(5) are not necessary. In (Wu 2015), the delta function was used in order to treat a 2D membrane embedded in 3D space.

    Besides that, there is no description of the initial conditions for the concentration fields to solve the reaction-diffusion equations. I think the description of the no-flux boundary condition is better put in the Methods rather than supplementary materials.

    (iii) As in the previous comment, the current model did not take into account the geometry of the system; namely, cytosol is in 3D and membrane is on 2D. Recent theoretical studies can handle the effect, and also the effect of confinement. I would appreciate it if the authors would make a comment on whether those issues are relevant or not for the conclusion of this work.

    (iv) I would appreciate it if the authors would describe the screening process more clearly. I did understand the first screening is a finite imaginary part and a positive real part at the first mode of spatial inhomogeneity in the eigenvalues. However, I did not understand the other processes clearly. The second screening is based on \lambda_N and I_Ratio, but its criteria is not clear. I think both quantities fluctuated in experimental results and I am not sure what to define numerical results match them.

    The third process is based on a fitting error using the fitting function of linear increase plus a constant. I am not sure why we need to exclude, for example, the bottom right example in Fig.S6 because it shows no oscillation until the cell length of 3um but then the gradient linearly increases. Please clarify how to justify the criteria. The same argument applies to the fourth screening process. It is not clear why the slope should be smaller than 2.

    (v) The authors claimed that the steeper gradient of MinD under glucose starvation results in cell division for shorter cells. I do not think the claim is convincing. It is necessary to measure the correlation between the length at the cell division and the gradient. It would also be nicer to show the correlation under other parameters. I think those studies truly support the authors' claim and the novelty of this work.

    (vi) The conclusion at Line 346 "This plasticity arises from spatial differences in molecular interactions between MinD and MinE, as demonstrated..." looks unclear to me. My understanding is that (i) by screening the randomly sampled parameters in the reaction-diffusion model, the authors found the parameters that "match" experimental results, and (ii) the parameters after screening show the correlation between them (k_dD-k_dE and k_D-k_ATP->ADP). The logic heavily relies on the reaction-diffusion model is quantitatively correct. First, I think it is better to explain the logic more explicitly, that is, the claim of the molecular interaction is not based on the experimental facts. Second, I personally think the reaction-diffusion model used in this work does not reproduce quantitatively the experimental results, as discussed in (iii) and also (iv). Please make some discussions on how to justify the comparison between the model and experiments.

    (vii) I did not capture the point why the authors can claim "... further distinguishing in vivo and in vitro observations. " at Line 350. I did not find the results comparing with vitro studies. I would appreciate a demonstration of vitro results and/or references.

    Minor comments:

    1. Line 214: It should be "Fange and Elf".
    2. I think it is better to show sampled points in Fig.4C and 4D to show how dense the authors sampled in the parameter space.

    REFERENCES:

    [1] Fischer-Friedrich, Elisabeth / Meacci, Giovanni / Lutkenhaus, Joe / Chaté, Hugues / Kruse, Karsten, "Intra- and intercellular fluctuations in Min-protein dynamics decrease with cell length", Proceedings of the National Academy of Sciences, 107, 6134-6139 (2010).

    [2] Meacci, Giovanni, "Physical Aspects of Min Oscillations in Escherichia Coli", PhD thesis (2006) available at https://www.pks.mpg.de/fileadmin/user_upload/MPIPKS/group_pages/BiologicalPhysics/dissertations/GiovanniMeacci2006.pdf

    Significance

    General assessment:

    I think the strength of this study is that it potentially shows the quantitative correlation between the MinD concentration gradient during the oscillation and the cell length when it divides. However, the current data of glucose starvation is not convincing enough. The model parts are interesting but their connection to the experiments is not clear in the current manuscript.

    Advance:

    The advance of this study is to measure the MinD concentration gradient under glucose starvation, and to compare the experimental results with the (simplified) model under a wide range of parameters. I do not think the advance in the current manuscript looks conceptual level because the conceptual conclusions are not really convincing from the results. In this respect, the advance of this work may be technical.

    Audience:

    As a theoretician working on biophysics, including the model of the Min system, I think a specialised audience would be interested in this study. People who are studying the mechanism of the Min oscillation and resulting cell division, particularly those who are interested in both experiments and models, would be interested in this work. For the broad audience, I do not think the novelty of this study is well described.

  6. Version published to 10.1101/2023.08.01.551406v1 on bioRxiv