Physical basis of the cell size scaling laws

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    This theoretical work deals with the problem of homeostasis of protein density within cells, relying on the Pump and Leak model. The model makes predictions both for growing and senescent cells, which they compare to experimental data on budding yeast. The work extends previous works and makes biologically-relevant predictions, which will be of interest to both theorists and experimentalists interested in cell physiology.

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Cellular growth is the result of passive physical constraints and active biological processes. Their interplay leads to the appearance of robust and ubiquitous scaling laws relating linearly cell size, dry mass, and nuclear size. Despite accumulating experimental evidence, their origin is still unclear. Here, we show that these laws can be explained quantitatively by a single model of size regulation based on three simple, yet generic, physical constraints defining altogether the Pump-Leak model. Based on quantitative estimates, we clearly map the Pump-Leak model coarse-grained parameters with the dominant cellular components. We propose that dry mass density homeostasis arises from the scaling between proteins and small osmolytes, mainly amino acids and ions. Our model predicts this scaling to naturally fail, both at senescence when DNA and RNAs are saturated by RNA polymerases and ribosomes, respectively, and at mitotic entry due to the counterion release following histone tail modifications. Based on the same physical laws, we further show that nuclear scaling results from a osmotic balance at the nuclear envelope and a large pool of metabolites, which dilutes chromatin counterions that do not scale during growth.

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  1. This Zenodo record is a permanently preserved version of a PREreview. You can view the complete PREreview at

    In the paper Cell size scaling laws: a unified theory by Rollin et al., a tuned simplified Pump-Leak model (PLM) in combination with order of magnitude estimations serves as physical basis for the derivation of various cell size scaling laws, a highly relevant biological topic with medical applications.

    The first sections introduce the simplified PLM together with the necessary assumptions and present the results of the order of magnitude estimations, highlighting the dominance of metabolites in the wet volume and the dominance of proteins in the dry volume.

    Under additional consideration of a growth model and an amino-acid biosynthesis model, the authors find a constant dry mass density during growth and dilution at senescence.

    A similar phenomenon with a different origin, mitotic swelling, can be explained taking into account Manning condensation.

    The last three sections deal with reasonable extensions and considerations that allow the authors to explain nuclear scaling and to identify influential parameters of the NC ratio, in particular the role of metabolites. Apart from relying on the simplified PLM, these sections on nuclear scaling are self-contained and therefore, given their abundance of insights, potentially more suitable for a separate research paper.

    Embedded in an excellent structure, and based on an introduction that is accessible for a general scientific audience, the authors carefully introduce simplifications and justifications. While physical basics seem to be properly explained for biologists, this does not always apply for the opposite case. Biological explanations are in places insufficiently provided to readers from a physics background, indicating that physicists are not the primary target group of this paper. At least in the discussion, more biological intuition is provided (e.g., the Sec. The nucleoskeletal theory). The authors' efforts to make explanations of their results intuitive are clearly visible but not in every aspect successful. That does not diminish the understanding of the authors' conclusions which are well supported and properly explained.

    Major comments

    • Schematic drawings of biological processes are barely understandable without the caption. I would strongly encourage to add legends to Figs. 1A, 2A, 4A to be able to understand the presented mechanisms at one glance.

    • To avoid confusion, it should be specified that the physical dimension of dry mass is volume.

    • In Fig. 2D, errorbars are shown together with more than one data point for each time. In general, errorbars should only be displayed together with the mean value. It is unclear whether the mean value is displayed. If so, it should be highlighted, for example in boxplot style. Otherwise, not an error bar, but a violin style plot would be appropriate.

    • Fig. 4B displays simulation results with parameters from experiments. This is correctly described in the respective caption. However, in the main text, the authors ambiguously refer to this figure in the context of an experimental verification.

    • Although Fig. 4B is certainly capable of showing a correlation between variations of NC ratio and variations of the NEP, mirroring one plot is a detour. Computing and/or displaying the cross correlation between the slopes would be more informative and support this statement more directly.

    • To conceptually link the Sec. Mitotic swelling with the following ones, it should already in Sec. Mitotic swelling be specified when effects concerning the nucleus (i.e., the relevance of chromatin) are considered, especially for non-biologist readers.

    • In Sec. Mitotic swelling, the 5th defining feature (nuclear envelope breakdown) is not sufficiently explained. Without further elaboration, it remains unclear why one could expect it to be an explanation for mitotic swelling.

    • The idea to include explanatory sections, e.g., on Manning condensation in the appendix, is very helpful.

    • When reporting that the actual NC ratio lies closer to NC1 than to NC2, the conclusion that metabolites are relevant since their role is considered in NC1 appears obvious but is not proven in the main text. The authors should explicitly point out that there is no underlying nonlinearity governing the relation between the influence of metabolites and the NC ratio, if necessary, by referring to additional calculations.

    Minor comments

    • Symbols that have superscripted subscripts, such as z_{A^f} , are hard to decipher, especially in nested fractions, e.g., Eq. 5.

    • The inset in Fig. 4D is too small to be properly readable in print.

    Competing interests

    The author declares that they have no competing interests.

  2. eLife assessment

    This theoretical work deals with the problem of homeostasis of protein density within cells, relying on the Pump and Leak model. The model makes predictions both for growing and senescent cells, which they compare to experimental data on budding yeast. The work extends previous works and makes biologically-relevant predictions, which will be of interest to both theorists and experimentalists interested in cell physiology.

  3. Reviewer #1 (Public Review):

    The authors made some biologically reasonable approximations of the Pump and Leak model. e.g., assuming the alpha_0 parameter to be zero. These approximations significantly simplify the model and make the results much more intuitive, e.g., Eq. 4 in the main text. The authors proposed an interesting and simple model of amino acid production, which is argued to be the primary determinant of cell volume. Combined with the gene expression model proposed recently by Lin and Amir, their model can nicely explain the homeostasis of protein density. Furthermore, by considering the saturation of DNA and mRNA by RNA polymerase and ribosome, the authors extended Lin and Amir's model by introducing protein degradation, which I think is the key to explaining cytoplasm dilution. The authors also discussed other applications of their model, including mitotic swelling and nuclear scaling. Below are my major comments:

    1. Eq. 2 is valid for stationary states where the cell volume is constant with time. However, many cells grow and divide, including yeast cells. I think the authors have implicitly neglected the effects of cell growth. The authors may want to mention this explicitly to avoid confusion.

    2. It's unclear how the authors go from Eq. S.21 to Eq. 2, although the authors mentioned it is straightforward. I think the dilute solution assumption is used without explicit mention, at least in section A of the SI.

    3. A slight deviation from equilibrium is implicitly assumed in Eq. S.22 I think since the flow is linearly proportional to the chemical potential difference. The authors may want to mention this explicitly since the linear assumption is not necessarily true for biological systems.

    4. A more general gene expression model is recently proposed by some of the authors of Ref. 30, in which the saturation of DNA by RNAPs is due to a high free RNAP concentration near the promoter (Wang and Lin, Nature Communications, 2021). I think the exact saturation mechanism is not very important to the conclusions. Still, I think it's good to let readers be aware that there are biologically more realistic saturation mechanisms.

    5. The success of the fitting in Figure 2E is intriguing but may not be a smoking gun evidence of the model's validity. All one needs is a protein number proportional to cell volume for tt**, as far as I understand. Alternative models incorporating the above features will be able to reproduce the fitting of Figure 2E as well, I think. For example, instead of adding protein degradation, one can alternatively assume that protein translation becomes much slower for t>t**, but amino acids are still produced at a constant rate. The time-dependences of amino acids and cell volume may not be important if one just wants to fit the data in Figure 2E since the cell volume dynamics are extracted from Figure 2B. The authors may want to discuss this point.

    6. On line 752, the estimation of the average charge of proteins is unclear to me. How did the authors obtain z_p = 0.8?

  4. Reviewer #2 (Public Review):

    The manuscript proposes a theoretical framework for the size scaling of cells. The main predictions are (1) the application of a nested pump-leak model to explain cell size scaling through an osmotic balance, (2) the role of metabolites in maintaining electroneutrality, and (3) the breakdown of this scaling law during specific phases of cell growth and senescence.

    Although the overall topic and approach are of significant interest, there are several issues with the presentation and claimed scope, detailed below.

    Major comments:

    1. The manuscript claims to provide a unified theory of cell size scaling, but quantitative agreement is only shown in a few specific cases (non-dividing yeast cells, mitotic swelling in mammalian cells, nuclear size scaling). Given the significant number of adjustable parameters in the model, the claim of a unified theory seems to be somewhat of a stretch. In addition, many of the approximations used (such as turgor pressure being negligible on p. 5) are valid in mammalian cells, but not in plant or yeast cells. For example, in walled cells, the rate of volume growth is dictated largely by cell-wall synthesis and turgor pressure (Rojas and Huang, 2018).

    2. The paper claims to supersede previous work: "Many theoretical papers have assumed a priori a linear phenomenological relation between volume and protein number in order to study cell size [30],[31],[32]. Our results instead emphasize that the proportionality is indirect, only arising from the scaling between amino-acid and protein numbers." However, the conclusions reached (e.g. NC1 in eq. 15) appear to recover those of previous work, at least in certain limiting cases. Moreover, this is not a fully accurate description of the previous work, since in some of the previous works the osmotic balance is given in terms of general macromolecules, not necessarily proteins, and the linear relationship was not assumed but rather derived based on osmotic balance. The authors should carefully explain the relationship of their work to the previous studies.

    3. The role of metabolites is an important point that should be further clarified. The authors state that "As a key consequence, we find that the NC ratio would be four times larger in the absence of metabolites". However, the formula obtained in the metabolite-dominated limit for NC1 in eq. 15 recovers previous results which were based solely on osmotic balance, without accounting for electroneutrality via metabolites. Why is electroneutrality violated in the absence of metabolites? Does this remain true if the chromatin and counterions are considered to be polyelectrolytes?

    4. Appendix H on the extension to scaling of other organelles contains no comparison to data. Is the size control of all membrane-bound organelles expected to behave according to the same principles, or is the theory applicable to a particular subset of organelles?

    5. It is stated several times that the size cell is "tightly regulated by active processes". The authors should define what they mean by "control" and "active" in this context. For example, one interpretation of the NC ratio size scaling result is that it is not under direct control, but rather is a consequence of the ratio of nuclear-bound proteins and is only controlled indirectly. (The authors themselves state that the relationship between volume and protein number is indirect.) If the NC ratio is actively controlled, this suggests that its maintenance at a certain value is important for the proper functioning of the cell. Is there evidence of this, or would the cell continue to function if the nuclear size could hypothetically be perturbed independently of the protein ratio?