Physical basis of the cell size scaling laws

Our theoretical approach to understand the various scaling laws associated to cell size is based on the Pump-Leak model (Tosteson and Hoffman, 1960 and Figure 1A). The Pump-Leak model is a coarse-grained model emphasizing the role of mechanical and osmotic balance. The osmotic balance involves two types of osmolytes, impermeant molecules such as proteins and metabolites, which cannot diffuse through the cell membrane, and ions, which cross the cell membrane and for which at steady state the incoming flux into the cell must equate the outgoing flux. For simplicity, we restrict ourselves to a two-ion Pump-Leak model where only cations are pumped outward of the cell. We justify in the ‘Discussion’, section ‘Physical grounds of the model’, why this minimal choice is appropriate for the purpose of this article. Within this framework, three fundamental equations determine the cell volume. (1) Electroneutrality: the laws of electrostatics ensure that in any aqueous solution such as the cytoplasm, the solution is neutral at length scales larger than the Debye screening length, that is, the electrostatic charge of any volume larger than the screening length vanishes. In physiological conditions, the screening length is typically on the nanometric scale. Therefore, the mean charge density of the cell vanishes in our coarse-grained description (Equation A.1). (2) Osmotic balance: balance of the chemical potential of water inside and outside the cell; the timescale to reach the equilibrium of water is of few seconds after a perturbation (Venkova et al., 2022; Milo and Phillips, 2015). Note that the second part of the equality in Equation 2 assumes that both the cellular and outer media are dilute solutions. This assumption may seem odd since the cytoplasm is crowded. Yet, we justify this approximation in the ‘Discussion’ (see section ‘Physical grounds of the model’). (3) Balance of ionic fluxes: the typical timescales of ion relaxation observed during a cell regulatory volume response after an osmotic shock are of the order of a few minutes (Venkova et al., 2022; Hoffmann et al., 2009). Together, this means that our quasi-static theory is designed to study cell volume variations on timescales larger than a few minutes. We emphasize that although Equation 2 and Equation 3 are stationary equations, the volume of the cell may vary, for example, through the biosynthesis of impermeant molecules $X$ (see Equation 4), but this is a slow process at the timescale of equilibration of water and ion fluxes and we use a quasistatic approximation. Mathematically, the three equations read (see Appendix 1, section 1.1 for the full derivations of these equations):

Figure 1

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where, $n^{+}$, $n^{-}$ are respectively the ionic concentrations of positive and negative ions inside the cell. The concentrations of cations and anions are equal in the external medium to enforce electroneutrality since the concentrations of non-permeant molecules in the outer medium are typically much lower than their ionic counterparts (Milo and Phillips, 2015). We call ${\displaystyle n_0}$ the corresponding concentrations, and the associated osmotic pressure in the outer medium reads ${\mathrm{\Pi }}_0=k_{B}T\cdot 2n_0$. The cell is modeled as a compartment of total volume ${\displaystyle n_0}$ divided between an excluded volume occupied by the dry mass $R$ and a wet volume. The cell contains ions and impermeant molecules such as proteins, RNA, free amino acids, and other metabolites. The number $X$, respectively the concentration $x=\frac{X}{V-R}$, of these impermeant molecules may vary with time due to several complex biochemical processes such as transcription, translation, plasma membrane transport, and degradation pathways. The average negative charge $-z$ of these trapped molecules induces a Donnan potential difference $U_c$ across the cell membrane. The Donnan equilibrium contributes to the creation of a positive difference of osmotic pressure $\mathrm{\Delta }\mathrm{\Pi }$ that inflates the cell. Cells have two main ways to counteract this inward water flux. They can either build a cortex stiff enough to prevent the associated volume increase, as done by plant cells. This results in the appearance of a hydrostatic pressure difference $\mathrm{\Delta }P$ between the cell and the external medium. Or they can pump ions outside the cell to decrease the internal osmotic pressure, a strategy used by mammalian cells. We introduce a pumping flux of cations $p$. Cations can also passively diffuse through the plasma membrane via ion channels with a conductivity $g^{+}$. In Equation 3, the pumping efficiency is measured by the dimensionless number ${\alpha }_0=e^{-\frac{p}{k_{B}T{g}^{+}}}$ where $T$ is the temperature and $k_B$ the Boltzmann constant. The pumping efficiency varies between ${\displaystyle 0}$ in the limit of ‘infinite pumping’ and ${\displaystyle 1}$ when no pumping occurs (see Appendix 1, section 1.1 for an explanation on the origin of this parameter).

Although proposed more than 60 years ago (Tosteson and Hoffman, 1960) and studied in depth by mathematicians (Mori, 2012) and physicists (Kay, 2017), little effort has been done to precisely map the coarse-grained parameters of the Pump-Leak model to microscopic parameters. We adopt here the complementary strategy and calculate orders of magnitude in order to simplify the model as much as possible, only keeping the leading order terms. We summarize in Table 1 the values of the Pump-Leak model parameters that we estimated for a ‘typical’ mammalian cell. Three main conclusions can be drawn: (1) pumping is important, as indicated by the low value of the pumping efficiency ${\alpha }_0\sim 0.14$. Analytical solutions presented in the main text will thus be given in the ‘infinite pumping’ limit, that is, ${\alpha }_0\sim 0$, corresponding to the scenario where the only ions present in the cell are the counterions of the impermeant molecules (see Appendix 1, section 2.1 for the general solutions). Though not strictly correct, this approximation gives a reasonable error of the order 10% on the determination of the volume due to the typical small concentration of free anions in cells (Table 1). This error is comparable to the typical volumetric measurement errors found in the literature. (2) Osmotic pressure is balanced at the plasma membrane of a mammalian cell since hydrostatic and osmotic pressures differ by at least three orders of magnitude. This result implies that even though the pressure difference $\mathrm{\Delta }P$ plays a significant role in shaping the cell, it plays a negligible role in fixing the volume (see Equation A.15 for justification). (3) The cellular density of impermeant species is high, $x\sim 120\mathrm{m}\mathrm{M}\mathrm{o}\mathrm{l}$, comparable with the external ionic density ${\displaystyle n_0}$.

In this limit, the volume of the cell hence reads (the complete expression is given in Appendix 1, section 2.1):

The wet volume of the cell is thus slaved to the number of impermeant molecules that the cell contains. While this conclusion is widely acknowledged, the question is to precisely decipher which molecules are accounted for by the number X. We first estimate the relative contributions of the cellular-free osmolytes to the volume of the cell and then compute their relative contributions to the dry mass of the cell. We provide a graphical summary of our orders of magnitudes in Figure 1B and C as well as the full detail of their derivations in Appendix 1, section 3.1. The conclusion is twofold. Metabolites and their counterions account for most of the wet volume of the cell, 78% of the wet volume against 1% for proteins. On the other hand, proteins account for most of the dry mass of mammalian cells, accounting for 60% of the cellular dry mass against 17% for metabolites.

We further note that metabolites are mainly amino acids and in particular three of them, glutamate, glutamine, and aspartate, account for 73% of the metabolites (Park et al., 2016). It is important to note that the relative proportion of free amino acids in the cell does not follow their relative proportion in the composition of proteins. For instance, glutamate represents 50% of the free amino acid pool while its relative appearance in proteins is only 6% (King and Jukes, 1969). This is evidence that some amino acids have other roles than building up proteins. In particular, we demonstrate throughout this article their crucial role on cell size and its related scaling laws.

These conclusions may appear surprising due to the broadly reported linear scaling between cell volume (metabolites) and dry mass (proteins), hence enforcing a constant dry mass density $\rho$ during growth. Theoretical papers often assume a linear phenomenological relation between volume and protein number in order to study cell size (Lin and Amir, 2018; Wu et al., 2022). Our results instead emphasize that the proportionality is indirect, only arising from the scaling between metabolites (mostly amino acids) and proteins. The dry mass density reads (to lowest order):

where ${\mathcal{M}}_a$, $z_{A^f}$ , and $A^f$ are respectively the average mass, charge, and number of amino acids; ${\displaystyle l_p}$, ${\displaystyle v_p}$_, and $P_{tot}$ the average length, excluded volume, and number of proteins. Note that density homeostasis is naturally achieved in the growth regime where $A^f$ is proportional to $P_{tot}$.

To further understand the link between amino acid and protein numbers, we build upon a recent model of gene expression and translation (Lin and Amir, 2018 and Figure 2A). The key feature of this model is that it considers different regimes of mRNA production rate ${\dot{M}}_j$ and protein production rate ${\dot{P}}_j$ according to the state of saturation of respectively the DNA by RNA polymerases (RNAPs) and mRNAs by ribosomes. For the sake of readability, we call enzymes both ribosomes and RNAPs, their substrates are respectively mRNAs and DNA and their products proteins and mRNAs. The scenario of the model is the following. Initially, the majority of enzymes are bound to their substrates and occupy a small fraction of all possible substrate sites. In this nonsaturated regime, that is, when the number of enzymes is smaller than a threshold value $P_p^{*}$ and $P_r^{*}$ (Equation A.37), the production rates of the products of type j read (Lin and Amir, 2018):

Figure 2

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Data tables extracted from Figure 3.A of Neurohr et al., 2019.

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Data tables extracted from Figure 3.B of Neurohr et al., 2019.

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Data tables extracted from Figure 3.C,D and S5C of Neurohr et al., 2019.

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Data tables adapted from Figure 3.H of Neurohr et al., 2019.

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Both production rates have two contributions. (1) A source term characterized by the rates k₀ and k_t at which the enzyme produces the product once it is bound to its substrate, times the average number of enzymes per substrate coding for the product of type j. This number is the fraction of substrates coding for product of type j – which can be identified as a probability of attachment (${\varphi }_j=\frac{g_j}{\sum g_j}$ and $\frac{M_j}{\sum M_j}$, where ${\displaystyle g_j}$, $M_j$ accounts for the number of genes and mRNAs coding for the product of type j) – multiplied by the total number of enzymes ($P_p$ and $P_r$). (2) A degradation term characterized by the average degradation times ${\tau }_m$ and ${\tau }_p$ of mRNAs and proteins. Note that we added a degradation term for proteins not present in Lin and Amir, 2018, which turns out to be of fundamental importance below. Although these timescales vary significantly between species their ratio remains constant, ${\tau }_m$ being at least one order of magnitude smaller than ${\tau }_p$ in yeast, bacteria, and mammalian cells (Milo and Phillips, 2015). This justifies a quasistatic approximation, ${\dot{M}}_j\sim 0$, during growth such that the number of mRNAs of type j adjusts instantaneously to the number of RNAPs, in the nonsaturated regime:

During interphase, the number of enzymes grows, increasingly more enzymes attach to the substrates up to the saturation value due to their finite size. In this regime, we use the same functional form for the production rates only replacing the average number of enzymes per substrate by their saturating values: ${\displaystyle g_j\cdot {\mathcal{N}}_p^{max}}$ for RNAPs and $M_j\cdot {\mathcal{N}}_r^{max}$ for ribosomes (see Appendix 1, section 4.1 and Equations A.35 and A.36), where ${\mathcal{N}}_p^{max}$ and ${\mathcal{N}}_r^{max}$ are the average maximal number of RNAPs and ribosomes per mRNAs and genes. Note that the model predicts that the saturation of DNA precedes that of mRNAs, whose number initially increases with the number of RNAPs (Equation 8) while the number of genes remains constant. We also highlight that a more general gene expression model was recently proposed (Wang and Lin, 2021), in which the saturation of DNA by RNAPs is due to a high free RNAP concentration near the promoter. Yet, we do not expect the exact saturation mechanism to change our conclusions. Once DNA is saturated, the number of mRNAs plateaus, leading to their saturation by ribosomes (see Appendix 1, section 4.1 and Equation A.41).

Our previous analysis has highlighted the fundamental importance of free amino acids on cell volume regulation Figure 1B. The production rate of free amino acids can be related to the number of enzymes catalyzing their biosynthesis using a linear process by assuming that the nutrients necessary for the synthesis are in excess:

where $k_{cat}$ is the rate of catalysis and $P_e$ the number of enzymes. The second term represents the consumption of amino acids to form proteins, with $P_{tot}=\sum P_j$. Although Equation 9 is coarse-grained, we highlight that, since glutamate and glutamine are the most abundant amino acids in the cell, it could in particular model the production of these specific amino acids from the Krebs cycle (Alberts et al., 2002). Note that we also ignored amino acid transport through the plasma membrane. The rationale behind this choice is twofold. (1) We do not expect any qualitative change when adding this pathway to our model since amino acid transport is also controlled by proteins. (2) We realized that the amino acids that actually play a role in controlling the volume, mainly glutamate, glutamine, and aspartate, are nonessential amino acids, hence that can be produced by the cell.

We now combine the Pump-Leak model, the growth model, and the amino acid biosynthesis model to make predictions on the variation of the dry mass density during interphase. A crucial prediction of the growth model is that as long as mRNAs are not saturated, that is, ${\displaystyle P_r<P_r^{\ast }}$, all the protein numbers scale with the number of ribosomes, $P_j\sim \frac{{\varphi }_j}{{\varphi }_r}\cdot P_r$. Moreover, the autocatalytic nature of ribosome formation makes their number grow exponentially (Equation 7), that is, $P_r=P_{r,0}\cdot e^{k_r\cdot t}$, where $k_r=k_t\cdot {\varphi }_r-\frac{1}{{\tau }_p}$ is the effective rate of ribosome formation (and also the rate of volume growth in this regime; Equation A.39) . The most important consequence of this exponential growth coupled to the equation modeling amino acid biosynthesis (Equation 9) is that it implies that both amino acids and total protein content scale with the number of ribosomes ultimately leading to a homeostatic dry mass density independent of time (see Appendix 1, section 4.1):

We emphasize that Equation 10 only applies far from its singularity since it was obtained assuming that the volume of the cell is determined by free amino acids, that is, ${\displaystyle \frac{{\varphi }_e}{l_p}\cdot \frac{k_{cat}}{k_r}>>1}$.

Not only does our model explain the homeostasis of the dry mass, but it also makes the salient prediction that this homeostasis naturally breaks down if the time spent in the G1 phase is too long. Indeed, after a time $t^{**}=\frac{1}{kr}\cdot ln\left(\frac{{\mathcal{N}}_r^{max}\cdot {\mathcal{N}}_p^{max}\cdot k_0\cdot {\tau }_m\cdot \sum g_j}{P_{r,0}}\right)$ (see Appendix 1, section 4.1), mRNAs become saturated by ribosomes, drastically changing the growth of proteins from an exponential growth to a plateau regime where the number of proteins remains constant. After the time $t^{**}+{\tau }_p$, all protein numbers reach their stationary values ${\displaystyle P_j^{stat}=k_t\cdot k_0\cdot {\tau }_p\cdot {\tau }_m\cdot {\mathcal{N}}_R^{max}\cdot {\mathcal{N}}_p^{max}\cdot g_j}$. In particular, the enzymes coding for amino acids also plateau implying the loss of the scaling between free amino acids and proteins as predicted by Equation 9. The number of amino acids then increases linearly with time, whereas the number of proteins saturates. In this regime, the volume thus grows linearly with time but the dry mass remains constant, leading to its dilution and the decrease in the dry mass density (see Appendix 1, section 4.1 and Equation A.45):

Finally, our model makes other important predictions related to the cell ploidy that we briefly enumerate. First, the cut-off $P_r^{*}$ (Equation A.41) at which dilution is predicted to occur depends linearly on the genome copy number $\sum g_j$. Intuitively, mRNAs are saturated only if DNA has previously saturated. At saturation, the RNA number is proportional to the genome size. As a consequence, the volume $V^{*}\propto P_r^{*}$ at which dilution occurs scales with the ploidy of the cell, a tetraploid cell is predicted to be diluted at twice the volume of its haploid homolog. On the other hand, by virtue of the exponential growth, the time $t^{**}$ (Equation A.42) at which the saturation occurs only depends logarithmically on the number of gene copies making the ploidy dependence much less pronounced timewise. Second, the growth rate in the linear regime scales with the ploidy of the cell, as opposed to the growth rate in the exponential regime. Indeed, in the saturated regime, the growth rate scales as $k_{cat}\cdot P_e^{stat}$ (see Appendix 1, section 4.1 and Equation A.43), where $P_e^{stat}$ is the number of enzymes catalyzing the reaction of amino acids biosynthesis after their numbers have reached their stationary values, while in the exponential regime, the growth rate $k_r={\varphi }_r\cdot k_t-\frac{1}{{\tau }_p}$ scales with the fraction of genes coding for ribosomes ${\varphi }_r$, which is independent of the ploidy.

Our main prediction, namely that the cell is diluted after the end of the exponential growth, is reminiscent of the intracellular dilution at senescence recently reported in fibroblasts, yeast cells, and more recently suspected in aged hematopoietic stem cells (Neurohr et al., 2019; Lengefeld et al., 2021). Here we quantitatively confront our theory to the data of Neurohr et al., 2019, where the volume, the dry mass, and the protein number were recorded during the growth of yeast cells that were prevented from both dividing and replicating their genome. Though our theory was originally designed for mammalian cells, it can easily be translated to cells with a cell wall provided that the hydrostatic pressure difference across the wall $\mathrm{\Delta }P$ is maintained during growth by progressive incorporation of cell wall components (see Appendix 1, section 2.1.2). Indeed, our conclusions rely on the fact that the cell volume is primarily controlled by small osmolytes whose concentration in the cell dominates the osmotic pressure, a feature observed to be valid across cell types (Park et al., 2016).

We first check the qualitative agreement between our predictions and the experiments. Two distinct growth regimes are observed, at least on the population level, in the volume data of nondividing yeast cells (Neurohr et al., 2019) an initial exponential growth followed by a linear growth (Figure 2C). The occurrence of linear or exponential growth has been the object of intense debate. One of the ambiguity may come from the fact that cells often divide before there is a clear distinction between the linear and the exponential regimes (${\displaystyle e^t\approx 1+t+\mathcal{O}(t^2),\text{ for }t<<1}$). Note that our model only requires the exit of the exponential growth regime to observe the dilution. Moreover, our results suggest that both regimes of growth are not equivalent for proper cell function. Cells that would remain in the ‘nonexponential’ growth regime for too long will be diluted. This may be at the root of the observed cell cycle defects (Neurohr et al., 2019) and be the cause of functional decline towards senescence in particular for fibroblasts (Neurohr et al., 2019) and hematopoietic stem cells (Lengefeld et al., 2021). We think that a more precise understanding of the relationship between dilution and cell cycle defects remains an exciting avenue for future research. Our theory also predicts that as long as protein number is constant the volume must grow linearly (Equations 9 and A.45). This is precisely what is observed in the experiments: cells treated with rapamycin exhibit both a constant protein content and a linear volume increase during the whole growth (see Figure S6.F in Neurohr et al., 2019). Finally, our predictions on the relationship between ploidy and dilution are in very good agreement with experiments as well. Indeed, while the typical time to reach the linear growth regime – of the order of 3 hr – seems independent of the ploidy of the cell, the volume at which dilution occurs is doubled (see Figure S7.A in Neurohr et al., 2019). Moreover, the growth rate during the linear regime scales with ploidy as the haploid cells growth rate is of order ${\displaystyle 129\mathrm{f}\mathrm{L}/\mathrm{h}}$ against ${\displaystyle 270\mathrm{f}\mathrm{L}/\mathrm{h}}$ for their diploid counterparts (Neurohr et al., 2019).

Encouraged by these qualitative correlations, we further designed a scheme to test our theory more quantitatively. Although our theory has a number of adjustable parameters, many of them can be combined or determined self-consistently as shown in Appendix 1, section 4.1.4. We end up fitting four parameters, namely ${\tau }_p$, $t^{**}$, ${\displaystyle k_r}$ and the initial cell volume ${\displaystyle v_1}$, using the cell volume data (Figure 2B). We detail in Appendix 1, section 4.1.5 the fitting procedure and the values of the optimal parameters. Interestingly, we find a protein degradation time ${\displaystyle {\tau }_p=1\mathrm{h}\mathrm{r}9\mathrm{m}\mathrm{i}\mathrm{n}}$, corresponding to an average protein half-life time: ${\displaystyle {\tau }_{1/2}\sim 48\mathrm{m}\mathrm{i}\mathrm{n}}$, which is very close to the value ${\displaystyle 43\mathrm{m}\mathrm{i}\mathrm{n}}$, measured in Belle et al., 2006. Moreover, we obtain a saturation time ${\displaystyle t^{\ast \ast }=2\mathrm{h}\mathrm{r}44\mathrm{m}\mathrm{i}\mathrm{n}}$, which remarkably corresponds to the time at which the dry mass density starts to be diluted (Figure 2E), thus confirming the most critical prediction of our model. We can then test our predictions on the two other independent datasets at our disposal, that is, the dry mass density, obtained from suspended microchannel resonator (SMR) experiments, and the normalized protein number, from fluorescent intensity measurements. We emphasize that the subsequent comparisons with experiments are done without any adjustable parameters. The agreement between theory and experiment is satisfactory (Figure 2D and E) and gives credit to our model. We underline that the value of the density of water that we used is 4% higher than the expected value, ${\rho }_w=1.04\mathrm{kg}/\mathrm{L}$ to plot Figure 2E. This slight difference originates from the fact that our simplified theory assumes that the dry mass is entirely due to proteins whereas proteins represent only 60% of the dry mass. This hypothesis is equivalent to renormalizing the density of water as shown in Appendix 1, section 4.1.4.

In summary, our theoretical framework combining the Pump-Leak model with a growth model and a model of amino acid biosynthesis provides a consistent quantitative description of the dry mass density homeostasis and its subsequent dilution at senescence without invoking any genetic response of the cell; the dilution is due to the physical crowding of mRNAs by ribosomes. It also solves a seemingly apparent paradox stating that the volume is proportional to the number of proteins although their concentrations are low in the cell without invoking any nonlinear term in the osmotic pressure (see ‘Discussion’ and Appendix 1, section 3.1.12).

Our previous results explain well the origin of the dilution of the cellular dry mass at senescence. But can the same framework be used to understand the systematic dry mass dilution experienced by mammalian cells at mitotic entry? Although this so-called mitotic swelling or mitotic overshoot is believed to play a key role in the change of the physicochemical properties of mitotic cells, its origin remains unclear (Son et al., 2015; Zlotek-Zlotkiewicz et al., 2015).

We first highlight five defining features of the mitotic overshoot. (1) It originates from an influx of water happening between prophase and metaphase, resulting in a typical 10% volume increase in the cells. (2) The swelling is reversible and cells shrink back to their initial volume between anaphase and telophase. (3) This phenomenon appears to be universal to mammalian cells, larger cells displaying larger swellings. (4) Cortical actin was shown not to be involved in the process, discarding a possible involvement of the mechanosensitivity of ion channels, contrary to the density increase observed during cell spreading (Venkova et al., 2022) (5) Nuclear envelope breakdown (NEB) alone cannot explain the mitotic overshoot since most of the swelling is observed before the prometaphase where NEB occurs (Son et al., 2015; Zlotek-Zlotkiewicz et al., 2015).

The dry mass dilution at mitotic overshoot is thus different from the cases studied in the previous section. First, it happens during mitosis when the dry mass is constant (Zlotek-Zlotkiewicz et al., 2015). Second, the 10% volume increase implies that we need to improve the simplified model used above, which considers only metabolites and proteins (and their counterions). Having in mind that ions play a key role in the determination of the cell volume (Figure 1B), we show how every feature of the mitotic overshoot can be qualitatively explained by our theory, based on a well-known electrostatic property of charged polymer called counterion condensation first studied by Manning, 1969. Many counterions are strongly bound to charged polymers (such as chromatin) because the electrostatic potential at their surface creates an attractive energy for the counterions much larger than the thermal energy $k_{B}T$. The condensed counterions partially neutralize the charge of the object and reduce the electrostatic potential. Condensation occurs up to the point where the attractive energy for the free counterions is of the order $k_{B}T$. The condensed counterions then do not contribute to the osmotic pressure given by Equation 2, which determines the cell volume. These condensed counterions act as an effective ‘internal’ reservoir of osmolytes. A release of condensed counterions increases the number of free cellular osmolytes and thus the osmotic pressure inside the cell. Therefore, it would lead to an influx of water in order to restore osmotic balance at the plasma membrane (Figure 3).

Figure 3

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But how to explain such a counterion release at mitotic overshoot? For linear polymers such as DNA, the condensation only depends on a single Manning parameter $u=\frac{l_b}{A}$; where ${\displaystyle l_b}$ is the Bjerrum length (Appendix 1—table 1) that measures the strength of the coulombic interaction and ${\displaystyle A}$ the average distance between two charges along the polymer. The crucial feature of Manning condensation is the increase in the distance between charges $A$ by condensing counterions and thus effectively decreasing ${\displaystyle u}$ down to its critical value equal to $u^{eff}=1$ (see Appendix 1, section 5.1 for a more precise derivation). Hence, the number of nominal elementary charges carried by a polymer of length $L_{tot}$ is $Q_{tot}=\frac{L_{tot}}{A}$. Due to condensation, the effective distance between charges increases to $A^{eff}=l_b$ such that the effective number of charges on the polymer is reduced to $Q^{eff}=\frac{Q_{tot}}{u}$. The number of counterions condensed on the polymer is $Q^{cond}=Q_{tot}\cdot (1-\frac{1}{u})$. The most important consequence of these equations is that they suggest that a structural modification of the chromatin could lead to a counterion release. Indeed, making the chromatin less negatively charged, that is, increasing ${\displaystyle A}$, is predicted to decrease ${\displaystyle u}$ and thus to lead to the decrease in $Q^{cond}$. Detailed numerical simulations of chromatin electrostatics show that this description is qualitatively correct (Materese et al., 2009).

Biologists have shown that chromatin undergoes large conformational changes at mitotic entry. One of them attracted our attention in light of the mechanism that we propose. It is widely accepted that the affinity between DNA and histones is enhanced during chromatin compaction by stronger electrostatic interactions thanks to specific covalent modifications of histone tails by enzymes. Some of these modifications such as the deacetylation of lysines add a positive charge to the histone tails, hence making the chromatin less negatively charged (Alberts et al., 2002). Moreover, histone tails are massively deacetylated during chromatin compaction (Zhiteneva et al., 2017), potentially meaning that this specific reaction plays an important role in counterion release and thus on the observed mitotic swelling. However, we underline that the idea that we propose is much more general and that any reaction modifying chromatin electrostatics is expected to impact the swelling. The question whether deacetylation of lysines is the dominant effect is left open here.

Is the proposed mechanism sufficient to explain the observed 10% volume increase? We estimate the effective charge of chromatin for a diploid mammalian cell to be $Q^{eff}=2\cdot {10}^9{\mathrm{e}}^{-}$ and the number of condensed monovalent counterions to be $Q^{cond}=8\cdot {10}^9$ (see Appendix 1, section 3.1.6 and 3.1.7). The Pump-Leak model framework predicts the subsequent volume increase induced by the hypothetical release of all the condensed counterions of the chromatin. We find an increase of order $\mathrm{\Delta }V\sim 100-150\mu {\mathrm{m}}^3$, which typically represents 10% of a mammalian cell size (see Appendix 1, section 3.1.8 and Equation A.29). Admittedly crude, this estimate suggests that chromatin counterion release can indeed explain the amplitude of mitotic swelling.

In summary, the combination of the Pump-Leak model framework with a well-known polymer physics phenomenon allows us to closely recapitulate the features displayed during mitotic swelling. In brief, the decondensation of the chromatin condensed counterions, hypothetically due to histone tail modifications, is sufficient to induce a 10% swelling. This implies that all mammalian cells swell during prophase and shrink during chromatin decondensation after anaphase; again, consistent with the dynamics of the mitotic overshoot observed on many cell types. Another salient implication is that the amplitude of the swelling is positively correlated with the genome content of the cells: cells having more chromatin are also expected to possess a larger ‘internal reservoir’ of osmolytes, which can participate in decondensation. This provides a natural explanation for the observed larger swelling of larger cells. For instance, Hela cells were shown to swell on average by 20%, in agreement with the fact that many of them are tetraploid. Admittedly, many other parameters enter into account and may disrupt this correlation such as the degree of histone tail modifications or the initial state of chromatin; The existence of a larger osmolyte reservoir does not necessarily mean that more ions are released.

Finally, we point out that the ideas detailed in this section can be tested experimentally using existing in vivo or in vitro methods. For example, we propose to massively deacetylate lysines during interphase by either inhibiting lysine acetyltransferases (KATs) or overexpressing lysine deacetylases (HDACS) in order to simulate the mitotic swelling outside mitosis. We also suggest to induce mitotic slippage or cytokinesis failure for several cell cycles, to increase the genome content, while recording the amplitude of swelling at each entry in mitosis (Gemble et al., 2022).

Another widely documented scaling law related to cell volume states that the volume of cell organelles is proportional to cell volume (Chan and Marshall, 2010; Cantwell and Nurse, 2019b). As an example, we discuss here the nuclear volume. We develop a generalized ‘nested’ Pump-Leak model that explicitly accounts for the nuclear and plasma membranes (see Figure 1A). Instead of writing one set of equation (Equations 1–3) between the interior and the exterior of the cell, we write the same equations both inside the cytoplasm and inside the nucleus (see Equation A.55). Before solving this nonlinear system of equations using combined numerical and analytical approaches, we draw general conclusions imposed by their structure. As a thought experiment, we first discuss the regime where the nuclear envelope is not under tension so that the pressure jump at the nuclear envelope $\mathrm{\Delta }{P}_n$ is much smaller than the osmotic pressure inside the cell ${\displaystyle \mathrm{\Delta }{P}_n<<{\mathrm{\Pi }}_0}$. The osmotic balance in each compartment implies that the two volumes have the same functional form as in the Pump-Leak model, with two contributions: an excluded volume due to dry mass and a wet volume equal to the total number of particles inside the compartment divided by the external ion concentration (see Equation A.56). It is noteworthy that the total cell volume, the sum of the nuclear and cytoplasmic volumes, is still given by Equation 4 as derived in the simple Pump-Leak model. This result highlights the fact that the Pump-Leak model strictly applies in the specific condition where the nuclear envelope is under weak tension. In addition, a crucial consequence of the osmotic balance condition at the nuclear envelope is that it leads to a linear scaling relation between the volumes of the two compartments:

where $V_i$, $R_i$, and $N_i^{tot}$ denote, respectively, the total volume, dry volume, and total number of osmolytes of compartment i, the index $i=n,c$ denoting either the nucleus, $n$, or the cytoplasm, $c$. Importantly, this linear scaling between the nucleus and the cytoplasm was reported repeatedly over the last century and is known as nuclear scaling (Webster et al., 2009; Cantwell and Nurse, 2019b). While this conclusion is emphasized in some recent papers (Lemière et al., 2022; Deviri and Safran, 2022), we point out that Equation 12 is only a partial explanation of the robustness of the nuclear-scaling law. To further understand this affirmation and also to motivate our work, we first consider the simpler case of a cell containing proteins, chromatin, and their counterions (no metabolites). For the sake of readability, we assume that the volume fraction occupied by the dry mass is the same in the nucleus and in the cytoplasm (see Appendix 1, section 6.1.1). The NC ratio is then given by the ratio of the wet volumes. The osmotic balance at the nuclear envelope reads

where z_p is the average charge of proteins, $q^{eff}$ is the concentration of the counterions of chromatin, and $p$ is the protein concentration either in the nucleus, subscript $n$, or in the cytoplasm, subscript $c$. The term $z_p\cdot p_n$ hence accounts for the concentration of counterions associated to the proteins trapped in the nucleus. The NC ratio can thus be expressed as

where the capital letters $P_i$ and $Q^{eff}$ now account for the number of proteins and chromatin counterions. It is noteworthy that although being permeable to the nuclear envelope, ions can still play a role in the NC ratio. A flawed reasoning would state that their permeability implies that their concentration is balanced at the nuclear envelope, and hence that their contribution to the NC ratio can be discarded. Here, the condition of electroneutrality inside the nucleus leads to the appearance of a nuclear difference of potential that effectively traps the ions inside the nucleus and thus creates an imbalance of ion concentration. Of course, this effect would be negligible if ${\displaystyle Q^{eff}<<(z_p+1)\cdot P_n}$. However, our estimate goes against this hypothesis as $Q^{eff}\sim (z_p+1)\cdot P_n$ for both mammalian and yeast cells (see Appendix 1, sections 3.1.6 and 3.1.10). The problem that arises is that $Q^{eff}$, which we remind is not negligible here, does not scale during growth. This would imply that the NC ratio decreases during growth (see Figure 4). We solve this apparent paradox in the next section by considering metabolites; a consideration that has largely been overlooked in the recent literature (Wu et al., 2022; Lemière et al., 2022; Deviri and Safran, 2022).

Figure 4

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Data table exctracted from Figure 3 of Finan et al., 2009.

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We now examine the influence of the metabolites on the NC ratio. Following the lines of our previous discussion, four different components play a role in volume regulation: chromatin (indirectly through its noncondensed counterions), proteins (mainly contributing to the dry volume), and metabolites and ions (mainly contributing to the wet volume). It is noteworthy that these components do not play symmetric roles in the determination of the NC ratio. This originates from the fact that metabolites are permeable to the nuclear membrane and that chromatin, considered here as a polyelectrolyte gel, does not contribute directly to the ideal gas osmotic pressure because its translational entropy is vanishingly small (de Gennes, 1979). The nested Pump-Leak model leads to highly nonlinear equations that cannot be solved analytically in the general case (see Equation A.55). Nevertheless, in the particular regime of monovalent osmolytes and high pumping $z_a=1$, $z_p=1$ and ${\alpha }_0=0$ corresponding to the case where there is no free anions in the cell, the equations simplify and are amenable to analytical results. This regime is physically relevant since it corresponds to values of the parameters close to the ones that we estimated (Table 1). For clarity, we first restrict our discussion to this particular limit. We will also discuss both qualitatively and numerically the influence of a change of the parameters later. In this scenario, the nested Pump-Leak model equations reduce to

where the first and second equations correspond to osmotic pressure balance in the two compartments; the third and fourth equations correspond to macroscopic electroneutrality in each compartment; and the fifth equation is the balance of the chemical potential of the cations and metabolites on each side of the nuclear envelope. ${\displaystyle p_i,n_i,a_i^f}$ respectively account for the concentrations of proteins, cations, and metabolites either in the cytoplasm – subscript c – or in the nucleus, subscript n. $q^{eff}$ accounts for the effective chromatin charge density. From these equations, we express the concentrations of cations in each compartment as functions of the extracellular concentration ${\displaystyle n_0}$ and the chromatin charge density $q^{eff}$ (Equation A.60) , leading to the following expression of the nuclear envelope potential:

A salient observation from Equation 16 is that the nuclear envelope potential difference $U_n$ is a proxy of the chromatin charge density. At low $q^{eff}$, $U_n=0$, that is, the respective concentrations of metabolites and cations are equal on each side of the membrane. Equation 15 also shows that the protein concentrations are equal in the two compartments. This implies that when the charge of chromatin is diluted, the volumes of the nucleus and of the cytoplasm adjust such that the NC ratio equals the ratio of protein numbers in the two compartments $N{C}_1=\frac{P_n}{P_c}$. In the Pump-Leak model, which considers a single compartment, a membrane potential appears as soon as there exist trapped particles in the compartment (see Appendix 1, section 6.1.2 and Equation A.58). In contrast, our extended nested Pump-Leak model predicts that in the case of two compartments, the system has enough degrees of freedom to adjust the volumes as long as ${\displaystyle q^{eff}}$ is small, thereby allowing the potential to be insensitive to the trapped charged proteins. At high values of the chromatin charge $Q^{eff}$, $U_n$ saturates to the value $-ln(2)$, which in physical units is equivalent to ${\displaystyle -17\mathrm{m}\mathrm{V}}$ at ${\displaystyle 300\mathrm{K}}$. Note that this lower bound for the potential is sensitive to the average charge of the proteins ${\displaystyle z_p}$ and can be lowered by decreasing this parameter. We also highlight that Equation 16 makes another testable prediction, namely, that the nuclear envelope potential is independent of the external ion concentration. In the literature, nuclear envelope potentials were recorded for several cell types (Mazzanti et al., 2001). They can vary substantially between cell types ranging from ${\displaystyle \sim 0\mathrm{m}\mathrm{V}}$ for Xenopus oocytes to ${\displaystyle -33\mathrm{m}\mathrm{V}}$ for Hela cells. This result is in line with our predictions. The Xenopus oocyte nucleus has a diameter roughly 20 times larger than typical somatic nuclei, but its chromatin content is similar (Dahl et al., 2004), resulting in a very diluted chromatin and a vanishing nuclear envelope potential. On the other hand, Hela cells are known to exhibit an abnormal polyploidy that may lead to a large chromatin charge density and a large nuclear membrane potential.

This last observation allows to understand the influence of the metabolites on the NC ratio. An increase in the number of metabolites in the cell $A_{tot}^f$ induces growth of the total volume (Equation A.56), leading to the dilution of the chromatin charge and a strong decrease in the nuclear membrane potential (Equation 16). In the limit where $A_{tot}^f$ is dominant, we thus expect the NC ratio to be set to the value $N{C}_1$. On the other hand, at low $A_{tot}^f$, metabolites do not play any role on the NC ratio, which is then given by ${\displaystyle N{C}_2>N{C}_1}$ (see Equation A.59 for the general formula), with

The actual NC ratio is intermediate between the two limiting behaviors (see Appendix 1—figure 1B and Equation A.65). Note that the regime $N{C}_2$ is equivalent to the regime given by Equation 14 found in our preliminary discussion.

During cell growth, the ratio $N{C}_1$ is constant, while the ratio $N{C}_2$ varies with time. Indeed, if nucleo-cytoplasmic transport is faster than growth, the protein numbers $P_n$ and $P_c$ are both proportional to the number of ribosomes in the exponential growth regime and the ratio $N{C}_1$ does not vary with time (see Appendix 1, section 6.1.5). On the other hand, the DNA charge $Q^{eff}$ is constant during G1 phase while $P_n$ grows with time, so $N{C}_2$ decreases with time. The fact that the NC ratio remains almost constant during growth (Neumann and Nurse, 2007; Pennacchio et al., 2022) suggests that cells are closer to the $N{C}_1$ regime and point at the crucial role of metabolites in setting the NC ratio (Figure 4 and Appendix 1—figure 1B). Importantly, these conclusions are overlooked in a large part of the existing literature that often assumes that metabolites do not play any role on the NC ratio due to their permeability at the nuclear envelope. We end this qualitative discussion by predicting the effect of a variation of the parameters ${\displaystyle z_p,z_a}$ and ${\alpha }_0$ that were so far assumed to be fixed. Our main point is that any parameter change that tends to dilute the chromatin charge also tends to increase the (negative) nuclear envelope potential and make the NC ratio closer to the regime $N{C}_1$ and farther from the regime $N{C}_2$. Consequently, when ${\displaystyle z_p}$ or ${\displaystyle z_a}$ are increased, the number of counterions carried by each protein or metabolite increases. This in turn results in a global growth of the volume and hence leads to the dilution of the chromatin charge and to the increase in the nuclear envelope potential difference. Any increase in the pumping parameter ${\alpha }_0$ (decrease in pumping efficiency) has a similar effect. It increases the number of ions in the cell, resulting again in the dilution of the chromatin charge. Note that in the absence of pumping (${\alpha }_0=1$), the Pump-Leak model predicts a diverging volume because this is the only way to enforce the balance of osmotic pressures at the plasma membrane (Equation A.11) (if there is no pressure difference at the membrane due to a cell wall).

Five crucial parameters have emerged from our analytical study: (1) $\frac{P_n}{P_c}$ ; (2) $\frac{A^f}{2P_n}$; (3) $\frac{Q^{eff}}{2P_n}$; (4) ${\alpha }_0$ ;and (5) ${\displaystyle z_p}$ and ${\displaystyle z_a}$. But what are the biological values of these parameters? We summarize our estimates in Appendix 1, section 3.1. Importantly, the ratio between chromatin (free) counterions and the number of nuclear trapped proteins (and their counterions) is estimated to be of order one (see Appendix 1, section 3.1 and Figure 4C). As a key consequence, we find that the NC ratio would be four times larger in the absence of metabolites (see Appendix 1—figure 1B). This nonintuitive conclusion sheds light on the indirect, yet fundamental, role of metabolites on the NC ratio, which have been overlooked in the literature.

We now turn to a numerical solution to obtain the normalized variations of the NC ratio during growth in the G1 phase for different parameters (Figure 4). Interestingly, variations of the NC ratio and variations of the nuclear envelope potential are strongly correlated, a feature that can be tested experimentally (Figure 4B). Moreover, we deduce from our numerical results that, in order to maintain a constant NC ratio during the cell cycle, cells must contain a large pool of metabolites (see Figure 4C). Our estimates point out that this regime is genuinely the biological regime throughout biology, thus providing a natural explanation on the origin of the nuclear scaling.

In summary, many of the predictions of our analysis can be tested experimentally. Experiments tailored to specifically modify the highlighted parameters are expected to change the NC ratio. For example, we predict that depleting the pool of metabolites by modifying amino acid biosynthesis pathways, that is, lowering $\frac{A^f}{2P_n}$, would lead to an increase in the NC ratio. Importantly, good metabolic targets in these experiments could be glutamate or glutamine because they account for a large proportion of the metabolites in the cell (Park et al., 2016). We also point out that cells with a smaller metabolic pool are expected to experience higher variations of the NC ratio during growth and thus larger fluctuations of this ratio at the population level (Figure 4B). These predictions could shed light on understanding the wide range of abnormal karyoplasmic ratio among cancer cells. Indeed, metabolic reprogramming is being recognized as a hallmark of cancer (Fujita et al., 2020) some cancer cells increase their consumption of the pool of glutamate and glutamine to fuel the TCA cycle and enhance their proliferation and invasiveness (Altman et al., 2016).

Moreover, disruption of either nuclear export or import is expected to change $\frac{P_n}{P_c}$ and thus the NC ratio. Numerical solutions of the equations displayed in Appendix 1—figure 1 show a natural decrease of the NC ratio due to the disruption of nuclear import. On the other hand, if nuclear export is disrupted, we expect an increase in the NC ratio. This is in agreement with experiments done recently in yeast cells (Lemière et al., 2022). The authors reported a transient decrease followed by an increase in the diffusivities in the nucleus. This is in line with what our theory would predict. The initial decay is due to the accumulation of proteins in the nucleus, resulting in an associated crowding. On the other hand, the following increase is due to the impingement of ribosome synthesis as this step requires nuclear export. Our model would then predict the loss of the exponential growth and a decoupling between protein and amino acid numbers that would drive the dilution of the nuclear content.

Finally, our framework also predicts that experiments that would maintain the five essential parameters unchanged would preserve the nuclear scaling. We thus expect that, as long as the nuclear envelope is not under strong tension, changing the external ion concentration does not influence the scaling directly. Experiments already published in the literature (Guo et al., 2017) show precisely this feature.

So far we have assumed that the osmotic pressure is balanced at the nuclear envelope, which is a key condition for the linear relationship between nuclear and cytoplasmic volume. But why should this regime be so overly observed in biology? We first address this question qualitatively. For simplicity in the present discussion, we assume that DNA is diluted so that the nuclear envelope potential is negligible. This implies that metabolites and ions are partitioned so that their concentrations are equal in the nucleoplasm and the cytoplasm, hence canceling their contribution to the osmotic pressure difference at the nuclear envelope. This allows to express the volume of the nucleus as

While the previous expression is not the exact solution of the equations, it qualitatively allows us to realize that the nuclear envelope hydrostatic pressure difference plays a role in the volume of the nucleus if it is comparable to the osmotic pressure exerted by proteins. This pressure is in the 1000 Pa range since protein concentration are estimated to be in the millimolar range (Appendix 1, section 3.1). We further estimated an upper bound for the nuclear pressure difference to be in the 10⁴ Pa range (Equation A.33). Admittedly crude, these estimates allow us to draw a threefold conclusion. (1) The nuclear pressure difference $\mathrm{\Delta }{P}_n$ can be higher than the cytoplasmic pressure difference $\mathrm{\Delta }{P}_c$, in part due to the fact that lamina has very different properties compared to cortical actin: it is much stiffer and its turnover rate is lower. This points out the possible role of nuclear mechanics in the determination of the nuclear volume contrary to the cortical actin of mammalian cells that does not play any direct role for the cell volume. (2) The typical hydrostatic pressure difference at which mechanical effects become relevant is at least two orders of magnitude lower for the nucleus than for the cytoplasm, for which it is of order ${\displaystyle {\mathrm{\Pi }}_0}$, (3) Assuming linear elasticity, small nuclear envelope extensions of 10% would be sufficient to impact nuclear volume. These conclusions stand in stark contrast to the observed robustness of the nuclear scaling, thus pointing out that the constitutive equation for the tension in the lamina is nonlinear. Biologically, we postulate that this nonlinearity originates from the folds and wrinkles that many nuclei exhibit (Lomakin et al., 2020). These folds could indeed play the effective role of membrane reservoirs, preventing the nuclear envelope tension to grow with the nuclear volume, hence setting the nuclear pressure difference to a small constant value, and maintaining cells in the scaling regime discussed in the previous sections. This conclusion is consistent with the results of Finan et al., 2009, which observed that the nucleus exhibits nonlinear osmotic properties.

To further confirm our conclusions quantitatively, we consider the thought experiment of nonadhered cells experiencing hypo-osmotic shock. This experiment is well adapted to study the mechanical role of nuclear components on nuclear volume because it tends to dilute the protein content while increasing the hydrostatic pressure by putting the nuclear envelope under tension. For simplicity, we ignore the mechanical contribution of chromatin that was shown to play a negligible role on nuclear mechanics for moderate extensions (Stephens et al., 2017). To gain insight into the nonlinear set of equations, we split the problem into two parts. First, we identify analytically the different limiting regimes of nuclear volume upon variation of the number of impermeant molecules $X_n$ present in the nucleus and the nuclear envelope tension ${\gamma }_n$. We summarize our results in a phase portrait (see Appendix 1, section 6.1.6 and Appendix 1—figure 1). Two sets of regimes emerge: those, studied above, where nuclear and cytoplasmic osmotic pressures are balanced, and those where the nuclear hydrostatic pressure matters. In the latter situations, the nuclear volume does not depend on the external concentration and saturates to the value (see Appendix 1, section 6.1.7):

where $s$ and $V_n^{iso}$ are respectively the fraction of membrane stored in the folds and the volume of the nucleus at the isotonic external osmolarity ${\displaystyle 2\cdot n_0^{iso}}$ is an effective adimensional modulus comparing the stretching modulus of the nuclear envelope $K$ with an osmotic tension that depends on the total number of free osmolytes contained by the nucleus $N_n^{tot}$. The saturation of the nuclear volume under strong hypo-osmotic shock originating from the pressure build up in the nucleus after the unfolding of the folds implies a significant decrease in the NC ratio and a loss of nuclear scaling (Figure 4D).

As a second step, we investigate the variations of $X_n=A_n^f+P_n$ after the shock. Our numerical solution again highlights the primary importance of considering the metabolites $A_n^f$ for the modeling of nuclear volume. Indeed, disregarding their contribution would lead to an overestimation of the number of trapped proteins. Additionally, $X_n$ would remain constant during the osmotic shock, resulting in the reduction of the effective modulus of the envelope (Equation 19). We would thereby overestimate the nuclear volume (Figure 4D, dashed line). In reality, since free osmolytes are mainly accounted for by metabolites that are permeable to the nuclear envelope, the number of free osmolytes in the nucleus decreases strongly during the shock. This decrease can easily be captured in the limit where metabolites are uncharged $z_a=0$. The balance of concentrations of metabolites in this regime implies that the number of free metabolites in the nucleus, $A_n^f$, passively adjusts to the NC ratio:

As mentioned earlier, the tension of the envelope is responsible for the decrease in the NC ratio. This in turn decreases the number of metabolites inside the nucleus, reinforcing the effect and thus leading to a smaller nuclear volume at saturation (Figure 4D). We find the analytical value of the real saturation by using Equation 19 with $N_n^{tot}=(z_p+1)\cdot P_n+Q^{eff}$, that is, no metabolites remaining in the nucleus.

Our investigations on the influence of the hydrostatic pressure term in the nested Pump-Leak model lead us to identify another key condition to the nuclear scaling, that is, the presence of folds at the nuclear envelope. Moreover, although not the purpose of this article, using our model to analyze hypo-osmotic shock experiments could allow a precise characterization of the nucleus mechanics.

To study the nuclear-scaling law, we developed a model for nuclear volume by generalizing the Pump-Leak model that includes both nuclear mechanics, electrostatics, and four different classes of osmolytes. The clear distinction between these classes of components is crucial according to our analysis and is new. (1) Chromatin, considered as a polyelectrolyte gel, does not play a direct role in the osmotic pressure balance because its translational entropy is vanishingly small. Yet, it plays an indirect role on nuclear volume through its counterions. This creates an asymmetry in our system of equations, leading to the unbalance of ionic concentrations across the nuclear envelope and to the appearance of a nuclear envelope potential related to the density of chromatin. (2) Proteins are considered trapped in the nucleus, their number being actively regulated by nucleo-cytoplasmic transport. (3) Metabolites are considered freely diffusable osmolytes through the nuclear envelope but not through the plasma membrane. Note that proteins that have a mass smaller than the critical value 30–60 kDa (Milo and Phillips, 2015) are not trapped in the nucleus as they can freely diffuse throughout the nuclear pores. This nevertheless represents more a semantic issue than a physical one, and permeant proteins are rigorously taken into account as metabolites in the model, but are negligible in practice due to the larger pool of metabolites. (4) Free ions are able to diffuse through the plasma membrane and the nuclear envelope.

As a consequence, we show that the nuclear scaling originates from two features. The first one is the balance of osmotic pressures at the nuclear envelope that we interpret as the result of the nonlinear elastic properties of the nucleus likely due to the presence of folds in the nuclear membrane of mammalian cells. Interestingly, yeast cells do not possess lamina such that the presence of nuclear folds may not be required for the scaling. In this regard, our model adds to a recently growing body of evidence suggesting that the osmotic pressure is balanced at the nuclear envelope in isotonic conditions (Deviri and Safran, 2022; Lemière et al., 2022; Finan et al., 2009). The second feature is the presence of the large pool of metabolites accounting for most of the volume of the nucleus. This explains why nuclear scaling happens during growth while the number of chromatin counterions does not grow with cell size.

Interestingly, although not the direct purpose of this article, our model offers a natural theoretical framework to shed light on the debated nucleoskeletal theory (Webster et al., 2009; Cantwell and Nurse, 2019b). Our results indicate that the genome size directly impacts the nuclear volume only if the number of (free) counterions of chromatin dominates the number of trapped proteins and the number of metabolites inside the nucleus. We estimate that this number is comparable to the number of trapped proteins while it is about 60 times smaller than the number of metabolites. This is in agreement with recent observations that genome content does not directly determine nuclear volume (Cantwell and Nurse, 2019b). Although not directly, chromatin content still influences nuclear volume. Indeed, nuclear volume (Equation A.61) is mainly accounted for by the number of metabolites, which passively adjusts according to (Equation 55) ; Jevtić and Levy, 2014. In the simple case, of diluted chromatin and no NE potential, metabolite concentration is balanced and $NC=\frac{P_n}{P_c}$, such that the metabolite number depends on two factors (Equation 20). The first one is the partitioning of proteins, $\frac{P_n}{P_c}$, that is biologically ruled by nucleo-cytoplasmic transport in agreement with experiments that suggest that the nucleo-cytoplasmic transport is essential to the homeostasis of the NC ratio (Cantwell and Nurse, 2019b). The second one is the total number of metabolites, ruled by the metabolism Equation 9, which ultimately depends upon gene expression (Appendix 1, section 4.1). This prediction is in line with genetic screen experiments done on fission yeast mutants (Cantwell and Nurse, 2019a). However, when the chromatin charge is not diluted, which is likely to occur for cells exhibiting high nuclear envelope potential such as some cancer cells, our theory predicts that the number of metabolites in the nucleus also directly depends on the chromatin content due to electrostatic effects. This highlights the likely importance of chromatin charge in the nuclear-scaling breakdown in cancer.

The nested Pump-Leak model predicts that the cell swells upon NEB if the nuclear envelope is under tension. NEB occurs at prometaphase and does not explain most of the mitotic swelling observed in Son et al., 2015; Zlotek-Zlotkiewicz et al., 2015, which occurs at prophase. Within our model based on counterion release, mitotic swelling is either associated with cytoplasm swelling if the released counterions leave the nucleus or with nuclear swelling if they remain inside. In the latter case, swelling at prophase would be hindered by an increase in nuclear envelope tension, and additional swelling would occur at NEB. This prediction can be tested by artificially increasing the nuclear envelope tension through strong uniaxial cell confinement (Berre et al., 2014), which would synchronize mitotic swelling with NEB.

Physically, why can such a wide range of biological phenomena be explained by such a simple theory? A first approximation is that we calculated the osmotic pressure considering that both the cytoplasm and the nucleus are ideal solutions. However, it is known that the cytoplasm and the nucleoplasm are crowded (Feig et al., 2015; McGuffee and Elcock, 2010). The qualitative answer again comes from the fact that small osmolytes constitute the major part of the free osmolytes in a cell so that steric and short-range attractive interactions are only a small correction to the osmotic pressure. We confirm this point by estimating the second virial coefficient that gives a contribution to the osmotic pressure only of order 2 kPa (see Appendix 1, section 3.1), typically two orders of magnitude smaller than the ideal solution terms (Table 1). However, note that we still effectively take into account excluded volume interactions in our theory through the dry volume $R$. Moreover, we show in Appendix 1, section 7.1 that although we use an ideal gas law for the osmotic pressure, the Donnan equilibrium effectively accounts for the electrostatic interactions. Finally, our theory can be generalized to take into account any ions species and ion transport law while keeping the same functional form for the expressions of the volume (Equation A.21) as long as only monovalent ions are considered. This is a very robust approximation because multivalent ions such as calcium are in the micromolar range. Together, these observations confirm that the minimal formulation of the Pump-Leak model that we purposely designed is well adapted to study cell size.

As a logical extension of our results, we suggest that our framework be used to explain the scaling of other membrane bound organelles such as vacuoles and mitochondria (Chan and Marshall, 2010). Provided that the organelles are constrained by the same physical laws, namely the balance of water chemical potential, the balance of ionic fluxes and the electroneutrality condition, we show in Appendix 1 (Equation A.83) that the incorporation of other organelles into our framework leads to the same equations as for the nucleus, thus pointing out that the origin of the scaling of other organelles may also arise from the balance of osmotic pressures. The precise experimental verification of such prediction for a specific organelle is left opened for future studies. We also propose that our theory be used to explain the scaling of membraneless organelles such as nucleoids (Gray et al., 2019). Indeed, the Donnan picture that we are using does not require membranes (Barrat and Joanny, 1996). However, we would have to add other physical effects in order to explain the partitioning of proteins between the nucleoid and the bacterioplasm.

Taken as a whole, our study demonstrates that cell size scaling laws can be understood and predicted quantitatively on the basis of a remarkably simple set of physical laws ruling cell size as well as a simple set of universal biological features. The new interpretations of previous empirical biological phenomena that our approach allows to provide indicates that this theoretical framework is fundamental to cell biology and will likely benefit the large community of biologists working on cell size and growth.

In this section, we derive and discuss the three physical constraints that we used throughout our article to study cell volume regulation. These results are classical and can be found in a reference textbook such as Sten-Knudsen, 2007.

The system of equation formed by Equations 1–3 is nonlinear and cannot be solved analytically in its full generality. One complexity arises from the difference of hydrostatic pressure $\mathrm{\Delta }P$. Intuitively, if the volume increases, the surface increases, which may in some situations increase the tension of the envelope and in turn impede the volume growth. Mathematically, Laplace law relates the difference of hydrostatic pressure to the tension $\gamma$ and the mean curvature of the interface, which simplifies to the radius of the cell ${\displaystyle R}$ in a spherical geometry. The difference of hydrostatic pressure then reads:

In the case where the interface exhibits a constitutive law that is elastic $\gamma =K\cdot \frac{S-S_0}{S_0}$, it is easy to see that $\mathrm{\Delta }P$ exhibits power of ${\displaystyle V}$, which makes the problem nonanalytical. However, we can get around this limitation in two biologically relevant situations:

• When $\mathrm{\Delta }P$ is negligible. As shown in Table 1, this happens for mammalian cells that do not possess cellular walls.

• When $\mathrm{\Delta }P$ is buffered by biological processes. We argue that this situation applies for yeasts and bacteria during growth. Indeed, if the volume increase is sufficiently slow, one can hypothesize that cells have time to add materials to their cellular walls such that the tension does not increase during growth.

We give the corresponding analytical expressions under these two hypotheses in the next two sections.

Throughout the main text, we used order of magnitudes to guide our investigations and justify our approximations. For the sake of readability, we gather all the parameter significations, values, and origins in Appendix 1—table 1.

We summarize here the equations derived and discussed in the main text (Equations 6 and 7). The rates of production of mRNAs and proteins in the nonsaturated and saturated regimes read:

The cut-off values – $P_p^{*}$, $P_r^{*}$ – above which the substrates become saturated are obtained by imposing continuities of the production rates at the transition: