Physical constraints and biological regulations underlie universal osmoresponses

When cell growth reaches a steady state, the proportions of all components, including free water volume, cell mass, and cell-wall volume, must be constant relative to the total cell volume to ensure homeostasis. Therefore, all growth rates in steady states of cell growth must be the same: ${\displaystyle {\mu }_f={\mu }_r={\mu }_{cw}}$. The consequence of cell-wall synthesis regulation can be seen directly from ${\displaystyle {\mu }_{cw}={\mu }_r}$: the turgor pressure at steady states is constant, ${\displaystyle \sigma ={\sigma }_c}$. Experimentally, the cell-wall strain was measured by applying an acute hyperosmotic shock to induce plasmolysis, and it is approximately constant as the external osmolarity increases (Misra et al., 2013; Rojas et al., 2014), suggesting a constant turgor pressure independent of external osmolarity, in concert with our model assumptions. The internal osmotic pressure at steady states is related to the external osmotic pressure through Equation 2,

Here, we have neglected the term ${\displaystyle {\mu }_f/k_w}$. Boer et al., 2011, show that an abrupt water flux occurs within hundreds of milliseconds after an osmotic shock, from which we can estimate the water permeability as ${\displaystyle k_w\sim 100{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}{\mathrm{a}\mathrm{t}\mathrm{m}}^{-1}}$ considering an osmotic shock with an amplitude ${\displaystyle \mathrm{\Delta }{\mathrm{\Pi }}_{out}=1}$ atm. Because the typical doubling times of microorganisms are from about 20 min to several hours, we estimate ${\displaystyle {\mu }_f/k_w\sim 10-100}$ Pa (Ye and Verkman, 1989; Boer et al., 2011), negligible compared with the typical cytoplasmic osmotic pressures, which can be several atmospheric pressures.

In steady states, the internal osmotic pressure is independent of time. Combining Equation 4 and the dynamics of the internal osmotic pressure, Equation 18c, we find the relationships between the protein density, the internal osmotic pressure, and the growth rate in the steady states:

The right-hand side of Equation 10a is a constant independent of external osmolarity (see its detailed expression in Section C of Appendix 1). In deriving Equation 10b, we have replaced ${\displaystyle {\rho }_p}$ by ${\displaystyle {\mathrm{\Pi }}_{in}}$ in Equation 8 using Equation 10a with the critical internal osmotic pressure ${\displaystyle {\mathrm{\Pi }}_{in,c}}$ proportional to ${\displaystyle {\rho }_c}$. Intriguingly, the relationship between the normalized growth rate (${\displaystyle {\mu }_r/{\mu }_r^{max}}$) and the normalized cytoplasmic osmotic pressure (${\displaystyle {\mathrm{\Pi }}_{in}/{\mathrm{\Pi }}_{in,c}}$), which we refer to as the growth curve in the following, has only one parameter ${\displaystyle H_r/(H_a+1)}$. Therefore, the growth curves of different organisms can be unified by a single formula, Equation 10b, and different organisms may have different values of ${\displaystyle H_r/(H_a+1)}$. Furthermore, Equation 10b predicts a critical external osmolarity ${\displaystyle {\mathrm{\Pi }}_{out,c}={\mathrm{\Pi }}_{in,c}-{\sigma }_c}$, beyond which cell growth is completely inhibited.

We test the validity of Equation 10b by fitting it to the experimental growth curves (Figure 2A). To do this, we calculate the internal osmotic pressure using Equation 9 given the values of the external osmotic pressure and the turgor pressure (Table 1). Intriguingly, the growth curves of multiple species can be well fitted by Equation 10b, from which we infer the parameters ${\displaystyle H_r/(H_a+1)}$ and ${\displaystyle {\mathrm{\Pi }}_{in,c}}$ (Table 1). We find that budding yeast cells exhibit notable resilience to high external osmolarities: their ${\displaystyle {\mathrm{\Pi }}_{in,c}}$ value is higher than those of Gram-positive bacteria, B. subtilis, and Gram-negative bacteria, E. coli. Further, budding yeast cells demonstrate a higher value of ${\displaystyle H_r/(H_a+1)}$, indicating a reduced susceptibility to growth rate reduction when exposed to mild increases in the external osmolarity. Meanwhile, the osmoadaptation capability of E. coli depends on the growth media, presumably arising from variations in metabolic fluxes and gene expressions (Cayley et al., 1991; Dai et al., 2018; Rojas et al., 2014).

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Data tables extracted from Figure 1 and Table 1 of Cayley et al., 1991.

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Data tables extracted from Figure 1A of Dai et al., 2018.

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Data tables extracted from Figure 1A of Rojas et al., 2014.

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Data tables extracted from Figure 3D of Rojas et al., 2017.

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To further reveal the functions of biological regulations, we study the steady-state properties of mutant cells in which either osmoregulation or cell-wall synthesis regulation is depleted. For mutant cells without osmoregulation, ${\displaystyle H_a=0}$ in Equation 4. In this case, the fraction of osmolyte-producing protein is constant with time, i.e., ${\displaystyle {\varphi }_a={\chi }_a^{max}}$. Comparing the dynamics of osmolyte and total protein mass, ${\displaystyle \dot{N_a}=k_a{\varphi }_a{m}_p}$ and ${\displaystyle \dot{m_p}=k_r{\varphi }_r{m}_p}$, one finds that the ratio of the number of osmolyte molecules and the total protein mass remains constant over time, irrespective of variations in external osmolarity (see the detailed derivation in Section C of Appendix 1). As the external osmolarity increases, the protein density of mutant cells quickly reaches the critical value ${\displaystyle {\rho }_c}$ according to Equation 10a with ${\displaystyle H_a=0}$. Therefore, the steady-state growth curve of the mutant cells terminates at an external osmolarity much smaller than wild-type (WT) cells (Figure 2B), in agreement with previous experiments (Brewster et al., 1993).

For mutant cells without the cell-wall synthesis regulation, ${\displaystyle H_{cw}=0}$; therefore, the cell-wall synthesis efficiency ${\displaystyle {\eta }_{cw}}$ equals 1 independent of time. Thus, the growth rate of the relaxed cell-wall volume is always equal to the growth rate of total protein mass (Equation 6 and Equation 7). Interestingly, in this case, the turgor pressure at steady states decreases with the increase of external osmolarity (Figure 2C and see the detailed proof in Section C of Appendix 1). The decreased turgor pressure lowers the internal osmotic pressure given the same ${\displaystyle {\mathrm{\Pi }}_{out}}$ according to Equation 9, leading to a lower protein density of mutant cells than WT cells according to Equation 10a. Therefore, mutant cells grow faster than WT cells under the same external osmolarity (Figure 2B). Nevertheless, the mutant cells are prone to plasmolysis at a threshold external osmolarity where the WT cells can maintain constant turgor pressure (see the vertical line in Figure 2C around 2 M extra external osmolarity). Reduced turgor pressure is detrimental to multiple biological processes, e.g., cytokinesis in fission yeast requires the participation of turgor pressure (Proctor et al., 2012).

To summarize, osmoregulation allows cells to grow in a wide range of external osmolarity conditions with a mild change in protein density. The cell-wall synthesis regulation allows cells to maintain a stable turgor pressure and avoid plasmolysis. Both regulatory mechanisms expand the range of external osmolarities that cells can adapt to.

Next, we study the dynamical behaviors of cellular properties in response to a constant osmotic shock: the external osmolarity changes abruptly and keeps its value for an infinitely long time. Intriguingly, we find that the dynamics of osmoresponse can be split into shock and adaptation periods (see insets of Figure 3C and D). The immediate water flow due to osmotic imbalance occurs in the shock period, during which the mass and osmolyte productions are negligible. Therefore, the ratio of the internal osmotic pressure and the protein density is invariant right before and after a shock period: ${\displaystyle {\mathrm{\Pi }}_{in}^i/{\rho }_p^i={\mathrm{\Pi }}_{in}^f/{\rho }_p^f}$, where the upper index ${\displaystyle i}$ (${\displaystyle f}$) means the state right before (after) the shock period. Given this condition, we introduce the normalized protein density ${\displaystyle {\tilde{\rho }}_p}$ as

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where the normalization factor ${\displaystyle {\overline{\rho }}_p\propto {\mathrm{\Pi }}_{in}}$ (see its detailed expression in Materials and methods) so that ${\displaystyle {\tilde{\rho }}_p}$ changes continuously across the shock period. Interestingly, we find that osmoresponse is governed by a two-dimensional dynamical system composed of ${\displaystyle {\tilde{\rho }}_p}$ and ${\displaystyle {\eta }_a\equiv {\varphi }_a/{\chi }_a^{max}}$ (Materials and methods):

Here, ${\displaystyle {\tilde{\rho }}_c={\rho }_c/{\overline{\rho }}_p}$ is the normalized critical protein density, and ${\displaystyle {\eta }_a}$ denotes the efficiency of osmoregulation. From the above equations, it is clear that the timescale of osmoregulation is set by the doubling time: it takes about the doubling time for the protein density and the fraction of osmolyte-producing protein to adapt to the new steady-state values. For walled cells, ${\displaystyle {\tilde{\rho }}_c}$ and ${\displaystyle {\overline{\rho }}_p}$ depend on time since ${\displaystyle {\mathrm{\Pi }}_{in}={\mathrm{\Pi }}_{out}+\sigma }$ and the turgor pressure ${\displaystyle \sigma }$ is time-dependent during osmoresponse processes (Figure 3A and B). For unwalled cells, such as mammalian cells and microbial cells with cell walls removed (i.e. protoplasts), ${\displaystyle {\tilde{\rho }}_c}$ is constant in a fixed environment (see detailed discussion on the transient dynamics of unwalled cells in Section D of Appendix 1).

Upon a constant hyperosmotic shock, the immediate water efflux leads to an instantaneous drop in turgor pressure and a rise in protein density (Figure 3A). The internal state of the cell, (${\displaystyle {\tilde{\rho }}_p}$, ${\displaystyle {\eta }_a}$), evolves toward the new equilibrium point, ${\displaystyle ({{\tilde{\rho }}_c}^{H_a/(H_a+1)},{{\tilde{\rho }}_c}^{-H_a/(H_a+1)})}$. One should note that the equilibrium point is time-dependent initially but eventually becomes fixed as the turgor pressure relaxes to the steady-state value (Figure 3C and Figure 3—video 1). Interestingly, the protein density increases initially and then decreases after the shock (Figure 3A). The decrease in protein density is because of the osmoregulation process, which is set by the doubling time (Equation 12a and Equation 12b). Meanwhile, we find that the initial increase of protein density is because of the suppressed growth of the relaxed cell-wall volume due to the low turgor pressure. Indeed, for unwalled cells, the protein density ${\displaystyle {\rho }_p}$ decreases immediately after the shock (Appendix 1—figure 2B). We note that the growth rate approaches the new steady-state value non-monotonically (Figure 3A) because of the spiral trajectory in the space of the internal state (Figure 3C), consistent with experimental observations from Rojas et al., 2014.

The phenomena are essentially the opposite for a constant hypoosmotic shock (Figure 3B and D, Figure 3—video 2). However, we find extremely fast cell growth after the hypoosmotic shock, with a growth rate peak occurring about 25 min after applying the shock, which we call the supergrowth phase (Knapp et al., 2019). One should note that 25 min is much shorter than the doubling time (about 2 hr) but comparable to the timescale of cell-wall synthesis regulation, which we set as ${\displaystyle {\tau }_{cw}^{+}=12.5}$ min in the simulations in Figure 3 (we will explain why we choose ${\displaystyle {\tau }_{cw}^{+}=12.5}$ min in the next section). Furthermore, applying a hypoosmotic shock to an unwalled cell does not induce a significant supergrowth phase compared with walled cells (Appendix 1—figure 2D).

We propose that supergrowth comes from the high turgor pressure caused by the hypoosmotic shock, which leads to fast cell-wall synthesis according to Equation 7. Rapid insertion of materials into the cell wall relaxes the turgor pressure and allows the cells to grow faster (Equation 2 and Equation 3). This idea is consistent with the observation that the growth rate and the growth rate of the relaxed cell-wall volume ${\displaystyle {\mu }_{cw}}$ reach their peaks simultaneously (Figure 3B). This observation also suggests that the timescale of supergrowth, i.e., the timing of growth rate peak, is set by the timescale of cell-wall synthesis regulation (${\displaystyle {\tau }_{cw}^{+}}$ in Equation 7). Notably, in the initial stage of the adaptation period, ${\displaystyle {\mu }_{cw}}$ approaches its target from below and reaches its target value at the growth rate peak (i.e. ${\displaystyle {\mu }_r(\sigma /{\sigma }_c{)}^{H_{cw}}}$) (the third panel of Figure 3B), after which ${\displaystyle {\mu }_{cw}}$ sticks to its target value and decreases accordingly because of the short relaxation time ${\displaystyle {\tau }_{cw}^{-}}$ (Equation 7). For comparison, we also show ${\displaystyle {\mu }_{cw}}$ and ${\displaystyle {\mu }_r(\sigma /{\sigma }_c{)}^{H_{cw}}}$ for the hyperosmotic shock in the third panel of Figure 3A. A detailed proof of the conditions for supergrowth, including the necessity of a cell wall and the regulation of cell-wall synthesis, is provided in Section E of Appendix 1.

Following the discussion above, we obtain an analytical expression of the growth rate peak after a hypoosmotic shock (see the detailed derivations in Section F of Appendix 1)

Here, all the variables on the right side are at the growth rate peak. Because the timescale of the osmoresponse process, which is around hours (Figure 3B), is much longer than the timescale of the supergrowth phase, which is about 20 min for S. pombe (Knapp et al., 2019), the turgor pressure at the growth rate peak can be well approximated by its immediate value after the shock. Therefore, the growth rate peak must increase as the amplitude of the hypoosmotic shock increases, which we confirm numerically in Figure 3E.

Next, we quantitatively compare our theoretical predictions regarding the supergrowth phase with experimental data. Knapp et al., 2019, applied an osmotic oscillation to fission yeast S. pombe during which the external osmolarity alternated between two values. They found cell growth was almost inhibited during the perturbation, while the protein and dry-mass densities increased. Surprisingly, cells grew unusually fast after the osmotic oscillation was removed and reached their maximum growth rate about 20 min after the end of the osmotic oscillation. The maximum growth rate can be twice the growth rate in the reference growth medium, and the elevation in growth rate can persist for two to three cell cycles. These observations are very similar to our results for a constant hypoosmotic shock (Figure 3B).

To test if our osmoresponse model captures the supergrowth phase for a periodic perturbation, we simulate WT S. pombe cells with the same protocols as the experiments (see details of simulations in Materials and methods). Intriguingly, our model successfully recapitulates the supergrowth phase and the gradually increasing protein density and dry-mass density during the perturbation (Figure 4). We confirm that cell-wall synthesis regulation is crucial for the emergence of the supergrowth phase since unwalled cells do not exhibit supergrowth after periodic perturbation (Appendix 1—figure 3). Interestingly, we find that an infinitely long periodic osmotic shock can be equivalently mapped to a constant osmotic shock (see the detailed discussions and proof in Section D of Appendix 1), which means that they have the same time-averaged growth rate and protein density in the steady states (Appendix 1—figure 2F).

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Knapp et al., 2019, measured the growth rate peaks vs. three different parameters of the osmotic oscillations: amplitude, period length, and number of periods. We first fit the growth rate peaks vs. the amplitudes (Figure 4D), from which we obtain ${\displaystyle H_{cw}=1.7}$, the sensitivity of the cell-wall synthesis efficiency to turgor pressure (Equation 7), and ${\displaystyle {\tau }_{cw}^{+}=12.5}$ min, the timescale in the upregulation of cell-wall synthesis efficiency (which is why we set ${\displaystyle {\tau }_{cw}^{+}=12.5}$ min in the previous section). Other model parameters are inferred from independent steady-state measurements, and we set the timescale in the downregulation of cell-wall synthesis efficiency as ${\displaystyle {\tau }_{cw}^{-}=0.1}$ min for simplicity (Table 1). We next fix the values of ${\displaystyle H_{cw}}$ and ${\displaystyle {\tau }_{cw}^{+}}$ and plot the predicted growth rate peaks vs. the period length (Figure 4B) and number of periods (Figure 4C). As a strong support of our model, our predictions quantitatively match the experimental data without any further fitting.

Two interesting features of the curve ${\displaystyle {\mu }^{sg}}$ vs. amplitude catch our attention: the non-monotonic behavior and the kink point at which the derivative is discontinuous (Figure 4D), which are conserved regardless of the number of periods (Figure 4—figure supplement 2). Therefore, we study the case of a single oscillation for simplicity, which is equivalent to a hyperosmotic shock of finite duration. For a mild hyperosmotic shock, during the period of hyperosmotic shock, the turgor pressure has almost recovered to the steady-state value ${\displaystyle {\sigma }_c}$ (Figure 3A). Therefore, switching from a long hyperosmotic period to the reference growth medium is equivalent to a constant hypoosmotic shock, where we have shown that the growth rate peak increases with the amplitude (Figure 3E). However, the crowding effect becomes more pronounced as the amplitude increases. Beyond the critical amplitude at the kink point, the cytoplasm is completely jammed during the hyperosmotic shock such that the cell states are precisely the same before and after the hyperosmotic shock, which means no supergrowth phase beyond this critical amplitude. Therefore, the curve ${\displaystyle {\mu }^{sg}}$ vs. amplitude must be non-monotonic (Figure 4E). Notably, for a very large ${\displaystyle H_r}$, cells can feel the crowding effect only when the cytoplasm is close enough to the critical protein density, shown as the abrupt decline of ${\displaystyle {\mu }^{sg}}$ (Figure 4F).

Finally, we remark that the significance of supergrowth is intimately related to the amount of recovered turgor pressure during the hyperosmotic shock ${\displaystyle \delta \sigma }$. We prove that the overshoot of turgor pressure after the removal of hyperosmotic shock (${\displaystyle {\sigma }^f-{\sigma }_c}$), which sets the growth rate peak, is mainly set by the recovered turgor pressure during the hyperosmotic shock (see the detailed discussions in Section G of Appendix 1). Indeed, ${\displaystyle {\mu }^{sg}}$, ${\displaystyle {\sigma }^f-{\sigma }_c}$, and ${\displaystyle \delta \sigma }$ are highly correlated as we change the amplitude (Figure 4E).

We define the fractions of osmolyte-producing protein and ribosomal proteins in the total proteome as ${\displaystyle {\varphi }_a=m_{p,a}/m_p}$ and ${\displaystyle {\varphi }_r=m_{p,r}/m_p}$, respectively. To model gene expression regulation, we introduce ${\displaystyle {\chi }_a}$ and ${\displaystyle {\chi }_r}$ as the fractions of ribosomes translating the osmolyte-producing protein and ribosomal proteins such that

Here, ${\displaystyle k_r}$ is proportional to the elongation speed of ribosomes on mRNAs divided by the protein mass of a single ribosome, which is affected by the global crowding effect as ${\displaystyle k_r=k_r^{max}{\eta }_r}$. Here, ${\displaystyle {\mu }_r}$ is the growth rate of total protein mass, which is also the growth rate of dry mass and bound volume in our model since they are all proportional. The osmolyte molecules are produced by the osmolyte-producing protein, with the rate given by

where ${\displaystyle k_a=k_a^{max}{\eta }_r}$ is the osmolyte production rate, including the crowding factor, and ${\displaystyle m_{p,a}}$ is the mass of osmolyte-producing protein. We summarize the dynamical equations involved in the osmoresponse model:

To describe the osmoregulation process using a two-dimensional dynamical system, we introduce the normalized protein density as

Combining Equation 11 and Equation 18a, we obtain the dynamical equation for ${\displaystyle {\tilde{\rho }}_p}$ as

Using Equation 15, we obtain the equation for ${\displaystyle {\eta }_a={\varphi }_a/{\chi }_a^{max}}$ as

where ${\displaystyle {\tilde{\rho }}_c={\rho }_c/{\overline{\rho }}_p}$. The unique equilibrium point for the internal state is

We employ the LSODA algorithm with automatic stiffness detection and switching (Petzold, 1983), implemented in SciPy (Virtanen et al., 2020), to solve Equation 18a–e. The parameters used for numerical simulations of walled cells are listed in Table 1.

In the main text, we mainly focus on the density of cytoplasmic proteins, defined as ${\displaystyle {\rho }_p=m_p/V_f}$, which is related to the dry-mass density, defined as ${\displaystyle {\hat{\rho }}_d=m_{tot,d}/V_{tot}}$. Here, the subscript ${\displaystyle tot}$ means the total mass and cell volume, encompassing regions outside the cytoplasm. Taking yeast as an example, the dry mass in the cell wall contributes to approximately 30% of the total dry mass (Lesage and Bussey, 2006), with mannoproteins constituting 40% of the dry mass in the cell wall (Klis et al., 2002). Therefore, a non-negligible proportion of proteins resides outside the cytoplasm in microorganisms. For simplicity, we assume that the fraction of protein mass ${\displaystyle m_{tot,p}}$ in the total dry mass ${\displaystyle m_{tot,d}}$ remains constant under different external osmolarities for a given species. Also, this constancy holds both inside and outside the cytoplasm such that

Further, we assume that the mass fraction of the cytoplasmic proteins in total proteins is constant, from which we infer the ratio between the total bound volume and the cytoplasmic bound volume: ${\displaystyle (V_{tot}-V_f)/V_b=m_{tot,p}/m_p}$. With these assumptions, ${\displaystyle {\rho }_p}$ and ${\displaystyle {\hat{\rho }}_d}$ are related as

Here, ${\displaystyle f}$ is the fraction of free volume in the cytoplasmic volume, ${\displaystyle f=V_f/V}$. In Appendix 1—table 1, we present the dry-mass fraction of protein ${\displaystyle m_p/m_d}$ and the cytoplasmic protein mass fraction ${\displaystyle m_p/m_{tot,p}}$ used in this work. The same parameters are used for S. pombe and S. cerevisiae for simplicity. Given Equation A2 and the value of ${\displaystyle f}$, which we discuss in the next section, we obtain the protein density from the dry-mass density.

Appendix 1—video 1

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Appendix 1—video 2

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In this section, all variables with the superscript ${\displaystyle \ast }$ are for the reference growth media we show in Table 1.

2. The free volume fraction of S. pombe

Molines et al., 2022, measured the immediate changes in cytoplasmic volume after a constant hyperosmotic shock with varying amplitudes, generating a curve of ${\displaystyle V^f/V^{\ast }}$ vs. ${\displaystyle \mathrm{\Delta }{\mathrm{\Pi }}_{out}}$, where ${\displaystyle \mathrm{\Delta }{\mathrm{\Pi }}_{out}}$ is the extra external osmolarity relative to the reference growth medium. As mentioned in the main text, there is a separation of timescales between the shock periods and the adaptation periods. The number of osmolyte molecules ${\displaystyle N_a}$ remains conserved during the shock periods. Consequently, the cytoplasmic volume right after the hyperosmotic shock is given by

(A3) ${\displaystyle V^f=V_f^f+V_b^{\ast }=\left[\frac{{\mathrm{\Pi }}_{in}^{\ast }}{{\mathrm{\Pi }}_{in}^f}{f}^{\ast }+(1-f^{\ast })\right]V^{\ast }}$

This relationship has been verified for various cell types (Zhou et al., 2009; Ting-Beall et al., 1993). Moreover, the relaxed cell-wall volume is also conserved during the shock periods. Using Equation A3, the turgor pressure after the shock ${\displaystyle {\sigma }^f}$ can be related to the turgor pressure of the reference growth medium by

(A4) ${\displaystyle {\displaystyle \frac{{\sigma }^f}{{\sigma }^{\ast }}=1+\left(1+{{ϵ}^{\ast }}^{-1}\right)\left(\frac{{\mathrm{\Pi }}_{out}^{\ast }+{\sigma }^{\ast }}{{\mathrm{\Pi }}_{out}^f+{\sigma }^f}-1\right)f^{\ast }.}}$

We note that the left-hand side of Equation A4 increases monotonically with ${\displaystyle {\sigma }^f}$, while the right-hand side decreases monotonically. Therefore, given the values of ${\displaystyle {ϵ}^{\ast }}$, ${\displaystyle f^{\ast }}$, and ${\displaystyle {\sigma }^{\ast }}$, ${\displaystyle {\sigma }^f}$ can be solved uniquely from Equation A4. In the case of an extreme hyperosmotic shock (i.e. a very large ${\displaystyle {\mathrm{\Pi }}_{out}^f}$), no positive solution of ${\displaystyle {\sigma }^f}$ can be found, corresponding to plasmolysis. After substituting ${\displaystyle {\mathrm{\Pi }}_{in}^f={\mathrm{\Pi }}_{out}^f+{\sigma }^f}$ into Equation A3, ${\displaystyle V^f/V^{\ast }}$ is obtained accordingly. Therefore, we infer the values of ${\displaystyle f^{\ast }}$ for S. pombe growing in YE5S media using the value of steady-state turgor pressure ${\displaystyle {\sigma }^{\ast }}$ in Table 1 (Lemière and Chang, 2023), the value of steady-state cell-wall strain ${\displaystyle {ϵ}^{\ast }}$(Atilgan et al., 2015), and the data of ${\displaystyle V^f/V^{\ast }}$ vs. ${\displaystyle \mathrm{\Delta }{\mathrm{\Pi }}_{out}}$ (Appendix 1—figure 1).

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Data tables extracted from Figure S1A of Molines et al., 2022.

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In this section, we discuss the conditions of the supergrowth phenomenon. Specifically, we focus on the scenario where the cell has undergone sufficient oscillatory osmotic cycles before switching back to the reference growth medium. Our conclusions are also valid for a constant hypoosmotic shock. By definition, the overall growth rate of the entire cytoplasmic volume is the weighted average of the dry-mass growth rate and the free-volume growth rate ${\displaystyle \mu =(1-f){\mu }_r+f{\mu }_f}$. We remark that the condition of supergrowth can be well approximated by the instantaneous condition: ${\displaystyle {\mu }_f>{\mu }_r}$ because ${\displaystyle {\mu }_r}$ only depends on the protein density, which changes much less significantly than the free-volume growth rate (Appendix 1—figure 7).

We rewrite the dynamics of the internal osmotic pressure, Equation 18c, in terms of the internal state ${\displaystyle ({\tilde{\rho }}_p,{\eta }_a)}$:

We first discuss the case of an unwalled cell. After sufficient cycles of osmotic oscillation, the internal state ${\displaystyle ({\tilde{\rho }}_p,{\eta }_a)}$ converges to a periodic steady state that lies on ${\displaystyle {\tilde{\rho }}_p{\eta }_a=1}$ (Appendix 1—figure 2E). After removing the oscillatory stimulus, the internal state leaves the curve ${\displaystyle {\tilde{\rho }}_p{\eta }_a=1}$ and returns to this curve after fully adapting to the reference growth media. During this adaptation process, the internal osmotic pressure constantly equals the external one because the turgor pressure is zero. Therefore, setting ${\displaystyle {\dot{\mathrm{\Pi }}}_{in}=0}$ in Equation A17, the free-volume growth rate takes the form of

We note that the internal state ${\displaystyle ({\tilde{\rho }}_p,{\eta }_a)}$ always follows a counterclockwise trajectory (Appendix 1—figure 2A and C). Therefore, the unwalled cell inevitably enters the region of ${\displaystyle {\tilde{\rho }}_p{\eta }_a<1}$ after removing the oscillatory stimulus (Appendix 1—figure 3A), and the supergrowth phenomenon cannot occur according to Equation A18 (the upper panel of Appendix 1—figure 3B).

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Next, we discuss a walled cell deficient in cell-wall synthesis regulation. In this case, the growth rate of the relaxed cell-wall volume is always equal to the growth rate of dry mass, ${\displaystyle {\mu }_{cw}={\mu }_r}$. According to Equation 18d, the time dependence of turgor pressure is given by

To maintain osmotic balance during the adaptation period, ${\displaystyle {\dot{\mathrm{\Pi }}}_{in}=\dot{\sigma }}$, the free-volume growth rate must satisfy

For the same reason as an unwalled cell, the internal state of a walled cell deficient in cell-wall synthesis regulation enters the region of ${\displaystyle {\tilde{\rho }}_p{\eta }_a<1}$ after removing the oscillatory stimulus (Appendix 1—figure 3A). According to Equation A20, there is no supergrowth phenomenon after the oscillation stimulus (the lower panel of Appendix 1—figure 3). In summary, cell-wall synthesis regulation is necessary for supergrowth.

Appendix 1—figure 4

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Here, we consider the case of a constant hypoosmotic shock and derive an analytical expression of the growth rate peak ${\displaystyle {\mu }^{sg}}$. As discussed in the main text, the abrupt water influx right after the shock rapidly stretches the cell wall, resulting in elevated turgor pressure. One should note that the ratio of the dry-mass growth rate before and after the shock is equal to that of the relaxed cell-wall volume, both set by the crowding effect,

Meanwhile, the timescale ${\displaystyle {\tau }_{cw}^{+}}$ governing the upregulation of cell-wall synthesis efficiency ${\displaystyle {\eta }_{cw}}$ to its target value ${\displaystyle (\sigma /{\sigma }_c{)}^{H_{cw}}}$ is considerably smaller than the timescale associated with the osmoregulation process. Consequently, during the timescale of supergrowth, cells can be considered effectively deficient in osmoregulation, producing osmolyte molecules and dry mass at the same rate. Thus, we can simplify the time dependence of the internal osmotic pressure as ${\displaystyle {\dot{\mathrm{\Pi }}}_{in}=({\mu }_r-{\mu }_f){\mathrm{\Pi }}_{in}}$. Meanwhile, the water flux is physically constrained to ensure osmotic balance: ${\displaystyle {\dot{\mathrm{\Pi }}}_{in}=\dot{\sigma }}$, leading to the expression:

We point out that the timescales of the relaxation dynamics of ${\displaystyle f}$, ${\displaystyle {\mathrm{\Pi }}_{in}}$, and ${\displaystyle \sigma }$ are all set by the doubling time (Figure 3—figure supplement 1); therefore, we can approximate them as constant for a short timescale. When the cell-wall synthesis efficiency ${\displaystyle {\eta }_{cw}={\mu }_{cw}/{\mu }_r}$ reaches its target value, which is also its maximum value (Figure 3), the ratio ${\displaystyle {\mu }_f/{\mu }_r}$ reaches its maximum value according to Equation A22, giving rise to the growth rate peak ${\displaystyle {\mu }^{sg}}$. Substituting ${\displaystyle {\mu }_{cw}={\mu }_r(\sigma /{\sigma }_c{)}^{H_{cw}}}$ into Equation A22 and employing the definition ${\displaystyle \mu =f{\mu }_f+(1-f){\mu }_r}$, we obtain the expression of ${\displaystyle {\mu }_{sg}}$:

All the variables on the right-hand side of Equation A23 are calculated at the growth rate peak.

Owing to the gradual restoration of turgor pressure, the turgor pressure at the growth rate peak differs from the immediate value after the hypoosmotic shock. Through direct simulations, we acquire the precise value of ${\displaystyle \sigma }$ at the growth rate peak. Given the value of ${\displaystyle \sigma }$, the corresponding values of ${\displaystyle {\mathrm{\Pi }}_{in}}$, ${\displaystyle {\mu }_r}$, and ${\displaystyle f}$ can be inferred. During the adaptation period, osmotic balance is maintained such that ${\displaystyle {\mathrm{\Pi }}_{in}={\mathrm{\Pi }}_{out}^f+\sigma }$. Given ${\displaystyle {\mathrm{\Pi }}_{in}}$, one can infer ${\displaystyle {\rho }_p}$ through Equation A10 since osmoregulation is effectively inactive, from which we calculate the ${\displaystyle {\eta }_r}$ factor, and the dry-mass growth rate ${\displaystyle {\mu }_r}$ is subsequently derived from Equation A6. Consequently, ${\displaystyle \sigma }$ is the only degree of freedom on the right-hand side of Equation A23. We compare this analytical expression with direct simulations of S. pombe at different upregulation timescales of the cell-wall synthesis process (Appendix 1—figure 4). As ${\displaystyle {\tau }_{cw}^{+}}$ tends to zero, the growth rate peak can be well approximated by the immediate value after the hypoosmotic shock ${\displaystyle {\sigma }^f}$. This extreme case is a good approximation for WT cells, since the timescale of the turgor pressure recovery process is set by the doubling time (Figure 3—figure supplement 1), much longer than ${\displaystyle {\tau }_{cw}^{+}}$. Indeed, a strong correlation exists between the growth rate peak and the turgor pressure right after the shock (Figure 3E).

During a single osmotic oscillation, the external osmolarity shifts to a hyperosmotic environment and then switches back to the reference growth medium (upper panel of Appendix 1—figure 5). We represent the turgor pressure jump resulting from the first osmolarity shift as ${\displaystyle {\sigma }^{f,1}-{\sigma }^{i,1}}$ and the corresponding jump from the second shift as ${\displaystyle {\sigma }^{f,2}-{\sigma }^{i,2}}$ (Appendix 1—figure 5). In Figure 4E, we show that the turgor pressure overshoot after the single oscillation ${\displaystyle {\sigma }^{f,2}-{\sigma }_c}$ can be approximated by the turgor recovery ${\displaystyle \delta \sigma }$ during the hyperosmotic period.