Stress granule formation via ATP depletion-triggered phase separation

Experimental studies have shown that SG assemble in response to multiple types of stress situations [21, 22, 23], and several pathways for SG formation have been identified. The most established is the ATP-dependent phosphorylation of the translation initiation factor eIF2α causing the arrest of RNA translation [16]. RNA is subsequently released from polysomes and aggregate with various proteins to form SG. There exist also other pathways that are independent of eIF2α phosphorylation [24], such as energy starvation [23]. Indeed, while various stress conditions causing SG assembly also cause a depletion of the cytoplasmic energy stores [21, 25, 26], energy starvation alone has been shown to trigger SG formation [23]. Here, we will focus exclusively on how changes in the energy level, and specifically the ATP concentration, can regulate SG formation. In other words, we will make the assumption that the ATP concentration directly regulates SG formation through ATP-dependent biochemical reactions.

In addition to this central premise, we will use two other pieces of biological observations to guide our modelling: first, during normal conditions, i.e. without imposed stress, the ATP concentration is at the normal level and SG are absent, or at least are so small that they are undetectable microscopically. Second, when external stress is imposed, ATP can fall by 50% [21, 26], in a time scale similar to that of SG formation [21, 25, 26], and SG assemble with sizes of the order of a micrometer [16]. We will therefore impose the following two constraints on our modelling: (1) a fall in the ATP level leads to the growth of SG, and (2) the response to the ATP level is switch-like, in the sense that a relatively mild change in the ATP level can lead to a very large change in SG size.

We now set out to construct minimal models that are compatible with these salient experimental findings on SG regulation.

At thermal equilibrium (k, h = 0), a finite, phase-separating system in the nucleation and growth regime (which is the regime relevant to our biological context) can only be in two stable steady-states: either the system is well-mixed (i.e., no granules) or a single granule enriched in P co-exists with the surrounding cytoplasm that is dilute in P [7]. Furthermore, due to surface tension, there exists a critical radius R_c below which droplets are no longer thermodynamically stable (equation (A.3) in appendix A). The critical radius can be estimated as a trade off between the surface energy (∝R²) that penalizes having two phases and the bulk free energy in the droplet (∝R³) that promotes droplet formation. As a result, in the early stage of phase separation when the mixture is homogeneous, droplets larger than R_c need to be nucleated. A quantitative analysis of nucleation is beyond the scope of this paper and we will assume that droplets are spontaneously nucleated when the system is inside the equilibrium phase boundary (figure 1(b)), potentially due to heterogeneous nucleation involving proteins or RNA as seeding platforms [35, 36]. Once multiple droplets are nucleated, droplets first grow by incorporation of phase-separating molecules P that diffuse from the cytoplasm to SG. This process, known as diffusion-limited growth [7], stops when the cytoplasmic concentration of P is depleted and reaches the saturation concentration ${\hat{P}}_{\mathrm{out}}$. The resulting multi-droplet system is always unstable and coarsen by Ostwald ripening [34]—the mechanism by which large droplets grow while small droplets dissolve—and/or coalescence of droplets upon encountering via diffusion [37]. Since the diffusion of protein complexes in the cytoplasm is strongly suppressed [38], we will ignore droplet diffusion and coalescence completely here and focus on Ostwald ripening. The phenomenon of Ostwald ripening is caused by (1) the Gibbs–Thomson effect that dictates that the solute concentration (P) outside a large droplet is smaller than that outside a small droplet (equation (7)); and (2) the dynamics of the solute in the dilute medium is well described by diffusion (equation (5a) with k = h = 0). These two effects combined lead to diffusive fluxes of solute from small droplets to large droplets and thus eventually a single droplet survives in a finite system [34].

Surprisingly, Ostwald ripening can be suppressed when non-equilibrium chemical reactions are present (k, h > 0) [10, 13, 39]. The physical reason lies in the new diffusive fluxes set up by the chemical reactions (figure 2). For instance, P is highly concentrated inside the droplets, and gets converted to S continuously in the interior. At the steady-state, the depletion of P inside the droplet has to be balanced by the influx of P from the outside medium. Intuitively, we expect that the depletion of P inside a droplet to scale like the droplet volume (∝R³), and the influx of P through its droplet boundary to scale like the surface area (∝R²). Taken together, we can see that as R increases, the depletion rate of a droplet will eventually surpass the influx of molecules from the medium and thus the growth will stop at a steady radius. As a result, multiple droplets of around the same size can co-exists in the system. We note that the above argument, although qualitatively correct, ignore pre-factors that also depend on the droplet radius R, which means that the quantitative calculation is more involved. Indeed, the influx of P actually scales like R instead of R² [13].

We now briefly summarize results from [13] for the stability of our multi-droplet system at steady-state. The overall concentration of P and S in the whole system, denoted by $\bar{P}$ and $\bar{S}$ respectively, are controlled solely by the chemical reaction rates:

$$\begin{eqnarray}&&\bar{P}=\displaystyle \frac{{h}_{i}(\alpha )}{{k}_{i}(\alpha )+{h}_{i}(\alpha )}\phi ,\ \ \bar{S}=\displaystyle \frac{{k}_{i}(\alpha )}{{k}_{i}(\alpha )+{h}_{i}(\alpha )}\phi ,\end{eqnarray} \tag{ 12 }$$

where we recall that $\phi \equiv \bar{P}+\bar{S}$ is the overall concentration of SG constituent molecules, whether phase-separating or soluble. In particular, $\bar{P}$ depends on the chemical reaction rate constants k_i, h_i, the ATP concentration α as well as the model under consideration (A, B or C, equations (2)–(4)). The qualitative behaviour of the system can be categorized into regimes based on the magnitude of the droplet radius R relative to the gradient length scale ξ (equation (11)).

In the small droplet regime ($R\ll \xi$), the stability diagram of a monodisperse multi-droplet system, obtained from [13], is shown in figure 3 for varying droplet radius R, forward rate constant k_i, and at fixed backward rate constant h_i. The droplet number density is variable in this figure, and depends on the nucleation process and potentially the coarsening kinetics. Droplet stability depends on the droplet radius R relative to the critical radii ${R}_{{\rm{c}}}\lt {R}_{l}\lt {R}_{u}$ [13]:

$$\begin{eqnarray}&&{R}_{{\rm{c}}}=\displaystyle \frac{{\hat{P}}_{{\rm{out}}}{l}_{{\rm{c}}}}{\bar{P}-{\hat{P}}_{{\rm{out}}}},\ \ {R}_{l}={\left(\displaystyle \frac{3{{Dl}}_{{\rm{c}}}{\hat{P}}_{{\rm{out}}}}{2{k}_{i}(\alpha ){\hat{P}}_{{\rm{in}}}}\right)}^{1/3},\ \ {R}_{u}=\sqrt{\displaystyle \frac{3D(\bar{P}-{\hat{P}}_{{\rm{out}}})}{{k}_{i}(\alpha ){\hat{P}}_{{\rm{in}}}}}.\end{eqnarray} \tag{ 13 }$$

Droplets whose radius R is smaller than the nucleus radius R_c (red irregular dashed line in figure 3) are thermodynamically unstable and dissolve. For ${R}_{{\rm{c}}}\lt R\lt {R}_{l}$ (black dashed line) the system is unstable against Ostwald ripening and the average droplet radius increases with time (upward arrows). Hence, for R < R_l, the system behaves qualitatively like an equilibrium system (without chemical reactions). On the contrary for ${R}_{l}\lt R\lt {R}_{u}$ (continuous red line, grey region), a mono-disperse system is stable against Ostwald ripening, and for R > R_u droplets shrink (downward arrows). The forward rate constant k_i is bounded by the critical values ${k}^{* },{k}_{{\rm{c}}}$:

$$\begin{eqnarray}&&{k}^{* }=\displaystyle \frac{2(\phi -{\hat{P}}_{{\rm{out}}})}{{\hat{P}}_{\mathrm{in}}}{h}_{i}(\alpha ),\ \ {k}_{{\rm{c}}}=\displaystyle \frac{\phi -{\hat{P}}_{{\rm{out}}}}{{\hat{P}}_{{\rm{out}}}}{h}_{i}(\alpha ).\end{eqnarray} \tag{ 14 }$$

If k_i > k_c all droplets dissolve due to the high conversion rate of phase-separating states P into soluble states S (equation (1)). If ${k}_{i}\lesssim {k}^{* }$ droplet radii R may be comparable or larger than ξ. Therefore the system enters the large droplet regime where the relations in equation (13) do not apply and Ostwald ripening may not be arrested. The interested reader is referred to [13] for a detailed analysis of this regime.

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We will now employ the formalism discussed to study the behaviour of the three minimal models introduced in section 2.

We will start with model B. In this model, ATP drives the S to P conversion (i.e., k_B = K_B and h_B = α H_B, equation (3)). As such, a reduction in ATP will naturally suppress this conversion and thus lead to a decrease in P and restrain phase separation. Therefore, depleting ATP cannot promote SG formation. As a result, we can eliminate this model since it contradicts our first biological constraint (section 1.1).

In model C, ATP drives both conversions (i.e., k_C = α K_C and h_C = α H_C, equation (4)). As a result, the overall concentrations $\bar{P}$ and $\bar{S}$ are independent of α (equation (12)). However α controls the multi-droplet stability as we shall see. We can distinguish qualitatively distinct regimes for model C depending on the relative magnitude of ${K}_{C}/{H}_{C}$ and the parameters

$$\begin{eqnarray}&&{\lambda }_{1}\equiv \displaystyle \frac{\phi -{\hat{P}}_{{\rm{out}}}}{{\hat{P}}_{\mathrm{in}}},\ \ {\lambda }_{2}\equiv \displaystyle \frac{\phi -{\hat{P}}_{{\rm{out}}}}{{\hat{P}}_{{\rm{out}}}}.\end{eqnarray} \tag{ 15 }$$

4.2.1.  ${K}_{C}/{H}_{C}\gt {\lambda }_{2}$ regime.

4.2.2.  ${\lambda }_{1}\ll {K}_{C}/{H}_{C}\lt {\lambda }_{2}$ regime (figure 4(a))

In this regime, using equations (4), (14) and (15) leads to ${k}^{* }\ll {k}_{C}\lt {k}_{{\rm{c}}}$, so droplets can form in the small droplet regime. The multi-droplet stability is described by equations (4) and (13). As shown in figure 4(a), droplets tend to shrink as the ATP concentration α increases, and droplets dissolve below the nucleus radius R_c (equation (13), red irregular dashed line). Systems with droplets larger than R_c are unstable against Ostwald ripening leading to an increase of the average droplet size (upward arrows). Above the critical radius R_l (black slanted dashed line) there is a region where monodisperse systems are stable (grey area) and this region is bounded by the maximal radius R_u (red continuous line) such that droplets larger than R_u shrink (downward arrows). Specifically, R_u has the following form (equations (4), (12) and (13))

$$\begin{eqnarray}&&{R}_{u}=\sqrt{\displaystyle \frac{3D}{\alpha {K}_{C}{\hat{P}}_{\mathrm{in}}}\left(\displaystyle \frac{{H}_{C}\phi }{{K}_{C}+{H}_{C}}-{\hat{P}}_{\mathrm{out}}\right)}\end{eqnarray} \tag{ 16 }$$

and we therefore find the scaling form

$$\begin{eqnarray}&&{R}_{u}\propto {\alpha }^{-1/2}.\end{eqnarray} \tag{ 17 }$$

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Namely, a fall of α increases the size of stable SG, thus satisfying our first constraint discussed in Section 1.1. However, the scaling of the maximal droplet radius R_u with the ATP concentration α is sub-linear. For instance, to decrease the maximal SG radius R_u by two-fold, a four-fold increase in the ATP concentration α is required. Therefore, depletion of ATP according to the scaling relation in equation (17) alone cannot account for the switch-like behaviour, which is our second biological constraint.

However, we cannot yet rule out this model because of another intriguing feature of this type of non-equilibrium phase-separating systems. When α is greater than a critical value ${\alpha }_{c}$ (vertical dashed line), even though the overall concentrations $\bar{P}$ and $\bar{S}$ remain constant, one can still eliminate droplets completely by quenching the stable radii below the nucleus radius R_c (${R}_{u}\lt {R}_{{\rm{c}}}$) (equation (13)). An estimate of an upper bound of R_c can be given by the smallest granule observed, which we take to be of the order 100 nm [16]. As we demonstrate in appendix A.1, the scaling law (equation (17)) remains valid until ${R}_{u}\simeq {R}_{{\rm{c}}}$ so we can use it to estimate the maximal SG size that would form upon varying α by a factor of two in the vicinity of ${\alpha }_{c}$. As a conservative estimate, if we assume that the tip of the phase boundary where α = α_c (figure 4(a)) corresponds to the ATP concentration in normal conditions, and that the corresponding droplet size is R_c, then a reduction of 50% of α can only lead to a maximal SG radius of around $100\times {2}^{1/2}\simeq 140\,\mathrm{nm}$ according to the upper bound law R_u. This radius is too small compared to experimental observations (section 1.1) and we thus rule out model C in this regime.

4.2.3.  ${K}_{C}/{H}_{C}\ll {\lambda }_{1}$ regime (figure 4(b))

In model A only the forward reaction rate k is amplified by an increase of the ATP concentration α and the backward rate h is constant (i.e., k_A = α K_A and h_A = H_A, equation (2)). We can again distinguish two qualitatively distinct regimes, depending on the magnitude of H_A relative to the parameter

$$\begin{eqnarray}&&\kappa \equiv \displaystyle \frac{D}{{l}_{{\rm{c}}}^{2}}{\left(\displaystyle \frac{\phi -{\hat{P}}_{\mathrm{out}}}{{\hat{P}}_{\mathrm{out}}}\right)}^{2}.\end{eqnarray} \tag{ 18 }$$

4.3.1.  ${H}_{A}\gg \kappa$ regime (figure 5(b))

In this regime, it can be shown that the nucleus radius R_c (equation (13)) is necessarily larger than the gradient length scale ξ (equation (11)). Therefore, since stable droplets are always larger than the nucleus, all droplets are also larger than ξ and the system is in the large droplet regime (section 3.2). As shown in figure 5(b), Although there exists a critical ATP concentration α_c beyond which all droplets dissolve (vertical dashed line), a multi-droplet system is always unstable and coarsen via Ostwald ripening (upward arrows). This is not a desirable feature for the formation of stable cytoplasmic organelles and we therefore discard model A in this regime.

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4.3.2.  ${H}_{A}\ll \kappa$ regime (figure 5(a))

As can be seen on figure 5(a), two qualitatively distinct regions can be distinguished based on the ATP concentration α. When α is much smaller than the critical value ${\alpha }^{* }\equiv {\lambda }_{1}{H}_{A}/{K}_{A}$ (equation (15)), it can be seen from equations (2), (14) and (15) that ${k}_{A}\ll {k}^{* }$. Hence large droplets may exist in the system (section 3.2) and no upper-bound radius exists.

If on the contrary $\alpha \gg {\alpha }^{* }$, then ${k}_{A}\gg {k}^{* }$ so the system is in the small droplet regime (section 3.2). Therefore, the nucleus radius R_c (red irregular dashed curve), the instability-stability transition radius R_l (black dashed curve) and the maximal droplet radius R_u (continuous red curve) are well approximated in this region by equation (13). In particular, Using equations (2), (12) and (13), the expression of the maximal droplet radius R_u is given by

$$\begin{eqnarray}&&{R}_{u}=\sqrt{\displaystyle \frac{1}{\alpha }\left(\displaystyle \frac{{H}_{A}\phi }{\alpha {K}_{A}+{H}_{A}}-{\hat{P}}_{\mathrm{out}}\right)}.\end{eqnarray} \tag{ 19 }$$

Similar to model C with ${\lambda }_{1}\ll {K}_{C}/{H}_{C}\lt {\lambda }_{2}$ (section 4.2.2), the maximal droplet radius R_u decreases with the ATP concentration α. A major difference however, is that the shrinkage is more pronounced due to the additional α in the denominator of $\bar{P}$ (equation (12)). This is due to the fact that increasing α accelerates the conversion $P{\to }^{k}S$, but not the reverse reaction. This leads to a reduction of the overall concentration $\bar{P}$ of phase separating-material in the system.

When α is greater than the critical value α_c (vertical dashed line), ${k}_{A}\gt {k}_{{\rm{c}}}$ (equation (14)) and all droplets dissolve (R_u = 0). This is due to the system being driven outside the phase-separating region by the chemical reactions ($\bar{P}\leqslant {\hat{P}}_{\mathrm{out}}$, figure 1(b), '$\square$' symbol). We find the expression of α_c by solving k_A(α_c) = k_c:

$$\begin{eqnarray}&&{\alpha }_{{\rm{c}}}=\left(\displaystyle \frac{\phi }{{\hat{P}}_{\mathrm{out}}}-1\right)\displaystyle \frac{{H}_{A}}{{K}_{A}}.\end{eqnarray} \tag{ 20 }$$

As a result, in model A droplet dissolution can be achieved by depleting P to the extent that the system crosses the equilibrium phase boundary. This suggests a stronger response than in model C which we quantify in appendix A.2. In particular we show that the ratio ${R}_{u}/{R}_{{\rm{c}}}$ for $\alpha \lesssim {\alpha }_{{\rm{c}}}$ does not have to be of order 1 as in model C but is a function of the system parameters ${\hat{P}}_{\mathrm{out}},{\hat{P}}_{\mathrm{in}},\phi ,D,{l}_{{\rm{c}}}$ and H_A. Therefore in the ${H}_{A}\ll \kappa$ regime of model A, droplets can be formed in a switch-like manner by a two-fold decrease of α, satisfying both our biological constraints (section 1.1).

In summary, we conclude that among the three minimal models introduced, model A in the ${H}_{A}\ll \kappa$ regime is the best suited to describe the physics of ATP-triggered SG formation.

We show in figure 6 the SG stability diagram for varying ATP concentration α and droplet radius R. In normal conditions α is at its basal value (blue arrow) leading to a low overall concentration $\bar{P}$ of phase-separating material. As a result, the system is outside the phase-separating region ($\bar{P}\lt {\hat{P}}_{\mathrm{out}}$, '$\square$' in the insert figure) and no droplets can exist. During stress conditions, ATP decreases two-fold (red arrow) leading to an increase of $\bar{P}$, thus taking the system inside the phase-separating region ($\bar{P}\gt {\hat{P}}_{\mathrm{out}}$, '$\bigtriangleup$'). SG form via the nucleation of small droplets of radius ≈ $50\,\mathrm{nm}$. Nuclei then grow (upward black arrow) leading to stable SG with much larger radii, between 0.5 and 2.3 μm (grey region), depending on the number of nucleated droplets. The relationship between the number of nucleation events and the radius of stable SG will be investigated in section 5.3. We note that as ATP increases and crosses the dissolution point α_c (equation (20)), the radius of the smallest stable droplet deceases from 0.5 to 0 μm. Therefore our model predicts that during normal conditions SG do not only shrink below detectable levels but fully dissolve.

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We thus find that the two biological constraints that SG must form when the ATP concentration decreases by two-fold, and in a switch-like manner (section 1.1), are both satisfied. Furthermore, the quantitative predictions for the stable SG radii are consistent with experimental observations. Therefore, model A reproduces salient experimental observations of SG formation and dissolution based on the ATP level.

Chemical reactions may destabilize the spherical shape of large droplets, leading to deformation or division into smaller droplets. This phenomenon, studied in a binary mixture [40] (P, S and no cytoplasm), is caused by the gradient of the P concentration being more pronounced near bulging regions, leading to a larger influx of P molecules in these regions. Hence, bulging regions grow by accumulation of material P, while depressed regions recede. This accentuates the shape deformation and can cause droplet division.

The solute concentration profiles in our ternary system (with cytoplasm), in the regime relevant for SG formation (small droplet regime, section 4.3.2), are identical to the profiles in a binary system [13]. We can therefore extend the results from [40] to our system and determine the radius R_s above which the droplet spherical shape becomes unstable:

$$\begin{eqnarray}&&{R}_{{\rm{s}}}={\left(\displaystyle \frac{33{{Dl}}_{{\rm{c}}}{\hat{P}}_{\mathrm{out}}}{k{\hat{P}}_{\mathrm{in}}}\right)}^{\tfrac{1}{3}}.\end{eqnarray} \tag{ 21 }$$

Using the parameters determined in appendix B we show R_s in figure 6 (dotted line). If SG are sufficiently large and the ATP is sufficiently depleted, droplets may deform and/or divide. We note however that, to our knowledge, no experimental work reported significant SG deformation or division during the stress phase.

So far we have restricted our analysis to the steady-states and stability of SG and ignored the dynamics of SG formation, which we will now analyse.

Let us consider the small droplet regime, which is the relevant regime for SG formation (section 4.3.2). As demonstrated in appendix C, the dynamics of droplet growth far from the steady-state is identical to that in an equilibrium system, i.e. without chemical reactions but with the same amount of P and S. The dynamics of SG formation can therefore be approximated by the dynamics of a multi-droplet system in equilibrium conditions (section 3.1), with the additional constraint that Ostwald ripening (OR) and diffusion-limited growth (DLG) are in such a way that they are arrested for droplet radii larger than R_l and R_u, respectively (equation (13)). We ignore for simplicity the nucleus radius (R_c = 0) and the effect of droplet deformation at large radii (section 5.2). In the equilibrium regimes (DLG and OR), at large times and far from the DLG-OR transition, the dynamics of the average droplet radius R is given by [41]

$$\begin{eqnarray}&&{R}_{\mathrm{DLG}}={({\beta }_{\mathrm{DLG}}t)}^{\tfrac{1}{2}},\ \ {R}_{\mathrm{OR}}={({\beta }_{\mathrm{OR}}t)}^{\tfrac{1}{3}},\end{eqnarray} \tag{ 22 }$$

with ${\beta }_{\mathrm{DLG}}=2D(\bar{P}-{\hat{P}}_{\mathrm{out}})/{\hat{P}}_{\mathrm{in}}$ and ${\beta }_{\mathrm{OR}}={{Dl}}_{{\rm{c}}}{\hat{P}}_{\mathrm{out}}/{\hat{P}}_{\mathrm{in}}$. We neglect the deviation from these laws at the DLG-OR transition, and assume that the system switches from the DLG regime to the OR regime at a critical average droplet radius R^* and time ${t}^{* }={({R}^{* })}^{2}/{\beta }_{\mathrm{DLG}}$. The dynamics for $t\gt {t}^{* }$ is therefore obtained by using the function ${R}_{\mathrm{OR}}(t)$ with the appropriate shift of the time variable t so that $R({t}^{* })={R}^{* }$. We obtain the following dynamics

$$\begin{eqnarray}R(t)=\left\{\begin{array}{ll}{({\beta }_{\mathrm{DLG}}t)}^{\tfrac{1}{2}} & \mathrm{if}\,t\leqslant {t}^{* },\,R\lt {R}_{u}\\ {[{\beta }_{\mathrm{OR}}(t-{t}^{* }+T)]}^{\tfrac{1}{3}}={\left[{\beta }_{\mathrm{OR}}\left(t-\displaystyle \frac{{({R}^{* })}^{2}}{{\beta }_{\mathrm{DLG}}}+\tfrac{{({R}^{* })}^{3}}{{\beta }_{\mathrm{OR}}}\right)\right]}^{\tfrac{1}{3}} & \mathrm{if}\,t\gt {t}^{* },\,R\lt {R}_{l},\end{array}\right.\end{eqnarray} \tag{ 23 }$$

where $T={({R}^{* })}^{3}/{\beta }_{\mathrm{OR}}$ is the solution of ${R}_{\mathrm{OR}}(T)={R}^{* }$, and the upper-bounds to R $({R}_{u},{R}_{l})$account for the stabilizing effect of the chemical reactions.

We now estimate the critical radius R^* at the DLG-OR transition. In the DLG regime droplets grow by depleting the cytoplasm from phase-separating material P. The droplet growth in this regime stops when the cytoplasmic concentration reaches the saturation concentration ${\hat{P}}_{\mathrm{out}}$ (equation (7) and figure 1). Since the total number of molecules P is constant, R^* is given by

$$\begin{eqnarray}&&{R}^{* }={\left(\displaystyle \frac{3(\bar{P}-{\hat{P}}_{\mathrm{out}})}{4\pi {\rho }_{n}{\hat{P}}_{\mathrm{in}}}\right)}^{\tfrac{1}{3}},\end{eqnarray} \tag{ 24 }$$

where we used the strong phase separation approximation ${\hat{P}}_{\mathrm{in}}\gg {\hat{P}}_{\mathrm{out}}$ and where ρ_n is the density number of nucleated droplets. ρ_n is also equal to the droplet number density throughout the DLG regime since we have assumed that droplet coalescence does not occur in our system (section 3).

As shown in figure 7 for the parameters determined in appendix B, the SG formation dynamics and the steady-state radius R (coloured circles) depend on the nucleus number density ρ_n. For large ρ_n, droplets grow first by DLG (solid coloured lines), then coarsen by OR (dotted coloured curves). The steady-state is reached when droplet radii are equal to R_l (vertical dashed line), above which OR is arrested. For smaller ρ_n, SG radii R increase by DLG until ${R}_{l}\lt R\lt {R}_{u}$ (vertical solid line) where OR is not active (grey region). Therefore the system is in a stable steady-state after the DLG. For even smaller ρ_n the DLG continues until droplet radii are equal to R_u (vertical continuous line), which is the maximal radius attainable. In this case, stable SG radii are equal to R_u and are independent on the number of SG in the system. The critical nucleus number density ${\rho }_{n}^{* }$ below which droplets grow exclusively by DLG is the solution of ${R}^{* }({\rho }_{n}^{* })={R}_{l}$ (equation (13)):

$$\begin{eqnarray}&&{\rho }_{n}^{* }=\displaystyle \frac{(\bar{P}-{\hat{P}}_{{\rm{out}}}){K}_{A}\alpha }{2\pi {{Dl}}_{{\rm{c}}}{\hat{P}}_{{\rm{out}}}}.\end{eqnarray} \tag{ 25 }$$

Using the physiologically relevant parameters estimated in appendix B we find ${\rho }_{n}^{* }\approx 6\times {10}^{-3}\mu {{\rm{m}}}^{-3}$.

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The time needed for nuclei growing exclusively in the DLG regime (${\rho }_{n}\lt {\rho }_{n}^{* }$) to reach the radii ${R}_{l}\approx 0.5\mu {\rm{m}}$ and R_u ≈ 2.2 μm are approximatively 2 min and 25 min, respectively. For nuclei growing exclusively in the OR regime (${\rho }_{n}\gg {\rho }_{n}^{* }$), the time needed to reach the radius ${R}_{l}\approx 0.5\mu {\rm{m}}$ is approximatively 30min. Therefore growth by DLG and OR are both consistent with SG formation time scales.

Our model provides the following experimental predictions:

• (i)  
  Since the $P\to S$ reaction is the one that requires the input of ATP, it is natural to relate the conversion to the ubiquitous ATP-driven phosphorylaton reaction. In other words, our model suggests that the soluble state of the SG assembling constituents corresponds to the phosphorylated form of these constituents.
• (ii)  
  We predict the existence of a concentration gradient of the phase-separating constituent P outside SG, with the gradient length scale of the form

  $$\begin{eqnarray}&&\xi =\sqrt{\displaystyle \frac{D}{\alpha {K}_{A}+{H}_{A}}}.\end{eqnarray} \tag{ 26 }$$

  Using biologically relevant parameters determined in appendix B we find ξ ≈ 15 μm, which is relevant in the context of the cell.
• (iii)  
  If the number density of nucleated droplets is large (${\rho }_{n}\gg {\rho }_{n}^{* }$, equation (25)), we predict a relationship between the stable SG radius R_l and the ATP level α (equation (13) and (2)):

  $$\begin{eqnarray}&&{R}_{l}\sim {\alpha }^{-1/3}.\end{eqnarray} \tag{ 27 }$$
• (iv)  
  Finally, if ATP is depleted beyond physiological conditions and SG are large enough (small droplet number density), we predict non-spherical SG or SG splitting due to shape instabilities (section 5.2).

The first prediction may be tested by screening the purified constituents of SG in an in vitro setting. The second, third and fourth predictions can be tested using imaging techniques with well regulated ATP concentration either in vivo or in vitro.

In model C, k = α K_C and h = α H_C (equation (4)) so the ratio k/h and Δ are constant. Therefore ${R}_{u}\propto {\alpha }^{-1/2}$ and we recover equation (17). Since this scaling is sub-linear we saw that it cannot explain SG formation and dissolution upon small variations of α. However when α approaches α_c the separation between R_c and R_u becomes small so the above approximations cease to be valid and a strong response regime exists: small variations of α lead to strong variations of R_u and R_c ($\alpha \approx {\alpha }_{{\rm{c}}}$ in figure A1).

Qualitatively, it can be seen from equation (A.1) that since we omitted the term $\propto {l}_{{\rm{c}}}/R$ in the determination of R_u we have overestimated R_u, and since we neglected the term $\propto {{kR}}^{3}$ in the determination of R_c we have underestimated R_c. Therefore the exact value of R_u is bounded from above by $\sqrt{3D{\rm{\Delta }}/({\hat{P}}_{\mathrm{in}}k)}$ while the exact value of R_c is bounded from below by ${R}_{{\rm{c}}}^{\mathrm{eq}}$ (equation (A.3)). The ratio ${R}_{u}/{R}_{{\rm{c}}}$ in the strong response regime is therefore also bounded (see figure A1):

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{u}}{{R}_{{\rm{c}}}}\lt {\left.\displaystyle \frac{\sqrt{3D{\rm{\Delta }}/({\hat{P}}_{\mathrm{in}}k)}}{{R}_{{\rm{c}}}^{{\rm{eq}}}}\right|}_{k={K}_{C}{\alpha }_{{\rm{c}}}}.\end{eqnarray} \tag{ A.6 }$$

At α = α_c the two fixed points R_c and R_u intersect at the radius R^* and since R_c and R_u are unstable and stable fixed points, respectively, we have

$$\begin{eqnarray}&&J({\alpha }_{{\rm{c}}},{R}^{* })=0,\end{eqnarray} \tag{ A.7 }$$

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{\rm{d}}J}{{\rm{d}}R}\right|}_{{\alpha }_{{\rm{c}}},{R}^{* }}=0\end{eqnarray} \tag{ A.8 }$$

and solving for α_c gives

$$\begin{eqnarray}&&{\alpha }_{{\rm{c}}}=\displaystyle \frac{4D{{\rm{\Delta }}}^{3}}{9{l}_{{\rm{c}}}^{2}{\hat{P}}_{\mathrm{out}}^{2}{\hat{P}}_{\mathrm{in}}{K}_{C}}.\end{eqnarray} \tag{ A.9 }$$

Using equations (4), (A.2) (A.6) and (A.9), we then find

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{u}}{{R}_{{\rm{c}}}}\lt \displaystyle \frac{3\sqrt{3}}{2}.\end{eqnarray} \tag{ A.10 }$$

In other words, the size of the stable droplets and nuclei are of the same order. This shows independently of the system parameters that the strong response regime in model C cannot account for the switch-like response observed experimentally.

We now concentrate on model A. Here the supersaturation Δ is no more constant since only the backward rate is constant (i.e. k = α K_A, h = H_A):

$$\begin{eqnarray}&&{\rm{\Delta }}=\displaystyle \frac{\phi }{1+\alpha \tfrac{{K}_{A}}{{H}_{A}}}.\end{eqnarray} \tag{ A.11 }$$

Therefore there exist a critical ATP concentration α_c above which Δ = 0 and all droplets dissolve (R_u = 0, equation (A.5)). The expression of α_c is found by solving Δ(α_c) = 0:

$$\begin{eqnarray}&&{\alpha }_{{\rm{c}}}=\displaystyle \frac{\phi -{\hat{P}}_{{\rm{out}}}}{{\hat{P}}_{{\rm{out}}}}\displaystyle \frac{{H}_{A}}{{K}_{A}}.\end{eqnarray} \tag{ A.12 }$$

We define ${\alpha }_{n}\equiv 4{\alpha }_{{\rm{c}}}/3$ and α_n/2 the ATP concentrations during normal and stress condition, respectively, in agreement with the biological constraint that ATP is depleted two-fold during stress conditions (section 1.1). Moreover we assumed these concentrations to be equidistant from α_c for simplicity. During normal conditions $\alpha ={\alpha }_{n}\gt {\alpha }_{{\rm{c}}}$ so no droplets can exist (${\rm{\Delta }},{R}_{u}\lt 0$). During stress condition $\alpha ={\alpha }_{n}/2\lt {\alpha }_{{\rm{c}}}$ and using equations (A.4), (A.5), (A.11) and (A.12), we find the size ratio ${R}_{u}/{R}_{{\rm{c}}}$ during stress conditions:

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{u}({\alpha }_{n}/2)}{{R}_{{\rm{c}}}({\alpha }_{n}/2)}=\displaystyle \frac{3}{4}\sqrt{\displaystyle \frac{D}{{l}_{{\rm{c}}}^{2}{H}_{A}}\displaystyle \frac{{({\hat{P}}_{{\rm{out}}})}^{2}}{(\phi -{\hat{P}}_{{\rm{out}}}){\hat{P}}_{\mathrm{in}}}{\left(\displaystyle \frac{\phi /{\hat{P}}_{{\rm{out}}}-2}{\phi /{\hat{P}}_{{\rm{out}}}+1}\right)}^{3}}.\end{eqnarray} \tag{ A.13 }$$

Therefore we find that in model A and contrarily to model C, the size ratio between stable droplets and nuclei is function of the system parameters and can be arbitrarily large. This can potentially provide the switch-like response observed experimentally which we discuss quantitatively in section 5.1 using physiologically relevant parameters.

We employ FRAP experiments [47] to estimate the diffusion coefficient of SG proteins. In these experiments a specific protein is tagged with a fluorescent marker, so that the local concentration of protein-markers is correlated with the local intensity of the fluorescence emission. A photobleaching laser beam is used to eliminate the fluorescence in a small region. Over time, the intensity in this region recovers as the non-fluorescent protein-markers are gradually replaced by fluorescent protein-markers, due to diffusion or other transport mechanisms. Assuming free diffusion of the protein-markers, the diffusion coefficient is given by $D\approx {w}^{2}/(4{t}_{1/2})$ with t_1/2 the half-time of fluorescence intensity recovery and w the radius of the bleached region [47].

We now consider FRAP results on TIA proteins. TIA proteins are key SG constituents, triggering SG formation when over-expressed, and inhibiting SG assembly when depleted [48]. Moreover TIA proteins harbour an intrinsically disordered domain, a class of protein domains that have raised recent interest for its ability to form phase-separated liquid droplets [49, 50, 51]. Hence, TIA protein is a prime candidate to explain SG formation based on phase separation.

FRAP experiments on TIA proteins show that the intensity of a bleached cytoplasmic region of radius ∼2 μm recovers with a half-time ${t}_{1/2}\simeq 2\,{\rm{s}}$ [33]. This leads to the protein diffusion coefficient

$$\begin{eqnarray}&&D\approx 0.5\,\mu {{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ B.2 }$$

Interestingly, the half-time recovery in experiments where a SG is bleached is roughly similar [33]. This indicates that SG proteins are highly mobile inside SG and supports our approximation of identical protein diffusion coefficients inside and outside SG (equations (5a) and (5b)). We note that such experiments are more complicated to interpret than cytoplasmic FRAP, because of the added parameters influencing the recovery time, such as the protein concentration ratio between SG and cytoplasm.

The global protein concentration in a cell is subject to fluctuations [52]. This can result in an increase of the overall concentration ϕ of SG constituent molecules, and potentially trigger undesired SG formation during non-stress conditions. This is due to the subsequent increase of the overall concentration $\bar{P}$ of phase-separating states (equation (12)), driving the system inside the phase-separating region ($\bar{P}\gt {\hat{P}}_{\mathrm{out}}$, figure 1). Let us define ϕ' the concentration of SG constituents such that, during normal conditions (α = α_n), $\bar{P}$ is identical to its value during stress conditions (α = α_n/2), thereby leading to undesired SG formation. In other words ϕ' is the solution of $\bar{P}({\alpha }_{n},\phi ^{\prime} )=\bar{P}({\alpha }_{n}/2,\phi )$ (equations (12) and (2)). The ratio ϕ'/ϕ should be large enough to prevent SG formation in normal conditions. From the definition of ϕ' and equations (12) and (B.1) we find

$$\begin{eqnarray}&&\displaystyle \frac{\phi ^{\prime} }{\phi }=\displaystyle \frac{4\phi /{\hat{P}}_{{\rm{out}}}-1}{2\phi /{\hat{P}}_{{\rm{out}}}+1}.\end{eqnarray} \tag{ B.3 }$$

The ratio ϕ'/ϕ is an increasing function of $\phi /{\hat{P}}_{\mathrm{out}}$, and reaches a plateau for $\phi /{\hat{P}}_{\mathrm{out}}\gg 1$ (continuous purple line in figure B1). Note that if $\phi /{\hat{P}}_{\mathrm{out}}\lt 1$ the system does not phase separate, since $\bar{P}\lt {\hat{P}}_{\mathrm{out}}$ in this case (equation (12), figure 1). If $\phi \gtrsim {\hat{P}}_{\mathrm{out}}$ then $\phi ^{\prime} \gtrsim \phi$ and even small fluctuations of ϕ cause unwanted SG formation. This scenario is not physiological and we discard it. If on the contrary $\phi /{\hat{P}}_{\mathrm{out}}\gg 1$ then $\phi ^{\prime} \approx 2\phi$. In this case, a two-fold increase of ϕ is necessary to bring $\bar{P}$ to its stress value. This scenario is therefore robust against concentration fluctuations. This prediction is consistent with experimental observations showing that a three-fold increase of the concentration of a given SG protein triggers SG formation [53]. Therefore we must impose

$$\begin{eqnarray}&&\displaystyle \frac{\phi }{{\hat{P}}_{\mathrm{out}}}\gg 1.\end{eqnarray} \tag{ B.4 }$$

We will see in appendix B.4 that $\phi /{\hat{P}}_{\mathrm{out}}$ is also maximally bounded.

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We use experimental results on TIA partitioning coefficient, i.e. the concentration ratio between SG and cytoplasm, to constrain further the model parameters. The combined concentrations of P + S inside and outside a SG of radius R are given by ${\hat{P}}_{\mathrm{in}}+S(R)$ and ${\hat{P}}_{\mathrm{out}}+S(R)$, respectively [13], and we recall that S(R) is the concentration of states S at the droplet interface (equation (8)). Note that the combined concentrations of P + S are homogeneous inside and outside, unlike the individual concentrations of P and S (equation (10) and figure 2). The partitioning coefficient a is therefore given by

$$\begin{eqnarray}&&a=\displaystyle \frac{{\hat{P}}_{\mathrm{in}}+S(R)}{{\hat{P}}_{\mathrm{out}}+S(R)}.\end{eqnarray} \tag{ B.5 }$$

The value of S(R) depends on the model parameters and the number of droplets in the system, and is bounded as follows [13]

$$\begin{eqnarray}&&\displaystyle \frac{{K}_{A}{\alpha }_{n}\phi }{{K}_{A}{\alpha }_{n}+{H}_{A}}\lt S(R)\lt \phi -{\hat{P}}_{\mathrm{out}}.\end{eqnarray} \tag{ B.6 }$$

This result with equations (B.1) and (B.4) leads to $S(R)\approx \phi$ and $a\simeq ({\hat{P}}_{\mathrm{in}}+\phi )/\phi$. Experimental observations on the partitioning coefficient a of TIA proteins is of order 10 [33], leading to the following constraint on the concentration parameters

$$\begin{eqnarray}&&{\hat{P}}_{\mathrm{in}}\approx 10\,\phi .\end{eqnarray} \tag{ B.7 }$$

From experimental observations we know that mature SG are much larger than the SG nuclei R_c (equation (13)) (section 1.1). Since R_u is the radius of the largest possible SG (equation (13)), the radius ratio r between mature SG and nuclei is upper-bounded as follows

$$\begin{eqnarray}&&r\lt \displaystyle \frac{{R}_{u}}{{R}_{{\rm{c}}}}.\end{eqnarray} \tag{ B.8 }$$

Using the expression of ${R}_{u}/{R}_{{\rm{c}}}$ obtained in appendix A (equation (A.13)), and imposing the constraints from equations (B.1), (B.4), and (B.7) on the ratio r, we find

$$\begin{eqnarray}&&r\lt \sqrt{\displaystyle \frac{3D{\hat{P}}_{\mathrm{out}}}{40{K}_{A}{\alpha }_{n}{l}_{{\rm{c}}}^{2}\phi }}.\end{eqnarray} \tag{ B.9 }$$

In appendix B.2 we have shown that $\phi /{\hat{P}}_{\mathrm{out}}\gg 1$ must be true to ensure robustness of the model against concentration fluctuations. Here we see that $\phi /{\hat{P}}_{\mathrm{out}}$ cannot be arbitrarily large since r would approach 0, as shown in figure B1 (green dashed line). In order to ensure both model robustness against concentration fluctuations and large radius ratio r between mature SG and SG nuclei, we choose the following constraint (dotted vertical line in figure B1)

$$\begin{eqnarray}&&\displaystyle \frac{\phi }{{\hat{P}}_{\mathrm{out}}}=10.\end{eqnarray} \tag{ B.10 }$$

Note that since r is also a function of other model parameters ($D,{K}_{A},{\alpha }_{n},{l}_{{\rm{c}}}$), the value of $\phi /{\hat{P}}_{\mathrm{out}}$ could in principle be much larger than 10, as long as the other parameters are adequate. We will see however that using physiologically relevant parameters determined in this section, the predicted ratio r is consistent with experimental observations. Therefore taking $\phi /{\hat{P}}_{\mathrm{out}}$ much larger than 10 would lead to an unrealistically small ratio r. This supports our choice in equation (B.10).

In our model SG form and dissolve in response to changes of the overall concentration $\bar{P}$ of the phase-separating states. To be consistent with experimental observations, the predicted time scales at which $\bar{P}$ varies must be shorter than the SG formation and dissolution time scales. This imposes constraints on the reaction rate constants K_A and H_A (equation (5)). In the model under consideration (model A) the variation over time of $\bar{P}$ is given by ${\rm{d}}\bar{P}/{\rm{d}}t=(-{K}_{A}\alpha +{H}_{A})\bar{P}+{H}_{A}\phi$ [13]. Let us assume for simplicity that the variation of $\bar{P}$ occurs at constant ATP concentration α.We can therefore solve for $\bar{P}$:

$$\begin{eqnarray}&&\bar{P}={c}_{1}+{c}_{2}{{\rm{e}}}^{-({K}_{A}\alpha +{H}_{A})t},\end{eqnarray} \tag{ B.11 }$$

with c_i functions of the parameters K_A, H_A, ϕ, α. The time scale at which $\bar{P}$ reaches chemical equilibrium is given by the inverse of K_A α + H_A. Since the ATP concentration α varies only by a factor 2 between stress and normal conditions (section 1.1), the increase and decrease of $\bar{P}$ occur on a similar time scale $\tau \approx 1/({K}_{A}{\alpha }_{n}+{H}_{A})$, with α_n the ATP concentration during normal conditions (equation (B.1)). This is consistent with the experimental observations that SG formation and dissolution times are within the same order of magnitude. As a rough estimate, τ must be smaller than the SG formation/dissolution time. Taking this time scale to be 15 min we get ${H}_{A}+{K}_{A}{\alpha }_{n}\gt {10}^{-3}\,{{\rm{s}}}^{-1}$. Since ${K}_{A}{\alpha }_{n}\gg {H}_{A}$ (from equations (B.1) and (B.4)) we obtain ${K}_{A}{\alpha }_{n}\gt {10}^{-3}\,{{\rm{s}}}^{-1}$. Taking an unnecessary large value of K_A α_n would lead to an ATP consumption that is unnecessary high. We therefore choose a conservative value of K_A α_n, just large enough to ensure that the time scale of $\bar{P}$ variation is fast enough:

$$\begin{eqnarray}&&{K}_{A}{\alpha }_{n}\approx 2\,\times \,{10}^{-3}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ B.12 }$$

We note that this rate constant is consistent with typical rates of protein phosphorylation [54, 55], which is the prototypical ATP-dependent reaction for protein post-transcriptional modification [30]. Using this result with equations (B.1) and (B.10) we find the backward reaction rate constant:

$$\begin{eqnarray}&&{H}_{A}\approx 1.5\,\times \,{10}^{-4}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ B.13 }$$

The capillary length l_c can be estimated from the expected SG nucleus radius R_c. The expression for R_c is given by equation (13):

$$\begin{eqnarray}&&{R}_{{\rm{c}}}=\displaystyle \frac{{\hat{P}}_{{\rm{out}}}{l}_{{\rm{c}}}}{{H}_{A}\phi /({H}_{A}+0.5{K}_{A}{\alpha }_{n})-{\hat{P}}_{{\rm{out}}}}\end{eqnarray} \tag{ B.14 }$$

where we used the ATP concentration during stress: α_n/2. Enforcing the constraints from equations (B.1) and (B.4) we find

$$\begin{eqnarray}&&{R}_{{\rm{c}}}\approx 3.3\,{l}_{{\rm{c}}}.\end{eqnarray} \tag{ B.15 }$$

The SG nucleus radius R_c is necessarily smaller than the radius of the smallest detectable SG ($\sim 100\,\mathrm{nm}$). On the other hand, there exist indications that SG are nucleated via heterogeneous nucleation involving RNA as seeding platforms [35, 36]. This implies that SG must reach a critical size before they become thermodynamically stable. Therefore SG nuclei are larger than the dimension of a single protein, which is of the order of 5 nm. We choose an intermediate value, R_c = 50 nm, leading to an estimation of the capillary length:

$$\begin{eqnarray}&&{l}_{c}\approx 15\,\mathrm{nm}.\end{eqnarray} \tag{ B.16 }$$

We have used theory and SG experimental results to determine physiologically relevant model parameters, or parameter constraints, summarized by equations (B.2), (B.7), (B.10), (B.12), (B.13) and (B.16). Note that the concentration parameters $\phi ,\,{\hat{P}}_{\mathrm{out}},\,{\hat{P}}_{\mathrm{in}}$, the ATP concentration ${\alpha }_{n}$, and the forward reaction constant K_A have not been explicitly expressed. Yet, the radius R_c of the SG nucleus, and the radii R_l and R_u that bound the stable region (equation (13)) are now explicitly determined from these relations.

Let us concentrate first on the situation where the droplet number density is small, so that the inter-droplet distance is much larger than the gradient length scale ξ. The term ${U}_{\mathrm{out}}^{(1)}$ in equation (10) must be zero to prevent diverging concentration P_out(r) far from droplets. Applying the boundary condition equation (7), we obtain the concentration profile of P in the cytoplasm:

$$\begin{eqnarray}&&{P}_{{\rm{out}}}(r)={P}_{\infty }-\displaystyle \frac{R}{r}\left({\rm{\Delta }}-\displaystyle \frac{{\hat{P}}_{{\rm{out}}}{l}_{c}}{R}\right){{\rm{e}}}^{-(r-R)/\xi },\end{eqnarray} \tag{ C.1 }$$

where ${P}_{\infty }$ is the cytoplasmic concentration of P far from droplets, or far-field concentration, and ${\rm{\Delta }}\equiv {P}_{\infty }-{\hat{P}}_{\mathrm{out}}$ is the supersaturation concentration. The influx of P material from the cytoplasm at the droplet interface ($4\pi {{DR}}^{2}{\rm{d}}{P}_{\mathrm{out}}/{\rm{d}}r{| }_{r=R}$) is

$$\begin{eqnarray}&&4\pi {DR}\left({\rm{\Delta }}-\displaystyle \frac{{\hat{P}}_{\mathrm{out}}{l}_{{\rm{c}}}}{R}\right),\end{eqnarray} \tag{ C.2 }$$

where we have used the small droplet condition ($R\ll \xi$). In this small droplet regime the concentration inside droplets is homogeneous (${P}_{\mathrm{in}}(r)={\hat{P}}_{\mathrm{in}}$). Therefore the depletion rate of P material inside a droplet by the chemical conversion $P{\to }^{k}S$ is given by

$$\begin{eqnarray}&&\displaystyle \frac{4\pi {R}^{3}}{3}{k}_{i}(\alpha ){\hat{P}}_{\mathrm{in}}.\end{eqnarray} \tag{ C.3 }$$

We can now write the total influx of P at the droplet interface:

$$\begin{eqnarray}&&J\,=\,4\pi {DR}\left({\rm{\Delta }}-\displaystyle \frac{{\hat{P}}_{\mathrm{out}}{l}_{{\rm{c}}}}{R}\right)-\displaystyle \frac{4\pi {R}^{3}}{3}{k}_{i}(\alpha ){\hat{P}}_{\mathrm{in}},\end{eqnarray} \tag{ C.4 }$$

where the first term accounts for the influx from the cytoplasm (equation (C.2)) and the second term accounts for the chemical conversion of P into S inside the droplet (equation (C.3)). At small droplet radius R, the reaction term ($\propto {k}_{i}$) in equation (C.4) is negligible compared to the diffusive term. The droplet growth dynamics can therefore be mapped onto the growth dynamics in an equilibrium system [34], where the supersaturation ${\rm{\Delta }}\equiv {P}_{\infty }-{\hat{P}}_{\mathrm{out}}$ is controlled by the chemical reaction rates, via the term ${P}_{{\rm{\infty }}}$. The largest possible droplet radius, ${R}_{u}$ (equation (13)), is independent of the system size. Hence at small enough droplet number density the presence of droplets does not affect the far-field concentration ${P}_{\infty }$. Therefore ${P}_{\infty }$ is equal to $\bar{P}$ (equation (12)), the overall concentration of P in the system. The value of the supersaturation Δ is then equal to that in the equilibrium condition, in the diffusion limited growth regime (${\rm{\Delta }}=\bar{P}-{\hat{P}}_{\mathrm{out}}$) [41].

If the droplet number density is large so that the inter-droplet distance is small compared to ξ, we can neglect the effect of the chemical reactions on the profiles and we recover the equilibrium profile [34]:

$$\begin{eqnarray}&&{P}_{{\rm{out}}}(r)={P}_{\infty }-\displaystyle \frac{R}{r}\left({\rm{\Delta }}-\displaystyle \frac{{\hat{P}}_{{\rm{out}}}{l}_{{\rm{c}}}}{R}\right).\end{eqnarray} \tag{ C.5 }$$

This result can be verified by expanding equation (10) for the small parameters R/ξ and L/ξ where L is the inter-droplet distance, and by imposing the boundary conditions (equations (6)–(9)). Similar to the case of large droplet number density (appendix C.1) the depletion rate of P inside a droplet is given by equation (C.3) and therefore the total influx of P at the droplet interface is given by equation (C.4). Again, for small droplet radius R the reaction term in equation (C.4) is negligible and the droplet growth dynamics can be mapped onto an equilibrium growth. Since the cytoplasmic profile is identical to that in an equilibrium system (equation (C.5)), the same must be true for Δ. Therefore the droplet growth dynamics is identical to that in the equilibrium condition, and if the number of droplet is large enough, Δ is small and the system undergo Ostwald ripening [41].

These equivalences between equilibrium and non-equilibrium regimes allow us to estimate SG growth dynamics far from the steady-state (section 5.3).