Chemical reaction-controlled phase separated drops:
Formation, size selection, and coarsening
Phase separation under non-equilibrium conditions is exploited by biological cells to organize their cytoplasm but remains poorly understood as a physical phenomenon. Here, we study a ternary fluid model in which phase-separating molecules can be converted into soluble molecules, and vice versa, via chemical reactions. We elucidate using analytical and simulation methods how drop size, formation, and coarsening can be controlled by the chemical reaction rates, and categorize the qualitative behavior of the system into distinct regimes. Ostwald ripening arrest occurs above critical reaction rates, demonstrating that this transition belongs entirely to the non-equilibrium regime. Our model is a minimal representation of the cell cytoplasm.
Phase separation is a ubiquitous phenomenon in our physical world, ranging from cloud formation to oil drop formation in water Bray (2002). Recently, it is also realised that phase separation is exploited in the cell cytoplasmic organization in the formation of non-membrane bound organelles called ribonucleoprotein (RNP) granules Brangwynne (2011); Hyman et al. (2014). RNP granules are a diverse set of structures that play important roles in the functioning of the cell, from RNA processing and stress response Anderson et al. (2009); Protter and Parker (2016), to cell division Zwicker et al. (2014) and germ line specification Brangwynne et al. (2009); Voronina et al. (2011). However, the mechanisms enabling the rapid and controlled assembly and disassembly of RNP granules have only begun to be investigated. Chemical reactions, e.g., in the form of ATP-driven enzymatic reactions that convert one protein state to another (e.g., unphosphorylated to phosphorylated) are prime candidates for the cell to manifest controlled phase separation. For instance, such a scheme has been proposed as a mean to induce localised phase separation in the C. elegans. embryo Lee et al. (2013); Saha et al. (2016); Weber et al. (2017), and to organise the centrosomes prior to cell division Zwicker et al. (2014). However the physics of non-equilibrium phase separation driven by chemical reactions has only started to be investigated. For instance, non-equilibrium processes have been discussed in the context of lipid domains in plasma membranes Turner et al. (2005); Fan et al. (2008). More recently, it has been realised that although in equilibrium phase separation, a multi-drop, finite system will invariably coarsen to a single condensed drop via Ostwald ripening, chemical reactions can arrest this ripening process completely in a binary fluid Glotzer et al. (1995); Zwicker et al. (2015). Here, we categorize comprehensively and under general conditions, how unimolecular reactions that convert a two-state molecule between a phase-separating state and a soluble state can control drop formation, coarsening, and size selection. We achieve this by generalizing and improving upon the assumptions adopted in Zwicker et al. (2015). Specifically, contrary to Zwicker et al. (2015), we analyse the regimes of large drops and non negligible supersaturation, include the presence of cytosol by going beyond the binary fluid restriction, and allow for arbitrary equilibrium concentrations inside and outside drops. Our model is arguably the minimal model relevant to the mechanism of chemical reaction-controlled phase separation in the cell cytoplasm.
Our ternary mixture consists of two molecular states, one phase-separating () and one soluble (), plus the solvent or cytosol (). States and can be converted into each other by the chemical reactions
where and are the reaction rate constants. The non-equilibrium nature of these reactions lies in the fact that both reaction rates are independent of the local concentrations and thus have to be driven by free energy consumption. In the context of the cell, these reactions can be, e.g., ATP-driven post-transcriptional protein modifications Alberts et al. (2008), that affect protein phase-separating behavior. For example, the phase separation of intrinsically disordered proteins can be regulated via their phosphorylation/dephosphorylation Li et al. (2012); Bah and Forman-Kay (2016).
At equilibrium (), a finite system will inevitably coarsen via Ostwald ripening Lifshitz and Slyozov (1961) and drop coalescence Siggia (1979). Here, we assume that drop diffusion is negligible so we will focus exclusively on the Ostwald ripening. In the cell context, this is motivated by the strong suppression of macromolecular diffusion in the cell cytoplasm Weiss et al. (2004). Ostwald ripening results from two effects: 1) the Gibbs-Thomson relation dictating that for a drop of size , the concentrations of solute inside and outside the drop next to the interface are and respectively, where is the capillary length and are the equilibrium phase coexistence concentrations (see Fig. 1a)); and 2) the concentration profile of the solute in the dilute phase is given by the steady-state solution to the diffusion equation (the quasi-static assumption). These two effects combined lead to a diffusive flux of solute from small drops to big drops Lifshitz and Slyozov (1961).
When chemical reactions are switched on, we assume that local thermal equilibrium remains valid so that the interface boundary conditions for are unchanged 111We have verified these conditions using simulation methods SI ().. is considered inert to phase separation in the sense that its concentration profile is continuous across the interface 222 This assumption is not essential and we describe the more general case where is discontinuous at the drop interface in SI ().. In addition, we assume that the concentration profiles inside and outside the drops are given by:
where and denote the concentration profiles of and inside and outside drops with subscripts “in” and “out”, respectively. For simplicity, we assume the same diffusion coefficient for both species and in both phases.
To see why Ostwald ripening can be arrested in our ternary mixture, we will now provide an intuitive argument based on a similar consideration for binary mixtures Zwicker et al. (2015). We consider a homogeneous system of total solute concentration where is the total concentration of in the system. If the supersaturation is positive drops can be nucleated, initiating phase separation (Fig. 1a)). At small the drop density is low and drops only interact with the far-field concentration. We focus only on the early growth regime so that the supersaturation remains close to : for a drop of radius , the diffusive profile leads to an influx of into the drop at the rate Lifshitz and Slyozov (1961)
At the same time, the chemical reactions inside the drop lead to a depletion of at the rate
As increases, the depletion rate will eventually surpass the influx from the medium, so that the balance between Eqs. (4) and (5) leads to a steady-state radius. In the limit of large so that we can ignore the term (but still small such that ), the steady-state is
In other words, we expect that in a multi-drop system, the size of all drops are given by Eq. (6). We shall see that this regime in fact corresponds to the upper bound of stable in a multi-drop system (Fig. 3).
In our argument above, we have neglected the reverse reaction and the the effect of the chemical reactions on the diffusive profiles, which, as we shall see, can significantly change the system’s behavior. We will now incorporate these effects into our analysis. We will also consider arbitrary supersaturations so that drops may be close to each other. As a result a far-field concentration may not exist, rendering the Lifshitz-Slyozov theory Lifshitz and Slyozov (1961) inapplicable. Consider a multi-drop system such that drops are on average a distance apart where is of the order with being the drop number density. For simplicity, we will first focus on two spherical subsystems of radius , each having a spherical drop in their center (Fig. 1b)). We assume that the concentrations and their gradients at the boundaries of the two subsystems match (Fig. 1c)). The rational for this approximation is that in a multi-drop system, the actual boundary conditions are influenced by many neighbouring drops and we treat these fluctuating boundary conditions in a mean-field manner by assuming spherical symmetry around the drops. In other words, the concentration outside a drop depends only on the distance from the drop centre. Moreover it is assumed that the two-drop system is stable (unstable) if the full multi-drop system is stable (unstable). The validity of this approximation will be verified later using Monte Carlo simulations. The corresponding boundary conditions, besides the Gibbs-Thomson at the drops’ interfaces, are
and the same apply to . The subscript denotes the drop index. Note that we use two different coordinate systems and , each having their respective drop’s center as the origin (Fig. 1b)).
Using the quasi-static approximation as in the equilibrium case, the steady-state concentration profiles of this two-drop system with radii and , such that , are SI ():
In the above, are independent of and are given in SI (), , denote the combined concentration inside and outside the -th drop, respectively, and are also independent of . Furthermore, , , and is the concentration gradient length scale. The profiles are given by for and for . Note that generally, is independent of only when , which we have assumed to be true here as we will focus on the case .
The volumetric growth rate of the -th drop in this two-drop system is Zwicker et al. (2014)
Given the drop growth rate above we can study the steady-state drop radius at which the two drops of the same size are in the steady-state ().
We can also analyse its stability by calculating the drops’ growth rates upon perturbing their sizes: and . Performing a linear stability analysis, we take and expand the growth rate with respect to :
The surface plot in Fig. 2 shows for a fixed backward reaction rate , the region of existence of the steady-state radius , delimited by the coloured outer surface. Above the black inner surface, the system consists of stable monodisperse drops whose sizes are controlled by the rate , the solute concentration and the drop number density (not fixed in Fig. 2). Outside the stable region but still within the outer surface, the monodisperse system is in an unstable steady-state and drops coarsen via Ostwald ripening. Outside the outer surface, drops always shrink.
Interestingly, there are qualitative changes in the system’s behaviour as varies with fixed as shown in Fig. 3, which describes multi-drop stability at fixed solute concentration . When (blue arrow), the system is in the Lifshitz-Slyozov regime and coarsen (upward arrows), while for (green arrow), the system can be stable (grey region), with co-existing drops of radius determined by . In other words, is the critical rate beyond which Ostwald ripening is arrested. Between and (black arrow), the system can also be stable, but with an upper bound on the radius. Beyond , no drops can exist in the system as all drops evaporate (downward arrows).
So far, our calculation has been based on our two-drop system with the mean-field matching assumption at the system boundaries. To test this assumption, we perform Monte Carlo simulations of our ternary model on a 2D lattice with multiple drops to detect the stability-instability boundary (black inner surface in Fig. 2 and dashed curve in Fig. 3) and compare the results with our predictions (see SI () for simulation details.) The good agreements are shown in the inset of Fig. 3 and in SI ().
We will now explain analytically the salient features of the stability diagram by focusing on distinct limits in the small supersaturation limit.
Upper bound on drop radius. We have seen that if , there is an upper bound on the drop radius. We focus here on the regime , which we will see is indeed the case when . We first analyse the limit of small drop number density so that the distance between drops is large: . By expanding with respect to the small parameters and we seek the set of such that the solutions to cease to exist. We find that the expression of this boundary is SI ():
Stability-instability boundary. For the stability-instability boundary we consider large so that . In this case, the small parameters are and . By expanding and around these small parameters, we solve for the steady-state and then seek the boundary of stability by looking at . The functional form of this boundary is SI ():
Critical reaction rate . The rate beyond which drops dissolve is the solution of and is maximally bounded as follow SI ():
Note that corresponds to the situation where the conversion is so strong that the system is outside the equilibrium phase-separating region (, see Fig. 1a)).
Lower and upper critical rates ( & ). Here we focus on the large drop limit so that the small parameters are and (since ). By expanding with respect to these two small parameters, we solve again for and investigate the corresponding stability by looking at . Specifically, we find SI ():
where is the transition rate from the stable to the unstable regime, and is the rate beyond which drops’ radii have an upper bound. Thus, the transition to the non-equilibrium regime, namely the arrest of Ostwald ripening, occurs at non-zero reaction rate (). This behavior has never been reported before in this system.
Finally, we note that given the richness of the system’s behaviour, the generic features of the stability diagram can vary according to , which we have explored in SI () and in the context of cellular response to environmental stresses wurtz_a17b ().
In summary, we have studied a phase-separating ternary fluid mixture with chemically active drops. We have categorised the qualitative behavior of the system into distinct regimes based on the reaction rates using a combination of analytical, numerical, and simulation methods. Our work is of direct importance to cytoplasmic organisation, and is also relevant to the control of emulsions in the engineering setting. Interesting future directions include the incorporation of drop coalescence into our coarsening picture, the study of potential shape instabilities in chemically active drops Zwicker et al. (2016), and the generalization of our formalism to many-component mixtures Sear and Cuesta (2003); Jacobs and Frenkel (2017).
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Chemical reaction-controlled phase separated drops:
Formation, size selection, and coarsening
Part I General theory
Appendix A Concentration Profiles and drop growth rates
with the drop label, the radius of the -th drop and the radius of each sub-system (see main text and Fig. 1b)). In a multi-drop system corresponds to the mean separation between drops and is related to the drop number density :
We denote the total solute concentration by where , are the total concentration of and , respectively. When phase separation does not occur, the system is homogeneous (), and by taking the volume integrals of Eqs. (S1)-(S4) over the whole system we have
with . When phase separation occurs and the system is at the steady-state, the concentration gradients must match eactly at the interface. Therefore the diffusion terms cancel out in the volume integrals of Eqs. (S1)-(S4) and we recover Eqs. (S6)-(S7). Later we will focus our analysis on small deviations from the steady-state and will approximate by Eqs. (S6)-(S7). Adding Eqs. (S1) + (S2) and Eqs. (S3) + (S4) gives
which we solve for two or three spatial dimensions, with spherical or circular symmetry, respectively:
with and constants, and is the number of spacial dimensions. Inside the drops (“in”), the total concentration must not diverge in the drop center (), therefore and is equal to a constant :
Outside the drop (phase “out”) and if the total concentration must be continuous at the boundary between the two sub-systems () therefore . In our study we will focus on small differences in drop radii () and we make the approximation that remains zero. Therefore is equal to a constant in both sub-systems:
We can express and in terms of the concentrations at the drops’ interfaces ():
and , . The concentrations and their gradients must be continuous at the sub-system boundaries () and we assume the Gibbs-Thomson relations hold at the interface (). This gives the following boundary conditions
where and are the equilibrium coexistence concentrations of at the interface (see main text Fig. 1a)) and is the capillary length. We solve the system Eqs. (S14)-(S19) here in spherical symmetry () or circular symmetry ():
is the gradient length scale, and are the 0-th order Bessel functions of the first and second kind, respectively, and is the imaginary unit . and are independent of and are solutions of the system:
The -th drop volumetric growth is Zwicker et al. (2014):
Appendix B Concentration jump of S at the drop interface
Appendix C Steady-state drop radius
A system with identical drop radii is at steady-state if the drop growths (Eq. (S31)) are zero. Therefore the steady-state condition is
and we solve for and :
where we have dropped the unnecessary upper script in ,
Appendix D Linear stability of the steady-state
We perturb the drop sizes about the steady-state:
with . We focus on the growth rate of the drop 1 (). Expanding for the small parameter :