key: cord-0786993-4zdvlnfh
authors: Bartolucci, G.; Adame-Arana, O.; Zhao, X.; Weber, C. A.
title: Controlling composition of coexisting phases via molecular transitions
date: 2021-03-01
journal: Biophysical journal
DOI: 10.1016/j.bpj.2021.09.036
sha: 08710fe100dd06c7e09966fe03e50b9ea26c105d
doc_id: 786993
cord_uid: 4zdvlnfh

Phase separation and transitions among different molecular states are ubiquitous in living cells. Such transitions can be governed by local equilibrium thermodynamics or by active processes controlled by biological fuel. It remains largely unexplored how the behavior of phase-separating systems with molecular transitions differs between thermodynamic equilibrium and cases where detailed balance of the molecular transition rates is broken due to the presence of fuel. Here, we present a model of a phase-separating ternary mixture where two components can convert into each other. At thermodynamic equilibrium, we find that molecular transitions can give rise to a lower dissolution temperature and thus reentrant phase behavior. Moreover, we find a discontinuous thermodynamic phase transition in the composition of the droplet phase if both converting molecules attract themselves with similar interaction strength. Breaking detailed-balance of the molecular transition leads to quasi-discontinuous changes in droplet composition by varying the fuel amount for a larger range of inter-molecular interactions. Our findings showcase that phase separation with molecular transitions provides a versatile mechanism to control properties of intra-cellular and synthetic condensates via discontinuous switches in droplet composition.

The ability to phase separate in aqueous solutions and to interconvert between distinct molecular states are two properties common among a large number of biomacromolecules. For example, DNA sequences composed of self-complementary strands phase-separate into DNA-rich condensates that coexist with a DNA-poor phase [1] [2] [3] . These sequences can also switch between an expanded ("open") DNA configuration and a collapsed ("closed") configuration where the self-complementary strands of the same DNA are base-paired. Another example are proteins that form phase-separated condensates in vitro [4] [5] [6] and in living cells [7, 8] . Proteins attract each other, as well as other macromolecules such as RNA and DNA, for example, by sticky amino-acid patterns such as RGG repeats [9] [10] [11] or charged domains [12, 13] . These interaction sites are considered to drive protein phase separation but also mediate conformational transitions where certain domains of the protein rearrange [14] .

In living cells, phase separation of macromolecules such as proteins and RNA can also be actively controlled by biological fuel. Hydrolysis of ATP by kinase and phosphatase, for example, enables the regulation of the phosphorylation states of proteins. The number of phosphoryl groups attached to a protein, in turn, affects its net charge [15] and may modify localized interaction sites along the sequence [16] . Thus, the phosphorylation state of proteins strongly determines their interactions and thereby their propensity to form aggregates and phase-separated compartments [17] [18] [19] .

To effectively account for the impact of fuel on chemical reactions and phase separation, various theoretical models were proposed with chemical reactions that are devoid of thermodynamic constraints and thus break detailed balance of the rates [20] [21] [22] [23] . These systems give rise to novel non-equilibrium phenomena such as bubbly-phase separation [22] and shape instabilities leading to droplet division [24] . Up to now, there has been a strong focus on establishing new universality classes and observing new non-equilibrium phenomena. However, the physical mechanisms of how transitions between different molecular states affect phase separation remain largely unexploreddespite their ubiquity in biological systems. In particular, it remains unclear how these systems respond to thermodynamic perturbations such as temperature and concentration changes, and how fuel-driven molecular transitions affect the phase separation behavior. This is mainly due to the lack of physical models for phase separation and chemical reactions with coarse-grained parameters that are linked to molecular properties of biological macromolecules.

To gain insight into how molecular transitions are driven by fuel control phase separation, we propose a thermodynamic model for polymers, such as DNA and proteins, which can phase-separate and undergo molecular transitions between two molecular states. At thermodynamic equilibrium, we show that the interplay between phase separation and the molecular transitions gives rise to reentrant phase behavior as a function of temperature. Moreover, we also find a discontinuous thermodynamic phase transition in droplet composition when changing the temperature. Similar behavior also exists away from thermodynamic equilibrium where consumption of fuel breaks detailed balance of the rates associated with the molecular transition. In this case, droplet composition can abruptly switch upon small changes in fuel level. In contrast to the corresponding thermodynamic system, the fuel-driven molecular transition releases the Gibbs phase rule constraint. This release allows abrupt switches in droplet composition within a range of molecular interaction parameters that is significantly broader compared to the system at thermodynamic equilibrium. Owing to the very general nature of our model, our findings suggest that many different phase-separating macromolecules that are capable of undergoing molecular transitions can switch the composition of dense phase via fuel-driven molecular transitions.

Here, we consider mixtures composed of unstructured macromolecules, such as DNA or proteins, and a solvent W . In our model, these macromolecules can undergo molecular transitions (e.g. changes in conformation or charge) between two states, referred to as A and B. The transition between the two states is described similarly to a chemical reaction with the scheme (Fig. 1a) . The corresponding transition rates depend on thermodynamic control parameters such as temperature T and macromolecules concentration, as well as fuel volume fraction φ F that drives the transition away from thermodynamic equilibrium. In our model, both components A and B can phaseseparate due to attractive interactions among them (Fig. 1b) .

We describe the mixture in the T -V -N i -ensemble making use of a free energy density f (T, φ i ) = behavior of unstructured polymers, we use a Flory-Huggins free energy density of the form

where k B is the Boltzmann constant, r i = ν i /ν W are the ratios of the molecular volumes relative to the solvent W , and i = A, B, W . For simplicity, we consider A and B having equal molecular volumes: ν A = ν B ≡ ν and thus r A = r B ≡ r, implying that the molecular transition conserves the volume of the system, V = ν i N i . Moreover, in our model, all molecular volumes are constant making the system incompressible at all times and leading to the incompressibility condition

For simplicity, we consider the fuel (and potential waste products) to be dilute or having a small molecular volume, thus not affecting the incompressibility condition. The first term in Eq. (1a) is the mixing entropy governing the tendency of the mixture to remain mixed [25, 26] . The second term describes the interactions among the molecules, where Ω ij denotes the interaction parameter between the components i and j. These interaction parameters depend on temperature. To lowest order, we can write [27] 

where e ij and s ij are temperature independent interaction energies and entropies per molecule.

The third term in Eq. (1a) accounts for the free energy needed to create the molecular components (e.g. via molecular transitions) and ω i (T ) denotes the internal free energy of component i (in units of k B T ). In particular, ω B − ω A corresponds to the internal free energy difference between the two molecular states A and B, which we express as

Here, e int and s int are the energetic and entropic differences between state A and B. As a prototypical example, we consider conformational transitions leading to an energy and entropy increase converting A to B, i.e. e int > 0 and s int > 0. Such a scenario corresponds for example to polymers switching from a tightly bound configuration to an unfolded one. For a dilute system, the temperature at which the internal free energy difference vanishes, ω B (T ) = ω A (T ), defines the melting temperature

above/below which B/A is the favored molecular state in the dilute limit where the interactions terms in Eq. (1a) are expanded up to the first order in volume fraction.

Chemical equilibrium is reached between molecular states if [28] [29] [30] 

where the exchange chemical potentials are given by [29] 

Here, f is the free energy density after using the incompressibility condition Eq. (1b). Exchange chemical potentials are related to the chemical potential differences difference between molecules and solvent, as explained in detail in Appendix A. Using the free energy density f (Eq. (1a)), the condition of chemical equilibrium can be written as

This relationship represents the non-ideal mass action law for the molecular transition depicted in Fig. 1 ; for a general discussion of non-ideal chemical reactions in multi-component mixtures, see Ref. [31] . The condition of chemical equilibrium (6), together with incompressibility Eq. (1b), reduces the number of independent variables from three to one (see Appendices A 1 and A 2 for details). Here, we consider the total volume fraction of A and B

which is conserved in the molecular transition, as independent variable. By means of Eqs. (6) and (7), we can recast Eq. (1a) in the form f = f (T, φ tot ), which can be used to determine the phase diagram via the common tangent construction (i.e., Maxwell construction). In fact, imposing the equivalence between exchange chemical potential and osmotic pressure in both phases, the conditions for phase coexistence of the phase I and II read [32] :

An example of Maxwell construction for the free energy density at chemical equilibrium is shown in Fig. 8 , in Appendix A 3.

Using irreversible thermodynamics, the kinetic equations for an incompressible, ternary mixture read (see Appendix C for the derivation)

where the inhomogeneous chemical potentials are given as [33] 

These chemical potentials are related to the free energy

viaμ i = ν i δF/δφ i , where the free energy density f is given by Eq. (1a). Moreover, Λ i is the diffusive mobility which depends on volume fraction. To ensure that the diffusion coefficient of A and B is constant in the dilute limit, we use the following scaling form, [34] , and consider Λ i,0 to be constant for simplicity. This choice indeed cancels the divergence stemming from the logarithmic terms in the free energy density. Thus, the diffusion constant in the dilute limit is given by D i = k B T Λ i,0 . For simplicity, we restrict ourselves to the special case of a zero

Onsager cross-coupling coefficient.

The kinetics of the chemical transition is captured by the reaction flux s(φ A , φ B ) which depends on the volume fraction of both molecular components. Here, we consider two different cases corresponding to different reaction fluxes. The first case is a system that relaxes toward thermodynamic equilibrium, suggesting the following form (for the derivation using linear response, see Appendix C):

where Λ s denotes the mobility for the molecular transition, which we consider to be constant for simplicity. If chemical potentials are homogeneous (μ i = const.) and the chemical potentials of A and B are equal (μ A =μ B ), the system is at thermodynamic equilibrium, self-consistently leading

The second case refers to a system, where the molecular transitions cannot relax toward thermodynamic equilibrium, i.e., detailed balance of the rates is broken [23, 35] . In living or active systems, this is often facilitated by a "fuel" component F which affects the balance between the two molecular states and is, to a good approximation, maintained by chemical reaction cycles [36, 37] .

Here, we consider a combination of the flux (12) and a second order chemical reaction that depends on the fuel volume fraction φ F (see Appendix D for details):

Here, k ← and k → are the independent rate constants of the backward and forward transition, respectively. This independence of rate constants implies that detailed balance of the rates corresponding to the molecular transition is broken; for a conceptual discussion see Ref. [23] . In contrast to Eq. (12), a system with a reaction flux s given by Eq. (13) cannot fulfill the two equilibrium conditions of equal and spatially constant chemical potentials. Thus, stationary solutions to Eqs. (9) using Eq. (13) are non-equilibrium steady states. Consistently with this, in the absence of fuel (φ F = 0), the reaction flux above reduces to Eq. (12) and the system can relax to thermodynamic equilibrium. In our model, the fuel level controls how far the system is maintained away from thermodynamic equilibrium.

Finally, we look for an equation for the fuel volume fraction φ F . To this end, we focus on the case where diffusion of fuel is fast compared to diffusion of the macromolecules A and B,

respectively. This limit is indeed reasonable for many biological systems since diffusivities for example between phase-separating macromolecules (proteins, RNA,...) and ATP differ by about two orders in magnitude [38, 39] . For simplicity, we consider the case of fuel being conserved,

i.e., it is maintained constant in time. This scenario applies to living cells under physiological conditions and in in vitro systems, where these conditions could be realized by encapsulated ATP or regeneration of ATP. Moreover, we assume that the fuel molecules interact in the same way with A and B. In this case, we can quasi-statically slave the fuel volume fraction φ F to the total concentration of A and B,

whereφ F denotes the average volume fraction of fuel that is constant in time. The choice above allows to capture the partitioning of fuel by accounting for the spatial correlations between fuel and the total volume fraction φ tot . The fuel partitioning coefficient P F , that is experimentally accessible, determines the values of the parameters α and β in Eq. (14) (see Appendix E for a definition of P F and the link to α and β).

In the following, we choose three parameter sets for α and β corresponding to three qualitatively different scenarios. First, the fuel partitions inside the φ tot -rich phase for α = 0 and β = 1/φ tot .

Second, fuel is enriched outside corresponding to α = −β = 1/(1 −φ tot ). And finally, we also consider the case of a homogeneous fuel for α = 1, β = 0. The latter case has been studied for example in Refs. [20, 40] .

In this section, we study the equilibrium phase diagrams as a function of the total volume fraction φ tot and a scaled temperature T /T 0 , with T 0 = −e BB /k B . In such phase diagrams, the binodal lines separate demixed and mixed thermodynamic states. Along the binodals, we also depict the composition in terms of the relative abundance of B molecules, φ B /φ tot (see the color code in Fig. 2 ). We then study how such phase diagrams are affected by the melting temperature T m . As a reference system, we consider a binary mixture composed of only B and W molecules (black lines in Figs. 2a-c). In all our studies, we choose r = 2 to account for differences in molecular volumes between macromolecules and water.

We first study the case of only attractive homotypic B-B interactions and neglect entropic contributions for simplicity (s ij = 0 in Eq. (1c)). Specifically, Ω BB = e BB and Ω ij = 0 otherwise.

Attractive homotypic interactions lead to phase coexistence between a φ tot -rich and a φ tot -poor phase, in the following referred to as the dense and dilute phase, respectively. Below the melting temperature T m (Eq. (3)), we find that coexisting phases are not only different in φ tot , but can also differ in the amount of B relative to A (see color code in Fig. 2a ). In particular, the dense branch of the binodal corresponding to large values of φ tot is rich in B components, while the composition of the dilute branch (low φ tot ) changes with temperature. The composition, defined as the relative amount of molecules in the B conformation, φ B /φ tot , remains rather uniform in the dense branch.

In fact, in this example, attraction among B molecules is the only interaction considered, thus the dense phase must be composed dominantly of B molecules. However, in the dilute and B-poor phase, the fraction of B is affected by temperature since the molecular transition, which favors A over B at low temperatures, dominates over the homotypic B-B interaction.

Increasing the melting temperature T m enhances the dominance of the molecular transition at low temperatures leading to a qualitative change of the thermodynamic phase diagram (Fig. 2b ).

In fact, we find a lower dissolution temperature (LDT), which implies a reentrant phase behavior.

Such behavior is manifested as the possibility that both increasing and decreasing the temperature leads to a phase transition from a demixed to a mixed state, respectively. The lower dissolution temperature is given by (for a derivation see Appendix B)

and is set by the competition between the homotypic interactions e BB and the molecular transition which is characterized by the melting temperature T m (Eq. (3)). For attractive interactions, Eq. (15) fulfills T d < T m , which is consistent with the fact that the majority of molecules have to be in the A state to undergo a phase transition to a mixed state for decreasing temperatures. A and the solvent, differences in molecular volumes of A and B with respect to the solvent (i.e.

r > 1) favors phase-separating B from A. In other words, two coexisting phases of different solvent content are entropically disfavoured. This entropic disadvantage implies that both phases will be up-shifted in φ tot leading to two coexisting phases rich in B and A, respectively, instead of B and solvent rich, as in the binary reference. This entropic disadvantage of phase-separating A with respect to B also explains the increase in critical temperature.

The qualitative difference among the phase diagrams can be summarized in terms of the internal entropy and energy differences between the two states A and B (Eq. (2)); see 

. We find a relationship for the transition line, −e int /e BB = 1 + s int r/2, which agrees with results from the Maxwell construction depicted in Fig. 2d . This relationship implies that the reentrant region widens with the internal entropy difference s int . Larger s int leads to more phase-separating B molecules and thus favors phase coexistence over mixing.

Here we discuss the case of attractive homotypic interactions for both molecular states, i.e., e AA < 0 and e BB < 0 and quantify the impact on the phase diagram of the relative A-A interaction strength e AA /e BB . For simplicity, we now fix T m to a value for which reentrant behavior was observed in the case of only B-B interactions, as discussed in the last section. Due to the attractive homotypic A-A interaction, we find an additional demixing region in the phase diagram at low temperatures. This region is disconnected from the reentrant region of the phase diagram, which is located at higher temperatures (Fig. 3a) . This new region at low temperatures has an upper critical solution temperature (UCST) and describes phase coexistence between φ tot -rich and φ totpoor phases that are both mainly composed of A. Thus, these phase diagrams have two UCSTs and one LDT. upon temperature changes between A-rich and B-rich. As mentioned, the triple line temperature is exactly the melting temperature T m (defined in Eq. (3)). The reason is that for e AA = e BB , both molecular states have the same phase separation propensity leaving it to the internal free energy balance to determine the composition of the dense phase. For total volume fractions larger than the dense binodal branch, there is also a discontinuous switch in A-B composition (see the black line in Fig. 4a) , however, the system is homogeneous in this case. When A-A and B-B attraction strengths approach each other, the total volume fraction of the dilute branch decreases, and the two denser branches, each corresponding to different A-B composition, merge (Fig. 4b) . Thus, also for the A-A attraction slightly weaker than the B-B attraction, crossing the temperature corresponding to the triple point leads to a jump in both composition and total volume fraction of the dense phase. For e AA = e BB , the jump only occurs in composition while the total volume fraction changes continuously.

The discontinuous change in the composition of the dense phase is tied to the existence of a triple point. This triple point in turn arises from the tendency of the ternary mixture to form three coexisting phases stemming from similar attractive interactions among A and B molecules, respectively. However, molecular transitions affect the dimensionality of the three phase coexistence domain in the phase diagram. Three phase coexistence is consistent with the Gibbs phase rule. In fact, due to the molecular transition, the system is reduced to an effective binary mixture solely characterized by the conserved variable φ tot . This reduction does not suppress the three phase coexistence region since the Gibbs rule for a binary mixture at fixed temperature and pressure allows for a maximum of three coexisting phases. However, in our case, the molecular transition at thermodynamic equilibrium reduces the dimensionality of this region by at least one, giving rise to a triple point or even a triple line.

The coexistence of three phases and thus also the triple point is controlled by heterotypic interactions e AB (see Fig. 4c ). In particular, for strong attractive A-B interactions, three phase coexistence is suppressed. Consistently, the triple point vanishes for increasing heterotypic interaction strength implying that the composition of the dense phase be cannot upon temperature changes.

Instead, reentrance forces phase separation to vanish within a narrow intermediate temperature window (green shaded domain in Fig. 4c ). For even larger heterotypic interaction strength, the composition of the dense phase changes continuously (orange line in Fig. 4c ).

In the last section, we focused on how temperature affects the composition of coexisting phases.

Up to now, there is no evidence that living systems control their temperature to regulate phase separation. Thus, we discuss how fuel-driven molecular transitions that are maintained away from thermodynamic equilibrium can control the composition inside droplets. For example, phosphorylation involving the hydrolysis of ATP is known to drive molecular transitions and thereby regulate protein phase separation [8, 18, 19] .

To this end, we consider two prototypical parameter sets: see Eq. (13) . Most importantly, the fuel-related contribution breaks detailed balance of the rates.

We numerically solve the kinetic Eqs. (9) and (13) in two dimensions with periodic boundary conditions combining the energy quadratization method [41] [42] [43] with with the stabilization method [44] (see Appendix F for details).

Breaking detailed balance of the rates allows the system to settle into non-equilibrium stationary states which differ from thermodynamic equilibrium in terms of droplet composition and droplet number density. To classify these non-equilibrium states, we numerically solve Eqs. (9) using Eq. (13) and initialize the system with one or two droplets of total volume fractions and A-Bcomposition corresponding to the thermodynamic phase diagrams. We find the emergence of various patterns ranging from equally sized droplets to rings and stripes ( Fig. 5 (a) For increasing effective backward rateφ F k ← , we often observe rings coexisting with droplets, see SI Movie II. For even larger effective ratesφ F k ← , rings break-up leading to the emergence of equally sized droplets (Fig. 5 (b) and SI Movie III). Equally sized droplets have been reported in a model using an Ginzburg-Landau type of free energy [40] . For fuel partitioning inside, droplets tend to strongly deviate from their spherical shape (see SI Movie IV) and deform into elongated domains reminiscent of stripes in pattern-forming reaction diffusion system [45] . Fuel partitioning also affects the boundary in the state diagram beyond which droplets dissolve.

For the case (ii) characterized by strong A-A homotypic interactions, i.e. e BB = e AA , e AB /e BB 1, the behavior in the k → -k ← state diagram changes significantly. For fuel partitioning outside, the composition is approximately independent of the rate constants and we exclusively find single, A-rich droplets that coexist with a solvent-rich phase (Fig. 5h) . The rate-independence implies that breaking detailed balance of the rates for the molecular transition does not enable control of phase separation in this setting. This changes if fuel is homogeneously distributed between the inside and outside or if fuel dominantly partitions inside the droplet phase ( Fig. 5 i,j) . In these cases, increasing the effective forward rate for low effective backward rates leads to a transition from an A-rich droplet to a B-rich droplet. In other words, for a large effective forward rate, this leads to a droplet of switched composition.

Between the two regions in the k → -k ← plane corresponding to single droplets of different composition (see Fig. 5 (i),(j)), there is a region where observe various kinds of patterns. In particular, for a low effective forward rate corresponding to the onset of pattern formation, we typically find that the shape of the A-rich domain deviates significantly from the initial single drop at thermodynamic equilibrium, while the composition of such domains remains close to the equilibrium value. Besides rings and equally sized drops, we also find patterns reminiscent of bubbly-phase separation [22] , see SI Movie V. For higher effective forward rates and low backward rates, we find patterns where three domains of different compositions stably coexist (see two triangles of different colors in Fig.   5 (i)-(j)). Representative stationary patterns composed of three domains are shown in Fig. 5 f,g, and SI Movies VI -VII. These patterns are spherically symmetric with a centered droplet enriched in B that is surrounded by a ring of smaller A-rich droplets of equal size. As mentioned above, for even higher effective forward rates, the system settles in a non-equilibrium stationary state composed of a single B-rich droplet.

For high values of k → and k ← in Fig.5 (h-j) and in contrast to (c-e), we find that the local composition of each domain is no more governed by the equilibrium values of a ternary mixture undergoing molecular transitions at thermodynamic equilibrium (as discussed in Sect. II B). The reason is that the fuel-mediated molecular transitions (detailed balance broken) dominate the thermodynamic ones and that A-A interactions are sufficiently strong. Surprisingly, the local composition of each domain in the fueled system is almost equal to the equilibrium value of a ternary mixture without molecular transitions. Contrary to the equilibrium system with molecular transitions that is effectively described by only one independent variable (i.e., φ tot ), a ternary mixture without transitions has two degrees of freedom (i.e., φ A and φ B ); the corresponding phase diagram for a fixed temperature is depicted in Fig. 6a . For ternary mixtures at thermodynamic equilibrium and parameters corresponding to the case (ii), there is a triangle in the φ A -φ B phase diagram where the vertices of the triangle correspond to the composition of each of the coexisting phases at thermodynamic equilibrium (shaded in orange in Fig. 6a ). In the presence of the detailed balance broken molecular transition, the composition in each domain is actually close to the vertices of the equilibrium triangle (see orange circles in Fig. 6a ). Changing the effective forward/backward transition rates amounts to moving the vector corresponding to the average volume fraction along the conserved trajectory φ B = φ tot − φ A . We now set the effective backward rateφ F k ← = 0 and k → = 2 10 3 t −1 D and consider variations of the effective forward rate induced by the average fuel amount. Specifically, in the absence of fuel (φ F = 0), an A-rich phase coexists with a solvent-rich phase at thermodynamic equilibrium (blue domain in phase diagram). Intermediate amounts of fuel induce a transition to a non-equilibrium stationary state comprised of three coexisting domains, see Fig. 6e . Thus, when the fuel is changed by an amount larger than this narrow window, the long-time, stationary state of the systems swaps from an A-rich to a B-rich domain. In general, this does not necessarily imply that the initial single droplet swaps its composition without dissolving. However, our results suggest the intriguing possibility, that under some conditions, a droplet may be able to change its composition without losing its identity related to the profile of the total volume fraction.

To test the possibility of a single droplet to kinetically swap its composition with time as the fuel is increased, we initially start with an A-rich droplet at thermodynamic equilibrium (φ F = 0) and gradually increase the average fuel volume fraction until it reaches a plateau value, and then gradually decrease it (Fig. 7a) . As the average fuel volume fraction is increased, a B-rich domain appears in the center of the initial A-rich droplet. This domain grows and splits into concentric rings enriched in A and B, respectively (Fig. 7d) . The outermost B-rich domain radially propagates inwards and outwards facilitating the formation of a final B-rich droplet. This inversion occurs approximately concomitant to the average fuel amount exceeding some specific value (Fig. 7e) .

Thus, the composition has indeed swapped with time compared to the initial state. As the fuel is gradually decreased, B material is consumed causing the droplet to shrink and a release of A material near its interface, see SI Movie VIII. This process forms an A-rich outer ring that relaxes with time to the spherical shape -the composition has swapped back to its initial value. These results highlight that the composition of a droplet can indeed be controlled and reversibly switched by fuel that breaks detailed balance of the rates without dissolving and re-nucleating the droplet at another position. 

tot , for various homo-and heterotypic interactions. These results show that for the temperature chosen, the region of the "active switch" is indeed much broader than the region corresponding to the"thermodynamic switch" (Fig. 7f) .

In this work, we derived and analyzed a theory of a three component mixture composed of a solvent and a macromolecule that can exist in two different molecular states. Our model accounts for phase separation of such macromolecules from the solvent and reversible transitions between the two molecular states. We considered the macromolecules as polymeric molecules and thus described the interactions among all three components by a Flory-Huggins mean field free energy.

We discussed two scenarios that essentially differ with respect to their kinetics of the molecular transition: a thermodynamic system that is fully governed by the minimization of the free energy, and a system where the flux of the molecular transition is influenced by the amount of fuel. The presence of fuel makes the transition flux independent of the model free energy and thus breaks detailed balance of the rates. In contrast to the thermodynamic system, the resulting stationary states exhibit a non-zero flux.

Our first key finding is that, at thermodynamic equilibrium, molecular transitions can control the composition of molecular states in both, the dilute and the dense phase as a function of temperature. Strikingly, for similarly strong self-interactions between each molecular state, we found a discontinuous switch of the dense phase between states where most macromolecules are either in one or the other state. Our second key finding is that a switch in composition can also be triggered in finite systems composed of macromolecule-rich droplets using fuel -a more likely control pathway in living cells in contrast to temperature. Most strikingly, we predict that such non-thermodynamic, fuel-related control of droplet composition is achievable within a much broader range of polymeric interaction parameters compared to the system at thermodynamic equilibrium.

This broad parameter range renders our reported fuel-controlled, switch-like transition in droplet composition as a highly relevant mechanism to control the functionality of droplet-like condensates in living cells. Most physicochemical properties and likely all biological functionality of intracellular droplets are downstream consequences of the distinct local composition of the dense phase relative to its environment. Hallmark examples are the partitioning of specific molecules into protein-RNA condensates leading for example to increased organism fitness [46] , controlled assembly hierarchies relevant for a functional tight-junction [47] , and initiation of transcription and splicing machinery [48] . Moreover, the droplet composition also determines the rheological properties of condensates, e.g. its viscoelasticity [49] , as well as its slow glass-like aging kinetics [50] .

Our reported mechanism for switching droplet composition is easily accessible parameter-wise and may represent a versatile and robust kinetic pathway of controlling the droplet composition-related functionality in living cells.

From a physics perspective, our results hint toward an unexplored question, namely what determines the number of coexisting distinct domains away from thermodynamic equilibrium. At thermodynamic equilibrium, the Gibbs phase rule sets an upper bound for the number of coexisting phases. In our specific case of an incompressible ternary mixture, the molecular transition has reduced the degrees of freedom to a single conserved variable. As a result, three-phase coexistence can only occur at a single point or line parametrized by the conserved variable. In contrast, in a ternary mixture without molecular transitions, there can be an extended two-dimensional domain in the phase diagram where three phases coexist (Fig. 6) . Thus, the molecular transition lowered the maximal dimensionality of the domain with three coexisting phases by one. Are there similar upper bounds in non-equilibrium systems, and if there are, can systems away from equilibrium increase or decrease this upper bound? Given our results, breaking detailed balance of the transition rates and thus maintaining the system away from thermodynamic equilibrium releases the system from the Gibbs phase rule and allows the formation of at least three phases, and potentially even more. To study the interplay between chemical reactions and phase separation, we consider the T -P -N i -ensemble and three different species, i = A, B, W . At equilibrium the total number of species s, the number of independent components c and the number of independent chemical reactions n r are related by [28] s = c + n r .

(A1)

In our case, we consider three species (s = 3) and by means of one chemical reaction (n r = 1; see Fig. 1 ), the number of independent components gets reduced by one and thus the dimensionality of the corresponding phase diagram reduces. As the two independent composition variables (c = 2), we consider the quantities conserved in the reaction Fig. 1 , namely N W (the number of solvent particles), and N tot = N A + N B (the total amount of A and B molecules). To find the chemical potentials corresponding to such conserved quantities, we start from the Gibbs free energy G(T, P, N i ) and recall that its variation corresponding to a chemical reaction vanishes at chemical equilibrium:

where µ i (T, P, N i ) = ∂G ∂N i . In the specific case of the molecular transition depicted in Fig. 1 , this gives

which simplifies the variation of G to

This result shows that at constant temperature and pressure the Gibbs free energy only depends on the conserved variables, i.e., the total particle number N tot and the solvent particle number N W . Moreover, we learn that the chemical potentials associated with these conserved quantities, N tot and N W , are µ A and µ W , respectively.

To study the system at equilibrium utilizing the familiar common tangent construction (or Maxwell construction), we perform the Legendre transform to the T -V -N i -ensemble, and switch to Helmholtz free energy F = G − P V , where the differential reads

In general, this transformation is difficult since the chemical equilibrium relation in Fig. 1 depends on pressure and thus expressing the free energy as a function of the conserved quantity introduces a non-trivial dependence on pressure.

For systems where volumes vary only slightly with pressure, we can use incompressibility as a further approximation. Incompressibility is guaranteed if molecular volumes, ν i = ∂V ∂N i | T,P,N j =i , are independent of temperature, pressure and composition. While incompressibility does not reduce the number of independent variables in the T -P -N i -ensemble, it provides a relation between N tot , N W , and the volume in the T -V -N i -ensemble and thereby allows to further reduce the number of independent variables by one.

However, this reduction via incompressibility requires a further condition, namely that molecular volumes are conserved in the molecular transition, Fig. 1 . In our case, this corresponds to ν A = ν B .

Thus, we obtain the following incompressibility relation connecting volume and particle numbers,

This relationship states that variations of the total volume are caused by variations of the conserved quantities. Using, ν A dN tot + ν W dN W = dV , leads to the following differential of the Helmholtz free energy:

Here, we introduced the exchange chemical potential, defined as

which characterizes the free energy difference for exchanging A with solvent W . The pressure

is not the mechanical pressure. This pressure quantifies the response to variations of the total volume while N tot is kept fixed. Thus, we can think of it as the pressure acting on a semi-permeable membrane which separates the system from a solute. This pressure is commonly referred to as osmotic pressure.

Conservation of molecular volumes, ν A = ν B , allows one to rewrite the chemical equilibrium condition, Eq. (A3), in terms of the exchange chemical potentials, leading to Eq. (4) in the main text.

Exploiting the chemical equilibrium condition (Eqs. (6) ) and incompressibility (7) 

Since the chemical potential must be equal in the two coexisting phases, for the system to phaseseparate at φ tot = 1, the chemical equilibrium relation must have solutions which are symmetric with respect to the exchange φ B → φ A = 1 − φ B , i.e.,

With the choice ω B − ω A = e int − T s int , we find for the lower transition temperature

For the phase diagram to be reentrant, this temperature must be positive. Thus, we derive the condition e int > e BB − e AA . Setting e AA = 0, explains the vertical line at the boundary of the "UCST" and "UCST and LDT" regions in Fig. 2(d) . We can also derive the condition for which the upper critical point merge with the lower first order transition point, T c = T d :

which, again in the case of e AA = e AB = 0, gives the line at the boundary of UCST and line in Fig. 2(d) .

Appendix C: Linear response theory for phase separation with molecular transitions

In a system where energy is conserved, the change of the system entropy,Ṡ, can be expressed in terms of the integrated rate of change of the total free energy density f tot = f + i=A,B κ i 2 |∇φ i | 2 + κ 2 ∇φ A · ∇φ B within the volume V and the free energy flux J f α through the volume boundaries ∂V [51] :

Using, ∂ t f tot = i=A,B (∂ t φ i )μ i /ν i , and the conservation law,

is the particle flux and the molecular transition between A and B implies (s i = −s j = s), we find for the entropy production

Partial integration leads to

where we have identified the free energy flux J f α = i=A,B j i,αμi . In our special case of molecular transitions conserving the molecular volume, we have ν A = ν B = ν. Thus, if we write to linear

the system entropy will increase and approach thermodynamic equilibrium. Here, Λ s denotes the mobility for the molecular transition and Λ i is the diffusive mobility, and we neglect cross couplings for simplicity. Moreover, we have used ν A = ν B = ν and absorbed it into the definition of the mobility Λ s . The equilibrium conditions areμ i = const. andμ A =μ B . Note that for an infinitely large thermodynamic system, we can neglect the spatial derivative in the chemical potentials, and express the equilibrium conditions in terms of the exchange chemical potentialsμ i = const. and (4)). 

Here, we imposed a chemical potential ∆µ = 0 that makes sure that the system cannot reach thermodynamic equilibrium withμ A =μ B (Eq. (4)). For homogeneous systems which obeyμ A − µ B + ∆µ < k B T , the leading order of the chemical flux can be written as

In the following, we impose the chemical potential by a fuel component of volume fraction φ F , and expand ∆µ(φ A , φ B , φ F ) to lowest order (φ i φ W ):

Defining k → = Λ s K→ k B T and k ← = Λ s K← k B T , we get the chemical flux shown in Eq. (13) .

We impose fuel conservation by keeping the spatial average of Eq. (14) fixed and equal toφ F . This implies α + βφ tot = 1, withφ tot being the total average volume fraction of macromolecules (see Eq. (7)). In Eq. (14) we notice the coefficient β encodes correlations between fuel φ F and total macromolecular material φ tot . Maximal spatial correlation between φ F and φ tot is reached maximizing β with the constraints α + βφ tot = 1 and 0 < φ F (x) < 1 everywhere in space. This leads to α = 0 and β = 1/φ tot . Maximal anti-correlation between φ F and φ tot is reached minimizing β with the same constraints, leading to α = −β = 1/(1 −φ tot ). Finally, no correlation between φ F and φ tot , i.e. fuel homogeneously distributed in the system, is achieved for β = 0 and, due to fuel conservation, α = 1. This explains the choices of α and β introduced at the end of Sec. II C.

At equilibrium and for the case where the fuel has only weak effects on the chemical flux (i.e., k ← 0, k → 0), we can make the connection between the coefficients α and β in Eq. (14) and the fuel partitioning even more explicit. We recall the definition of partitioning coefficient of the fuel component, P F = φ I F /φ II F , and of the total concentration, P tot = φ I tot /φ II tot . Here, I and II denote the dense and the dilute phase, respectively. We can express the fuel and total volume fractions in I and II, respectively, with the average fuel volume fractionφ F that is considered to be maintained at some constant value, and the conserved total volume fractionφ tot :

where i = {F, tot}. The partition degree [52] reads

where the phase-separated volume in the limit of dilute fuel is given as

Evaluating Eq. (14) inside and outside the dense phase, we find:

If the fuel partitions equally strong into both phases (P F = 1, and thus ξ F = 1), we get α = 1 and β = 0. Consistently, this corresponds to a homogeneous fuel profile, φ F (x) =φ F . For a fixed P tot > 1, the fuel partition coefficient P F determines the localization of the fuel. In particular, P F > 1, corresponds to fuel co-localizing with the total volume fraction φ tot with β > 0. On the contrary, when P F < 1, the fuel and the total volume fraction φ tot anti-co-localize with β < 0.

Appendix F: Numerical methods, interface tracking and parameter choices

We solve the kinetic equations (9) and (13) in two dimensions with periodic boundary conditions.

First, we use the energy quadratization method [41] [42] [43] to map the free energy of the system into a quadratic form. Then, we use the second-order finite difference method in space and the Crank-Nicolson method in time to discretize the partial differential equations. A stabilizing term [44] is added allowing larger time steps.

The interfaces between A-rich, B-rich, and solvent-rich domains, shown by dashed red and blue lines in Fig. 5 (a) and (f), are defined as the contour lines of the functions φ A (x) and φ B (x)

respectively, corresponding to the value φ * tot = (φ max tot − φ min tot )/2. Here, φ max tot and φ min tot correspond to the maxinum and the minimum value of φ tot (x), for a given time. When, instead, we are interested in identifying φ tot -rich and φ tot -poor phases, like in Fig. 7 (b) , without distinguishing between A and B conformations, we simply compare φ tot (x) with φ * tot .

We measure the effective rates in units of t D = L 2 /D A , which represents the time an A particle takes to diffuse across a length L = 100 κ A /(k B T ), where L corresponds to the system size. Moreover, the length scale κ A /(k B T ) approximates the droplet interface width [23] . In all simulations, we set Λ s = 5 t −1 D .

For the first prototypical parameter set (i) (e AA = 0.2 e BB , e AB = 0), we have chosen κ A = κ B and κ = 0. For the second prototypical parameter set (ii) (e AA = e BB , e AB = 0.2), we have chosen κ B = 5 κ A and κ = 2 κ A . This choice is motivated by the possibility of having active droplet inversion without dissolving the droplet and subsequently re-nucleating it. In particular, the hierarchy κ B > κ > κ A favors wetting of an A-rich layer onto a B-rich droplet. This is important in the second half of the fuel ramp (see Fig. 7 (a)), in order to produce A material not too far from the shrinking B-rich droplet, see SI movie VIII.

We obtained solutions to Eqs. (9) for a total time period up to 40 t D . Within this time, the majority of patterns shown in Fig. 5 appeared to be stable and stationary. For this movie the fuel partitions inside the dense phase and the effective rates areφ F k ← = 10 t −1 D andφ F k → = 15 t −1 D . Movie III: The initial equilibrium drop splits up into a ring and multiple drops. The ring get unstable wit time and a pattern composed of equally sized drops evolves. For this movie the fuel partitions inside the dense phase and the effective rates areφ F k ← = 15 t −1 D andφ F k → = 35 t −1 D . Movie IV: The initial equilibrium drop breaks into smaller fragments, subsequently three of them elongate and align, a bit reminiscent of stripes. For this movie the fuel partitions inside the dense phase and the effective rates areφ F k ← = 10 t −1 D andφ F k → = 50t −1 D .

Parameter set (ii), strong self interactions of the A species (e AA = e BB , e AB = 0.2e BB ):

Movie V: When the fuel is injected to the system holes appear in the drop. For this movie the fuel is constant in spaceφ F k ← = 45 t −1 D andφ F k → = 25 t −1 D . Movie VI: When the fuel is injected to the system composed of an initial A-rich drop, an internal A-rich ring forms and then becomes a central drop. The remaining A material forms drops that initially fill the space but, for later times, droplets far from the central B-rich drops dissolve. For this movie the fuel partitions inside the droplet and the effective rates areφ F k ← = 5 t −1 D and φ F k → = 35 t −1 D . Movie VII: When the fuel is injected to the system, in the center of the initial A-rich drop an internal B-rich drop forms. The remaining A material organizes into a ring wetting the B-rich central drop, and an outer layer of multiple drops. For this movie the fuel partitions inside the droplet and the effective rates areφ F k ← = 25 t −1 D andφ F k → = 40 t −1 D . Movie VIII: When the fuel is injected to the system, in the center of the initial A-rich drop becomes B-rich. When subsequently fuel is taken out the droplet switches back to its original composition. For this movie the is homogeneous in space and the effective rates areφ F k ← = 2.5 10 −5 t −1 D andφ F k → = 4 10 −4 t −1 D .

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