#  Controlling composition of coexisting phases via molecular transitions

Giacomo Bartolucci  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Strasse 38, 01187 Dresden, Germany  Center for Systems Biology
Dresden, Pfotenhauerstrasse 108, 01307 Dresden, Germany  Omar Adame-Arana
Department of Chemical and Biological Physics, Weizmann Institute of Science,
Rehovot 76100, Israel  Xueping Zhao  Max Planck Institute for the Physics of
Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany  Center for
Systems Biology Dresden, Pfotenhauerstrasse 108, 01307 Dresden, Germany
Christoph A. Weber***Corresponding author. Max Planck Institute for the
Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
Center for Systems Biology Dresden, Pfotenhauerstrasse 108, 01307 Dresden,
Germany

###### Abstract

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.

##  I Introduction

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
unexplored – despite 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.

##  II Model for phase separation and 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 $\phi_{\text{F}}$ that drives the transition away from thermodynamic
equilibrium. In our model, both components $A$ and $B$ can phase-separate due
to attractive interactions among them (Fig. 1b).

Figure 1:  Illustration of the model to study molecular transitions and phase
separation. In our model, we consider three components: $A$, $B$ and solvent
$W$. (a) We consider reversible molecular transitions between $A$ and $B$
which are controlled by temperature $T$ or fuel volume fraction
$\phi_{\text{F}}$. (b) Molecular transitions occur in the dense and dilute
phase, respectively. Depending on the molecular interactions of $A$ and $B$
molecules, the total volume fraction of both components and their composition
(i.e. the relative amount of $B$ molecules) change in each of the phases.

###  II.1 Free energy density

We describe the mixture in the $T$-$V$-$N_{i}$-ensemble making use of a free
energy density $f(T,\phi_{i})=F(T,V,N_{i})/V$, where $F$ denotes the Helmholtz
free energy depending on temperature $T$, volume $V$ and particle number
$N_{i}$ of component $i$. Particle numbers are related to volume fractions
$\phi_{i}={N_{i}\nu_{i}}/{V}$ with $\nu_{i}$ denoting the corresponding
molecular volumes. To describe the thermodynamic behavior of unstructured
polymers, we use a Flory-Huggins free energy density of the form

$f=\frac{k_{B}T}{\nu_{W}}\left[\sum_{i}\frac{\phi_{i}}{r_{i}}\ln\phi_{i}+\sum_{i,j}\Omega_{ij}(T)\phi_{i}\phi_{j}+\sum_{i}\omega_{i}(T)\phi_{i}\right]\,,$
(1a)
where $k_{B}$ is the Boltzmann constant, $r_{i}=\nu_{i}/\nu_{W}$ 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:
$\nu_{A}=\nu_{B}\equiv\nu$ and thus $r_{A}=r_{B}\equiv r$, implying that the
molecular transition conserves the volume of the system, $V=\sum\nu_{i}N_{i}$.
Moreover, in our model, all molecular volumes are constant making the system
incompressible at all times and leading to the incompressibility condition
$\phi_{W}=1-\phi_{A}-\phi_{B}\,.$ (1b) 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 $\Omega_{ij}$ denotes the interaction parameter between the
components $i$ and $j$. These interaction parameters depend on temperature. To
lowest order, we can write [27] $\Omega_{ij}=\frac{e_{ij}-s_{ij}T}{k_{B}T}\,,$
(1c)

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 $\omega_{i}(T)$ denotes the internal free energy of component
$i$ (in units of $k_{B}T$). In particular, $\omega_{B}-\omega_{A}$ corresponds
to the internal free energy difference between the two molecular states $A$
and $B$, which we express as

$\omega_{B}-\omega_{A}=\frac{e_{\text{int}}-s_{\text{int}}T}{k_{B}T}\,.$ (2)

Here, $e_{\text{int}}$ and $s_{\text{int}}$ 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. $e_{\text{int}}>0$ and $s_{\text{int}}>0$. 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,
$\omega_{B}(T)=\omega_{A}(T)$, defines the melting temperature

$T_{\text{m}}=e_{\text{int}}/s_{\text{int}}\,,$ (3)

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.

###  II.2 Thermodynamic equilibrium

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

$\bar{\mu}_{A}=\bar{\mu}_{B}\,,$ (4)

where the exchange chemical potentials are given by [29]

$\bar{\mu}_{i}=\nu_{i}\frac{\partial f}{\partial\phi_{i}}\,.$ (5)

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

$\frac{\phi_{B}}{\phi_{A}}=\exp\left[-r\left(\omega_{B}-\omega_{A}+2\sum_{i={A,B,W}}(\Omega_{Bi}-\Omega_{Ai})\phi_{i}\right)\right]\,.$
(6)

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 $B$

$\phi_{\text{tot}}=\phi_{A}+\phi_{B}\,,$ (7)

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
$f=f(T,\phi_{\text{tot}})$, 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]:

$\displaystyle\bar{\mu}_{A}(\phi_{\text{tot}}^{\text{I}})$
$\displaystyle=\bar{\mu}_{A}(\phi_{\text{tot}}^{\text{II}})\,,$ (8a)
$\displaystyle\bar{\mu}_{A}(\phi_{\text{tot}}^{\text{I}})$
$\displaystyle=\nu_{A}\frac{f(\phi_{\text{tot}}^{\text{II}})-f(\phi_{\text{tot}}^{\text{I}})}{\phi_{\text{tot}}^{\text{II}}-\phi_{\text{tot}}^{\text{I}}}\,.$
(8b)

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

###  II.3 Non-equilibrium thermodynamics

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

$\displaystyle\partial_{t}\phi_{A}$
$\displaystyle=\nabla\cdot\left(\Lambda_{A}\nabla\tilde{\mu}_{A}\right)+s(\phi_{A},\phi_{B})\,,$
(9a) $\displaystyle\partial_{t}\phi_{B}$
$\displaystyle=\nabla\cdot\left(\Lambda_{B}\nabla\tilde{\mu}_{B}\right)-s(\phi_{A},\phi_{B})\,,$
(9b)

where the inhomogeneous chemical potentials are given as [33]

$\tilde{\mu}_{i}=\bar{\mu}_{i}-\kappa_{i}\nabla^{2}\phi_{i}{-}\kappa\nabla^{2}\phi_{j}\,.$
(10)

These chemical potentials are related to the free energy

$F=\int\text{d}^{3}x\,\bigg{(}f+\sum_{i=A,B}\frac{\kappa_{i}}{2\,\nu}|\nabla\phi_{i}|^{2}+\frac{\kappa}{\nu}\nabla\phi_{A}\cdot\nabla\phi_{B}\bigg{)}$
(11)

via $\tilde{\mu}_{i}=\nu_{i}\delta F/\delta\phi_{i}$, where the free energy
density $f$ is given by Eq. (1a). Moreover, $\Lambda_{i}$ 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,
$\Lambda_{i}=\Lambda_{i,0}\phi_{i}\left(1-\phi_{A}-\phi_{B}\right)$ [34], and
consider $\Lambda_{i,0}$ 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
$D_{i}=k_{B}T\Lambda_{i,0}$. 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
$s(\phi_{A},\phi_{B})$ 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):

$s={\Lambda}_{s}\frac{\left(\tilde{\mu}_{B}-\tilde{\mu}_{A}\right)}{k_{B}T}\,,$
(12)

where ${\Lambda}_{s}$ denotes the mobility for the molecular transition, which
we consider to be constant for simplicity. If chemical potentials are
homogeneous ($\tilde{\mu}_{i}=\text{const.}$) and the chemical potentials of
$A$ and $B$ are equal ($\tilde{\mu}_{A}=\tilde{\mu}_{B}$), the system is at
thermodynamic equilibrium, self-consistently leading to
$\partial_{t}\phi_{A}=\partial_{t}\phi_{B}=0$ in Eqs. (9).

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 [35, 23]. 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 $\phi_{\text{F}}$
(see Appendix D for details):

$\displaystyle s$
$\displaystyle=\Lambda_{s}\frac{(\tilde{\mu}_{B}-\tilde{\mu}_{A})}{k_{B}T}+k_{\leftarrow}\phi_{B}\phi_{\text{F}}-k_{\rightarrow}\phi_{A}\phi_{\text{F}}\,.$
(13)

Here, $k_{\leftarrow}$ and $k_{\rightarrow}$ 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 ($\phi_{\text{F}}=0$), 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
$\phi_{\text{F}}$. 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
$\phi_{\text{F}}$ to the total concentration of $A$ and $B$,

$\phi_{\text{F}}(\bm{x},t)=\bar{\phi}_{\text{F}}\left(\alpha+\beta\,\phi_{\text{tot}}(\bm{x},t)\right)\,,$
(14)

where $\bar{\phi}_{\text{F}}$ 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 $\phi_{\text{tot}}$. The fuel partitioning coefficient
$P_{\text{F}}$, that is experimentally accessible, determines the values of
the parameters $\alpha$ and $\beta$ in Eq. (14) (see Appendix E for a
definition of $P_{\text{F}}$ and the link to $\alpha$ and $\beta$).

In the following, we choose three parameter sets for $\alpha$ and $\beta$
corresponding to three qualitatively different scenarios. First, the fuel
partitions inside the $\phi_{\text{tot}}$-rich phase for $\alpha=0$ and
$\beta=1/\bar{\phi}_{\text{tot}}$. Second, fuel is enriched outside
corresponding to $\alpha=-\beta=1/(1-\bar{\phi}_{\text{tot}})$. And finally,
we also consider the case of a homogeneous fuel for $\alpha=1,\beta=0$. The
latter case has been studied for example in Refs. [20, 40].

##  III Equilibrium phase diagrams

In this section, we study the equilibrium phase diagrams as a function of the
total volume fraction $\phi_{\text{tot}}$ and a scaled temperature $T/T_{0}$,
with $T_{0}=-e_{BB}/k_{\mathrm{B}}$. 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,
$\phi_{B}/\phi_{\text{tot}}$ (see the color code in Fig. 2). We then study how
such phase diagrams are affected by the melting temperature $T_{\text{m}}$. 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.

###  III.1 Reentrant phase behavior

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, $\Omega_{BB}=e_{BB}$ and $\Omega_{ij}=0$ otherwise. Attractive
homotypic interactions lead to phase coexistence between a
$\phi_{\text{tot}}$-rich and a $\phi_{\text{tot}}$-poor phase, in the
following referred to as the dense and dilute phase, respectively. Below the
melting temperature $T_{\text{m}}$ (Eq. (3)), we find that coexisting phases
are not only different in $\phi_{\text{tot}}$, 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
$\phi_{\text{tot}}$ is rich in $B$ components, while the composition of the
dilute branch (low $\phi_{\text{tot}}$) changes with temperature. The
composition, defined as the relative amount of molecules in the $B$
conformation, $\phi_{B}/\phi_{\text{tot}}$, 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 $T_{\text{m}}$ 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)

$T_{\text{d}}=T_{\text{m}}+{e_{BB}}/{s_{\text{int}}}\,,$ (15)

and is set by the competition between the homotypic interactions $e_{BB}$ and
the molecular transition which is characterized by the melting temperature
$T_{\text{m}}$ (Eq. (3)). For attractive interactions, Eq. (15) fulfills
$T_{\text{d}}<T_{\text{m}}$, 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.

Figure 2:  Molecular transitions can lead to a lower critical dissolution
temperature. The phase diagrams depict the mixed and demixed region in the
plane spanned by total volume fraction $\phi_{\text{tot}}$ and rescaled
temperature $T/T_{0}$. The composition of each phase is indicated by the color
code displayed on each branch of the binodal. The black line corresponds to
the binary $B$-$W$ mixture. For increasing $T_{\text{m}}$, the presence of
molecular transitions reshape the phase diagram. (a) For low $T_{\text{m}}$,
the demixed region is bound from above by an upper critical solution
temperature (UCST) similar to the binary case. (b) For
$T_{\text{m}}>-e_{BB}/s_{\text{int}}$, the phase diagram becomes reentrant,
i.e., bounded also from below by a lower dissolution temperature (LDT). (c)
Increasing $T_{\text{m}}$ further, the binodal is upshifted until the demixing
region shrinks into a point for $T_{\text{m}}=-r\,e_{BB}/(2k_{\text{B}})$.
Parameters for (a-c): $-e_{\text{int}}/e_{BB}=0.75,1.5,2.5$ (d) Influence
$e_{\text{int}}$ and $s_{\text{int}}$ on the phase behaviour, where the color
code represents the critical density. Increasing the interaction entropy
$s_{\text{int}}$, the interval of relative energies -$e_{\text{int}}/e_{BB}$
corresponding to the reentrant regime widens. All the results presented
correspond to $r=2$.

For even higher melting temperatures $T_{\text{m}}$, the upper critical
solution temperature $T_{\text{c}}$ (UCST) and critical volume fraction
$\phi_{\text{c}}$ exceed the corresponding values in the binary case (black
line in Fig. 2a-c). This is because below $T_{\text{m}}$, the dilute and dense
phases differ in composition, being enriched in $A$ and $B$, respectively. If
$T_{\text{m}}$ is larger than the critical temperature of the reference binary
system, this composition difference will emerge as soon as the homogeneous
system demixes into two phases. Even though, with our choice of interaction
parameters, $B$ effectively repels both $A$ 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
$\phi_{\text{tot}}$ 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 Fig. 2d. For $e_{\text{int}}<-e_{BB}$, the lower
dissolution temperature (Eq. (15)) vanishes, leading to phase diagrams with
only an upper critical solution temperature (UCST). This UCST arises from a
competition between $B$-$B$ interactions and the mixing entropy (Fig. 2a). If
the internal energy gain of the molecular transition exceeds the strength of
attractive $B$-$B$ interaction, i.e., $e_{\text{int}}\gtrsim-e_{BB}$
(corresponding to $T_{\text{m}}\gtrsim-e_{BB}/s_{\text{int}}$), there is a
finite lower dissolution temperature $T_{\text{d}}$ leading to _reentrant_
phase behavior. Reentrance arises from a competition between the attractive
interactions $e_{BB}$ and the energetic difference between the two molecular
states $e_{\text{int}}$ (see by inserting Eq. (3) into Eq. 15). Reentrance is
absent at low $T_{\text{m}}$ because $B$-$B$ interactions dominate the
molecular transition ($e_{\text{int}}<-e_{BB}$). Increasing $T_{\text{m}}$
corresponding to $e_{\text{int}}>-e_{BB}$ causes reentrant behavior because
the energetic difference between the two molecular states $e_{\text{int}}$
exceeds the strength of the attractive $B$-$B$ interaction. A further increase
of $T_{\text{m}}$ leads to a transition to a mixed region in the phase
diagram. We found that the disappearance of demixed states in the phase
diagram coincides with the upper critical point
$(\phi_{\text{c}},T_{\text{c}})$ reaching $\phi_{\text{tot}}=1$, which
corresponds to a system without solvent (see color coded $\phi_{\text{c}}$ in
Fig. 2d). This observation can be used to analytically calculate the
transition line between reentrant phase behavior (UCST and LDT) and mixed
states by $T_{c}(\phi_{\text{tot}}=1)=T_{\text{d}}$, where
$T_{c}(\phi_{\text{tot}}=1)=-r\,e_{BB}/(2k_{B})$ (for details see Appendix B).
We find a relationship for the transition line,
$-e_{\text{int}}/e_{BB}=1+s_{\text{int}}\,{r}/{2}$, 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
$s_{\text{int}}$. Larger $s_{\text{int}}$ leads to more phase-separating $B$
molecules and thus favors phase coexistence over mixing.

Figure 3:  Interplay between homo- and heterotypic interactions determine the
phase diagrams of phase-separating systems with molecular transitions.  (a,b)
$\phi_{\text{tot}}$ \- $T/T_{0}$ phase diagrams with the composition along the
binodal color coded, depicted for two prototypical cases discussed in the
paper. (a) A weak self interaction among the $A$ components creates a demixed
region at low temperature which leads to a second upper critical solution
temperature (UCST). Here, $e_{AA}=0.5\,e_{BB}$ and $e_{AB}=0$. (b) Increasing
the strength of $A$-$A$ homotypic interaction let the lower demixed region
merge with the upper reentrant region which generates a triple point. Here
$e_{AA}=0.9\,e_{BB}$ and $e_{AB}=0.1\,e_{BB}$. (c) Classification of phase
diagrams for different heteropypic $A$-$B$ and homotypic $A$-$A$ interactions,
keeping $e_{BB}$ and $T_{m}$ fixed. The classes are: phase diagrams with one
UCST, phase diagrams with reentrant behavior which either have one UCST and
one LDT or two UCSTs and one LDT. The $\Circle$ symbol marks phase diagrams
with a triple point (as shown in (b)) while $\otimes$ highlights cases in
which the triple point collapses giving rise to a jump in the composition of
the dense phase (see Fig. 4). Please note that for this classification, we
considered temperatures down to $T/T_{0}=0.1$. Here and for the rest of the
paper, we fix $T_{m}=0.615\,T_{0}$.

###  III.2  First order phase transition in droplet composition

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 $T_{\text{m}}$ 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
$\phi_{\text{tot}}$-rich and $\phi_{\text{tot}}$-poor phases that are both
mainly composed of $A$. Thus, these phase diagrams have two UCSTs and one LDT.

The phase diagram qualitatively changes when the strength of $A$-$A$
attraction is increased towards the value of $B$-$B$ interaction strength. In
this case, the two separated domains in the phase diagram in Fig. 3a merge.
The resulting phase diagrams exhibit either only one UCST (not shown) or one
UCST and a triple point; see Fig. 3b. At this triple point, three phases
coexist which differ in $\phi_{\text{tot}}$ and $A$-$B$ composition. The
precise value of $e_{AA}$ at which both domains merge is influenced by the
strength of the $A$-$B$ heterotypic interaction; see Fig. 3c.

Figure 4:  Systems with strong homotypic interactions exhibit a first order
phase transition in the composition of thre dense phase. (a) In the limiting
case of identical homotypic interaction strength combined with weak
heterotypic interactions, $e_{AA}=e_{BB}$, $e_{AB}/e_{BB}\ll 1$, the triple
point collapses into a point at which the composition of the dense phase
changes discontinuously. (b) Volume fraction of the three phases coexisting at
the triple point as a function of relative $A$-$A$ interaction strength,
$e_{AA}/e_{BB}$. For equal homotypic interactions, i.e. $e_{AA}=e_{BB}$, the
two upper branches merge giving rise to the discontinuous transition in
droplet composition. Here, $e_{AB}/e_{BB}$=0.2 and the color code depicts
composition of the phases. (c) Increasing the heterotypic interaction
strength, the droplet composition shows three distinct behaviors with
temperature: it can change discontinuously ($e_{AB}/e_{BB}=0.2$), or the
droplet first dissolves and then reforms with different composition
($e_{AB}/e_{BB}=0.4$), where the shaded region corresponds to no drops, or its
composition changes gradually ($e_{AB}/e_{BB}=1$). We used
$\phi_{\text{tot}}=0.5$ for all the three curves.

For $e_{AA}$ being equal to the value $e_{BB}$, the triple point temperature
coincides with both the melting temperature $T_{\text{m}}$ and the temperature
of the first order transition point where the two branches of different
$A$-$B$ composition meet (indicated by a cross in Fig. 4a). The result is a
triple line; see the black horizontal line in Fig. 4a. At the corresponding
triple line temperature, three phases coexist for any value of
$\phi_{\text{tot}}$ between the binodals. Crossing the triple line temperature
leads to a discontinuous phase transition of the composition of the dense
phase. In other words, in a finite system with a single droplet, the
composition of the droplet discontinuously switches upon temperature changes
between $A$-rich and $B$-rich. As mentioned, the triple line temperature is
exactly the melting temperature $T_{\text{m}}$ (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 $\phi_{\text{tot}}$. 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).

##  IV Kinetics of phase separation with detailed-balance broken molecular
transitions

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: (i) weak homotypic
interactions of the $A$ species and weak heterotypic $A$-$B$ interactions:
$e_{AA}/e_{BB}\ll 1$, $e_{AB}/e_{BB}\ll 1$ and (ii) strong attractive $A$-$A$
interactions and weak $A$-$B$ heterotypic interactions: $e_{AA}=e_{BB}$,
$e_{AB}/e_{BB}\ll 1$. At thermodynamic equilibrium, we observed reentrant
phase behavior for (i) and the discontinuous phase transition in droplet
composition for (ii). To account for the effects of fuel on the molecular
transition, we consider both a thermodynamic and a fuel-related contribution
to the reaction flux; 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).

Figure 5:  Molecular transition breaking detailed balance leads to non-
equilibrium stationary states. We monitor the emergence of patterns by
tracking the interface between $A$-rich (I), $B$-rich (II), and solvent rich
(III) domains (definition in App. F). As an example, such tracked interfaces
are shown by dashed line in (a) and (f), where the red and blue lines enclose
$A$-rich and $B$-rich domains, respectively. These domains are the analogue of
demixed phases at thermodynamic equilibrium. For weak $A$-$A$ self-
interactions ($e_{AA}=0.2\,e_{BB}$, $e_{AB}=0$), we display two prototypical
non-equilibrium stationary states due to the presence of the fuel (a,b) and we
determine the values of the effective rates corresponding to such out of
equilibrium patterns (c)-(e), where the color code indicates the composition
of the dense domain in terms of the relative $B$ amount, i.e.,
$\phi_{B}/\phi_{\text{tot}}$. Strikingly, fuel partitioning affects the
boundaries of the region corresponding to patterns, along with the dissolution
boundary. Here, $\phi_{\text{tot}}=0.35$, $T/T_{0}=0.55$. For strong $A$-$A$
self interactions ($e_{AA}=e_{BB}$, $e_{AB}=0.2\,e_{BB}$), stationary patterns
emerge that are composed of three domains distinct in composition (f) (g). In
the diagrams (i)-(j), such domains are indicated with two triangles of
different colors in which each color corresponds to the composition of the two
distinct dense phases. Here $\phi_{\text{tot}}=0.35$, $T/T_{0}=0.525$. The
rates are measured in units of the inverse diffusion time $t_{D}=L^{2}/D_{A}$,
with $L=100\sqrt{\kappa_{A}/(k_{\mathrm{B}}T)}$ (see App. F for details).
Scale bar: $20\sqrt{\kappa_{A}/(k_{\mathrm{B}}T)}$.

###  IV.1 Fuel controls non-equilibrium patterns

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
$A$-$B$-composition 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), (b), (f), (g), and SI movies I-VII). The emergence of
these patterns is determined by the partitioning of fuel into the droplets and
the strength of homotypic and heterotypic interaction of $A$ and $B$
molecules, respectively.

For the case (i) characterized by weak $A$-$A$ homotypic interactions, i.e.
$e_{AA}/e_{BB}\ll 1$ and $e_{AB}/e_{BB}<1$, we find that the
$\phi_{\text{tot}}$-rich phase is mainly composed of $B$ and that the inside
composition hardly varies due to the presence of fuel; Fig. 5 (c)-(e).
However, the fuel can induce the formation of patterns that are significantly
different from the initial single droplet at thermodynamic equilibrium (no
fuel). For certain fuel partitioning, we find that there is an extended region
in the $k_{\rightarrow}$-$k_{\leftarrow}$ state diagrams where patterns emerge
(gray line in Fig. 5 (c)-(e)). Strikingly, the location and the extent of
these regions are significantly influenced by fuel partitioning. In
particular, fuel partitioning inside $\phi_{\text{tot}}$-dense domains favors
the emergence of patterns for lower values of both effective forward and
backward rate. Close to the onset of pattern formation, we observe stable
ring-like patterns (Fig. 5 (a) and SI Movie I).

For increasing effective backward rate
$\bar{\phi}_{\text{F}}\,k_{\leftarrow}$, we often observe rings coexisting
with droplets, see SI Movie II. For even larger effective rates
$\bar{\phi}_{\text{F}}\,k_{\leftarrow}$, 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.
$e_{BB}=e_{AA}$, $e_{AB}/e_{BB}\ll 1$, the behavior in the
$k_{\rightarrow}$-$k_{\leftarrow}$ 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 $k_{\rightarrow}$-$k_{\leftarrow}$ 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 $k_{\rightarrow}$ and $k_{\leftarrow}$ 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.2). 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., $\phi_{\text{tot}}$), a ternary mixture without transitions
has two degrees of freedom (i.e., $\phi_{A}$ and $\phi_{B}$); 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 $\phi_{A}$-$\phi_{B}$ 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
$\phi_{B}=\phi_{\text{tot}}-\phi_{A}$. We now set the effective backward rate
$\bar{\phi}_{\text{F}}\,k_{\leftarrow}=0$ and
$k_{\rightarrow}=2\,10^{3}\,t_{D}^{-1}$ and consider variations of the
effective forward rate induced by the average fuel amount. Specifically, in
the absence of fuel ($\bar{\phi}_{\text{F}}=0$), 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.
6b-d for representative stationary states at different fuel levels. For an
even larger amount of fuel, the system transits to a non-equilibrium
stationary state where $B$-rich domains stably coexist with a solvent-rich
domain. Although the three phase coexistence regime occupies the majority of
the $\phi_{A}$-$\phi_{B}$-plane, coexistence of three domains is actually only
accessible over a narrow range of fuel values. To see this, compare the orange
regions along the conserved trajectory in Fig. 6a with the orange regions in
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.

Figure 6:  Breaking detailed balance of the molecular transition enables
controlling the number of distinct phases (a) Changing fuel in the systems can
be illustrated as a trajectory in the ternary phase diagram. Without fuel the
system demixes into two phases whose composition is set by thermodynamic
equilibrium; the corresponding average volume fraction is indicated by the
circled blue dot. The dashed lines connect the average volume fraction with
the values in the respective demixed domains (blue dots). Increasing fuel
pushes the average concentration from a region of two phase coexistence ($A$
and solvent rich, respectively) into a domain where three phase coexist at
equilibrium (orange shaded triangle). Increasing fuel further leads to two
domains, $A$ and solvent rich, respectively. Interestingly, the system demixes
into domains whose volume fractions are well approximated by the corresponding
equilibrium values. b)-d) $\phi_{A}$ and $\phi_{B}$ spatial profiles at the
stationary state for increasing fuel average volume fraction
$\bar{\phi}_{\text{F}}$. For moderate $\bar{\phi}_{\text{F}}$ the system
exhibits three phase coexistence (b), while for high values the composition of
the dense phase switches (c). (e) The number of phases along with their
density and composition as a function of the amount of fuel in the system. The
parameters are the same as the lower panel of Fig. 5, with fuel partitioning
inside. Scale bar: $20\sqrt{\kappa_{A}/(k_{\mathrm{B}}T)}$. Figure 7:
Fuelling leads to droplet composition switch in time for a wide range of
interaction parameters. The active switch in composition is achieved by
changing fuel in time according to the protocol in (a). (b) Time traces of
$\phi_{A}$, $\phi_{B}$ and $\phi_{\text{tot}}$ inside the dense phase show an
abrupt compositional switch with time, reminiscent of the discontinuous switch
observed at thermodynamic equilibrium when quasi-statically varying the
temperature. c)-e) Snapshots of $\phi_{A}$ $\phi_{B}$ profile evolving in time
within the first half of the fueling ramp shown in (a). The rescaled
temperature is kept constant to $T/T_{0}=0.525$. (f) We quantify the
compositional switch by determining the difference in droplet composition
between (1) the stationary state (corresponding to maximum fuel value in panel
(a) and (0) the initial equilibrium state: $\Delta
B=\phi_{B}^{(1)}/\phi_{\text{tot}}^{(1)}-\phi_{B}^{(0)}/\phi_{\text{tot}}^{(0)}$
with different homotypic and heterotypic interactions. The active
conformational switch occurs within a broader range of parameters compared to
the equilibrium system referred to as “Thermodynamic switch” (see white
symbols values). The parameters are the same as in the lower panel of Fig. 5,
with fuel being constant, $T/T_{0}=0.45$ and $\phi_{\text{tot}}=0.4$ and the
white circles were already shown in Fig. 3(c). Scale bar:
$20\sqrt{\kappa_{A}/(k_{\mathrm{B}}T)}$.

###  IV.2 Active switch of droplet composition

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 ($\bar{\phi}_{\text{F}}=0$) 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.

The control of phase composition by fuel which we report here is in principle
applicable to a rather broad class of macromolecules. In our model, their
interactions are characterized by the values of their homo- and heterotypic
interaction parameters. While the compositional switch controlled by
temperature at thermodynamic equilibrium (“thermodynamic switch”) is only
accessible within a narrow range of interaction parameters (see zoom in Fig.
3c), the switching via fuel (“active switch”) is possible in a significantly
broader range of interaction parameters, as shown in Fig. 7e. To characterize
this broader parameter range, we calculated the jump height of the
compositional change, $\Delta
B=\phi_{B}^{(1)}/\phi_{\text{tot}}^{(1)}-\phi_{B}^{(0)}/\phi_{\text{tot}}^{(0)}$,
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).

##  V Conclusion

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 intra-
cellular 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.

###### Acknowledgements.

We thank J. Bauermann, P. McCall, T. Harmon, L. Hubatsch and F. Jülicher for
fruitful discussions about the topic. Very special thanks goes to P. McCall,
K. Alameh, and J. Bauermann for very helpful feedback on the manuscript. We
thank A. Serrao, P. Schwintek , A. Kühnlein, C. Mast and D. Braun for
inspiring discussion on DNA gel formation and application to the Origin on
Life. O. Adame-Arana acknowledges funding from the Armando and Maria Jinich
postdoctoral fellowship for Mexican citizens. G. Bartolucci and C. Weber
acknowledge the SPP 2191 “Molecular Mechanisms of Functional Phase Separation”
of the German Science Foundation for financial support.

##  Appendix A Thermodynamic equilibrium

###  A.1 Conserved quantities and equilibrium condition of the molecular
transition

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]

$s=c+n_{r}\,.$ (16)

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 $N_{\text{tot}}=N_{A}+N_{B}$ (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:

$\text{d}G\,\big{\rvert}_{T,P}=\sum_{i}\mu_{i}\text{d}N_{i}=0\,,$ (17)

where $\mu_{i}(T,P,N_{i})=\frac{\partial G}{\partial N_{i}}$. In the specific
case of the molecular transition depicted in Fig. 1, this gives

$\mu_{A}(T,P,N_{i})=\mu_{B}(T,P,N_{i})\,,$ (18)

which simplifies the variation of $G$ to

$\displaystyle\text{d}G=-S\text{d}T+V\text{d}P+\mu_{A}\text{d}N_{\text{tot}}+\mu_{W}\text{d}N_{W}\,.$
(19)

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 $N_{\text{tot}}$ and the solvent particle number $N_{W}$. Moreover, we
learn that the chemical potentials associated with these conserved quantities,
$N_{\text{tot}}$ and $N_{W}$, are $\mu_{A}$ and $\mu_{W}$, 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 $F=G-PV$,
where the differential reads

$\text{d}F=-S\text{d}T-P\text{d}V+\mu_{A}\text{d}N_{\text{tot}}+\mu_{W}\text{d}N_{W}\,.$
(20)

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.

###  A.2 Equilibrium condition of the molecular transitions in incompressible
systems

For systems where volumes vary only slightly with pressure, we can use
incompressibility as a further approximation. Incompressibility is guaranteed
if molecular volumes, $\nu_{i}={\frac{\partial V}{\partial
N_{i}}}|_{T,P,N_{j\neq i}}$, 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
$N_{\text{tot}}$, $N_{W}$, 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 $\nu_{A}=\nu_{B}$. Thus, we obtain the
following incompressibility relation connecting volume and particle numbers,
$V=N_{A}\nu_{A}+N_{B}\nu_{B}+N_{W}\nu_{W}=N_{\text{tot}}\nu_{A}+N_{W}\nu_{W}$.
This relationship states that variations of the total volume are caused by
variations of the conserved quantities. Using,
$\nu_{A}\text{d}N_{\text{tot}}+\nu_{W}\text{d}N_{W}=\text{d}V$, leads to the
following differential of the Helmholtz free energy:

$\text{d}F=-S\text{d}T-\Pi\,\text{d}V+\bar{\mu}_{A}\,\text{d}N_{\text{tot}}\,.$
(21)

Here, we introduced the exchange chemical potential, defined as

$\bar{\mu}_{i}=\mu_{i}-\mu_{W}\frac{\nu_{i}}{\nu_{W}}\,,$ (22)

which characterizes the free energy difference for exchanging $A$ with solvent
$W$. The pressure $\Pi=\left(P-\frac{\mu_{W}}{\nu_{W}}\right)$ is not the
mechanical pressure. This pressure quantifies the response to variations of
the total volume while $N_{\text{tot}}$ 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, $\nu_{A}=\nu_{B}$, allows one to rewrite
the chemical equilibrium condition, Eq. (18), in terms of the exchange
chemical potentials, leading to Eq. (4) in the main text.

###  A.3 Free energy density and Maxwell construction

Exploiting the chemical equilibrium condition (Eqs. (6)) and incompressibility
(7), we can recast the free energy in Eq. (1a) in the form
$f=f(T,\phi_{\text{tot}})$. The existence of multiple solutions of the
chemical equilibrium relation in Eq. (6) can lead to the formation of a free
energy loop, as displayed in Fig. 8. We also show the common tangent
construction (or Maxwell construction), that is used to determine the phase
diagrams. For visualization purposes we display
$\tilde{f}=f-m\,\phi_{\text{tot}}$, where
$m=\left(f(\phi_{\text{tot}}^{I})-f(\phi_{\text{tot}}^{II})\right)/\left(\phi_{\text{tot}}^{I}-\phi_{\text{tot}}^{II}\right)$

Figure 8:  Stable and unstable free energy branches due to molecular
transitions.  Free energy $\tilde{f}$ of the mixture as a function of the
conserved quantity $\phi_{\text{tot}}$, where the tilde indicates that all
linear terms were subtracted for illustrative purposes. Molecular transitions
can lead to the emergence of multiple solutions for a single
$\phi_{\text{tot}}$ value in the demixing region (white). We distinguish three
branches: $1$ and $3$ are metastable and $2$ is a locally unstable branch. The
common tangent (Maxwell construction) is shown in black and separates the
demixing region from the mixed region (blue, red shaped). In the demixing
region, the system is homogeneous, the free energy convex and all branches
collapse.

##  Appendix B Lower transition temperature

Here we describe the criterion for the existence of the first order lower
transition point. This point appears when the two solutions of the phase
equilibrium coincide at $\phi_{\text{tot}}=1$, i.e., when the solution becomes
a polymer blend composed of $A$ and $B$ only. If the interaction entropies in
Eq. (1c) are negligible, the critical temperature below which such a mixture
demixes reads $k_{B}T_{c}=r(e_{AA}+e_{BB}-2\,e_{AB})/2$. Moreover,
incompressibility at $\phi_{\text{tot}}=1$ reads

$\displaystyle\phi^{\text{I}}_{A}=1-\phi^{\text{I}}_{B}=\phi^{\text{II}}_{B}\,,$
(23)
$\displaystyle\phi^{\text{I}}_{B}=1-\phi^{\text{I}}_{A}=\phi^{\text{II}}_{A}\,.$
(24)

Since the chemical potential must be equal in the two coexisting phases, for
the system to phase-separate at $\phi_{\text{tot}}=1$, the chemical
equilibrium relation must have solutions which are symmetric with respect to
the exchange $\phi_{B}\rightarrow\phi_{A}=1-\phi_{B}$, i.e.,

$\omega_{B}-\omega_{A}=e_{BB}-e_{AA}\,.$ (25)

With the choice $\omega_{B}-\omega_{A}=e_{\text{int}}-Ts_{\text{int}}$, we
find for the lower transition temperature

$T_{\text{d}}=\frac{e_{\text{int}}-(e_{BB}-e_{AA})}{s_{\text{int}}}\,.$ (26)

For the phase diagram to be reentrant, this temperature must be positive.
Thus, we derive the condition $e_{\text{int}}>e_{BB}-e_{AA}$. 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,
$T_{\text{c}}=T_{\text{d}}$:

$\displaystyle
e^{*}_{\text{int}}=\left(s^{*}_{\text{int}}\,\frac{r}{2}+1\right)e^{*}_{BB}+\left(s^{*}_{\text{int}}\,\frac{r}{2}-1\right)e^{*}_{AB}-s^{*}_{\text{int}}r\,e^{*}_{AB}\,,$
(27)

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,
$\dot{S}$, can be expressed in terms of the integrated rate of change of the
total free energy density
$f^{\text{tot}}=f+\sum_{i=A,B}\frac{\kappa_{i}}{2}|\nabla\phi_{i}|^{2}+\frac{\kappa}{2}\nabla\phi_{A}\cdot\nabla\phi_{B}$
within the volume $V$ and the free energy flux $J^{f}_{\alpha}$ through the
volume boundaries $\partial V$ [51]:

$T\dot{S}=-\int_{V}\text{d}^{3}x\,\partial_{t}f^{\text{tot}}-\int_{\partial
V}\text{d}\mathcal{S}_{\alpha}\,J^{f}_{\alpha}\,.$ (28)

Using,
$\partial_{t}f^{\text{tot}}=\sum_{i=A,B}\left(\partial_{t}\phi_{i}\right)\,\tilde{\mu}_{i}/\nu_{i}$,
and the conservation law,
$\partial_{t}\phi_{i}=-\partial_{\alpha}\left(j_{i,\alpha}\nu_{i}\right)+s_{i}$,
where $j_{i,\alpha}$ is the particle flux and the molecular transition between
$A$ and $B$ implies ($s_{i}=-s_{j}=s$), we find for the entropy production

$\displaystyle T\dot{S}$ $\displaystyle=-\int_{\partial
V}\text{d}\mathcal{S}_{\alpha}\,J^{f}_{\alpha}-\int_{V}\text{d}^{3}x\,s\left(\tilde{\mu}_{A}/\nu_{A}-\tilde{\mu}_{B}/\nu_{B}\right)+\int_{V}\text{d}^{3}x\,\sum_{i=A,B}\tilde{\mu}_{i}\,\partial_{\alpha}j_{i,\alpha}\,.$
(29)

Partial integration leads to

$\displaystyle T\dot{S}$
$\displaystyle=-\int_{V}\text{d}^{3}x\,s\left(\tilde{\mu}_{A}/\nu_{A}-\tilde{\mu}_{B}/\nu_{B}\right)-\int_{V}\text{d}^{3}x\,\sum_{i=A,B}j_{i,\alpha}\,\partial_{\alpha}\tilde{\mu}_{i}\,,$
(30)

where we have identified the free energy flux
$J^{f}_{\alpha}=\sum_{i=A,B}j_{i,\alpha}\,\tilde{\mu}_{i}$. In our special
case of molecular transitions conserving the molecular volume, we have
$\nu_{A}=\nu_{B}=\nu$. Thus, if we write to linear order

$\displaystyle s$
$\displaystyle=-\Lambda_{s}\,\left(\tilde{\mu}_{A}-\tilde{\mu}_{B}\right)\,,$
(31a) $\displaystyle j_{i,\alpha}$
$\displaystyle=-\Lambda_{i}\,\partial_{\alpha}\tilde{\mu}_{i}\,,$ (31b)

the system entropy will increase and approach thermodynamic equilibrium. Here,
${\Lambda}_{s}$ denotes the mobility for the molecular transition and
$\Lambda_{i}$ is the diffusive mobility, and we neglect cross couplings for
simplicity. Moreover, we have used $\nu_{A}=\nu_{B}=\nu$ and absorbed it into
the definition of the mobility ${\Lambda}_{s}$. The equilibrium conditions are
$\tilde{\mu}_{i}=\text{const.}$ and $\tilde{\mu}_{A}=\tilde{\mu}_{B}$. 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 potentials $\bar{\mu}_{i}=\text{const.}$ and
$\bar{\mu}_{A}=\bar{\mu}_{B}$ (Eq. (4)).

##  Appendix D Chemical flux for molecular transitions with broken detailed
balance

When detailed balance of the molecular transition rates is broken and
considering a non-linear dependence of the chemical flux on the chemical
potentials, the chemical flux in Eq. (13) can be written as [23, 24]

$\displaystyle s$
$\displaystyle=\Lambda_{s}\left[\exp\left(\frac{(\tilde{\mu}_{A}-\tilde{\mu}_{B})+\Delta\mu}{k_{B}T}\right)-1\right]\,.$
(32)

Here, we imposed a chemical potential $\Delta\mu\not=0$ that makes sure that
the system cannot reach thermodynamic equilibrium with
$\tilde{\mu}_{A}=\tilde{\mu}_{B}$ (Eq. (4)). For homogeneous systems which
obey $\bar{\mu}_{A}-\bar{\mu}_{B}+\Delta\mu<k_{B}T$, the leading order of the
chemical flux can be written as

$\displaystyle s$
$\displaystyle\simeq\Lambda_{s}\frac{(\tilde{\mu}_{A}-\tilde{\mu}_{B})}{k_{B}T}+\Lambda_{s}\frac{\Delta\mu}{k_{B}T}\,.$
(33)

In the following, we impose the chemical potential by a fuel component of
volume fraction $\phi_{\text{F}}$, and expand
$\Delta\mu(\phi_{A},\phi_{B},\phi_{\text{F}})$ to lowest order
($\phi_{i}\ll\phi_{W}$):

$\Delta\mu\simeq(K_{\leftarrow}\phi_{B}-K_{\rightarrow}\phi_{A})\phi_{\text{F}}\,.$
(34)

Defining $k_{\rightarrow}=\Lambda_{s}\frac{K_{\rightarrow}}{k_{B}T}$ and
$k_{\leftarrow}=\Lambda_{s}\frac{K_{\leftarrow}}{k_{B}T}$, we get the chemical
flux shown in Eq. (13).

##  Appendix E Derivation of equilibrium fuel profile

We impose fuel conservation by keeping the spatial average of Eq. (14) fixed
and equal to $\bar{\phi}_{\text{F}}$. This implies
$\alpha+\beta\bar{\phi}_{\text{tot}}=1$, with $\bar{\phi}_{\text{tot}}$ being
the total average volume fraction of macromolecules (see Eq. (7)). In Eq. (14)
we notice the coefficient $\beta$ encodes correlations between fuel
$\phi_{\text{F}}$ and total macromolecular material $\phi_{\text{tot}}$.
Maximal spatial correlation between $\phi_{\text{F}}$ and $\phi_{\text{tot}}$
is reached maximizing $\beta$ with the constraints
$\alpha+\beta\bar{\phi}_{\text{tot}}=1$ and $0<\phi_{\text{F}}(x)<1$
everywhere in space. This leads to $\alpha=0$ and
$\beta=1/\bar{\phi}_{\text{tot}}$. Maximal anti-correlation between
$\phi_{\text{F}}$ and $\phi_{\text{tot}}$ is reached minimizing $\beta$ with
the same constraints, leading to
$\alpha=-\beta=1/(1-\bar{\phi}_{\text{tot}})$. Finally, no correlation between
$\phi_{\text{F}}$ and $\phi_{\text{tot}}$, i.e. fuel homogeneously distributed
in the system, is achieved for $\beta=0$ and, due to fuel conservation,
$\alpha=1$. This explains the choices of $\alpha$ and $\beta$ introduced at
the end of Sec. II.3.

At equilibrium and for the case where the fuel has only weak effects on the
chemical flux (i.e., $k_{\leftarrow}\simeq 0,k_{\rightarrow}\simeq 0$), we can
make the connection between the coefficients $\alpha$ and $\beta$ in Eq. (14)
and the fuel partitioning even more explicit. We recall the definition of
partitioning coefficient of the fuel component,
$P_{\text{F}}=\phi_{\text{F}}^{\text{I}}/\phi_{\text{F}}^{\text{II}}$, and of
the total concentration,
$P_{\text{tot}}=\phi_{\text{tot}}^{\text{I}}/\phi_{\text{tot}}^{\text{II}}$.
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 fraction $\bar{\phi}_{\text{F}}$ that is considered to
be maintained at some constant value, and the conserved total volume fraction
$\bar{\phi}_{\text{tot}}$:

$\displaystyle\phi_{i}^{\text{I}}$
$\displaystyle=\xi_{i}\,P_{i}\,\bar{\phi}_{\alpha}\,,$ (35)
$\displaystyle\phi_{i}^{\text{II}}$
$\displaystyle=\xi_{i}\bar{\phi}_{\alpha}\,,$ (36)

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

$\xi_{i}=\frac{1}{1+(P_{i}-1)\frac{V^{\text{I}}}{V}}\,,$ (37)

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

${V^{\text{I}}}=V\frac{\bar{\phi}_{\text{tot}}-\phi_{\text{tot}}^{\text{II}}}{\phi_{\text{tot}}^{\text{I}}-\phi_{\text{tot}}^{\text{II}}}\,.$
(38)

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

$\displaystyle\alpha=\xi_{\text{F}}\left(1-\frac{P_{\text{F}}-1}{P_{\text{tot}}-1}\right)\,,$
(39)
$\displaystyle\beta=\frac{\xi_{\text{F}}}{\xi_{\text{tot}}}\frac{1}{\bar{\phi}_{\text{tot}}}\frac{P_{\text{F}}-1}{P_{\text{tot}}-1}\,.$
(40)

If the fuel partitions equally strong into both phases ($P_{\text{F}}=1$, and
thus $\xi_{\text{F}}=1$), we get $\alpha=1$ and $\beta=0$. Consistently, this
corresponds to a homogeneous fuel profile,
$\phi_{\text{F}}(x)=\bar{\phi}_{\text{F}}$. For a fixed $P_{\text{tot}}>1$,
the fuel partition coefficient $P_{\text{F}}$ determines the localization of
the fuel. In particular, $P_{\text{F}}>1$, corresponds to fuel co-localizing
with the total volume fraction $\phi_{\text{tot}}$ with $\beta>0$. On the
contrary, when $P_{\text{F}}<1$, the fuel and the total volume fraction
$\phi_{\text{tot}}$ anti-co-localize with $\beta<0$.

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

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
$L=100\sqrt{\kappa_{A}/(k_{\mathrm{B}}T)}$, where $L$ corresponds to the
system size. Moreover, the length scale $\sqrt{\kappa_{A}/(k_{\mathrm{B}}T)}$
approximates the droplet interface width [23]. In all simulations, we set
$\Lambda_{s}=5\,t_{D}^{-1}$.

For the first prototypical parameter set (i) ($e_{AA}=0.2\,e_{BB}$,
$e_{AB}=0$), we have chosen $\kappa_{A}=\kappa_{B}$ and $\kappa=0$. For the
second prototypical parameter set (ii) ($e_{AA}=e_{BB}$, $e_{AB}=0.2$), we
have chosen $\kappa_{B}=5\,\kappa_{A}$ and $\kappa=2\,\kappa_{A}$. 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 $\kappa_{B}>\kappa>\kappa_{A}$ 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.

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