#  Chemical Reactions regulated by Phase-Separated Condensates

Sudarshana Laha1,2, Jonathan Bauermann1,2, Frank Jülicher1,2,4, Thomas C.T.
Michaels5,6 and Christoph A. Weber1,2,7,8 1Max Planck Institute for the
Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
2Center for Systems Biology Dresden, Pfotenhauerstrasse 108, 01307 Dresden,
Germany
4Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany
5Department of Biology, Institute of Biochemistry, ETH Zurich, Otto Stern Weg
3, 8093 Zurich, Switzerland
6Bringing Materials to Life Initiative, ETH Zurich, Switzerland
7Faculty of Mathematics, Natural Sciences, and Materials Engineering:
Institute of Physics, University of Augsburg, Universitätsstr. 1, 86159
Augsburg, Germany
8Corresponding author<EMAIL_ADDRESS>
###### Abstract

Phase-separated liquid condensates can spatially organize and thereby regulate
chemical processes. However, the physicochemical mechanisms underlying such
regulation remain elusive as the intramolecular interactions responsible for
phase separation give rise to a coupling between diffusion and chemical
reactions at non-dilute conditions. Here, we derive a theoretical framework
that decouples the phase separation of scaffold molecules from the reaction
kinetics of diluted clients. As a result, phase volume and client partitioning
coefficients become control parameters, which enables us to dissect the impact
of phase-separated condensates on chemical reactions. We apply this framework
to two chemical processes and show how condensates affect the yield of
reversible chemical reactions and the initial rate of a simple assembly
process. In both cases, we find an optimal condensate volume at which the
respective chemical reaction property is maximal. Our work can be applied to
experimentally quantify how condensed phases alter chemical processes in
systems biology and unravel the mechanisms of how biomolecular condensates
regulate biochemistry in living cells.

##  I Introduction

Living cells are spatially organized by compartments such as organelles [1]
and protein-RNA condensates [2, 3]. While organelles like mitochondria are
enclosed by membranes, protein-RNA condensates lack a membrane and are instead
phase-separated biomolecular condensates that coexist with the cyto- or
nucleoplasm [4, 5, 6, 7]. Both types of compartments provide specific
physicochemical environments that are required for the occurrence of various
chemical reactions and related biological functions [8, 9]. While membrane-
bound organelles use active membrane pumps to create such specific
environments [10, 11, 12], phase-separated condensates differ in their
composition to the outside already at phase equilibrium [2, 13]. Owing to this
composition difference, reacting components partition differently between the
phases [14, 15, 16]. As a result, diffusion coefficients and reaction rate
coefficients are distinct to each phase [17, 18]. The phase coexistence
between a liquid condensate and its surrounding phase was suggested to
regulate various chemical reactions in living cells and model systems in the
field of systems biology, such as protein aggregation and phosphorylation [19,
20, 9, 21, 22, 23, O’Flynn:2021, 25].

However, when chemical reactions occur, they can give rise to diffusive fluxes
through the condensate interface separating the phases of different
composition [26, 27, 28, 29, 30]. Such diffusive fluxes affect phase
coexistence, i.e., their volumes and compositions. These changes, in turn,
create feedback on the kinetics of chemical reactions. This mutual coupling
between chemical and diffusive fluxes makes it difficult to dissect the
effects of phase-separated condensates on chemical reactions and establish
generic underlying principles.

In biology, a terminology was established for proteins that form biomolecular
condensates in living cells [8, 31, 32]. Such proteins were assigned to two
classes: scaffolds or clients. The scaffold components are thought to be the
main components that “create” the condensates, while the clients can simply
“visit” the condensate by partitioning without significantly affecting the
properties of the condensate. Clients typically participate in chemical
processes that are associated with biological functions [8, 33, 34]. Strictly
speaking, this class assignment is, however, not possible in general due to
the mutual coupling between phase separation and chemical processes.

To bridge this gap, we developed a theoretical framework to describe the
chemical reactions of diluted clients within phase-separated condensates. The
diluted clients interact with non-dilute scaffolds and solvents that phase-
separate into a condensate rich in scaffolds that coexist with its surrounding
scaffold-poor phase. However, consistent with the class assignment used in
cell biology, the clients cannot affect the properties of the condensates. To
illustrate the effects of condensates on chemical processes, we consider
simple but biologically relevant chemical processes. In particular, we
calculate the yield of reversible chemical reactions and the initial rates of
a simple assembly process (see Fig. 1(a,b)). Our key finding is that reaction
yields and initial rates can be maximal for distinct condensate volumes. The
possibility of tuning the system to this maximum suggests the relevance of
condensate size for the control of specific chemical reactions in living
cells.

Figure 1: Schematic representation of a system with dilute reacting clients in
the presence of a single spherical condensate. (a) The non-dilute scaffold
component can form a spherical condensate (phase I). The dilute clients
(reactant and product) partition between the condensed, scaffold-rich phase
and the scaffold-poor phase and undergo chemical reactions in each phase
indicated as I and II. In addition, clients are exchanged between two phases
via diffusion, maintaining the partition equilibrium of clients. (b) Overview
of the different considered chemical processes. We study the kinetics of each
reaction with and without a condensate to decipher how phase coexistence
affects chemical reactions. We distinguish between reactions that can relax
toward chemical equilibrium and reactions that are driven out of equilibrium
by the continuous supply of fuel energy, indicated by star symbols in (a) and
(b).

##  II Theoretical framework for reacting clients in coexisting phases

In this section, we discuss a general theoretical framework for chemical
reactions regulated by liquid-like, phase-separated condensates in an
incompressible system [30, 29]. This framework is derived for diluted
components that can undergo chemical reactions, which we refer to as clients
in the following. These clients are diluted relative to a scaffold component
and the solvent. The client volume fractions are non-zero such that chemical
reactions among clients can occur but are small compared to the volume
fractions of the scaffold and the solvent. The scaffold and solvent are non-
dilute and considered not to participate in the chemical reactions among the
clients; for simplicity, we neglect chemical reactions between scaffold and
solvent. The condensate is rich in this scaffold component and labeled by I.
The condensate phase coexists with the scaffold-poor phase II. For simplicity,
we focus on one spherical condensate embedded in a spherically symmetric
finite system; see Fig. 2(a). Using our framework, we will learn that clients’
kinetics depend on the coexisting phases, while the latter remains unaffected
by the clients.

###  II.1 General theoretical framework

We consider an incompressible mixture of $(N+2)$ components including a non-
reacting solvent ($i=0$), a non-reacting scaffold ($i=1$) and $N$ reacting
clients. Incompressibility corresponds to the case where each molecular volume
$\nu_{i}$ is constant. It implies that we can describe the mixture’s
composition by $(N+2)$ volume fraction fields $\phi_{i}(\textbf{x},t)$ with
$i=0,...,(N+1)$, where x denotes position and $t$ time. Such volume fraction
fields obey $\phi_{0}=1-\sum_{i=1}^{N+1}\phi_{i}$, which allows us to
substitute the solvent volume fraction $\phi_{0}$ in the following framework.
The time evolution of the volume fraction fields $\phi_{i}(\textbf{x},t)$ of
the remaining $(N+1)$ components ($i=1,...,(N+1)$) reads:

$\partial_{t}\phi_{i}=-\nabla\cdot\textbf{j}_{i}+s_{i}\,,$ (1a) with the
diffusive fluxes using linear response [35] given as:
$\textbf{j}_{i}=-\sum_{k}M_{ik}\nabla\bar{\mu}_{k}\,.$ (1b)

Here, $\bar{\mu}_{i}$ is the exchange chemical potential of component $i$,
each calculated relative to the solvent. Moreover, $M_{ik}$ is the volume
fraction-dependent, non-diagonal, and, in general, non-symmetric mobility
matrix. The non-diagonal terms in $M_{ik}$ are called cross-coupling
coefficients. Since the concentration of component $i$, $\nu_{i}\phi_{i}$ with
$\nu_{i}$ denoting its molecular volume, is the conjugate thermodynamic
quantities to the exchange chemical potential $\bar{\mu}_{i}$, the symmetric
Onsager matrix is $M_{ik}/\nu_{i}$ with Onsager’s reciprocal relationship
$M_{ik}\nu_{k}=M_{ki}\nu_{i}$.

The exchange chemical potentials can be expressed as follows [30]:

$\displaystyle\bar{\mu}_{i}(\\{\phi_{j}\\})$
$\displaystyle=\bar{\mu}^{0}_{i}+k_{B}T\log(\bar{\gamma}_{i}(\\{\phi_{j}\\})\phi_{i})-\kappa_{i}\nabla^{2}\phi_{i}\,,$
(2)

where $\bar{\mu}^{0}_{i}(T)$ is the exchange reference chemical potential,
$\bar{\gamma}_{i}(\\{\phi_{j}\\},T)$ denotes the exchange activity coefficient
and $T$ is temperature. The term proportional to $\log{\phi_{i}}$ originates
from the mixing entropy (Appendix A.1). While $\bar{\mu}^{0}_{i}(T)$ includes
component-specific internal free energies and a shift with respect to normal
conditions, the exchange activity coefficient
$\bar{\gamma}_{i}(\\{\phi_{j}\\},T)$ contains the interactions between
component $i$ and $j$. These exchange activity coefficients describe how phase
separation of the scaffold component affects the chemical kinetics and the
difference to the dilute mass action law in homogeneous systems. Moreover,
$\kappa_{i}$ characterizes the free energy penalties for gradients in volume
fractions $\phi_{i}$, where we have omitted cross-couplings for simplicity;
see Appendix A.1 for a more detailed discussion.

The form of the mobility matrix $M_{ik}$ in Eq. (1b) can be obtained by the
following argument: In the dilute limit of all components $i=1,...,(N+1)$
relative to solvent $i=0$, the diffusion matrix
$D_{il}=\sum_{k}M_{ik}\partial\bar{\mu}_{k}/\partial\phi_{l}$ in
$\boldsymbol{j}_{i}=-\sum_{l}D_{il}\nabla\phi_{l}+M_{ii}\kappa_{i}\nabla\nabla^{2}\phi_{i}$
is independent of composition. Thus, the mobility matrix $M_{ik}$ has to
cancel the composition dependence arising from the entropic contribution of
the exchange chemical potential gradient,
$-T\partial(\log{\phi_{i}})/\partial\phi_{i}$ (Eq. (2)). If we additionally
aim for a diagonal diffusion matrix without any cross-diffusion in the dilute
limit of all components $i=1,...,N+1$, i.e., $D_{ii}=m_{0i}k_{B}T$, we can
choose the following form for the mobility matrix [27]:

$\begin{split}&M_{ii}=m_{0i}\phi_{i}(1-\phi_{1}-\sum^{N+1}_{i=2}\phi_{i})+\sum^{N+1}_{\begin{subarray}{c}k\neq
i\,,\\\ k=1\end{subarray}}m_{ik}\phi_{i}\phi_{k}\,,\\\
&M_{ik}=-m_{ik}\phi_{i}\phi_{k}\,,\,\forall i\neq k\,,\end{split}$ (3)

with $m_{ik}$ being mobility coefficients that obey due to Onsager reciprocal
relationship, $m_{ik}\nu_{k}=m_{ki}\nu_{i}$.

The general reaction scheme between the chemical components $C_{i}$ can be
written as

$\sum_{i=2}^{N+1}\sigma_{i\alpha}^{+}C_{i}\rightleftharpoons\sum_{i=2}^{N+1}\sigma_{i\alpha}^{-}C_{i}\,,$
(4)

where the $\sigma_{i\alpha}^{\pm}$ are the stoichiometric coefficients for the
reactants and the products, respectively. Moreover, $\alpha=1,...,R$ labels
the chemical reaction, and $R$ is the total number of chemical reactions in
the system. The chemical reaction rates for the reactive clients, $s_{i}$, is
given as follows [30]:

$s_{i}=\nu_{i}\sum^{R}_{\alpha=1}k_{\alpha}\sigma_{i\alpha}H_{\alpha}\,,$ (5a)
where $\sigma_{i\alpha}=\sigma_{i\alpha}^{-}-\sigma_{i\alpha}^{+}$,
$k_{\alpha}$ is the composition-dependent reaction rate coefficient of
reaction $\alpha$ which can differ between the phases I/II, $\nu_{i}$ is the
molecular volume of component $i$ and $V$ is the system volume. As the system
is incompressible, the chemical reactions have to conserve volume, i.e.,
$\sum_{i}s_{i}=0$ implying that $\sum_{i}\sigma_{i\alpha}\nu_{i}=0$ for each
chemical reaction $\alpha$. Moreover, $H_{\alpha}$ is the reaction force which
can be expressed as follows:
$\begin{split}H_{\alpha}&=\prod^{N+1}_{i=2}\left[\exp{\left(\frac{\bar{\mu}_{i}^{0}+\tilde{\mu}^{+}_{F}-\kappa_{i}\nabla^{2}\phi_{i}}{k_{B}T}\right)}\bar{\gamma}_{i}\phi_{i}\right]^{\sigma_{i\alpha}^{+}}\\\
&\quad-\prod^{N+1}_{i=2}\left[\exp{\left(\frac{\bar{\mu}_{i}^{0}+\tilde{\mu}^{-}_{F}-\kappa_{i}\nabla^{2}\phi_{i}}{k_{B}T}\right)}\bar{\gamma}_{i}\phi_{i}\right]^{\sigma_{i\alpha}^{-}}\,.\end{split}$
(5b)

Here, the first terms correspond to the gain contributions for the products,
while the second terms are the product’s loss terms. Moreover,
$\tilde{\mu}^{\pm}_{F}$ denotes a fuel energy supply that maintains the system
away from equilibrium. This can be achieved by differing
$\tilde{\mu}^{\pm}_{F}$, between the phases I/II. Such a case could be
realized by chemostats for fuel components and their waste to which the fuel
turns over; see Ref. [30] for a more detailed discussion. Without fuel energy
supply ($\tilde{\mu}^{\pm}_{F}=0$) and when the system is at phase equilibrium
($\bar{\mu}_{i}^{\text{I}}=\bar{\mu}_{i}^{\text{II}}$), the reaction force
$H^{\text{I/II}}_{\alpha}$ is phase-independent [29].

###  II.2 Theoretical framework for diluted clients

####  II.2.1 Continuum description

When all clients $i=2,...,(N+1)$ are diluted relative to the solvent ($i=0$)
and scaffold ($i=1$) components, their exchange activity coefficients, defined
in Eq. (2), approach constant values for small client volume fractions. Such
values solely depend on the scaffold equilibrium volume fraction $\phi_{1}$
but not on the volume fraction of other clients $\\{\phi_{j}\\}$. For a mean-
field free energy including a mixing entropy and interactions up to second
order in volume fractions (Eq. (27)), the exchange activity coefficients
defined via Eq. (2) read for diluted clients (derivation see Appendix A):

$\bar{\gamma}_{i}=\frac{1}{(1-\phi_{1})^{r_{i}}}\exp{\bigg{[}r_{i}\phi_{1}(\chi_{1i}-\chi_{0i}-\chi_{01})\bigg{]}}\,,$
(6)

where $r_{i}=\nu_{i}/\nu_{0}$ denotes the ratio of the $i$-th component’s
molecule volume $\nu_{i}$ relative to the solvent molecular volume $\nu_{0}$.
Moreover, the strength of molecular interactions among the components is
characterized by interaction parameters, i.e., between scaffold and solvent
$\chi_{01}$, scaffold with clients $\chi_{1i}$, and solvent with clients
$\chi_{0i}$. The client-client interactions do not appear in Eq. (6) since
clients are dilute with respect to solvent and scaffold. When in addition, the
volume fraction of the scaffold becomes small ($\phi_{1}\to 0$ in Eq. (6)),
all exchange activity coefficients of clients approach unity. Note that Eq.
(6) is also valid when considering free energies beyond mean-field with
interaction terms beyond second order in volume fractions. The reason is that
it already captures the leading order coupling between clients with scaffold
and solvent components (see Appendix A).

For diluted clients in a system with non-dilute scaffold and solvent,
$\sum^{N+1}_{i=2}\phi_{i}\ll\phi_{1}$, cross-couplings between clients and
scaffolds/solvent vanish ($M_{1i}=0,i>1$), and the cross-couplings between
clients are negligible ($M_{ij}=0,i,j>1$). Thus, the mobility matrix (Eq.
(LABEL:eq:mob_matrix)) becomes for diluted clients ($i>1$):

$\displaystyle M_{11}=m_{01}\phi_{1}(1-\phi_{1})\,,$ (7a) $\displaystyle
M_{1i}=0\,,$ (7b) $\displaystyle
M_{ii}=m_{0i}\phi_{i}(1-\phi_{1})+m_{1i}\phi_{i}\phi_{1}\,,$ (7c)
$\displaystyle M_{i1}=0\,,$ (7d) $\displaystyle M_{ij}=0\,,\quad
i\not=j>1\,.\quad$ (7e)

Using Eq. (1), the time evolution of the scaffold volume fraction follows a
Cahn-Hilliard equation for a binary mixture [36]:

$\displaystyle\partial_{t}\phi_{1}$
$\displaystyle=\nabla\cdot\bigg{[}m_{01}\phi_{1}(1-\phi_{1})\bigg{(}k_{B}T\left(\frac{1}{\phi_{1}}+\frac{1}{\bar{\gamma}_{1}}\frac{\partial\bar{\gamma}_{1}}{\partial\phi_{1}}\right)\nabla\phi_{1}$
$\displaystyle\quad-\kappa_{1}\nabla\nabla^{2}\phi_{1}\bigg{)}\bigg{]}\,.$ (8)

The kinetic equations for the clients $i=2,...,(N+1)$ are given by:

$\partial_{t}\phi_{i}=-\nabla\cdot\textbf{j}_{i}+s_{i}(\\{\phi_{j}\\})\,,$
(9a) with the reaction rate $s_{i}(\\{\phi_{j}\\})$ and the client flux
$\textbf{j}_{i}=-\textbf{\emph{v}}_{i}(\phi_{1},\nabla\phi_{1})\phi_{i}-D_{i}(\phi_{1})\nabla\phi_{i}\,.$
(9b) We have neglected the contribution to the client flux
$\kappa_{i}\nabla\nabla^{2}\phi_{i}$ as the spatial transport of clients is
well-captured by the leading order diffusive flux that is proportional to
$\nabla\phi_{i}$; see Appendix B for more details.

Due to phase separation between scaffold components and solvent, clients are
effectively subject to a drift velocity
$\textbf{\emph{v}}_{i}(\phi_{1},\nabla\phi_{1})$. This effective drift arises
from cross-diffusion where clients are driven by gradients in scaffold
components, $\nabla\phi_{1}$ [27]. Moreover, clients diffuse with diffusion
coefficients $D_{i}(\phi_{1})$ that is set by the local scaffold volume
fraction (derivation see Appendix B):

$\displaystyle\textbf{\emph{v}}_{i}(\phi_{1},\nabla\phi_{1})$
$\displaystyle=D_{i}(\phi_{1})\frac{1}{\bar{\gamma}_{i}}\frac{\partial\bar{\gamma}_{i}}{\partial\phi_{1}}\nabla\phi_{1}\,,$
(9c) $\displaystyle D_{i}(\phi_{1})$
$\displaystyle=k_{B}T\bigg{[}m_{0i}(1-\phi_{1})+m_{1i}\phi_{1}\bigg{]}\,.$
(9d)

The effective drift velocities $\textbf{\emph{v}}_{i}$ depend on the local
diffusion coefficients $D_{i}(\phi_{1})$ highlighting once more their origin
in cross-diffusion. The effective drift velocities also depend on the
thermodynamic parameters such as molecular volumes $\nu_{i}$, encoded in
$r_{i}=\nu_{i}/\nu_{0}$, and the interaction parameters $\chi_{1i}$ and
$\chi_{0i}$. These dependencies come from the activity coefficients in Eq.
(9c) via:

$\frac{1}{\bar{\gamma}_{i}}\frac{\partial\bar{\gamma}_{i}}{\partial\phi_{1}}=r_{i}(\chi_{1i}-\chi_{0i}-\chi_{01})+\frac{r_{i}}{\left({1-\phi_{1}}\right)}\,.$
(10)

Note that the limit $\phi_{1}\to 1$ does not lead to a divergence of the drift
velocity because the system becomes homogeneous and $\nabla\phi_{1}$ vanishes.

Our derivation shows that the time-dependent spatial profiles of the scaffold
$\phi_{1}(\textbf{x},t)$ determine a spatiotemporal environment for clients
undergoing diffusion and chemical reactions. Note that the chemical reaction
rates $s_{i}$ for diluted clients are given by Eq. (5) with the exchange
activity coefficients $\bar{\gamma}_{i}$ shown in Eq. (5b) that solely depend
on the scaffold volume fraction. The term $\kappa_{i}\nabla^{2}\phi_{i}$ in
the reaction force $H_{\alpha}$ can be neglected as it solely creates a
positive up-shift of the diffusion coefficient (to lowest order when expanding
the exponential in Eq. (5b)).

A key hallmark of the chemical kinetics in a mixture that is composed of the
scaffold, solvent, and diluted clients is that the scaffold volume fraction
evolves according to Eq. (8) completely independent of the client volume
fraction. In other words, the clients do not affect the phase separation
between the scaffold component and solvent. However, the dynamic equations (9)
for the clients depend on the scaffold volume fraction. Once the scaffold
volume fraction has settled in a stationary state ($\partial_{t}\phi_{1}=0$ in
Eq. (8)), it exhibits a sigmoidal profile with the plateaus equal to the
scaffold volume fractions at phase equilibrium
$\phi^{\text{\text{I/II}}}_{1}$; see Fig. 2(b). The respective plateau values
determine the kinetic coefficients of clients, such as reaction rate
coefficients and diffusivities, in phase I and II. For a radially symmetric
system with the radial position $r$, this stationary profile is approximately
given as [26]

$\phi_{1}^{\text{int}}(r)\simeq\frac{1}{2}(\phi^{\text{I}}_{1}+\phi^{\text{II}}_{1})-\frac{1}{2}(\phi^{\text{I}}_{1}-\phi^{\text{II}}_{1})\tanh{\left((r-R)/\lambda^{\text{int}}\right)}\,,$
(11)

where the interface width is

$\lambda^{\text{int}}=\sqrt{\frac{\kappa}{(\chi_{01}-2)k_{B}T\nu_{0}r_{1}}}\,.$
(12)

For simplicity, we considered the limit $\lambda^{\text{int}}\ll R$, allowing
us to neglect the effects related to the Laplace pressure and the curvature of
the condensate [36].

Figure 2: Schematic representation of the system to study how phase
coexistence affects chemical reactions of dilute clients. (a) We consider a
spherical, condensate of radius $R$, which is rich in scaffold components
(phase I) coexisting with a scaffold-poor phase (phase II). The system is also
spherical with radius $L$. (b) Stationary spatial profiles of the scaffold
$\phi_{1}(r)$ in the dilute limit of clients; continuum model (grey dotted)
and thin interface model (solid orange).

####  II.2.2  Stationary scaffold in thin interface model

Here, we discuss an approximation of the continuous model for a scaffold
volume fraction profile that is stationary ($\partial_{t}\phi_{1}=0$ in Eq.
(8)) with an interface width that is thin compared to the scaffold-rich and
poor bulk phases, respectively, i.e., $\lambda^{\text{int}}/R\rightarrow 0$.
In this limit, the continuous profile given by Eq. (11) becomes a step profile
that jumps at the interface (Fig. 2(b); [37]). Stationarity of the scaffold
profile that is independent of the dilute clients implies that $\phi_{1}$ is
constant in each phase. Thus, the effective drift velocity
$\textbf{\emph{v}}_{i}(\phi_{1},\nabla\phi_{1})$ and higher order
contributions to the client flux, $\nabla\nabla^{2}\phi_{1}$, vanish in Eq.
(9b). In the following, we consider a single, spherical condensate of radius
$R$ in a radially symmetrical, spherical domain of diameter $2L$
($\textit{{r}}=(r,\theta,\varphi)$ denote the spherical coordinates).

Away from the interface and for a stationary scaffold profile
$\phi_{1}(\boldsymbol{r})$, the dynamical equation for the dilute reacting
clients in both phases I/II correspond to reaction-diffusion equations:

$\partial_{t}\phi_{i}(\textit{{r}},t)=D^{\text{I/II}}_{i}\nabla^{2}\phi_{i}(\textit{{r}},t)+s^{\text{I/II}}_{i}(\\{\phi_{j}(\textit{{r}},t)\\})\,,$
(13a) where index $i=2,...,(N+1)$, represents the individual clients in the
system. Here, phase I is located in $0<r<R$, while phase II extends in the
range, $R<r<L$. Moreover, $s_{i}^{\text{I/II}}$ are the chemical reaction
rates for the $i$-th client, which is given in Eq. (5) with reaction rate
coefficients $k_{\alpha}^{\text{I/II}}$ and fuel energy supply
$\tilde{\mu}^{\pm,\text{I/II}}_{F}$ being phase-dependent. Note that Eq. (13a)
is obtained without expanding around the phase equilibrium volume fractions
[26, 30]. Client diffusion following Fick’s law combined with, in general,
non-linear chemical reaction rates is a consequence of clients being diluted
with respect to the phase-separated scaffold and solvent components.
Furthermore, since the scaffold volume fraction $\phi_{1}^{\text{I/II}}$ is
homogeneous in each phase, the diffusion coefficients of clients are constants
that solely differ among the phases: $\displaystyle
D^{\text{I/II}}_{i}=\begin{cases}k_{B}T\bigg{(}m_{0i}(1-\phi^{\text{I}}_{1})+m_{1i}\phi^{\text{I}}_{1}\bigg{)}\,&\text{for}\,\,0<{r}<R\,\,,\\\
k_{B}T\bigg{(}m_{0i}(1-\phi^{\text{II}}_{1})+m_{1i}\phi^{\text{II}}_{1}\bigg{)}\,&\text{for}\,R<{r}<L\,.\end{cases}$
(13b) Note that the client diffusion constants $D^{\alpha}_{i}$ are
independent of client concentrations. Thus, they remain constant during the
chemical reaction kinetics of the clients, while client profiles vary in time
and space.

The reaction-diffusion equations of each component $i$ (Eq. (13a)) are coupled
via the boundary conditions at the interface. At the interface $r=R$ and the
system boundary $r=L$ (Fig. 2(a)), we can write the following boundary
conditions for the clients’ dynamic Eq. (13a):

  * •

Volume conservation of clients across the interface implies that the radial
fluxes across the interface,
$\boldsymbol{e}_{r}\cdot\boldsymbol{{\textit{{j}}}}_{i}$, inside ($r=R_{-}$)
and outside ($r=R_{+}$) of the interface are equal,

$\boldsymbol{e}_{r}\cdot\boldsymbol{{\textit{{j}}}}_{i}|_{r=R_{-}}=\boldsymbol{e}_{r}\cdot\boldsymbol{{\textit{{j}}}}_{i}|_{r=R_{+}}\,,$
(13c)

where
$\boldsymbol{{\textit{{j}}}}_{i}=-\boldsymbol{e}_{r}\,D_{i}^{\alpha}\partial_{r}\phi_{i}$
is the flux and $\boldsymbol{e}_{r}$ denotes the radial unit vector.

  * •

Local phase equilibrium at the interface leads to client volume fractions
inside and outside of the interface that satisfy the partition coefficient,

$P_{i}=\frac{\phi_{i}^{\text{}}|_{r=R_{-}}}{\phi_{i}^{\text{}}|_{r=R_{+}}}\,.$
(13d)

For diluted clients, the partition coefficients are constant and are
determined by the interaction strength of clients with scaffold and solvent
components, as well as the volume fraction difference of scaffold between
inside and outside.

  * •

For a system boundary at $r=L$ that is impermeable for the clients, the
diffusive flux of clients vanishes:

$\boldsymbol{e}_{r}\cdot\boldsymbol{{\textit{{j}}}}_{i}|_{r=L}=0\,.$ (13e)

  * •

In the center of the spherical condensate at $r=0$, the flux has to vanish for
each client:

$\boldsymbol{e}_{r}\cdot\boldsymbol{{\textit{{j}}}}_{i}|_{r=0}=0\,.$ (13f)

A solution of the volume fraction profile of a client at different time points
is shown in Fig. 4(a), where the jump at the interface is set by its partition
coefficient. The details of the considered chemical processes are discussed in
Sect. III.

####  II.2.3 Model at phase equilibrium

In this section, we discuss the case when clients diffuse fast compared to
their reaction, i.e., the reaction rate coefficients corresponding to the
linearized reaction rates are small compared to the system’s slowest diffusion
rates, $D_{i}^{\text{I/II}}/L^{2}$. In this case, the phases are homogeneous
and at phase equilibrium with respect to each other at all times during the
chemical kinetics. This limiting case reduces the mathematical complexity of
the theoretical description significantly, and chemical kinetics is governed
by ordinary differential equations [38, 39, 29], similar to the classical mass
action law kinetics in homogeneous systems.

At phase equilibrium, the client volume fractions $\phi^{\text{I/II}}_{i}$ in
each phase I/II are homogeneous and satisfy the partition coefficient,

$P_{i}=\frac{\phi^{\text{I}}_{i}}{\phi^{\text{II}}_{i}}\,.$ (14)

Using the exchange chemical potential (Eq. (2)), the partitioning coefficients
can be expressed at phase equilibrium in terms of exchange activity
coefficients,
$P_{i}=\bar{\gamma}_{i}^{\text{II}}/\bar{\gamma}_{i}^{\text{I}}$. For the
discussed case of diluted clients (Eq. (6)), the client partitioning
coefficients are constants and, thus, control parameters.

At phase equilibrium, the volume fractions of the clients in the two phases I
and II,

$\displaystyle\phi_{i}^{\text{I}}(t)$
$\displaystyle=P_{i}\,\zeta_{i}\,\bar{\phi}_{i}(t)\,,$ (15a)
$\displaystyle\phi_{i}^{\text{II}}(t)$
$\displaystyle=\zeta_{i}\,\bar{\phi}_{i}(t)\,,$ (15b) where
$\zeta_{i}(P_{i},V^{\text{I}})=\frac{1}{1+(P_{i}-1)\frac{V^{\text{I}}}{V}}\,$
(15c)

is the partitioning degree. The volume fractions of the clients in the two
phases can be expressed in terms of the average client volume fraction as

$\bar{\phi}_{i}(t)=\frac{1}{V}\Bigg{(}V^{\text{I}}\phi^{\text{I}}_{i}(t)+V^{\text{II}}\phi^{\text{II}}_{i}(t)\Bigg{)}\,,$
(16)

where $V^{\text{I/II}}$ denotes the respective phase volumes, and
$V=V^{\text{I}}+V^{\text{II}}$ is the total volume. At phase equilibrium, the
phase volume
$V^{\text{I}}=V(\bar{\phi}_{1}-\phi_{1}^{\text{II}})/(\phi_{1}^{\text{I}}-\phi_{1}^{\text{II}})$
is constant and set by the average scaffold volume fraction, $\bar{\phi}_{1}$.
Thus, for the client dynamics, $V^{\text{I}}$ is a control parameter.

The chemical reactions among clients change the average volume fractions of
clients in time:

$\frac{d}{dt}\bar{\phi}_{i}(t)=\bar{s}_{i}(\\{\bar{\phi}_{j}(t)\\})\,,$ (17)

where
$\bar{s}_{i}(t)=\left(V^{\text{I}}s^{\text{I}}_{i}+V^{\text{II}}s^{\text{II}}_{i}\right)/V$
are the average chemical reaction rates with the phase-dependent reaction
rates $s^{\text{I/II}}_{i}$ given in Eq. (5). Note that
$\phi_{i}^{\text{I/II}}$ in $s^{\text{I/II}}_{i}$ can be substituted by the
average volume fractions $\\{\bar{\phi}_{j}(t)\\}$ using Eqs. (15). Thus the
average reaction rates $\bar{s}_{i}$ solely depend on the average volume
fractions $\\{\bar{\phi}_{j}(t)\\}$.

Equation (17) governs the chemical reaction of client $i$ at phase
equilibrium. Compared to the thin interface model (Sect. II.2.2), the chemical
kinetics is governed by ordinary differential equations, and the diffusion
coefficients no longer determine the dynamics of the client volume fractions.
The remaining parameters at phase equilibrium are the reaction rate
coefficients $k^{\text{I/II}}_{\alpha}$, the fuel energies
$\tilde{\mu}^{\text{I/II}}_{F}$, the partition coefficients $P_{i}$, and the
phase volumes $V^{\text{I/II}}$. Note that Eq. (17) can describe chemical
kinetics maintained away from chemical equilibrium by fuel energy
$\tilde{\mu}^{\text{I/II}}_{F}$.

##  III Application of theoretical framework

In the following two sections, we apply the theoretical framework discussed in
Sect. II to reacting diluted clients in two coexisting scaffold-rich and poor
phases to two different types of chemical reactions (see Fig. 1(b) for an
overview). We determine the quantities relevant for each chemical kinetics,
such as stationary yields and initial rates. We compare such quantities with
and without condensates and distinguish systems that are maintained away from
chemical equilibrium by the fuel energy $\tilde{\mu}^{\text{I/II}}_{F}$ and
that can relax toward chemical equilibrium
($\tilde{\mu}^{\text{I/II}}_{F}=0$).

In Sect. III.1, we ask how coexisting phases alter the stationary states of
reversible chemical reactions, while in Sect. III.2, we analyze the effects of
coexisting phases on initial rates of irreversible assembly processes. To
illustrate the effects of coexisting phases on chemical reactions, we focus on
a single scaffold-rich condensate phase of radius $R$ that is located in the
center of a finite spherical symmetric container of radius $L$ (Fig. 2(a)).

Figure 3: Condensates strongly affect the steady state of bi-molecular
reactions ($A_{1}+A_{2}\rightleftharpoons B$). (a) Spatiotemporal profiles of
the product component, $\phi_{B}(r,t)$, relative to the conserved quantity
$\psi_{1}=(\bar{\phi}_{A_{1}}+\bar{\phi}_{A_{2}}+\bar{\phi}_{B})$. The system
initialized at phase equilibrium (light gray) approaches a non-equilibrium
steady state (blue, $\tilde{\mu}^{\text{I}}_{F}/k_{B}T=5$). Dark gray depicts
the profile at an intermediate time $t=1$. Time is rescaled by $t\to
t\,k^{\text{I}}$, where $k^{\text{I}}$ is the reaction rate coefficient in
phase I defined in Eq. (20). (b) The spatial volume fraction profiles of the
product at steady state for finite diffusivity values. When chemical reactions
are maintained away from equilibrium (solid,
$\tilde{\mu}^{\text{I}}_{F}/k_{B}T=5$), the profiles are spatially
heterogeneous, while product volume fractions are at thermodynamic equilibrium
(dashed blue line, $\tilde{\mu}^{\text{I}}_{F}/k_{B}T=0$). (c,d) The yield
$\mathcal{Y}$ relative to the case without condensate is maximal at a finite
volume $V^{\text{I},*}$. The maximal yield $\mathcal{Y}^{*}$ increases with
increasing fuel energy supply $\tilde{\mu}^{\text{I}}_{F}$ and the faster
diffusion compared to the chemical reactions ($\tilde{D}_{i}\to\infty$). (e,f)
The maximal yield $\mathcal{Y}^{*}$ increases when more substrate partitions
into the condensate and the faster the reactions are in the scaffold-rich
condensate compared to the scaffold-poor phase ($k^{\text{II}}/k^{\text{I}}\ll
1$). In this limit, the corresponding condensate volume decreases.

###  III.1 Reversible chemical reactions controlled by a scaffold-rich
condensate

Here, we study reversible two state transitions between $g$ deactivated
reactants $A_{i}$ (substrate) and the activated product $B$ with the reaction
scheme

$\begin{split}\,\sum^{g}_{i=1}A_{i}\ce{<=>[\ce{\tilde{\mu}^{\text{I/II}}_{F}}][\ce{\text{spontaneous}}]}B\,.\end{split}$
(18)

Here, $\tilde{\mu}^{\text{I/II}}_{F}$ is the fuel energy that maintains the
forward reaction continuously away from chemical equilibrium, while the
backward reaction can relax spontaneously toward chemical equilibrium.
Maintaining away from chemical equilibrium is realized by considering
different values of the fuel energy $\tilde{\mu}^{\text{I/II}}_{F}$ in each of
the phases. Special cases of the reaction scheme (18) are the uni-molecular
reaction for $g=1$ and the bi-molecular reaction for $g=2$. For simplicity, we
consider a single product. Below, we will discuss and compare the results
obtained for bi-molecular reactions (Fig. 3) to uni-molecular (Fig. 4).

The bi-molecular reaction ($g=2$) has two conserved quantities, which we
choose as $\psi_{1}=(\bar{\phi}_{A_{1}}+\bar{\phi}_{A_{2}}+\bar{\phi}_{B})$
and $\psi_{2}=(\bar{\phi}_{A_{1}}-\bar{\phi}_{A_{2}})$. In our work, we
consider incompressible systems and thereby restrict ourselves to volume-
conserving chemical reactions, requiring that the chemical reaction rates
obey:

$\displaystyle s^{\text{I/II}}_{A_{1}}$
$\displaystyle=s^{\text{I/II}}_{A_{2}}\,,$ (19a) $\displaystyle
s^{\text{I/II}}_{A_{1}}+s^{\text{I/II}}_{A_{2}}$
$\displaystyle=-s^{\text{I/II}}_{B}\,.$ (19b)

For diluted clients $A_{i}$ (substrates) and $B$ (product) at phase
equilibrium and for $g=2$, the chemical reaction rate of the product $B$ reads

$\displaystyle
s^{\text{I/II}}_{B}=k^{\text{I/II}}\frac{\nu_{B}}{\nu_{A_{i}}}\bigg{[}-\exp\bigg{(}\frac{\bar{\mu}^{0}_{B}}{k_{B}T}\bigg{)}(\bar{\gamma}_{B}\phi_{B})^{\text{I/II}}$
(20)
$\displaystyle\quad+\exp\bigg{(}\frac{\bar{\mu}^{0}_{A_{1}}+\bar{\mu}^{0}_{A_{2}}+\tilde{\mu}^{\text{I/II}}_{F}}{k_{B}T}\bigg{)}(\bar{\gamma}_{A_{1}}\phi_{A_{1}})^{\text{I/II}}(\bar{\gamma}_{A_{2}}\phi_{A_{2}})^{\text{I/II}}\bigg{]}\,,$

where molecular volumes obey $\nu_{A_{i}}/\nu_{B}=1/2$ for $i=1,2$ and
$k^{\text{I/II}}$ denote the reaction rate coefficients in phase I and II.
Moreover, $\bar{\gamma}^{\text{I/II}}_{i}$ are the constant exchange activity
coefficients of the clients $A_{i}$ and $B$, which depends on the constant
scaffold volume fraction $\phi^{\text{I/II}}_{1}$ in the respective phases.
Note that at phase equilibrium (Sect. II.2.3), the activity coefficients of
the two phases are coupled to each other via the partitioning coefficients
$P_{i}=\bar{\gamma}_{i}^{\text{II}}/\bar{\gamma}_{i}^{\text{I}}$
($i=A_{1},A_{2},B$) that are independent of client volume fraction for diluted
clients. In the case of the thin interface model (Sect. II.2.2), the position
and time-dependent client fields $\phi_{i}(r,t)$ satisfy partitioning at the
interface (Eq. (13d)) but not necessarily inside the phases.

Using the thin interface model for step-like and stationary scaffold profile,
we numerically calculated the volume fraction profiles of the activated
product $\phi_{B}(r,t)$. The results are shown in Fig. 3(a,b) for $g=2$ and in
Fig. 4(a,b) for $g=1$. We could also derive analytic solutions for uni-
molecular reactions ($g=1$); see Appendix D for details. Initializing the
concentrations in each phase at phase equilibrium, the chemical kinetics gives
rise to spatial gradients in each phase over time. For systems that are
maintained away from chemical equilibrium
($\tilde{\mu}_{F}^{\text{I/II}}\not=0$), gradients in the phases and, thereby,
fluxes between the phases persist in the non-equilibrium steady state.

####  III.1.1 Condensates and non-equilibrium driving strongly enhance the
yield of reversible reactions

To characterize the effects of phase-separated condensates on chemical
kinetics, we consider the yield at steady state ($t\to\infty$) of the average
volume fraction of the activated product, $\bar{\phi}_{B}(\infty)$. We define
the relative yield as the ratio of the yield in the presence of a condensate
of volume $V^{\text{I}}$ to the yield without a condensate ($V^{\text{I}}=0$):

$\mathcal{Y}=\frac{\bar{\phi}_{B}(\infty)}{\bar{\phi}_{B}(\infty)|_{V^{\text{I}}=0}}\,.$
(21)

The average volume fraction of each diluted client $\bar{\phi}_{i}(\infty)$ at
steady state is given by Eq. (16) at phase equilibrium, and for the thin
interface and continuum model, it is given by
$\bar{\phi}_{i}(\infty)=V^{-1}4\pi\int_{0}^{L}dr\,r^{2}\phi_{i}(r,\infty)$ and
$V=(4/3)\pi L^{3}$ (Fig. 2(a)). In summary, the relative yield characterizes
how much the compartment volume $V^{\text{I}}$, which is controlled by the
amount of scaffold $\bar{\phi}_{1}$, regulates the steady-state volume
fraction of the product.

Figure 4: Condensates affect the steady-state turnover of uni-molecular
reactions ($A_{1}\rightleftharpoons B$). (a) Spatiotemporal profile of the
product component, $\phi_{B}$, relative to the conserved quantity,
$\psi=(\phi_{A_{1}}+\phi_{B})$. The system initialized at phase equilibrium
(light gray) approaches a non-equilibrium steady state (blue,
$\tilde{\mu}^{\text{I}}_{F}/k_{B}T=5$). Dark gray depicts the profile at an
intermediate time $t=0.075$, where time is rescaled by $t\to t\,k^{\text{I}}$
with $k^{\text{I}}$ being the reaction rate coefficient in phase I. (b) The
spatial volume fraction profiles of the clients at steady state for a finite
diffusivity value, $\tilde{D}_{B}=0.01$. When chemical reactions are
maintained away from equilibrium (solid,
$\tilde{\mu}^{\text{I}}_{F}/k_{B}T=5$), the profiles are spatially
heterogeneous, while client volume fractions are homogeneous at thermodynamic
equilibrium (dashed, $\tilde{\mu}^{\text{I}}_{F}/k_{B}T=0$). (c,d) The yield
$\mathcal{Y}$ relative to the case without condensate is maximal at a finite
volume $V^{\text{I},*}$. The maximal yield $\mathcal{Y}^{*}$ increases with
increasing fuel energy supply $\tilde{\mu}^{\text{I}}_{F}$ and the faster
diffusivity (black dotted) compared to the chemical reactions. Importantly,
there is no maximum for the passive case corresponding to
$\tilde{\mu}^{\text{I}}_{F}=0$ for uni-molecular reactions (black dashed in
(c)) and the limit of no diffusivity (black solid in (d)). (e,f) The maximal
yield $\mathcal{Y}^{*}$ increases when less substrate $A_{1}$ partitions and
the larger the reaction rate coefficient in the condensate relative to the
scaffold-poor phase, $k^{\text{II}}/k^{\text{I}}\ll 1$. The condensate volume
corresponding to maximal yield gets smaller for decreasing
$k^{\text{II}}/k^{\text{I}}$ and partition coefficient $P_{A_{1}}$.

We find that for the passive case ($\tilde{\mu}_{F}^{\text{I/II}}=0$) and the
case with fuel energy supply ($\tilde{\mu}_{F}^{\text{I/II}}\not=0$), the
relative yield $\mathcal{Y}$ can increase due to the presence of a phase-
separated condensate, i.e., $\mathcal{Y}>1$; see Fig. 3(c). An exception is
the uni-molecular reaction ($g=1$), where $\mathcal{Y}$ monotonously decreases
with increasing condensate volume for $\tilde{\mu}_{F}^{\text{I/II}}=0$
(dashed line in Fig. 4(c)). The increase of the relative yield depends on the
value of fuel energy $\tilde{\mu}_{F}>0$. While the yield for passive systems
with bi-molecular reactions increases only weakly by having a condensate
(black dashed line in Fig. 3(c)), chemical reactions maintained away from
equilibrium give rise to a significantly more pronounced increase of the
relative yield $\mathcal{Y}$. Specifically, the yield can increase by 100-fold
already for a fuel energy supply inside the condensate
$\tilde{\mu}^{\text{I}}_{F}$ ($\tilde{\mu}^{\text{II}}_{F}=0$) that is a few
$k_{B}T$. For example, such an amount of free energy can be provided by the
hydrolysis of ATP [40].

####  III.1.2 Maximal yield in reversible reactions mediated by condensates

A hallmark feature of our framework where clients are diluted relative to
scaffold and solvent components is that the condensate volume $V^{\text{I}}$
is a control parameter. This means that $V^{\text{I}}$ can be varied without
being altered by the clients’ chemical kinetics. Most importantly, we expect
that the chemical kinetics of clients is affected when varying the condensate
volumes $V^{\text{I}}$. When increasing $V^{\text{I}}$, we find a maximum in
the relative yield (Fig. 3(c,d) and Fig. 4(c,d)). The yield maximum
$\mathcal{Y}^{*}$ corresponds to a specific optimal condensate volume
$V^{\text{I},*}$ at which
$\mathcal{Y}^{\prime}(V^{\text{I}})|_{V^{\text{I},*}}=0$. The existence of
such a maximum requires that the slope of the yield is positive at
$V^{\text{I}}=0$, and negative when the condensate volume is equal to the
system volume, $V^{\text{I}}=V$. These conditions are sufficient for a uni-
molecular reaction ($g=1$) and a bi-molecular reaction ($g=2$) since there is
at most one maximum in the interval $(0,V^{\text{I}}/V)$. The conditions for
both cases are given in the Appendix, Sect. C.1 and C.2.

For passive ($\tilde{\mu}_{F}^{\text{I}}=0$), uni-molecular ($g=1$) reactions
(see Fig. 4(c)), we find that there cannot be any maximum in yield. The yield
monotonously increases or decreases with condensate volume $V^{\text{I}}$. The
reason for this behavior is related to the linearity of the uni-molecular
reactions among diluted clients. Although partitioning of a reactant from one
to the other phase can alter the chemical kinetics in each phase, there is no
effect on the kinetics of the average composition and thereby on the relative
yield (Eq. (21)). This is because the average compositions are proportional to
the volume-weighted reaction rate coefficient,
$(V^{\text{I}}k^{\text{I}}+V^{\text{II}}k^{\text{II}})/V$, that changes
monotonously with condensate volume $V^{\text{I}}$.

A maximum exists when uni-molecular ($g=1$) reactions are maintained away from
equilibrium ($\tilde{\mu}_{F}^{\text{I}}\not=0$ in Fig. 4(c)), or when
considering bi-molecular reactions ($g=2$) (Fig. 3(c)). In both cases, a non-
linear volume dependence is introduced, giving rise to two competing effects
that lead to a maximum in the yield as a function of condensate volume. To
understand this competition, we discuss the effects of introducing a small
scaffold-rich condensate of volume $V^{I}/V\ll 1$. To obtain a maximum in the
relative yield, the tiny condensate has to promote the formation of product
$B$ for which we evaluate the yield $\mathcal{Y}$ (Eq. (21)). Product
formation is promoted by a condensate when the reaction rate coefficients
satisfy $k^{\text{I}}/k^{\text{II}}\gg 1$. Thus, increasing condensate volume
also increases the product yield in the entire system. However, increasing
condensate volume $V^{\text{I}}$ further toward $V^{\text{I}}/V=1$ leads to a
decrease in the relative yield $\mathcal{Y}$ at some point. This decrease in
$\mathcal{Y}$ results from a decrease in the rate of product formation $s_{B}$
due to a generic dilution effect inside the condensates when increasing the
condensate volume $V^{\text{I}}$ from a tiny volume to the volume of the
system $V$. Dilution arises from a decrease in substrate volume fraction
inside the condensate from $P_{A_{i}}\bar{\phi}_{A_{i}}\to\bar{\phi}_{A_{i}}$
for $V^{\text{I}}/V\to 1$. This effect is characterized by the partitioning
degree (Eq. (15c)) that decreases from 1 to $P_{A_{i}}^{-1}$ at
$V^{\text{I}}/V=1$.

We further observe that the relative yield $\mathcal{Y}$ is most pronounced in
the limit when client diffusion rates are fast compared to reaction rate
coefficients among clients (Fig. 3(d) and Fig. 4(d)). This case corresponds to
clients being at phase equilibrium during the reaction kinetics and is well-
fulfilled when the system size is smaller than the reaction length scales (see
Sect. 18 for a more detailed discussion). The reason why the yield is largest
at phase equilibrium can be understood by considering the opposite limit,
i.e., when client reactions become fast compared to diffusion of clients
(black solid line in Fig. 3(d) and Fig. 4(d)). In this case, there are still
two phases with different reaction rate coefficients $k^{\text{I}}$ and
$k^{\text{II}}$ because the scaffold component is phase-separated, providing
distinct domains for chemical reactions. However, as clients diffuse slowly
compared to their reactions, the bulk phases cannot follow the relative
partitioning at the interface. Thus, introducing a condensate is of no benefit
except for the differences in the reaction rate coefficients. This can only
lead to a monotonous increase in yield with condensate volume (for
$k^{\text{I}}>k^{\text{II}}$). Consistently, yields become equal in both
limits $V^{\text{I}}/V=0,1$ since there is no partitioning when there is no
condensate or when it occupies the full system volume $V$. Our studies show
that tiny condensates are optimal bioreactors. The volume corresponding to
maximal yield, $V^{\text{I},*}$ decreases and value of the maximal yield
$\mathcal{Y}$ increases when condensates favor more the formation of the
product ($k^{\text{II}}/k^{\text{I}}\to 0$ in Fig. 3(e,f) and 4(e,f)). We also
find that the larger the partitioning coefficients of the substrate $A_{i}$,
the smaller the volume at maximal yield, $V^{\text{I},*}$. Both trends support
the idea that biomolecular condensates in living cells, which are usually much
smaller than the system volume, can have significant effects on the yield of
reversible chemical reactions that are maintained away from chemical
equilibrium.

###  III.2 Assembly processes controlled by a scaffold-rich condensate

Figure 5: Condensates can speed up assembly processes. The temporal evolution
of the average volume fraction of the monomer $A$ is shown for different
assembly reaction orders, (a) $n=2$ and (b) $n=3$, for passive systems
($\tilde{\mu}_{F}^{\text{I}}=0$) and assembly processes promoted by a fuel
energy supply ($\tilde{\mu}_{F}^{\text{I}}\not=0$) (dashed and solid lines
respectively). Gray lines indicate the initial decay of components $A$,
$\bar{\phi}_{A}(t)=t\,\dot{\bar{\phi}}_{A}(t=0)$. The initial rate
$\dot{\bar{\phi}}_{A}(t=0)$ is analyzed in more detail in the following figure
panels. For both plots a and b, $V^{\text{I}}=0.008\,V$. Time is rescaled by
$t\to t\,k^{\text{I}}$, where $k^{\text{I}}$ is the reaction rate coefficient
in phase I defined in Eq. (23). (c) The relative initial rate of assembly
$\mathcal{R}$ (Eq. (24)) exhibits a maximum with condensate volume
$V^{\text{I}}$, for $n=2$ and $n=3$. The maximum results from a competition
between the condensate-mediated promotion of assembly and a dilution effect
for very large condensate volumes; see Sect. III.1.2 for details. (d)
Acceleration of the assembly process is most pronounced at phase equilibrium
(dotted line) where diffusion rates are fast compared to reaction rate
coefficients. This is evident in the slope of $\bar{\phi}_{A}(t)$ at early
times $t$. (e) The condensate volume corresponding to maximal assembly rates
decreases for larger partitioning of the monomer, $P_{A}$, while the ratio of
reaction rate coefficients, $k^{\text{II}}/k^{\text{I}}$, has almost no
impact. The assembly reaction order in (e) and (f) is $n=2$. (f) The maximum
amplification in assembly rates increases with decreasing ratio of reaction
rate coefficients, $k^{\text{II}}/k^{\text{I}}$. Partitioning of monomers in
the condensate phase doesn’t necessarily imply a higher value of
$\mathcal{R}$.

In this section, we use the framework developed in Sec. III to study a simple
model for the assembly of monomers to a nucleus composed of many monomers.
Such a process is abundant in living cells. Examples are the polymerization of
biofilaments and the aberrant aggregation of misfolded proteins. Condensates
were suggested to control the formation of such nuclei. Here, we study how the
initial assembly rate of nucleus formation is affected by a condensate of
volume $V^{\text{I}}$. We distinguish between a reversible assembly process
that can relax towards thermodynamic equilibrium and that is maintained away
by the fuel energy $\tilde{\mu}_{F}$. We choose the internal energy of the
nucleating components higher such that the backward pathway of the assembly
process is suppressed. In particular, at early times, this pathway is
negligible. We only include this pathway to obtain a stationary state at long
times that is consistent with thermodynamics, i.e., a state with a non-
vanishing volume fraction of assembly-prone monomers.

In this simple model for an assembly process, $n$ identical monomeric
components $A$ reversibly form a nucleus $A_{n}$ composed of $n$ monomers,
where $n$ is the assembly reaction order. The corresponding reaction scheme
is:

$n\,A\ce{<=>>[\ce{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}}\tilde{\mu}_{F}^{\text{I/II}}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}}}][\ce{\text{suppressed}}]}A_{n}\,,$
(22)

where $\tilde{\mu}_{F}^{\text{I/II}}$ is the phase-dependent fuel energy
maintaining the assembly process away from equilibrium. At phase equilibrium,
the corresponding chemical reaction rates are

$\begin{split}s^{\text{I/II}}_{A}&=-s^{\text{I/II}}_{A_{n}}\\\
&=k^{\text{I/II}}\frac{\nu_{A_{n}}}{\nu_{A}}\bigg{[}\exp\bigg{(}\frac{\bar{\mu}_{A_{n}}}{k_{B}T}\bigg{)}-\exp\bigg{(}\frac{n\bar{\mu}_{A}+\tilde{\mu}^{\text{I/II}}_{F}}{k_{B}T}\bigg{)}\bigg{]}\,.\end{split}$
(23)

where due to incompressibility the fraction of molecular volumes,
$\nu_{A_{n}}/{\nu_{A}}$, is chosen to be equal to the assembly reaction order
$n$. The conserved quantity of this assembly process is
$\psi=(\bar{\phi}_{A}+\bar{\phi}_{A_{n}})$. Using Eq. (23) in Eq. (13a), we
proceed to solve the system numerically. As initial condition, we use
$\bar{\phi}_{A_{n}}(t)=0.01\psi$.

####  III.2.1 Condensates and non-equilibrium driving can strongly accelerate
assembly processes

To characterize the effects of phase-separated condensates on the formation of
assemblies, we consider the initial (at time $t=0$) assembly rate of the
monomeric component $A$ relative to the case without condensates
($V^{\text{I}}=0$):

$\mathcal{R}=\frac{\dot{\bar{\phi}}_{A}(0)}{\dot{\bar{\phi}}_{A}(0)|_{V^{\text{I}}=0}}\,,$
(24)

where $\cdot=d/dt$ denotes time derivative. We compare initial assembly rates
for different assembly reaction orders $n=2$ and $n=3$, respectively. We find
that the initial rate of assembly increases with assembly reaction order $n$
(Fig. 5(a,b,c)). This trend applies to the case with and without condensates.
We can calculate the dynamics of the monomer volume fraction for intermediate
time scales, i.e., for timescales when a significant fraction of monomers have
assembled but the system has not yet reached the stationary state. We find
that such intermediate times the monomer volume fraction decays algebraically,
i.e., $\bar{\phi}_{A}(t)\propto t^{1/(1-n)}$, where the exponent of the power-
law is set by the assembly coefficient (inset of Fig. 5(a,b); details see
Appendix. E).

A key finding is that the presence of a fuel energy supply
$\tilde{\mu}_{F}^{\text{I}}$ strongly accelerates the assembly process; see
Figs. 5(a,b,d). To show this, we distinguish the case when assembly is
maintained away from equilibrium ($\tilde{\mu}_{F}^{\text{I}}\not=0$) and when
assembly occurs in a passive system ($\tilde{\mu}_{F}^{\text{I}}=0$). Since
the backward pathway is suppressed, in particular at early times, the
acceleration of assembly is a result of an effective increase of the monomer
chemical potential by the fuel energy $\tilde{\mu}^{\text{I}}_{F}$ inside the
condensates. These effects are most pronounced when diffusion is fast compared
to the assembly process (Fig. 5(d)).

####  III.2.2 Maximal acceleration of assembly processes mediated by
condensates

The acceleration of the assembly rate depends on condensate volume (Fig.
5(c)). The acceleration relative to the case without condensate is maximal at
a specific, optimal volume $V^{\text{I},*}$. At this volume, the relative
increase in assembly rate can be significant compared to the case without
condensates. This acceleration can be amplified even further to more than a
hundredfold if the system is maintained away from equilibrium
($\tilde{\mu}_{F}^{\text{I}}>0$).

The condensate volume at which the assembly rate is maximal, $V^{\text{I},*}$,
and the corresponding assembly rate $\mathcal{R}^{*}$ depend on the relative
assembly rate constants $k^{\text{II}}/k^{\text{I}}$ and the partition
coefficient of the monomers $P_{A}$ (Fig. 5(e,f)). We find that the optimal
volume $V^{\text{I},*}$ is mostly determined by monomer partitioning $P_{A}$
and that $V^{\text{I},*}$ decreases with increasing $P_{A}$. The maximal
assembly rate $\mathcal{R}^{*}$ is however strongly affected by both $P_{A}$
and $k^{\text{II}}/k^{\text{I}}$. It is larger the higher the assembly rate
constant inside, $k^{\text{I}}$, compared to outside, $k^{\text{II}}$, and the
smaller the partition coefficient of monomers $P_{A}$.

##  IV Conclusion

Unraveling the physicochemical principles of how condensates regulate chemical
processes is challenging due to the interplay between chemical reactions and
phase separation. To dissect the effects of condensates on chemical kinetics,
we developed a general theoretical framework for diluted clients undergoing
chemical reactions in a phase-separated environment where scaffold and solvent
components form a condensate. We derived the equations governing chemical
kinetics for diluted clients and showed that they undergo chemical reactions
in a spatially heterogeneous environment determined by the phase-separated
scaffold component. This environment gives rise to an effective drift that
originates from cross-diffusion. This drift drives the diffusive exchange
through the condensate interface while the clients follow reaction-diffusion
kinetics with phase-dependent transport coefficients. In the thin interface
limit, the drift gives rise to distinct boundary conditions at the condensate
interface. These boundary conditions entail the effects of how a condensate
affects chemical kinetics.

We illustrate the effects of a condensate on chemical processes by considering
two examples, namely a reversible reaction between a substrate and product and
an assembly process from a substrate to a product. We determined the product
yield for the reversible reactions and initial rates for the assembly process.
A key finding is that both quantities can be maximal at a specific condensate
volume. This maximum results from a competition of the condensate promoting
the formation of products and a dilution effect of the substrate inside the
condensate. We also found that when diffusion is fast compared to chemical
processes (i.e., the system is at phase equilibrium) and when maintaining
chemical processes away from chemical equilibrium, the effects of condensates
on chemical kinetics can be amplified significantly.

Regulating yields and initial speeds of chemical processes is key in
biological systems. For example, cells have to silence expression upon stress
[41] or control the formation of biofilaments [42] and aberrant aggregates
[43]. From a physical chemistry perspective, condensates have a great
propensity to provide switch-like mechanisms between regimes strongly
differing in the properties of chemical processes. This switch can for example
be achieved by cycles of condensate formation and dissolution. In particular,
since chemical processes in cells are maintained away from equilibrium, such
switches can be extraordinarily pronounced, according to our theoretical
studies.

Condensate-mediated switches may have also played an important role in the
molecular origin of life. The non-dilute environments leading to condensed
phases should have been abundant at Early Earth, as non-dilute conditions can
be easily created through drying [44, 45, 46] or freezing processes [47]. In
particular, in the presence of cycles (temperature, salt, etc.), switching the
speed of chemical processes could have provided selection mechanisms for
specific molecules that are based on the physical non-equilibrium conditions
[48].

Future applications of our theoretical framework might address questions in
systems with extremely many components, such as mixtures composed of DNA and
RNA differing in their sequences [49]. In such systems, most sequences are
diluted enabling to describe the reaction kinetics among sequences as diluted
clients. Moreover, our framework could be used to systematically study the
Turing patterns [50, 51] in systems with coexisting phases [52].

###### Acknowledgements.

We are grateful for the very helpful feedback from H. Vuijk for the detailed
discussions on various concepts discussed in the manuscript. We are also
grateful for the feedback of S. Gomez and G. Granatelli on the manuscript. We
thank Tyler S. Harmon for pointing out multi-step reactions as an interesting
application of the client-scaffold model and W. Pönisch for fruitful
discussions on the subject. We are also grateful for the discussions and the
collaborations on experimental realizations with D. Tang, A. Ghosh, and M.
Gao, as well as J. Boekhoven. F. Jülicher acknowledges funding by the
Volkswagen Foundation. T. C. T. Michaels thanks support from ETH Zurich and
the Swiss National Science Foundation (grant $200021\\_219703$). C. Weber
acknowledges the European Research Council (ERC) for financial support under
the European Union’s Horizon 2020 research and innovation programme (“Fuelled
Life” with Grant agreement No. 949021).

##  Appendix A Diluted clients in a phase-separated system

In this section, we derive the conditions for phase equilibrium for a mixture
composed of non-dilute scaffold and solvent components, and diluted clients
that undergo chemical reactions. Conceptionally, it is key to define the
meaning of client components being diluted in a chemically reactive mixture.

###  A.1 Thermodynamics and phase equilibrium of a $(N+2)$-component mixture

Due to the incompressibility condition $\sum_{i=0}^{N+1}\phi_{i}=1$, the
$(N+2)$-component mixture can be described by $(N+2)$ volume fractions
$\phi_{i}$, whereby $i=0$ denotes the solvent, $i=1$ denotes the scaffold, and
$i=2,...,(N+1)$ label the $N$ dilute clients components. The solvent volume
fraction can be substituted by the relationship,
$\phi_{0}=1-\sum_{i=1}^{N+1}\phi_{i}$, leading to $(N+1)$ independent volume
fractions. The thermodynamics of this $(N+2)$-component mixture is governed by
the Helmholtz free energy density of the form:

$f=f_{0}+\sum^{N+1}_{i,j=0}\frac{\tilde{\kappa}_{ij}}{2}\nabla\phi_{i}\cdot\nabla\phi_{j}\,,$
(25)

where $f_{0}$ is the homogeneous Helmholtz free energy density that solely
depends on the volume fractions of all components, $\phi_{i}$. The term
$\tilde{\kappa}_{ij}$ in Eq. (25) characterizes the free energy costs due to
gradients in volume fractions. For simplicity, we neglect cross couplings,
i.e., $\tilde{\kappa}_{ij}=0$ for $i\neq j$, using
$\tilde{\kappa}_{ij}=\delta_{ij}\kappa_{i}$. Note that the free energy costs
$\kappa_{i}$ also contribute to the chemical reaction rates $s_{i}$ (Eq. (5)).
However, since $\kappa_{i}>0$, this contribution solely shifts up the
diffusion coefficient
$D_{i}=M_{ii}\sum_{i}\partial\bar{\mu}_{i}/\partial\phi_{i}\to
D_{i}+\kappa_{i}$, and can thus be neglected in the following.

Phase equilibrium between two homogeneous phases I and II in the
($N+2$)-component mixture are governed by the balance of the exchange chemical
potentials $\bar{\mu}_{i}^{\text{\text{I/II}}}$ and the osmotic pressures
$\Pi^{\text{I/II}}$ between the phases:

$\displaystyle\bar{\mu}^{\text{I}}_{i}(\\{\phi^{\text{I}}_{j}\\})$
$\displaystyle=\bar{\mu}^{\text{II}}_{i}(\\{\phi^{\text{II}}_{j}\\})\,,$ (26a)
$\displaystyle\Pi^{\text{I}}$
$\displaystyle=\Pi^{\text{II}}+\frac{2\gamma}{R}\,,$ (26b)

The homogeneity of phases implies that the gradient-free energy cost vanishes
and thus the exchange chemical potential can be calculated via
$\bar{\mu}_{i}=\nu_{i}\partial f_{0}/\partial\phi_{i}$, and the osmotic
pressure is given by
$\Pi=-f_{0}+\sum_{i=1}^{N+1}\phi_{i}\bar{\mu}_{i}/\nu_{i}$. Here, $\nu_{i}$ is
the molecular volume of component $i$. The replacement of the solvent volume
fraction implies $(N+1)$ balance equations for the exchange chemical
potentials $\bar{\mu}_{i}$ with $i=1,...,(N+1)$ (Eq. (26a)). Moreover, the
exchange chemical potential measures the chemical potentials relative to the
solvent component, $i=0$. The volume of phase I is set by the average volume
fractions $\bar{\phi}_{i}$ by the relationship
$V^{\text{I}}/V=(\bar{\phi}_{i}-\phi^{\text{II}}_{i})/(\phi^{\text{I}}_{i}-\phi^{\text{II}}_{i})$,
with $V^{\text{II}}/V=1-V^{\text{I}}/V$, where $V$ is the system volume.

To illustrate the implications when $N$ reacting clients are diluted, we
consider the following mean-field, homogeneous free energy density of the
form:

$f_{0}=\frac{k_{B}T}{\nu_{0}}\bigg{[}\sum^{N+1}_{i=0}\frac{\phi_{i}}{r_{i}}\log(\phi_{i})+\sum^{N+1}_{i,j=0}\frac{\chi_{ij}}{2}\phi_{i}\phi_{j}+\sum^{N+1}_{i=0}\omega_{i}\phi_{i}\bigg{]}\,,$
(27)

where $r_{i}=\nu_{i}/\nu_{0}$ is the fraction of molecular volumes $\nu_{i}$.
Moreover, the first term represents the entropic contribution of all
components. The second term describes the mutual interactions among all
components with pair-wise interaction parameter, $\chi_{ij}$. The homogeneous
free energy density $f_{0}$ does not depend on the solvent volume fraction
$\phi_{0}$ due to $\phi_{0}=1-\sum_{i=1}^{N+1}\phi_{i}$. Note that $\phi_{0}$
has not been explicitly replaced in Eq. (27) for presentation purposes. We
remark that our framework for diluted and reacting clients is also valid for
free energies that take into account interactions beyond mean-field; the mean-
field free energy above is only chosen for simplicity.

Using Eq. (27), the exchange chemical potentials read

$\displaystyle\nu_{i}\partial f_{0}/\partial\phi_{i}$
$\displaystyle=k_{B}T\left[1-r_{i}+r_{i}(\omega_{i}-\omega_{0}+\chi_{0i})\right]$
(28) $\displaystyle\quad+k_{B}T\log(\bar{\gamma}_{i}\phi_{i})\,.$

Moreover, the exchange activity coefficients have the following form:

$\bar{\gamma}_{i}=\frac{1}{(1-\sum^{N+1}_{j=1}\phi_{j})^{r_{i}}}\exp\left[\sum^{N+1}_{j=1}r_{i}(\chi_{ij}-\chi_{0i}-\chi_{0j})\phi_{j}\right]\,.$
(29)

Comparing the two relationships above with the general form of the exchange
chemical potential Eq. (2), noting that the exchange chemical potential for
spatially heterogeneous system reads $\bar{\mu}_{i}=\nu_{i}\partial
f_{0}/\partial\phi_{i}-\kappa_{i}\nabla^{2}\phi_{i}$, the reference chemical
potentials are
$\bar{\mu}_{i}^{0}=k_{B}T[1-{r_{i}}+r_{i}(\omega_{i}-\omega_{0}+\chi_{0i})]$.

For an ideal solution and components having a non-zero molecular volume,
$\chi_{ij}=0$, and thus, the exchange activity coefficient
$\bar{\gamma}_{i}=(1-\sum^{N+1}_{j=1}\phi_{j})^{-r_{i}}$. Note that only for
ideal mixtures composed of point particles where all volume fractions except
the one of the solvent vanish to zero, $\bar{\gamma}_{i}=1$.

###  A.2 Thermodynamics of $N$ diluted clients

In this section, we derive approximate conditions for phase equilibrium (Eq.
(26)) and approximate expressions for the exchange activity coefficient (Eq.
(29)) when the $N$ reacting clients ($i=2,...,N+1$) are diluted compared to
scaffold component ($i=1$) and solvent ($i=0$). Diluted means that the client
volume fractions $\phi_{i}$ are much smaller than the volume fractions of
scaffold and solvent, $\phi_{1}$ and $\phi_{0}$, respectively:

$\phi_{i}\ll\phi_{0}\,,\phi_{1}\,,\quad i=2,...,(N+1)\,.$ (30)

####  A.2.1 Limit of vanishing client volume fractions

We first discuss the limit of zero volume fraction of the client components,
i.e., $\phi_{i}/\phi_{1}\to 0$ for $i=2,...,(N+1)$, and where the volume
fraction of the scaffold approaches $\phi_{1}\to(1-\phi_{0})$. From Eq. (29),
the client exchange activities for vanishing client volume fractions become:

$\bar{\gamma}_{i}|_{\\{\phi_{j}=0\\}}=\frac{1}{(1-\phi_{1})^{r_{i}}}\exp\left[r_{i}(\chi_{1i}-\chi_{0i}-\chi_{01})\phi_{1}\right]\,,$
(31)

where $\\{\phi_{j}=0\\}$ is an abbreviation for all client volume fractions
being zero, $\\{\phi_{2}=0,...,\phi_{N+1}=0\\}$. Moreover,
$r_{i}=\nu_{i}/\nu_{0}$, where $\nu_{i}$ are molecular volume of component
$i$.

At phase equilibrium (Eq. (26a)), the definition of the partitioning
coefficients, $P_{i}\equiv\phi_{i}^{\text{I}}/\phi_{i}^{\text{II}}$, can be
expressed in terms of the exchange activity coefficients,
$P_{i}=\bar{\gamma}_{i}^{\text{II}}/\bar{\gamma}_{i}^{\text{I}}$ [29]. For
vanishing client volume fraction, the partitioning coefficients solely depend
on the scaffold volume fraction $\phi_{1}^{\text{I/II}}$ and read:

$\displaystyle P_{i}|_{\\{\phi_{j}^{\text{I/II}}=0\\}}$
$\displaystyle=\bigg{(}\frac{1-\phi^{\text{I},0}_{1}}{1-\phi^{\text{II},0}_{1}}\bigg{)}^{r_{i}}$
(32)
$\displaystyle\times\exp[r_{i}(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})(\chi_{01}+\chi_{0i}-\chi_{1i})]\,.$

This expression can be rewritten using the phase equilibrium condition for the
scaffold component ($\bar{\mu}_{1}^{\text{I}}=\bar{\mu}_{1}^{\text{II}}$):

$\displaystyle P_{i}|_{\\{\phi_{j}^{\text{I/II}}=0\\}}$
$\displaystyle=\left(\frac{1-\phi_{1}^{\text{I}}}{1-\phi_{1}^{\text{II}}}\right)^{r_{i}-r_{i}/r_{1}}$
(33)
$\displaystyle\times\exp[r_{i}(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})(\chi_{0i}-\chi_{1i})+1-r_{1}^{-1}]\,.$

For equal molecular volumes of scaffold and solvent ($\nu_{0}=\nu_{1}$,
implying $r_{1}=1$, the partitioning coefficients in the dilute limit were
derived recently [38, 53]:

$P_{i}|_{\\{\phi_{j}^{\text{I/II}}=0\\}}=\exp[r_{i}(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})(\chi_{0i}-\chi_{1i})]\,.$
(34)

In summary, the exchange activity coefficients and the partitioning
coefficients each approach constant values for $\phi_{i}\to 0$ ($i=2,...,N+1$)
that are independent of the client volume fractions $\\{\phi_{i}\\}$.

####  A.2.2 Phase equilibrium conditions for small client volume fractions:
Systematic expansion

Before we perform a systematic expansion, we shortly review the physical and
mathematical reasoning of what equations should be expanded. A first choice
could be the dynamic equations governing diffusive transport and chemical
reactions (Eq. (1)). Expanding such equations is always possible around any
set of volume fractions. Expanding up to the $n$-th order, one obtains a set
of coupled partial differential equations of order $n$ in the volume
fractions. In particular, an expansion up to the first of the dynamic
equations would also linearize chemical reactions. The validity of this
linearization is restricted to spatial variations of client volume fractions
that are small deviations from the expansion point. Larger deviation from the
expansion point may lead to nonphysical results such as negative volume
fractions. Another reason against the expansion of the dynamic equations is
that the effects of clients being diluted should already be accessible on the
level of the thermodynamic conditions. Note that the conditions for phase
equilibrium govern the boundary conditions in the thin interface model (Sect.
II.2.2) and the dynamics of the average volume fractions when diffusion is
fast compared to chemical rates (Sect. II.2.3).

An alternative could be the expansion of the chemical potentials since their
spatial gradients drive diffusive fluxes and their differences between
chemical states drive chemical reactions. However, once the expansion point in
volume fractions approaches zero, each chemical potential diverges due to the
logarithmic dependence in $\phi_{i}$ (Eq. (2)). In other words, an expansion
of the chemical potentials around zero volume fraction is not defined.
Expanding only the non-diverging interaction terms remains conceptionally
unsatisfactory [38].

Another possibility is to expand the phase equilibrium conditions around zero
client volume fractions. At phase equilibrium, the equality of chemical
potentials cancels the divergence. Using Eq. (2) and the equilibrium
conditions (26a), one gets
$\log(\phi_{i}^{\text{I}}/\phi_{i}^{\text{II}})=\log(\bar{\gamma}_{i}^{\text{II}}/\bar{\gamma}_{i}^{\text{I}})$,
which has no divergence for vanishing average client volume fraction
($\bar{\phi}_{i}\to 0$) since $\phi_{i}^{\text{I}}\sim\bar{\phi}_{i}$ and
$\phi_{i}^{\text{II}}\sim\bar{\phi}_{i}$. In particular, for the client volume
fractions $\phi_{i}\to 0$ with $i=2,...,(N+1)$,
$\bar{\gamma}_{i}^{\text{II}}/\bar{\gamma}_{i}^{\text{I}}$ become the constant
partition coefficients $P_{i}|_{\\{\bar{\phi}_{j}=0\\}}$ (Eq. (32)) that do
not depend on the client volume fractions.

In summary, expanding the equilibrium conditions (26) or, equivalently, the
partitioning coefficients $P_{i}(\phi_{1},\phi_{2},...,\phi_{N+1})$ for small
average client volume fractions $\bar{\phi}_{i}$ appears as the most general
procedure. For clients $i=2,...,(N+1)$ with $\phi_{i}\to 0$, not only the
partition coefficient approaches a constant, but also the exchange activity
coefficients become constants in this limit (Eq. (31)). Thus, equivalently,
exchange activity coefficients can be expanded systematically.

Now we continue with the systematic expansion by expressing the equilibrium
volume fractions in each phase $\phi^{\text{I/II}}_{i}$ and the phase volume
$V^{\text{I}}$ as a function of the average volume fractions for the scaffold
$\bar{\phi}_{1}$, and the clients $\bar{\phi}_{i}$ (Eq. (16)):

$\begin{split}&\phi^{\text{I/II}}_{i}=h^{\text{I/II}}_{i}(\bar{\phi}_{1},\bar{\phi}_{2},..,\bar{\phi}_{N+1})\,,\\\
&\frac{V^{\text{I}}}{V}=g(\bar{\phi}_{1},\bar{\phi}_{2},..,\bar{\phi}_{N+1})\,.\end{split}$
(35)

In the following, we expand the client average volume fraction
$\\{\bar{\phi}_{i}\\}$ around zero average volume fraction
$\\{\bar{\phi}_{i}=0\\}$ ($i=2,..,N+1$). Consistently, the zeroth order of
this expansion will lead to the results discussed in Sect. A.2.1.

In linear order, the scaffold equilibrium volume fraction can be written as

$\phi^{\text{I/II}}_{1}=\phi^{\text{I/II},0}_{1}+\sum^{N+1}_{i=2}\alpha^{\text{I/II}}_{i}\bar{\phi}_{i}+\mathcal{O}(\bar{\phi}_{i})^{2}\,,$
(36a) where we abbreviated the zeroth order $\phi^{\text{I/II},0}_{1}$, and
the linear expansion coefficients are defined as
$\alpha^{\text{I/II}}_{i}={\partial
h^{\text{I/II}}_{1}}/{\partial\bar{\phi}_{i}}|_{\\{\bar{\phi}_{j}=0\\}}$.

The coefficients $\alpha^{\text{I/II}}_{i}$ are derived by linearizing both
sides of Eq. (26b). Moreover, we use the chemical potential balance (Eq.
(26a)), and express changes of the scaffold volume fractions
$\phi^{\text{I/II}}_{1}$ with client volume fractions $\phi^{\text{I/II}}_{i}$
in terms of $\bar{\phi}_{i}$ by utilizing the zeroth order partition
coefficients $P^{0}_{i}\equiv P_{i}|_{\\{\bar{\phi}_{j}=0\\}}$ ($i=2,...,N+1$)
[54]:

$\begin{split}&\alpha^{\text{I}}_{i}(\bar{\phi}^{0}_{1},\\{\bar{\phi}_{i}\\}=0)=a^{\text{I}}_{i}\frac{P^{0}_{i}}{1+(P^{0}_{i}-1)V^{\text{I},0}/V}\,,\\\
&\alpha^{\text{II}}_{i}(\bar{\phi}^{0}_{1},\\{\bar{\phi}_{i}\\}=0)=a^{\text{II}}_{i}\frac{1}{1+(P^{0}_{i}-1)V^{\text{I},0}/V}\,,\\\
\end{split}$ (36b)

where the coefficients

$\displaystyle a^{\text{I}}_{i}$
$\displaystyle=-\phi^{\text{I},0}_{1}\,\frac{1+\left(\Delta\chi_{i}+\frac{1-1/P^{0}_{i}}{r_{i}(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})}\right)\left(1-\phi^{\text{I},0}_{1}\right)}{\frac{\left(1-\phi^{\text{I},0}_{1}\right)}{r_{1}}+\phi^{\text{I},0}_{1}-2\chi_{01}\phi^{\text{I},0}_{1}\left(1-\phi^{\text{I},0}_{1}\right)}\,,$
(36c) $\displaystyle a^{\text{II}}_{i}$
$\displaystyle=-\phi^{\text{II},0}_{1}\,\frac{1+\left(\Delta\chi_{i}+\frac{P^{0}_{i}-1}{r_{i}(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})}\right)\left(1-\phi^{\text{II},0}_{1}\right)}{\frac{\left(1-\phi^{\text{II},0}_{1}\right)}{r_{1}}+\phi^{\text{II},0}_{1}-2\chi_{01}\phi^{\text{II},0}_{1}\left(1-\phi^{\text{II},0}_{1}\right)}\,.$
(36d)

In the equations above, we abbreviate

$\Delta\chi_{i}=\chi_{1i}-\chi_{0i}-\chi_{01}\,.$ (37)

Fig. 6(a,b) shows how the volume fraction of the scaffold component in each
phase, $\phi_{1}^{\text{I/II}}$, behaves for increasing average client volume
fraction $\bar{\phi}_{2}$. The horizontal dotted line is the zeroth order
$\phi_{1}^{\text{I/II},0}$, the dashed line the linear order
$\phi_{1}^{\text{I/II},1}$, and the orange line is the full numerical solution
to Eq. (26).

The client equilibrium volume fractions ($i=2,...,N+1$) can be expanded as
follows:

$\displaystyle\phi^{\text{I/II}}_{i}$
$\displaystyle=\sum^{N+1}_{j=2}\left.\frac{\partial
h^{\text{I/II}}_{i}}{\partial\bar{\phi}_{j}}\right|_{\\{\bar{\phi}_{l}=0\\}}\bar{\phi}_{j}$
(38)
$\displaystyle\quad+\frac{1}{2}\sum^{N+1}_{j,k=2}\left.\frac{\partial^{2}h^{\text{I/II}}_{i}}{\partial\bar{\phi}_{j}\partial\bar{\phi}_{k}}\right|_{\begin{subarray}{c}\\{\bar{\phi}_{l}=0\\}\\\
l\neq
i\end{subarray}}\bar{\phi}_{j}\bar{\phi}_{k}+\mathcal{O}(\bar{\phi}_{i})^{3}\,,$

where the zeroth order vanishes, i.e.,
$h^{\text{I/II}}_{i}(\bar{\phi}^{0}_{1},\\{\bar{\phi}_{i}=0\\})$. Thus, we
introduce the fraction ${\phi^{\text{I/II}}_{i}}/{\bar{\phi}_{i}}$ that is
finite when evaluating the limit of zero client volume fractions:

$\displaystyle\frac{\phi^{\text{I/II}}_{i}}{\bar{\phi}_{i}}$
$\displaystyle=\frac{\partial
h^{\text{I/II}}_{i}}{\partial\bar{\phi}_{i}}+\frac{1}{2}\sum^{N+1}_{j=2}\left.\frac{\partial^{2}h^{\text{I/II}}_{i}}{\partial\bar{\phi}_{i}\partial\bar{\phi}_{j}}\right|_{\\{\bar{\phi}_{l}\\}=0}\bar{\phi}_{j}+\mathcal{O}(\bar{\phi}_{i})^{2}\,.$
(39)

We see that the zeroth order in the expansion in client volume fractions gives
a linear order leading term. In other words, there is an order shift due to
exclusively expanding around zero client volume fractions. Thus, the $n-th$
order for the scaffold expansion (Eq. (36a)) corresponds to a $(n+1)-th$ order
in the client expansion.

Figure 6: Deviations to dilute limit of clients. We quantify the effects of a
non-diluted client component ($i=2$) on the equilibrium volume fractions of
scaffold $\phi^{\text{I/II}}_{1}$. (a,b) The zeroth order
$\phi_{1}^{\text{I/II},0}$ (dilute limit) is shown by the horizontal dotted
line, while the dashed line is the linear order correction
$\phi_{1}^{\text{I/II},1}$. The orange line corresponds to the full numerical
solution to Eq. (26), indicating a good agreement of the linear order
correction up to volume fraction around 0.1. For the shown curves, the zeroth
order becomes a good approximation for a client volume fraction below 0.01.
(c) We show the relative difference in the client’s exchange activity
coefficient, $\Delta\bar{\gamma}_{2}$ (Eq. (42a)), as a function of client
volume fraction ${\phi}_{2}$. The dashed line shows the relative linear
correction, ${\phi}_{2}/({\phi}_{2}^{*}\bar{\gamma}^{0}_{2})$ (Eq. (42b)), and
the blue solid line results from numerically solving Eq. (26). The parameters
are: $\chi_{01}=3$, $\chi_{02}=-1$, $\chi_{12}=0$, $\phi^{0}_{1}=0.05$. The
volume fraction scale (Eq. (43)) obtained for these parameters is
${\phi}_{2}^{*}=0.344$ and $\bar{\gamma}^{0}_{2}=0.95246$.
$\Delta\bar{\gamma}_{2}=1\%$ for $\bar{\phi}_{2}=3.27\cdot 10^{-3}$.

The expansion of the condensate phase volume $V^{\text{I}}$ can be written as:

$\displaystyle\frac{V^{\text{I}}}{V}$
$\displaystyle=\frac{V^{\text{I},0}}{V}+\sum^{N+1}_{i=2}\left.\frac{\partial
g}{\partial\bar{\phi}_{i}}\right|_{\\{\bar{\phi}_{i}\\}=0}\bar{\phi}_{i}+\mathcal{O}(\bar{\phi}_{i})^{2}$
$\displaystyle=\frac{\bar{\phi}^{0}_{1}-\phi^{\text{II},0}_{1}}{\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1}}+\sum^{N+1}_{i=2}\eta_{i}\bar{\phi}_{i}+\mathcal{O}(\bar{\phi}_{i})^{2}\,,$
(40a) where the coefficients $\displaystyle\eta_{i}$
$\displaystyle=-\frac{(\bar{\phi}^{0}_{1}-\phi^{\text{II},0}_{1})}{(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})^{2}}\alpha^{\text{I}}_{i}$
(40b)
$\displaystyle\quad+\bigg{[}\frac{(\bar{\phi}^{0}_{1}-\phi^{\text{II},0}_{1})}{(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})^{2}}-\frac{1}{(\phi^{\text{I},0}_{1}-\phi^{\text{II},0}_{1})}\bigg{]}\alpha^{\text{II}}_{i}\,.$

The expressions of $\alpha^{\text{I/II}}_{i}$ are obtained from Eq. (36b).

The exchange activity coefficient of all components can be expanded up to the
first order around $\\{\phi_{i}=0\\}$:

$\displaystyle\bar{\gamma}_{1}$
$\displaystyle=\frac{1}{(1-{\phi}^{0}_{1})^{r_{1}}}\exp{(-2r_{1}\chi_{01}{\phi}^{0}_{1})}$
(41a)
$\displaystyle+\sum^{N+1}_{i=2}r_{1}\frac{\exp{(-2r_{1}\chi_{01}{\phi}^{0}_{1})}}{(1-{\phi}^{0}_{1})^{r_{1}+1}}\left(1+(1-{\phi}^{0}_{1})\Delta\chi_{i}\right){\phi}_{i}$
$\displaystyle+\mathcal{O}(\phi_{i}^{2})\,,$ $\displaystyle\bar{\gamma}_{i}$
$\displaystyle=\frac{1}{(1-{\phi}^{0}_{1})^{r_{i}}}\exp{(r_{i}{\phi}^{0}_{1}\Delta\chi_{i})}$
(41b)
$\displaystyle+\sum^{N+1}_{j=2}r_{j}\frac{\exp{(r_{j}{\phi}^{0}_{1}\Delta\chi_{j})}}{(1-{\phi}^{0}_{1})^{r_{j}+1}}\left(1-2(1-{\phi}^{0}_{1})\chi_{0j}\right){\phi}_{j}$
$\displaystyle+\mathcal{O}(\phi_{j}^{2})\,,$

with clients labeled by $i=2,..,(N+1)$.

To illustrate the validity of the expansion, we consider one client ($i=2$) in
a homogeneous mixture that is also composed of solvent ($i=0$) and scaffold
($i=1$). To access when the zeroth order in the average client volume fraction
($\bar{{\phi}}_{2}=0$; see Sect. A.2.1)is a good approximation, we define the
linearly approximated exchange activity coefficient relative to its zeroth
order:

$\Delta\bar{\gamma}_{2}=\frac{\bar{\gamma}_{2}^{1}-\bar{\gamma}_{2}^{0}}{\bar{\gamma}_{2}^{0}}\,,$
(42a) where the first order of the client exchange activity coefficient is
$\bar{\gamma}_{2}^{1}=\bar{\gamma}_{2}^{0}+\bar{\phi}_{2}/\phi_{2}^{*}$ (42b)

with
$\bar{\gamma}_{2}^{0}={(1-{\phi}^{0}_{1})^{-r_{2}}}\exp{(r_{2}{\phi}^{0}_{1}\Delta\chi_{2})}$
being the the zeroth order according to Eq. (41b). Note that $\Delta\chi_{2}$
is given in Eq. (37). When decreasing the client volume fraction below the
volume fraction scale,

${\phi}_{2}^{*}=\left[{r_{2}}\frac{\exp{(r_{2}{\phi}^{0}_{1}\Delta\chi_{2})}}{(1-{\phi}^{0}_{1})^{r_{2}+1}}\left(1-2(1-{\phi}^{0}_{1})\chi_{02}\right)\right]^{-1}\,$
(43)

taking the zeroth order of the client activity coefficient becomes a good
approximation,i.e., $\bar{\gamma}_{2}\simeq\bar{\gamma}_{2}^{0}$ for
$\bar{\phi}_{2}\ll\phi_{2}^{*}$, respectively. In this case, the activity
coefficient is constant and equal to Eq. (31). Moreover, the partition
coefficient is also constant and equal to Eq. (32). Fig. 6(c) confirms that
below the volume scale, $\bar{\phi}_{2}<\phi_{2}^{*}$, the relative deviation
of the client activity coefficient becomes negligible, i.e.,
$\Delta\bar{\gamma}_{2}=1\%$ for
$\bar{\phi}_{2}=(\bar{\gamma}_{2}^{0}\phi_{2}^{*})/100$.

##  Appendix B Continuum model for diluted clients

Using linear response for the diffusive fluxes (Eq. (1b)) and the mobility
matrix (Eq. (7)), the fluxes of the scaffold ($i=1$) and the clients
$i=2,...,(N+1)$ are:

$\displaystyle\textbf{j}_{1}$
$\displaystyle=-m_{01}\phi_{1}(1-\phi_{1})\nabla\bar{\mu}_{1}\,,$ (44)
$\displaystyle\textbf{j}_{i}$
$\displaystyle=-\bigg{[}m_{0i}\phi_{i}(1-\phi_{1})+m_{1i}\phi_{i}\phi_{1}\bigg{]}\nabla\bar{\mu}_{i}\,.$

We express the exchange chemical potentials for scaffold $\bar{\mu}_{1}$ and
client $\bar{\mu}_{i}$ in terms of the exchange activity coefficients in the
limit of vanishing client volume fraction (Eq. (31)), which solely depends on
the scaffold volume fraction $\phi_{1}$:

$\displaystyle\nabla\bar{\mu}_{1}$
$\displaystyle=k_{B}T\bigg{[}\frac{1}{\phi_{1}}+\frac{1}{\bar{\gamma_{1}}}\frac{\partial\bar{\gamma}_{1}}{\partial\phi_{1}}\bigg{]}\nabla\phi_{1}-\kappa_{1}\nabla\nabla^{2}\phi_{1}\,,$
(45) $\displaystyle\nabla\bar{\mu}_{i}$
$\displaystyle=k_{B}T\bigg{[}\frac{1}{\bar{\gamma}_{i}}\frac{\partial\bar{\gamma}_{i}}{\partial\phi_{1}}\nabla\phi_{1}+\frac{1}{\phi_{i}}\nabla\phi_{i}\bigg{]}\,,$
(46)

where we have dropped the higher-order gradient contribution to the client
flux, $-\kappa_{i}\nabla\nabla^{2}\phi_{i}$. This contribution can be
neglected since the diluted clients $i=2,...,(N+1)$ cannot phase-separate and
do not form an interface. Thus, the spatial transport of clients is well
captured by the leading order diffusive flux that is proportional to
$\nabla\phi_{i}$. Using Eq. (31), we obtain Eqs. (8) and (9) in the main text.

##  Appendix C Condition for maximal yield for reversible chemical reactions

Here we give the solutions for reversible chemical reactions controlled by a
scaffold-rich condensate and the system being at phase equilibrium. We
determine the average product client volume fraction $\bar{\phi}_{B}(\infty)$
in the stationary state ($t\to\infty$) and compare it to the average volume
fraction of the homogeneous reference system where the compartment volume is
zero ($V^{\text{I}}=0$). We use such averages to obtain conditions for when
the relative yield is maximal.

###  C.1 Uni-molecular scheme

For an uni-molecular reaction ($g=1$) the average product client volume
fraction reads:

$\bar{\phi}_{B}(\infty)=\frac{\exp\left(\frac{\bar{\mu}^{0}_{A_{1}}}{k_{B}T}\right)\bar{\gamma}^{\text{I}}_{A_{1}}P_{A_{1}}\zeta_{A_{1}}\left[\sum_{\alpha}V^{\alpha}k^{\alpha}\exp\left(\frac{\tilde{\mu}^{\alpha}_{F}}{k_{B}T}\right)\right]}{\exp\left(\frac{\bar{\mu}^{0}_{A_{1}}}{k_{B}T}\right)\bar{\gamma}^{\text{I}}_{A_{1}}P_{A_{1}}\zeta_{A_{1}}\left[\sum_{\alpha}V^{\alpha}k^{\alpha}\exp\left(\frac{\tilde{\mu}_{F}^{\alpha}}{k_{B}T}\right)\right]+\exp\left(\frac{\bar{\mu}^{0}_{B}}{k_{B}T}\right)\bar{\gamma}^{\text{I}}_{B}P_{B}\zeta_{B}\left[\sum_{\alpha}V^{\alpha}k^{\alpha}\right]}\psi_{1}\,,$
(47)

where $\psi_{1}=(\bar{\phi}_{A_{1}}+\bar{\phi}_{B})$ is the conserved quantity
of the uni-molecular reaction. For vanishing condensate volume
($V^{\text{I}}=0$), the equation above simplifies:

$\bar{\phi}_{B}(\infty)|_{V^{\text{I}}=0}=\frac{\exp\left(\frac{\bar{\mu}^{0}_{A_{1}}+\tilde{\mu}_{F}}{k_{B}T}\right)\bar{\gamma}_{A_{1}}}{\exp\left(\frac{\bar{\mu}^{0}_{A_{1}}+\tilde{\mu}_{F}}{k_{B}T}\right)\bar{\gamma}_{A_{1}}+\exp\left(\frac{\bar{\mu}^{0}_{B}}{k_{B}T}\right)\bar{\gamma}_{B}}\psi_{1}\,.$
(48)

Figure 7: Graphical representation of the conditions for a maximum in product
yield $\mathcal{Y}$. The conditions are shown for the uni-molecular reaction
((a), $g=1$) and the bi-molecular reaction ((b), $g=2$). We vary the fraction
of the reaction rate coefficients $k^{\text{II}}/k^{\text{I}}$ and fuel supply
energy $\tilde{\mu}^{\text{I}}_{F}$. A yield maximum exits only in the orange
shaded domain where both conditions (Eqs. (LABEL:eq:cond_uni_max) for $g=1$)
and (Eqs. (57) for $g=2$ are satisfied.

To obtain conditions for the existence of a maximum in the relative yield
$\mathcal{Y}(V^{\text{I}})$ (Eq. (21)) as a function of the condensate volume
$V^{\text{I}}$, we consider the slope of the relative field,
$\mathcal{Y}^{{}^{\prime}}$. We evaluate this slope at the limits,
$V^{\text{I}}\rightarrow 0$ and $V^{\text{I}}\rightarrow 1$, respectively. If
the relative yield does not have multiple extrema, the conditions of a maximum
is given as:

$\begin{split}&\mathcal{Y}^{{}^{\prime}}(V^{\text{I}}/V)>0\quad\text{for}\quad
V^{\text{I}}/V\rightarrow 0\,,\\\
&\mathcal{Y}^{{}^{\prime}}(V^{\text{I}}/V)<0\quad\text{for}\quad
V^{\text{I}}/V\rightarrow 1\,.\end{split}$ (49)

These conditions can be expressed as:

$\begin{split}&\frac{k^{\text{II}}}{k^{\text{I}}}<\frac{\left(\exp{\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)-1}\right)}{(P_{A_{1}}-P_{B})}\quad\text{for}\quad
V^{\text{I}}/V\rightarrow 0\,,\\\
&\frac{k^{\text{II}}}{k^{\text{I}}}<\frac{\exp{\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)}\left(P_{A_{1}}-P_{B}\right)}{P_{A_{1}}P_{B}\left(\exp{\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)}-1\right)}\quad\text{for}\quad
V^{\text{I}}/V\rightarrow 1\,.\end{split}$ (50)

A graphical illustration of these conditions for a varying fraction of the
reaction rate coefficients $k^{\text{II}}/k^{\text{I}}$ and fuel supply energy
$\tilde{\mu}^{\text{I}}_{F}$ is shown in Fig. 7(a). If such conditions are
satisfied, a maximum in relative yield exists at a particular value of the
relative condensate volume $V^{\text{I}}/V$. When chemical reactions are not
maintained away from equilibrium ($\tilde{\mu}^{\text{I}}_{F}=0$), the
conditions reduce to $k^{\text{II}}(P_{A_{1}}-P_{B})<0$ and
$k^{\text{I}}(P_{A_{1}}-P_{B})>0$. Both conditions cannot be satisfied
simultaneously. This implies that for a uni-molecular reaction ($g=1$) at
equilibrium, there are no maxima in the relative yield as a function of
relative condensate volume $V^{\text{I}}/V$ (see Fig. 4(c)). However, for non-
vanishing fuel energy supply $\tilde{\mu}^{\text{I}}_{F}$, the yield as a
function of $V^{\text{I}}/V$ can develop a pronounced maximum. This maximum
already occurs at smaller volumes if more substrate $A_{1}$ partitions into
the condensate and if the condensate phase I favors the production of the
product ($k^{\text{II}}/k^{\text{I}}$ decreases) compared to phase II (see
Fig. 4(c-f)).

###  C.2 Bi-molecular scheme

For the second case of a fuel-driven bi-molecular reaction ($g=2$), the
reaction rates for the substrates $A_{i}$ read

$\begin{split}s^{\alpha}_{A_{i}}(\textit{{r}},t)&=-\frac{s^{\alpha}_{B}(\textit{{r}},t)}{2}\\\
&=k^{\alpha}\bigg{[}\exp\bigg{(}\frac{\bar{\mu}_{B}}{k_{B}T}\bigg{)}-\exp\bigg{(}\frac{\bar{\mu}_{A_{1}}+\bar{\mu}_{A_{2}}+\tilde{\mu}^{\alpha}_{F}}{k_{B}T}\bigg{)}\bigg{]}\,.\end{split}$
(51)

There are two conserved quantities, i.e.,
$\psi_{1}=(\bar{\phi}_{A_{1}}+\bar{\phi}_{A_{2}}+\bar{\phi}_{B})$ and
$\psi_{2}=(\bar{\phi}_{A_{1}}-\bar{\phi}_{A_{2}})$. The solution at steady
state $\bar{\phi}_{B}(\infty)$ obeys the condition
$\sum^{2}_{i=1}\bar{\mu}_{A_{i}}+\tilde{\mu}_{F}=\bar{\mu}_{B}$. Using the
conserved quantities, this condition can be written as:

$\displaystyle\bar{\phi}_{B}(\infty)$
$\displaystyle=\exp\bigg{(}\frac{\bar{\mu}^{0}_{A_{1}}+\bar{\mu}^{0}_{A_{2}}-\bar{\mu}^{0}_{B}}{k_{B}T}\bigg{)}\bigg{(}\frac{\psi_{1}+\psi_{2}-\bar{\phi}_{B}(\infty)}{2}\bigg{)}$
$\displaystyle\quad\times\bigg{(}\frac{\psi_{1}-\psi_{2}-\bar{\phi}_{B}(\infty)}{2}\bigg{)}\,.$
(52)

The root where $0<\bar{\phi}_{B}(\infty)<1$ is

$\bar{\phi}_{B}(\infty)=2C+\psi_{1}-\sqrt{4C^{2}+4C\psi_{1}+\psi^{2}_{2}}\,,$
(53)

where the coefficient

$\begin{split}C&=\exp\left(\frac{\bar{\mu}^{0}_{A_{1}}+\bar{\mu}^{0}_{A_{2}}-\bar{\mu}^{0}_{B}}{k_{B}T}\right)\frac{\bar{\gamma}^{\text{I}}_{B}P_{B}\zeta_{B}}{\prod^{2}_{i=1}\bar{\gamma}^{\text{I}}_{A_{i}}P_{A_{i}}\zeta_{A_{i}}}\\\
&\quad\times\left[\frac{\sum_{\alpha}V^{\alpha}k^{\alpha}}{\sum_{\alpha}V^{\alpha}k^{\alpha}\exp\left(\frac{\tilde{\mu}_{F}^{\alpha}}{k_{B}T}\right)}\right]\,.\end{split}$
(54)

To solve for the existence of a maximum in the relative yield, we determine
the slope of the Eq. (53). When evaluating at, $V^{\text{I}}\rightarrow/V0$,
we find:

$\begin{split}&\left[k^{\text{I}}\left(\exp\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)-1\right)+k^{\text{II}}(1-\sum_{i}P_{A_{i}}+P_{B})\right]\Bigg{[}2\exp\left(\frac{\bar{\mu}^{0}_{B}}{k_{B}T}\right)P_{B}\bar{\gamma}^{\text{I}}_{B}+\prod^{2}_{i=1}\exp\left(\frac{\bar{\mu}^{0}_{A_{i}}}{k_{B}T}\right)P_{A_{i}}\bar{\gamma}^{\text{I}}_{A_{i}}\\\
&\Bigg{(}\psi_{1}-\sqrt{\frac{4\exp\left(\frac{\bar{\mu}^{0}_{B}-2\sum_{i}\bar{\mu}^{0}_{A_{i}}}{k_{B}T}\right)\left(\exp\left(\frac{\bar{\mu}^{0}_{B}}{k_{B}T}\right)P_{B}\bar{\gamma}^{\text{I}}_{B}+\prod^{2}_{i=1}\exp\left(\frac{\bar{\mu}^{0}_{A_{i}}}{k_{B}T}\right)P_{A_{i}}\bar{\gamma}_{A_{i}}\psi_{1}\right)P_{B}\bar{\gamma}^{\text{I}}_{B}}{\prod_{i}(P_{A_{i}}\bar{\gamma}^{\text{I}}_{A_{i}})^{2}}+\psi_{2}^{2}}\Bigg{)}\Bigg{]}>0\,,\end{split}$
(55)

while when calculating at $V^{\text{I}}/V\rightarrow 1$, we obtain:

$\begin{split}&\left[k^{\text{I}}\exp\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)\left((P_{A_{1}}(P_{A_{2}}+P_{B}(P_{A_{2}}-1)))-P_{A_{2}}P_{B}\right)+k^{\text{II}}P_{A_{1}}P_{A_{2}}P_{B}\left(1-\exp\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)\right)\right]\\\
&\Bigg{[}-2\exp\left(\frac{\bar{\mu}^{0}_{B}}{k_{B}T}\right)\bar{\gamma}^{\text{I}}_{B}+\exp\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)\prod^{2}_{i=1}\exp\left(\frac{\bar{\mu}^{0}_{A_{i}}}{k_{B}T}\right)\bar{\gamma}_{A_{i}}\\\
&\left(-\psi_{1}+\sqrt{\frac{4\exp\left(\frac{\bar{\mu}^{0}_{B}-2\sum_{i}\bar{\mu}^{0}_{A_{i}}-\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)\left(\exp\left(\frac{\bar{\mu}^{0}_{B}-2\sum_{i}\bar{\mu}^{0}_{A_{i}}-\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)\bar{\gamma}^{\text{I}}_{B}+\prod^{2}_{i=1}\bar{\gamma}^{\text{I}}_{A_{i}}\right)\bar{\gamma}^{\text{I}}_{B}}{\prod_{i}(\bar{\gamma}^{\text{I}}_{A_{i}})^{2}}+\psi^{2}_{2}}\right)\Bigg{]}<0\,.\end{split}$
(56)

For the present choice of parameters, the terms within the second round
brackets in Eq. (55) and Eq. (LABEL:eq:condition_2), respectively, is always
positive. Thus, the sufficient conditions for the existence of a maximum
reduce to:

$k^{\text{I}}\left(\exp\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)-1\right)>k^{\text{II}}\left(\sum_{i}P_{A_{i}}-P_{B}-1\right)\quad\text{at}\quad
V^{\text{I}}/V\rightarrow 0\,,$ (57a)
$k^{\text{I}}\exp\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)\left((P_{A_{1}}(P_{A_{2}}+P_{B}(P_{A_{2}}-1)))-P_{A_{2}}P_{B}\right)>k^{\text{II}}\left(\exp\left(\frac{\tilde{\mu}^{\text{I}}_{F}}{k_{B}T}\right)-1\right)\quad\text{at}\quad
V^{\text{I}}/V\rightarrow 1\,.$ (57b)

In summary, if the conditions above are satisfied, then a maximum exists as a
function of condensate volume $V^{\text{I}}$. See Fig. 7(b) for a graphical
illustration of these conditions for varying fraction of the reaction rate
coefficients, $k^{\text{II}}/k^{\text{I}}$, and fuel supply energy
$\tilde{\mu}^{\text{I}}_{F}$. Interestingly, when
$\tilde{\mu}^{\text{I}}_{F}=0$, the conditions above can still be satisfied
for the considered case of a bi-molecular reaction ($g=2$).

##  Appendix D Analytic solutions of thin interface model for reversible uni-
molecular chemical reactions

Here, we provide analytical solutions for the stationary concentration fields
of diffusive clients undergoing uni-molecular reactions, with which the
results of Fig. 4(a-d) have been produced. From the stationary case of Eq.
(13), we can derive radial symmetric volume fractions for $A_{1}$ and $B$ in
three spatial dimensions as

$\begin{split}\phi^{\text{I}}_{A_{1}}(r)&=\frac{\beta(D^{\text{I}}_{A_{1}}C_{1}+D^{\text{I}}_{B}C_{2})}{(\alpha
D^{\text{I}}_{B}\exp\left(\tilde{\mu}^{\text{I}}_{F}/k_{B}T\right)+\beta
D^{\text{I}}_{A_{1}})}\\\ &+\frac{D^{\text{I}}_{B}(\alpha
C_{1}\exp\left(\tilde{\mu}^{\text{I}}_{F}/k_{B}T\right)-\beta C_{2})}{(\alpha
D^{\text{I}}_{B}\exp\left(\tilde{\mu}^{\text{I}}_{F}/k_{B}T\right)+\beta
D^{\text{I}}_{A_{1}})}\frac{\sinh({r/l^{\text{I}}})}{r/l^{\text{I}}}\,,\end{split}$
(58a)
$\begin{split}\phi^{\text{I}}_{B}(r)&=\frac{\alpha\exp\left(\tilde{\mu}^{\text{I}}_{F}/k_{B}T\right)(D^{\text{I}}_{A_{1}}C_{1}+D^{\text{I}}_{B}C_{2})}{(\alpha
D^{\text{I}}_{B}\exp\left(\tilde{\mu}^{\text{I}}_{F}/k_{B}T\right)+\beta
D^{\text{I}}_{A_{1}})}\\\ &-\frac{D^{\text{I}}_{A_{1}}(\alpha
C_{1}\exp\left(\tilde{\mu}^{\text{I}}_{F}/k_{B}T\right)-\beta C_{2})}{(\alpha
D^{\text{I}}_{B}\exp\left(\tilde{\mu}^{\text{I}}_{F}/k_{B}T\right)+\beta
D^{\text{I}}_{A_{1}})}\frac{\sinh({r/l^{\text{I}}})}{r/l^{\text{I}}}\,,\end{split}$
(58b)
$\begin{split}\phi^{\text{II}}_{A_{1}}(r)&=\frac{P_{B}\beta(D^{\text{II}}_{A_{1}}(C_{3}+rC_{4})+D^{\text{II}}_{B}(C_{5}+rC_{6}))}{r(P_{A_{1}}\alpha
D^{\text{II}}_{B}+P_{B}\beta D^{\text{II}}_{A_{1}})}\\\
&+\frac{D^{\text{II}}_{B}(P_{A_{1}}\alpha C_{3}-P_{B}\beta
C_{5}))}{(P_{A_{1}}\alpha D^{\text{II}}_{B}+P_{B}\beta
D^{\text{II}}_{A_{1}})}\frac{\exp(-{r/l^{\text{II}}})}{r}\\\
&+\frac{D^{\text{II}}_{B}(P_{A_{1}}\alpha C_{4}-P_{B}\beta
C_{6}))}{(P_{A_{1}}\alpha D^{\text{II}}_{B}+P_{B}\beta
D^{\text{II}}_{A_{1}})}\frac{\sinh({r/l^{\text{II}}})}{r/l^{\text{II}}}\,,\end{split}$
(58c)
$\begin{split}\phi^{\text{II}}_{B}(r)&=\frac{P_{A_{1}}\alpha(D^{\text{II}}_{A_{1}}(C_{3}+rC_{4})+D^{\text{II}}_{B}(C_{5}+rC_{6}))}{r(P_{A_{1}}\alpha
D^{\text{II}}_{B}+P_{B}\beta D^{\text{II}}_{A_{1}})}\\\
&-\frac{D^{\text{II}}_{B}(P_{A_{1}}\alpha C_{3}-P_{B}\beta
C_{5}))}{(P_{A_{1}}\alpha D^{\text{II}}_{B}+P_{B}\beta
D^{\text{II}}_{A_{1}})}\frac{\exp(-{r/l^{\text{II}}})}{r}\\\
&-\frac{D^{\text{II}}_{B}(P_{A_{1}}\alpha C_{4}-P_{B}\beta
C_{6}))}{(P_{A_{1}}\alpha D^{\text{II}}_{B}+P_{B}\beta
D^{\text{II}}_{A_{1}})}\frac{\sinh({r/l^{\text{II}}})}{r/l^{\text{II}}}\,,\end{split}$
(58d)

where
$\alpha=\exp(\bar{\mu}^{0}_{A_{1}}/k_{B}T)\bar{\gamma}^{\text{I}}_{A_{1}}$ and
$\beta=\exp(\bar{\mu}^{0}_{B}/k_{B}T)\bar{\gamma}^{\text{I}}_{B}$.
Furthermore, we have introduced the reaction-diffusion length scales

$l^{\alpha}=\sqrt{\frac{D^{\alpha}_{A_{1}}D^{\alpha}_{B}}{k^{\alpha}\left[D^{\alpha}_{A_{1}}\exp\left(\frac{\bar{\mu}^{0}_{B}}{k_{B}T}\right)\bar{\gamma}^{\alpha}_{B}+D^{\alpha}_{B}\exp\left(\frac{\bar{\mu}^{0}_{A_{1}}+\tilde{\mu}^{\alpha}_{F}}{k_{B}T}\right)\bar{\gamma}^{\alpha}_{A_{1}}\right]}}.$
(59)

Due to the linearity of Eq. (13), we can find these solutions as linear
combinations of solutions of the Laplace equation and the radial symmetric
spherical Bessel problem, leading in principle to eight coefficients $C_{i}$,
which have to be determined via boundary conditions. However, to avoid any
singularities at the droplet center, thereby fulfilling Eq. (13f), we have
omitted the $1/r$ solution of the Laplace equation and the Spherical Bessel
function of the second kind and zeroth order ($\propto\exp(-r)/r$) in phase I.
The remanding coefficients $C_{1},...,C_{6}$ are determined by the conditions
Eq. (13c)-Eq. (13e).

##  Appendix E Scaling laws of monomer assembly kinetic

The generic form of the dynamic equation of an irreversible assembly process
with assembly order $n$ reads for the average volume fractions (definition see
Eq. (16)) of the assembly $A_{n}$ and the monomers $A$:

$\dot{\bar{\phi}}_{A_{n}}(t)=k_{\text{eff}}\,(\psi-\bar{\phi}_{A_{n}}(t))^{n}=-\dot{\bar{\phi}}_{A}(t)\,.$
(60)

For the case when the scaffold component phase separates with a non-zero fuel
energy $\tilde{\mu}_{F}^{\text{I}}$ in phase I, we can use the assembly rate
given in Eq. (23) and write the effective rate at phase equilibrium as:

$\displaystyle k_{\text{eff}}$
$\displaystyle=\frac{\nu_{A_{n}}}{\nu_{A}}(\gamma_{A}^{\text{II}})^{n}\exp(\bar{\mu}^{0}_{A}/k_{B}T)\zeta_{A}^{n}$
(61)
$\displaystyle\quad\times[V^{I}k^{\text{I}}\exp(\tilde{\mu}_{F}^{\text{I}}/k_{B}T)+(V-V^{\text{I}})k^{\text{II}}]\,,$

where $\zeta_{i}$ is the partitioning degree defined in Eq. (15c). In a
homogeneous system that can be achieved in our model by $V^{\text{I}}=0$
(implying that $\zeta_{i}=1$),
$k_{\text{eff}}=({\nu_{A_{n}}}/{\nu_{A}})(\gamma_{A}(\phi_{1}))^{n}\exp(\bar{\mu}^{0}_{A}/k_{B}T)k$,
where $k$ denotes the assembly rate coefficient in the homogeneous system.

The solution of the generic dynamic equation (Eq. (60)) is given as:

$\bar{\phi}_{A}(t)=\left((n-1)\left(k_{\text{eff}}\,t+\frac{\psi}{n-1}\right)\right)^{-\frac{1}{(n-1)}}\,,$
(62)

where the conserved quantity of the assembly process is
$\psi=(\bar{\phi}_{A}+\bar{\phi}_{A_{n}})$.

Figure 8: Long-time behavior in assembly reactions. The long-time of the
decreasibg monomer voluem fraction $\phi_{A}(t)$ is shown for the assembly
orders $n=2$ and $3$. To illustrate the power-law decay, we use a log-log
representation, confirming the respective power laws of $t^{-1}$ and
$t^{-1/2}$. We use $\tilde{\mu}_{F}^{\text{I}}/k_{B}T=5$ as in Fig. 5(a,b),
where show a lin-lin representation of the same curve.

On long time-scales, we see that the average monomer volume fraction decays as
$\bar{\phi}_{A}(t)\simeq t^{-1/(n-1)}$. Specifically, for $n=2$,
$\bar{\phi}_{A}(t)\simeq t^{-1}$ while for $n=3$, $\bar{\phi}_{A}(t)\simeq
t^{-1/2}$. These trends are confirmed by solving the model for the reversible
assembly process in the presence of compartments (see section III.2); the
corresponding results are shown in Fig. 8. Note that for even longer times,
$\bar{\phi}/\psi$ will approach a non-zero plateau value due to the reversible
pathway from the assembly $A_{n}$ to the monomer $A$. However, by choosing the
reference chemical potential such that the assembly $A_{n}$ is strongly
favored over the monomer $A$, i.e.,
$\exp\left(\frac{\bar{\mu}^{0}_{A_{n}}}{k_{B}T}\right)\ll\exp\left(\frac{n\bar{\mu}^{0}_{A}}{k_{B}T}\right)$,
the plateau emerges at much longer time scales as depicted in Fig. 8.

##  Appendix F Parameters

Quantities |  Values
---|---
Fig.2b: $r_{0},r_{1}$ $\chi_{01}$ $\kappa_{1}$ |  $1,1$ $3$ $15.10^{-2}$
Fig.3: $r_{0},r_{1},r_{A_{1}/A_{2}},r_{B}$ $\psi_{1},\psi_{2}$ $\chi_{01},\chi_{0A_{1}},\chi_{0A_{2}},\chi_{0B}$ $P_{A_{1}},P_{A_{2}},P_{B}$ $\omega_{A_{1}},\omega_{A_{2}},\omega_{B}$ $k^{\text{II}}$ $\bar{\phi}_{B}(t=0)$ $\tilde{D}^{\text{I}}_{B},\tilde{D}^{\text{I}}_{A_{i}},\tilde{D}^{\text{II}}_{A_{i}}$ |  $1,1,1,2$ $1,0.2$ $3,2.5,2,0$ $10,10,200$ $-3.5,-3.5,1$ $0.01$ $0.01\psi_{1}$ $\frac{\tilde{D}^{\text{II}}_{B}}{10},\frac{\tilde{D}^{\text{II}}_{B}}{10},\frac{\tilde{D}^{\text{II}}_{B}}{10},\tilde{D}^{\text{II}}_{B}$
Fig.4: $r_{0},r_{1},r_{A_{1}/B}$ $\psi$ $\chi_{01},\chi_{0A_{1}},\chi_{0B}$ $P_{A_{1}},P_{B}$ $\omega_{A_{1}},\omega_{A_{2}},\omega_{B}$ $k^{\text{II}}$ $\bar{\phi}_{B}(t=0)$ $\tilde{D}^{\text{I}}_{B},\tilde{D}^{\text{I}}_{A_{1}},\tilde{D}^{\text{II}}_{A_{1}}$ |  $1,1,1$ $1$ $3,2.5,2$ $500,2$ $-4,0$ $0.01$ $10^{-4}\psi$ $\frac{\tilde{D}^{\text{II}}_{B}}{10},\frac{3}{10}\tilde{D}^{\text{II}}_{B},3\tilde{D}^{\text{II}}_{B}$
Fig.5: $r_{0},r_{1},r_{A},r_{A_{2}},r_{A_{3}}$ $\psi$ $\chi_{01},\chi_{0A},\chi_{0A_{2}},\chi_{0A_{3}}$ $P_{A},P_{A_{2}},P_{A_{3}}$ $\omega_{A},\omega_{A_{2}},\omega_{A_{3}}$ $k^{\text{II}}$ $\bar{\phi}_{A_{2/3}}(t=0)$ $\tilde{D}^{\text{I}}_{A_{2}},\tilde{D}^{\text{I}}_{A},\tilde{D}^{\text{II}}_{A}$ |  $1,1,1,2,3$ $10^{-4}$ $3,2.5,2,2$ $10,10,10$ $5.5,-7,-5$ $0.01$ $10^{-2}\psi$ $\frac{\tilde{D}^{\text{II}}_{A_{2}}}{10},\frac{\tilde{D}^{\text{II}}_{A_{2}}}{5},2\tilde{D}^{\text{II}}_{A_{2}}$
Fig.6: $r_{0},r_{1},r_{2}$ $\chi_{01},\chi_{02},\chi_{12}$ |  $1,1,1$ $3,-1,0$
Fig.7: Parameters |  Same as Figs. 4 and 3, respectively
Fig.8: Parameters |  Same as Figs. 5(a) and (b)

## References

  * [1] Edmund B. Wilson.  The structure of protoplasm.  Science, 10(237):33–45, 1899.
  * [2] Anthony A. Hyman, Christoph A. Weber, and Frank Jülicher.  Liquid-liquid phase separation in biology.  Annual Review of Cell and Developmental Biology, 30(1):39–58, 2014\.  PMID: 25288112.
  * [3] Yongdae Shin and Clifford P. Brangwynne.  Liquid phase condensation in cell physiology and disease.  Science, 357, 2017.
  * [4] Clifford P. Brangwynne, Christian R. Eckmann, David S. Courson, Agata Rybarska, Carsten Hoege, Jöbin Gharakhani, Frank Jülicher, and Anthony A. Hyman.  Germline p granules are liquid droplets that localize by controlled dissolution/condensation.  Science, 324(5935):1729–1732, 2009.
  * [5] Marina Feric, Nilesh Vaidya, Tyler S. Harmon, Diana M. Mitrea, Lian Zhu, Tiffany M. Richardson, Richard W. Kriwacki, Rohit V. Pappu, and Clifford P. Brangwynne.  Coexisting liquid phases underlie nucleolar subcompartments.  Cell, 165(7):1686–1697, 2016.
  * [6] Steven Boeynaems, Simon Alberti, Nicolas L. Fawzi, Tanja Mittag, Magdalini Polymenidou, Frederic Rousseau, Joost Schymkowitz, James Shorter, Benjamin Wolozin, Ludo Van Den Bosch, Peter Tompa, and Monika Fuxreiter.  Protein phase separation: A new phase in cell biology.  Trends in Cell Biology, 28(6):420–435, 2018.
  * [7] Iuliia A. Antifeeva, Alexander V. Fonin, Anna S. Fefilova, Olesya V. Stepanenko, Olga I. Povarova, Sergey A. Silonov, Irina M. Kuznetsova, Vladimir N. Uversky, and Konstantin K. Turoverov.  Liquid–liquid phase separation as an organizing principle of intracellular space: overview of the evolution of the cell compartmentalization concept.  Cellular and Molecular Life Sciences, 79(5):251, Apr 2022.
  * [8] Salman F Banani, Hyun O Lee, Anthony A Hyman, and Michael K Rosen.  Biomolecular condensates: organizers of cellular biochemistry.  Nature Reviews Molecular Cell Biology, 18(5):285–298, May 2017\.
  * [9] Simon Alberti.  Phase separation in biology.  Current Biology, 27(20):R1097–R1102, 2017.
  * [10] Peter L. Pedersen and Ernesto Carafoli.  Ion motive atpases. i. ubiquity, properties, and significance to cell function.  Trends in Biochemical Sciences, 12:146–150, 1987.
  * [11] Klaus W. Beyenbach and Helmut Wieczorek.  The v-type h+ atpase: molecular structure and function,physiological roles and regulation.  Journal of Experimental Biology, 209(4):577–589, Feb 2006.
  * [12] Rebecca Heald and Orna Cohen-Fix.  Morphology and function of membrane-bound organelles.  Current Opinion in Cell Biology, 26:79–86, 2014.  Cell architecture.
  * [13] Jonathon A. Ditlev, Lindsay B. Case, and Michael K. Rosen.  Who’s in and who’s out?compositional control of biomolecular condensates.  Journal of Molecular Biology, 430(23):4666–4684, 2018.  Phase Separation in Biology and Disease.
  * [14] Erica A. Frankel, Philip C. Bevilacqua, and Christine D. Keating.  Polyamine/nucleotide coacervates provide strong compartmentalization of mg2+, nucleotides, and rna.  Langmuir, 32(8):2041–2049, 2016.  PMID: 26844692.
  * [15] Jeffrey B. Woodruff, Beatriz Ferreira Gomes, Per O. Widlund, Julia Mahamid, Alf Honigmann, and Anthony A. Hyman.  The centrosome is a selective condensate that nucleates microtubules by concentrating tubulin.  Cell, 169(6):1066–1077.e10, 2017.
  * [16] William M. Aumiller and Christine D. Keating.  Experimental models for dynamic compartmentalization of biomolecules in liquid organelles: Reversible formation and partitioning in aqueous biphasic systems.  Advances in Colloid and Interface Science, 239:75–87, 2017.  Complex Coacervation: Principles and Applications.
  * [17] Shana Elbaum-Garfinkle, Younghoon Kim, Krzysztof Szczepaniak, Carlos Chih-Hsiung Chen, Christian R Eckmann, Sua Myong, and Clifford P Brangwynne.  The disordered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynamics.  Proc Natl Acad Sci U S A, 112(23):7189–7194, May 2015.
  * [18] Lars Hubatsch, Louise M Jawerth, Celina Love, Jonathan Bauermann, TY Dora Tang, Stefano Bo, Anthony A Hyman, and Christoph A Weber.  Quantitative theory for the diffusive dynamics of liquid condensates.  eLife, 10:e68620, oct 2021.
  * [19] Christopher A. Strulson, Rosalynn C. Molden, Christine D. Keating, and Philip C. Bevilacqua.  Rna catalysis through compartmentalization.  Nature Chemistry, 4(11):941–946, Nov 2012.
  * [20] Ekaterina Sokolova, Evan Spruijt, Maike M. K. Hansen, Emilien Dubuc, Joost Groen, Venkatachalam Chokkalingam, Aigars Piruska, Hans A. Heus, and Wilhelm T. S. Huck.  Enhanced transcription rates in membrane-free protocells formed by coacervation of cell lysate.  Proceedings of the National Academy of Sciences, 110(29):11692–11697, 2013.
  * [21] Björn Drobot, Juan M. Iglesias-Artola, Kristian Le Vay, Viktoria Mayr, Mrityunjoy Kar, Moritz Kreysing, Hannes Mutschler, and T-Y Dora Tang.  Compartmentalised rna catalysis in membrane-free coacervate protocells.  Nature Communications, 9(1):3643, Sep 2018.
  * [22] Karina K. Nakashima, Jochem F. Baaij, and Evan Spruijt.  Reversible generation of coacervate droplets in an enzymatic network.  Soft Matter, 14:361–367, 2018.
  * [23] Andreas M. Kueffner, Miriam Linsenmeier, Fulvio Grigolato, Marc Prodan, Remo Zuccarini, Umberto Capasso Palmiero, Lenka Faltova, and Paolo Arosio.  Sequestration within biomolecular condensates inhibits abeta 42 amyloid formation.  Chem. Sci., 12:4373–4382, 2021.
  * [24] Brian G. O’Flynn and Tanja Mittag.  A new phase for enzyme kinetics.  Nature Chemical Biology, 17(6):628–630, Jun 2021.
  * [25] Ludo L. J. Schoenmakers, N. Amy Yewdall, Tiemei Lu, Alain A. M. André, Frank. H.T. Nelissen, Evan Spruijt, and Wilhelm T. S. Huck.  In vitro transcription–translation in an artificial biomolecular condensate.  ACS Synthetic Biology, Jun 2023.
  * [26] Christoph A Weber, David Zwicker, Frank Jülicher, and Chiu Fan Lee.  Physics of active emulsions.  Reports on Progress in Physics, 82(6):064601, 2019.
  * [27] Stefano Bo, Lars Hubatsch, Jonathan Bauermann, Christoph A. Weber, and Frank Jülicher.  Stochastic dynamics of single molecules across phase boundaries.  Phys. Rev. Research, 3:043150, Dec 2021.
  * [28] David Zwicker.  The intertwined physics of active chemical reactions and phase separation.  Current Opinion in Colloid & Interface Science, page 101606, 2022\.
  * [29] Jonathan Bauermann, Sudarshana Laha, Patrick M. McCall, Frank Jülicher, and Christoph A. Weber.  Chemical kinetics and mass action in coexisting phases.  Journal of the American Chemical Society, 144(42):19294–19304, Oct 2022.
  * [30] Jonathan Bauermann, Christoph A. Weber, and Frank Jülicher.  Energy and matter supply for active droplets.  Annalen der Physik, 534(9):2200132, 2022.
  * [31] Jonathon A Ditlev, Lindsay B Case, and Michael K Rosen.  Who’s in and who’s out?compositional control of biomolecular condensates.  Journal of molecular biology, 430(23):4666–4684, 2018.
  * [32] Yifei Gao, Xi Li, Pilong Li, and Yi Lin.  A brief guideline for studies of phase-separated biomolecular condensates.  Nature Chemical Biology, 18(12):1307–1318, 2022.
  * [33] Jeffrey B Woodruff, Anthony A Hyman, and Elvan Boke.  Organization and function of non-dynamic biomolecular condensates.  Trends in biochemical sciences, 43(2):81–94, 2018.
  * [34] Andrew S. Lyon, William B. Peeples, and Michael K. Rosen.  A framework for understanding the functions of biomolecular condensates across scales.  Nature Reviews Molecular Cell Biology, 22(3):215–235, Mar 2021\.
  * [35] Frank Jülicher and Jacques Prost.  Generic theory of colloidal transport.  The European physical journal. E, Soft matter, 29:27–36, 05 2009\.
  * [36] Alan J. Bray.  Theory of phase-ordering kinetics.  Advances in Physics, 51:481 – 587, 1993.
  * [37] KR Elder, Martin Grant, Nikolas Provatas, and JM Kosterlitz.  Sharp interface limits of phase-field models.  Physical Review E, 64(2):021604, 2001.
  * [38] Christoph Weber, Thomas Michaels, and L Mahadevan.  Spatial control of irreversible protein aggregation.  Elife, 8:e42315, 2019.
  * [39] Thomas CT Michaels, L Mahadevan, and Christoph A Weber.  Enhanced potency of aggregation inhibitors mediated by liquid condensates.  Physical Review Research, 4(4):043173, 2022.
  * [40] Ron Milo and Rob Philips.  Cell biology by the numbers.  CRC Press, 2015.
  * [41] Zhenzhen Du, Kun Shi, Jordan S Brown, Tao He, Wei-Sheng Wu, Ying Zhang, Heng-Chi Lee, and Donglei Zhang.  Condensate cooperativity underlies transgenerational gene silencing.  Cell reports, 42(8), 2023.
  * [42] Amayra Hernández-Vega, Marcus Braun, Lara Scharrel, Marcus Jahnel, Susanne Wegmann, Bradley T Hyman, Simon Alberti, Stefan Diez, and Anthony A Hyman.  Local nucleation of microtubule bundles through tubulin concentration into a condensed tau phase.  Cell reports, 20(10):2304–2312, 2017.
  * [43] Avinash Patel, Hyun O Lee, Louise Jawerth, Shovamayee Maharana, Marcus Jahnel, Marco Y Hein, Stoyno Stoynov, Julia Mahamid, Shambaditya Saha, Titus M Franzmann, et al.  A liquid-to-solid phase transition of the als protein fus accelerated by disease mutation.  Cell, 162(5):1066–1077, 2015.
  * [44] Hadi M. Fares, Alexander E. Marras, Jeffrey M. Ting, Matthew V. Tirrell, and Christine D. Keating.  Impact of wet-dry cycling on the phase behavior and compartmentalization properties of complex coacervates.  Nature Communications, 11(1):5423, Oct 2020.
  * [45] Emre Tekin, Annalena Salditt, Philipp Schwintek, Sreekar Wunnava, Juliette Langlais, James Saenz, Dora Tang, Petra Schwille, Christof Mast, and Dieter Braun.  Prebiotic foam environments to oligomerize and accumulate rna.  ChemBioChem, 23(24):e202200423, 2022.
  * [46] Ivar Svalheim Haugerud, Pranay Jaiswal, and Christoph A Weber.  Nonequilibrium wet–dry cycling acts as a catalyst for chemical reactions.  The Journal of Physical Chemistry B, 2024.
  * [47] Hannes Mutschler, Aniela Wochner, and Philipp Holliger.  Freeze–thaw cycles as drivers of complex ribozyme assembly.  Nature Chemistry, 7(6):502–508, 2015.
  * [48] Giacomo Bartolucci, Adriana Calaca Serrao, Philipp Schwintek, Alexandra Kühnlein, Yash Rana, Philipp Janto, Dorothea Hofer, Christof B. Mast, Dieter Braun, and Christoph A. Weber.  Sequence self-selection by cyclic phase separation.  Proceedings of the National Academy of Sciences, 120(43):e2218876120, 2023.
  * [49] Giacomo Bartolucci, Ivar S. Haugerud, Thomas CT Michaels, and Christoph A Weber.  The interplay between biomolecular assembly and phase separation.  bioRxiv, pages 2023–04, 2023.
  * [50] Irving R Epstein and Bing Xu.  Reaction–diffusion processes at the nano-and microscales.  Nature nanotechnology, 11(4):312–319, 2016.
  * [51] Jacob Halatek and Erwin Frey.  Rethinking pattern formation in reaction–diffusion systems.  Nature Physics, 14(5):507–514, 2018.
  * [52] Lucas Menou, Chengjie Luo, and David Zwicker.  Physical interactions in non-ideal fluids promote turing patterns.  Journal of the Royal Society Interface, 20(204):20230244, 2023.
  * [53] Wolfram Pönisch, Thomas CT Michaels, and Christoph A Weber.  Aggregation controlled by condensate rheology.  Biophysical Journal, 122(1):197–214, 2023.
  * [54] Sudarshana Laha.  Chemical reactions controlled through compartmentalization: Applications to bottom-up design of synthetic life.  monograph, Technische Universität Dresden, June 2023.