††thanks: Contributed equally to this work††thanks: Contributed equally to
this work††thanks: To whom correspondence should be addressed; Email:
<EMAIL_ADDRESS>(M.Z.);
<EMAIL_ADDRESS>(H.S.C.)††thanks: To whom correspondence should be
addressed; Email:
<EMAIL_ADDRESS>(M.Z.);
<EMAIL_ADDRESS>(H.S.C.)

# Assembly of Model Postsynaptic Densities Involves Interactions
Auxiliary to Stoichiometric Binding

Yi-Hsuan Lin  Department of Biochemistry, University of Toronto, Toronto,
Ontario M5S 1A8, Canada  Molecular Medicine, The Hospital for Sick Children,
Toronto, Ontario M5G 0A4, Canada  Haowei Wu  Division of Life Science, State
Key Laboratory of Molecular Neuroscience, Hong Kong University of Science and
Technology, Clear Water Bay, Hong Kong, China  Bowen Jia  Division of Life
Science, State Key Laboratory of Molecular Neuroscience, Hong Kong University
of Science and Technology, Clear Water Bay, Hong Kong, China  Mingjie Zhang
Division of Life Science, State Key Laboratory of Molecular Neuroscience, Hong
Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
School of Life Sciences, Southern University of Science and Technology,
Shenzhen, China  Hue Sun Chan  Department of Biochemistry, University of
Toronto, Toronto, Ontario M5S 1A8, Canada

###### Abstract

The assembly of functional biomolecular condensates often involves liquid-
liquid phase separation (LLPS) of proteins with multiple modular domains,
which can be folded or conformationally disordered to various degrees. To
understand the LLPS-driving domain-domain interactions, a fundamental question
is how readily the interactions in the condensed phase can be inferred from
inter-domain interactions in dilute solutions. In particular, are the
interactions leading to LLPS exclusively those underlying the formation of
discrete inter-domain complexes in homogeneous solutions? We address this
question by developing a mean-field LLPS theory of two stoichiometrically
constrained solute species. The theory is applied to the neuronal proteins
SynGAP and PSD-95, whose complex coacervate serves as a rudimentary model for
neuronal postsynaptic densities (PSDs). The predicted phase behaviors are
compared with experiments. Previously, a three-SynGAP, two-PSD-95 ratio was
determined for SynGAP/PSD-95 complexes in dilute solutions. However, when this
3:2 stoichiometry is uniformly imposed in our theory encompassing both dilute
and condensed phases, the tie-line pattern of the predicted SynGAP/PSD-95
phase diagram differs drastically from that obtained experimentally. In
contrast, theories embodying alternate scenarios postulating auxiliary SynGAP-
PSD-95 as well as SynGAP-SynGAP and PSD-95-PSD-95 interactions in addition to
those responsible for stoichiometric SynGAP/PSD-95 complexes produce tie-line
patterns consistent with experiment. Hence, our combined theoretical-
experimental analysis indicates that weaker interactions or higher-order
complexes beyond the 3:2 stoichiometry, but not yet documented, are involved
in the formation of SynGAP/PSD-95 condensates, imploring future efforts to
ascertain the nature of these auxiliary interactions in PSD-like LLPS and
underscoring a likely general synergy between stoichiometric, structurally
specific binding and stochastic, multivalent “fuzzy” interactions in the
assembly of functional biomolecular condensates.

Short Title:
Interactions in SynGAP/PSD-95 Condensate

Statement of Significance

It has become increasingly clear that functional biomolecular condensates
underpinned by liquid-liquid phase separation (LLPS) are stabilized by dynamic
multivalent interactions as well as by structurally specific interactions. To
gain insights into the role of stoichiometric binding in biomolecular LLPS, we
develop theories for LLPS driven solely by interactions that stabilize
stoichiometric protein complexes in dilute solution and for alternate
scenarios in which auxiliary interactions also contribute. Application of our
formulations to experimental measurements of dilute- and condensed-phase
protein concentrations of the SynGAP/PSD-95 condensate model of postsynaptic
densities reveals that its assembly involves interactions auxiliary to those
stabilizing the 3:2 SynGAP/PSD-95 complex in dilute solution, exemplifying a
synergy between specific and stochastic interactions in the assembly of
biomolecular condensates.

INTRODUCTION

Biomolecular condensates composing of various proteins, nucleic acids, and
small molecules are important for biological functions Banani et al. (2017).
Examples include intracellular compartments, often referred to as
“membraneless organelles”, e.g., P-granules, nuclear speckles, and Cajal
bodies Brangwynne et al. (2009); Mao et al. (2011), and extracellular
materials such as the precursory tropolelastin coacervates of elastic
connective tissues Muiznieks et al. (2018). Some membraneless organelles, such
as nucleoli and stress granules, are organized further into subcompartments
Feric et al. (2016); Riback et al. (2020). Biomolecular condensates serve
diverse functions, including but not limited to gene regulation, cell growth,
and synaptic activities Chong and Forman-Kay (2016); Banani et al. (2017);
Zeng et al. (2018a); Kim et al. (2019). Membraneless organelles reduce
biomolecular noise in the cell Klosin et al. (2020) and provide regulated
local environments that facilitate specific biochemical processes, yet they
can also assemble/disassemble rapidly in response to environmental changes
Mitrea and Kriwacki (2016). Biophysically, liquid-liquid phase separation
(LLPS) is recognized as a major—albeit not the only McSwiggen et al.
(2019)—mechanism in the assembly, including subcompartmentalization Feric et
al. (2016), of many of these condensates Alberti et al. (2019).

Main ingredients of biomolecular condensates can be intrinsically disordered
proteins (IDPs), folded globular proteins, RNA, and/or proteins with folded
domains and intrinsically disordered regions (IDRs) Nott et al. (2015); Feric
et al. (2016); Banerjee et al. (2017); Cinar et al. (2019). When only parts of
the proteins (e.g., folded domains Li et al. (2012)) engage in significant
favorable interchain interactions whereas other parts (e.g., parts of or
entire IDRs) do not, LLPS energetics may be conceptualized using a “stickers
and linkers” picture Harmon et al. (2017, 2018); Choi et al. (2019). The
sticker/linker distinction is not always clear-cut, however, because every
part of a protein chain may contribute to LLPS-driving interactions. For
folded domains, the interactions can be—though not always—structurally
specific, or entail “folding upon binding” when IDR interacting partners are
involved Uversky (2002); Chen et al. (2015). In contrast, for interactions
among IDPs/IDRs, the molecular recognition mechanisms underlying their
sequence-dependent LLPS Nott et al. (2015); Lin et al. (2016); Schuster et al.
(2020); Martin et al. (2020); Das et al. (2020) are stochastic or “fuzzy”
Sharma et al. (2015) in that they involve diverse, dynamic conformations and
transient interactions Brady et al. (2017); Lin et al. (2017); Amin et al.
(2020); Pal et al. (2021a). However, some of the transient interactions may
entail fluctuations between disordered chain configurations with labile but
nonetheless specific fibril-like structures Murray et al. (2017); Kato and
McKnight (2018).

It has been known for some time that proteins participating in signaling
pathways are often organized structurally and sequentially in a modular
fashion, with folded domains such as SH2 and SH3 acting as modules playing key
regulatory roles Pawson (1995). Recently, multivalent interactions involving
some of these domains are found to be important for the assembly of
biomolecular condensates. Thus, LLPS can be a key physico-chemical mechanism
exploited by Nature for physiological regulation as LLPS’s essentially all-or-
none features entail sharp transitions among conformationally and hence
functionally distinct states of these proteins Li et al. (2012).

In view of the central role of multivalency in biomolecular condensates Li et
al. (2012); Banani et al. (2016); Harmon et al. (2017), network concepts such
as percolation transition Harmon et al. (2017, 2018); Choi et al. (2019);
Shillcock et al. (2020) and graph theory Sanders et al. (2020); Schmit et al.
(2020) have been applied to provide rationalizations and theoretical
formulations of LLPS and gelation Lin et al. (2018). In this perspective, the
nodes or “stickers” are folded or labile IDR interacting domains and the LLPS
or gelation transition is determined in large measure by the number of
connections (valence of attractive interactions) that can be effectuated
between the nodes. Recent applications of this approach include elucidating
the role of RNA-binding valence in creating sufficient connectivity of a
ribonucleoprotein network Sanders et al. (2020) in the assembly of stress
granules Yang et al. (2020); Guillén-Boixet et al. (2020) and the competition
between gelation and LLPS of the cancer-related protein SPOP with its
substrate Schmit et al. (2020).

Postsynaptic densities

With these general considerations in mind, the present work aims to gain
insight into the LLPS-driving interactions of two modular proteins, synaptic
Ras GTPase-activating protein (SynGAP) and postsynaptic density protein 95
(PSD-95), which are major constituents of postsynaptic densities (PSDs) Zeng
et al. (2016). Recent advances indicate that biomolecular condensates play
critical roles in neural function, probably because of their ability to
respond rapidly to stimuli, thus misregulated LLPS can lead to neurological
diseases Kaizuka and Takumi (2018); Ryan and Fawzi (2019); Chen et al. (2020);
Wu et al. (2020). In particular, phase separation is important for proper
communication between neurons as LLPS is involved in the assembly of synaptic
vesicles (SVs) attached on presynaptic plasma membranes Milovanovic and De
Camilli (2017); Milovanovic et al. (2018) as well as the PSDs beneath
postsynaptic membranes Zeng et al. (2016); Feng et al. (2018); Zeng et al.
(2018a, b).

Each PSD is a disc-shaped membraneless protein-rich compartment, an assembly
made up of nucleic acids and thousands of different types of proteins Zhu et
al. (2016); Kaizuka and Takumi (2018). These include RNA binding proteins
Zhang et al. (2012), transmembrane layer proteins such as N-methyl D-aspartate
receptor (NMDAR) Iacobucci and Popescu (2017), actins in the cytoskeleton
layer, scaffold proteins such as PDZ-domain-containing PSD-95s, which are
membrane-associated guanylate kinases (MAGUKs) Won et al. (2017), guanylate-
kinase (GK) associated protein (GKAP) Kim et al. (1997), SH3 and multiple
ankyrin repeat domain proteins (Shanks) Monteiro and Feng (2017), and Homer
family of adaptor proteins Brakeman et al. (1997), as well as the SynGAP
protein known to be highly enriched in PSDs Cheng et al. (2006); Zeng et al.
(2018a). Situated adjacent to postsynaptic membranes, PSDs can exchange
materials with the cytoplasm in synaptic spines and, accordingly, serves to
provide spatial and temporal organization of the neurotransmitter receptors at
the synapse Kaizuka and Takumi (2018). The functional importance of PSDs is
highlighted by intriguing recent observations that PSDs are downsized during
sleep de Vivo et al. (2017); Diering et al. (2017), which suggest that
synaptic strength is renormalized during sleep; and that a SynGAP-PSD-95
condensate model of PSD Zeng et al. (2016) can be disassembled by moderate
hydrostatic pressure Cinar et al. (2020, 2019), which points to a possible
biophysical origin of the high pressure neurological syndrome experienced by
divers Talpalar (2007).

A minimal experimental model of PSD constructed using SynGAP and PSD-95—both
of which are very abundant in PSDs—has been observed to self-organize into
highly condensed, PSD-like droplets in vitro, offering the first indication
that LLPS is a significant biophysical underpinning of PSD assembly Zeng et
al. (2016). Subsequently, more realistic in vitro models of PSD, encompassing
SynGAP and three scaffold proteins GKAP, Shank3, and Homer3 (in addition to
PSD-95) as well as NR2B as another glutamate receptor, have also been
constructed as a versatile research platform Zeng et al. (2018a). Here, as a
first step toward elucidating the statistical mechanics of PSD assembly, we
focus on the simpler two-component SynGAP–PSD-95 construct, the same system we
have utilized recently for studying the effect of hydrostatic pressure on PSD
stability Cinar et al. (2020). SynGAP and PSD-95 undergo LLPS together when
mixed but each of the individual components, SynGAP or PSD-95 (up to 100
$\mu$M concentration each), does not phase separate by itself Zeng et al.
(2016). It follows that the SynGAP–PSD-95 condensate is a complex coacervate,
the formation of which must involve favorable interactions between SynGAP and
PSD-95 molecules.

LLPS of stoichiometric SynGAP/PSD-95 complexes as a possible mechanism of PSD
assembly

Wildtype SynGAP has a long coiled-coil (CC) domain that can dock onto two
other such domains in an intertwining manner resulting in a highly stable
SynGAP trimer, which we term $\mathsf{S}_{3}$ hereafter. The SynGAP trimer
can, in turn, form a complex with two PSD-95 molecues Zeng et al. (2016). We
refer to this SynGAP–PSD-95 complex with a 3:2 stoichiometry, which is
observed in dilute solution of SynGAP and PSD-95, as
$\mathsf{S}_{3}\mathsf{P}_{2}$. Experimental studies indicate that formation
of the trimer $\mathsf{S}_{3}$ is necessary for PSD-like complex coacervation,
because a SynGAP mutant that abolishes SynGAP’s ability to trimerize does not
undergo LLPS in the presence of PSD-95 Zeng et al. (2016). Taken together,
these observations raise the tantalizing possibility that units of the
$\mathsf{S}_{3}\mathsf{P}_{2}$ complex act as the only nodes of favorable
interactions in a SynGAP/PSD-95 condensate. In other words, the condensate is
stabilized solely by interactions among the $\mathsf{S}_{3}\mathsf{P}_{2}$
complexes. Examples of how inter-$\mathsf{S}_{3}\mathsf{P}_{2}$ favorable
interactions may arise from known interactions among SynGAP and PSD-95 domains
that stabilize individual $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes Zeng et al.
(2016, 2018b) are provided in Fig. 1. Recent experiments indicate that the GK
domain of a $\mathsf{P}$ molecule interacts favorably with one of the PDZ
domains of another $\mathsf{P}$ molecule when that PDZ domain is bound to the
one of the PDZ binding domains (PBMs) of a $\mathsf{S}_{3}$ Zeng et al.
(2018b). These GK–PDZ-PBM interactions can lead to a network of favorably
interacting $\mathsf{S}_{3}\mathsf{P}_{2}$ (Fig. 1b and c). While such
interactions may lead merely to a linear array of favorably interacting
$\mathsf{S}_{3}\mathsf{P}_{2}$s (Fig. 1b) as previously envisioned Zeng et al.
(2018b), they may also result in a three-dimensional network of favorably
interacting $\mathsf{S}_{3}\mathsf{P}_{2}$s (Fig. 1c) and may therefore
stabilize a condensed phase of $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes. The
hypothetical scenario that SynGAP/PSD-95 condensate is stabilized entirely by
favorable interactions among units of $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes
represents a strong coupling between the
$\mathsf{S}_{3}+2\mathsf{P}\rightleftarrows\mathsf{S}_{3}\mathsf{P}_{2}$
binding equilibrium and LLPS-driving interactions. As such, it is instructive
to ascertain, theoretically, the implications of this assumed scenario on the
predicted LLPS properties. This knowledge would be useful, in general, not
only for studying the SynGAP/PSD-95 systems but also other coacervate systems
that involve a similar binding equilibrium, or a chemical equilibrium of
reversible reactions that takes an equivalent mathematical form as a binding
equilibrium. The theoretically predicted mesoscopic phase properties, such as
co-existence curves and phase diagrams, may then be compared against
corresponding experimental observations to assess whether the assumed
microscopic stoichiometric pattern of network interactions is indeed the case
or, alternatively, auxiliary nodes of favorable interactions in addition to
$\mathsf{S}_{3}\mathsf{P}_{2}$s are involved in stabilizing the condensed
coacervate phase.

Probing interaction networks in SynGAP/PSD-95 condensates by theory and LLPS
experiment

To this end, we endeavor to develop a theoretical formulation for protein
complex coacervation with an emphasis on the interactions among the folded
protein domains under constraints similar to those imposed by the modular
organization, multivalency and stoichiometry of the SynGAP/PSD-95 system. In
contrast, most of the recent attention of theoretical/computational
developments on biomolecular condensates have been on IDPs and IDRs. While
insights have been gained into multiple-component LLPS by several recent
theoretical/modeling studies (e.g., Refs. Feric et al. (2016); Frenkel and
Jacob (2017); Lin et al. (2017); Harmon et al. (2018); Lin et al. (2018); Pal
et al. (2021a)), and some folded domains such as the helicase domain in LAF-1
have been considered in explicit-chain simulations Dignon et al. (2018), the
preponderance of the efforts thus far are on systems with a single IDP/IDR
species. For instance, sequence-dependent IDP LLPS has been studied using
analytical theory Lin et al. (2016, 2020), coarse-grained explicit-chain
lattice Choi et al. (2019); Das et al. (2018a); Robichaud et al. (2019) and
continuum Dignon et al. (2018); Das et al. (2020); Nilsson and Irbäck (2020);
Schuster et al. (2020) as well as field-theoretic McCarty et al. (2019); Pal
et al. (2021a) simulations. These efforts have focused on how LLPS propensity
of heteropolymeric chain molecules depends on the sequence of charged monomers
Lin et al. (2016); Das et al. (2018b); Pal et al. (2021b) in general (as
characterized, e.g., by quantitative charge pattern parameters Das and Pappu
(2013); Sawle and Ghosh (2015)), effects of sequence patterns of hydrophobic
Statt et al. (2020), aromatic Martin et al. (2020), and other amino acid
residues Dignon et al. (2018); Das et al. (2020) on IDP phase behaviors, and
the impact of sequence patterns on the phase properties of synthetic polymers
Perry and Sing (2020); Sing and Perry (2020).

The microscopic complexity of the SynGAP/PSD-95 system makes modeling its LLPS
in atomic details with quantitative precision infeasible. In order to gain
insight into the essential biophysics of the system, we construct a highly
simplified model that treats each $\mathsf{S}_{3}$ or $\mathsf{P}$ molecule as
a single particle that can engage in certain interactions suggested by
experimental observations. The model is for LLPS driven largely by folded
domain association. As such, it is substantially different from the
aforementioned theories for IDP LLPS that take into account the conformational
diversity of IDP chains. In this respect, the conceptual underpinnings of the
present model are akin to those of patchy particle models of colloids Wertheim
(1986); Teixeira and Tavares (2017); Kamp et al. (2020), which have been
applied recently to study phase behaviors of folded proteins such as lens
crystallins Kastelic et al. (2016), ribonucleoprotein droplets in which both
the proteins and RNA are modeled as patchy articles Nguemaha and Zhou (2018),
as well as the relationship between patchy-particle and coarse-grained
explicit-chain models of biomolecular LLPSs including assembly of
multicomponent condensates Espinosa et al. (2020). The model developed herein
is structurally and energetically more simplified than patchy particle
formulations in that the valencies of $\mathsf{S}_{3}$ and $\mathsf{P}$ are
now treated in a mean-field manner without considering the interactions’
anisotropic directionality. This simplicity notwithstanding, our tractable
model provides a critical assessment of SynGAP/PSD-95 coacervation scenarios
because the model features a possible coupling between the formation of the
3:2 $\mathsf{S}_{3}\mathsf{P}_{2}$ complex and phase separation. We find that
the constraints imposed by an exclusively complex-driven phase separation
entail drastically different $\mathsf{S}_{3}$ and $\mathsf{P}$ compositions of
coexisting phases from those predicted by classical Flory-Huggins theories
Flory (1953); de Gennes (1979) in the absence of such coupling. Therefore,
whether SynGAP/PSD-95 coacervation is driven solely by interactions among
$\mathsf{S}_{3}\mathsf{P}_{2}$ complexes can be assessed by comparing our
theoretical predictions against experiment. With this recognition, we have now
conducted extensive experimental measurements of $\mathsf{S}_{3}$/$\mathsf{P}$
compositions of coexisting phases. The results are consistent with classical
Flory-Huggins theory but not the exclusively complex-driven LLPS theory,
indicating a hitherto unknown prevalence of auxiliary interactions beside
those among the $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes in the SynGAP/PSD-95
condensed phase. Details of these findings and their ramifications are
provided below.

MODELS, MATERIALS, AND METHODS

A general mean-field theoretical framework for stoichiometric complex-driven
LLPS

Before delving into the particulars of the SynGAP/PSD-95 system, it is useful
and instructive to first construct a general LLPS theory for a solution
containing two species of solutes $A$ and $B$ that are capable of forming a
stoichiometric complex $A_{x}B_{y}$ (where $x$, $y$ are two fixed positive
integers) and whose complex coacervation is contingent upon, i.e., coupled to,
the formation of such complexes. The formation and dissociation of the complex
may be expressed as a reversible binding or chemical reaction:

$xA+yB\rightleftarrows A_{x}B_{y}\;.$ (1)

Now let the total solution volume be $V$, and the number of solutes $A$ and
$B$ in the solution be, respectively, $n_{A}$ and $n_{B}$. Proceeding with a
Flory-Huggins (FH)-type lattice argument Flory (1953); Chan and Dill (1994),
we assume for simplicity that individual $A$ and $B$ solute molecules occupy
equal volume, denoted by $v_{\rm s}$, and discretize the solution volume
conceptually to a total of $M=V/v_{\rm s}$ lattice sites. Accordingly, a
lattice site in the system can be occupied by an $A$ or $B$ solute molecule it
its entirety, part of an $A_{x}B_{y}$ complex, or an entire solvent molecule.
In other words, each $A$, $B$, or solvent molecule takes up one lattice site,
whereas each $A_{x}B_{y}$ complex occupies a contiguous chunk of $x+y$ sites.

Consider a situation in which $m$ complexes are formed in the solution. The
binding reaction and the configurational freedom of such a state is described
by the partition function

$Z_{\rm bind}(m)=\frac{M!e^{m\epsilon_{\rm b}/k_{\rm
B}T}}{m!(n_{A}-xm)!(n_{B}-ym)!(M-n_{A}-n_{B})!}\;,$ (2)

where $-\epsilon_{\rm b}<0$ is the free energy associated with the favorable
complex-forming binding (forward) reaction in Eq. 1, $k_{\rm B}$ is Boltzmann
constant, and $T$ is absolute temperature. The free energy $\epsilon_{\rm b}$
does not account for the part of translational entropy described by the
combinatoric factors in Eq. 2. Nonetheless, in general $\epsilon_{\rm b}$ can
contain enthalpic as well as entropic components with entropic contributions
from, e.g., change in solvent orientational entropy associated with the
binding reaction. It should also be noted that for notational simplicity a
standard FH factor, $[(z-1)/M]^{m(x+y-1)}$, to account for the spatial
contiguity of each of the $A_{x}B_{y}$ complex (where $z$ is coordination
number of the lattice for a flexible, polymer-like complex and $z-1\rightarrow
1$ for a rigid complex) Chan and Dill (1994)) is omitted in Eq. 2 because the
factor may be formally absorbed into the entropic component of $\epsilon_{\rm
b}$ by redefining $\epsilon_{\rm b}\rightarrow\epsilon_{\rm b}+k_{\rm
B}T(x+y-1)\ln[(z-1)/M]$.

We further assume that LLPS of our model system is driven by short spatial
range contact-like interactions between individual bound solute $A$ or $B$
molecules from different $A_{x}B_{y}$ complexes. For each $m$-complex state,
the energetic contribution for these interactions is given by the following
free energy

$-k_{\rm B}T\ln Z_{\rm LLPS}(m)=-\frac{z\epsilon_{\rm
ps}}{M}\left(\begin{array}[]{c}m\\\ 2\end{array}\right)(x+y)^{2}+O(m^{3})\;,$
(3)

where $Z_{\rm LLPS}(m)$ is the pertinent partition function and $\epsilon_{\rm
ps}$ is the contact free energy of an $A$–$A$, $B$–$B$, or $A$–$B$ interaction
(which are assuned to carry the same $\epsilon_{\rm ps}$ for simplicity). In
situations where these interactions are relatively weak, as is apparent the
case for the SynGAP/PSD-95 system according to a recent analysis of
experimental data on the system’s sensitivity to hydrostatic pressure Cinar et
al. (2020), one may consider only the pairwise $O(m^{2})$ contribution in Eq.
3, which corresponds to the lowest-order interaction term in coordinate-space
polymer lattice cluster theory Baker et al. (1993).

It should be emphasized that the $\epsilon_{\rm ps}$ for $A$–$A$, $B$–$B$, and
$A$–$B$ contacts in the present formulation is only for inter-complex
interactions between different $A_{x}B_{y}$ units. The parameter
$\epsilon_{\rm ps}$ is not involved in the assembly of an individual
$A_{x}B_{y}$ complex, which is stipulated separately by a chemical equilibrium
[Eq. 1] as will be described further below. In the present mean-field
formulation, each unit of $A_{x}B_{y}$ interacts as a whole in an isotropic
manner. In this context, assigning different energies to inter-complex
$A$–$A$, $B$–$B$, and $A$–$B$ contacts would only amount, mathematically, to a
redefinition of $\epsilon_{\rm ps}$, which is effectively a weighted average
inter-complex energies over different types of contacts between the complexes’
$A$ and $B$ constituents. Going forward, it would be interesting in future
investigations to extend the present approach to stucturally and energetically
more realistic models that allow for anisotropic interactions, such as patchy
particle models Espinosa et al. (2020). Under those extended frameworks, it
would then be useful to study the effects of different inter-complex energies
for $A$–$A$, $B$–$B$, and $A$–$B$ contacts.

Here, with $Z_{\rm bind}$ and $Z_{\rm LLPS}$ in place, the total partition
function ${\cal Z}$ for our model solution system that takes into account all
possible $m$-complex states is given by

${\cal Z}=\sum_{m=0}^{m_{\rm max}}Z_{\rm bind}(m)Z_{\rm
LLPS}(m)\equiv\sum_{m=0}^{m_{\rm max}}e^{-Mf(m)}\;,$ (4)

where $m_{\rm max}=\lfloor{\rm min}\\{n_{A}/x,n_{B}/y\\}\rfloor$ is the
maximum number of $A_{x}B_{y}$ complexes possible for a given solution system
and $f(m)$ is the $m$-state free energy per lattice site in units of $k_{\rm
B}T$. Application of Stirling’s approximation for the factorials to Eqs. 2 and
3 leads to the following approximate expression for $Mf(m)$:

$\displaystyle Mf(m)$ (5)
$\displaystyle\approx(n_{A}-xm)\ln(n_{A}-xm)+(n_{B}-ym)\ln(n_{B}-ym)$
$\displaystyle\quad+(M-n_{A}-n_{B})\ln(M-n_{A}-n_{B})$
$\displaystyle\quad+m\ln m+m\left(x+y-1-\frac{\epsilon_{\rm b}}{k_{\rm
B}T}\right)$ $\displaystyle\quad-\chi\frac{[m(x+y)]^{2}}{M}-M\ln M\;,$

where $\chi=z\epsilon_{\rm ps}/(2k_{\rm B}T)$ is an effective FH parameter
Baker et al. (1993); Bawendi et al. (1987). In arriving at Eq. 5, we have
neglected $O(1/M)$ and $O([\ln M]/M)$ contributions to $f(m)$ because these
terms vanish in the $M\rightarrow\infty$ thermodynamic limit. Accordingly, the
$\chi m(x+y)^{2}$ term from Eq. 3 and terms from the $(2\pi n)^{1/2}$ part of
the Stirling approximation $n!\approx(2\pi n)^{1/2}n^{n}e^{-n}$ are not
included in Eq. 5.

Following common notation in FH phase separation theory, we define
$\phi_{A}\equiv n_{A}/M$ and $\phi_{B}\equiv n_{B}/M$ as overall volume
fractions for $A$ and $B$, respectively, $\psi\equiv m(x+y)/M$ as volume
fraction for $A_{x}B_{y}$, and $\psi_{A}\equiv\phi_{A}-x\psi/(x+y)$ and
$\psi_{B}\equiv\phi_{B}-y\psi/(x+y)$ as volume fractions, respectively, of
unbound $A$ and $B$ (i.e., which do not form $A_{x}B_{y}$). Using this
notation, Eq. 5 may be rewritten as

$\displaystyle f(\psi)=$
$\displaystyle\psi_{A}\ln\psi_{A}+\psi_{B}\ln\psi_{B}+\frac{\psi}{x+y}\ln\frac{\psi}{x+y}$
(6) $\displaystyle+(1-\phi_{A}-\phi_{B})\ln(1-\phi_{A}-\phi_{B})$
$\displaystyle+\frac{\psi}{x+y}\left(\ln{\cal
K}^{0}+x+y-1\right)-\chi\psi^{2}\;,$

where

${\cal K}^{0}\equiv\frac{e^{-\epsilon_{\rm b}/k_{\rm B}T}}{M^{x+y-1}}$ (7)

is a reduced (dimensionless) dissociation constant of $A_{x}B_{y}$ in dilute
solution (the “0” superscript symbolizes dilute solution), the unit of which
will be clarified below.

We are now ready to derive an approximate expression for the system described
by ${\cal Z}$ by replacing the summation in Eq. 4 with a single term at the
saddle point $\overline{\psi}$ of $f(\psi)$. In other words, the free energy
of the system

${\cal F}=-k_{\rm B}T\ln{\cal Z}\approx Mf(\overline{\psi})\;,$ (8)

where $\overline{\psi}$ is obtained by solving

$\displaystyle 0=\left.\frac{\partial
f(\psi)}{\partial\psi}\right|_{\psi=\overline{\psi}}=$
$\displaystyle-\frac{x}{x+y}\ln\overline{\psi}_{A}-\frac{y}{x+y}\ln\overline{\psi}_{B}$
(9) $\displaystyle+\frac{1}{x+y}\left[\ln\frac{\overline{\psi}}{x+y}+\ln{\cal
K}^{0}\right]-2\chi\overline{\psi}\;.$

Multiplying every term on the right hand side of Eq. LABEL:eq:dfdpsi=0 by
$x+y$ and taking exponential of all terms result in the equivalent condition

$\frac{\overline{\psi}_{A}^{\;x}\overline{\psi}_{B}^{\;y}}{[\overline{\psi}/(x+y)]}={\cal
K}^{0}e^{-2\chi(x+y)\overline{\psi}}\;,$ (10)

which can be further rewritten as

$\frac{[A]^{x}[B]^{y}}{[A_{x}B_{y}]}=K_{\rm d}^{0}e^{-2\chi v_{\rm
s}(x+y)^{2}[A_{x}B_{y}]}$ (11)

where $[A]$, $[B]$, $[A_{x}B_{y}]$ are the equilibrium concentrations,
respectively, of $A$, $B$, $A_{x}B_{y}$ and $K_{\rm d}^{0}$ is the
dissociation constant of $A_{x}B_{y}$ in dilute solution. These quantities are
given by

$\displaystyle[A_{x}B_{y}]$ $\displaystyle=\frac{\overline{\psi}}{v_{\rm
s}(x+y)}\equiv\frac{\overline{m}}{V}$ (12a) $\displaystyle[A]$
$\displaystyle=\frac{\overline{\psi}_{A}}{v}=[A_{\rm
T}]-x[A_{x}B_{y}]=\frac{n_{A}}{V}-x\frac{\overline{m}}{V}$ (12b)
$\displaystyle[B]$ $\displaystyle=\frac{\overline{\psi}_{B}}{v}=[B_{\rm
T}]-y[A_{x}B_{y}]=\frac{n_{B}}{V}-y\frac{\overline{m}}{V}$ (12c)
$\displaystyle K_{\rm d}^{0}$ $\displaystyle={\cal K}^{0}v_{\rm
s}^{x+y-1}=\frac{e^{-\epsilon_{\rm b}/k_{\rm B}T}}{V^{x+y-1}}\;,$ (12d)

where $[A_{\rm T}]$ and $[B_{\rm T}]$ are total concentrations of $A$ and $B$,
irrespective of whether they are unbound or part of an $A_{x}B_{y}$ complex.
As expected, when $\chi=0$, Eq. 11 reduces to the equilibrium equation for the
binding reaction in Eq. 1 for dilute solution. In contrast, when $\chi\neq 0$,
i.e., in the presence of the LLPS-driving interactions, the dissociation
constant of $A_{x}B_{y}$ defined by $[A]^{x}[B]^{y}/[A_{x}B_{y}]$ in Eq. 11
decreases exponentially as $\chi$ increases. This is because the individual
$A$ and $B$ solutes in an $A_{x}B_{y}$ complex cannot dissociate from one
another as long as the $A_{x}B_{y}$ complex is interacting with another
$A_{x}B_{y}$ complex via the LLPS-driving FH interaction. This constraint
results in an enhanced binding affinity between $A$ and $B$ relative to that
in a dilute solution where the LLPS-driving inter-complex interactions are
absent.

Experimental materials and methods

Protein expression and purification. The PSD-95 construct containing PDZ, SH3
and GK tandem (aa R306-L721 of the protein, referred to as PSD-95 in this
paper) was PCR amplified from human cDNA library (NCBI:
NP${\underline{\phantom{o}}}$001122299). The SynGAP construct containing a
trimeric coiled-coil (CC) domain and a PSD-95 PSG binding domain (PBM)
(encoding aa A1147-V1308, lacking 1192V-1193K and 1293E-1295G of the protein,
referred to as SynGAP in this paper) was PCR amplified from mouse cDNA library
(UniProt: J3QQ18) Zeng et al. (2016). The genes encoding PSD-95 and SynGAP was
individually cloned into vectors containing N-terminal His6-affinity tag.
Recombinant proteins were expressed in Escherichia coli BL21(DE3) cells in LB
medium at 16∘C overnight and purified using a nickel-NTA agarose affinity
column followed by size-exclusion chromatography (Superdex 200) with a column
buffer containing 50mM Tris, pH8.2, 100mM NaCl, 1mM EDTA, 1mM DTT. The fusion
tag of each protein was cleaved by HRV 3C protease and separated by another
step of size-exclusion chromatography.

Sedimentation-based assay of phase separation and phase diagram. All proteins
were pre-cleared through high-speed centrifugation (16873$\times$g 10 min) at
25∘C before sedimentation assay. Proteins were mixed at each designed
concentration in 50 $\mu$l total volume. After 10-min equilibration at room
temperature, the mixture was centrifuged at 16873$\times$g at 25∘C for 10 min.
The supernatant was collected, and pellet was thoroughly resuspended in 50
$\mu$l buffer. Proteins from supernatant and pellet fractions were analyzed
through SDS-PAGE with Coomassie blue R250 staining.

The phase diagram was constructed from the sedimentation-based phase
separation assay. Phase separation of a SynGAP/PSD-95 mixture was considered
to occur if the concentrations of SynGAP and PSD-95 recovered from the pellet
of the mixture were higher than the pellet fraction of the individual protein
alone in each sedimentation assay.

Quantification of protein concentration in the condensed phase. Concentrations
of PSD-95 in the condensed phase were measured according to the previous
reported protocol Zeng et al. (2018a). Briefly, Cy3-labeled PSD-95 was diluted
into the final concentration of 1% by mixing with unlabeled protein. 1%
Cy3-labeled PSD-95 was further mixed with SynGAP to form condensates. The
mixture was injected into a 96-well glass-bottomed plate (Thermo Fisher).
Confocal fluorescence images were captured using Zeiss LSM880 confocal
microscope with a 63$\times$ objective lens. The confocal slice images
spanning the middle of each droplet were used to quantify the concentration of
PSD-95.

To calculate concentration of PSD-95 in each droplet, a standard calibration
curve was generated. Cy3-labeled PSD-95 alone in dilute solutions at different
concentrations were added into a glass-bottomed plate. The fluorescence
intensity of each confocal slice image was measured using the same settings as
that for the droplet quantifications. According to the standard curve,
fluorescence intensity of PSD-95 in each condensate can be converted into
absolute molar concentration.

RESULTS AND DISCUSSION

A mean-field theory of stoichiometric complex-driven SynGAP/PSD-95 LLPS

We are now in a position to apply the general theory developed above to the
SynGAP/PSD-95 system, with the amino acid sequence information of the SynGAP
and PSD-95 constructs (157 and 416 residues, respectively) specified above
under the “Experimental materials and method” heading. For wildtype SynGAP,
because of the extremely strong propensity to trimerize Zeng et al. (2016), we
consider, as a very good approximation, that all SynGAP molecules exist as
part of $\mathsf{S}_{3}$ trimers in solution. Accordingly, we set
$A=\mathsf{S}_{3}$, $B=\mathsf{P}$, $x=1$, and $y=2$ (Fig. 2a) in the
formalism of the last section to obtain from Eq. LABEL:eq:fpsi the free energy
for the SynGAP/PSD-95 system:

$\displaystyle f(\phi_{\mathsf{S}_{3}},\phi_{\mathsf{P}})=$
$\displaystyle\overline{\psi}_{\mathsf{S}_{3}}\ln\overline{\psi}_{\mathsf{S}_{3}}+\psi_{\mathsf{P}}\ln\overline{\psi}_{\mathsf{P}}+\frac{\overline{\psi}}{3}\ln\frac{\overline{\psi}}{3}$
(13)
$\displaystyle+(1-\phi_{\mathsf{S}_{3}}-\phi_{\mathsf{P}})\ln(1-\phi_{\mathsf{S}_{3}}-\phi_{\mathsf{P}})$
$\displaystyle+\frac{\overline{\psi}}{3}\left[\ln{\cal
K}^{0}+2\right]-\chi\overline{\psi}^{2}\;,$

where

$\displaystyle\overline{\psi}_{\mathsf{S}_{3}}=$
$\displaystyle\phi_{\mathsf{S}_{3}}-\overline{\psi}/3,$ (14a)
$\displaystyle\overline{\psi}_{\mathsf{P}}=$
$\displaystyle\phi_{\mathsf{P}_{3}}-2\overline{\psi}/3,$ (14b)
$\displaystyle\frac{\overline{\psi}}{3}{\cal K}^{0}=$
$\displaystyle\overline{\psi}_{\mathsf{S}_{3}}\overline{\psi}_{\mathsf{P}}^{2}e^{6\overline{\psi}}\;$
(14c)

follow from Eq. 11. The LLPS-driving interaction network envisioned by this
formulation is schematically illustrated in Fig. 2b.

The corresponding phase-coexistence behaviors are solved by the numerical
method described in Ref. Lin et al. (2017). In general, for a solution with
overall average protein volume fractions
$(\phi_{\mathsf{S}_{3}}^{0},\phi_{\mathsf{P}}^{0})$, the free energy without
phase separation is given by

$f_{\rm bulk}=f(\phi_{\mathsf{S}_{3}}^{0},\phi_{\mathsf{P}}^{0})\;.$ (15)

Whether the system phase separates is determined by the difference between
$f_{\rm bulk}$ and the free energy $f_{\rm sep}$ of a phase-separated system
with two coexisting phases labeled by $\alpha$ and $\beta$, viz.,

$f_{\rm
sep}=vf(\phi_{\mathsf{S}_{3}}^{\alpha},\phi_{\mathsf{P}}^{\alpha})+(1-v)f(\phi_{\mathsf{S}_{3}}^{\beta},\phi_{\mathsf{P}}^{\beta})\;,$
(16)

where $v$ is the fraction of the total system volume in phase $\alpha$. By
definition, $v$ satisfies $0<v<1$ and the volume conservation conditions

$\displaystyle
v\phi_{\mathsf{S}_{3}}^{\alpha}+(1-v)\phi_{\mathsf{S}_{3}}^{\beta}$
$\displaystyle=\phi_{\mathsf{S}_{3}}^{0}\;,$ (17a) $\displaystyle
v\phi_{\mathsf{P}}^{\alpha}+(1-v)\phi_{\mathsf{P}}^{\beta}$
$\displaystyle=\phi_{\mathsf{P}}^{0}\;,$ (17b)

which, in turn, can be used to relate volume fractions in the $\beta$ phase to
those in the $\alpha$ phase, allowing $f_{\rm sep}$ for overall volume
fractions $\phi_{\mathsf{S}_{3}}^{0}$, $\phi_{\mathsf{P}}^{0}$ to be rewritten
as a function of $v$, $\phi_{\mathsf{S}_{3}}^{\alpha}$, and
$\phi_{\mathsf{P}}^{\alpha}$:

$\displaystyle\\!\\!\\!f_{\rm
sep}(v,\phi_{\mathsf{S}_{3}}^{\alpha},\phi_{\mathsf{P}}^{\alpha})=$
$\displaystyle vf(\phi_{\mathsf{S}_{3}}^{\alpha},\phi_{\mathsf{P}}^{\alpha})$
(18)
$\displaystyle+(1-v)f\left(\frac{\phi_{\mathsf{S}_{3}}^{0}\\!-\\!v\phi_{\mathsf{S}_{3}}^{\alpha}}{1-v},\frac{\phi_{\mathsf{P}}^{0}\\!-\\!v\phi_{\mathsf{P}}^{\alpha}}{1-v}\right)\;.$

The system will phase separate if there exists at least a set of
$(v,\phi_{\mathsf{S}_{3}}^{\alpha},\phi_{\mathsf{P}}^{\alpha})$ values for
which $f_{\rm sep}<f_{\rm bulk}$. In that case, the final protein volume
fractions—i.e. the binary coexistence phase boundary—are the volume fractions
that minimize $f_{\rm sep}$. This minimization is achieved numerically by
implementing a three-variable sequential least squares programming (SLSQP)
algorithm Kraft (1988) using the scipy.optimize.minimize function in Scipy, a
Python-based numerical package for scientific computation Jones et al.
(2001–). To ensure that there is no numerical errors in our calculation, we
check the optimized SLSQP solutions against the following equations for
chemical potential balance of coexisting phases to confirm that these
conditions are indeed satisfied:

$\displaystyle f_{\mathsf{S}_{3}}^{\prime\alpha}=$ $\displaystyle
f_{\mathsf{S}_{3}}^{\prime\beta}\;,$ (19a) $\displaystyle
f_{\mathsf{P}}^{\prime\alpha}=$ $\displaystyle
f_{\mathsf{P}}^{\prime\beta}\;,$ (19b) $\displaystyle\mu_{\rm w}^{\alpha}=$
$\displaystyle\mu_{\rm w}^{\beta}\;,$ (19c)

where

$\displaystyle f^{y}\equiv$ $\displaystyle
f(\phi_{\mathsf{S}_{3}}^{y},\phi_{\mathsf{P}}^{y}),$ (20a) $\displaystyle
f^{\prime y}_{x}\equiv$ $\displaystyle\left.\frac{\partial
f(\phi_{\mathsf{S}_{3}},\phi_{\mathsf{P}})}{\partial\phi_{x}}\right|_{(\phi_{\mathsf{S}_{3}},\phi_{\mathsf{P}})=(\phi_{\mathsf{S}_{3}}^{y},\phi_{\mathsf{P}}^{y})}\;,$
(20b) $\displaystyle\mu_{\rm w}^{y}=$ $\displaystyle
f^{y}-\phi_{\mathsf{S}_{3}}^{y}f^{\prime
y}_{\mathsf{S}_{3}}-\phi_{\mathsf{P}}^{y}f^{\prime y}_{\mathsf{P}},$ (20c)

with $x\in\\{\mathsf{S}_{3},\mathsf{P}\\}$ and $y\in\\{\alpha,\beta\\}$.

The trend of phase behaviors predicted by the free energy given by Eq. 13 for
this model of stoichiometric complex-driven LLPS is illustrated by the phase
diagram in Fig. 2c, which features an L-shape phase-separated regime (bound by
the light blue solid lines and the diagonal black solid line with slope
$=-1$). Apparently, a hallmark of stoichiometric complex-driven LLPS is a
trend of converging tie lines (dark blue dashed lines in Fig. 2c) from the
dilute, protein-depleted phase boundary (large light-blue “L” in Fig. 2c) to
the condensed, protein-rich phase boundary (small light-blue “L”). When the
overall concentrations (volume fractions) of $\mathsf{S}_{3}$ and $\mathsf{P}$
are in the dilute-solution stoichiometric ratio of 1:2
($\phi_{\mathsf{S}_{3}}^{0}/\phi_{\mathsf{P}}^{0}=1/2$), the tie line has
slope $=1/2$, coinciding exactly with the black solid line in Fig. 2c with the
same slope, indicating that this 1:2 stoichiometric ratio is maintained in
both the dilute and condensed phases. When
$\phi_{\mathsf{S}_{3}}^{0}/\phi_{\mathsf{P}}^{0}\neq 1/2$, the molecular
species ($\mathsf{S}_{3}$ or $\mathsf{P}$) that is in excess of the $1/2$
ratio has no effect on phase separation because they do not generate more
$\mathsf{S}_{3}\mathsf{P}_{2}$ complexes and, by model construction, only
interactions among $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes are capable of
driving LLPS. Consequently, the tie lines connecting coexisting dilute and
condensed phases are converging toward the
$\phi_{\mathsf{S}_{3}}/\phi_{\mathsf{P}}=1/2$ line, meaning that the condensed
phase is always closer than the dilute phase to the
$\phi_{\mathsf{S}_{3}}/\phi_{\mathsf{P}}=1/2$ stoichiometric ratio if
$\phi_{\mathsf{S}_{3}}^{0}/\phi_{\mathsf{P}}^{0}\neq 1/2$ because of the
larger number of favorable interactions among $\mathsf{S}_{3}\mathsf{P}_{2}$
complexes in the condensed phase. With this ramification of a hypothetical
stoichiometric complex-driven LLPS elucidated, we now proceed to test
experimentally whether the real SynGAP/PSD-95 system indeed undergoes such a
complex coacervation process.

Experimental measurements of the PSD-95 concentrations in the condensed phase
of SynGAP/PSD-95 mixtures

Our previous study showed that SynGAP and PSD-95 could undergo phase
separation together. In the dilute phase, the two proteins form a homogenous
3:2 complex Zeng et al. (2016). However, the binding mode of SynGAP and PSD-95
in the condensed phase is unknown. To explore whether the two proteins also
form a stoichiometric 3:2 complex in condensed phase, we measured
concentrations of the two proteins in dilute phase and in condensed phase
(labeled in Fig. 3a as “S” for supernatant and “P” for pellet, respectively,
as in Fig. 4 of ref. Zeng et al. (2016), not to be confused with the notation
$\mathsf{S}$ for SynGAP and $\mathsf{P}$ for PSD-95). SynGAP and PSD-95 were
mixed at different molar ratios. Sedimentation-based assay was used to assess
their distributions in the dilute and condensed phases by SDS-PAGE with
Coomassie blue staining (Fig. 3a). The percentages of SynGAP and PSD-95
recovered in the pellet were analyzed. A phase diagram of the SynGAP/PSD-95
system was then constructed according to the data obtained from the
sedimentation-based assay. Here, the phase diagram in Fig. 3b shows that
complex coacervation of SynGAP and PSD-95 is highly sensitive to the
concentrations of the two proteins. When concentrations of the two components
were higher than certain threshold values, phase separation of the mixtures
occurred persistently. Interestingly, when fixing the concentration of one
component at a value below its threshold value, increasing the concentration
of the other component could also induce phase separation. For instance, there
was no phase separation when SynGAP was at 30 $\mu$M and PSD-95 was at 20
$\mu$M. However, when the concentration of SynGAP was increased to 120 $\mu$M
with [PSD-95] unchanged at 20 $\mu$M, phase separation of the mixture was
observed (Fig. 3b). This observation suggests that SynGAP and PSD-95
interacted with each other not only through the PDZ domain of PSD-95 and the
PBM of SynGAP, but the interactions might also involve other sites of the two
proteins. To quantitatively measure the absolute concentration of PSD-95 in
the condensed phase so as to enable the theoretical analysis below that
provides a clearer answer to this basic question, we used a confocal
fluorescence image-based method developed in our earlier study Zeng et al.
(2018a) (Fig. 3c,d). The fluorescence intensity of PSD-95 in the condensed
droplets under each condition was converted into absolute concentration of the
protein based on the standard curve of Cy3-labeled PSD-95 obtained in dilute
solutions (Fig. 3c). The derived absolute concentrations of PSD-95 under the
conditions shown in the phase diagram of the PSD-95 and SynGAP mixtures (Fig.
3b) are shown by the heat map in Fig. 3e.

Experimental concentrations of SynGAP and PSD-95 in coexisting condensed and
dilute phases

The next step in our investigation is to analyze the experimental data to
determine whether they are consistent with the predicted behaviors in Fig. 2c
for the hypothetical scenario in which LLPS is driven solely by favorable
interactions among stoichiometric $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes.
Consider an aqueous solution of $\mathsf{S}$ and $\mathsf{P}$ with initial
concentrations $[\mathsf{S}^{0}]$ and $[\mathsf{P}^{0}]$. Using a notation
similar to that in Eqs. 16–18 for this system when it has undergone phase
separation, we define $v$ as the fraction of total volume in the dilute phase
and thus $1-v$ is the fraction of total volume in the condensed phase. It
follows from the conservation of $\mathsf{S}$ and $\mathsf{P}$ that

$\displaystyle[\mathsf{S}^{0}]=$ $\displaystyle v[\mathsf{S}^{\rm
dil}]+(1-v)[\mathsf{S}^{\rm cond}]$ (21a) $\displaystyle[\mathsf{P}^{0}]=$
$\displaystyle v[\mathsf{P}^{\rm dil}]+(1-v)[\mathsf{P}^{\rm cond}]\;$ (21b)

where quantities in the dilute and condensed phases are labeled, respectively,
by the superscripts “dil” and “cond”. Note that we have defined the
concentrations in Eqs. 21a and 21b purposely to correspond to those measured
in the experiments (Fig. 3) in that they refer to the total concentrations of
$\mathsf{S}$ and $\mathsf{P}$ molecules or their concentrations in the dilute
and condensed phases irrespective of their binding status. Nonetheless, based
on prior experiments Zeng et al. (2016) as noted in the introductory
discussion above, we recognize that virtually all $\mathsf{S}$ molecules are
in the form of $\mathsf{S}_{3}$ under the present experimental conditions such
that $[\mathsf{S}^{0}]\approx 3[\mathsf{S}_{3}^{0}]$, $[\mathsf{S}^{\rm
dil}]\approx 3[\mathsf{S}^{\rm dil}_{3}]$, and $[\mathsf{S}^{\rm cond}]\approx
3[\mathsf{S}^{\rm cond}_{3}]$ for the concentrations, whereas the volume
fraction $\phi_{\mathsf{S}}\approx\phi_{\mathsf{S}_{3}}$.

In order to construct a binary phase diagram, our goal here is to determine
$[\mathsf{S}^{\rm dil}]$, $[\mathsf{S}^{\rm cond}]$, $[\mathsf{P}^{\rm dil}]$,
and $[\mathsf{P}^{\rm cond}]$ from two sets of experiments: (i) the 121
centrifugation-based assays providing the fractional numbers of protein
molecules in the condensed phase, denoted here as

$\displaystyle\gamma_{\mathsf{S}}\equiv$
$\displaystyle\frac{(1-v)[\mathsf{S}^{\rm cond}]}{[\mathsf{S}^{0}]}\;,$ (22a)
$\displaystyle\gamma_{\mathsf{P}}\equiv$
$\displaystyle\frac{(1-v)[\mathsf{P}^{\rm cond}]}{[\mathsf{P}^{0}]}\;,$ (22b)

and (ii) the 25 confocal-microscopy measurements affording the condensed phase
concentration, $[\mathsf{P}^{\rm cond}]$, of PSD-95 ($\mathsf{P}$), which may
be identify, aside from a proportionality constant, to the volume fraction of
$\mathsf{P}$:

$\phi_{\mathsf{P}}^{\rm cond}\propto[\mathsf{P}^{\rm cond}]\;.$ (23)

Since the $\gamma_{\mathsf{S}}$ and $\gamma_{\mathsf{P}}$ in Eqs. 22a and 22b
determined from centriguation measurements (two experimental data points per
condition) are by themselves insufficient to determine both $[\mathsf{S}^{\rm
dil}]$ and $[\mathsf{P}^{\rm dil}]$ as well as $v$ (three variables) in Eqs.
21a and 21b, we can only make the full determination of dilute- and condensed-
phase protein concentrations for the 25 conditions for which confocal
microscopy data are also available, with

$\displaystyle[\mathsf{S}^{0}]=$ $\displaystyle 30,60,90,120,180\;\mu{\rm
M}\;,$ $\displaystyle[\mathsf{P}^{0}]=$ $\displaystyle
20,40,60,80,120\;\mu{\rm M}\;.$

For each of these given conditions, we may first use Eq. 22b to obtain

$v=1-\gamma_{\mathsf{P}}\frac{[\mathsf{P}^{0}]}{[\mathsf{P}^{\rm cond}]}\;$
(24)

by using $\gamma_{\mathsf{P}}$ from centrifugation measurements and
$[\mathsf{P}^{\rm cond}]$ from confocal microscopy. Once $v$ is determined,
$[\mathsf{P}^{\rm dil}]$ follows from Eq. 21b as

$[\mathsf{P}^{\rm dil}]=\frac{[\mathsf{P}^{0}](1-\gamma_{\mathsf{P}})}{v}\;,$
(25)

and the SynGAP concentrations in the dilute and condensed phases follow,
respectively, from Eqs. 21a and 22a as

$\displaystyle[\mathsf{S}^{\rm dil}]=$
$\displaystyle\frac{[\mathsf{S}^{0}](1-\gamma_{\mathsf{S}})}{v}\;,$ (26a)
$\displaystyle[\mathsf{S}^{\rm cond}]=$
$\displaystyle\gamma_{\mathsf{S}}\frac{[\mathsf{S}^{0}]}{1-v}\;,$ (26b)

by using the solved value of $v$ and $\gamma_{\mathsf{S}}$ from centrifugation
measurements.

General trend of binary SynGAP/PSD-95 phase behaviors inferred from
experimental data is not consistent with the stoichiometric complex-driven
LLPS scenario

The above-described analysis of experimental data yields the
$([\mathsf{P}^{\rm dil}],[\mathsf{S}^{\rm dil}])$ and $([\mathsf{P}^{\rm
cond}],[\mathsf{S}^{\rm cond}])$ data points in Fig. 4 (orange and blue dots)
as well as the tie lines connecting their coexisting values (dashed lines in
Fig. 4). Since the protein concentrations of the initial states in the
experiments are very close to those of the dilute phases and the dilute-phase
concentrations are not measured directly, there is appreciable scatter in the
$([\mathsf{P}^{\rm dil}],[\mathsf{S}^{\rm dil}])$ and $([\mathsf{P}^{\rm
cond}],[\mathsf{S}^{\rm cond}])$ data points. This is because when initial
concentrations of $\mathsf{S}$ and $\mathsf{P}$ are low, the droplets formed
through phase separation are very small. Thickness of individual droplet is
often smaller than confocal optical section thickness ($\sim 0.9\mu$m). In
such situations, the fluorescence intensity measured by confocal microscope is
not only affected by fluorescence signal in the droplet but also interfered by
the glass of the coverslip and labeled proteins in solution. This basic
limitation precludes an accurate quantification of the uncertainties in
measured condensed-phase concentrations when overall protein concentrations
are low, making it impractical to construct a smooth phase boundary from the
plotted data. In future investigations, this difficulty can possibly be
overcome by using solutions that lead to larger droplets, in which case the
quality of the data may be evaluated by comparing signal intensities for
different Z stacks obtained by confocal microscopy. For instance, if the
thickness of droplet is larger than the confocal optical section thickness,
the fluorescence intensities of several layers spanning the middle of each
droplet should be comparable (see, e.g., Fig. 3 of ref. Zeng et al. (2018a)).

Experimental limitations notwithstanding, one unmistakably clear message from
Fig. 4a is that the tie lines diverge from the protein-depleted phases (orange
data points in Fig. 4a) to the protein-rich phases (blue data points in Fig.
4a), a trend that is opposite to the converging tie lines from the dilute,
protein-depleted phase boundary the condensed, protein-rich phase boundary in
Fig. 2c. This information is of critical importance, despite noted
experimental uncertainties in measuring concentrations, because if the
stabilizing interactions in the SynGAP/PSD-95 coacervate are only those
between $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes, the relative concentrations
of $\mathsf{S}$ and $\mathsf{P}$ in the condensed phase should not change or
should not vary much even when experimental uncertainties are taken into
account. As discussed above, the converging tie lines and the L-shape phase
boundary in Fig. 2c are direct consequences of the premise, based on the
scenario illustrated in Fig. 2b, that the only LLPS-driving favorable
interactions are those among stoichiometric $\mathsf{S}_{3}\mathsf{P}_{2}$
complexes. In that hypothetical scenario, tie lines would have negative as
well as positive slopes (positive slopes when overall $\mathsf{S}$ and
$\mathsf{P}$ concentrations are relatively low, negative slopes when either
$\mathsf{S}$ or $\mathsf{P}$ concentration is high, see Fig. 2c). In contrast,
the slopes of the experimental tie lines in Fig. 4a are all positive, spanning
an approximate range from $0.24$ to $9.1$. It stands to reason, therefore,
based on the fact that the trend inferred from experimental data in Fig. 4a is
drastically different from that in Fig. 2c, it implies that favorable
interactions among stoichiometric $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes are
not the only (or not at all) the LLPS-driving interactions in the assembly of
the SynGAP/PSD-95 coacervate.

Theoretical analysis of experimental data suggests that non-stoichiometric
auxilliary interactions are involved in the assembly of SynGAP/PSD-95
coacervates

To rationalize the SynGAP/PSD-95 phase properties in Fig. 4, it is useful to
note that the trend of diverging tie lines in Fig. 4a is similar to that
predicted by simple FH model for certain interaction strengths, such as that
reported in Fig. 6a of ref. Lin et al. (2017). Accordingly, we now apply FH
approaches to assess alternate scenarios of SynGAP/PSD-95 coacervation (Fig.
5). Because SynGAP/PSD-95 LLPS must involve interaction between $\mathsf{S}$
and $\mathsf{P}$ ($\mathsf{S}$ or $\mathsf{P}$ does not individually phase
separate), one may first consider a simple FH model with free energy in units
of $k_{\rm B}TM$ given by

$\displaystyle f_{1}(\phi_{\mathsf{S}_{3}},\phi_{\mathsf{P}})=$
$\displaystyle\;\phi_{\mathsf{S}_{3}}\ln\phi_{\mathsf{S}_{3}}+\phi_{\mathsf{P}}\ln\phi_{\mathsf{P}}$
(27)
$\displaystyle+(1-\phi_{\mathsf{S}_{3}}-\phi_{\mathsf{P}})\ln(1-\phi_{\mathsf{S}_{3}}-\phi_{\mathsf{P}})$
$\displaystyle\quad-\chi_{\mathsf{S}\mathsf{P}}\phi_{\mathsf{S}_{3}}\phi_{\mathsf{P}}\;,$

where $\chi_{\mathsf{S}\mathsf{P}}$ is the favorable energetic (FH $\chi$)
parameter characterizing $\mathsf{S}$–$\mathsf{P}$ contacts as the only LLPS-
driving interactions in this scenario (Fig. 5a). An example of the binary
phase diagrams predicted by this model is provided in Fig. 5c. In contrast to
the converging tie lines in Fig. 2c, the tie lines in Fig. 5c exhibit some
degree of divergence from the protein-depleted to the protein-rich phase
boundary, but the divergence is not as prominent as that in Fig. 4a.
Recognizing that the simple free energy function
$f_{1}(\phi_{\mathsf{S}_{3}},\phi_{\mathsf{P}})$ in Eq. 27 does not provide
any bias toward formation of $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes although
such complexes are observed experimentally in dilute solution Zeng et al.
(2016), it is reasonable to test whether incorporating favorability toward
formation of $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes in the model would
result in predicted phase diagrams that conform better with the experimental
trend. We do so by making the following modifications to the free energy
$f(\phi_{\mathsf{S}_{3}},\phi_{\mathsf{P}})$ in Eq. 13 (which already contains
favorable energy for $\mathsf{S}_{3}\mathsf{P}_{2}$ formation):

$\displaystyle\chi\overline{\psi}^{2}\;{\rm[Eq.~{}\ref{eq:final}]}$
$\displaystyle\rightarrow\chi_{\mathsf{S}\mathsf{P}}\phi_{\mathsf{P}}\phi_{\mathsf{S}_{3}}+\frac{1}{2}[\chi_{\mathsf{S}\mathsf{S}}\phi_{\mathsf{S}}^{2}+\chi_{\mathsf{P}\mathsf{P}}\phi_{\mathsf{P}}^{2}]\;,$
(28a) $\displaystyle\frac{\overline{\psi}}{3}{\cal K}^{0}$
$\displaystyle=\overline{\psi}_{\mathsf{S}_{3}}\overline{\psi}_{\mathsf{P}}^{2}e^{6\overline{\psi}}{\rm[Eq.~{}\ref{eq:2step_Kd}]}\rightarrow\overline{\psi}_{\mathsf{S}_{3}}\overline{\psi}_{\mathsf{P}}^{2}\;,$
(28b)

wherein Eq. 28a stipulates that individual $\mathsf{S}_{3}$ trimers (not
complexed with $\mathsf{P}$ in the manner of the
$\mathsf{S}_{3}\mathsf{P}_{2}$ complex) and individual $\mathsf{P}$ molecules
may participate in LLPS-driving interactions and/or that higher-order
SynGAP/PSD-95 complexes in addition to $\mathsf{S}_{3}\mathsf{P}_{2}$ are
present when protein concentrations are high and these higher-order complexes
participate also in LLPS-driving interactiions because the two scenarios are
accounted for by the same highly coarse-grained, mean-field formulation in Eq.
28a (Fig. 5b). Moreover, unlike the stoichiometric complex-driven LLPS model
free energy in Eq. 13, by dropping the $\exp(6\overline{\psi})$ factor in Eq.
14c, Eq. 28b signifies that $\mathsf{S}_{3}\mathsf{P}_{2}$ formation is not
coupled to the LLPS-driving interactions provided by the FH $\chi$ parameters
(right hand side of Eq. 28a) in the modified formulation described by Eq. 28.
The modified formulation describes a system in which
$\mathsf{S}_{3}\mathsf{P}_{2}$ complexes are formed at low protein
concentrations, whereas phase separation at higher protein concentrations is
underpinned by a multitude of different $\mathsf{S}_{3}$ and $\mathsf{P}$
interactions.

As exemplified by the two example phase diagrams in Fig. 5d and e, the trend
of tie-line divergence predicted by this modified formulation is more akin to
the experimental results in Fig. 4a than that predicted by the simple FH model
in Fig. 5c, although the differences between the predicted tie-line patterns
among Fig. 5c–e are not dramatic. Between the two models in Fig. 5d and e with
the same strongly favorable $\mathsf{S}$–$\mathsf{P}$ interactions, it is
instructive to note that the model in Fig. 5e with favorable
$\mathsf{S}$–$\mathsf{S}$ and $\mathsf{P}$–$\mathsf{P}$ interactions
(consistent with experiment, these interactions are chosen to be
insufficiently strong by themselves for $\mathsf{S}$ or $\mathsf{P}$ to phase
separate individually) exhibits more tie-line divergence and thus captures
slightly better the tie-line trend in Fig. 4a than the model in Fig. 5d with
only favorable $\mathsf{S}$–$\mathsf{P}$ interactions. In particular, the
range of tie-line slopes, $([\mathsf{S}^{\rm cond}]-[\mathsf{S}^{\rm
dil}])/([\mathsf{P}^{\rm cond}]-[\mathsf{P}^{\rm dil}])$, for the experimental
results in Fig. 4a (slope $\approx 0.24$–$9.1$ as mentioned above) are quite
similar to that for the theoretical results in Fig. 5e (slope $\approx
0.28$–$12.0$, calculated by multiplying each volume-fraction slope by a factor
of 3). This model feature suggests that auxilliary $\mathsf{S}$–$\mathsf{S}$,
$\mathsf{P}$–$\mathsf{P}$, and $\mathsf{S}$–$\mathsf{P}$ interactions in
addition to those associated with $\mathsf{S}_{3}\mathsf{P}_{2}$ assembly,
rather than auxilliary $\mathsf{S}$–$\mathsf{P}$ interactions alone, are
likely in play in the phase separation of the experimental SynGAP/PSD-95
system as well. Nonetheless, the similarity of the tie lines in Fig. 5c–e and
the robustness of their overall pattern of divergence indicate that the
experimentally observed phase properties are largely underpinned by auxilliary
favorable $\mathsf{S}$–$\mathsf{P}$ interactions.

CONCLUSIONS

It has become increasing apparent that the assembly and disassembly of
biomolecular condensates are driven by stochastic, “fuzzy” interactions
involving intrinsically disordered protein regions (see, e.g., refs.
Brangwynne et al. (2009, 2015); Nott et al. (2015); Kim et al. (2019)) as well
as structurally specific interactions, including stoichiometric binding of
folded domains in some instances (see, e.g., refs. Guo et al. (2018); Kato and
McKnight (2018); Zeng et al. (2018a); Sanders et al. (2020); Zhu et al.
(2020)). Deciphering how these interactions impart molecular recognition and
how they may act in concert to achieve function is of fundamental importance
to molecular biology Fare et al. (2021); Schmit et al. (2021), as has been
emphasized in a recent study of the critical role of specific binding of the
GIT1 and $\beta$-Pix GTPase regulatory enzymes in their complex coacervation
Zhu et al. (2020). At a smaller length scale, a similar synergy between
structurally specific interactions and stochastic, dynamic multivalent
interactions has previously been indicated in the functional binding of IDPs
in discrete, binary fuzzy complexes Borg et al. (2007); Mittag et al. (2010).
In this context, the formation of the stoichiometric SynGAP/PSD-95 complex
$\mathsf{S}_{3}\mathsf{P}_{2}$ in dilute solution Zeng et al. (2016) raises
interesting questions regarding the role of these stoichiometric complexes in
SynGAP/PSD-95 LLPS. Intuitively, a possible scenario would be that LLPS is
driven solely by favorable interactions among $\mathsf{S}_{3}\mathsf{P}_{2}$
complexes. Theoretical considerations reveal that this scenario implies a
peculiar form of binary phase diagrams with an L-shape phase boundary and
converging tie lines. However, subsequent experiments indicate that the LLPS
properties of SynGAP/PSD-95 do not conform to this hypothetical scenario.
Instead, our pattern of phase behavior combined theoretical/experimental
approach suggests strongly that auxilliary interactions beside those
responsible for the formation of $\mathsf{S}_{3}\mathsf{P}_{2}$ are also
involved in stabilizing the SynGAP/PSD-95 condensate. Our analysis has thus
paved the way for investigating the nature of these auxilliary interactions.
It appears that such LLPS-driving auxilliary interactions are readily
available for certain folded proteins, as in the cases of lysozyme and
crystallin Cinar et al. (2019). It would be instructive to compare
SynGAP/PSD-95 with these simpler systems and explore also a possible
involvement of disordered regions Rana et al. (2021) in the formation of
SynGAP/PSD-95 condensates. For instance, based on the interface between the
third PDZ and SH3-GK domains in PSD-95 Zhang et al. (2013), the contacts
leading to higher order structures among PSD-95 molecules likely involve both
hydrophobic and ionic interactions. Moreover, bioinformatics considerations
Erdös et al. (2021) suggest that a stretch of chain sequence enriched in
arginines, lysines, and glutamic acids between the coiled-coil and PBM domains
of SynGAP is likely disordered. Such IDRs are potential participants in LLPS-
driving interactions as well. In this regard, recently employed techniques for
characterizing the roles of electrostatic and non-ionic interactions in
biomolecular condensates Krainer et al. (2021); Kim et al. (2021) would be
useful. In any event, having demonstrated the capability of the relatively
simple formulation developed here to gain fundamental insights into the
mechanism of complex coacervation of SynGAP and PSD-95, we hope that our
methodology will be useful for analyzing the role of stoichiometric complexes
in the assembly of other biomolecular condensates as well.

Author Contributions
Y.-H.L., M.Z., H.S.C. designed research, Y.-H.L., H.W., B.J. performed
research, Y.-H.L., H.W. analyzed data, and Y.-H.L., H.W., M.Z., H.S.C. wrote
the article.

Acknowledgements
We thank Julie Forman-Kay for helpful discussions. We also thank three
anonymous referees, whose constructive comments have helped improve the
clarity of our presentation. This work was supported by Canadian Institutes of
Health Research grant NJT-155930 and Natural Sciences and Engineering Research
Council of Canada Discovery grant RGPIN-2018-04351 to H.S.C. and National Key
R & D Program of China grant 2019YFA0508402 and RGC of Hong Kong grant
AoE-M09-12 to M.Z. as well as computational resources provided generously by
Compute/Calcul Canada.

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Figures

Figure 1: Schematics of hypothetical multiple SynGAP/PSD-95 configurations
that are based upon interactions among stoichiometric
$\mathsf{S}_{3}\mathsf{P}_{2}$ complexes. (a) Caricatures of the SynGAP trimer
($\mathsf{S}_{3}$, left) and PSD-95 ($\mathsf{P}$, right). For
$\mathsf{S}_{3}$, the pink circle represents the SynGAP coiled-coil (CC)
trimer, the three red circles represent the PDZ binding domains (PBMs) at the
C-termini of the three $\mathsf{S}$ chains, and the three yellow chains
represent the rest of each of the three $\mathsf{S}$ sequence N-terminal to
the CC domain. For $\mathsf{P}$, the blue circle represents the three PZD
domains and the green rectangle represents the C-terminal guanylate kinase
(GK) domain. The SH3 domain sequentially situated between these two domains is
not depicted explicitly. The blue curve represents the part of $\mathsf{P}$
N-terminal to the PZD domains. Domain organizations of SynGAP and PSD-95 are
described in more detail in Figs. 1 and 2 of Ref. Zeng et al. (2016). (b)
$\mathsf{S}_{3}\mathsf{P}_{2}$ multimerization by connecting each
$\mathsf{S}_{3}\mathsf{P}_{2}$ to two other $\mathsf{S}_{3}\mathsf{P}_{2}$s
results in a linear chain of $\mathsf{S}_{3}\mathsf{P}_{2}$ complexes. This
scenario corresponds to that in Fig. 7 of Ref. Zeng et al. (2018b). As
described in this reference, the binding between an $\mathsf{S}_{3}$ with two
$\mathsf{P}$ is mediated by PDZ-PBM contacts (shown by concentric red and blue
circles); favorable interaction between two $\mathsf{S}_{3}\mathsf{P}_{2}$s is
then effectuated by binding of a PDZ/PBM of one $\mathsf{S}_{3}\mathsf{P}_{2}$
to the GK (and SH3) of another $\mathsf{S}_{3}\mathsf{P}_{2}$ (indicated by a
contact between a green rectangle with a set of concentric red and blue
circles). (c) $\mathsf{S}_{3}\mathsf{P}_{2}$ multimerization by connecting
each $\mathsf{S}_{3}\mathsf{P}_{2}$ to three other
$\mathsf{S}_{3}\mathsf{P}_{2}$s results in a network of
$\mathsf{S}_{3}\mathsf{P}_{2}$ complexes. The notation for binding
interactions is the same as that in (b).

Figure 2: Prediction of a mean-field theory that assumes SynGAP/PSD-95 complex
coacervation is driven soley by favorable interactions among stoichiometric
$\mathsf{S}_{3}\mathsf{P}_{2}$ complexes (Eq. 13). (a) Simplified schematic
representation of $\mathsf{S}_{3}$, $\mathsf{P}$ and
$\mathsf{S}_{3}\mathsf{P}_{2}$. The three $\mathsf{S}_{3}$ PBMs and their
connecting chain segments to the CC trimer (large pink circle, see Fig. 1a)
are now shown as small red circles, the PDZ3-SH3-GK domains of $\mathsf{P}$
are now shown as a blue circle encased in a green rectangle, and the yellow
and blue N-terminal chain segments in Fig. 1a are not depicted explicitly.
Each $\mathsf{S}_{3}\mathsf{P}_{2}$ is enclosed by a dashed circle to
underscore its role as a unit of phase-separation-driving interaction in the
mean-field theory. (b) Schematic picture of a hypothetical SynGAP/PSD-95
condensed phase. The formulation in Eq. 13 stipulates that while unbound S3
and P may be present in the condensed phase, phase separation is only driven
by interactions between units of S3P2 (magenta dashed lines). These inter-unit
favorable interactions may include binding of a PDZ/PBM of one
$\mathsf{S}_{3}\mathsf{P}_{2}$ to the SH3-GK of another
$\mathsf{S}_{3}\mathsf{P}_{2}$ as envisioned in Fig. 1b and c. (c) Predicted
phase diagram for this hypothetical scenario based upon Eq. 13, using ${\cal
K}^{0}=10^{-10}$ and $\chi=1.5$ as illustration. The boundary of binodal phase
separation is shown by the light blue lines. Each of the dashed dark blue
lines is a tie line indicating a pair of coexisting phases on the boundary of
the phase-separated region. The overall volume constraint of
$\phi_{\mathsf{S}_{3}}+\phi_{\mathsf{P}}\leq 1$ is marked by the solid black
line with slope $=-1$ for $\phi_{\mathsf{S}_{3}}+\phi_{\mathsf{P}}=1$, whereas
the $\phi_{\mathsf{S}_{3}}:\phi_{\mathsf{P}}=1:2$ stoichiometric ratio of the
$\mathsf{S}_{3}\mathsf{P}_{2}$ complex is indicated by the black solid line
with slope $=1/2$.

Figure 3: Experimental measurements of SynGAP/PSD-95 phase separation. (a)
Sedimentation assay showing the dilute phase (marked by “S” at the top of the
columns) and the condensed phase (marked by “P” at the top of the columns)
distribution of PSD-95 and SynGAP mixed at different concentrations of PSD-95
and a fixed concentration of SynGAP at 90 $\mu$M. (b) Phase diagram of PSD-95
and SynGAP condensates. The hollow circles in phase diagram indicate no phase
separation and solid circles represent condensed phase formation. The results
enclosed in the dashed red box correspond to those from (a). The inclined
dashed line indicate the dilute-solution 3:2 ratio for [SynGAP]:[PSD-95]. (c)
Standard curve of Cy3-labeled PSD-95 obtained from dilute solutions. (d)
Confocal images showing an indicative confocal slice of condensed droplets of
each concentration of Cy3-labeled PSD-95 mixed with unlabeled SynGAP at a
fixed concentration at 180 $\mu$M. (e) Heat map plot showing the PSD-95
concentrations in condensed phase at different combinations of PSD-95 and
SynGAP concentrations. Each number represents the PSD-95 concentration
($\mu$M) in the condensed phase. The data enclosed in the dashed red box
correspond to those from (d).

Figure 4: Experimental trend of SynGAP/PSD-95 coexisting phases. Shown data
are inferred from confocal microscopy and centrifigation measurements as
described in the text. Initial (overall) concentrations
$([\mathsf{P}^{0}],[\mathsf{S}^{0}])$ are plotted as green data points. The
phase-separated dilute-phase $([\mathsf{P}^{\rm dil}],[\mathsf{S}^{\rm dil}])$
are plotted as orange data points, whereas the condensed-phase
$([\mathsf{P}^{\rm cond}],[\mathsf{S}^{\rm cond}])$ are plotted as blue data
points. Dashed lines are tie lines connecting coexisting phases. Results are
shown in linear (a) as well as log-log (b) scales for clarity, as the green
data points are not visible in the linear plot in (a) because they are very
close to the orange data points. Note that because the dilute-phase
$([\mathsf{P}^{\rm dil}],[\mathsf{S}^{\rm dil}])$ concentrations (orange data
points) are all very close to the origin, the slopes of tie lines in the log-
log plot in (b) are all approximately equal to one. The diversity of the
actual tie-line slopes as seen in (a) is manifested by the offsets of the log-
log tie lines (different intercepts) in (b).

Figure 5: Alternate scenarios of SynGAP/PSD-95 phase separation with
auxilliary interactions beyond those underpinning the assembly of
stoichiometric $\mathsf{S}_{3}\mathsf{P}_{2}$ in dilute solution. The
schematic representations of $\mathsf{S}_{3}$ and $\mathsf{P}$ here are the
same as those in Fig. 2a. (a) Favorable LLPS-driving interactions are
envisioned to be restricted to those between $\mathsf{S}_{3}$ and $\mathsf{P}$
(red dashed lines) as modeled by the $\chi_{\mathsf{S}\mathsf{P}}$ term in Eq.
27 for a simple FH model. (b) LLPS is envisioned to be driven also by
favorable interactions among $\mathsf{S}_{3}$ (blue dashed lines) and among
$\mathsf{P}$ (green dashed lines) as modeled by the
$\chi_{\mathsf{S}\mathsf{S}}$ and $\chi_{\mathsf{P}\mathsf{P}}$ terms in Eq.
28a. The bias afforded by this model toward formation of
$\mathsf{S}_{3}\mathsf{P}_{2}$ complexes in the dilute phase is not exhibited
in this schematic depiction of the condensed phase. (c)–(e) Mean-field FH
theory predictions for SynGAP/PSD-95 phase behaviors in the alternate scenario
in (a) [(c) and (d)] and the alternate scenario in (b) [(e)]. The black solid
lines with slopes $=1/2$ and $-1$, the dashed tie lines, and the light blue
phase boundaries in the three phase digrams carry the same meanings as those
in Fig. 2c. The FH parameters are: (c) $\chi_{\mathsf{S}\mathsf{P}}=10.0$ in
Eq. 27, (d) ${\cal K}^{0}=10^{-10}$, $\chi_{\mathsf{S}\mathsf{P}}=10.0$ and
$\chi_{\mathsf{S}\mathsf{S}}=\chi_{\mathsf{P}\mathsf{P}}=0$ in Eq. 28, and (e)
${\cal K}^{0}=10^{-10}$, $\chi_{\mathsf{S}\mathsf{P}}=4.0$ and
$\chi_{\mathsf{S}\mathsf{S}}=\chi_{\mathsf{P}\mathsf{P}}=3.5$ in Eq. 28.