Biomolecular condensates can form through the liquid–liquid phase separation (LLPS) of proteins and RNAs in cells. However, other states of organization, including mesostructured network microstructures and physical gels, have been observed, the physical mechanism of which are not well understood. We use the Polymer Reference Interaction Site Model liquid state integral equation theory to study the equilibrium behavior of (generally aperiodic in sequence) biomolecular condensates based on a minimal sticker–spacer associating polymer model. The role of polymer packing fraction, sequence, and the strength and range of intermolecular interactions on macromolecular scale spatial organization and phase behavior is studied for typical sticker–spacer sequences. In addition to the prediction of conventional LLPS, a sequence-dependent strongly fluctuating polymeric microemulsion homogeneous state is predicted at high enough concentrations beyond the so-called Lifshitz-like point, which we suggest can be relevant to the dense phase of microstructured biomolecular condensates. New connections between local clustering and the formation of mesoscopic microdomains, the influence of attraction range, compressibility, and the role of spatial correlations across scales, are established. Our results are also germane to understanding the polymer physics of dense solutions of nonperiodic and unique sequence synthetic copolymers and provide a foundation to create new theories for how polymer diffusion and viscosity are modified in globally isotropic and homogeneous dense polymeric microemulsions.

1.
A. A.
Hyman
,
C. A.
Weber
, and
F.
Jülicher
, “
Liquid-liquid phase separation in biology
,”
Annu. Rev. Cell Dev. Biol.
30
(
1
),
39
58
(
2014
).
2.
S.
Alberti
,
A.
Gladfelter
, and
T.
Mittag
, “
Considerations and challenges in studying liquid-liquid phase separation and biomolecular condensates
,”
Cell
176
(
3
),
419
434
(
2019
).
3.
S.
Ray
,
N.
Singh
,
R.
Kumar
,
K.
Patel
,
S.
Pandey
,
D.
Datta
,
J.
Mahato
,
R.
Panigrahi
,
A.
Navalkar
,
S.
Mehra
,
L.
Gadhe
,
D.
Chatterjee
,
A. S.
Sawner
,
S.
Maiti
,
S.
Bhatia
,
J. A.
Gerez
,
A.
Chowdhury
,
A.
Kumar
,
R.
Padinhateeri
,
R.
Riek
,
G.
Krishnamoorthy
, and
S. K.
Maji
, “
α-synuclein aggregation nucleates through liquid–liquid phase separation
,”
Nat. Chem.
12
(
8
),
705
716
(
2020
).
4.
P.
Li
,
S.
Banjade
,
H.-C.
Cheng
,
S.
Kim
,
B.
Chen
,
L.
Guo
,
M.
Llaguno
,
J. V.
Hollingsworth
,
D. S.
King
,
S. F.
Banani
,
P. S.
Russo
,
Q.-X.
Jiang
,
B. T.
Nixon
, and
M. K.
Rosen
, “
Phase transitions in the assembly of multivalent signalling proteins
,”
Nature
483
(
7389
),
336
340
(
2012
).
5.
Y.
Lin
,
D. S. W.
Protter
,
M. K.
Rosen
, and
R.
Parker
, “
Formation and maturation of phase-separated liquid droplets by RNA-binding proteins
,”
Mol. Cell
60
(
2
),
208
219
(
2015
).
6.
B. A.
Gibson
,
L. K.
Doolittle
,
M. W. G.
Schneider
,
L. E.
Jensen
,
N.
Gamarra
,
L.
Henry
,
D. W.
Gerlich
,
S.
Redding
, and
M. K.
Rosen
, “
Organization of chromatin by intrinsic and regulated phase separation
,”
Cell
179
(
2
),
470
484.e21
(
2019
).
7.
C. P.
Brangwynne
,
C. R.
Eckmann
,
D. S.
Courson
,
A.
Rybarska
,
C.
Hoege
,
J.
Gharakhani
,
F.
Jülicher
, and
A. A.
Hyman
, “
Germline P granules are liquid droplets that localize by controlled dissolution/condensation
,”
Science
324
(
5935
),
1729
1732
(
2009
).
8.
Y.
Shin
and
C. P.
Brangwynne
, “
Liquid phase condensation in cell physiology and disease
,”
Science
357
(
6357
),
eaaf4382
(
2017
).
9.
M.-T.
Wei
,
S.
Elbaum-Garfinkle
,
A. S.
Holehouse
,
C. C.-H.
Chen
,
M.
Feric
,
C. B.
Arnold
,
R. D.
Priestley
,
R. V.
Pappu
, and
C. P.
Brangwynne
, “
Phase behaviour of disordered proteins underlying low density and high permeability of liquid organelles
,”
Nat. Chem.
9
(
11
),
1118
1125
(
2017
).
10.
C. P.
Brangwynne
,
P.
Tompa
, and
R. V.
Pappu
, “
Polymer physics of intracellular phase transitions
,”
Nat. Phys.
11
(
11
),
899
904
(
2015
).
11.
L.
Jawerth
,
E.
Fischer-Friedrich
,
S.
Saha
,
J.
Wang
,
T.
Franzmann
,
X.
Zhang
,
J.
Sachweh
,
M.
Ruer
,
M.
Ijavi
,
S.
Saha
,
J.
Mahamid
,
A. A.
Hyman
, and
F.
Jülicher
, “
Protein condensates as aging Maxwell fluids
,”
Science
370
(
6522
),
1317
1323
(
2020
).
12.
T. M.
Franzmann
,
M.
Jahnel
,
A.
Pozniakovsky
,
J.
Mahamid
,
A. S.
Holehouse
,
E.
Nüske
,
D.
Richter
,
W.
Baumeister
,
S. W.
Grill
,
R. V.
Pappu
,
A. A.
Hyman
, and
S.
Alberti
, “
Phase separation of a yeast prion protein promotes cellular fitness
,”
Science
359
(
6371
),
eaao5654
(
2018
).
13.
A.
Patel
,
H. O.
Lee
,
L.
Jawerth
,
S.
Maharana
,
M.
Jahnel
,
M. Y.
Hein
,
S.
Stoynov
,
J.
Mahamid
,
S.
Saha
,
T. M.
Franzmann
,
A.
Pozniakovski
,
I.
Poser
,
N.
Maghelli
,
L. A.
Royer
,
M.
Weigert
,
E. W.
Myers
,
S.
Grill
,
D.
Drechsel
,
A. A.
Hyman
, and
S.
Alberti
, “
A liquid-to-solid phase transition of the ALS protein FUS accelerated by disease mutation
,”
Cell
162
(
5
),
1066
1077
(
2015
).
14.
S.
Ranganathan
and
E.
Shakhnovich
, “
The physics of liquid-to-solid transitions in multi-domain protein condensates
,”
Biophys. J.
121
(
14
),
2751
2766
(
2022
).
15.
G. L.
Dignon
,
R. B.
Best
, and
J.
Mittal
, “
Biomolecular phase separation: From molecular driving forces to macroscopic properties
,”
Annu. Rev. Phys. Chem.
71
(
1
),
53
75
(
2020
).
16.
J.-M.
Choi
,
F.
Dar
, and
R. V.
Pappu
, “
LASSI: A lattice model for simulating phase transitions of multivalent proteins
,”
PLoS Comput. Biol.
15
(
10
),
e1007028
(
2019
).
17.
M.
Rubinstein
and
A. V.
Dobrynin
, “
Solutions of associative polymers
,”
Trends Polym. Sci.
5
(
6
),
181
186
(
1997
).
18.
A. N.
Semenov
and
M.
Rubinstein
, “
Thermoreversible gelation in solutions of associative polymers. 1. Statics
,”
Macromolecules
31
(
4
),
1373
1385
(
1998
).
19.
M.
Kato
,
T. W.
Han
,
S.
Xie
,
K.
Shi
,
X.
Du
,
L. C.
Wu
,
H.
Mirzaei
,
E. J.
Goldsmith
,
J.
Longgood
, and
J.
Pei
, “
Cell-free formation of RNA granules: Low complexity sequence domains form dynamic fibers within hydrogels
,”
Cell
149
(
4
),
753
767
(
2012
).
20.
T. S.
Harmon
,
A. S.
Holehouse
,
M. K.
Rosen
, and
R. V.
Pappu
, “
Intrinsically disordered linkers determine the interplay between phase separation and gelation in multivalent proteins
,”
Elife
6
,
e30294
(
2017
).
21.
T.
Mittag
and
R. V.
Pappu
, “
A conceptual framework for understanding phase separation and addressing open questions and challenges
,”
Mol. Cell
82
,
2201
(
2022
).
22.
S.-F.
Li
and
M.
Muthukumar
, “
Theory of microphase separation in concentrated solutions of sequence-specific charged heteropolymers
,”
Macromolecules
55
(
13
),
5535
5549
(
2022
).
23.
S.
Ranganathan
and
E. I.
Shakhnovich
, “
Dynamic metastable long-living droplets formed by sticker-spacer proteins
,”
Elife
9
,
e56159
(
2020
).
24.
S.
Das
and
M.
Muthukumar
, “
Microstructural organization in α-synuclein solutions
,”
Macromolecules
55
(
11
),
4228
4236
(
2022
).
25.
M.
Kar
,
F.
Dar
,
T. J.
Welsh
,
L. T.
Vogel
,
R.
Kühnemuth
,
A.
Majumdar
,
G.
Krainer
,
T. M.
Franzmann
,
S.
Alberti
,
C. A. M.
Seidel
,
T. P. J.
Knowles
,
A. A.
Hyman
, and
R. V.
Pappu
, “
Phase-separating RNA-binding proteins form heterogeneous distributions of clusters in subsaturated solutions
,”
Proc. Natl. Acad. Sci. U. S. A.
119
(
28
),
e2202222119
(
2022
).
26.
L.
Leibler
, “
Theory of microphase separation in block copolymers
,”
Macromolecules
13
(
6
),
1602
1617
(
1980
).
27.
F. S.
Bates
and
G. H.
Fredrickson
, “
Block copolymer thermodynamics: Theory and experiment
,”
Annu. Rev. Phys. Chem.
41
(
1
),
525
557
(
1990
).
28.
D.
Schwahn
,
K.
Mortensen
,
H.
Frielinghaus
,
K.
Almdal
, and
L.
Kielhorn
, “
Thermal composition fluctuations near the isotropic Lifshitz critical point in a ternary mixture of a homopolymer blend and diblock copolymer
,”
J. Chem. Phys.
112
(
12
),
5454
5472
(
2000
).
29.
D.
Schwahn
,
K.
Mortensen
,
H.
Frielinghaus
, and
K.
Almdal
, “
Crossover from 3D Ising to isotropic Lifshitz critical behavior in a mixture of a homopolymer blend and diblock copolymer
,”
Phys. Rev. Lett.
82
(
25
),
5056
(
1999
).
30.
F. S.
Bates
,
W. W.
Maurer
,
P. M.
Lipic
,
M. A.
Hillmyer
,
K.
Almdal
,
K.
Mortensen
,
G. H.
Fredrickson
, and
T. P.
Lodge
, “
Polymeric bicontinuous microemulsions
,”
Phys. Rev. Lett.
79
(
5
),
849
(
1997
).
31.
K.
Mortensen
,
D.
Schwahn
,
H.
Frielinghaus
, and
K.
Almdal
, “
Ternary mixture of a homopolymer blend and diblock copolymer studied near the Lifshitz composition by small-angle neutron scattering
,”
J. Appl. Crystallogr.
33
(
3
),
686
689
(
2000
).
32.
T. L.
Morkved
,
P.
Stepanek
,
K.
Krishnan
,
F. S.
Bates
, and
T. P.
Lodge
, “
Static and dynamic scattering from ternary polymer blends: Bicontinuous microemulsions, Lifshitz lines, and amphiphilicity
,”
J. Chem. Phys.
114
(
16
),
7247
7259
(
2001
).
33.
Y.
Kriksin
and
I.
Erukhimovich
, “
Macrophase versus microphase separation in solutions of block copolymers: Lifshitz line in the energetic parameter’s space
,”
Macromol. Theory Simul.
29
(
6
),
2000044
(
2020
).
34.
D.
Düchs
,
V.
Ganesan
,
G. H.
Fredrickson
, and
F.
Schmid
, “
Fluctuation effects in ternary AB + A + B polymeric emulsions
,”
Macromolecules
36
(
24
),
9237
9248
(
2003
).
35.
K. S.
Schweizer
and
J. G.
Curro
, “
Integral-equation theory of the structure of polymer melts
,”
Phys. Rev. Lett.
58
(
3
),
246
(
1987
).
36.
K.
Schweizer
and
J.
Curro
, “
PRISM theory of the structure, thermodynamics, and phase transitions of polymer liquids and alloys
,” in
Atomistic Modeling of Physical Properties
(
Springer
,
2006
), pp.
319
377
.
37.
M.
Guenza
and
K. S.
Schweizer
, “
Fluctuations effects in diblock copolymer fluids: Comparison of theories and experiment
,”
J. Chem. Phys.
106
(
17
),
7391
7410
(
1997
).
38.
M.
Guenza
and
K. S.
Schweizer
, “
Local and microdomain concentration fluctuation effects in block copolymer solutions
,”
Macromolecules
30
(
14
),
4205
4219
(
1997
).
39.
K. S.
Schweizer
and
J. G.
Curro
,
Advances in Chemical Physics
(
John Wiley & Sons, Ltd
,
1997
), pp.
1
142
.
40.
E. F.
David
and
K. S.
Schweizer
, “
Integral equation theory of block copolymer liquids. II. Numerical results for finite hard‐core diameter chains
,”
J. Chem. Phys.
100
(
10
),
7784
7795
(
1994
).
41.
E. F.
David
and
K. S.
Schweizer
, “
Liquid state theory of thermally driven segregation of conformationally asymmetric diblock copolymer melts
,”
Macromolecules
30
(
17
),
5118
5132
(
1997
).
42.
K. A.
Kolbet
and
K. S.
Schweizer
, “
Microdomain scale organization and scattering patterns of associating polymer melts
,”
Macromolecules
33
(
4
),
1425
1442
(
2000
).
43.
K. A.
Kolbet
and
K. S.
Schweizer
, “
Real space structure of associating polymer melts
,”
Macromolecules
33
(
4
),
1443
1458
(
2000
).
44.
D.
Banerjee
and
K. S.
Schweizer
, “
Controlling effective interactions and spatial dispersion of nanoparticles in multiblock copolymer melts
,”
J. Polym. Sci., Part B: Polym. Phys.
53
(
16
),
1098
1111
(
2015
).
45.
I.
Lyubimov
,
D. J.
Beltran-Villegas
, and
A.
Jayaraman
, “
PRISM theory study of amphiphilic block copolymer solutions with varying copolymer sequence and composition
,”
Macromolecules
50
(
18
),
7419
7431
(
2017
).
46.
I.
Lyubimov
,
M. G.
Wessels
, and
A.
Jayaraman
, “
Molecular dynamics simulation and PRISM theory study of assembly in solutions of amphiphilic bottlebrush block copolymers
,”
Macromolecules
51
(
19
),
7586
7599
(
2018
).
47.
A.
Kulshreshtha
and
A.
Jayaraman
, “
Phase behavior and morphology of blends containing associating polymers: Insights from liquid-state theory and molecular simulations
,”
Macromolecules
55
(
20
),
9297
9311
(
2022
).
48.
J. C.
Shillcock
,
M.
Brochut
,
E.
Chénais
, and
J. H.
Ipsen
, “
Phase behaviour and structure of a model biomolecular condensate
,”
Soft Matter
16
(
27
),
6413
6423
(
2020
).
49.
J. C.
Shillcock
,
C.
Lagisquet
,
J.
Alexandre
,
L.
Vuillon
, and
J. H.
Ipsen
, “
Model biomolecular condensates have heterogeneous structure quantitatively dependent on the interaction profile of their constituent macromolecules
,”
Soft Matter
18
(
35
),
6674
6693
(
2022
).
50.
A.
Statt
,
H.
Casademunt
,
C. P.
Brangwynne
, and
A. Z.
Panagiotopoulos
, “
Model for disordered proteins with strongly sequence-dependent liquid phase behavior
,”
J. Chem. Phys.
152
(
7
),
075101
(
2020
).
51.
U.
Rana
,
C. P.
Brangwynne
, and
A. Z.
Panagiotopoulos
, “
Phase separation vs aggregation behavior for model disordered proteins
,”
J. Chem. Phys.
155
(
12
),
125101
(
2021
).
52.
K. A.
Burke
,
A. M.
Janke
,
C. L.
Rhine
, and
N. L.
Fawzi
, “
Residue-by-residue view of in vitro FUS granules that bind the C-terminal domain of RNA polymerase II
,”
Mol. Cell
60
(
2
),
231
241
(
2015
).
53.
A. C.
Murthy
,
W. S.
Tang
,
N.
Jovic
,
A. M.
Janke
,
D. H.
Seo
,
T. M.
Perdikari
,
J.
Mittal
, and
N. L.
Fawzi
, “
Molecular interactions contributing to FUS SYGQ LC-RGG phase separation and co-partitioning with RNA polymerase II heptads
,”
Nat. Struct. Mol. Biol.
28
(
11
),
923
935
(
2021
).
54.
Theory of Simple Liquids
,
4th ed.
, edited by
J.-P.
Hansen
and
I. R.
McDonald
(
Academic Press
,
Oxford
,
2013
), p.
i
.
55.
C.
Singh
and
K. S.
Schweizer
, “
Correlation effects and entropy‐driven phase separation in athermal polymer blends
,”
J. Chem. Phys.
103
(
13
),
5814
5832
(
1995
).
56.
T. B.
Martin
,
T. E. I.
Gartner
,
R. L.
Jones
,
C. R.
Snyder
, and
A.
Jayaraman
, “
pyPRISM: A computational tool for liquid-state theory calculations of macromolecular materials
,”
Macromolecules
51
(
8
),
2906
2922
(
2018
).
57.
Y.
Zhou
and
K. S.
Schweizer
, “
PRISM theory of local structure and phase behavior of dense polymer nanocomposites: Improved closure approximation and comparison with simulation
,”
Macromolecules
53
(
22
),
9962
9972
(
2020
).
58.
L. M.
Hall
and
K. S.
Schweizer
, “
Many body effects on the phase separation and structure of dense polymer-particle melts
,”
J. Chem. Phys.
128
(
23
),
234901
(
2008
).
59.
E. F.
David
and
K. S.
Schweizer
, “
Integral equation theory of block copolymer liquids. I. General formalism and analytic predictions for symmetric copolymers
,”
J. Chem. Phys.
100
(
10
),
7767
7783
(
1994
).
60.
Y.-H.
Lin
,
J. D.
Forman-Kay
, and
H. S.
Chan
, “
Sequence-specific polyampholyte phase separation in membraneless organelles
,”
Phys. Rev. Lett.
117
(
17
),
178101
(
2016
).
61.
Y.-H.
Lin
,
J.
Song
,
J. D.
Forman-Kay
, and
H. S.
Chan
, “
Random-phase-approximation theory for sequence-dependent, biologically functional liquid-liquid phase separation of intrinsically disordered proteins
,”
J. Mol. Liq.
228
,
176
193
(
2017
).
62.
J.
McCarty
,
K. T.
Delaney
,
S. P. O.
Danielsen
,
G. H.
Fredrickson
, and
J.-E.
Shea
, “
Complete phase diagram for liquid–liquid phase separation of intrinsically disordered proteins
,”
J. Phys. Chem. Lett.
10
(
8
),
1644
1652
(
2019
).
63.
M.
Tripathy
and
K. S.
Schweizer
, “
Theoretical study of the structure and assembly of Janus rods
,”
J. Phys. Chem. B
117
(
1
),
373
384
(
2013
).
64.
A. P.
Thompson
,
H. M.
Aktulga
,
R.
Berger
,
D. S.
Bolintineanu
,
W. M.
Brown
,
P. S.
Crozier
,
P. J.
in’t Veld
,
A.
Kohlmeyer
,
S. G.
Moore
,
T. D.
Nguyen
,
R.
Shan
,
M. J.
Stevens
,
J.
Tranchida
,
C.
Trott
, and
S. J.
Plimpton
, “
LAMMPS—A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales
,”
Comput. Phys. Commun.
271
,
108171
(
2022
).
65.
D.
Chandler
,
J. D.
Weeks
, and
H. C.
Andersen
, “
Van der Waals picture of liquids, solids, and phase transformations
,”
Science
220
(
4599
),
787
794
(
1983
).
66.
G. L.
Dignon
,
W.
Zheng
, and
J.
Mittal
, “
Simulation methods for liquid–liquid phase separation of disordered proteins
,”
Curr. Opin. Chem. Eng.
23
,
92
98
(
2019
).
67.
G. L.
Dignon
,
W.
Zheng
,
Y. C.
Kim
, and
J.
Mittal
, “
Temperature-controlled liquid–liquid phase separation of disordered proteins
,”
ACS Cent. Sci.
5
(
5
),
821
830
(
2019
).
68.
A. C.
Murthy
,
G. L.
Dignon
,
Y.
Kan
,
G. H.
Zerze
,
S. H.
Parekh
,
J.
Mittal
, and
N. L.
Fawzi
, “
Molecular interactions underlying liquid–liquid phase separation of the FUS low-complexity domain
,”
Nat. Struct. Mol. Biol.
26
(
7
),
637
648
(
2019
).
69.
J.
Ahlers
,
E. M.
Adams
,
V.
Bader
,
S.
Pezzotti
,
K. F.
Winklhofer
,
J.
Tatzelt
, and
M.
Havenith
, “
The key role of solvent in condensation: Mapping water in liquid-liquid phase-separated FUS
,”
Biophys. J.
120
(
7
),
1266
1275
(
2021
).
70.
M. H. J.
Hagen
and
D.
Frenkel
, “
Determination of phase diagrams for the hard‐core attractive Yukawa system
,”
J. Chem. Phys.
101
(
5
),
4093
4097
(
1994
).
71.
P. R.
Ten Wolde
and
D.
Frenkel
, “
Enhancement of protein crystal nucleation by critical density fluctuations
,”
Science
277
(
5334
),
1975
1978
(
1997
).
72.
D. F.
Rosenbaum
and
C. F.
Zukoski
, “
Protein interactions and crystallization
,”
J. Cryst. Growth
169
(
4
),
752
758
(
1996
).
73.
E.
Schöll-Paschinger
and
G.
Kahl
, “
Accurate determination of the phase diagram of model fullerenes
,”
Europhys. Lett.
63
(
4
),
538
(
2003
).
74.
N.
Gnan
,
F.
Sciortino
, and
E.
Zaccarelli
, in
Protein Self-Assembly: Methods and Protocols
, edited by
J. J.
McManus
(
Springer
,
New York
,
2019
), pp.
187
208
.
75.
R.
Nagarajan
and
K.
Ganesh
, “
Block copolymer self-assembly in selective solvents: Theory of solubilization in spherical micelles
,”
Macromolecules
22
(
11
),
4312
4325
(
1989
).
76.
R.
Nagarajan
and
K.
Ganesh
, “
Block copolymer self‐assembly in selective solvents: Spherical micelles with segregated cores
,”
J. Chem. Phys.
90
(
10
),
5843
5856
(
1989
).
77.
L.
Sawle
and
K.
Ghosh
, “
A theoretical method to compute sequence dependent configurational properties in charged polymers and proteins
,”
J. Chem. Phys.
143
(
8
),
085101
(
2015
).
78.
S.
Rekhi
,
D.
Sundaravadivelu Devarajan
,
M. P.
Howard
,
Y. C.
Kim
,
A.
Nikoubashman
, and
J.
Mittal
, “
Role of strong localized vs weak distributed interactions in disordered protein phase separation
,”
J. Phys. Chem. B
127
(
17
),
3829
3838
(
2023
).
79.
S. H.
Klass
,
M. J.
Smith
,
T. A.
Fiala
,
J. P.
Lee
,
A. O.
Omole
,
B.-G.
Han
,
K. H.
Downing
,
S.
Kumar
, and
M. B.
Francis
, “
Self-assembling micelles based on an intrinsically disordered protein domain
,”
J. Am. Chem. Soc.
141
(
10
),
4291
4299
(
2019
).
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