July 25, 2024

Differential Effects of Sequence-Local versus Nonlocal

Charge Patterns on Phase Separation and

Conformational Dimensions of Polyampholytes as

Model Intrinsically Disordered Proteins

Tanmoy PAL,1,† Jonas WESSÉN,1,† Suman DAS,1,2,† and Hue Sun CHAN1,∗

1Department of Biochemistry, University of Toronto, Toronto, Ontario M5S 1A8,
Canada

2Department of Chemistry, Gandhi Institute of Technology and Management,
Visakhapatnam, Andhra Pradesh 530045, India

To appear in "J. Phys. Chem. Lett."

†Contributed equally.

∗Correspondence information:
† Hue Sun CHAN. E-mail<EMAIL_ADDRESS>
† Tel: (416)978-2697; Fax: (416)978-8548
† Department of Biochemistry, University of Toronto, Medical Sciences Building
– 5th Fl.,
† 1 King’s College Circle, Toronto, Ontario M5S 1A8, Canada.

Abstract

Conformational properties of intrinsically disordered proteins (IDPs) are
governed by a sequence-ensemble relationship. To differentiate the impact of
sequence-local versus sequence-nonlocal features of an IDP’s charge pattern on
its conformational dimensions and its phase-separation propensity, the charge
“blockiness” $\kappa$ and the nonlocality-weighted sequence charge decoration
(SCD) parameters are compared for their correlations with isolated-chain radii
of gyration ($R_{\rm g}$s) and upper critical solution temperatures (UCSTs) of
polyampholytes modeled by random phase approximation, field-theoretic
simulation, and coarse-grained molecular dynamics. SCD is superior to $\kappa$
in predicting $R_{\rm g}$ because SCD accounts for effects of contact order,
i.e., nonlocality, on dimensions of isolated chains. In contrast, $\kappa$ and
SCD are comparably good, though nonideal, predictors of UCST because
frequencies of interchain contacts in the multiple-chain condensed phase are
less sensitive to sequence positions than frequencies of intrachain contacts
of an isolated chain, as reflected by $\kappa$ correlating better with
condensed-phase interaction energy than SCD.

Extensive recent research has elucidated myriad properties of biomolecular
condensates and their essential roles in diverse biological functions
(reviewed, e.g., in refs. (1; 2)). These membraneless compartments are
underpinned to a significant degree by liquid-liquid phase separation (LLPS)
of intrinsically disordered proteins (IDPs), though more complex thermodynamic
processes such as gelation/percolation and dynamic mechanisms—involving not
only IDPs but folded protein domains as well as nucleic acids—also contribute
prominently.PappuChemRev ; Mingjie2023 ; HXZhouRev2024  The propensity for an
IDP to undergo LLPS is dependent upon its sequence of amino acids.Nott15 ;
linPRL ; TanjaRohitNatChem2022  Many IDPs are enriched in charged and aromatic
residues. Accordingly, beside $\pi$-related interactions,robert ; Chen2015 ;
bauer2022 ; koby2023  electrostatics is an important driving force for many
aspects of IDP properties, as exemplified by its notable roles in IDP
conformational dimensions,rohit2013 ; kings2015  the stability of biomolecular
condensates,Nott15 ; knowles2022  and a condensate’s capability to selectively
recruit IDPs with different sequence charge patterns.Sabari2023

An IDP can exist and function as essentially isolated chain molecules and/or
collectively in a multiple-chain condensate, or certain intermediate
oligomeric configurations in between. With this in mind, analyzing the
relationship between the behaviors of isolated and condensed IDPs is
instrumental not only for inferring condensed-phase properties from those of
computationally and experimentally simpler isolated-chain systems. More
fundamentally, it offers insights into how sequence-encoded IDP properties are
modulated by IDP concentration. For homopolymers with short-spatial-range
interactions, it has been known since the 1950s via the Flory-Huggins (FH)
theory that the radius of gyration $R_{\rm g}$ of an isolated polymer
anticorrelates with the polymer’s LLPS propensity. This is because $R_{\rm g}$
decreases monotonically with the FH $\chi(T)$ interaction parameterFlory1951
at a given absolute temperature $T$ and the LLPS critical temperatureFlory1952
$T_{\rm cr}^{\rm FH}$ increases with $\chi(T)$ when the interactions are
purely enthalpic, i.e., $\chi(T)\sim 1/T$ and thus $T_{\rm cr}^{\rm
FH}=T\chi(T)/\chi_{\rm cr}$ where $\chi_{\rm cr}=(1+1/\sqrt{N_{\rm p}})^{2}/2$
and $N_{\rm p}$ is polymer chain length,KnowlesJPCL  as is verified by recent
simulations and more sophisticated polymer theories.Panag1998 ; WangWang2014
Inspired by recent interest in sequence-specific IDP LLPS, similar
relationships for heteropolymers are predicted between LLPS propensity and
isolated-chain $R_{\rm g}$ (ref. (22)), coil-globule transition,BestPNAS2018
and two-chain association in a dilute solution.Alan  Likewise, dilute-phase
$R_{\rm g}$ and demixing temperature was seen to correlate experimentally for
variants of the P domain of core stress-granule marker polyA-binding protein
undergoing heat-induced LLPS.Riback2017

The predicted dilute/condensed-phase correlations for heteropolymers are
significant but not perfect,lin2017 ; BestPNAS2018  as underscored by a recent
extensive simulation study of the human proteome.KrestenbioRxiv2024  The
imperfection is instructive about the physical chemistry of concentration-
dependent IDP interactions. We focus here on electrostatics as a first step.
Building on the substantial recent works on polyampholyte configurations and
LLPS,rohit2013 ; kings2015 ; lin2017 ; joanJPCL2019 ; joanPNAS ; Wessen2021 ;
Pal2021  we attend to a hitherto less explored consequence of sequence
specificity, namely how the impact of sequence-local patterns involving
charges proximate to one another differs from that of sequence-nonlocal
patterns encompassing charges far apart along the chain. The differing effects
of local versus nonlocal interactions has long been recognized in globular
proteins. These include folding stabilityDill1990  and the effects of
intrachain contact orderChanDill1990  on conformational ensembles as well as
folding kinetics.Dilletal1993 ; Plaxco1998 ; Chan98 ; AnnuRevPhysChem2011  But
much about the corresponding effects for IDPs remains to be elucidated. To
make progress, here we contrast the predictive powers of two common sequence
charge parameters, namely the “blockiness” measure $\kappa$ (ref. (13)) and
“sequence charge decoration” (SCD, ref. (14)). For a $N_{\rm p}$-bead
polyampholyte with charge sequence $\sigma_{\alpha}$ ($\alpha=1,2,\dots,N_{\rm
p}$), $\kappa$ captures primarily sequence-local aspects of the charge pattern
as its terms account for charge asymmetry only in blocks of $g=$ $5$ or $6$
consecutive beads relative to the overall charge asymmetry of the
polyampholyte in the form of $\kappa\sim$ $\sum_{\beta=1}^{N_{\rm
p}-g+1}[(\sum_{\alpha=\beta}^{\beta+g-1}\sigma_{\alpha})^{2}/(\sum_{\alpha=\beta}^{\beta+g-1}|\sigma_{\alpha}|)$
$-$ $(\sum_{\alpha=1}^{N_{\rm p}}\sigma_{\alpha})^{2}/(\sum_{\alpha=1}^{N_{\rm
p}}|\sigma_{\alpha}|)]^{2}$ (refs. (13; 37)). In contrast, ${\rm
SCD}\equiv\sum_{\alpha=2}^{N_{\rm
p}}\sum_{\beta=1}^{\alpha-1}\sigma_{\alpha}\sigma_{\beta}\sqrt{\alpha-\beta}/N_{\rm
p}$ takes into account nonlocal sequence charge pattern as it accords higher
weights for sequence-nonlocal pairs of charges.kings2015

Following several seminal studies,rohit2013 ; kings2015 ; lin2017 ;
joanJPCL2019  we consider $N_{\rm p}=50$ overall-neutral polyampholytes with
$\sigma_{\alpha}=\pm 1$ (in units of the protonic charge) and
$\sum_{\alpha=1}^{N_{\rm p}}\sigma_{\alpha}=0$. Substantiating a preliminary
analysis,suman2  we rigorously determine the joint distribution $P({\rm
SCD},\kappa)$ among all such sequences by first estimating the populated
region in the $({\rm SCD},\kappa)$-plane via a diversity-enhanced genetic
algorithmCamargoMolinaMandal2018  and then obtaining $P({\rm SCD},\kappa)$ in
the identified region using a Wang-Landau approachWangLandau2001  (all SCD
$<0$;Alan  see text and Fig. S1a of Supporting Information for details). The
resulting heat map for $P({\rm SCD},\kappa)$ in Fig. 1a exhibits a moderate
correlation between $-{\rm SCD}$ and $\kappa$ (Pearson correlation coefficient
$r=0.684$).

Among these 50mers, we select for further analysis 26 sequences with a broad
coverage in Fig. 1a and are illustrative of sequence variations of interest
(Fig. 1b). Beside the 12 “sv” sequences as examples of the 30 original sv
sequencesrohit2013  are the 4 “as” sequences as controls for their $-{\rm
SCD}$–$\kappa$ anticorrelation opposing the overall positive
correlation.suman2  To probe the differential effects of sequence-local versus
nonlocal charge patterns, we construct 4 “c$\kappa$” sequences with diverse
SCDs but essentially the same $\kappa$, and 4 “cSCD” sequences with diverse
$\kappa$s but essentially the same SCD. We also consider sequences obs1 and
ebs1 with odd and even numbers of charge blocks, respectively, to assess
effects of like versus opposite charges at the two chain ends. These sequences
and their $\kappa$ and $-$SCD values are listed in Table S1 and Fig. S1b of
Supporting Information and marked in Fig. 1a. By focusing on this set of
sequences with the same chain length, we address sequence charge patterns’
impact on the thermodynamics of polyampholytes. As such, further
investigations of dynamic and other material properties,koby2020 ;
MittalNatComm2024 ; Wingreen2024  broader questions about sequence specificity
for IDPs of different chain lengths,RuffMiMB2021 ; PappuJMB2021
polyampholytes in high salt,kings2018  and for sequences containing short
spatial range hydrophobic-like interactionsmoleculargrammar ; Zheng2020 ;
WessenDasPalChan2022 ; Mittal2023 ; MittalNatChem2024  and/or with high net
chargeskings2020 ; Caprin1_arXiv  are left to future studies.

Theories and computational models are available to address sequence-specific
LLPS of polyampholytes (see, e.g., refs. (53; 54; 55; 56; 57), reviewed in
ref. (58)). Here we apply three complementary methods: analytical random phase
approximation (RPA),Mahdi2000 ; Ermoshkin2003  field-theoretic simulation
(FTS),joanPNAS ; Parisi1983 ; Klauder1983 ; Fredrickson2006  and coarse-
grained molecular dynamics (MD) with the “slab” sampling method for phase
equilibriaPanagiotopoulos_Scaling_2017  (formulations described in Supporting
Information), which have afforded numerous physical insights.linPRL ;
joanJPCL2019 ; dignon18 ; Pal2021 ; WessenDasPalChan2022  Based on the same
path-integral polymer model, FTS is more accurate than RPA in principle
because it does not require an approximation like RPA and can be extended to
tackle nonelectrostatic interactions.WessenDasPalChan2022  Nonetheless, FTS is
limited by finite resolution, simulation box size, and treatments of excluded
volume.joanJPCL2019 ; Pal2021  Compared to RPA and FTS, coarse-grained MD
accounts better for structural and energetic features of IDPsZheng2020 ;
Mittal2023 ; SumanPNAS ; Mpipi ; Kresten2021  but is computationally more
costly. To focus on electrostatics, we adopt the “hard-core repulsion” MD
model with no nonelectrostatic attraction.suman2  The utility of combining the
complementary advantages of RPA, FTS, and MD is illustrated by recent studies
of the dielectric properties of condensatesWessen2021  and the effects of salt
and ATP on condensed polyampholytic and polyelectrolytic
biomolecules.Caprin1_arXiv

We employ all three methods for LLPS. $R_{\rm g}$s and pairwise bead-bead
contacts—which are not amenable to RPA currently—are computed by MD and the
following FTS approach: In a multiple-chain system, the root-mean-square
radius of gyration of the $i$th polymer ${R_{\rm
g}^{(i)}}^{2}=\langle\sum_{\alpha=1}^{N_{\rm p}}\sum_{\beta=1}^{N_{\rm
p}}(\bm{R}_{i,\alpha}-\bm{R}_{i,\beta})^{2}\rangle/{2N_{\rm p}^{2}}$ $=$ $\int
d\bm{r}\int d\bm{r}^{\prime}\langle\hat{\rho}_{\rm
c}^{(i)}(\bm{r})\hat{\rho}_{\rm
c}^{(i)}(\bm{r}^{\prime})\rangle(\bm{r}-\bm{r}^{\prime})^{2}/{2N_{\rm
p}^{2}}$, where $\langle\cdot\cdot\cdot\rangle$ denotes Boltzmann averaging,
$\bm{R}_{i,\alpha}$ is the position of the $\alpha$th bead along the chain,
and $\hat{\rho}_{\rm c}^{(i)}(\bm{r})\equiv\sum_{\alpha=1}^{N_{\rm
p}}\delta(\bm{r}-\bm{R}_{i,\alpha})$ is the position ($\bm{r}$)-dependent bead
center density. Since the correlation function
$G^{(i)}(|\bm{r}-\bm{r}^{\prime}|)$ $=$ $\langle\hat{\rho}_{\rm
c}^{(i)}(\bm{r})\hat{\rho}_{\rm c}^{(i)}(\bm{r}^{\prime})\rangle$ that depends
on the relative distance $|\bm{r}-\bm{r}^{\prime}|$ is amenable to FTS,Pal2021

${R_{\rm g}^{(i)}}^{2}=\frac{V}{2N_{\rm p}^{2}}\int
d\bm{r}G^{(i)}(|\bm{r}|)|\bm{r}|^{2}\;,$ (1)

where system volume $V=\int d\bm{r}$, can now be computed by FTS. Similarly,
with a generalized correlation $G_{\alpha,\beta}^{(i),(j)}(|\bm{r}|)$ between
the $\alpha$th bead of the $i$th chain and the $\beta$th bead of the $j$th
chain,

$\omega_{\alpha,\beta}^{i,j}=V\int_{0}^{2b}d\bm{r}\;G_{\alpha,\beta}^{i,j}(|\bm{r}|)$
(2)

is seen as the frequency of contact between the two beads, i.e., when their
centers are within a small distance (chosen here as $2b$ where $b$ is the
reference bond length between sequentially consecutive beads). Thus, through
appropriate choices of $i$ and $j$, intrachain contacts of an isolated chain
as well as intrachain and interchain contacts in the condensed phase (Fig. 1c)
can be computed by FTS via $G_{\alpha,\beta}^{(i),(j)}(|\bm{r}|)$. A
derivation of this formulation based on the general FTS
approachFredrickson2006 ; FredricksonGanesanDrolet2002 ;
RigglemanRajeevFredrickson2012 ; MiMB2023  is provided in the Supporting
Information.

As examples, Fig. 1d,e show the RPA, FTS, and MD phase diagrams for six
sequences. Phase diagrams for all 26 sequences in Fig. 1b are provided in Fig.
S2 of the Supporting Information. To facilitate comparisons, temperatures are
given as reduced temperature $T^{*}\equiv b/l_{\rm B}$ where $l_{\rm B}$ is
Bjerrum length. Because of the models’ different effective energy scales
arising from various approximations and treatments of excluded volume, the
critical temperatures, $T^{*}_{\rm cr}$s, predicted by RPA, FTS, and MD can be
substantially different for the same sequence (Fig. S1b in Supporting
Information). Nonetheless, the variation in $T^{*}_{\rm cr}$ predicted by the
models are well correlated (Fig. 1f, Pearson correlation coefficients
$r\gtrsim 0.95$), indicating that the models are capturing essentially the
same sequence-dependent trend of LLPS propensity. As to the relationship
between $T^{*}_{\rm cr}$ and $T^{*}$-dependent root-mean-square isolated-chain
$R_{\rm g}$, a $T^{*}$ sufficiently high for a large sequence-dependent
variation in $R_{\rm g}$ is chosen for each of the models in Fig. 1g.
Consistent with an earlier study on sv sequences,lin2017  $R_{\rm g}$s of
isolated polyampholytes are well correlated with their $T^{*}_{\rm cr}$s in
all three models (Fig. 1g). Notably, however, there is an appreciable $R_{\rm
g}$–$T^{*}_{\rm cr}$ scatter in the MD model involving the cSCD, c$\kappa$,
obs1, and ebs1 sequences which as a group is less conformative to the moderate
$-$SCD–$\kappa$ correlation than the sv sequences (Fig. 1a).

The impact of sequence-local versus nonlocal charge pattern on $R_{\rm g}$ and
$T^{*}_{\rm cr}$ is assessed by comparing the extent to which they are
(anti)correlated with $\kappa$ and SCD (Fig. 2). $R_{\rm g}$ depends on
$T^{*}$. Since the charge-pattern-dependent variation in $R_{\rm g}$ among
sequences with moderate to high $-$SCD values is small at low $T^{*}$ as they
adopt conformations with similarly high compactness, two $T^{*}$s are chosen
for each of the FTS and MD models in Fig. 2a,b,e,f with the higher $T^{*}$
producing ample $R_{\rm g}$ variations across the entire ranges of $-$SCD and
$\kappa$. Corresponding $R_{\rm g}$ data for additional $T^{*}$s are provided
in Fig. S3 of Supporting Information.

It is clear from Fig. 2a,b,e,f that $R_{\rm g}$ anticorrelates significantly
better with $-$SCD than $\kappa$. For the sv sequences, $R_{\rm g}$
anticorrelates reasonably well with both $-$SCD and $\kappa$. Indeed, the
significant $\kappa$–$R_{\rm g}$ scatter seen in both FTS and MD (Fig. 2e,f)
involves the as, cSCD, and c$\kappa$ sequences we introduced. Despite the
large variations in $\kappa$ among the as and cSCD sequences, their $R_{\rm
g}$s are very similar. For the c$\kappa$ sequences, despite their essentially
identical $\kappa$, their $R_{\rm g}$s are very different. By comparison, the
excellent $-$SCD–$R_{\rm g}$ anticorrelation is maintained when challenged by
these sequences (Fig. 1a,b). In this light, the good $\kappa$–$R_{\rm g}$
anticorrelation observed previously for the sv sequencesrohit2013  is largely
attributable to the good correlation between the $\kappa$ values of this
particular set of sv sequences and their $-$SCDs.

In contrast to $\kappa$ and SCD’s clearly different performance for $R_{\rm
g}$ (which is the original target of both parametersrohit2013 ; kings2015 ,
neither of these parameters was derived originally for multiple-chain
properties), their correlations with $T^{*}_{\rm cr}$ are more comparable
(Fig. 2c,d,g,h). Unlike the excellent $-$SCD–$R_{\rm g}$ anticorrelation, both
the $-$SCD–$T^{*}_{\rm cr}$ and $\kappa$–$T^{*}_{\rm cr}$ correlations are
good but not excellent. While $-$SCD is better than $\kappa$ with $T^{*}_{\rm
cr}$ for RPA and FTS (Fig. 2c,g), $\kappa$ is slightly better for MD (Fig.
2d,h). Notably, for all three models—RPA, FTS, and MD—the variation in
$T^{*}_{\rm cr}$ is larger among the c$\kappa$ than among the cSCD sequences.

The origin of the performances of $-$SCD and $\kappa$ for $R_{\rm g}$ is
explored by first considering how $R_{\rm g}$ is related to intrachain
contacts for a homopolymer with favorable short-spatial-range interactions
(model described in Supporting Information). In this baseline model, $R_{\rm
g}$s conditioned upon pairwise $\alpha,\beta$ contacts (Fig. 3a, top map,
upper triangle) indicate that more nonlocal (higher-order) contacts lead to
smaller $R_{\rm g}$. Sequence-local and nonlocal interactions thus have
different implications for $R_{\rm g}$. This basic observation offers a semi-
quantitative rationalization for SCD’s better performance with regard to
$R_{\rm g}$ because, as entailed by the original approximate analytical
theory,kings2015  nonlocal electrostatic interactions (larger
$|\alpha-\beta|$) are more heavily weighted in SCD than local interactions
(smaller $|\alpha-\beta|$).

Additional information is provided by contact patterns (Fig. 3a–d). The
intrachain contact frequencies $P^{[{\rm i}]}_{\alpha,\beta}$ and $P^{[{\rm
c}]}_{\alpha,\beta}$ are the average numbers of contacts between the
$\alpha$th and $\beta$th beads, respectively, of an isolated chain (at
infinite dilution) and a chain in the condensed phase, whereas the interchain
$C^{[{\rm c}]}_{\alpha,\beta}$ is the average number of contacts between the
$\alpha$th bead of one chain and the $\beta$th bead of another chain. By
definition, $P^{[{\rm i}]}_{\alpha,\beta}=P^{[{\rm i}]}_{\beta,\alpha}$,
$P^{[{\rm c}]}_{\alpha,\beta}=P^{[{\rm c}]}_{\beta,\alpha}$, and $C^{[{\rm
c}]}_{\alpha,\beta}=C^{[{\rm c}]}_{\beta,\alpha}$. For the baseline
homopolymer, the isolated-chain $P^{[{\rm i}]}_{\alpha,\beta}$’s pattern is
typical of Gaussian or self avoiding walk conformations, with substantially
lower frequencies for higher order contactsChanDill1990  (Fig. 3a, top map,
lower triangle). The condensed-phase intrachain pattern (Fig. 3a, bottom map,
lower triangle) shares a similar trend but with less variation in contact
frequency (cf. heat map scales for $-\ln P^{[{\rm i}]}_{\alpha,\beta}$ and
$-\ln P^{[{\rm c}]}_{\alpha,\beta}$ in Fig. 3a). In contrast, the interchain
$C^{[{\rm c}]}_{\alpha,\beta}$ is quite insensitive to $\alpha,\beta$ except
it is slightly higher when either $\alpha$, $\beta$, or both, are at or near
the chain ends (Fig. 3a, bottom map, upper triangle).

Polyampholyte contact data are illustrated here by an example sequence. The MD
(Fig. 3b) and FTS (Fig. 3c) patterns are quite well correlated, their
differences likely arise from the differing treatments of excluded volume in
the two approaches.joanJPCL2019 ; Pal2021  Data for the other 25 sequences in
Fig. 1b are in Fig. S4 of Supporting Information. In Fig. 3b, the $R_{\rm g}$
map exhibits sequence-specific features as well as a contact-order dependence
(top map, upper triangle) indicating differential sequence-local versus
nonlocal effects on $R_{\rm g}$. Similar to the homopolymer (Fig. 3a), the
intrachain pattern of an isolated polyampholyte (Fig. 3b,c top map, lower
triangle) is similar to that of a polyampholyte in the condensed phase (Fig.
3b,c bottom map, lower triangle). Unlike the homopolymer, the isolated-chain
intrachain pattern (Fig. 3b,c top map, lower triangle) is similar also to the
condensed-phase interchain pattern (Fig. 3b,c bottom map, upper triangle).
This feature, which is echoed by the comparison in Fig. 3e below, applies to
the other 25 sequences as well.

Averages of $P^{[{\rm i}]}_{\alpha,\beta}$, $P^{[{\rm c}]}_{\alpha,\beta}$, or
$C^{[{\rm c}]}_{\alpha,\beta}$ for a given $|\alpha-\beta|$ are illustrated
here using the homopolymer model and the example sequence (Fig. 3d,e). Similar
salient features are exhibited by the other 25 sequences (Fig. S5 of
Supporting Information). Condensed-phase interchain $C^{[{\rm
c}]}_{\alpha,\beta}$ is least sensitive to $|\alpha-\beta|$ for the
homopolymer (Fig. 3d) and for polyampholytes (Fig. 3e). Notably, the
polyampholyte interchain $C^{[{\rm c}]}_{\alpha,\beta}$ (orange curve) is
closer to the isolated-chain intrachain $P^{[{\rm i}]}_{\alpha,\beta}$ (green
curve) than to the condensed-phase intrachain $P^{[{\rm c}]}_{\alpha,\beta}$
(blue curve). The root-mean-square distance $R_{\alpha\beta}$ between beads
$\alpha$ and $\beta$ is highly sensitive to sequencerohit2013  and temperature
for an isolated polyampholyte (Fig. 3f) as its $R_{\rm g}$ decreases at low
$T^{*}$. In contrast, Fig. 3g shows that $R_{\alpha\beta}$ depends only weakly
on $T^{*}$. Depending on the sequence, $R_{\rm g}$ can increase or decrease
slightly with $T^{*}$ (Fig. S6 of Supporting Information). As in homopolymer
melts,Flory1949  condensed-phase polyampholytes adopt open, essentially
Gaussian-like conformations ($R_{\alpha\beta}/\sqrt{|\alpha-\beta|}\sim$
constant except for small $|\alpha-\beta|$ in Fig. 3g), a phenomenon also seen
in recent simulations of biomolecular condensates.MittalACS2023 ;
PappuNatComm2023 ; Johnson2024 ; Mittal2024

To gain further insights, we compare sequence-specific contact patterns such
as those in Fig. 3b by the following symmetrized form of the Kullback-Leibler
divergenceKL  between contact frequencies $\\{C^{(s)}_{\alpha\beta}\\}$ and
$\\{C^{(s^{\prime})}_{\alpha\beta}\\}$ (contact maps) of a pair of sequences
$s,s^{\prime}$:

$D_{s,s^{\prime}}=\sum_{\alpha=1}^{N_{\rm p}-d_{\rm
m}}\sum_{\beta=\alpha+d_{\rm m}}^{N_{\rm
p}}\Bigl{[}c^{(s)}_{\alpha\beta}-c^{(s^{\prime})}_{\alpha\beta}\Bigr{]}\ln\left[c^{(s)}_{\alpha\beta}/c^{(s^{\prime})}_{\alpha\beta}\right]\;,$
(3)

where $d_{\rm m}$ serves to exclude local contacts—which are often highly
populated—from overwhelming the contact pattern’s quantitative
characterization, and $c^{(s)}_{\alpha\beta}\equiv
C^{(s)}_{\alpha\beta}/\sum_{\alpha^{\prime}=1}^{N_{\rm p}-d_{\rm
m}}\sum_{\beta^{\prime}=\alpha+d_{\rm m}}^{N_{\rm
p}}C^{(s)}_{\alpha^{\prime}\beta^{\prime}}$ are normalized frequencies. The
comparisons of $P^{[{\rm i}]}_{\alpha,\beta}$ with $P^{[{\rm
c}]}_{\alpha,\beta}$ (Fig. 4a) and with $C^{[{\rm c}]}_{\alpha,\beta}$ (Fig.
4b) entail significant scatter. Nonetheless, the relatively low
$D_{s,s^{\prime}}$ values for $s=s^{\prime}$ (black circles) in Fig. 4b is
consistent with the impression from Fig. 3b,c that the patterns of isolated-
chain intrachain and condensed-phase interchain contacts are similar for a
given polyampholyte. This trend prevails for several other $d_{\rm m}$ values
and another isolated-chain $T^{*}_{\rm iso.}$ (Fig. S7 of Supporting
Information), underpinning correlations between sequence-dependent isolated-
chain and condensed-phase properties.

The relationship between isolated-chain and condensed-phase properties is
further elucidated by examining the total number of contacts made by a
polyampholyte. Whereas the number of intrachain contacts of an isolated chain
correlates poorly with that of a condensed-phase chain (Fig. 4c) because of
their significantly different $R_{\rm g}$s (see above), an excellent
correlation is seen between the number of intrachain contacts of an isolated
chain with the number of interchain contacts in the condensed phase (Fig. 4d),
echoed by the excellent correlation between isolated-chain and condensed-phase
potential energies (Fig. 4e). Similar trends are seen for other values of
$T^{*}_{\rm iso.}$ (Fig. S8 and Fig. S9a of Supporting Information).
$T^{*}_{\rm cr}$ anticorrelates reasonably well with isolated-chain potential
energy $E^{[{\rm i}]}$ (Fig. 4f), which expectedly correlates with the number
of interchain contacts (Fig. S9b in Supporting Information). Notably,
$E^{[{\rm i}]}$, in turn, anticorrelates quite well with $\kappa$ (Fig. 4h)
but not so well with $-$SCD (Fig. 4g), indicating that in some situations
$\kappa$ can be a better predictor of LLPS propensity, suggesting that
differential effects of sequence-local versus nonlocal interactions may be
less prominent for LLPS than for isolated-chain $R_{\rm g}$. This
understanding is underscored by the fitting coefficients $c_{\kappa}$ for
$\kappa$ and $c_{\rm SCD}$ for SCD in Fig. 4i,j indicating that variation in
$R_{\rm g}$ can be accounted for essentially entirely by SCD
($c_{\kappa}\approx 0$) but variation in $T^{*}_{\rm cr}$ is rationalized
approximately equally by $\kappa$ and SCD ($c_{\kappa}\approx c_{\rm SCD}$).
$\kappa$ is not a good general predictor for $R_{\rm g}$ though $\kappa$
correlates well with $E^{[{\rm i}]}$ because $R_{\rm g}$ is not determined
solely by $E^{[{\rm i}]}$. For instance, two chains each constrained by one
contact with the same energy but different contact orders can have very
different $R_{\rm g}$s.

To recapitulate, many IDPs can exist in dilute and condensed phases serving
different biological functions. Single-chain IDP $R_{\rm g}$s in dilute
solutions, readily accessible experimentally,SongSAXS2021  have been used to
benchmark MD potentials for sequence-dependent IDP properties.KrestenNat2024 ;
Alex2024  Isolated-chain contact maps,Zheng2023  related topological
constructs such as SCDMHuihuiKingsBJ2020  and energy maps,Devarajan2022  and
their relations with condensed-phase interchain contact mapsKresten2021 ;
bauer2022  have proven useful in recent computational analyses. For instance,
isolated-chain intrachain and condensed-phase interchain contacts are similar
for heterochromatin protein 1 paralogsPhan2023  and EWS sequencesJohnson2024
but not the TDP-43 C-terminal domain.mohanty2023  Here, our findings indicate
a fundamental divergence in the differential impact of local versus nonlocal
sequence patterns on isolated, single-chain and condensed-phase multiple-chain
properties. The differential impact is prominent for isolated-chain
conformational dimensions due to chain connectivity.ChanDill1990  It is
substantially less for LLPS propensity because the multiple-chain nature of
condensed-phase interactions dampens—though not entirely abolish—the effects
of contour separations between residues along a single connected sequence due
to the immense number of configurations in which residues from different
chains may interact. This is a fundamental factor in the dilute/condensed-
phase relationship of IDPs that needs to be taken into account when devising
improved sequence pattern parameters for the characterization of physical and
functional IDP molecular features.Sabari2023 ; Moses2021  Inasmuch as such
parameters’ aim is instant estimation of LLPS propensity, theoretical
quantities that can be numerically computed efficiently—such as the
$T^{*}_{\rm cr}$ predicted by RPA-related theoriesWessenDasPalChan2022 —may
just serve the purpose practically even if they are not closed-form
mathematical expressions.

Supporting Information
Methodological and formulational details, supporting table, and supporting
figures

Acknowledgements. This work was supported by Canadian Institutes of Health
Research (CIHR) grant NJT-155930 and Natural Sciences and Engineering Research
Council of Canada (NSERC) grant RGPIN-2018-04351 to H.S.C. We are grateful for
the computational resources provided generously by the Digital Research
Alliance of Canada.

Figures (Main Text)

Figure 1: Polyampholytes with representative variations in local and nonlocal
sequence charge patterns. (a) The SCD-$\kappa$ sequence-space population
distribution of 50-bead polyampholytes is depicted by a heat map (bottom
scale, region with zero population is in gray), wherein the 26 sequences
studied (b) are marked by symbols (a, top). (b) The positive (blue) and
negative (red) charge patterns of these sequences. (c) Coarse-grained MD
snapshots showing a conformation in the condensed phase (top) and an isolated
conformation (dotted box). (d,e) Phase diagrams in RPA (thin continuous
curves), FTS (symbols) (d), and MD (symbols) (e) for sequences c$\kappa$1,
c$\kappa$4, cSCD1, cSCD4, ebs1, and obs1. Thick solid and dashed lines
connecting symbols in (d) and (e) are merely guides for the eye. (f)
$T^{*}_{\rm cr}$s of all 26 sequences predicted by different theories. The
$T^{*}_{\rm cr}$–$T^{*}_{\rm cr}$ Pearson correlation coefficients for RPA-
FTS, RPA-MD, and FTS-MD are, respectively, $r=0.997$, $0.945$, and $0.948$.
Data symbols in (f) and (g) involving MD are identified by black edges. (g)
Correlation between $T^{*}_{\rm cr}$ and $R_{\rm g}$ in FTS (at $T^{*}=20$,
$r=-0.954$) or in MD (at $T^{*}=10$, $r=-0.842$). A complete list of
$T^{*}_{\rm cr}$s and phase diagrams for all sequences in (b) predicted by the
RPA, FTS, and MD models are provided in Figs. S1 and S2 of the Supporting
Information.

Figure 2: SCD and $\kappa$ as predictors for isolated-chain $R_{\rm g}$ and
multiple-chain LLPS critical temperature $T^{*}_{\rm cr}$. Symbols for
polyampholytes are those in Fig. 1a. (a,b,e,f) $R_{\rm g}$ computed by FTS
(a,e) and MD (b,f) versus $-$SCD (left) or $\kappa$ (right). Symbols with and
without black edges are for different temperatures ($T^{*}$s). $-$SCD–$R_{\rm
g}$ correlation coefficients are $r=-0.983$ for FTS at $T^{*}=20$ (a) and
$r=-0.991$ for MD at $T^{*}=10$ (b), the corresponding $\kappa$–$R_{\rm g}$
values are $r=-0.665$ (e) and $-0.627$ (f). (c,d,g,h) $T^{*}_{\rm cr}$ in FTS,
RPA [depicted, respectively, in (c,g) by symbols with and without black edges]
and MD (d,h) are plotted in units of the $T^{*}_{\rm cr}$ for sequence sv30
($T^{*}_{\rm cr,sv30}$). The $-$SCD–$T^{*}_{\rm cr}$ $r$ values are: $0.943$
for RPA, $0.932$ for FTS (c), and $0.834$ for MD (d). The corresponding
$\kappa$–$T^{*}_{\rm cr}$ values are $r=0.827$ for RPA, $0.841$ for FTS (g),
and $0.886$ for MD (h).

Figure 3: Contact patterns of polyampholytes as an isolated chain versus in
the phase-separated condensed phase. (a–c) Top heat maps (color scales above)
show isolated-chain contact frequencies (lower triangle) and $R_{\rm g}$
conditional upon intrachain $i,j$ contacts (upper triangle); bottom maps
(color scales below) show condensed-phase intrachain (lower triangle) and
interchain contact frequencies (upper triangle); $\alpha,\beta=1,2,\dots,50$
are bead labels along the chain. Contact frequencies $P^{[{\rm
i}]}_{\alpha,\beta}$, $P^{[{\rm c}]}_{\alpha,\beta}$, and $C^{[{\rm
c}]}_{\alpha,\beta}$ are defined in the text. Data are shown for (a) the
baseline homopolymer (MD, $T^{*}=2.0$) and (b) the sequence obs1 (MD,
$T^{*}=0.2$). The charge pattern of obs1 (as in Fig. 1b) is depicted along the
axes in (b). (c) Corresponding FTS contact maps for obs1 at $T^{*}=3.0$. (d,e)
Contact frequencies (color coded as indicated) in MD as functions of contact
order $|\alpha-\beta|$ for (d) the baseline homopolymer ($T^{*}=2.0$) and (e)
obs1 ($T^{*}=0.2$). (f,g) MD-simulated obs1 intrachain root-mean-square
distance $R_{\alpha\beta}$ (solid curves, left vertical scale) at select
temperatures for an isolated chain (f) and in the condensed phase (g); to
facilitate analysis (see text), $R_{\alpha\beta}/\sqrt{|\alpha-\beta|}$ plots
(dashed curves, right vertical scale) are also provided in (g).

Figure 4: Impact of local versus nonlocal sequence charge pattern on
polyampholyte contacts and interaction energies in the MD model. (a,b)
$D_{s,s^{\prime}}$ is symmetrized Kullback-Leibler divergence between the
normalized $|\alpha-\beta|\geq 4$ contact frequencies of sequences $s$ and
$s^{\prime}$ (see text). The scatter plots show $D_{s,s^{\prime}}$ versus
$s$–$s^{\prime}$ SCD difference. Each datapoint (translucent blue circle)
provides the $D_{s,s^{\prime}}$ between the intrachain contacts of an isolated
chain ($s$) and (a) the intrachain or (b) interchain contacts of a condensed-
phase chain ($s^{\prime}$) where $s,s^{\prime}=$ 25 of the sequences in Fig.
1b (all except sv1), with $T^{*}_{\rm iso.}=0.2$, $0.1\leq T^{*}_{\rm
cond.}\leq 0.3$, and $s=s^{\prime}$ datapoints as black circles. Red stars are
$D_{s,s^{\prime}}$ values for the baseline homopolymer at $T^{*}=2.0$. (c,d)
Scatter plots of number of intrachain contacts of isolated chains
$n_{c}^{[{\rm i}]}$ with number of (c) intrachain contacts $n_{c}^{[{\rm c}]}$
($r=-0.396$) or (d) interchain contacts $N_{c}^{[{\rm c}]}$ in the condensed
phase ($r=0.992$) for the 25 sequences analyzed in (a,b). (e–h) Scatter plots
of isolated-chain potential energy $E^{[{\rm i}]}$ (in units of $\epsilon$,
for all 26 sequences) with (e) condensed-phase potential energy $E^{[{\rm
c}]}$ ($r=0.998$), (f) $T^{*}_{\rm cr}$ ($r=-0.808$), (g) SCD ($r=0.505$), or
(h) $\kappa$ ($r=-0.855$). Datapoints for different sequences in (c–e,i) are
color-coded for SCD, with lighter color corresponding to larger $-$SCD (as in
Figs. S8 and S9). Symbols in (f–h) corresponds to those in Fig. 1a. All
quantities in (c–h) are computed for $T^{*}=0.2$ except a lower $T^{*}=0.04$
sufficient for LLPS of sv1 is used to obtain $E^{[{\rm c}]}$ for sv1 in (e).
(i,j) Best-fit linear combinations of $\kappa$ and SCD as predictors for (i)
$R_{\rm g}$ (in units of $R_{\rm g}$ of sv1, $R_{\rm g,sv1}$, at $T^{*}=10$)
and (j) $T^{*}_{\rm cr}$ for all sequences in Fig. 1a. (i) The dashed line is
$R_{\rm g}$ $=$ $1-(c_{\kappa}\kappa+c_{\rm SCD}{\rm SCD}/{\rm SCD}_{\rm
sv30})$ with $c_{\kappa}=-0.0017$, $c_{\rm SCD}=0.243$ ($r=-0.991$), and
SCDsv30 is the SCD of sv30. (j) The best-fit parameters are $a_{\kappa}=0.39$,
$c_{\rm SCD}=0.68$ for RPA ($r=0.996$), $c_{\kappa}=0.42$, $c_{\rm SCD}=0.65$
for FTS ($r=0.994$), and $c_{\kappa}=0.63$, $c_{\rm SCD}=0.54$ for MD
($r=0.962$).

July 4, 2024

Supporting Information

for

Differential Effects of Sequence-Local versus Nonlocal

Charge Patterns on Phase Separation and

Conformational Dimensions of Polyampholytes as

Model Intrinsically Disordered Proteins

Tanmoy PAL,1,† Jonas WESSÉN,1,† Suman DAS,1,2,† and Hue Sun CHAN1,∗

1Department of Biochemistry, University of Toronto, Toronto, Ontario M5S 1A8,
Canada

2Department of Chemistry, Gandhi Institute of Technology and Management,
Visakhapatnam, Andhra Pradesh 530045, India

†Contributed equally.

∗Correspondence information:
† Hue Sun CHAN. E-mail<EMAIL_ADDRESS>
† Tel: (416)978-2697; Fax: (416)978-8548
† Department of Biochemistry, University of Toronto, Medical Sciences Building
– 5th Fl.,
† 1 King’s College Circle, Toronto, Ontario M5S 1A8, Canada.

Methodological and Formulational Details

Estimating the sequence-space distribution $P({\rm SCD},\kappa)$

As outlined in the maintext, we investigate the relationship between the ${\rm
SCD}$ and $\kappa$ parameters and its ramifications on isolated-chain and
condensed-phase conformational properties by first calculating their
distribution $P({\rm SCD},\kappa)$ for electrically overall neutral 50mer
polyampholytes (sequences with 50 beads). Following convention in an earlier
work that used lysine (K) for the positively charged ($+1$) and glutamic acid
(E) for the negatively charged ($-1$) monomers/residues along the chain
sequence,rohit2013  we refer to the sequences under consideration as K/E
sequence in the following discussion, keeping in mind, however, that K and E
here refer only to $+1$ and $-1$ polymer beads with no sidechain structure, as
in several previous simplified models.kings2015 ; lin2017 ; joanJPCL2019 ;
suman2  Values for $\kappa$ are computed in the present work using the
$\kappa=(\kappa_{5}+\kappa_{6})/2$ expression defined in ref. (37).

The $P({\rm SCD},\kappa)$ distribution is defined in such a way that $P({\rm
SCD}^{\prime},\kappa^{\prime})\,\Delta{\rm SCD}\,\Delta\kappa$ is the number
of possible sequences with $({\rm SCD},\kappa)$ values in a small region
$\Delta{\rm SCD}\,\Delta\kappa$ (two-dimensional bin) in the vicinity of
$({\rm SCD}^{\prime},\kappa^{\prime})$. We compute $P({\rm SCD},\kappa)$ using
the Wang-Landau (WL) algorithm,WangLandau2001  which is a highly efficient
flat-histogram method for estimating multi-peaked densities of states.
However, before setting up the WL algorithm, prior knowledge about the the
mathematically possible $(\kappa,{\rm SCD})$ combinations is required since
the WL algorithm relies on checking that all $(\kappa,{\rm SCD})$ bins are
evenly sampled.

To find the region in the $({\rm SCD},\kappa)$-plane that can support 50mer
K/E sequences, we utilize a genetic algorithm (GA) that is capable of scanning
the space of sequences efficiently. Specifically, this GA takes an input
target point $({\rm SCD}_{\rm target},\kappa_{\rm target})$ in sequence space
and attempts to generate a new sequence $\\{\sigma_{\alpha}\\}$
($\alpha=1,2,\dots,N$, where the chain length $N=50$) with $({\rm
SCD},\kappa)$ that maximizes the fitness function

$f(\\{\sigma_{\alpha}\\})=-(\kappa-\kappa_{\rm target})^{2}-0.001({\rm
SCD}-{\rm SCD}_{\rm target})^{2}\;.$ (S1)

For a given target $({\rm SCD}_{\rm target},\kappa_{\rm target})$, the
algorithm proceeds as follows:

  1. 1.

Generate an initial “population” of $n_{\rm population}=100$ random
electrically neutral K/E sequences (each with 25 K and 25 E).

  2. 2.

Each sequence in the population is used to generate $n_{\rm offspring}=50$
“offspring” sequences. Every offspring sequence is generated from its parent
sequence using one of the following prescriptions:

     * •

Single flip (50%): A randomly selected pair of K/E residues are interchanged.

     * •

Cluster move (45%): A randomly selected block of same-charge residues is moved
one step (either left or right chosen at random), e.g., …EEEEEKKKK…
$\rightarrow$ …KEEEEEKKK… . As such, this is a special case of single flip. It
serves to enhance sampling of blocky sequences.

     * •

New sequence (5%): An entirely new electrically neutral sequence with no
relation to the parent sequence is randomly generated.

The three prescriptions are chosen at random with probabilities provided in
parentheses above. We do not use crossover to generate offspring sequences
(i.e. where multiple parent sequences are combined to generate offspring) as
initial exploratory runs did not reveal any major advantage when using
crossover.

  3. 3.

We next select $n_{\rm survivors}=30$ out the $n_{\rm population}\cdot n_{\rm
offspring}$ offspring sequences using a diversity enhanced survivor selection
algorithm. First, we evaluate the fitness $f_{i}$ of each offspring sequence
$\\{\sigma_{\alpha}^{(i)}\\}$, with $i=1,\dots,n_{\rm population}\cdot n_{\rm
offspring}$, according to Eq. (S1), and select the sequence corresponding to
the highest (least negative) $f_{i}$ as the first survivor. The fitness of the
remaining sequences $\\{\sigma_{\rm\alpha}^{(j\neq i)}\\}$ are then modified
as $f_{j}\rightarrow f_{j}-\mathrm{e}^{-10d_{ij}}$ where
$d_{ij}=\sum_{\alpha=1}^{N}\left(\sigma_{\alpha}^{(i)}-\sigma_{\alpha}^{(j)}\right)^{2}/4N$.
This step punishes sequences that are identical or near-identical to the first
selected survivor. The next survivor $i^{\prime}$ is then selected based on
the updated fitnesses and the remaining sequences acquires new punishments
depending on their similarity with $i^{\prime}$. This procedure is repeated
until $n_{\rm survivors}$ have been selected.

  4. 4.

The survivors now constitute the population of the next generation and steps
2–3 are iterated until either a sequence with $f_{i}\geq-10^{-5}$ has been
found or the 10th generation of iterations is reached.

The diversity enhanced survivor selection procedure prevents the algorithm
from being trapped in the vicinity of a local fitness maximum since sequences
are selected both according to having a high fitness and being dissimilar to
other sequences with high fitness. The method was introduced to scan for
experimentally viable parameter regions in extensions of the Standard Model in
particle physics,CamargoMolinaMandal2018  but the survivor selection procedure
was not outlined in full detail in ref. (38).

We coarse-grain the $({\rm SCD},\kappa)$-plane into $120\times 120$ bins that
cover the rectangular region ${\rm SCD}\in[{\rm SCD}_{\rm sv30},0]$ (i.e.,
$-{\rm SCD}\in[0,-{\rm SCD}_{\rm sv30}]$), $\kappa\in[0,1]$, where ${\rm
SCD}_{\rm sv30}$ is the SCD value of the diblock sequence sv30 in ref. (13).
The bin spanning the area $[-{\rm SCD}_{i},-{\rm SCD}_{i}+\Delta{\rm SCD}]$
and $[\kappa_{j},\kappa_{j}+\Delta\kappa]$ is indexed by $(i,j)$ (with
$i,j=1,\dots,120$). The GA is first run for targets $({\rm SCD}_{\rm
target},\kappa_{\rm target})$ on the sites of an evenly spaced $7\times 7$
grid (marked by white dots in the left panel of Fig. S1a). Every new bin
visited during the scan is recorded and the associated sequence is stored.
With the exception of the first target iteration, the above step 1 of the GA
is modified such that the initial population is generated as offspring of the
previously stored sequence with nearest ${\rm SCD}$ and $\kappa$ to the target
values instead of being randomly generated. The initial grid scan gives a
rough estimate of $({\rm SCD},\kappa)$ region populated by sequences. Next, we
iteratively run the GA for unvisited target bins adjacent to previously
visited bins to map out the boundary of the region. Fig. S1a shows in the left
panel the result of the GA scan, where bins for which a sequence was found are
shown in black. White bins indicate target bins for which no sequence was
found by the GA. For a few target bins, the GA was not able to find a sequence
but the subsequent WL scan (described below) revealed sequences populating
these bins. These mislabelled bins are shown in red in in the left panel of
Fig. S1a.

The goal of the WL algorithm is to calculate the quantity $g_{i,j}$
representing the number of sequences with ${\rm SCD}$ and $\kappa$ in bin
$(i,j)$. To avoid numerical errors associated with large numbers, we (as is
customary) formulate the algorithm in terms of $\ln(g_{i,j})$ rather than
$g_{i,j}$. Another central quantity in the WL algorithm is the histogram
$h_{i,j}$ over the number of visits in bin $(i,j)$. The WL algorithm operates
as follows:

  1. 1.

Initialize $h_{i,j}=\ln(g_{i,j})=0$ for all $(i,j)$ and set the update factor
$f=1$.

  2. 2.

Generate $n=100$ random electrically neutral 50mer K/E sequences,
$\\{\sigma_{\alpha}^{(a)}\\}$, $a=1,\dots,n$, $\alpha=1,2,\dots,N$, and
compute their associated ${\rm SCD}$ and $\kappa$ values.

  3. 3.

Perform the updates

$\displaystyle h_{i,j}$ $\displaystyle\rightarrow h_{i,j}+1,$ (S2)
$\displaystyle\ln(g_{i,j})$ $\displaystyle\rightarrow\ln(g_{i,j})+f,$

for the bin $(i,j)$ associated with each sequence.

  4. 4.

Check if $h_{i,j}$ is sufficiently flat. In this work, $h_{i,j}$ is deemed
flat if

$\frac{\langle h^{2}\rangle-\langle h\rangle^{2}}{\langle
h\rangle^{2}}<0.5\;,$ (S3)

where the averages are taken only over bins known to be populated by
sequences. If the flatness condition is satisfied we reset $h_{i,j}=0$ for all
$i,j$ and update $f$ as $f\rightarrow f/2$.

  5. 5.

Propose an update for every sequence $\\{\sigma_{\alpha}^{(a)}\\}$. The
updates and their associated probabilities used in this work are

     * •

Single flip (80%): Flip the charges of a randomly selected pair of oppositely
charged residues.

     * •

Multiple flips (20%): Perform $n_{\rm flip}$ single flips, where $n_{\rm
flip}$ is a uniformly distributed random integer between 2 and 10.

Compute ${\rm SCD}$ and $\kappa$ for all proposed sequences
$\\{\sigma_{\alpha}^{(a^{\prime})}\\}$ and accept the updates with
probabilities

$P_{\rm
acc}\left(\\{\sigma_{\alpha}^{(a)}\\}\rightarrow\\{\sigma_{\alpha}^{(a^{\prime})}\\}\right)=\max\left(1,\frac{g_{i,j}}{g_{i^{\prime},j^{\prime}}}\right),$
(S4)

where $(i,j)$ and $(i^{\prime},j^{\prime})$ refer to the bins associated with
$\\{\sigma_{\alpha}^{(a)}\\}$ and $\\{\sigma_{\alpha}^{(a^{\prime})}\\}$,
respectively. If a $({\rm SCD},\kappa)$ combination is encountered for which
the associated bin $(i,j)$ was estimated by the GA as empty, the bin is re-
labelled as non-empty and kept in all subsequent WL iterations.

  6. 6.

Repeat steps 3–5 until 25 WL “levels” have been completed, i.e., until
$h_{i,j}$ is flat with $f=2^{-24}$.

  7. 7.

Re-weight $g_{i,j}\rightarrow g_{i,j}/g_{\rm sv30}$ where $g_{\rm sv30}$ is
the value of $g_{i,j}$ for the bin associated with the di-block sequence sv30.
This step accounts for the double counting associated with reversed sequences
but relies on the discretization being sufficiently fine such that $g_{\rm
sv30}$ only received contributions from sv30 and its reverse.

Our version of the WL algorithm differs from more standard
implementationsIrback2013 ; IrbackWessen2015  since $n=100$ instantiations,
rather than $n=1$, of the system are evolved while modifying the same
$g_{i,j}$ and $h_{i,j}$. As such, our setup shares characteristics with the
replica-exchange Wang-Landau method,Vogel2013  although we do not e.g.,
constrain the individual sequences to subregions of the $({\rm
SCD},\kappa)$-plane. We observe a dramatic decrease in computation time when
$n=1\rightarrow 100$ and expect that the implementation can be made even more
efficient by parallel computing since the $n$ sequence updates in Step 5 can
be made independently in-between the $h_{i,j}$ and $g_{i,j}$ updates (see,
e.g., refs. (87; 88; 89) for more sophisticated parallel WL algorithm
implementations). The sought-after distribution, $P({\rm SCD},\kappa)\approx
g_{i,j}/\Delta{\rm SCD}\Delta\kappa$ up to discretization errors, is now shown
in the right panel of Fig. S1a as well as maintext Fig. 1a.

Given this $P({\rm SCD},\kappa)$, the moderate degree to which ${\rm SCD}$ and
$\kappa$ are correlated may be quantified by the Pearson correlation
coefficient

$r=\frac{\left\langle\left({\rm SCD}-\langle{\rm
SCD}\rangle\right)\left(\kappa-\langle\kappa\rangle\right)\right\rangle}{\sqrt{\langle{\rm
SCD}^{2}\rangle-\langle{\rm
SCD}\rangle^{2}}\sqrt{\langle\kappa^{2}\rangle-\langle\kappa\rangle^{2}}}\approx-0.684\;,$
(S5)

where $\langle\dots\rangle$ here denotes average over the GA/WL-sampled
$P({\rm SCD},\kappa)$ distribution (thus the $r$ value between $-$SCD and
$\kappa$ is $0.684$). The 26 overall-neutral polyampholyte sequences studied
in this work are given in maintext Fig. 1b and in Table S1, their ${\rm SCD}$
and $\kappa$ values are listed in Fig. S1 and plotted on the $-$SCD versus
$\kappa$ plane in maintext Fig. 1a.

Field-theoretic formulation of polyampholyte conformations and phase
separation

We study these model polyampholyte chain sequences using a field-theoretic
formulation—using field-theoretic simulation (FTS) and random phase
approximation (RPA)—as well as coarse-grained molecular dynamics (MD). As in
our previous works (reviewed in ref. (68)), the present field-theoretic
formulation is based on a Hamiltonian $\hat{H}$ that accounts for chain
connectivity, short-spatial-range excluded-volume repulsion and long-spatial-
range electrostatic interactions for a system of $n_{\rm p}$ polymers
(polyampholyte chains) each consisting of $N_{\rm p}$ monomer (beads).
$\hat{H}$ is given by

$\beta\hat{H}=\frac{3}{2b^{2}}\sum_{i=1}^{n_{\rm p}}\sum_{\alpha=1}^{N_{\rm
p}-1}\left(\bm{R}_{i,\alpha+1}-\bm{R}_{i,\alpha}\right)^{2}+\frac{v}{2}\int
d\bm{r}\hat{\rho}_{\mathrm{tot}}(\bm{r})^{2}+\frac{l_{\rm B}}{2}\int
d\bm{r}\int
d\bm{r}^{\prime}\frac{\hat{c}(\bm{r})\hat{c}(\bm{r}^{\prime})}{|\bm{r}-\bm{r}^{\prime}|}\;,$
(S6)

where $\beta\equiv 1/k_{\rm B}T$ ($k_{\rm B}$ is the Boltzmann constant and
$T$ is absolute temperature), $b$ is the reference root-mean-square bond
length between two adjacent monomers along the chain sequence when non-bonded
interactions are absent, ${\bm{R}}_{i,\alpha}$ is the position vector of the
$\alpha$th bead of the $i$th chain, $\hat{\rho}_{\mathrm{tot}}(\bm{r})$ is
bead number density (matter density), $\hat{c}(\bm{r})$ is charge density, $v$
is the excluded volume parameter and $l_{\rm
B}=e^{2}/(4\pi\epsilon_{0}\epsilon_{\rm r}k_{\rm B}T)$ is the Bjerrum length
that we use to quantify electrostatic interaction strength, with $e$ denoting
the protonic charge, $\epsilon_{0}$ and $\epsilon_{\rm r}$ are vacuum and
relative permittivities, respectively; and temperature in our field-
theoretical models is quantified by the reduced temperature $T^{*}\equiv
b/l_{\rm B}$. An electric charge $\sigma_{\alpha}$ is associated with the
$\alpha$th bead of each chain. In this work, we consider only overall charge
neutral polyampholyte sequences, thus $\sum_{\alpha=1}^{N_{\rm
p}}\sigma_{\alpha}=0$. To avoid potential singularities arising from modeling
polymer beads as point particles, we model each bead as a normalized Gaussian
distribution given by
$\Gamma(\bm{r})=\exp(-r^{2}/2\bar{a}^{2})/(2\pi\bar{a}^{2})^{3/2}$ where
$r^{2}\equiv|\bm{r}|^{2}$ (refs. (90; 67)). Accordingly, the bead number
density and charge density are given, respectively, by

$\displaystyle\hat{\rho}_{\mathrm{tot}}(\bm{r})$
$\displaystyle=\sum_{i=1}^{n_{\rm p}}\sum_{\alpha=1}^{N_{\rm
p}}\Gamma(\bm{r}-\bm{R}_{i,\alpha})\;,$ (S7) $\displaystyle\hat{c}(\bm{r})$
$\displaystyle=\sum_{i=1}^{n_{\rm p}}\sum_{\alpha=1}^{N_{\rm
p}}\sigma_{\alpha}\Gamma(\bm{r}-\bm{R}_{i,\alpha})\;.$

The field-theoretic model system defined by Eqs. S6 and S7 is analyzed using
random phase approximation (RPA) and field-theoretic simulation (FTS).

Random phase approximation (RPA). An approximate analytical theory, termed
RPA, can be derived from the partition function formulated by path integrals
based upon the Hamiltonian in Eq. S6. In the present study, all RPA
calculations are performed within the context of an implicit solvent polymer
field theoryPal2021  (no explicit solvent, unlike, e.g., in ref. (29)) with
contact-excluded volume interactions and unscreened Coulomb electrostatic
interactions as specified by Eq. S6 and, as mentioned, UV (short-
distance)-divergences are regulated by Gaussian-smeared beads involving the
function $\Gamma(\bm{r})$ as described above. In the present RPA calculations,
we use a fixed excluded volume parameter $v=0.0068b^{3}$ and a Gaussian
smearing length of ${\bar{a}}=b/\sqrt{6}$. The detailed mathematical
formulation and the computer code employed for our RPA calculations are
documented and available through our recent review.MiMB2023  Examples of phase
diagrams computed using FTS and RPA for the 26 polyampholyte sequences we
study (maintext Fig. 1b, Fig. S1b, Table S1) are provided in maintext Fig. 1d.
A complete list of these phase diagrams are documented in Fig. S2. The
critical temperatures of these sequences’ phase transitions in RPA and FTS are
tabulated in Fig. S1b, whereas the necessary details of our FTS methodology
are provided below.

Radius of gyration and contact maps of chain conformations in the field-
theoretic formulation. As a novel extension of our previous field-theoretic
formulation, we now connect the radius of gyration ($R_{\rm g}$) of a single
polymer chain to the polymer beads’ pair-correlation functionPal2021  as
follows:

$\displaystyle{R_{\rm g}}^{2}$ $\displaystyle=\frac{1}{2N_{\rm
p}^{2}}\left\langle\sum_{\alpha=1}^{N_{\rm p}}\sum_{\beta=1}^{N_{\rm
p}}(\bm{R}_{\alpha}-\bm{R}_{\beta})^{2}\right\rangle$ (S8)
$\displaystyle=\frac{1}{2N_{\rm p}^{2}}\int d\bm{r}\int
d\bm{r}^{\prime}\langle\hat{\rho}_{\rm c}(\bm{r})\hat{\rho}_{\rm
c}(\bm{r}^{\prime})\rangle(\bm{r}-\bm{r}^{\prime})^{2}\;,$

where the bead center density is defined here by $\hat{\rho}_{\rm
c}(\bm{r})=\sum_{\alpha=1}^{N_{\rm p}}\delta(\bm{r}-\bm{R}_{\alpha})$ and
$\langle\cdot\cdot\cdot\rangle$ denotes thermal averaging. Since the pair-
correlation function $G(|\bm{r}-\bm{r}^{\prime}|)=\langle\hat{\rho}_{\rm
c}(\bm{r})\hat{\rho}_{\rm c}(\bm{r}^{\prime})\rangle$ depends only on the
relative distance, we may perform a coordinate transformation such that
$\bm{R}=(\bm{r}+\bm{r}^{\prime})/2$ and
$\bm{\mathsf{r}}=\bm{r}-\bm{r}^{\prime}$, with the associated Jacobian’s
determinant $|J|=1$, and thus rewrite Eq. (S8) as

$\displaystyle{R_{\rm g}}^{2}$ $\displaystyle=\frac{1}{2N_{\rm p}^{2}}\int
d\bm{R}\int d\bm{\mathsf{r}}G(|\mathsf{r}|)|\mathsf{r}|^{2}$ (S9)
$\displaystyle=\frac{V}{2N_{\rm p}^{2}}\int d\bm{r}G(|\bm{r}|)|\bm{r}|^{2}\;,$

where $V=\int d\bm{R}$ is the volume of the system. Because we have elected to
use smeared Gaussian packets instead of point particles in our field-theoretic
formulation to regularize short-range divergences as noted above, we will
approximate the $\delta$-function defined pair-correlation function defined
above by smeared densities instead. Accordingly, in the formulation below for
calculating of $R_{\rm g}$, we replace $G(|\bm{r}-\bm{r}^{\prime}|)$ in Eq.
(S8) by
$\mathcal{C}(|\bm{r}-\bm{r}^{\prime}|)=\langle\hat{\rho}(\bm{r})\hat{\rho}(\bm{r}^{\prime})\rangle$
where $\hat{\rho}(\bm{r})=\sum_{\alpha=1}^{N_{\rm
p}}\Gamma(\bm{r}-\bm{R}_{\alpha})$ as in Eq. (S7). Mathematically,
$\mathcal{C}(r)$ (where $r=|\bm{r}|$) is expected to be different from $G(r)$
only in these functions’ variations over short distances $\lesssim\bar{a}$.
Since $G(r)$ is integrated and weighted by $|\bm{r}|^{2}=r^{2}$ in Eq. S9 for
${R_{\rm g}}^{2}$, the replacement of $G(r)$ by $\mathcal{C}(r)$ is not
numerically significant for the accuracy of the computed value of $R_{\rm g}$.
For our purpose, however, the smearing that replace $G(r)$ by $\mathcal{C}(r)$
is important as it allows $G(r)$ and thus $R_{\rm g}$ to be computed
approximately using FTS.

By construction, Eq. S9 is applicable only for a single chain and needs to be
extended for a multi-chain system. To do so, consider $n_{\rm p}$ identical
chains in a polyampholyte solution. We now choose one chain at random and
recognize it as “tagged”. We denote the rest of the $n_{\rm p}-1$ chains as
“rest”. The total bead density can now be rewritten as
$\hat{\rho}_{\mathrm{tot}}(\bm{r})=\hat{\rho}^{(\mathrm{t})}(\bm{r})+\hat{\rho}^{(\mathrm{r})}(\bm{r})$
where the superscripts “(t)” and “(r)” refer to “tagged” and “rest”,
respectively. With this setup, we can now substitute the smeared self-
correlation function of the tagged chain $\mathcal{C}^{(\mathrm{t})}(r)$ for
$G(r)$ in Eq. S9 to find the tagged chain’s ${R_{\rm g}}^{2}$ in the multi-
chain system, and can simply set $n_{\rm p}=1$ in this general formulation to
recover the formula for the ${R_{\rm g}}^{2}$ of a single isolated chain. For
clarity, we will denote this latter isolated-chain smeared self-correlation
function by $\mathcal{C}^{(\mathrm{isol})}(r)$.

Correlation functions can also be utilized to compute residue-residue (bead-
bead, or monomer-monomer) contact maps. To do so we only have to identify the
monomers’ sequence positions along the polymer chains by rewriting the total
bead center density for a multi-chain system as
$\hat{\rho}_{\mathrm{c,tot}}(\bm{r})=\sum_{\alpha=1}^{N_{\rm
p}}\hat{\rho}_{\mathrm{c},\alpha}^{\mathrm{(t)}}(\bm{r})+\sum_{\alpha=1}^{N_{\rm
p}}\hat{\rho}_{\mathrm{c},\alpha}^{\mathrm{(r)}}(\bm{r})$, where
$\hat{\rho}_{\mathrm{c},\alpha}^{\mathrm{(t/r)}}(\bm{r})$ is sequence-
position-specific density of the center of the $\alpha$th bead in the “tagged”
chain/“rest” chains. As specified above for Eq. S8, unlike the smeared
$\hat{\rho}(\bm{r})$ defined above, the bead center density
$\hat{\rho}_{\mathrm{c}}(\bm{r})$ (with subscript “c”) is defined by
$\delta$-function of position without $\Gamma$ smearing. We use unsmeared bead
center densities for the computation of contacts because contacts are defined
by spatial separations between bead centers. Now, the corresponding bead-
specific unsmeared correlation functions can be defined as
$G^{(x),(y)}_{\alpha,\beta}(|\bm{R}|)=\left\langle\hat{\rho}_{\mathrm{c},\alpha}^{(x)}(\bm{r})\hat{\rho}_{\mathrm{c},\alpha}^{(y)}(\bm{r}+\bm{R})\right\rangle$
where $\\{x,y\\}\in\\{\mathrm{t,r}\\}$. With this definition, we can compute
the average contact frequency between the $(x,\alpha)$ and $(y,\beta)$
monomers (beads) by spatially integrating the corresponding
$G^{(x),(y)}_{\alpha,\beta}(|\bm{R}|)$ from radial distance $|\bm{R}|=0$ up to
a suitably chosen cutoff in $|\bm{R}|$ for defining a contact, with the $x=y$
and $x\neq y$ cases accounting, respectively, for intrachain and interchain
contacts. For the present work, we adopt the definitions

$\displaystyle\omega_{\alpha,\beta}^{\mathrm{(t),(t)}}$
$\displaystyle=V\int_{0}^{2b}d\bm{r}\;G^{(\mathrm{t}),(\mathrm{t})}_{\alpha,\beta}(|\bm{r}|)\;,$
(S10a) $\displaystyle\omega_{\alpha,\beta}^{\mathrm{(t),(r)}}$
$\displaystyle=V\int_{0}^{2b}d\bm{r}\;G^{\mathrm{(t),(r)}}_{\alpha,\beta}(|\bm{r}|)\;,$
(S10b)

as intrachain (Eq. S10a) and interchain (Eq. S10b) contact frequencies,
wherein a radial cutoff distance of $2b$ is used for defining a bead-bead
spatial contact.

Key steps in the field-theoretic simulation (FTS). Using the expression for
the Hamiltonian in Eq. S6, the canonical partition function of our system of
interest is given by

$\mathcal{Z}=\frac{1}{n_{\rm p}!}\left(\prod_{i=1}^{n_{\rm
p}}\prod_{\alpha=1}^{N_{\rm p}}\int
d\bm{R}_{i,\alpha}\right)e^{-\beta\hat{H}}\;.$ (S11)

Following standard procedures,FredricksonGanesanDrolet2002 ; Fredrickson2006 ;
Pal2021 ; MiMB2023 ; WessenDasPalChan2022  we derive a field theory described
by a field Hamiltonian $H$ (different from $\hat{H}$) such that the partition
function itself remains same as Eq. S11 up to an inconsequential overall
multiplicative constant, viz.,

$\mathcal{Z}=\frac{V^{n_{\rm p}}}{n_{\rm
p}!}\int\mathcal{D}w\int\mathcal{D}\psi e^{-H[w,\psi]},$ (S12)

in which two fluctuating auxiliary fields $w$ and $\psi$ are introduced, with

$H[w,\psi]=\int
d\bm{r}\left[\frac{w(\bm{r})^{2}}{2v}+\frac{(\bm{\nabla}\psi(\bm{r}))^{2}}{8\pi
l_{\rm B}}\right]-n_{\rm p}\ln Q_{\rm p}[\breve{w},\breve{\psi}]\;,$ (S13)

where $\breve{\phi}(\bm{r})=\Gamma\star\phi\equiv\int
d\bm{r}^{\prime}\Gamma(\bm{r}-\bm{r}^{\prime})\phi(\bm{r}^{\prime})$,
$\phi=w,\psi$, with “$\star$” denoting this spatial convolution henceforth. In
Eq. S13, the single-chain partition function $Q_{\rm p}$ is given by

$Q_{\rm p}[\breve{w},\breve{\psi}]=\frac{1}{V}\int d\bm{R}_{N_{\rm
p}}\left(\prod_{\alpha=1}^{N_{\rm p}-1}\int
d\bm{R}_{\alpha}G^{0}(\bm{R}_{\alpha+1}-\bm{R}_{\alpha}|b)\right)e^{-\mathrm{i}\sum_{\alpha=1}^{N_{\rm
p}}\left[\breve{w}(\bm{R}_{\alpha})+\sigma_{\alpha}\breve{\psi}(\bm{R}_{\alpha})\right]}\;,$
(S14)

where $\mathrm{i}^{2}=-1$ is the imaginery unit and

$G^{0}(\bm{r}|b)=\left(\frac{3}{2\pi
b^{2}}\right)^{3/2}e^{-3|\bm{r}|^{2}/2b^{2}}.$ (S15)

Based on this particle-to-field reformulation, the equilibrium properties of
the system can now be studied through the dynamics of the fields. Since the
field Hamiltonian $H$ involves complex variables, we adoptFredrickson2006  a
“Complex Langevin” (CL) prescriptionParisi1983 ; Klauder1983  inspired by
stochastic quantizationParisiWu1981 ; ChanHalpern1986  for computing averages
for the system defined by Eq. S13 by introducing dependence on a fictitious
time to the fields, which then evolve dynamically in accordance to the
Langevin equations

$\displaystyle\frac{\partial w(\bm{r},t)}{\partial t}$
$\displaystyle=-\left[\mathrm{i}\tilde{\rho}_{\mathrm{tot}}(\bm{r},t)+\frac{w(\bm{r},t)}{v}\right]+\eta_{w}(\bm{r},t)\;,$
(S16) $\displaystyle\frac{\partial\psi(\bm{r},t)}{\partial t}$
$\displaystyle=-\left[\mathrm{i}\tilde{c}(\bm{r},t)-\frac{\bm{\nabla}^{2}\psi(\bm{r},t)}{4\pi
l_{\rm B}}\right]+\eta_{\psi}(\bm{r},t)\;,$

where $t$ is a fictitious time, $\eta_{w}$ and $\eta_{\psi}$ are fields of
real-valued random numbers drawn from a normal distribution of zero mean and
standard deviation that is nonzero only when $t=t^{\prime}$ and
$\bm{r}=\bm{r}^{\prime}$ [$\propto
2\delta(t-t^{\prime})\delta(\bm{r}-\bm{r}^{\prime})$, specifically,
$\langle\eta_{\phi}(\bm{r},t)\eta_{\phi^{\prime}}(\bm{r}^{\prime},t^{\prime})\rangle=2\delta_{\phi,\phi^{\prime}}\delta(\bm{r}-\bm{r}^{\prime})\delta(t-t^{\prime})$].
This is the basic Langevin formulation of FTS. In Eq. S16,
$\tilde{\rho}_{\mathrm{tot}}$ and $\tilde{c}$ are field operators for total
bead density and charge density, respectively, which may be expressed as

$\displaystyle\tilde{\rho}_{\mathrm{tot}}(\bm{r})$
$\displaystyle=\Gamma\star\frac{n_{\rm p}}{Q_{\rm p}V}\sum_{\alpha=1}^{N_{\rm
p}}q_{B}(\bm{r},\alpha)q_{F}(\bm{r},\alpha)e^{\mathrm{i}\breve{w}(\bm{r})+\mathrm{i}\sigma_{\alpha}\breve{\psi}(\bm{r})}\;,$
(S17) $\displaystyle\tilde{c}(\bm{r})$ $\displaystyle=\Gamma\star\frac{n_{\rm
p}}{Q_{\rm p}V}\sum_{\alpha=1}^{N_{\rm
p}}\sigma_{\alpha}q_{B}(\bm{r},\alpha)q_{F}(\bm{r},\alpha)e^{\mathrm{i}\breve{w}(\bm{r})+\mathrm{i}\sigma_{\alpha}\breve{\psi}(\bm{r})}\;,$

where the forward ($F$) and backward ($B$) chain propagators $q_{F}$ and
$q_{B}$ are constructed iteratively:

$\displaystyle q_{F}(\bm{r},\alpha+1)$
$\displaystyle=e^{-\mathrm{i}\breve{w}(\bm{r})-\mathrm{i}\sigma_{\alpha+1}\breve{\psi}(\bm{r})}\int
d\bm{r}^{\prime}G^{0}(\bm{r}-\bm{r}^{\prime}|b)q_{F}(\bm{r}^{\prime},\alpha)\;,$
(S18) $\displaystyle q_{B}(\bm{r},\alpha-1)$
$\displaystyle=e^{-\mathrm{i}\breve{w}(\bm{r})-\mathrm{i}\sigma_{\alpha-1}\breve{\psi}(\bm{r})}\int
d\bm{r}^{\prime}G^{0}(\bm{r}-\bm{r}^{\prime}|b)q_{B}(\bm{r}^{\prime},\alpha)\;,$

with
$q_{F}(\bm{r},1)=e^{-\mathrm{i}\breve{w}(\bm{r})-\mathrm{i}\sigma_{1}\breve{\psi}(\bm{r})}$
and $q_{B}(\bm{r},N_{\rm
p})=e^{-\mathrm{i}\breve{w}(\bm{r})-\mathrm{i}\sigma_{N_{\rm
p}}\breve{\psi}(\bm{r})}$.

In practice, the differential field evolution equations Eqs. S16 have to be
numerically solved in discretized space and discretized CL time. For the
present work, every FTS is conducted in a $32\times 32\times 32$ cubic grid
with a side length $32\bar{a}$ and we set $\bar{a}=b/\sqrt{6}$ following
previous works. A CL time step $dt=0.005$ and a semi-implicit CL time
integration schemesii  are employed. To compute the thermal average of any
thermodynamic observable by FTS, the particle-based thermal average has to be
replaced by an average over the field configurations of the corresponding
field operator. The field operators could be derived by introducing
appropriate “source” terms (fields) in the particle picture Hamiltonian (“$J$”
fields, see below), as is commonly practiced in theoretical particle
physics.IZ  If an observable $\hat{\mathcal{O}}(\bm{r})$ has a field operator
$\tilde{\mathcal{O}}(w,\psi)$, then in practice the equilibrium field average
of that specific operator is evaluated as an asymptotic CL time average
through

$\left\langle\hat{\mathcal{O}}(\\{\bm{R}_{i,\alpha}\\})\right\rangle=\left\langle\tilde{\mathcal{O}}[w,\psi]\right\rangle_{\rm
F}=\frac{\int\mathcal{D}w\int\mathcal{D}\psi\;\tilde{\mathcal{O}}(w,\psi)e^{-H[w,\psi]}}{\int\mathcal{D}w\int\mathcal{D}\psi\;e^{-H[w,\psi]}}=\frac{1}{M}\sum_{j=1}^{M}\tilde{\mathcal{O}}[w(\bm{r},t_{j}),\psi(\bm{r},t_{j})]\;,$
(S19)

where $M$ is the maximum number of field configurations considered
(configurations are labeled by $j$ in the above equation for selected values
$t_{j}$ of fictitious time $t$) and $\langle\cdot\cdot\cdot\rangle_{\rm F}$
denotes field averaging. The configurations at $t_{j}$s and the total number
of configurations $M$ used in the averaging are chosen such that the real part
of the field averaged quantity has a reasonably small fluctuation over
independent simulation runs and the corresponding imaginary part is
approximately zero. For all the FTS computation, we set the excluded volume
parameter $v=0.0068b^{3}$. Further details of our approach can be found in
ref. (68).

FTS operators for correlation functions in the computation of polymer radius
of gyration. As outlined above in the discussion preceding Eq. S10, to compute
equilibrium properties of individual polymer chains in a multiple-chain
system, we consider one chain, chosen at random, as tagged. The total bead
density in Eq. S7 can then be rewritten as

$\hat{\rho}_{\mathrm{tot}}(\bm{r})=\hat{\rho}^{\mathrm{(t)}}(\bm{r})+\hat{\rho}^{\mathrm{(r)}}(\bm{r})\;,$
(S20)

where the superscripts “(t)” and “(r)” denote ‘tagged’ and ‘rest’,
respectively, with $\hat{\rho}^{\mathrm{(t)}}(\bm{r})$ for the tagged chain
and $\hat{\rho}^{\mathrm{(r)}}(\bm{r})$ for the rest of the chains (untagged
chains) given by

$\displaystyle\hat{\rho}^{\mathrm{(t)}}(\bm{r})$
$\displaystyle=\sum_{\alpha=1}^{N_{\rm
p}}\Gamma\left(\bm{r}-\bm{R}^{\mathrm{(t)}}_{\alpha}\right)\;,$ (S21)
$\displaystyle\hat{\rho}^{\mathrm{(r)}}(\bm{r})$
$\displaystyle=\sum_{i=1}^{n_{\rm p}-1}\sum_{\alpha=1}^{N_{\rm
p}}\Gamma\left(\bm{r}-\bm{R}^{\mathrm{(r)}}_{i,\alpha}\right)\;.$

Field operators for total bead densities of “tagged” and “rest” chains are
identified by introducing the source terms $J^{\mathrm{(t)}}$ and
$J^{\mathrm{(r)}}$ in the original particle-picture Hamiltonian in Eq. S6 in
the following manner:

$\beta\hat{{H}}\rightarrow\beta\hat{{H}}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]=\beta\hat{H}-\int
d\bm{r}J^{\mathrm{(t)}}(\bm{r})\hat{\rho}^{\mathrm{(t)}}(\bm{r})-\int
d\bm{r}J^{\mathrm{(r)}}(\bm{r})\hat{\rho}^{\mathrm{(r)}}(\bm{r})\;.$ (S22)

The resulting source-dependent partition function is now given by

$\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]\sim\left(\prod_{\alpha=1}^{N_{\rm
p}}\int d\bm{R}^{\mathrm{(t)}}_{\alpha}\right)\left(\prod_{i=1}^{n_{\rm
p}-1}\prod_{\alpha=1}^{N_{\rm p}}\int
d\bm{R}^{\mathrm{(r)}}_{i,\alpha}\right)e^{-\beta\hat{{H}}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}\;.$
(S23)

From this last expression (Eq. S23), we obtain the formal relations

$\displaystyle\left\langle\hat{\rho}^{\mathrm{(t)}}(\bm{r})\right\rangle$
$\displaystyle=\left(\frac{1}{\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}\frac{\delta\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}{\delta
J^{\mathrm{(t)}}(\bm{r})}\right)_{J^{\mathrm{(t)}}=J^{\mathrm{(r)}}=0}\;,$
(S24)
$\displaystyle\left\langle\hat{\rho}^{\mathrm{(r)}}(\bm{r})\right\rangle$
$\displaystyle=\left(\frac{1}{\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}\frac{\delta\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}{\delta
J^{\mathrm{(r)}}(\bm{r})}\right)_{J^{\mathrm{(t)}}=J^{\mathrm{(r)}}=0}\;,$

where $\langle...\rangle$ denotes thermal averaging. In the corresponding
field theory representation, we have

$\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]\sim\int\mathcal{D}w\int\mathcal{D}\psi
e^{-{H}[w,\psi,J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}\;,$ (S25)

where

$\displaystyle{H}[w,\psi,J^{\mathrm{(t)}},J^{\mathrm{(r)}}]=$
$\displaystyle\int
d\bm{r}\left[\frac{w(\bm{r})^{2}}{2v}+\frac{(\bm{\nabla}\psi(\bm{r}))^{2}}{8\pi
l_{\rm B}}\right]$ (S26) $\displaystyle\quad\quad\quad\quad-\ln Q_{\rm
p}[\breve{w}-\mathrm{i}\breve{J}^{\mathrm{(t)}},\breve{\psi}]-(n_{\rm p}-1)\ln
Q_{\rm p}[\breve{w}-\mathrm{i}\breve{J}^{\mathrm{(r)}},\breve{\psi}]\;.$

In this equation (Eq. S26), $Q_{\rm p}$ has the same functional form as that
in Eq. S14. We can now apply the definitions Eq. S24 to Eq. S25 to arrive at
$\langle\hat{\rho}^{\mathrm{(t)}}(\bm{r})\rangle=\langle\tilde{\rho}^{\mathrm{(t)}}(\bm{r})\rangle_{\mathrm{F}}$
and
$\langle\hat{\rho}^{\mathrm{(r)}}(\bm{r})\rangle=\langle\tilde{\rho}^{\mathrm{(r)}}(\bm{r})\rangle_{\mathrm{F}}$
where

$\displaystyle\tilde{\rho}^{\mathrm{(t)}}(\bm{r})$
$\displaystyle=\frac{1}{n_{\mathrm{p}}}\tilde{\rho}_{\mathrm{tot}}(\bm{r})\;,$
(S27) $\displaystyle\tilde{\rho}^{\mathrm{(r)}}(\bm{r})$
$\displaystyle=\frac{n_{\mathrm{p}}-1}{n_{\mathrm{p}}}\tilde{\rho}_{\mathrm{tot}}(\bm{r})\;,$

and $\tilde{\rho}_{\mathrm{tot}}(\bm{r})$ is defined in Eq. S17 and, again,
$\langle...\rangle_{\mathrm{F}}$ represents average over field configurations.

We can also compute the self-correlation function of the tagged chain from the
partition function in Eq. S23 through the formal relation

$\mathcal{C}^{\mathrm{(t)}}(|\bm{r}-\bm{r}^{\prime}|)=\left\langle\hat{\rho}^{\mathrm{(t)}}(\bm{r})\hat{\rho}^{\mathrm{(t)}}(\bm{r}^{\prime})\right\rangle=\left(\frac{1}{\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}\frac{\delta^{2}\mathcal{Z}[J^{\mathrm{(t)}},J^{\mathrm{(r)}}]}{\delta
J^{\mathrm{(t)}}(\bm{r})\delta
J^{\mathrm{(t)}}(\bm{r}^{\prime})}\right)_{J^{\mathrm{(t)}}=J^{\mathrm{(r)}}=0}.$
(S28)

Applying this relation (Eq. S28) to Eq. S23 results in

$\mathcal{C}^{\mathrm{(t)}}(|\bm{r}-\bm{r}^{\prime}|)=\frac{\mathrm{i}}{v}\left\langle
w(\bm{r})\tilde{\rho}_{\mathrm{tot}}(\bm{r}^{\prime})\right\rangle_{\mathrm{F}}-\left\langle\tilde{\rho}^{\mathrm{(t)}}(\bm{r})\tilde{\rho}^{\mathrm{(r)}}(\bm{r}^{\prime})\right\rangle_{\mathrm{F}}\;,$
(S29)

where, following ref. (30), we have avoided computing a functional double
derivative of $Q_{\rm p}$.

When there is only a single isolated chain in the system, the self-correlation
function is obtained by setting $n_{\rm p}=1$ and
$\tilde{\rho}^{\mathrm{(r)}}(\bm{r}^{\prime})=0$ in the above derivation,
yielding

$\mathcal{C}^{\mathrm{(isol)}}(|\bm{r}-\bm{r}^{\prime}|)=\frac{\mathrm{i}}{v}\left\langle
w(\bm{r})\tilde{\rho}_{\mathrm{tot}}(\bm{r}^{\prime})\right\rangle_{\mathrm{F}}\;.$
(S30)

Utilizing translational invariance of our systems, we can write

$\displaystyle\left\langle\hat{\rho}(\bm{r})\hat{\rho}(\bm{r}^{\prime})\right\rangle$
$\displaystyle=\frac{1}{V}\int
d\bm{r}^{\prime\prime}\left\langle\hat{\rho}(\bm{r}+\bm{r}^{\prime\prime})\hat{\rho}(\bm{r}^{\prime}+\bm{r}^{\prime\prime})\right\rangle$
(S31)
$\displaystyle=\frac{1}{V}\int\frac{d\bm{k}}{(2\pi)^{3}}e^{-\mathrm{i}\bm{k}\cdot(\bm{r}-\bm{r}^{\prime})}\left\langle\check{\rho}(\bm{k})\check{\rho}(-\bm{k})\right\rangle,$

where, when the density smearing described above is applied,
$\check{\rho}(\bm{k})\equiv\int
d\bm{r}e^{{\mathrm{i}}\bm{k}\cdot\bm{r}}\hat{\rho}(\bm{r})$ is the Fourier
transform of the smeared density $\hat{\rho}(\bm{r})$. In that case, it
follows that $\check{\rho}(\bm{k})=\check{\Gamma}(\bm{k})\check{\rho}_{\rm
c}(\bm{k})$ where $\check{\Gamma}(\bm{k})=e^{-\bar{a}^{2}|\bm{k}|^{2}/2}$ is
the Fourier transform of the smearing function $\Gamma(\bm{r})$ and
$\check{\rho}_{\rm c}(\bm{k})$ is the Fourier transform of the bead center
density $\hat{\rho}_{\rm c}(\bm{r})$. This implies that when Gaussian smearing
is utilized in our formulation, the bead center-bead center pair correlation
function [$G(r)\rightarrow\mathcal{C}(r)$] can now be expressed as

$\left\langle\hat{\rho}_{\rm c}(\bm{r})\hat{\rho}_{\rm
c}(\bm{r}^{\prime})\right\rangle=\frac{1}{V}\int\frac{d\bm{k}}{(2\pi)^{3}}e^{-\mathrm{i}\bm{k}\cdot(\bm{r}-\bm{r}^{\prime})}\frac{\left\langle\check{\rho}(\bm{k})\check{\rho}(-\bm{k})\right\rangle}{\check{\Gamma}(\bm{k})^{2}}\;.$
(S32)

FTS operators for correlation functions in the computation of contact maps of
polymers. As noted above, the starting point for contact map computations is
the (unsmeared) bead center density correlation functions. As noted above in
the discussion preceding Eq. S10, we first rewrite the total bead center
density as

$\hat{\rho}_{\mathrm{c,tot}}(\bm{r})=\hat{\rho}_{\rm
c}^{\mathrm{(t)}}(\bm{r})+\hat{\rho}_{\rm c}^{\mathrm{(r)}}(\bm{r}),$ (S33)

where $\hat{\rho}_{\rm c}^{\mathrm{(t)}}(\bm{r})$ and $\hat{\rho}_{\rm
c}^{\mathrm{(r)}}(\bm{r})$ could be expressed in terms of the bead-position-
specific densities along the chains through

$\displaystyle\hat{\rho}_{\rm c}^{\mathrm{(t)}}(\bm{r})$
$\displaystyle=\sum_{\alpha=1}^{N_{\rm
p}}\hat{\rho}_{\mathrm{c},\alpha}^{\mathrm{(t)}}(\bm{r})=\sum_{\alpha=1}^{N_{\rm
p}}\delta\left(\bm{r}-\bm{R}^{\mathrm{(t)}}_{\alpha}\right)\;,$ (S34)
$\displaystyle\hat{\rho}_{\rm c}^{\mathrm{(r)}}(\bm{r})$
$\displaystyle=\sum_{\alpha=1}^{N_{\rm
p}}\hat{\rho}_{\mathrm{c},\alpha}^{\mathrm{(r)}}(\bm{r})=\sum_{i=1}^{n_{\rm
p}-1}\sum_{\alpha=1}^{N_{\rm
p}}\delta\left(\bm{r}-\bm{R}^{\mathrm{(r)}}_{i,\alpha}\right)\;.$

We now introduce monomer (residue)-specific source terms in the interaction
Hamiltonian Eq. S6:

$\beta\hat{{H}}\rightarrow\beta\hat{{H}}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]=\beta\hat{H}-\int
d\bm{r}\sum_{\alpha=1}^{N}J^{\mathrm{(t)}}_{\alpha}(\bm{r})\hat{\rho}^{\mathrm{(t)}}_{\mathrm{c},\alpha}(\bm{r})-\int
d\bm{r}\sum_{\alpha=1}^{N}J^{\mathrm{(r)}}_{\alpha}(\bm{r})\hat{\rho}^{\mathrm{(r)}}_{\mathrm{c},\alpha}(\bm{r})\;.$
(S35)

The corresponding partition function is

$\mathcal{Z}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]\sim\left(\prod_{\alpha=1}^{N_{\rm
p}}\int d\bm{R}^{\mathrm{(t)}}_{\alpha}\right)\left(\prod_{i=1}^{n_{\rm
p}-1}\prod_{\alpha=1}^{N_{\rm p}}\int
d\bm{R}^{\mathrm{(r)}}_{i,\alpha}\right)e^{-\beta\hat{{H}}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]}\;.$
(S36)

From this expression (Eq. S36), the average density of the $\alpha$th bead
center is formally

$\left\langle\hat{\rho}^{(x)}_{\mathrm{c},\alpha}(\bm{r})\right\rangle=\left(\frac{1}{\mathcal{Z}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]}\frac{\delta\mathcal{Z}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]}{\delta
J^{(x)}_{\alpha}(\bm{r})}\right)_{\\{J_{\alpha}^{\mathrm{(t)}}\\}=\\{J_{\alpha}^{\mathrm{(r)}}\\}=0}\;,$
(S37)

and the correlation function between any two bead center is given by

$G^{(x),(y)}_{\alpha,\beta}(|\bm{r}-\bm{r}^{\prime}|)=\left\langle\hat{\rho}^{(x)}_{\mathrm{c},\alpha}(\bm{r})\hat{\rho}^{(y)}_{\mathrm{c},\beta}(\bm{r}^{\prime})\right\rangle=\left(\frac{1}{\mathcal{Z}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]}\frac{\delta\mathcal{Z}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]}{\delta
J^{(x)}_{\alpha}(\bm{r})\delta
J^{(y)}_{\beta}(\bm{r})}\right)_{\\{J_{\alpha}^{\mathrm{(t)}}\\}=\\{J_{\alpha}^{\mathrm{(r)}}\\}=0}$
(S38)

where $\\{x,y\\}\in\\{\mathrm{t,r}\\}$. The corresponding partition function
in the field picture is then given by

$\mathcal{Z}[\\{J_{\alpha}^{\mathrm{(t)}}\\},\\{J_{\alpha}^{\mathrm{(r)}}\\}]\sim\int\mathcal{D}w\int\mathcal{D}\psi
e^{-{H}[w,\psi,\\{J_{\alpha}^{\mathrm{(t)}}\\},\\{J_{\alpha}^{\mathrm{(r)}}\\}]}$
(S39)

where

$\displaystyle{H}[w,\psi,\\{J_{\alpha}^{\mathrm{(t)}}\\},\\{J_{\alpha}^{\mathrm{(r)}}\\}]=$
$\displaystyle\int
d\bm{r}\left[\frac{w(\bm{r})^{2}}{2v}+\frac{(\bm{\nabla}\psi(\bm{r}))^{2}}{8\pi
l_{\rm B}}\right]-\ln Q_{\rm
p}[\breve{w}-\mathrm{i}\\{J_{\alpha}^{\mathrm{(t)}}\\},\breve{\psi}]$ (S40)
$\displaystyle-(n_{\rm p}-1)\ln Q_{\rm
p}[\breve{w}-\mathrm{i}\\{J_{\alpha}^{\mathrm{(r)}}\\},\breve{\psi}]\;,$

and the absence of breve in the source terms on the right hand side in the
above equation means that the source terms are not smeared. In Eq. S40,
$Q_{\rm p}$ has the same functional form as that in Eq. S14, which can be
written explicitly as

$\displaystyle Q_{\rm
p}[\breve{w}-\mathrm{i}\\{\breve{J}_{\alpha}\\},\breve{\psi}]$
$\displaystyle=\frac{1}{V}\int d\bm{R}_{N_{\rm p}}e^{-\mathrm{i}\Psi_{N_{\rm
p}}}\int d\bm{R}_{N_{\rm p}-1}G^{0}_{N_{\rm p},N_{\rm
p}-1}e^{-\mathrm{i}\Psi_{N_{\rm p}-1}}\cdot\cdot\cdot$ (S41)
$\displaystyle\quad\quad\quad\cdot\cdot\cdot\int
d\bm{R}_{\alpha}G^{0}_{\alpha+1,\alpha}e^{-\mathrm{i}\Psi_{\alpha}}\cdot\cdot\cdot\int
d\bm{R}_{1}G^{0}_{2,1}e^{-\mathrm{i}\Psi_{1}}\;,$

where
$\Psi_{\alpha}=\breve{w}(\bm{R}_{\alpha})+\sigma_{\alpha}\breve{\psi}(\bm{R}_{\alpha})+\mathrm{i}J_{\alpha}(\bm{R}_{\alpha})$,
and

$G^{0}_{\alpha+1,\alpha}\equiv G^{0}(\bm{R}_{\alpha+1}-\bm{R}_{\alpha}|b)$
(S42)

with $G^{0}$ defined by Eq. S15. Substituting Eq. S37 into Eq. S39 then
results in
$\langle\hat{\rho}^{\mathrm{(t)}}_{\mathrm{c},\alpha}(\bm{r})\rangle=\langle\tilde{\rho}^{\mathrm{(t)}}_{\mathrm{c},\alpha}(\bm{r})\rangle_{\rm
F}$ and
$\langle\hat{\rho}^{\mathrm{(r)}}_{\mathrm{c},\alpha}(\bm{r})\rangle=\langle\tilde{\rho}^{\mathrm{(r)}}_{\mathrm{c},\alpha}(\bm{r})\rangle_{\rm
F}$, with

$\displaystyle\tilde{\rho}^{\mathrm{(t)}}_{\mathrm{c},\alpha}(\bm{r})$
$\displaystyle=\frac{1}{Q_{\rm
p}[\breve{w},\breve{\psi}]V}q_{B}(\bm{r},\alpha)q_{F}(\bm{r},\alpha)e^{\mathrm{i}\breve{w}(\bm{r})+\mathrm{i}\sigma_{\alpha}\breve{\psi}(\bm{r})}\;,$
(S43) $\displaystyle\tilde{\rho}^{\mathrm{(r)}}_{\mathrm{c},\alpha}(\bm{r})$
$\displaystyle=\frac{n_{\rm p}-1}{Q_{\rm
p}[\breve{w},\breve{\psi}]V}q_{B}(\bm{r},\alpha)q_{F}(\bm{r},\alpha)e^{\mathrm{i}\breve{w}(\bm{r})+\mathrm{i}\sigma_{\alpha}\breve{\psi}(\bm{r})}\;.$

As discussed above in conjunction with eq. S10b, the numbers of interchain
contacts and thus interchain contact maps are determined in our formulation
via the cross correlation function between the $\alpha$th bead center of the
tagged chain and the $\beta$th bead center of the rest of the chains. This
cross correlation is readily obtained by taking double functional derivatives
of the partition function in Eq. S39 in accordance with Eq. S38 to yield

$G^{\mathrm{(t)},\mathrm{(r)}}_{\alpha,\beta}(|\bm{r}-\bm{r}^{\prime}|)=\langle\tilde{\rho}^{\mathrm{(t)}}_{\mathrm{c},\alpha}(\bm{r})\tilde{\rho}^{\mathrm{(r)}}_{\mathrm{c},\beta}(\bm{r}^{\prime})\rangle_{\rm
F}\;$ (S44)

for the calculation of interchain numbers of contacts in Eq. S10b. Similarly,
for the numbers of intrachain contacts and intrachain contact maps, we
compute, in accordance with Eq. S10a, the correlations between the beads of
the tagged chain as follows:

$\displaystyle G^{\mathrm{(t)},\mathrm{(t)}}_{\alpha,\beta}(|\bm{r}|)$
$\displaystyle=\frac{1}{V}\int
d\bm{R}\left\langle\hat{\rho}^{\mathrm{(t)}}_{\mathrm{c},\alpha}(\bm{R})\hat{\rho}^{\mathrm{(t)}}_{\mathrm{c},\beta}(\bm{r}+\bm{R})\right\rangle$
(S45) $\displaystyle=\frac{1}{V}\int
d\bm{R}\left(\frac{1}{\mathcal{Z}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]}\frac{\delta\mathcal{Z}[\\{J^{\mathrm{(t)}}_{\alpha}\\},\\{J^{\mathrm{(r)}}_{\alpha}\\}]}{\delta
J^{\mathrm{(t)}}_{\alpha}(\bm{R})\delta
J^{\mathrm{(t)}}_{\beta}(\bm{r}+\bm{R})}\right)_{\\{J_{\alpha}^{\mathrm{(t)}}\\}=\\{J_{\alpha}^{\mathrm{(r)}}\\}=0}$
$\displaystyle=\frac{1}{V}\int d\bm{R}\left(\frac{1}{Q_{\rm
p}[\breve{w}-\mathrm{i}\\{J_{\alpha}^{\mathrm{(t)}}\\},\breve{\psi}]}\frac{\delta^{2}Q_{\rm
p}[\breve{w}-\mathrm{i}\\{J_{\alpha}^{\mathrm{(t)}}\\},\breve{\psi}]}{\delta
J^{\mathrm{(t)}}_{\alpha}(\bm{R})\delta
J^{\mathrm{(t)}}_{\beta}(\bm{R}+\bm{r})}\right)_{\\{J_{\alpha}^{\mathrm{(t)}}\\}=0}$
$\displaystyle=\left\langle\tilde{G}^{\mathrm{(t)},\mathrm{(t)}}_{\alpha,\beta}(\bm{r})\right\rangle_{\rm
F}\;,$

where

$\displaystyle\tilde{G}^{\mathrm{(t)},\mathrm{(t)}}_{\alpha,\beta}(\bm{r})$
$\displaystyle=\frac{1}{V}\int d\bm{R}\left[\frac{1}{VQ_{\rm
p}[\breve{w},\breve{\psi}]}\int d\bm{R}^{\mathrm{(t)}}_{N_{\rm
p}}e^{-\mathrm{i}\Psi_{N_{\rm p}}(\bm{R}^{\mathrm{(t)}}_{N_{\rm p}})}\int
d\bm{R}^{\mathrm{(t)}}_{N_{\rm p}-1}G^{0}_{N_{\rm p},N_{\rm
p}-1}e^{-\mathrm{i}\Psi_{N_{\rm p}-1}(\bm{R}^{\mathrm{(t)}}_{N_{\rm
p}-1})}\cdots\right.$ (S46) $\displaystyle\left.\quad\quad\cdot\cdot\cdot\int
d\bm{R}^{\mathrm{(t)}}_{2}G^{0}_{3,2}e^{-\mathrm{i}\Psi_{2}(\bm{R}^{\mathrm{(t)}}_{2})}\int
d\bm{R}^{\mathrm{(t)}}_{1}G^{0}_{2,1}e^{-\mathrm{i}\Psi_{1}(\bm{R}^{\mathrm{(t)}}_{1})}\right]\delta(\bm{R}-\bm{R}^{\mathrm{(t)}}_{\alpha})\delta(\bm{R}+\bm{r}-\bm{R}^{\mathrm{(t)}}_{\beta})$
$\displaystyle=\frac{1}{VQ_{\rm
p}[\breve{w},\breve{\psi}]}\int\frac{d\bm{k}}{(2\pi)^{3}}\left[\frac{1}{V}\int
d\bm{R}^{\mathrm{(t)}}_{N_{\rm p}}e^{-\mathrm{i}\Psi_{N_{\rm
p}}(\bm{R}^{\mathrm{(t)}}_{N_{\rm p}})}\int d\bm{R}^{\mathrm{(t)}}_{N_{\rm
p}-1}G^{0}_{N_{\rm p},N_{\rm p}-1}e^{-\mathrm{i}\Psi_{N_{\rm
p}-1}(\bm{R}^{\mathrm{(t)}}_{N_{\rm p}-1})}\cdots\right.$
$\displaystyle\left.\quad\quad\cdot\cdot\cdot\int
d\bm{R}^{\mathrm{(t)}}_{2}G^{0}_{3,2}e^{-\mathrm{i}\Psi_{2}(\bm{R}^{\mathrm{(t)}}_{2})}\int
d\bm{R}^{\mathrm{(t)}}_{1}G^{0}_{2,1}e^{-\mathrm{i}\Psi_{1}(\bm{R}^{\mathrm{(t)}}_{1})}\right]e^{\mathrm{i}\bm{k}\cdot(\bm{R}^{\mathrm{(t)}}_{\beta}-\bm{R}^{\mathrm{(t)}}_{\alpha}-\bm{r})}$
$\displaystyle=\frac{1}{VQ_{\rm
p}}\int\frac{d\bm{k}}{(2\pi)^{3}}e^{-\mathrm{i}\bm{k}\cdot\bm{r}}Q^{(\alpha,\beta)}_{\rm
p}(\bm{k})\;,$

with
$\Psi_{\alpha}=\breve{w}(\bm{R}^{\mathrm{(t)}}_{\alpha})+\sigma_{\alpha}\breve{\psi}(\bm{R}^{\mathrm{(t)}}_{\alpha})$.
In the last Eq. S46, $Q^{(\alpha,\beta)}_{\rm p}(\bm{k})$ is a
$\bm{k}$-dependent (Fourier-transformed) single chain restricted partition
function, given by

$\displaystyle Q^{(\alpha,\beta)}_{\rm p}(\bm{k})$
$\displaystyle=\left[\frac{1}{V}\int d\bm{R}^{\mathrm{(t)}}_{N_{\rm
p}}e^{-\mathrm{i}\Psi_{N_{\rm p}}(\bm{R}^{\mathrm{(t)}}_{N_{\rm p}})}\int
d\bm{R}^{\mathrm{(t)}}_{N_{\rm p}-1}G^{0}_{N_{\rm p},N_{\rm
p}-1}e^{-\mathrm{i}\Psi_{N_{\rm p}-1}(\bm{R}^{\mathrm{(t)}}_{N_{\rm
p}-1})}\right.$ (S47) $\displaystyle\left.\quad\quad\cdot\cdot\cdot\int
d\bm{R}^{\mathrm{(t)}}_{2}G^{0}_{3,2}e^{-\mathrm{i}\Psi_{2}(\bm{R}^{\mathrm{(t)}}_{2})}\int
d\bm{R}^{\mathrm{(t)}}_{1}G^{0}_{2,1}e^{-\mathrm{i}\Psi_{1}(\bm{R}^{\mathrm{(t)}}_{1})}\right]e^{\mathrm{i}\bm{k}\cdot(\bm{R}^{\mathrm{(t)}}_{\beta}-\bm{R}^{\mathrm{(t)}}_{\alpha})}\;.$

This implies that in order to evaluate
$\tilde{G}^{\mathrm{(t)},\mathrm{(t)}}_{\alpha,\beta}(\bm{r})$ in Eq. S46, we
need to first evaluate $Q^{(\alpha,\beta)}_{\rm p}(\bm{k})$ at each $\bm{k}$
and then perform an inverse Fourier transform according to the last equality
in Eq. S46. Therefore, for a $N_{\rm L}\times N_{\rm L}\times N_{\rm L}$
spatial $(x,y,z)$ lattice used for FTS, ideally we should calculate the values
of $Q^{(\alpha,\beta)}_{\rm p}({\bm{k}})$ on a (reciprocal) ${\bm{k}}$-space
lattice of the same size, i.e., determine $Q^{(\alpha,\beta)}_{\rm
p}({\bm{k}})$ $(N_{\rm L})^{3}$ times for each $(\alpha,\beta)$ pair for each
field configuration (Eq. S47). This would be exceedingly computationally
intensive. For computational efficiency, we consider instead a slightly more
coarse-grained $\bm{k}$-space lattice of dimensions $N_{\bm{k}}\times
N_{\bm{k}}\times N_{\bm{k}}$ with $N_{\bm{k}}=N_{\rm L}/2$, such that the
resolution of its reciprocal space, i.e., the original $(x,y,z)$ space,
becomes $(\Delta k_{x},\Delta k_{y},\Delta
k_{z})\equiv\left(4\pi/L,4\pi/L,4\pi/L\right)$ with $L=N_{\rm L}\bar{a}$.
Nonetheless, we still compute the $\left(\prod_{\alpha}\int
d\bm{R}_{\alpha}\right)$ volume integrals in Eq. (S47) in the original $N_{\rm
L}\times N_{\rm L}\times N_{\rm L}$ spatial lattice with resolution $(\Delta
x,\Delta y,\Delta z)=(\bar{a},\bar{a},\bar{a})$. In this way, we reduce the
computational cost by $\sim 8$ times. Thus, for all the intrachain FTS contact
maps presented here, $N_{\rm L}=32$, $N_{\bm{k}}=16$, and the final
equilibrium field configurations of $32$ independent runs are used for field
averaging. As a check on the accuracy of this coarse-graining, we have also
computed contact maps with $N_{\bm{k}}=8$ reciprocal lattice and saw no
visible difference from the $N_{\bm{k}}=16$ results.

Volume integration in simulation boxes with periodic boundary conditions. To
perform volume integration of an isotropic function $f(r)$ in a $L\times
L\times L$ cubic box ($r\equiv|\bm{r}|$) with periodic boundary conditions,
the integration measure $dV$ has to be modified owing to the periodic boundary
conditions, as follows:

$\displaystyle dV(r)$ $\displaystyle=$
$\displaystyle\left\\{\begin{matrix}4\pi r^{2}dr\,,&0\leq r\leq L/2\,,\\\ 2\pi
r(3L-4r)dr\,,&L/2<r\leq\sqrt{2}L/2\,,\\\
2rL[3\pi-12g_{1}(r/L)+g_{2}(r/L)]dr\,,&\sqrt{2}L/2<r\leq\sqrt{3}L/2\,,\\\
0,&r>\sqrt{3}L/2\,,\end{matrix}\right.$ (S48)

where

$\displaystyle g_{1}(x)$ $\displaystyle=$
$\displaystyle\tan^{-1}\sqrt{4x^{2}-2}\,,$ $\displaystyle g_{2}(x)$
$\displaystyle=$ $\displaystyle
8x\left\\{\tan^{-1}\left[\frac{2x(4x^{2}-3)}{\sqrt{4x^{2}-2}(4x^{2}+1)}\right]\right\\}\,,$
(S49)

such that integration of any isotropic function $f(r)$ from $r=0$ to any upper
integration limit $r=r_{\rm max}$ on the periodic lattice is specified by

$\int_{|\bm{r}|\leq r_{\rm max}}d\bm{r}f(r)=\int_{0}^{r_{\rm max}}dV(r)f(r)\;$
(S50)

with $dV(r)$ defined in Eqs. S48 and S0.Ex1 above.

Numerical estimation of $T^{*}_{\rm cr}$ in FTS Binodal phase boundaries in
FTS are computed using the method described in ref. (68). In the present
study, to estimate the critical temperature $T^{*}_{\rm cr}$ from the FTS
binodals, we adopt—as in recent coarse-grained MD studies of biomolecular
condensatessuman2 ; dignon18 —the scaling approach outlined in ref. (61),
which assumes that low- and high-density phase concentrations, denoted by
$\rho_{\rm L}$ and $\rho_{\rm H}$, respectively, follow the relations

$\displaystyle\frac{\rho_{\rm H}+\rho_{\rm L}}{2}$ $\displaystyle=\rho_{\rm
cr}+A(T^{*}_{\rm cr}-T^{*})\;,$ (S51a) $\displaystyle\rho_{\rm H}-\rho_{\rm
L}$ $\displaystyle=\Delta\rho_{0}\left(1-\frac{T^{*}}{T^{*}_{\rm
cr}}\right)^{\nu}\;,$ (S51b)

where $\nu=0.325$, and $A$, $\Delta\rho_{0}$, critical density $\rho_{\rm
cr}$, and $T^{*}_{\rm cr}$ are free fitting parameters. Now, for each
sequence, we use the highest two simulated $T^{*}$ for fitting. First, we
estimate $\Delta\rho_{0}$ and $T^{*}_{\rm cr}$ by fitting the numerical values
of $(\rho_{\rm H}-\rho_{\rm L})$ to Eq. S51b. Next, we apply this fitted
$T^{*}_{\rm cr}$ to Eq. (S51a) to fit the numerical values of $(\rho_{\rm
H}+\rho_{\rm L})/2$ to yield fitted values for $\rho_{\rm cr}$ and $A$. These
fitted parameters are then applied to obtain $\rho_{\rm H}$ and $\rho_{\rm L}$
as functions of $T^{*}$ from the two relations in Eq. S51. In Fig. S2 (see
below) and maintext Fig. 1d, these fitted functions are used to construct
continuous curves through the fitted $(\rho_{\rm cr},T^{*}_{\rm cr})$ critical
point and the four simulated binodal $(\rho_{\rm L},T^{*})$ and $(\rho_{\rm
H},T^{*})$ datapoints with the two highest $T^{*}$, whereas the numerical FTS
$(\rho_{\rm L},T^{*})$ and $(\rho_{\rm H},T^{*})$ datapoints for the dilute
and dense branches of the binodals for the rest of the simulated $T^{*}$ are
simply connected by lines as guides for the eye.

Coarse-grained molecular dynamics (MD) model of polyampholyte conformations
and phase separation

The coarse-grained MD model here is essentially identical to the “hard-core
repulsion” model we used previously in ref. (37) for the simulation of
“sv”rohit2013  and “as”suman2  polyampholyte sequences. The only difference is
that the repulsive part of the Lennard-Jones (LJ) potential is set to zero for
$r>2^{1/6}a$ in the present study—where $a$ is the reference (equilibrium)
bond length between successive beads (monomers) along the chain
sequence—instead of for $r>a$ in ref. (37). MD simulation in the present work
is carried out using the protocol described in refs. (37; 48). A contact is
defined to exist between two monomers when their center-of-mass spatial
separation is within $2a$.

Table S1: All 26 polyampholyte sequences studied in this work. Following
previous notation,rohit2013 ; kings2015 ; lin2017 ; joanJPCL2019 ; suman2
positively and negatively charged beads along the sequences are symbolized
here, respectively, by “K” (lysine, charge $+1$) and “E” (glutamic acid,
charge $-1$).
Name |  Sequence
---|---
sv1 | EKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEKEK
sv2 | EEEKKKEEEKKKEEEKKKEEEKKKEEEKKKEEEKKKEEEKKKEEEKKKEK
sv10 | EKKKKKKEEKKKEEEEEKKKEEEKKKEKKEEKEKEEKEKKEKKEEKEEEE
sv12 | EKKEEEEEEKEKKEEEEKEKEKKEKEEKEKKEKKKEKKEEEKEKKKKEKK
sv15 | KKEKKEKKKEKKEKKEEEKEKEKKEKKKKEKEKKEEEEEEEEKEEKKEEE
sv17 | EKEKKKKKKEKEKKKKEKEKKEKKEKEEEKEEKEKEKKEEKKEEEEEEEE
sv20 | EEKEEEEEEKEEEKEEKKEEEKEKKEKKEKEEKKEKKKKKKKKKKKKEEE
sv23 | EEEEEKEEEEEEEEEEEKEEKEKKKKKKEKKKKKKKEKEKKKKEKKEEKK
sv24 | EEEEKEEEEEKEEEEEEEEEEEEKKKEEKKKKKEKKKKKKKEKKKKKKKK
sv28 | EKKKKKKKKKKKKKKKKKKKKKEEEEEEEEEEEEEEEEEEKKEEEEEKEK
sv29 | KEEEEKEEEEEEEEEEEEEEEEEEEEEKKKKKKKKKKKKKKKKKKKKKKK
sv30 | EEEEEEEEEEEEEEEEEEEEEEEEEKKKKKKKKKKKKKKKKKKKKKKKKK
as1 | KKKKKEKKKKKEKKKKEKKKKEKKEKEEEKEEEEKEEEEKEEEEKEEEEE
as2 | EEEEEEEEKKEKKKKEEEEEEEEEEEKKKKEEEKEKEKKKKKKKKKKKKK
as3 | KKKKEEEEEEEEEEEEEEEEEEKKEKKKKKKKKKKKKKKEKEEEEEKKKK
as4 | KKKKKKEEEEEEEEEEEEEEEKKKKKKKKKKKKKKKKKKEEEEEEEEEEK
c$\kappa$1 | KKKEEEEEEEEKKKKKKKKKKEEEEEEEEEEKKKKKKKKKEEKEEEKEKE
c$\kappa$2 | KKKEEEEEEEKKEEEEEKKKEKKKKKKKKEEEEEEEEEEEEKKKKKKKKK
c$\kappa$3 | KKKKKKKKKKKKKKKEKKKKKEKKEEEEKEEEEEKEEEEEKEEEEEEEEE
c$\kappa$4 | KKKKKKKKKKEKKKKKEKKKKEKKKKEEKEEEEEKEEEEEEEEEEEEEEE
cSCD1 | KKKKEKKKKEEKKKKKKKEEKKKKKEEEEKKEEEEEEEKEEKEEEKEEEE
cSCD2 | KEKKKKKKKKEEKKKKKKEEKKKKKEEEEEKKEEEEEEEEKKKEEEEEEE
cSCD3 | EEEEEEEKKKKKKKKKKKKKKKEEEKKKKKKKKKKEEEEEEEEEEEEEEE
cSCD4 | EEEEEKKKKKKKKKKKKKKKKKKKKEEEEEEEEEEEEEEEEEEEEKKKKK
obs1 | KKKKKKKKKKEEEEEEEEEEEEKKKKKEEEEEEEEEEEEEKKKKKKKKKK
ebs1 | EEEEEEEEEKKKKKKKKEEEEEEEEKKKKKKKKEEEEEEEEKKKKKKKKK

Supporting Figures

Fig. S1: Genetic-algorithm (GA) and Wang-Landau (WL) sampling of the
SCD–$\kappa$ sequence space of net-neutral 50mer polyampholytes. (a) Left–
Results of the diversity-enhanced GA scan. Black region represents the region
found by the GA to be populated by sequences. Bins containing target $({\rm
SCD},\kappa)$ values for which no sequence was found by the GA are shown in
white. Red bins indicate bins mislabelled by the GA as empty, i.e., bins for
which the GA was not able to find a sequence but that the subsequent WL
calculation revealed to be non-empty. Other regions where no sequence was
found by either GA or WL are in gray. Right– Joint distribution $P({\rm
SCD},\kappa)$ computed using the WL algorithm. (b) The 26 net-neutral
polyampholyte sequences studied in this work (same as those depicted in
maintext Fig. 1b), their $\kappa$, SCD values as well their critical
temperatures in the FTS, RPA, and MD models. Positively and negatively charged
monomers along the sequences are represented, respectively, by blue and red
beads.

Fig. S2: Phase diagrams of all 26 polyampholytes we consider in the FTS, RPA,
and MD models. The phase diagram for the baseline homopolymer model in MD is
shown as an inset in the bottom-right panel. Results in maintext Fig. 1d,e are
included here as well for completeness and to facilitate comparison.

Fig. S3: Variation of polyampholyte root-mean-square radius of gyration in the
FTS (top) and MD (bottom) models with SCD (left), $\kappa$ (right), and
temperature (as indicated). Unless specified otherwise, root-mean-square
radius of gyration, $\sqrt{{R_{\rm g}}^{2}}$, is denoted simply as $R_{\rm g}$
for notational simplicity in the present study. Results in maintext Fig.
2a,b,e,f are included here as well to facilitate comparison.

Fig. S4: Contact patterns of polyampholytes as an isolated chain versus in the
multiple-chain condensed phase. Results are presented in the same style as
that in maintext Fig. 3b,c for the sequence obs1. The corresponding results
for all the other 25 polyampholyte sequences are provided here. To facilitate
comparison, the obs1 contact patterns in maintext Fig. 3b,c are shown here
again as the last item in this supporting figure (1st page, to be cont’d).

Fig. S4: (2nd page, cont’d).

Fig. S4: (3rd page, cont’d).

Fig. S4: (4th page, cont’d).

Fig. S4: (5th page, cont’d).

Fig. S4: (6th page, cont’d).

Fig. S4: (7th page, cont’d). This is the last page of this supporting figure.

Fig. S5: Intrachain and interchain contact frequencies and intrachain root-
mean-square distance $R_{\alpha\beta}$ of polyampholytes, in essentially the
same style as that in maintext Fig. 3e–g for the sequence obs1. The
corresponding results for all the other 25 polyampholyte sequences are
provided here. The data here in columns (a) and (b) for other sequences
correspond, respectively, to the contact frequency data in maintext Fig. 3e
and 3f. Similarly, the data here in columns (c) and (d) correspond,
respectively to the $R_{\alpha\beta}$ and
$R_{\alpha\beta}/\sqrt{|\alpha-\beta|}$ data in maintext Fig. 3g, with the
dashed curves in Fig. 3g now replaced by solid curves in column (d). In (d),
positive slopes at large $|\alpha-\beta|$ observed for some sequences such as
sv28, sv29, sv30, as1, c$\kappa$4, and cSCD1 indicate that their condensed-
phase conformations are more expanded than Gaussian chains (1st page, to be
cont’d).

Fig. S5: (2nd page, cont’d).

Fig. S5: (3rd page, cont’d).

Fig. S5: (4th page, cont’d).

Fig. S5: (5th page, cont’d). This is the last page of this supporting figure.

Fig. S6: Temperature- and sequence-pattern-dependent relationship between
root-mean-square radii of gyration of isolated versus condensed-phase
polyampholyte chains in the MD model. Temperatures for root-mean-square
$R_{\rm g}$ values are color-coded as indicated on the right. Datapoints for
the same polyampholyte sequence (as labeled in different colors) are connected
by lines with matching color. Lines are otherwise merely guides for the eye.
Note that the range of $R_{\rm g}$ variation in the condensed phase ($\sim
0.3b$, vertical axis) is small compared to the corresponding variation for
isolated chains ($\sim 0.7b$, horizontal axis). Whereas isolated-chain $R_{\rm
g}$s uniformly increase with increasing $T^{*}$, condensed-phase $R_{\rm g}$s
for different sequences can increase or decrease (albeit only slightly) with
increasing $T^{*}$. The physical origin of this sequence-specific feature and
its possible biophysical ramifications should be further investigated in
future studies.

Fig. S7: Comparing contact patterns in the MD model using symmetrized
Kullback-Leibler divergences. (a) $D_{s,s^{\prime}}$ between the
$|\alpha-\beta|\geq d_{\rm m}$ intrachain contacts of two separately isolated
polyampholyte sequences $s$ and $s^{\prime}$ versus $s$–$s^{\prime}$ SCD
difference ($s,s^{\prime}=$ all 26 sequences in Fig. S1). Results are shown
for two temperatures $T^{*}=0.2$ and $0.5$ and different cutoffs $d_{\rm
m}=3$–$6$ for local contacts, with $s=s^{\prime}$ datapoints identified by
black circles and the value for baseline homopolymer at $T^{*}=2.0$ shown as a
red star. (b,c) Same as maintext Fig. 4a and Fig. 4b (which are for
$T^{*}_{\rm iso.}=0.2$ and $d_{\rm m}=4$), respectively, now including data
also for $T^{*}_{\rm iso.}=0.5$ and $d_{\rm m}=3,5$, and $6$.

Fig. S8: Variation in the number of isolated-chain intrachain contacts
$n_{c}^{[{\rm i}]}$ (horizontal axes) with the number of condensed-phase
intrachain contacts $n_{c}^{[{\rm c}]}$ (vertical axis, top panels) and the
number of condensed-phase interchain contacts $N_{c}^{[{\rm c}]}$ (vertical
axis, bottom panels) in the MD model. Results are present in the same style as
that for maintext Fig. 4c,d. In addition to the $T^{*}_{\rm iso.}=0.2$ data in
maintext Fig. 4c,d, corresponding data for $T^{*}_{\rm iso.}=0.5$, $1.0$, and
$5.0$ are also provided here.

Fig. S9: Relationship between condensed-phase contact frequencies and
interaction energies with phase-separation propensity in the MD model. (a)
Scatter plots of isolated-chain and condensed-phase potential energies
$E^{[{\rm i}]}$ and energies $E^{[{\rm c}]}$ (both in units of $\epsilon$) at
two simulation temperatures as indicated. The $T^{*}_{\rm iso.}=0.2$
datapoints are identical to those provided in maintext Fig. 4e, and are
included here to facilitate comparison with the $T^{*}_{\rm iso.}=0.5$
datapoints. (b) Critical temperature $T^{*}_{\rm cr}$ for all sequences except
sv1 in maintext Fig. 1b and Fig. S1. Symbols for sequences are the same as
those in maintext Fig. 1a.

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