Subcompartmentalization of polyampholyte species in organelle-like condensates is promoted by charge pattern mismatch and strong excluded-volume interaction
SSubcompartmentalization of polyampholyte species inorganelle-like condensates is promoted by charge pattern mismatchand strong excluded-volume interaction
Tanmoy Pal, Jonas Wessén, Suman Das, and Hue Sun Chan ∗ Department of Biochemistry, University of Toronto,Toronto, Ontario M5S 1A8, Canada (Dated: June 23, 2020)
Abstract
Polyampholyte field theory and explicit-chain molecular dynamics models of sequence-specificphase separation of a system with two intrinsically disordered protein (IDP) species indicate con-sistently that a substantial polymer excluded volume and a significant mismatch of the IDP sequencecharge patterns can act in concert, but not in isolation, to demix the two IDP species upon con-densation. This finding reveals an energetic-geometric interplay in a stochastic, “fuzzy” molecularrecognition mechanism that may facilitate subcompartmentalization of membraneless organelles. a r X i v : . [ q - b i o . B M ] J un ntroduction. —Liquid-liquid phase separation (LLPS) [1–5] in biomolecular condensates [6]has garnered intense interest in diverse areas of biomedicine, biophysics, and polymerphysics [7]. In the cellular environment, LLPS is a significant factor in the assembly ofcompartments, sometimes referred to as membraneless organelles, that act as hubs for bio-chemical processes and physiological regulation. These droplet-like structures coexist witha more dilute milieu. Examples include nucleoli, P-bodies, stress granules, cajal bodies, inan expanding list due to rapid experimental advance. Biomolecular condensates are criticalfor organismal function and thus their misregulation can cause diseases such as Alzheimer’s,Parkinson’s and amyotrophic lateral sclerosis [8, 9].Functional biomolecular LLPS often involves intrinsically disordered proteins (IDPs) andnucleic acids engaging in multivalent interactions [10, 11]. Although much remains to beelucidated, recent theories and computations have shed light on the physics of how LLPSsof IDPs are governed by their amino acid sequences. These efforts include analytical theory[12–14], explicit-chain lattice [15–17] and continuum molecular dynamics (MD) [18–20] sim-ulations, and field-theoretic simulation (FTS) [21–23] of LLPS, investigations of the relation-ship between LLPS propensity and single/double-chain properties [24, 25] as well as crystalsand filaments formation [26], and studies of the peculiar temperature [27] and pressure [28]dependence of biomolecular LLPS as well as finite-size scaling in droplet formation [29].Reviews of the emerging theoretical perspectives are available in Refs. [30–34].IDPs are enriched in charged and polar residues [35] and multivalent electrostatics is animportant driving force—among others [36]—for LLPS. One consistent finding from theory[12], chain simulation [16, 19] and FTS [22, 23] is that the LLPS propensity of a polyam-pholyte depends on its sequence charge pattern, which may be quantified by an intuitiveblockiness κ measure [37] or an analytic “sequence charge decoration” (SCD) parameter [38]that correlates with single-chain properties [37–39]. This perspective was applied to providea quantitative account [12] for the LLPS of RNA helicase Ddx4 [5].In contrast to simple laboratory systems that may contain only one IDP type (species),large numbers of different types of IDPs and other biomolecules interact in the cell while com-partmentalizing into a variety of different condensates. For some membraneless organelles,LLPS-mediated organization of intracellular space goes a step further by subcompartmen-talization [40]. Well-known examples include the nucleolus comprising of at least threesubcompartments enriched with distinct sets of proteins [15, 41] and stress granules with2 dense core surrounded by a liquid-like outer shell [42]. These phenomena of IDP com-partmentalization and subcompartmentalization raise intriguing physics questions as to thenature of the sequence-specific interactions that drive a subset of IDPs in a condensate tocoalesce among themselves while excluding other types of IDPs.Important insights into formation of subcompartments [15, 43] and general principlesof many-component phase behaviors [44] have been gained from simulated lattice systemswith energies assigned to favor or disfavor pairwise interactions between specific solute com-ponents. These approaches do not address, however, how those interactions arise fromelementary physical forces in a sequence-dependent manner. The first attempt to tackle thisfundamental question uses random phase approximation (RPA) [12, 45] to model LLPS oftwo polyampholytic IDP species. Sequence-specific molecular recognition is seen as arisingfrom elementary electrostatic interactions in a stochastic, “fuzzy” manner, in that the IDPspecies are predicted to demix upon LLPS when their sequence charge patterns are signif-icantly different (large difference in their SCD values), but tend to be miscible when theirSCD values are similar [46]. TABLE I. Hamiltonians used in this work; β = k B T , where k B is Boltzmann’s constant and T isabsolute temperature. ˆ H ˆ H ˆ H FTS: b β (cid:80) p,i,α | r p,i,α +1 − r p,i,α | v β (cid:82) d r (cid:82) d r (cid:48) ˆ ρ b ( r ) δ ( r − r (cid:48) )ˆ ρ b ( r (cid:48) ) l B β (cid:82) d r (cid:82) d r (cid:48) ˆ ρ c ( r )ˆ ρ c ( r (cid:48) ) | r − r (cid:48) | MD: K b (cid:80) p,i,α ( | r p,i,α +1 − r p,i,α |− a ) (cid:15) (cid:80) p,i,α, (cid:54) = q,j,γ (cid:104)(cid:16) r | r p,i,α − r q,j,γ | (cid:17) − (cid:16) r | r p,i,α − r q,j,γ | (cid:17) (cid:105) l B β (cid:80) p,i,α, (cid:54) = q,j,γ σ p,α σ q,γ | r p,i,α − r q,j,γ | Nonetheless, a definitive delineation of the roles of sequence charge pattern and poly-mer excluded volume in the mixing/demixing of polyampholytes upon LLPS is yet to beachieved because bead-bead excluded volume was not fully accounted for in RPA [46] andcomparisons with explicit-chain MD suggest that diminished excluded volume can lead toartefactually high LLPS propensity [19]. In this Letter, these issues are elucidated us-ing complementary methods of FTS and MD to model polyampholytes with short-rangeexcluded volume repulsion and long-range Coulomb interaction. By construction, FTS ismore accurate than RPA in the field-theoretic context if discretization and finite-volumeerrors can be neglected, whereas MD is more suitable for chemically realistic interactionsand its microscopic structural information is accessible. Although our models are both3ighly coarse-grained, the collective behaviors studied are not expected to be too sensitiveto microscopic details. Surprisingly, both models indicate that while charge pattern mis-match is necessary for polyampholyte demixing, the degree of demixing is highly sensitiveto the excluded-volume interaction strength, underscoring that excluded volume is a criticalorganizing principle not only for folded protein structures [47, 48] and disordered proteinconformations [49–51] but also for biomolecular condensates.Here we study binary mixtures of two species of fully charged, overall neutral bead-springpolyampholytes differing only in their charge patterns, defined by the set of positions r p,i,α of bead α on chain i of type (species) p with corresponding electric charges σ p,α for all i .The sequences considered (Fig. 1) are representative of the set of 50mer “sv sequences”, usedextensively for modeling [22, 24, 38, 46], that are listed in ascending κ values from the leastblocky, strictly alternating sv1 to the most blocky diblock sequence sv30 [37]. The FTS andMD Hamiltonians, ˆ H = ˆ H + ˆ H + ˆ H , are given in Table I. The chain connectivity term ˆ H takes the usual Gaussian form with Kuhn length b for FTS and the harmonic form withforce constant K b for MD (thus b corresponds to a ); the excluded-volume term ˆ H entailsa δ -function with strength v for FTS [22, 52] and a Lennard-Jones (LJ) potential with welldepth (cid:15)/ for MD [19]; whereas electrostatics is provided by ˆ H with Bjerrum length l B . E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K E K sv1
E K K K K K K E E K K K E E E E E K K K E E E K K K E K K E E K E K E E K E K K E K K E E K E E E E sv10
K K E K K E K K K E K K E K K E E E K E K E K K E K K K K E K E K K E E E E E E E E K E E K K E E E sv15
E E E E E E E E E E E K E E E E K E E K E E K E K K K K K K K K K K K K K K K K K K E E K K E E K E sv25
E K K K K K K K K K K K K K K K K K K K K K E E E E E E E E E E E E E E E E E E K K E E E E E K E K sv28
FIG. 1. Polyampholytes studied in this work. Blue/red beads of “K”s (lysines)/“E”s (glutamicacids) carry ± protonic charges. The sv labels are those of Ref. [37]. FTS. —The FTS interaction strengths are controlled by v and l B (Table I). Following stan-dard prescription, we express ˆ H and ˆ H in terms of ˆ ρ b ( r )= (cid:80) p ˆ ρ b ,p ( r ) , ˆ ρ c ( r )= (cid:80) p ˆ ρ c ,p ( r ) where ˆ ρ b ,p and ˆ ρ c ,p are, respectively, the microscopic bead (matter) and charge densitiesof polymer type p . The individual beads are modelled as normalized Gaussian distribu-tions Γ( r )=exp( − r / a ) / (2 πa ) / centered at positions r p,i,α [53, 54] such that ˆ ρ b ,p ( r )= (cid:80) i,α Γ( r − r p,i,α ) , ˆ ρ c ,p ( r )= (cid:80) i,α σ p,α Γ( r − r p,i,α ) . As in recent works [21, 22], we set the smear-ing length a = b/ √ .The canonical partition function for this system can be converted, through standard4ethods (see e.g. [55]), to that of a statistical field theory with Hamiltonian H [ w, Φ]= 1 β F (cid:40) − (cid:88) p n p ln Q p [i ˘ w, i ˘Φ]+ (cid:90) d r (cid:18) w v + ( ∇ Φ) πl B (cid:19)(cid:41) , (1)where β F ( (cid:54) = β ) is the reciprocal temperature in the field picture, n p is the number of moleculesof type p in the system, i = − . Here Q p [i ˘ w, i ˘Φ] is the partition function of a single polymer oftype p , subject to external chemical and electrostatic potential fields i ˘ w ≡ Γ (cid:63) i w and i ˘Φ ≡ Γ (cid:63) iΦ ,respectively, with ‘ (cid:63) ’ denoting spatial convolution.In our lattice simulations of this system, the continuum fields are approximated by discretefield variables defined on a simple cubic lattice (mesh) with periodic boundary conditions.Equilibrium dynamics entailed by β F H [ w, Φ] is simulated using a Complex-Langevin (CL)prescription [56–59], with CL-time evolution governed by ∂ϕ ( r ,t ) /∂t = − δβ F H/δϕ ( r ,t )+ η ϕ ( r ,t ) for ϕ = w, Φ , which we integrate numerically using the first-order semi-implicit methodof [60] with an appropriate real-valued Gaussian noise η ϕ , and ensemble averages, (cid:104) ... (cid:105) , arecomputed as asymptotic CL-time averages.Because the bead positions r p,i,α are traded in FTS for the w ( r ) , Φ( r ) fields as ther-modynamic degrees of freedom, information about spatial structure of the chains has to begleaned from functionals of { ˆ ρ b ,p } , with well-defined corresponding field operators, such asthe pair-distribution functions (PDFs), G p,q ( | r − r (cid:48) | )= (cid:104) ˆ ρ b ,p ( r ) ˆ ρ b ,q ( r (cid:48) ) (cid:105) , (2)between various p,q bead types. Both inter- ( p (cid:54) = q ) and intra ( p = q ) species PDFs are neededto characterize structural organization of different species. For instance, an intra species G p,p ( r ) peaking at small r and decays to at large r implies a relatively dense region, i.e.,a droplet, of p ; and demixing of two species p and q is signalled by G p,p ( r ) and G q,q ( r ) dominating over G p,q ( r ) at small r .We examine systematically the interplay of charge pattern and excluded volume in themixing/demixing of phase-separated polyampholytes by FTS of binary mixtures of foursequence pairs with p = sv28 ( − SCD = . ), q = sv1, sv10, sv15, sv25 ( − SCD = . , . , . , . ), bulk monomer densities ρ p = ρ q =0 . b − and a large l B =5 b to ensure LLPSin all cases, each at excluded-volume strengths v/b =0 . , . , . and . . The5 r/b . . . . . . . G p , q / b − sv28sv1 (a) v = 0 . b r/b sv28sv25 (b) v = 0 . b r/b sv28sv1 (c) v = 0 . b .
00 0 .
03 0 .
06 0 .
09 0 . v/b . . . . . . ξ p , q sv1sv10sv15sv25 (d) FIG. 2. FTS-computed PDFs and mixing parameter ξ p,q for binary sv sequence mixtures. (a–c)Each G p,p , G q,q (dashed, in color) and G p,q (solid, black) for the indicated v is computed usinga periodic mesh averaged over 30–40 independent runs (standard errors comparable to theplotting line width). Inset are illustrative snapshots of the real non-negative part of the densityoperators i n p δ ln Q p [i ˘ w, i ˘Φ] /δw ( r ) and i n q δ ln Q q [i ˘ w, i ˘Φ] /δw ( r ) for bead types p and q (see Supple-mentary Material) depicted in different colors; the component species in the same snapshot areshown separately on the side. (d) ξ p,q is computed using a periodic mesh (averaged over 70–80independent runs, solid lines) as well as the mesh (dashed lines) used for (a–c). Error barsrepresents standard errors of the mean. latter three v values are 5, 10 and 15 times the smallest v/b =0 . , often used in FTS asa relatively poor solvent condition [21–23] favorable to LLPS [61].PDFs indicate that significant charge pattern mismatch and strong v are both necessaryfor demixing. Representative results are shown in Fig. 2 (FTS details and all PDFs wecomputed are in the Supplementary Material). The strongest demixing is observed for sv28–sv1 with large charge pattern mismatch (SCDs differ by 15.58) at relatively high v values;e.g., for v =0 . b , G sv1 , sv28 ( r ) takes much lower values than G sv1 , sv1 ( r ) and G sv28 , sv28 ( r ) as r → (Fig.2a), indicating that some of the sv1 chains are expelled from the sv28-denseregion. Even when a single droplet is formed, it harbors sub-regions where either sv28 or sv1 dominates (snapshot in Fig.2a). However, when v decreases to . b , all three G s forsv28–sv1 share similar profiles, implying that the common droplet is well mixed (Fig.2c).In contrast, for sv28-sv25 with similar charge patterns (SCDs differ by . ), mixing in thephase-separated droplet remains substantial even at higher v (Fig. 2b). The general trendis summarized by the mixing parameter (Fig. 2d) ξ p,q ≡ ρ p ρ q G p,q (0)( ρ q ) G p,p (0)+( ρ p ) G q,q (0) , (3)which vanishes for two perfectly demixed species, because in that case at least one of the6actors in ˆ ρ b ,p ( r ) ˆ ρ b ,q ( r ) would be zero for any r , whereas ξ p,q =1 when ˆ ρ b ,p ( r ) ∝ ˆ ρ b ,q ( r ) , i.e.,when the species are perfectly mixed. MD. —While field theory affords deep physical insights, its ability to capture certainstructure-related features pertinent to polyampholyte LLPS, such as the interplay betweenexcluded volume and Coulomb interactions, can be limited [19]. Therefore, to assess therobustness of the above FTS-predicted trend, we now turn to coarse-grained explicit-chainMD for additional information.The complete coarse-grained MD potential is given in Table I. We simulate binary mix-tures of the same sv sequence pairs as with FTS, using a recently developed protocol involv-ing initial compression and subsequent expansion of a periodic simulation box to facilitateefficient equilibrium Langevin dynamics sampling for LLPS studies [18, 19, 62]. Each of ourMD systems contains 500 chains equally divided between the two sv sequences (250 chainseach). The LJ parameter (cid:15) that governs excluded volume is set at (cid:15) = l B /a (corresponding tothe “with 1/3 LJ” prescription in [19]), T ∗ ≡ ( β(cid:15) ) − is reduced temperature, and a stiff forceconstant K b =75 , (cid:15)/a for polymer bonds is employed as in [19, 62]. We compare resultsfrom using van der Waals radius r = a (as before [19]) and r = a / to probe the effect ofexcluded volume. Simulations are conducted at T ∗ =0 . and T ∗ =4 . , which is, respectively,below and above the LLPS critical temperatures of all sv sequences in Fig. 1, and at anintermediate T ∗ . Further details are in the Supplementary Material.A substantive difference between common FTS and MD is in their treatment of poly-mer excluded volume, as illustrated in Fig. 3a for the present models, wherein βV ex ( r ) is theexcluded-volume interaction, given by β ˆ H in Table I, for a pair of beads centered at r p,i,α and r q,j,γ , with r = | r p,i,α − r q,j,γ | . For our FTS model as well as several recent FTS studies [21–23], βV ex ( r )=( v/ (cid:82) d r Γ( r − r p,i,α )Γ( r − r q,j,γ )=( v/ πa ) / exp( − r / a ) is a Gaussian, which al-lows the beads to overlap completely ( r =0 ), albeit with a reduced yet non-negligible or evenmoderately high probability. In contrast, for MD, βV ex ( r )=4 / T ∗ [( r /r ) − ( r /r ) ] , whichentails a repulsive wall at ∼ r that is all but impenetrable, let alone an excluded-volume-violating complete overlap. Note that if the βV ex ( r ) for MD is shown for T ∗ =0 . (as forFTS) instead of T ∗ =0 . in Fig. 3a, the contrast would be even sharper between FTS andMD excluded-volume prescriptions.Despite this difference, MD and FTS predictions on sequence-pattern and excluded-volume dependent population mixing/demixing upon LLPS share the same trend, and are7 − . − . − . − . . β V e x r/b r/a FTSMD(a) − −
40 0 40 80 z/a . . . . . . ρ / a − (b) sv28sv1 r = a − −
40 0 40 80 z/a . . . . . . . (c) sv28sv25 r = a − −
20 0 20 40 z/a . . . . . . . . (d) sv28sv1 r = a / r/a G p , q / a − × − (e) r/a . . . . . × − (f) r/a × − (g) FIG. 3. MD-simulated LLPS of binary sv sequence mixtures. (a) Excluded volume interactionsin FTS (blue) for v/b = 0 . , . , . , and . (top to bottom) at T ∗ =0 . (i.e., l B =5 b as in Fig. 2) and in MD (brown) for r = a (solid) and r = a / (dashed) at T ∗ =0 . (insets showrelative sizes of the LJ spheres). (b)–(d) MD-simulated polyampholyte densities of binary mixtures, ρ ( z ) s for different sv sequences are colored differently (as indicated) here and in the snapshots (onthe side) of the rectangular periodic simulation boxes (wherein z is the vertical coordinate), eachharboring a condensed droplet. (e)–(g) G p,q of the MD systems in (b)–(d), respectively, (sameline style as Fig. 2a–c). Droplet snapshots (insets) are visualized [63] here with chains at periodicboundaries unwrapped. quantitatively similar in some respects. Results for the sv28–sv1 and sv28–sv25 pairs areshown in Fig. 3b–g for T ∗ =0 . to illustrate a perspective that is buttressed by additionalMD results for other sequence pairs and other T ∗ in the Supplementary Material.Fig. 3b–d show the average densities ρ ( z ) along the long axis, z , of the rectangularsimulation box. With full excluded volume and significant charge pattern mismatch, sv28and sv1 strongly demix (cf. blue and red curves in Fig. 3b). In contrast, without a significantcharge pattern mismatch, even with full excluded volume, sv28 and sv25 are quite well mixed(blue and green curves largely overlap in Fig. 3c); and, with reduced excluded volume, evensv28 and sv1 with significant charge pattern mismatch are well mixed (Fig. 3d).This trend is echoed by the correlation functions in Fig. 3e–g, each computed from 10,000MD snapshots. For the well-mixed cases in Fig. 3f,g, the MD-computed self ( G p,p , G q,q ) andcross ( G p,q ) correlations largely overlap, similar to those in Fig. 2b,c for FTS. For the sv28–sv1 pair with full excluded volume in MD, Fig. 2e shows that G p,q ( r ) is significantly smallerthan G p,p ( r ) and G q,q ( r ) for small r , as in Fig. 2a for FTS. Here, the MD G q,q for sv1 exhibits8 local maximum at r ≈ a corresponding to the distance between two sv1 density peaksin Fig. 3b. This feature reflects the anisotropic nature of the rectangular simulation boxadopted to facilitate efficient sampling [62]. Nonetheless, the geometric arrangement of sv28and sv1 in the MD system, as visualized by the snapshot in Fig. 3e, is consistent with thatin Fig. 2a for FTS in that an s28-enriched core (blue) is surrounded by an sv1-enrich (red)periphery in both cases. The other MD snapshots in Fig. 3f,g depict well-mixed droplets,similar to the corresponding FTS snapshots in Fig. 2b,c. Discussion. —Excluded volume has been shown to attenuate complex [61] and simple [22]coacervation and to promote demixing when applied differentially to molecular componentsin a condensate [43]. Here, FTS and MD both demonstrate a hitherto unrecognized stochas-tic molecular recognition principle, that a uniform excluded volume not discriminating be-tween polymer species can nonetheless promote demixing, and that a certain thresholdexcluded volume is required for heteropolymers with different sequence charge patterns todemix upon LLPS. Our MD results show clearly that sequences such as sv28 and sv1 thatare not obviously repulsive to each other can nevertheless demix, supporting RPA predic-tions that demixing of different species of overall neutral polyampholytes depends on chargepattern mismatch [46]. In light of the present finding, this success of RPA in [46] maybe attributed to the incompressibility constraint—which presupposes excluded volume—inits formulation. Surprisingly, although the FTS excluded volume repulsion we consider isexceedingly weak—the highest v only amounts to ∼ . k B T maximum and thus can easilybe overcome by thermal fluctuations (Fig. 3a), the demixing observed in FTS with this v issimilar to that in MD with a much stronger, more realistic excluded volume. While the the-oretical basis of this reassuring agreement, e.g., its possible relationship with the treatmentof chain entropy in FTS, remains to be ascertained, our observation that sv28 and sv1 donot demix at a lower v points to potential limitations of employing small v values in FTS.These basic principles offer new physical insights into subcompartmentalization of mem-braneless organelles, in terms of not only the sequence charge patterns of their constituentIDPs [46], but also of excluded volumes entailed by amino acid sidechains of various sizes,volume increases due to posttranslational modifications such as phosphorylations [64], pres-ence of folded domains, and the solvation properties of the IDP linkers connecting thesedomains [2, 43]. Guided by this conceptual framework, quantitative applications to real-lifebiomolecular condensates require further investigations to consider sequences that are not9ecessarily overall charge neutral [14], and to incorporate non-electrostatic driving forcesfor LLPS such as π -related [36] and hydrophobic [20, 65] interactions. Much awaits to bediscovered.We thank Yi-Hsuan Lin for insightful discussions, and gratefully acknowledge support byCanadian Institutes of Health Research grant NJT-155930, Natural Sciences and Engineer-ing Research Council of Canada Discovery grant RGPIN-2018-04351, and computationalresources of Compute/Calcul Canada.T.P. and J.W. contributed equally to this work.10 upplementary Material for “Subcompartmentalization of polyampholyte species in organelle-likecondensates is promoted by charge pattern mismatch andstrong excluded-volume interaction”FIELD THEORETIC SIMULATIONSField operators corresponding to pair distribution functions Our target observables in the field theoretic simulations (FTS) are the pair distributionfunctions (PDFs), denoted as G p,q ( | r − r (cid:48) | ) and defined in the main text in terms of themicroscopic bead densities ˆ ρ b ,p ( r ) . We show below how PDFs can be computed as ensembleaverages of certain corresponding field operators in the field theoretic context. Throughoutthis section of Supplementary Material, we let (cid:104) ... (cid:105) P and (cid:104) ... (cid:105) F denote, respectively, averagesover bead centers (i.e., in the “particle picture”) and averages over field configurations (i.e.,in the “field picture”).Using the notation in the main text, we begin by considering the canonical partitionfunction expressed as integrals over the positions of bead centers, r p,i,α , in the particlepicture, with an added source field J p ( r ) for each bead type density as is commonly practicedin field theory to facilitate subsequent calculation of averages of functionals of ˆ ρ : Z [ { J p } ]= (cid:32) (cid:89) p,i,α (cid:90) d r p,i,α (cid:33) e − β ˆ H − β ˆ H − β ˆ H + (cid:82) d r (cid:80) p ˆ ρ b ,p ( r ) J p ( r ) . (S1)To avoid notational clutter, overall multiplicative constant factors in Z that are immaterialto the quantities computed in this work are not included in the mathematical expressions inthe present derivation. Using Eq. (S1), averages of products of bead densities can formallybe computed using functional derivatives of Z with respect to the source fields J p , thenfollowed by setting J p =0 for all p . In particular, G p,q ( | r − r (cid:48) | ) ≡(cid:104) ˆ ρ b ,p ( r ) ˆ ρ b ,q ( r (cid:48) ) (cid:105) P = lim J p ,J q → Z δδJ p ( r ) δδJ q ( r (cid:48) ) Z. (S2)We may now turn Eq. (S1) into a statistical field theory (see, e.g., Ref. [55] for detailedformulation) while still keeping the source fields. To this end, without loss of generality, we11rst multiply the right hand side of (from the left) by unity (‘1’) in the form of (cid:90) D ρ b ( r ) δ [ ρ b − ˆ ρ b ] (cid:90) D ρ c ( r ) δ [ ρ c − ˆ ρ c ] , after which we can make the replacements ˆ ρ b , c → ρ b , c in β ˆ H , because of the δ -functionals,which are then expressed in their equivalent Fourier forms, δ [ ρ b − ˆ ρ b ]= (cid:90) D w ( r )e i (cid:82) d r w ( ρ b − ˆ ρ b ) , δ [ ρ c − ˆ ρ c ]= (cid:90) D Φ( r )e i (cid:82) d r Φ( ρ c − ˆ ρ c ) , where i = − , to allow for an explicit functional integrals over the ρ b ( r ) and ρ c ( r ) variablesintroduced by the above ‘1’ factor. Up to a multiplicative constant, the result of thoseintegrations is the formula Z [ { J p } ]= (cid:90) D w ( r ) (cid:90) D Φ( r )exp (cid:34)(cid:88) p n p ln Q p [i ˘ w − ˘ J p , i ˘Φ] − (cid:90) d r (cid:18) w v + ( ∇ Φ) πl B (cid:19)(cid:35) , (S3)where ˘ w ( r )=Γ (cid:63)w ( r ) ≡ (cid:82) d r (cid:48) Γ( r − r (cid:48) ) w ( r (cid:48) ) with Γ( r )=exp( − r / a ) / (2 πa ) / being the unit-normalized Gaussian distribution used to model a single bead centered at position r , asdiscussed in the main text. In Eq. (S3), the single-polymer partition function for a chain oftype p is defined as Q p [i ˘ w − ˘ J p , i ˘Φ] ≡ (cid:32) N p (cid:89) α =1 (cid:90) d r α (cid:33) exp (cid:34) − b N p − (cid:88) α =1 ( r α +1 − r α ) − N p (cid:88) α =1 (cid:16) i ˘ w ( r α ) − ˘ J ( r α )+i σ p,α ˘Φ( r α ) (cid:17)(cid:35) , where N p is the number of beads in a polymer of type p . The foregoing steps put us in aposition to derive field operators whose ensemble averages correspond to the PDFs. Considernow the field operator ˜ ρ b ,p ( r ) ≡ lim J p → n p δ ln Q p [i ˘ w − ˘ J p , i ˘Φ] δJ p ( r ) =i n p δ ln Q p [i ˘ w, i ˘Φ] δw ( r ) , so named ( ∼ ρ ) because (cid:104) ˜ ρ b ,p ( r ) (cid:105) F = (cid:104) ˆ ρ b ,p ( r ) (cid:105) P . [Incidentally, this ensemble average is eas-ily computed by exploiting the translation invariance of the model. Since (cid:104) ˆ ρ b ,p ( r ) (cid:105) P = (cid:104) ˆ ρ b ,p ( r + a ) (cid:105) P for any a , (cid:104) ˆ ρ b ,p ( r ) (cid:105) P = (cid:82) d r (cid:104) ˆ ρ b ,p ( r ) (cid:105) P /V = (cid:104) (cid:82) d r ˆ ρ b ,p ( r ) (cid:105) P /V = n p N p /V , where V is system volume. The last equality holds because (cid:82) d r ˆ ρ b ,p ( r )= n p N p holds identically.]Nonetheless, it should be emphasized that the correspondence between this field opera-tor and real-space bead density exists only at the level of their respective ensemble averages.Although individual spatial configurations of the real part [66] of ˜ ρ b ,p ( r ) that is non-negativeare highly suggestive and qualitatively consistent with the rigorous conclusions from PDFs(Fig. 2 in the main text), strictly speaking one cannot interpret ˜ ρ b ,p ( r ) in terms of the actual12ead positions for any single field configuration { w ( r ) , Φ( r ) } .We then compute Q p [i ˘ w, i ˘Φ] and ˜ ρ b ,p ( r ) using so-called forward- and backward chainpropagators q F ,p ( r ,α ) and q B ,p ( r ,α ) , constructed iteratively using the Chapman-Kolmogorovequations q F ,p ( r ,α +1)=e − i ˘ w ( r ) − i σ p,α +1 ˘Φ( r ) (cid:90) d r (cid:48) e − r − r (cid:48) ) / b q F ,p ( r (cid:48) ,α ) ,q B ,p ( r ,α − − i ˘ w ( r ) − i σ p,α − ˘Φ( r ) (cid:90) d r (cid:48) e − r − r (cid:48) ) / b q B ,p ( r (cid:48) ,α ) , while starting from q F ,p ( r , (cid:104) − i ˘ w ( r ) − i σ p, ˘Φ( r ) (cid:105) and q B ,p ( r ,N p )=exp (cid:104) − i ˘ w ( r ) − i σ p,N p ˘Φ( r ) (cid:105) .With q F ,p and q B ,p in place, we arrive at Q p [i ˘ w, i ˘Φ]= (cid:90) d r q F ,p ( r ,N p ) and ˜ ρ b ,p ( r )=Γ (cid:63) Q p [i ˘ w, i ˘Φ] N p (cid:88) α =1 q F ,p ( r ,α ) q B ,p ( r ,α )e i ˘ w ( r )+i σ p,α ˘Φ( r ) . For inter-species PDF, i.e., G p,q ( | r − r (cid:48) | ) with p (cid:54) = q , Eq. (S2) applied to Eq. (S3) leads directlyto G p,q ( | r − r (cid:48) | )= (cid:104) ˜ ρ b ,p ( r ) ˜ ρ b ,q ( r (cid:48) ) (cid:105) F , p (cid:54) = q. A direct application of Eq. (S2) to obtain the intra-species PDF G p,p ( | r − r (cid:48) | ) is also possible;but that procedure leads to an expression containing a double functional derivative, viz., ∼ δ ln Q p /δw ( r ) δw ( r (cid:48) ) , which is cumbersome to handle in numerical lattice simulations. Wetherefore obtain a simpler expression by performing the field redefinition w ( r ) → w ( r ) − i J p ( r ) instead before taking the second derivative. This alternate procedure results in G p,p ( | r − r (cid:48) | )= i v (cid:104) ˜ ρ b ,p ( r ) w ( r (cid:48) ) (cid:105) F − (cid:88) p (cid:54) = q (cid:104) ˜ ρ b ,p ( r ) ˜ ρ b ,q ( r (cid:48) ) (cid:105) F . That (cid:104) ˆ ρ b ,p ( r ) ˆ ρ b ,q ( r (cid:48) ) (cid:105) P depends only on | r − r (cid:48) | follows from translational and rotationalinvariance. In practice, we use knowledge of these symmetries to make computations ofthe PDFs more efficient. For instance, to calculate (cid:104) ˜ ρ b ,p ( r ) ˜ ρ b ,q ( r (cid:48) ) (cid:105) F , we can first calculate (cid:10)(cid:82) d a ˜ ρ b ,p ( r + a ) ˜ ρ b ,q ( r (cid:48) + a ) (cid:11) F /V , which can be conveniently executed in Fourier space, withaveraging over all possible directions of r − r (cid:48) . In this way, we obtain manifestly transla-tionally and rotationally invariant PDFs without spending computational time waiting fora droplet center of mass to explicitly visit all positions in the system or for a droplet to takeon all possible spatial orientations. In the calculation of G p,q ( | r − r (cid:48) | ) from lattice configura-tions, | r − r (cid:48) | is taken to be the shortest distance between positions r and r (cid:48) with periodicboundary conditions taken into account. 13 attice simulation and sequence-, excluded-volume-, andtemperature-dependent results FTS in the present study is performed on and lattices (meshes) with peri-odic boundary conditions and side-length V / =13 . b and V / =24 . b , respectively. TheComplex-Langevin (CL) evolution equations are integrated from random initial conditionsusing a step size ∆ t =0 . b in CL time for the mesh, and ∆ t =0 . b in CL timefor the mesh, with a Gaussian noise η ϕ satisfying (cid:104) η ϕ ( r ,t ) (cid:105) =0 and (cid:104) η ϕ ( r ,t ) η ϕ ( r (cid:48) ,t (cid:48) ) (cid:105) =2 δ ( r − r (cid:48) ) δ ( t − t (cid:48) ) . After an initial equilibration period of , steps, the systems are sam-pled every 1,000 steps until a total of ∼ , sample field configurations are obtained foreach run. These field configurations are used in the averages described above. For eachbinary sequence mixture and excluded-volume strength v , ∼ and ∼ independent runsare performed, respectively, for the and systems.Figs. S1 and S2 show PDFs of sv sequence pairs computed using, respectively, the and meshes under various excluded volume strengths v . Results are available for the highest v/b =0 . we simulated for the mesh but not for the mesh because equilibrationis problematic for the larger mesh at strong excluded volume. As discussed in the maintext, at the low temperature ( l B =5 b , T ∗ =0 . ) at which these simulations are conducted, ahallmark for the existence of a condensed droplet is the decay of the G p,p , G q,q , and G p,q functions to ≈ at r ≈ b ; and a significant demixing of the populations of the two sequencespecies is signaled by a substantially lower G p,q ( r ) ( p (cid:54) = q ), for small r ≈ , than both G p,p ( r ) and G q,q ( r ) in the same range of r . The trends exhibited by the two sets of results inFigs. S1 and S2 are consistent. They indicate robustly that both a significant differencein sequence charge pattern of the two polyampholyte species (difference decreases from thesv28-sv1 to the sv28-sv25 pair) and a substantial excluded volume (relatively large v values)are required for appreciable demixing. This observation corroborates the trend illustratedby the sv28-sv1 and sv28-sv25 examples and the ξ p,q measure presented in Fig. 2 of themain text. As a control, and not surprisingly, when FTS is conducted at a much highertemperature of T ∗ =20 ( l B =0 . b ) in Fig. S3, there is little sequence dependence—as seenby the very similar behaviors of all G p,p ( r ) , G q,q ( r ) , and G p,q ( r ) among the sequence pairsconsidered—and there is no droplet formation. Instead of converging to zero at large r as inFigs. S1 and S2, here all G ( r ) ’s converge to a finite (nonzero) value of (cid:104) ˆ ρ b ,p (cid:105) P (cid:104) ˆ ρ b ,q (cid:105) P ≈ . b − at large r in Fig. S3 for p (cid:54) = q as well as p = q , signaling a total lack of correlation betweenbead positions that are far apart. Illustrative snapshots of density field configurations
The dual requirements of a significant sequence charge pattern mismatch and a substantialgeneric excluded volume for demixing of two polyampholyte species in a condensed dropletare illustrated by the FTS snapshots for the sv28-sv1 pairs ( v/b =0 . and . ) and sv28-sv25 pairs ( v/b =0 . ) in Fig. 2a–c of the main text. Those snapshots present an overall14iew from the outside of the droplet. Thus, part of their interior structure is obscured, albeitthis limitation is partly remedied by the translucent color scheme. Further analyses to betterunderstand the interal structures of these FTS snapshots are provided by the cross-sectionalviews in Fig. S4. The contour plots in Fig. S4a for the sv28-sv1 system with a high genericexcluded volume strength show clearly that there is indeed a three-dimensional core withhighly enriched sv28 population surrounded by a shell with enriched sv1 population. Incontrast, the contour plots for the sv28-sv25 system at the same excluded volume strength(Fig. S4b) and the sv28-sv1 system at a low generic excluded volume strength (Fig. S4c)indicate that the two polyampholytes species are quite well mixed in the condensed dropletsof these two systems. Nonetheless, the patterns of the contours reveals that even for thesewell-mixed systems, sv28 is still slightly more enriched in the core and the other sv sequenceis slightly more enriched in a surrounding shell region. EXPLICIT-CHAIN COARSE-GRAINED MOLECULAR DYNAMICS(MD) SIMULATIONSMethodological details
All MD simulations are performed using the GPU version of HOOMD-blue simulationpackage [67, 68] as in Ref. [19]. We initially randomly place all the polyampholyte chainsinside a sufficiently large cubic simulation box of length a . The system is then energyminimized using the inbuilt FIRE algorithm to avoid any steric contact for a period of τ with a timestep of . τ , where τ ≡ (cid:112) ma /(cid:15) and m is the mass of each bead (representinga monomer, or residue). Each system is first initiated at a higher temperature—at a high T ∗ =4 . —for a period of , τ , where the reduced temperature T ∗ ≡ k B T /(cid:15) (see Table I in themain text and related discussion about the MD energy scale). The box is then compressedat T ∗ =4 . for a period of , τ using isotropic linear scaling until we reach a sufficientlyhigher density of ∼ . ma − which corresponds to a box size of a × a × a . Next, weexpand the simulation box length along one of the three Cartesian directions (labeled z ) 8times compared to its initial length to reach a final box length of a × a × a . Thebox expansion procedure is conducted at a sufficiently low temperature of T ∗ =0 . . Afterthat, each system is equilibrated again at the desired temperature for a period of , τ using Langevin dynamics with a weak friction coefficient of . m/τ [62]. Velocity-Verletalgorithm is used to propagate motion with periodic boundary conditions for the simulationbox. Production run is finally carried out for , τ and molecular trajectories are savedevery τ for subsequent analyses. For density distribution calculations, we first adjustthe periodic simulation box in such a way that its centre of mass is always at z =0 . Thesimulation box is then divided along the z -axis into 264 bins of size = a to produce a totaldensity profile as well as profiles for the two individual polyampholyte species in the binarymixture. 15ig. S5 shows the density profiles of six sv sequence pairs (the same sv pairs analyzedusing RPA in Ref. [46]). At a sufficiently low temperature of T ∗ =0 . , LLPS is observed forall systems simulated here, in that a droplet, manifested as a density plateau, is observed(left column of Fig. S5). At this low temperature, demixing of the two species in the bi-nary mixture is clearly observed for sv28-sv1 and sv28-sv10, and nearly complete mixing isobserved for sv28-sv24 and sv28-sv25. Intermediate behaviors that may be characterized aspartial demixing—with sv28 slightly enriched in the middle and the other sequence speciesslightly enriched on the two sides—are observed for sv28-sv15 and sv28-sv20. The trend isalso seen at intermediate temperatures ( T ∗ =1 . – . ). However, in some of these cases, oneof the polyampholytes either does not (e.g. sv1) or barely (e.g. sv15) phase separate, asindicated by the long “tails” of their density profile outside the central region (middle col-umn of Fig. S5). Not unexpectedly, at a high temperature of T ∗ =4 . , none of the simulatedsystems phase separates and the two species are mixed homogeneously throughout the simu-lation box (right column of Fig. S5). Representative snapshots of these systems are shown inFig. S6. As for the G p,q ( | r − r (cid:48) | ) in FTS, in the calculation of the MD-simulated G p,q ( | r − r (cid:48) | ) from configurations in the MD simulation box with periodic boundary conditions (Fig. 3e–gof the main text), | r − r (cid:48) | is taken to be the shorter distance of the two possible inter-beaddistances determined in the presence, or absence, of periodic boundary conditions. Cross-sectional views of MD and FTS droplet structures
The MD-simulated droplet snapshots at low temperature T ∗ =0 . in Fig. 3b–g of themain text underscore that demixing of two polyampholyte species in a condensed dropletrequires a significant mismatch in sequence charge pattern as well as a substantial excludedvolume repulsion. Because the beads (monomers) are represented in our MD drawingsas opague spheres, the bulk of those droplets below the surface of the image presentedcannot be visualized. To better illustrate that the observed mixing/demixing trend appliesnot only to the exterior of the presented image of those droplets but persists in the partsunderneath (as can be inferred by the behaviors of G p,p , G q,q , and G p,q in Fig. 3e–g of themain text), we prepare cut-out images of those droplets to reveal the spatial organizationin their “core” regions (Fig. S7). The spatial configurations of the MD droplets and theirgeneral trend of behaviors (Fig. S7, right column) are very similar to those exhibited bycross-sectional views of FTS droplets (contour plots in Fig. S4 and density plots in Fig. S7,left column), demonstrating once again the robustness of our observations. By construction,MD provides much more spatial details than FTS in this regard. Of particular future interestis the manner in which individual positively and negatively charged beads interact acrosspolyampholytes of different species. MD snapshots should be useful for elucidating thisissue. In contrast, although FTS snapshots—with their cloudy appearances—may show asimilar spatial organization of charge densities as that of MD, the field configurations do nottranslate into individual bead positions (Fig. S7, second row).16 . . . . . sv28sv1 v = 0 . b sv28sv1 v = 0 . b sv28sv1 v = 0 . b sv28sv1 v = 0 . b . . . . . sv28sv10 sv28sv10 sv28sv10 sv28sv10 . . . . . sv28sv15 sv28sv15 sv28sv15 sv28sv15 . . . . . sv28sv25 sv28sv25 sv28sv25 sv28sv25 G p , q / b − r/b FIG. S1. PDFs of binary mixtures of sv sequences computed by FTS using a mesh at l B =5 b ( T ∗ =0 . ) and various v . The plotting style follows that of Fig. 2 of the main text. Dashed bluecurves: G p,p ( r ) for sv28 ( − SCD= . ); dashed color curves: G q,q ( r ) for (top to bottom) sv1, sv10,sv15, and sv25 ( − SCD= . , . , . , and . , respectively); solid black curves: G p,q ( r ) . Theshaded region around each curve represents standard error of the mean among the ∼ independentruns for each system, which is mostly smaller than the width of the curve. . . . . . . . sv28sv1 v = 0 . b sv28sv1 v = 0 . b sv28sv1 v = 0 . b . . . . . . . sv28sv10 sv28sv10 sv28sv10 . . . . . . . sv28sv15 sv28sv15 sv28sv15 . . . . . . . sv28sv25 r/b sv28sv25 sv28sv25 G p , q / b − FIG. S2. PDFs of binary mixtures of sv sequences computed by FTS using a mesh at l B =5 b ( T ∗ =0 . ). Results for each system are from ∼ independent runs. The notation is otherwise thesame as that of Fig. S1. . . . . . . . . sv28sv1 v = 0 . b sv28sv1 v = 0 . b sv28sv1 v = 0 . b . . . . . . . . sv28sv10 sv28sv10 sv28sv10 . . . . . . . . sv28sv15 sv28sv15 sv28sv15 . . . . . . . . sv28sv25 r/b sv28sv25 sv28sv25 G p , q / b − FIG. S3. PDFs of binary mixtures of sv sequences computed by FTS using a mesh or a mesh at l B =0 . b ( T ∗ =20 . ). Dashed (dotted) blue curves: G p,p ( r ) for sv28 from a ( ) mesh;dashed (dotted) color curves: G q,q ( r ) for (top to bottom) sv1, sv10, sv15, and sv25 from a ( )mesh; solid (dotted) black curves: corresponding G p,q ( r ) obtained using a ( ) mesh. At thishigh temperature, the behaviors of all systems are very similar irrespective of the sequence chargepatterns or excluded volume interaction v values considered. The r/b scale is enlarged vis-à-visFigs. S1 and S2 to make the differences between the plotted curves here visible. a) sv28 sv1 (b) sv28 sv25 (c) sv28 sv1 − − x/b − − y / b v = 0 . b − − y/b − − z / b − − x/b − − z / b − − x/b − − v = 0 . b − − y/b − − − − x/b − − − − x/b − − v = 0 . b − − y/b − − − − x/b − − FIG. S4. Cross-sections of FTS droplets of binary sv sequence mixtures illustrating the interplayingroles of sequence charge pattern mismatch and generic excluded volume in mixing/demixing ofpolyampholyte species. Shown here are two-dimensional slides through the droplet center of massin the x – y (top), y – z (middle), and x – z (bottom) planes for the three FTS droplets depicted inFig. 2a–c of the main text. Density contours for the two sv sequence components p,q in a givenmixture are color coded as indicated by the labels at the top of the (a)–(c) columns. The contours forspecies p ( q ) are curves of constant bead density, where “bead density” here in a FTS snapshot meansthe real non-negative part of the density operator, viz., (cid:60) + ( ˜ ρ b ,p ( r )) = (cid:60) + (i n p δ ln Q p [i ˘ w, i ˘Φ] /δw ( r )) ( (cid:60) + ( ˜ ρ b ,q ( r )) = (cid:60) + (i n q δ ln Q q [i ˘ w, i ˘Φ] /δw ( r )) ), where (cid:60) + ( u ) ≡ [ (cid:60) ( u )+sign( (cid:60) ( u ))] / for any complexnumber u . (Among all snapshots considered, (cid:60) ( ˜ ρ b ,p ( r )) < − . b − occurs only for < of themesh points). The contours are evenly spaced from (cid:60) ( ˜ ρ b ,p ) , (cid:60) ( ˜ ρ b ,q ) = 0 [transparent] to (cid:60) + ( ˜ ρ b ,p )= max {(cid:60) + ( ˜ ρ b ,p ) } ( (cid:60) + ( ˜ ρ b ,q ) = max {(cid:60) + ( ˜ ρ b ,q ) } ) [opaque] where max {(cid:60) + ( ˜ ρ b ,p ) } ( max {(cid:60) + ( ˜ ρ b ,q ) } ) isthe maximum density of species p ( q ) in a given snapshot. . . . . . ρ / a − T ∗ = sv28sv1 T ∗ = sv28sv1 T ∗ = sv28sv1 . . . . . T ∗ = sv28sv10 T ∗ = sv28sv10 T ∗ = sv28sv10 . . . . . T ∗ = sv28sv15 T ∗ = sv28sv15 T ∗ = sv28sv15 . . . . . T ∗ = sv28sv20 T ∗ = sv28sv20 T ∗ = sv28sv20 . . . . . T ∗ = sv28sv24 T ∗ = sv28sv24 T ∗ = sv28sv24 − − −
50 0 50 100 1500 . . . . . T ∗ = sv28sv25 − − −
50 0 50 100 150 z/a T ∗ = sv28sv25 − − −
50 0 50 100 150 T ∗ = sv28sv25 ρ / a − FIG. S5. MD-simulated average density of binary mixtures of sv sequences along the z (long) axis ofthe simulation box at various temperatures. Solid curves: total bead density; color dashed curves:density of individual sv polyampholyte species. In addition to the four sv pairs studied using FTS,MD results for the sv28-sv20 ( − SCD = . , . ) and sv28-sv24 ( − SCD = . , . ) pairs areobtained to cover the six sv pairs studied using RPA [46]. a) sv28-sv1, T ∗ =1 . (b) sv28-sv1, T ∗ =4 . (c) sv28-sv25, T ∗ =2 . (d) sv28-sv25, T ∗ =4 . FIG. S6. Simulation snapshots of binary mixtures of sv sequences at the reduced temperatures indi-cated. Polyampholyte chains with charge sequences sv28, sv1, and sv25 are depicted, respectively,in blue, red, and green. TS MD sv28sv1sv28sv25sv28sv1 reducedexcludedvolume FIG. S7. Cross-sectional views of FTS and MD snapshots of binary mixtures of polyampholytes af-ford a consistent picture of sequence- and excluded-volume-dependent droplet organization. (Left)FTS density distributions on one of the two-dimensional planes in Fig. S4 through each droplet’scenter of mass. (Right) Corresponding cut-out views of the MD droplets shown inside the periodicsimulation boxes in Fig. 3b–d of the main text at one half of the box dimension extending perpen-dicularly into the page. Two different representations are used to visualize the sv28-sv1 droplet withfull excluded volume (top two rows; v =0 . b , r = a ). Upper row: sv1 and sv28 are depicted,respectively, in red and blue. Lower row: The negatively and positively charged beads in sv28are depicted, respectively, in red and blue, whereas the corresponding beads in sv1 are depictedin pink and cyan. 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