1. Physics of Living Systems
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Decoding the physical principles of two-component biomolecular phase separation

  1. Yaojun Zhang
  2. Bin Xu
  3. Benjamin G Weiner
  4. Yigal Meir
  5. Ned S Wingreen  Is a corresponding author
  1. Center for the Physics of Biological Function, Princeton University, United States
  2. Department of Physics, Princeton University, United States
  3. Department of Physics, Ben Gurion University of the Negev, Israel
  4. Department of Molecular Biology, Princeton University, United States
  5. Lewis-Sigler Institute for Integrative Genomics, Princeton University, United States
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Cite this article as: eLife 2021;10:e62403 doi: 10.7554/eLife.62403

Abstract

Cells possess a multiplicity of non-membrane-bound compartments, which form via liquid-liquid phase separation. These condensates assemble and dissolve as needed to enable central cellular functions. One important class of condensates is those composed of two associating polymer species that form one-to-one specific bonds. What are the physical principles that underlie phase separation in such systems? To address this question, we employed coarse-grained molecular dynamics simulations to examine how the phase boundaries depend on polymer valence, stoichiometry, and binding strength. We discovered a striking phenomenon – for sufficiently strong binding, phase separation is suppressed at rational polymer stoichiometries, which we termed the magic-ratio effect. We further developed an analytical dimer-gel theory that confirmed the magic-ratio effect and disentangled the individual roles of polymer properties in shaping the phase diagram. Our work provides new insights into the factors controlling the phase diagrams of biomolecular condensates, with implications for natural and synthetic systems.

Introduction

Eukaryotic cells are host to a multiplicity of non-membrane-bound compartments. Recent studies have shown that these compartments form via liquid-liquid phase separation (Brangwynne et al., 2009; Li et al., 2012; Molliex et al., 2015). The phase-separated condensates enable many central cellular functions – from ribosome assembly, to RNA regulation and storage, to signaling and metabolism (Shin and Brangwynne, 2017; Banani et al., 2017). Unlike conventional liquid-liquid phase separation, for example water-oil demixing, the underlying interactions that drive biomolecular phase separation typically involve strong one-to-one saturable interactions, often among multiple components (Ditlev et al., 2018). As a result, the phase diagrams of biomolecular condensates are complex and are sensitive to a variety of physical properties of the biomolecules, included number of binding sites, binding strengths, and additional nonspecific interactions. Importantly, these physical parameters can be subject to biological regulation, and can thus directly impact the organization and function of the condensates. It is therefore crucial to understand how the physical properties of the components shape the phase diagram of biomolecular condensates.

Biomolecular condensates typically contain tens to hundreds of types of molecules. Yet, when characterized in detail, only a small number of components are responsible for condensate formation (Ditlev et al., 2018). One class of such condensates are those formed by the association of two essential components. In the simplest case, each component consists of repeated domains/stickers that bind in a one-to-one fashion with the domains of the other component (Figure 1A and B; Choi et al., 2019; Xu et al., 2020). Such two-component condensates have been observed in both natural and engineered contexts. For example, the pyrenoid, an organelle responsible for carbon fixation in the alga Chlamydomonas reinhardtii, is a condensate of the CO2-fixing enzyme Rubisco with the linker protein Essential PYrenoid Component 1 (EPYC1). EPYC1 consists of five evenly-spaced Rubisco-binding regions, while Rubisco holoenzyme has eight specific binding sites for EPYC1. Multivalent interactions between Rubisco and EPYC1 are responsible for pyrenoid formation (Freeman Rosenzweig et al., 2017; Wunder et al., 2018; He et al., 2020). Promyelocytic leukemia (PML) nuclear bodies are condensates of PML proteins. PML is SUMOylated at three main positions and several minor sites. These modifications and a C-terminal SUMO Interaction Motif (SIM) found in most PML isoforms contribute to the formation of these bodies (Shen et al., 2006). Engineered polySUMO and polySIM proteins (10 repeats of Small Ubiquitin-like Modifier [SUMO] and SIM, respectively) phase separate when mixed together, but not as individual components (Banani et al., 2016; Ditlev et al., 2018).

Phase behavior of sticker and spacer associative polymers.

(A) Schematic of multivalent associative polymers. Each polymer consists of complementary domains (stickers) connected by flexible linkers (spacers). A and B denote the polymer type and m and n denote their valences (number of stickers). (B) Association of stickers drives phase separation, leading to the formation of a dense, network phase coexisting with a dilute phase of small oligomers (depicted by a dimer). (C) The phase diagram depends on variety of biologically tunable parameters. In this study, we focus on the effects of sticker-sticker binding strength, sticker:sticker concentration ratio (i.e. stoichiometry), and polymer valences. (D) Schematic of a representative 3D phase diagram of an An:Bn system as a function of temperature (inverse of binding strength) and A and B sticker concentrations. The dilute-phase concentration displays anomalous dependence on the binding strength and sticker concentrations in the strong binding regime. This is the ‘magic-ratio’ effect which we explore here in detail.

Previous simulations (Freeman Rosenzweig et al., 2017; Xu et al., 2020) of average cluster size in such two-component systems revealed a striking phenomenon – for sufficiently strong binding, the formation of large clusters is suppressed when the valence of one species equals or is an integral multiple of the valence of the other species, favoring the formation of small stable oligomers instead of a condensate. The phenomenon reminiscent of the exact filling of atomic shells leading to the unreactive noble gases was termed the ‘magic-number’ effect. A similar effect was found in a ternary system modeling the clustering of nephrin, Nck, and NWASP proteins which regulates cell-cell adhesion in podocyte cells of the kidney (Chattaraj et al., 2019). However, cluster size may reflect a sol-gel percolation transition rather than a thermodynamic phase transition (Harmon et al., 2017), and thus provides at best a qualitative measure of phase separation. Moreover, these previous studies focused on equal sticker stoichiometry, whereas biomolecular condensates cover a broad range of stoichiometries both in vitro (Li et al., 2012; Banani et al., 2016) and in vivo (Sanders et al., 2020).

Here, we directly delineate the full phase diagram of such two-component systems. Using coarse-grained molecular dynamics simulations, we explore systematically how phase boundaries depend on valence, stoichiometry, and binding strength of two associating polymers (Figure 1C and D). Our studies reveal an unanticipated effect – when the numbers of polymers of the two types have a rational stoichiometry (1:1, 1:2, etc.), phase separation can be strongly suppressed, which we call the ‘magic-ratio’ effect (Figure 1D, phase diagram at low temperatures). To understand the magic-ratio effects better, we develop a two-component sticker theory à la Semenov and Rubinstein (Semenov and Rubinstein, 1998). We model the system as dominated by polymer dimers in the dilute phase and by a condensate of independent stickers in the dense phase (Figure 1B). The resulting analytical theory captures the magic-ratio effect discovered in simulations, and allows us to disentangle the individual roles of valence, stoichiometry, specific-bond strength, and nonspecific attraction in determining the phase boundaries of two-component multivalent systems. Living cells regulate the valence and interactions of biomolecules through chemical modification, or on a slower timescale, tune the stoichiometry via synthesis/degradation or sequestration, and over evolutionary time, adapt the strength of specific and nonspecific interactions through mutation of molecular sequences. Understanding the individual roles of these biologically tunable variables thus brings new insights into possible cellular strategies for regulating the formation and dissolution of biomolecular condensates.

Results

Coarse-grained molecular-dynamics simulations

We perform coarse-grained molecular-dynamics simulations using LAMMPS (Plimpton, 1995) to determine the phase boundaries of two-component multivalent systems (Figure 2). Briefly, we model the two polymer species as flexible linear chains of beads connected by harmonic springs (Figure 2A). Each bead represents one associative domain/sticker of the polymer. To ensure associative domains of different polymer types bind in a one-to-one fashion, we impose a finite-ranged attractive interaction between beads of different types. This, however, could lead to more than one-to-one associations. Therefore, to avoid such unwanted associations, we impose strong repulsive interactions between beads of the same type over a large enough range to prevent other beads overlapping with a bound pair, thus preventing multiple-to-one binding (Figure 2B and Appendix 1—figure 1), see Appendix 1 for details.

Coarse-grained molecular-dynamics simulations of two-component multivalent associative polymers.

(A) The system consists of two types of polymers A (blue) and B (red) of varying lengths and concentrations. Depicted are A and B polymers of length 10, denoted as A10 and B10. Each polymer is modeled as a linear chain of spherical particles connected by harmonic bonds. Stickers of different types interact pairwisely through an attractive potential, while repulsion between stickers of the same type prevents them from overlapping and thus ensures one-to-one binding of stickers of different types (see Appendix 1 for details). (B) Snapshots of dimers formed by A10 and B10 with one-to-one bonds. (C) Snapshot of a simulation with 125 A10 and 125 B10 polymers. The system phase separates into a dense phase (middle region) and a dilute phase (two sides) in a 250 nm×50 nm×50 nm simulation box with periodic boundary conditions. (D) Same as C but with 138 A10 and 112 B10 polymers, yielding an overall sticker concentration ratio 1.23. (E) Sticker concentration profiles of A10:B10 systems at various overall sticker stoichiometries (total global sticker concentration fixed at 6.64 mM), each with the center of the dense phase aligned at x=0 and averaged over time and over ten simulation repeats (see Appendix 1). All simulations performed in LAMMPS (Plimpton, 1995).

To find the binodal phase boundaries, we simulate hundreds of polymers of types A and B with, respectively, m and n stickers (an Am:Bn system) in a box with periodic boundary conditions (Figure 2C and D). We initialize the system by constructing a dense slab of polymers in the middle of the box (Dignon et al., 2018). The system evolves and relaxes according to Langevin dynamics (Langevin, 1908). After the system has achieved equilibrium, two phases coexist: a dilute phase consisting of dimers and other small oligomers, and a dense phase of an interconnected polymer condensate. We measure the corresponding density profile (Figure 2E) and calculate the dilute- and dense-phase concentrations by averaging the density profile over the regions (x100nm or x100nm) and (10nmx10nm), respectively. See Appendix 1 for simulation details.

Effect of valence

It was shown previously that for equal sticker stoichiometry in the strong-binding regime, clustering is substantially suppressed when the number of binding sites on one polymer species is an integer multiple of the number of binding sites on the other, as this condition favors the assembly of small oligomers in which all binding sites are saturated (Freeman Rosenzweig et al., 2017; Xu et al., 2020). What does this magic-number effect imply for the actual phase diagram? To address this question, we fix the valence of polymer A at 14 and systematically vary the valence of polymer B from 5 to 16 while keeping the two sticker concentrations the same, that is, at equal global sticker stoichiometry.

Figure 3A and B show simulation results for the total sticker concentrations of the dilute and dense phases for A14:B5 to A14:B16 systems. In the strong binding regime, for magic-number cases, that is when the valence of B is 7 or 14, the dilute-phase concentration shows pronounced peaks (Figure 3A, black curve). What is the origin of the peak at A14:B14? Intuitively, when the dilute phase of the two-component system is dominated by dimers (for systems A14:B12 to A14:B16, as supported by cluster size analysis in Appendix 1—figure 2), each of these dimers has high translational entropy, whereas polymers in the dense condensate have low translational entropy. For A14:B14, all binding sites can pair up in a dimer just as well as in the condensate, so the energy per polymer is not necessarily lower in the condensate. Why then is the condensate still competitive with the dilute phase? In a dimer, the binding sites of A14 must match all the binding sites of B14, leading to a reduced overall conformational entropy. By comparison, the polymers in the condensate are more independent, binding to multiple members of the other species and enjoying a relatively higher overall conformational entropy. Because the translational entropy of each dimer decreases as their concentration goes up, the condensed phase eventually becomes more favorable and so the system phase separates with increasing concentration. Therefore, phase separation in A14:B14 is primarily driven by a competition between translational entropy and conformational entropy.

Simulations of associative polymers reveal a magic-number effect with respect to relative valence.

Total sticker concentrations (type A plus type B) in (A) dilute and (B) dense phases for simulated polymer systems at different binding strengths. U0 denotes the depth of the potential well, in units of kBT (see Appendix 1 for details). The valence of polymer A is 14, and the valence of polymer B ranges from 5 to 16. Global sticker stoichiometry is one and total global sticker concentration is 6.64 mM. Histograms of cluster size in (C) A14:B14 and (D) A14:B6 systems, for U0=14. ‘Counts’ refer to number of clusters. Cluster size is measured in stickers. Red dots indicate the dominant oligomer in the dilute phase.

By contrast, for A14:B13 and A14:B15, one of the stickers in the dimer cannot be paired, and for A14:B12 and A14:B16, two stickers per dimer cannot be paired. Therefore, forming a condensate not only increases the conformational entropy but more importantly lowers the energy of these systems. This significantly tilts the balance in favor of condensation. As a result, the dilute-phase concentration is sharply peaked at A14:B14, falling off rapidly for increasingly unequal polymer lengths. We note that the dense-phase concentration shows no such feature (Figure 3B), indicating that the peak at A14:B14 does not arise from differences in the internal structure of the dense phase.

The dilute phase of two-component systems is not always dominated by dimers (Appendix 1—figure 2). For example, the dilute phase of the A14:B7 system is dominated by fully-bonded trimers with 1 A14 and 2 B7, the dilute phase of A14:B8 is dominated by trimers with 1 A14 and 2 B8, which has two unpaired stickers per trimer, and the dilute phase of A14:B6 is dominated by oligomers with 3 A14 and 7 B6, which although fully-bonded is not small (Figure 3D). Consistent with the above logic, we find another peak in the dilute-phase concentration at A14:B7 (Figure 3A). More generally, in contrast to the magic-number systems, the dilute phases in other cases are dominated by oligomers which are not capable of being fully bonded (high energy) and/or not small (low translational entropy) (Appendix 1—figure 2). The dilute-phase concentration is therefore lower in these non-magic-number cases.

Effect of binding strength

How do the phase boundaries depend on the strength of binding? Figure 3A shows that, for non-magic-number systems, the dilute-phase concentration decreases monotonically with increasing binding strength, whereas for magic-number systems the dependence can be non-monotonic. This difference is attributed to the distinct underlying driving forces for phase separation. For non-magic-number systems, as clustering allows a larger fraction of binding sites to be paired, the stronger the binding, the more the energy is lowered by condensate formation. Therefore, the dilute-phase concentration drops as binding strength increases (or as temperature decreases). Such energy-dependence is expected for conventional phase-separation models, such as Flory, 1942; Huggins, 1941.

Interestingly, for the magic-number system A14:B14, the dilute-phase concentration first decreases with increasing binding strength in the weak binding regime, similar to non-magic-number systems. However, as the binding energy is increased further, most of binding sites pair up in both dilute and dense phases. Phase separation is then primarily driven by a competition between conformational and translational entropy. The pairing up of binding sites reduces the conformational entropy of both the dense and dilute phases. By contrast, the translational entropy of the dilute-phase components is almost unaffected. Consequently, the dilute phase becomes more competitive relative to the condensate, so the dilute phase boundary shifts to higher concentration.

By comparison, the dense-phase concentration increases monotonically with increasing binding strength for all systems (Figure 3B). This follows because the stronger the binding, the more stickers are paired, which tightens the condensate structure. We note that at substantially higher binding energies than studied here, essentially all the binding sites are satisfied in both magic-number and non-magic-number systems, and the phase boundaries become independent of binding energy.

Effect of sticker stoichiometry

How do the phase boundaries depend on overall sticker stoichiometry? Figure 4A and B show total sticker concentrations of the dilute and dense phases for magic-number systems A8:B8 to A14:B14 at different global sticker stoichiometries. For each system, the dilute-phase concentration peaks at equal sticker ratio, falls off initially as the ratio deviates from 1, and then curves back up. What is the origin of the peak at equal sticker stoichiometry? Recall that, in the strong binding regime, phase separation of magic-number systems is primarily driven by a competition between translational entropy and conformational entropy. Now consider starting with a system at equal sticker concentration, and adding more of one polymer species to the system. At the beginning, the added polymers readily enter the dense phase, which relaxes the conformational constraint that every sticker in the condensate has to pair with a partner. This increase of the conformational entropy of the condensate makes it more competitive, so the dilute-phase concentration decreases. However, as the ratio between the two polymers is increased further, it becomes possible to form a spectrum of dilute-phase oligomers which typically contain one extra polymer of the majority type (Appendix 1—table 1). These new oligomers have more relaxed structures than fully bonded dimers, which raises the conformational entropy of the dilute phase. Therefore, the dilute phase is favored over the condensate and its concentration curves back up.

Simulations of associative polymers reveal a magic-ratio effect with respect to polymer stoichiometry.

Sticker concentrations in (A) dilute and (B) dense phases for equal polymer length systems (i.e. An:Bn) at different global sticker stoichiometries. Sticker concentrations in (C) dilute and (D) dense phases for systems where polymer B is one sticker shorter than polymer A (i.e. An:Bn1) at different global sticker stoichiometries; black dots indicate cases where the number of polymers of each type is the same. Interaction strength U0=14 and total global sticker concentration 6.64 mM.

Figure 4A also reveals that the dilute-phase concentration decreases with increasing polymer valence. This follows in part because translational entropy in the dilute phase is per dimer center of mass, whereas conformational entropy in both phases scales with the number of stickers. The entropic gain of joining the dense phase is therefore more on a per sticker basis for longer polymers, so the dilute-phase concentration decreases with increasing valence. As a less apparent yet important point, Figure 4A also shows that increasing polymer valence enhances both the width and relative height of the peak in the dilute-phase concentration. The inferred phase diagram for the A8:B8 system at U0=14kBT is shown in Appendix 1—figure 3 together with the homogeneous gelation/percolation threshold obtained at U0=8kBT. We also report in Appendix 1—figure 5C the volume fraction of the polymers in the dense phase, which is ∼10%, comparable to the volume fraction of proteins in the cell cytoplasm.

Figure 4C and D show total sticker concentrations of the dilute and dense phases for unequal valence polymers A8:B7 to A14:B13 at different global sticker stoichiometries. The dilute phase boundary shows a symmetric minimum around equal stoichiometry for A8:B7, yet surprisingly, the phase boundary becomes asymmetric and then peaks at equal polymer stoichiometry with increasing polymer length (Figure 4C). What is the origin of these peaks? Taking the A14:B13 system as an example, its dilute phase is dominated by dimers with an unpaired A sticker. This strongly disfavors the dilute phase in the strong binding regime at equal sticker stoichiometry. However, as the overall A:B sticker stoichiometry increases, the excess As cannot be paired anyway. In particular, at equal polymer stoichiometry (denoted as black dots in Figure 4C), forming dimers is no longer energetically costly. Therefore, to the left of the A14:B13 peak at equal polymer stoichiometry, the dilute-phase concentration is low because dimers are energetically disfavored as more bonds can be satisfied in the condensate. By contrast, to the right of the peak, the dilute-phase concentration is low for a different reason – because the condensate is entropically favored, similar to the peak with respect to stoichiometry for magic-number systems. Eventually, the dilute-phase concentration curves back up due to formation of higher oligomers in the dilute phase, as discussed for magic-number systems.

We note that for all these systems the dense-phase concentration shows no such striking features. Rather, the concentration decreases monotonically as the global sticker stoichiometry departs from one and as the valence of polymers decreases (Figure 4B and D).

Effect of valence and stoichiometry

Above, we considered the role of both relative valence and relative stoichiometry. By plotting phase boundaries as joint functions of valence and stoichiometry, we obtain a unified picture: Figure 5A and B show the dilute- and dense-phase concentrations for systems A14:B12-16 at global sticker stoichiometries 14:12-16. Notably, the dilute-phase concentration is peaked along the diagonal (Figure 5A), that is at equal polymer stoichiometry, which we term the 'magic-ratio' effect because it occurs for rational ratios of associative polymers. Intuitively, all cases along the diagonal favor 1:1 polymer dimers: the dimers enjoy high translation entropy and there is no energy penalty involved in their formation. Thus, a dilute phase of dimers is strongly favored at equal polymer stoichiometry.

Simulations of associative polymers reveal a magic-ratio effect.

Sum of concentrations of stickers A and B in (A) dilute and (B) dense phases for systems A14:B12-16 at global sticker stoichiometries 14:12-16. Parameters: interaction strength U0=14 and total global sticker concentration 6.64 mM.

As for the dense phase concentration, it decreases monotonically as the global sticker stoichiometry departs from one and as the valence of polymers decreases (Figure 5B). This again indicates that the anomalous dependence of the dilute-phase concentration on valence and stoichiometry does not arise from special properties of the dense phase.

Dimer-gel theory

While our simulations have revealed that a magic-ratio effect influences the boundaries of phase separation for associating polymers, we desire a deeper understanding of the interplay of factors such as overall valence, stoichiometry, and interaction strength. To this end, we develop a mean field theory of two-component associative polymers à la Semenov and Rubinstein (Semenov and Rubinstein, 1998; Xu, 2018).

Specifically, we consider a system of A and B polymers as in our simulations. Each polymer is a linear chain of L1 or L2 stickers of type A or type B, respectively. Without loss of generality, we take L1L2. stickers of different types associate in a one-to-one fashion. Our simulations suggest that for polymers of similar valence close to equal polymer stoichiometry the dilute phase is dominated by dimers and the dense phase is a gel network. Therefore, we assume that polymers can associate either as dimers or, alternatively, as a condensate in which pairs of stickers bind independently. This assumption of independence is a mean field approximation, as it neglects correlations between stickers in the same chain, and thus only applies when the polymers strongly overlap, that is at densities above the semidilute regime (De Gennes, 1979).

The partition function of such a system can be divided into three parts: Z=ZniZsZns, where Zni, the partition function of a solution of non-interacting polymers, captures the translational and conformational entropy of the two polymer species, Zs captures specific interactions between associating stickers, and Zns captures all nonspecific interactions.

The corresponding free-energy density for the mixed non-interacting polymers is Semenov and Rubinstein, 1998:

(1) FnikBT=c1L1lnc1eL1+c2L2lnc2eL2,

where c1 and c2 are the concentrations of A and B polymers measured in terms of stickers. Note that the terms for the conformational entropy of non-interacting polymers are omitted in Equation 1, as they are linear in c1 and c2 and thus do not influence the phase boundaries.

To include specific interactions, we first consider the partition function Zs(Nd1,Nd2,Nb) for states with exactly Nd1 and Nd2 total numbers of stickers of A and B types in dimers (i.e. number of dimers equals Nd1/L1=Nd2/L2) and Nb additional sticker pairs,

(2) Zs(Nd1,Nd2,Nb)=P(Nd1,Nd2,Nb)W(Nd1,Nd2,Nb)exp(Nd1ϵd/L1+Nbϵb).

In Equation 2, P is the number of different ways that polymers and stickers can be chosen to pair up to form dimers and independent bonds,

(3) P(Nd1,Nd2,Nb)=(N1/L1Nd1/L1)(N2/L2Nd2/L2)(Nd1/L1)!(N1Nd1Nb)(N2Nd2Nb)Nb!,

where N1 and N2 are the total numbers of stickers of A and B types. (Note that in Equation (3) if L1>L2, the excess stickers of type A in dimers do not form additional bonds.) In Equation (2), W is the probability that all chosen polymers and stickers are, respectively, close enough to their specified partners in the non-interacting state to form dimers and independent bonds,

(4) W(Nd,Nb)=(vdV)Nd1L1(vbV)Nb,

where vd and vb are effective interaction volumes and V is the system volume. The last term in Equation (2) is the Boltzmann factor for specific interactions, where ϵd and ϵb are the effective binding energies of dimers and sticker pairs, in units of kBT.

The part of the free-energy density due to specific interactions is

(5) FskBT=-1VlnZs.

Using Stirling’s approximation lnN!=NlnN-N, we obtain

(6) FskBT=c1L1lnc1+(1L1)c1cd1L1ln(c1cd1)+(c1cd1cb)ln(c1cd1cb)c2L2lnc2+(1L2)c2cd2L2ln(c2cd2)+(c2cd2cb)ln(c2cd2cb)+cd1L1ln(ecd2L1Kd)+cbln(ecbKb),

where Kde-ϵd/vd and Kbe-ϵb/vb are, respectively, the dissociation constants of a dimer and of a pair of stickers. cd1 and cd2 are the concentrations of stickers of A and B types in dimers (so cd1/L1=cd2/L2), and cb is the concentration of independent bonds.

In the thermodynamic limit, Fs will be minimized with respect to cd1, cd2 and cb, which implies

(7) Kdcd2L1(c1cd1)L11(c2cd2)L21=(c1cd1cb)L1(c2cd2cb)L2,
(8) Kbcb=(c1cd1cb)(c2cd2cb).

Note that if cb in Equation (7) and cd1 and cd2 in Equation (8) are set to zero, these equations reduce to

(9) Kdρd=(ρ1ρd)(ρ2ρd),
(10) Kbcb=(c1cb)(c2cb),

where ρ1, ρ2, and ρd are the total concentrations of A and B polymers and dimers (measured in polymeric units), that is, ρ1=c1/L1, ρ2=c2/L2, and ρd=cd1/L1=cd2/L2. Equations (9) and (10) are consistent with the definitions of the dissociation constants of a dimer and of an independent bond, respectively.

The free-energy density due to nonspecific interactions can in general be written as a power expansion in the concentrations (Semenov and Rubinstein, 1998; De Gennes, 1979),

(11) FnskBT=12ijvijcicj+16ijkwijkcicjck,

where the sum is over all the species in the system, including free polymers/stickers, dimers and independent bonds, and vij and wijk are two- and three-body interaction parameters. For our simulation system, we derive a specific form of Fns by taking into account that (1) we are interested in the strong-binding regime where the magic-ratio effect is observed, (2) there is no nonspecific interaction between free polymers of different types in our simulation, and (3) nonspecific interactions are only important at high concentrations. The result is

(12) FnskBT=vb2max(c1,c2)2+wb6max(c1,c2)3,

where vb and wb are the two- and three-body interaction parameters for a solution of independent bonds. See Appendix 2 for details of the derivation.

Finally, substituting the conditions Equations (7) and (8) into Equation (6), we obtain the total free-energy density F=Fni+Fs+Fns,

(13) FkBT=c1L1lnc1cd1eL1+c1lnc1cd1cbc1cd1+cd1L1+c2L2lnc2cd2eL2+c2lnc2cd2cbc2cd2+cb+vb2max(c1,c2)2+wb6max(c1,c2)3,

where cd1, cd2, and cb are the solutions of Equations (7) and (8). Equations (7), (8) and (13) form a complete set which predicts the free-energy density of the two-component associative polymer system at given total global sticker concentrations, c1 and c2, of the two species.

Intuitively, in the strong-binding regime, that is when c1,c2Kd,Kb, polymers either associate as dimers or as independent bonds depending on their relative free energies. In the limit that dimers are preferred (ρd=min(ρ1,ρ2) and cb=0), the contribution from specific interactions is

(14) FsdimkBT=ρdlnKd+(ρ-ρd)lnρ-ρde-ρlnρe,

where ρ=max(ρ1,ρ2) is the concentration of the majority species in polymeric units. The terms on the right of Equation (14) reflect, respectively, the free-energy density due to dimer formation, translational entropy of leftover polymers, and loss of translational entropy of the majority species (in effect, the formation of each dimer removes the translation entropy of one free polymer). In the opposite limit that independent bonds are preferred (cb=min(c1,c2) and ρd=0),

(15) FsindkBT=cblnKb+(c-cb)lnc-cbe-clnce,

where c=max(c1,c2), and the terms are analogous to those in Equation (14). Numerical studies show that the full Fs(c1,c2) in Equation (6) is always well approximated by the lower of the two limiting values of Fs (Equation (14) and (15)).

In which regions of concentration space are dimers versus independent bonds preferred? For a magic-number system composed of two polymer species of valence L at equal sticker stoichiometry, Fsdim/kBT=ρln(Kde/ρ) and Fsind/kBT=cln(Kbe/c). Comparing the two expressions, dimers are favored at low concentrations, whereas a network of independent bonds is favored at high concentrations. The transition occurs when Fsdim=Fsind, that is at concentration c0=e(KbL/(KdL))1/(L-1). Away from equal stoichiometry, the transition occurs at a lower concentration cs=c0(s-1)s-1s-s, where s=max(c1,c2)/min(c1,c2)>1 (see Appendix 2 for details). As cs decreases rapidly with increasing s (Figure 6A inset, white curve), the preference for dimers over a gel exhibits a sharp peak around equal stoichiometry.

A dimer-gel theory predicts the magic-ratio effect.

Phase diagrams of (A) A8:B8 and (B) A8:B7 systems: one-phase region white, two-phase region green. The dilute- and dense-phase concentrations are connected by representative tie lines. The tie line along the direction of equal polymer stoichiometry is denoted with a black dot. Insets: fraction of stickers in dimers for (A) A8:B8 and (B) A8:B7 systems. White curve in A inset is the transition boundary between dimer- and independent bonds-dominated regions predicted by cs. Dashed white line in (B) inset denotes equal polymer stoichiometry. Sticker concentrations in (C) dilute and (D) dense phases for systems A8:B6-10 at global sticker stoichiometries 8:6-10. The total global sticker concentration is the same as in simulations, 6.64 mM. For details see Appendix 2. Parameters: vb=9×10-2mM-1, wb=7×10-3mM-2, Kb=3.8×10-3mM, and Kd values in Appendix 1—table 3.

To give a concrete example of the above analysis, we extract the values of Kd for dimers from simulations, choose a value of Kb for independent bonds close to the dissociation constant of a pair of stickers (see Appendix 2 for details), and numerically solve Equation (7) and (8) for cd1, cd2, and cb to find the fraction of stickers in dimers and independent bonds for all concentrations (c1,c2). We find that indeed for polymers of equal valence, dimers are favored at low concentrations and independent bonds at high concentrations. The dimer dominated region extends sharply to higher concentrations in a narrow zone around the diagonal, as quantitatively captured by cs (Figure 6A inset and Appendix 2—figure 1A). For polymers of similar but unequal valence, the dimer dominated region extends to higher concentrations along the direction of equal polymer stoichiometry (Figure 6B inset and Appendix 2—figure 1B).

Finally, to extract the binodal phase boundaries, we substitute the values of cd1, cd2, and cb into Equation (13) to first obtain the free energy as a function of c1 and c2. The free-energy landscape has two basins, one at small concentrations corresponding to the dilute dimer-dominated phase, and one at high concentrations corresponding to the dense independent-bond-dominated gel-phase (Appendix 2—figure 2). We locate the phase boundaries by applying convex-hull analysis to this free-energy landscape (see Appendix 2).

Does the dimer-gel theory capture the magic-ratio effect revealed by our simulations? Figure 6A and B show the phase diagrams of A8:B8 and A8:B7 systems. In both cases, the phase boundaries on the dilute side extend sharply into the two-phase region along the direction of equal polymer stoichiometry (tie lines along this direction are denoted by black dots). Figure 6C and D show the dilute- and dense-phase concentrations for systems A8:B6-10 at global sticker stoichiometries 8:6-10. Notably, the dilute-phase concentrations are substantially shifted up around the diagonal, verifying the magic-ratio effect observed in simulations (Figure 5A).

One of the major assumptions of the dimer-gel theory is a mean-field approximation. Mean-field theory ignores correlations in binding between stickers in the same chain, and therefore has been applied to long chains in the weak binding regime (such that not every sticker is bound) (Prusty et al., 2018; Choi et al., 2020b). Our dimer-gel theory bypasses this stringent requirement by explicitly assuming the dilute-phase components to be dimers, and only considers stickers to associate independently in the dense phase. This approximation captures a key feature of the dense phase, namely that a single polymer binds to multiple partners. Nevertheless, because stickers belonging to the same polymer are tethered together with relatively short linkers in our simulations, correlations in binding exist (Appendix 2—figure 5A). Therefore, what should be considered to be ‘independent’ is not individual stickers but rather segments of the binding correlation length (∼1.8 stickers). The dense phase of a valence 14 system is thus more accurately described by the theory at valence 14/1.88. We therefore present results for valence eight systems in Figure 6. (The theoretical phase diagrams and the dilute- and dense-phase concentrations for valence 14 systems also verify the magic-ratio effect (Appendix 2—figure 4)).

The dimer-gel theory has only a handful of parameters: the valences L1 and L2 of polymers A and B, the dissociation constants Kd and Kb of dimers and independent bonds, and the nonspecific interaction parameters vb and wb. How are the phase boundaries and the magic-ratio effects determined collectively by these parameters? If valence is increased while keeping all other parameters fixed in the theory, for equal valence polymers we find that the dilute-phase concentration decreases, while the dense-phase concentration increases, and the peak with respect to stoichiometry is enhanced in terms of the dilute-phase peak-to-valley ratio (Appendix 2—figure 6A and B). If valence is increased for unequal valence polymers, we observe that the shape of the dilute phase boundary transitions from a shoulder to a peak (Appendix 2—figure 6C and D). All these features are consistent with the simulation results in Figure 4.

For the theory to agree quantitatively with the phase boundaries from simulations, we find that smaller values of nonspecific interaction parameters are necessary for higher valence systems (Appendix 2—figure 6C and D). Intuitively, this follows because higher valence polymers have more backbone bonds, which bring bound sticker pairs closer together in the dense phase – effectively reducing the nonspecific repulsion between them. Finally, the dimer-gel theory also predicts that the magic-ratio effect disappears in the weak-binding regime (Appendix 2—figure 7), consistent with our simulation results (Figure 3).

Discussion

Intracellular phase separation is driven by multivalent interactions between macromolecules. These interactions are separated into two classes (Ditlev et al., 2018; Pak et al., 2016): (1) specific interactions, such as binding between protein domains, are relatively strong and involve specific partners and (2) nonspecific interactions, such as electrostatic and hydrophobic interactions, which are much weaker, more generic, and non-saturable. Multivalent systems with specific interactions allow for ‘orthogonal’ condensates to form: the specific interactions holding together one class of droplets will typically not interfere with those holding together another class. Motivated by the key role of specific interactions in intracellular phase separation, we focused on exploring the effects of specific interactions on the phase boundaries of two-component associative polymers. Specifically, we combined coarse-grained molecular dynamics simulations and analytical theory to examine the individual roles of valence, stoichiometry, and binding strength on the phase boundaries. In particular, we identified a magic-ratio effect: for sufficiently strong binding, phase separation is strongly suppressed at equal polymer stoichiometry.

The magic-ratio effect occurs exclusively in the strong-binding regime. Are specific protein-protein, protein-RNA, and RNA-RNA interactions strong enough to lead to the magic-ratio effect? The onset of the effect in our simulations occurs around U0=9kBT (Figure 3A), which corresponds to a sticker-sticker dissociation constant Kd=0.4mM. This value is consistent with the onset Kd of 1–2.5 mM estimated from 3D lattice simulations with one polymer and one rigid component (Xu et al., 2020). For comparison, the measured Kd for a SUMO protein domain with a SIM peptide is 0.01 mM (Banani et al., 2016) and the Kd for an SH3 domain and a PRM peptide is 0.35 mM (Li et al., 2012). Thus for systems as strongly interacting as SUMO-SIM or SH3-PRM, the magic-ratio effect in principle should manifest in their phase diagrams. However, the magic-ratio effect has not been observed in these systems (Li et al., 2012; Banani et al., 2016), possibly due to size and linker length mismatch between the two associating polymers. Furthermore, real biological systems are more complex than our simple model. For example, there can be multiple-to-one binding, multiple components, and the spacers/linkers can also play nontrivial roles (Banjade et al., 2015; Harmon et al., 2017). Currently, the in vivo relevance of the effects explored in this work remains an open question. Magic-ratio effects could also manifest in other experimental systems, such as non-biological polymers, DNA origami (Hu and Niemeyer, 2019), or patchy colloid systems (Bianchi et al., 2011). As an inverse problem, the magic-ratio effect could be exploited to determine the relative valence of associating biomolecules by measuring their phase diagram.

The magic-ratio effect allows for novel mechanisms of regulation. Chemical modifications, such as phosphorylation or SUMOylation, which change the effective valence of one component into or out of a magic-ratio condition could shift the phase boundary as a means of condensate regulation. Cells may also have evolved to avoid magic ratios so as to better promote condensate formation. For example, EPYC1 has valence five and Rubisco has valence eight, and the geometry of binding sites on Rubisco and the length of linkers in EPYC1 are such that they disfavor fully-bonded Rubisco-EPYC1 dimers even at equal polymer stoichiometry, which suppresses the magic-ratio effect (He et al., 2020). However, active removal of a terminal EPYC1 binding site, for example by phosphorylation (Turkina et al., 2006), would dramatically change the valence ratio to 1:2, which would then favor stable trimer formation, as previously suggested (Freeman Rosenzweig et al., 2017). We hope that our work will stimulate exploration of magic-ratio effects in both natural and synthetic multivalent, multicomponent systems.

The simulations and theory presented here are aimed at providing conceptual insights into the phase separation of associating polymers that form one-to-one specific bonds. Quantitative descriptions of related real systems will likely require additional features, such as details of molecular shape and flexibility, linker lengths, as well as range and type of interactions. For example, while the magic-ratio effect is robust with respect to the strength of nonspecific interactions and linker length, these variables do strongly influence phase boundaries. The dilute-phase concentrations in our simulations are ∼mM, while the reported values for biological systems are typically tens of μM or less. The discrepancy is likely due to different strengths of nonspecific attraction, different length scales of steric replusion between stickers, and/or different lengths and flexibilities of the linkers (Bhandari et al., 2021). Indeed, increasing the nonspecific attraction in our simulations by a small amount 0.07kBT leads to a 50% reduction in the dilute-phase concentration (Appendix 1—figure 4A). Reducing the steric repulsion between beads of the same type has a similar effect (Appendix 1—figure 5A). More significantly, increasing the mean linker length from 4.7 nm to 5.9 nm leads to a more than 10-fold reduction in the dilute-phase concentration (Appendix 1—figure 6A). On the other hand, the dense-phase concentration strongly depends on the steric repulsion — increasing the sticker size from 2.5 to 2.9 nm decreases the dense phase concentration by a factor of 2 (Appendix 1—figure 5B). This is consistent with results from previous studies on the role of linkers: a self-avoiding random coil linker which occupies a large volume can substantially lower the dense-phase concentration and even prevent phase separation (Harmon et al., 2017). Future work will explore the interplay between specific and nonspecific interactions, and other molecular properties, and their roles in determining the physical properties of droplets, such as surface tension, viscosity, and rate of exchange between phases.

Appendix 1

Modeling two-component multivalent associative polymers

We perform coarse-grained molecular-dynamics simulations using LAMMPS (Plimpton, 1995) to simulate two-component multivalent associative polymers. Individual polymers are modeled as linear chains of spherical particles connected by harmonic bonds (Appendix 1—figure 1A, type A polymer in blue and type B polymer in yellow). Bonds are modeled using a harmonic potential (Appendix 1—figure 1C, left)

(16) Ub(r)=k(r-rb)2,

where rb = 4.5 nm is the mean bond length, k=20kBT/rb2 is the bond stiffness, kB is the Boltzmann constant, and T = 300 k is room temperature.

Appendix 1—figure 1
Coarse-grained molecular-dynamics simulations of two-component multivalent associative polymers.

(A) Polymers are modeled as linear chains of spherical particles connected by harmonic bonds. Depicted are A8 (blue) and B8 (yellow). (B) Snapshot of a dimer of A8 and B8 formed in the strong-binding regime. (C) Neighboring stickers in a polymer are connected through a harmonic potential (left). Stickers of the same type interact pairwisely through a repulsive potential (middle). Stickers of different types interact pairwisely through an attractive potential (right). (D) Interaction energy between three particles (one A and two B stickers) as a function of their separation distances. Simultaneous binding of two B stickers to one A sticker is energetically highly disfavored (lower left region) compared to one-to-one binding (dark blue regions).

Stickers of the same type interact through a softened, truncated Lennard-Jones potential (Appendix 1—figure 1C, middle) 11See LAMMPS manual at https://lammps.sandia.gov/doc/Manual.html for details about this potential.

(17) Ur(r)=4ϵλ{[(1-λ)2+(rσ)6]-2-[(1-λ)2+(rσ)6]-1},r<rc,

where ϵ=0.15kBT, λ=0.68, σ=3.5nm, and rc=5nm. These parameters effectively lead to a sticker of diameter d3nm and a weak attractive tail of depth 0.06kBT. The weak attractive tail is employed solely to promote a more compact dense condensate.

Stickers of different types interact through an attractive potential (Appendix 1—figure 1C, right)

(18) Ua(r)=-12U0(1+cosπrr0),r<r0

where U0=14kBT is used in all simulations in the main text, except as indicated in Figure 3. The attraction cut-off distance is r0=2nm. Note that due to the strong repulsion between stickers of the same type, simultaneous binding of two stickers of one type to a sticker of the other type is energetically highly disfavored (Appendix 1—figure 1D). This ensures one-to-one binding of stickers of different types (Appendix 1—figure 1B). Across our simulations, on average the fraction of stickers that have more than one partner is less than 0.001%.

Phase equilibration and data recording

Each system consists of n1 and n2 polymers of types A and B, respectively. The number of polymers are determined by their valences/lengths (L1 and L2) and global A:B sticker stoichiometry (s) through

(19) n1=round(NsL1(1+s)),
(20) n2=round(NL2(1+s)).

The round function is used as we can only simulate an integer number of polymers. For all simulations except those in Figure 5 in the main text, we use N=2500, so the total number of stickers is around 2500.

Simulations are equilibrated using a Langevin thermostat in the NVT ensemble at T = 300 K in a box of size 250 nm×50 nm×50 nm with periodic boundary conditions, that is the system evolves according to Langevin, 1908:

(21) md2ridt2=-γdridt-riU(r1,,rN)+f.

where ri is the coordinate of particle i, m is its mass, γ is the friction coefficient, f is random thermal noise, and the energy U(r1,,rN) contains all interactions between particles, including harmonic bonds, nonspecific, and specific interactions (Equations (16–18)).

To promote phase equilibrium and ensure that only a single dense condensate is formed, we first initialize the simulation by confining polymers in the region 50nm<x<50nm. The attractive interaction between A and B stickers (Equations (18)) is gradually switched on from U0 = 0 to 14 over 108 time steps. The Langevin thermostat is applied using a damping factor τ=m/γ=125ns, step size dt=2.5ns, and mass of particle m=3534.3 ag during this time period. These parameters give the particle the right diffusion coefficient D=kBT/(3πηd) for times longer than τ, where η is the water viscosity 0.001 kg/m/s and d the diameter of the particle. This annealing procedure leads to the formation of a dense phase close to its equilibrated concentration. The confinement is then removed, and the system is equilibrated for 108 more time steps to allow the formation of dilute phase and relaxation of the dense phase. After these procedures, we switch to smaller τ=10 ns, dt=0.5 ns, and m=282.7 ag for data recording (D remains the same). The system is relaxed for another 108 steps. The relaxation time of the system depends on the sticker-sticker bond lifetime; to ensure that the dilute and dense phases are in equilibrium, the above choice of relaxation time before recording corresponds to ∼2000 bond lifetimes. We then recorded the positions of all particles every 106 steps for 400 recordings. For each choice of valence and stoichiometry, we performed 10 simulation replicates with different random seeds. We also checked whether there are systematic deviations between the first and second halves of the recorded simulations, and found consistent results between the two halves.

To test the effect of finite size on the phase boundaries, simulations in Figure 5 in the main text are performed with N=5000 and a box of size 315 nm×63 nm×63 nm, that is both total number of stickers and box volume are doubled while the total global sticker concentration remains the same (6.64 mM). Procedures for equilibration and data recording are the same (including the initial confinement region) except systems are relaxed for 5×108 steps at dt=0.5ns before recording, as the larger system requires a longer relaxation time.

Determining the phase boundaries

To determine the phase boundaries, we need to obtain the concentrations in the dilute and dense phases. The data we recorded for each system contains 4000 snapshots of polymer configurations (10 replicates and 400 time points each). To measure concentrations, we first group polymers into clusters in each snapshot. Connected stickers are grouped into one cluster: two stickers of the same type are connected if they are neighbors in the same polymer, and two stickers of different types are connected if they are within the attraction distance r0 = 2 nm. In most of our simulations, in each snapshot, we observe one large cluster which contains most of polymers, and a few to tens of very small clusters (Appendix 1—figure 2). There is a clear gap between the sizes of these large and small clusters. We define the large cluster as the dense phase, and the smaller clusters as constituents of the dilute phase. In cases where the separation between the dense and dilute phases is unclear, we discard the data set. Appendix 1—figure 2 shows the size distribution of clusters pooled over all snapshots. Appendix 1—table 1 lists all the dilute-phase components and their total sticker percentages in the A14:B14 system at global sticker stoichiometry 1.21.

Appendix 1—figure 2
Histograms of cluster size in stickers in (A–L) A14:B5-16 systems at equal global sticker stoichiometry.

Parameters: specific binding strength U0=14kBT and total global sticker concentration 6.64 mM. Red dots indicate the dominant oligomer in the dilute phase.

Appendix 1—table 1
List of all dilute-phase components and sticker percentages for A14:B14 system at global sticker stoichiometry 1.21.
Comp1A141A14+1B142A14+1B142A14+2B143A14+2B144A14+3B145A14+3B14
frac0.10%37.0%7.67%0.38%19.1%16.6%0.38%
comp5A14+4B146A14+4B146A14+5B147A14+5B148A14+6B1410A14+7B1417A14+13B14
frac10.8%0.85%0.42%0.80%3.45%0.48%1.99%

To find the dilute- and dense-phase concentrations, we calculate the center of mass of the dense cluster for each snapshot, and recenter the simulation box to this center of mass. We then compute the sticker concentration histogram along the x axis with a bin size 1/50 of box length. The resulting concentration profile has high values in the middle corresponding to the dense-phase concentration, and low values on the two sides corresponding to the dilute-phase concentration. The dilute- and dense-phase concentrations are calculated by averaging the concentration profile over the regions (x100nm or x100nm) and (10nmx10nm), respectively.

Testing the effect of finite simulation size on the phase boundaries

To test the effect of finite size on the phase boundaries, we compare the dilute- and dense-phase concentrations of systems with N=2500 to systems with N=5000 (Appendix 1—table 2). The systems being compared have the same valence, stoichiometry, and total sticker concentration. Simulations with different total numbers of particles show consistent results, suggesting the effect of finite size is minor.

Appendix 1—table 2
Comparison of dilute- and dense-phase concentrations for systems with different total numbers of particles.
Valence (Stoichiometry)A14:B14 (14:14)A14:B13 (14:14)A14:B13 (14:13)
cdil (mM) for N=25000.50 ± 0.060.08 ± 0.020.39 ± 0.02
cdil (mM) for N=50000.45 ± 0.030.12 ± 0.020.34 ± 0.03
cdil (mM) for N=250028.0 ± 0.227.76 ± 0.0925.80 ± 0.08
cden (mM) for N=500027.93±0.0527.82±0.0625.82±0.06

Determining the percolation threshold

Phase transitions in associative polymeric systems can be thought as phase separation aided percolation, that is, the dense phase of an associative polymer system is a percolating network Semenov and Rubinstein, 1998; Harmon et al., 2017; Choi et al., 2020a. What would be the percolation threshold in our associative polymer system if the density remained homogeneous? To answer this question, we determined the percolation threshold of an A8:B8 system at a weak binding strength U0=8kBT, which avoids phase separation. Briefly, for a given sticker concentration (cA,cB), we perform one simulation at U0=8kBT and analyze the size of clusters for all snapshots recorded after the system equilibrates. We judge whether polymers are in a sol- or gel-state based on the following gelation criterions: First, for each snapshot if the largest cluster contains more than 70% of the stickers and the second largest cluster contains less than 10% of the stickers in the system, we label this snapshot as having a percolating cluster. Second, for a given system if more than 50% of snapshots have a percolating cluster, we label this (cA,cB) point as a ‘gel’ state. The systems do not meet the gelation criterions are labeled as a ‘sol’ state (Appendix 1—figure 3). It is clear that the dilute/dense phases of the A8:B8 system formed at a strong binding strength U0=14kBT are in a sol-/gel-state. By inspection, the same clear dichotomy applies to the other systems we simulated.

Appendix 1—figure 3
Phase diagram and percolation threshold of A8:B8 system.

Phase boundaries (red dots) are measured from MD simulations at U0=14kBT. The complete two-phase region (green) and one-phase region (white) are extrapolated based on the phase boundaries from simulations. Sol- (gray dots) and gel- (black dots) states from simulations at U0=8kBT where the system remains homogeneous are identified based on the gelation criterions. Percolation threshold (dashed line) is interpolated from the labeled sol- and gel-states.

Measuring the dissociation constants

The phase boundaries of associative polymers from theory are very sensitive to model parameters. Here, we extract the dimer and sticker dissociation constants Kd and Kb from simulations. These values are then utilized in the dimer-gel theory to obtain the free-energy density landscape and predict the phase boundaries. Our simulations are performed in the strong binding regime. At our chosen binding strength of U0=14kBT, the long lifetime of each bond means that dimers of valence ≥4 never dissociate in our simulations. This prevents us from directly extracting the dimer dissociation constants for long polymers. We therefore use a reweighting method Frenkel and Smit, 2001 to obtain the dissociation constant for these dimers.

Briefly, we perform simulations with 1 polymer of type A and 1 polymer of type B of valences L1 and L2 in a cubic box of side 20L1 nm with periodic boundary conditions. The systems are equilibrated using a Langevin thermostat in the NVT ensemble, all the parameters are the same as the ones used for recording the data, except here we use U0=7kBT which is half of the original value, in order to allow dimers to dissociate.

Theoretically, the dissociation constant of a dimer is defined as:

(22) Kd1=VUAB<0eβUA(r1 A,,rL1 A)eβUB(r1 B,,rL2 B)eβUAB(r1 A,,rL1 A,r1 B,,rL2 B)dr1 AdrL1 Adr1 B,drL2 BeβUA(r1 A,,rL1 A)eβUB(r1 B,,rL2 B)dr1 AdrL1 Adr1 B,drL2 B,

where β=1/kBT, and (r1 A,,rL1 A) and (r1 B,,rL2 B) are the coordinates of stickers in polymers A and B. UA(UB) contain all interactions within the polymer A(B), including bond potentials Ub (Equations (16)) and nonspecific interactions Ur (Equations (17)). UAB contains all the specific interactions between polymers A and B, that is the sum of Uas (Equations (18)). Integration is over the entire volume V. In the numerator, the integration is further confined to the region where UAB<0. Note that for a dissociated dimer UAB=0.

To link the simulation with the definition of Kd, we define three variables in the simulation:

(23) w1=eβEAB,w2=1,w3=eβEAB,ifEAB<0;w1=w2=w3=0,ifEAB=0,

where EAB=UAB(r1 A,,rL1 A,r1 B,,rL2 B). We then have,

(24) Kd1(U0=0kBT)=Cw1eβUAeβUBeβUABdr1 AdrL1 Adr1 B,drL2 B=C~w1,
(25) Kd1(U0=7kBT)=Cw2eβUAeβUBeβUABdr1 AdrL1 Adr1 B,drL2 B=C~w2,
(26) Kd1(U0=14kBT)=Cw3eβUAeβUBeβUABdr1 AdrL1 Adr1 B,drL2 B=C~w3,

where w1, w2, and w3 are the mean values of w1, w2, and w3 obtained by averaging over a simulation with U0=7kBT. C and C~ are constants and are the same in all three equations. Therefore,

(27) Kd(U0=0kBT)w1=Kd(U0=7kBT)w2=Kd(U0=14kBT)w3.

On the other hand, it can be shown that the binding probability w2 for U0=7kBT is

(28) w2=w2eβUAeβUBeβUABdr1 AdrL1 Adr1 B,drL2 BeβUAeβUBeβUABdr1 AdrL1 Adr1 B,drL2 B=11+Kd(U0=7)(VKd1(U0=0)).

Combining Equations (27) and (28), we have

(29) Kd(U0=0kBT)=1+w1w2w1V,
(30) Kd(U0=7kBT)=1+w1w2w2V,
(31) Kd(U0=14kBT)=1+w1w2w3V.

Appendix 1—table 3 shows a list of dissociation constants from simulations. The reweighting method provides very accurate sticker-sticker and dimer-dimer dissociation constants, as confirmed by comparing with theory and direct simulations for stickers and polymers of length L=2. The dissociation constants obtained by the reweighting method are used in the dimer-gel theory to predict the phase boundaries (Appendix 2).

Appendix 1—table 3
List of dissociation constants from simulations.
A1:B1 (t)A1:B1 (d)A1:B1 (r)A2:B2 (d)A2:B2 (r)
Kd(U0=0kBT)(mM)49.6*48.9*12.4
Kd(U0=7kBT)(mM)1.88*1.90*0.31
Kd(U0=14kBT)(mM)5.7e-35.9e-35.8e-37.7e-67.1e-6
A4:B(r)A4:B4 (r)A6:B5 (r)A6:B6 (r)
Kd(U0=0kBT)(mM)4.323.291.811.52
Kd(U0=7kBT)(mM)4.5e-22.2e-24.3e-32.2e-3
Kd(U0=14kBT)(mM)3.77e-91.38e-116.95e-153.00e-17
A8:B6 (r)A8:B7 (r)A8:B8 (r)A8:B9 (r)A8:B10 (r)
Kd(U0=0kBT)(mM)1.151.010.900.790.73
Kd(U0=7kBT)(mM)9.0e-44.5e-42.5e-41.5e-41.0e-4
Kd(U0=14kBT)(mM)4.22e-181.26e-207.07e-231.77e-237.59e-24
A14:B12 (r)A14:B13 (r)A14:B14 (r)A14:B15 (r)A14:B16 (r)
Kd(U0=0kBT)(mM)0.370.320.300.320.27
Kd(U0=7kBT)(mM)1.4e-66.7e-73.8e-72.6e-71.5e-7
Kd(U0=14kBT)(mM)2.74e-351.08e-378.85e-401.80e-405.16e-41
  1. *footnotetext: (t) theoretical value, ({d}) direct simulation, and (r) reweighting method.

Effects of nonspecific interactions and linker length

In the main text, we focused on the effects of binding strength, sticker stoichiometry, and polymer valences on the phase boundaries of two-component systems. Here, we explore the effects of nonspecific interactions and linker length. Increasing the nonspecific attraction between stickers of same type leads to decreased/increased dilute-/dense-phase concentrations (Appendix 1—figure 4). The additional attractive interaction is modeled by a cosine-squared potential

(32) U(r)={ϵ,r<σ,ϵcos(π(rσ)2(rcσ))2,σr<rc,0,rrc.
Appendix 1—figure 4
Strength of nonspecific attractive interactions strongly influences simulated dilute-phase boundary.

Total sticker concentrations in (A) dilute and (B) dense phases for A14:B14 system at different global sticker stoichiometries with the additional attractive interaction strength ϵ=0kBT (blue) and ϵ=0.07kBT (red) (σ=3.5nm and rc=5nm in Equation (32)). Parameters: specific interaction strength U0=14kBT, total global sticker concentration 6.64 mM.

The cosine-squared potential is applied together with the softened, truncated Lennard-Jones potential (Equation (17)). Similarly, reducing the range of repulsion between stickers of the same type leads to decreased/increased dilute-/dense-phase concentrations (Appendix 1—figure 5). Here, the repulsive interaction potential between same type of stickers is replaced by the standard repulsive Lennard-Jones potential

(33) Ur(r)=4ϵ[(σr)12(σr)6+14],r<21/6σ.
Appendix 1—figure 5
Range of nonspecific repulsive interactions strongly influences phase boundaries in simulations.

Sticker concentrations in (A) dilute and (B) dense phases and (C) volume fraction of polymers in dense phase for A8:B8 system at different global sticker stoichiometries with standard repulsive Lennard-Jones potential (Equation (33)) σ = 2.9 nm (blue) and σ = 2.5 nm (red). Parameters: repulsive interaction strength ϵ=1kBT, specific interaction strength U0=12kBT with cut-off distance r0 = 1.5 nm, and total global sticker concentration is 6.64 mM. Note that the overlapping volume between two bound stickers is counted once in the volume fraction calculation, that is, two perfectly overlapping stickers only occupy a volume of one sticker.

The phase boundaries are also sensitive to the linker length. Here, we model the repulsive interactions between same type stickers by Equation (33), the bond between stickers by an expanded FENE potential

(34) Ub(r)=-12KR02ln[1-(r-σR0)2],r<σ+R0,

and attraction between different types of stickers by Equation (18). Increasing of mean bond length from 4.7 to 5.9 nm leads to a decrease of the dilute-phase concentration by more than a factor of 10 (Appendix 1—figure 6).

Appendix 1—figure 6
Linker length strongly influences phase boundaries in simulations.

Sticker concentrations in (A) dilute and (B) dense phases for A8:B8 system at different global sticker stoichiometries with linkers modeled by a FENE potential (Equation (34)) with K=0.3kBT and R0 = 7 nm (mean linker length 4.7 nm, blue) and K=0.15kBT and R0 = 14 nm (mean linker length 5.9 nm, red). Parameters: repulsive interaction strength ϵ=1kBT and length scale σ = 3 nm, specific interaction strength U0=12kBT with cut-off distance r0 = 1.5 nm, and total global sticker concentration 6.64 mM.

Appendix 2

Derivation of free-energy density for nonspecific interactions

The free-energy density due to nonspecific interactions can be written as a power expansion in the concentrations Semenov and Rubinstein, 1998; De Gennes, 1979,

(35) FnskBT=12ijvijcicj+16ijkwijkcicjck,

where the sum is over all the species in the system, including free polymers/stickers, dimers and independent bonds, and vij and wijk are two- and three-body interaction parameters.

In the strong-binding regime where the magic-ratio effect is observed, bound stickers strongly overlap, so the size of a bound pair is almost that of a free sticker. For simplicity, we therefore assume the interactions between dimers to be the same as between free polymers of the same type (denoted as vd and wd), and the interactions between independent bonds to be the same as between free stickers of the same type (denoted as vb and wb). When independent bonds are preferred (i.e. when ρd0), the free-energy density for nonspecific interactions is

(36) FnsindkBT=vb2[(c1cb)2+(c2cb)2+2(c1cb)cb+2(c2cb)cb+cb2]+wb6[(c1cb)3+(c2cb)3+3(c1cb)2cb+3(c1cb)cb2+3(c2cb)2cb+3(c2cb)cb2+cb3],

where c1, c2, and cb are the total concentrations of polymer A, B, and independent bonds in sticker units. c1-cb and c2-cb are therefore the concentrations of free sticker A and B. Note that in our simulations, there is no nonspecific interaction between free polymers of different types. Therefore, all v and w terms involving different free species are 0. Equation (36) simplifies to

(37) FnsindkBT=vb2(c12+c22-cb2)+wb6(c13+c23-cb3).

Similarly, when dimers are preferred (i.e. when cb0), the free-energy density for nonspecific interactions is

(38) FnsdimkBT=vd2(ρ12+ρ22-ρd2)+wd6(ρ13+ρ23-ρd3),

where ρ1, ρ2, and ρd are the total concentrations of polymer A, B, and dimers in polymeric units.

As nonspecific interactions are only important at high concentrations, we simply set Fns=Fnsind. Further, in the strong-binding regime, cbmin(c1,c2), so

(39) FnskBT=vb2max(c1,c2)2+wb6max(c1,c2)3.

Determining model parameters

The developed dimer-gel theory has only four parameters for a given system AL1:BL2: the dissociation constants of dimers and independent bonds Kd and Kb, and the nonspecific interaction parameters vb and wb. The four parameters together determine the competitiveness of the dilute dimer-phase with the dense gel-phase.

We have extracted the values of Kd from simulations (Appendix 1—table 3). Physically, we expect Kb to be close to the sticker-sticker dissociation constant 5.7μM (Appendix 1—table 3). It is not exactly the same because the bonds in the condensate are tethered by the backbones of the polymers. We expect the nonspecific interaction parameters to be approximately vb=2B2 and wb=3B3, where B2 and B3 are the second and third virial coefficients Katsura, 1959: vb=6.8×10-2mM-1 and wb=2.2×10-3mM-2 for hard spheres of diameter 3 nm (the size of a sticker in the simulations), respectively.

The predicted phase boundaries are sensitive to these parameters. We thus tune Kb, vb, and wb around their estimated values to match the dilute- and dense-phase concentrations of the A8:B8 system from simulation. Specifically, we yield Kb=3.8×10-3mM, vb=9×10-2mM-1, and wb=7×10-3mM-2. These parameters are used for A8:B6-10 systems. Further discussions on how model parameters affect phase boundaries can be found in later sections.

Derivation of the transition concentration cs

In the strong-binding regime, the free-energy density contributions from specific interactions in the dimer-dominated and independent bond-dominated limit are, respectively,

(40) FsdimkBT=ρlnKd+(ρ+ρ)lnρ+ρeρ+lnρ+e,
(41) FsindkBT=clnKb+(c+c)lnc+cec+lnc+e,

where ρ+=max(c1/L1,c2/L2), ρ-=min(c1/L1,c2/L2), c+=max(c1,c2), and c-=min(c1,c2). Equations (40) and (41) are the same as Equations (14) and (15).

At given (c1,c2), whether the system will form dimers or independent bonds depends on their relative free energies. For equal valence polymers L1=L2=L, letting c+=sc-, Equations (40) and (41) become

(42) FsdimkBT=ρln(s1)s1eKdssρ,
(43) FsindkBT=cln(s1)s1eKbssc.

Comparing the two expressions, dimers are favored at low concentrations (Fsdim<Fsind), and independent bonds are favored at high concentrations (Fsind<Fsdim). The transition occurs when Fsdim=Fsind, i.e. at the concentrations c-(s)=c0(s-1)s-1s-s and correspondingly c+(s)=c0s(s-1)s-1s-s where c0=e(KbL/(KdL))1/(L-1). The boundary between dimer- and independent bond-dominated regions is described by (c+(s),c-(s)) and (c-(s),c+(s)), respectively, in the lower and upper halves of the (c1,c2) plane (Appendix 2–figure 1A, white curve).

Solving reaction Equations (7) and (8)

The high powers in Equation (7) and the small value of Kd make it difficult to find numerical solutions of cd and cb accurately. To overcome this difficulty, we define a variable λ=c1-cd1-cb when ρ1ρ2 and λ=c2-cd2-cb when ρ1>ρ2, and rewrite Equations (7) and (8) in terms of λ. Specifically,

(44) cb=λ(c2L1c1L2+λL2)[KbL1+λ(L1L2)]1,cd1=c1λcb,KdL2(c1cbλ)(1+λKb1)L21(λ+cb)L11=KbcbλL11

for ρ1ρ2, and

(45) cb=λ(c1L2c2L1+λL1)[KbL2+λ(L2L1)]1,cd2=c2λcb,KdL1(c2cbλ)(1+λKb1)L11(λ+cb)L21=KbcbλL21

for ρ1>ρ2. We then solve Equations (44) and (45) using the MatLab function vpasolve with the constraints 0<λ<c1 and 0<λ<c2, respectively. vpasolve provides all solutions within the specified range. When multiple solutions coexist, we take the one with the lowest Fs. Numerical solutions of ρd and cb for A8:B8 and A8:B7 systems are shown in Appendix 2–figure 1, where the fraction of stickers in dimers is defined as ρd/min(ρ1,ρ2) and fraction of stickers in independent bonds is defined as cb/min(c1,c2).

Appendix 2—figure 1
Fraction of stickers in dimers (A) and in independent bonds (B) for A8:B8 system.

White curve in (A) is the transition boundary between dimer- and independent bonds-dominated regions predicted by c+(s) and c-(s). Fraction of stickers in dimers (C) and in independent bonds (D) for A8:B7 system. Dashed white line denotes equal polymer stoichiometry.

Determining phase boundaries and tie lines

We obtain the free-energy landscape by substituting the numerical solutions of ρd and cb into Equation (13) (Appendix 2–figure 2). We locate the phase boundaries by applying convex-hull analysis to this free-energy landscape using the MatLab function convhull (Figure 6A and B).

Appendix 2—figure 2
Free-energy density as a function of global sticker concentrations (c1,c2) for (A) A8:B8 and (B) A8:B7 systems.

White curves highlight the basins in the dilute dimer-dominated and dense gel-dominated regions.

To find the tie line going through a given initial concentration point (c1in,c2in), we adopt a modified vector method Marcilla Gomis, 2011. We first draw a line though this point along a direction defined by an angle α, and then find the crossing points between this line and the phase boundaries, that is (c1dil,c2dil) and (c1den,c2den). The ‘true’ α is the one that minimizes the free-energy density of mixing pF(c1dil,c2dil)+(1-p)F(c1den,c2den), where p=rden/(rdil+rden) is the volume fraction of dilute phase and 1-p is the volume fraction of dense phase, and rdil and rden are the distances between the point (c1in,c2in) and the two points (c1dil,c2dil) and (c1den,c2den), respectively.

In order to compare with the simulated dilute- and dense-phase concentrations, we use the initial concentrations from simulations for the specified system at given stoichiometry s: c1=cts/(1+s) and c2=ct/(1+s), where the total sticker concentration is c= 6.64 mM. We then find the tie line going through this initial concentration point and the corresponding (c1dil,c2dil) and (c1den,c2den). For simplicity, we show the total dilute- and dense-phase concentrations cdil=c1dil+c2dil and cden=c1den+c2den in Figure 6C and D.

A simple approximation for the free-energy density F

Finding the numerical solutions of Equations (7) and (8) becomes difficult with increasing valence. From the full solutions of cd and cb (Appendix 2–figure 1), we see that in the dimer-dominated region cb is almost 0, and in the independent bond-dominated region cd is almost 0. Therefore, we can approximate the free-energy density as the lower value of the two limiting cases

(46) F=Fni+min(Fsdim,Fsind)+Fns,

where

(47) FsdimkBT=ρdlnKd+ρdlnρde+(ρ1ρd)lnρ1ρde+(ρ2ρd)lnρ2ρdeρ2lnρ2eρ1lnρ1e,
(48) FsindkBT=cblnKb+cblncbe+(c1cb)lnc1cbe+(c2cb)lnc2cbec1lnc1ec2lnc2e,

and

(49) ρd=12[ρ1+ρ2+Kd(ρ1+ρ2+Kd)24ρ1ρ2],
(50) cb=12[c1+c2+Kb(c1+c2+Kb)24c1c2],

are solutions of Equations (9) and (10). Equation (46) provides a very good approximation to the full expression for F (Equation (13)). The phase diagrams derived from Equation (13) and (46) are almost identical for A14:B14 (Appendix 2–figure 3). Results in Appendix 2–figure 4 are obtained with this approximation (Equations (46–50)) as there are difficulties solving Equations (7) and (8) numerically for A14:B12-16 systems due to their high valences.

Appendix 2—figure 3
Comparison of phase diagrams derived from the full expression for F (Equation (13)) and the approximate expression (Equation (46)) shown on a (A) linear and (B) log scale for A14:B14 system.
Appendix 2—figure 4
A dimer-gel theory predicts the magic-ratio effect.

Phase diagrams of (A) A14:B14 and (B) A14:B13 systems: one-phase region white, two-phase region green. The dilute- and dense-phase concentrations are connected by representative tie lines. The tie line along the direction of equal polymer stoichiometry is denoted with a black dot. Inset: enlarged dilute-phase boundaries. Sticker concentrations in (C) dilute and (D) dense phases for systems A14:B12-16 at global sticker stoichiometries 14:12-16 and total sticker concentration 6.64 mM. Parameters: vb=7×10-2mM-1, wb=5×10-3mM-2, Kb=3.8×10-3mM, and Kd in Appendix 1—table 3.

Correlated binding in the dense phase

In the dimer-gel theory, we assume that stickers of different types can associate independently in the dense phase. However, as stickers belonging to the same polymer are tethered together, neighboring stickers in one polymer are more likely to bind to neighboring stickers in another polymer, that is there are correlations in binding. To quantify this correlation, we first identify consecutive segments in a polymer that bind to consecutive segments in another polymer. (For example, if in polymer 1 of type A, stickers 1, 2, 3, and 4 bind to stickers 2, 4, 3, and 5 of one polymer of type B, sticker 5 binds to sticker 8 of a second polymer of type B, and stickers 6, 7, and 8 bind to stickers 1, 2, and 3 of a third polymer of type B, then there are three individual segments in polymer 1 of type A with lengths 4, 1, and 3.) Clearly, what should be considered to be ‘independent’ are not individual stickers but rather these consecutively bound segments. To quantify the length of these segments, we measure the probability p(l) that a bound sticker is in a segment of length l. Appendix 2—figure 5 shows the probability distribution p(l) for simulated A8:B8 and A14:B14 systems at equal stoichiometry. The mean length of ‘independent’ segments is 1.8 for both cases.

Appendix 2—figure 5
Binding between stickers of different types in the dense phase is correlated.

Probability distribution p(l) for finding a bound sticker to be in a consecutively bound segment of length l in the dense phase of A8:B8 and A14:B14 systems at equal stoichiometry.

Effects of model parameters on phase boundaries

The dimer-gel theory has only a handful of parameters: the valences L1 and L2 of polymers A and B, the dissociation constants Kd and Kb of dimers and independent bonds, and the nonspecific interaction parameters vb and wb. These parameters together determine the competitiveness of the dilute dimer-phase with the dense gel-phase. We first fix Kb=3.8×10-3mM, vb=9×10-2mM-1, and wb=7×10-3mM-2 (see previous section Determining Model Parameters for how these parameters are derived, and note that the values of Kd are taken directly from simulations [Appendix 1—table 3]), and explore the dependence of phase boundaries on the valences L1, L2 and on the stoichiometry.

Appendix 2–figure 6A and B show phase diagrams and dilute-phase concentrations for A4:B4 to A14:B14 systems. For these equal valence systems, the dilute-/dense-phase concentrations decreases/increases with increasing valence, and the magic-ratio effect with respect to stoichiometry is enhanced with increasing valence in terms of dilute-phase peak-to-valley ratio. Appendix 2–figure 6C and D show phase diagrams and dilute-phase concentrations for A4:B3 to A14:B13 systems. Note that the shape of the dilute phase boundary transitions from a shoulder to a peak with increasing valence. All these features are consistent with the simulation results in Figure 4. Appendix 2–figure 7B shows the dilute-phase concentrations for A8:B6-10 systems at equal sticker stoichiometry. The dilute-phase concentration is sharply peaked at A8:B8, consistent with the simulation results in Figure 3.

Appendix 2—figure 6
Effects of model parameters on the phase boundaries in the strong-binding regime.

(A) Phase diagrams and (B) dilute-phase concentrations at different global stoichiometries for A4:B4 to A14:B14 systems. (C) Phase diagrams and (D) dilute-phase concentrations at different global stoichiometries for A4:B3 to A14:B13 systems. Black dots indicate equal polymer stoichiometries. Parameters: Kb=3.8×10-3mM, values of Kd in Appendix 1—table 3 for binding strength U0=14kBT, vb=9×10-2mM-1 and wb=7×10-3mM-2 for all systems except for A14:B14* and A14:B13* where vb=7×10-2mM-1 and wb=5×10-3mM-2. Total global sticker concentration 6.64 mM.

Appendix 2—figure 7
The magic-ratio effect disappears in the weak-binding regime.

(A) Phase diagrams for A8:B8 system with different binding strengths U0=7kBT (gray) and U0=14kBT (black). (B) Dilute-phase concentrations for A8:B6-10 systems at equal global sticker stoichiometry with different binding strengths U0=7kBT (gray) and U0=14kBT (black). Parameters: Kb=1.88mM for U0=7kBT, Kb=3.8×10-3mM for U0=14kBT, values of Kd in Appendix 1—table 3, vb=9×10-2mM-1, and wb=7×10-3mM-2. Total global sticker concentration 6.64 mM.

However, the dilute-phase concentration of A14:B14 does not quantitatively agree with simulation results (Appendix 2–figure 6B vs. Figure 4A). Also, the dilute-phase concentrations for unequal valence systems do not decrease with increasing valence (Appendix 2–figure 6D vs. Figure 4C). What is the origin of these discrepancies? Intuitively, the dense phase properties are determined by Kb, vb, and, wb. Using the same values of these parameters for systems with different valences means that we are treating the dense phases of these systems as exactly equivalent. However, there are more polymer backbone bonds in higher valence systems. These backbone bonds, from a mean-field point of view, act like attractive potentials between stickers, which effectively reduces the nonspecific repulsion between stickers. Therefore, the dense phases of higher valence systems are energetically favored, and we expect correspondingly lower values of vb and wb for valence 14 systems compared to valence 8 systems. Indeed, we find that somewhat smaller nonspecific interaction parameters, vb=7×10-2mM-1 and wb=5×10-3mM-2, lead to quantitative agreement of the dilute- and dense-phase boundaries for A14:B14 and A14:B13 systems with the simulation results (Appendix 2–figure 6B and D, curves labeled with *, compared to Figure 4A and C). We thus use these parameters for A14:B12-16 systems in Appendix 2–figure 4.

Finally, the dimer-gel theory also predicts that the magic-ratio effect disappears in the weak-binding regime (Appendix 2–figure 7A and B, gray curves), consistent with the simulation results (Figure 3A).

Data availability

Codes for generating data in this manuscript can be found at https://github.com/yaojunz/matlab-lammps-phasediagram/tree/codes (copy archived at https://archive.softwareheritage.org/swh:1:rev:eff8367b00e1d12c17542bb9d03d85960a3e53e8/).

References

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    1. De Gennes P-G
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    Scaling Concepts in Polymer Physics
    Cornell University Press.
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    1. Frenkel D
    2. Smit B
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    Understanding Molecular Simulation: From Algorithms to Applications
    Elsevier.
    1. Langevin P
    (1908)
    Sur la théorie Du mouvement brownien on the theory of brownian motion
    Comptes Rendus De l'Académie Des Sciences 146:530–533.
  3. Book
    1. Xu B
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Decision letter

  1. Pierre Sens
    Reviewing Editor; Institut Curie, PSL Research University, CNRS, France
  2. Aleksandra M Walczak
    Senior Editor; École Normale Supérieure, France
  3. Rohit V Pappu
    Reviewer; Washington University in St Louis, United States

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Acceptance summary:

In this work, Yaojun Zhang and coworkers develop a model based on concepts of polymer physics to study the formation of high density condensates of associating polymers. This occurs through a liquid-liquid phase separation process driven by the one-to-one association of specific stickers along the polymer chains. The authors provide an elaboration based on computations and a generalised theory to explain how a so-called magic number and / or magic ratio effect contributes to modulating the driving forces for phase transitions of multivalent proteins. In the case of two associating polymer species, the magic number effect occurs in the regime of strong specific interactions when the valence of one species is equal to or an integral multiple of the valence of the second species, allowing for the formation of small oligomers in the dilute phase. Here, the authors show that there is a magic ratio effect, whereby suppression of phase separation can be realized, again in the regime of strong, specific interactions, when the ratios of valencies are rational numbers. These findings, based on coarse-grained molecular dynamics simulations are recapitulated by a mean field theory and intuitively explained in terms of the interplay between conformational entropy and translational entropy costs. This work lays down conceptual foundations that will be helpful in understanding how signalling may be regulated by epigenetic changes to interactions among and stoichiometries of multivalent molecules.

Decision letter after peer review:

Thank you for submitting your article "Decoding the physical principles of two-component biomolecular phase separation" for consideration by eLife. Your article has been reviewed by two peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Aleksandra Walczak as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Rohit V Pappu (Reviewer #2).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission. As explained below, both reviewers found your work important and valuable, as theory / computation have a lot to offer in terms of forging a conceptual understanding for the benefit of the biological soft matter community and a broader audience. However, there are a number of issues that need to be resolved and important comments that need to be addressed.

We would like to draw your attention to changes in our policy on revisions we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, when editors judge that a submitted work as a whole belongs in eLife but that some conclusions require a modest amount of additional new data, as they do with your paper, we are asking that the manuscript be revised to either limit claims to those supported by data in hand, or to explicitly state that the relevant conclusions require additional supporting data.

Our expectation is that the authors will eventually carry out the additional experiments and report on how they affect the relevant conclusions either in a preprint on bioRxiv or medRxiv, or if appropriate, as a Research Advance in eLife, either of which would be linked to the original paper.

Summary:

In this work, the authors provide an elaboration based on computations and a generalised theory to explain how a so-called magic number and / or magic ratio effect contributes to modulating the driving forces for phase transitions of multivalent proteins. In the case of two associating polymer species, the magic number effect occurs in the regime of strong specific interactions when the valence of one species is equal to or an integral multiple of the valence of the second species, allowing for the formation of small oligomers in the dilute phase. Here, the authors show that there is a magic ratio effect, whereby suppression of phase separation can be realized, again in the regime of strong, specific interactions, when the ratios of valencies are rational numbers. These findings, based on Langevin dynamics simulations are recapitulated by a mean filed theory and intuitively explained in terms of the interplay between conformational entropy and translational entropy costs. This work lays down conceptual foundations that will be helpful in understanding how signaling may be regulated by epigenetic changes to interactions among and stoichiometries of multivalent molecules.

Essential revisions:

The following list of comments is rather long, but I believe that most of them can be addressed without extensive additional work:

Regarding the presentation

1) The average reader of eLife is likely to become confused by the various terms used. Monomers are used somewhat interchangeably to refer to the specific modules within an associative polymer or to an individual associative polymer. I recommend the use of the term sticker or domain (the former is more precise) to refer to the individual modules within the associative polymers. Then, the authors can explain clearly that they are assessing the impact of: (a) varying the valence of stickers whilst keeping the global stoichiometry of stickers fixed; (b) varying the global stoichiometry (concentration) of stickers whilst keeping the valence fixed; and (c) for both (a) and (b) they are also titrating the strengths of inter-sticker interactions. A box defining the key terms and a figure showing exactly what is being titrated and how would be very helpful to the authors' cause.

2) The main important features of Figure 4A and B should be explained more clearly. In particular, Figure 4B is not discussed in the text.

Regarding the model

3) Spacers are not likely to be passive. This point has been made in a few papers that deploy the stickers-and-spacers formalism. Please see: https://elifesciences.org/articles/30294. Accordingly, there will be an interplay between the effects of spacers and the triad of effects due to stickers. The effects of are largely ignored. It would be important to highlight this fact because even for the two-component system that motivates the current work, there are known effects due to spacers that cannot be ascribed to being non-specific effects. Please see: http://www.pnas.org/content/112/47/E6426.long.

4) Specific, inter-sticker interactions should be anisotropic (please see Choi, Holehouse and Pappu, 2020a). In the model used for the simulations, the inter-sticker interactions are isotropic. To work around this problem, the potential includes a repulsive term between stickers of the same type. This creates a conundrum because it introduces auxiliary energy and length scales. Therefore, the effective heterotypic inter-sticker interaction energies will be different from what is prescribed by the values for U0. Of additional importance is the diluting effect of the homotypic repulsive interactions. This could have non-trivial many-body consequences and further, it will impact the cluster size distributions, and the pressure. It would help if the authors were to furnish two details from analysis of their simulations: How does the internal pressure scale with the energy and length scales associated with repulsive homotypic interactions? And how do these repulsive interactions impact the U0 dependence of the magic ratio effect? One set of calculations should suffice.

Regarding the nature of the transition

5) The dimer gel theory is very interesting. Following Harmon et al., 2017 and 2018, Choi et al., 2019, and the seminal work of Semenov and Rubinstein, 1998, it is important to recognize that phase transitions in associative polymeric systems are best thought of as phase separation aided percolation. Accordingly, it would help to calculate the percolation threshold from the simulations and from the theory as well. For the computational work, this will help us understand how U0, the magic ratio, and valence impact the interplay between density transitions (aka phase separation) and percolation. Furthermore, there are two new generalizations of the work of Semenov and Rubinstein that are relevant to the theoretical work here. Please see: https://doi.org/10.1021/acs.macromol.8b00661 and http://dx.doi.org/10.1103/PhysRevE.102.042403. A comparison of these generalizations and the dimer gel theory introduced here would be very helpful and insightful, especially in the context of bond cooperativity.

6) Throughout, but most notable in the subsection “Dimer-gel Theory Results”, there seems to be confusion of phase separation (resulting from a thermodynamic instability) and gelation (a connectivity transition). The two cases need to be delineated more clearly.

7) Are the phase boundaries (both theoretical and from simulations) spinodals or binodals?

Regarding the simulations

8) The MD protocol is unusual. Why include harmonic bonding potentials (instead of FENE)? Why use a softened LJ potential with an attractive tail? Typically Monte Carlo moves are needed to enforce 1:1 binding (pioneered by Arlette Baljon) or a carefully-designed combination of 2-body and 3-body potentials (pioneered by Francesco Sciortino). The model in the current manuscript may also succeed at enforcing 1:1 binding, but this needs to be proven and shown not to introduce other biases.

9) The phase separation analysis of the MD results could be explained more rigorously. What checks have been done to confirm phase separation rather than a sol-gel transition or other artifact of simulation preparation. Have you confirmed that the two resultant phases are in chemical and mechanical equilibrium? Do the simulation pressure or structure factor show signatures of phase separation?

Regarding the Results and Discussion

10) The mean-field assumption allows for useful analytical results, but is not expected to be valid for the types of systems simulated and discussed. The chains of such short length are not overlapping over most of the concentration regime discussed and locations of stickers is highly correlated instead of mean-field. Further, as would be expected, simulations by C. E. Sing. and A. Alexander-Katz (Macromolecules 2011) have shown sticker-spacer descriptions do not extend to the case where every bead is sticky.

11) A primary limitation of the work is the very short chains used in the MD simulations and necessary for the results to be useful. In a true polymeric limit and for most real systems, the numbers of beads and stickers are an order of magnitude larger than discussed in the paper. At this large N limit, this magic ratio effect becomes rather trivial. The extent to which the results presented are a feature of the discrete limit of very short oligomers should be properly discussed.

12) The results in Figure 3 raise some questions. The results in the left column show a different non-monotonic dependence of the dilute phase concentration on the A:B monomer concentration ratio (best referred to as the concentration ratio of A : B stickers) when compared to the dense phase concentration. First, given that there are two species, how are concentrations in the dilute and dense phases being reduced to a single number? Second, there ends up being a regime where the dilute phase concentration of stickers increases and yet the dense phase concentration of stickers also increases, leading to what seems like a conservation of the width of the two-phase regime. Likewise, the opposite scenario obtains in a different regime, wherein the dilute phase concentration decreases and over on the right column, we see a decrease in the dense phase concentration. Does this imply that the width of the two-phase regime, quantifiable as the log10(cdense/cdilute) stays fixed as the A:B sticker concentration ratio changes? This aspect needs some thinking and further analysis or clarification. I am missing something because the inference I draw from the increase in the concentration of the dilute arm is of weakening the driving forces for phase separation. However, if that is accompanied by an increase in the concentration of the dense phase, then it would imply that despite the apparent diminution in the driving forces for phase separation, an increase in the concentration of the dense phase results in equalizing the chemical potential with the dilute phase. The converse also seems to obtain from the results. And this is perplexing.

13) The dense and dilute phase concentrations never seem to be more than an order of magnitude apart. This too is perplexing. Additionally, the concentrations, even in the dilute phase are in the millimolar range. And there is a perplexing conversion of an interaction energy of 9kT to dissociation constants, which ends up being rather high, in the millimolar range. This is confusing because the authors propose that these estimates are on a par with SUMO-SIM and SHR-PRM interactions, which is not the case as these interactions are in the low μM rather than millimolar range. There appears to an issue here with converting units into molar scales. I recommend the usage of volume fractions. For polymeric systems, volume fractions are the most robust route for quantifying concentrations – a point emphasized by Flory back in the early days of polymer science.

14) Finally, given the audience, i.e., readers of eLife, the nagging concern will pertain to the regime and systems where the magic number and magic ratio effects are manifest. This might be relevant and / or applicable in engineered systems. This is alluded to in the Discussion. It might also be the case that given the rather strong interactions required to observe the magic number and / or magic ratio effect, this might not be realizable in vivo. The challenge at high interaction strengths will be problems with solubility, the formation of precipitates and / or the onset of the glass transition. In fact, the latter has been speculated as being important in biology, see https://doi.org/10.1016/j.sbi.2016.05.002 and https://elifesciences.org/articles/09347. Indeed, the simulations of Choi et al., 2019, show this onset of glassy behavior as interaction strengths increase or temperature is lowered. Might it be the case that the magic number or ratio effects are masked by the drive to avoid precipitation and / or glassy behavior? Alternatively, as the authors suggest, the magic number / ratio effect might be a way to regulate phase behavior. Some elaboration of these issues, beyond what is currently offered, might be beneficial to the reader, especially if can be accompanied by pointers to specific systems that motivate experimental investigations.

https://doi.org/10.7554/eLife.62403.sa1

Author response

Essential revisions:

The following list of comments is rather long, but I believe that most of them can be addressed without extensive additional work:

Regarding the presentation

1) The average reader of eLife is likely to become confused by the various terms used. Monomers are used somewhat interchangeably to refer to the specific modules within an associative polymer or to an individual associative polymer. I recommend the use of the term sticker or domain (the former is more precise) to refer to the individual modules within the associative polymers. Then, the authors can explain clearly that they are assessing the impact of: (a) varying the valence of stickers whilst keeping the global stoichiometry of stickers fixed; (b) varying the global stoichiometry (concentration) of stickers whilst keeping the valence fixed; and (c) for both (a) and (b) they are also titrating the strengths of inter-sticker interactions. A box defining the key terms and a figure showing exactly what is being titrated and how would be very helpful to the authors' cause.

We agree with the reviewers that it is more suitable to use stickers/domains to refer to specific modules within the associative polymers for the readers of eLife, and have made corresponding changes throughout the manuscript. We have also added Figure 1 along with its caption to serve as a guideline of our work.

2) The main important features of Figure 4A and B should be explained more clearly. In particular, Figure 4B is not discussed in the text.

We agree – we have added more discussion concerning Figure 4 in the text:

“As for the dense phase concentration, it decreases monotonically as the global sticker stoichiometry departs from 1 and as the valence of polymers decreases (Figure 5B). This again indicates that the anomalous dependence of the dilute-phase concentration on valence and stoichiometry does not arise from special properties of the dense phase.”

Regarding the model

3) Spacers are not likely to be passive. This point has been made in a few papers that deploy the stickers-and-spacers formalism. Please see: https://elifesciences.org/articles/30294. Accordingly, there will be an interplay between the effects of spacers and the triad of effects due to stickers. The effects of are largely ignored. It would be important to highlight this fact because even for the two-component system that motivates the current work, there are known effects due to spacers that cannot be ascribed to being non-specific effects. Please see: http://www.pnas.org/content/112/47/E6426.long.

We thank the reviewers for highlighting important recent work on the role of spacers in phase separation. To address this and the following point regarding the length scale of homotypic repulsion, we have added the following sentences in the Discussion:

“Furthermore, real biological systems are more complex than our simple model. For example, there can be multiple-to-one binding, multiple components, and the spacers/linkers can also play nontrivial roles Banjade et al., 2015; Harmon et al., 2017.”

“On the other hand, the dense-phase concentration strongly depends on the steric repulsion – increasing the sticker size from 2.5nm to 2.9nm decreases the dense phase concentration by a factor of 2 (Appendix 1—figure 5B). This is consistent with results from previous studies on the role of linkers: a self-avoiding random coil linker which occupies a large volume can substantially lower the dense-phase concentration and even prevent phase separation Harmon et al., 2017.”

4) Specific, inter-sticker interactions should be anisotropic (please see Choi et al., 2020a). In the model used for the simulations, the inter-sticker interactions are isotropic. To work around this problem, the potential includes a repulsive term between stickers of the same type. This creates a conundrum because it introduces auxiliary energy and length scales. Therefore, the effective heterotypic inter-sticker interaction energies will be different from what is prescribed by the values for U0. Of additional importance is the diluting effect of the homotypic repulsive interactions. This could have non-trivial many-body consequences and further, it will impact the cluster size distributions, and the pressure. It would help if the authors were to furnish two details from analysis of their simulations: How does the internal pressure scale with the energy and length scales associated with repulsive homotypic interactions? And how do these repulsive interactions impact the U0 dependence of the magic ratio effect? One set of calculations should suffice.

Ur(r)=4ϵ[(σr)12(σr)6+14], r<21/6σ,

We agree with the reviewers that biological sticker-sticker interactions are anisotropic. It is this anisotropy and volume exclusion between stickers that lead to one-to-one association. The model we developed is highly simplified, keeping the key property (one-to-one association) but removing system specific details. It is therefore a conceptual model. That said, our approach does not introduce additional energy or length scales: the repulsive interactions between like stickers correspond to steric repulsion which must be present in any model, and the energy scale of this repulsion is essentially irrelevant, since the repulsive interactions are effectively in the regime of “hard-sphere” repulsion. Quantitative descriptions of real systems will likely require additional features, for example using sticky patches to model anisotropic binding. To address the reviewers’ question about the dependence on the length scale of repulsion, we performed new simulations for the A8:B8 system. Here, we used the standard repulsive LJ potential for interactions between stickers of the same typewhere ϵ=1kBT. To examine the role of the repulsive length scale, we employed two choices of sticker diameters σ = 2.9 nm (second virial coefficient matched to the soft LJ potential used in the manuscript) and a smaller bead σ = 2.5 nm. The bond potential and the attraction between stickers of different types are the same as previously employed (i.e., Equations 16 and 18 in the manuscript), except we used binding parameters r0 = 1.5 nm and U0=12kBT which lead to weaker binding. The dilute- and dense-phase sticker concentrations and dense-phase volume fraction for bead diameter σ = 2.9 nm are shown in Author response image 1, and for σ = 2.5 nm are shown in Author response image 2.

Author response image 1
Author response image 2

(Note that the overlapping volume between two bound stickers is counted once in the volume fraction calculation, i.e., two perfectly overlapping stickers would occupy a volume of one sticker.) Note also that because we used a harmonic potential for backbone linker bonds which has a well-defined mean bond length 4.7 nm (not sensitive to the choice of σ), the changes in the dilute- and dense-phase concentrations purely reflect the effect of the homotypic repulsive interactions. (We show in point 13 below how these quantities depend on linker length.)Comparing the two cases, the reviewers are certainly correct that there is a diluting effect of the homotypic repulsive interactions. A stronger repulsion/larger repulsive length scale results in a lower concentration and volume fraction in the dense phase. The onset binding strength U0 for the magic-number/magic-ratio effect also becomes higher. For σ = 2.9 nm there is almost no magic-ratio effect at U0=12kBT and the effect is there at U0=14kBT (similar to what is shown in Figure 3 in the manuscript), whereas for σ = 2.5 nm the effect is already quite significant at U0=12kBT.

Related to this effect, if we were to model the linkers explicitly and if the linkers fall into the category of self-avoiding random coil (SARC) (Harmon et al., 2017), then, qualitatively, compared to a Flory random coil (FRC) linker, the SARC linker effect can be viewed as increasing the repulsive length scale of stickers (assume the SARC and FRC linkers have the same mean end-to-end distance), and therefore could lead to weakened magic-number/magic-ratio effects.

Please see the additional remarks we have added to the Discussion noted in our response to point 3.

Regarding the nature of the transition

5) The dimer gel theory is very interesting. Following Harmon et al., 2017 and 2018, Choi et al., 2019, and the seminal work of Semenov and Rubinstein, 1998, it is important to recognize that phase transitions in associative polymeric systems are best thought of as phase separation aided percolation. Accordingly, it would help to calculate the percolation threshold from the simulations and from the theory as well. For the computational work, this will help us understand how U0, the magic ratio, and valence impact the interplay between density transitions (aka phase separation) and percolation. Furthermore, there are two new generalizations of the work of Semenov and Rubinstein that are relevant to the theoretical work here. Please see: https://doi.org/10.1021/acs.macromol.8b00661 and http://dx.doi.org/10.1103/PhysRevE.102.042403. A comparison of these generalizations and the dimer gel theory introduced here would be very helpful and insightful, especially in the context of bond cooperativity.

We agree with the reviewers that the dense phase of associative polymers is a percolating network, and we thank the reviewer for pointing us to relevant theories. To our knowledge, and as argued by Semenov and Rubenstein, a bulk percolation threshold (aka homogeneous gelation transition) only exists at high temperatures/weak binding where phase separation doesn’t happen. Our current study, however, is focused on the low temperature regime. At these low temperatures, while we expect the percolation threshold cp to lie between the dilute- and dense-phase concentrations, if we were to simulate a system at cp, the system would automatically phase separate. That said, following the reviewers’ suggestion we can gain some insight into a characteristic density for homogeneous percolation in our model by characterizing the percolation threshold in the weak binding regime U0=8kBT, which avoids phase separation but also prevents many-to-one binding. To present these results, we have incorporated a new section and figure in Appendix 1 (subsection “Determining the percolation threshold”, Appendix 1—figure 3).

We have also highlighted the relevant theories that generalize the work of Semenov and Rubenstein in the Discussion:

“One of the major assumptions of the dimer-gel theory is a mean-field approximation. […] This approximation captures a key feature of the dense phase, namely that a single polymer binds to multiple partners.”

6) Throughout, but most notable in the subsection “Dimer-gel Theory Results”, there seems to be confusion of phase separation (resulting from a thermodynamic instability) and gelation (a connectivity transition). The two cases need to be delineated more clearly.

Because the dense phase of associative polymers is a gel, we used “gel” interchangeably with “dense phase” in some cases. To avoid possible confusion, we have made the following changes:

“However, cluster size may reflect a sol-gel percolation transition rather than a thermodynamic phase transition Harmon et al., 2017, and thus provides at best a qualitative measure of phase separation.”

“Comparing the two expressions, dimers are favored at low concentrations, whereas a network of independent bonds is favored at high concentrations.”

7) Are the phase boundaries (both theoretical and from simulations) spinodals or binodals?

The phase boundaries are binodals, and we have now clarified this point: “To find the binodal phase boundaries, we simulate hundreds of polymers of types A and B…” and “Finally, to extract the binodal phase boundaries, we substitute…”

Regarding the simulations

8) The MD protocol is unusual. Why include harmonic bonding potentials (instead of FENE)? Why use a softened LJ potential with an attractive tail?

Ur(r)=4ϵ[(σr)12(σr)6+14], r<21/6σ.
Ub(r)=12KR02ln[1(rσR0)2], r<σ+R0.
Ua(r)=12U0(1+cosπrr0), r<r0,

We agree with the reviewers that our MD potentials are not the most standard choices. Part of our reason for using a harmonic potential and a softened LJ potential was to speed up the MD simulations, as the softened potentials allow a larger integration time step. To demonstrate the robustness of our conclusions to the choice of MD potentials, we have performed new simulations for the A8:B8 system using FENE and standard LJ potentials. Briefly, the interaction between stickers of the same type is purely repulsive We take ϵ=1kBT and σ=3 nm, which results in a repulsive potential comparable to our previous softened LJ potential. Neighboring stickers are connected by a FENE potentialWe take K=0.3kBT/nm2 and R0=7 nm, which (together with the repulsive interaction) results in the same mean bond length 4.7 nm as for our previous harmonic potential. (This potential resembles a 20 amino acid disordered FCS linker of persistence length 0.8 nm.) The attractive interaction between different types of stickers has the same form as in our manuscript: where r0=1.5 nm is the cutoff distance. For U0=14kBT, we show the total dilute- and dense-phase sticker concentrations in Author response image 3.

Author response image 3

The results are qualitatively unchanged from the results presented in Figure 3 in our manuscript. Quantitatively, the dilute-phase concentrations drop by a factor of 2 and the dense-phase concentrations are comparable to the reported values. We conclude that our conceptual results on the magic-number/magic-ratio effects are robust to the detailed choice of potentials.

Typically Monte Carlo moves are needed to enforce 1:1 binding (pioneered by Arlette Baljon) or a carefully-designed combination of 2-body and 3-body potentials (pioneered by Francesco Sciortino). The model in the current manuscript may also succeed at enforcing 1:1 binding, but this needs to be proven and shown not to introduce other biases.

Testing for biases is certainly important. We always checked the number of bound partners for all the stickers in our simulations. A sticker is considered to be a partner of another sticker if the two are of different types and are within the cutoff distance r0 of the attractive potential. Across our simulations, the average fraction of stickers that have more than one partner is less than 0.001%.

Regarding other possible biases, we note that Langevin dynamics can be viewed as moves to achieve thermodynamic equilibrium. We carefully designed the initial state of the system, and ran long enough simulations for the system to fully equilibrate, and are thus confident that there is no bias against equilibrium.

One possible “bias” is that stickers overlap upon binding, which leads to a reduction in volume. In principle, this could enhance the effect of “crowding promoted binding” in the dense phase, i.e., binding being further favored by the resulting reduction in volume. However, the volume fraction of the dense phase, ∼10%, is not high, so this effect is very modest.

9) The phase separation analysis of the MD results could be explained more rigorously. What checks have been done to confirm phase separation rather than a sol-gel transition or other artifact of simulation preparation. Have you confirmed that the two resultant phases are in chemical and mechanical equilibrium? Do the simulation pressure or structure factor show signatures of phase separation?

Phase separation and a sol-gel transition may be ambiguous in the high temperature regime and/or close to a critical point. In our case, we are working in the strongly condensed regime, i.e., low temperature and adequate polymer concentrations, and have excluded systems where the dilute- and dense-phase concentrations become comparable (e.g., A8:B8 system at A:B sticker ratio 8:6). For all the systems reported in the manuscript, there is a clearly-defined, large dense-phase cluster located in a small region of the simulation box and a dilute-phase of polymers/oligomers dispersed throughout the rest of the box (representative snapshots are shown in Figure 2C and 2D), it is thus clear these systems are phase separating (and thus undergoing dense-phase gelation driven by phase separation).

Langevin dynamics ensures a Boltzmann distribution. We have carefully designed the initial state of the system (a fully connected dense phase) to prevent the system becoming stuck in a local minimum (such as dispersed dense droplets that take a long time to merge). We have then run a long simulation to allow the dense phase to equilibrate with the dilute phase. The relaxation time of the system depends on the sticker-sticker bond lifetime τ. To ensure that the dilute and dense phases are in equilibrium, all simulations were equilibrated ∼2000τ before data was recorded. We also checked whether there are systematic deviations between the first and second halves of the recorded simulation, and find consistent results between the two halves.

We have added a few sentences to make the discussion of phase separation more rigorous:

“Across our simulations, on average the fraction of stickers that have more than one partner is less than 0.001%.” and “The relaxation time of the system depends on the sticker-sticker bond lifetime; to ensure that the dilute and dense phases are in equilibrium…”

Regarding the Results and Discussion

10) The mean-field assumption allows for useful analytical results, but is not expected to be valid for the types of systems simulated and discussed. The chains of such short length are not overlapping over most of the concentration regime discussed and locations of stickers is highly correlated instead of mean-field. Further, as would be expected, simulations by C. E. Sing. and A. Alexander-Katz (Macromolecules 2011) have shown sticker-spacer descriptions do not extend to the case where every bead is sticky.

We agree with the reviewers that the mean-field assumption is not valid over the entire range of possible concentrations as the chains may not overlap. However, to locate the dilute and dense-phase boundaries with our convex-hull analysis, we only need our dense-phase mean-field approximation to be valid in the concentration regime comparable to the dense-phase concentrations. The mean-field approximation is still not fully quantitative at dense-phase concentrations as we found correlations in binding in our simulations (correlation length ~1.8 beads) that go beyond mean-field theory. Therefore, a simulated A14:B14 system has an effective valence ~8 and therefore behaves more like an A8:B8 system in theory. Nevertheless, the theory works better for systems with longer linker lengths (or smaller beads), where binding is less correlated in the dense phase.

11) A primary limitation of the work is the very short chains used in the MD simulations and necessary for the results to be useful. In a true polymeric limit and for most real systems, the numbers of beads and stickers are an order of magnitude larger than discussed in the paper. At this large N limit, this magic ratio effect becomes rather trivial. The extent to which the results presented are a feature of the discrete limit of very short oligomers should be properly discussed.

Our work focuses on proteins with complementary domains (“stickers”) instead of intrinsically disordered proteins which could have very large numbers of effective stickers. Examples of the kinds of systems that motivated our work include SUMO-SIM (Banani et al., 2016), SHR-PRM (Li et al., 2012), and Rubisco-EPYC1 (Rosenzweig et al., 2017), which are all short in the sense of having <10-15 stickers per polymer. That said, the magic-number/magic-ratio effects extend to large N and, in fact, the effects become stronger with increasing polymer valences (Figure 3 in the manuscript).

12) The results in Figure 3 raise some questions. The results in the left column show a different non-monotonic dependence of the dilute phase concentration on the A:B monomer concentration ratio (best referred to as the concentration ratio of A : B stickers) when compared to the dense phase concentration. First, given that there are two species, how are concentrations in the dilute and dense phases being reduced to a single number?

The concentrations shown in Figure 3 are the sum of the A and B sticker concentrations, which we now refer to as the total sticker concentration.

Second, there ends up being a regime where the dilute phase concentration of stickers increases and yet the dense phase concentration of stickers also increases, leading to what seems like a conservation of the width of the two-phase regime. Likewise, the opposite scenario obtains in a different regime, wherein the dilute phase concentration decreases and over on the right column, we see a decrease in the dense phase concentration. Does this imply that the width of the two-phase regime, quantifiable as the log10(cdense/cdilute) stays fixed as the A:B sticker concentration ratio changes? This aspect needs some thinking and further analysis or clarification.

These are certainly correct statements. In fact, the nonmonotonic dependence of the dilute-phase concentration on A:B sticker ratio is a hallmark of the magic-number/magic-ratio effect. Taking the simulation in point 8 as an example, cdense/cdilute vs. A:B sticker concentration ratio (plotted on a log-log scale) first increases and then decreases as the A:B ratio deviates from 1. This nonmonotonicity is entirely due to the magic-number/magic-ratio effect raising the dilute-phase concentration at equal A:B ratio.

Author response image 4

I am missing something because the inference I draw from the increase in the concentration of the dilute arm is of weakening the driving forces for phase separation. However, if that is accompanied by an increase in the concentration of the dense phase, then it would imply that despite the apparent diminution in the driving forces for phase separation, an increase in the concentration of the dense phase results in equalizing the chemical potential with the dilute phase. The converse also seems to obtain from the results. And this is perplexing.

For conventional polymer phase separation (such as systems described by Flory-Huggins mean-field theory), the increase of the dilute-phase concentration is connected to a weakening of the driving forces for phase separation and hence a lowered dense-phase concentration. However, this is not always the case for more complicated systems, in particular when phase separation is not simply driven by a competition between energy and entropy. For our two-component system, phase separation is essentially driven by a competition between translational and conformational entropy. Briefly, as the A:B sticker ratio approaches 1, the dilute-phase concentration increases. This means that the chemical potential of dimers in the dilute phase increases. As the dense phase is in equilibrium with the dilute phase, the chemical potential in the dense phase necessarily also increases. Mechanistically, why does the dense phase have a higher concentration and a higher chemical potential at A:B ratio 1? In fact, this is only true in the strong binding regime. As all the stickers have to be paired when binding is strong at A:B ratio 1, this leads to a contraction of the dense phase, and thus a high concentration. This same contraction leads to conformational frustration in the dense phase, reduces the conformational entropy, and thus raises the chemical potential.

13) The dense and dilute phase concentrations never seem to be more than an order of magnitude apart. This too is perplexing. Additionally, the concentrations, even in the dilute phase are in the millimolar range.

We agree with the reviewers that our dense- and dilute-phase concentration ratios as well as the dilute-phase concentrations are not in the range typical of most of biological systems. We have explored how these quantities depend on the simulation parameters. In particular, we found that the dilute-phase concentration is very sensitive to the linker length. Taking U0=12kBT,K=0.3kBT/nm2, and R0=7nm (a 20 amino acid FRC linker of persistence length 0.8 nm, mean bond length 4.7 nm), all other parameters are the same as listed in point 8, see Author response image 5, where as U0=12kBT,K=0.15kBT/nm2, and R0=14 nm (a 40 amino acid FRC linker of persistence length 0.8 nm, mean bond length 5.9nm) yields are shown in Author response image 6

Author response image 5
Author response image 6

Thus, upon increasing the linker length ~25%, the dilute-phase concentration drops more than a factor of 10, while the dense-phase concentration remains almost unchanged. Therefore, higher dense-phase to dilute-phase concentration ratios and lower dilute-phase concentrations are achievable with minimal modifications. We acknowledge that many parameter choices in the manuscript are at least in part motivated by promoting simulation speed. We have incorporated the above results in our manuscript “Reducing the steric repulsion between beads of the same type has a similar effect (Appendix 1—figure 5A). More significantly, increasing the mean linker length from 4.7 nm to 5.9 nm leads to a more than 10-fold reduction in the dilute-phase concentration (Appendix 1—figure 6A)…” and a new section Effects of nonspecific interactions and linker length in Appendix 1.

And there is a perplexing conversion of an interaction energy of 9kT to dissociation constants, which ends up being rather high, in the millimolar range. This is confusing because the authors propose that these estimates are on a par with SUMO-SIM and SHR-PRM interactions, which is not the case as these interactions are in the low μM rather than millimolar range.

Kd=[0r04πr2exp[Ua(r)kBT]dr]1,

We designed our potentials to fall in the range of typical sticker-sticker Kd values. The sticker-sticker dissociation constant can be found fromwhere r0 is the cutoff distance for an attractive potential. U0 is the minimum of Ua(r). Importantly, Kd is not only determined by U0 but also by the volume of the binding pocket. We found Kd=0.4mM for U0=9kBT and r0=2nm based on the above expression. The reported sticker-sticker dissociation constant for SUMO-SIM is Kd=0.01mM (Banani et al., 2016) and for SHR-PRM is Kd=0.35mM (Li et al., 2012). The magic-number/magic-ratio effects are enhanced with increasing binding strength. We find that Kd0.4mM provides a rough threshold for the onset of these effects, although this threshold also depends on valence of the polymers, sticker size, and bond length, etc.

There appears to an issue here with converting units into molar scales. I recommend the usage of volume fractions. For polymeric systems, volume fractions are the most robust route for quantifying concentrations – a point emphasized by Flory back in the early days of polymer science.

We agree with the reviewers that volume fraction is a sensible quantity for characterizing dense-phase concentration, the dense-phase volume fractions are ~10% in the simulations in our manuscript. We have accordingly added this quantification in the text where it will be most helpful. “We also report in Appendix 1—figure 5C the volume fraction of the polymers in the dense phase, which is ~10%, comparable to the volume fraction of proteins in the cell cytoplasm.”

14) Finally, given the audience, i.e., readers of eLife, the nagging concern will pertain to the regime and systems where the magic number and magic ratio effects are manifest. This might be relevant and / or applicable in engineered systems. This is alluded to in the Discussion. It might also be the case that given the rather strong interactions required to observe the magic number and / or magic ratio effect, this might not be realizable in vivo. The challenge at high interaction strengths will be problems with solubility, the formation of precipitates and / or the onset of the glass transition. In fact, the latter has been speculated as being important in biology, see https://doi.org/10.1016/j.sbi.2016.05.002 and https://elifesciences.org/articles/09347. Indeed, the simulations of Choi et al., 2019, show this onset of glassy behavior as interaction strengths increase or temperature is lowered.

We share the same uncertainty as the reviewers whether the magic-number/magic-ratio effects can manifest and be utilized by cells as a way to regulate condensates. From a purely theoretical point of the view, as long as the sticker-sticker dissociation constant is below a certain (submillimolar) threshold, there is a very real chance to observe these effects. In this sense, many condensate scaffold components satisfy this criterion. However, real biological systems are of course far more complex than our simple model. There can be multiple-to-one binding, multiple components, and, as has been pointed out by the reviewers, the spacers can also play important, but complicated roles. So the in vivo relevance of the effects explored here remains an open question.

Might it be the case that the magic number or ratio effects are masked by the drive to avoid precipitation and / or glassy behavior? Alternatively, as the authors suggest, the magic number / ratio effect might be a way to regulate phase behavior. Some elaboration of these issues, beyond what is currently offered, might be beneficial to the reader, especially if can be accompanied by pointers to specific systems that motivate experimental investigations.

We thank the reviewers for sharing these interesting thoughts, and we have modified the text accordingly “However, the magic-ratio effect has not been observed in these systems Li et al., 2012; Banani et al., 2016, possibly due to size and linker length mismatch between the two associating polymers.[…] Currently, the in vivo relevance of the effects explored in this work remains an open question.”

https://doi.org/10.7554/eLife.62403.sa2

Article and author information

Author details

  1. Yaojun Zhang

    Center for the Physics of Biological Function, Princeton University, Princeton, United States
    Contribution
    Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-4587-6834
  2. Bin Xu

    Department of Physics, Princeton University, Princeton, United States
    Contribution
    Conceptualization, Writing - review and editing
    Competing interests
    No competing interests declared
  3. Benjamin G Weiner

    Department of Physics, Princeton University, Princeton, United States
    Contribution
    Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-1995-8660
  4. Yigal Meir

    1. Department of Physics, Princeton University, Princeton, United States
    2. Department of Physics, Ben Gurion University of the Negev, Beersheba, Israel
    3. Department of Molecular Biology, Princeton University, Princeton, United States
    Contribution
    Conceptualization, Writing - review and editing
    Competing interests
    No competing interests declared
  5. Ned S Wingreen

    1. Department of Molecular Biology, Princeton University, Princeton, United States
    2. Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, United States
    Contribution
    Conceptualization, Supervision, Funding acquisition, Writing - original draft, Writing - review and editing
    For correspondence
    wingreen@princeton.edu
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-7384-2821

Funding

National Science Foundation (PHY 1734030)

  • Yaojun Zhang
  • Bin Xu
  • Benjamin G Weiner
  • Yigal Meir
  • Ned S Wingreen

National Science Foundation (PHY 1521553)

  • Benjamin G Weiner
  • Ned S Wingreen

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

We thank Guanhua He, Martin Jonikas, and Daniel Lee for insightful suggestions and comments. This work was supported in part by the National Science Foundation, through the Center for the Physics of Biological Function (PHY-1734030), and NSF grant PHY-1521553.

Senior Editor

  1. Aleksandra M Walczak, École Normale Supérieure, France

Reviewing Editor

  1. Pierre Sens, Institut Curie, PSL Research University, CNRS, France

Reviewer

  1. Rohit V Pappu, Washington University in St Louis, United States

Publication history

  1. Received: August 24, 2020
  2. Accepted: February 18, 2021
  3. Version of Record published: March 11, 2021 (version 1)

Copyright

© 2021, Zhang et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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Further reading

    1. Neuroscience
    2. Physics of Living Systems
    Zhenyong Wu et al.
    Research Article Updated

    All membrane fusion reactions proceed through an initial fusion pore, including calcium-triggered release of neurotransmitters and hormones. Expansion of this small pore to release cargo is energetically costly and regulated by cells, but the mechanisms are poorly understood. Here, we show that the neuronal/exocytic calcium sensor Synaptotagmin-1 (Syt1) promotes expansion of fusion pores induced by SNARE proteins. Pore dilation relied on calcium-induced insertion of the tandem C2 domain hydrophobic loops of Syt1 into the membrane, previously shown to reorient the C2 domain. Mathematical modelling suggests that C2B reorientation rotates a bound SNARE complex so that it exerts force on the membranes in a mechanical lever action that increases the height of the fusion pore, provoking pore dilation to offset the bending energy penalty. We conclude that Syt1 exerts novel non-local calcium-dependent mechanical forces on fusion pores that dilate pores and assist neurotransmitter and hormone release.

    1. Physics of Living Systems
    Myron Barber Child VI et al.
    Research Article Updated

    Three-dimensional eukaryotic genome organization provides the structural basis for gene regulation. In Drosophila melanogaster, genome folding is characterized by somatic homolog pairing, where homologous chromosomes are intimately paired from end to end; however, how homologs identify one another and pair has remained mysterious. Recently, this process has been proposed to be driven by specifically interacting ‘buttons’ encoded along chromosomes. Here, we turned this hypothesis into a quantitative biophysical model to demonstrate that a button-based mechanism can lead to chromosome-wide pairing. We tested our model using live-imaging measurements of chromosomal loci tagged with the MS2 and PP7 nascent RNA labeling systems. We show solid agreement between model predictions and experiments in the pairing dynamics of individual homologous loci. Our results strongly support a button-based mechanism of somatic homolog pairing in Drosophila and provide a theoretical framework for revealing the molecular identity and regulation of buttons.