The complementary nature of our multiple methodologies allows us to focus sharply on the electrostatic aspects of the hydrolysis-independent role of ATP in biomolecular condensation by comparing ATP’s effects with those of simple salt. Here, Caprin1 and pY-Caprin1 are modeled minimally as heteropolymers of charged and neutral beads in rG-RPA and FTS. ATP and ATP-Mg are modeled as simple salts (single-bead ions) in rG-RPA, whereas they are modeled with more structural complexity as short charged polymers (multiple-bead chains) in FTS, although the latter models are still highly coarse-grained. Despite this modeling difference, rG-RPA and FTS both rationalize experimentally observed ATP- and NaCl-modulated reentrant LLPS of Caprin1 and a lack of a similar reentrance for pY-Caprin1 as well as a prominent colocalization of ATP with the Caprin1 condensate. Consistently, the same contrasting trends in the effect of NaCl on Caprin1 and pY-Caprin1 are also seen in our coarse-grained MD simulations, although polymer field theories tend to overestimate LLPS propensity (Shen and Wang, 2017). The robustness of the theoretical trends across different modeling platforms underscores electrostatics as a significant component in the diverse roles of ATP in the context of its well-documented ability to modulate biomolecular LLPS via hydrophobic and π-related effects (Patel et al., 2017; Kota et al., 2024; Kang et al., 2019). Analyses of these other nonelectrostatic effects are mostly beyond the scope of the present work but their impact is nevertheless illustrated by the Flory-Huggins interactions augmented to rG-RPA to quantitatively account for experimental data and our MD simulation of the arginine-to-lysine Caprin1 mutants. These findings are detailed below.

The 103-residue Caprin1 is a highly charged IDR with 19 charged residues [Figure 1a and Appendix 1, Appendix 1—figure 1]: 15 R, 1 K, and 3 aspartic acids (D); fraction of charged residues = 19/103 = 0.184 and NCPR = 13/103 = 0.126. With a substantial positive net charge, Caprin1’s phase behaviors are markedly different from those of polyampholytic IDRs with nearly zero net charge such as Ddx4 to which early sequence-specific LLPS theories were targeted (Nott et al., 2015; Lin et al., 2016). Instead, Caprin1 behaves like chemically synthesized polyelectrolytes (Dobrynin and Rubinstein, 2005). In contrast, when most or all of the 7 tyrosines (Y) in the Caprin1 IDR are phosphorylated (pY), negative charges are added to produce a near-net-neutral polyampholyte. Mass spectrometry indicates that the experimental sample of highly phosphorylated Caprin1 consists mainly of a mixture of IDRs with 6 or 7 phosphorylations (Appendix 1—figure 2). We refer to this experimental sample as pY-Caprin1 below. For simplicity, we use only the Caprin1 IDR with 7 pYs to model the behavior of this experimental sample in our theoretical/computational formulations, partly to avoid the combinatoric complexity of sequences with 5 or 6 pYs. Accordingly, since the charge of a pY is ≈ –2 at the experimental pH = 7.4, –14 charges are added to Caprin1 for our model pY-Caprin1, resulting in a polyampholyte with a very small NCPR = −1/103 = −0.00971 (Figure 1b). Both the experimental pY-Caprin1 (NCPR ≈ ±1/103 = ±0.00971) and model pY-Caprin1 are expected to exhibit phase properties similar to other polyampholytic IDRs.

Figure 1

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Experimental data points and numerical data for the theoretical curves in Figure 1c and d.

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While sequence-specific RPA has been applied successfully to model electrostatic effects on the LLPSs of various polyampholytic IDRs (Lin et al., 2018; Lin et al., 2016; Lin and Chan, 2017; Das et al., 2020; Wessén et al., 2021), RPA is less appropriate for polyelectrolytes with large NCPR (Mahdi and Olvera de la Cruz, 2000; Ermoshkin and Olvera de la Cruz, 2003; Orkoulas et al., 2003) because of its treatment of polymers as ideal Gaussian chains (Muthukumar, 2017). Traditionally, theories for polyelectrolytes tackle their peculiar conformations by various renormalized blob constructs (Dobrynin and Rubinstein, 2005; Mahdi and Olvera de la Cruz, 2000), two-loop polymer field theory (Muthukumar, 1996), modified thermodynamic perturbation theory (Budkov et al., 2015), and renormalized Gaussian fluctuation (RGF) theory (Shen and Wang, 2017; Shen and Wang, 2018), among others. As such, these formulations are mostly designed for homopolymers, making them difficult to apply directly to heteropolymeric biopolymers. In order to analyze Caprin1 and pY-Caprin1 LLPSs, we utilize rG-RPA (Lin et al., 2020), which combines Gaussian chains of effective (renormalized) Kuhn length with the key idea of RGF (Sawle and Ghosh, 2015).

Figure 1c and d show that the salt- and temperature (${\displaystyle T}$)-dependent phase diagrams predicted by rG-RPA with an augmented Flory-Huggins (FH) mean-field ${\displaystyle \chi (T)={ϵ}_h/T^{\ast }+{ϵ}_s}$ parameter for nonelectrostatic interactions, where ${ϵ}_h$ and ${ϵ}_s$ are the enthalpic and entropic contributions, respectively, and ${\displaystyle T^{\ast }}$ is reduced temperature (Lin et al., 2016; Lin et al., 2020; Eq. 10 of Lin et al., 2016 and ‘rG-RPA+FH’ theory in Appendix 1), are in reasonable agreement with experiment using bulk [Caprin1] (initial overall concentration) ≈ 200µM. (Concentrations are provided in molarity and also as mass density in Figure 1 and subsequent figures.) The rG-RPA+FH results in Figure 1c indicate that (i) Caprin1 undergoes LLPS below 20 °C with 100 mM NaCl, and that (ii) LLPS propensity, quantified by the upper critical solution temperature (UCST), increases with [NaCl]. These predictions are consistent with experimental data, including the observation that Caprin1 does not phase separate at room temperature without salt, ATP, RNA, or other proteins, though Caprin1 LLPS can be triggered by adding wildtype (WT) and phosphorylated FMRP and/or RNA (overall [Caprin1] ≳ 10 μM) (Kim et al., 2019), NaCl (Wong et al., 2020), or ATP (overall [Caprin1] = 400 μM) (Kim et al., 2021). The trend here is also in line with other theories of polyelectrolytes (Shen and Wang, 2018). In contrast, rG-RPA+FH results in Figure 1d for pY-Caprin1 shows decreasing LLPS propensity with increasing [NaCl], consistent with experimental data and the expected salt dependence of LLPS of nearly net-neutral polyampholytic IDRs such as Ddx4 (Lin et al., 2016).

Interestingly, the decrease in some of the condensed-phase [pY-Caprin1]s with decreasing ${\displaystyle T}$ (orange and green symbols for ≲ 20 °C in Figure 1d trending toward slightly lower [pY-Caprin1]) may suggest a hydrophobicity-driven lower critical solution temperature (LCST)-like reduction of LLPS propensity as temperature approaches ∼ 0 °C as in cold denaturation of globular proteins (Lin et al., 2018; Dignon et al., 2019a), though the hypothetical LCST is below 0 °C and therefore not experimentally accessible. If that is the case, the LLPS region would resemble those with both an UCST and an LCST (Cinar et al., 2019). As far as simple modeling is concerned, such a feature may be captured by a FH model wherein interchain contacts are favored by entropy at intermediate to low temperatures and by enthalpy at high temperatures, thus entailing a heat capacity contribution in ${\displaystyle \chi (T)}$, with ${\displaystyle {ϵ}_h\to {ϵ}_h(T)}$, ${\displaystyle {ϵ}_s\to {ϵ}_s(T)}$ (Lin et al., 2018; Dill et al., 1989; Kaya and Chan, 2003), beyond the temperature-independent ${ϵ}_h$ and ${ϵ}_s$ used in Figures 1c, d , 2. Alternatively, a reduction in overall condensed-phase concentration can also be caused by formation of heterogeneous locally organized structures with large voids at low temperatures even when interchain interactions are purely enthalpic (Figure 4 of Statt et al., 2020).

Figure 2

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Numerical plotting data for the theory-predicted phase diagrams in Figure 2.

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Figure 1c and d, though informative, are computed by a restricted rG-RPA+FH that assumes a spatially uniform [Na⁺]. For a more comprehensive physical picture, we now examine possible differences in salt concentration between the IDR-dilute and condensed phases by applying unrestricted rG-RPA+FH to compute two-dimensional salt-Caprin1/pY-Caprin1 phase diagrams (Figure 2).

As stated in Materials and methods and Appendix 1, here we define ‘counterions’ and ‘salt ions’, respectively, as the small ions with charges opposite and identical in sign to that of the net charge, ${\displaystyle Q}$, of a given polymer. For the Caprin1/NaCl system, since Caprin1’s net charge is positive, Na⁺ is salt ion and Cl^– is counterion. Overall electric neutrality of the system implies that the concentrations (${\displaystyle \rho }$’s) of polymer (${\displaystyle {\rho }_{\mathrm{p}}}$), counterions (${\displaystyle {\rho }_{\mathrm{c}}}$), and salt ions (${\displaystyle {\rho }_{\mathrm{s}}}$) are related by

where ${\displaystyle z_{\mathrm{s}}}$ and ${\displaystyle z_{\mathrm{c}}}$ are, respectively, the valencies of salt ions and counterions. For Caprin1 and pY-Caprin1, ${\displaystyle Q=+13}$ and –1, respectively, and ${\displaystyle (z_{\mathrm{s}},z_{\mathrm{c}})=(1,1),(1,2),\mathrm{a}\mathrm{n}\mathrm{d}(2,4)}$ are models for different small-ion species in the system. Specifically, in Figure 2, we identify the ${\displaystyle z_{\mathrm{s}}=1}$ salt ion as Na⁺ (Figure 2a–f) and the ${\displaystyle z_{\mathrm{c}}=1}$ counterion as Cl^– (Figure 2a–d), the ${\displaystyle z_{\mathrm{c}}=2}$ counterion as (ATP-Mg)²⁻ (Figure 2g and h), the ${\displaystyle z_{\mathrm{s}}=2}$ salt ion as Mg²⁺ and the ${\displaystyle z_{\mathrm{c}}=4}$ counterion as ATP⁴⁻ (Figure 2i–l). As mentioned above, in the present rG-RPA formulation, (ATP-Mg)²⁻ and ATP⁴⁻ are modeled minimally as a single-bead ion. They are represented by charged polymer models with more structural complexity in the FTS models below.

Notably, Figure 2a and b (${\displaystyle z_{\mathrm{s}}=z_{\mathrm{c}}=1}$) predicts that Caprin1 does not phase separate without Na⁺, consistent with experiment, indicating that monovalent counterions alone (Cl^– in this case) are insufficient for Caprin1 LLPS. When [Na⁺] is increased, the system starts to phase separate at a small [Na⁺] ≲ 0.1 M, with LLPS propensity increasing to a maximum at [Na⁺] ∼ 1 M before decreasing at higher [Na⁺], in agreement with experiment (Figure 3a, blue data points) and consistent with Caprin1 LLPS propensity increasing with [NaCl] from 0.1 to 0.5 M (Figure 1c). The predicted reentrant dissolution of Caprin1 condensate at high [Na⁺] in Figure 2a is consistent with measurement up to [Na⁺] ≈ 4.6 M indicating a significant decrease in LLPS propensity when [Na⁺] ≳ 2.5 M (Figure 3a), though the gradual decreasing trend suggests that complete dissolution of condensed droplets is not likely even when NaCl reaches its saturation concentration of ∼6 M.

Figure 3

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Numerical values of the experimental data plotted in Figure 3a.

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The negative tieline slopes in Figure 2a and b predict that Na⁺ is partially excluded from the Caprin1 condensate. This ‘salt partitioning’ is most likely caused by Caprin1’s net positive charge and is consistent with published research on polyelectrolytes with monovalent salt (Shen and Wang, 2018; Eisenberg and Mohan, 1959; Zhang et al., 2016). Here, the rG-RPA predicted trend is consistent with our experiment showing significantly reduced [Na⁺] in the Caprin1-condensed phase compared to the Caprin1-dilute phase (Table 1), although the larger experimental reduction of [Na⁺] in the Caprin1 condensed droplet relative to our theoretical prediction remains to be elucidated. In this regard, a similar experimental trend of Na⁺ tielines was observed recently for the IDP A1-LCD (WT) with a positive (+8) net charge (Posey et al., 2024). In contrast, for the near-neutral, very slightly negative model pY-Caprin1 (Figure 2c and d), rG-RPA predicts LLPS at [Na⁺] ≈ 0, and the positive tieline slopes indicate that [Na⁺] is higher in the condensed than in the dilute phase. Consistent with Figure 1d, Figure 2c shows that pY-Caprin1 LLPS propensity always decreases with increasing [Na⁺].

Interestingly, a different salt dependence of Caprin1 LLPS is predicted when the salt ion remains monovalent but the monovalent counterion Cl⁻ is replaced by a divalent ${\displaystyle z_{\mathrm{c}}=2}$ anion modeling (ATP-Mg)^2– (as a single-bead ion) under the simplifying assumption that ATP^4– and Mg²⁺ do not dissociate in solution. The corresponding rG-RPA results (Figure 2e–h) indicate that, in the presence of divalent counterions (needed for overall electric neutrality of the Caprin1 solution), Caprin1 can undergo LLPS without the monovalent salt (Na⁺) ions (LLPS regions extend to [Na⁺] = 0 in Figure 2e and f; that is, ${\displaystyle {\rho }_{\mathrm{s}}=0}$, ${\displaystyle {\rho }_{\mathrm{c}}>0}$ in Equation 1), because the configurational entropic cost of concentrating counterions in the Caprin1 condensed phase is lesser for divalent (${\displaystyle z_{\mathrm{c}}=2}$) than for monovalent (${\displaystyle z_{\mathrm{c}}=1}$) counterions as only half of the former are needed for approximate electric neutrality in the condensed phase.

Other predicted differences between monovalent (Figure 2a and b) and divalent (Figure 2e and f) counterions’ impact on Caprin1 LLPS include: (i) The maximum condensed-phase [Caprin1] at low [Na⁺] is lower with monovalent than with divalent counterions ([Caprin1] ∼ 40 mM vs. ∼ 70 mM). (ii) The [Na⁺] at the commencement of reentrance (i.e., at the maximum condensed-phase [Caprin1]) is much higher with monovalent than with divalent counterions ([Na⁺] ∼ 1 M vs. ∼ 0.1 M). (iii) [Na⁺] is depleted in the Caprin1 condensate with both monovalent and divalent counterions when overall [Na⁺] is high (negative tieline slopes for [Na⁺] ≳ 2 M in Figure 2a and e). However, for lower overall [Na⁺], [Na⁺] is slightly higher in the Caprin1 condensate with divalent but not with monovalent counterions (slightly positive tieline slopes for [Na⁺] ≲ 2 M in Figure 2e and f). This prediction suggests that under physiological [Na⁺] = 150∼170 mM, monovalent positive salt ions such as Na⁺ can be attracted, somewhat counterintuitively, into biomolecular condensates scaffolded by positively charged polyelectrolytic IDRs in the presence of divalent counterions. This phenomenon most likely arises from the attraction of the positively charge monovalent salt ions to the negatively charged divalent counterions in the protein-condensed phase because although the three negatively charged D residues in Caprin1 can attract Na⁺, it is notable that Na⁺ is depleted in condensed Caprin1 when the counterion is monovalent (Figure 2a).

For the ${\displaystyle z_{\mathrm{s}}=2}$, ${\displaystyle z_{\mathrm{c}}=4}$ case in Figure 2i–l modeling (ATP-Mg)²⁻ complex dissociating completely in solution into Mg²⁺ salt ions and ATP⁴⁻ counterions (modeled as single-bead ions), rG-RPA predicts Caprin1 LLPS with ATP⁴⁻ (Figure 2k and l) in the absence of Mg²⁺ (the LLPS region includes the horizontal axes in Figure 2i and j), likely because the configurational entropy loss of tetravalent counterions in the Caprin1 condensate is less than that of divalent and monovalent counterions. Tetravalent counterions also increase the theoretical maximum condensed-phase [Caprin1] to ≳ 120 mM. At the commencement of reentrance (maximum condensed-phase [Caprin1] in Figure 2i and j), [Mg²⁺] ∼ 0.4 M, which is intermediate between the corresponding [Na⁺] ∼ 1.0 and 0.1 M, respectively, for monovalent and divalent counterions with ${\displaystyle (z_{\mathrm{s}},z_{\mathrm{c}})=(1,2)}$ and (1, 1). All tieline slopes for Mg²⁺ and ATP⁴⁻ in Figure 2i–l are significantly positive, except in an extremely high-salt region with [Mg²⁺] > 8 M, indicating that [(ATP-Mg)²⁻] is almost always substantially enhanced in the Caprin1 condensate. These observations from analytical theory will be corroborated by FTS below with the introduction of structurally more realistic models of (ATP-Mg)²⁻, ATP⁴⁻ together with the possibility of simultaneous inclusion of Na⁺, Cl⁻, and Mg²⁺ in the FTS models of Caprin1/pY-Caprin1 LLPS systems. Despite the tendency for polymer field theories to overestimate LLPS propensity and condensed-phase concentrations quantitatively because they do not account for ion condensation (Shen and Wang, 2017)—which can be severe for small ions with more than ±1 charge valencies as in the case of condensed [Caprin1] ≳ 120 mM in Figure 2i–l, our present rG-RPA-predicted semi-quantitative trends are consistent with experiments indicating [ATP-Mg]-dependent reentrant phase behavior of Caprin1 (Figure 3a, red data points, and Figure 3b) and that [Mg²⁺] as well as [ATP⁴⁻] are significantly enhanced in the Caprin1 condensate by a factor of ∼5–60 for overall [ATP-Mg] = 3–30 mM (Table 2).

To gain deeper insights, we extend the widely-utilized coarse-grained explicit-chain MD model for biomolecular condensates (Dignon et al., 2018; Das et al., 2020; Silmore et al., 2017) to include explicit small cations and anions (Materials and methods). ATP-mediated LLPS of short basic peptides was studied recently using all-atom simulations indicating ATP engaging in electrostatic and cation-π bridging interactions (Kota et al., 2024). Here, we limit the small ions in our coarse-grained MD simulations of Caprin1 and pY-Caprin1 LLPS to Na⁺ and Cl^–, focusing on the physical origins of reentrance or lack thereof as well as the effects of ariginine-to-lysine (RtoK) mutations on Caprin1. Coarse-grained models allow for the study of larger systems (IDPs of longer chain lengths and more IDPs in the system), though they cannot provide insights into more subtle structural and energetic effects as in all-atom simulations (Kota et al., 2024; Krainer et al., 2021; MacAinsh et al., 2024). For computational efficiency, here we neglect solvation effects that can arise from the directional hydrogen bonds among water molecules (see, e.g., Shimizu and Chan, 2001) by treating other aspects of the aqueous solvent implicitly as in most, although not all (Lin et al., 2023; Wessén et al., 2021) applications of the methodology (Dignon et al., 2018). Several coarse-grained interaction schemes were used in recent MD simulations of biomolecular LLPS (Dignon et al., 2018; Das et al., 2020; Joseph et al., 2021; Regy et al., 2021; Dannenhoffer-Lafage and Best, 2021; Tesei et al., 2021; Wessén et al., 2022a; Wessén et al., 2023; Dignon et al., 2019b). Since we are primarily interested in general principles rather than quantitative details of the phase behaviors of Caprin1 and its RtoK mutants, here we adopt the Kim-Hummer (KH) energies for pairwise amino acid interactions derived from contact statistics of folded protein structures (Dignon et al., 2018), which can largely capture the experimental effects of R vs K on LLPS (Das et al., 2020).

Consistent with experiment (Figure 3) and rG-RPA (Figure 2a–d), explicit-ion coarse-grained MD results in Figure 4 show [NaCl]-dependent reentrant phase behavior for Caprin1 but not for pY-Caprin1 (non-monotonic and monotonic trends indicated, respectively, by the grey arrows in Figure 4a and b). In other words, the critical temperature ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$, which is defined as the maximum temperature (UCST) of a given phase diagram (binodal, or coexistence curve), increases then decreases with addition of NaCl for Caprin1 but ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$ always decreases with increasing [NaCl] for pY-Caprin1. Moreover, consistent with the rG-RPA-predicted tielines in Figure 2a–d (negative slopes for Caprin1 and positive slopes for pY-Caprin1), Figure 4e and g show that Na⁺ is slightly depleted in the Caprin1 condensed droplet, exhibiting the same trend as that in experiment (Figure 3a, blue data points; and Table 1) but is enhanced in the pY-Caprin1 droplet (Figure 4f and h). Because model temperatures in Figure 4a and b and subsequent MD results are given in units of the MD-simulated ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$ of WT Caprin1 at [NaCl] = 0 (denoted as ${\displaystyle T_{\mathrm{c}\mathrm{r}}^0}$ here), the ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$’s of systems with higher or lower LLPS propensities than WT Caprin1 at zero [NaCl] is characterized, respectively, by ${\displaystyle T_{\mathrm{c}\mathrm{r}}/T_{\mathrm{c}\mathrm{r}}^0>1}$ or < 1.

Figure 4

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Numerical plotting data for the coarse-grained molecular dynamics-simulated curves in Figure 4.

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Figure 4e and g show that [Cl^–] is enhanced while [Na⁺] is depleted in the Caprin1 droplet. By comparison, Figure 4f and h show that both [Cl^–] and [Na⁺] are enhanced in the pY-Caprin1 droplet with an excess of [Na⁺] to balance the negatively charged pY-Caprin1 (Figure 4h). The enhancement of [Cl^–] in the Caprin1 condensed phase depicted in Figure 4f and h is further illustrated in Figure 5a–d by comparing the entire simulation box with a condensed droplet in the middle (Figure 5a) with individual distributions of the Caprin1 IDR (Figure 5b), Na⁺ (Figure 5c), and Cl⁻ (Figure 5d). A similar trend, also attributed to charge effects, was observed in explicit-water, explicit-ion MD simulations in the presence of a preformed condensate of the N-terminal RGG domain of LAF-1 with a positive net charge (Zheng et al., 2020). For Caprin1, Figure 5e and f suggests that, as counterion, Cl^– can coordinate two positively charged R residues and thereby stabilize indirect counterion-bridged interchain contacts among polycationic Caprin1 molecules to promote LLPS, consistent with an early lattice-model analysis of generic polyelectrolytes (Orkoulas et al., 2003) and a recent atomic simulation study of A1-LCD (MacAinsh et al., 2024).

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To assess the extent to which Cl^–-mediated bridging interactions (as illustrated in Figure 5f) contribute to condensation of polyelectrolytic IDRs, we examine the relative positions of positively charged arginine residues (Arg⁺) and negatively charged counterions (Cl^–) of a Caprin1 solution under phase-separation conditions in which essentially all Caprin1 molecules are in the condensed phase, using 4000 frames (MD snapshots) of an equilibrated salt-free ([NaCl] = 0) ensemble of 100 WT Caprin1 chains (net charge per chain = +13) with 1300 Cl^– counterions at ${\displaystyle T<T_{\mathrm{c}\mathrm{r}}^0}$ as an example (Figure 6). For simplicity, we focus on Arg⁺–Cl^– interactions because the overwhelming majority (15/16) of the positively charged residues in Caprin1 are arginines. The computed radial distribution function, ${\displaystyle \rho (r)}$, of Cl^– around a given Arg⁺ exhibits a sharp peak at small $r$ that drops to a minium at ${\displaystyle r\approx 11}$ Å (Figure 6a), indicating a strong spatial association between the oppositely charged Arg⁺ and Cl^– as expected. Indeed, within the ensemble we analyze, 5,121,148/(4000×1300) = 98.5% of the Cl^– ions are within 11 Å of an Arg⁺. We next enumerate putative bridging interactions involving two Arg⁺s on different Caprin1 chains and one Cl^– (Figure 6b) by identifying three-bead configurations in which the distance of Cl^– to each of the two Arg⁺ is ≤ 11 Å (within the dominant small-r peak of ${\displaystyle \rho (r)}$ in Figure 6a), which implies that the distance between the two Arg⁺s is ≤ 22 Å. In our ensemble, 4,519,387/(4000×1300) = 86.9% of the Cl^– counterions are identified to be in one or more of a total of 25,112,331 such putative bridging interaction configurations. This means that, on average, each Cl^– is involved in 25,112,331/4,519,387 = 5.56 configurations, and thus are coordinating ≈ 4 Arg⁺s because there are 6 (≈5.56) ways of pairing 4 Arg⁺s. Figure 6c shows the distribution of putative bridging configurations with respect to Arg⁺–Arg⁺ distance ${\displaystyle R}$. Spatial distributions of Cl^– in these configurations are provided in Figure 6d and e, which are quite similiar to those of isolated Arg⁺–Cl^––Arg⁺ systems for ${\displaystyle R\lesssim 14}$ Å (Figure 6f and g). Among the putative bridging configurations, we make an energetic distinction between true bridging and neutralizing (screening) configurations. Physically, a true bridging configuration may be defined by an overall favorable (< 0) sum of (i) unfavorable Coulomb potential between two Arg⁺ and (ii) the favorable Coulomb potential between the Cl^– and one of the Arg⁺s that is farther away from the Cl^–. By the same token, a neutralizing (screening) configuration may be defined by a corresponding overall unfavorable or neutral (≥ 0) sum of these two Coulomb potentials (i.e., the farther Arg⁺–Cl^– distance is larger than the Arg⁺–Arg⁺ distance). In this regard, and in more general terms, Cl^– ions in bridging and neutralizing interactions may be considered, respectively, as a ‘strong-attraction promoter’ and a ‘weak-attraction suppressor’ of LLPS (Nguemaha and Zhou, 2018; Ghosh et al., 2019).

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Numerical values of the coarse-grained molecular dynamics-simulated data plotted in Figure 6a and c.

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In the present analysis, we group putative bridging configurations by ${\displaystyle R}$ in bins of 2 Å (Figure 6c). Accordingly, we may classify Cl^– positions satisfying the above condition of favorable (< 0) sum of Coulomb potentials for all ${\displaystyle R}$ values within the 2 Å range of the bin as in true bridging configurations (79.6%), those Cl^– positions satisfying the above condition of unfavorable (≥ 0) sum of Coulomb potentials for all ${\displaystyle R}$ values in the 2 Å range as in neutralizing configurations (7.4%), and those that satisfy neither as ‘intermediate’ configurations (13.0%). Even with this more stringent criterion, ≈80% of putative bridging configurations are true bridging configurations. Because on average a Cl^– counterion known to be involved in at least one putative bridging configuration is on average participating in ∼ 5–6 such configurations, the probability that it is involved in at least one true bridging configuration is very high, at ≈ 1.0 − (0.2)⁵ = 99.97%. Thus, even without taking into consideration bridging interactions involving lysines, we may reasonably conclude that an overwhelming majority (≈ 87%) of Cl⁻ counterions in the coarse-grained MD system considered are engaged in condensation-driving true bridging interactions coordinating pairs of Arg⁺ on different Caprin1 chains. Similar extensive Cl⁻ and Na⁺ bridging interactions are observed in a recent all-atom molecular dynamics study of LLPS of short peptides under a variety of overall salt concentrations (MacAinsh et al., 2024).

We apply our MD methodology also to four RtoK Caprin1 variants, termed 15Rto15K, 4Rto4K_N, 4Rto4K_M, and 4Rto4K_C (Appendix 1—figure 1), which involve 15 or 4 RtoK substitutions (Wong et al., 2020). The simulated phase diagrams in Figure 7 exhibit reentrant phase behaviors for all three 4Rto4K variants. While these results are consistent with experiments showing LLPS of these 4Rto4K variants commencing at different nonzero [NaCl]s (Wong et al., 2020), the simulated reentrant dissolution is not observed experimentally, probably because the actual [NaCl] needed is beyond the experimentally investigated or physically possible range of salt concentration. Simulated reentrant phase behaviors are also seen for 15Rto15K; but as will be explained below, its much lower simulated UCST is consistent with no experimental LLPS for this variant (Wong et al., 2020). Since our main focus here is on general physical principles, we do not attempt to fine-tune the MD parameters for a quantitative match between simulation and experiment. Experimentally, only WT exhibits a clear trend toward reentrant dissolution of condensed droplets (with a LLPS propensity plateau at [NaCl] ≈ 1.55−2.5 M, Figure 3a, blue data points), whereas the LLPS of 4Rto4K_M and 4Rto4K_C commences at [NaCl] ≈ 1.3 M, LLPS propensity then increases with [NaCl] (a trend consistent with the MD-predicted increasing LLPS propensity at low [NaCl]s in Figure 7b and c), but no sign of reentrant dissolution is seen up to the maximum [NaCl] = 2 M investigated experimentally for the RtoK variants (Figure 9B of Wong et al., 2020). In contrast, the MD phase diagrams in Figure 7 show a maximum LLPS propensity (highest ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$) at [NaCl] ≈ 0.5 M. This qualitative agreement with quantitative mismatch suggests that real Caprin1 LLPS is somewhat less sensitive to small monovalent ions than that stipulated by the present MD model. This question should be tackled in future studies by considering, for example, alternate pairwise amino acid interaction energies (Das et al., 2020; Dignon et al., 2018; Joseph et al., 2021; Regy et al., 2021; Dannenhoffer-Lafage and Best, 2021; Tesei et al., 2021; Wessén et al., 2022a; Wessén et al., 2023) and their temperature dependence (Cinar et al., 2019; Dignon et al., 2019a).

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Numerical values of the coarse-grained molecular dynamics-simulated data points plotted as filled circles in the phase diagrams in Figure 7.

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Limitations notwithstanding, the MD-simulated trend agrees largely with experiment. Predicted LLPS propensities quantified by the ${\displaystyle T_{\mathrm{c}\mathrm{r}}\mathrm{s}}$ in Figure 7 follow the rank order of WT > 4Rto4K_M > 4Rto4K_N ≈ 4Rto4K_C > 15Rto15K, which is essentially identical to that measured experimentally, viz., WT > 4Rto4K_M > 4Rto4K_C > 4Rto4K_N > 15Rto15K (Figure 9B of Wong et al., 2020). In comparing theoretical and experimental LLPS, a low theoretical ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$ can practically mean no experimental LLPS when the theoretical ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$ is below the freezing temperature of the real system (Lin et al., 2016; Brady et al., 2017). Figure 7a shows that even the highest ${\displaystyle T_{\mathrm{c}\mathrm{r}}}$ for 15Rto15K (at model [NaCl] = 480 mM) is essentially at the same level as ${\displaystyle T_{\mathrm{c}\mathrm{r}}^0}$ for WT at [NaCl] = 0 (${\displaystyle T_{\mathrm{c}\mathrm{r}}/T_{\mathrm{c}\mathrm{r}}^0\approx 1}$). This MD prediction is consistent with the combined experimental observations of no LLPS for 15Rto15K up to at least [NaCl] = 2 M and no LLPS for WT Caprin1 at [NaCl] = 0 (Figure 9B and C of Wong et al., 2020).

We next turn to modeling of Caprin1 or pY-Caprin1 LLPS modulated by both ATP-Mg and NaCl. Because tackling such many-component LLPS systems using rG-RPA or explicit-ion MD is numerically challenging, here we adopt the complementary FTS approach (Fredrickson, 2006) outlined in Materials and methods for this aspect of our investigation. FTS is based on complex Langevin dynamics (Parisi, 1983; Klauder, 1983), which is related to an earlier formulation for stochastic quantization (Parisi and Wu, 1981; Chan and Halpern, 1986) and has been applied extensively to polymer solutions (Fredrickson et al., 2002; Fredrickson, 2006). Recently, FTS has provided insights into charge-sequence-dependent LLPS of IDRs (McCarty et al., 2019; Pal et al., 2021; Wessén et al., 2022a; Lin et al., 2019b; Lin et al., 2023). The starting point of FTS is identical to that of rG-RPA. FTS invokes no RPA and is thus advantageous over rG-RPA in this regard, although it is still limited by the lattice size used for simulation and its restricted treatment of excluded volume (Pal et al., 2021). Here we apply the protocol detailed in Pal et al., 2021; Lin et al., 2023.

Going beyond the single-bead model for (ATP-Mg)²⁻ in our analytical rG-RPA theory (Figure 2), we now adopt a 6-bead polymeric representation of (ATP-Mg)²⁻ (Figure 8a) in which four negative and two positive charges serve to model ATP⁴⁻ and Mg²⁺ respectively. Modeling (ATP-Mg)²⁻ as a short charged polymer enables application of existing FTS formulations for multiple charge sequences to systems with IDRs and (ATP-Mg)²⁻. While the model in Figure 8a does not capture structural details, its charge distribution does correspond roughly to that of the chemical structure of (ATP-Mg)²⁻. In developing FTS models involving IDR, (ATP-Mg)²⁻, and NaCl, we first assume for simplicity that (ATP-Mg)²⁻ does not dissociate and consider systems consisting of any given overall concentrations of IDR and (ATP-Mg)²⁻ wherein all positive and negative charges on the IDR and (ATP-Mg)²⁻ are balanced, respectively, by Cl⁻ and Na⁺ to maintain overall electric neutrality (Figure 8a).

Figure 8

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Figure 8—source data 1

Numerical values of all theoretical and experimental data points plotted in Figure 8.

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LLPS of FTS systems can be monitored by correlation functions (Pal et al., 2021). Here, we compute intra-species IDR self-correlation functions ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)}$ (Figure 8b and c) and inter-species cross-correlation functions ${\displaystyle G_{\mathrm{p}\mathrm{q}}(r)}$ between the IDR and (ATP-Mg)²⁻ or NaCl (Figure 8d and e) at three different overall [(ATP-Mg)²⁻] = 10⁻⁴b⁻³, 0.03b⁻³, and 0.5b⁻³, where b may be taken as the peptide virtual bond length ≈ 3.8 Å (Materials and methods). The correlation functions in Figure 8b–e are normalized by overall densities ${\displaystyle {\rho }_{\mathrm{p}}^0}$ of the IDR and ${\displaystyle {\rho }_{\mathrm{q}}^0}$ for (ATP-Mg)²⁻, Na⁺ or Cl⁻, wherein density is the bead density for the given molecular species in units of ${\displaystyle b^{-3}}$. LLPS of the IDR is signaled by ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)/({\rho }_{\mathrm{p}}^0{)}^2}$ in Figure 8b and c dropping below the unity baseline (dashed) at large distance ${\displaystyle r}$ because it implies a spatial region with depleted IDR below the overall concentration, which is possible only if the IDR is above the overall concentration in at least another spatial region. In other words, ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)/({\rho }_{\mathrm{p}}^0{)}^2<1}$ for large ${\displaystyle r}$ indicates that IDR concentration is heterogeneous and thus the system is phase separated. For small ${\displaystyle r}$, ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)/({\rho }_{\mathrm{p}}^0{)}^2}$ is generally expected to increase because IDR chain connectivity facilitates correlation among residues local along the chain. On top of this, LLPS propensity may be quantified by ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)/({\rho }_{\mathrm{p}}^0{)}^2}$ for small ${\displaystyle r}$ because a higher value indicates a higher tendency for different chains to associate and thus a higher LLPS propensity (Pal et al., 2021).

[(ATP-Mg)²⁻]-modulated reentrance is predicted by FTS for Caprin1 but not for pY-Caprin1: When [(ATP-Mg)²⁻]/b⁻³ varies from 10⁻⁴ to 0.03 to 0.5, small-r values of the Caprin1 ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)}$ in Figure 8b initially increase then decrease, whereas the corresponding small-r values of the pY-Caprin1 ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)}$ in Figure 8c decrease monotonically, consistent with rG-RPA (Figure 2g, h, k, l) and experiment (Figure 3). The inter-species cross-correlations in Figure 8d, e show further that when an IDR condensed phase is present at [(ATP-Mg)²⁻] = 0.03b⁻³ (as indicated by large-${\displaystyle r}$ behaviors of ${\displaystyle G_{\mathrm{p}\mathrm{p}}(r)/({\rho }_{\mathrm{p}}^0{)}^2}$ in Figure 8b, c), (ATP-Mg)²⁻ is colocalized with Caprin1 or pY-Caprin1 (high value of ${\displaystyle G_{\mathrm{p}\mathrm{q}}/{\rho }_{\mathrm{p}}^0{\rho }_{\mathrm{q}}^0}$ for small ${\displaystyle r}$) in the IDR-condensed droplet. By comparison, the variation of [Na⁺] and [Cl^–] is much weaker. For Caprin1, Cl^– is enhanced over Na⁺ in the Caprin1 condensed phase (small-${\displaystyle r}$ ${\displaystyle G_{\mathrm{p}\mathrm{q}}/{\rho }_{\mathrm{p}}^0{\rho }_{\mathrm{q}}^0}$ of the former larger than the latter in Figure 8d), but the reverse is seen for pY-Caprin1 (Figure 8e). This FTS-predicted difference, most likely arising from the positive net charge on Caprin1 and the smaller negative net charge on pY-Caprin1, is consistent with the MD results in Figure 4e-h and Appendix 1—figure 3.

The propensities for (ATP-Mg)²⁻, Na⁺, and Cl^– to associate with each residue ${\displaystyle i}$ along the Caprin1 IDR (${\displaystyle i=1,2,\dots ,103}$) in FTS are quantified by the residue-specific integrated correlation ${\displaystyle {\mathcal{G}}_{\mathrm{p}\mathrm{q}}^{(i)}/{\rho }_{\mathrm{p},i}^0{\rho }_{\mathrm{q}}^0}$ in Figure 8f, which is the integral of the corresponding ${\displaystyle G_{\mathrm{p}\mathrm{q}}^{(i)}(r)}$ from ${\displaystyle r=0}$ to a relative short cutoff distance ${\displaystyle r=r_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}}$ to provide a relative contact frequency for residue $i$ and ionic species q to be in spatial proximity (Materials and methods and Appendix 1). Notably, the residue-position-dependent integrated correlation for (ATP-Mg)²⁻ varies significantly, exhibiting much larger values near the N-terminal and a little before the C-terminal but weaker correlation elsewhere (Figure 8f, red symbols). The two regions of high integrated correlation (i.e., favorable association) coincide with regions with high sequence concentration of positively charged residues. This FTS prediction is remarkably similar to the experimental NMR finding that binding between (ATP-Mg)²⁻ and Caprin1 occurs strongly at the arginine-rich N- and C-terminal regions, as indicated by the volume ratio ${\displaystyle V/V_0}$ data in Figure 1C of Kim et al., 2021 that quantifies the ratio of peaks in NMR spectra in the presence and absence of trace amounts of ATP-Mn. For comparison with the FTS results, this set of experimental data is replotted as ${\displaystyle 1-V/V_0}$ in Figure 8f (grey symbols, right vertical axis) to illustrate the similarity in experimental and theoretical trends because ${\displaystyle 1-V/V_0}$ is expected to trend with contact frequency. Corresponding FTS results for Na⁺ and Cl⁻ in Figure 8f exhibit much less residue-position-dependent variation, with Cl⁻ displaying only slightly enhanced association in the same arginine-rich regions, and Na⁺ showing even less variation, presumably because the positive charges on Caprin1 are already essentially neuralized by the locally associated (ATP-Mg)²⁻ or Cl^– ions. The theory-experiment agreement in Figure 8f regarding ATP-Caprin1 interactions indicates once again that electrostatics is an important driving force underlying many aspects of experimentally observed Caprin1–(ATP-Mg)²⁻ association.

The above FTS-predicted trends are further illustrated in Figure 9 by field snapshots. Such FTS snapshots are generally useful for visualization and heuristic understanding (McCarty et al., 2019; Pal et al., 2021; Wessén et al., 2022a), including insights into subtler aspects of spatial arrangements exemplified by recent studies of subcompartmentalization entailing either co-mixing or demixing in multiple-component LLPS that are verifable by explicit-chain MD (Pal et al., 2021; Wessén et al., 2022a). Now, trends deduced from the correlation functions in Figure 8 are buttressed by the representative snapshots in Figure 9: As the bead density of (ATP-Mg)²⁻ is increased from ${\displaystyle {10}^{-4}{b}^{-3}}$ to ${\displaystyle 0.03b^{-3}}$ to ${\displaystyle 0.5b^{-3}}$, the spatial distribution of Caprin1 evolves from an initially dispersed state to a concentrated droplet to a (reentrant) dispersed state again (Figure 9a), whereas the initial dense pY-Caprin1 droplet becomes increasingly dispersed monotonically (Figure 9b). Colocalization of (ATP-Mg)²⁻ with both the Caprin1 (Figure 9c) and pY-Caprin1 (Figure 9d) droplets is clearly visible at [(ATP-Mg)²⁻] = 0.03b⁻³, though the degree of colocalization is appreciably higher for Caprin1 than for pY-Caprin1. This is likely because the positive net charge of Caprin1 is more attractive to (ATP-Mg)²⁻. By comparison, variations in Na⁺ and Cl⁻ distribution between Caprin1/pY-Caprin1 dilute and condensed phases are not so discernible in Figure 9e–h, consistent with the small differences in the corresponding FTS correlation functions (Figure 8d and e).

Figure 9

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We have also assessed the generality of the results in Figures 8 and 9 by considering three variations in the molecular species treated by FTS: (i) Caprin1 or pY-Caprin1 with only Na⁺ and Cl^– but no (ATP-Mg)²⁻ (Appendix 1—figure 4), (ii) Caprin1 with (ATP-Mg)²⁻ and either Na⁺ or Cl^– (but not both) to maintain overall charge neutrality or pY-Caprin1 with (ATP-Mg)²⁻ and Na⁺ as counterion but no Cl^– (Appendix 1—figure 5), and (iii) Caprin1 or pY-Caprin1 with ATP⁴⁻, Mg²⁺, Na⁺ and Cl⁻ (Appendix 1—figure 6). Despite these variations in FTS models, Appendix 1—figures 4–6 consistently show reentrant behavior for Caprin1 but not pY-Caprin1 and Appendix 1—figures 5 and 6 both exhibit colocalization of ATP with condensed Caprin1, suggesting that these features are robust consequences of the basic electrostatics at play in Caprin1/pY-Caprin1 + ATP-Mg + NaCl systems.

Our theoretical predictions are also largely in agreement with recent computational studies on salt concentrations in the dilute versus condensed phases (Zheng et al., 2020) and salt-dependent reentrant behaviors (Krainer et al., 2021) of other biomolecular condensates, including explicit-water, explicit-ion atomic simulations with preformed condensates of the N-terminal RGG domain of LAF-1 (Zheng et al., 2020) and of the highly positive proline-arginine 25-repeat dipeptide PR₂₅ (Garaizar and Espinosa, 2021).

A recent study examines salt-dependent reentrant LLPSs of full-length FUS (WT and G156E mutant), TDP-43, bromodomain-containing protein 4 (Brd4), sex-determining region Y-box 2 (Sox2), and annexin A11 (Krainer et al., 2021). Unlike the requirement of a nonzero monovalent salt concentration for Caprin1 LLPS, LLPS is observed for all these six proteins with KCl, NaCl, or other salts at concentrations as low as 50 mM. Also unlike Caprin1, their protein condensates dissolve at intermediate salt then re-appear at higher salt, a phenomenon the authors rationalize by a tradeoff between decreasing favorability of cation-anion interactions and increasing favorability of cation-cation, cation-π, hydrophobic, and other interactions with increasing monovalent salt (Krainer et al., 2021).

Two reasons may account for this difference. First, Caprin1 does not phase separate at low salt because it is a relatively strong polyelectrolyte (NCPR = +13/103 = +0.126). By comparison, five of the six proteins in Krainer et al., 2021 are much weaker polyelectrolytes or not at all, with NCPR = +14/526 = +0.0266, +13/526 = +0.0247, −7/80 = −0.0875, 0, and +3/326 = +0.00920, respectively, for FUS (WT, mutant), TDP-43, Brd4, and A11. Apparently, their weak electrostatic repulsions can be overcome by favorable nonelectrostatic interactions alone to enable LLPS.

Second, compared to Caprin1, the proteins in Krainer et al., 2021 are either significantly larger (WT and mutant FUS) or significantly more hydrophobic and aromatic (the other four proteins), both properties are conducive to LLPS. For instance, although Sox2’s NCPR = +14/88 = +0.159 is higher than that of Caprin1, among Sox2’s amino acid residues, 21/88 = 23.9% are large hydrophobic or aromatic residues leucine (L), isoleucine (I), valine (V), methionine (M), phenylalanine (F), or tryptophan (W), and 17/88 = 19.3% are large aliphatic residues L, I, V, or M. This amino acid composition suggests that hydrophobic or π-related interactions in Sox2 can be sufficient to overcome electrostatic repulsion to effectuate LLPS at zero salt. In contrast, the Caprin1 IDR contains merely one L; only 10/103 = 9.7% of the residues of Caprin1 are in the L, I, V, M, F, W hydrophobic/aromatic category and only 6/103 = 5.8% are in the L, I, V, M aliphatic category. The corresponding aliphatic fractions of TDP-43, Brd4 and A11, at 21/80 = 26.3%, 33/132 = 25%, and 90/326 = 27.6%, respectively, are also significantly higher than that of Caprin1.

Effects of salts on LLPS, including partitioning of salt into polymer-rich phases, are of long-standing interest in polymer physics (Jedlinska and Riggleman, 2023). In the biomolecular condensate context, the versatile functional roles of salts are highlighted by the interplay between electrostatic and cation-π interactions (Hazra and Levy, 2023; Kim et al., 2017), salts’ modulating effects on heat-induced LLPSs of RNAs (Wadsworth et al., 2023), their regulation of condensate liquidity (Morishita et al., 2023), and even their potential impact in extremely high-salt exobiological environments (Fetahaj et al., 2021). While some of these recent studies focus primarily on salts’ electrostatic screening effects without changing the signs of the effective polymer charge-charge interaction (Hazra and Levy, 2023), effective attractions between like charges bridged by salt or other oppositely-charged ions (Orkoulas et al., 2003) as illustrated by Caprin1 (Figures 5f and 6) and a recent study of A1-LCD (MacAinsh et al., 2024) are likely needed to account for phenomena such as salt-induced dimerization of highly charged, medically relevant arginine-rich cell-penetrating short peptides (Tesei et al., 2017; Liu et al., 2022). In this regard, it should be noted that positively and negatively charged salt ions can also coordinate with backbone carbonyls and amides, respectively, in addition to coordinating with charged amino acid sidechains (MacAinsh et al., 2024). The impact of such effects, which are not considered in the present coarse-grained models, should be ascertained by further investigations using atomic simulations (MacAinsh et al., 2024; Rauscher and Pomès, 2017; Zheng et al., 2020).

In view of Caprin1’s polyelectrolytic nature, the mildly negative tieline slopes in Figure 2a and b are consistent with rG-RPA predictions for a fully charged polyelectrolyte (Figure 10a of Lin et al., 2020). This depletion of monovalent salt in the condensed phase is similar to that observed in the complex coacervation of oppositely charged polyelectrolytes (Radhakrishna et al., 2017; Li et al., 2018; Herrera et al., 2023). By comparison, the positive rG-RPA tieline slopes for polyampholytic pY-Caprin1 (Figure 2c and d), confirmed by MD in Figure 4f and h, are appreciably steeper than that predicted for fully charged (±1) diblock polyampholytes by rG-RPA and the essentially flat tielines predicted by FTS (Fig. 10b of Lin et al., 2020 and Figure 7 of Danielsen et al., 2019). Whether this difference originates from the presence of divalently charged (−2) phosphorylated sites in pY-Caprin1 remains to be elucidated. In any event, tieline analysis is generally instrumental for revealing details, such as stoichiometry, of the interactions driving multiple-component biomolecular (Lin et al., 2022; Posey et al., 2024; Qian et al., 2022); rG-RPA should be broadly useful as a computationally efficient tool for this purpose (Lin et al., 2020).

Our rG-RPA prediction that the maximum condensed-phase [Caprin1] at low [Na⁺] is substantially higher with divalent than with monovalent counterions is in line with early findings that higher-valency counterions are more effective in bridging polyelectrolyte interactions to favor LLPS (Olvera de la Cruz et al., 1995) and recent observations that salt ions with higher valencies enhance biomolecular LLPS (Lenton et al., 2021; Crabtree et al., 2023). The possibility that this counterion/salt effect on LLPS may be exploited more generally for biological functions and/or biomedical applications remains to be further explored. In this regard, while recognizing that ATP can engage in π-related interactions (Kota et al., 2024; Kang et al., 2018; Kang et al., 2019), our electrostatics-based perspective of ATP-dependent reentrant phase behaviors is consistent with recent observations on polylysine LLPS modulated by enzymatically catalyzed ATP turnovers (Herrera et al., 2023; Nakashima et al., 2018). More broadly, differential effects of salt ions on biomolecular LLPS can also arise from the sizes and charge densities of the ions—properties related to the Hofmeister phenomena (Hribar et al., 2002; Lo Nostro and Ninham, 2012)—even for ions with the same valency (Posey et al., 2024). These features should be addressed in future theoretical models as well.

Beyond the above comparisons, further experimental testing of other aspects of our theoretical predictions should be pursued, especially those pertaining to pY-Caprin1. Future theoretical efforts should address a broader range of scenarios by independent variations of [ATP⁴⁻], [Mg²⁺], [Na⁺], [Cl⁻] and to account for nonelectrostatic aspects of ATP-Mg dissociation (Wessén et al., 2022b) with predictions such as tieline slopes analyzed in detail to delineate effects of sizes, charge densities (Posey et al., 2024), and configurational entropy of salt ions (Adhikari et al., 2018) as well as solvent quality (Li et al., 2021). In addition to our basic modeling constructs, the impact of excluded volume and solvent/cosolute-mediated temperature-dependent effective interactions should be incorporated. Excluded volume is known to affect LLPS (Danielsen et al., 2019), demixing of IDP species in condensates (Pal et al., 2021), and partition of salt ions in polymer LLPS (Li et al., 2018). Moreover, LCST can be driven not only by hydrophobicity (Cinar et al., 2019; Lin et al., 2018; Dignon et al., 2019a) but also by electrostatics, as suggested by experiment on complex coacervates of oppositely charged polyelectrolytes (Ali et al., 2019). Bringing together these features into a comprehensive formulation will afford a more accurate physical picture.

To recapitulate, we have employed three complementary theoretical and computational approaches to account for the interplay between sequence pattern, phosphorylation, counterion, and salt in the phase behaviors of IDPs. Application to the Caprin1 IDR and its phosphorylated variant pY-Caprin1 provides physical rationalization for a variety of trends observed in experiments, including reentrance behaviors and very substantial ATP colocalization. These findings support a significant—albeit not exclusive—role of electrostatics in these biophysical phenomena, providing physical insights into effects of sequence-specific charge-charge interactions on ATP-modulated physiological functions of biomolecular condensates such as regulation of ion concentrations. The approach developed here should be of general utility as a computationally efficient tool for hypothesis generation, design of new experiments, exploration and testing of biophysical scenarios, as well as a starting point for more sophisticated theoretical/computational modeling.

The low complexity 607–709 domain of Caprin1 was expressed and purified as before (Kim et al., 2019; Kim et al., 2021). WT Caprin1 was used in all experiments except those on [NaCl] dependence reported in Table 1 and Figure 3a, for which a double mutant was used because residue pairs N623-G624 and N630-G631 in WT Caprin1 form isoaspartate (IsoAsp) glycine linkages over time which alters the charge distribution of the IDR (Wong et al., 2020).

Phosphorylation of the WT Caprin1 IDR was performed as described in our prior study (Kim et al., 2019) by using the kinase domain of mouse Eph4A (587-896) (Wiesner et al., 2006) with an N-terminal His-SUMO tag.

We established phase diagrams for Caprin1 and pY-Caprin1 by measuring the protein concentrations in dilute and condensed phases across a range of [NaCl]s (Figure 1c and d). Initially homogenizing the two phases of the demixed samples into a milky dispersion through vortexing, ∼200 μL aliquots were then incubated in a PCR thermocycler with a heated lid at 90 °C, in triplicate, for a minimum of one hour. During incubation, the condensed phase settled and formed a clear phase at the bottom. For concentration measurements, the samples were diluted in 6 M GdmCl and 20 mM NaPi (pH 6.5). The dilute phase (top layer) was analyzed through a tenfold dilution of 10 μL samples, and the condensed phase (bottom layer) was analyzed through 250- to 500-fold dilution of 2 or 10 μL samples. Notably, using a positive displacement pipettor (Eppendorf) and tips was essential for accurately pipetting the viscous condensed phase.

Inductively coupled plasma optical emission spectroscopy (ICP-OES) measurements of [Na⁺] were performed using a Thermo Scientific iCAP Pro ICP-OES instrument in axial mode. ICP-OES was also used to determine [ATP] and [Mg²⁺] (Table 2). The detection of phosphorus and magnesium served as proxies for quantifying ATP and Mg²⁺ levels, respectively. Standard curves were prepared using solutions with known [ATP] and [Mg²⁺], ranging from 0 to 90 ppm for ATP and 0–25 ppm for Mg²⁺.

A 6 mM solution of double-mutant Caprin1 IDR (see above) in buffer (25 mM sodium phosphate, pH 7.4) was prepared by exchanging (3 times) the purified protein after size exclusion chromatography using centrifugal concentrators (3 kDa, EMD Millipore). Caprin samples for turbidity measurements were prepared by taking 0.5 μL of the above solution and diluting it into buffer (25 mM sodium phosphate, pH 7.4) containing varying [NaCl]s ranging from 0 to 4.63 M, in a sample volume of 9 μL, so as to achieve [Caprin1] of 300 μM. After rigorous mixing, 5 μL samples were loaded into a μCuvette G1.0 (Eppendorf). OD600 measurements (Figure 3a) were recorded three times using a BioPhotometer D30 (Eppendorf).

Turbidity assays were conducted using the method we described previously (Wong et al., 2020).

As detailed in Wessén et al., 2022a; Lin et al., 2023, an example of the sequence-specific polymer theories (Lin et al., 2016; Lin et al., 2020) is that for a solution with a single species of charged heteropolymers in ${\displaystyle n_{\mathrm{p}}}$ copies, ${\displaystyle n_{\mathrm{c}}}$ counterions (same type), and ${\displaystyle n_{\mathrm{s}}}$ salt ions (same type, but different from the counterions). Each polymer chain has ${\displaystyle N}$ monomers (residues) with charge sequence ${\displaystyle |\sigma ⟩=[{\sigma }_1,{\sigma }_2,...{\sigma }_N{]}^{\mathrm{T}}}$ in vector notation, where ${\displaystyle {\sigma }_i\in \{0,\pm 1,-2\}}$ is the charge of the ith residue. The counterions and salt ions are monomers carrying $z_{\mathrm{c}}$ and $z_{\mathrm{s}}$ charges, respectively. The particle-based partition function is given by

where ${\displaystyle n_{\mathrm{w}}}$ denotes the number of water (solvent) molecules, ${\displaystyle {\mathbf{R}}_{\alpha ,i}}$ is the position vector of the ith residue of the αth polymer, ${\displaystyle {\mathbf{r}}_a}$ is the position vector of the ath small ion. ${\displaystyle \mathcal{T}}$ accounts for polymer chain connectivity modeled by a Gaussian elasticity potential with Kuhn length $l$. ${\displaystyle \mathcal{U}}$ describes the interactions among all molecular components of the system, here consisting only of Coulomb electrostatics (el) and excluded-volume (ex) for simplicity, viz., ${\displaystyle \mathcal{U}={\mathcal{U}}_{\mathrm{e}\mathrm{l}}+{\mathcal{U}}_{\mathrm{e}\mathrm{x}}}$. Their interaction strengths are governed by the Bjerrum length ${\displaystyle l_{\mathrm{B}}}$ and the two-body excluded volume parameter ${\displaystyle v_2}$. By introducing conjugate fields ${\displaystyle \psi (\mathbf{r})}$, ${\displaystyle w(\mathbf{r})}$ and applying the Hubbard-Stratonovich transformation, the system defined by the particle-based partition function in Equation 2 is recast as a field theory of ${\displaystyle \psi ,w}$ in which their interactions with polymer, salt, and counterion are described, respectively, by single-molecule partition functions ${\displaystyle {\mathcal{Q}}_{\mathrm{p}}}$, ${\displaystyle {\mathcal{Q}}_{\mathrm{s}}}$, and ${\displaystyle {\mathcal{Q}}_{\mathrm{c}}}$. For instance,

where ${\displaystyle {\mathcal{H}}_{\mathrm{p}}}$ is the single-polymer Hamiltonian and the chain label α is dropped.

Following Sawle and Ghosh, 2015; Lin et al., 2020, ${\displaystyle {\mathcal{H}}_{\mathrm{p}}}$ can be separated into a Gaussian-chain Hamiltonian with an effective (renormalized) Kuhn length ${\displaystyle l_1=xl}$ and a remaining term, ${\displaystyle {\mathcal{H}}_{\mathrm{p}}={\mathcal{H}}_{\mathrm{p}}^0+{\mathcal{H}}_{\mathrm{p}}^1}$, where

with ${\displaystyle {\mathrm{i}}^2=-1}$. By requiring the observable polymer square end-to-end distance be properly quantified by ${\displaystyle {\mathcal{H}}_{\mathrm{p}}^0}$, ${\displaystyle x}$ can be approximated by variational theory (Sawle and Ghosh, 2015). RPA can then be applied to the renormalized Gaussian (rG) chain system with ${\displaystyle l\to xl}$ and a corresponding scaling of the contour length to arrive at an improved theory, rG-RPA, for sequence-specific LLPS.

The MD model in this work augments a class of implicit-water coarse-grained models (Dignon et al., 2018; Das et al., 2020) that utilize a ‘slab’ approach for efficient equilibration (Silmore et al., 2017) by incorporating explicit small ions. As before (Das et al., 2020), the total MD potential energy ${\displaystyle U_{\mathrm{T}}}$ is the sum of long-spatial-range electrostatic (el) and short-spatial-range (sr) interactions of the Lennard-Jones (LJ) type as well as bond interactions, that is, ${\displaystyle U_{\mathrm{T}}=U_{\mathrm{e}\mathrm{l}}+U_{\mathrm{s}\mathrm{r}}+U_{\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}}$. With small ions, the electrostatic component is given by a sum of polymer-polymer (pp), polymer-ion (pi), and ion-ion (ii) contributions: ${\displaystyle U_{\mathrm{e}\mathrm{l}}=U_{\mathrm{e}\mathrm{l},\mathrm{p}\mathrm{p}}+U_{\mathrm{e}\mathrm{l},\mathrm{p}\mathrm{i}}+U_{\mathrm{e}\mathrm{l},\mathrm{i}\mathrm{i}}}$. Details of these terms are provided in Appendix 1.

FTS is useful for sequence-specific multiple-component LLPSs encountered in biomolecular settings. The new applications developed here are based on recent advances (see, e.g., McCarty et al., 2019; Danielsen et al., 2019; Pal et al., 2021; Fredrickson et al., 2002; Lin et al., 2023; Fredrickson, 2006). Consider the field theoretic Hamiltonian

where ${\displaystyle {\mathcal{Q}}_m}$ is single-molecule partition function (here ${\displaystyle m}$ labels the components in the system, Equation 3) and the breves denote convolution with ${\displaystyle \mathrm{\Gamma }}$, i.e., for a generic field ${\displaystyle \varphi }$, ${\displaystyle \breve{\varphi }(\mathbf{r})=\mathrm{\Gamma }\star \varphi (\mathbf{r})\equiv \int d{\mathbf{r}}^{\prime }\mathrm{\Gamma }(\mathbf{r}-{\mathbf{r}}^{\prime })\varphi ({\mathbf{r}}^{\prime })}$; here ${\displaystyle \varphi =w,\psi }$, and ${\displaystyle \mathrm{\Gamma }}$ is a Gaussian smearing function (Lin et al., 2023). FTS utilizes the Complex-Langevin (CL) method (Parisi, 1983; Klauder, 1983) by introducing an artifical CL time variable (${\displaystyle t}$), viz., ${\displaystyle w(\mathbf{r})\to w(\mathbf{r},t)}$, ${\displaystyle \psi (\mathbf{r})\to \psi (\mathbf{r},t)}$ and letting the system evolve in CL time in accordance with a collection of Langevin equations

where the Gaussian noise ${\displaystyle {\eta }_{\varphi }(\mathbf{r},t)}$ satisfies ${\displaystyle ⟨{\eta }_{\varphi }(\mathbf{r},t){\eta }_{{\varphi }^{\prime }}({\mathbf{r}}^{\prime },t^{\prime })⟩=2{\delta }_{\varphi ,{\varphi }^{\prime }}\delta (\mathbf{r}-{\mathbf{r}}^{\prime })\delta (t-t^{\prime })}$. Thermal averages of thermodynamic observables are then computed as asymptotic CL time averages of the corresponding field operators. Spatial information about condensation and proximity of various components is readily gleaned from density-density correlation functions (Pal et al., 2021; Lin et al., 2023),

where $m,n$ are labels for the components in the model system. For instance, $m$ may represent all polymer beads (denoted ‘p’) irrespective of the sequence positions of the beads [${\displaystyle {\hat{\rho }}_{\mathrm{p}}(\mathbf{r})=\sum _{\alpha =1}^{n_{\mathrm{p}}}\sum _{i=1}^N\mathrm{\Gamma }(\mathbf{r}-{\mathbf{R}}_{\alpha ,i})}$], and ${\displaystyle n}$ may represent all six beads in our ATP-Mg model (Figure 8a). One may also define

where ${\displaystyle (i)}$ represents the ith residue along a protein chain [${\displaystyle {\hat{\rho }}_{\mathrm{p},i}(\mathbf{r})\equiv \sum _{\alpha =1}^{n_{\mathrm{p}}}\mathrm{\Gamma }(\mathbf{r}-{\mathbf{R}}_{\alpha ,i})}$ is the density of the ith residue among all the protein chains], and q = (ATP-Mg)²⁻, Na⁺, or Cl⁻. With this definition, residue-specific relative contact frequencies are estimated by integrating Equation 8 over a spherical volume within a small inter-component distance ${\displaystyle r_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}}$:

For the normalized ${\displaystyle {\mathcal{G}}_{\mathrm{p}\mathrm{q}}^{(i)}/{\rho }_{\mathrm{p},\mathrm{i}}^0{\rho }_{\mathrm{q}}^0}$ plotted in Figure 8f, ${\displaystyle {\rho }_{\mathrm{p},\mathrm{i}}^0}$ and ${\displaystyle {\rho }_{\mathrm{q}}^0}$ are bulk (overall) densities, respectively, of the ith protein residue and of (ATP-Mg)²⁻ or small ions, and ${\displaystyle r_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}\approx 1.5b}$ is used to characterize contacts. Further details are provided in Appendix 1.

The following is a more extensive summary to supplement the brief outline in Materials and methods of the maintext provided under the heading ‘sequence-specific theory of heteropolymer phase separation’. In general, sequence-specific polymer field theories (Lin et al., 2016; Lin et al., 2020) are constructed to model systems of polymers with various salt and counterions. Further details are available from our recent publications (e.g. Lin et al., 2023; Wessén et al., 2022a and references therein).

Using the same notation for the partition function ${\displaystyle \mathcal{Z}}$ in Equation 2, with ${\displaystyle \mathcal{T}+\mathcal{U}}$ being the Hamiltonian in units of the product ${\displaystyle k_{\mathrm{B}}T}$ of Boltzmann’s constant ${\displaystyle k_{\mathrm{B}}}$ and absolute temperature ${\displaystyle T}$, the connectivity term ${\displaystyle \mathcal{T}}$ is given by

with ${\displaystyle [\mathbf{R}]}$ being shorthand for ${\displaystyle [\{{\mathbf{R}}_{\alpha ,i}\}]}$. Considering the case when the total potential energy ${\displaystyle \mathcal{U}}$ is taking one of its simplest forms, in that it serves only to model the two-body (pairwise) interactions among polymer residues, salt ions, and counterions, we further confine the interaction types in our formulation to Coulomb electrostatics (el) and excluded-volume (ex). As stated in the maintext, their interaction strengths are governed by the Bjerrum length ${\displaystyle l_{\mathrm{B}}}$ (${\displaystyle l_{\mathrm{B}}=e^2/(4\pi {ϵ}_0{ϵ}_{\mathrm{r}}{k}_{\mathrm{B}}T)}$, where ${\displaystyle e}$ is protonic charge, ${\displaystyle {ϵ}_0}$ is vacuum permittivity, and ${\displaystyle {ϵ}_{\mathrm{r}}}$ is relative permittivity), and the two-body excluded volume parameter $v_2$. We now provide the precise field-theoretic forms for the Coulomb electrostatic potential ${\displaystyle {\mathcal{U}}_{\mathrm{e}\mathrm{l}}}$ and two-body excluded volume interaction ${\displaystyle {\mathcal{U}}_{\mathrm{e}\mathrm{x}}}$ terms in ${\displaystyle \mathcal{U}={\mathcal{U}}_{\mathrm{e}\mathrm{l}}+{\mathcal{U}}_{\mathrm{e}\mathrm{x}}}$ by introducing the number density operators

for the monomers (residues or beads) of the polymer (p), salt (s), and counterion (c), respectively, where ${\displaystyle \delta }$ represents the Dirac ${\displaystyle \delta }$ distribution. Solvent (water) degrees of freedom [${\displaystyle n_{\mathrm{w}}}$ in Equation 2 for ${\displaystyle \mathcal{Z}}$ in the maintext] are not included in Equation S2 above because they are only used for the incompressibility constraint in our rG-RPA formulation as an approximate treatment of excluded volume, and solvents are not treated explicitly at all in the present field-theoretic simulation (FTS), that is, the ${\displaystyle n_{\mathrm{w}}}$ factor is dropped for FTS. The corresponding charge density operators for the number density operators in Equation S2 are

For ${\displaystyle {\hat{c}}_{\mathrm{p}}}$, ${\displaystyle {\sigma }_i=+1}$ for arginine and lysine, ${\displaystyle {\sigma }_i=-1}$ for aspartic and glutamic acids, ${\displaystyle {\sigma }_i=-2}$ for phosphorylated tyrosine, and ${\displaystyle {\sigma }_i=0}$ for all other amino acid residues in the Caprin1/pY-Caprin1 sequences studied (as stated in the maintext). ${\displaystyle {\mathcal{U}}_{\mathrm{e}\mathrm{l}}}$ and are now given by

As mentioned in the maintext, two conjugate fields, ${\displaystyle \psi (\mathbf{r})}$ for Coulomb interaction and ${\displaystyle w(\mathbf{r})}$ for excluded volume, are then introduced to linearize the density operators that are quadratic in ${\displaystyle {\mathcal{U}}_{\mathrm{e}\mathrm{l}}}$ and ${\displaystyle {\mathcal{U}}_{\mathrm{e}\mathrm{x}}}$ by applying the Hubbard-Stratonovich transformation (Stratonovich, 1958; Hubbard, 1959), resulting in a reformulated partition function ${\displaystyle {\mathcal{Z}}^{\prime }\equiv (n_{\mathrm{p}}!n_{\mathrm{s}}!n_{\mathrm{c}}!n_{\mathrm{w}}!)\mathcal{Z}}$ [with ${\displaystyle \mathcal{Z}}$ given by Equation 2 of maintext] expressed as a functional integral over the fields ${\displaystyle \psi }$ and ${\displaystyle w}$:

where ${\displaystyle {\mathcal{Q}}_{\mathrm{p}}}$, ${\displaystyle {\mathcal{Q}}_{\mathrm{s}}}$, and ${\displaystyle {\mathcal{Q}}_{\mathrm{c}}}$ are single-molecule partition functions of polymer, salt ion, and counterion, respectively, which are all functionals of ${\displaystyle \psi }$ and ${\displaystyle w}$:

wherein ${\displaystyle \mathrm{i}}$ is the imaginery unit, that is, ${\displaystyle {\mathrm{i}}^2=-1}$.

The ${\displaystyle {\mathcal{Z}}^{\prime }}$ in Equation S5 can be analyzed via various field theoretic approaches. Two approaches are utilized in the present work: (i) one-loop perturbation expansion is employed to derive analytical theories based upon random-phase-approximation (RPA), and (ii) field-theoretic simulation (FTS) is conducted to compute observables numerically.

Coarse-grained MD simulations are performed with the GPU version of HOOMD-Blue software (Anderson et al., 2020; Anderson et al., 2008) using the slab method that has been developed recently to allow for simulations of relatively large number of polymers (Silmore et al., 2017) and applied to liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) (Dignon et al., 2018). This general MD protocol has been utilized extensively, (Joseph et al., 2021; Dignon et al., 2019b) including by our group (Das et al., 2020; Das et al., 2018a).

Within the methodological framework of this coarse-grained simulation protocol, we introduce explicit small ions into our present simulations because they are necessary to account for subtle experimental observations that are not readily reproduced by using implicit-ion electrostatic screening. Simulations in the present study are performed with 100 chains of the Caprin1 IDR (wildtype or variants) at four salt concentrations ([NaCl]): (i) at [NaCl] = 0 where the system is neutralized by adding appropriate number (1300) of chloride (Cl⁻) ions, (ii) neutralized and at [NaCl] = 200 mM by adding 15,000 pairs of explicit Na⁺ and Cl⁻ ions, (iii) neutralized and at [NaCl] = 480 mM by adding the same number of 15,000 pairs of explicit Na⁺ and Cl⁻ ions, and (iv) neutralized and at [NaCl] = 960 mM again with 15,000 pairs of explicit Na⁺ and Cl⁻ ions. As will be described below, specific small-ion concentrations in (ii), (iii), and (iv) are implemented by varying the size of the final simulation box. A similar procedure is used for simulation of pY-Caprin1 IDR phase behaviors under these four [NaCl] values. Because each pY-Caprin1 IDR has a net −1 charge, the only difference with the Caprin1 IDR simulation is that neutralization of the pY-Caprin1 chain requires 100 Na⁺ ions instead of 1300 Cl^– ions. The amino acid sequences simulated using coarse-grained MD in the present study are provided in Appendix 1—figure 1. Note that the experimental pY-Caprin1 sample is highly phosphorylated, consisting mainly of a mixture of Caprin1 IDRs with six or seven phosphorlations, with only a very small fraction of IDRs with five phosphorylations, and essentially no population with fewer than five phosphorylations (Appendix 1—figure 2). As stated in the maintext, for the sake of simplicity in our theoretical/computational models, we use only the Caprin1 IDR with all seven tyrosines phosphorylated (referred to simply as pY-Caprin1 in Appendix 1—figure 1) to model the behaviors of this experimental sample, partly to avoid the combinatoric complexity of sequences with five or six phosphorlations, which would entail 21 and 7 possible different sequences, respectively, with currently unknown population fractions.