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# Molecular design of self-coacervation phenomena in block polyampholytes

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved March 12, 2019 (received for review January 12, 2019)

## Significance

By combining field-theoretic simulations and molecular-dynamics simulations, we show how the charge sequence of block polyampholytes affects their solution-phase behavior and accessible chain conformations. We find a striking effect of like charge block length and connectivity on their self-coacervation or liquid–liquid phase separation. Charge patterning on smaller lengthscales allows for more expanded chain configurations and increased stability in dilute solution. These findings may provide insight into the condensation of intrinsically disordered proteins in vivo.

## Abstract

Coacervation is a common phenomenon in natural polymers and has been applied to synthetic materials systems for coatings, adhesives, and encapsulants. Single-component coacervates are formed when block polyampholytes exhibit self-coacervation, phase separating into a dense liquid coacervate phase rich in the polyampholyte coexisting with a dilute supernatant phase, a process implicated in the liquid–liquid phase separation of intrinsically disordered proteins. Using fully fluctuating field-theoretic simulations using complex Langevin sampling and complementary molecular-dynamics simulations, we develop molecular design principles to connect the sequenced charge pattern of a polyampholyte with its self-coacervation behavior in solution. In particular, the lengthscale of charged blocks and number of connections between oppositely charged blocks are shown to have a dramatic effect on the tendency to phase separate and on the accessible chain conformations. The field and particle-based simulation results are compared with analytical predictions from the random phase approximation (RPA) and postulated scaling relationships. The qualitative trends are mostly captured by the RPA, but the approximation fails catastrophically at low concentration.

Polyampholytes are charged copolymers containing positive, negative, and neutral segments. The sequence of those segments can be annealed (i.e., depend on the pH of solution in the case of weak polyampholytes) or quenched randomly or in a specific charge pattern (e.g., alternating or blocky). Polyampholytes exbibit richer phase behavior and chain conformations than polyelectrolytes, their uniformly charged analogues, due to the repulsion between like-charged segments stretching the chain and the attraction between oppositely charged segments collapsing the chain (1⇓⇓⇓⇓⇓–7).

Block polyampholytes, in particular, are an attractive system to study the conformational regimes (8⇓–10) and phase separation (11⇓⇓–14) at different ionic strengths as a function of polyampholyte concentration, fractional charge, net charge, and sequence pattern. The molecular design principles developed should give insight into the more complicated sequences of intrinsically disordered proteins (IDPs) (1, 15⇓⇓⇓–19) and solution environments in vivo affecting the liquid–liquid phase separation responsible for membraneless organelles (20⇓⇓⇓⇓⇓⇓⇓⇓⇓–30).

The liquid–liquid phase separation of block polyampholytes into a polymer-rich phase and a dilute supernatant coexisting primarily of small salt ions is an example of single-component coacervation or self-coacervation. This self-coacervation phenomenon is strikingly similar to the complex coacervation of oppositely charged polyelectrolytes, due to the tendency for polyelectrolytes to form charge-neutral dimers in dilute solution (8, 31). Polyampholytes, however, have chain conformations and phase behavior that are sensitive not only to their total charge, but also to the patterning of charges along the polymer, where self-coacervation is suppressed in charge-scrambled analogues (24, 32). Simple analytical theories based on the random phase approximation (RPA) have partially accounted for the phase behavior of sequence-specific electrostatic interactions for a few specific IDP charge patterns (12⇓–14) and other simple patterns (6, 7, 33), although the RPA is known to break down at low concentration, so it is unable to reliably predict the dilute branch of polyampholyte-phase diagrams. It should also be noted that the widely applied Voorn–Overbeek model of coacervation (34) neglects the connectivity of charges to the polymeric backbone (31, 35, 36) and thus is not useful for understanding sequence-dependent self-coacervation phenomena.

The phase behavior and chain conformations of nearly charge-neutral alternating, random, and diblock polyampholytes have been studied extensively through scaling arguments, RPA, as well as molecular simulations highlighting the attractive electrostatic fluctuations that cause collapse into globular configurations and phase separation at low concentrations (1, 5⇓⇓⇓–9, 19, 33, 37, 38). These approaches have qualitatively matched experiments on synthetic polyampholytes where the overall charge and sequence of charges are difficult to control (3, 4, 39⇓⇓–42), so the accuracy of these approaches is yet unknown. So far, however, the only approximation-free phase diagrams revealing both dilute and concentrated branches of the two-phase coexistence envelopes for polyampholytes in solution have been developed for the symmetric diblock polyampholyte (31) and a few model IDP sequences (43), but general design rules are missing to understand the role of charge sequence along a polyampholyte on self-coacervation behavior, independent of variables such as total charge, solvent quality, and molecular weight.

Here, we use field-theoretic simulations to construct complete, approximation-free phase diagrams of block polyampholytes as a function of sequence. The phase diagrams are approximation-free in that they contain no uncontrollable approximations (i.e., analytical or numerical approximations such as closures, mean-field approximations, interaction cutoffs, or simplifications such as neglecting the finite polymer density in the dilute phase), outside of the choice of coarse-grained model. This allows for self-coacervation phenomena to be elucidated as a function of the number of blocks, block asymmetry, and charge asymmetry. In this study, explicit counterions and added salt ions are neglected, but, as we show elsewhere, the inclusion of explicit counterions has only a weak effect on the phase diagrams or structure of charge-neutral polyampholytes in solution, primarily through enhanced electrostatic screening that serves to slightly reduce the effective electrostatic strength. Manipulating the lengthscale and placement of charged patterns are shown to have a strong effect on the structure of the phases, particularly on chain conformations in the dilute branch of phase coexistence. Sequence modulation on the scale of the electrostatic correlation length disrupts self-coacervation, with phase separation maximized for polyampholytes with long runs of like-charged residues and a minimum number of connection points between oppositely charged segments.

## Model and Methodology

### Molecular Model.

In the field theory, we use a coarse-grained model (31, 44⇓⇓–47) of sequence-defined polyampholytes as continuous Gaussian chains, with all pairs of statistical segments interacting through a weak contact excluded volume parameter, v, and charged segments interacting via a Coulomb potential screened by a uniform background dielectric of Bjerrum length,

To ensure that the chemical potentials and pressures calculated in the field-theoretic simulations are insensitive to the computational grid and are free of UV divergences (48, 49), the statistical segments are smeared over a finite volume by convolution with a normalized Gaussian profile,

This interaction energy can be equivalently written in a particle-based representation of bead–spring chains with a nonbonded pair potential between beads separated by a distance r as**2**. While well-defined, such a model will not realistically capture liquid structure and electrostatic correlations on angstrom-level monomer and ion-length scales, as segment and ion potentials of mean force are harshly repulsive at close distances. For weakly charged polyelectrolytes with charge spacing greater than the Bjerrum length, this aspect of the model should not influence the results reported here, since structural and electrostatic correlations occur on mesoscopic scales beyond the segment size. In the case of strongly charged polyampholytes, it would seem unlikely that the soft repulsions in the present model would provide a realistic description of phenomena such as counterion condensation and ion pairing. Thus, the reader is cautioned that the trends reported here may not be applicable outside the weakly charged regime.

### Field-Theoretic Simulations.

Field-theoretic (FTS-CL) and molecular-dynamics (MD) simulations of the corresponding field theory and particle models, respectively, are used to sample the energetic landscape prescribed by the interaction energy of Eqs. **1** and **2** and provide a framework for examining the sequence effects on the structure and thermodynamics of block polyampholytes. The canonical partition function of the model specified in Eq. **1**, integrated over segment coordinates, can be converted via an exact Hubbard–Stratonovich transformation to a complex-valued statistical field theory (44):*,* one elementary charge per statistical segment) chain of

The nonbonded interactions among statistical segments are consequently decoupled, and the segments interact only with auxiliary fields

Electrostatically driven phase separation in charge-neutral polyampholytes cannot be described at the simplest mean-field level; there is no Coulombic contribution to the mean-field free energy of a bulk system with periodic boundary conditions due to global electroneutrality (31, 44⇓–46). Phase separation is only obtained by considering field fluctuations around the electroneutral state (8, 31), through either a Gaussian approximation (RPA) formalism (31, 46, 47) (*SI Appendix*) or field-theoretic simulations that fully sample the field configurations and incorporate all higher-order fluctuation effects (31).

The functional integrals are taken over real-valued *SI Appendix*.

Phase-equilibrium conditions are constructed through the explicit computation of the osmotic pressure *SI Appendix*) of the chemical potential and pressure in an analogous manner.

### MD.

The same coarse-grained molecular model used for the field-theoretic simulations can be used to construct MD simulations (43). For the particle simulations, a discrete Gaussian chain model with

To evaluate the electrostatic potential using mesh-based Ewald summation for Gaussian-smeared charges (55), the real space contribution can be written,

As explored in detail elsewhere, particle MD simulation provides an opportunity to supplement the FTS-CL simulations with structural information regarding single-chain conformations (43).

#### Chain Conformation Analysis.

The conformations of the polymer chains are analyzed by using shape parameters that are defined by means of the invariants of the gyration tensor S.*i*th Cartesian component of the chain center of mass and

#### Single-Chain Structure Factor.

The single-chain structure factor (58) of an isolated polyampholyte chain in the dilute supernatant is given by

## Results and Discussion

Work on the sequence effects or molecular design of complex coacervates or self-coacervates have focused thus far on differences in charge pattern that mimic those found in IDPs (43), relying on characterizing a sequence by a charge pattern metric such as

### Excluded Volume.

Solvent quality is often a function of concentration, temperature, and even sequence due to the arrangement of hydrophilic and hydrophobic residues. Even specific ion effects can impact solvent quality, depending on the particular ion identity. Since the solvent quality may change as a function of such variables, we first present the effect of solvent quality/excluded volume on the phase diagram of the diblock polyampholyte *SI Appendix*).

Increasing the excluded volume parameter

In addition to shifting toward higher electrostatic strengths, the two-phase region of the phase window narrows with increasing B. This narrowing occurs on both the concentrated and dilute branches. Recall that the coexistence conditions are dictated by chemical, *SI Appendix* for extrapolation procedure) shifting to lower values.

RPA predictions shown in Fig. 1 qualitatively capture the narrowing of the two-phase window with increasing excluded volume, including quantitatively capturing the behavior at high C. However, due to the catastrophic failure on the dilute branch, the RPA underestimates the effects on the critical point, a failure that worsens with increasing B.

Even at a relatively weak electrostatic strength, *E* = 1,000 (recall *SI Appendix*, Fig. S5). Thus, the excluded volume effects on the isolated chains are comparatively weak, and the polyampholytes corresponding to different B values have nearly identical structure (*SI Appendix*, Figs. S2–S4). This confirms that the effects of excluded volume on isolated polyampholytes in the dilute phase are relatively unimportant; even though the local densities are comparable, the dilute-phase effects on the binodal are due to propagation of chemical and mechanical equilibrium conditions due to osmotic pressure and chemical potential increases in the concentrated phase.

As argued above, the phase behavior is dominated by the concentrated phase, where overlapping polyampholytes with increasing excluded volume is strongly penalized, resulting in coacervates of lower concentration. An increase in B leads to a sharp decrease in the isothermal compressibility (

The structure factors indicate that charge and mass fluctuations are practically decoupled (and are strictly so in RPA), such that

### Block Length and Number.

The clustering of like charges into local patches amplifies electrostatic fluctuations, while the scrambling of those charges diminishes phase separation (13, 32). We probe this sequence effect on the self-coacervation behavior by tuning the “blockiness” or block length and number in an alternating positively and negatively charged block polyampholyte, where the blocks are of uniform, equal, and opposite valences. We increase the block number

The number and length of the block charge pattern has a substantial effect on the phase diagram, with shorter blocks significantly suppressing phase separation (Fig. 3). There is a dramatic increase in the critical electrostatic strength, which increases by an order of magnitude from the diblock polyampholyte (

While there is a minor effect on the concentrated branch that is primarily due to the difference in critical electrostatic strength, the concentrated phase is relatively agnostic to the charge pattern. This is because in the coacervate phase, the characteristic lengthscales of density, *SI Appendix*, Fig. S6) and nearly the same peak wavevector in the electrostatic structure factor (*SI Appendix*, Fig. S7) for all block lengths at

The largest effect, however, is on the dilute branch, where the binodal curve sweeps inward to higher polymer concentration over three orders of magnitude, significantly narrowing the two-phase region as the number of blocks is increased and the length of the charge repeat is decreased. Recall the RPA prediction of the electrostatic correlation length in units of *B*). This impingement creates additional electrostatic screening that stabilizes increased concentration of the polyampholytes in the dilute branch (supernatant), shifting the binodal curve to higher E and C. The catastrophic failure of RPA in predicting the dilute binodal branches is seen again in Fig. 3 and in the dilute electrostatic structure factors of Fig. 4*B*.

The thermodynamic properties of the polyampholytes in the dilute phase are further influenced by the chain conformations accessible by each sequence. The diblock polyampholyte has little choice but to collapse to a charge-neutral globule. However, as the number of blocks is increased, the charge neutrality can be satisfied on a more local scale, allowing for swelling and greater conformational freedom. The density–density structure factor from FTS-CL (Fig. 4*A*), as well as the configuration snapshots (Fig. 4), single-chain structure factor (*SI Appendix*, Fig. S8), and gyration tensor metrics (*SI Appendix*, Fig. S9) from MD, all capture this increase in the number of conformational states accessible and show an increase in

### Block Asymmetry.

To understand the importance of block number on the phase behavior and structure of block polyampholytes, we split the chain into an increasing number of smaller blocks. However, it is important to distinguish the effects of block size and symmetry from the number of A–C connecting junctions. This can be achieved by interrogating a triblock polyampholyte of form

The block-asymmetry effect is far more subtle than the changes to phase behavior from changing the number of blocks and is not properly captured by the *SI Appendix*, Fig. S10). Again, we see that the local charge patterning produces the same lengthscale for electrostatic fluctuations, *SI Appendix*, Fig. S11).

Interestingly, the dilute branch saturates to the same binodal concentrations at high E for all of the triblock polyampholytes, but distinct from the diblock and tetrablock polyampholytes. Thus, the dilute behavior at high electrostatic strengths is dictated by the number of connecting points between oppositely charged residues. This is a remarkable result, as even small patches of oppositely charged segments can induce a significant difference in phase behavior at strong electrostatic strengths. The additional connection point must affect the local packing of a single coil, which, in the strong electrostatic limit, results in a reduced set of accessible chain conformations. As the electrostatic strength is lowered (decreasing E) toward the critical point, the block asymmetry has an even greater influence on the phase behavior. Near the critical point, the behavior transitions continuously from a symmetric triblock (*A* and *SI Appendix*, Figs. S12 and S13) and more isotropic configurations (*SI Appendix*, Fig. S13), as the larger coherent blocks penalize extended conformations.

The dilute solutions of the block-asymmetric polyampholytes further have similar *B*), which decreases slightly with block asymmetry. Interestingly, this is the opposite trend of that predicted by RPA, possibly a consequence of miscapturing intramolecular correlations, as has been recently suggested (36, 64). However, one should expect

### Charge Asymmetry.

There is another symmetry that can be affected by the patterning of the individual sequence: charge density along the chain. Here, we maintain overall charge neutrality, but increase the charge density of one block with a compensating reduction in block length. We compare a charge-asymmetric polyampholyte

We find only a very weak effect on the phase diagram by altering the charge-density symmetry, reinforcing the importance of connectivity and balanced total charge and suggesting the relative unimportance in how finely the charge is packed within blocks, at a fixed total polymeric charge (Fig. 7). The result is a near-identical critical electrostatic strength. The two-phase region is shifted by charge-density asymmetry to slightly lower polymer concentration throughout the phase diagram, suggesting stronger correlations that induce electrostatic attractions at lower polymer concentration in the dilute phase, but more electrostatic screening in the concentrated phase.

As expected, there is catastrophic failure of the RPA on the dilute branch, but, interestingly, the RPA also fails to capture behavior of the charge-asymmetric block polyampholyte at high C, where it is expected to be valid. The origin of this failure is unknown: The RPA is accurate in predicting the osmotic pressure (*SI Appendix*, Fig. S14) and structure of the coacervate phase (Fig. 8*B* and *SI Appendix*, Fig. S16), but fails to predict the chemical potential of the charge-asymmetric polyampholyte, showing deviations between RPA and FTS-CL, even at high C (*SI Appendix*, Fig. S15), possibly due to more intense local ion-pair correlations.

In the coacervate, the charge-symmetric and -asymmetric polyampholytes maintain the same total density correlations, but the charge-asymmetric case has stronger intensity of electrostatic correlations, likely due to the increased localized charge density, with a slightly smaller

As with previous alterations of the molecular structure, the differences in the thermodynamics correlate with conformational changes in the dilute supernatant. FTS-CL shows that the dilute-phase total-density structure factor, *SI Appendix*, Fig. S17), but the single-chain MD simulations show the existence of significantly more compact globules (Fig. 8 and *SI Appendix*, Figs. S18 and S19). Evidently, in the asymmetric case, the more intense electrostatic correlations between blocks have overcome the Gaussian excluded volume repulsions to a larger extent than in the charge-symmetric polyampholyte. This conclusion is supported by the longer lengthscale and more intense electrostatic correlations of the charge-asymmetric polyampholyte in the dilute solution. However, RPA fails to predict the dominant lengthscale of those correlations. FTS-CL shows a longer lengthscale

## Conclusions

Approximation-free phase diagrams have been developed for the effects of excluded volume, block length and number, block symmetry, and charge symmetry on the self-coacervation behavior of block polyampholytes. This phase behavior and structural insights into both dilute and concentrated phases are made possible by a unique combination of field-theoretic and MD simulations of the same underlying molecular model. [Note: The FTS-CL simulations and RPA predictions use continuous Gaussian chains, while the MD particle simulations use discrete Gaussian chains; the discretization of the chain is expected to be unimportant (43, 45, 46)].

Block length in particular has a large effect on the critical electrostatic strength for phase separation due to impingement on the electrostatic correlation length and increased charge screening as the block length decreases. Manipulating the asymmetry of the block placement or size highlights the importance of the number of chain-connection points and ability to control phase separation by the patterning lengthscale.

As seen in complex coacervation of oppositely charged polyelectrolytes (65), we have found a strong effect of charge pattern on the thermodynamics and materials structure. However, simplified charge sequence metrics like

The sequence-specific effects have the largest influence on the dilute phase. The charge distribution alters charge–charge correlations and dramatically influences coil dimensions and their propensity for self-aggregation, thus impacting the dilute branch of the coexistence region and the overall stability of polyampholyte in the solution.

## Acknowledgments

This work was supported by the Materials Research Science and Engineering Centers (MRSEC) Program of the National Science Foundation (NSF) through Award DMR-1720256 (IRG-3). J.-E.S. received partial support from NSF Award MCB-1716956. J.M. and J.-E.S. were partially supported by the National Institutes of Health Grant R01AG05605. All simulations were performed by using computational facilities of the Extreme Science and Engineering Discovery Environment (XSEDE; supported by NSF Project TG-MCA05S027) and of the Center for Scientific Computing from the California NanoSystems Institute, Materials Research Laboratory: NSF MRSEC Grant DMR-1720256 and NSF Grant CNS-1725797.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: ghf{at}ucsb.edu or kdelaney{at}ucsb.edu.

Author contributions: S.P.O.D., J.M., K.T.D., and G.H.F. designed research; S.P.O.D. and J.M. performed research; S.P.O.D. analyzed data; and S.P.O.D., J.M., J.-E.S., K.T.D., and G.H.F. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1900435116/-/DCSupplemental.

Published under the PNAS license.

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