Coacervation drives morphological diversity of mRNA encapsulating nanoparticles

Abstract

The spatial arrangement of components within an mRNA encapsulating nanoparticle has consequences for its thermal stability, which is a key parameter for therapeutic utility. The mesostructure of mRNA nanoparticles formed with cationic polymers has several distinct putative structures: here, we develop a field theoretic simulation model to compute the phase diagram for amphiphilic block copolymers that balance coacervation and hydrophobicity as driving forces for assembly. We predict several distinct morphologies for the mesostructure of these nanoparticles, depending on salt conditions and hydrophobicity. We compare our predictions with cryogenic-electron microscopy images of mRNA encapsulated by charge altering releasable transporters. In addition, we provide a graphics processing unit-accelerated, open-source codebase for general purpose field theoretic simulations, which we anticipate will be a useful tool for the community.

I. INTRODUCTION

The widespread deployment of lipid nanoparticle mRNA vaccines^1,2 has accelerated efforts to diversify polymer-based drug delivery technologies. A number of physical properties, including high thermal stability and structural robustness,² are desirable for therapeutics, but we currently lack the necessary design principles to ensure these features on theoretical grounds.³ Among the many candidate technologies,² amphiphilic polymers that encapsulate mRNA via coacervation have emerged as a potentially fruitful route to building nanoparticles with delivery specificity.^1,2,4–6 mRNA encapsulating nanoparticles are also of interest as simplified models for liquid–liquid phase separation in biological systems.⁷ Here, we study the structural properties of mRNA encapsulating polymeric nanoparticles using field theoretic simulations and cryogenic-electron microscopy (cryoEM) to elucidate the physiochemical driving forces that underlie several distinct mesostructures.

While ionizable lipid nanoparticles have been the dominant modality for non-viral gene delivery vectors,⁸ a number of cationic synthetic polymers have emerged as alternatives for mRNA encapsulation and delivery.^4,9–11 In this work, we focus on charge altering releasable transports (CARTs), a class of copolymers formed from a cationic block and lipid block that exhibit a remarkable self-immolative degradation to neutral small molecules, hypothesized to aid in the endosomal escape of mRNA,⁴ although we study only the thermodynamics of the pre-degradation polymers in this work. Distinct formulations of CART polymers have been demonstrated to produce delivery vehicles that allow for the targeting of distinct cell types or even preferentially target individual organs.^4,5,12 However, assessing the impact of tunable parameters, such as block length and hydrophobicity of the encapsulating polymers on the self-organization of the resulting nanoparticles, remains a largely empirical effort due to the absence of predictive models.³ Furthermore, the incipient understanding of tissue selectivity has not yet been unambiguously connected to the structural features of delivery nanoparticles.¹²

The physical mechanism that drives the assembly of CARTs features both a hydrodynamic driving force and electrostatic interactions between polyanionic mRNA and the charged block of the encapsulating polymers⁴ via a process known as coacervation. Coacervation is an active area of research, both experimentally and theoretically.¹³ Because coacervation is a fluctuation-induced effect, mean-field theories incorrectly predict that no assembly of oppositely charged polymers should occur.^14,15 Analytical theories that go beyond mean-field, such as the random phase approximation (RPA), make corrections that resolve the deficiency of mean-field theory and predict correct scaling behavior.^16–18 However, it remains challenging within analytical theory to simultaneously incorporate the effects of hydrophobicity, charge, and sequence, and more sophisticated models that represent dynamical fluctuations are needed.^19,20

While some progress on coacervation dynamics has been made using atomistic^14,21 or coarse-grained^13,22 models, the computational costs of simulating mesostructures are prohibitive for many systems of interest, so we turn to field theoretic simulation²³ of a polymer field theory.²⁴ Numerical simulation of an appropriate polymer field theory offers an appealing alternative to molecular models for systems that balance coacervation with hydrophobic driving forces, although to our knowledge this specific setting has not been previously studied with fully fluctuating field theories. Because polymer field theories map the density of each species to a continuous field, they can be employed to predict both the spatial organization of components and also thermodynamic properties derivable from the computationally accessible partition function.²⁴ There is a vast literature on the numerical simulation of polymer field theories, discussed at length in the recent monograph Ref. 23, which has led to algorithms that make field theoretic simulation computationally efficient and demonstrations that numerical simulations of polymer field theories yield excellent agreement with experimentally determined morphology in a variety of contexts.^25–27 Despite its suitability for the self-organization of polymers such as CART-mRNA nanoparticles, previous work on coacervation has limited itself to the setting of a “good” solvent to isolate electrostatic effects,¹³ used methods that are difficult to sample, such as density-explicit field theories,^28–30 modeled charged interactions using mean field approaches that do not capture coacervation,^29,31–33 or resorted to simpler models of charged interaction.³⁴

Because we focus on the generic physical properties that drive encapsulation in this work, we develop a field theoretic representation of the CART-mRNA nanoparticle system, a highly coarse-grained representation, that enables simulations at the “mesoscale.” We find that this model predicts morphological features also present in experimental cryoEM. We observe a transition to a dense coacervate phase, the specific morphology of which depends strongly on both salt concentration and polymer hydrophobicity. Despite minimal modeling assumptions, the morphologies that we observe are in excellent agreement with experimental cryoEM micrographs. The methodology and open-source graphics processing unit (GPU)-accelerated simulation package that we present here are both broadly applicable to charged polymer mixtures in the solvent; we anticipate, in particular, that our software package will be useful for the polymer physics community because existing open-source tools for field theoretic simulation do not include fluctuations beyond static self-consistent field theory.³⁵

II. THEORETICAL FRAMEWORK

A. Preliminaries

This work builds heavily on prior work focused on field theoretic simulation.^23,24 We use a modified version of Düchs et al.’s multi-species exchange model³⁶ with the coacervation model derived by Delaney and Fredrickson.¹⁵ We follow Villet and Fredrickson³⁷ for efficient numerical schemes for the modified diffusion equation, smearing to regularize ultraviolet divergences, and pressure and chemical potential operators. We utilize the “soft-explicit” solvent model,³⁸ which has proved useful for systems with both charged and hydrophobic interactions.³⁹ This approach can be viewed in the larger context of auxiliary field methods, which have previously been used to simulate systems with charged and hydrophobic interactions.^29,31–33 Alternatively, this problem can also be overcome with density explicit field theories.^28,29 For completeness, we provide derivations with only small changes from these aforementioned papers.

As emphasized above, the formal derivation of the model is relatively standard and builds on Refs. 15, 24, 36, and 37; it is included here so that this work is relatively self-contained. Throughout, we consider a mixture of charged and neutral polymers in the solvent with explicit salt. There are P total polymer species, and each species has n_p polymers, S solvent species, and two salt species. The system has M different types of monomers that make up the polymer and solvent. An individual polymer takes coordinates $\mathit{r}\in \mathrm{\Omega }\subset {\mathbb{R}}^d$, a domain with volume V and periodic boundary conditions. In this work, d = 2, although all equations are general for any d, and we emphasize that our code base also treats three-dimensional systems. We rescale all lengths relative to a single reference polymer length N and a reference bond stretching parameter b. For example, the volume $\tilde{V}$ becomes $V=\tilde{V}/{R_{\text{g}}}^d$, where $R_{\text{g}}=b/\sqrt{N/6}$ and d is the system dimensionality. An unscaled position vector x is rescaled in V as r ≔ x/R_g.

Interactions between the monomers are governed by three distinct physical effects: Flory–Huggins pairwise interactions, Coulomb interactions for charged species, and bonded stretching interactions for polymer components. We model bonded interactions with a continuous Gaussian chain, where the stretching potential for a single polymer is a functional over a space curve representing the polymer,

$$U_0[\mathit{r}]=\frac{3k_{\text{B}}T}{2}{\int }_0^{N_p/N}\mathrm{d}s|\frac{\mathrm{d}\mathit{r}(s)}{\mathrm{d}s}{|}^2.$$ (1)

Here, s represents the scaled length variable indicating the parametric position along the polymer. Within this model, the partition function for an isolated chain can be computed analytically as a Gaussian functional integral. We define the single chain partition function as

$$z_0={\int }_{\mathrm{\Omega }}\mathcal{D}\mathit{r}{e}^{-\beta U_0[\mathit{r}]}.$$ (2)

We compute this integral over all possible curves for the polymer type in question. For a non-interacting system of single chains,

$${\mathcal{Z}}_{\text{ideal}}=\prod _i^P\frac{{\left(z_{0_i}{g}_N^{(i)}V\right)}^{n_i}}{n!}\prod _j^{S+2}\frac{V}{{\lambda }_j^{d}N!},$$ (3)

where λ is the thermal wavelength for a given species and $g_N^{(i)}$ is the effective thermal wavelength for the Gaussian chain associated with species i.

Before incorporating interactions, it is useful to define a density operator for any monomer species in our system. Each polymer of type p has a monomer sequence, which we represent with a parameter m_p(s) that indicates the monomer type at each point s along p. Combining the density for both polymers and solvents, the density operator is

$$\begin{array}{ll}\hfill {\overline{\rho }}_m(\mathit{r})& \equiv {\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\sum _{p=1}^P\sum _{j=1}^{n_p}{\int }_0^{N_j/N}\mathrm{d}s\delta (\mathit{r}-{\mathit{r}}_j(s))\delta (m-m_p(s))\hfill \\ \hfill & +\sum _{s=1}^S\sum _{l=1}^{n_s}\delta (\mathit{r}-{\mathit{r}}_l)\delta (m-m_s).\hfill \end{array}$$ (4)

Because the interparticle interactions are formally divergent without regularization, we regularize the density and the charge density with a Gaussian kernel,

$$\mathrm{\Gamma }(\mathit{r})={(2\pi a^2)}^{-d/2}\exp \left(-\frac{1}{2a^2}{\mathit{r}}^T\mathit{r}\right).$$ (5)

This function is a normalized, d-dimensional symmetric Gaussian with width a. With this smoothing, we can write the smoothed monomer density as

$${\rho }^{(m)}(\mathit{r})\equiv \mathrm{\Gamma }\ast {\overline{\rho }}^{(m)}(\mathit{r})={\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(|\mathit{r}-{\mathit{r}}^{\prime }|){\overline{\rho }}^{(m)}(\mathit{r})\mathrm{d}\mathit{r}.$$ (6)

Similarly, we write the smoothed charge density for a polymer species p as

$${\rho }_{\text{C}}^{(p)}(\mathit{r})=\mathrm{\Gamma }\ast {\overline{\rho }}_{\text{C}}^{(p)}(\mathit{r})$$ (7)

with

$${\overline{\rho }}_{\text{C}}^{(p)}(\mathit{r})=\sum _{n=1}^{n_p}{\int }_0^{N_p/N}{\sigma }_n(s)\delta (\mathit{r}-{\mathit{r}}_n^{(p)}(s))\mathrm{d}s,$$ (8)

where $\sigma :[0,N_p/N]\to \mathbb{R}$ gives the charge along the polymer. We denote the vector of all monomer densities $\mathit{\rho }:\mathrm{\Omega }\to {\mathbb{R}}^M$, and the total charge density is ${\rho }_{\text{C}}(\mathit{r})\equiv {\sum }_{i=0}^M{Z}_i{\rho }_i(\mathit{r})$, where Z_i is the charge of species i.

To non-dimensionalize the equations, we also rescale the FH interactions ${\tilde{\chi }}_{ij}$ as ${\chi }_{ij}={\tilde{\chi }}_{ij}{N}^2/{R_{\text{g}}}^d$. The Coulomb charge–charge interaction relies on the rescaled Bjerrum length E = 4Z²πN²ℓ_B/R_g, where ℓ_b is the unscaled Bjerrum length. We simplify the parameter E by setting the reference charge Z = 1 and scale charges explicitly in the system if needed. Throughout, we assume that E does not depend on the system composition due to the screening from the explicit salt in the system. Ignoring a finite constant, the charge–charge interaction is

$$\beta U_{\text{C}}=\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{d}{\mathit{r}}^{\prime }\sum _i^M\sum _j^M\mathrm{\Gamma }\ast {\mathit{\rho }}_i(\mathit{r})\frac{Z_i{Z}_{j}E}{|\mathit{r}-{\mathit{r}}^{\prime }|}\mathrm{\Gamma }\ast {\mathit{\rho }}_j({\mathit{r}}^{\prime }).$$ (9)

This term simplifies if we define ${\rho }_{\text{C}}(\mathit{r})\equiv {\sum }_{i=0}^M{Z}_i{\rho }_i(\mathit{r})$ to

$$\beta U_{\text{C}}=\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{d}{\mathit{r}}^{\prime }\mathrm{\Gamma }\ast {\rho }_{\text{C}}(\mathit{r})\frac{E}{|\mathit{r}-{\mathit{r}}^{\prime }|}\mathrm{\Gamma }\ast {\rho }_{\text{C}}({\mathit{r}}^{\prime }).$$ (10)

Reintroducing the FH terms, we can write the full Hamiltonian with the densities,

$$\begin{array}{ll}\hfill \beta U[\mathit{\rho },{\mathit{\rho }}_{\text{C}}]& ={\int }_{\mathrm{\Omega }}\mathit{\rho }{(\mathit{r})}^T\mathit{\chi }\mathit{\rho }(\mathit{r})\mathrm{d}\mathit{r}\mathrm{d}{\mathit{r}}^{\prime }+\frac{E}{2}{\int }_{\mathrm{\Omega }}\frac{{\rho }_{\text{C}}(\mathit{r}){\rho }_{\text{C}}({\mathit{r}}^{\prime })}{|\mathit{r}-{\mathit{r}}^{\prime }|}\mathrm{d}\mathit{r}\mathrm{d}{\mathit{r}}^{\prime }-\beta U_0.\hfill \end{array}$$ (11)

Here, $\mathit{\chi }\in {\mathbb{R}}^{M\times M}$ is the Flory–Huggins (FH) interaction matrix and U₀ is a constant arising from the polymer self-interaction. Unfortunately, the Hamiltonian in this form is not immediately useful, as it is not practical to integrate over all density configurations. To this, we introduce an auxiliary field that will make these integrals tractable.

B. Flory–Huggins pair interaction

After regularization and rescaling, the total Flory–Huggins interaction for a pair of species is

$$\beta U_{ij}={\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{d}{\mathit{r}}^{\prime }{\chi }_{ij}\mathrm{\Gamma }\ast {\mathit{\rho }}_i(\mathit{r})\delta (\mathit{r}-{\mathit{r}}^{\prime })\mathrm{\Gamma }\ast {\mathit{\rho }}_j({\mathit{r}}^{\prime }).$$ (12)

A diagonal basis for the Flory–Huggins matrix yields the linear combinations of species that are physically distinct, and hence this basis is a natural one for developing a field theory.³⁶ We follow the derivation of Düchs et al.³⁶ closely. The FH matrix is given by

$$\mathit{\chi }=\left[\begin{array}{ccc}\hfill 2{\chi }_{11}\hfill & \hfill \dots \hfill & \hfill {\chi }_{1M}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\chi }_{M1}\hfill & \hfill \dots \hfill & \hfill 2{\chi }_{MM}\hfill \end{array}\right].$$ (13)

The matrix will generally have a size of M × M because each pair of monomer types can have a unique interaction potential. Often (including in the model presented here), this matrix is degenerate because multiple species have identical interaction profiles, and the non-degenerate matrix will be a smaller one where multiple monomer types can all be associated with the same chemical potential field. Our code base automatically identifies these cases and groups the relevant species. There are also less trivial degenerate FH matrices where more complex decompositions can be conducted, but we do not address them here.

Because χ is a real, symmetric matrix, we can diagonalize it with the decomposition,

$${\mathit{\rho }}^T(\mathit{r})\mathit{\chi }\mathit{\rho }(\mathit{r})={\left({\mathit{b}}^T\mathit{\rho }(\mathit{r})\right)}^T\mathit{B}{\mathit{b}}^T\mathit{\rho }(\mathit{r}),$$ (14)

where b is the matrix of eigenvectors of χ and B is the corresponding diagonal matrix of eigenvalues. In addition, we define a vector γ s.t. γ_i takes value i (value $\sqrt{-1}$) if B_i > 0 and 1 otherwise. In this diagonal representation,

$$\begin{array}{ll}\hfill U_{\text{FH}}& =\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{\Gamma }\ast {\mathit{\rho }}^T(\mathit{r})\mathit{\chi }\mathrm{\Gamma }\ast \mathit{\rho }(\mathit{r})=\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\sum _{i=1}^M{B}_i{\left({\mathit{b}}^T\mathrm{\Gamma }\ast \mathit{\rho }(\mathit{r})\right)}_i^2.\hfill \end{array}$$ (15)

Conveniently, this representation also makes the partition function Gaussian,

$$\mathcal{Z}={\mathcal{Z}}_{\text{ideal}}{\mathcal{Z}}_{\text{C}}\times \int \mathcal{D}{\left({\mathit{b}}^T\mathit{\rho }\right)}_i{e}^{-\frac{\beta }{2}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{d}{\mathit{r}}^{\prime }{\left({\mathit{b}}^T\mathit{\rho }(\mathit{r})\right)}_i{B}_i{\delta }_{\mathit{r},{\mathit{r}}^{\prime }}{\left({\mathit{b}}^T\mathit{\rho }({\mathit{r}}^{\prime })\right)}_i},$$ (16)

we sum the repeated indices in this expression from i = 1…M. Here, ${\mathcal{Z}}_{\text{ideal}}$ is the ideal gas partition function, and ${\mathcal{Z}}_{\text{C}}$ is the contribution from the charged part of the Hamiltonian, which is discussed later.

We use a Hubbard–Stratonovich transform to integrate out the spatial density, which for a generic pair potential u takes the form

$$\begin{array}{ll}\hfill & e^{-\frac{{\gamma }^2\beta }{2}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }\rho (\mathit{r})u(\mathit{r}-{\mathit{r}}^{\prime })\rho ({\mathit{r}}^{\prime })}\hfill \\ \hfill & =\frac{\int \mathcal{D}\omega e^{-\frac{1}{2\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }\omega (\mathit{r})u^{-1}(\mathit{r}-{\mathit{r}}^{\prime })\omega ({\mathit{r}}^{\prime })-\gamma {\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\omega (\mathit{r})\rho (\mathit{r})}}{\int \mathcal{D}\omega e^{-\frac{1}{2\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }\omega (\mathit{r})u^{-1}(\mathit{r}-{\mathit{r}}^{\prime })\omega ({\mathit{r}}^{\prime })}}.\hfill \end{array}$$ (17)

For each of B_i, we carry out a Hubbard–Stratonovich transform, and by making the substitution back into the partition function, we obtain

$$\begin{array}{ll}\hfill \mathcal{Z}& ={\mathcal{Z}}_{\text{ideal}}{\mathcal{Z}}_{\text{C}}\int \mathcal{D}{\left({\mathit{b}}^T\mathit{\rho }\right)}_i\hfill \\ \hfill & \times \frac{\int \mathcal{D}{\omega }_i{e}^{-\frac{1}{2\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }{\omega }_i(\mathit{r})\frac{1}{B_i}\delta (\mathit{r}-{\mathit{r}}^{\prime }){\omega }_i({\mathit{r}}^{\prime })-{\gamma }_i{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\omega }_i(\mathit{r}){\left({\mathit{b}}^T\mathrm{\Gamma }*\mathit{\rho }(\mathit{r})\right)}_i}}{\int \mathcal{D}{\omega }_i{e}^{-\frac{1}{2\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }{\omega }_i(\mathit{r})\frac{1}{B_i}\delta (\mathit{r}-{\mathit{r}}^{\prime }){\omega }_i({\mathit{r}}^{\prime })}}.\hfill \end{array}$$ (18)

Using the constraint imposed by the δ-function and moving the convolution from the density to the conjugate field, which is valid because the convolution is only a function of r, we arrive at

$$\begin{array}{ll}\hfill \mathcal{Z}& ={\mathcal{Z}}_{\text{ideal}}{\mathcal{Z}}_{\text{C}}\times \int \mathcal{D}{\left({\mathit{b}}^T\mathit{\rho }\right)}_i\hfill \\ \hfill & \times \frac{\int \mathcal{D}{\omega }_i{e}^{-\frac{1}{2B_i\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\omega }_i{(\mathit{r})}^2-{\gamma }_i{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{\Gamma }*{\omega }_i(\mathit{r}){\left({\mathit{b}}^T\mathit{\rho }(\mathit{r})\right)}_i}}{\int \mathcal{D}{\omega }_i{e}^{-\frac{1}{2B_i\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\omega }_i{(\mathit{r})}^2}}.\hfill \end{array}$$ (19)

This equation can be further simplified by making a Wick rotation μ_i = γ_iω_i to give the final result,

$$\begin{array}{ll}\hfill \mathcal{Z}& ={\mathcal{Z}}_{\text{ideal}}{\mathcal{Z}}_{\text{C}}\prod _{i=1}^M\int \mathcal{D}{\left({\mathit{b}}^T\mathit{\rho }\right)}_i\hfill \\ \hfill & \times \frac{\int \mathcal{D}{\mu }_i{e}^{-\frac{{\gamma }_i^2}{2B_i\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\mu }_i{(\mathit{r})}^2-{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{\Gamma }*{\mu }_i(\mathit{r}){\left({\mathit{b}}^T\mathit{\rho }(\mathit{r})\right)}_i}}{\int \mathcal{D}{\mu }_i{e}^{-\frac{{\gamma }_i^2}{2B_i\beta }{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\mu }_i{(\mathit{r})}^2}}.\hfill \end{array}$$ (20)

C. Charge–charge interactions

We now compute the contribution to the partition function from charge–charge interactions,

$${\mathcal{Z}}_{\text{C}}=\int \mathcal{D}{\rho }_{\text{C}}{e}^{\beta U_{\text{C}}}=\int \mathcal{D}{\rho }_{\text{C}}{e}^{\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }\mathrm{\Gamma }*{\rho }_{\text{C}}(\mathit{r})\frac{E}{|\mathit{r}-{\mathit{r}}^{\prime }|}\mathrm{\Gamma }*{\rho }_{\text{C}}({\mathit{r}}^{\prime })}.$$ (21)

We again use a Hubbard–Stratonovich transform, noting that the functional inverse of $\frac{E}{|\mathit{r}-{\mathit{r}}^{\prime }|}$ is $\delta (\mathit{r}-{\mathit{r}}^{\prime })\frac{{\nabla }^2}{E}$, to give

$${\mathcal{Z}}_{\text{C}}=\int \mathcal{D}{\rho }_{\text{C}}\frac{\int \mathcal{D}\phi e^{-\frac{1}{2E}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }\phi (\mathit{r})\delta (\mathit{r}-{\mathit{r}}^{\prime }){\nabla }^2\phi ({\mathit{r}}^{\prime })-i{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\phi (\mathit{r})\mathrm{\Gamma }*{\rho }_{\text{C}}(\mathit{r})}}{\int \mathcal{D}\phi e^{-\frac{1}{2E}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{r}}^{\prime }\phi (\mathit{r})\delta (\mathit{r}-{\mathit{r}}^{\prime }){\nabla }^2\phi ({\mathit{r}}^{\prime })}}.$$ (22)

We contract over the delta function, make a Wick rotation, and move the convolution to obtain

$${\mathcal{Z}}_{\text{C}}=\int \mathcal{D}{\rho }_{\text{C}}\frac{\int \mathcal{D}\phi e^{-\frac{1}{2E}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\nabla }^2\phi {(\mathit{r})}^2-i{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathrm{\Gamma }*\phi (\mathit{r}){\rho }_{\text{C}}(\mathit{r})}}{\int \mathcal{D}\phi e^{-\frac{1}{2E}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\nabla }^2\phi {(\mathit{r})}^2}}.$$ (23)

Hence, the partition function involves an integral over the field φ. We can take the integral over ρ and ρ_C to simplify our equation and give us the single chain partition function,

$$\mathcal{Z}={\mathcal{Z}}_{\text{ideal}}{\mathcal{Z}}_{\text{C}}\prod _{i=1}^M{\mathcal{Z}}_i={\mathcal{Z}}_{\text{ideal}}\int \mathcal{D}\phi \prod _{i=1}^M\int \mathcal{D}{\mu }_i{e}^{-H[\{{\mu }_i\},\phi ]}$$ (24)

with the field theoretic Hamiltonian,

$$\begin{array}{ll}\hfill H[\{{\mu }_i\},\phi ]& =\sum _{i=1}^M\frac{{\gamma }_i^2}{2B_i}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\mu }_i^2(\mathit{r})+\frac{1}{2E}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}|\nabla \phi (\mathit{r}){|}^2\hfill \\ \hfill & -\sum _{j=1}^{P+S+2}{n}_j\log Q_j[\{{\mu }_i\},\phi ].\hfill \end{array}$$ (25)

The equations we obtained are written in the diagonal basis of the chemical potential fields μ_i, each of which couples to a linear combination of polymer species. In the basis of the enumerated species,

$$\mathit{\psi }(\mathit{r})=\mathit{b}\mathit{\mu }(\mathit{r})+\mathit{Z}\phi (\mathit{r}).$$ (26)

Our model assumes no substantive FH interactions among the salt species, so ψ_S = Z_Sφ(r).

The single particle partition function for species j is

$$Q_j=\frac{1}{V}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{e}^{-\mathrm{\Gamma }*{\psi }_j(\mathit{r})/N}$$ (27)

with corresponding density

$${\rho }_j(\mathit{r})=\frac{\partial \log Q_j}{\partial {\psi }_j(\mathit{r})}=\frac{C_j}{Q_j}{e}^{-\mathrm{\Gamma }*{\psi }_j(\mathit{r})/N},$$ (28)

where concentration C_j ≔ n_j/V.

For the polymer species, we use the single chain partition function,

$$Q_j=\frac{1}{V}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{q}_j\left(\frac{N_j}{N},\mathit{r}\right),$$ (29)

where q_j evolves according to the modified diffusion equation (MDE),

$$\frac{\partial q_j(s,\mathit{r})}{\partial s}={\nabla }^2{q}_j(s,\mathit{r})-\mathrm{\Gamma }*{\psi }_j(s,\mathit{r})q_j(s,\mathit{r}),$$ (30)

and ψ_j(s, r) is a position-dependent field corresponding to the monomer type parametrically at position s along the curve of the polymer. We solve the MDE with the following initial condition:

$$q_j(0,\mathit{r})=1.$$ (31)

The corresponding density is

$${\rho }_m(\mathit{r})=\frac{C_j}{Q_j}{\int }_0^{N_j/N}\mathrm{d}s{q}_j(s,\mathit{r})q_j^{†}(s,\mathit{r})\delta (m-m(s)).$$ (32)

The adjoint $q_j^{†}(s,\mathit{r})$ is defined analogously to q_j(s, r), although starting at the opposite polymer end. That is, $q_j^{†}(s,\mathit{r})$ is the solution of

$$\begin{array}{c}-\frac{\partial q_j^{†}(s,\mathit{r})}{\partial s}={\nabla }^2{q}_j^{†}(s,\mathit{r})-\mathrm{\Gamma }*{\psi }_j(s,\mathit{r})q_j^{†}(s,\mathit{r}),\\ q_j^{†}\left(\frac{N_j}{N},\mathit{r}\right)=1.\end{array}$$ (33)

If a monomer type exists in multiple polymers or solvents, then we can simply calculate the amount each species contributes to that monomer’s density and sum to get the total density.

Previous derivations stop at this point, but we can analytically solve the homogeneous part of the free energy to avoid having to iteratively update the average value of the fields. The average field values do not affect all observables, but the free energy and chemical potential do depend on them. While the homogeneous solution can be approximated during the complex Langevin update scheme, we opt to instead solve exactly the homogeneous component and set the average field value for all fields to zero. Conveniently, because of charge conservation ⟨φ⟩ = 0, there is no electrostatic free energy for the homogeneous case and no need to add any analytic term. The homogeneous solution does contribute to the free energy for the FH terms, so we define a new vector $\mathit{c}=\frac{1}{V}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}\mathit{\rho }$. We can use the normal definition of FH interaction and the trivial solution for the homogeneous case to add the proper analytic correction. Using this correction, we can constrain all the fields s.t. ⟨μ_i⟩ = 0; the Hamiltonian becomes

$$\begin{array}{ll}\hfill H[\{{\mu }_i\},\phi ]& =\sum _{i=1}^M\frac{{\gamma }_i^2}{2B_i}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\mu }_i^2(\mathit{r})+\frac{1}{2E}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}|\nabla \phi (\mathit{r}){|}^2\hfill \\ \hfill & -\sum _{j=1}^{P+S+2}{n}_j\log Q_j[\{{\mu }_i\},\phi ]+\frac{V{\mathit{c}}^T\mathit{\chi }\mathit{c}}{2}.\hfill \end{array}$$ (34)

When comparing different simulations with the same concentration and interaction strengths, the last term can be ignored.

D. Numerical evaluation of the field theoretic Hamiltonian

Equilibrating the system requires repeated computation of the density of all species, which involves solving the pair of MDEs (30) and (33). Efficient numerical schemes for such PDEs are both well-studied and widely used.³⁷ We employ standard integration schemes using a splitting scheme together with the pseudospectral method.

The splitting scheme we use alternates updates of the linear part of the MDE with expensive evaluations of the Laplacian operator,

$$q(s+\mathrm{\Delta }s,\mathit{r})=e^{-\frac{\psi (s,\mathit{r})}{2}\mathrm{\Delta }s}{e}^{{\nabla }^2\mathrm{\Delta }s}{e}^{-\frac{\psi (s,\mathit{r})}{2}\mathrm{\Delta }s}q(s,\mathit{r})+O(\mathrm{\Delta }{s}^3).$$ (35)

The operator $e^{{\nabla }^2\mathrm{\Delta }s}$ is calculated in Fourier space, which is the diagonal basis for the Laplacian. We use a method based on Richardson extrapolation⁴⁰ that has a fourth order error. We calculate two estimates of q_j(s + Δs, r), one with a single step of Δs and one with two steps of Δs/2, and use the extrapolation,

$$q(s+\mathrm{\Delta }s,\mathit{r})=\frac{4q_{\mathrm{\Delta }s/2}(s+\mathrm{\Delta }s,\mathit{r})-q_{\mathrm{\Delta }s}(s+\mathrm{\Delta }s,\mathit{r})}{3},$$ (36)

to obtain an approximation that is O(Δs⁴) with three evaluations of the MDE.

E. Field theoretic simulation

Because coacervation arises due to fluctuations, mean-field approximations do not capture the underlying physics accurately. The mean-field solution for a soluble polyampholyte is always a homogeneous charge neutral solution, inconsistent with the phenomenology of coacervation. Monte Carlo methods and complex Langevin are both options for sampling fluctuations in the local density of each species. Building on many previous successful applications,^4,15,37 we have opted to use complex Langevin.

While it is well-known that complex Langevin lacks rigorous theoretical foundations and strong convergence guarantees, numerical results in the polymer field literature are in good agreement with both analytical and experimental results. In principle, the complex Langevin algorithm is relatively straightforward, essentially coupling gradient descent to a noise term. The main difference is that for complex Langevin, all of the chemical potential fields are promoted to being fully complex, allowing sampling and descent over both parts of the plane. The noise injected during sampling can be either complex or real, but in this case, the noise will all be purely real before Wick rotation. Previous work²⁴ has shown that any choice of noise should correctly sample the distribution, but the choice to only use real noise has a physical interpretation. For each field, sampling over the real axis is the goal, but it is generally easiest to do so by finding a saddle point at a different point in imaginary space and using Cauchy’s theorem to verify that this gives the same result as the real integral. As such, it is natural to expect that the real goal is to sample the real axis and condition this sampling on finding the corresponding maximum value of the imaginary axis. This regime corresponds directly to injecting noise only into the real part of the field. This approach is known as the “standard” complex Langevin and has been chosen here consistent with previous work.²⁴ With that in mind, we will write down the general form of the complex Langevin, where we have promoted (ω_i, φ) to be complex fields such as z = w + iv, where w and v are both purely real,

$$\frac{\partial }{\partial t}\mathbf{w}(t)=-\lambda \mathrm{R}\mathrm{e}\left[\frac{\partial H(\mathit{z}(t))}{\partial \mathit{z}}\right]+\mathit{\eta }(t),$$ (37)

$$\frac{\partial }{\partial t}\mathbf{v}(t)=-\lambda \mathrm{I}\mathrm{m}\left[\frac{\partial H(\mathit{z}(t))}{\partial \mathit{z}}\right].$$ (38)

t is a fictitious time variable constructed for sampling purposes, and λ is a time step scale. In practice, the actual equations are a series of vector equations, where we can write the force on all fields as

$$\mathbf{F}(\mathit{\mu }(t,\mathit{r}))=\frac{\partial H[\mathit{\mu }(\mathit{r}),\phi (\mathit{r})]}{\partial \mathit{\mu }(t,\mathit{r})}=\left(\frac{{\mathit{\gamma }}^2}{\mathit{B}}\right)\odot \mathit{\mu }(t,\mathit{r})-{\mathit{b}}^T\left(\mathrm{\Gamma }\ast \mathit{\rho }\right)(t,\mathit{r})$$ (39)

and

$$F(\phi (t,\mathit{r}))=\frac{\partial H[\mathit{\mu }(\mathit{r}),\phi (\mathit{r})]}{\partial \phi (t,\mathit{r})}=\frac{1}{E}{\nabla }^2\phi (\mathit{r})-{\rho }_{\text{C}}(\mathit{r}),$$ (40)

where ⊙ represents the Hadamard product. We can also define the Fourier transformed forces, which avoid calculating convolutions and gradients directly as

$$\begin{array}{ll}\hfill \widehat{\mathbf{F}}(\mathit{\mu }(t,\mathit{k}))& =\mathcal{F}\left(\frac{\partial H[\mathit{\mu }(\mathit{r}),\phi (\mathit{r})]}{\partial \mathit{\mu }(t,\mathit{r})}\right)=\left(\frac{{\mathit{\gamma }}^2}{\mathit{B}}\right)\odot \widehat{\mathit{\mu }}(t,\mathit{k})-{\mathit{b}}^T\widehat{\mathit{\rho }}(t,\mathit{k})\hfill \end{array}$$ (41)

and

$$\widehat{\mathbf{F}}(\phi (t,\mathit{k}))=\mathcal{F}\left(\frac{\partial H[\mathit{\mu }(\mathit{r}),\phi (\mathit{r})]}{\partial \phi (t,\mathit{r})}\right)=\frac{1}{E}{\mathit{k}}^2\widehat{\phi }(\mathit{k})-{\widehat{\rho }}_{\text{C}}(\mathit{k}).$$ (42)

We can finally write down the update step for a first-order Euler–Maruyama as

$$\frac{\partial \mathit{\mu }(t,\mathit{k})}{\partial t}=-{\mathit{\lambda }}_{\mu }\widehat{\mathbf{F}}(\mathit{\mu }(t,\mathit{k}))+\mathit{\gamma }\odot {\widehat{\mathit{\eta }}}_{\mu }(t,\mathit{k}),$$ (43)

$$\frac{\partial \phi (t,\mathit{k})}{\partial t}=-{\lambda }_{\phi }\widehat{\mathbf{F}}(\phi (t,\mathit{k}))+i{\widehat{\eta }}_{\phi }(t,\mathit{k}).$$ (44)

Because some of our fields have been Wick rotated, their corresponding noise is rotated in the same way. The noise terms are Fourier transforms of Gaussian white noise with the following statistics:

$$\begin{array}{c}〈{\eta }_{i\mu }(t,\mathit{r})〉=〈{\eta }_{\phi }(t,\mathit{r})〉=0,\\ 〈{\eta }_{i\mu }(t,\mathit{r}){\eta }_{i^{\prime }\mu }(t^{\prime },{\mathit{r}}^{\prime })〉=\frac{2{\lambda }_{i\mu }{\beta }_{i\mu }}{\mathrm{\Delta }V}{\delta }_{i,i^{\prime }}{\delta }_{t,t^{\prime }}{\delta }_{\mathit{r},{\mathit{r}}^{\prime }},\\ 〈{\eta }_{\phi }(t,\mathit{r}){\eta }_{\phi }(t^{\prime },{\mathit{r}}^{\prime })〉=\frac{2{\lambda }_{\phi }{\beta }_{\phi }}{\mathrm{\Delta }V}{\delta }_{t,t^{\prime }}{\delta }_{\mathit{r},{\mathit{r}}^{\prime }}.\end{array}$$ (45)

β_iμ and β_φ are the temperatures corresponding to each field. They are not without physical consequences on the distribution reached, especially in the case of coacervation.

In practice, the system in the Fourier basis is very stiff, and EM1 or any Runge–Kutta method usually requires prohibitively small time steps to be practical. Previous studies³⁷ have examined other integration schemes, and exponential time differencing (ETD) has generally proven to be the preferred method for integrating these equations. The general form for ETD is

$$\widehat{w}(t+\mathrm{\Delta }t)=\widehat{w}(t)-\frac{1-e^{-\lambda \mathrm{\Delta }tc(\mathit{k})}}{c(\mathit{k})}\widehat{F}(\mathbf{w}(t))+{\left(\frac{1-e^{-2\lambda \mathrm{\Delta }tc(\mathit{k})}}{2\lambda \mathrm{\Delta }tc(\mathit{k})}\right)}^2\widehat{\eta }(t),$$ (46)

where

$$c(\mathit{k})\equiv \frac{\partial \widehat{F}(\mathit{k})}{\partial \widehat{w}(\mathit{k})}{\left.\right|}_{\widehat{w}(\mathit{k})=0}.$$ (47)

This equation uses all the same variables as the EM scheme listed above. c(k) is somewhat troublesome as it requires an analytical approximation to properly rescale the relaxation rates for each field. In practice, we only need to know the partial differential between density and fields because the other components of force are trivially solvable. The full derivation is a tedious extension of previous work,⁴¹ but the final result is

$$\frac{\partial \rho (\mathit{k},[\widehat{\mu }])}{\partial {\widehat{\mu }}_i(\mathit{k})}{\left.\right|}_{\{\widehat{\mu }(\mathit{k})\}=0}={\gamma }_i^{2}C\sum _j{\widehat{g}}_{Ij},$$ (48)

$${\widehat{g}}_{ii}=-{\left(f_{i+1}-f_i\right)}^2{g}_D\left(\left(f_{i+1}-f_i\right)k^2\right),$$ (49)

$${\widehat{g}}_{Ij}=\frac{1}{k^4}\left(e^{-|f_{i+1}-f_{j+1}|k^2}-e^{-|f_{i+1}-f_j|k^2}\right.$$ (50)

$$\left.+e^{-|f_i-f_j|k^2}-e^{-|f_i-f_{j+1}|k^2}\right),$$ (51)

where ${\widehat{g}}_D$ is the Debye function,

$${\widehat{g}}_D(k^2)=\frac{2}{k^4}\left(e^{-k^2}+k^2-1\right).$$ (52)

The values {f₀, f₁, …, f_n+1} are the set of break points along a polymer with n blocks where a block of one monomer type gives way to another. By convention, f₀ = 0 and f_n+1 = N for all polymers.

To validate our implementation of the field theoretic simulation algorithms, we made an extended comparison to results on coacervation from Ref. 15 in Appendix A. We compared the chemical potential for a diblock polyampholyte to their results by deriving the chemical potential operator and computing it using partition functions obtained from field theoretic simulation. In addition, we used the Gibbs ensemble approach to determine the phase coexistence line between the supernatant and coacervate phases in this system, which required deriving the osmotic pressure operator.

F. Soft-explicit solvent

The system under consideration requires balancing attractive FH interactions with the electrostatic interactions that drive coacervation in a unified model, a case that has not been previously considered in the literature. Reference 15 studied coacervation in a model with charge–charge interactions but purely repulsive FH interactions and an implicit solvent. Other works^28,31 have opted to use density-explicit field theories, which easily handle this problem but are much less numerically unstable for complex Langevin.²³ An interesting technical challenge arises because implicit treatment of the solvent field requires that all FH interactions be repulsive after diagonalization of the fields.³⁶ Due to the attractive interactions among the components in our system, this is not possible, and only an explicit solvent can be used. However, the addition of a field that preserves the total density at every location in space also requires an infinite FH parameter,³⁶ which makes the system undesirably stiff.

This issue motivates an alternative approach that replaces the hard constraint on the density with a weaker one, which we refer to as a soft-explicit model.³⁸ This relaxed model can be interpreted as an explicit solvent model where the total density is only weakly enforced, with the degree of enforcement being controlled by the self-interaction FH parameter (B_ii). In the limit of all B_ii → ∞, there is no penalty on the field of the total volume, and the hard solvent behavior is reestablished. An alternative interpretation is that the simulation now includes one additional solvent species, an extra implicit co-solvent that fills the extra space left in the uneven density. In this case, the B_ii → ∞ limit can be interpreted as the implicit solvent being infinitely repulsive to all other species and, thus, enforcing total density. This approach allows us to simultaneously incorporate attractive interactions and charge–charge interactions, expanding the scope of field theoretic simulation to other biomaterial systems (Fig. 1).

FIG. 1.

Description of system composition and schematic of mixture in real space. (a) CART preparation and molecular structure containing an initiator (R1) lipophilic carbonate block with a lipid side chain (R2) and a hydrophilic cationic block derived from N-hydroxyethyl α-amino acids (R3 for amino acid side chain). (b) Sequence of all components (L: lipid block, A: cationic hydrophilic block, and B: anionic hydrophobic block). (c) Schematic showing a mixture of solvent, salt, and polymers.

III. RESULTS AND DISCUSSION

A variety of experimental measurements, including dynamic light scattering (DLS)¹² and ζ-potential measurements,⁴ show that nanoparticles made with CARTs and similar polymeric materials form stable, predominantly spherical structures.⁴² These measurements, however, do not provide detailed insight into the microstructure (i.e., the spatial organization of the components within the nanoparticles). Understanding the connection between polymer properties and microstructure would be helpful in making predictions about nanoparticle stability. As shown in Fig. 2, our model predicts a lamellar morphology for assembled nanoparticles in a range of conditions. Recent cryoEM data also support this viewpoint.

FIG. 2.

Morphological dependence on salt concentration and hydrophobicity: (a) Phase diagrams with parameter cutoffs for homogeneous: O_lipid < 1.15 and 1 < O_ionic < 1.0016, lipid-core: O_ionic < 1.001, coacervate: O_lipid < 2.4, and lamellar: all others. (b) Representative lipid-core phase. (c) Coacervate-core phase. (d) Lamellar phase.

We choose to examine morphology along a range of salt concentrations (C_s) and lipid hydrophobicity (B). Salt concentrations are experimentally variable, and FH parameters are difficult to determine experimentally, so we choose to examine how behavior changes as they are varied. These two parameters essentially control the two possible driving forces for assembly (hydrophobicity and coacervation), so by changing them, we can see how morphology is dependent on each effect. If we arrange the species in the order (Fig. 3)

$$\mathit{\rho }=\left[\begin{array}{cccc}\hfill {\rho }_{+}\hfill & \hfill {\rho }_{-}\hfill & \hfill {\rho }_{\text{s}}\hfill & \hfill {\rho }_{\text{l}}\hfill \end{array}\right],$$ (53)

FIG. 3.

Two order parameters used to characterize the different morphologies. (a) Lipid density order parameter. (b) Charged species order parameter.

then the FH matrix for the system is

$$\mathit{\chi }=\left[\begin{array}{cccc}\hfill T\hfill & \hfill T\hfill & \hfill T\hfill & \hfill T+B\hfill \\ \hfill T\hfill & \hfill T\hfill & \hfill T\hfill & \hfill T+B\hfill \\ \hfill T\hfill & \hfill T\hfill & \hfill T\hfill & \hfill T+B\hfill \\ \hfill T+B\hfill & \hfill T+B\hfill & \hfill T+B\hfill & \hfill T\hfill \end{array}\right].$$ (54)

Because all the non-lipid species have identical interaction profiles, the matrix is degenerate. We can write down the non-degenerate form as

$$\overline{\mathit{\chi }}=\left[\begin{array}{cc}\hfill T\hfill & \hfill T+B\hfill \\ \hfill T+B\hfill & \hfill T\hfill \end{array}\right],$$ (55)

for which the corresponding density ordering is

$$\tilde{\mathit{\rho }}=\left[\begin{array}{cc}\hfill \{{\rho }_{+},{\rho }_{-},{\rho }_{\text{s}}\}\hfill & \hfill {\rho }_{\text{l}}\hfill \end{array}\right].$$ (56)

This description elucidates the structure of the interactions—there is a total interaction term denoted T that controls the strength of the penalty associated with the total density, and there is a variable term B that controls the strength of the hydrophobic interaction. We set T = 1.5, which provided enough stiffness to restrain density variation. To recapture hard explicit models such as those in,³⁶ we could take T → ∞, although the simulation degrades in stability for large values of T. The parameter B is modulated by the choice of the lipid in the CART polymer, and we discuss its implications in detail below.

A. Microphase structure and phase diagram

Examining the morphology as we modify B and C_s, we see four distinct morphologies in Fig. 2. These were distinguished with two order parameters,

$$\begin{array}{c}{O}_{\text{lipid}}(B,C_{\text{s}})=\frac{1}{V}{\int }_{\mathrm{\Omega }}{\rho }_{\text{l}}\frac{3{\rho }_{\text{l}}+\delta }{{\rho }_{\text{l}}+{\rho }_{+}+{\rho }_{-}+\delta }\mathrm{d}\mathit{r},\\ O_{\text{ionic}}(B,C_{\text{s}})=\frac{1}{V}{\int }_{\mathrm{\Omega }}\left({\rho }_{+}+{\rho }_{-}\right)\exp \left(-\frac{1+\delta }{|{\rho }_{+}-{\rho }_{-}|+\delta }\right)\mathrm{d}\mathit{r},\end{array}$$ (57)

in this equation, we use δ = 10⁻⁴ to regularize fluctuations in the density that arise during complex Langevin simulation. These two parameters distinguish the separation between the lipid and charged species. The lipid parameter measures the relative density of the lipid compared to all species and will increase as the lipid aggregates. The ionic parameter measures the degree to which the positive and negative charges are separated; larger values correspond to salt fields strongly overlapping. Both are scaled by the local density of the species of interest to better correlate their effects with their representative species. We chose cutoffs such that the morphologies were distinguished as clearly as possible, although the transition is continuous, so the boundaries are somewhat arbitrary.

These parameters can be used to distinguish between four morphologies: one homogeneous and three condensed spatial arrangements that are connected by continuous transitions. The main difference between the separated phases is in the structure of the dense phase. In regions of high-salt and hydrophobic lipids, lipid segregation is the driving factor. Lipid segregation creates lipid-core micelles with a corona of the bound cation and a second corona of mRNA that is weakly attracted to the cationic block. In cases where the lipid is weakly hydrophobic and the salt concentration is low, we see phase separation driven by coacervation where the coacervate forms a dense phase with a corona of the bound lipid. For cases when there is low salt and hydrophobic lipid, both coacervation and hydrophobicity are driving forces, so we have a lamellar phase where both dense phases are in contact with the solvent. There is no coexistence between the lipid-core, lamellar, and coacervate phases because they require different conditions to exist, and they exist in a continuum as the parameters are varied. Even though smooth change between them is possible, they still represent distinctive morphologies because they have varied structure factors, real densities, and driving forces.

We also computed the structure factor for each species,

$$S(\mathit{q})=\frac{1}{V}{\int }_V{e}^{-i\mathit{q}\cdot (\mathit{r}-{\mathit{r}}^{\prime })}\left(⟨\rho (\mathit{r})\rho ({\mathit{r}}^{\prime })⟩-⟨\rho (\mathit{r})⟩⟨\rho ({\mathit{r}}^{\prime })⟩\right)\mathrm{d}\mathit{r}\mathrm{d}{\mathit{r}}^{\prime },$$ (58)

which provides insight into relevant length scales of features associated with each component. The structure factors, shown in Fig. 4, evince differences between these three morphologies. The lipid core nanoparticle is the most distinctive, with a series of peaks well correlated with each of the three distinct nanoparticle morphologies. The spacing of these peaks depends on the micelle density, but the cation and mRNA have significantly decreased long-range correlation, which is characteristic of the phase. Both of the other two morphologies have a significant long-range correlation of the charged species that is indicative of coacervation. The main distinguishing feature between the lamellar phase and the coacervate-core phase is the presence of a shorter range shoulder for the lipid–lipid correlation. This is only present in the lamellar phase because the coacervate core phase has a significantly less dense and organized lipid region. In interpreting these structure factors, long range correlations are less reliable as they start to be affected by aberrations from the periodic box.

FIG. 4.

Comparison of structure factor for a representative member of each morphology. (a) Structure factor for the coacervate phase. (b) Structure factor for the lamellar phase (note higher correlation at low q values for lipid). (c) Structure factor for lipid-core phase.

B. Experimental cryoEM

We conducted cryoEM imaging of CART-mRNA nanoparticles to elucidate their microstructure and compared it with the field theoretic simulations we conducted. We prepared a diblock copolymer with a combination of MTC-dodecyl carbonate for the hydrophobic block and N-hydroxyethyl glycine-derived α-amino ester for the charged cationic block. This diblock copolymer was combined with mRNA in a +/− charge ratio of 10:1 in 1 × PBS buffer at pH = 5.5, upon which rapid formation of nanoparticles was observed.⁴ The nanoparticles were then imaged using cryoEM to assess the organization within a nanoparticle. The structures shown in Fig. 5 show clear lamellar ordering, a feature also observed in simulations with low salt and high hydrophobicity.

FIG. 5.

(a) Top: representative cryoEM micrograph of a CART/mRNA NP with a +/− ratio of 10/1. Below is the chemical structure of the CART, a dodecyl lipid carbonate block with a N-hydroxyethyl glycine-derived cationic block. (b) Simulated CART-mRNA mixture with higher polymer density (63% polymer by mass) at conditions corresponding to lamellar phase. (c) Line profile for the green region in the experimental image. (d) Line profile from the simulated mixture. Images are rescaled pixel intensity of the microscope image or the underlying densities.

There are a few notable differences between the experimental system and simulation due to simulation constraints. The size of a full nanoparticle is larger than the simulations that we can presently conduct. Nevertheless, the mesoscale structure is well captured by our model. Experimentally, the mRNA is significantly longer than the cationic block of the CART polymer; however, the numerical stability of our field theoretic simulation degrades as the individual chains become long. We did not observe significant morphological changes in the morphology for increasing mRNA block length in simulations stable enough to converge. With this in mind, we chose the mRNA length to match the cationic block length, which provided high stability across conditions for the phase diagram Fig. 2. Nevertheless, the model accurately captures clear lamellar structure in the CART-mRNA nanoparticles, resolving an open question about the internal arrangement of components in this model.

IV. CONCLUSIONS

To determine the possible mesostructures for mRNA-CART nanoparticles, we built a complete and descriptive field theoretic model that incorporates hydrophobicity, explicit solvent and salt effects, and charge–charge interactions. The complexity of this model is a barrier to analytical treatment, so we relied on field theoretic simulation to compute a phase diagram that depends only on the generic and controllable parameters of these materials. Our results agree qualitatively with experimentally determined cryoEM, broadly showing lamellar mesoscale organization of nanoparticle contents. Furthermore, our results demonstrate that the constituent lipids strongly inform the morphology, which we believe will have implications for nanoparticle thermostability and efficacy as a delivery vehicle. In particular, we hypothesize that intermediate hydrophobicity will be advantageous for delivery in CART-based nanoparticles because the coacervate phase is best protected from the solvent under conditions such as those depicted in Fig. 2(c). We hope to experimentally test these suggested design principles with assays focused on mRNA expression levels in live cells.

The model that we have developed is highly coarse-grained, which does place some inherent limitations on the conclusions that we can draw. In particular, treatment of the microscopic molecular properties of the polymers, such as their variable rigidities, is not possible within the free Gaussian chain formalism typically used for field theoretic simulation. In addition, this work is limited to linear polymers, although there are some suggestions that more complex architecture may yield more effective CART nanoparticles.⁴³ In future work, we may extend our code to branched polymers.

Because the driving forces balance coacervation and hydrophobicity, just as in biomolecular materials, including intrinsically disordered proteins, the toolkit developed here may be useful for studying so-called membraneless organelles.⁴⁴ Furthermore, the lack of free and open-source tools for advanced simulation techniques using field theoretic approaches has been a barrier to the adoption of these techniques.⁴⁵ Our codebase is freely available and open-source and should serve to promote future work in field theoretic simulation.

V. METHODS: EXPERIMENTAL PREPARATION OF CARTS

CART/mRNA nanoparticle preparation: A dodecyl-glycine CART was synthesized according to the literature precedent.⁴ CART/mRNA nanoparticles were formulated by mixing 17.72 μl of a PBS buffer at pH = 5.5 with 1 μl of a 1 mg/ml Fluc RNA solution (Trilink) and then adding 1.28 μl of a stock solution of 2 mM CART to the resulting solution to end up with a +/− charge ratio of 10/1. The solution was mixed for 20 s and immediately vitrified.

Cryogenic-electron microscopy (cryoEM) Grid Preparation and Imaging: 3 μl of the CART/mRNA solution was applied to a Quantifoil R2/1 grid with a 2 nm ultrathin carbon backing. Prior to loading, grids were glow discharged for 15 s at 10 mA to increase hydrophilicity. Vitrification was carried out by Vitrobot (ThermoFisher). Grid preparation was performed at 100% humidity, and the grids were blotted for 3 s at a blot force setting of two prior to plunging into liquid ethane. CryoEM samples were clipped and then imaged on a ThermoFisher Glacios CryoEM. Images were recorded using SerialEM software in low dose mode with a K3 camera with an exposure time of 3 s and a defocus of −3 μm.

VI. METHODS: NUMERICAL ALGORITHMS FOR COMPLEX LANGEVIN SIMULATION

Scripts to run all numerical experiments in this paper are available at https://github.com/rotskoff-group/polycomp.git. These include full parameter sets and documentation. We benchmarked our code extensively against analytical results and previously published numerical results for coacervation that employ field theoretic simulation.¹⁵ Details of these benchmarks are given in Appendix A.

While the best source of reference for replication remains our example code base, we are also providing simulation parameters here for completeness. The primary simulations are those used to generate the plot in Fig. 2(a), so unless otherwise specifically noted, these parameters refer to those used in generating that figure. For those simulations, the system reference length is N = 5, and all lengths will be scaled by N. Our simulations include a diblock lipid–cation polymer where each block has a length of 1 and anionic mRNA with a length of 1. All charged species have a linear charge density of ±1 depending on whether they are anionic or cationic. The polymer integration uses an integration width of 1/30 so that a polymer with a length of 1 will have 30 integration points. Each polymer has a concentration of C = 0.3, and the solvent concentration is C = 1.5, which gives a total monomer density of 2.4 (not including salt). The salt concentration in the simulation is C_s × N, where C_s are the reported values. These values were selected so that both phases would be generally visible in a given simulation. As per the main text, the simulations use a total repulsion term of T = 1.5 and variable lipid hydrophobic interactions. The smearing constant for all species is 0.163.

The box size is 30 × 30 with 512 × 512 grid points. For the thin needle simulations, the box size is 120 × 7.5, and the grid size is 2048 × 64. The relaxation rate for all chemical potential fields is 1/dV, which is 0.0032 for these simulations. The relaxation rate for the charged field is 2.5/dV or 0.0080. For the chemical potentials, we use two temperatures to accelerate sampling, a hotter temperature of 3 × 10⁻³ and a colder temperature of 1 × 10⁻⁵. The temperature for the charged fields is 1. All of these temperatures are fictitious, although the charged temperature has physical consequences. The dielectric constant is uniform for the entire system and is E = 1 × 10⁴. This value was chosen to be large because the explicit salts are intended to provide screening for the system. Systems were equilibrated for 1.8 × 10⁵ steps, with the first 3.6 × 10⁴ conducted at the higher temperature. Density plots are generated by averaging over 6000 time steps, and the reported plots are always for the last 6000 steps.

The simulations run exclusively on GPUs, and various hardware was used to conduct the simulations. To give a general sense of the computational cost involved, we bench marked the performance on an NVIDIA A40 GPU, where the simulation described above took 12.4 h to complete. That configuration will also require ∼1 GB of RAM, which is not prohibitive but can offer a cap for the size of simulations as the system gets larger.

Updating the fields and evaluating the free energy both require evaluations of the single-chain partition function, which is the most computationally difficult part of the Hamiltonian to evaluate (30). This equation requires evaluating both Laplacian and linear terms, so it is solved by a pseudospectral decomposition where the Laplacian update is done in reciprocal space while the other updates are done in real space. To improve the accuracy of this operation, we use a fourth order Richardson extrapolation scheme, which requires only three evaluations of each step.

For the actual sampling, we use a complex Langevin scheme. This scheme builds trajectories in fictitious times in which the actual trajectory has no physical interpretation, but the statistical averages should converge to the correct values. Because field theoretic simulations aim to find a contour on the imaginary axis that has a local maximum for smooth integration, the sampling only happens on the real axis. Under this scheme, noise is injected into the real axis to sample it, while the imaginary axis relaxes to find local maxima that correspond to the states that are sampled over.

Our actual system updates happen in Fourier space. This leads to a stiff system, so we follow previous work and implement an exponential time differencing (ETD) scheme to update the fields for the complex Langevin time stepping. The ETD scheme works by analytically solving the linear portion of a derivative to give an estimate of the relaxation rate of each mode. With this estimate in hand, we can essentially rescale the time step for each mode of our Fourier space representation. This allows all the modes to relax efficiently because their effective time step matches their relaxation rate, avoiding the need for the slower modes to relax at the same rate as the faster modes. Here, we implement a first order ETD scheme, following previous work. Extensions to higher order ETD methods are possible but not explored at this point.⁴⁶

APPENDIX A: COMPARISON WITH DELANEY AND FREDRICKSON

To validate our implementation, we made direct comparisons with both previous simulations of coacervation in implicit solvent and analytical results.¹⁵ To make these comparisons, we implemented a chemical potential operator, an osmotic pressure operator, and the Gibbs ensemble. These operators are described in detail here before the direct comparisons are presented with previous results. We are including the derivation of these operators in completeness here because they are required to make the strongest comparisons to previous work and because our formulation includes minor differences from previous works that should be made explicit.

1. Free energy

The free energy has essentially already been defined, but we will explicitly express the ideal gas term so that we can take proper derivatives even between different system states. Doing so gives us the free energy functional,

$$\begin{array}{ll}\hfill F(T)& =kT\left(\right.\sum _j^{P+S+2}\left(-n_j\log z_{j0}-n_j\log V+n_j\log n_j-n_j\right)\hfill \\ \hfill & +{⟨H[\{{\mu }_i\},\phi ]⟩}_{\text{S}}\left.\right),\hfill \end{array}$$ (A1)

$$F(T)=kT\left(\sum _j\left(-n_j\log z_{j0}+n_j\log C_j-n_j\right)+{⟨H[\{{\mu }_i\},\phi ]⟩}_{\text{S}}\right).$$ (A2)

Here, the average ${⟨H[\{{\mu }_i\},\phi ]⟩}_{\text{S}}$ is the average over a well equilibrated simulation that approximates the infinite functional integral in question.

2. Chemical potential operator

The chemical operator is defined for each species of polymer, solvent, and salt as

$${\mu }_j=\frac{\partial F(T)}{\partial n_j},$$ (A3)

which requires solving

$$\begin{array}{ll}\hfill \frac{\partial H}{\partial n_j}& =\frac{\partial }{\partial n_j}\left(\sum _{i=1}^M\frac{{\gamma }_i^2}{2B_i}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}{\mu }_i^2(\mathit{r})+\frac{1}{2E}{\int }_{\mathrm{\Omega }}\mathrm{d}\mathit{r}|\nabla \phi (\mathit{r}){|}^2\right.\hfill \\ \hfill & \left.-\sum _{j=1}^{P+S+2}{n}_j\log Q_j[\{{\mu }_i\},\phi ]+\frac{V{\mathit{c}}^T\mathit{\chi }\mathit{c}}{2}\right).\hfill \end{array}$$ (A4)

The first two terms do not depend on this derivative at all, and the single species partition is trivial. The remaining analytic term is the only one that poses any difficulty. This is mainly because the vectors in that term are indexed via monomer type rather than species, so we have to make the expansion,

$$\frac{\partial \mathbf{c}}{\partial n_j}=\left[\begin{array}{c}\hfill \frac{\partial C_1}{\partial n_j}\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{\partial C_i}{\partial n_j}\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{\partial C_M}{\partial n_j}\hfill \end{array}\right]=\frac{1}{V}\left[\begin{array}{c}\hfill \frac{\partial n_1}{\partial n_j}\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{\partial n_i}{\partial n_j}\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{\partial n_M}{\partial n_j}\hfill \end{array}\right]=\frac{1}{V}\left[\begin{array}{c}\hfill \frac{{\kappa }_{j1}{N}_j}{N}\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{{\kappa }_{ji}{N}_j}{N}\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{{\kappa }_{jM}{N}_j}{N}\hfill \end{array}\right]=\frac{\vec{\kappa }}{V},$$ (A5)

where κ_ji is the fraction of the j-th species that is comprised of the i-th monomer type. If the j-th species is a solvent or a homopolymer with monomer type i, then κ_ji = 1 and all other κ_ji′ = 0. Similarly, if the j-th species is a diblock copolymer with equal fractions of species 1 and 2, then κ_j1 = κ_j2 = 1/2, with all other κ_ji′ = 0.

Now we can compute,

$$\frac{\partial }{\partial n_j}\frac{V{\mathbf{c}}^T\mathit{\chi }\mathbf{c}}{2}=\frac{V{\vec{\kappa }}^T\mathit{\chi }\mathbf{c}}{2V}+\frac{V{\mathbf{c}}^T\mathit{\chi }\vec{\kappa }}{2V},$$ (A6)

where because χ is symmetric, ${\vec{\kappa }}^T\mathit{\chi }\mathbf{c}={\mathbf{c}}^T\mathit{\chi }\vec{\kappa }$, and

$$\frac{\partial }{\partial n_j}\frac{V{\mathbf{c}}^T\mathit{\chi }\mathbf{c}}{2}={\vec{\kappa }}^T\mathit{\chi }\mathbf{c}.$$ (A7)

With this, we can write down the total free energy, where the only other terms are easy analytic derivatives of the free energy,

$${\mu }_j=\frac{\partial F(T)}{\partial n_j}=kT\left(-\log z_{j0}+\log C_j-{⟨\log Q_j[\{{\mu }_i\},\phi ]⟩}_{\text{S}}+{\vec{\kappa }}^T\mathit{\chi }\mathbf{c}\right).$$ (A8)

In practice the log z_j0 term is always the same between different simulations and will never contribute in a meaningful way to comparing states, so it will be ignored.

3. Pressure operator

Another operator that we will need to equilibrate our system is the osmotic pressure, which represents changes in the amount of volume in the system (and, thus, implicit solvent). This operator is generically defined as

$$\mathrm{\Pi }=-\frac{\partial F(T)}{\partial V}.$$ (A9)

This operator provides multiple complexities, as we shall soon see, but we can start with the relatively easier term associated with the homogenous field. We will start by noting that $\frac{\partial C_j}{\partial V}=-C_j/V$ then proceed to

$$\frac{\partial }{\partial V}\frac{V{\mathbf{c}}^T\mathit{\chi }\mathbf{c}}{2}=\frac{{\mathbf{c}}^T\mathit{\chi }\mathbf{c}}{2}-{\mathbf{c}}^T\mathit{\chi }\mathbf{c}=-\frac{{\mathbf{c}}^T\mathit{\chi }\mathbf{c}}{2}.$$ (A10)

a. Hamiltonian derivative

The Hamiltonian derivative is significantly more complicated. Most of this derivation will closely follow a previous derivation³⁷ with modifications to generalize across system dimensionality, polymer sequence, and charge. We will ignore the FH contribution term as we have already handled it, and it does not have to be sampled over. For everything else, we note that

$$\begin{array}{ll}\hfill \beta {\mathrm{\Pi }}_{ex}& =-\frac{\partial }{\partial V}\log \frac{{\mathcal{Z}}_{\text{C}}}{{\mathcal{Z}}_0}\hfill \\ \hfill & =-\frac{\partial }{\partial V}\log \frac{{\prod }_i\int \mathcal{D}{\mu }_i\int \mathcal{D}\phi e^{-H[\{{\mu }_i\},\phi ]}}{{\prod }_i\int \mathcal{D}{\mu }_i{e}^{-\frac{{\gamma }_i^2}{2B_i}{\int }_{\mathrm{\Omega }}d\mathit{r}{\mu }_i{(\mathit{r})}^2}\int \mathcal{D}\phi e^{-\frac{1}{2E}{\int }_{\mathrm{\Omega }}d\mathit{r}|\nabla \phi (\mathit{r}){|}^2}}.\hfill \end{array}$$ (A11)

As in the previous derivation, we need to rescale all {μ_i}, φ, such that the $\frac{\partial }{\partial V}$ with respect to the denominator vanishes. It was previously shown that the correct rescaling for μ is $\mu (\mathit{r})=V^{-\frac{1}{2}}w(\mathit{r})$, and we will posit and demonstrate that the correct rescaling for φ is φ(r) = V^−1/2df(r), where d is the system dimensionality. We will now show that this is the correct rescaling by splitting the logarithm and taking each individual derivative,

$$\begin{array}{ll}\hfill \beta {\mathrm{\Pi }}_{ex}& =\frac{\partial }{\partial V}\left(-\log \left(\prod _i\int \mathcal{D}{w}_i\int \mathcal{D}f{e}^{-H[\{V^{-1/2}{w}_i\},V^{-1/2d}f]}\right)\right.\hfill \\ \hfill & +\sum _i\log \left(\int \mathcal{D}{w}_i{e}^{-\frac{{\gamma }_i^2}{2B_i}{\int }_{\mathrm{\Omega }}d\mathit{r}{(V^{-1/2}{w}_i(\mathit{r}))}^2}\right)\hfill \\ \hfill & \left.+\log \int \mathcal{D}f{e}^{-\frac{1}{2E}{\int }_{\mathrm{\Omega }}d\mathit{r}{|\nabla (V^{-1/2d}f(\mathit{r}))|}^2}\right)\hfill \\ \hfill & =\frac{\prod _i\int \mathcal{D}{w}_i\int \mathcal{D}f\frac{\partial H[\{V^{-1/2}{w}_i\},V^{-1/2d}f]}{\partial V}{e}^{-H[\{V^{-1/2}{w}_i\},V^{-1/2d}f]}}{\prod _i\int \mathcal{D}{w}_i\int \mathcal{D}f{e}^{-H[\{V^{-1/2}{w}_i\},V^{-1/2d}f]}}\hfill \\ \hfill & +\frac{\partial }{\partial V}\sum _i\log \int \mathcal{D}{w}_i{e}^{-\frac{{\gamma }_i^2}{2B_{i}V}{\int }_{\mathrm{\Omega }}d\mathit{r}{w}_i{(\mathit{r})}^2}\hfill \\ \hfill & +\left.\frac{\partial }{\partial V}\log \int \mathcal{D}f{e}^{-\frac{1}{2E}{\int }_{\mathrm{\Omega }}d\mathit{r}{|\nabla (V^{-1/2d}f(\mathit{r}))|}^2}\right).\hfill \end{array}$$ (A12)

Both of the last two terms are identically zero. They are both Gaussian integrals that have no V dependence. While this is more difficult to see for the φ term if we use the convention that z = V^−1/dr, we can do the intermediate derivative explicitly, which is

$$\begin{array}{ll}\hfill \frac{\partial }{\partial V}|\nabla \left(V^{-\frac{1}{2d}}f(\mathit{r})\right){|}^2& =\frac{\partial }{\partial V}{\left(\sum _i^d{V}^{\frac{1}{d}}\frac{\partial }{\partial {\mathit{z}}_i}{V}^{-\frac{1}{2d}}f(\mathit{r})\right)}^2\hfill \\ \hfill & =2\nabla \left(V^{-\frac{1}{2d}}f(\mathit{r})\right)\left(\sum _i^d\frac{\partial }{\partial {\mathit{z}}_i}f(\mathit{r})\right)\frac{\partial }{\partial V}{V}^{\frac{1}{2d}}\hfill \\ \hfill & =2\nabla \left(V^{-\frac{1}{2d}}f(\mathit{r})\right)\left(\sum _i^d\frac{\partial }{\partial {\mathit{z}}_i}f(\mathit{r})\right)\frac{1}{2d}{V}^{\frac{1}{2d}-1}\hfill \\ \hfill & =\frac{1}{Vd}|\nabla \left(V^{-\frac{1}{2d}}f(\mathit{r})\right){|}^2.\hfill \end{array}$$ (A13)

If we propagate this through, the extra term of 1/V will make the entire Gaussian function lose all V dependence, and thus the term will be zero. With this, we can now drop the last two terms and recognize that the first term is actually just a sampled operator, namely,

$$\beta {\mathrm{\Pi }}_{ex}={⟨\frac{\partial H[\{V^{-1/2}{w}_i\},V^{-1/2d}f]}{\partial V}⟩}_{\text{S}}.$$ (A14)

At this point, we still need to actually calculate the operator, but we can simply calculate its value for a given simulation state and then sample over its value across a simulation. The operator in question is

$$\begin{array}{ll}\hfill \frac{\partial H[\{V^{-1/2}{w}_i\},V^{-1/2d}f]}{\partial V}& =\frac{\partial }{\partial V}\left(\sum _{i=1}^M\frac{{\gamma }_i^2}{2B_{i}V}{\int }_{\mathrm{\Omega }}d\mathit{r}{w}_i^2(\mathit{r})\right.\hfill \\ \hfill & +\frac{1}{2E}\int d\mathit{r}|\nabla (V^{-1/2d}f(\mathit{r})){|}^2\hfill \\ \hfill & \left.-\sum _j^{P+S+2}{C}_{j}V\log Q_j[\{{\mu }_i\},\phi ]\right).\hfill \end{array}$$ (A15)

Taking a short detour, we will often exploit the general derivative of any extensive property that includes a spatial integral, any extensive variable of the form

$$G=\int d\mathit{r}g(\mathit{r})$$ (A16)

will have a dependence on the total volume of the system because the spatial integral will vary with the total volume of the system even if the underlying function g(r) is invariant to changes in system volume. For any function G that can be defined such that

$$\frac{\partial G}{\partial m_p}=\frac{\partial V}{\partial m_p}\frac{\partial }{\partial V}\int d\mathit{r}g(\mathit{r}).$$ (A17)

We can make the change of variables $\mathit{z}=V^{-\frac{1}{d}}\mathit{r}$, and the area/volume of integration will become invariant under the derivative,

$$\begin{array}{ll}\hfill \frac{\partial }{\partial V}\int Vd\mathit{z}g\left(V^{\frac{1}{d}}\mathit{z}\right)& =\int d\mathit{z}g\left(V^{\frac{1}{d}}\mathit{z}\right)+\int Vd\mathit{z}\frac{\partial g\left(V^{\frac{1}{d}}\mathit{z}\right)}{\partial V}\hfill \\ \hfill & =\frac{1}{V}\int d\mathit{r}g(\mathit{r})+\int d\mathit{r}\frac{\partial g(\mathit{r})}{\partial V},\hfill \end{array}$$ (A18)

where the second term will drop out if g(r) exhibits no volume dependence. Taking each component one at a time and using this special chain/product rule, we have for any i,

$$\begin{array}{ll}\hfill \frac{\partial }{\partial V}\frac{{\gamma }_i^2}{2B_{i}V}{\int }_{\mathrm{\Omega }}d\mathit{r}{w}_i^2(\mathit{r})& =-\frac{{\gamma }_i^2}{2B_i{V}^2}{\int }_{\mathrm{\Omega }}d\mathit{r}{w}_i^2(\mathit{r})\hfill \\ \hfill & +\frac{{\gamma }_i^2}{2B_i{V}^2}{\int }_{\mathrm{\Omega }}d\mathit{r}{w}_i^2(\mathit{r})=0.\hfill \end{array}$$ (A19)

In addition, for the φ term,

$$\begin{array}{ll}\hfill \frac{\partial }{\partial V}\frac{1}{2E}{\int }_{\mathrm{\Omega }}d\mathit{r}|\nabla V^{\frac{1}{2d}}\phi (\mathit{r}){|}^2& =\frac{1}{2EV}{\int }_{\mathrm{\Omega }}d\mathit{r}|\nabla \phi (\mathit{r}){|}^2\hfill \\ \hfill & -\frac{1}{2E}{\int }_{\mathrm{\Omega }}d\mathit{r}\frac{1}{V}|\nabla \phi (\mathit{r}){|}^2=0.\hfill \end{array}$$ (A20)

With both of the field terms removed, we can proceed to the partition functions. We already know that

$$\frac{\partial }{\partial V}{n}_j\log Q_j[\{{\mu }_i\},\phi ]=n_j{Q}_j^{-1}\frac{\partial Q_j}{\partial V}$$ (A21)

and can begin with the simpler solvent case where $Q_j=\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{e}^{\mathrm{\Gamma }*\left(-{\psi }_j(\mathit{r})\right)}$. In this case, we do not need to worry about propagating the volume derivative across the MDE, but we still do have to handle the smearing term. We begin by rewriting the partition function with appropriate rescaling [note that we will define $p(\mathit{r})\equiv V^{-\frac{1}{2}}\mu (\mathit{r})$],

$$Q_l=\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{e}^{\mathrm{\Gamma }*\left(-V^{-\frac{1}{2}}{p}_l(\mathit{r})-Z_l{V}^{-\frac{1}{2d}}f(\mathit{r})\right)/N}.$$ (A22)

If we collapse everything in the exponential to W(V, r), then we get

$$\frac{\partial }{\partial V}{Q}_l=\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}\frac{\partial }{\partial V}{e}^{-W(V,\mathit{r})}=\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{e}^{-W(V,\mathit{r})}\frac{\partial }{\partial V}(-W(V,\mathit{r})).$$ (A23)

Now we can handle the final term,

$$\begin{array}{ll}\hfill \frac{\partial W(V,\mathit{r})}{\partial V}& =\frac{1}{N}\frac{\partial }{\partial V}{\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }\frac{e^{-\frac{|\mathit{r}-{\mathit{r}}^{\prime }{|}^2}{2{\alpha }^2}}}{{(2\pi )}^{\frac{d}{2}{\alpha }^d}}{V}^{-\frac{1}{2}}{p}_l(\mathit{r})+Z_l{V}^{-\frac{1}{2d}}f(\mathit{r})=\frac{1}{N}{\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }\frac{1}{V}\left(1-\frac{|\mathit{r}-{\mathit{r}}^{\prime }{|}^2}{d{\alpha }^2}\right)\frac{e^{-\frac{|\mathit{r}-{\mathit{r}}^{\prime }{|}^2}{2{\alpha }^2}}}{{(2\pi )}^{\frac{d}{2}{\alpha }^d}}{V}^{-\frac{1}{2}}{p}_l(\mathit{r})\hfill \\ \hfill & +Z_l{V}^{-\frac{1}{2d}}f(\mathit{r})+\frac{e^{-\frac{|\mathit{r}-{\mathit{r}}^{\prime }{|}^2}{2{\alpha }^2}}}{{(2\pi )}^{\frac{d}{2}{\alpha }^d}}\frac{\partial }{\partial V}\left(V^{-\frac{1}{2}}{p}_l(\mathit{r})-Z_l{V}^{-\frac{1}{2d}}f(\mathit{r})\right)=\frac{1}{VN}{\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }\left(1-\frac{|\mathit{r}-{\mathit{r}}^{\prime }{|}^2}{d{\alpha }^2}\right)\frac{e^{-\frac{|\mathit{r}-{\mathit{r}}^{\prime }{|}^2}{2{\alpha }^2}}}{{(2\pi )}^{\frac{d}{2}{\alpha }^d}}{V}^{-\frac{1}{2}}{p}_l(\mathit{r})\hfill \\ \hfill & +Z_l{V}^{-\frac{1}{2d}}f(\mathit{r})+\mathrm{\Gamma }\ast \left(-\frac{1}{2}{V}^{-\frac{1}{2}}{p}_l(\mathit{r})+-\frac{1}{2d}{Z}_l{V}^{-\frac{1}{2d}}f(\mathit{r})\right)=\frac{1}{NV}\left({\mathrm{\Gamma }}_2-\frac{1}{2}\mathrm{\Gamma }\right)\ast \left({\psi }_{\left(l,/,j\right)},-,Z_{\left(l,/,j\right)},\phi \right)+\left({\mathrm{\Gamma }}_2-\frac{1}{2d}\mathrm{\Gamma }\right)\ast Z_l\phi .\hfill \end{array}$$ (A24)

This is the same term identified previously¹⁵ but with generalizations for dimensionality and on a slightly different basis. Where we have defined ${\mathrm{\Gamma }}_2(\mathit{r})=(1-\frac{|\mathit{r}{|}^2}{d{\alpha }^2})\mathrm{\Gamma }(\mathit{r})$. Conveniently, Γ₂(k) = α²|k|²Γ(k) in Fourier space, regardless of dimensionality. This is also the correct solution for the salts with the usual salt condition of ψ(r) = 0.

Putting everything together, we get the solvent that

$$\begin{array}{ll}\hfill & \frac{\partial }{\partial V}{C}_{l}V\log Q_l[\{{\mu }_i\},\phi ]\hfill \\ \hfill & =-C_{l}V{Q}_l^{-1}\frac{1}{NV}\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{e}^{-W(V,\mathit{r})}\hfill \\ \hfill & \times \left(\left({\mathrm{\Gamma }}_2-\frac{1}{2}\mathrm{\Gamma }\right)\ast \left({\psi }_{\left(l,/,j\right)},-,Z_{\left(l,/,j\right)},\phi \right)(\mathit{r})+\left({\mathrm{\Gamma }}_2-\frac{1}{2d}\mathrm{\Gamma }\right)\ast Z_l\phi (\mathit{r})\right)\hfill \\ \hfill & =-\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{\rho }_l(\mathit{r})\left(\left({\mathrm{\Gamma }}_2-\frac{1}{2}\mathrm{\Gamma }\right)\ast \left({\psi }_{\left(l,/,j\right)},-,Z_{\left(l,/,j\right)},\phi \right)(\mathit{r})\right.\hfill \\ \hfill & \left.+\left({\mathrm{\Gamma }}_2-\frac{1}{2d}\mathrm{\Gamma }\right)\ast Z_l\phi (\mathit{r})\right).\hfill \end{array}$$ (A25)

Conveniently, this is a product of the convolved field and the density, which are all commonly used operators, so this extra operator is fairly cheap to compute.

b. Modified diffusion derivative

While the polymer case is quite similar, we do have to calculate how the MDE changes with changes in volume, which is somewhat involved. Again, this derivation is nearly identical to work by Villet and Fredrickson,³⁷ done in full for clarity. The two modifications from previous works are generalization for any system dimensionality to match our 2D simulation and the inclusion of a charged term to recapitulate the result quoted for the 3D charged case in Appendix B of Delaney and Fredrickson.¹⁵ We begin by writing down our MDE with the correctly rescaled variables, namely,

$$\begin{array}{ll}\hfill \frac{\partial }{\partial s}q(V^{1/d}\mathit{z},s,W(s,\mathit{r}))& =V^{-2/d}{\nabla }_{\mathit{z}}^{2}q(V^{1/d}\mathit{z},s,W(s,\mathit{r}))\hfill \\ \hfill & -W(s,V^{1/d}\mathit{z})q(V^{1/d}\mathit{z},s,W(s,\mathit{r})).\hfill \end{array}$$ (A26)

We then take the derivative of both sides with respect to V,

$$\frac{{\partial }^{2}q}{\partial s\partial V}={\nabla }^2\frac{\partial q}{\partial V}-\frac{2}{dV}{\nabla }^{2}q-W(s,\mathit{r})\frac{\partial q}{\partial V}-\frac{\partial W(s,\mathit{r})}{\partial V}q$$ (A27)

and rearrange to get a nonhomogeneous equation in $\frac{\partial q}{\partial V}$,

$$\left(\frac{\partial }{\partial s}-{\nabla }^2+W(s,\mathit{r})\right)\frac{\partial q}{\partial V}=-\frac{2}{dV}{\nabla }^{2}q-\frac{\partial W(s,\mathit{r})}{\partial V}q$$ (A28)

with the homogeneous condition that $\frac{\partial q(\mathit{r},0)}{\partial V}=0$.

This can be solved with the associated Green’s function,

$$\frac{\partial g}{\partial s}+{\mathcal{L}}_{\mathit{r}}g=0;{\mathcal{L}}_{\mathit{r}}\equiv -{\nabla }_{\mathit{r}}^2+W(s,\mathit{r});g(\mathit{r},{\mathit{r}}^{\prime },0)=\delta (\mathit{r}-{\mathit{r}}^{\prime }),$$ (A29)

where the periodic boundary conditions can be formally expressed as

$$g(\mathit{r},{\mathit{r}}^{\prime },s,W(s,\mathit{r}))=e^{-{\mathcal{L}}_{\mathit{r}}s}\delta (\mathit{r}-{\mathit{r}}^{\prime }).$$ (A30)

Our original propagator can be related to the Green’s function as

$$q(\mathit{r},s,W(s,\mathit{r}))={\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }g(\mathit{r},{\mathit{r}}^{\prime },s;W(s,\mathit{r})).$$ (A31)

Because q^† has the inverse causality of q but otherwise is identical, there is no problem in identifying

$$q^{†}(\mathit{r},s^{*}-s,W(s,\mathit{r}))={\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }g({\mathit{r}}^{\prime },\mathit{r},s;W(s,\mathit{r})).$$ (A32)

From here we can construct a solution to the nonhomogeneous equation using a forcing function, F(r, s),

$$\frac{\partial f(\mathit{r},s)}{\partial s}+{\mathcal{L}}_{\mathit{r}}=F(\mathit{r},s),$$ (A33)

$$e^{-{\mathcal{L}}_{\mathit{r}}s}\frac{\partial }{\partial s}(e^{{\mathcal{L}}_{\mathit{r}}s}f(\mathit{r},0))={\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }F(\mathit{r},s)\delta (\mathit{r}-{\mathit{r}}^{\prime }),$$ (A34)

$$f(\mathit{r},s)-e^{-{\mathcal{L}}_{\mathit{r}}s}f(\mathit{r},0)={\int }_0^{s}d{s}^{\prime }{\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }F({\mathit{r}}^{\prime },s^{\prime })e^{-{\mathcal{L}}_{\mathit{r}}(s-s^{\prime })}\delta (\mathit{r}-{\mathit{r}}^{\prime }).$$ (A35)

We can simplify with our initial condition and write our Green’s function explicitly as

$$f(\mathit{r},s)={\int }_0^{s}d{s}^{\prime }{\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }F({\mathit{r}}^{\prime },s^{\prime })g(\mathit{r},{\mathit{r}}^{\prime },s-s^{\prime }).$$ (A36)

Next, we replace Green’s function with our inverse propagator to get

$$\begin{array}{ll}\hfill {\int }_{\mathrm{\Omega }}d\mathit{r}f(\mathit{r},s)& ={\int }_0^{s}d{s}^{\prime }{\int }_{\mathrm{\Omega }}d{\mathit{r}}^{\prime }F({\mathit{r}}^{\prime },s^{\prime }){\int }_{\mathrm{\Omega }}d\mathit{r}g(\mathit{r},{\mathit{r}}^{\prime },s-s^{\prime })\hfill \\ \hfill & ={\int }_0^{s}d{s}^{\prime }{\int }_{\mathrm{\Omega }}d\mathit{r}F(\mathit{r},s^{\prime })q^{†}(\mathit{r},s,W(s,\mathit{r})).\hfill \end{array}$$ (A37)

By using the previously established definition for the forcing term, we can now write

$$\begin{array}{ll}\hfill {\int }_{\mathrm{\Omega }}d\mathit{r}\frac{\partial q(\mathit{r},s)}{\partial V}& ={\int }_0^{s}d{s}^{\prime }{\int }_{\mathrm{\Omega }}d\mathit{r}{q}^{†}(\mathit{r},s^{\prime },W(s,\mathit{r}))\hfill \\ \hfill & \ast \left(\frac{2}{dV}{\nabla }^2+\frac{\partial W(s,\mathit{r})}{\partial V})\right)q(\mathit{r},s^{\prime },W(s,\mathit{r})),\hfill \end{array}$$ (A38)

which conveniently is closely related to the density operator,

$$\frac{\partial Q[W]}{\partial V}=-\frac{Q}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{\int }_0^{\frac{N_j}{N}}ds\frac{2}{Vd}{\rho }_{\nabla j}(\mathit{r},s)+{\rho }_j(s)\frac{\partial W(\mathit{r},s)}{\partial V},$$ (A39)

where

$${\rho }_{\nabla j}(\mathit{r},s)\equiv Q^{-1}q(\mathit{r},s){\nabla }^2{q}^{†}(\mathit{r},s)$$ (A40)

and ρ uses the conventional definition of

$${\rho }_j(\mathit{r},s)\equiv Q^{-1}q(\mathit{r},s)q^{†}(\mathit{r},s).$$ (A41)

At this point we have everything to finish the derivative; we can use the same $\frac{\partial W}{\partial V}$ as in the solvent case to get

$$\begin{array}{ll}\hfill & \frac{\partial }{\partial V}{C}_{j}V\log Q_j[\{{\mu }_i\},\phi ]\hfill \\ \hfill & =-\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{\int }_0^{\frac{N_j}{N}}ds\left[\frac{2}{d}{\rho }_{\nabla j}(\mathit{r},s)+{\rho }_j(\mathit{r},s)\right.\hfill \\ \hfill & \left.\times \left(\left({\mathrm{\Gamma }}_2-\frac{1}{2}\mathrm{\Gamma }\right)\ast \left({\psi }_{\left(l,/,j\right)},-,Z_{\left(l,/,j\right)},\phi \right)(\mathit{r},s)+\left({\mathrm{\Gamma }}_2-\frac{1}{2d}\mathrm{\Gamma }\right)\ast Z_l\phi (\mathit{r},s)\right)\right].\hfill \end{array}$$ (A42)

c. Ideal gas and final result

The only thing that we have left to calculate is the ideal gas derivative, which is straightforward. We just have

$$\frac{\partial }{\partial V}{n}_j\log V=\frac{n_j}{V}=C_j.$$ (A43)

With this, we can write down our final operator, namely,

$$\begin{array}{ll}\hfill \beta \mathrm{\Pi }& =\sum _j^{P+S+2}{C}_j-\frac{{\mathbf{c}}^T\mathit{\chi }\mathbf{c}}{2}+\sum _j^P\frac{1}{V}{\int }_{\mathrm{\Omega }}d\mathit{r}{\int }_0^{\frac{N_j}{N}}ds\left[\frac{2}{d}{\rho }_{\nabla j}(\mathit{r},s)\right.\hfill \\ \hfill & +{\rho }_j(\mathit{r},s)\left(\left({\mathrm{\Gamma }}_2,-,\frac{1}{2},\mathrm{\Gamma }\right),\ast ,\left({\psi }_{\left(l,/,j\right)},-,Z_{\left(l,/,j\right)},\phi \right),(\mathit{r},s)\right.\hfill \\ \hfill & \left.\left.+,\left({\mathrm{\Gamma }}_2-\frac{1}{2d}\mathrm{\Gamma }\right),\ast ,Z_j,\phi ,(\mathit{r},s)\right)\right]+\sum _j^{S+2}\frac{1}{NV}{\int }_{\mathrm{\Omega }}d\mathit{r}{\rho }_j(\mathit{r})\hfill \\ \hfill & \times \left(\left({\mathrm{\Gamma }}_2-\frac{1}{2}\mathrm{\Gamma }\right)\ast \left({\psi }_{\left(l,/,j\right)},-,Z_{\left(l,/,j\right)},\phi \right)(\mathit{r})+\left({\mathrm{\Gamma }}_2-\frac{1}{2d}\mathrm{\Gamma }\right)\ast Z_j\phi (\mathit{r})\right).\hfill \end{array}$$ (A44)

4. Gibbs ensemble

With both the osmotic pressure and chemical potentials in hand, we can now build our Gibbs ensemble. In this work, we only used the Gibbs ensemble to validate our model by recreating previous results, so we will restrict ourselves to models that only consider implicit solvents and are free of explicit salt. This condition simplifies all the previous equations by neglecting any contribution not related to the polymer. This also simplifies the Gibbs ensemble because there is only one charge neutral move that is possible. Because each simulation must be charge neutral, every move must also be charge neutral. With mixtures of three or more charged species, it is not always obvious how to correctly determine the best charge neutral move, but for a two component mixture, there is only one option. With these preliminaries noted, we can proceed to write down our Gibbs ensemble dynamics.

We begin by instantiating two simulations that can be at any arbitrary state but ideally will be selected to be near the two pure phases we are trying to demonstrate coexistence for. These simulations will each have a volume (V_I and V_II) and concentration (C_Ij and C_IIj for each species). We can also determine the amount of particles in each simulation, for convenience denoted m_Ij and m_IIj, for which m_Ij = C_IjV_I, and so on. The total mass and volume are conserved so that V_I + V_II = V_T, and m_Ij + m_IIj = m_Tj for all species. The joint free energy of the system can be written as F_I(n_Ij, V_I, T) + F_II(n_IIj, V_II, T) = F(n_Tj, V_T, T). The system will be in equilibrium if

$$\frac{\partial F}{\partial n_{Ij}}=({\mu }_I-{\mu }_{II})=0$$ (A45)

for all j and

$$-\frac{\partial F}{\partial V_I}=({\mathrm{\Pi }}_I-{\mathrm{\Pi }}_{II})=0,$$ (A46)

where μ and Π are the chemical potentials and osmotic pressure calculated previously. This equilibrium condition can be used to write simple equations of motion,

$$\frac{\partial m_{Ij}}{\partial t}={\mu }_{\mathit{II}j}-{\mu }_{Ij},$$ (A47)

$$\frac{\partial V_I}{\partial t}={\mathrm{\Pi }}_I-{\mathrm{\Pi }}_{\mathit{II}j}$$ (A48)

with first order update scheme

$$m_{Ij}(t+\mathrm{\Delta }t)=m_{Ij}(t)+\mathrm{\Delta }{t}_j\left({\mu }_{\mathit{II}j}-{\mu }_{Ij}\right),$$ (A49)

$$V_I(t+\mathrm{\Delta }t)=V_I(t)+\mathrm{\Delta }{t}_V\left({\mathrm{\Pi }}_I-{\mathrm{\Pi }}_{II}\right).$$ (A50)

Our update scheme operates on the mass and volumes, and the concentrations can be trivially recovered from their values for each cell. In principle, the update step for each species can be chosen separately, but for the simple two-component case, we will replace the two m_Ij’s with a single mass and chemical potential for the charge neutral pair m_I = Z₁m_I1 − Z₂m_I2 and μ_I = Z₁μ_I1 − Z₂μ_I2. For the polyampholyte, the polymer is charge neutral, and there is only one species.

The update scheme is allowed to proceed until a stationary point is found, at which point phase coexistence is proven. In principle, higher order update schemes could be used to achieve faster convergence. Because the Gibbs ensemble is not a significant focus of this work and is only used to demonstrate that our model is consistent with known results, we have not examined these potential improvements.

5. Comparison to previous work

With all the mathematical preliminaries handled, we now present the direct comparisons between our model and previous simulation methods of coacervates.¹⁵ We present two results here using the same conditions as Delaney’s previous work, which show good agreement with our current methods (Fig. 6).

FIG. 6.

Chemical potential vs concentration for use in phase equilibrium conditions. Simulation is a diblock polyampholyte with B = 1.0, E = 400, and a = 0.2. Comparison is against Fig. 5(b) in Ref. 15.

First, we were able to make a close match on the chemical potential of a polyampholyte simulation at different polyampholyte concentrations. While there is some variation between the two plots, this can be attributed to differences in dynamic parameters and random number generation. The estimates of variance likely underestimate the true variance due to non-ergodic sampling that is difficult to capture with standard methods used to identify the variance compared to the true underlying distribution.

We also made comparisons against the final phase concentration predictions made by running Gibbs ensemble simulations as shown in Fig. 7. Here, we show close agreement for the density of the coacervate phase and reasonably good agreement on the supernatant phase. Measuring the exact density of the supernatant phase is particularly difficult due to the small concentrations in question. These small values exacerbate the variance of the noise relative to the signal and amplify issues with non-ergodic sampling. Still, we believe that we have shown sufficiently close agreement overall to indicate that our simulation method correctly recapitulates previously reported results. Without access to the underlying code used to generate these previous results, there is no practical way to discern small differences in performance, and we have shown that we achieve the same results as previous work.

FIG. 7.

Plots of phase coexistence for a symmetric diblock polyampholyte at various values of E. Comparison is to Fig. 6 of Ref. 15 using a = 0.2 and B = 1.^47,48

APPENDIX B: NOTATION GLOSSARY

1. System-wide constants

M	Number of monomer species
P	Number of polymer types
S	Number of solvent types
N	Reference polymer length
B	Reference bond stretching parameter
R g	Reference radius of gyration
D	System dimensionality
V	Rescaled volume
ΔV	Rescaled volume of unit cell
A	Smearing constant
E	Rescaled bjerrum length

2. Spatial variables and functions

r	Rescaled position vector
S	Scaled polymer space curve coordinate
ωi(r)	Chemical potential field in the diagonalized basis
μi(r)	ωi after Wick rotation
φ(r)	Electrical potential field
ψi(r)	Field experienced by monomer type i
ρi(r)	Smeared density vector for monomer type i
${\overline{\rho }}_i(\mathit{r})$	Unsmeared density vector for monomer type i
qp(r, s)	Forward chain propagator for polymer type p
$q_p^{†}(\mathit{r},s)$	Reverse chain propagator for polymer type p
Γ	Gaussian smearing kernel

3. Indexed parameters

n p	Number of polymers of species p
N p	Length of polymers of species p
C j	Reduced concentration of species j
χ ij	Rescaled Flory–Huggins interaction between monomers of types i and j
B i	Diagonalized rescaled Flory–Huggins interaction for the ith eigenvector
mp(s)	Monomer identity of polymer type p at s
σ(s)	Charge density at s
m s	Monomer identity of solvent type s
Z m	Charge per monomer of type m
γ i	ωi Wick rotation variable
Q j	Single unit partition function for species type j

4. Dynamics variables

λjμ or λφ	Step size for corresponding dynamics
ηjμ or ηφ	Random noise injected into corresponding dynamics
βjμ or βφ	Fictitious temperature for corresponding dynamics
T	Fictitious time
Δt	Fictitious time step
c(k)	Analytical approximation of linear response of force to fields

5. Analytic constant terms and partition functions

b	Matrix of eigenvalues to transform species basis to diagonalized basis
κ ij	Fraction of the i-th species composed of the j-th monomer type
C	Average concentration vector
$\mathcal{Z}$	Total partition function
${\mathcal{Z}}_{\text{ideal}}$	Ideal gas contribution to partition function
${\mathcal{Z}}_{\text{C}}$	Charge contribution to partition function
${\mathcal{Z}}_i$	Diagonal field i contribution to the partition function

6. Gibbs ensemble variable

μ i	Chemical potential for species i
m i	Total mass of species i
Π	Osmotic pressure
V	Volume
F	Free energy
Δt	Fictitious time step

All species are in either simulation I or II. The corresponding net values are denoted with a Δ.

APPENDIX C: DEFAULT COMPUTATIONAL PARAMETERS

N	Reference length	3
C solvent	Solvent concentration	1.5
C polymer	Polymer concentration	0.3
C salt	Salt concentration	(0–5) N
α	Smearing length	0.163
Lx, Ly	Box side length	30
ΔLx, ΔLy	Box grid fineness	15/256
Δs	Polymer integration width	1/30
λ μ	Chemical relaxation rate	0.0032
β μ	Fictitious chemical temperature	(1 × 10−5–1 × 10−5)
λ φ	Electrostatic relaxation rate	0.008
β φ	Electrostatic fictitious temperature	1
T	Total FH parameter	1.5
B	Lipid FH parameter	(0–6)
E	Dielectric constant	1 × 104
Z +	Positive monomer charge (all species)	1
Z −	Negative monomer charge (all species)	−1
t	Simulation time steps	1.8 × 105

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Emmit K. Pert: Conceptualization (equal); Formal analysis (equal); Software (lead); Writing – original draft (lead); Writing – review & editing (equal). Paul J. Hurst: Investigation (equal); Methodology (equal); Writing – review & editing (equal). Robert M. Waymouth: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Grant M. Rotskoff: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.