Field-theoretic Simulations Research Articles

Accelerating Langevin Field-Theoretic Simulation of Polymers with Deep Learning

Accelerating Langevin Field-Theoretic Simulation of Polymers with Deep Learning

Field-Theoretic Simulation Method to Study the Liquid-Liquid Phase Separation of Polymers.

Liquid-liquid phase separation (LLPS) is a process that results in the formation of a polymer-rich liquid phase coexisting with a polymer-depleted liquid phase. LLPS plays a critical role in the cell through the formation of membrane-less organelles, but it also has a number of biotechnical and biomedical applications such as drug confinement and its targeted delivery. In this chapter, we present a computational efficient methodology that uses field-theoretic simulations (FTS) with complex Langevin (CL) sampling to characterize polymer phase behavior and delineate the LLPS phase boundaries. This approach is a powerful complement to analytical and explicit-particle simulations, and it can serve to inform experimental LLPS studies. The strength of the method lies in its ability to properly sample a large ensemble of polymers in a saturated solution while including the effect of composition fluctuations on LLPS. We describe the approaches that can be used to accurately construct phase diagrams of a variety of molecularly designed polymers and illustrate the method by generating an approximation-free phase diagram for a classical symmetric diblock polyampholyte.

Direct calculation of the functional inverse of realistic interatomic potentials in field-theoretic simulations.

We discuss the functional inverse problem in field-theoretic simulations for realistic pairwise potentials such as the Morse potential (widely used in particle simulations as an alternative to the 12-6 Lennard-Jones one), and we propose the following two solutions: (a) a numerical one based on direct inversion on a regular grid or deconvolution and (b) an analytical one by expressing attractive and repulsive contributions to the Morse potential as higher-order derivatives of the Dirac delta function; the resulting system of ordinary differential equations in the saddle-point approximation is solved numerically with appropriate model-consistent boundary conditions using a Newton-Raphson method. For the first time, exponential-like, physically realistic pair interactions are analytically treated and incorporated into a field-theoretic framework. The advantages and disadvantages of the two approaches are discussed in detail in connection with numerical findings from test simulations for the radial distribution function of a monatomic fluid at realistic densities providing direct evidence for the capability of the analytical method to resolve structural features down to the Angstrom scale.

Numerical Techniques for Applications of Analytical Theories to Sequence-Dependent Phase Separations of Intrinsically Disordered Proteins.

Biomolecular condensates, physically underpinned to a significant extent by liquid-liquid phase separation (LLPS), are now widely recognized by numerous experimental studies to be of fundamental biological, biomedical, and biophysical importance. In the face of experimental discoveries, analytical formulations emerged as a powerful yet tractable tool in recent theoretical investigations of the role of LLPS in the assembly and dissociation of these condensates. The pertinent LLPS often involves, though not exclusively, intrinsically disordered proteins engaging in multivalent interactions that are governed by their amino acid sequences. For researchers interested in applying these theoretical methods, here we provide a practical guide to a set of computational techniques devised for extracting sequence-dependent LLPS properties from analytical formulations. The numerical procedures covered include those for the determination of spinodal and binodal phase boundaries from a general free energy function with examples based on the random phase approximation in polymer theory, construction of tie lines for multiple-component LLPS, and field-theoretic simulation of multiple-chain heteropolymeric systems using complex Langevin dynamics. Since a more accurate physical picture often requires comparing analytical theory against explicit-chain model predictions, a commonly utilized methodology for coarse-grained molecular dynamics simulations of sequence-specific LLPS is also briefly outlined.

Field theory description of ion association in re-entrant phase separation of polyampholytes.

Phase separation of several different overall neutral polyampholyte species (with zero net charge) is studied in solution with two oppositely charged ion species that can form ion pairs through an association reaction. Hereby, a field theory description of the system, which treats polyampholyte charge sequence dependent electrostatic interactions as well as excluded volume effects, is given. Interestingly, analysis of the model using random phase approximation and field theoretic simulation consistently shows evidence of a re-entrant polyampholyte phase separation at high ion concentrations when there is an overall decrease of volume upon ion association. As an illustration of the ramifications of our theoretical framework, several polyampholyte concentration vs ion concentration phase diagrams under constant temperature conditions are presented to elucidate the dependence of phase separation behavior on the polyampholyte sequence charge pattern as well as ion pair dissociation constant, volumetric effects on ion association, solvent quality, and temperature.

Direct free energy evaluation of classical and quantum many-body systems via field-theoretic simulation

Free energy evaluation in molecular simulations of both classical and quantum systems is computationally intensive and requires sophisticated algorithms. This is because free energy depends on the volume of accessible phase space, a quantity that is inextricably linked to the integration measure in a coordinate representation of a many-body problem. In contrast, the same problem expressed as a field theory (auxiliary field or coherent states) isolates the particle number as a simple parameter in the Hamiltonian or action functional and enables the identification of a chemical potential field operator. We show that this feature leads a “direct” method of free energy evaluation, in which a particle model is converted to a field theory and appropriate field operators are averaged using a field-theoretic simulation conducted with complex Langevin sampling. These averages provide an immediate estimate of the Helmholtz free energy in the canonical ensemble and the entropy in the microcanonical ensemble. The method is illustrated for a classical polymer solution, a block copolymer melt exhibiting liquid crystalline and solid mesophases, and a quantum fluid of interacting bosons.

Is the “Bricks-and-Mortar” Mesophase Bicontinuous? Dynamic Simulations of Miktoarm Block Copolymer/Homopolymer Blends

A new mesophase in binary blends of A–b–(BA′)3 miktoarm star block copolymers and A homopolymers has recently been discovered experimentally and explored with field-theoretic simulations. This mesophase has been reported to consist of aperiodic discrete domains of A embedded in a continuous matrix of B up to very high concentrations of A. Because of the material’s potential as a thermoplastic elastomer, a deeper understanding of its structural and dynamic-mechanical properties, including its domain connectivity, linear rheological behavior, response to shear, and response to uniaxial tension, is warranted. These properties are explored here using dissipative particle dynamics in three dimensions, for the first time. These simulations establish that the so-called “bricks-and-mortar” phase, while appearing discrete in two dimensions, is bicontinuous. The simulations focusing on dynamics establish that the role of molecular bridging dominates the mechanical behavior and outweighs the influence of microphase segregation (contributions from the interfacial tension alone) even at the highest homopolymer concentrations we study. Additionally, it appears that the bricks-and-mortar phase is sensitive to the application of sufficiently high shear, leading to nonisotropic mechanical responses, which has ramifications for the processability of such materials. We find that upon application of shear the phase becomes closer in structure to its speculated discrete nature. Molecular simulations on our longest accessible timescales show that the material is unable to relax back to its original structure, suggesting that the morphology observed depends heavily on the material process pathway.

Order-to-Disorder Transition of Cylinder-Forming Block Copolymer Films Confined within Neutral Interfaces

We present the phase behavior of cylinder-forming block copolymer (BCP) films confined within neutral interfaces through theory and experiment. The Langevin field-theoretic simulation (L-FTS) is utilized to account for the compositional fluctuation effect on the order-to-disorder transition (ODT), and the ultraviolet divergence in the L-FTS result is eliminated by renormalization of the Flory–Huggins parameter χ, allowing accurate comparison of thickness-dependent (χN)ODT, where N is the polymerization index. The L-FTS is accelerated by utilizing a deep learning method, and we managed to achieve a 50–60% reduction in simulation time. The L-FTS results show that (χN)ODT decreases as the film thickness decreases at a very large value of the invariant polymerization index N̅, and BCPs undergo a phase transition from disordered to cylindrical phases through a spherical phase as predicted by the self-consistent field theory calculation. By performing the L-FTS at experimentally relevant low N̅, we show that the BCPs undergo a direct transition from disordered to cylindrical phases, and the (χN)ODT value of cylinder-forming BCPs confined within neutral interfaces increases as the film thickness decreases. To verify and complement the theoretical results, we investigate the ODT temperature (TODT) of cylinder-forming polystyrene-b-poly(2-vinylpyridine) (PS-b-P2VP) films supported on a neutral mat of P(S-r-2VP) random copolymers using grazing-incidence small-angle X-ray scattering. Our film experiments with low-N̅ PS-b-P2VP confirm the direct transition from disordered to cylindrical phases and the decrease in TODT with decreasing thickness below the onset thickness above which the TODT levels off.

Quantitative Comparison of Field-Update Algorithms for Polymer SCFT and FTS

A wide range of field-update algorithms for polymer self-consistent field theory (SCFT) and field theoretic simulations (FTSs) are analyzed. We provide the first direct comparison between Anderson ...

Disordered Micelle Regime in a Conformationally Asymmetric Diblock Copolymer Melt

We report results from Monte Carlo field-theoretic simulations (MC-FTS) of a compositionally asymmetric diblock copolymer melt (fA = 0.2) at an invariant degree of polymerization of N = 104 for bo...

Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation.

Field-theoretic simulations (FTS) provide an efficient technique for investigating fluctuation effects in block copolymer melts with numerous advantages over traditional particle-based simulations. For systems involving two components (i.e., A and B), the field-based Hamiltonian, , depends on a composition field, , that controls the segregation of the unlike components and a pressure field, , that enforces incompressibility. This review introduces researchers to a promising variant of FTS, in which fluctuates while tracks its mean-field value. The method is described in detail for melts of AB diblock copolymer, covering its theoretical foundation through to its numerical implementation. We then illustrate its application for neat AB diblock copolymer melts, as well as ternary blends of AB diblock copolymer with its A- and B-type parent homopolymers. The review concludes by discussing the future outlook. To help researchers adopt the method, open-source code is provided that can be run on either central processing units (CPUs) or graphics processing units (GPUs).

Field-theoretic simulations beyond δ-interactions: Overcoming the inverse potential problem in auxiliary field models.

Modern field-theoretic simulations of complex fluids and polymers are constructed around a particle-to-field transformation that brings an inverse potential u-1 in the model equations. This has restricted the application of the framework to systems characterized by relatively simple pairwise interatomic interactions; for example, excluded volume effects are treated through the use of δ-function interactions. In this study, we first review available nonbonded pair interactions in field-theoretic models and propose a classification. Then, we outline the inverse potential problem and present an alternative approach on the basis of a saddle-point approximation, enabling the use of a richer set of pair interaction functions. We test our approach by using as an example the Morse potential, which finds extensive applications in particle-based simulations, and we calibrate u-1 with results from a molecular dynamics simulation. The u-1 thus obtained is consistent with the field-theoretic model equations, and when used in stand-alone self-consistent field simulations, it produces the correct fluid structure starting from a random initial state of the density field.

Liquid-Liquid Phase Separation in the Presence of Macromolecular Crowding and State-dependent Kinetics.

Biomolecular condensates formed via liquid–liquid phase separation (LLPS) are increasingly being shown to play major roles in cellular self-organization dynamics in health and disease. It is well established that macromolecular crowding has a profound impact on protein interactions, particularly those that lead to LLPS. Although synthetic crowding agents are used during in vitro LLPS experiments, they are considerably different from the highly crowded nucleo-/cytoplasm and the effects of in vivo crowding remain poorly understood. In this work, we applied computational modeling to investigate the effects of macromolecular crowding on LLPS. To include biologically relevant LLPS dynamics, we extended the conventional Cahn–Hilliard model for phase separation by coupling it to experimentally derived macromolecular crowding dynamics and state-dependent reaction kinetics. Through extensive field-theoretic computer simulations, we show that the inclusion of macromolecular crowding results in late-stage coarsening and the stabilization of relatively smaller condensates. At a high crowding concentration, there is an accelerated growth and late-stage arrest of droplet formation, effectively resulting in anomalous labyrinthine morphologies akin to protein gelation observed in experiments. These results not only elucidate the crowder effects observed in experiments, but also highlight the importance of including state-dependent kinetics in LLPS models, and may help in designing further experiments to probe the intricate roles played by LLPS in self-organization dynamics of cells.

A Simple Explicit-Solvent Model of Polyampholyte Phase Behaviors and Its Ramifications for Dielectric Effects in Biomolecular Condensates.

Biomolecular condensates such as membraneless organelles, underpinned by liquid-liquid phase separation (LLPS), are important for physiological function, with electrostatics, among other interaction types, being a prominent force in their assembly. Charge interactions of intrinsically disordered proteins (IDPs) and other biomolecules are sensitive to the aqueous dielectric environment. Because the relative permittivity of protein is significantly lower than that of water, the interior of an IDP condensate is expected to be a relatively low-dielectric regime, which aside from its possible functional effects on client molecules should facilitate stronger electrostatic interactions among the scaffold IDPs. To gain insight into this LLPS-induced dielectric heterogeneity, addressing in particular whether a low-dielectric condensed phase entails more favorable LLPS than that posited by assuming IDP electrostatic interactions are uniformly modulated by the higher dielectric constant of the pure solvent, we consider a simplified multiple-chain model of polyampholytes immersed in explicit solvents that are either polarizable or possess a permanent dipole. Notably, simulated phase behaviors of these systems exhibit only minor to moderate differences from those obtained using implicit-solvent models with a uniform relative permittivity equals to that of pure solvent. Buttressed by theoretical treatments developed here using random phase approximation and polymer field-theoretic simulations, these observations indicate a partial compensation of effects between favorable solvent-mediated interactions among the polyampholytes in the condensed phase and favorable polyampholyte-solvent interactions in the dilute phase, often netting only a minor enhancement of overall LLPS propensity from the very dielectric heterogeneity that arises from the LLPS itself. Further ramifications of this principle are discussed.

Fluctuation correction for the order-disorder transition of diblock copolymer melts.

The order-disorder transition (ODT) of diblock copolymer melts is evaluated for an invariant polymerization index of N¯=104, using field-theoretic simulations (FTS) supplemented by a partial saddle-point approximation for incompressibility. For computational efficiency, the FTS are performed using the discrete Gaussian-chain model, and results are then mapped onto the continuous model using a linear approximation for the Flory-Huggins χ parameter. Particular attention is paid to the complex phase window. Results are found to be consistent with the well-established understanding that the gyroid phase extends down to the ODT. Furthermore, our simulations are the first to predict that the Fddd phase survives fluctuation effects, consistent with experiments.

Fundamental Challenges and Outlook in Simulating Liquid–Liquid Phase Separation of Intrinsically Disordered Proteins

Intrinsically disordered proteins (IDPs) populate an ensemble of dynamic conformations, making their structural characterization by experiments challenging. Many IDPs undergo liquid-liquid phase separation into dense membraneless organelles with myriad cellular functions. Multivalent interactions in low-complexity IDPs promote the formation of these subcellular coacervates. While solution NMR, Förster resonance energy transfer (FRET), and small-angle X-ray scattering (SAXS) studies on IDPs have their own challenges, recent computational methods draw a rational trade-off to characterize the driving forces underlying phase separation. In this Perspective, we critically evaluate the scope of approximation-free field theoretic simulations, well-tempered ensemble methods, enhanced sampling techniques, coarse-grained force fields, and slab simulation approaches to offer an improved understanding of phase separation. A synergy between simulation length scale and model resolution would reduce the existing caveats and enable theories of polymer physics to elucidate finer details of liquid-liquid phase separation (LLPS). These computational advances offer promise for rigorous characterization of the IDP proteome and designing peptides with tunable material and self-assembly properties.

Coexistence of Polymeric Microemulsion with Homopolymer-Rich Phases

Recent field-theoretic simulations of symmetric ternary blends of A- and B-type homopolymers with AB diblock copolymer [Vorselaars, B.; Phys. Rev. Lett., 2020, 125, 117801]predicted three-phase coe...

Instability of the Microemulsion Channel in Block Copolymer-Homopolymer Blends.

Field theoretic simulations are used to predict the equilibrium phase diagram of symmetric blends of AB diblock copolymer with A- and B-type homopolymers. Experiments generally observe a channel of bicontinuous microemulsion (BμE) separating the ordered lamellar (LAM) phase from coexisting homopolymer-rich (A+B) phases. However, our simulations find that the channel is unstable with respect to macrophase separation, in particular, A+B+BμE coexistence at high T and A+B+LAM coexistence at low T. The preference for three-phase coexistence is attributed to a weak attractive interaction between diblock monolayers.

Dehydration entropy drives liquid-liquid phase separation by molecular crowding

Complex coacervation driven liquid-liquid phase separation (LLPS) of biopolymers has been attracting attention as a novel phase in living cells. Studies of LLPS in this context are typically of proteins harboring chemical and structural complexity, leaving unclear which properties are fundamental to complex coacervation versus protein-specific. This study focuses on the role of polyethylene glycol (PEG)—a widely used molecular crowder—in LLPS. Significantly, entropy-driven LLPS is recapitulated with charged polymers lacking hydrophobicity and sequence complexity, and its propensity dramatically enhanced by PEG. Experimental and field-theoretic simulation results are consistent with PEG driving LLPS by dehydration of polymers, and show that PEG exerts its effect without partitioning into the dense coacervate phase. It is then up to biology to impose additional variations of functional significance to the LLPS of biological systems.

Field theoretic approach for block polymer melts: SCFT and FTS.

This perspective addresses the development of polymer field theory for predicting the equilibrium phase behavior of block polymer melts. The approach is tailored to the high-molecular-weight limit, where universality reduces all systems to the standard Gaussian chain model, an incompressible melt of elastic threads interacting by contact forces. Using mathematical identities, this particle-based version of the model is converted to an equivalent field-based version that depends on fields rather than particle coordinates. The statistical mechanics of the field-based model is typically solved using the saddle-point approximation of self-consistent field theory (SCFT), which equates to mean field theory, but it can also be evaluated using field theoretic simulations (FTS). While SCFT has matured into one of the most successful theories in soft condensed matter, FTS are still in its infancy. The two main obstacles of FTS are the high computational cost and the occurrence of an ultraviolet divergence, but fortunately there has been recent groundbreaking progress on both fronts. As such, FTS are now well poised to become the method of choice for predicting fluctuation corrections to mean field theory.