Application of Monte Carlo simulation in addressing ...

CIS-01653; No of Pages 15 Advances in Colloid and Interface Science xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

Advances in Colloid and Interface Science journal homepage: www.elsevier.com/locate/cis

Application of Monte Carlo simulation in addressing key issues of complex coacervation formed by polyelectrolytes and oppositely charged colloids Jie Xiao a, Yunqi Li b, Qingrong Huang a,⁎ a b

Department of Food Science, Rutgers, the State University of New Jersey, 65 Dudley Road, New Brunswick, NJ 08901, USA Key Laboratory of Synthetic Rubber & Laboratory of Advanced Power Sources, Changchun Institute of Applied Chemistry (CIAC), Chinese Academy of Sciences, Changchun, PR China

a r t i c l e

i n f o

Available online xxxx Keywords: Monte Carlo simulation Complex coacervation Polyelectrolyte Macroion Electrostatic interaction

a b s t r a c t This paper reviews the recent advance of Monte Carlo (MC) simulation in addressing key issues of complex coacervation between polyelectrolytes and oppositely charged colloids. Readers were first supplied with a brief overview of current knowledge and experimental strategies in the study of complex coacervation. In the next section, the general MC simulation procedures as well as representative strategies applied in complex coacervation were summarized. The unique contributions of MC simulation in either capturing delicate features, easing the experimental trials or proving the concept were then elucidated through the following aspects: i) identify phase boundary and decouple interaction contributions; ii) clarify composition distribution and internal structure; iii) predict the influences of physicochemical conditions on complex coacervation; iv) delineate the mechanisms for “binding on the wrong side of the isoelectric point”. Finally, current challenges as well as prospects of MC simulation in complex coacervation are also discussed. The ultimate goal of this review is to provide readers with basic guideline for synergistic design of experiments in combination with MC simulation, and deliver convincing interpretation and reliable prediction for the structure and behavior in polyelectrolyte– macroion complex coacervation. © 2016 Elsevier B.V. All rights reserved.

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief overview of complex coacervation . . . . . . . . . . . . . . . . . . . 2.1. Driving forces and stability mechanisms . . . . . . . . . . . . . . . . 2.2. Phase boundary of complex coacervation . . . . . . . . . . . . . . . 2.3. Theoretical models for complex coacervation . . . . . . . . . . . . . 2.4. Experimental techniques to characterize complex coacervation . . . . . 2.5. Application status and fundamental research issues . . . . . . . . . . . Apply Monte Carlo simulation to study complex coacervation . . . . . . . . . 3.1. Modeling molecules in complex coacervate . . . . . . . . . . . . . . 3.1.1. Modeling of large macroions . . . . . . . . . . . . . . . . . 3.1.2. Modeling of flexible polyelectrolytes . . . . . . . . . . . . . 3.1.3. Modeling of small ions and solvent . . . . . . . . . . . . . . 3.2. Energy functions guided MC simulation . . . . . . . . . . . . . . . . 3.3. Equilibration and production runs of MC simulation . . . . . . . . . . MC in addressing key issues of complex coacervation . . . . . . . . . . . . . 4.1. Key parameters in MC simulation to characterize complex coacervation . 4.2. Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Identify phase boundary and disentangle interaction contributions 4.2.2. Clarify composition distribution and internal structure . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

⁎ Corresponding author. Tel.: +1 848 932 5514; fax: +1 732 932 6776. E-mail address: [email protected] (Q. Huang).

http://dx.doi.org/10.1016/j.cis.2016.05.010 0001-8686/© 2016 Elsevier B.V. All rights reserved.

Please cite this article as: Xiao J, et al, Application of Monte Carlo simulation in addressing key issues of complex coacervation formed by polyelectrolytes and oppositely c..., Adv Colloid Interface Sci (2016), http://dx.doi.org/10.1016/j.cis.2016.05.010

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2

J. Xiao et al. / Advances in Colloid and Interface Science xxx (2016) xxx–xxx

4.2.3. Predict influences of physicochemical conditions on complex coacervation . 4.2.4. Delineate mechanisms for “binding on the wrong side of the isoelectric point” 5. Concluding remarks: current problems and prospects . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Complex coacervation describes the phenomenon of two oppositely charged macromolecules in solvent separating into two liquid phases. Each phase contains both macromolecules in partition and/or chemical equilibrium [1,2]. Bungenberg de Jong and Kruyt coined this name to distinguish it from the coacervation of a single polymer after their pioneering investigation in the mixture of gelatin–gum arabic [3]. The resultant complexes have been roughly divided into three types according to their physical states: (i) water soluble complex; (ii) aggregated water-soluble complexes or complex coacervates; and (iii) insoluble amorphous precipitates or complex precipitation [4]. Among which, the water-soluble complex is often regarded as “precursor” or “primary complex” of coacervates [5], and our scope of discussion will restrain within these two physical states, complex precipitation will not be covered due to the scale limitation of simulation. In a broad sense, complex coacervation not only takes place among oppositely charged polyelectrolytes [6,7], but also occurs when one component is charged colloid (also referred to as macroions e.g. proteins, dendrimers, micelles, and nanoparticles [8–14]). In this review, we would like to extend our scope of discussion within the broad concept of complex coacervations formed among polyelectrolytes and macroions, which are relevant to many physiological processes and to a wide range of applications. Actually, ever since the pioneering work of Bungenberg de Jong and Kruyt in 1929, complex coacervates have experienced diversified, multi-directional developments. And rapid progresses reflecting the convergence of applications, instrumentations and simulations have been well recognized during the last few decades. For a long period of time, our understanding towards various types of complex coacervation systems relies heavily on the interpretation of experimental data obtained through experimental techniques. Nowadays, simulation reports based on molecular dynamic modeling, ligand-docking simulation and Monte Carlo (MC) simulation provide alternative ways to reveal the formation mechanisms and properties at spatial and temporal scales that are difficult to be experimentally observed [15]. Our discussion stream will focus on the particular contribution of MC simulation, while other closely relevant simulation methods will be mentioned as needed. We would like to first present the readers with a concise overview of current knowledge and experimental strategies in the field of complex coacervation. Constant research focuses within the regime of fundamental principles will then be brought forward. Afterward, readers are provided with basic knowledge of common operation protocol of MC simulation in the field of complex coacervation. These two parts are provided to supply users with the information required to understand the existing fundamental research objectives as well as case studies discussed later on. In the following section we will focus on the contributions of MC simulation in addressing some of the key research issues, and this part will be illustrated through analytical parameters involved as well as representative case studies.

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

0 0 0 0 0

interaction considered to tune complex coacervation under circumstances where components bring charges of opposite signs. While other interactions, such as hydrogen bonding and hydrophobic interaction, may contribute to the coacervation process under conditions where electrostatic interaction is suppressed (e.g. by high salt, at pH range where microions bring the same charges, or polyions interact with non-ionic polymers) [16–18]. In such cases, extraordinary challenges for interpretation and simulation are inevitable [19]. The stability of complex coacervation formed thereafter can then be qualitatively predicted based on the detailed information of interactions. Stability mechanisms, such as: charge neutralization [20–25], charge regulation [26], minimization of electrostatic interaction [27] and minimization of free energy (or energy functions engaged in computer simulations) [28–30], counterion condensation [31–34] have all been proposed to explain the stabilization of complex coacervation under certain conditions. 2.2. Phase boundary of complex coacervation Our current understanding towards physical state prior to and during the coacervation process is largely shaped by the combined experimental approaches of titration, light scattering and phase contrast microscopy [9,22,35,36]. One of the most detailed descriptions of phase transition processes was proposed specifically for oppositely charged protein and polysaccharide system by Kaibara et al. [9] via correlating them with a series of experimentally observed pH values. As shown in Fig. 1, the onset of soluble complex is identified as pHcrit where turbidity shows a sharp increase within 0.1 units. The formation of soluble primary complexes then proceeds until “pH′crit”, followed by a region (pHcrit b pH b pHpre) of stable intrapolymer complexes. The aggregation of primary complexes starts at “pHpre” where scattering intensity begins to rise. Subsequently, a maximum in scattering intensity at pHφ is observed coincident with the appearance of turbidity and the first microscopic observation of coacervate droplets. Afterward, turbidity increases abruptly, and microcoacervate droplets experienced morphological changes at “pHmorph”, followed by the transformation to solid precipitates or flocks at pHprecip. Since complexation is electrostatically driven to a large extent and coacervation tends to be maximum when the stoichiometry of the

2. Brief overview of complex coacervation 2.1. Driving forces and stability mechanisms At the molecular interaction level, electrostatic interaction among components is the most widely accepted driving force and the major

Fig. 1. Schematic pH profile of turbidity (τ) and scattering intensity (I90). Arrows indicate specific pH points, pHcrit, pH'crit, pHφ, pHpre, pHmorph, and pHprecip. Adapted and modified with permission from Ref. [9]. Copyright 2000, American Chemical Society.



macroion charges is equal to one. The phase boundary is thus highly sensitive to whichever physicochemical conditions that may affect the binding initiation all the way up to the eventually charge equivalence phase separate process. Generally speaking, there is a narrow window of physicochemical conditions where the complexes are formed and stay in liquid phase [37], identification of which relies on tracing of changes rooting from either structure or interaction energy.

2.3. Theoretical models for complex coacervation There have been several attempts to construct and refine theoretical models to account for particular features of complex coacervation. The major challenge is to find accurate approximations for the free energy, which sensitively depends on the as yet poorly characterized spatial correlations of components in complex coacervates [37,38]. Voorn and Overbeek [39] proposed the first theoretical description of complex coacervation by estimating the total free energy of mixing as a sum of random mixing entropy terms and electrostatic interaction potential in a Debye–Hückel approximation. Subsequent theoretical models were developed by Veis et al. [40] by including non-electrostatic interaction effects into the Voorn–Overbeek theory. Later on, the random phase approximation and loop expansions theory for weakly charged polyelectrolytes was developed to make up the inherent weak spatial correlations of the Voorn–Overbeek approximation [41–44]. Recently, Biesheuvel and Stuart [45–47] extended the Voorn–Overbeek approach by describing the electrostatic free energy term in more detail. They described ionizable groups along the polymer chains with pHdependent charges. Ions in system are part of the diffuse layer around the polymers in cylindrical geometry, and either the linearized or the full Poisson–Boltzmann equation is solved to calculate their contributions. While this expression accurately describes weak correlations, it is a poor approximation for the much stronger spatial correlations in systems containing macro-ions. Thus apart from these theoretic models particularly considered complex coacervation between oppositely charged flexible chains, other models, such as self-consistent field theory for complex coacervation of flexible chains and spherical macro-ions was recently proposed [48,49]. Overall, the presence of macroions, proteins or colloids creates discrete interface, which violate mean field based theoretical approximations and make theoretical models lose integrity and robust in different complex systems. It is still full of challenge to present theoretical models that can cover variant complex systems, transfer conclusions to different systems still like treading on thin ice.

2.4. Experimental techniques to characterize complex coacervation Experimental techniques and their relevance for understanding and characterization of complex coacervates have involved: titration method to identify the phase boundary [35]; static light scattering to yield quantitative information on both molecular weight and dimensional parameters (e.g. fractal dimension (Df) and radius of gyration (Rg)); facilitated with hydrated radius (Rh) extracted from dynamic light scattering, the ratio of Rg/Rh infers the nucleation and growth mechanism for complex phase separation [50,51]. Small angle neutron scattering (SANS) is capable of accessing structural parameters, including Rg and Df, at much detailed spatial scales than that from static light scattering techniques [52]. Enthalpy changes (△H) and Gibbs free energy changes (ΔG) derived from isothermal titration calorimetry (ITC) enable the estimation of binding constant, number of bounded macroions, saturation binding and evaluation of possible driving forces [53,54]. For detailed information on basic principle as well as experimental practices of these research methods readers are referred to two comprehensive reviews compiled by Cooper et al. [8] and Kayitmazer et al. [50].

3

2.5. Application status and fundamental research issues Complex coacervates may have different hierarchical structures (length scale ranging from nanometers to micrometers) beyond assembled complexes, such as coacervates, multilayers, and polyelectrolyte brushes [8]. Utilizing them to application-oriented ends has always been research hot zones. Tremendous research practices have been applied in areas such as microencapsulation vehicles (for oils, proteins, peptides, genes, small molecular weight bioactives, live cells, etc. [21,55–59]), cell seeding scaffold [60], fat replacers [61], flavor-masking, protein separation, enzymes immobilization, rheological properties tuning agent in food processing [62,63], and emulsion stabilizer [64–68]. Their widely adjustable compatibilities and readily tunable stimuli responsibilities highlight them as versatile formulations. Fundamental understanding and cutting-edge application of complex coacervation rapidly accumulate in recent years. Here, we summarized the key issues, i.e. research targets related to complex coacervation as following: a) Identification of molecular interactions and thus driving forces for complex formation; b) establish phase diagrams defining the various stages of phase separation; c) identify special correlation between components as well as binding induced conformational changes during complex formation process; d) predict influences of intrinsic and external physicochemical conditions on the phase behaviors and the properties of the resulting products; e) develop ubiquitous physical models and approaches to explore rational characteristics for variant complex systems.

3. Apply Monte Carlo simulation to study complex coacervation Monte Carlo simulations have been widely applied to study phase and aggregation behaviors, interactions and structures in liquid media. For example, physical gelation in polymer solutions [69– 75], adsorption of polyelectrolyte onto membranes [76–79], various phase transitions and phase equilibrium of polymer solutions [80], protein ligand complex structures [81], folding and unfolding of biomacromolecules [82–84], specific charge distribution on the surface nanoparticles [85], polyelectrolyte–macroion complexation [86] and many other systems have been subjected to MC simulations. Despite the vast amount of sampling applications for particular subjects, herein, we present major considerations of MC simulation in the study of complex coacervation. The key concept involved is to generate an ensemble of representative configurations under specific thermodynamic conditions for a complex coacervation system via Markov procedures. Consequently, conformation, structure, index of aggregation and thermodynamic observables, such as the size of macromolecules, the structure factor, the secondary Virial coefficient and the mixing free energy, may be calculated under the frame of statistical thermodynamics. As a liquid–liquid phase separation, complex coacervation spans spatial scale from tens to hundreds of nanometers. In such scale, millions of atoms in MC simulation are to be tracked, stored and their free volumes are exhaustively sampled. It is thus imperative to use alternative approaches to reduce the population of particles in simulation, and coarse grain modeling is the most popular and successful one. Our objective in this section is not to explain the fundamental algorithms but to highlight their specific operation procedure in modeling, free energy calculation and sampling in the context of complex coacervation. For detailed algorithmic approaches and fundamental discussion associated with sampling of the representative space, determination of the number of microstates associated with a macromolecule, the probability of occurrence of a specific conformation, and the ensemble averages of observables, readers are recommended with the textbook [87,88] and other reviews [89,90] as references.


4


3.1. Modeling molecules in complex coacervate Modeling and parameterization of molecules are always the start point for simulation. These molecular models then are damped into a simulation lattice with finite and reasonable size, and periodic boundary condition is generally applied on the lattice so to study various phase behaviors. For complex coacervation, it would involve the modeling of up to three distinct types of molecules: (i) large charged macroions, typically proteins, nanoparticles and their surface modified forms, or micelles; (ii) linear polyelectrolytes; (iii) the accompanying electrolytes, usually counter-ions and added salt in forms of cations or anions and solvent. Depending on the target of simulation and requirements for simulation accuracy and efficiency, different modeling strategies may be applied to each type of molecules.

3.1.1. Modeling of large macroions The surface properties and topology of a protein in simulation may be maximized representation using atomic model or minimized represented using hard sphere model. Atomic model of a protein requires thousands of vectors, while the hard sphere model of a protein only has one vector. In practice, the parameters of protein models are usually derived from native or crystal structures deposited in Protein Data Bank (www.rcsb.org). Coarse graining can be performed at four levels: atomic and residual [91], cluster of residues with similar properties [92], patches composited either by cluster of residues [93,94] or overlap of electrostatic force [95], or homogenous spheres [96]. Representative coarse graining models for proteins are depicted in Fig. 2. In these models, partial charge of a modeled particle, regardless of it being an atom, a residue or a sphere, is critical to build complex and study complex coacervation. Partial charges can be assigned according to disassociation equilibrium of Lewis groups or the solutions of Poisson–Boltzmann equation at selected pH and salt concentrations with the assistance of program DelPhi or UHBD, etc. [97,98]. Depending on the residues distribution and titrational state, the charge patches can be further divided as positive, negative and even hydrophobic patches. Overall, the level of resolution and the degree of flexibility of proteins vary from several particles per amino acid to one particle per protein, depending on the size of the object and motions that need to be described [99]. Besides the surface properties of proteins, which make each has unique feature distinct from other proteins, the topology of proteins is another important identifier. Many proteins have topologies far from a sphere. Unlike proteins, other macroions including micelle and nanoparticle are usually modeled as spheres with uniform distributed surface properties. Either penetrable or impenetrable surface layer, and the diameter, charge density and charge distribution of the spheres can be adjusted according to modeled target.

3.1.2. Modeling of flexible polyelectrolytes Synthesized or natural polyelectrolytes (PEs) are normally modeled as linear chains of beads (or segments) in freely jointed chain model, freely rotation chain or wormlike chain model [100]. The rigidity of chain can be easily tuned by the angle and dihedral energy term, while nonbond energy terms related parameters like van der Waals radius and partial charge of each bead can be exactly computed from the chemical structure of monomer in PE under the frame of general molecular force field [101]. Note that depending on the nature of PE, strong or weak, partial charge on beads or charge distribution on polymer chains was either fixed or adjusted based on the dissociation of functional groups that are sensitive to pH variations. 3.1.3. Modeling of small ions and solvent There are implicit and explicit modeling for small ions and solvent [102]. In the explicit modeling, counter-ions or salt ions in the surrounding solution are considered in the model system, they are usually described as hard spheres of radius Rion with the valency of zion, that are free to move in any direction. In the implicit modeling, both small ions and solvent are treated in a force field, with change in parameters such as the relative dielectric permittivity εs (liquid media), the Debye screening length (ionic strength) and the disassociation fraction chargeable groups (pH), while the same uniform static dielectric permittivity (εr) is assigned to all charged species interior in consideration of their dipole-charge interactions. 3.2. Energy functions guided MC simulation The core of MC simulation is the design of efficient energy function to guide simulation to a state close to real case [103]. A general form of the energy function for the simulation of complex coacervation can be written as: E ¼ Enonbond þ Elocal

ð1Þ

The nonbonded energy term between atoms (or particles), is often described as a sum of excluded-volume potential presented via a Lennard-Jones potential term and the electrostatic interaction potential computed by Coulombic force: " Enonbonds ¼ ∑

nonbond

4εij

σ ij r ij

12

6 #! qi q j σ ij þ r ij 4πε0 εs r ij

ð2Þ

Here the interaction potential particle pair i and j denote either a polyelectrolyte segment, a unit in macroion, an ion or a solvent molecule. Σij = (σi + σj)/2 and εij = (εiεj)/2 with σi and εi are the van

Fig. 2. Representative coarse grained MC modeling strategies for protein: (A) Atomic model for human serum albumin (HSA); (B) amino acid residue level model for HSA; (C) protein modeled based on characteristic distributions of surface charge and hydrophobic patches. Red, blue and green colored spheres represent positive, negative and exposed hydrophobic patches; (D) protein symbolized as soft sphere with positive and negative charge patches randomly distributed on surface. Reprinted with permission from Ref. [91] Copyright 2014, American Chemical Society; Ref [115] Copyright 2013, American Chemical Society; Ref [94] Copyright 2012, American Chemical Society, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)



der Waals radii and the Lennard-Jones well-depth of particle i, respectively. rij stands for their separation distance, ε0is the vacuum permittivity, εs is the relative permittivity in a continuous media, qi = zie and qj = zje (e is the elementary charge, z is valency value) denote the partial charges on particles i and j, respectively. In simulation, to avoid attractive electrostatic interaction induced particle overlap, a hard-core overlap restriction is usually imposed [104]: 8 < uij r ij ¼ :

∞ r ij bRi þ R J qi q j r ≥R þ R j 4πε 0 εs r ij ij i

ð3Þ

Alternatively, a more accurate modeling of electrostatic interaction using DLVO potential [91] [20] has been used, it provides better description for complex coacervation. The visualization of the overlap of electrostatic interaction for macro-ion can be assisted by Delphi electrostatic modeling, which solves the Poisson–Boltzmann equation as a function of pH and ionic strength. The electrostatic interaction of ions and solvent molecules can be explicitly counted from all pairs using Coulomb potential or implicitly approximated from an integration of continuous media along contourion condensation interface, i.e., Stern double layer. Apart from the van der Waals interaction and the electrostatic interaction, other terms such as hydrophobic interaction [91], π–π stacking [105] and hydrogen bonding [106] can also be considered. The development of an accurate and computation efficient energy function is still in development. The local energy term Elocal, normally contains bonds, angles, dihedral angles and improper dihedral potentials [107], if considered, is to ensure molecules in the simulation always has reasonable geometry. The parameters of energy terms can be derived from either from quantum mechanics computation, or generalized molecular/coarse grained force field such as CHARMM, AMBER, OPLS/ SA, and MARTINI. 3.3. Equilibration and production runs of MC simulation Typical MC simulation consists of the pre-equilibration phase (usually in the order of at least 105 trial moves per particle) and the production runs [104]. Equilibrium properties of complex coacervation systems can be generated using the traditional metropolis MC algorithm in the canonical (with a constant number of particles, volume, and temperature) or grand canonical ensemble guided by the energy function. The appropriate coordinate space was sampled by randomly modifying subjects' positions and orientations using ergodic sampling, such as (i) randomly translation; (ii) pivot rotation of a part of the PE chain; (iii) translation of the entire molecule, and (iv) slithering move at either end of the whole chain [94,108]. When the motion in simulation involves large amount of particles, synergistic relaxation is needed, the step and the fraction of motions also require careful tune

5

to guarantee acceptable sampling efficiency and the ergodity of simulation trajectories. Following each elementary random move, the change in Hamiltonian (△H) or energy (△E) of the simulation configuration is counted. It is the key factor to determine the acceptance or rejection of a movement according to the widely Metropolis sampling criterion [109], i.e., the probability to accept is min{1,exp(−ΔE/kBT)} [110]. The energy computation is the most time consuming part, especially at the presence of long-range electrostatic interaction. Truncation at sufficient scale to accelerate the computation is widely used in MC simulation [89], while it is estimated using Particle-Mesh Ewald summation with higher accuracy in molecular dynamics simulation [111]. On the other side, smoothening energy landscape of complex is also helpful to locate the near-native complex structure at the deepest funnel. Replica exchange Monte Carlo simulation [91,112], umbrella sampling, accelerated MC [113], and using softened excluded volume effect such as the usage of truncated and shifted Lennard-Jones potential [114] are ways to avoid simulation trapped in local minima and facilitate simulation reaching the most favorite configurations for a given energy function. Overall, MC simulation set up is always a combined consideration of the accuracy and efficiency of system description. For parameters involved in MC modeling and their setting references, readers are referred to Table 1. 4. MC in addressing key issues of complex coacervation From the simulation settings described above, we get to know that MC simulation can provide information at the levels of residues, molecules, aggregates of molecules/colloids and the associated phase behaviors. The complexity of simulation can cover system with one macroion and one polyelectrolyte to system comprising of multiple macroions and polyelectrolyte chains, while the investigation scope may range from one physical event to several physical states. The correlation of these pieces of information and stringent comparison with experiments make it perfect to arouse molecular mechanism for a given phase behavior and the formation of well-controlled assemblies (Fig. 3). In this section, some of the most widely applied analytical strategies in MC simulation will be first presented and further illustrated through a number of case studies (Table 2). Nevertheless, the following section needs to be prefaced by the disclaimer that the MC simulation studies cited below cover only tip of the iceberg, which is by no means exhaustive. 4.1. Key parameters in MC simulation to characterize complex coacervation MC simulation characterizes complex coacervation through a number of parameters: configurations/snapshots under thermodynamic

Table 1 General setting parameters in MC simulation Parameters

Consideration basis for setting

Parameters

Consideration basis for setting

Simulation box length (L) or cell radius (Rcell)

Dimension and concentration of inclusive subjects Simulation scenario Simulation scenario Intrinsic property of protein Crystal structure of protein

No. of mobile ion (Nion)

Simulation scenario

Mobile ion radius (Rion) Mobile ion charge (Zion) Bulk relative permittivity (εs) Effective dielectric constant for protein, PE or micelle (εr) Force constants (K) Backbone angle/dihedral terms of PE (θ/φ/ψ) pH Temperature (T)

Simulation scenario Simulation scenario Solvent property Assumption Distribution in solvent continuous media Simulation scenario Local structure, rigidity Simulation scenario Simulation scenario

Ionic strength (I)

Simulation scenario

No. of macroions (Nm) No. of PE chains (NPE) Macroion radius (RM) Equivalent radius for protein residues (Rres) No. of segments in each PE chain (Nseg) Radius for each segment in PE chain (Rseg) Partial charge for amino acid residue bead (Zres) Surface charge density for nanoparticles/micelles (ó) Partial charge in PE segment (Zseg)

Intrinsic property of PE Intrinsic property of PE pK values of ionizable residues Dissociation or protonation of functional groups Dissociation of charged groups in PE


6


Fig. 3. Work flow chart of MC simulation in study of complex coacervation.

equilibrium; energy and energy terms to address driving forces for the phase separation; spatial correlation of molecules around macroions and orientation enhanced correlation coefficient to clarify microstructure in the aggregates and coacervates; radii of gyration or the mean end-to-end distance to track the conformation changes of PE; interaction enhanced pair correlation function, second virial coefficient, and aggregation of molecules to describe the associated states, etc. To determine the contribution of each kind of elementary interactions (mainly nonbond interaction), comparison of one of the above parameters that have the best correlation with experimental observation can be carried out. It can be used to construct a phase diagram and comparable with that which is experimentally determined. The “switch on” of the energy term, or its weights can be readily changed during MC simulation. The best match in the statistical parameter then is decision of the final simulation strategy and the presentation of mechanism. In the MC simulation of complex coacervation, the criteria to identify the association of components are critical. It has three forms: i) the cutoff of free energy; ii) the cutoff of distance for interacted pairs; or iii) the lifetime of binding state. Using the first criterion, the association of two molecules/colloids can be counted when their interaction E b ΓkBT, where the cutoff Γ = − 1 representing random associations, more negative values standing for stronger association [91,94]. Besides, a dynamic definition of the transition from free to bound state is depicted by calculating the probabilities of finding a specified PE a given distance from the center of mass of the macroions. The probability can be counted either from the histogram in pair distribution function (PDF, an alternative form for the correlation function), or the lifetime of bound state in a simulation trajectory, a higher probability indicates that they prefer aggregated state [42,95]. With regard to the conformational changes, the coacervates formed can be analogous to percolated network formed through the bridging

effect of associated PE and macroion, the extent of which can be monitored through counting the geometric percolated aggregates formed in a simulation configuration. Under thermodynamic equilibrium, the number of percolated samples out of the sampled configurations (i.e. percolation probability) as well as the volume fraction of the largest aggregation simulation box can be calculated to a give a rough picture of composition distribution [20,94,115]. The conformational changes of PE before and after absorption onto macroions can be derived through calculating parameters that characterizing its extension and shape, such as the average angle between three consecutive polyelectrolyte segments, the root-mean-square end-to-end distance [42], and the mean-square radius of gyration (bRg2 N) of PE [116]. The microstructure, i.e. the distribution and the association of molecules in simulation configurations can also be viewed through the pair correlation function (PCF) and the second virial coefficient (B2) [115]. A value of PCF larger than 1 indicates the enrichment of two molecules at given separation distance, while less than 1 suggests the depletion of the two molecules at given correlation length. A negative or positive B2 value indicates attractive or repulsive interaction, respectively, with the magnitude of B2 value proportional to the binding affinity or the repelling strength. 4.2. Case studies 4.2.1. Identify phase boundary and disentangle interaction contributions Identification of the phase boundary between soluble and insoluble complexes under defined external conditions and disentangle the contribution from each interaction have been a constant research subject for each complex coacervation system of interest. In one of our recent work [94], we considered not only the classical complex association between proteins and PE chains, but also the contributions from intermolecular self-aggregation of proteins and entanglement of


Components

Modeling set up

Items in energy function

Calculated items for statistical analysis

Contribution of MC simulation

Ref.

Serum albumin; model polycation

Protein is simplified as a soft sphere with charge patches on surface; polycations are soft beads connected at fixed bond lengths with partial charge at the center of mass.

Van der Waals interaction; electrostatic interaction potential

Clarify how nonspecific interactions such as electrostatic and van der Waals interactions affect complex formation

[94]

10 polycations and a variable amount of polyanions

Polyions are modeled as freely joined charged hard spheres connected by harmonic bonds

Electrostatic interaction via screened Coulomb potential and bond potential Electrostatic interaction among particles and beads

Distribution of intermolecular interactions under different pH values; percolation probability; the volume fraction of the largest aggregate; correlation functions of PE segments and proteins Cluster size probability distribution; partial structure factors of positive beads

Cluster composition of systems containing polycation and different amounts of polyanion

[47]

Probability of occupation of monomer as a function of the monomer position; maximum number of adsorbed NPs at PE saturation

Changes in conformation and structure of complexes when ionic concentration, pH, NP surface charge density, and PE intrinsic rigidity are modified

[116]

Electrostatic interaction potential; bond potential and angle potential

Microions and PE distribution probability; structure factors

[104]

Van der Waals potential; electrostatic interaction potential; hard-sphere potential involving only charge patches Electrostatic interaction potential; bending energy of polyelectrolyte Electrostatic interaction potential

Percolation probability; the volume fraction of largest aggregate

The complexation, phase separation, and redissolution behavior of macroion solution with addition of PE; effects of macroion radius, chain length, and chain flexibility on the phase separation behavior Effects of pH, the ratio of protein to polysaccharide on complex coacervation; the distribution of binding sites

Effects of PE chain stiffness, charge mobility, and charge sequences on binding to proteins

[133]

Driving forces for the adsorption of charged spherical proteins onto oppositely charged surface-grafted polymers; effects of fundamental variables on particle adsorption. The influence from the specific distribution of charge and hydrophobic patches in protein surfaces on the self-association of proteins and their complex coacervation Complexation responses as ionic strength, protein net charge, and protein charge distribution; the distribution profile of PE beads at the protein surface.

[134]

Positively charged NPs are modeled as impenetrable spheres and positive charge znp is nanoparticles and rod-like or concentrated into a point located on their centers; PE is modeled as joined hard sphere Nm = 200 with strictly linear or flexible chain flexible linear PE and each monomer can either carry a charge zm = −1 or be uncharged. Macroion solution with Large unconnected spheres represent macroions; connected spheres different amount of represent PE; small unconnected spheres represent small cations oppositely charged PE chains and anions

Bovine serum albumin (BSA) and pectin

Protein is modeled as a soft sphere with charge patches, the partial charge at each residue is calculated according to acid–base equilibrium; pectin is modeled as freely jointed chain of soft beads

BSA and polyanion

BSA is modeled as impenetrable, uniformly charged sphere; PE monomer is modeled as hard sphere with negative or neutral charge

Model proteins and surface-grafted charged polymers

Protein is modeled as a spherical particle with net charge placed in the central lattice; polymer segments are flexible with the polymerization degree of 30

BSA, BLGA, BLGB and pectin

Proteins are modeled based on distributions of surface charge and hydrophobic patches; pectin is modeled as a string of freely joined soft beads

Lysozyme and model PE

Protein is represented by a hard sphere with embedded pH-dependent discrete charge; PE is modeled as sequence of negatively charged hard spheres

One macroion and one oppositely charged PE chain

Macroion is modeled as impenetrable and uniformly charged sphere; PE chains are represented as freely jointed hard spheres

β-lactoglobulin (BLG), α-lactalbumin; one model PE

Protein with Nres residues and Nch charged groups are modeled as rigid bodies; polyanion is modeled as a chain of Nmon = 20 spheres with radius Rmon

Lysozyme, a-lactalbumin, β-lactoglobulin; flexible PE

Proteins are modeled as rigid bodies in full atomistic detail; PE is modeled as a chain of Nmon = 21 charged hard spheres of radii Rmon and charges of qmon = −e

Lysozyme, a-chymotrypsinogen, and calbindin D9k Albumin, goat, bovine α-lactalbumin, β-lactoglobulin, insulin, κ-casein, lysozyme, pectin methylesterase; flexible PE

Proteins were modeled in atomic model or amino acid model

Proteins are modeled as rigid bodies in full atomistic detail; PE is modeled as a chain of Nmon = 21 charged hard spheres of radius Rmon with fixed valency zmon = −1

Electrostatic interaction potential; hydrophobic potential; and van der Waals interaction Excluded volume potential; electrostatic and hydrophobic interaction potentials Electrostatic interaction potential; bending energy of polyelectrolyte chains Electrostatic interaction potential; monomer-charge and monomer-residue excluded volume potential Electrostatic interaction potential; bond interaction potential among PE Εlectrostatic interaction potential; van der Waals interactions Electrostatic interaction potential; bond interaction potential among PE

Critical conditions for complex formation; electrostatic potential models of protein created at critical binding conditions The free energy of moving a protein from the bulk to a absorption layer; the partition coefficient of proteins between the bulk from the surface and a position at z layers from surface Percolation probability; pair correlation function profiles of proteins and pectin; second virial coefficient

Running coordination numbers; the probability of contact between the protein and PE beads; the distribution of PE beads in a plane from the protein center Simulated polyelectrolyte chain scattering functions Influence of ionic concentration, intrinsic chain rigidity, PE contour length, size of the macroion on the scattering property of complex Electrostatic interaction energy; critical ionic Correlate differences in complexation behavior to strengths; center of mass distribution for differences in distribution of “charge patches” polyelectrolyte pI values for proteins; dipole moment number; protein capacitance at pI; the contribution to free energy of interaction from the charge-induced charge term and the ion-dipole term Free energy of interaction for proteins with different counterion valencies; second virial coefficients for protein as a function of pH Charge capacitance; dipole moment number; minimum free energy at the protein–PE separation distance R

[20]

[115]

[42]

[137]



Table 2 Non-exhaustive case studies of MC simulation in addressing key issues in complex coacervation

[108]

Characterize the charge fluctuations effect of protein when subject to perturbation by polyelectrolyte

[142]

Describe interactions between protein molecules in solution; fluctuations of residue charges in two proteins at their IP can result in an attractive force Relevance of protein capacitance to the potential of mean force between protein and PE; the importance of charge regulation in complex formation at vicinity of pI.

[145]

[143]

7

8


PS chains. Three types of aggregates were distinguished by setting threshold intermolecular interaction potentials and defining a morphological definition of entanglement. As presented in Fig. 4 (A), the MC simulation was able to distinguish: i) the classical protein–PE complex formed through strong intermolecular interactions (− 8 kBT cutoff, presented in the green color); ii) complex further expanded through protein self-aggregation (purple) and iii) aggregates enlarged through entanglement of PE chains (presented in orange color). Such approach helps to decouple the contribution of protein self-aggregation and polysaccharide chain entanglement on the formation of complex coacervation. As mentioned in previous section, complete phase diagram can be grasped by calculating percolation probability and volume fraction of the largest aggregate based on MC simulation configurations. One of the representative results is shown in Fig. 4 (B), the percolation probability curve calculated at given pH values and the cutoff of intermolecular interaction (Γ) displayed similar upswing and plateau tendency as that of the volume fraction of the largest aggregate, and they showed slightly different extent of decline at high pH values. Both curves delivered consistent phase boundaries in comparison with pH titration experiments of the corresponding systems [94,115]. Furthermore, by calculating percolation probabilities and volume fractions of the largest aggregates of protein (PRO)–polysaccharides (PS) complex, PRO–PRO self-aggregation and PS chain entanglement under different pH and Γ, their respective contributions along with the entire phase boundaries were disentangled. As it is shown in Fig. 4 (B), the PRO–PS complex dominated complex formation under weak intermolecular association (Γ = − 2). Under strong electrostatic attraction (Γ = −5, −8), PRO–PS dominated complex formation at low pH values, while PRO–PRO and PS–PS entanglement exhibited visible contribution at the high pH region. When Γ was set to −10, contribution from PRO self-aggregation was negligible, while PS–PS aggregates still made visible contribution at low and high pH values. In another work, delicate differences in phase boundary of complex formed by different protein (two isomers of beta-lactoglobulin (BLGA, BLGB) and BSA) and pectin were presented through PCF profiles against separation distance at different pH values [115] (Fig. 4(C)). While no significant difference was found between PCFPS–PRO curves of BLGA– pectin and BLGB–pectin complexes, the PCF profiles were sensitive enough to reflect the distinct profile for BSA–pectin complex and, compared to BLG–pectin complex, BSA resulted in complex coacervation with narrow pH window. By further comparison of PCFPRO–PRO (r) and PCFPS–PS (r) curves among three proteins, the contribution of hydrophobic interactions, self-association proneness of protein in complexes were able to be disentangled.

4.2.2. Clarify composition distribution and internal structure The composition and internal structure of complex coacervates deal with structural dimension of multi-length scales in accordance with events of binding, soluble complex formation and phase separation. Structural features of interest have included: the domain of the protein active in the binding process, the number of macroions bound per polymer chain, the dimensions of complex, number of macroions in an aggregate, local arrangement of polyelectrolyte segments in their binding site, etc. Experimental achievements in proposing and refining possible models for composition distribution and internal structure of complex coacervate have included the combined usage of light scattering, rheology, cryo-TEM, confocal scanning laser microscopy, small-angle neutron scattering, etc. [50,117–121]. Recent research attempts of introducing MC simulation into the solution gallery lead to disclosure of the composition and structural information in a more detailed and intuitive manner. One of the representative case studies was conducted by Kayitmazer and coworkers [95]. In their study, full atomistic MC simulation and electrostatic modeling (Delphi) were applied to reveal the binding conformation of one synthetic polyanion on the positive domain of BSA. The conformational snapshots corresponding to minimize binding energy at 200 K, 400 K, 600 K, and 1000 K MC steps (Fig. 5) clearly demonstrated that the bound decamer preserved some degree of conformational freedom in its binding site. The observed multiple conformations further triggered the hypothesis that the variety of charge sequence arrangements in PE may be complementary to protein charge patterns to give rise to local binding selectivity, which can be quite instructive in our understanding of the local association of protein and PE. Apart from identifying local binding confirmation, MC can be robust in differentiating the composition and conformation distribution within complex coacervate system involving several molecular. The delicate composition and conformational differences between complex coacervation systems formed by protein (BSA, BLGA, BLGB) and pectin at different pH values were illustrated by coarse-grained MC simulation. By setting threshold potentials and defining morphological definition for each type of aggregation, schematic diagrams with rich information on composition and special correlation were obtained (Fig. 6). For all the three systems, complex coacervate with broadly spatial expansion and thus high percolation probability was observed at intermediate pH (pH = 3.5). At low (pH = 2) or high pH (pH 6.0), where proteins interact with neutral or negatively charged pectin chains, protein–pectin complexes were highly suppressed. And the entanglement of pectin chains played non-negligible role in the extension of complex coacervation.

Fig. 4. (A) Equilibrium configuration of protein–polysaccharides complex at pH 10.5. Polysaccharide chains are presented as cyan beads, proteins are modeled as large isolated spheres with red and blue dots representing positive and negative charge patches, respectively; (B) effects of pH on percolation probability (left) and the volume fraction of largest aggregates (right) at different intermolecular interaction cutoffs. Black squares, red spheres, and blue triangles represent aggregates formed between PRO–PS (type I), PRO–PRO (type II) and PS– PS (type III). (C) Pair correlation function profiles from BLGA, BLGB, and BSA and pectin complex systems presented by three kinds of correlations at different pH values. Reprinted with permission from Ref [94], Copyright 2012, American Chemical Society, and Ref [115], Copyright 2013, American Chemical Society. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)



9

Fig. 5. Full-atom MC simulation showing the mobility of the bound synthetic polyanion resides on the positive domain of BSA. Red and blue correspond to the negative and positive potentials, respectively. Each conformation corresponds to the minimized binding energy at given Monte Carlo steps: (a) 200 K, (b) 400 K, (c) 600 K, and (d) 1000 K MC steps. Reprinted with permission from Ref. [95], Copyright 2010, American Chemical Society. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The capacity of MC in depicting composition and structure is even utilized to capture the dynamic process corresponding to the “colloid titration” experimental approach. Hayashi et al. investigated the

evolution of complexes in solutions containing polycations with increasing amounts of polyanions by MC simulations [47,122]. The authors defined the correlation between individual polycation beads

Fig. 6. Snapshots of simulation configurations under thermodynamic equilibrium. Red, blue, and green spheres are positive charge, negative charge, and exposed hydrophobic patches in protein surface; purple bubbles represent the excluded volume of the whole protein, and pectin beads are cyan. The black colored out the largest protein–pectin aggregate in each configuration, and the yellow colored out the aggregate attaching to the largest aggregate through protein self-association. Reprinted with permission from Ref [115], Copyright @ 2013, American Chemical Society. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


10


as the bead structure factor, which is a product of polycation form factor and a center-of-mass structure factor, to characterize the structure of the systems. From the simulations, the composition of complexes formed was presented by distribution functions that show the probabilities of all possible combinations of polyions. By defining the threshold of cluster formation, the cluster size probability distribution was also calculated. Configurationally snapshots obtained through MC simulation clearly elaborated the structural evolution process as: with only polycations, a homogeneous polyelectrolyte solution stretched by electrostatic repulsion was observed; when five polyanion chains are added (Nneg = 5), five separate complexes were discernible, each complex containing two polycations and one polyanion; with Nneg = 9, the snapshot showed two complexes, one containing nine polycations and eight polyanions and another made up of the remaining two polyions of opposite charges; finally, with Nneg = 10, the solution underwent large structural changes and several neutral complexes were found. Adapted similar MC simulation and analysis approaches as Hayashi et al., Skepo and Linse [104] investigated the structural evolution in solutions containing macroions with increasing amounts of polyanions. Simulation snapshots were able to track the sequence of states starting from homogeneous macroion solution to stable solution containing macroion–PE complexes to two-phase system comprising of one macromolecules rich phase and one poor in them to a single and stable macroion–PE solution (re-dissolution). The structure factors derived from the simulation at different stoichiometric charge ratios

help to finely depict the spatial correlations of macroion–macroion, macroion–PE segment, and segment–segment. 4.2.3. Predict influences of physicochemical conditions on complex coacervation Accumulated experimental evidences have proved the strong dependency of complex coacervates on multiple external factors (i.e., pH, temperature, ionic strength, and mixing ratio etc.) and intrinsic characteristics (i.e., molecular weight, surface charge density, chain rigidity and charge distribution etc.) [36,123–127]. Although common features or tendency of complex coacervates under some of the above-mentioned parameters have been summarized as empirical principles, distinctive responses to extrinsic stimuli still widely existed. Compared to experimental approaches, MC simulation can be both effective and highly sensitive [86,128–134]. The effect of BSA–pectin ratio on the pHc values and phase boundaries was delivered through the percolation concepts and simulation configurations [20]. MC simulation not only presented almost identical phase boundaries as that obtained from turbidimetric titration curves, but also elucidated internal structural of complex coacervates along the complex coacervation formation pH window which will be otherwise implicit via sole titration or light scattering methods (Fig. 7). The complexes between PE and spherical nanoparticles (NPs) are research subjects of a large number of MC simulations reported in the literatures [135–137]. Stoll and coworkers [116] employed MC

Fig. 7. (A) Percolation probability (left) and the volume fraction of largest aggregates (right) as a function of the protein/polysaccharide ratio and pH at different intermolecular association strength cutoffs; (B) simulation configurations under thermodynamic equilibrium. Each column shows configurations at different BSA/pectin ratios of 5:1, 1:1 and 1:5, and each row presents configurations from pH 2.0, 3.5, 4.7 to 5.5. In each snapshot, gray lines show the periodic boundaries, cyan beads, red/blue spheres, and green bulbs represent BSA beads, positive/negative charge patches and the excluded volume of proteins, respectively. Adapted with permission from Ref. [20]. Copyright 2013 Elsevier Ltd. All rights reserved. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)



simulation in investigating the effects of ionic strength (Ci), PE intrinsic rigidity, NP surface charge density (σ) and (pH − pK0) on complex formed between a weak PE and several NPs. To determine the favorite binding regions, they defined a parameter ηi as the ratio of the number of times a monomer i is in contact with a NP over the number of MC steps of the production period. In this way, variation of the probability of occupation of monomer ηi as a function of the monomer position i for PE in presence of different numbers of NPs and values of σ at fixed Ci was derived. The maximum numbers of adsorbed NPs at PE saturation as a function of Ci for both rod-like and flexible PEs were also obtained. Besides, the 〈R2G〉of PE was calculated to quantitatively characterize flexible PE conformational changes. Through the snapshots saved from MC runs, it is extremely straightforward that the chain flexibility of PE largely affected the number of NPs adsorbed, flexible chains absorbed significantly less NPs than that of rod-like PEs, while chain flexibility promoted the contact probability of monomer segments with NPs (Fig. 8 (A)). When σ = +10 mC/m2 was applied, 〈R2G〉of PE decreased continuously as the (pH − pK0) decreased and reached the conformation of isolated chain when pH − pK0 ≤ − 0.25. Further increasing the surface charge density of the NPs immediately led to formation of more compact structures with decrease of the number of adsorbed NPs at high pH − pK0 values. Besides, when 2 NPs were absorbed, dumbbell conformations were obtained with various separation distances that varied with the degree of ionization (α) (Fig. 8 (B)). As a follow-up work of Skepo and Linse [104], MC simulation was further applied to distinguish the effects of the macroion radius (RM), chain length (Nseg), and chain flexibility (lp) on the phase separation propensity in PE–macroion system at equal stoichiometric charge ratio. Simulation snapshots (Fig. 9) clearly demonstrated that the propensity of phase separation increased as the macroions were made

11

smaller at fixed charge and as the polyelectrolytes were made longer. Moreover, the tendency for phase separation was smaller for a semiflexible chain and larger for flexible and rigid chains. 4.2.4. Delineate mechanisms for “binding on the wrong side of the isoelectric point” Although complex coacervation is commonly accepted as the result of electrostatic interactions between oppositely charged components, a number of systems still witness the formation of soluble complexes when components carry the same net charge. Currently, two competing explanations exist for the experimentally observed phenomenon of “binding on the wrong side of the isoelectric point”. One explanation relies on the heterogeneously charged protein surface that created regions of repulsive and attractive potential simultaneously, i.e. the hypothesis of “charge anisotropy of proteins” [138–140]. The second explanation proposed the possibility of “charge regulation” effect of protein [141], that is, electrostatic potential from a neighboring molecule will perturb the titrating amino acid residues and reverse their charges. Thus the charge regulation mechanism is a consequence of the protein's ability to regulate its charge in the presence of charged PE. Recent applications of MC simulation in protein–polysaccharides complexation systems not only prove the contribution from both proposed mechanisms but also provide valuable insights on relative significance of both observed effects. As an example, de Vries et al. [108] related differences in complexation behaviors to differences in distribution of “charge patches” in whey proteins with the aid of MC simulations. Simulation was carried out in a model system consisting of a single model protein (β-lactoglobulin or α-lactalbumin) and a single model PE chain at the vicinity of each protein's isoelectric point. Positively or negatively charged groups

Fig. 8. (A) Equilibrated conformations of (a) a rod-like and (b) flexible PE at α = 1.00 in the presence of nanoparticles vs ionic concentration &&Ci. (B) Equilibrated conformations of a weak PE in the presence of 16 nanoparticles at Ci = 0.001 M as a function of (pH − pK0) and surface charge densities σ. Reprinted with permission from Ref. [116]. Copyright 2006, American Chemical Society.


12


Fig. 9. Typical configurations for different solutions of macroions and oppositely charged polyelectrolytes at the stoichiometric charge ratio σ = 1. The systems are (a) the reference system and the reference system but (b) RM = 10 Å, (c) RM = 20 Å, (d) Npe = 5 and Nseg = 40, (e) Npe = 40 and Nseg = 5, (f) lp = 42 Å, and (g) lp = 1480 Å. Reprinted with permission from Ref. [104]. Copyright 2003, American Chemical Society.

were assigned with constant degree of dissociation, respectively. In this way, only charge patch effect, not charge regulation effect, is considered. From the saved configurations of MC simulations, the average electrostatic interaction energy of the PE with protein for each run was computed, and the corresponding critical ionic strengths below which soluble complexes can form on the wrong side of the protein's pI were estimated. Consistent with experimental results, it was found that calculated critical ionic strengths are much larger for β-lactoglobulin than for α-lactalbumin, indicating that α-lactalbumin complexes are much more strongly formed. The distributions of PE center of mass around both protein surface were also computed, which clearly demonstrated that stronger complexation was formed due to localized binding to a single large positive “charge patch” on α-lactalbumin, as opposed to multiple smaller charge patches on β-lactoglobulin. Protein model with discrete charges leads to a stronger complexation as compared to a homogeneous surface charge distribution which was also verified by other MC simulation case studies [42]. Meanwhile, Da Silva and co-workers [142] propose the “charge regulation mechanism” on the basis of MC simulation combined with perturbation theory. In their approach, physicochemistry properties, such as pI, charge capacitance and dipole moment number of model proteins were first defined and calculated from MC simulations of a single protein in solutions of low salt concentrations. In the following MC simulation, potentials of mean force for proteins at their respective pI and PE were calculated. Their results suggested that for the three model proteins (lysozyme, α-lactalbumin and β-lactoglobulin), the induced ionization of amino acid residues due to the PE led to up to several kTs' stronger electrostatic interaction than the ion–dipole interaction. Utilizing a similar protocol, Da Silva and Jonsson [143] continued this work by investigating interactions between a flexible PE chain and eight wild type proteins at their respective pI, respectively. Calculation based on MC simulation was plotted as the potential of mean force between the centers of mass of proteins and PE together with the corresponding separation. Combined with the calculated

capacitance value for each investigated protein, a clear trend was observed: large capacitance resulted in more stable complexes. The authors were also able to present a qualitative picture of the relative significance of “charged patches”, i.e. ion-dipole interactions, versus the charge regulation term based on this study: protein charge reversal will be the major driving force for most cases while for patchiness to have a strong contribution, the patches need very high charge densities. Similar conclusion was also derived from the complex formed between a PE brush and a protein at its IP point by de Vos et al. [141] via a Scheutjens–Fleer self consistent field analysis model. 5. Concluding remarks: current problems and prospects Combined with tailored statistical analysis strategies, MC simulation is making increasingly important and unique contribution in interpreting experimental data and unveiling fundamental mechanisms in the field of complex coacervation. As concluding remarks, we proposed precautions as well as prospects in the following aspects: i) The accuracy and efficiency of MC simulation are sensitive to both energy function guiding the simulation and the modeling approaches applied to approximate the inclusive molecular species. Although appropriate simplification in both sides will implement simulation at large temporal and spatial scales, one of the most attractive advantages of MC simulation, precautions should be taken when interpreting the delivered results. On one hand, use of arbitrary energy functions could be incomplete, additional forces not present in the simulation might turn out to play non-negligible roles. One the other hand, a high degree of coarse graining compresses the computational cost at the expense of sacrificing structural resolution. For instance, electrostatic interactions represented by including a point charge on each charged unit cannot capture the full electrostatic



properties (e.g. multipolar moments) of a molecule. While the use of whole-chain persistence lengths with bead-spring models, in the absence of solvation, leads to overestimation of chain flexibility allowing PEs to wrap around proteins. In the implicit solvent models, the condensation effect and the regulation of charges during complexation might be excluded. After all, the simulation accuracy was judged by their ability to reproduce experimentally accessible properties. The gap between both directions could be bridged through refining energy function or modeling to replicate critical details of complex coacervation and make properties tractable. ii) The success of MC simulation is usually realized through calculating quantitative analysis indicators based on sampled putative complex geometries, the effectiveness of which could be of vital importance for revealing more elaborate features and provide robust predictions. Adopting sophisticated algorithms and concepts in other academic fields and verifying their applicability could be a direction with tremendous promising potentials. On the other hand, since one protein bears new feature distinct from others, and factors to affect the interaction, conformation, structure and phase behavior are complicated, none of the correlation of any factor–property pairs is simple and straightforward, dedicated development in MC strategies to predict system-dependent feature in a general consensus frame is still a huge challenge. iii) We have to realize that MC simulation is an underutilized tool owning to the small group of available software packages (such as CHARMM, MCPRO, PROFASI and CAMPARI) in the biomacromolecular simulation community. The flourish of MC simulation in complex coacervation investigation heavily relies on the new contributions from software development. Moreover, although MC simulations allow us to sample the most probable macromolecular states, they do not provide information about temporal evolution. A worthwhile approach to fulfill the potential of MC simulations is to use it as a presampling tool in an approximate manner, then followed by molecular dynamic simulations starting from representative structures from the most dominantly populated clusters within the MC ensemble [144]. Such a hierarchical strategy offers an easy way to expand boundaries of both simulation approaches.

We have to bear in mind that a MC simulation approach requires experimental data for tuning and validation before it can be confidently applied. Appropriate algorithm analysis incorporated afterward determines the efficiency as well as quality of MC simulation. It is our ultimate goal that this review will provide readers with basic guideline for synergistically design of experimental, analytical and MC modeling methods for the purpose of accurate interpretation, prediction and even control of polyelectrolyte–macroion complex coacervation systems. Acknowledgements Financial support from the China Scholarship Council for Jie Xiao was acknowledged. We are also grateful for the support of the National Natural Science Foundation of China (21374117), the 100 Talents Program of Chinese Academy of Sciences for Yunqi Li. References [1] Kizilay E, Kayitmazer AB, Dubin PL. Complexation and coacervation of polyelectrolytes with oppositely charged colloids. Adv Colloid Interface Sci 2011;167:24–37. [2] Burgess DJ. Practical analysis of complex coacervate systems. J Colloid Interface Sci 1990;140:227–38.

13

[3] Jong HGB, Kruyt HR. Coacervation (partial miscibility in colloid systems). Proc K Ned Akad Wet 1929;32:849–56. [4] Spruijt E, Westphal AH, Borst JW, Cohen Stuart MA, van der Gucht J. Binodal compositions of polyelectrolyte complexes. Macromolecules 2010;43: 6476–84. [5] Zintchenko A, Rother G, Dautzenberg H. Transition highly aggregated complexes– soluble complexes via polyelectrolyte exchange reactions: kinetics, structural changes, and mechanism. Langmuir 2003;19:2507–13. [6] Priftis D, Xia XX, Margossian KO, Perry SL, Leon L, Qin J, et al. Ternary, tunable polyelectrolyte complex fluids driven by complex coacervation. Macromolecules 2014; 47:3076–85. [7] Dautzenberg H, Gao YB, Hahn M. Formation, structure, and temperature behavior of polyelectrolyte complexes between ionically modified thermosensitive polymers. Langmuir 2000;16:9070–81. [8] Cooper CL, Dubin PL, Kayitmazer AB, Turksen S. Polyelectrolyte–protein complexes. Curr Opin Colloid Interface Sci 2005;10:52–78. [9] Kaibara K, Okazaki T, Bohidar HB, Dubin PL. pH-induced coacervation in complexes of bovine serum albumin and cationic polyelectrolytes. Biomacromolecules 2000; 1:100–7. [10] Harada A, Kataoka K. Chain length recognition: core–shell supramolecular assembly from oppositely charged block copolymers. Science 1999;283:65–7. [11] Wang YL, Kimura K, Dubin PL, Jaeger W. Polyelectrolyte-micelle coacervation: effects of micelle surface charge density, polymer molecular weight, and polymer/ surfactant ratio. Macromolecules 2000;33:3324–31. [12] Wang YL, Kimura K, Huang QR, Dubin PL, Jaeger W. Effects of salt on polyelectrolyte-micelle coacervation. Macromolecules 1999;32:7128–34. [13] Zhang H, Dubin PL, Ray J, Manning GS, Moorefield CN, Newkome GR. Interaction of a polycation with small oppositely charged dendrimers. J Phys Chem B 1999;103: 2347–54. [14] Mahmoudi M, Lynch I, Ejtehadi MR, Monopoli MP, Bombelli FB, Laurent S. Protein– nanoparticle interactions: opportunities and challenges. Chem Rev 2011;111: 5610–37. [15] Adcock SA, McCammon JA. Molecular dynamics: survey of methods for simulating the activity of proteins. Chem Rev 2006;106:1589–615. [16] Matsudo T, Ogawa K, Kokufuta E. Complex formation of protein with different water-soluble synthetic polymers. Biomacromolecules 2003;4:1794–9. [17] Gao JY, Dubin PL. Binding of proteins to copolymers of varying hydrophobicity. Biopolymers 1999;49:185–93. [18] Li Y, Huang Q. Protein–polysaccharide complexes for effective delivery of bioactive functional food ingredients. In: CM S, Chen H, RY Y, editors. Nanotechnology and functional foods Singapore. Wiley-Blackwell; 2015. p. 224–46 [Chapter 14]. [19] Chourpa I, Ducel V, Richard J, Dubois P, Boury F. Conformational modifications of alpha gliadin and globulin proteins upon complex coacervates formation with gum arabic as studied by Raman microspectroscopy. Biomacromolecules 2006;7:2616–23. [20] Li Y, Zhao Q, Huang Q. Understanding complex coacervation in serum albumin and pectin mixtures using a combination of the Boltzmann equation and Monte Carlo simulation. Carbohydr Polym 2014;101:544–53. [21] Weinbreck F, Nieuwenhuijse H, Robijn GW, de Kruif CG. Complex formation of whey proteins: exocellular polysaccharide EPS B40. Langmuir 2003;19:9404–10. [22] Mekhloufi G, Sanchez C, Renard D, Guillemin S, Hardy J. pH-induced structural transitions during complexation and coacervation of beta-lactoglobulin and acacia gum. Langmuir 2005;21:386–94. [23] Gummel J, Boue F, Clemens D, Cousin F. Finite size and inner structure controlled by electrostatic screening in globular complexes of proteins and polyelectrolytes. Soft Matter 2008;4:1653–64. [24] De Santis S, Ladogana RD, Diociaiuti M, Masci G. Pegylated and thermosensitive polyion complex micelles by self-assembly of two oppositely and permanently charged diblock copolymers. Macromolecules 2010;43:1992–2001. [25] Xu Y, Mazzawi M, Chen K, Sun L, Dubin PL. Protein purification by polyelectrolyte coacervation: influence of protein charge anisotropy on selectivity. Biomacromolecules 2011;12:1512–22. [26] Boral S, Bohidar HB. Effect of ionic strength on surface-selective patch bindinginduced phase separation and coacervation in similarly charged gelatin-agar molecular systems. J Phys Chem B 2010;114:12027–35. [27] Weinbreck F, Tromp RH, de Kruif CG. Composition and structure of whey protein/ gum arabic coacervates. Biomacromolecules 2004;5:1437–45. [28] Biesheuvel PM, Cohen Stuart MA. Electrostatic free energy of weakly charged macromolecules in solution and intermacromolecular complexes consisting of oppositely charged polymers. Langmuir 2004;20:2785–91. [29] Pawar N, Bohidar HB. Statistical thermodynamics of liquid–liquid phase separation in ternary systems during complex coacervation. Phys Rev E 2010;82:036107. [30] Veis A. A review of the early development of the thermodynamics of the complex coacervation phase separation. Adv Colloid Interface Sci 2011;167:2–11. [31] Seijo M, Ulrich S, Filella M, Buffle J, Stoll S. Modeling the surface charge evolution of spherical nanoparticles by considering dielectric discontinuity effects at the solid/ electrolyte solution interface. J Colloid Interface Sci 2008;322:660–8. [32] Seijo M, Ulrich S, Filella M, Buffle J, Stoll S. Modeling the adsorption and coagulation of fulvic acids on colloids by Brownian dynamics simulations. Environ Sci Technol 2009;43:7265–9. [33] Ulrich S, Seijo M, Carnal F, Stoll S. Formation of complexes between nanoparticles and weak polyampholyte chains. Monte Carlo Simul Macromol 2011;44:1661–70. [34] Huber K, Scheler U. New experiments for the quantification of counterion condensation. Curr Opin Colloid Interface Sci 2012;17:64–73. [35] Watanabe H, Fujiyama A, Hattori M, Taylor TD, Toyoda A, Kuroki Y, et al. DNA sequence and comparative analysis of chimpanzee chromosome 22. Nature 2004; 429:382–8.


14


[36] Weinbreck F, de Vries R, Schrooyen P, de Kruif CG. Complex coacervation of whey proteins and gum arabic. Biomacromolecules 2003;4:293–303. [37] de Kruif CG, Weinbreck F, de Vries R. Complex coacervation of proteins and anionic polysaccharides. Curr Opin Colloid In 2004;9:340–9. [38] Nguyen TT, Shklovskii BI. Complexation of a polyelectrolyte with oppositely charged spherical macroions: giant inversion of charge. J Chem Phys 2001;114: 5905–16. [39] Overbeek JTG, Voorn MJ. Phase separation in polyelectrolyte solutions. Theory of complex coacervation. J Cell Comp Physiol 1957;49:7–26. [40] Veis A, Aranyi C. Phase separation in polyelectrolyte systems. I. Complex coacervates of gelatin. J Phys Chem 1960;64:1203–10. [41] Borue VY, Erukhimovich IY. A statistical theory of weakly charged polyelectrolytes — fluctuations, equation of state, and microphase separation. Macromolecules 1988;21:3240–9. [42] Castelnovo M, Joanny JF. Complexation between oppositely charged polyelectrolytes: beyond the random phase approximation. Eur Phys J E 2001;6:377–86. [43] Kudlay A, Ermoshkin AV, de la Cruz MO. Phase diagram of charged dumbbells: a random phase approximation approach. Phys Rev E 2004;70. [44] Castelnovo M, Joanny JF. Phase diagram of diblock polyampholyte solutions. Macromolecules 2002;35:4531–8. [45] Biesheuvel PM, Stuart MAC. Cylindrical cell model for the electrostatic free energy of polyelectrolyte complexes. Langmuir 2004;20:4764–70. [46] Biesheuvel PM, Stuart MAC. Electrostatic free energy of weakly charged macromolecules in solution and intermacromolecular complexes consisting of oppositely charged polymers. Langmuir 2004;20:2785–91. [47] Hayashi Y, Ullner M, Linse P. Complex formation in solutions of oppositely charged polyelectrolytes at different polyion compositions and salt content. J Phys Chem B 2003;107:8198–207. [48] Lee J, Popov YO, Fredrickson GH. Complex coacervation: a field theoretic simulation study of polyelectrolyte complexation. J Chem Phys 2008;128. [49] Allen RJ, Warren PB. Complexation and phase behavior of oppositely charged polyelectrolyte/macroion systems. Langmuir 2004;20:1997–2009. [50] Kayitmazer AB, Seeman D, Minsky BB, Dubin PL, Xu YS. Protein–polyelectrolyte interactions. Soft Matter 2013;9:2553–83. [51] Xia JL, Dubin PL, Dautzenberg H. Light-scattering, electrophoresis, and turbidimetry studies of bovine serum-albumin poly(dimethyldiallylammonium chloride) complex. Langmuir 1993;9:2015–9. [52] Berret JF, Herve P, Aguerre-Chariol O, Oberdisse J. Colloidal complexes obtained from charged block copolymers and surfactants: a comparison between small-angle neutron scattering, cryo-TEM, and simulations. J Phys Chem B 2003;107:8111–8. [53] Girard M, Turgeon SL, Gauthier SF. Thermodynamic parameters of betalactoglobulin–pectin complexes assessed by isothermal titration calorimetry. J Agric Food Chem 2003;51:4450–5. [54] Harnsilawat T, Pongsawatmanit R, McClements DJ. Characterization of betalactoglobulin–sodium alginate interactions in aqueous solutions: a calorimetry, light scattering, electrophoretic mobility and solubility study. Food Hydrocoll 2006;20:577–85. [55] Weinbreck F, Minor M, De Kruif CG. Microencapsulation of oils using whey protein/ gum arabic coacervates. J Microencapsul 2004;21:667–79. [56] Peniche C, Howland I, Carrillo O, Zaldivar C, Arguelles-Monal W. Formation and stability of shark liver oil loaded chitosan/calcium alginate capsules. Food Hydrocoll 2004;18:865–71. [57] Burgess DJ, Ponsart S. Beta-glucuronidase activity following complex coacervation and spray drying microencapsulation. J Microencapsul 1998;15:569–79. [58] Zhang L, Guo J, Peng XH, Jin Y. Preparation and release behavior of carboxymethylated chitosan/alginate microspheres encapsulating bovine serum albumin. J Appl Polym Sci 2004;92:878–82. [59] Cui F, Yang K, Li Y. Investigate the binding of catechins to trypsin using docking and molecular dynamics simulation. PLoS One 2015;10, e0125848. [60] Toh YC, Ho ST, Zhou Y, Hutmacher DW, Yu H. Application of a polyelectrolyte complex coacervation method to improve seeding efficiency of bone marrow stromal cells in a 3D culture system. Biomaterials 2005;26:4149–60. [61] Laneuville SI, Paquin P, Turgeon SL. Formula optimization of a low-fat food system containing whey protein isolate-xanthan gum complexes as fat replacer. J Food Sci 2005;70:S513-S9. [62] Imeson AP, Ledward DA, Mitchell JR. On the nature of the interaction between some anionic polysaccharides and proteins. J Sci Food Agric 1977;28:661–8. [63] Dickinson E. Stability and rheological implications of electrostatic milk protein– polysaccharide interactions. Trends Food Sci Technol 1998;9:347–54. [64] Dickinson E. Interfacial structure and stability of food emulsions as affected by protein–polysaccharide interactions. Soft Matter 2008;4:932–42. [65] Liu S, Low NH, Nickerson MT. Entrapment of flaxseed oil within gelatin–gum arabic capsules. J Am Oil Chem Soc 2010;87:809–15. [66] Gu YS, Decker EA, McClements DJ. Influence of iota-carrageenan on droplet flocculation of beta-lactoglobulin-stabilized oil-in-water emulsions during thermal processing. Langmuir 2004;20:9565–70. [67] Schmitt C, Turgeon SL. Protein/polysaccharide complexes and coacervates in food systems. Adv Colloid Interf 2011;167:63–70. [68] Salminen H, Weiss J. Electrostatic adsorption and stability of whey protein–pectin complexes on emulsion interfaces. Food Hydrocoll 2014;35:410–9. [69] Li Y, Shi T, An L, Lee J, Wang X, Huang Q. Effects of polar group saturation on physical gelation of amphiphilic polymer solutions. J Phys Chem B 2007;111: 12081–7. [70] Li Y, Shi T, Sun Z, An L, Huang Q. Investigation of sol–gel transition in pluronic F127/ D2O solutions using a combination of small-angle neutron scattering and Monte Carlo simulation. J Phys Chem B 2006;110:26424–9.

[71] Li Y, Huang Q, Shi T, An L. How does solvent molecular size affect the microscopic structure in polymer solutions? J Chem Phys 2006;125:044902. [72] Li Y, Sun Z, Su Z, Shi T, An L. The effect of solvent size on physical gelation in triblock copolymer solutions. J Chem Phys 2005;122:194909. [73] Li Y, Sun Z, Shi T, An L. Conformation studies on sol–gel transition in triblock copolymer solutions. J Chem Phys 2004;121:1133–40. [74] Edgecombe S, Linse P. Monte Carlo simulation of polyelectrolyte gels: effects of polydispersity and topological defects. Macromolecules 2007;40:3868–75. [75] Schneider S, Linse P. Monte Carlo simulation of defect-free cross-linked polyelectrolyte gels. J Phys Chem B 2003;107:8030–40. [76] Duan XZ, Zhang R, Li YQ, Yang YB, Shi TF, An LJ, et al. Effect of polyelectrolyte adsorption on lateral distribution and dynamics of anionic lipids: a Monte Carlo study of a coarse-grain model. Eur Biophys J Biophy 2014;43:377–91. [77] Duan XZ, Li YQ, Zhang R, Shi TF, An LJ, Huang QR. Compositional redistribution and dynamic heterogeneity in mixed lipid membrane induced by polyelectrolyte adsorption: effects of chain rigidity. Eur Phys J E 2014;37:71. [78] Duan X, Zhang R, Li Y, Shi T, An L, Huang Q. Monte Carlo study of polyelectrolyte adsorption on mixed lipid membrane. J Phys Chem B 2013;117:989–1002. [79] Duan X, Li Y, Zhang R, Shi T, An L, Huang Q. Regulation of anionic lipids in binary membrane upon the adsorption of polyelectrolyte: a Monte Carlo simulation. AIP Adv 2013;3:062128. [80] Binder K. Monte Carlo methods for the study of phase transitions and phase equilibria. Eur Phys J B 2008;64:307–14. [81] Zhang Z, Schindler CEM, Lange OF, Zacharias M. Application of enhanced sampling Monte Carlo methods for high-resolution protein–protein docking in rosetta. PLoS One 2015;10. [82] Shimada J, Kussell EL, Shakhnovich EI. The folding thermodynamics and kinetics of crambin using an all-atom Monte Carlo simulation. J Mol Biol 2001;308:79–95. [83] Nivon LG, Shakhnovich EI. All-atom Monte Carlo simulation of GCAA RNA folding. J Mol Biol 2004;344:29–45. [84] Mitternacht S, Luccioli S, Torcini A, Imparato A, Irback A. Changing the mechanical unfolding pathway of FnIII(10) by tuning the pulling strength. Biophys J 2009;96: 429–41. [85] Seijo M, Ulrich S, Filella M, Buffle J, Stoll S. Effects of surface site distribution and dielectric discontinuity on the charging behavior of nanoparticles. A grand canonical Monte Carlo study. Phys Chem Chem Phys 2006;8:5679–88. [86] Akinchina A, Linse P. Monte Carlo simulations of polyion–macroion complexes. 1. Equal absolute polyion and macroion charges. Macromolecules 2002;35:5183–93. [87] Binder K, Heermann D. Monte Carlo simulation in statistical physics. 5th ed. New York: Springer-Verlag Berlin Heidelberg; 2010. [88] Robert C, Casella G. Monte Carlo statistical methods. 2nd ed. New York: SpringerVerlag; 2004. [89] Vitalis A, Pappu RV. Methods for Monte Carlo simulations of biomacromolecules. Ann Rep Comp Chem 2009;5:49–76. [90] Paquet E, Viktor HL. Molecular dynamics, Monte Carlo simulations, and Langevin dynamics: a computational review. Biomed Res Int 2015. [91] Li YQ, An LJ, Huang QR. Replica exchange Monte Carlo simulation of human serum albumin–catechin complexes. J Phys Chem B 2014;118:10362–72. [92] Li YQ, Huang QR. Influence of protein self-association on complex coacervation with polysaccharide: a Monte Carlo study. J Phys Chem B 2013;117:2615–24. [93] Jonsson M, Linse P. Structure and thermodynamics of protein–polymer solutions: effects of spatially distributed hydrophobic surface residues. J Phys Chem B 2005; 109:15107–17. [94] Li YQ, Shi TF, An LJ, Huang QR. Monte Carlo simulation on complex formation of proteins and polysaccharides. J Phys Chem B 2012;116:3045–53. [95] Kayitmazer AB, Quinn B, Kimura K, Ryan GL, Tate AJ, Pink DA, et al. Protein specificity of charged sequences in polyanions and heparins. Biomacromolecules 2010;11: 3325–31. [96] Skepo M, Linse P, Arnebrant T. Coarse-grained modeling of proline rich protein 1 (PRP-1) in bulk solution and adsorbed to a negatively charged surface. J Phys Chem B 2006;110:12141–8. [97] Chen KM, Xu YS, Rana S, Miranda OR, Dubin PL, Rotello VM, et al. Electrostatic selectivity in protein–nanoparticle interactions. Biomacromolecules 2011;12:2552–61. [98] Grymonpre KR, Staggemeier BA, Dubin PL, Mattison KW. Identification by integrated computer modeling and light scattering studies of an electrostatic serum albumin-hyaluronic acid binding site. Biomacromolecules 2001;2:422–9. [99] Monticelli L, Kandasamy SK, Periole X, Larson RG, Tieleman DP, Marrink SJ. The MARTINI coarse-grained force field: extension to proteins. J Chem Theory Comput 2008;4:819–34. [100] Li YQ, Huang QR, Shi TF, An LJ. Effects of chain flexibility on polymer conformation in dilute solution studied by lattice Monte Carlo simulation. J Phys Chem B 2006; 110:23502–6. [101] Lund M, Jonsson B. On the charge regulation of proteins. Biochemistry-Us 2005;44: 5722–7. [102] Chen JH, Brooks CL, Khandogin J. Recent advances in implicit solvent-based methods for biomolecular simulations. Curr Opin Struct Biol 2008;18:140–8. [103] Mackerell AD. Empirical force fields for biological macromolecules: overview and issues. J Comput Chem 2004;25:1584–604. [104] Skepo M, Linse P. Complexation, phase separation, and redissolution in polyelectrolyte–macroion solutions. Macromolecules 2003;36:508–19. [105] Cala O, Pinaud N, Simon C, Fouquet E, Laguerre M, Dufourc EJ, et al. NMR and molecular modeling of wine tannins binding to saliva proteins: revisiting astringency from molecular and colloidal prospects. FASEB J 2010;24:4281–90. [106] Yui T, Shiiba H, Tsutsumi Y, Hayashi S, Miyata T, Hirata F. Systematic docking study of the carbohydrate binding module protein of cel7A with the cellulose I alpha crystal model. J Phys Chem B 2010;114:49–58.


J. Xiao et al. / Advances in Colloid and Interface Science xxx (2016) xxx–xxx [107] Moreira IS, Fernandes PA, Ramos MJ. Computational determination of the relative free energy of binding — application of alanine scanning mutagenesis. In: Sokalski WA, editor. Molecular materials with specific interactions — modeling and design. The Netherlands: Springer; 2007. p. 314–5 Chapter 6. [108] de Vries R. Monte Carlo simulations of flexible polyanions complexing with whey proteins at their isoelectric point. J Chem Phys 2004;120:3475–81. [109] Chib S, Greenberg E. Understanding the Metropolis–Hastings algorithm. Am Stat 1995;49:327–35. [110] Zeng QH, Yu AB, Lu GQ. Multiscale modeling and simulation of polymer nanocomposites. Prog Polym Sci 2008;33:191–269. [111] Juraszek J, Bolhuis PG. Effects of a mutation on the folding mechanism of betahairpin. J Phys Chem B 2009;113:16184–96. [112] Woods CJ, Ng MH, Johnston S, Murdock SE, Wu B, Tai K, et al. Grid computing and biomolecular simulation. Philos T R Soc A 2005;363:2017–35. [113] Mitsutake A, Sugita Y, Okamoto Y. Generalized-ensemble algorithms for molecular simulations of biopolymers. Biopolymers 2001;60:96–123. [114] Chen WD, Zhao HC, Liu LJ, Chen JZ, Li YQ, An LJ. Effects of excluded volume and hydrodynamic interaction on the deformation, orientation and motion of ring polymers in shear flow. Soft Matter 2015;11:5265–73. [115] Li Y, Huang Q. Influence of protein self-association on complex coacervation with polysaccharide: a Monte Carlo study. J Phys Chem B 2013;117:2615–24. [116] Ulrich S, Seijo M, Laguecir A, Stoll S. Nanoparticle adsorption on a weak polyelectrolyte. Stiffness, pH, charge mobility, and ionic concentration effects investigated by Monte Carlo simulations. J Phys Chem B 2006;110:20954–64. [117] Bohidar H, Dubin PL, Majhi PR, Tribet C, Jaeger W. Effects of protein–polyelectrolyte affinity and polyelectrolyte molecular weight on dynamic properties of bovine serum albumin-poly(diallyldimethylammonium chloride) coacervates. Biomacromolecules 2005;6:1573–85. [118] Sanchez C, Mekhloufi G, Schmitt C, Renard D, Robert P, Lehr CM, et al. Selfassembly of beta-lactoglobulin and acacia gum in aqueous solvent: structure and phase-ordering kinetics. Langmuir 2002;18:10323–33. [119] Weinbreck F, Rollema HS, Tromp RH, de Kruif CG. Diffusivity of whey protein and gum arabic in their coacervates. Langmuir 2004;20:6389–95. [120] Wang XY, Li YQ, Wang YW, Lal J, Huang QR. Microstructure of beta-lactoglobulin/ pectin coacervates studied by small-angle neutron scattering. J Phys Chem B 2007;111:515–20. [121] Berret JF. Stoichiometry of electrostatic complexes determined by light scattering. Macromolecules 2007;40:4260–6. [122] Hayashi Y, Ullner M, Linse P. A Monte Carlo study of solutions of oppositely charged polyelectrolytes. J Chem Phys 2002;116:6836–45. [123] Li Y, Huang Q, Shi T, An L. Effects of chain flexibility on polymer conformation in dilute solution studied by lattice Monte Carlo simulation. J Phys Chem B 2006;110: 23502–6. [124] Ru QM, Wang YW, Lee J, Ding YT, Huang QR. Turbidity and rheological properties of bovine serum albumin/pectin coacervates: effect of salt concentration and initial protein/polysaccharide ratio. Carbohydr Polym 2012;88:838–46. [125] Souza CJF, Garcia-Rojas EE. Effects of salt and protein concentrations on the association and dissociation of ovalbumin–pectin complexes. Food Hydrocoll 2015;47: 124–9.

15

[126] Wang X, Wang YW, Ruengruglikit C, Huang Q. Effects of saft concentration on formation and dissociation of beta-lactoglobulin/pectin complexes. J Agric Food Chem 2007;55:10432–6. [127] Seyrek E, Dubin PL, Tribet C, Gamble EA. Ionic strength dependence of protein– polyelectrolyte interactions. Biomacromolecules 2003;4:273–82. [128] Laguecir A, Stoll S, Kirton G, Dubin PL. Interactions of a polyanion with a cationic micelle: comparison of Monte Carlo simulations with experiment. J Phys Chem B 2003;107:8056–65. [129] Wallin T, Linse P. Monte Carlo simulations of polyelectrolytes at charged micelles .1. Effects of chain flexibility. Langmuir 1996;12:305–14. [130] Carlsson F, Linse P, Malmsten M. Monte Carlo simulations of polyelectrolyte–protein complexation. J Phys Chem B 2001;105:9040–9. [131] Akinchina A, Linse P. Monte Carlo simulations of polyion–macroion complexes. 2. Polyion length and charge density dependence. J Phys Chem B 2003;107:8011–21. [132] Carlsson F, Malmsten M, Linse P. Protein–polyelectrolyte cluster formation and redissolution: a Monte Carlo study. J Am Chem Soc 2003;125:3140–9. [133] Cooper CL, Goulding A, Kayitmazer AB, Ulrich S, Stoll S, Turksen S, et al. Effects of polyelectrolyte chain stiffness, charge mobility, and charge sequences on binding to proteins and micelles. Biomacromolecules 2006;7:1025–35. [134] Johansson HO, Van Alstine JM. Modeling of protein interactions with surfacegrafted charged polymers. Correlations between statistical molecular modeling and a mean field approach. Langmuir 2006;22:8920–30. [135] Jonsson M, Linse P. Polyelectrolyte–macroion complexation. I. Effect of linear charge density, chain length, and macroion charge. J Chem Phys 2001;115: 3406–18. [136] Jonsson M, Linse P. Polyelectrolyte–macroion complexation. II. Effect of chain flexibility. J Chem Phys 2001;115:10975–85. [137] Carnal F, Laguecir A, Stoll S, Laguecir A. Simulations and scattering functions of polyelectrolyte–macroion complexes. Colloid Polym Sci 2004;283:317–28. [138] Jȍbstl E, O'Connell J, Fairclough JPA, Williamson MP. Molecular model for astringency produced by polyphenol/protein interactions. Biomacromolecules 2004;5: 942–9. [139] Tanford C, Kirkwood JG. Theory of protein titration curves .1. General equations for impenettrable spheres. J Am Chem Soc 1957;79:5333–9. [140] Chong SH, Ham S. Impact of chemical heterogeneity on protein self-assembly in water. Proc Natl Acad Sci U S A 2012;109:7636–41. [141] de Vos WM, Leermakers FAM, de Keizer A, Stuart MAC, Kleijn JM. Field theoretical analysis of driving forces for the uptake of proteins by like-charged polyelectrolyte brushes: effects of charge regulation and patchiness. Langmuir 2010;26:249–59. [142] da Silva FLB, Lund M, Jonsson B, Akesson T. On the complexation of proteins and polyelectrolytes. J Phys Chem B 2006;110:4459–64. [143] da Silva FLB, Jonsson B. Polyelectrolyte–protein complexation driven by charge regulation. Soft Matter 2009;5:2862–8. [144] De Mori GMS, Colombo G, Micheletti C. Study of the villin headpiece folding dynamics by combining coarse-grained Monte Carlo evolution and all-atom molecular dynamics. Proteins 2005;58:459–71. [145] Lund M, Jonsson B. A mesoscopic model for protein–protein interactions in solution. Biophys J 2003;85:2940–7.


Application of Monte Carlo simulation in addressing ...

Application of Monte Carlo simulation in addressing ...

Suggest Documents

MONTE CARLO SIMULATION - Assakkaf

Direct Simulation Monte Carlo

Application of Monte Carlo Simulation in Treatment ...

Monte Carlo Simulation of Characteristic

Monte Carlo simulation of transport in

Monte Carlo simulation of electromigration in

Monte Carlo Simulation - Google Sites

Monte Carlo Simulation Study - JSTS

Monte Carlo Simulation Generation Through

Replica Monte Carlo Simulation (Revisited)

application monte carlo method on simulation of disaster ... - IAARC

Application of Monte Carlo simulation to SEM ... - Wiley Online Library

1 Application of Monte Carlo Simulation to Analytical ...

the application of monte carlo simulation to evaluate ...

Monte Carlo Simulation in Financial Valuation

Monte Carlo Simulation of Multicomponent ... - Semantic Scholar

The Monte Carlo Simulation of Radiation Transport

MONTE CARLO SIMULATION OF ELECTRON ... - UW-Madison

Monte-Carlo Simulation of Electricity Transmission ...

Monte Carlo Ray Tracing Simulation of

A Monte Carlo Simulation of Reproducible Hysteresis

Monte-Carlo based simulation of surface light

Monte Carlo simulation of spontaneous miniature excitatory ...

MONTE CARLO SIMULATION OF ELECTRON ... - CiteSeerX