Polyampholyte physics: Liquid–liquid phase separation and biological condensates

Here we review the current understanding of molecular interactions that govern liquid–liquid phase separation (LLPS) of biological condensates. The connection between sequence, chain conformation, and phase separation of intrinsically disordered proteins (IDPs) and their model polyampholytes is discussed. In particular, we highlight how the charge pattern influences the conformation and phase behavior of natural IDPs. We then describe recent results from theoretical treatments of polyampholytes implementing random phase approximation, field-theoretic simulations, and transfer matrix theory that show an increase in charge segregation results in an increased tendency to phase separate.     

Microphase separation in polyelectrolytic diblock copolymer melt: weak segregation limit

The authors present a generalized theory of microphase separation for charged-neutral diblock copolymer melt. The stability limit of the disordered phase for salt-free melt has been calculated using random phase approximation (RPA) and self-consistent-field theory (SCFT). Explicit analytical free energy expressions for different classical ordered microstructures (lamellar, cylinder, and sphere) are presented. The authors demonstrate that the chemical mismatch required for the onset of microphase separation (chi*N) in charged-neutral diblock melt is higher and the period of ordered microstructures is lower than those for the corresponding neutral-neutral diblock system. Theoretical predictions on the period of ordered structures in terms of Coulomb electrostatic interaction strength, chain length, block length, and chemical mismatch between blocks are presented. SCFT has been used to go beyond the stability limit, where electrostatic potential and charge distribution are calculated self-consistently. Stability limits calculated using RPA are in perfect agreement with the corresponding SCFT calculations. Limiting laws for the stability limit and the period of ordered structures are presented and comparisons are made with an earlier theory. Also, transition boundaries between different morphologies have been investigated.     

Preparation of zwitterionic sulfobetaine end‐functionalized poly(ethylene glycol)‐b‐poly(caprolactone) diblock copolymers and examination of their thermogelling properties

To investigate thermogelling behavior, in this study, we prepared a methoxy poly(ethylene glycol)‐b‐poly(ε‐caprolactone) diblock copolymer (MPC) with varying hydrophobic poly(ε‐caprolactone) (PCL) lengths and an MPC featuring a zwitterionic sulfobetaine (MPC‐ZW) at the chain end of the PCL segment. The terminal zwitterionic sulfobetaine was stoichiometrically modified to the terminal MPC diblock copolymer. The introduction of the zwitterionic end group lowered the crystallization enthalpies of the PCL block segments and increased the solubility of the diblock copolymer. The MPC and MPC‐ZW copolymers thus obtained formed translucent emulsions at room temperature when prepared as 20 wt %. When the temperature was increased above room temperature, MPC and MPC‐ZW exhibited a sol‐to‐gel phase transition. The phase transition and the gelation time of MPC and MPC‐ZW were affected by the length of the hydrophobic segments and the zwitterionic end group. Furthermore, introducing a zwitterionic end group into the PCL segment altered the onset temperature of gelation. Thus, we conclude that zwitterionic end groups introduced into PCL segments of distinct lengths could serve as key determinants in the thermogelling behavior of copolymers. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2014 , 52, 2185–2191     

Molecular dynamics simulations of polyelectrolyte-polyampholyte complexes. Effect of solvent quality and salt concentration

Complexation between polyelectrolyte and polyampholyte chains in poor solvent conditions for the polyelectrolyte backbone has been studied by molecular dynamics simulations. In a poor solvent a polyelectrolyte forms a necklace-like structure consisting of polymeric globules (beads) connected by strings of monomers. The simulation results can be explained by assuming the existence of two different mechanisms leading to the necklace formation. In the case of weak electrostatic interactions, the necklace formation is driven by optimization of short-range monomer-monomer attraction and electrostatic repulsion between charged monomers on the polymer backbone. In the case of strong electrostatic interactions, the necklace structure appears as a result of counterion condensation. While the short-range attractions between monomers are still important, the correlation-induced attraction between condensed counterions and charged monomers and electrostatic repulsion between uncompensated charges provide significant contribution to optimization of the necklace structure. Upon forming a complex with both random and diblock polyampholytes, a polyelectrolyte chain changes its necklace conformation by forming one huge bead. The collapse of the polyelectrolyte chain occurs due to the neutralization of the polyelectrolyte charge by polyampholytes. In the case of the random polyampholyte, the more positively charged sections of the chain mix with negatively charged polyelectrolyte forming the globular bead while more negatively charged chain sections form loops surrounding the collapsed core of the aggregate. In the case of diblock polyampholyte, the positively charged block, a part of the negatively charged block, and a polyelectrolyte chain form a core of the aggregate with a substantial section of the negatively charged block sticking out from the collapsed core of the aggregate. In both cases the core of the aggregate has a layered structure that is characterized by the variations in the excess of concentration of monomers belonging to polyampholyte and polyelectrolyte chains throughout the core radius. These structures appear as a result of optimization of the net electrostatic energy of the complex and short-range attractive interactions between monomers of the polyelectrolyte chain.     

A monte carlo study of weak polyampholytes: stiffness and primary structure influences on titration curves and chain conformations

The conformation and titration curves of weak polyampholytes are examined using Monte Carlo simulations with screened Coulomb potentials in the Grand Canonical ensemble. Two different types of monomers are considered. Depending on the solution pH, monomers A are weak acidic sites that can either be negatively charged or uncharged (as carboxylic groups), whereas monomers B are weak basic sites that can either be positively charged or uncharged (as amino groups). The influence of the chain stiffness, primary structure, and ionic concentration on the acid/base properties of the polyampholyte chains are systematically investigated. By adjusting the pH values, titration curves and then the fractions of positively and negatively ionized charged monomers are calculated. Stiffness influence is estimated by comparing two models of chain: a fully flexible and a rod-like polyampholyte. Different primary structures such as statistical (diblock, octablock, and alternating) and random polyampholytes are also considered. We demonstrate that the primary structure plays important roles in the acid/base properties as well as the charge distribution along the polymer backbone of a statistical rod-like polyampholyte. When flexible polyampholytes are considered, polyampholyte conformations promote the attractive electrostatic interactions between positively and negatively charged monomers, hence leading to more or less compact conformations and acid/base properties relatively different in comparison to the rod-like polyampholytes. Various conformations such as extended, globular, and pearl-necklace conformations are found in good agreement with the literature by adjusting the interaction parameter between monomers and monomer stoichiometry.     

Diblock polyampholytes grafted onto spherical particles: Monte Carlo simulation and lattice mean-field theory

Spherical brushes composed of diblock polyampholytes (diblock copolymers with oppositely charged blocks) grafted onto solid spherical particles in aqueous solution are investigated by using the primitive model solved with Monte Carlo simulations and by lattice mean-field theory. Polyampholyte chains of two compositions are considered: a copolymer with a long and a short block, A100B10, and a copolymer with two blocks of equal length, A50B50. The B block is end-grafted onto the surface, and its charge is varied, whereas the charge of the A block is fixed. Single-chain properties, radial and lateral spatial distributions of different types, and structure factors are analyzed. The brush structure strongly depends on the charge of the B block. In the limit of an uncharged B block, the chains are stretched and form an extended polyelectrolyte brush. In the other limit with the charges of the blocks compensating each other, the chains are collapsed and form a polyelectrolyte complex surrounding the particles. At intermediate charge conditions, a polyelectrolyte brush and a polyelectrolyte complex coexist and constitute two substructures of the spherical brush. The differences of the brush structures formed by the A100B10 and A50B50 polyampholytes are also analyzed. Finally, a comparison of the predictions of the two theoretical approaches is made.     

利用TDGL方法模拟嵌段共聚物链长和浓度对A/B/A-B三组分体系相形态演化的影响

从Flory-Huggins自由能出发,得到了适合TDGL模拟的自由能泛函.在自由能中保留了各序参量的耦合项,并且还保留了链长的信息.利用自由能中的链长的信息,模拟了不同嵌段共聚物链长以及不同浓度下体系的形态变化,在均聚物的链长和浓度确定的条件下,存在一个使体系的相区尺寸最小的最佳嵌段共聚物链长.同时,在嵌段共聚物和均聚物的链长都一定的情况下,研究了不同量的嵌段共聚物对体系相行为的影响,发现嵌段共聚物的浓度不同,体系的结构存在很大的区别.此外,如果均聚物A和B的浓度不同,其相区的结构也不同.     

嵌段共聚物和星型均聚物共混体系相行为的自洽场理论研究

用自洽场理论方法(Self-consistent field theory, SCFT)计算了嵌段共聚物AB和三等臂星型均聚物A共混体系的微相形态. 为了简化计算, 着重讨论了固定嵌段共聚物本体的相形态(如层状相)时, 所加入的均聚物的体积分数及均聚物与嵌段共聚物链长之比对体系相形态的影响; 并结合体系的熵和相互作用能的变化, 讨论了星型均聚物在体系微相结构中的分布.     

Phase behaviors of cyclic diblock copolymers

A spectral method of self-consistent field theory has been applied to AB cyclic block copolymers. Phase behaviors of cyclic diblock copolymers, such as order-disorder transition, order-order transition, and domain spacing size, have been studied, showing good consistency with previous experimental and theoretical results. Compared to linear diblocks, cyclic diblocks are harder to phase separate due to the topological constraint of the ring structure. A direct disorder-to-cylinder transition window is observed in the phase diagram, which is significantly different from the mean field phase diagram of linear diblock copolymers. The domain spacing size ratio between cyclic and linear diblock copolymers is typically close to 0.707, indicating in segregation that the cyclic polymer can be considered to be made up of linear diblocks with half of the original chain length.     

Nonlinear depression of the lower critical solution temperatures in aqueous solutions of thermo-sensitive random copolymers

We develop a theoretical model of cooperative hydration to clarify the molecular origin of the observed nonlinear depression of the lower critical solution temperature (LCST) in the aqueous solutions of thermosensitive random copolymers and find the monomer composition at which LCST shows a minimum. Phase diagrams of poly(N-isopropylacrylamide-co-N,N-diethylacrylamide) copolymer solutions are theoretically derived on the basis of the theory of cooperative hydration by introducing the microscopic structure parameter η which characterizes the distribution of the monomer sequences along the chains. We compared them with the experimental data of LCST of random copolymers with various monomer compositions and also of the diblock copolymers with equimolar monomer composition. The transition temperature shifts to lower than those of homopolymer counterparts when the monomer sequence of the chains has an alternative tendency. On the contrary, for the blocky polymers such as diblock copolymers, the transition temperature remains almost the same as those of the homopolymers. Thus, the nonlinear effect in phase separation appears when the average block length of the copolymers is shorter than the average sequence length of the cooperative hydration. The degree of hydration is calculated as a function of the temperature and polymer concentration for varied distribution of the copolymer compositions. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2013, 51, 1112–1123     

The structure and phase transitions in polymer blends,diblock copolymers and liquid crystalline polymers: The Landau-Ginzburg approach

The polymer systems are discussed in the framework of the Landau-Ginzburg model. The model is derived from the mesoscopic Edwards Hamiltonian via the conditional partition function. We discuss flexible, semiflexible and rigid polymers. The following systems are studied: polymer blends, flexible diblock and multi-block copolymer melts, random copolymer melts, ring polymers, rigid-flexible diblock copolymer melts, mixtures of copolymers and homopolymers and mixtures of liquid crystalline polymers. Three methods are used to study the systems: mean-field model, self consistent one-loop approximation and self consistent field theory. The following problems are studied and discussed: the phase diagrams, scattering intensities and correlation functions, single chain statistics and behavior of single chains close to critical points, fluctuations induced shift of phase boundaries. In particular we shall discuss shrinking of the polymer chains close to the critical point in polymer blends, size of the Ginzburg region in polymer blends and shift of the critical temperature. In the rigid-flexible diblock copolymers we shall discuss the density nematic order parameter correlation function. The correlation functions in this system are found to oscillate with the characteristic period equal to the length of the rigid part of the diblock copolymer. The density and nematic order parameter measured along the given direction are anticorrelated. In the flexible diblock copolymer system we shall discuss various phases including the double diamond and gyroid structures. The single chain statistics in the disordered phase of a flexible diblock copolymer system is shown to deviate from the Gaussian statistics due to fluctuations. In the one loop approximation one shows that the diblock copolymer chain is stretched in the point where two incompatible blocks meet but also that each block shrinks close to the microphase separation transition. The stretching outweights shrinking and the net result is the increase of the radius of gyration above the Gaussian value. Certain properties of homopolymer/copolymer systems are discussed. Diblock copolymers solubilize two incompatible homopolymers by forming a monolayer interface between them. The interface has a positive saddle splay modulus which means that the interfaces in the disordered phase should be characterized by a negative Gaussian curvature. We also show that in such a mixture the Lifshitz tricritical point is encountered. The properties of this unusual point are presented. The Lifshitz, equimaxima and disorder lines are shown to provide a useful tool for studying local ordering in polymer mixtures. In the liquid crystalline mixtures the isotropic nematic phase transition is discussed. We concentrate on static, equilibrium properties of the polymer systems.     

嵌段序列对聚合物囊泡形成过程影响的Monte Carlo模拟

采用Monte Carlo模拟方法研究了具有相同链长和组分比的不同嵌段序列的AB两嵌段共聚物与ABA三嵌段共聚物在选择性溶剂中形成囊泡的动力学过程. 模拟结果表明, AB两嵌段共聚物囊泡的形成与ABA三嵌段共聚物囊泡的形成的动力学过程不同. 在慢速退火条件下, ABA三嵌段共聚物囊泡是通过亲水链段向胶束的表面和中心扩散而形成的, 而AB两嵌段共聚物囊泡则由片层弯曲闭合而形成. 相对而言, 退火速度对AB两嵌段共聚物囊泡形成的动力学过程没有显著影响, 其改变仅影响亲水链段与疏水链段发生相分离的难易程度. 当退火速度较快时, 亲水链段和疏水链段发生相分离的速度较快且相分离发生在囊泡形成之前; 而当退火速度较慢时亲水链段和疏水链段之间的相分离在囊泡形成之后仍在进行.     

对称长链和近对称短链两嵌段聚合物混合体系相行为研究

利用自洽平均场理论(SCMFT)系统地研究了对称长链和近对称短链两嵌段聚合物混合体系在纳米尺度下的自组装行为.体系中具有较高聚合度的对称长链熔体处于层状相,聚合度较低的近对称短链熔体处于无序相,而其混合体系却随着两种成分的不同比例呈现出有序-无序相转变、有序-有序相转变及有序-无序两相共存等复杂的相行为,计算结果与近期类似体系的实验有着较好的吻合.同时与两种对称的两嵌段聚合物混合体系的计算结果进行了比较,得出这两种体系的异同之处.     

良溶剂中接枝型聚两性电解质单链构象的布朗动力学模拟

采用布朗动力学研究了在良溶剂中荷电平衡的接枝聚两性电解质(GPA)的单链构象转变行为,讨论了主链链长、支链数及电荷密度对GPA分子链构象转变的影响.研究发现,随着静电相互作用的增强,GPA分子链构象转变过程由线团、主链与支链间的折叠、链段塌缩和电荷配对形成偶极子与四极子等4个阶段构成.与线型聚两性电解质不同,GPA存在的额外支链间空间排斥与静电排斥作用随着分子结构的变化而改变,并影响构象转变行为.在强静电相互作用下,良溶剂中的GPA链由于溶剂化作用会再伸展,以保证偶极子完全配对成四极子.减小主链长度或电荷密度或增加支链数目都会增大体系的排斥力和主链的刚性,阻滞分子链的塌缩,并使得分子链再伸展的幅度增大.     

Physicochemical and biological properties of polyampholytes: Quaternized derivatives of poly(4-vinylpyridine)

The modification of poly(4-vinylpyridine) with ω-bromocarboxylic acids and alkyl bromides yields three types of polyampholytes: polyampholytes containing both cationic and anionic groups in each monomer unit (polybetaines), polyampholytes containing betaine and cationic units, and polyampholytes containing betaine units and side cetyl radicals. Their complex formation with liposomes formed from zwitterionic (electroneutral) phosphatidylcholine and anionic diphosphatidylglycerol (cardiolipin) is investigated. The method for fixation of polymers on the liposomal membrane and the stability of the formed complexes are determined by the chemical structure of macromolecules. For the most part, polyelectrolytes are electrostatically adsorbed on the membrane and are fully removed from it with an increase in the salt concentration in the surrounding solution. An exception is the polybetaine obtained through the modification of poly(4-vinylpyridine) with ω-bromobutyric acid, which irreversibly binds to liposomes probably owing to the incorporation of macromolecular fragments into the hydrophobic part of the lipid bilayer. The insertion of side cetyl radicals into polybetaine molecules stabilizes their complexes with liposomes in the presence of salts. The cytotoxicity of the synthesized polyampholytes is one to two orders of magnitude lower than that of a cationic polymer with the same degree of polymerization.     

Pressure-induced formation of diblock copolymer "micelles" in supercritical fluids. A combined study by small angle scattering experiments and mean-field theory. I. The critical micellization density concept

We developed a simple mean-field theory to describe polymer and AB diblock copolymer phase separation in supercritical (SC) fluids. The highly compressible SC fluid has been described by using a phenomenological hole theory, properly extended to consider the solvent/polymer/vacancy pseudoternary mixture. The model has been applied to describe the phase behavior of AB-diblock copolymers under the assumption of a strong solvent selectivity for just one copolymer chain. In our model the solvent selectivity is a strong function of the external pressure because in compressible fluids vacancies reduce the number of favorable solvent-polymer contacts. The combined effect of the pressure on the average solvent quality and selectivity for a single polymer chain makes the phase behavior of a diblock copolymer in SC fluids quite complex. Small angle neutron and x-ray scattering (SANS and SAXS) measurements have been performed on SC-CO2 solutions of different AB-diblock copolymers containing a perfluorinated chain. The data obtained over a wide range of pressure and temperature confirm our theoretical predictions.     

Aqueous dispersion polymerization: a new paradigm for in situ block copolymer self-assembly in concentrated solution

Reversible addition-fragmentation chain transfer polymerization has been utilized to polymerize 2-hydroxypropyl methacrylate (HPMA) using a water-soluble macromolecular chain transfer agent based on poly(2-(methacryloyloxy)ethylphosphorylcholine) (PMPC). A detailed phase diagram has been elucidated for this aqueous dispersion polymerization formulation that reliably predicts the precise block compositions associated with well-defined particle morphologies (i.e., pure phases). Unlike the ad hoc approaches described in the literature, this strategy enables the facile, efficient, and reproducible preparation of diblock copolymer spheres, worms, or vesicles directly in concentrated aqueous solution. Chain extension of the highly hydrated zwitterionic PMPC block with HPMA in water at 70 °C produces a hydrophobic poly(2-hydroxypropyl methacrylate) (PHPMA) block, which drives in situ self-assembly to form well-defined diblock copolymer spheres, worms, or vesicles. The final particle morphology obtained at full monomer conversion is dictated by (i) the target degree of polymerization of the PHPMA block and (ii) the total solids concentration at which the HPMA polymerization is conducted. Moreover, if the targeted diblock copolymer composition corresponds to vesicle phase space at full monomer conversion, the in situ particle morphology evolves from spheres to worms to vesicles during the in situ polymerization of HPMA. In the case of PMPC(25)-PHPMA(400) particles, this systematic approach allows the direct, reproducible, and highly efficient preparation of either block copolymer vesicles at up to 25% solids or well-defined worms at 16-25% solids in aqueous solution.     

Stiff chain persistence lengths in liquid crystalline copolyesters

This paper describes a continuation of our X-ray diffraction work on the structure of the mesomorphic copolyesters prepared from p-hydroxybenzoic acid (HBA) and 2-hydroxy-6-naphthoic acid (HNA). The X-ray patterns of meltspun fibers of these copolymers show a series of aperiodic meridional maxima, and these are predicted by a model consisting of stiff extended chains of completely random copolymer sequence. The calculated intensity data are independent of the chain length for the model, except that the width of the peak at d≃2.1Å decreases with increasing chain length. For 58/42 copoly(HBA/HNA) the best agreement obtained is for a chain length of 11 monomers, and this corresponds to a correlation or persistence length for the stiff chain conformation in the solid state. This effect has been modeled for an infinite chain by deriving experimental monomer length distribution functions, from a survey of the conformations of models of typical random chain sequences. The distribution function is then incorporated into the intensity calculations and leads to prediction of peak widths that are comparable to those observed.     

Synthesis of zwitterionic block copolymers via RAFT polymerization

A series of poly [2-(dimethylamino)ethyl methacrylate (DMA)-sodium acrylate (SA)] diblock copolymers were synthesized using reversible addition-fragmentation chain transfer (RAFT) polymerization. The polymerization exhibits controlled characters: well-controlled molecular weight, narrow molecular weight distribution, molecular weight increasing with polymerization time. The zwitterionic diblock copolymers show rich solution behaviors. Dynamic light scattering (DLS) indicated the formation of micelles and reverse micelles of copolymers is affected by net charge density of copolymers. Microcalorimetry studies showed that the lower critical solution temperature (LCST) increases with incorporation of hydrophilic segments in buffer.     

Effect of chain stiffness on the morphology of diblock copolymer melts

The authors have studied the microphase separation of symmetric diblock copolymers with variable block stiffness and different block chain lengths using coarse-grained molecular dynamics simulations. The simulation results show that for symmetric diblock copolymers, a combination of chain length and relative stiffness between the blocks may play the major role in determining the equilibrium morphology of the system. When the variation in stiffness between blocks is small, the equilibrium morphology of the diblock system is found to be lamellar; this is also the case for systems with small chain lengths, regardless of the difference in block stiffness. However, in systems with longer chains with modest variation in stiffness between the blocks, an ordered cylindrical phase is formed in which the stiffer blocks form cylinders completely surrounded by the flexible components. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2556–2565, 2009