Dynamics of Collapse of Flexible Polyelectrolytes in Poor Solvents

Collapse and expansion kinetics of a single polyelectrolyte chain with hydrodynamic interactions

We investigate the collapse and expansion dynamics of a linear polyelectrolyte (PE) with hydrodynamic interactions. Using dissipative particle dynamics with a bead-spring PE model, long-range electrostatics, and explicit ions, we examine how the timescales of collapse tcol and expansion texp depend on the chain length N and obtain scaling relationships tcol ∼ Nα and texp ∼ Nβ. For neutral polymers, we derive values of α = 0.94 ± 0.01 and β = 1.97 ± 0.10. Interestingly, the introduction of electrostatic interaction markedly shifts α to α ≈ 1.4 ± 0.1 for salt concentrations within c = 10-4 to 10-2 M. A reduction in the ion-to-monomer size ratio noticeably reduces α. On the other hand, the expansion scaling remains approximately constant, β ≈ 2, regardless of the salt concentration or ion size considered. We find β > α for all conditions considered, implying that expansion is always slower than collapse in the limit of long polymers. This asymmetry is explained by distinct kinetic pathways of collapse and expansion processes.

Hydrodynamic Effects on the Collapse Kinetics of Flexible Polyelectrolytes

Understanding the collapse kinetics of polyelectrolytes (PEs) is crucial for comprehending various biological and industrial phenomena. Despite occurring in an aqueous environment, previous computational studies have overlooked the influence of hydrodynamic interactions (HIs) facilitated by fluid motion. Here, we directly compute the Navier-Stokes equation to investigate the collapse kinetics of a highly charged flexible PE. Our findings reveal that HI accelerates PE collapse induced by hydrophobicity and multivalent salt. In the case of hydrophobicity, HI induces long-range collective motion of monomers, accelerating the coarsening of local clusters through either Brownian-coagulation-like or evaporation-condensation-like processes, depending on the strength of hydrophobicity with respect to electrostatic interaction. Regarding multivalent salt, HI does not affect the condensation dynamics of multivalent ions but facilitates quicker movement of local dipolar clusters along the PE, thereby expediting the collapse process. These results provide valuable insights into the underlying mechanisms of HI in PE collapse kinetics.

Insights into the Characterization of the Self-Assembly of Different Types of Amphiphilic Molecules Using Dynamic Light Scattering

The self-assembly of cetyltrimethylammonium bromide, sodium dodecylsulfate, Triton X-100, and sulfobetaine surfactants in aqueous solutions was examined by dynamic light scattering, both in the presence and absence of 0.1 M NaCl salt, across various temperatures. For each surfactant, critical parameters, such as concentration and phase transition temperatures, of micelles were determined by monitoring changes in the hydrodynamic diameter with concentration and temperature. Additionally, we explored the self-assembly behavior of these surfactants when they are introduced alongside polystyrene nanoparticles. Our findings enabled the elucidation of surfactant molecule adsorption mechanisms onto polystyrene nanoparticle surfaces. Furthermore, by analyzing variations in the z-average diameter and zeta potential, we were able to establish the Krafft point, a parameter that remains imperceptible when polystyrene nanoparticles are absent from the solution.

Polyelectrolytes: From Seminal Works to the Influence of the Charge Sequence

We propose a selected tour of the physics of polyelectrolytes (PE) following the line initiated by de Gennes and coworkers in their seminal 1976 paper. The early works which used uniform charge distributions along the PE backbone achieved tremendous progress and set most milestones in the field. Recently, the focus has shifted to the role of the charge sequence. Revisited topics include PE complexation and polyampholytes (PA). We develop the example of a random PE in poor solvent forming pearl-necklace structures. It is shown that the pearls typically adopt very asymmetric mass and charge distributions. Individual sequences do not necessarily reflect the ensemble statistics and a rich variety of behaviors emerges (specially for PA). Pearl necklaces are dynamic structures and switch between various types of pearl-necklace structures, as described for both PE and PA.

Kinetics of charged polymer collapse in poor solvents

Extensive molecular dynamics simulations, using simple charged polymer models, have been employed to probe the collapse kinetics of a single flexible polyelectrolyte (PE) chain under implicit poor solvent conditions. We investigate the role of the charged nature of PE chain (A), valency of counterions (Z) on the kinetics of such PE collapse. Our study shows that the collapse kinetics of charged polymers are significantly different from those of the neutral polymer and that the finite-size scaling behavior of PE collapse times does not follow the Rouse scaling as observed in the case of neutral polymers. The critical exponent for charged PE chains is found to be less than that of neutral polymers and also exhibits dependence on counterion valency. The coarsening of clusters along the PE chain suggests a multi-stage collapse and exhibits opposite behavior of exponents compared to neutral polymers: faster in the early stages and slower in the later stages of collapse.

The Measurement of Information and Free Energy in Mechanical-Force-Driven Coil-Globule Transitions

We study the role of information (the relative entropy) for polymers undergoing coil-globule transitions driven by a time-dependent force. Pulling experiments at various speeds are performed by Brownian dynamics simulations. We obtain the work distributions for the forward and time-reversed backward processes and information stored at the end of the nonequilibrium pulling processes. We present the systematic method to measure the information from the pulling experiments and extract the information by analyzing slowly relaxing modes. When the information is incorporated, the work distributions modified by the information allow access to the proper free energy via the formulation of the generalized fluctuation theorems even if the initial states of the forward and time-reversed backward processes are out of equilibrium. This demonstrates that the work-information conversion works well for a single-molecule system with many degrees of freedom.

Ion Condensation onto Ribozyme Is Site Specific and Fold Dependent

The highly charged RNA molecules, with each phosphate carrying a single negative charge, cannot fold into well-defined architectures with tertiary interactions in the absence of ions. For ribozymes, divalent cations are known to be more efficient than monovalent ions in driving them to a compact state, although Mg²⁺ ions are needed for catalytic activities. Therefore, how ions interact with RNA is relevant in understanding RNA folding. It is often thought that most of the ions are territorially and nonspecifically bound to the RNA, as predicted by the counterion condensation theory. Here, we show using simulations of Azoarcus ribozyme, based on an accurate coarse-grained three-site interaction model with explicit divalent and monovalent cations, that ion condensation is highly specific and depends on the nucleotide position. The regions with high coordination between the phosphate groups and the divalent cations are discernible even at very low Mg²⁺ concentrations when the ribozyme does not form tertiary interactions. Surprisingly, these regions also contain the secondary structural elements that nucleate subsequently in the self-assembly of RNA, implying that ion condensation is determined by the architecture of the folded state. These results are in sharp contrast to interactions of ions (monovalent and divalent) with rigid charged rods, in which ion condensation is uniform and position independent. The differences are explained in terms of the dramatic nonmonotonic shape fluctuations in the ribozyme as it folds with increasing Mg²⁺ or Ca²⁺ concentration.

Intermolecular Interactions in Polyelectrolyte and Surfactant Complexes in Solution

Polyelectrolytes are an important class of polymeric materials and are increasingly used in complex industrial formulations. A core use of these materials is in mixtures with surfactants, where a combination of hydrophobic and electrostatic interactions drives unique solution behavior and structure formation. In this review, we apply a molecular level perspective to the broad literature on polyelectrolyte-surfactant complexes, discussing explicitly the hydrophobic and electrostatic interaction contributions to polyelectrolyte surfactant complexes (PESCs), as well as the interplay between the two molecular interaction types. These interactions are sensitive to a variety of solution conditions, such as pH, ionic strength, mixing procedure, charge density, etc. and these parameters can readily be used to control the concentration at which structures form as well as the type of structure in the bulk solution.

Forced-rupture of cell-adhesion complexes reveals abrupt switch between two brittle states

Cell adhesion complexes (CACs), which are activated by ligand binding, play key roles in many cellular functions ranging from cell cycle regulation to mediation of cell extracellular matrix adhesion. Inspired by single molecule pulling experiments using atomic force spectroscopy on leukocyte function-associated antigen-1 (LFA-1), expressed in T-cells, bound to intercellular adhesion molecules (ICAM), we performed constant loading rate (r_f) and constant force (F) simulations using the self-organized polymer model to describe the mechanism of ligand rupture from CACs. The simulations reproduce the major experimental finding on the kinetics of the rupture process, namely, the dependence of the most probable rupture forces (f*s) on ln r_f (r_f is the loading rate) exhibits two distinct linear regimes. The first, at low r_f, has a shallow slope, whereas the slope at high r_f is much larger, especially for a LFA-1/ICAM-1 complex with the transition between the two occurring over a narrow r_f range. Locations of the two transition states (TSs) extracted from the simulations show an abrupt change from a high value at low r_f or constant force, F, to a low value at high r_f or F. This unusual behavior in which the CACs switch from one brittle (TS position is a constant over a range of forces) state to another brittle state is not found in forced-rupture in other protein complexes. We explain this novel behavior by constructing the free energy profiles, F(Λ)s, as a function of a collective reaction coordinate (Λ), involving many key charged residues and a critical metal ion (Mg²⁺). The TS positions in F(Λ), which quantitatively agree with the parameters extracted using the Bell-Evans model, change abruptly at a critical force, demonstrating that it, rather than the molecular extension, is a good reaction coordinate. Our combined analyses using simulations performed in both the pulling modes (constant r_f and F) reveal a new mechanism for the two loading regimes observed in the rupture kinetics in CACs.

Charge fluctuation effects on the shape of flexible polyampholytes with applications to intrinsically disordered proteins

Random polyampholytes (PAs) contain positively and negatively charged monomers that are distributed randomly along the polymer chain. The interaction between charges is assumed to be given by the Debye-Huckel potential. We show that the size of the PA is determined by an interplay between electrostatic interactions, giving rise to the polyelectrolyte effect due to net charge per monomer (σ) and an effective attractive PA interaction due to charge fluctuations, δσ. The interplay between these terms gives rise to non-monotonic dependence of the radius of gyration, R _g , on the inverse Debye length, κ, when PA effects are important ( δ σ σ > 1 ). In the opposite limit, R _g decreases monotonically with increasing κ. Simulations of PA chains, using a charged bead-spring model, further corroborate our theoretical predictions. The simulations unambiguously show that conformational heterogeneity manifests itself among sequences that have identical PA parameters. A clear implication is that the phases of PA sequences, and by inference intrinsically disordered proteins (IDPs), cannot be determined using only the bare PA parameters (σ and δσ). The theory is used to calculate the changes in R _g on N, the number of residues for a set of IDPs. For a certain class of IDPs, with N between 24 and 441, the size grows as R _g ∼ N ^0.6, which agrees with data from small angle X-ray scattering experiments.

Multivalent ions induce lateral structural inhomogeneities in polyelectrolyte brushes

Subtle details about a polyelectrolyte's surrounding environment can dictate its structural features and potential applications. Atomic force microscopy (AFM), surface forces apparatus (SFA) measurements, and coarse-grained molecular dynamics simulations are combined to study the structure of planar polyelectrolyte brushes [poly(styrenesulfonate), PSS] in a variety of solvent conditions. More specifically, AFM images provide a first direct visualization of lateral inhomogeneities on the surface of polyelectrolyte brushes collapsed in solutions containing trivalent counterions. These images are interpreted in the context of a coarse-grained molecular model and are corroborated by accompanying interaction force measurements with the SFA. Our findings indicate that lateral inhomogeneities are absent from PSS brush layers collapsed in a poor solvent without multivalent ions. Together, AFM, SFA, and our molecular model present a detailed picture in which solvophobic and multivalent ion-induced effects work in concert to drive strong phase separation, with electrostatic bridging of polyelectrolyte chains playing an essential role in the collapsed structure formation.

Toroidal Condensates by Semiflexible Polymer Chains: Insights into Nucleation, Growth and Packing Defects

Deciphering the principles of DNA condensation is important to understand problems such as genome packing and DNA compaction for delivery in gene therapy. DNA molecules condense into toroids and spindles upon the addition of multivalent ions. Nucleation of a loop in the semiflexible DNA chain is critical for both the toroid and spindle formation. To understand the structural differences in the nucleated loop, which cause bifurcation in the condensation pathways leading to toroid or spindle formation, we performed molecular dynamics simulations using a coarse-grained bead-spring polymer model. We find that the formation of a toroid or a spindle is correlated with the orientation of the chain segments close to the loop closure in the nucleated loop. Simulations show that toroids grow in size when spindles in solution interact with a pre-existing toroid and merge into it by spooling around the circumference of the toroid, forming multimolecular toroidal condensates. The merging of spindles with toroids is facile, indicating that this should be the dominant pathway through which the toroids grow in size. The Steinhardt bond order parameter analysis of the toroid cross section shows that the chains pack in a hexagonal fashion. In agreement with the experiments there are regions in the toroid with good hexagonal packing and also with considerable disorder. The disorder in packing is due to the defects, which are propagated during the growth of toroids. In addition to the well-known crossover defect, we have identified three other forms of defects, which perturb hexagonal packing. The new defects identified in the simulations are amenable to experimental verification.

Continuous, size and shape-control synthesis of hollow silica nanoparticles enabled by a microreactor-assisted rapid mixing process

Hollow silica nanoparticles (HSNPs) were synthesized using a microreactor-assisted system with a hydrodynamic focusing micromixer. Due to the fast mixing of each precursor in the system, the poly(acrylic acid) (PAA) thermodynamic-locked (TML) conformations were protected from their random aggregations by the immediately initiated growth of silica shells. When altering the mixing time through varying flow rates and flow rate ratios, the different degrees of the aggregation of PAA TML conformations were observed. The globular and necklace-like TML conformations were successfully captured by modifying the PAA concentration at the optimized mixing condition. Uniform HSNPs with an average diameter ∼30 nm were produced from this system. COMSOL numerical models was established to investigate the flow and concentration profiles, and their effects on the formation of PAA templates. Finally, the quality and utility of these uniform HSNPs were demonstrated by the fabrication of antireflective thin films on monocrystalline photovoltaic cells which showed a 3.8% increase in power conversion efficiency.

Regimes of electrostatic collapse of a highly charged polyelectrolyte in a poor solvent

We perform extensive molecular dynamics simulations of a highly charged, collapsed, flexible polyelectrolyte chain in a poor solvent for the case when the electrostatic interactions, characterized by the reduced Bjerrum length l_B, are strong. We find the existence of several sub-regimes in the dependence of the gyration radius of the chain R_g on l_B characterized by R_g ∼ l. In contrast to a good solvent, the exponent γ for a poor solvent crucially depends on the size and valency of the counterions. To explain the different sub-regimes, we generalize the existing counterion fluctuation theory by including a more complete account of all possible volume interactions in the free energy of the polyelectrolyte chain. We also show that the presence of condensed counterions modifies the effective attraction among the chain monomers and modulates the sign of the second virial coefficient under poor solvent conditions.

Comparing Solvophobic and Multivalent Induced Collapse in Polyelectrolyte Brushes

Coarse-grained molecular dynamics enhanced by free-energy sampling methods is used to examine the roles of solvophobicity and multivalent salts on polyelectrolyte brush collapse. Specifically, we demonstrate that while ostensibly similar, solvophobic collapsed brushes and multivalent-ion collapsed brushes exhibit distinct mechanistic and structural features. Notably, multivalent-induced heterogeneous brush collapse is observed under good solvent polymer backbone conditions, demonstrating that the mechanism of multivalent collapse is not contingent upon a solvophobic backbone. Umbrella sampling of the potential of mean-force (PMF) between two individual brush strands confirms this analysis, revealing starkly different PMFs under solvophobic and multivalent conditions, suggesting the role of multivalent "bridging" as the discriminating feature in trivalent collapse. Structurally, multivalent ions show a propensity for nucleating order within collapsed brushes, whereas poor-solvent collapsed brushes are more disordered; this difference is traced to the existence of a metastable PMF minimum for poor solvent conditions, and a global PMF minimum for trivalent systems, under experimentally relevant conditions.

Polyelectrolyte conformational transition in aqueous solvent mixture influenced by hydrophobic interactions and hydrogen bonding effects: PAA-water-ethanol

Molecular dynamics simulations of poly(acrylic acid) PAA chain in water-ethanol mixture were performed for un-ionized and ionized cases at different degree-of-ionization 0%, 80% and 100% of PAA chain by Na(+) counter-ions and co-solvent (ethanol) concentration in the range 0-90vol% ethanol. Aspects of structure and dynamics were investigated via atom pair correlation functions, number and relaxation of hydrogen bonds, nearest-neighbor coordination numbers, and dihedral angle distribution function for back-bone and side-groups of the chain. With increase in ethanol concentration, chain swelling is observed for un-ionized chain (f=0) and on the contrary chain shrinkage is observed for partially and fully ionized cases (i.e., f=0.8 and 1). For un-ionized PAA, with increase in ethanol fraction ϕeth the number of PAA-ethanol hydrogen bonds increases while PAA-water decreases. Increase in ϕeth leads to PAA chain expansion for un-ionized case and chain shrinkage for ionized case, in agreement with experimental observations on this system. For ionized-PAA case, chain shrinkage is found to be influenced by intermolecular hydrogen bonding with water as well as ethanol. The localization of ethanol molecules near the un-ionized PAA backbone at higher levels of ethanol is facilitated by a displacement of water molecules indicating presence of specific ethanol hydration shell, as confirmed by results of the RDF curves and coordination number calculations. This behavior, controlled by hydrogen bonding provides a significant contribution to such a conformational transition behavior of the polyelectrolyte chain. The interactions between counter-ions and charges on the PAA chain also influence chain collapse. The underlying origins of polyelectrolyte chain collapse in water-alcohol mixtures are brought out for the first time via explicit MD simulations by this study.

Energy landscape analyses of disordered histone tails reveal special organization of their conformational dynamics

Histone tails are highly flexible N- or C-terminal protrusions of histone proteins which facilitate the compaction of DNA into dense superstructures known as chromatin. On a molecular scale histone tails are polyelectrolytes with high degree of conformational disorder which allows them to function as biomolecular "switches", regulating various genetic processes. Unfortunately, their intrinsically disordered nature creates obstacles for comprehensive experimental investigation of both the structural and dynamical aspects of histone tails, because of which their conformational behaviors are still not well understood. In this work we have carried out ∼3 microsecond long all atom replica exchange molecular dynamics (REMD) simulations for each of four histone tails, H4, H3, H2B, and H2A, and probed their intrinsic conformational preferences. Our subsequent free energy landscape analysis demonstrated that most tails are not fully disordered, but show distinct conformational organization, containing specific flickering secondary structural elements. In particular, H4 forms β-hairpins, H3 and H2B adopt α-helical elements, while H2A is fully disordered. We rationalized observed patterns of conformational dynamics of various histone tails using ideas from physics of polyelectrolytes and disordered systems. We also discovered an intriguing re-entrant contraction-expansion of the tails upon heating, which is caused by subtle interplay between ionic screening and chain entropy.

Probing solvation decay length in order to characterize hydrophobicity-induced bead-bead attractive interactions in polymer chains

In this paper, we quantitatively demonstrate that exponentially decaying attractive potentials can effectively mimic strong hydrophobic interactions between monomer units of a polymer chain dissolved in aqueous solvent. Classical approaches to modeling hydrophobic solvation interactions are based on invariant attractive length scales. However, we demonstrate here that the solvation interaction decay length may need to be posed as a function of the relative separation distances and the sizes of the interacting species (or beads or monomers) to replicate the necessary physical interactions. As an illustrative example, we derive a universal scaling relationship for a given solute-solvent combination between the solvation decay length, the bead radius, and the distance between the interacting beads. With our formalism, the hydrophobic component of the net attractive interaction between monomer units can be synergistically accounted for within the unified framework of a simple exponentially decaying potential law, where the characteristic decay length incorporates the distinctive and critical physical features of the underlying interaction. The present formalism, even in a mesoscopic computational framework, is capable of incorporating the essential physics of the appropriate solute-size dependence and solvent-interaction dependence in the hydrophobic force estimation, without explicitly resolving the underlying molecular level details.

Charge density of divalent metal cations determines RNA stability

RNA molecules are exquisitely sensitive to the properties of counterions. The folding equilibrium of the Tetrahymena ribozyme is measured by nondenaturing gel electrophoresis in the presence of divalent group IIA metal cations. The stability of the folded ribozyme increases with the charge density (zeta) of the cation. Similar scaling is found when the free energy of the RNA folded in small and large metal cations is measured by urea denaturation. Brownian dynamics simulations of a polyelectrolyte show that the experimental observations can be explained by nonspecific ion-RNA interactions in the absence of site-specific metal chelation. The experimental and simulation results establish that RNA stability is largely determined by a combination of counterion charge and the packing efficiency of condensed cations that depends on the excluded volume of the cations.

Ion-mediated nucleic acid helix-helix interactions

Salt ions are essential for the folding of nucleic acids. We use the tightly bound ion (TBI) model, which can account for the correlations and fluctuations for the ions bound to the nucleic acids, to investigate the electrostatic free-energy landscape for two parallel nucleic acid helices in the solution of added salt. The theory is based on realistic atomic structures of the helices. In monovalent salt, the helices are predicted to repel each other. For divalent salt, while the mean-field Poisson-Boltzmann theory predicts only the repulsion, the TBI theory predicts an effective attraction between the helices. The helices are predicted to be stabilized at an interhelix distance approximately 26-36 A, and the strength of the attractive force can reach -0.37 k(B)T/bp for helix length in the range of 9-12 bp. Both the stable helix-helix distance and the strength of the attraction are strongly dependent on the salt concentration and ion size. With the increase of the salt concentration, the helix-helix attraction becomes stronger and the most stable helix-helix separation distance becomes smaller. For divalent ions, at very high ion concentration, further addition of ions leads to the weakening of the attraction. Smaller ion size causes stronger helix-helix attraction and stabilizes the helices at a shorter distance. In addition, the TBI model shows that a decrease in the solvent dielectric constant would enhance the ion-mediated attraction. The theoretical findings from the TBI theory agree with the experimental measurements on the osmotic pressure of DNA array as well as the results from the computer simulations.

Counterion charge density determines the position and plasticity of RNA folding transition states

The self-assembly of RNA structure depends on the interactions of counterions with the RNA and with each other. Comparison of various polyamines showed that the tertiary structure of the Tetrahymena ribozyme is more stable when the counterions are small and highly charged. By monitoring the folding kinetics of the ribozyme as a function of polyamine concentration, we now find that the charge density of the counterions determines the positions of the folding transition states. The transition state ensemble (TSE) between U and N moves away from the native state as the counterion valence and charge density increase, as predicted by the Hammond postulate. The TSE is broader and less structured when the RNA is refolded in polyamines rather than Mg2+. That the charge density of the counterions determines the plasticity of the TSE demonstrates the importance of interactions among condensed counterions for the self-assembly of RNA structures. We propose that the major barrier to RNA folding is dominated by entropy changes when counterion charge density is low and enthalpy differences when it is high.

Folding of the Tetrahymena ribozyme by polyamines: importance of counterion valence and size

Polyamines are abundant metabolites that directly influence gene expression. Although the role of polyamines in DNA condensation is well known, their role in RNA folding is less understood. Non-denaturing gel electrophoresis was used to monitor the equilibrium folding transitions of the Tetrahymena ribozyme in the presence of polyamines. All of the polyamines tested induce near-native structures that readily convert to the native conformation in Mg(2+). The stability of the folded structure increases with the charge of the polyamine and decreases with the size of the polyamine. When the counterion excluded volume becomes large, the transition to the native state does not go to completion even under favorable folding conditions. Brownian dynamics simulations of a model polyelectrolyte suggest that the kinetics of counterion-mediated collapse and the dimensions of the collapsed RNA chains depend on the structure of the counterion. The results are consistent with delocalized condensation of polyamines around the RNA. However, the effective charge of the counterions is lowered by their excluded volume. The stability of the folded RNA is enhanced when the spacing between amino groups matches the distance between adjacent phosphate groups. These results show how changes in intracellular polyamine concentrations could alter RNA folding pathways.

Metastable intermediates in the condensation of semiflexible polymers

Motivated by results from an earlier Brownian dynamics simulation for the collapse of a single, stiff polymer in a poor solvent [B. Schnurr, F. C. MacKintosh, and D. R. M. Williams, Europhys. Lett. 51, 279 (2000)] we calculate the conformational energies of the intermediate (racquet) states suggested by the simulations. In the absence of thermal fluctuations (at zero temperature) the annealed shapes of these intermediates are well-defined in certain limits, with their major structural elements given by a particular case of Euler's elastica. In appropriate units, a diagram emerges that displays the relative stability of all states, tori, and racquets. We conclude that, in marked contrast to the collapse of flexible polymers, the condensation of semiflexible or stiff polymers generically proceeds via a cascade through metastable intermediates, the racquets, towards a ground state, the torus or ring, as seen in the dynamical simulations.