Role of chain stiffness on the conformation of single polyelectrolytes in salt solutions

Conformation of single polyelectrolytes in tetravalent salt solutions is investigated under the framework of a coarse-grained model, using Langevin dynamics simulations. The chain size, studied by the radius of gyration, shows three different variational behaviors with salt concentration, depending on the chain stiffness. According to the size variations, polyelectrolytes of fixed chain length are classified into three categories: (1) flexible chain, for which the variation shows a curve similar to a tilted L, (2) semiflexible chain, whose curve resembles U, and (3) rigid chain, for which the curve is a straight line. The wormlike chain model with persistence length predicted by the Odijk-Skolnick-Fixman theory is found to be able to qualitatively describe the end-to-end distance at low salt concentration not only for semiflexible and rigid chains but also for flexible chain. In a low salt region, a flexible polyelectrolyte extends more significantly than a semiflexible chain, in reference of the size of their uncharged counterparts, and in a high salt region, regardless of chain stiffness, a chain attains a dimension comparable to that of its neutral polymer. The chain stiffness influences both the local and the global chain structures. A flexible chain exhibits a zigzagged local structure in the presence of salt ions, and the condensed structure is a disordered, random globule. A semiflexible chain is locally smooth, and the condensed structure is orderly packed, taking a form such as hairpin or toroid. Moreover, the chain stiffness can also affect the nature of the coil-globule transition. The transition occurred in a discrete manner for semiflexible chain, whereas it occurred in a continuous way for flexible chain. This discrete feature happened not only at low salt concentration when a semiflexible chain collapsed but also at high salt concentration when the collapsed chain is reexpanded. At the end, the effects of chain stiffness and salt concentration on the conformation of single polyelectrolytes are summarized in a schematic state diagram.     

Electrostatic rigidity of polyelectrolytes from reparametrization invariance

The persistence length l_p of a polyelectrolyte chain can be represented as l_p = l_O + l_e where l_O is the bare persistence and I_e is the electrostatic contribution coming from the effects of electrostatic chain self-interactions. Using a reparametrization-invariant path integral model of semiflexible polymers we find that I_e depends on the ionic strength I as I_e ∼ I^−1/2. This result accords with experimental observations and recent Monte Carlo simulations. Reparametrization-invariance is apparently an essential constrainet in selecting acceptable models of semiflexible polymers.     

Viscoelastic properties of an isolated polymer chain with arbitrary flexibility,chain length and hydrodynamic interactions from 4‐dimensional Dirac propagator

Kholodenko's theory of semiflexible polymer chains, the conformation and properties of which are obtained from the Dirac propagator, shows applicability to dilute solutions of semiflexible polymers of arbitrary persistence and contour lengths by calculating the static scattering function and the squared end‐to‐end distance of the polymer chain. In the present work, the theory is extended and applied to obtain the intrinsic viscosity with consideration of hydrodynamic interactions. The intrinsic viscosity formula is derived as function of chain length and persistence length. The hydrodynamic interactions are also taken into account following the Kirkwood and Riseman scheme. From this calculation, we obtain the general expression for the intrinsic viscosity and diffusion coefficients covering the whole range of chain flexibilities without confusion with the excluded volume effects. Calculated limiting values of hydrodynamical observables are in complete agreement with those known for random coils and rigid rods.     

Computer simulations of diffusion and dynamics of short-chain polyelectrolytes

Brownian dynamics simulations are conducted to investigate the diffusional and dynamic properties of polyelectrolytes in dilute salt-free solutions. The polyelectrolyte molecule is represented by a bead-spring chain in a primitive model. The long-range hydrodynamic and Coulomb interactions are both taken into consideration through the Ewald summations for the first time. The major finding of our simulations is that the dependence of the long-time chain diffusivity on the Coulomb interaction strength is very different from that of the Kirkwood short-time diffusivity, which simply shows a trend nearly opposite to the chain size. When ignoring the hydrodynamic interaction (HI), the coupling effect between the chain and its counterions gives rise to a noticeable increase in the long-time diffusivity at intermediate electrostatic interaction strengths. However, the incorporation of HI suppresses this effect to a degree that one can no longer discern it. Moreover, the rotational relaxation is found to show a dependence opposite to that of the gyration radius relaxation.     

Simulations and scattering functions of polyelectrolyte–macroion complexes

Using Monte Carlo simulations of complex formation between a polyelectrolyte chain and an oppositely charged macroion, we calculated the scattering function of the polyelectrolyte chain. We investigated the case of the isolated polyelectrolyte chain and studied the effect and influence of key parameters such as the ionic concentration of the solution, polyelectrolyte length and intrinsic rigidity on the scattering function. Then, we focused on the polyelectrolyte–macroion complex by calculating the structure factor S(q) of the adsorbed polyelectrolyte chain. Typical conformations ranging from coils, extended chains to solenoids are revealed and the corresponding S(q) analysed. The effects of ionic concentration, chain length and intrinsic rigidity and relative size ratio between the polyelectrolyte and the macroion are investigated. Important effects on the structure factor of the adsorbed polyelectrolyte are observed when the macroion is partially or totally wrapped by the polyelectrolyte. Distance correlations between the polyelectrolyte monomer positions at the surface of the macroion induce the formation of peaks in the fractal regime of S(q). For semiflexible chains, when solenoid conformations are observed, the position of the peaks in the fractal regime corresponds directly to the separation distance between the turns. The formation of a protruding tail in solution is also observed through the formation in the fractal regime of a linear domain.This revised version was published online in November 2004 with corrections to the authors.     

Kinetics of interior loop formation in semiflexible chains

Loop formation between monomers in the interior of semiflexible chains describes elementary events in biomolecular folding and DNA bending. We calculate analytically the interior distance distribution function for semiflexible chains using a mean field approach. Using the potential of mean force derived from the distance distribution function we present a simple expression for the kinetics of interior looping by adopting Kramers theory. For the parameters, that are appropriate for DNA, the theoretical predictions in comparison with the case are in excellent agreement with explicit Brownian dynamics simulations of wormlike chain (WLC) model. The interior looping times (tauIC) can be greatly altered in the cases when the stiffness of the loop differs from that of the dangling ends. If the dangling end is stiffer than the loop then tauIC increases for the case of the WLC with uniform persistence length. In contrast, attachment of flexible dangling ends enhances rate of interior loop formation. The theory also shows that if the monomers are charged and interact via screened Coulomb potential then both the cyclization (tauc) and interior looping (tauIC) times greatly increase at low ionic concentration. Because both tauc and tauIC are determined essentially by the effective persistence length [lp(R)] we computed lp(R) by varying the range of the repulsive interaction between the monomers. For short range interactions lp(R) nearly coincides with the bare persistence length which is determined largely by the backbone chain connectivity. This finding rationalizes the efficacy of describing a number of experimental observations (response of biopolymers to force and cyclization kinetics) in biomolecules using WLC model with an effective persistence length.     

Titration of hydrophobic polyelectrolytes using Monte Carlo simulations

The conformation and titration curves of weak (or annealed) hydrophobic polyelectrolytes have been examined using Monte Carlo simulations with screened Coulomb potentials in the grand canonical ensemble. The influence of the ionic concentration pH and presence of hydrophobic interactions has been systematically investigated. A large number of conformations such as extended, pearl-necklace, cigar-shape, and collapsed structures resulting from the subtle balance of short-range hydrophobic attractive interactions and long-range electrostatic repulsive interactions between the monomers have been observed. Titration curves were calculated by adjusting the pH-pK(0) values (pK(0) represents the intrinsic dissociation constant of an isolated monomer) and then calculating the ionization degree alpha of the polyelectrolyte. Important transitions related to cascades of conformational changes were observed in the titration curves, mainly at low ionic concentration and with the presence of strong hydrophobic interactions. We demonstrated that the presence of hydrophobic interactions plays an important role in the acid-base properties of a polyelectrolyte in promoting the formation of compact conformations and hence decreasing the polyelectrolyte degree of ionization for a given pH-pK(0) value.     

Diblock polyampholytes grafted onto spherical particles: effect of stiffness, charge density, and grafting density

The structure of spherical brushes formed by symmetric diblock polyampholytes end-grafted onto small spherical particles in aqueous solution is examined within the framework of the so-called primitive model using Monte Carlo simulations. The properties of the two blocks are identical except for the sign of their charges. Three different chain flexibilities corresponding to flexible, semiflexible, and stiff blocks are considered at various polyampholyte linear charge densities and grafting densities. The link between the two blocks is flexible at all conditions, and the grafted segments are laterally mobile. Radial and lateral spatial distribution functions of different types and single-chain properties are analyzed. The brush structure strongly depends on the chain flexibility. With flexible chains, a disordered polyelectrolyte complex is formed at the surface of the particle, the complex becoming more compact at increasing linear charge density. With stiff blocks, the inner blocks are radially oriented. At low linear charged density, the outer blocks are orientationally disordered, whereas at increasing electrostatic interaction the two blocks of a polyampholyte are parallel and close to each other, leading to an ordered structure referred to as a polyampholyte star. As the grafting density is increased, the brush thickness responds differently for flexible and nonflexible chains, depending on a different balance between electrostatic interactions and excluded volume effects.     

Scattering properties of a single semiflexible polyelectrolyte

The structure factor of a single semiflexible polyelectrolyte has been calculated as a function of chain length, intrinsic backbone stiffness, and salt concentration. Because of the insignificant coupling of the intrinsic stiffness and electrostatic persistence length, we carry out our calculations in the flexible limit. Within the variational scheme adopted here, we obtain fractal dimensions consistent with our earlier calculations of the configurational properties. As the chain length is increased, the electrostatic interaction is progressively screened, leading to the crossover regions. In the first crossover, the effective fractal dimension, D_eff, is as low as 1, and in the second crossover D_eff is 5/3, although the radius of gyration exponent is 2/5. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2644–2652, 2001     

Langevin dynamics of semiflexible polyelectrolytes: rod-toroid-globule-coil structures and counterion distribution

We have investigated the nature of counterion condensation on uniformly charged semiflexible polyelectrolyte chains and the concomitant configurations by monitoring the role of chain stiffness, chain length, counterion valency, and the strength of electrostatic interaction. The counterion condensation is seen to follow the adsorption process and the effective polymer charge increases with chain stiffness. Size and shape, as calculated through the radius of gyration, effective persistence length, and hydrodynamic radius, are studied. Stable coil-like, globular, folded-chain, toroidal, and rodlike configurations are possible at suitable combinations of values of chain stiffness, chain length, electrostatic interaction strength, and the valency of counterion. For high strengths of electrostatic interactions, sufficiently stiff polyelectrolytes form toroids in the presence of multivalent counterions, whereas flexible polyelectrolytes form disordered globules. The kinetic features of the nucleation and growth of toroids are monitored. Several metastable structures are found to frustrate the formation of toroids. The generic pathway involves the nucleation of one primary loop somewhere along the chain contour, followed by a growth process where the rest of the chain is folded continuously on top of the primary loop. The dependence of the average radii of toroids on the chain length is found to be roughly linear, in disagreement with existing scaling arguments.     

Persistence lengths and structure factors of wormlike polymers under confinement

The behavior of semiflexible chains modeling wormlike polymers such as DNA and actin in confined spaces was explored by coarse-grained Monte Carlo simulations. The persistence length P, mean end-to-end distance R2, mean radius of gyration Rg2, and the size ratio R2/Rg2 were computed for chains in slits, cylinders, and spheres. It was found that the intrinsic persistence length of a free chain undergoes on confinement substantial alteration into the apparent persistence length. The qualitative differences were found in trends of the apparent persistence lengths between slits and cylinders on one side and spheres on the other side. The quantities P, R2, Rg2, and R2/Rg2 display similar dependences upon squeezing the chains in nanopores. The above quantities change nonmonotonically with confinement in slits and cylinders, whereas they drop smoothly with decreasing radius of a sphere. For elongation of a chain in a cylinder, two regimes corresponding to strong and moderate confinements were found and compared to experiments and predictions of the blob and Odijk theories. In a spherical cavity, the toroidal chain structure with a hole in the center was detected under strong confinements. The scattering form factor S(q) computed for semiflexible confined chains revealed three regimes of behavior in a slit and a cylinder that matched up well with the scaling theory. The complex form of the function S(q) computed for a sphere was interpreted as a sign of the toroidal structure. A reasonable agreement was found between the simulations and measurements of DNA and actin filaments, confined in nano- and microfluidic channels and spherical droplets, pertaining to the changes of the persistence lengths, chain elongation, and toroidal structure formation.     

海藻酸钠在KCl水溶液中的粘度行为

通过测定海藻酸钠水溶液的特性粘数及在低高于强度的条件下其浓度与比浓粘度关系曲线上的峰值，系统地研究了KCl浓度在2×10（－5）－0.5mol·L（-1）范围内对海藻酸钠溶液粘度行为的影响．根据Odijk－Skolnick－Fixman理论和Rinaudo的处理方法，从理论上对海藻酸钠溶液的粘度行为进行了探讨．研究结果表明：聚电解质溶液的电粘滞效应可使用静电相关长度得到合理解释     

Electrostatic recognition and induced fit in the kappa-PVIIA toxin binding to Shaker potassium channel

Brownian dynamics (BD) and molecular dynamics (MD) simulations and electrostatic calculations were performed to study the binding process of kappa-PVIIA to the Shaker potassium channel and the structure of the resulting complex. BD simulations, guided by electrostatic interactions, led to an initial alignment between the toxin and the channel protein. MD simulations were then carried out to allow for rearrangements from this initial structure. After approximately 4 ns, a critical "induced fit" process was observed to last for approximately 2 ns. In this process, the interface was reorganized, and side chains were moved so that favorable atomic contacts were formed or strengthened, while unfavorable contacts were eliminated. The final complex structure was stabilized through electrostatic interactions with the positively charged side chain of Lys7 of kappa-PVIIA deeply inserted into the channel pore and other hydrogen bonds and by hydrophobic interactions involving Phe9 and Phe23 of the toxin. The validity of the predicted structure for the complex was assessed by calculating the effects of mutating charged and polar residues of both the toxin and the channel protein, with the calculated effects correlating reasonably well with experimental data. The present study suggests a general binding mechanism, whereby proteins are pre-aligned in their diffusional encounter by long-range electrostatic attraction, and nanosecond-scale rearrangements within the initial complex then lead to a specifically bound complex.     

Protein-polyelectrolyte cluster formation and redissolution: a Monte Carlo study

Aqueous solutions of proteins and oppositely charged polyelectrolytes were studied at different polyelectrolyte chain length, ionic strength, and protein-protein interaction potential as a function of the polyelectrolyte concentration. One of the protein models used represented lysozyme in aqueous environment. The model systems were solved by Monte Carlo simulations, and their properties were analyzed in terms of radial distribution functions, structure factors, and cluster composition probabilities. In the system with the strongest electrostatic protein-polyelectrolyte interaction the largest clusters were formed near or at equivalent amount of net protein charge and polyelectrolyte charge, whereas in excess of polyelectrolyte a redissolution appeared. Shorter polyelectrolyte chains and increased ionic strength lead to weaker cluster formation. An inclusion of nonelectrostatic protein-protein attraction promoted the protein-polyelectrolyte cluster formation.     

Ejection dynamics of semiflexible polymers out of a nanochannel

Using theoretical analysis and three-dimensional Langevin dynamics simulations, we investigate the influence of chain rigidity on the ejection dynamics of polymers from a nanochannel. We find that there exist two distinct dynamical regimes divided by a critical chain length for both flexible and semiflexible chains. At the short chain regime, semiflexible chains eject faster than flexible chains of the same chain length due to the longer occupying length. In contrast, at the long chain regime, semiflexible chains eject slower than flexible ones as the effective entropic driving force decreases. Based on these results, we propose that the nanochannels could be used to separate flexible and semiflexible chains effectively.     

Molecular dynamics simulations of polyelectrolyte multilayering on a charged particle

Molecular dynamics simulations of polyelectrolyte multilayering on a charged spherical particle revealed that the sequential adsorption of oppositely charged flexible polyelectrolytes proceeds with surface charge reversal and highlighted electrostatic interactions as the major driving force of layer deposition. Far from being completely immobilized, multilayers feature a constant surge of chain intermixing during the deposition process, consistent with experimental observations of extensive interlayer mixing in these films. The formation of multilayers as well as the extent of layer intermixing depends on the degree of polymerization of the polyelectrolyte chains and the fraction of charge on its backbone. The presence of ionic pairs between oppositely charged macromolecules forming layers seems to play an important role in stabilizing the multilayer film.     

Chain stiffness, salt valency, and concentration influences on titration curves of polyelectrolytes: Monte Carlo simulations

Monte Carlo simulations have been used to study two different models of a weak linear polyelectrolyte surrounded by explicit counterions and salt particles: (i) a rigid rod and (ii) a flexible chain. We focused on the influence of the pH, chain stiffness, salt concentration, and valency on the polyelectrolyte titration process and conformational properties. It is shown that chain acid-base properties and conformational properties are strongly modified when multivalent salt concentration variation ranges below the charge equivalence. Increasing chain stiffness allows to minimize intramolecular electrostatic monomer interactions hence improving the deprotonation process. The presence of di and trivalent salt cations clearly promotes the chain degree of ionization but has only a limited effect at very low salt concentration ranges. Moreover, folded structures of fully charged chains are only observed when multivalent salt at a concentration equal or above charge equivalence is considered. Long-range electrostatic potential is found to influence the distribution of charges along and around the polyelectrolyte backbones hence resulting in a higher degree of ionization and a lower attraction of counterions and salt particles at the chain extremities.     

Simulation of nonlinear shear rheology of dilute salt-free polyelectrolyte solutions

Brownian dynamics simulations are used to conduct a systematic analysis of the nonlinear shear rheology of dilute polyelectrolyte solutions, exploring its relationship to shear rate, Bjerrum length, and concentration. A simple coarse-grained bead-spring chain model that incorporates explicit counterions is used. It is found that the polyelectrolyte chains exhibit a shear thinning behavior at high shear rate (as characterized by bead Peclet number Pe) that is independent of the electrostatic strength due to the stripping of ions from close proximity to the chain caused by the flow. In contrast, at low values of Pe, the viscosity increases monotonically with increasing Bjerrum length over the range studied here, in contrast to the nonmonotonic trend displayed by the chain size. Furthermore, at fixed Bjerrum length, the reduced viscosity increases monotonically with concentration. The mechanism underlying these observations is essentially the primary electroviscous effect; the ion cloud surrounding a polyelectrolyte chain deforms in flow, causing a significant increase in viscosity as concentration increases. Finally, the authors have also considered the role of hydrodynamic interactions in these simulations, finding that for low concentration studies in shear flow, these do not qualitatively affect the results.     

Dressed ion theory of size-asymmetric electrolytes: effective ionic charges and the decay length of screened Coulomb potential and pair correlations

The effects of ionic size asymmetry on long-range electrostatic interactions in electrolyte solutions are investigated within the primitive model. Using the formalism of dressed ion theory we analyze correlation functions from Monte Carlo simulations and the hypernetted chain approximation for size asymmetric 1:1 electrolytes. We obtain decay lengths of the screened Coulomb potential, effective charges of ions, and effective permittivity of the solution. It is found that the variation of these quantities with the degree of size asymmetry depends in a quite intricate manner on the interplay between the electrostatic coupling and excluded volume effects. In most cases the magnitude of the effective charge of the small ion species is larger than that of the large species; the difference increases with increasing size asymmetry. The effective charges of both species are larger (in absolute value) than the bare ionic charge, except for high asymmetry where the effective charge of the large ions can become smaller than the bare charge.     

Packaging of a flexible polyelectrolyte inside a viral capsid: effect of salt concentration and salt valence

The effect of salt on the location and structure of a flexible polyelectrolyte confined inside a viral capsid and the Donnan equilibrium of the salt across the capsid have been examined using a coarse-grained model solved by Monte Carlo simulations. The polyelectrolyte was represented by a linear jointed chain of charged beads, and the capsid was represented by a spherical shell with embedded charges. At low salt concentration, the polyelectrolyte was strongly adsorbed onto the inner capsid surface, whereas at high salt concentration it was located preferentially in the central part of the capsid. Under the condition of equal Debye screening length, the electrostatic screening increased as the valence of the polyelectrolyte counterion was increased. The distribution of the small cations and anions was unequal across the capsid. An excess of polyelectrolyte counterions occurred inside the capsid, and the excess increased with the salt concentration. A simplified representation of the small ions through the use of the screened Coulomb potential provided only a qualitatively correct picture; the electrostatic screening originating from the small ions was exaggerated.