Folding transition of a single semiflexible polyelectrolyte chain through toroidal bundling of loop structures

We consider how the DNA coil-globule transition progresses via the formation of a toroidal ring structure. We formulate a theoretical model of this transition as a phenomenon in which an unstable single loop generated as a result of thermal fluctuation is stabilized through association with other loops along a polyelectrolyte chain. An essential property of the chain under consideration is that it follows a wormlike chain model. A toroidal bundle of loop structures is characterized by a radius and a winding number. The statistical properties of such a chain are discussed in terms of the free energy as a function of the fraction of unfolded segments. We also present an actual experimental observation of the coil-globule transition of single giant DNA molecules, T4 DNA (165.5 kbp), with spermidine (3+), where intrachain phase segregation appears at a NaCl concentration of more than 10 mM. Both the theory and experiments lead to two important points. First, the transition from a partially folded state to a completely folded state has the characteristics of a continuous transition, while the transition from an unfolded state to a folded state has the characteristics of a first-order phase transition. Second, the appearance of a partially folded structure requires a folded structure to be less densely packed than in the fully folded compact state.     

Emergence of multiple tori structures in a single polyelectrolyte chain

We investigated the collapsed structure of a weakly charged wormlike chain under a moderate concentration of 1:1 electrolyte solution. By assuming a torus as a grand state, we found that the size of a torus is determined by the balance between surface energy and electrostatic energy, which leads to a finite torus thickness almost independent of the chain contour length. Owing to this unique characteristic, a long charged wormlike chain forms multiple tori structure as a collapsed product, which is never seen with a neutral wormlike chain. These features were confirmed by a Monte Carlo simulation.     

Coil-globule transition: Self-assembly of a single polymer chain

The coil collapse problem is of interest not only because it represents the simplest model of protein folding, but also because of its fundamental importance as related to polymer nanostructures and fractionation. It is extremely difficult to observe the coil-to-globule transition experimentally because at finite concentrations in a poor solvent, the macromolecules tend to aggregate due to phase separation when the collapsed state is being achieved. In the mid-1980s, two-stage kinetics of a single-chain collapse was proposed theoretically.^1,2 The first successful experimental observation of a two-stage coil-to-globule transition was achieved by quenching a dilute solution of polystyrene (PS) in cyclohexane.³ By using a thinnest capillary tube cell with a wall thickness of 0.01 mm and a diameter of 5 mm for dynamic light scattering, two relaxation times, τ_crum for the crumpled globule state and τ_eq for the compact globule state, were determined⁴ for the first time. The relaxation times were much slower than expected. From the size of the crumpled globule and that of the compact globule and by assuming the intraglobular density to be uniform, the volume fraction of the PS chain in the crumpled globule state, ϕ_crum, and that in the compact globule state, ϕ_comp, can be estimated, with ϕ_crum = 0.02 and ϕ_comp ∼ 0.24-0.4 at 28°C for polystyrene in cyclohexane. The results imply that a single-chain globule contains a large amount of solvent. It should also be noted that ϕ_comp is temperature dependent, i.e., one would have to go to hypothetically low temperatures in order to squeeze out all the solvent (cyclohexane) in the compact PS globule. The single-chain coil collapse state could be achieved under equilibrium conditions by using a high molecular weight, M_w ∼ 1.08 × 10⁷ g/mol; M_w/M_n < 1.06) poly(N-isopropylacrylamide) (PNIPAM) in water,<⁵ even though the ten million molecular weight for PNIPAM was substantially lower than that for polystyrene (M_w ∼ 50 × 10⁶ g/mole).⁶ Under equilibrium conditions, it was feasible to determine both the hydrodynamic radius R_h and the radius of gyration R_g. The ratio of R_g/R_h changed from 1.45 to 0.77, clearly demonstrating the transition from the theta coil state to the compact globule state. At the maximum value of the scaled expansion factor α_s³ |τ| M_w^1/2, R_g/R_h = 1.33 where α_s = R_g/R_g (θ) and τ = |T-θ| / θ with θ being the theta temperature. In the compact globule, R_g/R_h was of the order of 0.7, implying that the PNIPAM compact globule in water still contained ∼80% water, of the same order of magnitude as the PS compact globule in cyclohexane at 7° below its theta temperature (35°C).     

On the formation of rings-on-a-string conformations in a single polyelectrolyte chain: A possible scenario

Recent single-molecular observations have revealed that a single giant DNA molecule assumes (micro) phase separated structures upon the addition of condensing agents. Electron and atomic force microscopy have clearly shown the coexistence of ordered tori and disordered coil structures within a single DNA molecule. Motivated by these experimental findings, we theoretically investigated the collapse transition of a single polyelectrolyte chain driven by the addition of condensing guest molecules. We found that the transition behavior critically depends on the degree of the surviving charge inside the torus. When the torus is charged, even slightly, "rings-on-a-string" structures are expected for a sufficiently long chain, owing to the combinational entropy of segment state distribution along the chain and the unique property of the stability of charged torus.     

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     

Mean field theory for the intermolecular and intramolecular conformational transitions of a single flexible polyelectrolyte chain

The diMarzio theory has been extended to elucidate the intermolecular and intramolecular phase segregations of a single flexible chain polyelectrolyte in dilute salt-free solutions. At the long chain limit, this theory yields the formalism obtained from the more sophisticated Edward Hamiltonian for polyelectrolyte problems. The calculated phase diagram exhibits the features of a first-order phase transition, with continuous and discontinuous transitions separated by a critical point. Under the discontinuous transition, the polyelectrolyte chain exhibits coexistent expanded and collapsed conformational states, same as intermolecular phase segregation. For a limiting long chain, the mean chain size at critical point is roughly 90% of the size of an ideal chain. Such a result implies that partial contraction within a chain molecule is required to collapse a flexible polyelectrolyte chain. Moreover, the theory predicts that for a longer chain, intramolecular segregated conformations differ significantly from intermolecular segregated conformations, but the difference becomes small for shorter chains. Besides, the charge needed to induce intramolecular segregation is smaller than that of intermolecular segregation for a given chain length. These findings are consistent with previous literature results.     

Coil-globule transition of a single short polymer chain: an exact enumeration study

The authors present an exact enumeration study of short self-avoiding walks in two as well as in three dimensions that addresses the question, "what is the shortest walk for which the existence of all the three scaling regimes--coil, globule, and the theta--could be demonstrated." Even though they could easily demonstrate the coil and the globule phase from free energy considerations, they could demonstrate the existence of a theta temperature only by using a scaling form for the distribution of gyration radius. That even such short walks have a scaling behavior is an unexpected result of this work.     

Mineralization of single flexible polyelectrolyte molecules

Conformation of a single flexible polyelectrolyte molecule with a hydrophobic backbone in aqueous solution is effected by the interplay of the short-range intramolecular attraction and the long-range Coulomb repulsion. The conformation can be frozen if the molecule is trapped by a solid substrate. With this approach, we prepared the range of single molecule templates from poly(2-vinylpyridine) (P2VP) deposited on the surface of Si-wafer or mica in different conformations from an elongated wormlike coil to compact globule. Pd(+2) was coordinated by P2VP via an ion exchange reaction exposing the samples to palladium acetate acidic aqueous solution. In the next step, Pd(+2) was reduced by dimethylamine borane. This route results in wire-shaped metallic nanoparticle assembles of about 2-5 nm in diameter and 50-700 nm in length. The conformation and size of the underlaying polyelectrolyte molecules determine the dimensions of nanoparticles.     

Single flexible hydrophobic polyelectrolyte molecules adsorbed on solid substrate: transition between a stretched chain, necklace-like conformation and a globule

Single flexible polyelectrolyte molecules of poly(2-vinylpyridine) undergo conformational transition from a stretched wormlike coil to a necklace-like globule, and to a compact globule depending on pH and ionic strength in aqueous solution in good agreement with recent theoretical reports. AFM investigations allow the visualization of details of the chain conformation on mica and the extraction of quantitative statistics of molecular dimensions.     

Encapsulation of a polyelectrolyte chain by an oppositely charged spherical surface

Using the ground state dominance approximation and a variational theory, we study the encapsulation of a polyelectrolyte chain by an oppositely charged spherical surface. The electrostatic attraction between the polyelectrolyte and the surface and the entropy loss of the encapsulated polyelectrolyte chain dictate the optimum conditions for encapsulation. Two scenarios of encapsulation are identified: entropy-dominated and adsorption-dominated encapsulation. In the entropy-dominated encapsulation regime, the polyelectrolyte chain is delocalized, and the optimum radius of the encapsulating sphere decreases with increasing the attraction. In the adsorption-dominated encapsulation regime, the polyelectrolyte chain is strongly localized near the surface, and the optimum radius increases with increasing the attraction. After identifying a universal encapsulation parameter, the dependencies of the optimum radius on the salt concentration, surface charge density, polymer charge density, and polymer length are explored.     

Calcium induced volume transition in polyelectrolyte gels

The osmotic properties and the small angle neutron scattering (SANS) behaviour of fully neutralized sodium polyacrylate gels are investigated in the presence of calcium ions. Analysis of the SANS response displays three characteristic length scales, two of which are of thermodynamic origin, while the third, associated with the frozen-in structural inhomogeneities, is static. The SANS results are consistent with direct osmotic observations which indicate that the thermodynamic properties cannot be adequately described by a single correlation length. The concentration dependence of the osmotic pressure displays a power law behaviour with an exponent that decreases with increasing calcium concentration.     

The effect of polyelectrolyte chain length on layer-by-layer protein/polyelectrolyte assembly--an experimental study

The effect of polyelectrolyte chain length on the formation of multilayered assemblies of alternating globular proteins and linear polyanions prepared by the layer-by-layer electrostatic adsorption technique was investigated. The systems studied were albumin/sodium poly(styrenesulfonate), immunoglobulin G/sodium poly(styrenesulfonate), albumin/sodium dextran sulfate, and albumin/heparin. The formation of assemblies was followed using FTIR multiple internal reflection spectroscopy. While the amount of polyelectrolyte adsorbed on the first (primary) protein layer did not depend on its molecular weight, the effect of polyelectrolyte chain length was clearly observed in the following steps of alternating adsorption. Some short-chain polyanion molecules were removed from the surface when a next protein layer was adsorbed from solution. The short polyanion chains were not able to make a sufficient number of ion pairs for stable interaction with additional protein molecules and left the surface as soluble protein/polyanion complexes. The most pronounced effect could be seen with sodium poly(styrenesulfonate) of Mw up to ca. 2 x 10(4), but a detectable effect could be traced even up to Mw ca. 8 x 10(4). Such a pronounced effect, however, was not observed with dextran sulfate. The effect of molecular weight of heparin was clearly observed but all heparins tested, regardless of their molecular weight, effectively assembled with albumin to form multilayer.     

Dynamics of a single copolymer chain

The dynamics of a single copolymer chain in a solution containing identical chains is studied. The intermediate scattering function is calculated, and two different modes are identified and analyzed as a function of wave vector q. The first mode is characteristic of the copolymer system and leads to a peculiar constant relaxation frequency as q goes to zero. The second mode represents the diffusion of the chain in the solution.     

Phase transitions of a single polyelectrolyte in a poor solvent with explicit counterions

Conformational properties of a single flexible polyelectrolyte chain in a poor solvent are studied using constant temperature molecular dynamics simulation. The effects of counterions are explicitly taken in to account. Structural properties of various phases and the transition between these phases are studied by tracking the values of asphericity, radius of gyration, fraction of condensed counterions, number of non-bonded neighbours, and Coulomb interaction energies. From our simulations, we find strong evidence for a first-order phase transition from extended to collapsed phase consistent with earlier theoretical predictions. We also identify a continuous phase transition associated with the condensation of counterions and estimate the critical exponents associated with the transition. Finally, we argue that previous suggestions of existence of an independent intermediate phase between extended and collapsed phases is only a finite size effect.     

A study on the structural transition of a single polymer chain by parallel tempering molecular dynamics simulation

The structural transition of a single polymer chain with chain length of 100,200 and 300 beads was investigated by parallel tempering MD simulation.Our simulation results can capture the structural change from random coil to orientationally ordered structure with decreasing temperature.The clear transition was observed on the curves of radius of gyration and global orientational order parameter P as the function of temperature,which demonstrated structural formation of a single polymer chain.The linear relationships between three components of square radius of gyration R_gx²,R_gx²,R_gz² and global orientational order P can be obtained under the structurally transformational process.The slope of the linear relationship between x（or y-axis） component R_gx²(or R_gy²) and P is negative,while that of RL as the function of P is positive.The absolute value of slope is proportional to the chain length.Once the single polymer chain takes the random coil or ordered configuration,the linear relationship is invalid.The conformational change was also analyzed on microscopic scale.The polymer chain can be treated as the construction of rigid stems connecting by flexible loops.The deviation from exponentially decreased behavior of stem length distribution becomes prominent,indicating a stiffening of the chain arises leading to more and more segments ending up in the trans state with decreasing temperature.The stem length N_tr is about 21 bonds indicating the polymer chain is ordered with the specific fold length.So,the simulation results,which show the prototype of a liquid-crystalline polymer chain,are helpful to understand the crystallization process of crystalline polymers.     

Structure of transition state in catalysis of chain transfer to a monomer

Translated from Teoreticheskaya Éksperimental'naya Khimiya, Vol. 26, No. 1, p. 128, January–February, 1990.     

Roles of chain conformation and interpenetration in the growth of a polyelectrolyte multilayer

In the present study, we have investigated the growth of a multilayer formed by poly(sodium 4-styrene sulfonate) (PSSS) and poly(diallyldimethylammonium chloride) (PDDA) at different salt concentrations by use of quartz crystal microbalance with dissipation (QCM-D). The frequency change (Deltaf) demonstrates that the exponential growth mode gradually becomes dominant as NaCl concentration (C(NaCl)) increases. On the other hand, the dissipation change (DeltaD) reveals that the deposition is dominated by chain conformation at C(NaCl) < 1.0 M, where the change of the characteristic growth parameter agrees well with the results fit with Debye length. At C(NaCl) > or = 1.0 M, the growth is not determined by chain conformation but by chain interpenetration.     

Stepwise growth of a single polymer chain

By monitoring the modulation of an ionic current passing through a nanoreactor formed from a protein pore, the step-by-step growth of an individual polymer chain was monitored. The observation of polymer growth at the single-molecule level will be useful for studying the kinetics of chain growth or the movement of polymers under confinement. It might also be used to synthesize "molecular fishing lines" in situ, for applications in stochastic sensing.