Flory Radius Research Articles

Long-term survivable liposomes in seawater.

Molecular robotics is a research topic to develop a bio-molecular device that can equip sensors, circuits, and actuators. One of the objectives for this is to work them in various biological environments. Liposome-type molecular robot (liposome robot) can encapsulate chemicals and has high biocompatibility, so it is expected to use them in those environments. However, the robot is hardly implemented due to the lack of stability. In this study, we proposed that stable liposomes work in seawater as an example of implementation. To improve the stability, we considered the use of polyethylene glycol-modified lipids (PEG-modified lipids). PEGs immobilized on the substrate are known to form two conformations, a mushroom and a brush, depending on the density. At low density, PEGs form mushroom structures because there are plenty of spaces on the lipid membrane. When the distance between PEG chains is beyond the Flory radius of the grafted PEG, adjacent PEGs try to leave a space between each other. This results in the conformational change of PEGs to brush. We hypothesized that the kind and conformational differences affect liposome stability. We here focused on two PEG-modified lipids; lipid-PEG2000 (PEG2000) and lipid-PEG5000 (PEG5000). We prepared four types of liposomes in seawater; (1) without, with (2) PEG2000, (3) PEG5000, and (4) the 1:1 mixture of PEG2000 and PEG5000. For each PEG condition, we prepared sub-types of the liposomes which were expected to form brush and mushroom conformations. We microscopically observed them and evaluated the survival rate for two weeks after preparation. For each brush PEG condition, the average survival rate was more than 10% for 14 days. Opposite this, the rate was less than 10% in the other conditions. Based on them, we attempted to define the optimized condition for stable liposomes in seawater.

Protein Adsorption on Grafted Zwitterionic Polymers Depends on Chain Density and Molecular Weight

AbstractThis study demonstrates that protein adsorption on end‐grafted, zwitterionic poly(sulfobetaine) (pSBMA) thin films depends on the grafting density, molecular weight, and ionic strength. Zwitterionic polymers exhibit ultralow nonspecific fouling (protein adsorption) and excellent biocompatibility. This picture contrasts with a recent report that soluble pSBMA chains bind proteins and alter the protein folding stability. To address this apparent contradiction, the dependence of protein adsorption on the chain grafting parameters is investigated: namely, the grafting density, molecular weight, and ionic strength. Studies compared the adsorption of phosphoglycerate kinase and positively charged lysozyme versus the scaled grafting parameter s/2RF, where s is the distance between grafting sites and RF is the Flory radius. Plots of the adsorbed protein amount versus s/2RF exhibit a bell‐shaped curve, with a maximum near s/2RF ≈ 1 and an amplitude that decreases with ionic strength. This behavior is qualitatively consistent with theoretical models for colloid interactions with weakly attractive, grafted chains. The results confirm that proteins do adsorb to pSBMA thin films, and they suggest an underlying mechanism. Comparisons with polymer models further identify design rules for pSBMA films that effectively repel protein.

Thermoresponsive poly(glycidyl ether) brushes on gold: Surface engineering parameters and their implication for cell sheet fabrication.

Cell sheet engineering as a scaffold-free approach towards tissue engineering resembles a milestone in regenerative medicine. The fabrication of confluent cell sheets maintains the extracellular matrix of cells which serves as the physiological cell scaffold. Thermoresponsive poly(glycidyl ether)s are highly cell-compatible and brushes thereof promote cell adhesion and growth without modification with additional cell adhesive ligands. Thus, a direct correlation of temperature-dependent serum protein adsorption and cell response with surface design parameters such as grafting density and molecular weight became accessible. Hence, surface engineering parameters of well-defined poly(glycidyl ether) monolayers for reproducible cell sheet fabrication have been identified. These design guidelines may also prove beneficial in the development of other brush-like thermoresponsive coatings for cell sheet engineering.

Adsorption of Plasma Proteins onto PEGylated Lipid Bilayers: The Effect of PEG Size and Grafting Density.

Lipid bilayers grafted with polyethylene glycol (PEG) of different sizes (Mw = 750, 2000, and 5000) and grafting densities (1.6-25 mol % of PEGylated lipid in dipalmitoylphosphatidylcholine (DPPC) lipid molecules) were simulated with human serum albumin (HSA) using coarse-grained force fields. At low enough grafting density, the PEG has a conformation similar to that of an isolated chain in water, and its Flory radius RF is smaller than the distance between the grafting points (d), which is the so-called "mushroom" regime. In contrast, densely grafted PEG chains (RF > d) extend like brushes, with brush layer thickness given by the Alexander-de Gennes theory. A nearly spherical HSA added to this simulation migrates to the bilayer surface because of the charge interactions between anion residues of HSA and cationic cholines of DPPC, but this HSA-bilayer binding can be sterically suppressed by the PEG chains to an extent that depends on the PEG size and grafting density. In particular, regardless of the extent of the coverage of the PEG on the bilayer, the binding between HSAs and bilayers is suppressed by the PEG layer in a brush but not in a mushroom, indicating that the attractive force between proteins and bilayers can overcome the steric effect of the PEG layer in the mushroom state or in the transition region from mushroom to brush. This helps explain and clarify experiments that show much less adsorption of plasma proteins onto the particle or membrane surface when PEGs are in the brush rather than in the mushroom state.

Chemically Realistic Tetrahedral Lattice Models for Polymer Chains: Application to Polyethylene Oxide.

To speed up the generation of an ensemble of poly(ethylene oxide) (PEO) polymer chains in solution, a tetrahedral lattice model possessing the appropriate bond angles is used. The distance between noncovalently bonded atoms is maintained at realistic values by generating chains with an enhanced degree of self-avoidance by a very efficient Monte Carlo (MC) algorithm. Potential energy parameters characterizing this lattice model are adjusted so as to mimic realistic PEO polymer chains in water simulated by molecular dynamics (MD), which serves as a benchmark. The MD data show that PEO chains have a fractal dimension of about two, in contrast to self-avoiding walk lattice models, which exhibit the fractal dimension of 1.7. The potential energy accounts for a mild hydrophobic effect (HYEF) of PEO and for a proper setting of the distribution between trans and gauche conformers. The potential energy parameters are determined by matching the Flory radius, the radius of gyration, and the fraction of trans torsion angles in the chain. A gratifying result is the excellent agreement of the pair distribution function and the angular correlation for the lattice model with the benchmark distribution. The lattice model allows for the precise computation of the torsional entropy of the chain. The generation of polymer conformations of the adjusted lattice model is at least 2 orders of magnitude more efficient than MD simulations of the PEO chain in explicit water. This method of generating chain conformations on a tetrahedral lattice can also be applied to other types of polymers with appropriate adjustment of the potential energy function. The efficient MC algorithm for generating chain conformations on a tetrahedral lattice is available for download at https://github.com/Roulattice/Roulattice .

Models for Self‐Avoiding Polymer Chains on the Tetrahedral Lattice

A hierarchy of models for self‐avoiding polymer chains on the tetrahedral lattice is introduced. The chain comprises a concatenation of identical atoms. The models (SAWn), are characterized by the degree of self‐avoidance (specified by the integer n), which is controlled by systematic variation of the closest distance allowed between atom pairs that are not covalently bonded. SAW1, possessing the lowest degree of self‐avoidance, is the simple self‐avoidance model (i.e., no two atoms of the chain occupy the same site) that has been routinely employed in studies of fundamental phenomena. The results of Monte Carlo calculations are presented that show the influence of n on such properties of the chain as Flory radius, distribution of dihedral angles, and entropy loss due to self avoidance. Algorithms are developed that allow the efficient generation of large ensembles of chain conformations, which are necessary especially for a reliable calculation of the entropy loss induced by self‐avoidance.

Effect of void structure on the toughness of double network hydrogels

AbstractIntroduction of soft filler in a hard body, which is one of the common toughening methods of hard polymeric materials, was applied for further toughening of robust double network (DN) hydrogels composed of poly(2‐acrylamido‐2‐methylpropanesulfonic acid) gels (PAMPS gels) as the first component and polyacrylamide (PAAm) as the second component. The fracture energy of the DN gels with the void structure (called void‐DN gels) became twice when the volume fraction of void was 1–3 vol % and the void diameter was much larger than the Flory radius of the PAAm chains. Such toughening was induced by wider range of internal fracture of the PAMPS network derived from partial stress concentration near void structure. Considering the mechanical tests and the dynamic light scattering results, it is implied that the absence of the load‐bearing PAAm structure inside the void is important for the toughening. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 49: 1246–1254, 2011

Polymer Dynamics in Nanochannels of Porous Silicon: A Neutron Spin Echo Study

Neutron spin echo spectra of poly(ethylene oxide) (PEO) melts confined in porous silicon (the mean pore diameter: 13 nm) were recorded at Q = 0.05, 0.08, and 0.11 Å−1 and successfully analyzed in terms of a two-state model, where chains adsorbed to the pore walls exhibit much slower internal dynamics than in the bulk and their centers-of-mass do not move, while free chains have bulklike internal dynamics and diffuse within an infinite cylinder in the center of the pore. The radius of this cylinder was found to be 1.4 nm for PEO 3 kg/mol (3k); this corresponds to the thickness of the adsorbed layer to be approximately equal to the Flory radius. As opposed to PEO 10k, for which details on the center of mass diffusion could not be discerned, for PEO 3k the diffusion rate along the pore was found to be smaller than that in the radial direction.

Lipopolymer gradient diffusion in supported bilayer membranes

We measure the gradient diffusion coefficient of a model lipopolymer in supported lipid bilayer membranes from Fourier-transform post-electrophoresis relaxation. The experiments and accompanying quantitative interpretation furnish the concentration dependence of the gradient diffusion coefficient. In striking contrast to the recent measurements of the self-diffusion coefficient from fluorescence recovery after photobleaching, the lipopolymer gradient diffusion coefficient increases with concentration. We interpret the enhancement at small but finite concentrations using the Scalettar-Abney-Owicki (SAO) statistical mechanical theory (1988) and the Bussell-Koch-Hammer (BKH) hydrodynamic theory (1995), which are customarily adopted to model membrane protein dynamics. The SAO theory furnishes an effective disc radius and soft repulsive interaction radius that are comparable to the Flory radius of the unperturbed polyethylene glycol chains. On the other hand, the BKH theory predicts a gradient diffusion coefficient that decreases with disc/membrane protein concentration. Thus, in contrast to membrane proteins, we conclude that lipopolymer hydrodynamic interactions are weak because the principal disturbances are in the low-viscosity aqueous phase. Accordingly, lipopolymer interactions are dominated by thermodynamic interactions among polymer chains. Interestingly, our experiments suggest that increasing (decreasing) the polymer molecular weight should increase (decrease) the relaxation rate of lipopolymer concentration fluctuations.

Concentration dependence of lipopolymer self-diffusion in supported bilayer membranes.

Self-diffusion coefficients of poly(ethylene glycol)2k-derivatized lipids (DSPE-PEG2k-CF) in glass-supported DOPC phospholipid bilayers are ascertained from quantitative fluorescence recovery after photobleaching (FRAP). We developed a first-order reaction-diffusion model to ascertain the bleaching constant, mobile fraction and lipopolymer self-diffusion coefficient D(s) at concentrations in the range c ≈ 0.5-5 mol%. In contrast to control experiments with 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(7-nitro-2-1,3-benzoxadiazol-4-yl) (ammonium salt) (DOPE-NBD) in 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), the lipopolymer self-diffusion coefficient decreases monotonically with increasing concentration, without a distinguishing mushroom-to-brush transition. Our data yield a correlation D(s) = D(0)/(1 + αc), where D(0) ≈ 3.36 µm(2) s(-1) and α ≈ 0.56 (with c expressed as a mole percent). Interpreting the dilute limit with the Scalettar-Abney-Owicki statistical mechanical theory for transmembrane proteins yields an effective disc radius a(e) ≈ 2.41 nm. On the other hand, the Bussell-Koch-Hammer theory, which includes hydrodynamic interactions, yields a(e) ≈ 2.92 nm. As expected, both measures are smaller than the Flory radius of the 2 kDa poly(ethylene glycol) (PEG) chains, R(F) ≈ 3.83 nm, and significantly larger than the nominal radius of the phospholipid heads, a(l) ≈ 0.46 nm. The diffusion coefficient at infinite dilution D(0) was interpreted using the Evans-Sackmann theory, furnishing an inter-leaflet frictional drag coefficient b(s) ≈ 1.33 × 10(8) N s m(-3). Our results suggest that lipopolymer interactions are dominated by the excluded volume of the PEG-chain segments, with frictional drag dominated by the two-dimensional bilayer hydrodynamics.

Crossover behavior in static and dynamic properties of a single DNA molecule from three to quasi-two dimensions

We studied the conformation and dynamics of a single DNA molecule in a thin slit by a fluorescent microscope. In a slit thinner than the Flory radius in three dimensions, the length of the major axis, the translational self-diffusion coefficient and the rotational relaxation time in a dilute solution show the apparent dependence on the thickness of the slit. The observed dependence is in agreement with that predicted by blob theory, despite the number of blobs is very small. The radial distribution of the segments around the center of mass of a single molecule was also studied and compared with that calculated for a Gaussian and an excluded volume chain. The influence of the polymer concentration on the geometrical confinement by slits was also studied in a semidilute solution near the overlap concentration c∗. The confinement effect is found to be not so serious near c∗ and is only significant in the so-called "two-dimensional pancake" region.

Conformational Studies of Covalently Grafted Poly(ethylene glycol) on Modified Solid Matrices Using X-ray Photoelectron Spectroscopy

Amine functionalized poly(ethylene glycols) (PEGs) with molecular weights 2000 and 4000 Da were covalently grafted onto carboxy modified hydrophilic Sephadex derivatives and hydrophobic polystyrene derivatives using anhydrous amine conjugation methods. Varying PEG surface concentration and layer thickness were achieved by controlling the reaction parameters and were analyzed by X-ray photoelectron spectroscopy (XPS). C-O intensities obtained from high resolution C 1s scans were correlated using the standard overlay model to study the grafting kinetics as well as conformational properties of grafted polymer chains. A detailed and systematic comparison of PEG layer thickness and distance between grafted chains with the Flory radius of surface grafted PEG resulted in valuable information regarding conformational behavior of the polymer. The influence of the nature of the solid matrix on grafting kinetics and conformational properties of the grafted polymer chain was also established from the XPS results.

Flory radius of polymers in a periodic field: An exact analytic theory

We found an exact expression for the Flory radius R (F) of Gaussian polymers placed in an external periodic field. This solution is expressed in terms of the two parameters eta and a that describe the reduced strength of an external field and the period of the field to the polymer gyration radius ratio, respectively. R (F) is found to be a decaying function of eta for any values of a . Provided that the gyration radius is of the order of the period of an external field or less, the ground-state (GS) approximation of the exact result for R (F) is shown to give qualitatively incorrect results. In addition to the "ground-state" contribution, the exact solution for R (F) contains an additional term that is overlooked by the GS approximation. This term gives rise to the fact that R (F) as a function of eta exhibits power law behavior (rather than exponential decay obtained from the GS result) once eta exceeds the threshold value eta(con) .

Polymers in an external periodic field: The effect of the excluded volume interactions

By developing and making use of the "transfer operator" formalism, we calculate the number density and average Flory end-to-end distance of the polymers placed in an external periodic field. The considered mathematical problem is of immediate relevance for such realistic physical systems as the homopolymers immersed in the host structure of alternating layers that have different affinities for homopolymers (e.g., lamellar microphases of copolymers, ripple morphology of the mixed brush, and lipidwater systems). In contrast to the conventional ground state dominance approximation, the developed method makes it possible to calculate the characteristic size (Flory radius R(F)) of the polymers in the direction of applied external periodic field, with the effect of the excluded volume taken into account. The excluded volume interactions are shown to qualitatively change the behavior of R(F) as a function of the reduced field strength theta relative to the case of ideal Gaussian polymers. In particular, in the limit of strong fields theta>>1 the average Flory radius R(F) is found to saturate to its minimal value, which is calculated as a function of the excluded volume parameter u. This finding is in distinct contrast to the result for the Flory radius R(F) in the case of ideal polymers where R(F) approaches zero as the interaction parameter theta increases.

Confinement effect of chain dynamics in micrometer thick layers of a polymer melt below the critical molecular weight

Polymer melts confined in micrometer thick layers were examined with the aid of field-cycling NMR relaxometry. It is shown that chain dynamics under such moderate confinement conditions are perceptibly different from those observed in the bulk material. This is considered to be a consequence of the corset effect, which predicts a crossover between Rouse and reptationlike dynamics for molecular weights below the critical value at confinement length scales much larger than 10RF, where RF is the Flory radius of the bulk polymer coil [Fatkullin et al., New J. Phys. 6, 46 (2004)]. For the polymer species studied, a perfluoropolyether with a molecular weight of 11 000, the Flory radius is of the order 10 nm, so that the experiment refers to the far end of the predicted crossover region from confined to bulk chain dynamics. Remarkably the confinement effect is shown to reach polymer-wall distances of the order 100 Flory radii.

Conformational properties and dynamics of protein-like chains confined in an infinite cylinder

In this paper the conformational properties and dynamics of protein-like lattice chains confined in an infinite cylinder are investigated by using Monte Carlo simulation method and the hybrid (3 1 0) lattice model. When the diameter of infinite cylinder D is greater than 0.5–0.7 R F, the shape of protein-like chains remains unchanged, here Flory radius R F is R F = 3.0 N 0.6. When D < 0.5 R F, the confinement of the cylinder help to form the helices, however, it hinders to form the coils, and the shape of protein-like chains changes from sphere-like structure to rod-like one. Different confinements affect distinctly to form the secondary structure of proteins. There exist a different dynamics behavior between protein-like chains confined in an infinite cylinder and general protein-like chains without confinement. These investigations may provide some insights into the influence of crowding condition to the conformation properties and dynamics of proteins.

The shape of a flexible polymer in a cylindrical pore

We calculate the mean end-to-end distance R of a self-avoiding polymer encapsulated in an infinitely long cylinder with radius D. A self-consistent perturbation theory is used to calculate R as a function of D for impenetrable hard walls and soft walls. In both cases, R obeys the predicted scaling behavior in the limit of large and small D. The crossover from the three-dimensional behavior (D --> infinity) to the fully stretched one-dimensional case (D --> 0) is nonmonotonic. The minimum value of R is found at D approximately 0.46R(F), where R(F) is the Flory radius of R at D --> infinity. The results for soft walls map onto the hard wall case with a larger cylinder radius.

Interaction Forces and Morphology of a Protein-Resistant Poly(ethylene glycol) Layer

The molecular interactions on a protein-resistant surface coated with low-molecular-weight poly(ethylene glycol) (PEG) copolymer brushes are investigated using the extended surface forces apparatus. The observed interaction force is predominantly repulsive and nearly elastic. The chains are extended with respect to the Flory radius, which is in agreement with qualitative predictions of scaling theory. Comparison with theory allows the determination of relevant quantities such as brush length and adsorbed mass. Based on these results, we propose a molecular model for the adsorbed copolymer morphology. Surface-force isotherms measured at high resolution allow distinctive structural forces to be detected, suggesting the existence of a weak equilibrium network between poly(ethylene glycol) and water—a finding in accordance with the remarkable solution properties of PEG. The occurrence of a fine structure is interpreted as a water-induced restriction of the polymer's conformational space. This restriction is highly relevant for the phenomenon of PEG protein resistance. Protein adsorption requires conformational transitions, both in the protein as well as in the PEG layer, which are energetically and kinetically unfavorable.

Effective force between colloidal particles with end-grafted long polymer chains dissolved in a high-molecular weight solvent

We aim at the determination of the effective force between colloidal particles clothed each by f end-grafted long polymer chains of polymerization degree N, which are immersed in a melt of short chains of polymerization degree P< N. We assume that short and probe chains have the same chemical nature. The clothed particles are small enough to be considered as star-polymers. To compute the expected force, we consider two regimes in the ( P, f)-plane: (i) high-grafting density f>f*= P ; and (ii) small-grafting density ( f< f*). We compute, in any case, the repulsive effective force, F( h), as a function of the inter-particle distance h. This force originates from the excluded volume interactions, where long probe chains are swollen by the presence of shorter ones of P units. For regime (i), we find that the force decays as k B TA fh −1· aP f <h<R F , with the universal amplitude A f ∼ f 3/2, as for low-molecular weight solvents case. Here, R F∼ aP −1/5 f 1/5 N 3/5 is the common Flory radius of the two star-polymers. For regime (ii), we show that, for higher distances aP 1/4< h< R F, the force decreases according to the same power law as for regime (i). However, below the cross-over distance aP 1/4·( aP 1/6 f 1/6< h< aP 1/4), the excluded volume interactions are strongly screened by the presence of mobile P-chains, and the force decays rather as k B TB f h −2 (mean-field behavior), with the new universal amplitude B f ∼ f 4/3. For f= f*, the mean-field distance-regime disappears, and then, the two forces relative to regimes (i) and (ii) are the same.

The confined-to-bulk dynamics transition of polymer melts in nanoscopic pores of solid matrices with varying pore diameter

The confinement of polymer melts in nanoscopic pores leads to chain dynamics significantly different from bulk behaviour. This so-called ‘corset effect’ occurs both above and below the critical molecular mass and induces dynamic features as predicted for reptation. The confined-to-bulk dynamics crossover is treated analytically on the basis of general thermodynamic relations connected to the fluctuation of the number of particles (Kuhn segments) in a given volume. Bulk behaviour is shown to occur only if the pore diameter complies with the limit dpore≫(b3/kBTκT)1/3RF≈10RF, where b is the Kuhn segment length, κT the isothermal compressibility, T the temperature, kB the Boltzmann constant and RF the Flory radius. For smaller pores, the confined polymer chains reptate along their own contours in tubes with an effective diameter , where ρs is the number density of Kuhn segments. From the theoretical point of view, the crucial factors on which the corset effect is based are (i) impenetrable pore walls, (ii) low compressibility and (iii) the uncrossability of polymer chains.