Helical Wormlike Chain Research Articles

Solution SAXS measurements over a wide temperature range to determine the unperturbed chain dimensions of polystyrene and a cyclic amylose derivative

The mean-square radii of gyration <S2> of two polystyrene (PS) samples with weight-average molar masses Mw of 2.18 × 104 and 3.88 × 104 in toluene and 2-butanone and of a cyclic amylose tris(phenylcarbamate) (cATPC) with a Mw of 4.73 × 104 in tetrahydrofuran were determined by synchrotron radiation small-angle X-ray scattering measurements over a wide range of temperatures from −77 °C to 70 °C. Both PS and cATPC are sufficiently soluble to enable SAXS measurements even at −77 °C in the solvents used. The <S2> of cATPC does not depend on temperature over the range investigated here. This result may be reasonable for such rigid ring polymers. In contrast, the radii of PS depend on temperature to a significant degree, whereas the second virial coefficient is mostly temperature independent. The resulting characteristic ratio C∞ for PS in toluene decreases monotonically with increasing temperature, as predicted both by the rotational isomeric state (RIS) and by (helical) wormlike chain models. However, C∞ in 2-butanone exhibits a minimum ∼10 °C and increases with increasing temperature, suggesting that the RIS energy parameters should be affected by the intermolecular interactions between the polymer and solvent. Mean-square radii of gyration <S2> for two polystyrene samples in toluene and 2-butanone and a cyclic amylose tris(phenylcarbamate) (cATPC) sample in tetrahydrofuran were determined by SAXS between −77 and 70 °C. Whereas <S2> of cATPC does not depend on temperature over the range investigated, those for polystyrene depend to a significant degree. The characteristic ratio for polystyrene in toluene monotonically decreases with increasing temperature. However, that in 2-butanone has a minimum ∼10 °C, suggesting that the RIS energy parameters are affected by the intermolecular interactions between the polymer and solvent.

Chain Conformation of Poly[2-(methacryloyloxy)ethyltrimethylammonium chloride] in Aqueous Sodium Chloride Solutions

A series of poly[2-(methacryloyloxy)ethyltrimethylammonium chloride] (PMTAC) in aqueous NaCl solutions with a weight-average molecular weight (Mw) of between 6.0 × 103 and 1.5 × 105 were characterized by synchrotron radiation small-angle X-ray scattering (SAXS) and dynamic light scattering (DLS) measurements at 298 K. The hydrodynamic radius (RH) of PMTAC initially decreased with increasing salt concentration (Cs) and then leveled off at Cs = 0.5–1.0 M. The values of z-average root-mean-square radius of gyration (⟨S2⟩z1/2) simply increased with a decrease in Cs for Mw > 2 × 104 g/mol. In contrast, the values of ⟨S2⟩z1/2 for Mw < 2 × 104 g/mol increased with an increase in Cs probably because of a change in local conformation of PMTAC chain. To study local chain conformation of PMTAC, scattering functions and Mw dependences of ⟨S2⟩z1/2 and RH for PMTAC were analyzed in terms of the helical wormlike chain (HW) model which is a simple continuous elastic wire model with bending and torsional energies. The cha...

Statistical mechanics of helical wormlike chain model

We investigate the statistical mechanics of polymers with bending and torsional elasticity described by the helical wormlike model. Noticing that the energy function is factorizable, we provide a numerical method to solve the model using a transfer matrix formulation. The tangent-tangent and binormal-binormal correlation functions have been calculated and displayed rich profiles which are sensitive to the combination of the temperature and the equilibrium torsion. Their behaviors indicate that there is no finite temperature Lifshitz point between the disordered and helical phases. The asymptotic behavior at low temperature has been investigated theoretically and the predictions fit the numerical results very well. Our analysis could be used to understand the statics of dsDNA and other chiral polymers.

Effects of the Nucleoid Protein HU on the Structure, Flexibility, and Ring-Closure Properties of DNA Deduced from Monte Carlo Simulations

The histone-like HU (heat unstable) protein plays a key role in the organization and regulation of the Escherichia coli genome. The nonspecific nature of HU binding to DNA complicates analysis of the mechanism by which the protein contributes to the looping of DNA. Conventional models of the looping of HU-bound duplexes attribute the changes in biophysical properties of DNA brought about by the random binding of protein to changes in the effective parameters of an ideal helical wormlike chain. Here, we introduce a novel Monte Carlo approach to study the effects of nonspecific HU binding on the configurational properties of DNA directly. We randomly decorated segments of an ideal double-helical DNA with HU molecules that induce the bends and other structural distortions of the double helix find in currently available X-ray structures. We find that the presence of HU at levels approximating those found in the cell reduces the persistence length by roughly threefold compared with that of naked DNA. The binding of protein has particularly striking effects on the cyclization properties of short duplexes, altering the dependence of ring closure on chain length in a way that cannot be mimicked by a simple wormlike model and accumulating at higher-than-expected levels on successfully closed chains. Moreover, the uptake of protein on small minicircles depends on chain length, taking advantage of the HU-induced deformations of DNA structure to facilitate ligation. Circular duplexes with bound HU show much greater propensity than protein-free DNA to exist as negatively supercoiled topoisomers, suggesting a potential role of HU in organizing the bacterial nucleoid. The local bending and undertwisting of DNA by HU, in combination with the number of bound proteins, provide a structural rationale for the condensation of DNA and the observed expression levels of reporter genes in vivo.

Optical Measurement of Mechanical Forces Inside Short DNA Loops

Knowledge of the mechanical properties of double-stranded DNA (dsDNA) is essential to understand the role of dsDNA looping in gene regulation and the mechanochemistry of molecular machines that operate on dsDNA. Here, we use a newly developed tool, force sensors with optical readout, to measure the forces inside short, strained loops composed of both dsDNA and single-stranded DNA. By varying the length of the loops and their proportion of dsDNA, it was possible to vary their internal forces from 1 pN to >20 pN. Surprisingly, internal loop forces changed erratically as the amount of dsDNA was increased for a given loop length, with the effect most notable in the smallest loop (57 nucleotides). Monte Carlo simulations based on the helical wormlike chain model accurately predict internal forces when more than half of the loop is dsDNA but fail otherwise. Mismatches engineered into the double-stranded regions increased flexibility, suggesting that Watson-Crick basepaired dsDNA can withstand high compressive forces without recourse to multibase melts. Fluorescence correlation spectroscopy further excluded transient melting (microsecond to millisecond duration) as a mechanism for relief of compressive forces in the tested dsDNAs. DNA loops with integrated force sensors may allow the comprehensive mapping of the elasticity of short dsDNAs as a function of both sequence and salt.

Temperature Coefficients of Unperturbed Chain Dimensions for Polystyrene and Poly(α-methylstyrene)

The temperature coefficient d ln 0/dT of the unperturbed mean-square end-to-end distance 0 with T the absolute temperature was determined from light scattering measurements for atactic polystyrene (a-PS) with the fraction of racemic diads fr=0.59 and atactic poly(α-methylstyrene) (a-PαMS) with fr=0.72, both in toluene (good solvent) and in the range of temperature 15.0–55.0 °C, on the assumption that the binary-cluster integral and hence the gyration-radius expansion factor do not depend on T. It is then found that d ln 0/dT is definitely negative for a-PS with a weak helical nature but almost vanishes for a-PαMS with a strong helical nature. The results are consistent with the helical wormlike chain theory prediction.

First Cumulant of the Dynamic Structure Factor for Polymers in Θ Solvents. Effects of Chain Stiffness and Local Chain Conformation

The first cumulant Ω(k) of the dynamic structure factor as a function of the magnitude k of the scattering vector was determined from dynamic light scattering (DLS) measurements for two atactic poly(α-methylstyrene) (a-PαMS) samples with weight-average molecular weights Mw=3.22×106 and 5.46×106 in cyclohexane at 30.5 °C (Θ) and for one polyisobutylene (PIB) sample with Mw=6.63×106 in isoamyl isovalerate at 25.0 °C (Θ). The translational diffusion coefficient was also determined from DLS measurements for the a-PαMS sample with the larger Mw and for the PIB sample in the same solvent conditions. The mean-square radius of gyration was determined from static light scattering measurements for the PIB sample. From a comparison of the present data for a-PαMS and PIB along with the previous ones for atactic polystyrene and poly(n-hexyl isocyanate) with the helical wormlike chain theory, it may be concluded that the universality predicted by the Gaussian chain theory for the plot of η0Ω(k)/kBTk3 against 1/2k cannot be realized for flexible polymers even with very high Mw (\\simeq106–107), and that the height of the plateau in the k3-region depends on chain stiffness and local chain conformation, where η0 is the solvent viscosity, kB the Boltzmann constant, and T the absolute temperature.

Nuclear Magnetic Relaxation of Poly(n-hexyl isocyanate) in Dilute Solution. Effects of Chain Stiffness

The spin–lattice relaxation time T1, spin–spin relaxation time T2, and nuclear Overhauser enhancement NOE were determined from nuclear magnetic relaxation measurements on five well-fractionated samples of poly(n-hexyl isocyanate) in the range of weight-average molecular weight Mw from 1.65×104 to 5.71×104 in n-hexane at 25 °C. It is found that T1 increases and T2 decreases monotonically with increasing Mw in the range of Mw examined, while NOE is almost independent of Mw. Such behavior may be regarded as arising from effects of chain stiffness. A comparison is then made of the present data for T1, T2, and NOE with the theory developed for the generalized Kratky–Porod (KP) wormlike chain as a special case of the helical wormlike chain, and it is shown that the theory may almost quantitatively explain the data, using the established values of the KP model parameters. A comparison is also made of the present data with the theoretical values calculated from the equations of Woessner for the equivalent, rigid prolate spheroid.

Application of a New Cell Model for the Equation of State to 20 Polymer Melts and a Blend Melt

In a previous publication, we reported a new cell model for the equation of state of polymer melts. The equation contains external degrees of freedom explicitly as a function of molecular weight of repeating unit. In this paper, we apply the theory to 20 polymer melts and a blend melt. The linear lengths of the coarse-grained particle (CGP) of several polymers coincide with those obtained by scattering experiments. In addition, we estimated the characteristic quantity T* of the equation of state per repeating unit using van der Waals constants and the critical temperature of the repeating unit or monomer. The values, T*, agreed with those from our result per repeating unit. This result suggests the quantitative validity of our theory. We strongly suggest that our theory must be related to the conformer model by Matsuoka and the helical wormlike chain model by Yamakawa, because T* must be dependent on the chemical structure and stiffness of the polymers.

Second and Third Virial Coefficients of Polystyrene with Benzyl Ends near the ϑ Point

Second and third virial coefficients (A2 and A3, respectively) for cyclohexane solutions of linear polystyrene (benzyl-PS) samples, synthesized with benzyllithium (n-butyllithium/tetramethylethylenediamine/toluene ligand complex) as an initiator, were determined as functions of temperature and weight-average molecular weight Mw ranging from 710 to 7650. At the ϑ temperature (34.5 °C), where A2 = 0 for sufficiently high Mw, A2 decreased with lowering Mw, in contrast to the sharp increase reported for low molecular weight polystyrene with a butyl group (butyl-PS) at one of the chain ends. This molecular weight dependence of negative A2 for benzyl-PS was explained, without consideration of end effects, by the recent theory of Yamakawa and Yoshizaki for the helical wormlike chain with three-segment interactions, while the upswing of positive A2 for butyl-PS was ascribed primarily to the butyl end effect that surpasses the ternary cluster effect. The A3 values for both benzyl- and butyl-PS were positive and in...

Effects of excluded volume and polydispersity on solution properties of lentinan in 0.1 M NaOH solution

Seven lentinan fractions of various weight-average molecular weights (M(w)), ranging from 1.45 x 10(5) to 1.13 x 10(6) g mol(-1) were investigated by static light scattering and viscometry in 0.1M NaOH solution at 25 degrees C. The intrinsic viscosity [eta] - M(w) and radius of gyration s(2)(z) (1/2) - M(w) relationships for lentinan in 0.1M NaOH solution were found to be represented by [eta] = 5.1 x 10(-3)M(w) (0.81) cm(3) g(-1) and s(2)(z) (1/2) = 2.3 x 10(-1)M(w) (0.58) nm, respectively. Focusing on the effects of the M(w) polydispersity with the Schulz-Zimm distribution function, the data of M(w), s(2)(z) (1/2), and [eta] was analyzed on the basis of the Yoshizaki-Nitta-Yamakawa theory for the unperturbed helical wormlike chain combined with the quasi-two-parameter (QTP) theory for excluded-volume effects. The persistence length, molecular weight per unit contour length, and the excluded-volume strength were determined roughly to be 6.2 nm, 980 nm(-1), and 0.1, respectively. Compared with the theoretical value calculated by the Monte Carlo model, the persistence length is longer than that of the single (1 --> 3)-beta-(D)-glucan chain. The results revealed that lentinan exists as single-stranded flexible chains in 0.1M NaOH solution with a certain degree of expansion due to the electrostatic repulsion from the interaction between the OH(-) anions and lentinan molecules.

Nuclear Magnetic Relaxation of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The spin–lattice relaxation time T1 and nuclear Overhauser enhancement NOE were determined for three samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with weight-average degree of polymerization xw = 2, 3, and 67.1 in cyclohexane at 30 °C. A comparison is made of the present data for T1 and NOE with the helical wormlike (HW) chain theory, and it is shown that the theory may explain well the data for large xw (≳ 10). For smaller xw, the rigid sphere model having the radius equal to the apparent root-mean-square radius of gyration of the HW chain fails to give a quantitative explanation of T1 in contrast to the case of the relaxation time τΓ determined from dynamic depolarized light scattering measurements, indicating that the nuclear magnetic relaxation and dynamic depolarized light scattering cannot be described in terms of a common single relaxation time. However, it is shown that there is an effective (mean) magnetic relaxation time τM approximately equal to 0.6τΓ. From a comparison of the present results for T1 for a-PαMS with previous ones for typical flexible polymers, it is also shown that the behavior of T1 for them, including a-PαMS, may be consistently explained by the use of τM (proportional to τΓ).

Dynamic Depolarized Light Scattering by Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The power spectrum JΓ of the excess depolarizedcomponent of scattered light was determined from frequency-domain dynamic depolarized light scattering measurements for 10 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr=0.72, in the range of weight-average degree of polymerization xw from 2 to 67.1 in cyclohexane at 30.5 °C (Θ). It is found that, as in the cases of atactic polystyrene (a-PS), atactic poly(methyl methacrylate) (a-PMMA), and isotactic (i-) PMMA, JΓ may be well represented in terms of a single Lorentzian independently of xw, and that the relaxation time τΓ as defined as the reciprocal of the half-width at half-maximum of JΓ evaluated at infinite dilution increases with increasing xw and levels off to its asymptotic value in the limit of xw→∞, being consistent with the theoretical prediction on the basis of the helical wormlike (HW) chain model. A comparison is made of the present data for τΓ with the HW theory, and it is shown that the theory may explain satisfactorily the data in the range of xw ≳ 10, although semiquantitatively. For xw ≲ 10, the rigid sphere model having the radius equal to the apparent root-mean-square radius of gyration of the HW chain may give a good explanation of τΓ. From a comparison of the present results for τΓ for a-PαMS with previous ones for a-PS and a- and i-PMMAs, it is shown that there is good correlation between static and dynamic chain stiffness, the latter being defined as the ratio of the relaxation time associated with the local motion of the long chain to that of its isolated repeat unit (monomer), as predicted by the HW theory.

Some Comments on the Analysis of the Third Virial Coefficient for Polymer Chains

An analysis is made of experimental data for the third virial coefficient A3 of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) in the Θ solvent cyclohexane at 30.5 °C and in three good solvents, toluene, 4-tert-butyltoluene, and n-butyl chloride, at 25.0 °C on the basis of the helical wormlike (HW) chain model. It is found that A3 at Θ, which is denoted by A3,Θ, becomes a constant of 5.0×10−4 cm6 mol/g3 independent of the weight-average molecular weight Mw for Mw ≳ 104 and deviates from it for Mw ≲ 104 because of effects of chain ends. The observed dependence of A3,Θ on Mw may be well explained by the HW theory that takes account of the effects. For the three good-solvent systems, the behavior of the factor g defined by A3/[A2(HW)]2Mw is examined as a function of the cubed gyration-radius expansion factor αS3, where A2(HW) is the part of the second virial coefficient A2 without the effects of chain ends. It is found that the data points for a-PαMS samples with Mw ≳ 105 in all the three good solvents form nearly a single-composite curve, as predicted by the HW theory, and that they follow the Stockmayer–Casassa theoretical curve for αS3 ≳ 2 but deviate upward from it for αS3 ≲ 2 because of effects of three-segment interactions. Effects of chain stiffness are of minor importance for an explanation of this deviation. Some literature data for polystyrene are also examined.

Conformational statistics of bent semiflexible polymers

This paper extends previous methods for obtaining the probability distribution function of end-to-end distance for semiflexible polymers, and presents a general formalism that can generate conformational statistics of any continuum filament model of semiflexible chains with internal bends and twists. In particular, our focus is distribution functions for chains composed of straight or helical segments connected with discrete bends or twists. Prior polymer theories are not able to fully account for the effects of these internal shape discontinuities. We use the operational properties of the noncommutative Fourier transform for the group of rigid-body motions in three-dimensional space. This general method applies to various stiffness models of semiflexible chainlike macromolecules. Examples are given which apply the stiffness parameters defined in the Kratky–Porod model, Yamakawa helical wormlike chain model, and revised Marko–Siggia double-helix model to chains with intrinsic bends or twists in their undeformed (minimal energy) state. We demonstrate how the location and magnitude of internal bends in the chain affect the distribution of end-to-end distances for each of these models. This capability allows one to study the entropic effects of intrinsic shape changes (e.g., bend angle) in various models, and may lead to coarse-grained continuum mechanical models of processes that occur during transcription regulation.

Mean-Square Optical Anisotropy of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The mean-square optical anisotropy 〈Γ2〉 was determined from anisotropic light scattering measurements for 10 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with the fraction of racemic diads fr = 0.72 in the range of weight-average degree of polymerization xw from 2 to 67.1 in cyclohexane at 30.5 °C (ϑ). A comparison is made of the present data with the helical wormlike (HW) chain theory with the values of the model parameters previously determined from the mean-square radius of gyration and with those of the local polarizability tensor α0 per repeat unit properly assigned on the basis of the bond and group polarizabilities. The HW theory value of 〈Γ2〉/xw in the limit of xw → ∞ is found to be somewhat larger than the experimental one. It is then shown that if α0 is replaced by 0.90α0, the HW theory may reproduce quantitatively the experimental results for all the samples except those with xw ≤ 4 for which effects of chain ends become appreciable. A comparison is also made of the present results for a-PαMS with previous ones for atactic polystyrene (fr = 0.59), atactic poly(methyl methacrylate) (a-PMMA) (fr = 0.79), isotactic PMMA (fr ≃ 0.01), and poly(n-hexyl isocyanate), and the behavior of 〈Γ2〉 is shown to depend remarkably on chain stiffness and local chain conformation.

A Monte Carlo study of effects of chain stiffness and chain ends on dilute solution behavior of polymers. II. Second virial coefficient

A Monte Carlo (MC) study is made of the second virial coefficient A2 for polymers using two freely rotating chains, each of bond angle 109°, with the Lennard-Jones 6-12 intramolecular and intermolecular potentials between beads in a cutoff version for the number of bonds in the chain ranging from 6 to 1000 in the Θ and good-solvent conditions. It is found that effects of chain ends on A2 are appreciable for small molecular weight M, as was expected, and that the second virial coefficient A2,Θ at the Θ temperature, at which the ratio 〈S2〉/M of the mean-square radius of gyration 〈S2〉 to M becomes a constant independent of M for very large M, remains slightly negative even for such large (but finite) M where the effects of chain ends disappear. Such behavior of A2,Θ, which cannot be explained within the framework of the binary cluster theory, is shown to be understandable if possible effects of three-segment interactions are considered. The present MC data for A2 (along with the previous ones for 〈S2〉) may then be consistently explained by the existent theory based on the helical wormlike chain model only if a minor correction is made to the theoretical A2,Θ in almost the same range where the effects of chain ends are appeciable. The present MC data are also compared with experimental data, and it is shown that the latter may also be similarly explained.

Ultracentrifugal Characterization of Poly(ethylene terephthalate) in Dilute Solutions

The well-fractionated poly(ethylene terephthalate) (PET) samples ranging in weight average molecular weight Mw from 0.9 to 16×104 have been studied by sedimentation velocity and equilibrium, and viscometry in hexafluoroisopropanol (HFIP) and m-cresol at 30°C. The second virial coefficient A2 determined in HFIP by the sedimentation equilibrium method is found to have high dependence on Mw expected for the helical wormlike chains in very good solvents. The constant Φ1/3P−1 in the Mandelkern-Flory equation is 2.54×106 for PET in HFIP and 2.57×106 in m-cresol, being in excellent agreement with that reported for several polymer-solvent systems. The dependences of sedimentation coefficient s0 on Mw agree with also those expected for good solvents. The Mark-Houwink-Sakurada relations established with HFIP and m-cresol as the solvents are [η]=2.95×10−4Mw0.72 in HFIP at 30°C (3.1×104<Mw) [η]=4.97×10−3Mw0.48 in HFIP at 30°C (3.1×104>Mw) [η]=4.04×10−4Mw0.68 in m-cresol at 30°C (2.4×104<Mw)The values of steric factor σ estimated from the unperturbed chain dimensions range from 1.3 to 1.5, irrespective of the solvent nature, and such small σ values suggest that the extent of steric hindrance to the internal rotation of PET chain is significantly low.

Scattering Function of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The scattering function was determined for four samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr = 0.72, in the range of weight-average molecular weight Mw from 1.27 × 103 to 6.46 × 104 in cyclohexane at 30.5 °C (ϑ) in the range of the magnitude k of the scattering vector smaller than 1 Å-1 by the use of a point-focusing small-angle X-ray scattering (SAXS) camera. The Kratky function Fs as a function of k increases monotonically with increasing k for Mw ≲ 5 × 103 and exhibits a plateau in the range of 0.15 Å-1 ≲ k ≲ 0.3 Å-1 for Mw ≳ 5 × 103. For k ≲ 0.5 Å-1, all the experimental data may be well explained by the corresponding helical wormlike (HW) chain theory. For k ≳ 0.5 Å-1, however, agreement between theory and experiment is only qualitative. A comparison is made of the present SAXS data for a-PαMS with the previous data for atactic polystyrene with fr = 0.59, atactic poly(methyl methacrylate) (a-PMMA) with fr = 0.79, isotactic PMMA with fr ≃ 0.01, and syndiotactic (s-) PMMA with fr = 0.92. It is then found that the HW theory may give a good explanation of the difference in the behavior of Fs for k ≲ 0.3 Å-1 between the four polymers except s-PMMA, for which the HW model parameters have not been determined.

Transport Coefficients of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The intrinsic viscosity [η] was determined for 25 samples of atactic oligo- and poly(α-methylstyrene)s (a-PαMS), each with the fraction of racemic diads fr = 0.72, in the range of weight-average molecular weight Mw from 2.94 × 102 (dimer) to 3.22 × 106 in cyclohexane at 30.5 °C (ϑ). The translational diffusion coefficient D was also determined from dynamic light-scattering measurements for 21 of them in the range of Mw from 5.30 × 102 (tetramer) to 3.22 × 106 under the same solvent condition. It is found that the double-logarithmic plots of [η] and MwD against Mw follow their respective asymptotic straight lines of slope 1/2 for Mw ≳ 2 × 105, but deviate upward and downward, respectively, from them with decreasing Mw for smaller Mw. From an analysis of these transport coefficients on the basis of the helical wormlike (HW) chain model, it is shown that the above Mw dependence of [η] and D may be well explained by the corresponding HW theories with the values of the model parameters consistent with those pr...