Scaling laws in simple and complex proteins: size scaling effects associated with domain number and folding class

The native states of the most compact globular proteins have been described as being in the so-called “collapsed-polymer regime,” characterized by the scaling law R _g ~ n ^ν, where R _g is radius of gyration, n is the number of residues, and ν ≈ 1/3. However, the diversity of folds and the plasticity of native states suggest that this law may not be universal. In this work, we study the scaling regimes of: (i) one to four-domain protein chains, and (ii) their constituent domains, in terms of the four major folding classes. In the case of complete chains, we show that size scaling is influenced by the number of domains. For the set of domains belonging to the all-α, all-β, α/β, and α?+?β folding classes, we find that size-scaling exponents vary between 0.3?≤?ν?≤?0.4. Interestingly, even domains in the same folding class show scaling regimes that are sensitive to domain provenance, i.e., the number of domains present in the original intact chain. We demonstrate that the level of compactness, as measured by monomer density, decreases when domains originate from increasingly complex proteins.     

Size scaling behaviour in protein domains belonging to the all-α, all-β, α / β, and α?+?β folding classes

We studied the size scaling behaviour in an ensemble of 8,614 non-redundant protein domains belonging to the all-α, all-β, α / β, and α + β folding classes. We find that the most compact structural domains can be characterized by an effective exponent ν _eff  = 0.39 ± 0.01, which is larger than the value for “collapsed-polymers,” i.e., ν = 1/3. We also show that the global ν _eff -exponent is an average of the scaling regimes for short and long compact chains, where the values change from ν _eff ≈ 0.37 to ν _eff ≈ 0.45 at chain length of ca. 269. A transition from short-chain to long-chain scaling behaviour is found in all major folding classes, over a window of chain lengths between 216 and 269 residues. In addition, variations in scaling exponent with respect to folding class indicates that the smallest domains in the (all-β) and (α / β) families appear to be more compact structures than the smallest (all-α)- and (α + β)-domains.     

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).     

Electronic spectra of the linear cationic chains NC<Subscript>2<Emphasis Type="Italic">n</Emphasis></Subscript>N<Superscript>+</Superscript> (<Emphasis Type="Italic">n</Emphasis> = 1–7): an ab initio study

The ground-state equilibrium geometries of the linear carbon chain cations NC_2n N⁺ (n = 1–7) have been investigated with B3LYP, CAM-B3LYP, and RCCSD(T) calculations. The ground state (X²П_g/u) and excited state (1²П_u/g) have been optimized by using the complete active space self-consistent field method. The present study reveals that these linear cations generally have the characteristic of bond length alternation in both electronic states. The vertical excited energies for the dipole-allowed (1, 2, 3)²П_u/g ← X²П_g/u transitions as well as the dipole-forbidden 1²Φ_u/g ← X²П_g/u transitions have been computed with the complete active space second-order perturbation theory. The calculated transition energies of 1²П_u/g ← X²П_g/u for NC_2n N⁺ (n = 1–6) in the gas phase are 2.26, 2.09, 1.91, 1.72, 1.56, and 1.39 eV, respectively, which mutually agree well with the available experimental values of 2.11, 2.07, 1.88, 1.67, 1.49, and 1.34 eV. Moreover, the corresponding absorption wavelengths are predicted to have the significant nonlinear size dependence, which is different from the bands origin in NC_2n N (n = 1–7).     

Polymers in 2D confinement: A nanoscale mechanism for thermo‐mechanical reinforcement

Acrylate‐clay nanocomposites, a 2D confined system, exhibited unusual increase of thermo‐mechanical properties. The nature of this reinforcement can be ascribed to chain dynamics modification and therefore investigated via dynamic mechanical analysis. Transmission electron microscopy and dynamic light scattering showed a strong nanoconfined regime, 2R_h ≫ d₀₀₁, where R_h is the polymer's hydrodynamic radius and d₀₀₁ is the clay gallery spacing. The geometrical constraints to polymer dynamics led to significant enhancement of the thermo‐mechanical properties. Adding only 1 wt% nanoclay, the glass transition temperature increased significantly, ΔT_g = T_g − T_g,bulk ~ 10°C, and the dynamic modulus E′ increased 10‐fold. Analysis of dynamic mechanical spectra showed an increase of relaxation time τ, ie, polymer dynamics retardation. Furthermore, the mechanical damping tan δ was strongly attenuated evidencing the reduction of viscous dissipation. The activation energy E_a of the α‐transition increased as the confined macromolecules needed to overcome higher energy barriers to achieve configurational rearrangements. The considerable increase of mechanical modulus cannot be explained by polymer composite models, rather it was associated to a “nano‐effect,” scaling with the degree of confinement as E/E_matrix ~ (2R_h/d₀₀₁)ⁿ. This study paves the road for further understanding of polymer dynamics under 2D confinement and the reinforcement mechanism of thermo‐mechanical properties.     

Equation of state for polymers based on glass transition data

This paper addresses a method for predicting the participating constants in equation of state (EOS) for compressed polymeric fluids using two scaling constants, the surface tension γ _g and the molar density ρ _g, both at the glass transition point. The theoretical EOS undertaken is the one attributed to Tao and Mason. The second virial coefficients are calculated from a two-parameter corresponding states correlation, which is constructed with two constants as scaling parameters, i.e., the surface tension γ _g and the molar density ρ _g. This new correlation has been applied to the Tao–Mason (TM) EOS to predict the volumetric behavior of several polymer melts. The operating temperature range is from 291.25 to 603.4 K and pressures of up to 202.5 MPa. A collection of 516 data points has been examined for the aforementioned polymers. The average absolute deviation between the calculated densities and the experimental ones is of the order of 0.44%.     

Relaxational mode structure for optical probe diffusion in high molecular weight hydroxypropylcellulose

We studied translational diffusion of dilute monodisperse spheres (diameters 14 < d < 455 nm) in aqueous 1 MDa hydroxypropylcellulose (0 ≤ c ≤ 7 g/L) at 25°C using quasielastic light scattering. Spectra are highly bimodal. The two spectral modes (“slow,” “fast”) have different physical properties. Probe behavior differs between small (d < R_h) and large (d ≥ R_g) probes; R_h and R_g are the matrix polymer hydrodynamic radius and the radius of gyration, respectively. We examined the dependences of spectral lineshape parameters on d, c, scattering vector q, and viscosity η for all four probe-size and mode-type combinations. We find three time scale-separated modes: (1) a large-probe slow mode has properties characteristic of particle motion in a viscous medium; (2) a large-probe fast mode and small-probe slow modes share the same time scale, and have properties characteristic of probe motion coupled to internal chain dynamics; and (3) a small-probe fast mode has properties that can be attributed to the probe sampling local chain relaxations. In the analysis, we also attempted to apply the coupling/scaling (CS) model of Ngai and Phillies [Ngai, K. L., Phillies, G. D. J. J. Chem. Phys., 105 , 8385 (1996)] to analyze our data. We find that the second mode is described by the coupling/scaling model for probe diffusion; the first and third modes do not follow the predictions of this model. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 3087–3100, 1998     

Perfluoroalkylphosphines and arsines obtained by Pd-catalyzed cross-coupling reaction with organoheteroatom stannanes

Pd-catalyzed cross-coupling reaction of organoheteroatom stannanes containing elements of the groups 15 (P, As) and 16 (Se) with perfluoroalkyl iodides (R_fI) was studied. Herein, a one-pot two-step reaction to form P–R_f, As–R_f and Se–R_f bonds is reported. The stannanes n-Bu₃SnZPh_n (Z = P, As, Se; n = 1–2) were generated in situ by the reaction of the Ph_nZ⁻ anion with n-Bu₃SnCl. The cross-coupling reactions of these stannanes with R_fI afforded C-heteroatom products, new perfluoroalkylarsines and perfluoroalkylselenides in good yields (47–90%) and perfluoroalkylphosphines in moderate yields (15–48%).     

Certification of a pure reference material for the ginsenoside Rg 1

A pure certified reference material (CRM) for the ginsenoside Rg ₁ was prepared from roots of Panax ginseng by extraction and separation of ginsenosides. The mass fraction of the main component (ginsenoside Rg ₁ ) in the reference material was determined and its uncertainty was assessed from various input quantities, such as organic impurities, residual moisture, residual solvent, ash, and insoluble matters. To measure these input quantities, HPLC/CAD, Karl Fischer (KF) coulometry, gravimetry, and GC/FID were used. Homogeneity and long-term stability of the reference material are discussed. The certified mass fraction of Rg ₁ in the reference material is 0.974 ± 0.006 (k = 2) with a shelf life of 1 year.     

Fractal character of the SAXS correlation volume in poly(ethylene glycol)/silica hybrid wet gels

Small-angle X-ray scattering (SAXS) was employed to study the nanostructural properties of poly(ethylene glycol)(PEG)/silica hybrid wet gels prepared from hydrolysis of tetraethoxysilane (TEOS) in mixtures with PEG/(PEG + TEOS) molar ratio in the nominal range between 0 and 0.8. The SAXS pattern was found to be well fitted by the scattering from a polymeric particle of radius of gyration R _g with an internal structure of primary silica particles described by a polymeric constraint of a mass-fractal of dimension D. R _g increases with the PEG quantity while D first increases from 2.24 for the pure TEOS-derived gel to reach values between 2.46 and 2.40 with the additions of PEG. The correlation volume V _c as determined by SAXS for such a polymeric structure behaves as a volume-fractal and accordingly scales with R _g as V _c ~ R _g ^D . An overall mean value of about 1 nm could be estimated as an upper limit for the radius of the primary silica particle building up the structure of all the wet gels, independent of the PEG addition.     

Scaling analysis of cotton cellulose/LiCl·DMAc solution using light scattering and rheological measurements

Semidilute solution of cotton lint (CC1) in 8 wt % LiCl/N,N‐dimethylacetamide was investigated using static light scattering (SLS) and rheological measurements. The reduced osmotic modulus estimated by SLS measurements for CC1 solutions are proportional to c^1.16 in the semidilute region. From the exponent of 1.16, de Gennes' scaling theory derives the relationship between radius of gyration, R_g, and molecular weight, M_w, of CC1 as R_g ∝ M^0.62 This corresponds to the Mark‐Houwink‐Sakurada exponent of 0.86. This exponent is very close to that estimated from scaling analysis of zero shear rate viscosity, that is 0.85. Apparent radius of gyration, R_g,app, estimated by SLS measurements for CC1 solutions are proportional to c^?0.5 in the semidilute region. R_g,app indicates the mesh size of polymer entanglement in the semidilute region. On the assumption of the Gaussian behavior of CC1 molecule in the semidilute region, the exponent of ?0.5 gives the relationship between the molar mass between entanglements, M_e, and c as following relationship: M_e ∝ c^?1. This agrees with the concentration dependence on plateau modulus estimated from the dynamic viscoelastic measurements. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2155–2160, 2006     

Oxalic acid-assisted preparation of LiFePO4/C cathode material for lithium-ion batteries

LiFePO₄/C composite is synthesized by oxalic acid-assisted rheological phase method. Fe₂O₃ and LiH₂PO₄ are chosen as the starting materials, sucrose as carbon sources, and oxalic acid as the additive. The crystalline structure and morphology of the products are characterized by X-ray diffraction and field emission scanning electron microscopy. The charge–discharge kinetics of LiFePO₄ electrode is investigated using cyclic voltammetry and electrochemical impedance spectroscopy. It is found that the introduction of appropriate amount of oxalic acid leads to smaller particle sizes, more homogeneous size distribution, and some Fe₂P produced in the final products, resulting in reduced polarization, impedance, and improved Li⁺ ion diffusion coefficient. The best cell performance is delivered by the sample with R = 1.5 (R of the molar ratio of oxalic acid to LiH₂PO₄). Its discharge capacity is 154 mAh g⁻¹ at 0.2 C rate and 120 mAh g⁻¹ at 5.0 C rate. At the same time, it exhibits an excellent cycling stability; no obvious decrease even after 1,000 cycles at 1.0 C rate.     

Study of capacitive properties in supercapacitor for copolymer of aniline with m-phenylenediamine

The copolymers were synthesized with different molar ratios of m-phenylenediamine to aniline (R for short) by a chemical oxidation method. The products were first used as electrochemical activity materials of the supercapacitor. Capacitive behaviors of the prepared copolymers in 1 mol·L⁻¹ H₂SO₄ electrolyte were examined by electrochemical impedance spectroscopy, cyclic voltammeter, and galvanostatic charge/discharge. The relationship of molar ratios with capacitive property of the prepared products was investigated too. The results showed that the product with R of 2:98 displayed better electrochemical properties than that of the other products. Compared with the synthesized polymer in the absence of m-phenylenediamine, the polymerized copolymer with R of 2:98 exhibited the initial specific capacitance value of 475 F·g⁻¹, which increased by nearly 10.1% than that of the former at a current density of 200 mA·g⁻¹ in 1 mol·L⁻¹ H₂SO₄ electrolyte in the potential range of −0.3 to 0.7 V. The discharge specific capacitance value of the copolymer remained 300 F·g⁻¹ after 1,000 cycles, exhibiting a good cycling performance and the structure stability.     

Conformation of nonideal hyperbranched polymer in ABn (n = 2, 4) type polymerization

The conformation of hyperbranched polymers from one pot polymerization with AB_n (n = 2, 4) type monomers, applying the reactive 3D bond fluctuation lattice model, are systematically studied using scaling relation R ～ N^λ, where R is the radius of gyration or the hydrodynamic radius of a hyperbranched polymer with the degree of polymerization N. The exponent λ was calculated at various monomer concentrations and group conversions. When the concentration of monomers with the equal reactivity of B groups increases from 0.1 to 0.9, the exponents λ_g and λ_h (corresponding to the radius of gyration and hydrodynamic radius, respectively) are in the ranges of 0.51–0.37 and 0.41–0.34 at the full conversion of A groups. Especially, we find that λ_g decreases linearly with the reaction conversion increasing. The ratio of z‐average radius, R_gz/R_hz, ranges from 1.08 to 1.32 and indicates that hyperbranched polymer is soft macromolecule with penetrable structure. In the case of AB₂ type monomer with unequal reactivities, λ displays complicated dependence on the reaction conversion and the reactivity ratio. The results of our simulation are consistent with those of experiments and theories, and valuable in better understanding the fundamental properties of hyperbranched polymers. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 610–616, 2010     

Kinetic partitioning mechanism as a unifying theme in the folding of biomolecules

We describe a unified approach to describe the kinetics of protein and RNA folding. The underlying conceptual basis for this framework relies on the notion that biomolecules are topologically frustrated due to their polymeric nature and due to the presence of conflicting energies. As a result, the free energy surface (FES) has, in addition to the native basin of attraction (NBA), several competing basins of attraction. A rough FES results in direct and indirect pathways to the NBA, i.e., a kinetic partitioning mechanism (KPM). The KPM leads to a foldability principle according to which fast folding sequences are characterized by the folding transition temperature T _F being close to the collapse transition temperature T _θ, at which a transition from the random coil to the compact structure takes place. Biomolecules with T _θ ≈ T _F , such as small proteins and tRNAs, are expected to fold rapidly with two-state kinetics. Estimates for the multiple time scales in KPM are also given. We show that experiments on proteins and RNA can be understood semi-quantitatively in terms of the KPM. Received: 14 January 1997 / Accepted: 23 January 1997     

T型混合反应器内共沉淀法制备吡啶硫酮铜

在T型混合反应器内通过喷射共沉淀方法合成了吡啶硫酮铜(CPT).利用扫描电镜和小角度光散射实验手段研究了分散液中吡啶硫酮铜的形貌和大小。通过喷射共沉淀方法获得的初级吡啶硫酮铜颗粒为棒状, 在分散液中这些棒状颗粒易发生团聚, 形成分形维数为2.1的团聚体。用平均回转半径(R_g)表征分散液中团聚体的平均大小, 考察了反应温度和反应物化学计量比(吡啶硫酮钠/硫酸铜)对团聚体大小的影响, 随着反应温度的降低, 团聚体的回转半径逐渐减小;过量的吡啶硫酮钠也有利于降低团聚体的大小, 当吡啶硫酮钠过量量达到~25%时, 进一步增大吡啶硫酮钠的过量量, 团聚体的回转半径不再发生明显变化。     

Comparison of the molecular dimensions of di-alkyl and aryl-alkyl-substituted polysilanes in solution using SEC/LS

Polysilanes provide an opportunity for exceptional control of the chain hindrances to rotation through the choice of substituents on each backbone silicon. Two alkyl substituents on each silicon result in a large characteristic ratio of at least 19 for poly (di-n-hexylsilane), determined by extrapolation of intrinsic viscosities. Bulky aromatic substituents provide even more hindrance to backbone rotations, and can be expected to result in a more extended polymer chain. Direct measurement of the dimensions of these polymers by scattering techniques has been limited by the small quantities available, and by the polydispersity of samples. The recent introduction of light-scattering detectors for size exclusion chromatography enables the simultaneous measurement of light scattered at as many as 15 scattering angles as the fractionated polymer elutes from the column. Determination of both M and the root-mean-square radius of gyration R_g of narrow fractions eluting from a column allows determination of the R_g M relation over as much as a decade in M with less than a milligram of sample. Values of R_g and M across the distribution have been determined for alkyl and aryl substituted polysilanes with this technique. Estimation of R_g,0/M unperturbed by long-range interactions is made by an extrapolation procedure. The dependence of R_g,0 on M across the distribution is compared among the different substituents and with other measurements reported for these polymers. © 1994 John Wiley & Sons, Inc.     

Determination of organic micro-pollutants such as personal care products, plasticizers and flame retardants in sludge

In this study, a method for the determination of organic micro-pollutants, i.e. personal care products such as synthetic musk fragrances, household bactericides, organophosphate flame retardants and plasticizers, as well as phthalates in sludge, has been developed. This method is based on lyophilisation and accelerated solvent extraction followed by clean-up steps, i.e. solid phase extraction and size exclusion chromatography. The determination is performed by gas chromatography coupled to mass spectrometry. Stable isotope-labelled compounds such as musk xylene (MX D₁₅), tri-n-butylphosphate (TnBP D₂₇) and triphenylphosphate (TPP D₁₅) were used as internal standards. Recovery rates were determined to be 36–114% (with typical relative standard deviation of 5% to 23%) for the target compounds. The limit of detection was 3–30 ng g⁻¹, and the limit of quantification was 10–100 ng g⁻¹ dry matter.     

Determination of glyphosate and aminomethylphosphonic acid in aqueous soil matrices: A critical analysis of the 9‐fluorenylmethyl chloroformate derivatization reaction and application to adsorption studies

The assessment of the environmental fate of glyphosate and its degradation product (aminomethylphosphonic acid) is of great interest given the widespread use of the herbicide. Studies of adsorption–desorption and transport processes in soils require analytical methods with sensitivity, accuracy, and precision suitable for determining the analytes in aqueous equilibrium solutions of varied complexity. In this work, the effect of factors on the yield of the derivatization of both compounds with 9‐fluorenylmethyl chloroformate for applying in aqueous solutions derived from soils was evaluated through factorial experimental designs. Interference effects coming from background electrolytes and soil matrices were established. The whole method had a linear response up to 640 ng/mL (R² > 0.999) under optimized conditions for high‐performance liquid chromatography with fluorescence detection. Limits of detection were 0.6 and 0.4 ng/mL for glyphosate and aminomethylphosphonic acid, respectively. The relative standard deviation was 4.4% for glyphosate (20 ng/mL) and 5.9% for aminomethylphosphonic acid (10 ng/mL). Adsorption of compounds on four different soils was assessed. Isotherm data fitted well the Freundlich model (R² > 0.97). K_f constants varied between 93 ± 3.1 and 2045 ± 157 for glyphosate and between 99 ± 4.1 and 1517 ± 56 (μg^1‐1/n mL^1/n g^–1) for aminomethylphosphonic acid, showing the broad range of applicability of the proposed method.     

Parakristalline Gitterstörungen in Kettenrichtung von Cellulose I und II

Zusammenfassung Röntgenweitwinkelmessungen an Ramie sowie an einer Viskosefaser zeigen, daß in Kettenrichtung parakristalline Störungen vong=1,1% bzw.g=1,5% vorliegen. Die mittleren Kristallitgrößen in Kettenrichtung betragen¯D ₀₁₀=485 Å bzw.¯D ₀₁₀=112 Å, senkrecht dazu etwa¯D=52 Å bzw.¯D=41 Å. Aus Röntgenkleinwinkelmessungen ergeben sich kleinste Fibrillendurchmesser von 2R=52 Å bzw. 2R=45 Å. Die gefundenen Fibrillendurchmesser verhalten sich wie 1248.Für die Viskosefaser wird eine Langperiode vonP=168 Å gemessen, woraus mit¯D ₀₁₀=112 Å folgt, daß die ungeordneten Bereiche in Kettenrichtung eine Länge von etwa 56 Å haben.

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Summary Wide angle X-ray diffraction studies on ramie and on a viscose fibre show that paracrystalline distortions exist in chain direction havingg-values ofg=1.1% andg=1.5%, respectively. The mean crystallite sizes in chain direction are¯D ₀₁₀=485 Å and¯D ₀₁₀=112 Å, respectively. Perpendicular to the chain direction mean crystallite sizes of approximately¯D=52 Å and¯D=41 Å, respectively, are measured.Small angle X-ray diffraction studies give the diameters of the cellulose fibrils, the smallest of which are 2R=52 Å and 2R=45 Å, respectively. The diameters of the fibrils are in the ratio of 1248.A long period ofP=168 Å is measured for the viscose fibre implying a length of 56 Å for the disordered domains in chain direction.