Mean-square helical hydrophobic moments in partially ordered proteins

Helical hydrophobic moment ratios, 〈h²〉/〈H²〉, have been evaluated for 34 polypeptides under conditions where the helix content is dictated solely by the short-range interactions operative in aqueous media. The mean-square helical hydrophobic moment is denoted by 〈h²〉, and 〈H²〉 is the averaged of the squared hydrophobicites. This ratio would be one in absence of any correlation in the hydrophobicities of amino acid residues in helices. The 〈h²〉/〈H²〉 tend to be substantially larger than values of the analogous ratio formulated for the mean-square dipole moments of typical synthetic polymers. For 24 of the 34 polypeptide chains considered, 〈h²〉/〈H²〉 is found to be greater than one, indicating a tendency to form helices with amphiphilic character. The ratio is exceptionally large in the case of the δ-hemolysins. It is also large for two other surface-active peptides, for two of the four apolipoproteins examined, and for myohemerythrin. A much smaller 〈h²〉/〈H²〉 is found for melittins. If melittins is to form helices with large 〈h²〉/〈H²〉, the configurational statistics must be governed by effects in addition to those short-range interactions that occur when water is the solvent.     

Mean-square hydrophobic moment for partially helical polypeptides

A mean-square helical hydrophobic moment, 〈h²〉, is defined for polypeptides in analogy to the mean-square dipole moment, 〈μ²〉, for polymer chains. For a freely jointed polymer chain, 〈μ²〉 is given by Σm, where m_i denotes the dipole moment associated with bond i. In the absence of any correlations in the hydrophobic moments of individual amino acid residues in the helix, 〈h²〉 is specified by ΣH, where H_i denotes the hydrophobicity of residue i. The tendency for correlations in orientations of residue hydrophobic moments in helices therefore dictates the size of 〈h²〉/〈H²〉, where 〈H²〉 denotes the average value of ΣH for all helices. The value of 〈h²〉/〈H²〉 will be greater than one in amphiphilic helices. A necessary prerequisite for this diagnostic usage of 〈h²〉/〈H²〉 is that the residue hydrophobic moment be oriented prependicular to the principal axis of the helix. Matrix-generation schemes are formulated that permit rapid evaluation of 〈h²〉 and 〈H²〉. The behavior of 〈h²〉/〈H²〉 is illustrated by calculations performed for model sequential copolypeptides.     

A Monte Carlo study of the excluded-volume effect on the amylosic chain conformation

A Monte Carlo procedure was used to determine the effect of excluded volume on the conformational dimensions of amylosic chains. The excluded volume was introduced into the model by assuming that hard spheres, which cannot overlap each other, exist at the center of mass of each glucose unit in the chain sequence. Monte Carlo chains, which were generated to be distributed consistent with the potential energy of nonbonded nearest-neighbor interactions, underwent self-intersections, and the attrition rate in the generation of self-avoiding chains was found to obey an exponential decay law with increasing chain length x. Thus, in order to generate effectively a large number of self-avoiding chains with long sequences, we used the Wall–Erpenbeck s-p method of chain enrichment [F. T. Wall and J. J. Erpenbeck (1959) J. Chem. Phys. 30 , 634–637]. By examination of the radial distribution of the end-to-end distance and the chain-length dependence of the quantity 〈r²〉/xl² (〈r²〉 is the mean square end-to-end distance and l is the virtual bond length), it was found that unperturbed amylosic chains change in overall conformation from a non-Gaussian chain having a helical character to Gaussian as x is increased, whereas perturbed chains do not show Gaussian behavior even at x = 500. For the perturbed chains, 〈r²〉 can be expressed by the equation 〈r²〉 = ax^b in the range of 100 ≤ x ≤ 500, where a and b > 1 are constants. From comparisons of the persistence vectors and perspective drawings of representative unperturbed and perturbed chains, we felt the local conformation of the amylosic chains, i.e., the local helical character, is also affected by the long-range excluded-volume interaction.     

Nonradiative energy transfer in oligopeptide chains generated by a monte-cralo mehod including long-range interactions

A Monte-Carlo method including long-range interactions is used to oligopeptide chains in random-coil state. The chains are composed of 4, 9, or 14 repeating units and are labeled with the luminopheres tyrosine or tryptophan. Interactions with a solvent (water) are taken into account in the calculations through modifications of the semiempirical potential-energy functions. The chains represent oligopeptides composed of hydrophobic or hydrophilic amino acid residues. Various properties relavent to the interpretaiton of nonradiative enrgy-transfer experiments, such as the average value of the orientation factor for dipole-dipole interaction of the luminophores, 〈k²〉, the distribution function of the distances between the luminophores f(r_l), the efficiences of energy transfer in the static and dyamic averaging regimes, 〈T〉_s amnd 〈T〉_d, as well as the fluorescence decay I(t) of the donor luminophore in various averaging conditions, are computed. It is shown that, for all chains considered, 〈k²〉 is not vary far form 0.67 and that 〈T〉_s and 〈T〉_d have completely different values. Due to the small extent of correlation between the distances r_l and the mutual orientations of the lumninophores, the decay kinetics 〈I(t)_s corresponding to a static averaging regime can be expressed in terms of distribution functions f(r_l). These results are in agrrement with those obtained previously for the unperturbed chain model.     

The spatial distributions of randomly coiling polynucleotides

Statistical mechanical averages of vectors and tensors characterizing the allowed configurations of randomly coiling polynucleotides have been calculated for chains of 2⁰–2¹⁰ repeating units. Specifically, the persistence vector p = 〈 r 〉 has been obtained as a function of chain length. Configurational averages of the Cartesian tensors formed from the displacement vector ρ = r – p have been computed up to and including the tensor of seventh rank. From these tensors the three-dimensional spatial distributions of end-to-end vectors have been constructed to provide comprehensive pictures of the directional tendencies of the randomly coiling polynucleotide. The elements of the third and fourth moment tensors, however, give sufficient information to represent accurately the qualitative features of the spatial distributions. For long chains, more than 2⁶ (64) repeating units, the spatial distributions assume spherically symmetric shapes that can be adequately characterized by one-dimensional radial distribution functions. These radial distribution functions agree well with the radial distributions calculated from Monte Carlo samples containing more than 5000 chains. The constraints of fixed bond lengths, fixed bond angles, and hindered internal rotations severely distort the spatial distributions of short polynucleotide chains to mushroom-shaped volumes. These skewed distributions provide information useful to the analysis of small, single-stranded loops, bulges, and circles. The formation of these structures requires the termini of the polynucleotides to lie within specifically delineated volumes with respect to coordinate systems affixed to the first bonds of the chains. The extent to which these loop closure volumes overlap the three-dimensional spatial distributions provides estimates of loop formation that are much more reliable than earlier studies based upon the radial distribution function.     

Total intensity light scattering from short,optically anisotropic wormlike chains

Simple exact equations are derived for intensity light scattering from optically anisotropic wormlike chains in the absence of excluded volume. The results are valid at low scattering angles (q² 〈R²_G〉 ? 1) for all sormilke chains from rigid rods to random couils. The present work and an earlier theory [Nagai, K. (1972) Polym. J. 3 , 67–83] appear to be equivalent, although they were both derived using different methods. The present work is primarily concerned with short wormlike chains, since intensity light scattering from short fragments may provide valuable information about DNA flexibility. By using the results of this work to reanalyze some older light-scattering studies [Godfrey, J. E. & Eisenberg, H. (1976) Biophys. Chem. 5 , 301–318], it is shown that anisotropy corrections to polarized light-scattering measurements have been overcorrected in the past. It can be anticipated that future light-scattering experiments will determine the base-pair anisotropy.     

NMR study of the structure of short-chain peptides in solution.

The electron-mediated spin–spin coupling constant J between the amide NH and the α-CH protons in the dipeptide fragment C^α? CO(NH? C^αH)R? C′ONH? C^α is dependent on the dihedral angle of rotation (Φ) around the N? C bond. Measurement of J in a series of zwitterionic dipeptides H₃N⁺? CHR₁? CONH? CHR₂? CO₂^? (which is conformationally similar to the dipeptide fragment) in TFA solution shows that J is independent of R₁, but dependent on the steric bulk of R₂. The data are interpreted in terms of a model that assumes that what we measure is an average value of J? a thermal average over all the possible rotamers. The groups R₁ and R₂ are, in most cases, sterically kept apart by the trans and planar amide bonds, and hence the independence of J of R₁. This model is consistent with the theoretical calculations done on the dipeptide fragment. The effect of the structural characteristics of the side chains (e.g., the effect of lengthening and branching the side chains) on the J values in dipeptides is discussed in the light of the existing results of theoretical calculations. Study of 〈J〉 values in tripeptides (C₆H₅CH₂OCONH? CHR₁? CONH? CHR₂? CO₂CH₃, essentially three linked peptide units) shows that electrostatic interaction between the two amide bonds modifies the potential energy surface and the 〈J〉 value of a dipeptide subunit in the tripeptides. Also in some cases, direct steric interaction between the two side chains in the two adjacent dipeptide subunits in the tripeptide affects the potential energy surfaces of the individual dipeptide subunits and hence the 〈J〉 values. The influence of the structural characteristics of the side chains of individual amino acids on structure formation at or beyond the dipeptide level is discussed at various points. The J(NH? ^αCH) values of CH₃CONH? CHR? CONH₂ and CH₃CONH? CHR? CO₂CH₃ with the same R are quite different for R = valine, leucine, phenylalanine, methionine, but equal for R = glycine. This, coupled with the fact that one of the carboxamide NH resonances has a chemical shift different from its counterpart in simple amides like CH₃CONH₂ and the other carboxamide NH has the same chemical shift as its counterpart in CH₃CONH₂, suggest the presence of a hydrogen bond in dipeptide CH₃CONH? CHR? CONH₂ with carboxamide NH as the donor. Theoretical evidence for two seven-membered hydrogen-bonded rings with the carboxamide NH as donor and the acetyl oxygen as acceptor is summarized. Our data cannot suggest the number of such hydrogen-bonded rings, nor can they conclude the relative proportion of these rings in a particular dipeptide. A discussion of the difficulty of interpretation is presented and the data are discussed under certain simplifying assumptions.     

Measurement of diffusion coefficients of meningococcal polysaccharide by optical mixing spectroscopy. I. A preliminary characterization on the aggregation of the group C polysaccharide

The angular distribution of scattered intensity and decay times of concentration fluctuations have been measured by means of digital photon counting and single-clipped photon correlation for solutions of Group C meningococcal polysaccharides at 31°C. The z-average diffusion coefficient 〈D〉_z and its second moment 〈D²〉_z have been determined from the time-dependent correlation function using the cumulant expansion technique. Very low observed values of 〈D〉_z and the tremendous width of the polydispersity index, which is the z-average normalized variance, suggest a higher degree of aggregation than the monomer–dimer type self-association at finite concentrations.     

Analysis of derivative binding isotherms. Theoretical considerations

The basic theoretical groundwork for the use of derivative binding isotherms in the analysis of ligand binding is presented. The derivative binding isotherm is defined as Γ (Y) = df/dy where f = fractional degree of saturation and y = natural logarithm of the free ligand concentration. Since Γ (y) is a positive function, which goes to zero as y → ±∞, the mean value of y, 〈y〉, and the second and third moments, μ₂ and μ₃ about 〈y〉 are well defined. For a macromolecular system consisting of N equivalent and independent binding sites, Γ (y) is a symmetrical bell-shaped function with one maximum. The maximum occurs when y = ?ln K_assoc; μ₂ = π²/3, and μ₃ = 0. For multiple sets of independent binding sites, Γ (y) is a superposition of Γ-type functions. If the sets are sufficiently well separated in binding free energy, multiple extrema may be seen at positions corresponding to the logarithms of the dissociation constants for the individual sets. In any case, 〈y〉 is equal to the mean value of the logarithms of the dissociation constants for the sets; μ₂ > π²/3 and equal to π²/3 plus the variance of the logarithms of the dissociation constants about their mean value; and μ₃ is, except by coincidence, not equal to zero and equals the third moment of the distribution of logarithms of the dissociation constants about their mean value. Analysis of Γ(y) for the case of cooperative interactions within a set of binding sites was investigated by examining (1) the Hill model (whose mathematical representation is equivalent to that used to describe antibody heterogeneity except that in the latter case the parameter a, the Sips, constant, is constrained (0 < a ≤1);(2) a common model for cooperativity in which the cooperative free energy is a linear function of the fraction bound; and (3) a general representation of cooperative interactions within a set of sites in terms of ?(f), a smooth function that gives the interaction free energy in units of RT. For the Hill model (or Sips model) Γ(y) is a symmetrical function with one maximum at y = (?1)/a)lnK, μ₂ = π²/3a²; and μ₃ = 0. For the case in which the cooperative free energy is a linear function of f [?(f) = cf], 〈y〉 = ?ln K₀ + (c/2); μ₂ = (π²/3) + c[(c/12) + 1] where c > ?4; and μ₃ = 0. General expressions for the moments in terms of ?(f) are derived. In general, μ₂ < (π²/3) for positive cooperativity and μ₂ > (π₂/3) for negative for negative cooperativity. Γ(y) will be symmetrical if and only if the cooperative free energy is introduced symmetrically about f = 0.5.     

Interpretation of energy-transfer experiments by theoretical studies of model compounds using semiempirical potential functions. I. Three-linked aromatic peptide units

It is shown that theoretical conformational analysis, based on the evaluation of semiempirical potential functions, can be used to compute the quantities relevant to the interpretation of energy-transfer experiments. The relevant properties are computed for a segment of a polypeptide chain with the sequence Tyr-Tyr. In particular, the average value of the orientation factor 〈κ²〉 and its distribution ?(κ²) are examined. It appears that the degrees of freedom for rotation of the side chains are not sufficient to randomize completely the orientation factor of the transition dipoles. Two additional degrees of freedom, namely the torsion angles around the valence bonds of the backbone, ψ₁ and ?₂, bring 〈κ²〉 close to the value that corresponds to randomly oriented transition dipoles.     

Solution properties and chain flexibility of pullulan in aqueous solution

Molecular characteristics for pullulan, a polysaccharide produced by a fungus Aureobasidium pullulans, were measured by light scattering, viscometry, and gel-permeation chromatography. From the experimental data the Mark-Houwink-Sakurada viscosity equation in water at 25°C was determined for samples having the molecular weight M ranging from 48 × 10³ to 2.18 × 10⁶ g mol^?1 as [η] = (1.91 ± 0.02) × 10^?2M_w^0.67±0.01 (in cm³ g^?1); and as molecular weight decreased, the slope of the viscosity equation decreased, although the molecular weight values below 30 × 10³ g mol^?1 evaluated by gel-permeation chromatography were somewhat unreliable. The unperturbed dimensions 〈R²〉₀^1/2 of pullulan were estimated by determining the expansion factor α_s, from the theoretical combination of theories for the interpenetration function Ψ and those for α_s. The 〈R²〉₀/M value estimated from this procedure in 6.7 × 10^?17 cm² mol g^?1. We concluded that the polysaccharide chain that is linked by the α-1,6-glucosidic linkage behaves like a flexible chain in aqueous solution.     

Fluctuations of an alpha-helix.

Fluctuations in backbone dihedral angles in the α-helical conformation of homopolypeptides are studied based on an assumption that the conformational energy function of a polypeptide consisting of n amino-acid residues can be approximated by a 2n-dimensional parabola around the minimum point in the range of fluctuations. A formula is derived that relates 〈Δθ_iΔθ_j〉, the mean value of the product of deviations of dihedral angles ?_i and ψ_i (collectively designated by θ_i) from their energy minimum values, with a matrix inverse to the second derivative matrix F ,_n of the conformational energy function at the minimum point. A method of calculating the inverse matrix F _n^?1 explicitly is given. The method is applied to calculating 〈Δθ_iΔθ_j〉 for the α-helices of poly(L -alanine) and polyglycine. The autocorrelations 〈(Δ?_i)²〉 and 〈(Δψ_i)²〉 at 300°K are found to be about 66 deg² and 49 deg², respectively, for poly(L -alanine), and 84 deg² and 116 deg², respectively, for polyglycine. The length of correlations of fluctuations along the chain is found for both polypeptides to be about eight residues long.     

Solvent Effects on the Shape of a Tetramer

Abstract

The solvent effect on the shape of a tetramer with increasing temperature is analyzed. For this purpose models of an isolated chain and a chain immersed in a solvent have been simulated by Molecular Dynamics. A solvent model represented by stochastic forces has been tested against molecular dynamics results. The behaviour of the mean-square end-to-end distance 〈R ²〉 and 〈l ₁ ³ S ²〉 with increasing temperature are shown. where l ₁ is the longest eigenvalue of the moment of inertia tensor and S is the radius of gyration. All the chain models studied show different behaviour of these quantities at low temperature compared to high temperature where the shape of the tetramer is temperature insensitive. The main solvent effect is to pospone the transition to higher temperature. The stochastic solvent model qualitatively agrees with molecular dynamics results.     

Evolvability and Single-Genotype Fluctuation in Phenotypic Properties: A Simple Heteropolymer Model

Experiment showed that the response of a genotype to mutation, i.e., the magnitude of mutational change in a phenotypic property, can be correlated with the extent of phenotypic fluctuation among genetic clones. To address a possible statistical mechanical basis for such phenomena at the protein level, we consider a simple hydrophobic-polar lattice protein-chain model with an exhaustive mapping between sequence (genotype) and conformational (phenotype) spaces. Using squared end-to-end distance, R_N², as an example conformational property, we study how the thermal fluctuation of a sequence's R_N² may be predictive of the changes in the Boltzmann average 〈R_N²〉 caused by single-point mutations on that sequence. We found that sequences with the same ground-state (R_N²)₀ exhibit a funnel-like organization under conditions favorable to chain collapse or folding: fluctuation (standard deviation σ) of R_N² tends to increase with mutational distance from a prototype sequence whose 〈R_N²〉 deviates little from its (R_N²)₀. In general, large mutational decreases in 〈R_N²〉 or in σ are only possible for some, though not all, sequences with large σ values. This finding suggests that single-genotype phenotypic fluctuation is a necessary, though not sufficient, indicator of evolvability toward genotypes with less phenotypic fluctuations.     

Configuration-dependent properties of randomly coiling polynucleotide chains. I. A comparison of theoretical energy estimates

Various theoretical estimates of the conformational energy associated with polynucleotides in solution have been compared with each other and also with the experimentally observed conformations found in X-ray crystallographic investigations of low-molecular-weight nucleic acid analogs. In view of the disparities between these data, certain configuration-dependent properties (i.e., the mean-square unperturbed end-to-end distance 〈r²〉₀ and the average vicinal nmr coupling constant 〈J〉) appropriate to randomly coiling polynucleotides described by either the energy estimates or by the crystallographically preferred conformations have also been calculated and compared with the known solution behavior of polynucleotide chains. Both the theoretical energy surfaces and the X-ray data show good agreement with the nmr coupling constant indications of the preferred rotations about the O-C and C-C bonds of the chain backbone. The principal discrepancies between the theoretical methods and X-ray data arise in their ability to predict successfully the preferred rotations about the two phosphodiester bonds of the chain backbone and the unperturbed dimensions of randomly coiling polynucleotide chains.     

Human Microtubule-Associated-Protein Tau Regulates the Number of Protofilaments in Microtubules: A Synchrotron X-Ray Scattering Study

Microtubules (MTs), a major component of the eukaryotic cytoskeleton, are 25 nm protein nanotubes with walls comprised of assembled protofilaments built from αβ heterodimeric tubulin. In neural cells, different isoforms of the microtubule-associated-protein (MAP) tau regulate tubulin assembly and MT stability. Using synchrotron small angle x-ray scattering (SAXS), we have examined the effects of all six naturally occurring central nervous system tau isoforms on the assembly structure of taxol-stabilized MTs. Most notably, we found that tau regulates the distribution of protofilament numbers in MTs as reflected in the observed increase in the average radius 〈R^MT〉 of MTs with increasing Φ, the tau/tubulin-dimer molar ratio. Within experimental scatter, the change in 〈R^MT〉 seems to be isoform independent. Significantly, 〈R^MT〉 was observed to rapidly increase for 0 < Φ < 0.2 and saturate for Φ between 0.2-0.5. Thus, a local shape distortion of the tubulin dimer on tau binding, at coverages much less than a monolayer, is spread collectively over many dimers on the scale of protofilaments. This implies that tau regulates the shape of protofilaments and thus the spontaneous curvature C_o^MT of MTs leading to changes in the curvature C^MT (=1/R^MT). An important biological implication of these findings is a possible allosteric role for tau where the tau-induced shape changes of the MT surface may effect the MT binding activity of other MAPs present in neurons. Furthermore, the results, which provide insight into the regulation of the elastic properties of MTs by tau, may also impact biomaterials applications requiring radial size-controlled nanotubes.     

Characterization of group C meningococcal polysaccharide by light-scattering spectroscopy. III. Determination of molecular weight, radius of gyration, and translational diffusional coefficient.

Group-specific polysaccharides isolated by means of a cetavlon procedure are immunogenic in man and induce protective immunity against meningococcal meningitis. Minute quantities of the polymers in solution can act as vaccines. We now report the first characterization of a fractionated (C-1) group C polysaccharide in 0.4KM KCl and 0.05M sodium acetate by means of light-scattering spectroscopy. Independent measurements of refractive index increments, absolute scattered intensities, angular scattering intensities and line widths as a function of scattering angles and delay times at different concentrations using incident wavelengths of 632.8 nm from a He–Ne laser and of 488 nm from an argon–ion laser yield information on aggregation properties, molecular weight (M_r), radius of gyration 〈r⁰_g〉^1/2_z, translational diffusion coefficient 〈D〉⁰_z, and second virial coefficients A₂ and B₂ of C-1 polysaccharide. At relatively high ionic strength (0.04M KCl + 0.05M sodium acetate), we obtain for the C-1 polysaccharide in solution M_r = 5.15 × 10⁵, 〈r²_g〉^1/2_z = 345 Å, A₂ = 1.25 × 10^?4 ml/g, 〈D〉 = 1.16 × 10^?7 cm²/sec with a corresponding Stokes radius of 240 Å and B₂ = 4.4 ml/g. A₂ and B₂ are the second virial coefficients from intensity- and diffusion-coefficient measurements. The C-1 polysaccharide aggregates in solution and behaves hydrodynamically like random coils. Viscosity and sedimentation studies further confirm our conclusions that the fractioned C-1 polysaccharide aggregates in solution and EDTA can partially break up those aggregates. However, the system remains polydisperse even after adding an excess amount of EDTA. The weight-average molecular weight of the C-1 polysaccharide in solution depends upon ionic strength and exhibits a minimum at ～0.2M KCl. Finally, viscosity, light-scattering, and sedimentation results all show that the aggregated macromolecular system behaves like random-coiled polymers with no measurable shape factors.     

Intrinsic viscosities of cyclic and linear lamda DNA

The ratio of the intrinsic viscosities of the linear and circular forms of λ DNA, [η]L /[η]c, has been measured as a function of ionic strength in the range [Na⁺] = 0.6. M–0.03MCorrections were made for the presence of uncyclizable linear contaminant in circular preparations. By combining data in the literature on the ionic strength dependence of linear DNA of various molecular weights with that obtained here, it was possible to determine the expansion parameter εL as a function of [Na⁺]. εL is defined by the relation 〈L²〉 = b²N^1+εL, where 〈L¹〉 is the mean-square end-to-end distance of a chain of N segments of length b. The empirical relation εL = 0.05 ? 0.11 log [Na⁺] for native NaDNA at 25°C is found. When εL = 0, [η]L /[η]c extrapolates to 1.6, in good agreement with the theoretical prediction of 1.55. As εL increases, [η]L /[η]c increases, in agreement with a theory of Bloomfield and Zimm.