Equivalence of two Fourier methods for biological sequences

 Two methods for defining Fourier power spectra for DNA sequences or other biological sequences are compared. The first method uses indicator sequences for each letter. The second method by Silverman and Linsker assigns to each letter a vertex of a regular tetrahedron in space, and this can be generalized to any dimension. While giving different Fourier transforms, it is shown that the power spectra of the two methods are essentially the same. This is also true if one replaces the Fourier transform in both methods with another linear transform, such as the Walsh transform. Received 4 December 1995     

Biophysical approaches to membrane protein structure determination.

Recently, there have been several technical advances in the use of solution and solid-state NMR spectroscopy to determine the structures of membrane proteins. The structures of several isolated transmembrane (TM) helices and pairs of TM helices have been solved by solution NMR methods. Similarly, the complete folds of two TM beta-barrel proteins with molecular weights of 16 and 19 kDa have been determined by solution NMR in detergent micelles. Solution NMR has also provided a first glimpse at the dynamics of an integral membrane protein. Structures of individual TM helices have also been determined by solid-state NMR. A combination of NMR with site-directed spin-label electron paramagnetic resonance or Fourier transform IR spectroscopy allows one to assemble quite detailed protein structures in the membrane.     

The effect of boundary conditions on epicardial potential distributions

This study presents a comparison of semi-analytical and numerical solution techniques for solving the passive bidomain equation in simple tissue geometries containing a region of subendocardial ischaemia. When the semi-analytical solution is based on Fourier transforms, recovering the solution from the frequency domain via fast Fourier transforms imposes a periodic boundary condition on the solution of the partial differential equation. On the other hand, the numerical solution uses an insulation boundary condition. When these techniques are applied to calculate the epicardial surface potentials, both yield a three well potential distribution which is identical if fibre rotation within the tissue is ignored. However, when fibre rotation is included, the resulting three-well distribution rotates, but through different angles, depending on the solution method. A quantitative comparison between the semi-analytical and numerical solutiontechniques is presented in terms of the effect fibre rotation has on the rotation of the epicardial potential distribution. It turns out that the Fourier transform approach predicts a larger rotation of the epicardial potential distribution than the numerical solution. The conclusion from this study is that it is not always possible to use analytical or semi-analytical solutions to check the accuracy of numerical solution procedures. For the problem considered here, this checking is only possible when it is assumed that there is no fibre rotation through the tissue.     

Vibrational circular dichroism spectra of three conformationally distinct states and an unordered state of poly(L-lysine) in deuterated aqueous solution

Fourier transform ir vibrational circular dichroism (VCD) spectra in the amide I′ region of poly(L-lysine) in D₂O solutions have confirmed the existence of three distinct conformational states and an unordered conformational state in this homopolypeptide. Characteristic VCD spectra are presented for the right-handed α-helix, the antiparallel β-sheet, an extended helix conformation previously referred to as the so-called “random coil,” and a completely unordered conformation characterized by the absence of any amide I′ VCD. VCD for the antiparallel β-sheet in solution and the unordered chain conformation are presented for the first time. Each of the four different VCD spectra is unique in appearance and lends weight to the view that VCD has the potential to become a sensitive new probe of the secondary structure of proteins in solution.     

Characterization of the stable, acid-induced, molten globule-like state of staphylococcal nuclease.

Titration of a salt-free solution of native staphylococcal nuclease by HCl leads to an unfolding transition in the vicinity of pH 4, as determined by near- and far-UV circular dichroism. At pH 2-3, the protein is substantially unfolded. The addition of further HCl results in a second transition, this one to a more structured species (the A state) with the properties of an expanded molten globule, namely substantial secondary structure, little or no tertiary structure, relatively compact size as determined by hydrodynamic radius, and the ability to bind the hydrophobic dye 1-anilino-8-naphthalene sulfonic acid. The addition of anions, in the form of neutral salts, to the acid-unfolded state at pH 2 also causes a transition leading to the A state. Fourier transform infrared analysis of the amide I band was used to compare the amount and type of secondary structure in the native and A states. A significant decrease in alpha-helix structure, with a corresponding increase in beta or extended structure, was observed in the A state, compared to the native state. A model to account for such compact denatured states is proposed.     

Infrared spectroscopic characterization of the structural changes connected with the E1----E2 transition in the Ca2+-ATPase of sarcoplasmic reticulum

The Ca2+-transporting ATPase (EC 3.6.1.38) of sarcoplasmic reticulum alternates between several conformational states during ATP-dependent Ca2+ transport. The E1 conformation is stabilized by 0.1 mM Ca2+ and the E2 conformation by vanadate in a Ca2+-free medium. Fourier transform infrared spectroscopy reveals significant differences between the two states that indicate differences in the protein secondary structure. The two states and the corresponding spectra can be interconverted reversibly by changing the Ca2+ concentration of the medium. The infrared spectral changes indicate the appearance of a new alpha-helical substructure connected with the E1----E2 conversion accompanied by small changes in beta-turns, while the beta-sheet content remains essentially unchanged. There are also differences between the E1 and E2 states in the C = O stretching vibrations of the ester carbonyl groups of phospholipids in intact sarcoplasmic reticulum that are not observed under identical conditions in isolated sarcoplasmic reticulum lipid dispersions. These observations imply an effect of proteins on the structure of the interfacial regions of the phospholipids that is dependent on the conformational state of the Ca2+-ATPase. The CH2- and CH3-stretching frequencies of the membrane lipids are not affected significantly by the E1----E2 transition. The Fourier transform infrared spectra of sarcoplasmic reticulum vesicles in the presence of 20 mM Ca2+ suggest the stabilization of a protein conformation similar to the E2 state except for differences in the behavior of COO- and phospholipid ester C = O groups that may reflect charge effects of the bound Ca2+.     

Two-dimensional Fourier representation used in the bioelectric forward problem.

The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.     

A method for two-dimensional registration and construction of the two-dimensional atlas of gene expression patterns in situ

We apply the fast redundant dyadic wavelet transform to the spatial registration of two-dimensional gene expression patterns of 736 Drosophila melanogaster embryos. This method is superior to the Fourier transform or windowed Fourier transform because of its ability to reduce noise and is of high resolution. In registration of the dataset we use two cost functions based on computing the Euclidean or Mahalanobis distance. The algorithm shows a high level of accuracy. For early temporal classes the cost function based on Mahalanobis distance gives better results. We have reported a method for construction of an integrated dataset elsewhere. In this paper the method is extended to the two-dimensional case. The procedure for data assembly provides for the preservation of some aspects of the nuclear structure of a two-dimensional gene expression pattern. It is based on creating an averaged model that reproduces the spatial distribution of nuclei over the embryo image. The average concentrations of each protein in each averaged nucleus are computed from the series of embryos of the same age.     

Structure and dynamics of lipid-associated states of apocytochrome c.

Apocytochrome c (apocyt c), which in aqueous solution is largely unstructured, acquires an alpha-helical conformation upon association with lipid membranes. The extent of alpha-helix induced in apocyt c is lipid-dependent and this folding process is driven by both electrostatic and hydrophobic lipid-protein interactions. The structural and dynamic properties of apocyt c in lipid membranes were investigated by attenuated total reflection Fourier transform infrared spectroscopy combined with amide H-D exchange kinetics. Apocyt c acquires a higher content of alpha-helical structure with negatively charged membranes than with zwitterionic ones. For all membranes studied here, the helices of these partially folded states of apocyt c have a preferential orientation perpendicular to the plane of the lipid membrane. The H-D exchange revealed that a small fraction of amide protons of apocyt c, possibly associated with a stable folded domain protected by the lipid, remained protected from exchange over 20 min. However, a large fraction of amide protons exchanged in less than 20 min, indicating that the helical states of apocyt c in lipid membranes are very dynamic.     

Isolation and partial purification of cytochrome P-450 from uninduced rat liver

Carbon-13 Fourier transform nmr has been used to make the first observation of a carbon-13-iron-57 spin-spin coupling constant in a protein, sperm whale carbonyl myoglobin enriched to 90% in both iron-57 and carbon-13. The coupling constant, 27.1±0.2 Hz, is found to be essentially identical to that of a model compound, supporting the view that the carbonyl is not tilted with respect to the heme plane in solution. Such carbon-13-iron-57 couplings, and the resultant iron-57 chemical shifts obtained from decoupling experiments, should provide valuable new tools for studying the different affinity states of tetrameric hemoglobins.     

Spectroelectrochemical characterization of the [NiFe] hydrogenase of Desulfovibrio vulgaris Miyazaki F

The active site in the [NiFe] hydrogenase of Desulfovibrio vulgaris Miyazaki F has been investigated by Fourier transform infrared (FTIR) spectroscopy. Analysis of the spectra allowed the three diatomic inorganic ligands to Fe in this enzyme to be identified as one CO molecule and two CN(-) molecules. Furthermore, pH-dependent redox titrations were performed to determine the midpoint potentials as well as the pK value of the respective reactions and revealed that each single-electron redox transition is accompanied by a single-proton transfer step. The comparison of these spectra with those published for other [NiFe] hydrogenases shows that the electronic structure of the active sites of these enzymes and their redox processes are essentially the same. Nevertheless, differences with respect to the frequency of the CO band and the pH dependence of the Ni-R states have been observed. Finally, the frequency shifts of the bands in the IR spectra were interpreted with respect to the electronic configuration of the redox intermediates in the catalytic cycle.     

Optical spectroscopic differentiation of various equilibrium denatured states of horse cytochrome c

Thermal unfolding of cytochrome c (cyt c) from several states has been studied using equilibrium spectroscopic techniques. CD in the uv, vibrational circular dichroism, infrared, and uv-vis absorption spectra measured at various temperatures, pHs, salt concentrations, and GuHCl concentrations are used to show the conformational as well as heme structural differences between native and various denatured states. The difference in thermal denaturation behaviors of cyt c starting from acid denatured, molten globule (MG), and the A and native states are explored. Different final high temperature states were observed for cytochrome c unfolding from four different initial states (native, MG, A, and acid denatured state) by electronic CD, Fourier transform infrared (FTIR), and vibrational CD (VCD). Consistent with this, different thermal unfolding pathways for the MG and A states are suggested by the FTIR and VCD data for this process.     

A Hybrid Technique for the Periodicity Characterization of Genomic Sequence Data

Many studies of biological sequence data have examined sequence structure in terms of periodicity, and various methods for measuring periodicity have been suggested for this purpose. This paper compares two such methods, autocorrelation and the Fourier transform, using synthetic periodic sequences, and explains the differences in periodicity estimates produced by each. A hybrid autocorrelation—integer period discrete Fourier transform is proposed that combines the advantages of both techniques. Collectively, this representation and a recently proposed variant on the discrete Fourier transform offer alternatives to the widely used autocorrelation for the periodicity characterization of sequence data. Finally, these methods are compared for various tetramers of interest in C. elegans chromosome I.     

On the computation of molecular surface correlations for protein docking using fourier techniques

The computation of surface correlations using a variety of molecular models has been applied to the unbound protein docking problem. Because of the computational complexity involved in examining all possible molecular orientations, the fast Fourier transform (FFT) (a fast numerical implementation of the discrete Fourier transform (DFT)) is generally applied to minimize the number of calculations. This approach is rooted in the convolution theorem which allows one to inverse transform the product of two DFTs in order to perform the correlation calculation. However, such a DFT calculation results in a cyclic or "circular" correlation which, in general, does not lead to the same result as the linear correlation desired for the docking problem. In this work, we provide computational bounds for constructing molecular models used in the molecular surface correlation problem. The derived bounds are then shown to be consistent with various intuitive guidelines previously reported in the protein docking literature. Finally, these bounds are applied to different molecular models in order to investigate their effect on the correlation calculation.     

Frequency analysis of infrared absorption and vibrational circular dichroism of proteins in D2O solution.

The IR absorption frequencies as derived from second derivatives of the Fourier transform IR spectra of the amide I' bands of globular proteins in D2O are compared to those obtained from band fitting of the vibrational circular dichroism (VCD) spectra. The two sets of frequencies are in very good agreement, yielding consistent ranges where amide I' VCD and IR features occur. Use of VCD to complement the IR allows one to add sign information to the frequency information so that features occurring in the overlapping frequency ranges that might arise from different secondary structures can be better discriminated. From this comparison, it is clear that correlation just of the frequency of a given IR transition to secondary structure can lead to a nonunique solution. Different sign patterns were identified for correlated groups of globular proteins in restricted frequency ranges that have been previously assigned to defined secondary structural elements. Hence, different secondary structural elements must contribute band components to a given frequency range.     

Fourier transform coupled tryptophan scanning mutagenesis identifies a bending point on the lipid-exposed δM3 transmembrane domain of the Torpedo californica nicotinic acetylcholine receptor

The nicotinic acetylcholine receptor (nAChR) is a member of a family of ligand-gated ion channels that mediate diverse physiological functions, including fast synaptic transmission along the peripheral and central nervous systems. Several studies have made significant advances toward determining the structure and dynamics of the lipid-exposed domains of the nAChR. However, a high-resolution atomic structure of the nAChR still remains elusive. In this study, we extended the Fourier transform coupled tryptophan scanning mutagenesis (FT-TrpScanM) approach to gain insight into the secondary structure of the δM3 transmembrane domain of the Torpedo californica nAChR, to monitor conformational changes experienced by this domain during channel gating, and to identify which lipid-exposed positions are linked to the regulation of ion channel kinetics. The perturbations produced by periodic tryptophan substitutions along the δM3 transmembrane domain were characterized by two-electrode voltage clamp and (125)I-labeled α-bungarotoxin binding assays. The periodicity profiles and Fourier transform spectra of this domain revealed similar helical structures for the closed- and open-channel states. However, changes in the oscillation patterns observed between positions Val-299 and Val-304 during transition between the closed- and open-channel states can be explained by the structural effects caused by the presence of a bending point introduced by a Thr-Gly motif at positions 300-301. The changes in periodicity and localization of residues between the closed-and open-channel states could indicate a structural transition between helix types in this segment of the domain. Overall, the data further demonstrate a functional link between the lipid-exposed transmembrane domain and the nAChR gating machinery.     

Chicken liver bile acid-binding protein is in a compact partly folded state at acidic pH. Its relevance to the interaction with lipid membranes

Chicken liver bile acid-binding protein (formerly known as chicken liver basic fatty acid-binding protein) binds to anionic lipid membranes acquiring a partly folded state [Nolan, V., Perduca, M., Monaco, H., Maggio, B., and Montich, G. (2003) Biochim. Biophys. Acta 1611, 98-106]. To understand the mechanisms of its interactions with membranes, we have investigated the presence of partly folded states in solution. Using fluorescence spectroscopy of the single Trp residue, circular dichroism in the far- and near-UV, Fourier transform infrared spectroscopy, and size-exclusion chromatography, we found that L-BABP was partly unfolded at pH 2.5 and low ionic strength, retaining some of its secondary structure. Addition of 0.1 M NaCl at pH 2.5 or decreasing the pH to 1.5 produced a more compact partly folded state, with a partial increase of secondary structure and none of tertiary structure. Fluorescence emission spectra of this state indicate that the Trp residue is within an environment of low polarity, similar to the native state. This environment is not produced by the insertion of the Trp into soluble aggregates as revealed by size-exclusion chromatography, fluorescence anisotropy, and infrared spectroscopy. The presence of partly folded states under acidic conditions in solution suggests the possibility that membrane binding of L-BABP occurs via this state.     

Nucleation and growth of insulin fibrils in bulk solution and at hydrophobic polystyrene surfaces

A technique was developed for studying the nucleation and growth of fibrillar protein aggregates. Fourier transform infrared and attenuated total reflection spectroscopy were used to measure changes in the intermolecular beta-sheet content of bovine pancreatic insulin in bulk solution and on model polystyrene (PS) surfaces at pH 1. The kinetics of beta-sheet formation were shown to evolve in two stages. Combined Fourier transform infrared, dynamic light scattering, atomic force microscopy, and thioflavin-T fluorescence measurements confirmed that the first stage in the kinetics was related to the formation of nonfibrillar aggregates that have a radius of 13 +/- 1 nm. The second stage was found to be associated with the growth of insulin fibrils. The beta-sheet kinetics in this second stage were used to determine the nucleation and growth rates of fibrils over a range of temperatures between 60 degrees C and 80 degrees C. The nucleation and growth rates were shown to display Arrhenius kinetics, and the associated energy barriers were extracted for fibrils formed in bulk solution and at PS surfaces. These experiments showed that fibrils are nucleated more quickly in the presence of hydrophobic PS surfaces but that the corresponding fibril growth rates decrease. These observations are interpreted in terms of the differences in the attempt frequencies and energy barriers associated with the nucleation and growth of fibrils. They are also discussed in the context of differences in protein concentration, mobility, and conformational and colloidal stability that exist between insulin molecules in bulk solution and those that are localized at hydrophobic PS interfaces.