The pair-functional method for direct solution of molecular structures. I. Statistical principles.

The pair-functional principle shows how to construct a unique statistical ensemble of strongly interacting atoms that corresponds to any feasible measured set of X-ray intensities. The ensemble and all its distribution functions are strictly periodic in the crystal lattice, so that each unit cell has exactly the same arrangement of atoms at all times. The mean particle density in the cell is uniform because the ensemble has undefined phases and the origin is not fixed. The atoms in this maximum-entropy ensemble interact through pairwise additive periodic statistical forces within the unit cell. The ensemble average pair-correlation function is matched to the observed originless Patterson function of the crystal. The derived pairing force then becomes approximately proportional to the Ornstein-Zernicke direct correlation function of the ensemble. The atoms have a many-body Boltzmann distribution and the logarithm of the likelihood of any particular conformation is related to its total pairing potential. The pairing potential of a group of atoms acts like a local field in the cell. This property is used in the pair-functional method. Molecular structures can be solved by a direct search in real space for clusters of atoms with high pair potentials. During a successful search, the atoms move from their original random positions to form larger and larger clusters of correctly formed fragments. Finally, every atom belongs to a single cluster, which is the correct solution.

The pair-functional method for direct solution of molecular structures. II. Small-molecule tests.

The new pair-functional direct method has been implemented and tested. Like the Patterson function, the pairing force has valuable imaging properties at high resolution. Two simple iterative algorithms were designed to refine on the total pair potential and the normalized intensity correlation coefficient of an atomic model. The first algorithm is a peak-picking method which selects the best-paired high peaks from a density map and then uses the strong reflections to generate a new Fourier filtered map. The second algorithm, the pair-and-square method, uses a tangent formula step instead of the Fourier and is a little more efficient. Computational experiments on a point-atom grid model, with perfect data, reached exact ab initio solutions for up to 600 atoms. Point-atom models were also solved by searching for reduced structures that contained as few as one quarter of the atoms. Seeded searches, guided by a small known fragment, solved up to 30000 atoms on the grid. Realistic tests on actual molecules showed that Sheldrick's [Acta Cryst. (1990), A46, 467-473] test structures of 50-200 atoms can be solved under a variety of conditions.

The pair-functional method. III. The pairing forces.

The theory of the pair-functional ensemble is developed to provide estimates of the pairing forces from experimental X-ray intensities. The statistical mechanics of the grand ensemble leads to a diagram expansion for the forces, in terms of the direct correlation function of the fluid ensemble combined with a series of small higher-order corrections. A simpler treatment, based on a biased Gaussian probability distribution, gives approximate formulae, valid for reflections of any type in all space groups. The role of symmetry is analysed. The entropy of an asymmetrical ensemble can always be increased by averaging it over equivalent positions of the atoms in the true space group, with the result that the atoms naturally tend to adopt the highest symmetry compatible with the data. In a cell with different types of atom, the atoms experience a single force function but they interact with a strength proportional to the products of their scattering factors. Numerical estimates are given for typical cases.

Analysis of repeated motifs in the talin rod.

The amino acid sequence of the rod portion of talin contains strong periodic patterns with a long period of 32 to 34 residues superimposed on short periods of 7 and 7/2 residues. The rod includes 50 to 60 copies of an irregular repeated motif approximately 34 residues long. The motif itself consists of three sections: a short "leader" segment of about six residues, which has a high proportion of the prolines and acidic residues; a relatively well-conserved hydrophobic "core" pattern of approximately 21 residues; and a highly variable "linker" region of seven residues which joins onto the next leader. The core section sequence has many of the characteristics of an amphipathic helix. The extensive hydrophobic side of this postulated helix has a characteristic surface pattern of large and small hydrophobic residues (mainly Leu and Ala), with a strong periodicity of seven residues. It also has a narrow hydrophilic edge with a highly variable sequence. The core sequence is unlike either a normal helical coiled coil or a leucine zipper, because it contains several helical ridges and grooves. The helical cores probably form a tightly packed hydrophobic central strand for the fibrous tail. The leader and linker sections are highly variable in length, so that the spacing between the starting points of adjacent cores varies between 20 and 40 residues. The most common spacing is 34, and many spacings are close to this length.

Entropy phase dynamics.

The entropy-dynamics method seeks maxima for the entropy of the electron density for N atoms in a crystal cell, when the Fourier amplitudes are fixed, but their phases are unknown. By analogy with molecular dynamics, the effective potential energy is the negative entropy V = -NS. The kinetic energy is proportional to the squared velocities of the electron densities at grid points in the map. It reduces to a sum of Fourier-mode rotor energies. Each rotor angle experiences a couple equal to the phase gradient of S, and local dynamical equilibrium yields a Boltzmann distribution of S. Discrete phase angles (e.g. signs) are treated as quantized rotor modes. The distributions depend on a popularity function of the entropy histogram. Trial calculations have been made of phase averages and correlations in a centrosymmetric projection of the membrane protein bacteriorhodopsin. The maximum-entropy solution and the correct solution do not always coincide.

Secondary structure-based profiles: use of structure-conserving scoring tables in searching protein sequence databases for structural similarities.

The profile method, for detecting distantly related proteins by sequence comparison, has been extended to incorporate secondary structure information from known X-ray structures. The sequence of a known structure is aligned to sequences of other members of a given folding class. From the known structure, the secondary structure (alpha-helix, beta-strand or "other") is assigned to each position of the aligned sequences. As in the standard profile method, a position-dependent scoring table, termed a profile, is calculated from the aligned sequences. However, rather than using the standard Dayhoff mutation table in calculating the profile, we use distinct amino acid mutation tables for residues in alpha-helices, beta-strands or other secondary structures to calculate the profile. In addition, we also distinguish between internal and external residues. With this new secondary structure-based profile method, we created a profile for eight-stranded, antiparallel beta barrels of the insecticyanin folding class. It is based on the sequences of retinol-binding protein, insecticyanin and beta-lactoglobulin. Scanning the sequence database with this profile, it was possible to detect the sequence of avidin. The structure of streptavidin is known, and it appears to be distantly related to the antiparallel beta barrels. Also detected is the sequence of complement component C8, which we therefore predict to be a member of this folding class.

Paramyosin gene (unc-15) of Caenorhabditis elegans. Molecular cloning, nucleotide sequence and models for thick filament structure.

Paramyosin is a major structural component of thick filaments isolated from many invertebrate muscles. The Caenorhabditis elegans paramyosin gene (unc-15) was identified by screening with specific antibodies an "exon-expression" library containing lacZ/nematode gene fusions. Short probes recovered from the library were used to identify bacteriophage lambda and cosmid clones that encompass the entire paramyosin (unc-15) gene. From these clones, numerous subclones containing epitopes reacting with anti-paramyosin sera were obtained, providing strong evidence that the initial cloned fragment was, in fact, derived from the structural gene for paramyosin. The complete nucleotide sequence of a 12 x 10(3) base-pair region spanning the gene was obtained. The gene is composed of ten short exons encoding a protein of 866 [corrected] amino acid residues. Paramyosin is highly similar to residues 267 to 1089 of myosin heavy chain rods. For most of its length, paramyosin appears to form an alpha-helical coiled-coil and shows the expected heptad repeat of hydrophobic amino acid residues and the 28-residue repeat of charged amino acids characteristic of myosin heavy chain rods. However, paramyosin differs from myosin in having non-helical extensions at both the N and C termini and an additional "skip" residue that interrupts the 28-residue repeat. The distribution of charges along the length of the paramyosin rod is also significantly different from that of myosin heavy chain rods. Potential charge-mediated interactions between paramyosin rods and between paramyosin and myosin rods were calculated using a model successfully applied previously to the analysis of the myosin rod sequences. Myosin rods aligned in parallel show optimal charge-charge interactions at multiples of 98 residue staggers (i.e. at axial displacements of multiples of 143 A). Paramyosin rods, in contrast, appear to interact optimally at parallel staggers of 493 residues (i.e. at axial displacements of 720 A) but show only weak interaction peaks at 98 or 296 residues. Similar calculations suggest optimal interactions between paramyosin molecules and myosin rods and in their anti-parallel alignments. The implications of these results for the structure of the bare zone and the assembly of nematode thick filaments are discussed.

Profile analysis: detection of distantly related proteins.

Profile analysis is a method for detecting distantly related proteins by sequence comparison. The basis for comparison is not only the customary Dayhoff mutational-distance matrix but also the results of structural studies and information implicit in the alignments of the sequences of families of similar proteins. This information is expressed in a position-specific scoring table (profile), which is created from a group of sequences previously aligned by structural or sequence similarity. The similarity of any other sequence (target) to the group of aligned sequences (probe) can be tested by comparing the target to the profile using dynamic programming algorithms. The profile method differs in two major respects from methods of sequence comparison in common use: (i) Any number of known sequences can be used to construct the profile, allowing more information to be used in the testing of the target than is possible with pairwise alignment methods. (ii) The profile includes the penalties for insertion or deletion at each position, which allow one to include the probe secondary structure in the testing scheme. Tests with globin and immunoglobulin sequences show that profile analysis can distinguish all members of these families from all other sequences in a database containing 3800 protein sequences.

A model to account for the elastic element in muscle crossbridges in terms of a bending myosin rod.

We advance a structural model to account for the rapid elastic element seen in mechanical transient experiments on vertebrate skeletal muscle (A.F. Huxley & Simmons 1971 Nature, Lond. 233, 533-538). In contrast to other crossbridge models, ours does not envisage a myosin rod made up of two rigid portions connected by a hinge, but rather a gradually bending rod portion connecting the heads to the thick filament shaft. We propose that, in relaxed muscle, the subfragment 2 (S2) portion of the myosin rod is bound to the thick filament shaft by ionic interactions analogous to those between the light meromyosin (LMM) portions of the rod that constitute the body of the shaft. These interactions probably involve the alternating zones of positive and negative charge seen in myosin rod amino acid sequences. As the crossbridge cycle that generates tension begins, we propose that part of S2 detaches from the thick filament shaft and bends to enable the myosin head to attach to actin. When tension develops in the crossbridge, the S2 is straightened and more of it becomes detached from the shaft so that the junction between S2 and the myosin heads moves 3-4 nm axially. As tension declines at the end of the crossbridge stroke, we propose that S2 rebinds to the thick filament shaft and that this provides the restoring force to return the junction of the heads and S2 to its original axial position. Thus this movement would have the characteristics of an elastic element; detailed calculations indicate that it would have properties similar to those observed experimentally. Furthermore, this model can account for the radial attractive force seen in rigor and in contracting muscle, the decrease in stiffness when interfilament spacing is increased in skinned muscle, and the increased rate of proteolysis observed at the S2-LMM junction in contracting muscle.

Temperature dependence of the extinction coefficient of fused silica for CO(2) laser wavelengths.

The mechanism of the interaction of CO(2) laser radiation with fused silica is determined by the absorption depth of the radiation in the material. The extinction coefficient of a number of high-purity fused silica samples has been measured by the transmission method by fabricating samples approximately 30 microm in thickness. Results obtained agree with the latest reported values. In addition, samples were heated by an auxiliary CO(2)laser and the extinction coefficient determined as a function of temperature for six CO(2) laser lines. No significant difference in the extinction coefficient was observed for samples from different makers of the high-purity silica. Measurements were also conducted on a silica-rich glass Vycor and a significant difference was observed.

Solvation energy in protein folding and binding.

We have developed a method for calculating the stability in water of protein structures, starting from their atomic coordinates. The contribution of each protein atom to the solvation free energy is estimated as the product of the accessibility of the atom to solvent and its atomic solvation parameter. Applications of the method include estimates of the relative stability of different protein conformations, estimates of the free energy of binding of ligands to proteins and atomic-level descriptions of hydrophobicity and amphiphilicity.

Hydrophobicity and amphiphilicity in protein structure.

The importance of the hydrophobic interaction in stabilizing native protein structure has long been appreciated. However, more than other component forces, this one has resisted quantitative description. We present two approximate methods of assessing the hydrophobic component to the free energy of protein folding. Both are expressed in terms of what can be called hydrophobic moments of the protein. The first method is intended to yield an approximate value for the hydrophobic energy. This energy is calculated from a set of atomic coordinates in terms of the hydrophobicity (or 0th hydrophobic moment) of each amino acid residue and its accessibility or lack of it to aqueous solvent. The second method considers the first moment of the hydrophobicity of a group of residues, the hydrophobic moment. Segments of secondary structure in folded proteins tend to have hydrophobic moments that oppose each other. For example, alpha-helices on the protein surface tend to have one hydrophobic face and one hydrophilic face, with the hydrophilic face out towards the solvent. This pattern of organization is often apparent from a computer model of the protein that shows the magnitude and direction of the hydrophobic moment of each segment of secondary structure. Examples are given for the incorrectly folded structures of Novotný et al [J Mol Biol 177:787, 1984] and for the correct structures to which they correspond.

Confidence limits for homology in protein or gene sequences. The c-myc oncogene and adenovirus E1a protein.

We describe new tests, of general application, for deciding whether two proteins or DNA sequences are significantly homologous, in cases where the relationship is neither evidently true nor evidently false. Ralston and Bishop's comparison of the c-myc oncogene with the adenovirus E1a protein is discussed as an example. When the comparison matrix test is used to establish a homology between two sequences it is necessary that the number of high scores exceeds the expected mean level for random sequences by a statistically significant margin. The mean level itself is found from the double matching probability distribution. In examples where the number of high scores is larger than expected, but the highest score is not in itself exceptional, the variance of the numbers of scores expected for unrelated sequences is an important factor. We have analysed these variances by several methods. A simple binomial distribution gives only a rather inaccurate and low first estimate, but we derive a more rigorous and accurate statistical treatment, to take account of the correlations between scores in different parts of the comparison matrix. The theory is exact for random DNA or protein sequences with fluctuating compositions, selected by random draws from an infinite pool. In the more realistic situation, where sequences of fixed composition are formed by random permutations of the original sets, the deviations are smaller, and have been analysed by computer simulation. We find that although the relationship proposed by Ralston & Bishop, between the c-myc oncogene and adenovirus E1a proteins, appears to be significant in the binomial approximation, it is not supported by the full analysis. We conclude that, in general, great care is needed to establish any weak homology on the basis of comparisons that include no truly exceptional high scores, but merely have an enhanced number of scores at the upper end of the expected distribution.

Repetitive zinc-binding domains in the protein transcription factor IIIA from Xenopus oocytes.

The 7S particle of Xenopus laevis oocytes contains 5S RNA and a 40-K protein which is required for 5S RNA transcription in vitro. Proteolytic digestion of the protein in the particle yields periodic intermediates spaced at 3-K intervals and a limit digest containing 3-K fragments. The native particle is shown to contain 7-11 zinc atoms. These data suggest that the protein contains repetitive zinc-binding domains. Analysis of the amino acid sequence reveals nine tandem similar units, each consisting of approximately 30 residues and containing two invariant pairs of cysteines and histidines, the most common ligands for zinc. The linear arrangement of these repeated, independently folding domains, each centred on a zinc ion, comprises the major part of the protein. Such a structure explains how this small protein can bind to the long internal control region of the 5S RNA gene, and stay bound during the passage of an RNA polymerase molecule.

A method for measuring the non-random bias of a codon usage table.

We describe a new statistical method for measuring bias in the codon usage table of a gene. The test is based on the multinomial and Poisson distributions. The method is used to scan DNA sequences and measure the strength of codon preference. For E. Coli we show that the strength of codon preference is related to levels of gene expression. The method can also be used to compare base triplet frequencies with those expected from the base composition. This second type of codon bias test is useful for distinguishing coding from non-coding regions.

Sequence comparison by exponentially-damped alignment.

We describe a new method of comparing sequences, based on the Needleman-Wunsch sequence alignment algorithm, which can detect similarities that are interrupted by insertions or deletions in either sequence. The sequences are compared by calculating for each pair of residues a score which represents the best local alignment bringing those residues into correspondence; smooth localisation is achieved by reducing the contribution of distant parts of the alignment path by a factor which decreases exponentially with their distance from the point in question. The calculated values are used to draw a line graph in which regions of local similarity are shown by diagonal lines. Examples are shown of the application of the method to nucleic acid and amino acid sequences.