Calculations of protein volumes: sensitivity analysis and parameter database

MOTIVATION: The precise sizes of protein atoms in terms of occupied packing volume are of great importance. We have previously presented standard volumes for protein residues based on calculations with Voronoi-like polyhedra. To understand the applicability and limitations of our set, we investigated, in detail, the sensitivity of the volume calculations to a number of factors: (i) the van der Waals radii set, (ii) the criteria for including buried atoms in the calculations or atom selection, (iii) the method of positioning the dividing plane in polyhedra construction, and (iv) the set of structures used in the averaging. RESULTS: We find that different radii sets have only moderate affects to the distribution and mean of volumes. Atom selection and dividing plane methods cause larger changes in protein atoms volumes. More significantly, we show how the variation in volumes appears to be clearly related to the quality of the structures analyzed, with higher quality structures giving consistently smaller average volumes with less variance.     

The interpretation of protein structures: total volume, group volume distributions and packing density

Following the general procedure of Bernal &; Finney (1967) using Voronoi polyhedra, volumes occupied by all the atoms, or groups of atoms, in lysozyme and ribonuclease S have been estimated from the atomic co-ordinates provided by crystal structure studies. The average packing density for the interior of both proteins is close to 0.75, which is in the middle of the range found for crystals of most small organic molecules. For all atom types the mean packing densities fall between 0.7 and 0.8 with standard deviations between ± 0.1 and ± 0.2. It is suggested that simple geometrical packing considerations may provide useful criteria in guiding and evaluating trial structures in theoretical studies of protein folding, especially the association of distant parts of a peptide chain.Packing densities averaged over a relatively small number of atoms (5 to 15) appear to vary substantially in different parts of the same protein. Low densities representing packing defects may permit relatively easy motions, for example in an active site. Surrounding areas of high density may serve as relatively incompressible regions which transmit or correlate motions over considerable distances.The total volume of each of these two proteins as derived from the co-ordinate list appears to be larger than that estimated from the partial specific volume by 7 to 10%. If this volume difference is attributed to a change in the packing of water in a monolayer surrounding the protein, it would correspond to an average decrease, relative to bulk water, of 1 to 2 ų per water molecule in this monolayer. Such a change is about half of the decrease that occurs on the melting of ice.     

CX,an algorithm that identifies protruding atoms in proteins

MOTIVATION: A simple and fast algorithm is described that calculates a measure of protrusion (cx) for atoms in protein structures, directly useable with the common molecular graphics programs. RESULTS: A sphere of predetermined radius is centered around each non-hydrogen atom, and the volume occupied by the protein and the free volume within the sphere (internal and external volumes, respectively) are calculated. Atoms in protruding regions have a high ratio (cx) between the external and the internal volume. The program reads a PDB file, and writes the output in the same format, with cx values in the B factor field. Output structure files can be directly displayed with standard molecular graphics programs like RASMOL, MOLMOL, Swiss-PDB Viewer and colored according to cx values. We show the potential use of this program in the analysis of two protein-protein complexes and in the prediction of limited proteolysis sites in native proteins. AVAILABILITY: The algorithm is implemented in a standalone program written in C and its source is freely available at ftp.icgeb.trieste.it/pub/CX or on request from the authors.     

The interpretation of protein structures: estimation of static accessibility

A program is described for drawing the van der Waal's surface of a protein molecule. An extension of the program permits the accessibility of atoms, or groups of atoms, to solvent or solute molecules of specified size to be quantitatively assessed. As defined in this study, the accessibility is proportional to surface area. The accessibility of all atoms in the twenty common amino acids in model tripeptides of the type Ala-X-Ala are given for defined conformation. The accessibilities are also given for all atoms in ribonuclease-S, lysozyme and myogoblin. Internal cavities are defined and discussed. Various summaries of these data are provided. Forty to fifty per cent of the surface area of each protein is occupied by non-polar atoms. The actual numerical results are sensitive to the values chosen for the van der Waal's radii of the various groups. Since there is uncertainty over the correct values for these radii, the derived numbers should only be used as a qualitative guide at this stage.     

Functional units in rainbow trout (Salmo gairdneri, Richardson) liver: III. Morphometric analysis of parenchyma, stroma, and component cell types

Hepatic stroma and parenchyma with its component cell types were quantitatively described in adult male and female actively-spawning 5-year-old rainbow trout (Salmo gairdneri, Richardson). Point-count morphometry of glycol methacrylate sections estimated volume compartments for stroma and parenchyma. Veins composed 85% of the stroma while arteries and bile ducts occupied approximately 6-7% each. Parenchyma accounted for 95% of hepatic volume. Point-count morphometry of transmission electron micrographs estimated volume compartments as well as numerical and surface density measurements for parenchymal components. Within the hepatic parenchymal compartment, hepatocytes occupied 85% and showed significant sex differences. Female hepatocytes were significantly more numerous but were smaller, only 60% of the volume of male hepatocytes. Since hepatocyte nuclear volume was equal in both sexes, differences were due to reduced cytoplasmic volume in females. Perisinusoidal macrophages of females occupied larger volumes of their respective parenchymal compartments, and their larger mean cytoplasmic volumes suggested activation. Biliary epithelial cells of preductules and ductules were numerous. Ratios of numerical density of hepatocytes to biliary epithelial cells were consistent with a tubular arrangement of hepatocytes. Factors possibly mediating the sexual dimorphism are discussed.     

A parametric comparison of diagnostic accuracy with three ordinal diagnostic groups

Many medical diagnostic studies involve three ordinal diagnostic groups in which the diagnostic accuracy can be summarized by the volume or partial volume under a Receiver Operating Characteristic (ROC) surface. We study in this paper the statistical comparison of diagnostic accuracy from multiple diagnostic tests when three ordinal diagnostic groups are involved. Under the assumption that the multiple diagnostic tests follow a multivariate normal distribution within each diagnostic group, we provide the asymptotic variance and covariance for the maximum likelihood estimates of the volumes under the ROC surfaces from multiple diagnostic tests and propose statistical tests to test whether the diagnostic accuracy as measured by the volume under the ROC surface is the same for multiple diagnostic tests. We also propose a confidence interval estimate to the difference of two volumes under two ROC surfaces. Our approach depends crucially on the assumptions of normal distributions on diagnostic tests, which might not be robust when such assumptions are violated. Finally, we apply our proposed methodology to a real data set of 118 subjects to compare the diagnostic accuracy of early stage Alzheimer's disease (AD) from multiple neuropsychological tests.     

Retinoic acid induces nonuniform alveolar septal growth after right pneumonectomy.

To determine whether all-trans retinoic acid (RA) enhances compensatory lung growth in fully mature animals, adult male dogs (n = 4) received 2 mg x kg(-1) x day(-1) po RA 4 days/wk beginning the day after right pneumonectomy (R-PNX, 55-58% resection). Litter-matched male R-PNX controls (n = 4) received placebo. After 4 mo, the remaining lung was fixed by tracheal instillation of fixatives at a constant airway pressure for detailed morphometric analysis. After RA treatment compared with placebo, lung volume was slightly but not significantly lower. Volume density of septum to lung was 37% higher because of a 50 and 25% higher volume density of capillary and septal tissue, respectively. Mean septal thickness was 27% higher. Absolute volumes of endothelial cells and capillary blood were 31-37% higher, whereas epithelial and interstitial volumes were not different between groups. Absolute alveolar-capillary surface areas did not differ between groups, and alveolar septal surface-to-volume ratio was 20% lower in RA-treated animals. RA treatment exaggerated interlobar differences in morphometric indexes and caused alveolar capillary morphology to revert to a more immature state. Thus RA treatment during early post-R-PNX adaptation preferentially enhanced alveolar capillary and endothelial cell volumes consistent with formation of new capillaries, but the associated septal distortion precluded a corresponding increase in gas-exchange surface or morphometric estimates of lung diffusing capacity.     

Determining the minimum number of types necessary to represent the sizes of protein atoms.

MOTIVATION: Traditionally, for packing calculations people have collected atoms together into a number of distinct 'types'. These, in fact, often represent a heavy atom and its associated hydrogens (i.e. a united atom). Also, atom typing is usually done according to basic chemistry, giving rise to 20-30 protein atom types, such as carbonyl carbons, methyl groups, and hydroxyl groups. No one has yet investigated how similar in packing these chemically derived types are. Here we address this question in detail, using Voronoi volume calculations on a set of high-resolution crystal structures. RESULTS: We perform a rigorous clustering analysis with cross-validation on tens of thousands of atom volumes and attempt to compile them into types based purely on packing. From our analysis, we are able to determine a 'minimal' set of 18 atom types that most efficiently represent the spectrum of packing in proteins. Furthermore, we are able to uncover a number of inconsistencies in traditional chemical typing schemes, where differently typed atoms have almost the same effective size. In particular, we find that tetrahedral carbons with two hydrogens are almost identical in size to many aromatic carbons with a single hydrogen. AVAILABILITY: Programs available from http://geometry.molmovdb.org. CONTACT: JerryTsai@TAMU.edu; neil.voss@yale.edu; Mark.Gerstein@yale.edu SUPPLEMENTARY INFORMATION: Available at http://geometry.molmovdb.org.     

Distributions of water around amino acid residues in proteins

The atomic co-ordinates from 16 high-resolution (less than or equal to 1.7 A = 0.1 nm), non-homologous proteins have been used to study the distributions of water molecule sites around the 20 different amino acid residues. The proportion of residues whose main-chain atoms are in contact with water molecules was fairly constant (between 40% and 60%), irrespective of the nature of the side-chain. However, the proportion of residues whose side-chain atoms were in contact with water molecules showed a clear (inverse) correlation with the hydrophobicity of the residue, being as low as 14% for leucine and isoleucine but greater than 80% for asparagine and arginine. Despite the problems in determining accurate water molecule sites from X-ray diffraction data and the complexity of the protein surface, distinct non-random distributions of water molecules were found. These hydration patterns are consistent with the expected stereochemistry of the potential hydrogen-bonding sites on the polar side-chains. The water molecules around apolar side-chains lie predominantly at van der Waals' contact distances, but most of these have a primary, shorter contact with a neighbouring polar atom. Further analysis of these distributions, combined with energy minimization techniques, should lead to improved modelling of protein structures, including their primary shells of hydration.     

Atoms-in-molecules study of the genetically encoded amino acids. II. Computational study of molecular geometries

The geometries of the 20 genetically encoded amino acids were optimized at the restricted Hartree-Fock level of theory using the 6-31+G* basis set. A detailed comparison showed the calculated geometries to be in excellent agreement with those determined by X-ray crystallography. The study demonstrated that the geometric parameters for the main-chain group and for the bonds and common functional groups of the side-chains exhibit a high degree of transferability among the members of this set of molecules. This geometric transferability is a necessary prerequisite for the corresponding transferability of their electron density distributions and hence of their bond and atomic properties. The transferability of the electron distributions will be demonstrated and exploited in the following paper of this series, which uses the topology of the electron density to define an atom within the quantum theory of atoms in molecules. Particular features of the geometries of the amino acids are discussed. It has been shown, for example, how the apparent anomaly of the Calpha-N bond length in a peptide being shorter than in the charged species Calpha-NH3+ is resolved when the charge separation is gauged by the differences in the charges of the Calpha and N atoms as opposed to the use of formal charges. A compilation of literature sources on experimental geometries covering each member of the 20 amino acids is presented. A set of rules for labeling the atoms and bonds, complementing the generally accepted IUPAC-IUB rules, is proposed to uniquely identify every atom and bond in the amino acids.     

Hydration of amino acid side chains: dependence on secondary structure.

Energy calculations have been used to study the hydration sites around the polar groups of serine, threonine and tyrosine side chains. These hydration sites depend not only on the hybridization of the polar group but also on the local secondary structure, the chi 1 side chain torsion angle and the position of the hydroxyl hydrogen atom. For tyrosine side chains, two solvent sites are found approximately in the plane of the ring. Even for serine and threonine side chains only two minimum energy sites are found in general of which one is in an expected position within hydrogen bonding of the hydroxyl hydrogen atom (unless this is blocked from interaction with solvent molecules by, for example, Oi-4 or Oi-3. The position of the second of these sites depends not only on the position of the hydroxyl oxygen but also on neighbouring main chain atoms to which it can also hydrogen bond. There is good agreement with the solvent distributions obtained from crystallographic data.     

Monte Carlo calculations of ion distributions surrounding the oligonucleotide d(ATATATATAT)2 in the B, A, and wrinkled D conformations.

We calculated the uni-univalent ion distributions around the oligonucleotide d(AT)5.d(AT)5 in the A, B and wrinkled D conformation using the Metropolis Monte Carlo method. All atoms were included in the oligonucleotide model with partial charges and hard sphere radii assigned to each atom. The univalent counter- and coions were modeled as hard spheres with radius 0.3 nm. The solvent was assigned a dielectric constant of 80, corresponding to a temperature of 298K. The counterion distribution surrounding each of the conformers and the distribution surrounding an impenetrable cylinder, were calculated for four salt concentrations. We found significant counterion density in the major groove of the A DNA while fewer counterions occupied the grooves of B DNA. In the wrinkled D DNA, where groove occupancy is sterically hindered, the ion distributions were identical to the distributions surrounding the impenetrable, cylindrical model. This suggests that excluded volume effects significantly influence the details of the ion distributions near the oligomer, while the detailed charge distributions of the oligomer affects the ion distributions only minimally. Although substantial variation in counterion density was observed near the oligomers of differing conformations, the total number of counterions located within a cylinder surrounding the oligomer bounded radially by 2.4 nm was independent of the conformation of the oligomer. Therefore, for this model system, the local univalent counterion distributions are extremely sensitive to the geometry of the oligonucleotide whereas the extent of neutralization of the oligoanion is insensitive to the conformation of the oligomer.     

A selective histochemical method for the quantitative estimation of elastic fibers by computerized morphometric analysis. Effect of colchicin treatment

A morphometric technique is reported that uses a new selective staining of the elastic system fibers in skin biopsy specimens to facilitate the quantitative evaluation of the volume fraction occupied by these elastic fibers in the tissue. The study of elastic fibers in the dermis of 30 patients, before and after six months of treatment with Colchicin, was carried out with a Quantimet 720 system. Preelastic (oxytalan and elaunin) fibers and mature elastic fibers were quantitated separately. Compared to the average volume fraction (surface occupied by the elastic fibers) before treatment with Colchicin (1.449 +/- 0.64%), the mean values after treatment were significantly increased (2.076 +/- 0.61%). The same results were found for the preelastic fibers: 0.807 +/- 0.51% before treatment and 1.025 +/- 0.54% after treatment. These results demonstrate the advantages of our monochromatic staining method for automatic quantitation of elastic fibers as well as the possibilities of the quantitative study of the elastic fibers in human dermis. This methodology should be applicable to other inherited or acquired diseases affecting skin elastic fibers as well as to other tissues containing elastic fibers.     

Quantification of protein surfaces,volumes and atom-atom contacts using a constrained Voronoi procedure

MOTIVATION: Geometric representations of proteins and ligands, including atom volumes, atom-atom contacts and solvent accessible surfaces, can be used to characterize interactions between and within proteins, ligands and solvent. Voronoi algorithms permit quantification of these properties by dividing structures into cells with a one-to-one correspondence with constituent atoms. As there is no generally accepted measure of atom-atom contacts, a continuous analytical representation of inter-atomic contacts will be useful. Improved geometric algorithms will also be helpful in increasing the speed and accuracy of iterative modeling algorithms. RESULTS: We present computational methods based on the Voronoi procedure that provide rapid and exact solutions to solvent accessible surfaces, volumes, and atom contacts within macromolecules. Furthermore, we define a measure of atom-atom contact that is consistent with the calculation of solvent accessible surfaces, allowing the integration of solvent accessibility and inter-atomic contacts into a continuous measure. The speed and accuracy of the algorithm is compared to existing methods for calculating solvent accessible surfaces and volumes. The presented algorithm has a reduced execution time and greater accuracy compared to numerical and approximate analytical surface calculation algorithms, and a reduced execution time and similar accuracy to existing Voronoi procedures for calculating atomic surfaces and volumes.     

Structural characteristics of immature rat Sertoli cells in vivo and in vitro.

The structural properties of pelleted prepubertal Sertoli cells (pre-culture pelleted cells) from 19-day-old rats and of similar cells cultured for 7 days were compared with Sertoli cells from the intact animal (testis tissue from 19- and 26-day-old rats, the in vivo groups). Sertoli cells from freshly isolated pellets and those cultured for 7 days were similar in cell and nuclear volumes to their in vivo counterparts. Cell volumes, organelle volumes, and organelle volume densities of newly isolated Sertoli cells were similar to those of sectioned cells taken from the 19-day-old in vivo group, indicating that the procedure for isolation does not grossly alter Sertoli cells. Mean height of cells cultured for 7 days was significantly lower than that of cells from intact animals at 19 and 26 days of age. In vivo, Sertoli cells of 26-day-old animals displayed increased organelle volumes and organelle surface areas compared with those from 19-day-old animals; volume densities and surface densities remained relatively constant, indicating that in vivo, organelle growth is in proportion to growth of the cell. Most organelle volume and surface densities were not significantly different when 19-day-old in vivo cells and pre-culture pelleted cells were compared. Many organelle volume and surface density values were significantly less in cells grown in culture for 7 days as compared to freshly isolated pelleted cells. After 7 days of culture, most Sertoli cell organelles were significantly less in both volume density and surface density, as compared to the in vivo cell groups (19 or 26 day). This indicates that in vitro the organelles do not develop in proportion to the growth of the cell. After 7 days in culture, the absolute volumes and surface areas of the organelles remained generally unchanged as compared to cells from 19-day-old animals. The data show that Sertoli cells grow in volume in vitro like their in vivo counterparts; however, their subcellular features, although well maintained, do not develop in proportion to the cell. This suggests that short-term cultures are a more ideal system in which to study biochemical responses. Also, cultured prepubertal Sertoli cells are most appropriately used to study prepubertal Sertoli cell function. This is the first study to quantify developmental changes in Sertoli cell structure in vivo as well as to compare them with cellular changes occurring in vitro.     

Statistical and energetic analysis of side-chain conformations in oligopeptides

The distributions of side-chain conformations in 258 crystal structures of oligopeptides have been analyzed. The sample contains 321 residues having side chains that extend beyond the C beta atom. Statistically observed preferences of side-chain dihedral angles are summarized and correlated with stereochemical and energetic constraints. The distributions are compared with observed distributions in proteins of known X-ray structures and with computed minimum-energy conformations of amino acid derivatives. The distributions are similar in all three sets of data, and they appear to be governed primarily by intraresidue interactions. In side chains with no beta-branching, the most important interactions that determine chi 1 are those between the C gamma H2 group and atoms of the neighboring peptide groups. As a result, the g- conformation (chi 1 congruent to -60 degrees) occurs most frequently for rotation around the C alpha-C beta bond in oligopeptides, followed by the t conformation (chi 1 congruent to 180 degrees), while the g+ conformation (chi 1 congruent to 60 degrees) is least favored. In residues with beta-branching, steric repulsions between the C gamma H2 or C gamma H3 groups and backbone atoms govern the distribution of chi 1. The extended (t) conformation is highly favored for rotation around the C beta-C gamma and C gamma-C delta bonds in unbranched side chains, because the t conformer has a lower energy than the g+ and g- conformers in hydrocarbon chains. This study of the observed side-chain conformations has led to a refinement of one of the energy parameters used in empirical conformational energy computations.     

Validation of equilibration and chromium reduction methods for deuterium measurements of fluid volumes.

Determinations of fluid volumes are of importance for correct treatment of patients subjected to shock and trauma. Gas isotope ratio mass spectrometry (GIRMS) is an advanced method for analysis of stable isotopes. These can be used as tracers for measurement of various fluid volumes. In the current in vitro study, deuterium was used to determine different volumes of water simulating a range of body fluid volumes from neonates to adults. A high-precision scale gave control weights (i.e., volumes), and two methods, equilibration (EQ) and chromium reduction (CR), were compared by use of a GIRMS. The coefficient of variation was <1% when using both EQ (0.45%) and CR (0.79%). The variability was greater at small volumes, and, when regression equations for the relation between measured and calculated volumes were used as formulas, the deviation was 0.4% using EQ and 2.8% using CR at the volume of 1,000 ml. At larger volumes, the deviation when using CR approached 1%. These variations are better than previously published data using other methods. It was concluded that GIRMS is a suitable technique for fluid volume determinations in neonates as well as in adult patients, using deuterium as a tracer. EQ and CR methods were both regarded to give acceptable variabilities in this in vitro study. GIRMS may in the future increasingly be used clinically for accurate measurements of body fluid volumes.     

Myocardial capillaries in the fetal and the neonatal rat: a morphometric analysis of the maturing myocardial capillary bed

Developing myocardial capillaries from 16-day-gestation fetus to adult undergo several morphological changes including a thinning of the lateral extensions of the capillary endothelial cells, the formation of a basal lamina, and an increase in the number of plasmalemmal vesicles. A decrease in the extracellular space, an increase in the number of capillaries, and a decrease in the capillary diameter were also observed during the developmental period. In view of these ultrastructural changes, a morphometric analysis was made on the developing myocardial wall to demonstrate specific quantitative changes. The volumes which were occupied by capillary endothelial cells, capillary lumina, extracellular space, and myocardial myocytes within a reference volume of myocardium were measured; and we found that 8% of the reference myocardial volume was occupied by capillary endothelial cells, 85% was occupied by myocardial myocytes, 4% was occupied by capillary lumina, and, except for a significant change in extracellular space at 16 days gestation, 3% was occupied by extracellular space. Each volume ratio was found to be nearly constant throughout the studied period. In contrast to this constancy in the volume ratios, other parameters which were measured demonstrated significant changes during the developmental period studied. These overall changes include a 135% increase in capillary density, a 63% increase in luminal surface area of capillary endothelial cells, a 24% decrease in capillary diameter, a 12% decrease in diffusion distance, and a 35% decrease in the diameter of the erythrocyte population.(ABSTRACT TRUNCATED AT 250 WORDS)     

A fragment library based on Gaussian mixtures predicting favorable molecular interactions

Here, a protein atom-ligand fragment interaction library is described. The library is based on experimentally solved structures of protein-ligand and protein-protein complexes deposited in the Protein Data Bank (PDB) and it is able to characterize binding sites given a ligand structure suitable for a protein. A set of 30 ligand fragment types were defined to include three or more atoms in order to unambiguously define a frame of reference for interactions of ligand atoms with their receptor proteins. Interactions between ligand fragments and 24 classes of protein target atoms plus a water oxygen atom were collected and segregated according to type. The spatial distributions of individual fragment - target atom pairs were visually inspected in order to obtain rough-grained constraints on the interaction volumes. Data fulfilling these constraints were given as input to an iterative expectation-maximization algorithm that produces as output maximum likelihood estimates of the parameters of the finite Gaussian mixture models. Concepts of statistical pattern recognition and the resulting mixture model densities are used (i) to predict the detailed interactions between Chlorella virus DNA ligase and the adenine ring of its ligand and (ii) to evaluate the "error" in prediction for both the training and validation sets of protein-ligand interaction found in the PDB. These analyses demonstrate that this approach can successfully narrow down the possibilities for both the interacting protein atom type and its location relative to a ligand fragment.