Conformational energy of the 5-membered ring. Implication of geometric and energetic properties in the conformational characteristics

In a previous article (8) a geometrical study of the five-membered ring showed that: a) for the case of the 20 symmetrical C₂ and C_s conformations, the pseudorotation formulae for the torsion angles are a geometrical property of the ring; b) geometrical considerations alone are unable to define the puckering amplitude, the bond angle values, and the pathway between two symmetrical conformations. Here we examine how the energy equations enable us to define the deformation amplitude _m, establish the bond angles expressions and check the energy invariability along the pseudorotation circuit. The problem is next developed fully in the case where the bond and torsional energy only are considered: the literal expression¹ of _m is then given as a function of the bond angle which cancels out the bond angle energy. A numerical application is carried out on cyclopentane and the values of the parameters K^t, K¹ and used in the Conformational energy calculations are considered.Notations used 1 _i bond lengths 1 in the case of the regular ring - _i torsional angles - _i bond angles - 3/5 = 108 - 4/5 = 144 - , _{i i} – = complement to the 108 bond angle _i - _T - E Conformational energy of the 5-membered ring - E Conformational energy difference between planar and deformed ring - A _n Coefficients of the energy development in terms of - E _i ^l Bond energy relative to atom i (associated with angle _i) - K _i ^l Bond constant relative to atom i (associated with angle _i) - E _i ^l Torsional energy relative to the i ^th bond (associated with angle _i) - k _i ^l Torsional constant relative to the i ^th bond (associated with angle _i) - _i Angle _i value corresponding to zero bond energy E _i ^l (when the 5 atoms of the ring are identical, _i ) - r _ij Distance between atoms i and j - q _i Charge carried by atom i - e Constant of proportionality including the effective dielectric constant - A _ij, B_ij, d_ij Coefficients dependent on the nature of the atoms i and j and accounted for in the Van der Waals energy and hydrogen bond expressions - S (r _ij) Electrostatic contribution to the hydrogen bond energy - P Pseudorotation phase angle - _m Maximum torsional angle value characterising the deformation amplitudeM     

Functional aspects of shell geometry in some British land snails

Allometric relationships between the area of the shell mouth and live body weight are examined in 19 species of British land snails. Within species, the rate of increase of mouth area on weight is usually less than the isometric expectation, in spite of the logarithmic spiral pattern of growth in most species. It is suggested that this deviation is due mainly to changes in density with size. Two species which conform to isometric expectation alter the geometry of the shell as they pow.
Between species, the rate of increase in mouth area on weight in adults is much greater than isometric expectation, and the range of mouth areas at standard weight is considerable. These deviations are almost entirely accounted for by differences in shell geometry between large and small species, the former having higher rates of whorl expansion and smaller or non–existent umbilicuses.
The range of loading (weight per unit mouth area) on resting adult snails is thus much less than would be expected. It is suggested that large species are limited in the range of possible shell geometries by the need to minimize loading, while in small species other forces such as desiccation and predation may also be important: the range of geometries is generally larger. Observations on ecology and behaviour tend to support these conclusions.     

Computed tomography and biomechanical analysis of fossil long bones.

Computerized transverse axial scanning (computed tomography) is a relatively new radiographic technique designed to recover precise cross-sectional images (tomograms) of 3-dimensional objects. This highly sensitive process permits tissues of similar density to be separated and displayed unambiguously. These special features are, therefore, ideal for analyzing the cross-sectional geometry of intact fossil long bones, even when they are highly mineralized and their medullary cavities are occluded by matrix. In order to demonstrate the utility of this method in assessing the complex relationship between fossil structure and probable function, geometrical and biomechanical properties of midshaft tomograms of femora and tibiae have been analyzed for a comparative primate sample consisting of Megaladapis edwardsi (an extinct giant prosimian from Madagascar), Indri indri (the largest extant prosimian), and Homo sapiens.     

The membrane geometry of the prolamellar body

Summary When etioplasts are exposed to light, the branched tubular lattice of the prolamellar body becomes the large flat discs of the primary lamellae. We show that in spite of the very different apparent morphology of these two membranous structures, their membranes have similar average curvature and inside-outside surface areas. This implies that the packing or molecular organization of the lipids and proteins can be similar in the two structures, and that consequently the transformation does not require much high energyflip-flop of membrane components from one side to the other. We also discuss the relative physiological stability of the two structures.     

Measurement of three-dimensional structure of plants with a simple device and estimation of light capture of individual leaves

1. A simple device called a 'pocometer' (POlar COordinate METER) was developed to measure three-dimensional structure of plants. It consists of a tape-measure to measure distance and two protractors to measure zenith angle and azimuth angle.
2. The pocometer can determine locations of points within a few metres distance with a resolution of less than 1cm. Location of any point on a plant can be measured in 10 to 30s depending on the ease of pulling the tape measure to the point of interest.
3. A system to use data obtained with the pocometer to calculate plant light capture was developed. The degree of shading at any point on a plant is estimated by checking obstruction by other plant parts of the view toward the sky at that point.
4. Photon flux density (PFD) on leaf surfaces was estimated for Aucuba japonica , a broad-leaved evergreen shrub, using the above system. The estimated PFDs for individual leaves of a plant corresponded to the sensor-measured PFDs with correlation coefficients of 0·67 to 0·92.     

Scale-dependent foraging and patch choice in fractal environments

Many spatially complex environments are fractal, and consumers in these environments face scale-dependent trade-offs between encountering high densities of small resource patches versus low densities of large resource patches. I address the effects of these trade-offs on foraging by incorporating scale-dependent encounter of resources in fractal landscapes into classical optimal foraging theory. This model is then used to predict optimal scales of perception (foraging scale) and patch choice in response to spatial features of landscapes. The model predicts that, for a given density of resources, landscapes with greater extent and fractal dimension and that contain patchy (low fractal dimension) resources favour large foraging scales and specialization on a small proportion of resource patches. Fragmented (low fractal dimension) landscapes of small extent with dispersed (high fractal dimension) resources favour smaller foraging scales and generalists that use a large proportion of available resource patches. These predictions synthesize the results of other spatially explicit consumer–resource models into a simple framework and agree reasonably well with results of several empirical studies. This study thus places optimal foraging theory in a spatial context and suggests evolutionary mechanisms of consumers' responses to important spatial phenomena (e.g. habitat fragmentation, resource aggregation). This revised version was published online in July 2006 with corrections to the Cover Date.     

Solution structure of N-terminal segment of hepatitis B virus surface antigen Pre-S1

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A geometrical constraint approach for reproducing the native backbone conformation of a protein

It is known that the backbone conformation of a protein can be reproduced with precision once a correct contact map (two-dimensional representation showing residue pairs in contact) is given as geometrical constraints. There is, however, no way to infer the correct contact map for a protein of unknown structure. We started with one-dimensional constraints using the quantity N14 (the number of neighboring residues within the radius of 14 Å). Since the plot of N14 along a chain shows a good correlation with the corresponding amino acid sequence, the N14 profile obtained from the X-ray structure is predictable from the sequence. Construction of backbone conformations under a given N14 profile was carried out in the following two steps: (1) a contact map from the N14 profile was produced by taking the product of N14 values of every two residues; (2) backbone conformations were generated by applying the distance geometry technique to distance constraints given by the contact map. If present, disulfide bonds in a protein, as well as the secondary structure, were treated as additional constraints, and both cases with or without the additional information were examined. The method was tested for 11 proteins of known structure, and the results indicated that the reproduced conformation was fairly good, using an X-ray structure for comparison, for small proteins of less than 80 residues long. The basic assumption and effectiveness of the present method were compared with those of previous studies employing the geometrical constraint approach. It has become clear that the specific, one-dimensional information (e.g., N14 profile) is more effective than nonspecific, two-dimensional constraints, such as average interresidue distances between particular types of amino acids. © 1993 Wiley-Liss, Inc.     

Statistical geometry based prediction of nonsynonymous SNP functional effects using random forest and neuro-fuzzy classifiers

There is substantial interest in methods designed to predict the effect of nonsynonymous single nucleotide polymorphisms (nsSNPs) on protein function, given their potential relationship to heritable diseases. Current state-of-the-art supervised machine learning algorithms, such as random forest (RF), train models that classify single amino acid mutations in proteins as either neutral or deleterious to function. However, it is frequently the case that the functional effect of a polymorphism on a protein resides between these two extremes. The utilization of classifiers that incorporate fuzzy logic provides a natural extension in order to account for the spectrum of possible functional consequences. We generated a dataset of single amino acid substitutions in human proteins having known three-dimensional structures. Each variant was uniquely represented as a feature vector that included computational geometry and knowledge-based statistical potential predictors obtained though application of Delaunay tessellation of protein structures. Additional attributes consisted of physicochemical properties of the native and replacement amino acids as well as topological location of the mutated residue position in the solved structure. Classification performance of the RF algorithm was evaluated on a training set consisting of the disease-associated and neutral nsSNPs taken from our dataset, and attributes were ranked according to their relative importance. Similarly, we evaluated the performance of adaptive neuro-fuzzy inference system (ANFIS). The utility of statistical geometry predictors was compared with that of traditional structural and evolutionary attributes employed by other researchers, revealing an equally effective yet complementary methodology. Among all attributes in our feature set, the statistical geometry predictors were found to be the most highly ranked. On the basis of the AUC (area under the ROC curve) measure of performance, the ANFIS and RF models were equally effective when only statistical geometry features were utilized. Tenfold cross-validation studies evaluating AUC, balanced error rate (BER), and Matthew's correlation coefficient (MCC) showed that our RF model was at least comparable with the well-established methods of SIFT and PolyPhen. The trained RF and ANFIS models were each subsequently used to predict the disease potential of human nsSNPs in our dataset that are currently unclassified (http://rna.gmu.edu/FuzzySnps/).     

Similar chemistry, but different bond preferences in inter versus intra-protein interactions

Proteins fold into a well-defined structure as a result of the collapse of the polypeptide chain, while transient protein-complex formation mainly is a result of binding of two folded individual monomers. Therefore, a protein-protein interface does not resemble the core of monomeric proteins, but has a more polar nature. Here, we address the question of whether the physico-chemical characteristics of intraprotein versus interprotein bonds differ, or whether interfaces are different from folded monomers only in the preference for certain types of interactions. To address this question we assembled a high resolution, nonredundant, protein-protein interaction database consisting of 1374 homodimer and 572 heterodimer complexes, and compared the physico-chemical properties of these interactions between protein interfaces and monomers. We performed extensive statistical analysis of geometrical properties of interatomic interactions of different types: hydrogen bonds, electrostatic interactions, and aromatic interactions. Our study clearly shows that there is no significant difference in the chemistry, geometry, or packing density of individual interactions between interfaces and monomeric structures. However, the distribution of different bonds differs. For example, side-chain-side-chain interactions constitute over 62% of all interprotein interactions, while they make up only 36% of the bonds stabilizing a protein structure. As on average, properties of backbone interactions are different from those of side chains, a quantitative difference is observed. Our findings clearly show that the same knowledge-based potential can be used for protein-binding sites as for protein structures. However, one has to keep in mind the different architecture of the interfaces and their unique bond preference.