On surface geometry coupled to morphogen

An expression is derived for both the Gauss and the Mean curvature of a surface, in terms of three simple cell parameters. The surface is thought of as composed of a single-cell thick sheet of cells joined laterally. The three cellular parameters involved are the ratios of (linear) basal to apical dimension in two orthogonal directions, S1 and S2, and the cell thickness "h". These three parameters may be envisioned as functions of a morphogen or morphogens which vary from point to point over the (middle) surface. As an example, the "reaction-diffusion" equations which are often used to describe pattern-formation in early development can be seen as possible candidates for these morphogens, when the resultant surface deformations are given when the dependence of the three cellular parameters are specified as a function of morphogen concentration. The coupling back of the surface deformations to the set of reaction-diffusion equations is simply given, and is through the dependence on geometry of the Laplacian operator which enters these equations.     

A model of morphogenetic pattern formation

A model for the morphogenetic movement of surfaces composed of cellular monolayers is proposed. The cells are presumed joined at their lateral surfaces. An otherwise unspecified substance called a "morphogen" is introduced which is the agent of change in the individual cell (or cell-like region). The distribution of these cellular deformations define a surface (the middle surface, through the middle of the cell heights) via equations given for the Gauss and Mean curvatures of the surface defined at each point. The Gauss curvature as a function of the morphogen level determines the metric of the surface "g(u, v)" in conformal co-ordinates u, v. A unique equation for the morphogen distribution over the survace is presented which has the property of size invariance, that is, the model "regulates" without need of further arguments. The two resulting coupled equations for the metric and the morphogen, eqns (4) and (2), both non-linear equations, are to be solved self-consistently, once the individual cell deformation as a function of morphogen is given. The surface geometry determines the morphogen distribution, and the morphogen distribution in turn affects the surface geometry. Extension of the model to two or more morphogens is straightforward, and the key property of "regulation" or size invariance of the model is retained. Numerical integration of the two coupled equations is carried out in the case of axial symmetry, and the results presented by the case that individual cells deform by changing the ratio of their apical to basal areas, as well as their heights. Gastrulation in small regulating holoblastic eggs (e.g. starfish, sea urchin and amphioxus) is discussed in light of the present model.     

A comparison of the influence of global functional loads vs. local contact anatomy on articular cartilage thickness at the knee

Cartilage contact geometry, along with joint loading, can play an important role in determining local articular cartilage tissue stress. Thus individual variations in cartilage thickness can be associated with both individual variations in joint loading associated with activities of daily living as well as individual differences in the anatomy of the contacting surfaces of the joint. The purpose of this study was to isolate the relationship between cartilage thickness predicted by individual variations in contact surface geometry based on the radii of the femur and tibia vs. cartilage thickness predicted by individual variations in joint loading. Knee magnetic resonance (MR) images and the peak knee adduction moments during walking were obtained from 11 young healthy male subjects (age 30.5+/-5.1 years). The cartilage thicknesses and surface radii of the femoral and tibial cartilage were measured in the weight-bearing regions of the medial and lateral compartments of three-dimensional models from the MR images. The ratio of contact pressure between the medial and lateral compartments was calculated from the radii of tibiofemoral contact surface geometries. The results showed that the medial to lateral pressure ratios were not correlated with the medial to lateral cartilage thickness ratios. However, in general, pressure was higher in the lateral than medial compartments and cartilage was thicker in the lateral than medial compartments. The peak knee adduction moment showed a significant positive linear correlation with medial to lateral thickness ratio in both femur (R(2)=0.43,P<0.01) and tibia (R(2)=0.32,P<0.01). The results of this study suggest that the dynamics of walking is an important factor to describe individual differences in cartilage thickness for normal subjects.     

Prediction of the viscosity radius and the size exclusion chromatography behavior of PEGylated proteins

Size exclusion chromatography (SEC) was used to determine the viscosity radii of equivalent spheres for proteins covalently grafted with poly(ethylene glycol) (PEG). The viscosity radius of such PEGylated proteins was found to depend on the molecular weight of the native protein and the total weight of grafted PEG but not on PEG molecular weight, or PEG-to-protein molar grafting ratio. Results suggest grafted PEG's form a dynamic layer over the surface of proteins. The geometry of this layer results in a surface area-to-volume ratio approximately equal to that of a randomly coiled PEG molecule of equivalent total molecular weight. Two simple methods are given to predict the viscosity radius of PEGylated proteins. Both methods accurately predicted (3% absolute error) the viscosity radii of various PEG-proteins produced using three native proteins, alpha-lactalbumin (14.2 kDa MW), beta-lactoglobulin dimer (37.4 kDa MW), and bovine serum albumin (66.7 kDa MW), three PEG reagents (2400, 5600, and 22500 MW), and molar grafting ratios of 0 to 8. Accurate viscosity radius prediction allows calculation of the distribution coefficient, K(av), for PEG-proteins in SEC. The suitability of a given SEC step for the analytical or preparative fractionation of different PEGylated protein mixtures may therefore be assessed mathematically. The methods and results offer insight to several factors related to the production, purification, and uses of PEGylated proteins.     

Image texture features for cell nuclei based on absorbance surface curvature

The absorbance values of a cell nucleus, as measured at each pixel location, can be considered as forming a surface in a three-dimensional space. The principal curvatures of differential geometry can then be evaluated for this surface. From these curvatures, two new variables are computed that, when averaged over all pixels in the nucleus, are viewed as image texture measures. These two measures were found to be effective for differentiating insect populations and compared well with other features used in the TICAS system. This paper presents the results of comparing the performance of these two features with the performance of other features for several populations of cells.     

Determining stereo-specific 1H nuclear magnetic resonance assignments from distance geometry calculations

Stereo-specific 1H nuclear magnetic resonance assignments can be obtained following distance geometry structure calculations. The key to this method is to allow stereo-related atoms or methyls to float between pro-R and pro-S configurations, the final configuration being determined by the experimental constraints. Resonances from stereo-related pairs are given initial random assignments (either pro-R or pro-S) for identifying nuclear Overhauser effects (NOEs). A list of distance constraints using these assignments is compiled and a series of structures calculated where the chirality of non-C alpha chiral centers is not constrained; no pseudoatom corrections are required. Calculated structures are both locally and globally well-determined since the assignments rely upon the structure determination rather than the structure quality relying upon stereo-specific assignments. The method represents a global approach to determining stereo-specific assignments versus previously reported methods where only intraresidue NOEs and J-coupling information are used.     

The radial decline of nerve impulses in a restricted cylindrical extracellular space

In order to increase the potentials recorded extracellularly from nerve fibres, peripheral nerves are often placed in restricted space with cylindrical geometry. Equations are derived for computing the potentials expected at the surface of the cylinder, based on the potentials at the external surface of a small nerve fibre located on the long axis of the cylinder. These equations are evaluated numerically, using two formulae for a nerve impulse given in the literature. In both cases there is little attenuation for cylinders with radii less than 0.5 mm, but the potential declines approximately as a power of radius b for 1<b<10 mm. Various factors which might affect these results under different experimental conditions are discussed.     

Twisted hyperboloid (strophoid) as a model of β-barrels in proteins

Using a least-squares fitting procedure, polypeptide backbones of one parallel and seven antiparallel β-barrels were approximated with various curved surfaces. Although the hyperboloid gave better approximations to all the β-barrel backbones than the ellipsoid, elliptical cylinder or catenoid, the best approximations were obtained with a novel surface, a twisted hyperboloid (strophoid). The root-mean-square errors between individual β-barrels and the fitted strophoid surfaces ranged from 0.75 Å to 1.64 Å. The parameters which determine the strophoid surface allow groups of β-barrel shapes to be defined according to their barrel twists (i.e. angles subtended by directions of the long axis of cross-section at the top and the bottom of the barrel), course of elliptical cross-sections (either monotonically increasing along the barrel axis, as in cones, or having a middle “waist”, as in hyperboloids), and types of backbone curvatures (either convex or concave). The curvatures at individual points of strophoid surface are local, variable quantities related to the local helicity (coil) of the polypeptide backbone, in contrast to values of β-sheet twist (i.e. dihedral angles subtended by adjacent β-strands) known to be virtually identical in all the β-sheets. The variability found in parameters such as barrel shapes and curvatures suggests that simple models (isotropically stressed surfaces, principle of minimal surface tension) proposed in the past to account for β-barrel shapes are not sufficient. Rather, the complex nature of best-fit theoretical surfaces points to an important role played by a local variability of the forces involved.     

Solution structures of the rabbit neutrophil defensin NP-5

Solution structures of the rabbit neutrophil defensin NP-5 have been determined by 1H nuclear magnetic resonance (n.m.r.) spectroscopy and distance geometry techniques. This 33 amino acid peptide is part of the oxygen-independent mammalian defense system against microbial infection. The structures were generated from 107 n.m.r. derived inter-residue proton-proton distance constraints. A distance geometry algorithm was then used to determine the range of structures consistent with these distance constraints. These distance geometry calculations employed an improved algorithm that allowed the chirality constraints to be relaxed on prochiral centers when it was not possible to make stereo-specific assignments of protons on these centers. This procedure gave superior results compared with standard distance geometry methods and also produced structures that were more consistent with the original n.m.r. data. Analysis of the NP-5 structures shows that the overall folding of the peptide backbone is well defined by the n.m.r. distance information but that the side-chain group conformations are generally less well defined.     

Three-dimensional solution structure of Ca(2+)-loaded porcine calbindin D9k determined by nuclear magnetic resonance spectroscopy.

The three-dimensional solution structure of native, intact porcine calbindin D9k has been determined by distance geometry and restrained molecular dynamics calculations using distance and dihedral angle constraints obtained from 1H NMR spectroscopy. The protein has a well-defined global fold consisting of four helices oriented in a pairwise antiparallel manner such that two pairs of helix-loop-helix motifs (EF-hands) are joined by a linker segment. The two EF-hands are further coupled through a short beta-type interaction between the two Ca(2+)-binding loops. Overall, the structure is very similar to that of the highly homologous native, minor A form of bovine calbindin D9k determined by X-ray crystallography [Szebenyi, D. M. E., & Moffat, K. (1986) J. Biol. Chem. 261, 8761-8776]. A model structure built from the bovine calbindin D9k crystal structure shows several deviations larger than 2 A from the experimental distance constraints for the porcine protein. These structural differences are efficiently removed by subjecting the model structure to the experimental distance and dihedral angle constraints in a restrained molecular dynamics protocol, thereby generating a model that is very similar to the refined distance geometry derived structures. The N-terminal residues of the intact protein that are absent in the minor A form appear to be highly flexible and do not influence the structure of other regions of the protein. This result is important because it validates the conclusions drawn from the wide range of studies that have been carried out on minor A forms rather than the intact calbindin D9k.     

Effect of trabecular curvature on the stiffness of trabecular bone

Simplified structural models of trabecular bone have been used to model various forms of trabecular variability. The structural effects of variability of direction, length and thickness of the trabeculae have been studied using 'lattice-type' finite element models. However, many of the trabeculae are not perfectly straight, and have a small degree of curvature. The objective of this study is to quantify the influence of small curvatures of the trabeculae on the effective modulus of trabecular bone, in the principal material direction. An analytical analysis of the effect of curvature on a single trabecula is performed, utilizing the concept of cellular-solid models. Closed-form expressions are derived for the effect of curvature on the flexibility in the principal material direction. For comparison, expressions are derived for the flexibility of a straight oblique element, representing angular variability. A quantitative comparison is presented, which is dependent on the thickness of the trabeculae. It was found that small curvatures have a large effect on the stiffness of the trabecular structure. This effect is largest for thin trabeculae, and decreases for thick trabeculae. The stiffness of the trabecular structure can be reduced by a factor of up to four for thin trabeculae and up to two for thick trabeculae, even for small curvatures. The flexibility of curved elements is found to be larger than the flexibility of oblique elements with similar eccentricities. Thus it seems that curvature might play a role in determining the effective modulus of trabecular bone.     

Determination of the upper and lower limits of the mechanistic stoichiometry of incompletely coupled fluxes. Stoichiometry of incompletely coupled reactions

A rationale is formulated for the design of experiments to determine the upper and lower limits of the mechanistic stoichiometry of any two incompletely coupled fluxes J1 and J2. Incomplete coupling results when there is a branch at some point in the sequence of reactions or processes coupling the two fluxes. The upper limit of the mechanistic stoichiometry is given by the minimum value of dJ2/dJ1 obtained when the fluxes are systematically varied by changes in steps after the branch point. The lower limit is given by the maximum value of dJ2/dJ1 obtained when the fluxes are varied by changes in steps prior to the branch point. The rationale for determining these limits is developed from both a simple kinetic model and from a linear nonequilibrium thermodynamic treatment of coupled fluxes, using the mechanistic approach [Westerhoff, H. V. & van Dam, K. (1979) Curr. Top. Bioenerg. 9, 1-62]. The phenomenological stoichiometry, the flux ratio at level flow and the affinity ratio at static head of incompletely coupled fluxes are defined in terms of mechanistic conductances and their relationship to the mechanistic stoichiometry is discussed. From the rationale developed, experimental approaches to determine the mechanistic stoichiometry of mitochondrial oxidative phosphorylation are outlined. The principles employed do not require knowledge of the pathway or the rate of transmembrane leaks or slippage and may also be applied to analysis of the stoichiometry of other incompletely coupled systems, including vectorial H+/O and K+/O translocation coupled to mitochondrial electron transport.     

Role of the membrane cortex in neutrophil deformation in small pipets.

The simplest model for a neutrophil in its "passive" state views the cell as consisting of a liquid-like cytoplasmic region surrounded by a membrane. The cell surface is in a state of isotropic contraction, which causes the cell to assume a spherical shape. This contraction is characterized by the cortical tension. The cortical tension shows a weak area dilation dependence, and it determines the elastic properties of the cell for small curvature deformations. At high curvature deformations in small pipets (with internal radii less than 1 micron), the measured critical suction pressure for cell flow into the pipet is larger than its estimate from the law of Laplace. A model is proposed where the region consisting of the cytoplasm membrane and the underlying cortex (having a finite thickness) is introduced at the cell surface. The mechanical properties of this region are characterized by the apparent cortical tension (defined as a free contraction energy per unit area) and the apparent bending modulus (introduced as a bending free energy per unit area) of its middle plane. The model predicts that for small curvature deformations (in pipets having radii larger than 1.2 microns) the role of the cortical thickness and the resistance for bending of the membrane-cortex complex is negligible. For high curvature deformations, they lead to elevated suction pressures above the values predicted from the law of Laplace. The existence of elevated suction pressures for pipets with radii from 1 micron down to 0.24 micron is found experimentally. The measured excess suction pressures cannot be explained only by the modified law of Laplace (for a cortex with finite thickness and negligible bending resistance), because it predicts unacceptable high cortical thicknesses (from 0.3 to 0.7 micron). It is concluded that the membrane-cortex complex has an apparent bending modulus from 1 x 10(-18) to 2 x 10(-18) J for a cortex with a thickness from 0.1 micron down to values much smaller than the radius of the smallest pipet (0.24 micron) used in this study.     

Hydration of DMPC and DPPC at 4 degrees C produces a novel subgel phase with convex-concave bilayer curvatures

Hydration of dimyristoyl- and dipalmitoylphosphatidylcholines at 4 degrees C results in the formation of a characteristic subgel phase designated Pcc. Examination of the phase by freeze-fracture electron microscopy shows convex-concave deformations of the planar bilayer which are of two types. A smaller type with a radius of curvature of about 20 nm predominates in DMPC, and a larger type with about 70 nm radii of curvatures dominates in DPPC. The Pcc phase can also be formed in samples hydrated at temperatures above the main phase transition if the dispersion is frozen slowly and subsequently incubated at 4 degrees C for several days. The subgel Pcc phase was distinguished from the subgel Lc phase by the temperature of transition, packing of the acyl chains on the basis of wide-angle X-ray diffraction, and 2H-NMR spectra characteristic of a 'solid-ordered' phase. Vibrational spectra of the carbonyl and phosphate regions are consistent with a partially reduced hydration state. The origin of the convex-concave bilayer deformation is believed to result from constraints imposed by limiting hydration of the headgroup and a frustration arising from the spontaneous curvature of both monolayers.     

Three-Dimensional Geometrical Modelling of Wild Boar Head by Reverse Engineering Technology

In this paper, a wild boar head was taken as the bionic research object for the development of new ridgers, a kind of plough. The reverse engineering technology was adopted to obtain the surface geometrical information of the head. Several three-dimensional （3D） point clouds of the head were captured first using a non-touch laser scanner, and an integrated point cloud was generated by aligning these point clouds using UG/Imageware. Then, the digital surface model of the head was rebuilt by means of CATIA. The characteristic curves of the surface model were analyzed. The results show that the average error between the rebuilt surface and the point cloud is -0.431 ram. The max curvature of the ridge on the neb of the head is 0.187 mm^-1, and the max and rain Gauss curvatures on the surface are 0.008 mm^-2 and -0.002 mm^-2. These geometrical information are the essential parameters for biomimetics study of the ridger.     

Palaeomagnetic stratigraphy of Pliocene continental deposits of the Bolivian Altiplano

Fluvio-lacustrine post-Miocene deposits of the Bolivian Altiplano have been sampled in 400 m of section in the La Paz and Avo Avo basins in order to provide a magnetostratigraphic scheme for chronostratigraphic interpretations. Stable characteristics components were isolated after stepwise AF or thermal demagnetizations. The directions of cleaned magnetization were compared to the normal and reverse dipole field directions at the site. The plot of the mean directions calculated for each stratigraphic level define successive magnetozones which can be identified on the standard geomagnetic polarity scale with the help of the K/Ar ages obtained on tuff layers. The deposition of the middle and upper part of the La Paz Formation covers the Gauss epoch. The Gauss/Matuyama (2.48 Ma) and the Gilbert/Gauss (3.4 Ma) limits are located in the top and the middle of the formation. The occurrence of the first glacial/interglacial oscillation can be placed in the early Matuyama (ca 2.2 Ma) in agreement with the oxygen isotopic record from the equatorial ocean.     

Analytical model of corneal surgery

We present a model of the human cornea in order to study the changes in its shape resulting from surgical operations (e.g., radial keratotomy). A simple closed-form solution is given for a thin linearly elastic spherical shell model of the cornea. We assume axisymmetry and isotropy in the shell surface. The surgery is modeled by permitting Young's modulus and shell thickness to depend on position. The analytical nature of the solution permits principal shell curvatures to be explicitly calculated. The model is used to investigate the effect of surgery on corneal flattening and the associated sensitivity to intraocular pressure changes.     

Equilibrium Shape Equation and Geometrically Permissible Condition for Two-Component Lipid Bilayer Vesicles

Equilibrium shapes of vesicles composed of a mixture of partially miscible amphiphiles are investigated. To take into account the influences of the composition, a simple phenomenological coupling between the co mposition and the curvatures, including the mean curvature and the Gauss curvature of the membrane surface, is suggested. By minimizing the potential functional, the general shape equation is obtained and solved analytically for vesicles with simple shapes. Besides, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are put forward. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. The close relations between the predictions and existing experimental phenomena published recently are shown.     

Determination of nucleic acid backbone conformation by 1H NMR.

In DNA or RNA duplexes, the six-bond C3'-O3'-P-O5'-C5'-C4'-C3' backbone linkage connecting adjacent residues contains six torsion angles (epsilon, zeta, alpha, beta, gamma, delta) but only four protons. This seriously limits the ability to define the backbone conformation by NMR using purely 1H-1H distance geometry (DG) methods. The problem is further compounded by the inability to assign two of the four backbone protons, namely the poorly resolved H5' and H5' protons, and invariably leads to DG structures with poorly defined backbone conformations. We have developed and tested a reliable method to constrain the beta, gamma, and epsilon (and indirectly alpha and zeta) backbone torsion angles by lower-bound NOE distances to unassigned H5'/H5' resonances combined with either 1H line widths or the conservative use of sigma J measurements; the method relies only on 1H 2-D NMR data, does not involve any structural assumptions, and leads to much improved backbone convergence among DG structures. The C4'-C5' torsion angle gamma is constrained by lower-bound NOE distances from H2' and from H6/H8 to any H5'/H5', as well as by sigma JH4, coupling measurements in the 3.9-4.4 ppm region; delta is constrained by H1'-H4' NOE distances and by H3'-H4' and H3'-H2' J couplings in COSY data; epsilon is partially constrained by H3' line width and/or further constrained by subtracting the minimum possible sigma JH3'-H from the observed sigma JH3' (COSY) to arrive at the maximum possible JH3'-P, which is then converted to H3'-P distance bounds. The angle beta is partially constrained via H5'-P and H5'-P distance bounds consistent with the maximum H5'-P and H5'-P J couplings derived from the observed H5' and H5' line widths, while alpha and zeta are indirectly constrained by lower distance bounds on the observed (n)H1' to (n + 1)H5'/H5' NOEs combined with the prior partial constraints on beta, gamma, delta, and epsilon. The combined effects of these additional constraints in determining distance geometry structures have been demonstrated using a 12-base duplex, [d(GCCGTTAACGGC)]2. Coordinate RMSDs per atom between structures refined with these constraints from random-embedded DG structures, from ideal A-DNA, and from B-DNA starting structures were less than 0.4 A for the central 8 base pairs indicating good convergence. All backbone angles for the central 8 base pairs are very well constrained with less than 10 degrees variation in any of the 48 torsion angles.