The interspecific mass–density relationship and plant geometry

We present an a priori theoretical framework for the interspecific allometric relationship between stand mass and plant population density. Our model predicts a slope of −\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\frac{1}{3}\end{equation*}\end{document} between the logarithm of stand mass and the logarithm of stand density, thus conflicting with a previously assumed slope of −½. Our model rests on a heuristic separation of resource-limited living mass and structural mass in the plant body. We point out that because of similar resource requirements among plants of different sizes, a nonzero plant mass–density slope is primarily defined by structural mass. Specifically, the slope is a result of (i) the physical size-dependent relationship between stem width and height, (ii) foliage-dependent demands of conductance, and (iii) the cumulative nature of structural mass. The data support our model, both when the potential sampling bias of taxonomic relatedness is accounted for and when it is not. Independent contrasts analyses show that observed relationships among variables are not significantly different from the assumptions made to build the model or from its a priori predictions. We note that the dependence of the plant mass–density slope on the functions of structural mass provides a cause for the difference from the zero slope found in the animal population mass–density relationship; for the most part, animals do not have a comparable cumulative tissue type.

The making of the architecture of the plant cell wall: How cells exploit geometry

Cell wall deposition is a key process in the formation, growth, and differentiation of plant cells. The most important structural components of the wall are long cellulose microfibrils, which are synthesized by synthases embedded in the plasma membrane. A fundamental question is how the microfibrils become oriented during deposition at the plasma membrane. The current textbook explanation for the orientation mechanism is a guidance system mediated by cortical microtubules. However, too many contraindications are known in secondary cell walls for this to be a universal mechanism, particularly in the case of helicoidal arrangements, which occur in many situations. An additional construction mechanism involves liquid crystalline self-assembly [A. C. Neville (1993) Biology of Fibrous Composites: Development Beyond the Cell Membrane (Cambridge Univ. Press, Cambridge, U.K.)], but the required amount of bulk material that is able to equilibrate thermally is not normally present at any stage of the wall deposition process. Therefore, we have asked whether the complex ordered texture of helicoidal cell walls can be formed in the absence of direct cellular guidance mechanisms. We propose that they can be formed by a mechanism that is based on geometrical considerations. It explains the genesis of the complicated helicoidal texture and shows that the cell has intrinsic, versatile tools for creating a variety of textures. A compelling feature of the model is that local rules generate global order, a typical phenomenon of life.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Specific atomic groups and RNA helix geometry in acceptor stem recognition by a tRNA synthetase

Oligonucleotides that recapitulate the acceptor stems of tRNAs are substrates for aminoacylation by many tRNA synthetases in vitro, even though these substrates are missing the anticodon trinucleotides of the genetic code. In the case of tRNAAla a single acceptor stem G⋅U base pair at position 3·70 is essential, based on experiments where the wobble pair has been replaced by alternatives such as I⋅U, G⋅C, and A⋅U, among others. These experiments led to the conclusion that the minor-groove free 2-amino group (of guanosine) of the G⋅U wobble pair is essential for charging. Moreover, alanine-inserting tRNAs (amber suppressors) that replace G⋅U with mismatches such as G⋅A and C⋅A are partially active in vivo and can support growth of an Escherichia coli tRNAAla knockout strain, leading to the hypothesis that a helix irregularity and nucleotide functionalities are important for recognition. Herein we investigate the charging in vitro of oligonucleotide and full-length tRNA substrates that contain mismatches at the position of the G⋅U pair. Although most of these substrates have undetectable activity, G⋅A and C⋅A variants retain some activity, which is, nevertheless, reduced by at least 100-fold. Thus, the in vivo assays are much less sensitive to large changes in aminoacylation kinetic efficiency of 3·70 variants than is the in vitro assay system. Although these functional data do not clarify all of the details, it is now clear that specific atomic groups are substantially more important in determining kinetic efficiency than is a helical distortion. By implication, the activity of mutant tRNAs measured in the in vivo assays appears to be more dependent on factors other than aminoacylation kinetic efficiency.

Geometry of circular vectors and pattern recognition of shape of a boundary

This paper deals with pattern recognition of the shape of the boundary of closed figures on the basis of a circular sequence of measurements taken on the boundary at equal intervals of a suitably chosen argument with an arbitrary starting point. A distance measure between two boundaries is defined in such a way that it has zero value when the associated sequences of measurements coincide by shifting the starting point of one of the sequences. Such a distance measure, which is invariant to the starting point of the sequence of measurements, is used in identification or discrimination by the shape of the boundary of a closed figure. The mean shape of a given set of closed figures is defined, and tests of significance of differences in mean shape between populations are proposed.

On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

Isotropic isotopy and symplectic null sets

Capacity is an important numerical invariant of symplectic manifolds. This paper studies when a subset of a symplectic manifold is null, i.e., can be removed without affecting the ambient capacity. After examples of open null sets and codimension-2 non-null sets, geometric techniques are developed to perturb any isotopy of a loop to a hamiltonian flow; it follows that sets of dimension 0 and 1 are null. For isotropic sets of higher dimensions, obstructions to the perturbation are found in homotopy groups of the orthogonal groups.

Convex polytopes and quantization of symplectic manifolds

Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered.

Fos and Jun bend the AP-1 site: effects of probe geometry on the detection of protein-induced DNA bending.

The effect of Fos and Jun binding on the structure of the AP-1 recognition site is controversial. Results from phasing analysis and phase-sensitive detection studies of DNA bending by Fos and Jun have led to opposite conclusions. The differences between these assays, the length of the spacer between two bends and the length of the sequences flanking the bends, are investigated here using intrinsic DNA bend standards. Both an increase in the spacer length as well as a decrease in the length of flanking sequences resulted in a reduction in the phase-dependent variation in electrophoretic mobilities. Probes with a wide separation between the bends and short flanking sequences, such as those used in the phase-sensitive detection studies, displayed no phase-dependent mobility variation. This shape-dependent variation in electrophoretic mobilities was reproduced by complexes formed by truncated Fos and Jun. Results from ligase-catalyzed cyclization experiments have been interpreted to indicate the absence of DNA bending in the Fos-Jun-AP-1 complex. However, truncated Fos and Jun can alter the relative rates of inter- and intramolecular ligation through mechanisms unrelated to DNA bending, confounding the interpretation of cyclization data. The analogous phase- and shape-dependence of the electrophoretic mobilities of the Fos-Jun-AP-1 complex and an intrinsic DNA bend confirm that Fos and Jun bend DNA, which may contribute to their functions in transcription regulation.