Ralph O. Erickson

Kinematics of plant growth

Abstract Many of the concepts and equations which have been used in the study of compressible fluids can be applied to problems of plant development. Growth field variables, i.e. functions of position in the plant and of time, can be specified in either Eulerian (spatial) or Lagrangian (material) terms. The two specifications coincide only when the spatial distribution of the variable is steady, and steady patterns are most likely to emerge when an apex is chosen as origin of the co-ordinate system. The growth field itself can be described locally by the magnitude and orientation of the principal axes of the rate of strain tensor and by the vorticity tensor. Material derivatives can be calculated if the temporal and spatial variation in both growth velocity, u (rate of displacement from a material origin), and the variable of interest are known. The equation of continuity shows the importance of including both growth velocity, u, and growth rate, ▽ ·u in estimates of local biosynthesis and transport rates in expanding tissue, although the classical continuity equation must be modified to accommodate the compartmentalized distributions characteristic of plant tissue. Relatively little information on spatial variation in plant organs can be found in the botanical literature, but the current availability of interactive computer graphics equipment suggests that analysis of the spatial distribution of growth rates at least is no longer difficult.

Tubular Packing of Spheres in Biological Fine Structure

The symmetrical arrangements of monomers into such cylindrical structures as microfilaments of actin, flagella of bacteria, microtubules of many organisms, and the protein coats of viruses can be specified by citing the index numbers of two or three sets of contact parastichies, or helical ranks of monomers, as has been done in classical studies of phyllotaxis. This specification has the form k(m, n) or k(m, n, m+n), where m, n, and (m+n) are parastichy numbers specifying screw displacements, and k is the jugacy, or frequency of rotational symmetry. For simple structures, k = 1. This notation has the advantage of terseness and of indicating the basic isometries of these helically symmetrical structures. Theoretical models of the packing of spheres whose centers lie on the surface of a cylinder have been investigated geometrically. Their symmetry properties are discussed. Parameters of these models, such as the angular divergence, α, the longitudinal displacement between successive spheres, h, the radius of the cylinder, and the angles of inclination of the parastichies, have been computed for representative patterns. The ultrastructural symmetry of several biological structures of this sort has been inferred by comparison with these models. Actin, for example, has the symmetry (1, 2), Salmonella flagella, 2(2, 3, 5), the tobacco mosaic virus, (1, 16, 17) and the microtubules of many higher organisms, (6, 7, 13).

Nucleic acids in relation to cell division in Lilium longiflorum

Methods are described for preparing cell suspensions of Lilium microsporocytes, microspores and pollen grains; for obtaining cell counts of these suspensions; and for their analysis for pentose nucleic acid (PNA) and desoxypentose nucleic acid (DNA). The results of these analyses have been calculated to nucleic acid content in μμg per microsporocyte, microspore or pollen grain, and the results related to logarithm of flower bud length, an index of the developmental status of the cells, and of their temporal relationship to meiosis, microspore mitosis and opening of the flower. DNA content per cell drops sharply at the end of meiosis, with the formation of four microspores from each microsporocyte. It then increases gradually during the microspore interphase between meiosis and the microspore mitosis. At microspore mitosis DNA content doubles rapidly. In the development of the resulting binucleate pollen grain, from microspore mitosis until the opening of the flower, there is a further gradual increase of DNA content. PNA content of these cells follows the same pattern up to microspore mitosis at a level about twice that of DNA, increases sharply at mitosis, and continues to increase rapidly at a rate nine times that for DNA in the maturing pollen grain. The absolute amounts of DNA and PNA are great. At the time of anthesis the two-celled pollen grain contains about 375 μμg of DNA and 1705 μμg of PNA.

Tubular arrays of spheres: Geometry, continuous and discontinuous contraction, and the role of moving dislocations☆

Abstract Microtubules, bacterial flagella, viral capsids, and other biological structures are tubular packings of subunits. The subunits form helical rows or parastichies. Such arrays are modeled as tubular packings of spheres. Parastichies along which spheres are in contact provide a symbol for a particular tubule. A tubule with hexagonal packing and k -fold rotational symmetry about its axis has the symbol k ( m ; m + n ; n ) where km , k ( m + n ), and kn represent the three sets of contact parastichies. A tubule with rhombic packing has two sets of contact parastichies and the symbol k ( m ; n ). The symbol determines the chirality. Two processes are described by which tubules can be interconverted: continuous contraction and discontinuous contraction. In continuous contraction of a hexagonally packed tubule, contacts along km -, k ( m + n )-, or kn -parastichies are broken uniformly throughout the tubule. The tubule becomes rhombic and undergoes twisting and change of length and radius. Continuation of the process converts the intermediate rhombic packing into a new hexagonal packing. Any tubule can be converted to any other having the same rotational symmetry k , by one or more steps of continuous contraction, but not to a tubule with different k . With respect to change in length, continuous contraction from one hexagonal packing to another may be either monotonic or non-monotonic. A step of discontinuous contraction of a hexagonally packed tubule is mediated by passage of an edge dislocation through the tubule, by glide or climb. The presence of a single edge dislocation in a tubule divides it into two parts with two different packings. Passage of the dislocation to one end or the other of the tubule converts the entire tubule into one packing or the other. Any tubule may be converted to any other, regardless of k , by one or more steps of discontinuous contraction. Maps showing possible paths of continuous and discontinuous contraction summarize the relationships among tubules. The analysis will provide a useful basis for studying particular biological cases of contraction.

Phyllotaxis in Xanthium Shoots Altered by Gibberellic Acid

Gibberellic acid treatment of vegetative Xanthium shoots induced a change in phyllotaxis and almost doubled the rate of leaf production. Phyllotaxis in control plants displayed a 2,3 contact parastichy pattern; that of the treated plants could be approximated with a 3,5 pattern. Thus, the Xanthium apex switched to a new mode of growth and a higher order of phyllotactic leaf arrangement not seen in untreated plants. It may be inferred from these experiments that gibberellic acid plays a role in determining the site of leaf initiation.

Polyhedral cell Shapes

Many authors have discussed the shapes of cells of the parenchyma of plants, and remarked on their similarity of form to bubbles in foam, to the grains of certain metal alloys, and to compressed lead shot, balls of “Plasticene,” and of peas, which have imbibed water while packed in a closed container. In all of these cases the “cells” have a great variety of polyhedral forms, and are closely packed so as to fill space. I will review some of the studies of cell shapes which have been published, and the ideas which have been proposed to account for these shapes. A partial enumeration of the possible polyhedra will be discussed, and some consideration will be given to the packing of these polyhedra.