scholarly journals Spatial variations of growth within domes having different patterns of principal growth directions

2014 ◽  
Vol 56 (4) ◽  
pp. 611-623 ◽  
Author(s):  
Jerzy Nakielski
Keyword(s):  
Growth Rate ◽  
Coordinate System ◽  
Growth Rates ◽  
Relative Rate ◽  
Spatial Variations ◽  
Coordinate Systems ◽  
Dome A ◽  
Rate Of Growth ◽  
Natural Coordinate System ◽  
Natural Coordinate

Growth rate variations for two paraboloidal domes: A and B, identical when seen from the outside but differing in the internal pattern of principal growth directions, were modeled by means of the growth tensor and a natural coordinate system. In dome A periclinal trajectories in the axial plane were given by confocal parabolas (as in a tunical dome), in dome B by parabolas converging to the vertex (as in a dome without a tunica). Accordingly, two natural coordinate systems, namely paraboloidal for A and convergent parabolic for B, were used. In both cases, the rate of growth in area on the surfaces of domes was assumed to be isotropic and identical in corresponding points. It appears that distributions of growth rates within domes A and B are similar in their peripheral and central parts and different only in their distal regions. In the latter, growth rates are relatively large; the maximum relative rate of growth in volume is around the geometric focus in dome A, and on the surface around the vertex in dome B.

scholarly journals Modeling of spatial variations of growth within apical domes by means of the growth tensor. L Growth specified on dome axis

2014 ◽  
Vol 53 (1) ◽  
pp. 17-28 ◽  
Author(s):  
Zygmunt Hejnowicz ◽  
Jerzy Nakielski ◽  
Krystyna Hejnowicz
Keyword(s):  
Growth Rates ◽  
Distal Region ◽  
Spatial Variations ◽  
Coordinate Systems ◽  
Prolate Spheroidal ◽  
Dome A ◽  
Lower Growth ◽  
Apical Dome ◽  
Curvilinear Coordinate ◽  
Natural Coordinate

By using the growth tensor and a natural curvilinear coordinate system for description of the distribution of growth in plant organs, three geometric types of shoot apical domes (parabolic, elliptical and hyperbolic) were modeled. It was assumed that apical dome geometry remains unchanged during growth and that the natural coordinate systems are paraboloidal and prolate spheroidal. Two variants of the displacement velocity fields V were considered. One variant is specified by a constant relative elemental rate of growth along the axis of the dome. The second is specified by a rate increasing proportionally with distance from the geometric focus of the coordinate systems (and the apical dome). The growth tensor was used to calculate spatial variations of growth rates for each variant of each dome type. There is in both variants a clear tendency toward lower growth rates in the distal region of the dome. A basic condition for the existence of a tunica is met.

scholarly journals Modeling of spatial variations of growth within apical domes by means of the growth tensor. II. Growth specified on dome surface

2014 ◽  
Vol 53 (3) ◽  
pp. 301-316 ◽  
Author(s):  
Zygmunt Hejnowicz ◽  
Jerzy Nakielski ◽  
Krystyna Hejnowicz
Keyword(s):  
Growth Rate ◽  
Coordinate System ◽  
Growth Rates ◽  
Relative Rate ◽  
Spatial Variations ◽  
Curvilinear Coordinate System ◽  
Volumetric Growth ◽  
Shoot Apices ◽  
Curvilinear Coordinate ◽  
Area Growth

Variations of the elemental relative rate of growth are modeled for parabolic, elliptic and hyperbolic domes of shoot apices by using the growth tensor in a suitable curvilinear coordinate system when the mode of area growth on the dome surface is known. Variations of growth rates within the domes are obtained in forms of computer-made maps for the following variants of growth on the dome surface: (1) constant meridional growth rate, (2) isotropic area growth, (3) anisotropy of area growth which becomes more intensive with increasing distance from the vertex. In variants 1 and 2 a maximum of volumetric growth rate appears in the center of the dome. Such a distribution of growth seems to be unrealistic. However, the corresponding growth tensors are interesting because they can be used in combination with other growth tensors to get the expected minimum volumetric growth rate in the dome center.

scholarly journals Variations of growth in shoot apical domes of spruce seedlings: A study using the growth tensor

2014 ◽  
Vol 56 (4) ◽  
pp. 625-643 ◽  
Author(s):  
Jerzy Nakielski
Keyword(s):  
Surface Layer ◽  
Coordinate System ◽  
Parabolic System ◽  
Growth Rates ◽  
Distal Part ◽  
Seedling Age ◽  
Rate Of Growth ◽  
Natural Coordinate System ◽  
Natural Coordinate ◽  
Peripheral Regions

Variations of the relative elemental rate of growth within apical domes, for the case when dome geometry changes during development, were modeled. It was ascertained that: 1) the domes of spruce seedlings have a paraboloidal shape; 2) the shape is maintained during growth, but the domes become higher and wider; 3) the relative elemental rate of growth in area on dome surface is isotropic, as indicated by analysis of cell packets in the surface layer. These data were used in modeling by means of the growth tensor and natural coordinate system. Growth of the dome was considered a superposition: 1) of relatively fast steady shape growth, where the isotropy of growth in area on the surface of the dome, was determined, and 2) of relatively slow isogonic growth, which does not disturb the isotropy. The convergent parabolic system was selected as the natural coordinate system. Distributions of the growth rates in the form of computer-made maps for three domes differing in age, were obtained. It appears that the growth rates within the dome are relatively high in the distal part and smaller in the central and peripheral regions. This variation decreases progressively with seedling age when the dome becomes wider. The relative elemental rate of growth in volume, averaged for the whole dome, also decreases.

scholarly journals Trajectories of principal directions of growth, natural coordinate system in growing plant organ

2014 ◽  
Vol 53 (1) ◽  
pp. 29-42 ◽  
Author(s):  
Zygmunt Hejnowicz
Keyword(s):  
Cell Wall ◽  
Coordinate System ◽  
Symmetric Part ◽  
Coordinate Systems ◽  
Plant Organ ◽  
Different Types ◽  
Natural Coordinate System ◽  
Principal Directions ◽  
Periclinal Walls ◽  
Natural Coordinate

In symplasticly growing organs the principal directions of growth, which are indicated by the eigenvectors of the symmetric part of the growth tensor, can be associated with each positional point and joined up to form a network of orthogonal trajectories, unless the growth is isotropic. The trajectories represent a natural coordinate system suitable for description of growing organs. These trajectories often can be recognized in patterns of nonrandom alignments in the cell wall network: these alignments are normal to anticlinal and periclinal walls. Coordinate systems that fit the trajectories in different types of growing organ are listed.

A population birth-and-mutation process, I: explicit distributions for the number of mutants in an old culture of bacteria

1974 ◽  
Vol 11 (03) ◽  
pp. 437-444 ◽  
Author(s):  
Benoit Mandelbrot
Keyword(s):  
Growth Rate ◽  
Mutation Rate ◽  
Growth Rates ◽  
Random Variable ◽  
Mutation Process ◽  
Limit Condition ◽  
Rate Of Growth ◽  
Mathematical Techniques ◽  
First Time

Luria and Delbrück (1943) have observed that, in old cultures of bacteria that have mutated at random, the distribution of the number of mutants is extremely long-tailed. In this note, this distribution will be derived (for the first time) exactly and explicitly. The rates of mutation will be allowed to be either positive or infinitesimal, and the rate of growth for mutants will be allowed to be either equal, greater or smaller than for non-mutants. Under the realistic limit condition of a very low mutation rate, the number of mutants is shown to be a stable-Lévy (sometimes called “Pareto Lévy”) random variable, of maximum skewness ß, whose exponent α is essentially the ratio of the growth rates of non-mutants and of mutants. Thus, the probability of the number of mutants exceeding the very large value m is proportional to m –α–1 (a behavior sometimes referred to as “asymptotically Paretian” or “hyperbolic”). The unequal growth rate cases α ≠ 1 are solved for the first time. In the α = 1 case, a result of Lea and Coulson is rederived, interpreted, and generalized. Various paradoxes involving divergent moments that were encountered in earlier approaches are either absent or fully explainable. The mathematical techniques used being standard, they will not be described in detail, so this note will be primarily a collection of results. However, the justification for deriving them lies in their use in biology, and the mathematically unexperienced biologists may be unfamiliar with the tools used. They may wish for more details of calculations, more explanations and Figures. To satisfy their needs, a report available from the author upon request has been prepared. It will be referred to as Part II.

The Relative Growth of Parts in Palaemon Carcinus

1930 ◽  
Vol 7 (2) ◽  
pp. 165-174
Author(s):  
M. A. TAZELAAR
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Linear Measurements ◽  
Relative Growth ◽  
Male And Female ◽  
Relative Growth Rates ◽  
Rate Of Growth ◽  
The Difference ◽  
Rates Of Growth

Linear measurements of certain appendages and the carapace of P. carcinus were made and plotted in various ways. The following conclusions were drawn: 1. The cheliped shows heterogonic growth in both male and female, but more markedly in the male, the values of k being: male 1.8 and female 1.48 2. The pereiopods in both male and female are slightly heterogonic. The relative growth rates are graded from p3 to p5, that of p3 being slightly greater than that of p5 3. Of the ordinary pereiopods the rate of growth of p1 is the smallest in the male, but the largest in the female. 4. The difference between the rates of growth of p1 and p3 in male and female is greatest where the rate of growth in the heterogonic organ, the cheliped, is most excessive in the male. 5. The growth of the 3rd maxilliped is slightly negatively heterogonic, the value of k in the male being 0.93 and in the female 0.95. Hence there seems to be a correlation between the marked heterogony in the cheliped on the growth rate of neighbouring appendages. In those immediately posterior to the cheliped the growth rate is increased and in those anterior decreased.

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