Modelling the growth of plants with a uniform growth logistics.

BACKGROUND AND AIMS: The increment model has previously been used to describe the growth of plants in general. Here, we examine how the same logistics enables the development of different superstructures. METHODS: Data from the literature are analyzed with the increment model. Increments are growth-invariant molecular clusters, treated as heuristic particles. This approach formulates the law of mass action for multi-component systems, describing the general properties of superstructures which are optimized via relaxation processes. RESULTS: The daily growth patterns of hypocotyls can be reproduced implying predetermined growth invariant model parameters. In various species, the coordinated formation and death of fine roots are modeled successfully. Their biphasic annual growth follows distinct morphological programs but both use the same logistics. In tropical forests, distributions of the diameter in breast height of trees of different species adhere to the same pattern. Beyond structural fluctuations, competition and cooperation within and between the species may drive optimization. CONCLUSION: All superstructures of plants examined so far could be reproduced with our approach. With genetically encoded growth-invariant model parameters (interaction with the environment included) perfect morphological development runs embedded in the uniform logistics of the increment model.

On the structure-bounded growth processes in plant populations.

If growing cells in plants are considered to be composed of increments (ICs) an extended version of the law of mass action can be formulated. It evidences that growth of plants runs optimal if the reaction-entropy term (entropy times the absolute temperature) matches the contact energy of ICs. Since these energies are small, thermal molecular movements facilitate via relaxation the removal of structure disturbances. Stem diameter distributions exhibit extra fluctuations likely to be caused by permanent constraints. Since the signal-response system enables in principle perfect optimization only within finite-sized cell ensembles, plants comprising relatively large cell numbers form a network of size-limited subsystems. The maximal number of these constituents depends both on genetic and environmental factors. Accounting for logistical structure-dynamics interrelations, equations can be formulated to describe the bimodal growth curves of very different plants. The reproduction of the S-bended growth curves verifies that the relaxation modes with a broad structure-controlled distribution freeze successively until finally growth is fully blocked thus bringing about "continuous solidification".

The general growth logistics of cell populations.

An increment model based on thermodynamics lays bare that the cell size distributions of archaea, prokaryotes and eukaryotes are optimized and belong to the same universal class. Yet, when a cell absorbs mass or signals are processed, these conditions are disturbed. Relaxation re-installs ideal growth conditions via an exponential process with a rate that slows down with the cell size. In a growing ensemble, a distribution of relaxation modes comes in existence, exactly defined by the universal cell size distribution. The discovery of nano-mechanic acoustic activities in cells led us to assume that in a growing ensemble acoustic signals may contribute significantly to the transmission of essential information about growth-induced disturbances to all cells, initiating that way coordinated relaxation. The frequency increases with the cell number shortening the period between successive signals. The completion of rearrangements occurring at a constant rate is thus progressively impaired, until cellular growth stops, totally. Due to this phenomenon, the so-called "relaxation-frequency-dispersion" cell colonies should exhibit a maximum cell number. In populations with large cell numbers, subsystems, behaving similar-like colonies, should form network-like patterns. Based on these ideas, we formulate equations that describe the growth curves of all cell types, verifying that way the general nature of the growth logistics.

Stationary cell size distributions and mean protein chain length distributions of Archaea, Bacteria and Eukaryotes described with an increment model in terms of irreversible thermodynamics.

In terms of an increment model irreversible thermodynamics allows to formulate general relations of stationary cell size distributions observed in growing colonies. The treatment is based on the following key postulates: i) The growth dynamics covers a broad spectrum of fast and slow processes. ii) Slow processes are considered to install structural patterns that operate in short periods as temporary stationary states of reference in the sense of irreversible thermodynamics. iii) Distortion during growth is balanced out via the many fast processes until an optimized stationary state is achieved. The relation deduced identifies the numerous different stationary patterns as equivalents, predicting that they should fall on one master curve. Stationary cell size distributions of different cell types, like Hyperphilic archaea, E. coli (Prokaryotes) and S. cerevisiae (Eukaryotes), altogether taken from the literature, are in fact consistently described. As demanded by the model they agree together with the same master curve. Considering the "protein factories" as subsystems of cells the mean protein chain length distributions deduced from completely sequenced genomes should be optimized. In fact, the mean course can be described with analogous relations as used above. Moreover, the master curve fits well to the patterns of different species of Archaea, Bacteria and Eukaryotes. General consequences are discussed.

Stained native collagen: interpretation of the small angle X-ray scattering and electron microscopic pictures in terms of the primary structure.

The average axial electron-density distribution of stained collagen fibers derived from X-ray small angle pattern, is shown to be substantially related to the projection of the polar amino acid residues onto the axis of the microfibril taking into consideration the staggering of neighbouring molecules. In order to arrive ultimately at a quantitative representation of the experimental data, an additional periodic fluctuation of the density in direction of the fibril axis is required. The necessity of this regular microphase structure for an understanding of known physical properties of collagen fibers will be indicated.