On the shapes of leaves

The properties of a function G(x, y) of 2-dimensional Cartesian coordinates x, y can be used to obtain complex shapes through a nonlinear iterative procedure. It is conjectured that at least some biological shapes are determined by this procedure, and those of leaves are considered as a test of the hypothesis. It is shown that this procedure will generate the shapes of even very complicated leaves such as those of the holly tree to a considerable degree of accuracy. Certain issues of evolutionary biology appear in our view in a clearer light according to this procedure, and these are discussed towards the end of the paper. © 1994 Wiley-Liss, Inc.     

Describing dispersal under habitat constraints: A randomization approach in lesser kestrels

Animal dispersal is usually studied with capture-mark-reencounter data, which provide information on realized dispersal but rarely on underlying processes. In this context, the unreliable assumption of all habitat being available is usually made when describing and analysing dispersal patterns. However, actual settlement options may be constrained by the spatial distribution of appropriate patches, so an important task to understand movement patterns is to adequately describe dispersal when the dispersers’ options are constrained by the sites that are available to them. Using a long-term monitored population of the migratory lesser kestrel, we show how randomization procedures can be used to describe dispersal strategies in such situations. This species breeds colonially in discrete patches, most individuals (83%) disperse from their natal colony, and dispersers tend to move short distances (median=7.2 km). Observed patterns (natal dispersal rates and median dispersal distances of birds emigrating from their natal colony) were compared with those expected from two null models of random settlement of individuals: in any colony available in the whole population, or within the subpopulation (cluster of colonies) of origin. Our simulations indicate that philopatry to the natal colony was much higher than expected under both null expectations, and observed distances were much lower than expected in the whole population. When individuals were constrained to settle within their natal subpopulation in the simulations, dispersal distances were longer than expected in females, but were higher or lower in males depending on year. Dispersal was not only constrained by the spatial distribution of settlement options, but specific hypotheses arise that can be helpful to design and conduct further research. These results challenge previous interpretations of observed dispersal patterns, which may not reflect free decisions of individuals but environmental or social constraints. We suggest using simulation procedures as a routine to advance in the understanding of dispersal ecology and evolution.     

The spatial distribution of soils across Europe: A fractal approach

In this work we investigate the applicability of fractal modeling to the soil surface system in Europe, as described by the European Soil Database (V2.0) and the soil classification of the World Reference Base [FAO, 1999. World Reference Base for Soil Resources. ISSS–ISRIC–FAO, Rome, World Soil Resources Report No. 84]. Firstly, we observe the power law patterns of the distribution of pedotaxa as well as their pedorichness–area relationships. After this, we consider the 45 most abundant soils that cover 92% of the surface of Europe and estimate the box-counting fractal dimension of the areas covered by each of the pedotaxa within three orders of magnitude. Our analysis shows a remarkable scaling behavior of soil distribution and strongly suggests the fractal nature of pedotaxa distribution across Europe. These findings lead us to propose that considering the pedotaxa fractal dimension could play an important role in the analysis of the complexity of European soil systems.     

Detecting clusters of different geometrical shapes in microarray gene expression data

MOTIVATION: Clustering has been used as a popular technique for finding groups of genes that show similar expression patterns under multiple experimental conditions. Many clustering methods have been proposed for clustering gene-expression data, including the hierarchical clustering, k-means clustering and self-organizing map (SOM). However, the conventional methods are limited to identify different shapes of clusters because they use a fixed distance norm when calculating the distance between genes. The fixed distance norm imposes a fixed geometrical shape on the clusters regardless of the actual data distribution. Thus, different distance norms are required for handling the different shapes of clusters. RESULTS: We present the Gustafson-Kessel (GK) clustering method for microarray gene-expression data. To detect clusters of different shapes in a dataset, we use an adaptive distance norm that is calculated by a fuzzy covariance matrix (F) of each cluster in which the eigenstructure of F is used as an indicator of the shape of the cluster. Moreover, the GK method is less prone to falling into local minima than the k-means and SOM because it makes decisions through the use of membership degrees of a gene to clusters. The algorithmic procedure is accomplished by the alternating optimization technique, which iteratively improves a sequence of sets of clusters until no further improvement is possible. To test the performance of the GK method, we applied the GK method and well-known conventional methods to three recently published yeast datasets, and compared the performance of each method using the Saccharomyces Genome Database annotations. The clustering results of the GK method are more significantly relevant to the biological annotations than those of the other methods, demonstrating its effectiveness and potential for clustering gene-expression data. AVAILABILITY: The software was developed using Java language, and can be executed on the platforms that JVM (Java Virtual Machine) is running. It is available from the authors upon request. SUPPLEMENTARY INFORMATION: Supplementary data are available at http://dragon.kaist.ac.kr/gk.     

Chloroplast movements in fern leaves: correlation of movement dynamics and environmental flexibility of the species

The movements of chloroplasts in response to varying levels and wavelengths of incident light were investigated in leaves of four fern species: Adiantum capillus-veneris, Adiantum caudatum, Adiantum diaphanum and Pteris cretica. In all of the species studied blue light induced chloroplast redistribution resulting in face and profile patterns that were typical of low and high fluence rates, respectively. Fluence rate response characteristics and the kinetics of transmission changes accompanying these blue-light-induced movements were similar to those observed in the leaves of higher plants. Only in A. capillus-veneris was the distribution of chloroplasts affected by red light. The response was of the weak-light type, irrespective of the light intensity. The most effective fluence rate for red light was found to be below 7·2 μmol m^–2 s^–1 (1 W m^–2). The effect of red light was far-red reversible, indicating phytochrome involvement. Chloroplast responses were more dynamic in A. capillus-veneris and P. cretica, the two species that exhibited higher environmental flexibility.     

A geometrical constraint approach for reproducing the native backbone conformation of a protein

It is known that the backbone conformation of a protein can be reproduced with precision once a correct contact map (two-dimensional representation showing residue pairs in contact) is given as geometrical constraints. There is, however, no way to infer the correct contact map for a protein of unknown structure. We started with one-dimensional constraints using the quantity N14 (the number of neighboring residues within the radius of 14 Å). Since the plot of N14 along a chain shows a good correlation with the corresponding amino acid sequence, the N14 profile obtained from the X-ray structure is predictable from the sequence. Construction of backbone conformations under a given N14 profile was carried out in the following two steps: (1) a contact map from the N14 profile was produced by taking the product of N14 values of every two residues; (2) backbone conformations were generated by applying the distance geometry technique to distance constraints given by the contact map. If present, disulfide bonds in a protein, as well as the secondary structure, were treated as additional constraints, and both cases with or without the additional information were examined. The method was tested for 11 proteins of known structure, and the results indicated that the reproduced conformation was fairly good, using an X-ray structure for comparison, for small proteins of less than 80 residues long. The basic assumption and effectiveness of the present method were compared with those of previous studies employing the geometrical constraint approach. It has become clear that the specific, one-dimensional information (e.g., N14 profile) is more effective than nonspecific, two-dimensional constraints, such as average interresidue distances between particular types of amino acids. © 1993 Wiley-Liss, Inc.     

Mathematical modelling of the formation of coated vesicles. A theoretical geometrical approach.

The mechanism by which flat hexagonal lattices of clathrin trimers transform into pentagonal/hexagonal spheres remains a mystery. In light of the geometrical nature of this process we have pursued a mathematical approach to the question. Through the geometrical analysis of flat hexagonal lattices we have discovered three possible forms of transformation to introduce curvature into the centre of the lattice: hub-centre transformation; hub-edge transformation; fringe transformation. Hub-edge and fringe transformations are used first to close the lattice while introducing localized curvature at the edges of the lattice. Hub-centre transformation is used after closure to relax the severely localized curvature generated during closure. This scheme not only maximizes the size of the coated vesicle generated, but also minimizes the number of transformations, thus minimizing the energy expended.     

Analytical solutions of Poisson's equation for realistic geometrical shapes of membrane ion channels.

Analytical solutions of Poisson's equations satisfying the Dirichlet boundary conditions for a toroidal dielectric boundary are presented. The electric potential computed anywhere in the toroidal conduit by the analytical method agrees with the value derived from an iterative numerical method. We show that three different channel geometries, namely, bicone, catenary, and toroid, give similar potential profiles as an ion traverses along their central axis. We then examine the effects of dipoles in the toroidal channel wall on the potential profile of ions passing through the channel. The presence of dipoles eliminates the barrier for one polarity of ion, while raising the barrier for ions of the opposite polarity. We also examine how a uniform electric field from an external source is affected by the protein boundary and a mobile charge. The channel distorts the field, reducing it in the vestibules, and enhancing it in the constricted segment. The presence of an ion in one vestibule effectively excludes ions of the same polarity from that vestibule, but has little effect in the other vestibule. Finally, we discuss how the solutions we provide here may be utilized to simulate a system containing a channel and many interacting ions.     

MartiTracks: a geometrical approach for identifying geographical patterns of distribution

Panbiogeography represents an evolutionary approach to biogeography, using rational cost-efficient methods to reduce initial complexity to locality data, and depict general distribution patterns. However, few quantitative, and automated panbiogeographic methods exist. In this study, we propose a new algorithm, within a quantitative, geometrical framework, to perform panbiogeographical analyses as an alternative to more traditional methods. The algorithm first calculates a minimum spanning tree, an individual track for each species in a panbiogeographic context. Then the spatial congruence among segments of the minimum spanning trees is calculated using five congruence parameters, producing a general distribution pattern. In addition, the algorithm removes the ambiguity, and subjectivity often present in a manual panbiogeographic analysis. Results from two empirical examples using 61 species of the genus Bomarea (2340 records), and 1031 genera of both plants and animals (100118 records) distributed across the Northern Andes, demonstrated that a geometrical approach to panbiogeography is a feasible quantitative method to determine general distribution patterns for taxa, reducing complexity, and the time needed for managing large data sets.     

Plant morphogenesis: A geometrical model for the ramification

A geometrical model is proposed that describes the emergence of a primordium at the shoot apex in Dicotyledons. It is based on recent fundamental results on plant morphogenesis, viz.:

the emergence is preceded by the reorganization of the microtubules of the cortical cytoskeleton, leading to a new orientation of the synthesis of the cell wall microfibrils;

the resulting global stress is related to the general orientation of the cell growth;

The model sums up the continuous interactions that link the microtubules, the microfibrils and the cell growth axis. The paper tries to answer three essential questions:

Why does the principal stem shifts its growth direction after each lateral emergence?

Why do the three axes involved in any ramification (namely the old and the new principal stems and the lateral emergence) exhibit a plane configuration whereas this is an essentially three dimensional phenomenon?

Does phyllotaxis exclusively depend upon the local emergence of a primordium?

An interactive procedure between empirical botanical knowledge and the mathematical model leads to an insight of the compatibility mechanisms that link the various microtubules and microfibrils networks, and the apical dome restoration. A geometrical formalism allows a redefinition of both the “generating centre” and the “organizing centre”, and their field effect.     

Generalization of the theory of transition times in metabolic pathways: a geometrical approach.

Cell metabolism is able to respond to changes in both internal parameters and boundary constraints. The time any system variable takes to make this response has relevant implications for understanding the evolutionary optimization of metabolism as well as for biotechnological applications. This work is focused on estimating the magnitude of the average time taken by any observable of the system to reach a new state when either a perturbation or a persistent variation occurs. With this aim, a new variable, called characteristic time, based on geometric considerations, is introduced. It is stressed that this new definition is completely general, being useful for evaluating the response time, even in complex transitions involving periodic behavior. It is shown that, in some particular situations, this magnitude coincides with previously defined transition times but differs drastically in others. Finally, to illustrate the applicability of this approach, a model of a reaction mediated by an allosteric enzyme is analyzed.     

Modeling of movement-related potentials using a fractal approach

In bio-signal applications, classification performance depends greatly on feature extraction, which is also the case for electroencephalogram (EEG) based applications. Feature extraction, and consequently classification of EEG signals is not an easy task due to their inherent low signal-to-noise ratios and artifacts. EEG signals can be treated as the output of a non-linear dynamical (chaotic) system in the human brain and therefore they can be modeled by their dimension values. In this study, the variance fractal dimension technique is suggested for the modeling of movement-related potentials (MRPs). Experimental data sets consist of EEG signals recorded during the movements of right foot up, lip pursing and a simultaneous execution of these two tasks. The experimental results and performance tests show that the proposed modeling method can efficiently be applied to MRPs especially in the binary approached brain computer interface applications aiming to assist severely disabled people such as amyotrophic lateral sclerosis patients in communication and/or controlling devices.     

Biochemistry of fern spore germination: protease activity in ostrich fern spores

Protease activities were detected in quiescent and germinating spores of the ostrich fern (Matteuccia struthiopteris [L.] Todaro). Peak endopeptidase, aminopeptidase, and carboxypeptidase activities were detected 12 to 24 hours after spores began imbibing under light. There was a correlation between activities of proteases, the onset of a decline in levels of soluble protein, and an increase in levels of free amino acids. The earliest visible event of spore germination, breakage of the spore coat and protrusion of a rhizoid cell, was observed after peak protease activity, 48 to 72 hours after the start of imbibition. Results of this study demonstrate similarities in the pattern of protease activities during germination of ostrich fern spores to those of some seeds.     

Cell wall polysaccharides from fern leaves: evidence for a mannan-rich Type III cell wall in Adiantum raddianum

Primary cell walls from plants are composites of cellulose tethered by cross-linking glycans and embedded in a matrix of pectins. Cell wall composition varies between plant species, reflecting in some instances the evolutionary distance between them. In this work the monosaccharide compositions of isolated primary cell walls of nine fern species and one lycophyte were characterized and compared with those from Equisetum and an angiosperm dicot. The relatively high abundance of mannose in these plants suggests that mannans may constitute the major cross-linking glycan in the primary walls of pteridophytes and lycophytes. Pectin-related polysaccharides contained mostly rhamnose and uronic acids, indicating the presence of rhamnogalacturonan I highly substituted with galactose and arabinose. Structural and fine-structural analyses of the hemicellulose fraction of leaves of Adiantum raddianum confirmed this hypothesis. Linkage analysis showed that the mannan contains mostly 4-Man with very little 4,6-Man, indicating a low percentage of branching with galactose. Treatment of the mannan-rich fractions with endo-β-mannanase produced characteristic mannan oligosaccharides. Minor amounts of xyloglucan and xylans were also detected. These data and those of others suggest that all vascular plants contain xyloglucans, arabinoxylans, and (gluco)mannans, but in different proportions that define cell wall types. Whereas xyloglucan and pectin-rich walls define Type I walls of dicots and many monocots, arabinoxylans and lower proportion of pectin define the Type II walls of commelinoid monocots. The mannan-rich primary walls with low pectins of many ferns and a lycopod indicate a fundamentally different wall type among land plants, the Type III wall.