Finding the center of a phyllotactic pattern

The calculation of divergence angles between primordia in a plant apex depends on the point used as the center of the apex. In mathematically ideal phyllotactic patterns, the center is well defined but there has not been a precise definition for the center of naturally occurring phyllotactic patterns. A few techniques have been proposed for estimating the location of the center but without a precise definition for the center the accuracy of these methods cannot be known. This paper provides a precise definition that can be used as the center of a phyllotactic pattern and a numerical method which can accurately find it. These tools will make it easier to compare theory against experiment in phyllotaxis.     

A stochastic approach to phyllotactic patterns analysis

A statistical method is presented to characterize the degree of order in phyllotactic systems. We developed equations allowing the theoretical estimation of the number of leaves regularly distributed (spiral or verticillate) in a partially random phyllotactic system. The equations are simple and accurate enough to make quantitative predictions concerning the organization of different phyllotactic patterns (verticillate, distichous, spiral and random). This method can bring out patterns that are not visible a priori on a planar representation of the shoot apex. As a case study, the method was applied to the quantitative analysis of the sho mutants recently produced by Itoh et al. [2000. SHOOT ORGANIZATION genes regulate shoot apical meristem organization and the pattern of leaf primordium initiation in Rice. Plant Cell 12, 2161-2174]. By using our method, it was possible to predict the number of leaves distributed in distichous or random patterns on these phyllotactic mutants.     

Statistical recognition of random and regular phyllotactic patterns

AIMS: A statistical method used in ecology is adapted to characterize the degree of order in phyllotactic systems. SCOPE: The test consists of subdividing a planar projection of the stem apical meristem into 16 sectors and counting the number of primordia appearing in each. By dividing the sum of squared deviations by the mean number of primordia per sector the chi-square (chi2) is obtained. When there are a total number of 20 primordia, if the chi2 is less than 6.26, the phyllotaxis is spiral; if it is between 6.26 and 27.5 the phyllotaxis is random; and if it is greater than 27.5, the phyllotaxis is distichous or whorled (level of significance alpha = 5 %). It is also possible to remove one or more sectors. If there are k sectors, the two critical values delimiting the random zone will be found in a chi2 table for k - 1 degrees of freedom. CONCLUSIONS: The method is applied to the analysis of sho mutants described by Itoh et al. in 2000 (Plant Cell 12: 2161-2174). The results obtained are in agreement with the theoretical analysis showing that a whorled or spiral phyllotactic system may contain a certain number of randomly distributed elements without losing its regular global structure.     

Floral development and phyllotactic variation in Ceratophyllum demersum (Ceratophyllaceae)

The floral development of staminate and pistillate flowers of Ceratophyllum demersum was observed, with particular focus on the phyllotactic variation in staminate flowers, using scanning electronic microscopy (SEM). We discerned patterns of development of some important new morphological features, e.g., the difference and discontinuity between the organ initiation in stamens and that in bracts (or tepals) and the initial presence of a mucilaginous appendage on each pistil. Female flowers are considered to be very specialized through reduction. In male flowers stamen initiation changes between early and late floral development. The four or five stamens in the outermost whorl initiate first on the abaxial and lateral sides of the floral apex and only later on the adaxial side (unidirectional). Later the inner stamens initiate spirally, and this is the main pattern in the stamen initiation. Members of each whorl differ among themselves in time of initiation and in ultimate size. The phyllotactic variation in staminate flowers of Ceratophyllum, suggested by previous studies, is derived from the variation in stamen number and the difference of stamen initiation between the early and later stages. The development in Ceratophyllum has some similarities to those of ANITA plants except for Nymphaeales.     

Complexity and information in regular and random phyllotactic patterns.

In biology, the theory of information has been used to study the degree of order of many living systems. Different concepts of entropy have been applied to the analysis of phyllotaxis. In the present paper we will determine the degree of order of disorganized patterns by using informational entropy concepts deduced from the work of Brillouin, Shannon, and Yagil. As case studies, we will apply these concepts of entropy to the disorganized patterns found in mutants of Arabidopsis. The calculation of entropy gives a precise idea of the degree of order of a phyllotactic system.     

Geometrical relationships specifying the phyllotactic pattern of aquatic plants

The complete range of various phyllotaxes exemplified in aquatic plants provide an opportunity to characterize the fundamental geometrical relationships operating in leaf patterning. A new polar-coordinate model was used to characterize the correlation between the shapes of shoot meristems and the arrangements of young leaf primordia arising on those meristems. In aquatic plants, the primary geometrical relationship specifying spiral vs. whorled phyllotaxis is primordial position: primordia arising on the apical dome (as defined by displacement angles θ ≤ 90° during maximal phase) are often positioned in spiral patterns, whereas primordia arising on the subtending axis (as defined by displacement angles of θ ≥ 90° during maximal phase) are arranged in whorled patterns. A secondary geometrical relationship derived from the literature shows an inverse correlation between the primordial size?:?available space ratio and the magnitude of the Fibonacci numbers in spiral phyllotaxis or the number of leaves per whorl in whorled phyllotaxis. The data available for terrestrial plants suggest that their phyllotactic patterning may also be specified by these same geometrical relationships. Major exceptions to these correlations are attributable to persistent embryonic patterning, leaflike structures arising from stipules, congenital splitting of young primordia, and/or non-uniform elongating of internodes. The geometrical analysis described in this paper provides the morphological context for interpreting the phenotypes of phyllotaxis mutants and for constructing realistic models of the underlying mechanisms responsible for generating phyllotactic patterns.     

Leaf asymmetry as a developmental constraint imposed by auxin-dependent phyllotactic patterning

In a majority of species, leaf development is thought to proceed in a bilaterally symmetric fashion without systematic asymmetries. This is despite the left and right sides of an initiating primordium occupying niches that differ in their distance from sinks and sources of auxin. Here, we revisit an existing model of auxin transport sufficient to recreate spiral phyllotactic patterns and find previously overlooked asymmetries between auxin distribution and the centers of leaf primordia. We show that it is the direction of the phyllotactic spiral that determines the side of the leaf these asymmetries fall on. We empirically confirm the presence of an asymmetric auxin response using a DR5 reporter and observe morphological asymmetries in young leaf primordia. Notably, these morphological asymmetries persist in mature leaves, and we observe left-right asymmetries in the superficially bilaterally symmetric leaves of tomato (Solanum lycopersicum) and Arabidopsis thaliana that are consistent with modeled predictions. We further demonstrate that auxin application to a single side of a leaf primordium is sufficient to recapitulate the asymmetries we observe. Our results provide a framework to study a previously overlooked developmental axis and provide insights into the developmental constraints imposed upon leaf morphology by auxin-dependent phyllotactic patterning.     

The PAR network: redundancy and robustness in a symmetry-breaking system

To become polarized, cells must first ‘break symmetry’. Symmetry breaking is the process by which an unpolarized, symmetric cell develops a singularity, often at the cell periphery, that is used to develop a polarity axis. The Caenorhabditis elegans zygote breaks symmetry under the influence of the sperm-donated centrosome, which causes the PAR polarity regulators to sort into distinct anterior and posterior cortical domains. Modelling analyses have shown that cortical flows induced by the centrosome combined with antagonism between anterior and posterior PARs (mutual exclusion) are sufficient, in principle, to break symmetry, provided that anterior and posterior PAR activities are precisely balanced. Experimental evidence indicates, however, that the system is surprisingly robust to changes in cortical flows, mutual exclusion and PAR balance. We suggest that this robustness derives from redundant symmetry-breaking inputs that engage two positive feedback loops mediated by the anterior and posterior PAR proteins. In particular, the PAR-2 feedback loop stabilizes the polarized state by creating a domain where posterior PARs are immune to exclusion by anterior PARs. The two feedback loops in the PAR network share characteristics with the two feedback loops in the Cdc42 polarization network of Saccharomyces cerevisiae.     

Cooperative symmetry-breaking by actin polymerization in a model for cell motility

Polymerizing networks of actin filaments are capable of exerting significant mechanical forces, used by eukaryotic cells and their prokaryotic pathogens to change shape or to move. Here we show that small beads coated uniformly with a protein that catalyses actin polymerization are initially surrounded by symmetrical clouds of actin filaments. This symmetry is broken spontaneously, after which the beads undergo directional motion. We have developed a stochastic theory, in which each actin filament is modelled as an elastic brownian ratchet, that quantitatively accounts for the observed emergent symmetry-breaking behaviour. Symmetry-breaking can only occur for polymers that have a significant subunit off-rate, such as the biopolymers actin and tubulin.     

Recognizing Sequences of Sequences

The brain''s decoding of fast sensory streams is currently impossible to emulate, even approximately, with artificial agents. For example, robust speech recognition is relatively easy for humans but exceptionally difficult for artificial speech recognition systems. In this paper, we propose that recognition can be simplified with an internal model of how sensory input is generated, when formulated in a Bayesian framework. We show that a plausible candidate for an internal or generative model is a hierarchy of ‘stable heteroclinic channels’. This model describes continuous dynamics in the environment as a hierarchy of sequences, where slower sequences cause faster sequences. Under this model, online recognition corresponds to the dynamic decoding of causal sequences, giving a representation of the environment with predictive power on several timescales. We illustrate the ensuing decoding or recognition scheme using synthetic sequences of syllables, where syllables are sequences of phonemes and phonemes are sequences of sound-wave modulations. By presenting anomalous stimuli, we find that the resulting recognition dynamics disclose inference at multiple time scales and are reminiscent of neuronal dynamics seen in the real brain.     

Synthetic Turing protocells: vesicle self-reproduction through symmetry-breaking instabilities

The reproduction of a living cell requires a repeatable set of chemical events to be properly coordinated. Such events define a replication cycle, coupling the growth and shape change of the cell membrane with internal metabolic reactions. Although the logic of such process is determined by potentially simple physico-chemical laws, modelling of a full, self-maintained cell cycle is not trivial. Here we present a novel approach to the problem that makes use of so-called symmetry breaking instabilities as the engine of cell growth and division. It is shown that the process occurs as a consequence of the breaking of spatial symmetry and provides a reliable mechanism of vesicle growth and reproduction. Our model opens the possibility of a synthetic protocell lacking information but displaying self-reproduction under a very simple set of chemical reactions.     

Spontaneous symmetry-breaking energy functions and the emergence of orientation selective cortical cells

The mammalian visual cortex is comprised of hypercolumns of orientation selective cells. The developmental process by which cells are generated with receptive fields tuned to a variety of orientations has so far remained a mystery. We present a model for the production of orientation selective cells that requires no external stimuli and a minimum of input parameters. The process involves spontaneous symmetry-breaking in an energy function that governs the maturation of the cortical cells in a multi-layer network of Hebb-type feedforward neurons. An important feature is that the symmetry breaking occurs for each cell separately and is not due to global organizing effects. We present examples of receptive field profiles calculated with the symmetry-breaking procedure and note that the results seem robust and may be useful in the study of development in several types of cortical cells. The inclusion of long range lateral (intra-layer) correlations in the energy function could result in the development of cell groups with correlated preferred orientations that resemble the hypercolumns seen in the visual cortex.     

Beyond Fibonacci patterns and the golden angle: phyllotactic variations and their cellular origin

Journal of Plant Research -     

Chiral symmetry-breaking transition in growth front of crystal phase of 1,1'-binaphthyl in its supercooled melt

Although the theory of spontaneous chiral symmetry-breaking in open systems was proposed some time ago, experimental realization of this phenomenon has not been achieved. In this article, we note that the crystal growth front of 1,1'-binaphthyl shows many of the characteristics of an open system in which chiral symmetry-breaking has occurred. We studied the temperature profiles of the crystallizing surface and obtained X-ray diffraction data of the crystals grown from the melt under controlled conditions. The data show that, under appropriate conditions, the observed bimodal probability distribution of enantiomeric excess (ee) with maxima approximately 60% is due exclusively to chiral crystals and not due to racemic crystals of 1,1'-binaphthyl that can also form at large supercooling. The mass independence of the ee shows that the growing front maintains a constant ee, which is a clear signature of open systems in steady state. Chirality 16:131-136, 2004.     

基于编码序列、基因间序列和氨基酸序列构建的系统发生关系比较

以7种古菌、46种细菌和10种真核生物的基因组为样本,考虑碱基间的短程关联和长程关联作用,得到编码序列的密码对和基因间序列的三联体对中不同位点的二核苷酸频率,据此构建了基于编码序列和基因间序列的系统发生关系。无论是基于编码序列还是基因间序列对信息进行聚类,古菌或真核均被聚在一支上,表明聚类参数的选择是合适的;与基于氨基酸序列构建的系统发生关系进行两两比较,发现大部分硬壁菌的编码序列与基因间序列之间,以及编码序列与氨基酸序列之间的进化都存在较大差异。通过分析认为,只有综合考虑这三类序列的进化信息,才可能得到更自然的系统发生关系。