Phyllotaxis Patterns Research Articles

Biomechanical Model of Morphogenesis of Spiral Phyllotaxis Patterns and its Computer’s Visualization

Biomechanical Model of Morphogenesis of Spiral Phyllotaxis Patterns and its Computer’s Visualization

Phyllotaxis transition over the lifespan of a palm tree using Magnetic Resonance Imaging (MRI) and Terrestrial Laser Scanning (TLS): the case of Jubaea chilensis

BackgroundJubaea chilensis (Molina) Baillon, is a uniquely large palm species endemic to Chile. It is under threatened status despite its use as an ornamental species throughout the world. This research seeks to identify the phyllotaxis of the species based on an original combination of non-destructive data acquisition technologies, namely Magnetic Resonance Imaging (MRI) in saplings and young individuals and Terrestrial Laser Scanning (TLS) in standing specimens, and a novel analysis methodology.ResultsTwo phyllotaxis parameters, parastichy pairs and divergence angle, were determined by analyzing specimens at different developmental stages. Spiral phyllotaxis patterns of J. chilensis progressed in complexity from parastichy pairs (3,2) and (3,5) in juvenile specimens and (5,3), (8,5) and (8,13) for adult specimens. Divergence angle was invariable and averaged 136.9°, close to the golden angle. Phyllotactic pattern changes associated with establishment phase, the adult vegetative and the adult reproductive phases were observed. Both technologies, MRI and TLS proved to be adequate for the proposed analysis.ConclusionsUnderstanding phyllotactic transitions may assist identification of developmental stages of wild J. chilensis specimens. The proposed methodology may also be useful for the study of other palm species.

Phyllotaxis-inspired nanosieves with multiplexed orbital angular momentum

Nanophotonic platforms such as metasurfaces, achieving arbitrary phase profiles within ultrathin thickness, emerge as miniaturized, ultracompact and kaleidoscopic optical vortex generators. However, it is often required to segment or interleave independent sub-array metasurfaces to multiplex optical vortices in a single nano-device, which in turn affects the device’s compactness and channel capacity. Here, inspired by phyllotaxis patterns in pine cones and sunflowers, we theoretically prove and experimentally report that multiple optical vortices can be produced in a single compact phyllotaxis nanosieve, both in free space and on a chip, where one meta-atom may contribute to many vortices simultaneously. The time-resolved dynamics of on-chip interference wavefronts between multiple plasmonic vortices was revealed by ultrafast time-resolved photoemission electron microscopy. Our nature-inspired optical vortex generator would facilitate various vortex-related optical applications, including structured wavefront shaping, free-space and plasmonic vortices, and high-capacity information metaphotonics.

Development of EEG Data-driven Generative Art Application for Real-time and Dynamic Interaction

Generative art is produced by procedural techniques. It has obtained a lot of attention since the beginning of computer graphics. Many works of art are inspired by nature, among which phyllotaxis is as well. It is a combination of mathematics and the beauty of nature. Not only can it be seen everywhere in nature, but also often appear in man-made objects, becoming part of culture or religion. This paper presents the development of an interactive generative art application that is created from a phyllotaxis pattern by using the user’s Electroencephalogram (EEG) data. When people are using it, it will allow them to more easily relax and achieve the function of art therapy. We tried to use EEG data to make an interactive installation art that creates phyllotaxis patterns that are projected on the wall. Everyone has a different state, the generated patterns are also different from person to person, which creates interesting interactive contents. In addition, sound can also be changed by EEG data to become dynamic and real-time contents.

Multidimensional generalization of phyllotaxis

The regular spiral arrangement of various parts of biological objects (leaves, florets, etc.), known as phyllotaxis, could not find an explanation during several centuries. Some quantitative parameters of the phyllotaxis (the divergence angle being the principal one) show that the organization in question is, in a sense, the same in a large family of living objects, and the values of the divergence angle that are close to the golden number prevail. This was a mystery, and explanations of this phenomenon long remained “lyrical”. Later, similar patterns were discovered in inorganic objects. After a series of computer models, it was only in the XXI century that the rigorous explanation of the appearance of the golden number in a simple mathematical model has been given. The resulting pattern is related to stable fixed points of some operator and depends on a real parameter. The variation of this parameter leads to an interesting bifurcation diagram where the limiting object is the SL(2;Z)-orbit of the golden number on the segment [0,1]. We present a survey of the problem and introduce a multidimensional analog of phyllotaxis patterns. A conjecture about the object that plays the role of the golden number is given.

The unified rule of phyllotaxis explaining both spiral and non-spiral arrangements.

Leaf-like appendages of different plant groups are arranged in common phyllotaxis patterns categorized into two types: spiral and non-spiral arrangements. The adaptive reason for this morphological convergence is unknown. In the non-spiral arrangement, the divergence angle between successive leaves is a simple fraction of 360°, e.g. distichy, decussate and whorled phyllotaxis. In the spiral arrangement, the divergence angle of nascent leaves at the shoot apex is fixed at the golden angle 137.5°, whereas those of the developed leaves varies within a sequence of Fibonacci fractions, such as 1/3, 2/5, 3/8, 5/13, etc. The optimality of the golden angle has been shown recently by assuming that the pattern of developed leaves varies during growth in a manner depending on the divergence angle of nascent leaves. Here we propose a unified rule of phyllotaxis to explain both types of arrangement: the developed leaves form vertical rows along the stem. In the non-spiral arrangement, nascent to developed leaves always follow this rule, so that the number of leaf rows is kept constant irrespective of stem growth. In the spiral arrangement, developed leaves attain this rule by adjusting the divergence angle from the golden angle. The spiral arrangement is adaptive in that the number of leaf rows varies during growth depending on shoot thickness.

Whole-heart coronary MR angiography using a 3D cones phyllotaxis trajectory.

To develop a 3D cones steady-state free precession sequence with improved robustness to respiratory motion while mitigating eddy current artifacts for free-breathing whole-heart coronary magnetic resonance angiography. The proposed sequence collects cone interleaves using a phyllotaxis pattern, which allows for more distributed k-space sampling for each heartbeat compared to a typical sequential collection pattern. A Fibonacci number of segments is chosen to minimize eddy current effects with the trade-off of an increased number of acquisition heartbeats. For verification, phyllotaxis-cones is compared to sequential-cones through simulations, phantom studies, and in vivo coronary scans with 8 subjects using 2D image-based navigators for retrospective motion correction. Simulated point spread functions and moving phantom results show less coherent motion artifacts for phyllotaxis-cones compared to sequential-cones. Assessment of the right and left coronary arteries using reader scores and the image edge profile acutance vessel sharpness metric indicate superior image quality and sharpness for phyllotaxis-cones. Phyllotaxis 3D cones results in improved qualitative image scores and coronary vessel sharpness for free-breathing whole-heart coronary magnetic resonance angiography compared to standard sequential ordering when using a steady-state free precession sequence.

Spatial regularity control of phyllotaxis pattern generated by the mutual interaction between auxin and PIN1.

Phyllotaxis, the arrangement of leaves on a plant stem, is well known because of its beautiful geometric configuration, which is derived from the constant spacing between leaf primordia. This phyllotaxis is established by mutual interaction between a diffusible plant hormone auxin and its efflux carrier PIN1, which cooperatively generate a regular pattern of auxin maxima, small regions with high auxin concentrations, leading to leaf primordia. However, the molecular mechanism of the regular pattern of auxin maxima is still largely unknown. To better understand how the phyllotaxis pattern is controlled, we investigated mathematical models based on the auxin–PIN1 interaction through linear stability analysis and numerical simulations, focusing on the spatial regularity control of auxin maxima. As in previous reports, we first confirmed that this spatial regularity can be reproduced by a highly simplified and abstract model. However, this model lacks the extracellular region and is not appropriate for considering the molecular mechanism. Thus, we investigated how auxin maxima patterns are affected under more realistic conditions. We found that the spatial regularity is eliminated by introducing the extracellular region, even in the presence of direct diffusion between cells or between extracellular spaces, and this strongly suggests the existence of an unknown molecular mechanism. To unravel this mechanism, we assumed a diffusible molecule to verify various feedback interactions with auxin–PIN1 dynamics. We revealed that regular patterns can be restored by a diffusible molecule that mediates the signaling from auxin to PIN1 polarization. Furthermore, as in the one-dimensional case, similar results are observed in the two-dimensional space. These results provide a great insight into the theoretical and molecular basis for understanding the phyllotaxis pattern. Our theoretical analysis strongly predicts a diffusible molecule that is pivotal for the phyllotaxis pattern but is yet to be determined experimentally.

A stochastic multicellular model identifies biological watermarks from disorders in self-organized patterns of phyllotaxis.

Exploration of developmental mechanisms classically relies on analysis of pattern regularities. Whether disorders induced by biological noise may carry information on building principles of developmental systems is an important debated question. Here, we addressed theoretically this question using phyllotaxis, the geometric arrangement of plant aerial organs, as a model system. Phyllotaxis arises from reiterative organogenesis driven by lateral inhibitions at the shoot apex. Motivated by recurrent observations of disorders in phyllotaxis patterns, we revisited in depth the classical deterministic view of phyllotaxis. We developed a stochastic model of primordia initiation at the shoot apex, integrating locality and stochasticity in the patterning system. This stochastic model recapitulates phyllotactic patterns, both regular and irregular, and makes quantitative predictions on the nature of disorders arising from noise. We further show that disorders in phyllotaxis instruct us on the parameters governing phyllotaxis dynamics, thus that disorders can reveal biological watermarks of developmental systems.

Acacia (wattle) and Cananga (ylang-ylang): from spiral to whorled and irregular (chaotic) phyllotactic patterns – a pictorial report

Phyllotaxis, i.e., the arrangement of leaves around the stem and leaf-like organs inside flowers is regular in most vascular plants. Thus, developmental models usually explain regular phyllotactic patterns such as Fibonacci spirals and decussate/whorled patterns that obey Hofmeister’s rule: primordia form as far away as possible from previously initiated primordia. However, flowering plants showing at first Fibonacci spirals or whorled phyllotaxes may switch to other patterns that lack an obvious order and thus may be called irregular or even chaotic. Vegetative shoot tips of various Australian wattles (<em>Acacia</em> spp., Leguminosae in eudicots) and flower buds of ylang-ylang (<em>Cananga odorata</em>) and other Annonaceae (basal angiosperms) provide examples of irregular patterning. This pictorial report provides food for thought for scientists interested in phyllotaxis patterns beyond the usual spiral and whorled patterns. Emphasis is given on irregular phyllotaxes that occur in wild-type plants, mainly correlated with geometrical parameters such as leaf and stamen primordia that are very small as compared to the size of their apical meristems. They call for additional explanatory models, combining auxin-driven development with geometrical constraints and biophysical processes.

Phyllotaxis triangular unit; phyllotactic transitions as the consequences of the apical wedge disclinations in a crystal-like pattern of the units

The complexity of helical phyllotaxis expressed in diverse phyllotactic patterns and the phenomenon of phyllotactic transformations were investigated theoretically. Two categories of transformations were distinguished: continuous and discontinuous, the latter being interpreted as the consequences of wedge disclinations present at the shoot apex when the number of initials changes. Phyllotaxis triangular unit was described as the common element of all patterns of helical phyllotaxis. It is proposed that the unit plays the same role in a process of the formation of phyllotactic pattern as the crystal basic unit does in a process of crystal growth.

Structural Behaviour of Single Layer Space Frame with Member Arrangement Following Spiral Phyllotaxis

The regular arrangement of the plant organs is botanically known as phyllotaxis. Among the types of phyllotaxis found in nature, spiral phyllotaxis is the most frequently found pattern. The unique arrangement of this pattern in plants has benefited most of the plant structure as it enables them to grow more efficiently and compatible for physical constraints such as moisture and amount of light falls. In this research, the unique spiral phyllotaxis pattern has been adopted as idea for member arrangement pattern for structural space frame system. Two type of spiral phyllotaxis pattern and one regular model are generated. Analysis using finite element software LUSAS has been carried out in order to determine the effect of member arrangement following spiral phyllotaxis on stresses and displacement in the space frames.

Local auxin biosynthesis regulation by PLETHORA transcription factors controls phyllotaxis in Arabidopsis

Lateral organ distribution at the shoot apical meristem defines specific and robust phyllotaxis patterns that have intrigued biologists and mathematicians for centuries. In silico studies have revealed that this self-organizing process can be recapitulated by modeling the polar transport of the phytohormone auxin. Phyllotactic patterns change between species and developmental stages, but the processes behind these variations have remained unknown. Here we use regional complementation experiments to reveal that phyllotactic switches in Arabidopsis shoots can be mediated by PLETHORA-dependent control of local auxin biosynthesis.

Repulsion pressure model and numerical simulation for spiral phyllotactic patterns of plants

We present biologically motivated simple models to study phyllotaxis patterns. A simplified MaxMin-principle is applied for determining new-born primordium appearance. We propose a new repulsion pressure model to interpret the angle movement of each primordium influenced by the repulsion from all other primordia or from the nearest left and right neighbors. For a large range of growing velocities, both mechanisms can generate uniformly packed patterns under suitable settings of repulsion parameters. Measurement of the pattern uniformity and demonstration of robustness are also provided.

Spiral phyllotaxis: The natural way to construct a 3D radial trajectory in MRI

While radial 3D acquisition has been discussed in cardiac MRI for its excellent results with radial undersampling, the self-navigating properties of the trajectory need yet to be exploited. Hence, the radial trajectory has to be interleaved such that the first readout of every interleave starts at the top of the sphere, which represents the shell covering all readouts. If this is done sub-optimally, the image quality might be degraded by eddy current effects, and advanced density compensation is needed. In this work, an innovative 3D radial trajectory based on a natural spiral phyllotaxis pattern is introduced, which features optimized interleaving properties: (1) overall uniform readout distribution is preserved, which facilitates simple density compensation, and (2) if the number of interleaves is a Fibonacci number, the interleaves self-arrange such that eddy current effects are significantly reduced. These features were theoretically assessed in comparison with two variants of an interleaved Archimedean spiral pattern. Furthermore, the novel pattern was compared with one of the Archimedean spiral patterns, with identical density compensation, in phantom experiments. Navigator-gated whole-heart coronary imaging was performed in six healthy volunteers. High reduction of eddy current artifacts and overall improvement in image quality were achieved with the novel trajectory.

Phyllotaxis instability – exploring the depths of first available space

The theoretical analysis of the consequences of the phyllotactic pattern being propagated according to the first available space rule has revealed that all monojugate patterns, with the exception of the main Fibonacci pattern, should become developmentally unstable in their low expressions. This fact explains why the main Fibonacci pattern plays the dominant role among other patterns of spiral phyllotaxis. The probability that the pattern becomes unstable varies for different patterns, which likely makes them more or less frequent, and thus easier or more difficult to encounter in nature. The unstable pattern inevitably transforms into another, as the computer simulations show. Theoretically predicted instability of low order phyllotaxis may be treated as one of the causes of natural ontogenetic transitions, occurring in plants. This, however, still does not explain why in nature some patterns with high order of phyllotaxis also change, quite readily one into the other, in shoot apical meristem’s ontogeny.

Quantum monodromy and pattern formation

Hamiltonian monodromy is known to be the first obstruction to the existence of global action coordinates in integrable systems. Its manifestation in quantum systems can be seen as characteristic defects of the regular lattice formed by the joint eigenvalues of mutually commuting quantum operators. The relation between topology of singular fibers of classical integrable fibrations and patterns formed by joint spectrum of corresponding quantum systems is discussed. The notion of the sign of ‘elementary monodromy defect’ is introduced on the basis of ‘cut and glue’ construction of the lattice defects. Special attention is paid to non-elementary defects which generically appear in phyllotaxis patterns and can be associated with plant morphology.

Diversity and Lability of Floral Phyllotaxis in the Pluricarpellate Families of Core Laurales (Gomortegaceae, Atherospermataceae, Siparunaceae, Monimiaceae)

Floral phyllotaxis of Laurales (Magnoliidae) is poorly and sometimes conflictingly documented, especially in the pluricarpellate families of the core Laurales (Gomortegaceae, Atherospermataceae, Siparunaceae, Monimiaceae). In this study four types of floral phyllotaxis were recovered: Fibonacci spiral, simple-whorled (decussate), complex-whorled, and irregular. Whorled and spiral phyllotaxis co‐occur in all families except Gomortegaceae and even vary within a species in some Mollinedioideae (Monimiaceae). Complex‐whorled floral phyllotaxis with two or more organs in a position where only one is expected and changes in merism are especially prominent in Atherospermataceae and Monimiaceae. The most elaborate complex‐whorled phyllotaxis pattern (leading to 8‐merous whorls) is present in flowers with a flat floral base. Presence of a hyperstigma is correlated with double positions in the perianth. Flowers with low organ number commonly have simple‐whorled phyllotaxis; flowers with high organ number have compl...

Stem fasciated, a Recessive Mutation in Sunflower (Helianthus annuus), Alters Plant Morphology and Auxin Level

Plant lateral organs such as leaves arise from a group of initial cells within the flanks of the shoot apical meristem (SAM). Alterations in the initiation of lateral organs are often associated with changes in the dimension and arrangement of the SAM as well as with abnormal hormonal homeostasis. A mutation named stem fasciated (stf) that affects various aspects of plant development, including SAM shape and auxin level, was characterized in sunflower (Helianthus annuus). F1, F2 and F3 generations were obtained through reciprocal crosses between stf and normal plants. For the genetic analysis, a chi2 test was used. Phenotypic observations were made in field-grown and potted plants. A histological analysis of SAM, hypocotyl, epicotyl, stem and root apical meristem was also conducted. To evaluate the level of endogenous indole-3-acetic acid (IAA), a capillary gas chromatography-mass spectrometry-selected ion monitoring analysis was performed. stf is controlled by a single nuclear recessive gene. stf plants are characterized by a dramatically increased number of leaves and vascular bundles in the stem, as well as by a shortened plastochron and an altered phyllotaxis pattern. By histological analysis, it was demonstrated that the stf phenotype is related to an enlarged vegetative SAM. Microscopy analysis of the mutant's apex also revealed an abnormal enlargement of nuclei in both central and peripheral zones and a disorganized distribution of cells in the L2 layer of the central zone. The stf mutant showed a high endogenous free IAA level, whereas auxin perception appeared normal. The observed phenotype and the high level of auxin detected in stf plants suggest that the STF gene is necessary for the proper initiation of primordia and for the establishment of a phyllotactic pattern through control of both SAM arrangement and hormonal homeostasis.

Cell Differentiation and Organ Initiation at the Shoot Apical Meristem

Plants continuously generate organs at the flanks of their shoot apical meristems (SAMs). The patterns in which these organs are initiated, also called patterns of phyllotaxis, are highly stereotypic and characteristic for a particular species or developmental stage. This stable, predictable behaviour of the meristem has led to the idea that organ initiation must be based on simple and robust mechanisms. This conclusion is less evident, however, if we consider the very dynamic behaviour of the individual cells. How dynamic cellular events are coordinated and how they are linked to the regular patterns of organ initiation is a major issue in plant developmental biology.