Morphogenesis in biological systems involves interacting biochemical and biophysical mechanisms that operate at a range of temporal and spatial scales. In particular, recent experiments indicate that phyllotactic patterns (the arrangement and shapes of leaf buds at plant apices) arise from nonhomogeneous growth controlled by multiple mechanisms. This work develops models for phyllotactic patterning based on experimental insights on the interaction between biochemistry and biomechanics in plant morphogenesis, and derives equations governing defects in phyllotactic patterns. Data will be gathered on the shape of leaf buds, wavelet methods applied to analyze the patterns, the model tested experimentally with gels that mimic biological growth, and the packing properties of the patterns studied. Models for the growth of individual plant cells will also be extended to models for the growth of cells in tissue. Mathematical techniques involved include multiple scales analysis and tools from dynamical systems theory. Collaborators include Isaac Chenchiah, Todd Cooke, and Alan Newell.

Biologists are becoming increasingly aware that mechanical stimuli interact with biochemical pathways in plants as well as animals. Clarifying the mechanisms and properties of these intricate interactions, while experimentally very challenging, has the potential to impact our understanding of many physiological processes. This project focuses on modeling the biochemical and biomechanical processes involved in the patterning of leaf buds at plant apices, a topic that has intrigued scientists for centuries and that is currently the subject of much experimental work. Patterns observed on plants present unique variations on planforms of ripples, hexagons, or squares observed in many laboratory and natural systems such as cloud formations, animal coats, and fluid convection experiments. These variations provide us with potential clues on the competition and cooperation between mechanisms. The modeling and mathematical analysis is complemented by the development of simple experimental systems in which the models can be tested on gels in a laboratory, with more control over parameters than in biological systems. The involvement of undergraduate and graduate students is an integral part of both the theoretical and experimental components of the project.

Project Report

Phyllotaxis, the arrangement on plants of leaves or leaf analogs such as pinecone bracts, sunflower florets, or cactus spines, has fascinated natural scientists for at least two millenia. On many plants, the phylla are arranged so that they lie on families of sprials, and the numbers of spirals in each family are typically members of the Fibonacci sequence 1,2,3,5,8,13,21,55,89,.... On other plants, phyllotaxis is characterized by a "whorl" pattern whereby the phylla come in groups that alternate in angle. Understanding the mechanisms behind the formation of these patterns and why so few classes of patterns are observed in nature is a challenge that involves questions of how biochemical and biophysical processes interact in biological systems, as well as how optimal packing may be achieved by these processes. The work supported by this grant built on an approach developed by the PI and colleagues in which dynamical models capture both the biochemical and biophysical processes at work near a plant's growth tip to produce phylla. A central research question concerned how to model the intricate interplay between biochemical processes that lead to growth and the resulting mechanical stresses that arise in the plant body. Focusing on the mathematical description of growth in biological materials, we developed a general mathematical foundation that encompasses a wide range of biological materials and growth laws. Phyllotactic patterns are produced in a annular generative region near a plant shoot tip, and as the radius of this annulus increases or decreases, the pattern may undergo transitions. Simulating the equations of our model with a changing radius of the annular region, we probed these transitions. We found examples of smooth transitions along a Fibonacci progression as well as reasons and situations under which a Fibonacci progression can be interrupted via instabilities. A surprising result is that the progression along a Fibonacci sequence is less smooth (and involves more defects) if the radius of the generative annulus is increasing than when the radius is decreasing. We were also able to find conditions that lead to the transition from whorls to Fibonacci spirals or from whorls to other whorls. A fascinating result from analyzing both results from simulations and 3d images of real plant apices is that the patterns via the laws of nonlinear dynamics coincide with simulations based rules of optimal packing. We found ways to mathematically characterize the optimality of the packing. We also found that one of the ways to characterize this optimality may be generalized to any dynamical system, from discrete rotation maps to symbolic dynamics (which have applications in a wide variety of fields) and defined a new notion of optimal topological transitivity to capture the degree of optimality in general dynamical systems. In a related study of optimality, we found that our coordinatization of phyllotactic patterns, which uses results from number theory, extends naturally to optimal coding theory. Laboratory experiments were also part of the body of work. An undergraduate chemistry student was able to produce patterns in gel systems that mimic the interaction of biochemical and biophysical processes in plants. Also, observations of plant apex geometry gave some evidence that the geometry of the plant tip plays a role in the phyllotactic pattern that is chosen. Involved in the work for this grant were four graduate students at Colorado State University and three undergraduate students. Results from this work were included in an undergraduate honors project, a Masters thesis and a PhD thesis.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Mary Ann Horn
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Colorado State University-Fort Collins
Fort Collins
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