A mathematical model of auxin-mediated radial growth in trees

Wood Formation Modeling - A Research Review and Future Perspectives

Wood formation has received considerable attention across various research fields as a key process to model. Historical and contemporary models of wood formation from various disciplines have encapsulated hypotheses such as the influence of external (e.g., climatic) or internal (e.g., hormonal) factors on the successive stages of wood cell differentiation. This review covers 17 wood formation models from three different disciplines, the earliest from 1968 and the latest from 2020. The described processes, as well as their external and internal drivers and their level of complexity, are discussed. This work is the first systematic cataloging, characterization, and process-focused review of wood formation models. Remaining open questions concerning wood formation processes are identified, and relate to: (1) the extent of hormonal influence on the final tree ring structure; (2) the mechanism underlying the transition from earlywood to latewood in extratropical regions; and (3) the extent to which carbon plays a role as "active" driver or "passive" substrate for growth. We conclude by arguing that wood formation models remain to be fully exploited, with the potential to contribute to studies concerning individual tree carbon sequestration-storage dynamics and regional to global carbon sequestration dynamics in terrestrial vegetation models.

Biophysical modelling of intra-ring variations in tracheid features and wood density of Pinus pinaster trees exposed to seasonal droughts

Process-based models that link seasonally varying environmental signals to morphological features within tree rings are essential tools to predict tree growth response and commercially important wood quality traits under future climate scenarios. This study evaluated model portrayal of radial growth and wood anatomy observations within a mature maritime pine (Pinus pinaster (L.) Aït.) stand exposed to seasonal droughts. Intra-annual variations in tracheid anatomy and wood density were identified through image analysis and X-ray densitometry on stem cores covering the growth period 1999-2010. A cambial growth model was integrated with modelled plant water status and sugar availability from the soil-plant-atmosphere transfer model MuSICA to generate estimates of cell number, cell volume, cell mass and wood density on a weekly time step. The model successfully predicted inter-annual variations in cell number, ring width and maximum wood density. The model was also able to predict the occurrence of special anatomical features such as intra-annual density fluctuations (IADFs) in growth rings. Since cell wall thickness remained surprisingly constant within and between growth rings, variations in wood density were primarily the result of variations in lumen diameter, both in the model and anatomical data. In the model, changes in plant water status were identified as the main driver of the IADFs through a direct effect on cell volume. The anatomy data also revealed that a trade-off existed between hydraulic safety and hydraulic efficiency. Although a simplified description of cambial physiology is presented, this integrated modelling approach shows potential value for identifying universal patterns of tree-ring growth and anatomical features over a broad climatic gradient.

A general formalism for tissue morphogenesis based on cellular dynamics and control system interactions

Morphogenesis is a key process in developmental biology. An important issue is the understanding of the generation of shape and cellular organisation in tissues. Despite of their great diversity, morphogenetic processes share common features. This work is an attempt to describe this diversity using the same formalism based on a cellular description. Tissue is seen as a multi-cellular system whose behaviour is the result of all constitutive cells dynamics. Morphogenesis is then considered as a spatiotemporal organization of cells activities. We show how this formalism relies on Reaction-Diffusion/Positional Information approach and how it permits to generalize its modelling possibilities. Three quite different applications for concrete morphogenetic processes are presented. The first one is a model for epithelial invagination, the second is a model of cellular differentiation by local cell-cell signalling. The last example is the secondary radial growth of conifer trees. From the mathematical point of view, different modelling tools are used according to the specificity of each process.

Modeling of branching patterns in plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations.Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients-we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.

Modelling of auxin transport affected by gravity and differential radial growth

When a tree stem deviates from verticality, as a result of different environmental factors, patterns of differential radial growth appear. Higher rates of wood production have been observed on the lower side of the tree and lower rates in the opposite side. Biological studies on plant hormones have shown that the concentration of auxin induces radial growth. They also have demonstrated the redistribution of auxin transport in response to gravity. Auxin is then designated as a mediator for differential growth. This paper presents a model for three-dimensional (3-D) auxin transport in conifer trees, which includes gravity dependence. We obtain realistic heterogeneous patterns of auxin distribution over the tree. Then, we propose a law of growth based on auxin concentration to simulate successive differential radial growths. The predicted growths are compared with experimental results of reconstruction of 3-D annual growth of Radiata pine.

Patterned cell development in the secondary phloem of dicotyledonous trees: a review and a hypothesis

The secondary phloem of dicotyledonous trees and shrubs is constructed of sieve tube cells (S) and their companion cells, as well as parenchyma (P) and fibre (F) cells. Different species have characteristic sequences of these S, P and F cells within the radial files of their phloem. The sequences are recurrent, and are evidence of rhythmic cell determination and differentiation. A model was devised to account for the sequences found in various dicot tree species. It is based on the pattern of radial displacement of cells through a gradient of morphogen which supports secondary phloem development. According to this model, each tree species shows a particular pattern of post-mitotic cellular displacement along each radial file as a result of a corresponding sequence of periclinal division in the cambial initial and its descendents. The divisions and displacements ensure that at each timestep (equivalent to an interdivisional interval) each cell resides in a specific location within the morphogenic gradient. Cells then emerge from the post-mitotic zone of cell determination, having acquired different final positional values. These values lie above a series of thresholds that permit the respective determination and subsequent differentiation of one or other of the three cell types S, P and F. The recurrent nature of the sequences of the three cell types within each radial cell file, as well as their tangential banding, are a consequence of a shared rhythmic spatio-temporal pattern of periclinal cambial divisions. With a single set of morphogen parameters required for cell determination, and using three positions for cambial cell divisions, all the cellular sequences of secondary phloem illustrated in the literature can be accounted for.

Cellular Modelling of Secondary Radial Growth in Conifer Trees: Application to Pinus Radiata (D. Don)

The radial growth of conifer trees proceeds from the dynamics of a merismatic tissue called vascular cambium or cambium. Cambium is a thin layer of active proliferating cells. The purpose of this paper was to model the main characteristics of cambial activity and its consecutive radial growth. Cell growth is under the control of the auxin hormone indole-3-acetic. The model is composed of a discrete part, which accounts for cellular proliferation, and a continuous part involving the transport of auxin. Cambium is modeled in a two-dimensional cross-section by a cellular automaton that describes the set of all its constitutive cells. Proliferation is defined as growth and division of cambial cells under neighbouring constraints, which can eliminate some cells from the cambium. The cell-growth rate is determined from auxin concentration, calculated with the continuous model. We studied the integration of each elementary cambial cell activity into the global coherent movement of macroscopic morphogenesis. Cases of normal and abnormal growth of Pinus radiata (D. Don) are modelled. Abnormal growth includes deformed trees where gravity influences auxin transport, producing heterogeneous radial growth. Cross-sectional microscopic views are also provided to validate the model's hypothesis and results.

Defect coarsening in a biological system: the vascular cambium of cottonwood trees

We present micrographic evidence for the annihilation of topological defect pairs and defect-mediated coarsening in the vascular cambium of cottonwood trees (Populus deltoides). We also show that a recently published mathematical model of cell orientation dynamics in the cambium reproduces many qualitative features of the defect coarsening process.

A mathematical model of pattern formation in the vascular cambium of trees

The beautiful patterns apparent in wood grain have their origin in the alignment of fusiform initial cells in the vascular cambium of trees. We develop a mathematical model to describe the orientation of fusiform initial cells, and their interaction with the plant hormone indole-3-acetic acid (auxin). The model incorporates the following four assumptions: (1) auxin is actively transported parallel to the long axis of the initials, (2) auxin diffuses perpendicular to the long axis of the initials, (3) the initials tend to orient parallel to the flux of auxin through the cambium, and (4) adjacent initials tend to orient parallel to one another. Each assumption is justified on the basis of available evidence and cast in mathematical form. Our main result is a pair of nonlinear differential equations that describe the coupling between the distribution of auxin in the cambium and the orientation of fusiform initials. Numerical solutions to the equations show qualitative resemblance to the wood grain patterns observed at branch junctions, wounds and knots, and topological defects.