Mathematical Modeling of the Dynamics of Shoot-Root Interactions and Resource Partitioning in Plant Growth

Regulation of tissue growth in plants - A mathematical modeling study on shade avoidance response in Arabidopsis hypocotyls

Introduction

Plant growth is a plastic phenomenon controlled both by endogenous genetic programs and by environmental cues. The embryonic stem, the hypocotyl, is an ideal model system for the quantitative study of growth due to its relatively simple geometry and cellular organization, and to its essentially unidirectional growth pattern. The hypocotyl of Arabidopsis thaliana has been studied particularly well at the molecular-genetic level and at the cellular level, and it is the model of choice for analysis of the shade avoidance syndrome (SAS), a growth reaction that allows plants to compete with neighboring plants for light. During SAS, hypocotyl growth is controlled primarily by the growth hormone auxin, which stimulates cell expansion without the involvement of cell division.

Methods

We assessed hypocotyl growth at cellular resolution in Arabidopsis mutants defective in auxin transport and biosynthesis and we designed a mathematical auxin transport model based on known polar and non-polar auxin transporters (ABCB1, ABCB19, and PINs) and on factors that control auxin homeostasis in the hypocotyl. In addition, we introduced into the model biophysical properties of the cell types based on precise cell wall measurements.

Results and Discussion

Our model can generate the observed cellular growth patterns based on auxin distribution along the hypocotyl resulting from production in the cotyledons, transport along the hypocotyl, and general turnover of auxin. These principles, which resemble the features of mathematical models of animal morphogen gradients, allow to generate robust shallow auxin gradients as they are expected to exist in tissues that exhibit quantitative auxin-driven tissue growth, as opposed to the sharp auxin maxima generated by patterning mechanisms in plant development.

OPT-ing out: Root-shoot dynamics are caused by local resource capture and biomass allocation, not optimal partitioning

Combining plant growth analysis with a simple model of local resource capture and biomass allocation applied to exemplary experimental data, showed that dynamic changes in allocation to roots when nutrients are scarce is caused by disparities in growth rates between roots and shoots. Whole-plant growth rates also change but are not caused by an adaptive allocation response. Allocation and whole-plant growth rate are interdependent, not independent, traits. Following a switch in nutrient availability or partial biomass removal, convergence of allocation and growth rate trajectories does not reflect goal-seeking behaviour, but the constraints imposed by finite resource availability. Optimal root-shoot allocations are unnecessary to maximise whole-plant growth rate. Similar growth rates are attainable with different allocations. Changes in allocation cannot maintain or restore a superior whole-plant growth rate. Roots and shoots do not have to be tightly coordinated but can operate semiautonomously, as their modular construction permits. These findings question the plausibility of the prevailing general explanation of plants' root-shoot allocation responses, 'optimal partitioning theory' (OPT). Local allocation models, not OPT, explain the origins of variability in root-shoot trade-offs in individuals, the allocation of biomass at global and ecosystem scales and inform selection for allocation plasticity in crop breeding.

Overcoming the Challenges to Enhancing Experimental Plant Biology With Computational Modeling

The study of complex biological systems necessitates computational modeling approaches that are currently underutilized in plant biology. Many plant biologists have trouble identifying or adopting modeling methods to their research, particularly mechanistic mathematical modeling. Here we address challenges that limit the use of computational modeling methods, particularly mechanistic mathematical modeling. We divide computational modeling techniques into either pattern models (e.g., bioinformatics, machine learning, or morphology) or mechanistic mathematical models (e.g., biochemical reactions, biophysics, or population models), which both contribute to plant biology research at different scales to answer different research questions. We present arguments and recommendations for the increased adoption of modeling by plant biologists interested in incorporating more modeling into their research programs. As some researchers find math and quantitative methods to be an obstacle to modeling, we provide suggestions for easy-to-use tools for non-specialists and for collaboration with specialists. This may especially be the case for mechanistic mathematical modeling, and we spend some extra time discussing this. Through a more thorough appreciation and awareness of the power of different kinds of modeling in plant biology, we hope to facilitate interdisciplinary, transformative research.

Overcoming Physiological Bottlenecks of Leaf Vitality and Root Development in Cuttings: A Systemic Perspective

Each year, billions of ornamental young plants are produced worldwide from cuttings that are harvested from stock plants and planted to form adventitious roots. Depending on the plant genotype, the maturation of the cutting, and the particular environment, which is complex and often involves intermediate storage of cuttings under dark conditions and shipping between different climate regions, induced senescence or abscission of leaves and insufficient root development can impair the success of propagation and the quality of generated young plants. Recent findings on the molecular and physiological control of leaf vitality and adventitious root formation are integrated into a systemic perspective on improved physiologically-based control of cutting propagation. The homeostasis and signal transduction of the wound responsive plant hormones ethylene and jasmonic acid, of auxin, cytokinins and strigolactones, and the carbon-nitrogen source-sink balance in cuttings are considered as important processes that are both, highly responsive to environmental inputs and decisive for the development of cuttings. Important modules and bottlenecks of cutting function are identified. Critical environmental inputs at stock plant and cutting level are highlighted and physiological outputs that can be used as quality attributes to monitor the functional capacity of cuttings and as response parameters to optimize the cutting environment are discussed. Facing the great genetic diversity of ornamental crops, a physiologically targeted approach is proposed to define bottleneck-specific plant groups. Components from the field of machine learning may help to mathematically describe the complex environmental response of specific plant species.

Mathematical modeling of diurnal patterns of carbon allocation to shoot and root in Arabidopsis thaliana

We developed a mathematical model to simulate dynamics of central carbon metabolism over complete diurnal cycles for leaves of Arabidopsis thaliana exposed to either normal (120 µmol m-2 s-1) or high light intensities (1200 µmol m- 2 s-1). The main objective was to obtain a high-resolution time series for metabolite dynamics as well as for shoot structural carbon formation (compounds with long residence time) and assimilate export of aerial organs to the sink tissue. Model development comprised a stepwise increment of complexity to finally approach the in vivo situation. The correct allocation of assimilates to either sink export or shoot structural carbon formation was a central goal of model development. Diurnal gain of structural carbon was calculated based on the daily increment in total photosynthetic carbon fixation, and this was the only parameter for structural carbon formation implemented in the model. Simulations of the dynamics of central metabolite pools revealed that shoot structural carbon formation occurred solely during the light phase but not during the night. The model allowed simulation of shoot structural carbon formation as a function of central leaf carbon metabolism under different environmental conditions without structural modifications. Model simulations were performed for the accession Landsberg erecta (Ler) and its hexokinase null-mutant gin2-1. This mutant displays a slow growth phenotype especially at increasing light intensities. Comparison of simulations revealed that the retarded shoot growth in the mutant resulted from an increased assimilate transport to sink organs. Due to its central function in sucrose cycling and sugar signaling, our findings suggest an important role of hexokinase-1 for carbon allocation to either shoot growth or assimilate export.

A Simple Method for Estimating Field Crop Evapotranspiration from Pot Experiments

Pot experiments are a low-cost and easy-to-use technique for studies of soil evaporation and plant transpiration in controlled environments. However, little attention has been paid to the applicability of evapotranspiration (ET) measured in pot experiments to the field. The objective of this study was to determine whether a pot experiment can be used for measuring field ET. Evapotranspiration experiments with winter wheat and summer maize were conducted in pots and lysimeters under various water-deficit conditions. The measured ET values in the pot experiments under different water conditions were considerably different from those of the lysimeters. Causes of such differences in ET were analyzed, and a series of corrections were proposed to eliminate the effects of different crop densities, representative areas per plant, and soil moisture conditions on pot experiment results. After these corrections, the discrepancy in the total ET of wheat-maize seasons between pots and lysimeters was greatly reduced from a maximum of 117% to only approximately 10%. The relative mean square errors (RMSEs) for daily ET values also decreased from a maximum value of 4.56 mm to less than 1.5 mm for the wheat season and from a maximum value of 6.02 mm to approximately 2 mm for the maize season. Possible measures were proposed to further improve the accuracy of the corrected ET obtained from pot experiments. In sum, pot experiments can serve as a feasible tool for estimating ET in the field just with a few routine measurements at regions where large-scale weighing lysimeters, an eddy covariance device, and even meteorological data are not available. The proposed corrections can also be used for upscaling small-scale ET measurements to a large scale.

A Dynamic Multi-Tissue Flux Balance Model Captures Carbon and Nitrogen Metabolism and Optimal Resource Partitioning During Arabidopsis Growth

Plant metabolism is highly adapted in response to its surrounding for acquiring limiting resources. In this study, a dynamic flux balance modeling framework with a multi-tissue (leaf and root) diel genome-scale metabolic model of Arabidopsis thaliana was developed and applied to investigate the reprogramming of plant metabolism through multiple growth stages under different nutrient availability. The framework allowed the modeling of optimal partitioning of resources and biomass in leaf and root over diel phases. A qualitative flux map of carbon and nitrogen metabolism was identified which was consistent across growth phases under both nitrogen rich and limiting conditions. Results from the model simulations suggested distinct metabolic roles in nitrogen metabolism played by enzymes with different cofactor specificities. Moreover, the dynamic model was used to predict the effect of physiological or environmental perturbation on the growth of Arabidopsis leaves and roots.

Diurnal periodicity of assimilate transport shapes resource allocation and whole-plant carbon balance

Whole-plant carbon balance comprises diurnal fluctuations of photosynthetic carbon gain and respiratory losses, as well as partitioning of assimilates between phototrophic and heterotrophic organs. Because it is difficult to access, the root system is frequently neglected in growth models, or its metabolism is rated based on generalizations from other organs. Here, whole-plant cuvettes were used for investigating total-plant carbon exchange with the environment over full diurnal cycles. Dynamics of primary metabolism and diurnally resolved phloem exudation profiles, as proxy of assimilate transport, were combined to obtain a full picture of resource allocation. This uncovered a strong impact of periodicity of inter-organ transport on the efficiency of carbon gain. While a sinusoidal fluctuation of the transport rate, with minor diel deflections, minimized respiratory losses in Arabidopsis wild-type plants, triangular or rectangular patterns of transport, found in mutants defective in either starch or sucrose metabolism, increased root respiration at the end or beginning of the day, respectively. Power spectral density and cross-correlation analysis revealed that only the rate of starch synthesis was strictly correlated to the rate of net photosynthesis in wild-type, while in a sucrose-phosphate synthase mutant (spsa1), this applied also to carboxylate synthesis, serving as an alternative carbon pool. In the starchless mutant of plastidial phospho-gluco mutase (pgm), none of these rates, but concentrations of sucrose and glucose in the root, followed the pattern of photosynthesis, indicating direct transduction of shoot sugar levels to the root. The results demonstrate that starch metabolism alone is insufficient to buffer diurnal fluctuations of carbon exchange.

Photosynthetic Entrainment of the Circadian Clock Facilitates Plant Growth under Environmental Fluctuations: Perspectives from an Integrated Model of Phase Oscillator and Phloem Transportation

Plants need to avoid carbon starvation and resultant growth inhibition under fluctuating light environments to ensure optimal growth and reproduction. As diel patterns of carbon metabolism are influenced by the circadian clock, appropriate regulation of the clock is essential for plants to properly manage their carbon resources. For proper adjustment of the circadian phase, higher plants utilize environmental signals such as light or temperature and metabolic signals such as photosynthetic products; the importance of the latter as phase regulators has been recently elucidated. A mutant of Arabidopsis thaliana that is deficient in phase response to sugar has been shown, under fluctuating light conditions, to be unable to adjust starch turnover and to realize carbon homeostasis. Whereas, the effects of light entrainment on growth and survival of higher plants are well studied, the impact of phase regulation by sugar remains unknown. Here we show that endogenous sugar entrainment facilitates plant growth. We integrated two mathematical models, one describing the dynamics of carbon metabolism in A. thaliana source leaves and the other growth of sink tissues dependent on sucrose translocation from the source. The integrated model predicted that sugar-sensitive plants grow faster than sugar-insensitive plants under constant as well as changing photoperiod conditions. We found that sugar entrainment enables efficient carbon investment for growth by stabilizing sucrose supply to sink tissues. Our results highlight the importance of clock entrainment by both exogenous and endogenous signals for optimizing growth and increasing fitness.

CN-Wheat, a functional-structural model of carbon and nitrogen metabolism in wheat culms after anthesis. I. Model description

Background and Aims Improving crops requires better linking of traits and metabolic processes to whole plant performance. In this paper, we present CN-Wheat, a comprehensive and mechanistic model of carbon (C) and nitrogen (N) metabolism within wheat culms after anthesis. Methods The culm is described by modules that represent the roots, photosynthetic organs and grains. Each of them includes structural, storage and mobile materials. Fluxes of C and N among modules occur through a common pool and through transpiration flow. Metabolite variations are represented by differential equations that depend on the physiological processes occurring in each module. A challenging aspect of CN-Wheat lies in the regulation of these processes by metabolite concentrations and the environment perceived by organs. Key Results CN-Wheat simulates the distribution of C and N into wheat culms in relation to photosynthesis, N uptake, metabolite turnover, root exudation and tissue death. Regulation of physiological activities by local concentrations of metabolites appears to be a valuable feature for understanding how the behaviour of the whole plant can emerge from local rules. Conclusions The originality of CN-Wheat is that it proposes an integrated view of plant functioning based on a mechanistic approach. The formalization of each process can be further refined in the future as knowledge progresses. This approach is expected to strengthen our capacity to understand plant responses to their environment and investigate plant traits adapted to changes in agronomical practices or environmental conditions. A companion paper will evaluate the model.