To be practical, a multi-scale model should generate well-constrained predictions despite significant parameter uncertainty . It is desirable that a multi-scale model has certain modularity in its design such that individual modules are responsible for modeling specific spatial aspects of the system . Imaging techniques can validate multi-scale models such that simulations can reliably guide experimental studies. To illustrate the challenges of multi-scale modeling, we highlight an example that encompasses molecular and cellular scales. At the molecular scale, models can treat some biomolecules as diffusive, but others, such as membrane-bound receptors, can be spatially restricted . Separately, at the cellular scale, mathematical models describe dynamics of cell networks where the mechanical pressures exerted on the cell walls are important factors for cell growth and division . In models describing plant development in a two-dimensional cross-section geometry, cells are often modeled as polygons defined by walls between neighboring cells. The spatial position of a vertex, where the cell walls of three neighboring cells coalesce, is a convenient variable for mathematical modeling of the dynamics of cellular networks . Mutations and deletions of the genes encoding the biomolecules can be modeled by changing parameters. By inspecting the effects of such modifications on the dynamics of the cellular networks, dutch buckets the relationship between genotypes and phenotypes can be predicted.
For example, Fujita et al. model integrates the dynamics of cell growth and division with the spatio-temporal dynamics of the proteins involved in stem cell regulation and simulates shoot apical meristem development in wild type and mutant plants . Quantitative measures of plant morphology are critical to understand function. Vogel was the first to provide quantitative data that showed how shape changes in leaves reduce drag or friction in air or water flows. He found that single broad leaves reconfigure at high flow velocities into cone shapes to reduce flutter and drag . More recent work discovered that the cone shape is significantly more stable than other reconfigurations such as U-shapes . Subsequent experimental studies on broad leaves, compound leaves, and flowers also support rapid repositioning in response to strong currents as a general mechanism to reduce drag . It is a combination of morphology and anatomy, and the resultant material properties, which lead to these optimal geometric re-configurations of shape. From a functional perspective, it is highly plausible that leaf shape and surface-material properties alter the boundary layer of a fluid/gas over the leaf surface or enhance passive movement that can potentially augment gas and heat exchange. For example, it has been proposed that the broad leaves of some trees flutter for the purpose of convective and evaporative heat transfer . Any movement of the leaf relative to the movement of the air or water may decrease the boundary layer and increase gas exchange, evaporation, and heat dissipation .
Each of these parameters may be altered by the plant to improve the overall function of the leaf . The growth of the plant continuously modifies plant topology and geometry, which in turn changes the balance between organ demand and production. At the organismal scale, the 3D spatial distribution of plant organs is the main interface between the plant and its environment. For example, the 3D arrangement of branches impacts light interception and provides the support for different forms of fluxes and signals that control plant functioning and growth . Mathematics has been likened to “biology’s next microscope,” because of the insights into an otherwise invisible world it has to offer. Conversely, biology has been described as “mathematics’ next physics,” stimulating novel mathematical approaches because of the hitherto unrealized phenomena that biology studies . The scale of the needed interplay between mathematics and plant biology is enormous and may lead to new science disciplines at the interface of both: ranging from the cellular, tissue, organismal, and community levels to the global; touching upon genetic, transcriptional, proteomic, metabolite, and morphological data; studying the dynamic interactions of plants with the environment or the evolution of new forms over geologic time; and spanning quantification, statistics, and mechanistic mathematical models. Research is becoming increasingly interdisciplinary, and undergraduate, graduate, and post-graduate groups are actively trying to bridge the gap between mathematics and biology skill sets. While many graduate programs have specialization tracks under the umbrella of mathematics or biology-specific programs, increasingly departments are forming specially designed graduate groups for mathematical/quantitative biology1,2 to strengthen the interface between both disciplines.Citizen science, which is a method to make the general public aware of scientific problems and employ their help in solving them3 , is an ideal platform to initiate a synthesis between plant biology and mathematics because of the relatively low cost and accessibility of each field.
Arguably, using citizen science to collect plant morphological diversity has already been achieved, but has yet to be fully realized. In total, it is estimated that the herbaria of the world possess greater than 207 million voucher specimens4 , representing the diverse lineages of land plants collected over their respective bio-geographies over a time span of centuries.Digital documentation of the millions of vouchers held by the world’s botanic gardens is actively underway, allowing for researchers and citizens alike to access and study for themselves the wealth of plant diversity across the globe and centuries . The developmental changes in plants responding to environmental variability and microclimatic changes over the course of a growing season can be analyzed by studying phenology. Citizen science projects such as the USA National Phenology Network5 or Earthwatch6 and associated programs such as My Tree Tracker7 document populations and individual plants over seasons and years, providing a distributed, decentralized network of scientific measurements to study the effects of climate change on plants.Citizen science is also enabled by low-cost, specialized equipment. Whether programming a camera to automatically take pictures at specific times or automating a watering schedule for a garden, the maker movement—a do-it yourself cultural phenomenon that intersects with hacker culture—focuses on building custom, programmable hardware, whether via electronics, robotics, 3D-printing, or time-honored skills such as metal- and woodworking. The focus on programming is especially relevant for integrating mathematical approaches with plant science experiments. The low-cost of single-board computers like Raspberry Pi, HummingBoard, or CubieBoard is a promising example of how to engage citizen scientists into the scientific process and enable technology solutions to specific questions. Simply bringing mathematicians and plant biologists together to interact, to learn about new tools, approaches, and opportunities in each discipline is a major opportunity for further integration of these two disciplines and strengthen new disciplines at the interface of both. This white paper itself is a testament to the power of bringing mathematicians and biologists together, grow bucket resulting from a National Institute for Mathematical and Biological Synthesis workshop titled “Morphological Plant Modeling: Unleashing Geometric and Topologic Potential within the Plant Sciences,” held at the University of Tennessee, Knoxville, September 2– 4, 2015. Other mathematical institutes such as the Mathematical Biology Institute at Ohio State University, the Statistical and Applied Mathematical Sciences Institute in Research Triangle Park , the Institute for Mathematics and Its Applications at University of Minnesota, and the Centre for Plant Integrative Biology at the University of Nottingham have also hosted workshops for mathematical and quantitative biologists from the undergraduate student to the faculty level. There are efforts to unite biologists and mathematics through initiatives brought forth from The National Science Foundation, including Mathematical Biology Programs and the Joint DMS/NIGMS Initiative to Support Research at the Interface of the Biological and Mathematical Sciences . Outside of the Mathematics and Life Sciences Divisions, the Division of Physics houses a program on the Physics of Living Systems. Societies such as The Society for Mathematical Biology and the Society for Industrial and Applied Mathematics Life Science Activity Group15 are focused on the dissemination of research at the intersection of math and biology, creating many opportunities to present research and provide funding.
We emphasize the importance that funding opportunities have had and will continue to have in the advancement of plant morphological modeling.Ultimately, mathematicians, computational scientists, and plant biology must unite at the level of jointly collecting data, analyzing it, and doing science together. Open and timely data sharing to benchmark code is a first step to unite these disciplines along with building professional interfaces to bridge between the disciplines . A number of platforms provide open, public access to datasets, figures, and code that can be shared, including Dryad, Dataverse, and Figshare. Beyond the ability to share data is the question of open data formats and accessibility. For example, in remote sensing research it is unfortunately common that proprietary data formats are used, which prevents their use without specific software. This severely limits the utility and community building aspects of plant morphological research. Beyond datasets, making code openly available, citable, and user-friendly is a means to share methods to analyze data. Places to easily share code include web-based version controlled platforms like Bitbucket or Github and software repositories like Sourceforge. Furthermore, numerous academic Journals already accept publications that focus on methods and software to accelerate new scientific discovery . Meta-analysis datasets provide curated resources where numerous published and unpublished datasets related to a specific problem can be accessed by researchers. The crucial element is that data is somehow reflective of universal plant morphological features, bridging the gap between programming languages and biology, as seen in the Root System Markup Language and OpenAlea . Bisque is a versatile platform to store, organize, and analyze image data, providing simultaneously open access to data and analyses as well as the requisite computation . CyVerse is a similar platform, on which academic users get 100 GB storage for free and can create analysis pipelines that can be shared and reused . For example, DIRT is an automatic, high throughput computing platform that the public can use hosted on CyVerse using the Texas Advanced Computing Center resources at UT Austin that robustly extracts root traits from digital images. The reproducibility of these complex computational experiments can be improved using scientific workflows that capture and automate the exact methodology followed by scientists . We emphasize here the importance of adopting open science policies at the individual investigator and journal level to continue strengthening the interface between plant and mathematically driven sciences.Plant morphology is a mystery from a molecular and quantification point of view. Hence, it fascinates both mathematical and plant biology researchers alike. As such, plant morphology holds the secret by which predetermined variations of organizational patterns emerge as a result of evolutionary, developmental, and environmental responses. The persistent challenge at the intersection of plant biology and mathematical sciences might be the integration of measurements across different scales of the plant. We have to meet this challenge to derive and validate mathematical models that describe plants beyond the visual observable. Only then we will be able to modify plant morphology through molecular biology and breeding as means to develop needed agricultural outputs and sustainable environments for everybody. Cross-disciplinary training of scientists, citizen science, and open science are inevitable first steps to develop the interface between mathematical-driven and plant biology-driven sciences. The result of these steps will be new disciplines, that will add to the spectrum of researchers in plant biology. Hence, to unleash the potential of geometric and topological approaches in the plant sciences, we need an interface familiar with both plants and mathematical approaches to meet the challenges posed by a future with uncertain natural resources as a consequence of climate change. The implications for the IR, however, are ambiguous because the degree and direction of recall bias depends on a number of factors that can offset each other. Overestimation is likely to be greatest when surveys ask about hours worked per person per plot, but can be substantially smaller or even underestimated when focusing on total household hours per farm. At this level, the authors conclude that labor productivity might be underestimated in Tanzania and overestimated in Ghana . How can this brief review of the recent literature on measurement error guide our empirical analysis of Brazil? First, we recognize that these are serious concerns.