Working with longitudinal data: quantifying developmental processes using function‐valued trait modeling

For plants, growth and development are not merely pathways to maturity reproduction. They also strategies for responding dynamically environmental inputs. Plants able integrate complex signals alter their developmental rates or durations accordingly. These changes can result in different trajectories consequently functional anatomy, morphology, physiology, allowing plants take advantage of conducive environments respond stress. Data from single time points often miss important aspects ontogeny, variation be obscured annuals determinate organs that arrive at similar endpoints despite varying trajectories. Instead, studying the entirety organismal ontogeny lead better fitness yield predictions uncover novel loci—even final sizes—that would otherwise detected (Baker et al., 2015). Additionally, shapes heritable, subject selection 2015; Kulbaba 2017), incorporated into process-based models (Wang 2019). findings underscore importance quantifying only biologists, but advancing our understanding ecological evolutionary processes as well contributing crop improvement. Despite its significance, is underrepresented literature, unaddressed. One reason incorporating adds experimental cost; however, automated phenotyping making data collection increasingly feasible. Here we address a more generalizable challenge: analysis. Historically, there have been number alternatives handling data. include interval, integral, function-valued trait (FVT) approaches (summarized by Chiariello 1989). Interval estimate mean values between two successive employed when sparse. Integral use area under curve, sometimes with few points. focus on FVT approaches, which flexible succinctly describe complex, nonlinear Pairing modeling downstream analyses provides an appealing solution improves upon interval integral approaches. In framework, viewed continuously changing traits, individual plotted over approximate curves. Mathematical functions used these curves, i.e., relationship one another function (Kingsolver 2001). trajectories; this approach applied diverse traits such responses increasing temperature, pathogen load, herbivore abundance (Gomulkiewicz 2018). has multiple advantages traditional analyses. It effective simplification step: curves estimated observations described just parameters. yields intuitive biologically interpretable results effectively capture dynamics, easily visualized. However, unlike reduction steps, intermediate times collection, capturing continuous nature development. collected (e.g., even vs. odd days) thus comparable Critically, minimizes pseudoreplication increases statistical power many circumstances (Stinchcombe Kirkpatrick, 2012). Above all, integrates across variables durations, influence genetic controls than size, estimated. Function-valued requires fitting biology, sigmoidal logistic, Gompertz, Richards’) commonly work biological leaf anthocyanin accumulation. indeterminate plant may approximated exponential functions. fluctuate periodically fit wave other appropriate functions, including Legendre polynomials. Finding suitable form first step involve some trial error. Alternatively, basis expansions without relying specified (Griswold 2008). There methods finding best replicates. Frequentist least squares (LS) fast easy, LS optimization typically good fits, provided raw (Fig. 1; see Appendices S1 S2 R code Supporting Information https://github.com/rlbaker5/AJB_FVTmodeling2021). Bayesian several strengths compared frequentist they come cost considerable complexity computational time. Hierarchical account underlying structure leverage population-level information inferring parameters individuals (Gelman Hill, 2006). incorporate propagate error explicitly (Rouder Lu, 2005). Their comprehensive enables estimation growth-curve relative procedures heritability estimates 2018a). Finally, covariates models. example, examining height variation, nontarget carbon assimilation impact FVT, function. This mathematically factors out variance target architecture incredibly terms main First, FVTs known Deviations among interpretable; indicate shifts early onset, longer duration, decreased rate reference population, genotype, environment Second, describing analyses, requiring no specialized (Table Appendix S2). construct phylogenetic relationships (Goolsby, 2015), reconstruct ancestral states (Hadjipantelis 2013), examine phenotypic plasticity via reaction norms (Chevin Hoffmann, test (Kulbaba identify regulatory modules characterize single- multiple-QTL mapping 2015, 2018b, 2019), generate gene coexpression networks candidate discovery predict phenotypes based genotypic 2018b). Multiple pitfalls occur during experiment. Among most common insufficient instance, inadvertently start later necessary ontogeny. Non-FVT ignore missing data, insufficiently characterized problematic Under certain conditions, help Baker al. (2018a) demonstrated could truncated not, enabled analysis field sizes were recorded. Even sufficient collected, noisy cause problems model fitting. Hodge Doust (2021 [Preprint]) found difficulties Setaria phytomers. developed second-order (2FVT) leverages organism-level ontogenetic averaging derived fits all phytomers improve phytomer-level fits. criticism “parameters data” replicates individually do genotype population level (Stinchcombe, hinted Stinchcombe (2010), side-step shortcoming producing replicate while higher order Perhaps fundamental problem occurs genotypes follow forms Li 2020). case, above applied, “area curve” independent specific (example S3). time-point make little distinction cannot decomposed rates, timing critical A second alternative random regression approach, employ (Campbell As becomes customizable, cost-effective, user-friendly, will need powerful, intuitive, integrated any Quantitatively addressing empowers further integration dynamics theory. Understanding interplay genetics, environment, organism throughout quantifiable framework improved phenotypes. efforts support continued conservation programs sustainable agricultural systems. The authors thank Editor-in-Chief, Dr. P. K. Diggle, J. (U. Toronto), anonymous reviewer feedback manuscript. Drs. D. Kliebenstein, M. Tang (U.C. Davis), A. (Oklahoma State University, Stillwater) valuable discussion. Y. Yang C. Hoagland Controlled Environment Phenotyping Facility (Purdue University) assisted phenotyping. Robert L Baker: Conceptualization (equal); curation Formal Investigation Methodology Project administration Visualization Writing – original draft review & editing (equal). Diane R. Wang: Validation R.L.B. D.R.W. contributed stages conception, implementation, analysis, interpretation drafting revising All available Information. accessed https://github.com/rlbaker5/AJB_FVTmodeling2021. Please note: publisher responsible content functionality supporting supplied authors. 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