Abstract. Given a distribution ρ on persistence diagrams and observations X1, ...Xn iid ∼ ρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1, ...Xn. If the underlying measure ρ is a combination of Dirac masses ρ = 1 m m i=1 δZ i then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

Identification of genes that control root system architecture in crop plants requires innovations that enable high-throughput and accurate measurements of root system architecture through time. We demonstrate the ability of a semiautomated 3D in vivo imaging and digital phenotyping pipeline to interrogate the quantitative genetic basis of root system growth in a rice biparental mapping population, Bala × Azucena. We phenotyped >1,400 3D root models and >57,000 2D images for a suite of 25 traits that quantified the distribution, shape, extent of exploration, and the intrinsic size of root networks at days 12, 14, and 16 of growth in a gellan gum medium. From these data we identified 89 quantitative trait loci, some of which correspond to those found previously in soil-grown plants, and provide evidence for genetic tradeoffs in root growth allocations, such as between the extent and thoroughness of exploration. We also developed a multivariate method for generating and mapping central root architecture phenotypes and used it to identify five major quantitative trait loci (r 2 = 24-37%), two of which were not identified by our univariate analysis. Our imaging and analytical platform provides a means to identify genes with high potential for improving root traits and agronomic qualities of crops.Oryza sativa | QTL | three-dimensional | live root imaging | multivariate analysis R oot systems are high-value targets for crop improvement because of their potential to boost or stabilize yields in saline, dry, and acid soils, improve disease resistance, and reduce the need for unsustainable fertilizers (1-7). Root system architecture (RSA) describes the spatial organization of root systems, which is critical for root function in challenging environments (1-10). Modern genomics could allow us to leverage both natural and engineered variation to breed more efficient crops, but the lack of parallel advances in plant phenomics is widely considered to be a primary hindrance to developing "next-generation" agriculture (3,11,12). Root imaging and analysis have been particularly intractable: Decades of phenotyping efforts have failed to identify genes controlling quantitative RSA traits in crop species. Several factors confound RSA gene identification, including polygenic inheritance of root traits, soil opacity, and a complex 3D morphology that can be influenced heavily by the environment. Most phenotyping efforts have relied on small numbers of basic measurements to extrapolate system-wide traits. For example, given the length and mass of a few sample roots and the excavated root system mass, one can estimate the total root length, volume, and average root width of the entire root system (13,14). Other common measurements involve measuring the root surface exposed on a soil core or pressed against a transparent surface to estimate root coverage at a certain soil horizon. In these cases, the choices of sample roots and phenotyping standards, the size and shape of the container, and the limitations of 2D descriptions of 3D struct...

The ability to nondestructively image and automatically phenotype complex root systems, like those of rice (Oryza sativa), is fundamental to identifying genes underlying root system architecture (RSA). Although root systems are central to plant fitness, identifying genes responsible for RSA remains an underexplored opportunity for crop improvement. Here we describe a nondestructive imaging and analysis system for automated phenotyping and trait ranking of RSA. Using this system, we image rice roots from 12 genotypes. We automatically estimate RSA traits previously identified as important to plant function. In addition, we expand the suite of features examined for RSA to include traits that more comprehensively describe monocot RSA but that are difficult to measure with traditional methods. Using 16 automatically acquired phenotypic traits for 2,297 images from 118 individuals, we observe (1) wide variation in phenotypes among the genotypes surveyed; and (2) greater intergenotype variance of RSA features than variance within a genotype. RSA trait values are integrated into a computational pipeline that utilizes supervised learning methods to determine which traits best separate two genotypes, and then ranks the traits according to their contribution to each pairwise comparison. This trait-ranking step identifies candidate traits for subsequent quantitative trait loci analysis and demonstrates that depth and average radius are key contributors to differences in rice RSA within our set of genotypes. Our results suggest a strong genetic component underlying rice RSA. This work enables the automatic phenotyping of RSA of individuals within mapping populations, providing an integrative framework for quantitative trait loci analysis of RSA.

BackgroundCharacterizing root system architecture (RSA) is essential to understanding the development and function of vascular plants. Identifying RSA-associated genes also represents an underexplored opportunity for crop improvement. Software tools are needed to accelerate the pace at which quantitative traits of RSA are estimated from images of root networks.ResultsWe have developed GiA Roots (General Image Analysis of Roots), a semi-automated software tool designed specifically for the high-throughput analysis of root system images. GiA Roots includes user-assisted algorithms to distinguish root from background and a fully automated pipeline that extracts dozens of root system phenotypes. Quantitative information on each phenotype, along with intermediate steps for full reproducibility, is returned to the end-user for downstream analysis. GiA Roots has a GUI front end and a command-line interface for interweaving the software into large-scale workflows. GiA Roots can also be extended to estimate novel phenotypes specified by the end-user.ConclusionsWe demonstrate the use of GiA Roots on a set of 2393 images of rice roots representing 12 genotypes from the species Oryza sativa. We validate trait measurements against prior analyses of this image set that demonstrated that RSA traits are likely heritable and associated with genotypic differences. Moreover, we demonstrate that GiA Roots is extensible and an end-user can add functionality so that GiA Roots can estimate novel RSA traits. In summary, we show that the software can function as an efficient tool as part of a workflow to move from large numbers of root images to downstream analysis.

This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional probabilities. This provides a theoretical basis for a statistical treatment of persistence diagrams, for example computing sample averages and sample variances of persistence diagrams. We first prove that the space of persistence diagrams with the Wasserstein metric is complete and separable. We then prove a simple criterion for compactness in this space. These facts allow us to show the existence of the standard statistical objects needed to extend the theory of topological persistence to a much larger set of applications.

We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance between their persistence diagrams and the second in terms of their total persistence.

Interest in the structure and function of physical biological networks has spurred the development of a number of theoretical models that predict optimal network structures across a broad array of taxonomic groups, from mammals to plants. In many cases, direct tests of predicted network structure are impossible given the lack of suitable empirical methods to quantify physical network geometry with sufficient scope and resolution. There is a long history of empirical methods to quantify the network structure of plants, from roots, to xylem networks in shoots and within leaves. However, with few exceptions, current methods emphasize the analysis of portions of, rather than entire networks. Here, we introduce the Leaf Extraction and Analysis Framework Graphical User Interface (LEAF GUI), a user-assisted software tool that facilitates improved empirical understanding of leaf network structure. LEAF GUI takes images of leaves where veins have been enhanced relative to the background, and following a series of interactive thresholding and cleaning steps, returns a suite of statistics and information on the structure of leaf venation networks and areoles. Metrics include the dimensions, position, and connectivity of all network veins, and the dimensions, shape, and position of the areoles they surround. Available for free download, the LEAF GUI software promises to facilitate improved understanding of the adaptive and ecological significance of leaf vein network structure.

For many bacterial viruses, the choice of whether to kill host cells or enter a latent state depends on the multiplicity of coinfection. Here, we present a mathematical theory of how bacterial viruses can make collective decisions concerning the fate of infected cells. We base our theory on mechanistic models of gene regulatory dynamics. Unlike most previous work, we treat the copy number of viral genes as variable. Increasing the viral copy number increases the rate of transcription of viral mRNAs. When viral regulation of cell fate includes nonlinear feedback loops, very small changes in transcriptional rates can lead to dramatic changes in steady-state gene expression. Hence, we prove that deterministic decisions can be reached, e.g., lysis or latency, depending on the cellular multiplicity of infection within a broad class of gene regulatory models of viral decision-making. Comparisons of a parameterized version of the model with molecular studies of the decision structure in the temperate bacteriophage lambda are consistent with our conclusions. Because the model is general, it suggests that bacterial viruses can respond adaptively to changes in population dynamics, and that features of collective decision-making in viruses are evolvable life history traits.