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A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper, we analyse the rate of convergence of gradient descent for smooth unconstrained multiobjective optimization, and we do it for non-convex, convex, and strongly convex vector functions. These global rates are shown to be the same as for gradient descent in single-objective optimization and correspond to appropriate worst-case complexity bounds. In the convex cases, the rates are given for implicit scalarizations of the problem vector function. [ABSTRACT FROM AUTHOR]

The readily available global rock phosphate ( P) reserves may run out within the next 50-130 years, causing soils to have a reduced P concentration which will affect plant P uptake. Using a combination of mathematical modelling and experimental data, we investigated potential plant-based options for optimizing crop P uptake in reduced soil P environments. By varying the P concentration within a well-mixed agricultural soil, for high and low P (35.5-12.5 mg L−1 respectively using Olsen's P index), we investigated branching distributions within a wheat root system that maximize P uptake. Changing the root branching distribution from linear (evenly spaced branches) to strongly exponential (a greater number of branches at the top of the soil) improves P uptake by 142% for low- P soils when root mass is kept constant between simulations. This causes the roots to emerge earlier and mimics topsoil foraging. Manipulating root branching patterns, to maximize P uptake, is not enough on its own to overcome the drop in soil P from high to low P. Further mechanisms have to be considered to fully understand the impact of P reduction on plant development. [ABSTRACT FROM AUTHOR]

Management and irrigation of plants increasingly relies on accurate mathematical models for the movement of water within unsaturated soils. Current models often use values for water content and soil parameters that are averaged over the soil profile. However, many applications require models to more accurately represent the soil-plant-atmosphere continuum, in particular, water movement and saturation within specific parts of the soil profile. In this paper a mathematical model for water uptake by a plant root system from unsaturated soil is presented. The model provides an estimate of the water content level within the soil at different depths, and the uptake of water by the root system. The model was validated using field data, which include hourly water content values at five different soil depths under a grass/herb cover over 1 year, to obtain a fully calibrated system for plant water uptake with respect to climate conditions. When compared quantitatively to a simple water balance model, the proposed model achieves a better fit to the experimental data due to its ability to vary water content with depth. To accurately model the water content in the soil profile, the soil water retention curve and saturated hydraulic conductivity needed to vary with depth. [ABSTRACT FROM AUTHOR]

Background: Single-cell sequencing (sc-Seq) experiments are producing increasingly large data sets. However, large data sets do not necessarily contain large amounts of information., Results: Here, we formally quantify the information obtained from a sc-Seq experiment and show that it corresponds to an intuitive notion of gene expression heterogeneity. We demonstrate a natural relation between our notion of heterogeneity and that of cell type, decomposing heterogeneity into that component attributable to differential expression between cell types (inter-cluster heterogeneity) and that remaining (intra-cluster heterogeneity). We test our definition of heterogeneity as the objective function of a clustering algorithm, and show that it is a useful descriptor for gene expression patterns associated with different cell types., Conclusions: Thus, our definition of gene heterogeneity leads to a biologically meaningful notion of cell type, as groups of cells that are statistically equivalent with respect to their patterns of gene expression. Our measure of heterogeneity, and its decomposition into inter- and intra-cluster, is non-parametric, intrinsic, unbiased, and requires no additional assumptions about expression patterns. Based on this theory, we develop an efficient method for the automatic unsupervised clustering of cells from sc-Seq data, and provide an R package implementation., (© 2023. BioMed Central Ltd., part of Springer Nature.)

Dearomatization is an effective method to transform readily available N-heterocycles into partially saturated motifs. Manipulation of dihydro-derivatives holds great potential and provides access to a variety of semi-saturated N-heterocyclic building blocks. However, current strategies are limited in scope and the use of sensitive reagents restricts the applicability in synthetic laboratories. Herein, we report the synthesis of a broad variety of N-substituted 1,4- and 1,2-dihydropyridines by very mild and selective reduction with amine borane for the first time., (© 2021 The Authors. Angewandte Chemie International Edition published by Wiley-VCH GmbH.)

Many common diseases have a complex genetic basis in which large numbers of genetic variations combine with environmental factors to determine risk. However, quantifying such polygenic effects has been challenging. In order to address these difficulties we developed a global measure of the information content of an individual's genome relative to a reference population, which may be used to assess differences in global genome structure between cases and appropriate controls. Informally this measure, which we call relative genome information (RGI), quantifies the relative "disorder" of an individual's genome. In order to test its ability to predict disease risk we used RGI to compare single-nucleotide polymorphism genotypes from two independent samples of women with early-onset breast cancer with three independent sets of controls. We found that RGI was significantly elevated in both sets of breast cancer cases in comparison with all three sets of controls, with disease risk rising sharply with RGI. Furthermore, these differences are not due to associations with common variants at a small number of disease-associated loci, but rather are due to the combined associations of thousands of markers distributed throughout the genome. Our results indicate that the information content of an individual's genome may be used to measure the risk of a complex disease, and suggest that early-onset breast cancer has a strongly polygenic component.

Two major breast cancer sub-types are defined by the expression of estrogen receptors on tumour cells. Cancers with large numbers of receptors are termed estrogen receptor positive and those with few are estrogen receptor negative. Using genome-wide single nucleotide polymorphism genotype data for a sample of early-onset breast cancer patients we developed a Support Vector Machine (SVM) classifier from 200 germline variants associated with estrogen receptor status (p<0.0005). Using a linear kernel Support Vector Machine, we achieved classification accuracy exceeding 93%. The model indicates that polygenic variation in more than 100 genes is likely to underlie the estrogen receptor phenotype in early-onset breast cancer. Functional classification of the genes involved identifies enrichment of functions linked to the immune system, which is consistent with the current understanding of the biological role of estrogen receptors in breast cancer.

Because of the complexity of gene-phenotype relationships machine learning approaches have considerable appeal as a strategy for modelling interactions. A number of such methods have been developed and applied in recent years with some modest success. Progress is hampered by the challenges presented by the complexity of the disease genetic data, including phenotypic and genetic heterogeneity, polygenic forms of inheritance and variable penetrance, combined with the analytical and computational issues arising from the enormous number of potential interactions. We review here recent and current approaches focusing, wherever possible, on applications to real data (particularly in the context of genome-wide association studies) and looking ahead to the further challenges posed by next generation sequencing data.