Two genes controlling the purple pericarp trait in wheat, TaPpm1 and TaPpb1, are identified and the mechanism by which they co-regulate anthocyanin synthesis is proposed.

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We study the applicability of the Peaceman-Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.

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In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forward-backward envelope to be a level-bounded and Kurdyka-Lojasiewicz function with an exponent of 1 2 ; these results are important for the efficient minimization of the forward-backward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some difference-of-convex regularized least squares problems by minimizing a suitably constructed forward-backward envelope. Our preliminary numerical results on randomly generated instances of large-scale 1−2 regularized least squares problems [37] illustrate that an implementation of this approach with a limited-memory BFGS scheme usually outperforms standard first-order methods such as the nonmonotone proximal gradient method in [35].

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To protect above‐ground plant organs from excessive water loss, their surfaces are coated by waxes. The genes involved in wax formation have been investigated in detail in Arabidopsis but scarcely in crop species. Here, we aimed to isolate and characterize a CER1 enzyme responsible for formation of the very long‐chain alkanes present in high concentrations especially during late stages of wheat development. On the basis of comparative wax and transcriptome analyses of various wheat organs, we selected TaCER1‐1A as a primary candidate and demonstrated that it was located to the endoplasmic reticulum, the subcellular compartment for wax biosynthesis. A wheat nullisomic‐tetrasomic substitution line lacking TaCER1‐1A had significantly reduced amounts of C33 alkane, whereas rice plants overexpressing TaCER1‐1A showed substantial increases of C25–C33 alkanes relative to wild type control. Similarly, heterologous expression of TaCER1‐1A in Arabidopsis wild type and the cer1 mutant resulted in increased levels of unbranched alkanes, iso‐branched alkanes and alkenes. Finally, the expression of TaCER1‐1A was found activated by abiotic stresses and abscisic acid treatment, resulting in increased production of alkanes in wheat. Taken together, our results demonstrate that TaCER1‐1A plays an important role in wheat wax alkane biosynthesis and involved in responding to drought and other environmental stresses.