The falling number (FN) reflects the level of alpha amylase activity in grain/flour and serves as an international standard of grain quality. The lower the FN the lower the quality of grain and its price. The main factor responsible for high α amylase levels and low FN values is visible or hidden preharvest sprouting (PHS) under adverse weather conditions. The decisive role in obtaining consistently high FNs belongs to breeding for preharvest sprouting resistance (PHSR) and directly for the FN. In recent decades, significant progress has been made in understanding biochemical and molecular processes accompanying PHS and the genetic control (PHSR) and the FN, as well as in the development of methods of breeding for PHSR and the FN. However, breeding for these traits and the corresponding theoretical studies have not yet been properly developed or reflected in the Russian literature. This review illustrates the genetic complexity and context dependence of the PHSR and FN traits. A specific feature of breeding for high PHSR and the FN is that it involves eliminating late maturity α-amylase (LMA) genotypes from early hybrid generations and choosing parents for crossing; the choice of genetically distinct donors of target traits; methods of evaluating varieties and lines for PHSR and the FN under different environmental conditions; determining F2 population sizes with regard to the combination of PHSR, FN, high performance, grain quality, and other commercial traits. New methods and approaches in breeding have opened new prospects. They include the doubled haploid (DH) method, allowing homozygotes to be obtained from early hybrid generations, and DNA technologies, which permit a genetically reasonable selection of the PHSR and FN donors, pyramiding of desirable alleles, and efficient selection of desired offspring using closely linked molecular markers.
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