Geometry applied to breeding common beans (Phaseolus vulgaris)

Abstract

ABSTRACT. The primary components of common bean (Phaseolus vulgaris) grain yield (W) are the number of pods (X), the number of grains per pod (Y), and the weight of the grains (Z). In 1964, Grafius suggested using geometry in plant breeding; W corresponds to the volume of a parallelepiped with three axes, X, Y, and Z. Because the cube is the largest parallelepiped by volume, maximum yield is obtained when the relative contributions of X, Y, and Z are the same. We evaluated individual plants of a 'Talismã' x 'L.59583' cross in two sowing periods. The sum of squares of deviations from the ideal plant (GI), i.e., the plant in which the X, Y, and Z contributions were the same, was estimated. Mean and variance genetic components, and genetic and phenotypic correlations between the characteristics were also estimated. Good concordance was observed in the magnitude and direction of the genetic and phenotypic correlation estimates of the paired characteristics. However, a low GI heritability (h 2 r = 6.7%) indicated that success due to selection should be small. Ninety-four progenies of 'Pérola' x 'ESAL 686' crosses were also evaluated, where X, Y, Z, and W were obtained and GI was estimated. The h 2 estimate was higher, but still low (h 2 = 39.0%). Therefore, the selection of individuals to obtain plants in which the X, Y, and Z products tend to the cube is unfeasible, because the sums of X, Y, and Z vary between individuals. In addition, the GI h 2 value was low.

Research Article Genetic control of number of flowers and pod set in common bean.

“…It is inferred that plants with a greater number of flowers tend to have less percentage of the pod set. This fact has also been verified for correlations of the three primary components of grain yield of bean (X, Y, and Z) with W (Lima and Ramalho, 2016). The relative contribution of X, Y, and Z varies according to environmental conditions, as occurred in this study.…”

“…The relative contribution of X, Y, and Z varies according to environmental conditions, as occurred in this study. Lima and Ramalho (2016) discuss this question from the perspective of geometry. According to these authors, grain yield (W) is as if it were a parallelepiped, in which products of the axes X, Y, and Z would be equal to W. Since the largest volume of the parallelepiped is the cube, the contribution of X, Y, and Z would have to be the same to obtain the maximum W.…”

Simultaneous selection in beans for stability and high agronomic performance

“…Within this context, increasing the grain yield of new common bean cultivars would offer a nutritional and affordable product to a large portion of the global population. However, the grain yield of common bean has dominant effects (Lima and Ramalho, 2016), suggesting difficulties in selection. Breeding programs for Brazilian black and carioca common bean have led to a reduction in the genetic diversity of agronomic traits (Cardoso et al, 2013).…”

Research Article Genetic control of traits related to grain filling in the common bean

“…Grain yield (W) is the product of X.Y.Z. If one of the axis is influenced by any environmental factor, another one may compensate by supporting yield (Lima and Ramalho, 2016). Thus, it is practically unfeasible to try to increase RATE and NDPM simultaneously, unless there is a favorable environmental condition with no stress throughout the cycle.…”

“…They are those involved in the expression of the primary components of yield. In the common bean, for example, they are the number of pods/plant, the number of seeds/pod, and 100-seed weight (Zilio et al, 2011;Lima and Ramalho, 2016). Among these traits, the weight of 100 seeds is directly associated with dry matter accumulation during the period of grain filling.…”