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AGRICULTURAL BIOLOGY,
2015, V. 50, № 5, pp. 590-599 (SEL’SKOKHOZYAISTVENNAYA BIOLOGIYA)
ISSN 2412-0324 (English ed Online)
ISSN 0131-6397 (Russian ed. Print) ISSN 2313-4836 (Russian ed. Online)
Math modeling and instrumental methods in breeding
UDC 634.11:631.541:575:51-76 doi: 10.15389/agrobiology.2015.5.590rus
doi: 10.15389/agrobiology.2015.5.590eng
THE ASSESSMENT OF THE VARIETY AND ROOTSTOCK GENOTYPES INTERACTION IN APPLE (Malus domestica Borkh.) GRAFTED TREES USING BIOMETRIC METHODS
I.A. DRAGAVTSEVA1, V.A. DRAGAVTSEV1, I.L. EFIMOVA1, S.N. SHCHEGLOV2, V.V. DOMOZHIROVA1, A.S. MORENETS1
1 North-Caucasian Zonal Research Institute of Orcharding and Viniculture, Federal Agency of Scientific Organizations, 39, ul. 40 let Pobedy, Krasnodar, 350901 Russia, e-mail i_d@list.ru, dravial@mail.ru, vetch-dv@yandex.ru, efimiril@mail.ru;
2Kubanian State University, 149, ul. Stavropol’skaya, Krasnodar, 350040 Russia, e-mail gold_finch@mail.ru Acknowledgements:
Supported by Russian Foundation for Basic Research (grant № 13-01-96519-r_yug_a) and the Administration of
Krasnodar Krai
Received February 16, 2015
Abstract
In grafted fruit plants with vegetative propagation the scion (variety) and rootstock influence each other in newly-formed variety-rootstock combination. Under intensive technologies of fruit production based on maximal realization of grafted fruit plants’ bio potential it is very important to obtain accurate knowledge about quantitative traits which characterize productivity and biometrical parameters of fruit plants. To forecast the productivity of apple (Malus domestica Borkh.) trees, we studied the possibility of math modeling for the prediction of variety and rootstock influence on formation of quantitative traits in grafted plants using formulas offered by biometric genetics to estimate the same indexes in parent forms and their F1 hybrids. The data of apple trees productivity obtained in Prikubanskaya zone of Krasnodar region in 1983 to 2003 years (a total of 22 years) were analyzed. We studied the variety-rootstock combinations (VRC) of four apple varieties (Idared, Golden Delicious, Jonathan, Korah) as scions and seven rootstocks (I-48-1, I-47-55, I-48-46, M2 , M3, M4, M7) with regard to yields, the width of the crown from North to South and from West to East, the tree height and trunk diameter. An impact of the year conditions, the genotypes of the variety and the rootstock and their interaction on the yield of the variety-rootstock combinations was proven using math statistics. It was shown that the conditions of the year have the greatest effect (37 % of the total variance). Strength of the variety influence on the VRC yield was determined to be expressed with a several years interval. The rate of the rootstock impact and the cumulative effect of the scion and the rootstock were found to be roughly equial. For the first time for biological objects, which are characterized by non-linear relationship of traits, it is revealed that the forecasting models of VRC productivity based on multiple linear regression analysis with a linearized model is more effective and promising approach which takes into account the inadequacy of linear models previously considered. The histogram of residuals showed their normal distribution that is in conformity with correct use of the applied regression analysis. It provides a basis for adequate non-linear (quadratic) model of the yield production in each variety-rootstock combination as related to morphological and anatomical characteristics of grafted trees. Thus, based on theoretical analysis and the 22 year survey, we suggested the mathematical models for the variety and rootstock genotypes influence on quantitative traits in a grafted plant, primarily its productivity. It has been developed for the first time. This model enables more accurate control of stable and effective yield production in perennial crops.
Keywords: math modeling, methods of the math statistics, biometric genetics, fruit crops, apple tree, variety, rootstock, variety-rootstock combinations, regularities of the influence of variety and rootstock, the quantitative traits of the grafted plant, the yield forecast for variety-rootstock combinations, productivity, management.
In fruit plants that propagate vegetatively, the scion and rootstock are known to have a mutual effect on a newly formed scion-rootstock combination [1-5]. When using intensive technologies based on the maximum realization of
the biological potential of understock, obtaining accurate knowledge of quantitative traits that characterize the yield and biometrics of the obtained combinations is actual. In horticulture, extensive experimental data have been accumulated on the effect of rootstocks on the viability and adaptability to growing conditions, longevity, productivity, production quality and other properties of grafted plants [6-13]. At the same time, gardeners need to know about quantitative changes of agronomic traits in various combinations [14, 15].
In world practice, the ability of stocks to influence the size of plants and improve the productivity of understock trees is detected by means of long-term costly empirical testing [16-20]. The novelty of our approach is to predict the interaction between the genotypes of the scion and rootstock using biometric methods. Analysis of published sources [21-37] demonstrates the lack of the information on such studies, and the proposed approach is used in gardening for the first time.
Our purpose was to develop techniques for assessing the interaction between the genotypes of the scions and apple rootstocks based on the analysis of biometric parameters to identify the most economically promising combinations.
Technique. The research has been carried out in the Kuban area of the Krasnodar Territory at Experimental Production Farm Tsentral’noe of the North Caucasus Regional Research Institute of Horticulture and Viticulture (SKZNIISiV) within 22 years (1982-2003). Scion-rootstock combinations of apple (Malus domestic Borkh.) were studied based on the four varieties (scions) (Idared, Golden Delicious, Jonathan, Korah) and seven rootstocks (I-48-1, I-47-55, I-48-46, М2, М3, М4, М7). The yield of understock trees was estimated annually using the weight method. Biometric studies were performed in 1983 and 1985. In combination of Korah and Idared varieties with the I-48-1, I-47-55, I-48-46, M2, M3, M4, M7 stocks, the width of the crown in the direction from north to south and from west to east (cm), height (cm), the diameter of the trunk (cm) were determined.
Statistical calculations were performed using the StatSoft Statistica v. 10.0 and Statgraphics XVI program [38]. The method of three-way analysis of variance was used to assess the contribution of the study year, the scion varieties, the rootstock and their interaction in various combinations to the variability of the yields. The two-way analysis of variance was used to verify the significance of differences between rootstocks, scions and the presence of the «rootstock-scion» interaction in the years of research. Average long-term data were analyzed using the Box & Whisker plot graphical representations of the data. The proportion of the effect of the study year, scion and rootstock varieties and their interactions on the morphological and anatomical characteristics of ubderstock trees were also determined by analysis of variance. To assess the strength of association of traits studied, Spearman and Pearson correlation analysis was used; to construct the models for yield forecasting in scion-rootstock combinations, multiple linear regression analysis was used.
Results. Analysis of variance is one of the most used methods in biology [39, 40]. Two-factor analysis of variance (Table 1) showed that the conditions of the year, the scion variety, the rootstock and their combination affected the yield significantly. The conditions of the year had the greatest effect (31.7 %), the variety and the stock accounted for about 1.0 %. The cumulative effect of the year and variety (17.9 %) was the second most important, whereas the combined effect of the year and the stock was almost 7.5 times lower (2.4 %). We succeeded to detect a slight but statistically significant effect of the combination of variety and rootstock (0.5 %). Interestingly, the cumulative effect of all three factors was the third most important (4.9 %). This once again confirms the complexity of the
mathematical modeling of laws describing the role of the scion and rootstock in the formation of quantitative traits.
1. Assessment of the effect of the conditions of the year, the scion and rootstock on the yield of obtained apple combinations by analysis of variance (EPF Tsentral’noe of SKZNIISiV, Krasnodar, 1982-2003)
Variability df mS F a2 The proportion of the effect in the total variance, %
Inter-year 21 235629.45 394.2* 465.41 31.7
Inter-variety 3 38581.43 64.6* 13.68 0.9
Inter-rootstock 6 22935.62 38.4* 14.07 1.0
Year x variety 63 33730.03 56.4* 262.43 17.9
Year x rootstock 126 3194.65 5.3* 36.00 2.4
Variety x rootstock 18 3671.20 6.1* 7.75 0.5
Year x variety x rootstock 378 1906.06 3.2* 72.54 4.9
Residual 10494 597.69 597.69 40.7
Note. df is a degree of freedom, mS is the mean square, Fis the Fisher’s variance ratio, and a2 is a variance. * р < 0.01.
Fig. 1. The proportion of variety (scion) (1), rootstock (2) and combination of these factors (3) effect on the yield of apple scion-rootstock combinations on years of research (EPF Tsentral’noe of SKZNIISiV, Krasnodar, 1982-2003).
Two-way analysis of variance showed a significant effect of the scion, rootstock and their interaction on the yield. This regularity was traced in all the years of research and in an average sample. The proportion of the scion effect on the yield ranged from 2.6 (the year of 2000) to 56.6 % (the year of 1984). The year of 2002 was an exception. The proportion of the rootstock effect was from 1.6 (the year of 2002) to 20.2 % (the year of 1982). Special conditions of 2002 that caused a sharp decline in variability in different scion-rootstock combinations should be noted. The effect of stock
was not found in 1991 and 1994.
Fig. 2. Distribution of the statistical characteristics of yield in scion-rootstock combinations in the Idared (А), Golden Delicious (B), Jonathan (C), and Korah (D) apple tree varieties: 1 - I-48-1, 2 - I-47-55, 3 - I-48-46, 4 - М2, 5 -М3, 6 - М4, 7 - М7 rootstocks (Krasnodar, EPF Tsentral’noe of SKZNIISiV, 1982-2003).
Annual effect of the rootstock varieties on the pro-
ductivity of scion-rootstock combinations was pronounced every few years (Fig. 1). Perhaps this is due to the periodicity of fruiting that is characteristic of apple trees, climatic conditions and their combination.
The shares of the effect and interaction of variety and rootstock on the yield were approximately equivalent. Average long-term data are shown by Box & Whisker plot graphical representations (Fig. 2). In the graphs of this type, points represent the arithmetic mean, the borders of the rectangle represent the error of the mean, the lines outside the borders of the rectangle represent confidence intervals which makes it possible to graphically describe the statistical characteristics of the samples. It is evident that the best yield was provided by I-48-46, I-48-1 and M4 rootstocks in Idared, by I-48-46, M2 and M4 rootstocks in Golden Delicious variety, by M4, I-48-46 and M2 rootstocks in Jonathan variety, and by I-48-46, M2 and M4 rootstocks in Korah variety (see Fig. 2).
The results of analysis of variance based on biometric studies demonstrated a very strong effect of the year conditions on all the traits studied. The proportion of variance ranged from 9.9 (height) to 63.4 % (trunk diameter). The effect of variety genotype was also found for all of the analyzed features. The variability due to this factor ranged from 0.1 (trunk diameter) to 17.8 % (yield). The effect of the stock appeared to be 3 times lower than in the variety, but statistically significant ranging from 0.8 (crown width from north to south) to 5.9 % (trunk diameter). The combined effect of the scion and the year conditions was from 2.0 (height) to 32.0 % (yield), and the combined effect of the rootstock genotype with the year conditions varied from 1.8 (crown width from north to south and yield) to 5.7 % (height). The crown width from west to east was the exception (the effect on this feature has not been determined).
In our opinion, detection of statistically significant interactions involving the variety and rootstock genotypes which was shown for the first time for all traits without exception was of the most importance. The cumulative effect of the variety and rootstock genotypes was from 2.1 (crown width from north to south and yield) to 7.2 % (height), and the cumulative effects of the variety and rootstock genotypes and of the year conditions varied from 0.9 (trunk diameter) to 14.0 % (height). Therefore, the effects of the variety and rootstock genotypes and of environmental conditions were approximately similar.
The effect of the year condition on the traits studied is of interest, and this assessment is important for creating effective selection programs. The interaction of the genotype with the environment in a population (set of varieties) results in the changes in the total phenotypic, genotypic and additive variance that can be estimated if the tests are performed at different places in the same year or within several years in the same locality. Where the test genotypes do not change the grades of efficiency in all environments, then the interaction of «genotype-environment» is zero (absent). In the experiment performed in the same location or within one year, the variance of genotype—environment interactions will be mixed with a random variation. The impossibility of their separation results in the displacement of the estimates of genetic parameters. Even if the error variance is estimated under ideal experimental conditions, genotypic variance and the variance of genotype—environment interaction remain combined which results in the wrong conclusions. In this regard, the need to perform all the genetic and selection experiments for a number of years or in a few locations is obvious.
Calculation of relative values of the various components of genotype— environment interactions in practical selection should provide a more rational distribution of resources in experiments. In particular, it will make it possible to pre-
diet the importance of setting tests in a larger number of locations or in a single location for a longer period of time. For the genotype reliable ranking the tests performed in the same environment but with a larger number of replications (as in our case) are as informative as the costly experiments in various environments. In addition, the study of the genotype and environment interaction helps to select varieties or hybrids with more adaptive capabilities.
Until now, the main methods for the study of the genotype-environment interaction effects are the ones based on the analysis of variance and the regression analysis that provide a sufficiently reliable estimate of the variability. So, it is possible to trace the nature of the variation of additive and non-additive gene effect depending on the growing conditions in the analysis of combining ability in diallel crosses. More attention should be paid to the contribution of the nonadditive component in the effects of the genotype—environment interaction as an additively functioning gene does not interact neither with the genes in the cell, nor with the varying environmental factors.
The first step in constructing a mathematical model was to find correlations between traits. To do this, we used a pooled biometric sample of trees of various scion-rootstock combinations. This sample (about 800 trees) suggests that the traits will be subject to the normal distribution (this condition is mandatory for the use of parametric methods). The normality of trait distribution was estimated by x2 test (x2 = 561.77, p < 0.01 for tree crown width from north to south; x2 = 1924.13, p < 0.01 for tree crown width from west to east; x2 = 50.28, p < 0.01 for height; x2 = 109.83, p < 0.01 for trunk diameter; X2 = 1594.72; p < 0.01 for yield).
2. Trait correlation coefficients in scion-rootstock apple tree combinations (EPF Tsentral’noe of SKZNIISiV, Krasnodar, 1982-2003)
Crown width Height Trunk diameter Yield
Trait north to south west to east
Crown width north to south 0.80* 0.49* -0.46* 0.47*
west to east 0.76* 0.42* -0.27* 0.47*
Height 0.39* 0.39* -0.07* 0.10*
Trunk diameter -0.51* -0.28* -0.03 -0.13*
Yield 0.39* 0.35* 0.03 -0.08*
Note. Pearson correlation coefficients are above the diagonal, Spearman correlation coefficients are below the
diagonal. * Association of traits is reliable at the 5 % significance level.
The analysis was performed using Pearson’s correlation test parametric and non-parametric Spearman rank correlation test (Table 2). Pearson correlation coefficient was more effective as it is intended for such a distribution. Spearman’s correlation coefficient was calculated for the two pre-ranked variables. An inadequate result obtained with it can be attributed to frequent same values. When ranking these, there is a problem of tied ranks. In this case, a special rule applies according to which the objects with the same values are attributed to the same average rank. In the presence of similar (tied) ranks, the Spearman rank correlation formula cannot be used.
Mathematical modeling to analyze the relationship of the scion and rootstock requires the detection of significant variables of the system and setting links between them, so that the model yielded the same result of behavior as the object under study. This model is able to predict the behavior of the system in different environments, especially those poorly understood, such as the interaction of the scion and the rootstock in a new two-component fruit plant. When it comes to phenomena and processes with a complex structure and diversity of inherent relationships, this analysis is very complicated.
For further studies, the following linear models were used that suggest the observed values to be interconnected by dependence:
yi = b + bxi + ci, (1)
Where b), bi are unknown parameters (coefficients of the equation), c is independent normally distributed random variables with a zero expected value and a ct2 variance. This procedure was needed to construct a bi, b0 model and confidence intervals for bi, b0 according to the observations of x, yi in the best way, and to test the hypothesis on the significance of the equation and the regression coefficients, as well as to assess the adequacy of the obtained dependence.
A multiple regression model with several predictors was used:
yi = bixii + bx2i +•••+ bpXpi + b) + Ci, (2)
where bo, bi, b, ... bp are unknown model parameters.
A linear multiple regression in the Statgraphics XVI software package resulted in the following model:
Yield = -30.0i49 + 0.i83946 x crown width from north to south +
+ 0.i37848 x crown width from west to east — 0.i45484 x height + (3)
+ 0.564993 x trunk diameter.
The analysis of results show a weak association of the response and the predictors (R2 = 0.28), the constructed linear regression adequately described the association of the response and the predictors, and the absolute term was statistically significant.
There are several reasons to assume the linearity of trait association in regression analysis. Often, such an assumption is the simplest, so it becomes a natural to start the analysis with it. Many mathematical methods adapted to the solution of linear problems, forcing the use of linear circuits, even in cases where there are serious grounds to expect that the real association is significantly different from the linear one. Moreover, as a rule, all dependencies in the surrounding nature are nonlinear. Nevertheless, there are dependencies, the linearity of which in the considered applications is practically significant with any reasonable degree of accuracy. In the construction of mathematical models, the assumption of linearity is much more likely to have a distinct nature of the assumption, though it is not always stated as such. Therefore, when modeling the association of the processes and phenomena studied, nonlinear regression models are appropriate to be considered along with linear regression models. Typically, the need for a nonlinear regression appears if the researcher obtains data about the inadequacy of the linear model, and some nonlinear terms are added to the equation to clarify it [38].
In general, the regression model can be expressed as follows:
Y = F(Xi, X2, Xn). (4)
To model the association of the yield and morpho-physiological characteristics of plant stocks the nonlinear coupling of these features that we have discovered should be taken into account.
Physiologists have found that the association of the productivity of an object and the degree of its physiological arousal is expressed by regression excitation equation:
Y = b + bX + bX2, (5) where b) is an absolute term, bi and b are regression coefficients, Y and X are the values that characterize productivity and excitation, respectively. The nonlinearity of the model is expressed by the X2 term. This model is called nonlinear in the variables. It allows linearization which can be done by the replacement: X = Xi, X2 = X2. The equation takes the following form:
Y = b + biXi + bX2. (6)
Using the Fixed Nonlinear Regression module, multiple linear regres-
sion analysis of the linearized model was performed. Due to the software limitations on the number of variables, the trait of the crown width from north to south had to be excluded from the analysis, and the second trait of this category (crown width from west to east) was left in the analysis. Of the various options for linearizing transformations, the best one was:
X = X2. (7)
To assess the adequacy of the resulting model, a histogram of residual was constructed (Fig. 3). Residual is the difference of observed and fitted (or predicted by the model) values. One of the conditions for the correct use of regression analysis is the matching of residuals obtained and the normal distribution. Figure 3 shows that this term was satisfied, that is we constructed an adequate nonlinear (quadratic) model of the dependence between the yield in variety-rootstock combinations and morphological and anatomical features of grafted trees. The resulting model was as follows:
Yield = 171.3953 — 0.4599 x crown width from west to east —
— 0.7876 x height + 5.5432 x trunk diameter + 0,0008 x crown width from (8)
west to east2 + 0.0009 x height2 — 0.0628 x trunk diameter2.
We note that the adjusted multiple regression equation with a linearized model has a coefficient of determination R2 = 0.53 which refers to the average power of association of the response and the predictors. In other words, the proposed model describes 53 % of the original variation, allowing further improving the efficiency of the yield prediction in the grafted fruit trees.
Thus, we first used biometric methods to identify the Fig. 3. ffiiitognm °f resMrnls in toe e'ratoatira of toe тог- interaction between the stock rectoess of the regression analysis used to model the asso- interaction between the stock ciation of the yield and morphological and physiological and scion genotypes in apple features of grafted apple trees according to long-term ob- varieties. It was found that the
m2atl2°003)Krasnodar, EPF Tsentral’noe of ™, studied traits are significantly
affected by the conditions of the year (37 % of the total variance). The power of the grafted variety effects on the yield of rootstock-scion combinations has pronounced intervals of several years. The proportion of the rootstock effect and the cumulative effect of variety and rootstock on the analyzed traits were approximately equivalent. For the first time, for the biological objects which are characterized by nonlinear trait associations, it was found that multiple linear regression analysis with the linearized model was more effective in the construction of prognostic models of the yield of scion-rootstock combinations. It is a promising methodological approach which takes into account the inadequacy of the previously considered linear models. The correctness of regression analysis application was confirmed by the histogram of residuals. Their distribution was proven to correspond to normal. It provides a basis for the construction of an adequate nonlinear (quadratic) model for the dependence of the productivity of scion-rootstock combination on morphological and anatomical features of grafted trees. The approach to the mathematical modeling of the regularities describing the effect of the scion and the
rootstock genotype on the formation of quantitative traits in grafted plants (primarily its yield), which was for the first time theoretically proven and developed by us using the data from 22 years of observations, will allow to more accurately and reasonably control the stability and efficiency of the production process in perennial crops.
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