- Email: [email protected]
Quantitative Inheritance
Quantitative Inheritance J Gai, Nanjing Agricultural University, Nanjing, China
© 2013 Elsevier Inc. All rights reserved.
This article is a revision of the previous edition article by E Pollak, volume 3, pp 1595–1597, © 2001, Elsevier Inc.
Glossary Additive-dominance-epistasis genetic model The genetic effects of genes in a population composed of additive effect of a homozygous locus, dominance effect of a heterozygous locus, and epistasis effect between two loci. Akaike information criterion (AIC) A measure of the relative goodness of fit of a statistical model based on the principle of entropy maximization. Analysis of variance (ANOVA) A statistical procedure in which the observed variance of a variable is partitioned into components attributable to different sources of variation, then the experimental factors are tested for their significance and the related multiple means are compared with each other for their significant differences. Combining ability The capacity of an individual to transmit superior performance to its offspring. The general combining ability is the average performance of a particular inbred in a series of hybrid combinations, whereas specific combining ability refers to the performance of a combination of specific inbred in a particular cross. Expectation–maximization (EM) An iterative algorithm for finding maximum likelihood estimates of parameters in statistical models, which comprises an expectation (E) step and a maximization (M) step. If the maximization step is replaced with conditional maximization, the
Quantitative Trait and Quantitative Inheritance The resemblance between parent organisms and their offspring or among the offspring is normally understood as due to inheri tance. People observe the inheritance usually through their phenotypic traits or characteristics. A trait is a distinct phenotypic character of an organism that can be measured, such as plant height of wheat in centimeters, or described, such as seed color of wheat in red or white. The traits can be classified into quanti tative ones and qualitative ones, depending on the scale of the traits. The scale of quantitative traits usually is continuous, either measurable such as 1000-seed weight or countable such as number of seeds per plant, while the scale of qualitative traits is immeasurable or uncountable but attributable to some dis tinct categories such as resistant or sensitive to some disease of a plant. The observed data for a quantitative trait composed of a variable on a quantitative scale while those for a qualitative trait composed of a variable on an attribute scale. The transmission of hereditary characteristics from parent organisms to their off spring is through the transmission of gene(s) rather than the characteristics themselves. The genetic constitution of a quanti tative trait is usually different from that of a qualitative trait. Therefore, the inheritance of a quantitative trait may be different
18
algorithm is called expectation and conditional maximization (ECM). Gene network The state or relationship among a complex of genes. Heritability The proportion of phenotypic variance attributable to genetic variance. Since the genetic variance comprises additive, dominance, and epistatic variances, the proportion of phenotypic variance accounted for by total genetic variance is defined as heritability in broad sense, while that accounted for by additive and additive × additive epistasis variance only is defined as heritability in narrow sense. Inbreeding coefficient and coancestry coefficient Inbreeding coefficient is the probability for two alleles on a locus in an individual to be identical by descent. Coancestry coefficient is the probability of two alleles from individual A and B, respectively, to be identical by decent. The coancestry coefficient between parents A and B of an individual is equal to the inbreeding coefficient of the individual. Path coefficient The standardized partial regression coefficient in multiple linear regression analysis. Quantitative trait locus (QTL) Stretches of DNA containing or linked to the genes that underlie a quantitative trait. Mapping regions of the genome that contain genes involved in specifying a quantitative trait is done using molecular tags such as SSRs and SNPs.
from that of a qualitative trait. A qualitative trait inherited qua litatively means the offspring having their distribution on the same attribute scale as their parents, while a quantitative trait that is usually inherited quantitatively means that the offspring are distributed on the same continuous scale as their parents. However, sometimes a trait measured quantitatively is not necessarily inherited quantitatively but may be inherited quali tatively if the offspring are distributed in different categories on a continuous scale, such as leaf blight lesion lengths on rice leaves less than or larger than 6 cm would constitute two distinct categories (resistant and sensitive to leaf blight, respectively) in a segregation population. It is usually known that the qualitative inheritance involves only a few genes, while the quantitative inheritance involves multiple genes.
Historical Hypotheses on Quantitative Inheritance Mendelian Inheritance The rules of transmission of hereditary characteristics from parent organisms to their offspring, known as the Law of Segregation and the Law of Independent Assortment, were first discovered by Gregor Johann Mendel in 1865 and 1866.
Brenner’s Encyclopedia of Genetics, 2nd edition, Volume 6
doi:10.1016/B978-0-12-374984-0.01250-X
Quantitative Inheritance The primary genetic laws were ‘rediscovered’ in 1900, and were initially very controversial. When they were integrated with the chromosome theory of inheritance by Thomas Hunt Morgan in 1915, they became the core of classical genetics. Originally, the Mendelian laws were thought to fit the inheri tance of qualitative traits only and not of quantitative traits. In 1889, Francis Galton reported that the filial body height regressed to that of their parents in human being in his ‘Natural inheritance’. Then Karl Pearson used mathematics to study inheritance and evolution of organisms, publishing a series of articles on ‘Mathematical contributions to the the ory of evolution’ during the 1890s–1900s. There was a serious controversy on the genetics of quantitative traits between the Mendelian school headed by Bateson and the Biometrical school headed by Pearson with the focus on whether contin uous variation could be inherited and whether continuous or discontinuous variation caused evolution. Nilsson-Ehle pro posed the ‘multiple-factor hypothesis’ of quantitative trait with his study on the inheritance of wheat seed color in 1909, which means that multiple Mendelian genes are responsible for quantitative traits. East confirmed Nilsson Ehle’s hypothesis with his study on the corolla size of tobacco in 1913. This hypothesis ended the controversy between the two schools and brought the inheritance of quantitative traits into Mendelian inheritance.
Classical Quantitative Inheritance Based on multiple-factor Mendelian inheritance, Fisher estab lished the additive-dominance genetic model (g = a + d, p = g + e, where g and p are genotypic and phenotypic effects, a and d are additive and dominance effects, and e is random error, respec tively, in Fisher’s notation) for quantitative traits to separate the gene effect and the genetic variance into the respective parts, that is, additive and dominance, in ‘The correlation between relatives on the supposition of Mendelian inheritance’ in 1918. Later on during 1924–34, Haldane explained the genetic changes of quantitative traits under natural and artificial selection in a series of articles of ‘A mathematical theory of natural and artificial selection’. Wright overviewed the mating systems and then defined the inbreeding coefficient and path coefficient in a series of articles on ‘Systems of mating’. With the former, various mating systems of populations could be linked to each other for joint analysis of quantitative inheritance; and with the latter, the genetic effects of quantitative traits under various mating systems could be estimated. A very important concept of herit ability in quantitative inheritance was established and further defined as heritability in a broad sense and heritability in a narrow sense by Lush in his ‘Animal Breeding Plan’ in 1940s, which was relevant in studying selection efficiency and the expected genetic gain of selection procedures for quantitative traits. After that, Sprague and Tatum established the concept of combining ability of a parent in 1942, including general and specific combining ability, indicating the quantitative inheri tance performed at heterozygous generation. Malécot defined the coancestry coefficient (coefficient of parentage) as the prob ability of a pair of the same alleles from both parents being identical by descent in 1948, with which the general relationship of covariance between relatives was established. All these con cepts helped in the establishment of the classical quantitative
19
inheritance. Mather and his colleagues extended Fisher’s simple genetic model into that with epistasis (g = d + h + i + j + l, where d, h, i, j, and l are additive, dominance, additive by additive epis tasis, additive by dominance epistasis, and dominance by dominance epistasis effects, respectively, in Mather’s notation), and then developed a series of procedures to detect and verify the genetic model and related genetic effects. They mainly focused on the genetic populations derived from a biparental cross, including triple test cross (TTC) genetic design. Their representa tive publication is ‘Biometrical Genetics’. Meanwhile, based on Fisher’s simple genetic model, Kempthorne, Comstock, and Falconer also extended it into that with epistasis (g = a + d + aa + ad + dd, where aa, ad, and dd are additive by additive epistasis, additive by dominance epistasis, and dominance by dominance epistasis effects, respectively, in Kempthorne’s notation based on Fisher), and then developed a series of procedures to detect and verify genetic models. Their approach was mainly based on the variance–covariance analysis among relatives and their genetic materials were mainly random mating population and its deri vatives with different degrees of inbreeding. A series of mating designs were developed for studying quantitative inheritance, such as NC I, NC II, and NC III. ‘An Introduction to Genetics Statistics’ by Kempthorne in 1957 and ‘Introduction to Quantitative Genetics’ by Falconer in 1961 are their representa tive publications. There was controversy between the Birmingham school headed by Mather and Jinks and the Iowa– North Carolina–Edinburgh school headed by Kempthorne and others. In fact, the two schools worked on different approaches to the same problem. The former mainly used biparental F2 as reference population with their derived populations to detect the genetic models and estimate the first-order genetic parameters, while the latter used random mating population as a reference population with their derived populations to detect the genetic models and estimate the relative importance of various genetic effects based on second-order genetic parameters. However, the latter was more likely to cover a wide range of genetic popula tions since an inbreeding coefficient was used to link random mating population to populations with certain degree of inbreeding and, therefore, connecting more tightly with improv ing breeding procedures for cross-pollinated and self-pollinated crops, as well as animals. In addition, both schools had realized the importance of genotype � environment interaction in the inheritance and performance of quantitative traits. At this stage, the classical quantitative genetics was well established. The major characteristics of this stage were to establish statistical or biome trical procedures for designed genetic populations to detect genetic models and estimate the relative importance of the genetic components. Under this consideration, the effect of each gene was assumed very small but the number of genes was assumed very large, called polygenes or minor genes, in the quantitative genetic system, and therefore, the detected genetic effect was due to the collective performance of polygenes or minor genes. Thus, the polygenes or minor genes could be detected collectively but not individually.
Generalized Major Gene and Polygene Mixed Inheritance Based on Biometrical Analysis A number of genetic phenomena indicated that the effects of individual genes in the genetic system may differ from each
20
Quantitative Inheritance
other. Thus, during the 1970s–80s, the concept and analytical procedure of a single major gene plus polygene mixed inheri tance model were raised by Elston and Stewart, Elkind and Cahaner, and others. Based on it, Gai and his group extended the concept of the single major gene plus polygene mixed inheritance into the hypothesis of the generalized major gene plus polygene mixed inheritance model of quantitative traits since late 1990s. The hypothesis comprises the following points: (1) A quantitative trait may be controlled by multiple genes with different effects, large or small, but not necessarily equal. (2) The gene with a large effect performs as the major gene, while the gene with relatively very small effect, which cannot be detected individually in the experimental condi tions, performs as the minor gene or polygene. (3) The distinction between major and minor genes is relative to and depends on the experimental error or precision, and in one environment a gene may perform as a major gene but in another environment as a minor one. In addition, a gene may perform as a major gene under a precise experiment, while it cannot be detected individually and may perform as a minor one under a less precise experiment. Therefore, for a quantitative trait, the genetic system with both major and minor gene(s) is a general model, and pure major gene inheri tance and pure minor gene inheritance are only specific cases to the general model. Of course, the above statement is appro priate only under experimental conditions for applied study, and theoretically, it is not reasonable to set a critical value of gene effect for classifying it into major or minor since the gene effect of quantitative traits varies continuously. Based on the above theory, the segregation analysis procedures of quanti tative traits were established, including the following concepts and steps: (1) The frequency distribution of a segregating population is composed of several component distributions of major gene genotypes. Each component meets the assump tion of normality and is modified with both polygenes and environments. Therefore, the whole distribution is a mixture of several components meeting the assumption of normality. (2) The maximum likelihood functions are established from a single segregating generation or jointly from multiple generations for various genetic models. In a function, the proportion, mean, and variance of each component distribu tion are included and are to be estimated from the experimental data. (3) The expectation–maximization (EM) or iterated expectation and conditional maximization (ECM) algorithm is used to estimate the parameters of component distributions for various genetic models. (4) After the maximum likelihood analyses of all types of genetic models for a given set of experimental data, the Akaike information criterion (AIC) and a set of tests, including the uniformity test, the Smirnov test, and the Kolmogorov test, are used to choose an optimal genetic model and pick up its corresponding esti mates of component distributions. (5) The estimates of genetic parameters, including the additive, dominance, and epistasis effects of major gene(s), the total additive, domi nance, and epistasis effects of minor genes, and the major gene heritability and the polygene heritability are calculated from the estimates of component distributions of the optimal genetic model by using the method of least squares according to the relationship between the genetic parameters and the component distribution parameters of a given genetic model.
Generalized Major Gene and Polygene Mixed Inheritance Based on QTL Mapping The capacity of segregation analysis (biometrical analysis) is limited in detecting the number of major genes. Up to now, the analytical procedure worked out is limited only for up to four major genes plus polygenes mixed inheritance model. But for tunately, the quantitative trait locus (QTL) mapping using molecular markers has made the study on quantitative inheri tance much easier and broader. Here, a QTL is not necessarily limited as a gene, but might include several genes in terms of DNA segments. Originally, QTL mapping was aimed at detect ing additive major genes, then it expanded to include epistatic QTL pairs and QTL � environment interactions. Along with improved mapping and experiment precision, the small-effect major QTLs could also be detected. Among the quantitative traits, some performed to have major QTLs only, some per formed to have no major QTLs detected, but most of the traits performed to have a few or several large-effect major QTLs and a relatively large number of small-effect major QTLs. Recently, in mapping QTLs responsible for aluminum tolerance and other traits in soybean, Gai and his group found that the sum of contributions from mapped additive QTLs and epistatic QTL pairs was much less than the total genetic variation estimated from genotypic variances in ana lysis of variance (ANOVA). The remaining part of genetic variation was attributed to those QTLs not detected in the mapping procedure. In this way, the total phenotypic variance (100%) was partitioned into QTL, QTL � environment, and environment variance components. In terms of the trait of aluminum tolerance, heritability from the joint analysis of multiple environment data was 77.80% and additive QTL effects contributed about 22.30%, while epistasis QTL pairs contributed about 14.86% to phenotypic variance. Thus, 40.64%, a substantial part of the phenotypic variance, was due to unmapped QTLs. Here, the unmapped QTLs were mainly QTLs with very small effects because the saturation level of the used genetic linkage map should be able to detect most of the major QTLs, including small-effect major QTLs. Here, they used unmapped QTL collective (or unmapped minor QTL collective) to represent this part of genetic varia tion, which in fact accounted for a major portion of both the genetic and the phenotypic variation in the trait. The same results have been observed in the QTL mapping studies for a number of traits, such as oil and fatty acid contents, protein, and protein component contents in soybeans. Accordingly, the above QTL mapping results support further the hypothesis of generalized major gene plus minor gene mixed inheritance.
A Further Hypothesis: Gene Network of Quantitative Inheritance At early stages of genetic studies, people observed the inheritance of traits individually and recognized one gene responsible for one trait or several genes responsible for one trait, with the gene or genes working independently, especially for qualitative traits. Afterward, the phenomena of one gene responsible for more than one trait and interaction between/among genes were observed and then inheritance of a group of traits was concerned. In the book ‘Biometrical Genetics’, Mather distinguished traits as
Quantitative Inheritance supercharacter and subcharacter, and noticed two subcharacters may not be independent in their genetic control. That means a group of genes is responsible for multiple traits. This is the primordial understanding of the gene network of inheritance. Now genomic sequencing has helped to estimate the number of protein-coding genes in a species, for example, Schmutz et al. predicted 46 430 protein-coding genes in soybean genome. It seems that the number of genes in a species is limited but the number of phenotypic traits, including supercharacters and subcharacters is almost not limited because during the lifetime there might be huge number of time points for evaluation of unlimited number of phenotypic traits. All kinds of phenotypic performance should be related to the limited number of genes. Therefore, all the genes composed of the gene network of a species and the phenotypic measurement are just a marginal projection of the gene network onto the trait. There might be some genes showing a major role on the marginal projection and, therefore, may be detected as major genes; some genes showing a small role on the marginal projection may be recognized as small-effect major genes, while a great number of genes showing very small role on the marginal projection may be considered as a collective undetected minor genes or polygenes. With this perspective, the hypothesis of gene networks of quantitative inheritance needs to be demonstrated with future experimental practices.
See also: Epistasis; Population Genetics; Quantitative Genetics; QTL Mapping.
21
Further Reading Elkind Y and Cahaner A (1986) A mixed model for the effects of single gene, polygenes and their interaction on quantitative traits. 1. The model and experimental design. Theoretical and Applied Genetics 72: 377–383. Elston RC and Stewart J (1973) The analysis of quantitative traits for simple genetic models from parental, F1 and backcross data. Genetics 73: 695–711. Falconer DS (1961) Introduction to Quantitative Genetics, 1st edn. New York: Ronald Press; 4th edn. (1996, with Mackey TFC) Harlow, UK: Longman. Fisher RA (1918) The correlation between relatives on the supposition of Mendelian inheritance. Philosophical Transactions of the Royal Society of Edinburgh 52: 399–433. Gai JY (2006) Segregation analysis on genetic system of quantitative traits in plants. Frontiers of Biology in China 1: 85–92. Gai JY and Wang JK (1998) Identification and estimation of a QTL model and its effects. Theoretical and Applied Genetics 97: 1162–1168. Gai JY, Zhang YM, and Wang JK (2003) Genetic System of Quantitative Traits in Plants. Beijing: Science Press House. Haldane JBS (1924) A mathematical theory of natural and artificial selection. Transactions of the Cambridge Philosophical Society 23: 19–41. Kempthorne O (1957) An Introduction to Genetic Statistics. New York: Wiley. Korir PC, Qi B, Wang Y, et al. (2011) A study on relative importance of additive, epistasis and unmapped QTL for aluminium tolerance at seedling stage in soybean. Plant Breeding 130(5): 551–562. Lush JL (1945) Animal Breeding Plans, 3rd edn. Ames, IA: Iowa State College Press. Mather K (1949) Biometrical Genetics, 1st edn. London: Methuen; 3rd edn. (1982, with Jinks JL) London: Chapman and Hall. Schmutz J, Cannon SB, Schlueter J, et al. (2010) Genome sequence of the palaeopolyploid soybean. Nature 463: 178–183. Sprague GF and Tatum LA (1942) General vs. specific combining ability in single crosses of corn. Journal of the American Society of Agronomy 34: 923–932. Wright S (1921) Systems of mating. I. The biometric relations between parent and offspring. Genetics 6: 111–123.
© 2013 Elsevier Inc. All rights reserved.
This article is a revision of the previous edition article by E Pollak, volume 3, pp 1595–1597, © 2001, Elsevier Inc.
Glossary Additive-dominance-epistasis genetic model The genetic effects of genes in a population composed of additive effect of a homozygous locus, dominance effect of a heterozygous locus, and epistasis effect between two loci. Akaike information criterion (AIC) A measure of the relative goodness of fit of a statistical model based on the principle of entropy maximization. Analysis of variance (ANOVA) A statistical procedure in which the observed variance of a variable is partitioned into components attributable to different sources of variation, then the experimental factors are tested for their significance and the related multiple means are compared with each other for their significant differences. Combining ability The capacity of an individual to transmit superior performance to its offspring. The general combining ability is the average performance of a particular inbred in a series of hybrid combinations, whereas specific combining ability refers to the performance of a combination of specific inbred in a particular cross. Expectation–maximization (EM) An iterative algorithm for finding maximum likelihood estimates of parameters in statistical models, which comprises an expectation (E) step and a maximization (M) step. If the maximization step is replaced with conditional maximization, the
Quantitative Trait and Quantitative Inheritance The resemblance between parent organisms and their offspring or among the offspring is normally understood as due to inheri tance. People observe the inheritance usually through their phenotypic traits or characteristics. A trait is a distinct phenotypic character of an organism that can be measured, such as plant height of wheat in centimeters, or described, such as seed color of wheat in red or white. The traits can be classified into quanti tative ones and qualitative ones, depending on the scale of the traits. The scale of quantitative traits usually is continuous, either measurable such as 1000-seed weight or countable such as number of seeds per plant, while the scale of qualitative traits is immeasurable or uncountable but attributable to some dis tinct categories such as resistant or sensitive to some disease of a plant. The observed data for a quantitative trait composed of a variable on a quantitative scale while those for a qualitative trait composed of a variable on an attribute scale. The transmission of hereditary characteristics from parent organisms to their off spring is through the transmission of gene(s) rather than the characteristics themselves. The genetic constitution of a quanti tative trait is usually different from that of a qualitative trait. Therefore, the inheritance of a quantitative trait may be different
18
algorithm is called expectation and conditional maximization (ECM). Gene network The state or relationship among a complex of genes. Heritability The proportion of phenotypic variance attributable to genetic variance. Since the genetic variance comprises additive, dominance, and epistatic variances, the proportion of phenotypic variance accounted for by total genetic variance is defined as heritability in broad sense, while that accounted for by additive and additive × additive epistasis variance only is defined as heritability in narrow sense. Inbreeding coefficient and coancestry coefficient Inbreeding coefficient is the probability for two alleles on a locus in an individual to be identical by descent. Coancestry coefficient is the probability of two alleles from individual A and B, respectively, to be identical by decent. The coancestry coefficient between parents A and B of an individual is equal to the inbreeding coefficient of the individual. Path coefficient The standardized partial regression coefficient in multiple linear regression analysis. Quantitative trait locus (QTL) Stretches of DNA containing or linked to the genes that underlie a quantitative trait. Mapping regions of the genome that contain genes involved in specifying a quantitative trait is done using molecular tags such as SSRs and SNPs.
from that of a qualitative trait. A qualitative trait inherited qua litatively means the offspring having their distribution on the same attribute scale as their parents, while a quantitative trait that is usually inherited quantitatively means that the offspring are distributed on the same continuous scale as their parents. However, sometimes a trait measured quantitatively is not necessarily inherited quantitatively but may be inherited quali tatively if the offspring are distributed in different categories on a continuous scale, such as leaf blight lesion lengths on rice leaves less than or larger than 6 cm would constitute two distinct categories (resistant and sensitive to leaf blight, respectively) in a segregation population. It is usually known that the qualitative inheritance involves only a few genes, while the quantitative inheritance involves multiple genes.
Historical Hypotheses on Quantitative Inheritance Mendelian Inheritance The rules of transmission of hereditary characteristics from parent organisms to their offspring, known as the Law of Segregation and the Law of Independent Assortment, were first discovered by Gregor Johann Mendel in 1865 and 1866.
Brenner’s Encyclopedia of Genetics, 2nd edition, Volume 6
doi:10.1016/B978-0-12-374984-0.01250-X
Quantitative Inheritance The primary genetic laws were ‘rediscovered’ in 1900, and were initially very controversial. When they were integrated with the chromosome theory of inheritance by Thomas Hunt Morgan in 1915, they became the core of classical genetics. Originally, the Mendelian laws were thought to fit the inheri tance of qualitative traits only and not of quantitative traits. In 1889, Francis Galton reported that the filial body height regressed to that of their parents in human being in his ‘Natural inheritance’. Then Karl Pearson used mathematics to study inheritance and evolution of organisms, publishing a series of articles on ‘Mathematical contributions to the the ory of evolution’ during the 1890s–1900s. There was a serious controversy on the genetics of quantitative traits between the Mendelian school headed by Bateson and the Biometrical school headed by Pearson with the focus on whether contin uous variation could be inherited and whether continuous or discontinuous variation caused evolution. Nilsson-Ehle pro posed the ‘multiple-factor hypothesis’ of quantitative trait with his study on the inheritance of wheat seed color in 1909, which means that multiple Mendelian genes are responsible for quantitative traits. East confirmed Nilsson Ehle’s hypothesis with his study on the corolla size of tobacco in 1913. This hypothesis ended the controversy between the two schools and brought the inheritance of quantitative traits into Mendelian inheritance.
Classical Quantitative Inheritance Based on multiple-factor Mendelian inheritance, Fisher estab lished the additive-dominance genetic model (g = a + d, p = g + e, where g and p are genotypic and phenotypic effects, a and d are additive and dominance effects, and e is random error, respec tively, in Fisher’s notation) for quantitative traits to separate the gene effect and the genetic variance into the respective parts, that is, additive and dominance, in ‘The correlation between relatives on the supposition of Mendelian inheritance’ in 1918. Later on during 1924–34, Haldane explained the genetic changes of quantitative traits under natural and artificial selection in a series of articles of ‘A mathematical theory of natural and artificial selection’. Wright overviewed the mating systems and then defined the inbreeding coefficient and path coefficient in a series of articles on ‘Systems of mating’. With the former, various mating systems of populations could be linked to each other for joint analysis of quantitative inheritance; and with the latter, the genetic effects of quantitative traits under various mating systems could be estimated. A very important concept of herit ability in quantitative inheritance was established and further defined as heritability in a broad sense and heritability in a narrow sense by Lush in his ‘Animal Breeding Plan’ in 1940s, which was relevant in studying selection efficiency and the expected genetic gain of selection procedures for quantitative traits. After that, Sprague and Tatum established the concept of combining ability of a parent in 1942, including general and specific combining ability, indicating the quantitative inheri tance performed at heterozygous generation. Malécot defined the coancestry coefficient (coefficient of parentage) as the prob ability of a pair of the same alleles from both parents being identical by descent in 1948, with which the general relationship of covariance between relatives was established. All these con cepts helped in the establishment of the classical quantitative
19
inheritance. Mather and his colleagues extended Fisher’s simple genetic model into that with epistasis (g = d + h + i + j + l, where d, h, i, j, and l are additive, dominance, additive by additive epis tasis, additive by dominance epistasis, and dominance by dominance epistasis effects, respectively, in Mather’s notation), and then developed a series of procedures to detect and verify the genetic model and related genetic effects. They mainly focused on the genetic populations derived from a biparental cross, including triple test cross (TTC) genetic design. Their representa tive publication is ‘Biometrical Genetics’. Meanwhile, based on Fisher’s simple genetic model, Kempthorne, Comstock, and Falconer also extended it into that with epistasis (g = a + d + aa + ad + dd, where aa, ad, and dd are additive by additive epistasis, additive by dominance epistasis, and dominance by dominance epistasis effects, respectively, in Kempthorne’s notation based on Fisher), and then developed a series of procedures to detect and verify genetic models. Their approach was mainly based on the variance–covariance analysis among relatives and their genetic materials were mainly random mating population and its deri vatives with different degrees of inbreeding. A series of mating designs were developed for studying quantitative inheritance, such as NC I, NC II, and NC III. ‘An Introduction to Genetics Statistics’ by Kempthorne in 1957 and ‘Introduction to Quantitative Genetics’ by Falconer in 1961 are their representa tive publications. There was controversy between the Birmingham school headed by Mather and Jinks and the Iowa– North Carolina–Edinburgh school headed by Kempthorne and others. In fact, the two schools worked on different approaches to the same problem. The former mainly used biparental F2 as reference population with their derived populations to detect the genetic models and estimate the first-order genetic parameters, while the latter used random mating population as a reference population with their derived populations to detect the genetic models and estimate the relative importance of various genetic effects based on second-order genetic parameters. However, the latter was more likely to cover a wide range of genetic popula tions since an inbreeding coefficient was used to link random mating population to populations with certain degree of inbreeding and, therefore, connecting more tightly with improv ing breeding procedures for cross-pollinated and self-pollinated crops, as well as animals. In addition, both schools had realized the importance of genotype � environment interaction in the inheritance and performance of quantitative traits. At this stage, the classical quantitative genetics was well established. The major characteristics of this stage were to establish statistical or biome trical procedures for designed genetic populations to detect genetic models and estimate the relative importance of the genetic components. Under this consideration, the effect of each gene was assumed very small but the number of genes was assumed very large, called polygenes or minor genes, in the quantitative genetic system, and therefore, the detected genetic effect was due to the collective performance of polygenes or minor genes. Thus, the polygenes or minor genes could be detected collectively but not individually.
Generalized Major Gene and Polygene Mixed Inheritance Based on Biometrical Analysis A number of genetic phenomena indicated that the effects of individual genes in the genetic system may differ from each
20
Quantitative Inheritance
other. Thus, during the 1970s–80s, the concept and analytical procedure of a single major gene plus polygene mixed inheri tance model were raised by Elston and Stewart, Elkind and Cahaner, and others. Based on it, Gai and his group extended the concept of the single major gene plus polygene mixed inheritance into the hypothesis of the generalized major gene plus polygene mixed inheritance model of quantitative traits since late 1990s. The hypothesis comprises the following points: (1) A quantitative trait may be controlled by multiple genes with different effects, large or small, but not necessarily equal. (2) The gene with a large effect performs as the major gene, while the gene with relatively very small effect, which cannot be detected individually in the experimental condi tions, performs as the minor gene or polygene. (3) The distinction between major and minor genes is relative to and depends on the experimental error or precision, and in one environment a gene may perform as a major gene but in another environment as a minor one. In addition, a gene may perform as a major gene under a precise experiment, while it cannot be detected individually and may perform as a minor one under a less precise experiment. Therefore, for a quantitative trait, the genetic system with both major and minor gene(s) is a general model, and pure major gene inheri tance and pure minor gene inheritance are only specific cases to the general model. Of course, the above statement is appro priate only under experimental conditions for applied study, and theoretically, it is not reasonable to set a critical value of gene effect for classifying it into major or minor since the gene effect of quantitative traits varies continuously. Based on the above theory, the segregation analysis procedures of quanti tative traits were established, including the following concepts and steps: (1) The frequency distribution of a segregating population is composed of several component distributions of major gene genotypes. Each component meets the assump tion of normality and is modified with both polygenes and environments. Therefore, the whole distribution is a mixture of several components meeting the assumption of normality. (2) The maximum likelihood functions are established from a single segregating generation or jointly from multiple generations for various genetic models. In a function, the proportion, mean, and variance of each component distribu tion are included and are to be estimated from the experimental data. (3) The expectation–maximization (EM) or iterated expectation and conditional maximization (ECM) algorithm is used to estimate the parameters of component distributions for various genetic models. (4) After the maximum likelihood analyses of all types of genetic models for a given set of experimental data, the Akaike information criterion (AIC) and a set of tests, including the uniformity test, the Smirnov test, and the Kolmogorov test, are used to choose an optimal genetic model and pick up its corresponding esti mates of component distributions. (5) The estimates of genetic parameters, including the additive, dominance, and epistasis effects of major gene(s), the total additive, domi nance, and epistasis effects of minor genes, and the major gene heritability and the polygene heritability are calculated from the estimates of component distributions of the optimal genetic model by using the method of least squares according to the relationship between the genetic parameters and the component distribution parameters of a given genetic model.
Generalized Major Gene and Polygene Mixed Inheritance Based on QTL Mapping The capacity of segregation analysis (biometrical analysis) is limited in detecting the number of major genes. Up to now, the analytical procedure worked out is limited only for up to four major genes plus polygenes mixed inheritance model. But for tunately, the quantitative trait locus (QTL) mapping using molecular markers has made the study on quantitative inheri tance much easier and broader. Here, a QTL is not necessarily limited as a gene, but might include several genes in terms of DNA segments. Originally, QTL mapping was aimed at detect ing additive major genes, then it expanded to include epistatic QTL pairs and QTL � environment interactions. Along with improved mapping and experiment precision, the small-effect major QTLs could also be detected. Among the quantitative traits, some performed to have major QTLs only, some per formed to have no major QTLs detected, but most of the traits performed to have a few or several large-effect major QTLs and a relatively large number of small-effect major QTLs. Recently, in mapping QTLs responsible for aluminum tolerance and other traits in soybean, Gai and his group found that the sum of contributions from mapped additive QTLs and epistatic QTL pairs was much less than the total genetic variation estimated from genotypic variances in ana lysis of variance (ANOVA). The remaining part of genetic variation was attributed to those QTLs not detected in the mapping procedure. In this way, the total phenotypic variance (100%) was partitioned into QTL, QTL � environment, and environment variance components. In terms of the trait of aluminum tolerance, heritability from the joint analysis of multiple environment data was 77.80% and additive QTL effects contributed about 22.30%, while epistasis QTL pairs contributed about 14.86% to phenotypic variance. Thus, 40.64%, a substantial part of the phenotypic variance, was due to unmapped QTLs. Here, the unmapped QTLs were mainly QTLs with very small effects because the saturation level of the used genetic linkage map should be able to detect most of the major QTLs, including small-effect major QTLs. Here, they used unmapped QTL collective (or unmapped minor QTL collective) to represent this part of genetic varia tion, which in fact accounted for a major portion of both the genetic and the phenotypic variation in the trait. The same results have been observed in the QTL mapping studies for a number of traits, such as oil and fatty acid contents, protein, and protein component contents in soybeans. Accordingly, the above QTL mapping results support further the hypothesis of generalized major gene plus minor gene mixed inheritance.
A Further Hypothesis: Gene Network of Quantitative Inheritance At early stages of genetic studies, people observed the inheritance of traits individually and recognized one gene responsible for one trait or several genes responsible for one trait, with the gene or genes working independently, especially for qualitative traits. Afterward, the phenomena of one gene responsible for more than one trait and interaction between/among genes were observed and then inheritance of a group of traits was concerned. In the book ‘Biometrical Genetics’, Mather distinguished traits as
Quantitative Inheritance supercharacter and subcharacter, and noticed two subcharacters may not be independent in their genetic control. That means a group of genes is responsible for multiple traits. This is the primordial understanding of the gene network of inheritance. Now genomic sequencing has helped to estimate the number of protein-coding genes in a species, for example, Schmutz et al. predicted 46 430 protein-coding genes in soybean genome. It seems that the number of genes in a species is limited but the number of phenotypic traits, including supercharacters and subcharacters is almost not limited because during the lifetime there might be huge number of time points for evaluation of unlimited number of phenotypic traits. All kinds of phenotypic performance should be related to the limited number of genes. Therefore, all the genes composed of the gene network of a species and the phenotypic measurement are just a marginal projection of the gene network onto the trait. There might be some genes showing a major role on the marginal projection and, therefore, may be detected as major genes; some genes showing a small role on the marginal projection may be recognized as small-effect major genes, while a great number of genes showing very small role on the marginal projection may be considered as a collective undetected minor genes or polygenes. With this perspective, the hypothesis of gene networks of quantitative inheritance needs to be demonstrated with future experimental practices.
See also: Epistasis; Population Genetics; Quantitative Genetics; QTL Mapping.
21
Further Reading Elkind Y and Cahaner A (1986) A mixed model for the effects of single gene, polygenes and their interaction on quantitative traits. 1. The model and experimental design. Theoretical and Applied Genetics 72: 377–383. Elston RC and Stewart J (1973) The analysis of quantitative traits for simple genetic models from parental, F1 and backcross data. Genetics 73: 695–711. Falconer DS (1961) Introduction to Quantitative Genetics, 1st edn. New York: Ronald Press; 4th edn. (1996, with Mackey TFC) Harlow, UK: Longman. Fisher RA (1918) The correlation between relatives on the supposition of Mendelian inheritance. Philosophical Transactions of the Royal Society of Edinburgh 52: 399–433. Gai JY (2006) Segregation analysis on genetic system of quantitative traits in plants. Frontiers of Biology in China 1: 85–92. Gai JY and Wang JK (1998) Identification and estimation of a QTL model and its effects. Theoretical and Applied Genetics 97: 1162–1168. Gai JY, Zhang YM, and Wang JK (2003) Genetic System of Quantitative Traits in Plants. Beijing: Science Press House. Haldane JBS (1924) A mathematical theory of natural and artificial selection. Transactions of the Cambridge Philosophical Society 23: 19–41. Kempthorne O (1957) An Introduction to Genetic Statistics. New York: Wiley. Korir PC, Qi B, Wang Y, et al. (2011) A study on relative importance of additive, epistasis and unmapped QTL for aluminium tolerance at seedling stage in soybean. Plant Breeding 130(5): 551–562. Lush JL (1945) Animal Breeding Plans, 3rd edn. Ames, IA: Iowa State College Press. Mather K (1949) Biometrical Genetics, 1st edn. London: Methuen; 3rd edn. (1982, with Jinks JL) London: Chapman and Hall. Schmutz J, Cannon SB, Schlueter J, et al. (2010) Genome sequence of the palaeopolyploid soybean. Nature 463: 178–183. Sprague GF and Tatum LA (1942) General vs. specific combining ability in single crosses of corn. Journal of the American Society of Agronomy 34: 923–932. Wright S (1921) Systems of mating. I. The biometric relations between parent and offspring. Genetics 6: 111–123.