An Introduction to Quantitative Genetics

Transcription

1 An Introduction to Quantitative Genetics Mohammad Keramatipour MD, PhD ac ir 1 Mendel s work Laws of inheritance Basic Concepts Applications Predicting outcome of crosses Phenotype of offsprings Essentials needed Essentials needed Parental phenotype / genotype Laws of inheritance Statistical tools Plant & animal breeding Genetic counseling for human diseases 2 Review on Statistical Concepts Review on Statistical Concepts Probability? Accuracy of probability / prediction Sample size / random sampling error Calculating probabilities The sum rule Predict the occurrence of mutually exclusive events The product rule Predict the probability of independent events The binomial expansion equation Predict t the probability bilit of an unordered d combination of events 3 Testing hypothesis What do we really test? t? Can we really prove the hypothesis? Concept of goodness of fit The chi square test t Concept of P value The degrees of freedom 4 Terminology Importance of Quantitative Genetics Quantitative genetics: Study of traits that can be described numerically and their variation within population Characters/traits: Mendelian: monogenic Non-Mendelian Such as multifactorial (complex) Phenotype: observable characteristic, trait or character Genes & environment 5 In medicine: Many complex human diseases (traits) Understanding the role of genetics and environment in such diseases help in prevention, counseling and even treatment In agriculture/ animal breeding: Many economically important traits are quantitative Selective breeding/ role of environmental variation In evolution Natural selection requires heritable variation 6

2 Spectrum of Characters Few characters are purely Mendelian, polygenic or environmental Most characters depend on some mixture of major and minor genetic determinants, together with environmental influences Single Gene Traits Properties: Different phenotype result from alternative ti genotypes of a single gene Well-suited for analysis using crosses and pedigree Number of genotypes & phenotypes are small Relationship between genotype and phenotype is simple 7 8 Complex Traits Multiple genetic and environmental factors are involved Environmental influence, much more than single gene traits Complex inheritance: Single genotype many yp possible phenotypes Single phenotype caused by many possible genotypes Effects of segregation g of alleles of one gene may be masked by effects of other genes Such traits can not be studied with usual pedigree methods Traits with Complex Inheritance Three categories of traits show complex inheritance: Continuous traits (quantitative): No few discrete phenotypic variation with clear breaks Measurable characters blood pressure, body mass index, height, weight Meristic traits: Phenotype is found by counting Such as number of fingerprint ridges, number of bristles on a fly 9 10 Traits with Complex Inheritance A Bit of History Galton experiments: Threshold traits (discontinuous, dichotomous, discrete, qualitative): Have only two or few classes Either present or absent (affected, non-affected) Each individual has an underlying risk or liability to express it If the underlying liability is high enough (over the threshold), the trait will be expressed Francis Galton & studying family resemblances, quantifying observation and applying statistical analysis Hereditary Talent and Character,article in 1865, and then a book Hereditary Genius in : Galton compared the physical attributes of parents and children and established the degree of correlation between relative By 1900 the idea of continuous or quantitative characters introduced 11 12

3 A Bit of History Around 1900, there were two camps: ( ) Biometricians: Continuous or quantitative traits Mendelians: Discrete traits Are quantitative traits inherited in the same way as discrete traits? 1918, Reconciliation! Reconciliation: Multiple loci (genes) contribute to variation R. A. Fisher, 1918: Traits governed by a large number of Mendelian factors (polygenic) display the continuous nature, quantitative variation and family correlation Falconer extended this model to dichotomous characters Results of Fisher s and Falconer s analysis: Polygenic Theory of Inheritance for discrete traits Human heights Quantitative Traits Polygenic Theory of Discontinuous Traits Falconer s Polygenic Threshold Model Liability for a trait (disease) is multifactorial/polygenic and follow a Normal distribution ib ti in the population There is a threshold that whenever exceeded the trait expresses Medical applications: Recurrence risk in relatives of a proband with such disorders Studying Quantitative Traits They show a continuum of phenotypic variation (range of phenotype) Can not be described by small number of discrete categories Alternative way to describe a population p is to use a frequency distribution Shows the proportion of individuals that fall within a certain range of phenotype Frequency histogram Frequency Distribution Type of frequency distribution Symmetric (bell-shaped) Asymmetric (skewed) Positive Negative Bimodal symmetry 17 18

4 The Normal Distribution The normal distribution curve: A distribution for an infinite sample in which the trait of interest varies in a symmetric way around an average value First recognized in seventeenth century by an English mathematician, de Moivre Normal distribution can be characterized by two parameters: The mean The variance (or its square root, the standard deviation) 19 Normal Distribution of Quantitative Traits The figure shows a simple model for distribution in the population of a character determined by one, two, three or many loci: Mean value of the character is 100 unit All alleles have frequency of 0.5 and additive/co-dominant dominant effects Each upper case allele l adds 5 units to the value and each lower case allele subtracts 5 units 20 Mean & Variance The mean shows what the average individual looks like But all individuals don t look the same Symbolically: y X = ƒ i X / N The variance show how much individuals differ, or vary from each other (indirectly by showing how much on average they vary from the mean) The sum of squared deviation from the mean divided id d by the degrees of freedom Variance Variances are additive Total variance for a trait can be predicted by adding variances for different factors contribute to the trait But is difficult to use Mean and variance can explain a normal distribution Symbolically: V x = ƒ i (X X) 2 / (N-1) Standard Deviation (SD) SD is the square root of variance Standard deviation shows the proportion of individuals in a normal distribution with certain differences from the mean Comparing Two Variables Comparing two variables: Comparing between traits and genetic relationship: do tall parents produce tall offspring? Comparing the occurrence of two traits: do obese animals have larger hearts? Comparison between traits and environmental factors: insecticide resistance and exposure to insecticides 23 24

5 Comparing Two Variables Covariance & Correlation Coefficient Correlation coefficient: statistic that show how traits are correlated Correlation concerns the strength of association (relationship) between the values of two variables Ranges from -1 to +1 0 means no correlation Positive value: shows direct correlation Negative value: shows inverse correlation The absolute number measure the strength of correlation Correlations do not imply cause-and-effect To measure correlation coefficient i (r), we need first to determine covariance 25 Covariance: describe the degree of variation between two variables within a group See this example: Consider mother s and offspring s weight at 5 years among cattle Measuring r: r (x, y) = CoV (x, y) /SD x SD y r = Interpreting r Assumption in Calculating r Is there a true association? Test the null hypothesis Significance of r is directly related to sample size and the degree of freedom Here the df is N-2 (one less than df for variance) Similar to chi square values, r values can show the significance levels 27 This approach is valid only if these assumptions are met: No sampling bias (Values of X and Y must have been obtained by an unbiased sampling) Scores of X and Y follow a normal distribution Relationship between X and Y is linear If a cause and effect relationship established, regression analysis predict how much one variable will change in response to the other 28 Predicting Population r Significance of a correlation coefficient depends on: The size of the coefficient The size of the sample (and degree of freedom) The standard error of the correlation coefficient: SE r = (1 r 2 ) / N More About r It s accepted that the square of the correlation coefficient tells us how much of the variation in one variable can be explained by variation in the other The overlap or interaction effects Some of the variation in the variable is explained by variation in different factors acting together, Meaning what? Interpretation t ti of r this way: The null hypothesis: P r = 0 (population r ) SE r = 1 / N Significance at 5% and 1% level??? 29 How do you weigh up the size of correlation coefficient? Assume an r 1 = 0.6 and another r 2 = 0.3 How do you compare this? How strong is r 1 in comparison to r 2? 30

6 Regression Analysis It determines the nature of the relationship (between the two variables) and enables us to make prediction from that One way is to reduce the data to a straight line, called line of best fit or regression line Regression Equation Regression equation is an equation representing the regression line Y = bx + a b: regression coefficient (represent the slope of the regression line) a: the Y intercept Y Y & X are the variables Computing b and a b = CoV (X,Y) / V x, a = Y - bx Regression to the Mean Comparing the heights of sons and their fathers by Distribution Francis Galton: Taller-than-average average fathers tended to have taller-than-average average sons The sons tended to be nearer the average height of all men than their fathers He called this: regression to the mediocrity among fathers Distribution among children 33 Regression to the Mean From figure in previous slide: For each class of fathers, mean for the children is halfway ay between the mothers value and the population mean For each class of children, mean for the fathers is half way between the children s value and the population mean The distribution in children is the same as the distribution in the fathers Caution: regression to the mean is a purely statistical phenomenon not a genetic mechanism Assumptions to this model: Random mating No dominance 34 Heritability (h 2 ) Definition: the proportion of phenotypic variation within a group of individuals that is due to genetic variation Heritability values are relevant only to particular groups raised in a particular environment Heritability values change between 0 to 1 1 means all phenotypic variability is due to genetic variation 0 means all the variation is due to environmental effects Aim: analysis of the genetic and environmental components that affect quantitative traits 35 Partitioning Heritability Assumptions: Genetic and environmental factors are the only components that determine a trait Genetic and environmental factors are independent of each other Then: phenotypic variance is due to the additive effects of genetic variance and environmental variance [ V p =V E +V G ] Heritability of a trait is the proportion p of the total variance that is genetic, that is V G / V p 36

7 Measuring Heritability Comparing the variation in traits between genetically identical and genetically disparate groups can determine V E and V G Example: Inbreeding in mice to develop homogeneous strains V G =0, so V p = V E Try an example: determine V E and dv G for weight in mice sfsadf Heritability in Human Heritability of human traits can be estimated from the correlation between measurements of that trait among relatives of known degrees of relatedness (parents, children, MZ and DZ twins, ) Using twin studies h 2 = (Variance in DZ pairs - variance in MZ pairs) / Variance in DZ pairs Examples: h 2 = 0.8 for stature, & h 2 = for body mass index When heritability is known, it is easy to determine V E Broad Sense Heritability Heritability is the relative amount of phenotypic variation that is due to genetic variation Broad sense heritability: It takes into account all genetic variation that may affect the phenotype But genes affect the phenotype in various ways Additive effect Dominant/ recessive effect Epistasis Narrow Sense Heritability It is possible to subdivide H 2 B as V G = V A + V D + V I V A variance due to additive alleles V D variance due to alleles with dominant/recessive pattern of inheritance V I variance due to genes that interact in an epistatic manner For quantitative traits V D and V I are very small in comparison to V A so narrow sense heritability is used as h 2 N = V A / V p So: continue in next slide 39 ` Narrow sense heritability is an inaccurate measure of actual heritability 40 Narrow Sense Heritability Limitations of Heritability How to estimate it?? There are different methods A common strategy is: Measuring a quantitative traits among groups of genetically related individuals Calculate the correlation Then: h 2 N = r obs / r exp Expected correlation is based on the known genetic relationship?? At the end get an average of heritabilities obtained 41 Difficulties in measurement and interpretation Genetic and environmental factors are not independent because genetic and social (environmental) disadvantages go together and the equation [ V p = V E + V G ], cannot be accurate Correlation between relatives may not simply reflect their familial genetic relationship because they share their environment as well as their genes It is not accurate to extend the h 2 obtained from twins or from an ethnic group to the whole population or to another population If socioeconomic conditions change, even h 2 obtained from the same group is not applicable again 42

8 Example Heritabilities Dermal Ridge in Human Human fingerprint is a quantitative trait Explaining fingerprints: Bonnevie method for counting dermal ridge 43 Heritability of Ridge Count in Human Genetic Contribution Dermal ridge count has a genetic component Determining the genetic contribution in the variation of dermal ridge counts Measuring the heritability of human fingerprint patterns Results Interpret the data: Average heritability is 0.97 (very close to one) Nearly all of the variation in fingerprint pattern is due to genetic variation Conceptual Questions Explain the difference between a continuous trait and a discontinuous trait. Give several examples of quantitative traits. At the molecular level, explain why quantitative traits often exhibit a continuum of phenotype within a population. p How does the environment help produce this continuum? What is a normal distribution? What is a frequency distribution? What is polygenic inheritance? Discuss the issues that make polygenic inheritance difficult to study?

9 Conceptual Questions What does it mean when a correlation coefficient is negative? Can you think of examples? If an r value equals 0.5 and N=4, how do you interpret your results? What if N=500? When a correlation coefficient is statistically significant, what do you conclude about the two variables? What do the results mean with regard to cause and effect? Explain the meaning of heritability. Why is a heritability value valid only for a particular population p of individuals raised in a particular environment? The broad sense heritability for a trait is 1.0. Explain what this value means. Would you conclude that t the environment is unimportant in the outcome of this trait? Conceptual Questions Assume in a fairly large population of people living in a particular part of Iran, everyone cares about good nutrition. All the members of this population eat very nutritional foods, and their diets are very similar to each other. With regard to height, how do you think this population would compare to the general population in the following categories? Mean height Heritability for height Genetic variation for alleles that affect height Conceptual Questions Are the following statements regarding heritability true or false? Heritability applies to a specific population raised in a particular environment Heritability in the narrow sense takes into account all types of genetic variance Heritability is a measure of the amount that genetic contribute to the outcome of a trait The following are data that describe the 6-week weights of mice and their offspring of the same sex: Calculate the correlation coefficient. Answer: r (parent, offspring) = 0.27 Parent Offspring The average bristle numbers in two strains of flies were 35 and 42. The genetic variance for bristle number calculated for both strains was 0.8. What is the minimum number of genes that affect bristle number? Answer: at least eight genes Solution: following equation shows the relation between genetic variance and the minimum number of genes affecting a quantitative trait: n = D 2 / 8V G D: difference between mean value of the trait in two strains VG: genetic variance for the trait, calculated using data from both strains The variance for fathers (in square inches) was 112, the variance for sons was 122, and the covariance was 144. The mean height for fathers was 68 in., and the mean height for sons was 69 in. If a father had a height of 70 in., what is the most probable height of his son? Answer: 71.6 in

10 The following graph shows the ranges of blood pressures for a selected population of people. The red line depicts the frequency distribution of the systolic pressure for the entire population. Several individuals with high blood pressure were identified, and the blood pressure of their relatives were determined. This frequency distribution is depicted d with a blue line (includes only their relatives not themselves). What do these data suggest with regard to a genetic basis for high blood pressure? What statistical approach could you use to determine the heritability for this trait? The correlation for height were determined for 15 pairs of individuals with the following genetic relationships: Mother / daughter: 0.36 Mother / granddaughter: 0.17 Sister / Sister: 0.39 Sister / sister (fraternal twins): 0.40 Sister / sister (identical twins: 0.77 What is the average heritability for height in this group of females? Answer: 0.75 Selective Breeding Selective breeding or artificial selection? Selective Breeding Selective breeding or artificial selection? Artificial Selection & Heritability Measuring narrow sense heritability in artificial selection experiments: h N2 = R / S R: response to selection in the offspring, R = Mean of the offspring mean of the starting population S: selection differential in the parents, S = Mean of the parents mean of the starting population Narrow sense heritability gives the ability to predict the outcome of selective breeding

11 Conceptual Questions From an agricultural point of view, discuss the advantages and disadvantages of selective breeding. What about making hybrids strains? Why narrow sense heritability is important in agricultural genetics? When artificial selection is practiced over many generations, it is common for the trait to reach a plateau in which further selection has little effect on the outcome of fth the trait. Explain why? Discuss whether a natural population of wolves or a domesticated t d population of German shepherds h is more likely to have a higher heritability for the trait of size. 61 The narrow sense heritability for potato weight in a starting population of potatoes is 0.42, and the mean weight is 1.4 lb. If a breeder crosses two individuals with average potato weights of 1.9 and 2.1 lb, respectively, what is the predicted average weight of potatoes in the offspring? h N2 = R / S h N2 = (Mean of the offspring mean of the starting population) / (Mean of the parents mean of fthe starting population) Answer: 1.65 lb A farmer wants to increase the average body weight in a herd of cattle. He begins with a herd having a mean weight of 595 kg and chooses individuals to breed that have a mean weight of 625 kg. Twenty offspring were obtained, having the following weights (kg), 612, 587, 604, 589, 615, 641, 575, 611, 610, 598, 589, 620, 617, 577, 609, 633, 588, 599, 601, and 611. Calculate l the realized heritability in this herd with regard to body weight. Answer: h N2 = 0.3 Conceptual Questions Regarding genetic drift, specify the True or False statements: Over the long run, genetic drift will lead to allele fixation or loss. When a new mutation occurs within a population, genetic drift is more likely to cause the loss of the new allele rather than the fixation of the new allele. Genetic drift promotes genetic diversity between populations. Genetic drift promotes genetic diversity in large populations. 64 Conceptual Questions In genetic drift, what is drifting? Why is this an appropriate term to describe this phenomenon? Why is genetic drift more significant in small populations? Why does it take longer for genetic drift to cause allele fixation in large population than in small ones? Describe what happens to allele frequencies during the bottleneck effect. Discuss the relevance of this effect with regard to species that are approaching extinction. A group of four birds flies to a new location and initiated the formation of a new colony. Three of them are homozygous DD,, and one bird is heterozygous Dd (for a particular locus). What is the probability that the d allele will become fixed in the population? If fixation occurs, how long will it take? How will the growth of the population, from generation to generation, affect the answers to parts a and b? Briefly explain please

12 Thank you for listening, any comments? 67