Molecular Biotechnology: Principles and Applications - Glick, B., Pasternak, J. 2002

Molecular Biotechnology of Microbiological Systems
Human Molecular Genetics
Detection and Assessment of Human Genetic Linkage

Prior to the advent of Recombinant DNA technology in the early 1980s, detecting and assessing human genetic linkage was a complex, highly labor-intensive Procedure that also frequently proved unsuccessful. Researchers faced A number of obstacles in this endeavor. First, the genetic status of parents is usually unknown, making it difficult to distinguish between recombinant and non-recombinant progeny. Second, The small size of most families reduces the statistical Significance of the obtained results. The presence of a single X chromosome in males significantly simplifies the estimation of genetic distances between Gene loci. In this case, all alleles of genes located on the X chromosome are phenotypically expressed. Sons of women who are double heterozygous for X-linked loci receive either a recombinant or a non-recombinant X chromosome. If the phase in which the alleles of two gene loci exist in the mothers is known, distinguishing between recombinant and non-recombinant types among the sons is straightforward. The paternal genotype is irrelevant in this case, since sons inherit solely the maternal X chromosome. Sometimes, the allele phase in a doubly heterozygous mother can be determined from her father's phenotype. For example, if the mother's father (the grandfather) exhibits two recessive X-linked traits, while the mother herself shows the dominant traits, then the mother is doubly heterozygous and the alleles in question are in the cis phase, i.e., AB/ab (Fig. 20.8). This method of linkage detection is based on counting two-locus phenotypes in sons from A large number of doubly heterozygous women with a known allele phase. In this case, the proportion of Chromosomes recombinant for two specific gene loci (the recombination fraction) will equal the sum of recombinant chromosomes (R) divided by the total number of chromosomes—both recombinant and non-recombinant (NR):

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Fig. 20.8. X-chromosome mapping. In this case, the genetic phase of two or more X-linked loci in a daughter (Mother) is established based on data concerning the X-linked alleles of her father (Grandfather). This information, in turn, is used to determine which of her sons (Sons) inherited a recombinant (R) or non-recombinant (NR) chromosome. In this example, the grandfather carries two recessive genes at loci A and B on the X chromosome, his daughter is a double heterozygote, and the alleles in question are in the cis phase. The X chromosome displays the alleles of loci A and B, while the Y chromosome is depicted as a shorter bar.

However, this approach has several drawbacks. First, it is not always possible to determine the grandfather's genotype; consequently, the phase of the alleles in the putatively doubly heterozygous mother remains unknown. Second, not all mothers in a large sample of families will be heterozygous for the exact same two loci. Despite all efforts, prior to the 1980s, it was not possible to construct a sufficiently extended and unambiguous linkage map of the human X chromosome based on counting recombinant and non-recombinant chromosomes. At that time, only a few loci were known and far too few alleles had been identified.

Maximum likelihood linkage analysis: the logarithm of the odds ratio (LOD score)

In addition to the method that determines the recombination frequency between two loci through the direct counting of recombinant and non-recombinant chromosomes, it was necessary to develop a more general, indirect method that could: 1) rigorously distinguish between independent assortment and linkage; 2) not rely strictly on data regarding the allele phase of doubly heterozygous parents; 3) accumulate information derived from a large number of diverse families; and 4) allow estimation of the recombination fraction whenever linkage was detected. Such a method, which is widely used today, was developed by Morton in 1955.

When studying linkage, the recombination fraction is denoted by the Greek letter theta (θ). Morton's method compares the probability L(θ) that two loci are linked in siblings (i.e., localized on the same chromosome and situated close to each other) with the probability L(0.50) that the two loci are unlinked (i.e., located on different chromosomes or far apart on the same chromosome), for any given recombination fraction θ. In the case of linkage, since the recombination fraction is unknown, it can take any value in the range from 0 to 0.5 (0 ≤ θ < 0.50). If two loci assort independently, however, θ = 0.50 by definition. In other words, when half of the Gametes produced by a heterozygous parent contain novel genetic combinations, the two loci reside either on non-homologous chromosomes or far enough apart on the same chromosome that they appear to be located on separate chromosomes. Consequently, if L(θ) = L(0.50), the two loci are unlinked. The base-10 logarithm of The ratio of these two probabilities, i.e., log[L(θ)/L(0.50)], represents the log-of-odds ratio, referred to as the LOD score. The LOD score is denoted by the letter Z; Z(θ) is the LOD score for a given value of θ, where 0 ≤ θ < 0.50.

L(θ) can be determined if the probability of obtaining a specific combination of recombinant and non-recombinant chromosomes for the siblings of each studied family is known. The probability that offspring inherit a non-recombinant chromosome from a heterozygous parent is ¹/₂(1—θ) + ¹/₂(1—θ), or 1—θ, while the probability that they receive a recombinant chromosome is ¹/₂θ + ¹/₂θ, or θ. For example, in a family with five children, the probability for each child to inherit a non-recombinant chromosome from the heterozygous parent is K(1—θ)⁵, where (1—θ) is the probability of obtaining a non-recombinant chromosome, the exponent 5 is the number of siblings with a non-recombinant chromosome (more precisely, the number of non-recombinant chromosomes among the siblings), and K is a coefficient. If all chromosomes are identical—that is, all non-recombinant or all recombinant—then K = 1 (i.e., 5!/5!0!, or n!/n!0!, where n is the number of siblings in the given family). In a family with four children, the probability that all of them receive a recombinant chromosome from the heterozygous parent is θ⁴. Furthermore, the probability that in a family of nine children, five receive non-recombinant chromosomes and four receive recombinant ones is K(1—θ)⁵(θ)⁴, where K = 126, i.e., 9!/5!4!. The LOD score is expressed as a ratio of quantities that share identical coefficients. These coefficients in the numerator and denominator cancel out and are therefore omitted in linkage analysis.

Let us illustrate the calculation of the LOD score using a single family's sibship as an example (Fig. 20.9). The designations B and O in Fig. 20.9 correspond to the ABO Blood group alleles ABO*B and ABO*O. Filled symbols indicate an autosomal dominant disorder with complete penetrance—nail-Patella syndrome (NPS). The primary Clinical Features of NPS include nail Dysplasia of the fingers and toes and hypoplasia or absence of the patella. The NPS gene is designated NPS1, and its recessive ("normal") and dominant ("pathological") alleles are NPS1*N and NPS1*D, respectively. NPS represents a highly suitable trait for linkage studies because it is diagnosed early in life, does not affect viability or reproductive fitness, and is present at birth.

Fig. 20.9. Inheritance of the nail-patella syndrome gene and ABO blood group genes. Filled symbols denote individuals with hereditary nail-patella syndrome, while open symbols indicate individuals lacking the traits of this disorder. Letters beneath each symbol designate the ABO blood group alleles (Abbreviations are used: O corresponds to ABO*O, and B to ABO*B).

Father I-2 (Fig. 20.9) is heterozygous for the NPS locus, as his children include both affected and unaffected individuals. He is also heterozygous for the ABO locus (ABO*B/ABO*O) since his children exhibit both O and B phenotypes, while the genotype of his spouse (I-1) is ABO*O/ABO*O. Consequently, the father is diheterozygous for these two autosomal loci (NPS1*N/NPS1*D; ABO*B/ABO*O). If the ABO and NPS1 loci are linked, the phase of their alleles in the father is initially unknown (unknown phase state). It could be either ABO*B NPS1*D/ABO*O NPS1*N (phase 1) or ABO*B NPS1*N/ABO*O NPS1*D (phase 2), or, in abbreviated form, B D/O N (phase 1) or B N/O D (phase 2).

Assuming that the ABO and NPS loci are linked and their alleles in the father are in phase 1 (ABO*B NPS1*D/ABO*O NPS1*N), children II-1, II-2, II-4, II-6, II-7, II-8, II-9, and II-10 inherited from him a non-recombinant chromosome, either ABO*B NPS1*D or ABO*O NPS1*N (Fig. 20.10). All children inherited the ABO*O NPS1*N chromosome from the mother (I-1) because she is homozygous for both loci (ABO*O NPS1*N/ABO*O NPS1*N). In this case, the mother's genetic contribution is known and does not affect the linkage analysis. Based on the premise that the father's alleles are in phase 1, each of his children II-3, II-5, and II-11 inherited a recombinant chromosome. Consequently, the probability of such a combination of non-recombinant and recombinant chromosomes for this family is (1-θ)⁸(θ)³.

Fig. 20.10. Genetic Organization of nail-patella syndrome and ABO blood group alleles in members of the pedigree shown in Fig. 20.9, assuming linkage between these two loci. Abbreviations for ABO blood group alleles are used: O corresponds to ABO*O, and B corresponds to ABO*B. The recessive ("normal") and dominant ("pathological") alleles of the hereditary nail-patella syndrome locus are designated as N and D, respectively. The father's genotype (I-2) can be in either of two phases (phase 1, phase 2). The father's chromosomes and the chromosomes inherited from him by the children are highlighted in blue, while the mother's chromosomes (I-1) and those inherited from her are highlighted in light brown. Paternal chromosomes that are non-recombinant (NR) or recombinant (R) for phase 1 and phase 2 are indicated.

It is equally likely that the father's alleles are in phase 2, i.e., ABO*B NPS1*N/ABO*O NPS1*D. In that case, children II-3, II-5, and II-11 inherited non-recombinant chromosomes from him, whereas each of the remaining children inherited a recombinant chromosome (Fig. 20.10). The probability of this combination for the given family is (1-θ)³(θ)⁸.

Since both phases are equally probable for the father's genotype, the overall probability L(θ) of the chromosome combination observed in his children's pedigree is ¹/₂(1-θ)⁸(θ)³ + ¹/₂(1-θ)³(θ)⁸. Next, the value of this expression is calculated for various θ. Typically, the following set of θ values is used: 0, 0.001, 0.05, 0.10, 0.2, 0.3, 0.4, and 0.50, or, if time permits, the entire spectrum of θ values from 0 to 0.50 can be used. Then, the logarithm of the odds ratio is calculated for each θ (except 0.50) relative to the probability at θ = 0.50. For example, for θ = 0.10, the ratio L(0.10)/L(0.50) equals

The common logarithm of 0.441 is —0.356; this is the lod score for this ratio. In other words, Z(0.10) = —0.356.

If the phase of the alleles in the father is known, the probability L(θ) for the family is also known. For example, if genotype I-2 is in phase 1 (ABO*B NPS1*D/ABO*O NPS1*N), then, as noted above, the probability L(θ) for this family will be (1-θ)⁸(θ)³, and Z(θ = 0.10) will be

Conversely, if genotype I-2 is in phase 2 (ABO*B NPS1*N/ABO*O NPS1*D), then Z(0.10) will be

For the unknown phase state, the Z values for the pedigree shown in Fig. 20.9 range from —5.993 at θ = 0.001 to +0.029 at θ = 0.45 (Table 20.1). If the sibs have inherited at least one recombinant chromosome and θ = 0, then Z = -∞. As seen from Table 20.1, the lod score reaches its maximum (Z_max) at θ close to 0.30. Performing additional calculations for θ between 0.20 and 0.40 yields Z_max = +0.214 at θ = 0.276.

Table 20.1. Z values at various θ for the pedigree shown in Fig. 20.9, for the unknown phase condition

The value of Z_max = +0.214 does not provide conclusive evidence for linkage between the ABO and NPS1 loci. It has been conventionally established that two autosomal loci can be considered linked only if the maximum LOD score is greater than or equal to +3.000, which corresponds to a linkage odds ratio of 1,000 to 1 or higher. For X-linked genes known to reside on the same chromosome, the threshold Z_max value indicating linkage is greater than or equal to +2.000, corresponding to odds in favor of linkage of 100 to 1 or higher. Conversely, if Z = -2.000, linkage between the two loci is excluded, as there is only a 1 in 100 chance favoring linkage.

To detect linkage, it is necessary to calculate the Z score at various θ values across different families and determine its maximum. Converting the likelihood ratio for each family into a base-10 logarithm allows the resulting Z(∞) values to be summed. To establish linkage between the ABO and NPS1 loci, 25 pedigrees were analyzed—including several with large sibships—yielding a value of Z(0.10) = +31.235 (Table 20.2); since this exceeds +3.000, the two loci are indeed linked.

The value of θ at which Z reaches its maximum provides a rough estimate of the recombination fraction for the two linked loci. In the initial study determining linkage between the ABO and NPS1 loci, the exact Z_max value was not calculated, but the Z score at θ = 0.10 was the highest among all Z values computed for various θ, leading to the Conclusion that the distance between these two loci is approximately 10 cM. It must be emphasized that the analysis of various NPS pedigrees revealed that different alleles of the ABO system are linked to the NPS1 locus. In other words, there is no specific linkage between a particular ABO allele and the NPS1 locus. Unless proven otherwise, one can speak only of genetic linkage between loci rather than between specific alleles. It should also be noted that the LOD score method does not determine the autosomal localization of two linked loci. As we will see, establishing that the ABO and NPS1 loci are located on the long arm (q) of chromosome 9 in the region between bands 34 and 34.2 (i.e., 9q34–9q34.2) required additional studies.

Table 20.2. Cumulative Z values at various ∞ for the ABO and NPS1 loci¹⁾

¹⁾ Adapted from Renwick and Schulze, Ann. Hum. Genet. 28:379–392, 1965, with modifications.