haemophilia A and F V G1691A mutation during continuous infusion of F VIII after intramural jejunal bleeding-successful thrombolysis under heparin therapy. Eur J

20210 G-->A mutation in the 3'-untranslated region of the prothrombin gene and

Kluijtmans LA & van den Heuvel LP. A candidate genetic risk factor for vascular disease: a common mutation in methylenetetrahydrofolate reductase. Nat Genet.

Zotz RB. Prothrombin and factor V mutations in women with a history of thrombosis during pregnancy and the puerperium. N Engl J Med. 2000 Feb

relationships and inhibitor development. Semin Thromb Hemost. 2003

Recombinant versus plasma-derived factor VIII products and the development of inhibitors in previously untreated patients with severe hemophilia A: The CANAL

Bunschoten EP. Influence of the type of F8 gene mutation on inhibitor development in a single centre cohort of severe haemophilia A patients. Haemophilia. 2011 Mar;

Human methylenetetrahydrofolate reductase: isolation of cDNA, mapping and

Preston FE & Peake IR. HLA class II profile: a weak determinant of factor VIII

#### **4. Acknowledgements**

The author/s wish to thank the patients and their families for participating in the studies associated with the present work as well as the attending clinical and laboratory specialists for providing us with the relevant details of the medical history and the laboratory tests. This research was supported by grant No. DO02-69 of the Ministry of Education, Youth and Science of Republic of Bulgaria.

#### **5. References**

The author/s wish to thank the patients and their families for participating in the studies associated with the present work as well as the attending clinical and laboratory specialists for providing us with the relevant details of the medical history and the laboratory tests. This research was supported by grant No. DO02-69 of the Ministry of Education, Youth and

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**1. Introduction**

**0**

**4**

*Korea*

Myung-Hoon Chung *Hongik University, Jochiwon*

**Population Evolution in Hemophilia**

Gene defects cause diseases. It is reasonable to argue that a disease is related to many genes and a gene affects on many diseases. This multiple-to-multiple mapping between genes and diseases makes it difficult to understand the roles of genes completely. However, in some case, only one gene is involved in one disease. A good example of one-to-one mapping is

Biologists found that there are approximately 3 billion nitrogenous bases in 44+2 human DNA. Most sequences of these bases are irrelevant to genetic inheritance. Less than 3% of whole pair sequences are known to determine characteristic features of all human genes (Lander et al., 2001; Venter et al., 2001). Since the diversity of human's inherited characteristics is huge, it

Single gene effect with multiple alleles such as blood groups (Chung et al., 1997), color blindness and hemophilia (Lee et al., 2001) can be studied by using the automata equations. Their solutions have some analogy with fixed points of renormalization group equations in physics (Mekjian, 1991; Perelson & Weisbuch, 1997), and lead into the Hardy-Weinberg

We extended the automata equations to investigate multiple gene effects on population evolution with any number of loci and alleles in the presence of mutation and selection (Chung et al., 2003). As results of the study, we present the generalized Hardy-Weinberg formula and a simulation program on the Internet (Chung, 2007). The program explores simultaneous control of parameters that affect the behavior of gene variations in a population. We note that Hampe *et al.* (1998) studied population evolution in a different view point from ours. The advantage of our approach is that we do not need a large RAM memory because we do not treat individual person, but consider groups characterized by genes. One more good point beyond the work of Hampe *et al.* (1998) is that we achieve quickly equilibrium state. Using the simulation program, we find that the mortality rates due to gene inheritance are greatly enhanced for multiple gene cases. Another user friendly simulation program provides a convenient scheme for the most common cases (Quardokus, 2000). However, it is not easy

Human beings inherit not only biological genes but also social status from their parents. A few examples of the social inheritance are family name, nationality, or wealth. One of the outstanding questions is how or whether this social inheritance influences the standard biological gene evolution. We believe that this type of interaction may play a crucial

hemophilia related to gene F8 (National Hemophilia Foundation, 1998).

can only be explained by cooperation of multiple genes.

to extend this scheme to include more sophisticated situations.

formula (Haldane, 1935; Hedrick, 1985; Li, 1976).

## **Population Evolution in Hemophilia**

Myung-Hoon Chung

*Hongik University, Jochiwon Korea*

#### **1. Introduction**

50 Hemophilia

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Independent molecular entities. J Exp Med 1973, 138: 1015.

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formation. Thromb Haemost. 1995 Dec;74(6):1402-6.

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for prothrombin gene 20210 A allele in a thrombophilic family without clinical

Schwaab R, Olek K & Kazazian HH. Haemophilia A: database of nucleotide substitutions, deletions, insertions and rearrangements of the factor VIII gene.

Leone G. The risk of recurrent deep venous thrombosis among heterozygous carriers of both factor V Leiden and the G20210A prothrombin mutation. N Engl J

del Rio E, Fontcuberta J & Baiget M. The prothrombin 20210 allele influences clinical manifestations of hemophila A in patients with intron 22 inversion and

Connor JM, Hoyer LW, Yoshioka A, Peake IR, Olek K, Kazazian HH, Lavergne J-M, Giannelli F, Antonarakis SE & Cooper DN. Haemophilia A: database of nucleotide substitutions, deletions, insertions and rearrangements of the factor VIII gene,

analysis of the combination of hemophilia A and factor V(LEIDEN). Blood. 1997

structural stability in factor VIII and factor VIIIa. J Biol Chem. 2008 Apr

Czeizel AE, Gelman-Kohan Z, Scarano G, Bianca S, Ettore G, Tenconi R, Bellato S, Scala I, Mutchinick OM, López MA, de Walle H, Hofstra R, Joutchenko L, Kavteladze L, Bermejo E, Martínez-Frías ML, Gallagher M, Erickson JD, Vollset SE, Mastroiacovo P, Andria G & Botto LD. Geographical and ethnic variation of the 677C>T allele of 5,10 ethylenetetrahydrofolate reductase (MTHFR): findings from over 7000 newborns from 16 areas worldwide. J Med Genet. 2003 Aug;40(8):619-25.

Recurrent mutations in haemophilia A give evidence for CpG mutation hotspots.

Characterization of five partial deletions of the factor VIII gene. Proc Natl Acad Sci

Gene defects cause diseases. It is reasonable to argue that a disease is related to many genes and a gene affects on many diseases. This multiple-to-multiple mapping between genes and diseases makes it difficult to understand the roles of genes completely. However, in some case, only one gene is involved in one disease. A good example of one-to-one mapping is hemophilia related to gene F8 (National Hemophilia Foundation, 1998).

Biologists found that there are approximately 3 billion nitrogenous bases in 44+2 human DNA. Most sequences of these bases are irrelevant to genetic inheritance. Less than 3% of whole pair sequences are known to determine characteristic features of all human genes (Lander et al., 2001; Venter et al., 2001). Since the diversity of human's inherited characteristics is huge, it can only be explained by cooperation of multiple genes.

Single gene effect with multiple alleles such as blood groups (Chung et al., 1997), color blindness and hemophilia (Lee et al., 2001) can be studied by using the automata equations. Their solutions have some analogy with fixed points of renormalization group equations in physics (Mekjian, 1991; Perelson & Weisbuch, 1997), and lead into the Hardy-Weinberg formula (Haldane, 1935; Hedrick, 1985; Li, 1976).

We extended the automata equations to investigate multiple gene effects on population evolution with any number of loci and alleles in the presence of mutation and selection (Chung et al., 2003). As results of the study, we present the generalized Hardy-Weinberg formula and a simulation program on the Internet (Chung, 2007). The program explores simultaneous control of parameters that affect the behavior of gene variations in a population. We note that Hampe *et al.* (1998) studied population evolution in a different view point from ours. The advantage of our approach is that we do not need a large RAM memory because we do not treat individual person, but consider groups characterized by genes. One more good point beyond the work of Hampe *et al.* (1998) is that we achieve quickly equilibrium state. Using the simulation program, we find that the mortality rates due to gene inheritance are greatly enhanced for multiple gene cases. Another user friendly simulation program provides a convenient scheme for the most common cases (Quardokus, 2000). However, it is not easy to extend this scheme to include more sophisticated situations.

Human beings inherit not only biological genes but also social status from their parents. A few examples of the social inheritance are family name, nationality, or wealth. One of the outstanding questions is how or whether this social inheritance influences the standard biological gene evolution. We believe that this type of interaction may play a crucial

**2. Model**

**2.1 Symbols**

gene. Hence, we denote gene

relevant *Li* genes as

denote for a female genotype,

*<sup>T</sup>*(*F*) *<sup>a</sup>* = (*C*(1)

*<sup>T</sup>*(*M*) *<sup>a</sup>* = (*C*(1)

Similarly, we denote for a male genotype,

hemophilia increases the number of the male patients and the female carriers, thus clearly showing that the social effects significantly influence the inheritance of the biological gene.

Population Evolution in Hemophilia 53

The fundamental concept for the population evolution is the gene, which allows multiple loci, multiple allele inheritance with recombination and mutation. The *l*-th gene in the *i*-th chromosome is labeled by two indices (*i*, *l*). We denote *n*(*i*,*l*) as the number of alleles for the

*<sup>G</sup>*(*i*,*l*) *<sup>a</sup>* ,

where 1 <sup>≤</sup> *<sup>a</sup>* <sup>≤</sup> *<sup>n</sup>*(*i*,*l*) and *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , 22, *<sup>X</sup>*,*Y*, *<sup>m</sup>*,*<sup>s</sup>* including sex chromosome *<sup>X</sup>*,*Y*, mitochondria *m*, and social gene *s*. Here, we note that the social inheritance is treated on an equal footing with the biological one. However, the social genes may have different rules

*<sup>a</sup>* <sup>=</sup> *<sup>G</sup>*(*i*,1) *<sup>a</sup>*<sup>1</sup> *<sup>G</sup>*(*i*,2) *<sup>a</sup>*<sup>2</sup> ··· *<sup>G</sup>*(*i*,*Li*) *aLi* ,

*Ma* <sup>=</sup> *<sup>G</sup>*(*m*,1) *<sup>a</sup>*<sup>1</sup> *<sup>G</sup>*(*m*,2) *<sup>a</sup>*<sup>2</sup> ··· *<sup>G</sup>*(*m*,*Lm*) *aLm* .

*Sa* <sup>=</sup> *<sup>G</sup>*(*s*,1) *<sup>a</sup>*<sup>1</sup> : *<sup>G</sup>*(*s*,2) *<sup>a</sup>*<sup>2</sup> : ··· : *<sup>G</sup>*(*s*,*Ls*) *aLs* .

: *C*(*i*) *<sup>a</sup>*<sup>1</sup> , *<sup>C</sup>*(*i*) *<sup>a</sup>*<sup>2</sup> :

where the chromosome pair allele would run from 1 to (*n*(*i*) <sup>+</sup> <sup>1</sup>)*n*(*i*)/2 with *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , 22, *<sup>X</sup>*,

Integrating all, a female genotype *<sup>T</sup>*(*F*) *<sup>a</sup>* and a male genotype *<sup>T</sup>*(*M*) *<sup>a</sup>* are expressed in terms of chromosome pairs, mitochondria and a series of social genes having multiple alleles. We

*<sup>p</sup>*<sup>22</sup> , *<sup>C</sup>*(22)

*<sup>p</sup>*<sup>22</sup> , *<sup>C</sup>*(22)

*<sup>q</sup>*<sup>22</sup> : *<sup>C</sup>*(*X*)

*<sup>q</sup>*<sup>22</sup> : *<sup>C</sup>*(*X*)

*pX* , *<sup>C</sup>*(*X*)

*pX* , *<sup>C</sup>*(*Y*)

*qX* : *Mpm* : *Sps* ).

*qY* : *Mpm* : *Sps* ).

with *i* = 1, 2, ··· , 22, *X*,*Y*. We write mitochondria *Ma* containing relevant *Lm* genes as

*<sup>a</sup>* containing

*<sup>l</sup>*=<sup>1</sup> *<sup>n</sup>*(*i*,*l*) <sup>≡</sup> *<sup>n</sup>*(*i*) with *<sup>i</sup>* <sup>=</sup>

of reproduction and mutation rates, when compared with the biological genes. The string of genes models each chromosome. We denote chromosome *C*(*i*)

*C*(*i*)

Also we write social gene string *Sa* containing relevant *Ls* genes as

The number of distinguishable strings is given by ∏*Li*

1, 2, ··· , 22, *X*,*Y*, *m*,*s*. It is useful to introduce chromosome pair

while *Y* chromosome and mitochondria do not need the pairing.

*<sup>p</sup>*<sup>1</sup> , *<sup>C</sup>*(1)

*<sup>p</sup>*<sup>1</sup> , *<sup>C</sup>*(1)

*<sup>q</sup>*<sup>1</sup> : ··· : *<sup>C</sup>*(22)

*<sup>q</sup>*<sup>1</sup> : ··· : *<sup>C</sup>*(22)

role in explaining appearance or disappearance of certain physical or social traits in some communities. In order to study population evolution with inheritance of social status, we treat inheritable social traits as *social genes* that behave similar to Mendelian genes. This approach allows us to handle the biological and the social genes in a unified way and to examine the mutual influence between the social and the biological genes.

One of the distinct features of the social gene is that it causes preferential non-random mating. Non-random mating exists even without the social gene concepts.  However, with the social restriction and prejudice, the tendency of non-random mating is expected to become more prominent and complicated.

The population evolution depends on four dominant factors imposed by nature with genes. These factors are (1) mating, (2) mutation, (3) reproduction, and (4) selection. The author proposed automata equations in order to describe the effects of these factors in the population evolution. These equations were used to determine the equilibrium population ratios in multiple gene inheritance, where arbitrary numbers of loci and alleles were allowed in the presence of natural selection, mutation and recombination. A user-friendly numerical simulation program was proposed for estimation of the infant mortality rate for fatal diseases (Chung et al., 2003). In order to incorporate the social genes in the inheritance scheme, we generalize the scheme to include non-random mating explicitly.

It is worthwhile to mention other population evolution studies, which are different from our approach. First of all, Cavalli-Sforza and Feldman (1981) developed a mathematically-rich theory of cultural transmission and evolution. Gene-culture coevolution refers to the evolutionary phenomena that arise from interactions between the biological and social inheritance systems (Aoki, 2001). In genetics, there exist current efforts to locate genes that contribute to diseases or to valuable traits (Piccolboni & Gusfield, 2003). Furthermore, it is essential to analyze the structure of populations on the basis of genetic data (Santafe et al., 2008). It is also well-known that the coalescent method (Hudson, 1991) is use to determine mutation rates (Thomson et al., 2000) and recombination rates (Fu & Li, 1999) in the way of statistical inference (Rosenberg & Nordborg, 2002). Studies on substructured populations introduce the similar feature of social gene, for instance, age (Charlesworth, 1994) or last name. We note that many studies related to mating already exist. The convergence of the multilocus systems under selection with a random mating was investigated (Nagylaki et al., 1999). In the population models for the diploid ancestral process with a random mating, the convergence criterium was proved (Möhle et al., 2003). Non-random mating has been found to play a significant role in the models of the population genetics. In the work related with a non-random mating (Hausken & Hirshleifer, 2001), the truthful signalling hypothesis was used in the mating competition theory. Strategic mating between males and females was also considered by Alpern and Reyniers (2005) and Radcliffe and Rass (1999).

The main purpose of this chapter is to provide a theoretical scheme and a simulation tool to handle the social and the biological genes in a unified way. We present a generalized numerical simulation tool to account for the role of the non-random mating induced by the social genes in addition to mutation, recombination and selection. We expect that the scheme will allow one to examine closely the impact of the social genes on the biological ones and vice versa. For example, we study hemophilia thoroughly without and with a social gene. Simulation results show that a medical screening to prevent birth of females with fatal hemophilia increases the number of the male patients and the female carriers, thus clearly showing that the social effects significantly influence the inheritance of the biological gene.

#### **2. Model**

2 Hemophilia

role in explaining appearance or disappearance of certain physical or social traits in some communities. In order to study population evolution with inheritance of social status, we treat inheritable social traits as *social genes* that behave similar to Mendelian genes. This approach allows us to handle the biological and the social genes in a unified way and to examine the

One of the distinct features of the social gene is that it causes preferential non-random mating. Non-random mating exists even without the social gene concepts.  However, with the social restriction and prejudice, the tendency of non-random

The population evolution depends on four dominant factors imposed by nature with genes. These factors are (1) mating, (2) mutation, (3) reproduction, and (4) selection. The author proposed automata equations in order to describe the effects of these factors in the population evolution. These equations were used to determine the equilibrium population ratios in multiple gene inheritance, where arbitrary numbers of loci and alleles were allowed in the presence of natural selection, mutation and recombination. A user-friendly numerical simulation program was proposed for estimation of the infant mortality rate for fatal diseases (Chung et al., 2003). In order to incorporate the social genes in the inheritance scheme, we

It is worthwhile to mention other population evolution studies, which are different from our approach. First of all, Cavalli-Sforza and Feldman (1981) developed a mathematically-rich theory of cultural transmission and evolution. Gene-culture coevolution refers to the evolutionary phenomena that arise from interactions between the biological and social inheritance systems (Aoki, 2001). In genetics, there exist current efforts to locate genes that contribute to diseases or to valuable traits (Piccolboni & Gusfield, 2003). Furthermore, it is essential to analyze the structure of populations on the basis of genetic data (Santafe et al., 2008). It is also well-known that the coalescent method (Hudson, 1991) is use to determine mutation rates (Thomson et al., 2000) and recombination rates (Fu & Li, 1999) in the way of statistical inference (Rosenberg & Nordborg, 2002). Studies on substructured populations introduce the similar feature of social gene, for instance, age (Charlesworth, 1994) or last name. We note that many studies related to mating already exist. The convergence of the multilocus systems under selection with a random mating was investigated (Nagylaki et al., 1999). In the population models for the diploid ancestral process with a random mating, the convergence criterium was proved (Möhle et al., 2003). Non-random mating has been found to play a significant role in the models of the population genetics. In the work related with a non-random mating (Hausken & Hirshleifer, 2001), the truthful signalling hypothesis was used in the mating competition theory. Strategic mating between males and females was also

mutual influence between the social and the biological genes.

mating is expected to become more prominent and complicated.

generalize the scheme to include non-random mating explicitly.

considered by Alpern and Reyniers (2005) and Radcliffe and Rass (1999).

The main purpose of this chapter is to provide a theoretical scheme and a simulation tool to handle the social and the biological genes in a unified way. We present a generalized numerical simulation tool to account for the role of the non-random mating induced by the social genes in addition to mutation, recombination and selection. We expect that the scheme will allow one to examine closely the impact of the social genes on the biological ones and vice versa. For example, we study hemophilia thoroughly without and with a social gene. Simulation results show that a medical screening to prevent birth of females with fatal

#### **2.1 Symbols**

The fundamental concept for the population evolution is the gene, which allows multiple loci, multiple allele inheritance with recombination and mutation. The *l*-th gene in the *i*-th chromosome is labeled by two indices (*i*, *l*). We denote *n*(*i*,*l*) as the number of alleles for the gene. Hence, we denote gene

$$G\_a^{(iJ)}$$

where 1 <sup>≤</sup> *<sup>a</sup>* <sup>≤</sup> *<sup>n</sup>*(*i*,*l*) and *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , 22, *<sup>X</sup>*,*Y*, *<sup>m</sup>*,*<sup>s</sup>* including sex chromosome *<sup>X</sup>*,*Y*, mitochondria *m*, and social gene *s*. Here, we note that the social inheritance is treated on an equal footing with the biological one. However, the social genes may have different rules of reproduction and mutation rates, when compared with the biological genes.

The string of genes models each chromosome. We denote chromosome *C*(*i*) *<sup>a</sup>* containing relevant *Li* genes as

$$\mathbf{C}\_{a}^{(i)} = \mathbf{G}\_{a\_1}^{(i,1)} \mathbf{G}\_{a\_2}^{(i,2)} \cdots \mathbf{G}\_{a\_{L\_l}}^{(i,L\_l)} \, .$$

with *i* = 1, 2, ··· , 22, *X*,*Y*. We write mitochondria *Ma* containing relevant *Lm* genes as

$$M\_{\mathfrak{a}} = G\_{\mathfrak{a}\_1}^{(m,1)} G\_{\mathfrak{a}\_2}^{(m,2)} \cdots \cdot G\_{\mathfrak{a}\_{L\_m}}^{(m,L\_m)} \cdot \cdot$$

Also we write social gene string *Sa* containing relevant *Ls* genes as

$$S\_{\mathfrak{a}} = G\_{\mathfrak{a}\_1}^{(s,1)} : G\_{\mathfrak{a}\_2}^{(s,2)} : \cdots : G\_{\mathfrak{a}\_{L\_{\mathfrak{a}}}}^{(s,L\_{\mathfrak{s}})} .$$

The number of distinguishable strings is given by ∏*Li <sup>l</sup>*=<sup>1</sup> *<sup>n</sup>*(*i*,*l*) <sup>≡</sup> *<sup>n</sup>*(*i*) with *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , 22, *X*,*Y*, *m*,*s*. It is useful to introduce chromosome pair

$$: \mathsf{C}^{(i)}\_{a\_1} \mathsf{C}^{(i)}\_{a\_2}:$$

where the chromosome pair allele would run from 1 to (*n*(*i*) <sup>+</sup> <sup>1</sup>)*n*(*i*)/2 with *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , 22, *<sup>X</sup>*, while *Y* chromosome and mitochondria do not need the pairing.

Integrating all, a female genotype *<sup>T</sup>*(*F*) *<sup>a</sup>* and a male genotype *<sup>T</sup>*(*M*) *<sup>a</sup>* are expressed in terms of chromosome pairs, mitochondria and a series of social genes having multiple alleles. We denote for a female genotype,

$$T\_a^{(F)} = (\mathbb{C}\_{p\_1}^{(1)}, \mathbb{C}\_{q\_1}^{(1)} : \cdots : \mathbb{C}\_{p\_{22}}^{(22)}, \mathbb{C}\_{q\_{22}}^{(22)} : \mathbb{C}\_{p\_X}^{(X)}, \mathbb{C}\_{q\_X}^{(X)} : M\_{p\_m} : \mathbb{S}\_{p\_s}).$$

Similarly, we denote for a male genotype,

$$T\_a^{(M)} = (\mathbb{C}\_{p\_1}^{(1)}, \mathbb{C}\_{q\_1}^{(1)} : \cdots : \mathbb{C}\_{p\_{22}}^{(22)}, \mathbb{C}\_{q\_{22}}^{(22)} : \mathbb{C}\_{p\_X}^{(X)}, \mathbb{C}\_{q\_Y}^{(Y)} : M\_{p\_W} : \mathbb{S}\_{p\_s}).$$

For the social genes, mutation rates are mainly determined by the society and will be different for each case. In consequence, all genotype mutation rates are written in terms of the gene

Population Evolution in Hemophilia 55

assumption that randomly chosen half of the father's chromosomes and half of the mother's chromosomes are delivered to their baby, however only the mother's mitochondria becomes the baby's mitochondria. For the social genes, the rule of reproduction will be different on a case-by-case basis. Assuming equal preference for each genotype, the reproduction

*<sup>b</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>c</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>a</sup>* ) can be calculated. In fact, the reproduction coefficients

<sup>4</sup> to *<sup>ζ</sup>*(*i*).

<sup>4</sup> to *<sup>ζ</sup>*(*i*).

<sup>4</sup> to *<sup>ζ</sup>*(*i*).

<sup>4</sup> to *<sup>ζ</sup>*(*i*).

<sup>2</sup> to *<sup>ζ</sup>*(*F*).

<sup>2</sup> to *<sup>ζ</sup>*(*F*).

<sup>4</sup> , <sup>1</sup>

*<sup>q</sup>* <sup>→</sup> *<sup>C</sup>*(*X*) *<sup>s</sup>* , *<sup>C</sup>*(*Y*)

Finally, the reproduction coefficients *ζ*(*s*) for social genes will be determined by a case-by-case

22 ∏ *i*=1 *ζ*(*i*) ]*ζ*(*S*) *ζ*(*m*) *ζ*(*s*) .

*<sup>s</sup>* , *<sup>C</sup>*(*i*)

*<sup>b</sup>* ) will make

*<sup>p</sup>* , *<sup>C</sup>*(*Y*) *q* →

*<sup>b</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>c</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>a</sup>* ) are calculated with the

*<sup>t</sup>* ) is determined by the following simple

<sup>2</sup> , 1. Also, *<sup>ζ</sup>*(*F*)(*C*(*X*) *<sup>v</sup>* , *<sup>C</sup>*(*X*) *<sup>w</sup>* <sup>×</sup> *<sup>C</sup>*(*X*)

*<sup>t</sup>* ) is determined as follows:

mutation rates. After mutation, the effective parent population *<sup>P</sup><sup>n</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>c</sup>* )=[

*<sup>q</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

In reproduction, the reproduction coefficients *<sup>ξ</sup>*(*T*(*F*)

*<sup>ξ</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>w</sup>* <sup>×</sup> *<sup>C</sup>*(*i*)

*<sup>p</sup>* , *<sup>C</sup>*(*i*)

*<sup>v</sup>* , *<sup>C</sup>*(*i*)

If (*s*, *t*)=(min(*v*, *p*), max(*v*, *p*)), then add <sup>1</sup>

If (*s*, *t*)=(min(*v*, *q*), max(*v*, *q*)), then add <sup>1</sup>

If (*s*, *t*)=(min(*w*, *p*), max(*w*, *p*)), then add <sup>1</sup>

If (*s*, *t*)=(min(*w*, *q*), max(*w*, *q*)), then add <sup>1</sup>

Hence, *ζ*(*i*) is given by one of four values, 0, <sup>1</sup>

If (*s*, *t*)=(min(*v*, *p*), max(*v*, *p*)), then add <sup>1</sup>

If (*s*, *t*)=(min(*w*, *p*), max(*w*, *p*)), then add <sup>1</sup>

*<sup>p</sup>* , *<sup>C</sup>*(*Y*)

<sup>2</sup> to *<sup>ζ</sup>*(*M*).

<sup>2</sup> to *<sup>ζ</sup>*(*M*).

For mitochondria, *<sup>ζ</sup>*(*m*)(*Mv* <sup>×</sup> *Mp* <sup>→</sup> *Ms*) is determined as follows:

*<sup>t</sup>* ) is similarly determined:

Furthermore, *<sup>ζ</sup>*(*M*)(*C*(*X*) *<sup>v</sup>* , *<sup>C</sup>*(*X*) *<sup>w</sup>* <sup>×</sup> *<sup>C</sup>*(*X*)

If (*s*, *t*)=(*v*, *q*), then add <sup>1</sup>

If (*s*, *t*)=(*w*, *q*), then add <sup>1</sup>

If *s* = *v*, then add 1 to *ζ*(*m*).

offsprings.

coefficients *<sup>ξ</sup>*(*T*(*F*)

Here, *<sup>ζ</sup>*(*i*) = *<sup>ζ</sup>*(*i*)(*C*(*i*)

Start with *ζ*(*i*) = 0.

Start with *ζ*(*F*) = 0.

Start with *ζ*(*M*) = 0.

Start with *ζ*(*m*) = 0.

consideration.

are written as

algorithm:

*<sup>C</sup>*(*X*) *<sup>s</sup>* , *<sup>C</sup>*(*X*)

We find that the number of different genotypes is given by [∏<sup>22</sup> *i*=1 1 <sup>2</sup> (<sup>1</sup> <sup>+</sup> *<sup>n</sup>*(*i*))*n*(*i*)] <sup>1</sup> <sup>2</sup> (1 + *n*(*X*))*n*(*X*)*n*(*m*)*n*(*s*) for female, and [∏<sup>22</sup> *i*=1 1 <sup>2</sup> (<sup>1</sup> <sup>+</sup> *<sup>n</sup>*(*i*))*n*(*i*)]*n*(*X*)*n*(*Y*)*n*(*m*)*n*(*s*) for male. Note that, for several given genes, the number of distinguishable genotypes increases dramatically.

For a genotype *<sup>T</sup>*(*S*) *<sup>a</sup>* with *<sup>S</sup>* <sup>=</sup> *<sup>F</sup>* or *<sup>M</sup>*, we introduce four kinds of population ratios at the *<sup>n</sup>*-th generation: adult, birth, parent before mutation, and effective parent after mutation, which are denoted by *An*(*T*(*S*) *<sup>a</sup>* ), *Bn*(*T*(*S*) *<sup>a</sup>* ), *Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>b</sup>* ), and *<sup>P</sup>*�*n*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>b</sup>* ), respectively. Here, the population ratios denote the frequencies of specific genotypes or genotype pairs of parents. We normalize so that the sums of any population ratios are equal to 1 as shown in Table 1. We shall notice the relationships between these population ratios later.

The population evolution is governed by the four factors: (1) mating, (2) mutation, (3) reproduction, and (4) selection. We introduce some symbols used in the population evolution as follows in each step.

In mating, *<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*) *<sup>b</sup>* ) denotes the probability that a genotype *<sup>T</sup>*(*F*) *<sup>a</sup>* adult woman mates with a *T*(*M*) *<sup>b</sup>* adult man, while *<sup>ω</sup>*(*T*(*M*) *<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ) is the probability that a genotype *<sup>T</sup>*(*M*) *<sup>b</sup>* adult man mates with a *<sup>T</sup>*(*F*) *<sup>a</sup>* adult woman. This mating probability reflects social and cultural effects. For the random mating case, the probability *<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*) *<sup>b</sup>* ) is simply *An*(*T*(*M*) *<sup>b</sup>* ), and *<sup>ω</sup>*(*T*(*M*) *<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ) = *An*(*T*(*F*) *<sup>a</sup>* ). Notice that *<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*) *<sup>b</sup>* ) �<sup>=</sup> *<sup>ω</sup>*(*T*(*M*) *<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ).

In mutation, the genotype mutation rates *<sup>μ</sup>*(*T*(*S*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>b</sup>* ) are written in terms of the gene mutation rates *<sup>μ</sup>*(*G*(*i*,*l*) *<sup>c</sup>* <sup>→</sup> *<sup>G</sup>*(*i*,*l*) *<sup>d</sup>* ) and the frequency of recombination due to chromosomal crossover. It is known that the frequency of recombination between two locations depends on their distance. Mutation rates between chromosomes are given by

$$\mu(\mathbb{C}\_a^{(i)} \to \mathbb{C}\_b^{(i)}) = \prod\_{l=1}^L \mu(G\_{a\_l}^{(i,l)} \to G\_{b\_l}^{(i,l)}).$$

The mutation rates of mitochondria *μ*(*Ma* → *Mb*) and the mutation rates of social gene string *μ*(*Sa* → *Sb*) are given as the above similarly. Since each chromosome behaves independently, we find the mutation rate of a specific genotype *<sup>T</sup>*(*S*) *<sup>a</sup>* with the sex index *<sup>S</sup>* <sup>=</sup> *<sup>F</sup>* or *<sup>M</sup>* as

$$\mu(T\_a^{(\mathcal{S})} \to T\_b^{(\mathcal{S})}) = [\prod\_{i=1}^{22} \eta^{(i)}] \eta^{(\mathcal{S})} \mu(M\_{a\_m} \to M\_{b\_m}) \mu(\mathcal{S}\_{a\_s} \to \mathcal{S}\_{b\_s}) \mu$$

where *<sup>η</sup>*(*i*) = *<sup>η</sup>*(*i*)(*C*(*i*) *<sup>v</sup>* , *<sup>C</sup>*(*i*) *<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*i*) *<sup>p</sup>* , *<sup>C</sup>*(*i*) *<sup>q</sup>* ) is given by

$$\eta^{(i)} = \begin{cases} \mu(\mathsf{C}\_{v}^{(i)} \to \mathsf{C}\_{p}^{(i)})\mu(\mathsf{C}\_{w}^{(i)} \to \mathsf{C}\_{p}^{(i)}) & \text{for } p = q \\\ \mu(\mathsf{C}\_{v}^{(i)} \to \mathsf{C}\_{p}^{(i)})\mu(\mathsf{C}\_{w}^{(i)} \to \mathsf{C}\_{q}^{(i)}) + \mu(\mathsf{C}\_{v}^{(i)} \to \mathsf{C}\_{q}^{(i)})\mu(\mathsf{C}\_{w}^{(i)} \to \mathsf{C}\_{p}^{(i)}) & \text{for } p \neq q \end{cases}$$

Furthermore, *<sup>η</sup>*(*F*)(*C*(*X*) *<sup>v</sup>* , *<sup>C</sup>*(*X*) *<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*X*) *<sup>p</sup>* , *<sup>C</sup>*(*X*) *<sup>q</sup>* ) is the same as *η*(*i*), and

$$
\mu^{(M)}(\mathbb{C}\_v^{(X)}, \mathbb{C}\_w^{(Y)} \to \mathbb{C}\_p^{(X)}, \mathbb{C}\_q^{(Y)}) = \mu(\mathbb{C}\_v^{(X)} \to \mathbb{C}\_p^{(X)})\mu(\mathbb{C}\_w^{(Y)} \to \mathbb{C}\_q^{(Y)}).
$$

for several given genes, the number of distinguishable genotypes increases dramatically. For a genotype *<sup>T</sup>*(*S*) *<sup>a</sup>* with *<sup>S</sup>* <sup>=</sup> *<sup>F</sup>* or *<sup>M</sup>*, we introduce four kinds of population ratios at the *<sup>n</sup>*-th generation: adult, birth, parent before mutation, and effective parent after mutation, which are

population ratios denote the frequencies of specific genotypes or genotype pairs of parents. We normalize so that the sums of any population ratios are equal to 1 as shown in Table 1. We

The population evolution is governed by the four factors: (1) mating, (2) mutation, (3) reproduction, and (4) selection. We introduce some symbols used in the population evolution

man mates with a *<sup>T</sup>*(*F*) *<sup>a</sup>* adult woman. This mating probability reflects social and cultural

crossover. It is known that the frequency of recombination between two locations depends on

*L* ∏ *l*=1

The mutation rates of mitochondria *μ*(*Ma* → *Mb*) and the mutation rates of social gene string *μ*(*Sa* → *Sb*) are given as the above similarly. Since each chromosome behaves independently, we find the mutation rate of a specific genotype *<sup>T</sup>*(*S*) *<sup>a</sup>* with the sex index *<sup>S</sup>* <sup>=</sup> *<sup>F</sup>* or *<sup>M</sup>* as

*<sup>q</sup>* ) is given by

*<sup>q</sup>* ) + *<sup>μ</sup>*(*C*(*i*)

*<sup>v</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

*<sup>q</sup>* ) is the same as *η*(*i*), and

*<sup>q</sup>* ) = *<sup>μ</sup>*(*C*(*X*) *<sup>v</sup>* <sup>→</sup> *<sup>C</sup>*(*X*)

*i*=1 1

<sup>2</sup> (<sup>1</sup> <sup>+</sup> *<sup>n</sup>*(*i*))*n*(*i*)]*n*(*X*)*n*(*Y*)*n*(*m*)*n*(*s*) for male. Note that,

*<sup>b</sup>* ), and *<sup>P</sup>*�*n*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* ) denotes the probability that a genotype *<sup>T</sup>*(*F*) *<sup>a</sup>* adult woman mates

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ) is the probability that a genotype *<sup>T</sup>*(*M*)

*<sup>b</sup>* ) �<sup>=</sup> *<sup>ω</sup>*(*T*(*M*)

*<sup>d</sup>* ) and the frequency of recombination due to chromosomal

*bl* ).

*μ*(*Mam* → *Mbm* )*μ*(*Sas* → *Sbs* ),

*<sup>p</sup>* ) for *p* = *q*

*<sup>q</sup>* )*μ*(*C*(*i*)

*<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

*<sup>p</sup>* )*μ*(*C*(*Y*) *<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*Y*)

*<sup>p</sup>* ) for *p* �= *q*

*<sup>q</sup>* ).

*<sup>μ</sup>*(*G*(*i*,*l*) *al* <sup>→</sup> *<sup>G</sup>*(*i*,*l*)

<sup>2</sup> (<sup>1</sup> <sup>+</sup> *<sup>n</sup>*(*i*))*n*(*i*)] <sup>1</sup>

*<sup>b</sup>* ), respectively. Here, the

*<sup>b</sup>* ) is simply *An*(*T*(*M*)

*<sup>b</sup>* ) are written in terms of the gene

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ).

<sup>2</sup> (1 +

*<sup>b</sup>* adult

*<sup>b</sup>* ), and

We find that the number of different genotypes is given by [∏<sup>22</sup>

shall notice the relationships between these population ratios later.

effects. For the random mating case, the probability *<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

their distance. Mutation rates between chromosomes are given by

*<sup>a</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

22 ∏ *i*=1 *η*(*i*) ]*η*(*S*)

*<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

*<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

*<sup>p</sup>* , *<sup>C</sup>*(*X*)

*<sup>p</sup>* , *<sup>C</sup>*(*Y*)

*<sup>b</sup>* ) =

*<sup>μ</sup>*(*C*(*i*)

*<sup>b</sup>* )=[

*<sup>p</sup>* )*μ*(*C*(*i*)

*<sup>p</sup>* )*μ*(*C*(*i*)

*<sup>p</sup>* , *<sup>C</sup>*(*i*)

*<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

(*C*(*X*) *<sup>v</sup>* , *<sup>C</sup>*(*Y*) *<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*X*)

*<sup>μ</sup>*(*T*(*S*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*S*)

*<sup>v</sup>* , *<sup>C</sup>*(*i*)

*<sup>v</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

*<sup>v</sup>* <sup>→</sup> *<sup>C</sup>*(*i*)

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ) = *An*(*T*(*F*) *<sup>a</sup>* ). Notice that *<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

In mutation, the genotype mutation rates *<sup>μ</sup>*(*T*(*S*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*S*)

*i*=1 1

*n*(*X*))*n*(*X*)*n*(*m*)*n*(*s*) for female, and [∏<sup>22</sup>

as follows in each step. In mating, *<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

mutation rates *<sup>μ</sup>*(*G*(*i*,*l*) *<sup>c</sup>* <sup>→</sup> *<sup>G</sup>*(*i*,*l*)

where *<sup>η</sup>*(*i*) = *<sup>η</sup>*(*i*)(*C*(*i*)

⎧ ⎨ ⎩

*<sup>μ</sup>*(*C*(*i*)

*<sup>μ</sup>*(*C*(*i*)

Furthermore, *<sup>η</sup>*(*F*)(*C*(*X*) *<sup>v</sup>* , *<sup>C</sup>*(*X*) *<sup>w</sup>* <sup>→</sup> *<sup>C</sup>*(*X*)

*η*(*M*)

*η*(*i*) =

with a *T*(*M*)

*<sup>ω</sup>*(*T*(*M*)

denoted by *An*(*T*(*S*) *<sup>a</sup>* ), *Bn*(*T*(*S*) *<sup>a</sup>* ), *Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* adult man, while *<sup>ω</sup>*(*T*(*M*)

For the social genes, mutation rates are mainly determined by the society and will be different for each case. In consequence, all genotype mutation rates are written in terms of the gene mutation rates. After mutation, the effective parent population *<sup>P</sup><sup>n</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>b</sup>* ) will make offsprings.

In reproduction, the reproduction coefficients *<sup>ξ</sup>*(*T*(*F*) *<sup>b</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>c</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>a</sup>* ) are calculated with the assumption that randomly chosen half of the father's chromosomes and half of the mother's chromosomes are delivered to their baby, however only the mother's mitochondria becomes the baby's mitochondria. For the social genes, the rule of reproduction will be different on a case-by-case basis. Assuming equal preference for each genotype, the reproduction coefficients *<sup>ξ</sup>*(*T*(*F*) *<sup>b</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>c</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>a</sup>* ) can be calculated. In fact, the reproduction coefficients are written as

$$\xi(T\_a^{(F)} \times T\_b^{(M)} \to T\_c^{(S)}) = [\prod\_{i=1}^{22} \xi^{(i)}] \xi^{(S)} \mathcal{T}^{(m)} \mathcal{T}^{(s)}.$$

Here, *<sup>ζ</sup>*(*i*) = *<sup>ζ</sup>*(*i*)(*C*(*i*) *<sup>v</sup>* , *<sup>C</sup>*(*i*) *<sup>w</sup>* <sup>×</sup> *<sup>C</sup>*(*i*) *<sup>p</sup>* , *<sup>C</sup>*(*i*) *<sup>q</sup>* <sup>→</sup> *<sup>C</sup>*(*i*) *<sup>s</sup>* , *<sup>C</sup>*(*i*) *<sup>t</sup>* ) is determined by the following simple algorithm:

Start with *ζ*(*i*) = 0. If (*s*, *t*)=(min(*v*, *p*), max(*v*, *p*)), then add <sup>1</sup> <sup>4</sup> to *<sup>ζ</sup>*(*i*). If (*s*, *t*)=(min(*v*, *q*), max(*v*, *q*)), then add <sup>1</sup> <sup>4</sup> to *<sup>ζ</sup>*(*i*). If (*s*, *t*)=(min(*w*, *p*), max(*w*, *p*)), then add <sup>1</sup> <sup>4</sup> to *<sup>ζ</sup>*(*i*). If (*s*, *t*)=(min(*w*, *q*), max(*w*, *q*)), then add <sup>1</sup> <sup>4</sup> to *<sup>ζ</sup>*(*i*).

Hence, *ζ*(*i*) is given by one of four values, 0, <sup>1</sup> <sup>4</sup> , <sup>1</sup> <sup>2</sup> , 1. Also, *<sup>ζ</sup>*(*F*)(*C*(*X*) *<sup>v</sup>* , *<sup>C</sup>*(*X*) *<sup>w</sup>* <sup>×</sup> *<sup>C</sup>*(*X*) *<sup>p</sup>* , *<sup>C</sup>*(*Y*) *q* → *<sup>C</sup>*(*X*) *<sup>s</sup>* , *<sup>C</sup>*(*X*) *<sup>t</sup>* ) is similarly determined:

Start with *ζ*(*F*) = 0. If (*s*, *t*)=(min(*v*, *p*), max(*v*, *p*)), then add <sup>1</sup> <sup>2</sup> to *<sup>ζ</sup>*(*F*). If (*s*, *t*)=(min(*w*, *p*), max(*w*, *p*)), then add <sup>1</sup> <sup>2</sup> to *<sup>ζ</sup>*(*F*). Furthermore, *<sup>ζ</sup>*(*M*)(*C*(*X*) *<sup>v</sup>* , *<sup>C</sup>*(*X*) *<sup>w</sup>* <sup>×</sup> *<sup>C</sup>*(*X*) *<sup>p</sup>* , *<sup>C</sup>*(*Y*) *<sup>q</sup>* <sup>→</sup> *<sup>C</sup>*(*X*) *<sup>s</sup>* , *<sup>C</sup>*(*Y*) *<sup>t</sup>* ) is determined as follows: Start with *ζ*(*M*) = 0. If (*s*, *t*)=(*v*, *q*), then add <sup>1</sup> <sup>2</sup> to *<sup>ζ</sup>*(*M*). If (*s*, *t*)=(*w*, *q*), then add <sup>1</sup> <sup>2</sup> to *<sup>ζ</sup>*(*M*). For mitochondria, *<sup>ζ</sup>*(*m*)(*Mv* <sup>×</sup> *Mp* <sup>→</sup> *Ms*) is determined as follows: Start with *ζ*(*m*) = 0. If *s* = *v*, then add 1 to *ζ*(*m*).

Finally, the reproduction coefficients *ζ*(*s*) for social genes will be determined by a case-by-case consideration.

frequencies.

populations. The birth populations are determined in the third equation. The fractional adult populations at the next generation after selection and normalization are given by the fourth equation. These automata equations enable us to calculate the evolution of the genotype

Population Evolution in Hemophilia 57

For a given initial normalized set of adult populations, the automata equations will produce a fixed point of *<sup>A</sup>*∗(*T*(*S*) *<sup>a</sup>* ) eventually. We note that the fixed point has a global stability, and is a function of mutation rates and disadvantage factors. We present the entire library for solving the population equation on the Internet (Chung, 2007). In fact, written in C# language, the single reusable library of Science.dll contains Science.Biology.PopulationGenetics, with which one can simulate all cases. It is open to the public, and runs on a personal computer with Windows operating system. Any number of loci and alleles, and any values of mutation rates

It is worth mentioning that these population equations are not universal. In fact, if we introduce age as a social gene, these equations should be modified because only certain aged

While it is difficult to obtain detailed information on the biological genes of a prospective marriage partner, it is easy to do so on the social genes. Thus, the mating involving social

In the population equation for mating of Eq. (1) which is analogous to detailed balance in the

Introducing a so-called *mating factor* Γ*ab* which should not be negative, we rewrite the parent

Here, Γ*ab* are input parameters in this model. However, the condition of Eq. (5) assures that

*An*(*T*(*F*) *<sup>a</sup>* )Γ*ab* <sup>=</sup> 1 for 1 <sup>≤</sup> *<sup>b</sup>* <sup>≤</sup> *<sup>n</sup>*(*M*)

These restrictions on Γ*ab* assume that all adults marry and reproduce offsprings. When some genotype adults have handicaps for marriage, it is reflected in the disadvantage factor in

*<sup>b</sup>* ) = <sup>1</sup> <sup>=</sup> <sup>∑</sup>*<sup>a</sup>*

*<sup>ω</sup>*(*T*(*M*)

*<sup>b</sup>* ) = *An*(*T*(*F*) *<sup>a</sup>* )Γ*abAn*(*T*(*M*)

*<sup>b</sup>* ) = 1 for 1 <sup>≤</sup> *<sup>a</sup>* <sup>≤</sup> *<sup>n</sup>*(*F*)

Γ*ab* ≥ 0. (9)

*b*

*An*(*T*(*M*)

*An*(*T*(*F*) *<sup>a</sup>* ) = ∑

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ) = 1. (5)

*<sup>b</sup>* ). (6)

, (7)

, (8)

*<sup>b</sup>* ), (10)

*<sup>b</sup>* ) = <sup>∑</sup>*<sup>a</sup>*

Monte Carlo simulation, the sum of probabilities must be 1. In fact, we require

*<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

*Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

<sup>Γ</sup>*abAn*(*T*(*M*)

*n*(*F*) ∑ *a*=1

*n*(*M*) ∑ *b*=1

Since we obtain the same equation from Eqs. (7) and (8) as

*An*(*T*(*F*) *<sup>a</sup>* )Γ*abAn*(*T*(*M*)

and disadvantage factors are allowed simultaneously.

adults can marry and reproduce offsprings.

genes will be a non-random mating in general.

∑ *b*

the elements of Γ*ab* should satisfy

∑ *ab*

**2.3 Non-random mating**

population as

selection.

Table 1. Symbols used in the paper. The index *S* in genotype represents female with *S* = *F*, and male with *S* = *M*.

In selection, since human beings with faulty genes have lower survival rate, we introduce disadvantage factor *<sup>δ</sup>*(*T*(*S*) *<sup>a</sup>* ) for each genotype, which is given by a value between 0 and 1. A larger value of disadvantage factor means less chance of survival. The value of 1 represents the complete extinction. The terminology of fitness can alternatively be used to replace the disadvantage factor.

#### **2.2 Population equation**

For given population ratios *An*(*T*(*S*) *<sup>a</sup>* ) at the *<sup>n</sup>*-th generation, our prime concern is "what are the next generation population ratios?". To answer this question, we introduce the four main effects on population evolution: mating, mutation, reproduction, and selection. Based on these four events, using the parameters in relation to probability as explained in the previous subsection, we formulate automata equations for population evolution:

• Mating

$$P\_{\boldsymbol{\pi}}(T\_{\boldsymbol{a}}^{(\mathcal{F})} \times T\_{\boldsymbol{b}}^{(M)}) = A\_{\boldsymbol{\pi}}(T\_{\boldsymbol{a}}^{(\mathcal{F})})\omega(T\_{\boldsymbol{a}}^{(\mathcal{F})} \to T\_{\boldsymbol{b}}^{(M)}) = A\_{\boldsymbol{\pi}}(T\_{\boldsymbol{b}}^{(M)})\omega(T\_{\boldsymbol{b}}^{(M)} \to T\_{\boldsymbol{a}}^{(\mathcal{F})}),\tag{1}$$

• Mutation

$$\tilde{P}\_n(T\_a^{(F)} \times T\_b^{(M)}) = \sum\_{c,d} P\_n(T\_c^{(F)} \times T\_d^{(M)}) \mu(T\_c^{(F)} \to T\_a^{(F)}) \mu(T\_d^{(M)} \to T\_b^{(M)}),\tag{2}$$

• Reproduction

$$B\_{\boldsymbol{\eta}}(T\_{\boldsymbol{a}}^{(\mathcal{S})}) = \sum\_{\boldsymbol{b}, \boldsymbol{c}} \tilde{P}\_{\boldsymbol{\eta}}(T\_{\boldsymbol{b}}^{(\mathcal{F})} \times T\_{\boldsymbol{c}}^{(\mathcal{M})}) \xi(T\_{\boldsymbol{b}}^{(\mathcal{F})} \times T\_{\boldsymbol{c}}^{(\mathcal{M})} \to T\_{\boldsymbol{a}}^{(\mathcal{S})}),\tag{3}$$

• Selection

$$A\_{n+1}(T\_a^{(S)}) = \frac{B\_n(T\_a^{(S)})\{1 - \delta(T\_a^{(S)})\}}{1 - \sum\_b B\_n(T\_b^{(S)})\delta(T\_b^{(S)})}.\tag{4}$$

The first equation is similar to the detailed balance in Monte Carlo simulation. The second equation shows that new populations are written as the sum of mutations from old

Name Symbol Property Adult population *An*(*T*(*S*) *<sup>a</sup>* ) <sup>∑</sup>*<sup>a</sup> An*(*T*(*S*) *<sup>a</sup>* ) = <sup>1</sup> Birth population *Bn*(*T*(*S*) *<sup>a</sup>* ) <sup>∑</sup>*<sup>a</sup> Bn*(*T*(*S*) *<sup>a</sup>* ) = <sup>1</sup>

Disadvantage factor *<sup>δ</sup>*(*T*(*S*) *<sup>a</sup>* ) 0.0 <sup>≤</sup> *<sup>δ</sup>*(*T*(*S*) *<sup>a</sup>* ) <sup>≤</sup> 1.0

Table 1. Symbols used in the paper. The index *S* in genotype represents female with *S* = *F*,

In selection, since human beings with faulty genes have lower survival rate, we introduce disadvantage factor *<sup>δ</sup>*(*T*(*S*) *<sup>a</sup>* ) for each genotype, which is given by a value between 0 and 1. A larger value of disadvantage factor means less chance of survival. The value of 1 represents the complete extinction. The terminology of fitness can alternatively be used to replace the

For given population ratios *An*(*T*(*S*) *<sup>a</sup>* ) at the *<sup>n</sup>*-th generation, our prime concern is "what are the next generation population ratios?". To answer this question, we introduce the four main effects on population evolution: mating, mutation, reproduction, and selection. Based on these four events, using the parameters in relation to probability as explained in the previous

subsection, we formulate automata equations for population evolution:

*<sup>b</sup>* ) = *An*(*T*(*F*) *<sup>a</sup>* )*ω*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

*Pn*(*T*(*F*) *<sup>c</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>c</sup>* )*ξ*(*T*(*F*)

*An*+1(*T*(*S*) *<sup>a</sup>* ) = *Bn*(*T*(*S*) *<sup>a</sup>* ){<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*(*T*(*S*) *<sup>a</sup>* )}

The first equation is similar to the detailed balance in Monte Carlo simulation. The second equation shows that new populations are written as the sum of mutations from old

<sup>1</sup> <sup>−</sup> <sup>∑</sup>*<sup>b</sup> Bn*(*T*(*S*)

*<sup>P</sup>n*(*T*(*F*)

*<sup>b</sup>* ) <sup>∑</sup>*a*,*<sup>b</sup> Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* ) <sup>∑</sup>*a*,*<sup>b</sup> <sup>P</sup><sup>n</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* ) <sup>∑</sup>*<sup>b</sup> <sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* ) <sup>∑</sup>*<sup>b</sup> <sup>μ</sup>*(*T*(*S*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*S*)

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>c</sup>* ) <sup>∑</sup>*<sup>c</sup> <sup>ξ</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* ) = *An*(*T*(*M*)

*<sup>d</sup>* )*μ*(*T*(*F*) *<sup>c</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* )*μ*(*T*(*M*)

*<sup>b</sup>* )*δ*(*T*(*S*) *<sup>b</sup>* )

*<sup>b</sup>* )*ω*(*T*(*M*)

*<sup>d</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>c</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>a</sup>* ), (3)

. (4)

*<sup>b</sup>* ) = 1

*<sup>b</sup>* ) = 1

*<sup>b</sup>* ) = 1

*<sup>b</sup>* ) = 1

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*S*) *<sup>c</sup>* ) = <sup>1</sup>

*<sup>b</sup>* <sup>→</sup> *<sup>T</sup>*(*F*) *<sup>a</sup>* ), (1)

*<sup>b</sup>* ), (2)

Parents population *Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

Effective parents population *<sup>P</sup><sup>n</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

Mating probability *<sup>ω</sup>*(*T*(*F*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*M*)

Genotype mutation rate *<sup>μ</sup>*(*T*(*S*) *<sup>a</sup>* <sup>→</sup> *<sup>T</sup>*(*S*)

Reproduction coefficient *<sup>ξ</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

and male with *S* = *M*.

disadvantage factor.

• Mating

• Mutation

• Reproduction

• Selection

**2.2 Population equation**

*Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>P</sup><sup>n</sup>*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

*<sup>b</sup>* ) = ∑ *c*,*d*

*Bn*(*T*(*S*) *<sup>a</sup>* ) = ∑

*b*,*c*

populations. The birth populations are determined in the third equation. The fractional adult populations at the next generation after selection and normalization are given by the fourth equation. These automata equations enable us to calculate the evolution of the genotype frequencies.

For a given initial normalized set of adult populations, the automata equations will produce a fixed point of *<sup>A</sup>*∗(*T*(*S*) *<sup>a</sup>* ) eventually. We note that the fixed point has a global stability, and is a function of mutation rates and disadvantage factors. We present the entire library for solving the population equation on the Internet (Chung, 2007). In fact, written in C# language, the single reusable library of Science.dll contains Science.Biology.PopulationGenetics, with which one can simulate all cases. It is open to the public, and runs on a personal computer with Windows operating system. Any number of loci and alleles, and any values of mutation rates and disadvantage factors are allowed simultaneously.

It is worth mentioning that these population equations are not universal. In fact, if we introduce age as a social gene, these equations should be modified because only certain aged adults can marry and reproduce offsprings.

#### **2.3 Non-random mating**

While it is difficult to obtain detailed information on the biological genes of a prospective marriage partner, it is easy to do so on the social genes. Thus, the mating involving social genes will be a non-random mating in general.

In the population equation for mating of Eq. (1) which is analogous to detailed balance in the Monte Carlo simulation, the sum of probabilities must be 1. In fact, we require

$$\sum\_{b} \omega(T\_a^{(F)} \to T\_b^{(M)}) = \sum\_{a} \omega(T\_b^{(M)} \to T\_a^{(F)}) = 1. \tag{5}$$

Introducing a so-called *mating factor* Γ*ab* which should not be negative, we rewrite the parent population as

$$P\_{\boldsymbol{\eta}}(T\_{\boldsymbol{a}}^{(\mathcal{F})} \times T\_{\boldsymbol{b}}^{(M)}) = A\_{\boldsymbol{\eta}}(T\_{\boldsymbol{a}}^{(\mathcal{F})}) \Gamma\_{\boldsymbol{a}\boldsymbol{b}} A\_{\boldsymbol{\eta}}(T\_{\boldsymbol{b}}^{(M)}).\tag{6}$$

Here, Γ*ab* are input parameters in this model. However, the condition of Eq. (5) assures that the elements of Γ*ab* should satisfy

$$\sum\_{a=1}^{n^{(F)}} A\_{\hbar}(T\_a^{(F)}) \Gamma\_{ab} = 1 \text{ for } 1 \le b \le n^{(M)},\tag{7}$$

$$\sum\_{b=1}^{n^{(M)}} \Gamma\_{ab} A\_n(T\_b^{(M)}) = 1 \text{ for } 1 \le a \le n^{(F)},\tag{8}$$

$$
\Gamma\_{ab} \ge 0.\tag{9}
$$

These restrictions on Γ*ab* assume that all adults marry and reproduce offsprings. When some genotype adults have handicaps for marriage, it is reflected in the disadvantage factor in selection.

Since we obtain the same equation from Eqs. (7) and (8) as

$$\sum\_{ab} A\_{\hbar}(T\_a^{(F)}) \Gamma\_{ab} A\_{\hbar}(T\_b^{(M)}) = 1 = \sum\_a A\_{\hbar}(T\_a^{(F)}) = \sum\_b A\_{\hbar}(T\_b^{(M)}),\tag{10}$$

Here all the free input parameters of the mating factors are given by zero. The *<sup>n</sup>*(*M*) <sup>+</sup> *<sup>n</sup>*(*F*) <sup>−</sup> <sup>1</sup>

Population Evolution in Hemophilia 59

where we omit presenting other Γ*ab*, which have similar expressions in terms of the genotype

We consider a new and more general approach by introducing the method of the linear

When non-random mating is involved, there are *<sup>n</sup>*(*F*) · *<sup>n</sup>*(*M*) unknown values of <sup>Γ</sup>*ab*. Since the number of the unknowns is bigger than the number of the restriction equations of Eqs. (7) and (8), there is no unique way to determine the mating factors Γ*ab*. However, if we add a theoretical restriction such as maximization or minimization of a specific population, a unique process becomes possible. We introduce an additional condition of maximizing *Z*, which is

*<sup>β</sup>abPn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

where the parameters *βab* represent the tendency of mating. In the process of maximizing *Z*, a given positive (negative) value of *<sup>β</sup>ab* will produce a bigger (smaller) value of *Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*)

This problem of maximizing the objective function *Z* with the constraints of Eqs. (7)-(9) is well known, and can be solved as a linear programming problem, for example by the simplex

The nature of *βab* values can be understood if we use practical demographic data, for example, incomes or divorce rates for parent groups. For instance, for the society that wants to reduce the divorce rates, it is possible to find the mating factors Γ*ab* for the minimal divorce rate. If *βab* values represent incomes, we find the mating factors for the maximum income of the society.

For hemophilia, the locus of the relevant gene called F8 is on *X* chromosome, and the gene has

the model of hemophilia, it is worthwhile to present all input parameters explicitly. First of

. For male, there are two genotypes: normal *XY*, and disease *X*�

which corresponds to inbreeding (outbreeding) between *<sup>T</sup>*(*F*) *<sup>a</sup>* type female and *<sup>T</sup>*(*M*)

*An*(*T*(*M*) <sup>1</sup> )

(<sup>1</sup> <sup>−</sup> *An*(*T*(*F*)

<sup>1</sup> ) *An*(*T*(*M*) <sup>1</sup> )

, (14)

*<sup>b</sup>* ), (16)

. For female, three genotypes exist: normal *XX*, carrier

*<sup>b</sup>* ),

*Y*. In

*<sup>b</sup>* type male.

), (15)

factors should be determined by the constraints. As a result, we can determine Γ*ab* as

<sup>Γ</sup><sup>21</sup> <sup>=</sup> <sup>1</sup>

*Z* = ∑ *ab*

··· ,

frequencies.

programming.

• General non-random mating

method (Press et al., 1992).

**3. Hemophilia**

, and disease *X*�

*XX*�

written in terms of the parent populations as

In this paper, we consider only medical screening.

two alleles: normal *X* and abnormal *X*�

*X*�

*An*(*T*(*F*) <sup>2</sup> )

<sup>Γ</sup><sup>11</sup> <sup>=</sup> <sup>1</sup>

the constraint equations of Eqs. (7) and (8) are not linearly independent. We note that the number of constraints is given by *<sup>n</sup>*(*M*) <sup>+</sup> *<sup>n</sup>*(*F*) <sup>−</sup> 1. Hence, the number of free input parameters is given by *<sup>n</sup>*(*F*)*n*(*M*) <sup>−</sup> *<sup>n</sup>*(*M*) <sup>−</sup> *<sup>n</sup>*(*F*) <sup>+</sup> 1.

Although there are many ways to assign free parameters, we present here four cases for the mating factors.

• Random mating

A trivial but important solution would be the random mating for which all of Γ*ab* are given by 1.

• Selective mating

If only one specific genotype is preferred completely, say *T*(*F*) <sup>1</sup> and *<sup>T</sup>*(*M*) <sup>1</sup> , we let the free parameters of mating factors all zero as

$$(\Gamma\_{ab}) = \begin{pmatrix} \Gamma\_{11} & \Gamma\_{12} \ \Gamma\_{13} & \cdots \ \Gamma\_{1n^{(M)}} \\ \Gamma\_{21} & 0 & 0 & 0 \\ \Gamma\_{31} & 0 & 0 & 0 \\ \vdots & & \ddots & \vdots \\ \Gamma\_{n^{(F)}1} & 0 & 0 & \cdots & 0 \end{pmatrix}. \tag{11}$$

In order to satisfy the constraints, other *<sup>n</sup>*(*M*) <sup>+</sup> *<sup>n</sup>*(*F*) <sup>−</sup> 1 factors should be given by

$$
\begin{pmatrix} \Gamma\_{ab} \end{pmatrix} = \begin{pmatrix} \frac{-1 + A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(F)}) + A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(M)})}{A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(F)})A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(M)})} & \frac{1}{A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(F)})} & \frac{1}{A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(F)})} & \cdots & \frac{1}{A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(F)})} \\ \frac{1}{A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(M)})} & 0 & 0 & 0 \\ \frac{1}{A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(M)})} & 0 & 0 & 0 \\ \vdots & & \ddots & \vdots \\ \vdots & & \ddots & \vdots \\ \frac{1}{A\_{\boldsymbol{u}}(\boldsymbol{T}\_{1}^{(M)})} & 0 & 0 & \cdots & 0 \end{pmatrix}. \tag{12}
$$

We note that the mating factors depend on population ratios at each generation. It should be emphasized from <sup>Γ</sup><sup>11</sup> that the population of *An*(*T*(*F*) <sup>1</sup> ) + *An*(*T*(*M*) <sup>1</sup> ) must be greater than 1 at the initial stage, and remains greater than 1 until the population ratios arrive at the equilibrium in this non-random mating.

• Hierarchical mating

For the case of *n*(*F*) = *n*(*M*), we can define the mating factors as

$$(\Gamma\_{ab}) = \begin{pmatrix} \Gamma\_{11} & 0 & 0 & \cdots & 0 \\ \Gamma\_{21} \ \Gamma\_{22} & 0 & & & 0 \\ & 0 & \Gamma\_{32} \ \Gamma\_{33} & & & 0 \\ & \vdots & & \ddots & & \vdots \\ & 0 & 0 & \cdots & \Gamma\_{n^{(\mathcal{F})}n^{(M)}} \end{pmatrix}. \tag{13}$$

the constraint equations of Eqs. (7) and (8) are not linearly independent. We note that the number of constraints is given by *<sup>n</sup>*(*M*) <sup>+</sup> *<sup>n</sup>*(*F*) <sup>−</sup> 1. Hence, the number of free input parameters

Although there are many ways to assign free parameters, we present here four cases for the

A trivial but important solution would be the random mating for which all of Γ*ab* are given by

Γ<sup>11</sup> Γ<sup>12</sup> Γ<sup>13</sup> ··· Γ1*n*(*M*) Γ<sup>21</sup> 00 0 Γ<sup>31</sup> 00 0

. ... .

1 *An*(*T*(*F*) <sup>1</sup> )

. ... .

<sup>1</sup> ) + *An*(*T*(*M*)

. .

We note that the mating factors depend on population ratios at each generation. It should be

initial stage, and remains greater than 1 until the population ratios arrive at the equilibrium

Γ<sup>11</sup> 0 0 ··· 0 Γ<sup>21</sup> Γ<sup>22</sup> 0 0 0 Γ<sup>32</sup> Γ<sup>33</sup> 0

. ... .

000 ··· Γ*n*(*F*)*n*(*M*)

Γ*n*(*F*)<sup>1</sup> 0 0 ··· 0

. .

1 *An*(*T*(*F*)

00 0

00 0

0 0 ··· 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

<sup>1</sup> ) ··· <sup>1</sup>

*An*(*T*(*F*) <sup>1</sup> ) ⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

<sup>1</sup> ) must be greater than 1 at the

. (13)

. (12)

. .

<sup>1</sup> and *<sup>T</sup>*(*M*)

<sup>1</sup> , we let the free

. (11)

If only one specific genotype is preferred completely, say *T*(*F*)

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

. .

In order to satisfy the constraints, other *<sup>n</sup>*(*M*) <sup>+</sup> *<sup>n</sup>*(*F*) <sup>−</sup> 1 factors should be given by

<sup>1</sup> )+*An*(*T*(*M*) <sup>1</sup> )

<sup>1</sup> )*An*(*T*(*M*) <sup>1</sup> )

1 *An*(*T*(*M*) <sup>1</sup> )

1 *An*(*T*(*M*) <sup>1</sup> )

> . .

1 *An*(*T*(*M*) <sup>1</sup> )

(Γ*ab*) =

−1+*An*(*T*(*F*)

*An*(*T*(*F*)

For the case of *n*(*F*) = *n*(*M*), we can define the mating factors as

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

. .

(Γ*ab*) =

is given by *<sup>n</sup>*(*F*)*n*(*M*) <sup>−</sup> *<sup>n</sup>*(*M*) <sup>−</sup> *<sup>n</sup>*(*F*) <sup>+</sup> 1.

parameters of mating factors all zero as

(Γ*ab*) =

in this non-random mating. • Hierarchical mating

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

emphasized from <sup>Γ</sup><sup>11</sup> that the population of *An*(*T*(*F*)

mating factors. • Random mating

• Selective mating

1.

Here all the free input parameters of the mating factors are given by zero. The *<sup>n</sup>*(*M*) <sup>+</sup> *<sup>n</sup>*(*F*) <sup>−</sup> <sup>1</sup> factors should be determined by the constraints. As a result, we can determine Γ*ab* as

$$\Gamma\_{11} = \frac{1}{A\_{\hbar}(T\_1^{(M)})},\tag{14}$$

$$\Gamma\_{21} = \frac{1}{A\_{\boldsymbol{\eta}}(T\_2^{(F)})} (1 - \frac{A\_{\boldsymbol{\eta}}(T\_1^{(F)})}{A\_{\boldsymbol{\eta}}(T\_1^{(M)})}),\tag{15}$$

where we omit presenting other Γ*ab*, which have similar expressions in terms of the genotype frequencies.

• General non-random mating

We consider a new and more general approach by introducing the method of the linear programming.

When non-random mating is involved, there are *<sup>n</sup>*(*F*) · *<sup>n</sup>*(*M*) unknown values of <sup>Γ</sup>*ab*. Since the number of the unknowns is bigger than the number of the restriction equations of Eqs. (7) and (8), there is no unique way to determine the mating factors Γ*ab*. However, if we add a theoretical restriction such as maximization or minimization of a specific population, a unique process becomes possible. We introduce an additional condition of maximizing *Z*, which is written in terms of the parent populations as

$$Z = \sum\_{ab} \beta\_{ab} P\_n(T\_a^{(F)} \times T\_b^{(M)}) \, \tag{16}$$

where the parameters *βab* represent the tendency of mating. In the process of maximizing *Z*, a given positive (negative) value of *<sup>β</sup>ab* will produce a bigger (smaller) value of *Pn*(*T*(*F*) *<sup>a</sup>* <sup>×</sup> *<sup>T</sup>*(*M*) *<sup>b</sup>* ), which corresponds to inbreeding (outbreeding) between *<sup>T</sup>*(*F*) *<sup>a</sup>* type female and *<sup>T</sup>*(*M*) *<sup>b</sup>* type male. This problem of maximizing the objective function *Z* with the constraints of Eqs. (7)-(9) is well known, and can be solved as a linear programming problem, for example by the simplex method (Press et al., 1992).

The nature of *βab* values can be understood if we use practical demographic data, for example, incomes or divorce rates for parent groups. For instance, for the society that wants to reduce the divorce rates, it is possible to find the mating factors Γ*ab* for the minimal divorce rate. If *βab* values represent incomes, we find the mating factors for the maximum income of the society. In this paper, we consider only medical screening.

#### **3. Hemophilia**

For hemophilia, the locus of the relevant gene called F8 is on *X* chromosome, and the gene has two alleles: normal *X* and abnormal *X*� . For female, three genotypes exist: normal *XX*, carrier *XX*� , and disease *X*� *X*� . For male, there are two genotypes: normal *XY*, and disease *X*� *Y*. In the model of hemophilia, it is worthwhile to present all input parameters explicitly. First of

In fact, we determine the mutation rate *μ*(*X* → *X*�

let *<sup>β</sup>* <sup>=</sup> 0.0 and *<sup>α</sup>* <sup>=</sup> 3.3 <sup>×</sup> <sup>10</sup>−<sup>5</sup> for hemophilia.

10−<sup>5</sup> shows that the equilibrium adult population is calculated as

*XX* : *XX*� : *X*�

*XY* : *X*�

*Y*) = 0.1, and *δ*(*X*�

*XY* : *X*�

*XX* : *XX*� : *X*�

data of birth population, *XY* : *X*�

**3.2 Random mating with** *δ* = 0.1

) = *δ*(*XY*) = 0, *δ*(*X*�

above, we find the equilibrium population:

study the effect of the non-random mating below.

**3.3 Non-random mating with** *δ* = 0.1 **in minimizing P***n*(**XX**�

*δ*(*XX*�

prevalence.

becomes more plausible.

) as about 3.3 <sup>×</sup> <sup>10</sup>−<sup>5</sup> using the demographic

) = 1 with the same mutation rates as the

*Y* = 0.999125 : 0.000875, (18)

×**X**� **Y**)

*Y* = 0.9999 : 0.0001. The population equation at *α* = 3.3 ×

*Y* = 1 : 0, (17)

*X*� = 0.999868 : 0.000132 : 0,

where the value of 0.000132 was rather insensitive on the value of *β*. We found that this low sensitivity originates from the severe disadvantage factor in hemophilia. Although there is a report for existence of a gene self-repairing mechanism in the process of evolution (Avise, 1993), we assume that self-repairing is very rare. Throughout this section hereafter, we simply

Population Evolution in Hemophilia 61

It is shown that these equilibrium population ratio values can be easily achieved not only analytically but also numerically (Lee et al., 2001). Numerical simulations show that these values become stable after about 100 generations, having no dependence on the initial state.

We consider the case where male patients of hemophilia can be treated so that some of them could lead normal lives, although a genetic defect remains intact. However, since female baby patients are dead during the pregnancy, it is assumed that there is still no cure for female patients as it stands at present. In fact, when the disadvantage factors are given by *δ*(*XX*) =

*X*�

in the random mating case where all Γ*ab* = 1. These numbers will be used as references to

Recently an interesting article (Stonebraker et al., 2010) on hemophilia prevalence in different countries was published. The prevalence (per 100,000 males) for high income countries was 12.8 ± 6.0 (mean ± SD) whereas it was 6.6 ± 4.8 for the rest of the world. Within a country, there was a strong trend of increasing prevalence over time: the prevalence for Canada ranged from 10.2 in 1989 to 14.2 in 2008 and for the United Kingdom it ranged from 9.3 in 1974 to 21.6 in 2006. The data are consistent with the fact that the cure results in increasing hemophilia

Since rapid advances in molecular genetics have highlighted the potential use of genetic testing to screen for adult-onset chronic diseases (Burke et al., 2001), medical screening

We consider an artificial circumstance, where society-wide medical screening is available to control the mating factors. For example, we consider the case of hemophilia with a medical screening. The goal in this consideration is to find the equilibrium populations in that society. We assume that the medical screening helps male patients avoid marrying disease carrier

*X*� = 0.998121 : 0.001879 : 0,

all, for the random mating, all of mating factors are given by 1 as

$$
\Gamma\_{ab} = 1.
$$

The mutation rates are given by

$$
\mu(X \to X') = \alpha, \ \mu(X \to X) = 1 - \alpha,
$$

$$
\mu(X' \to X) = \beta, \ \mu(X' \to X') = 1 - \beta.
$$

The reproduction coefficients are found as

$$\begin{aligned} \xi(XX \times XY \to XX) &= 1, \quad \xi(XX \times XY \to XX') = 0, \quad \xi(XX \times XY \to X'X') = 0, \\ \xi(XX \times X'Y \to XX) &= 0, \quad \xi(XX \times X'Y \to XX') = 1, \quad \xi(XX \times X'Y \to X'X') = 0, \\ \xi(XX' \times XY \to XX) &= 0.5, \quad \xi(XX' \times XY \to XX') = 0.5, \quad \xi(XX' \times XY \to X'X') = 0, \\ \xi(XX' \times Y'Y \to XX) &= 0, \quad \xi(XX' \times X'Y \to XX') = 0.5, \quad \xi(XX' \times X'Y \to X'X') = 0.5, \\ \xi(XX' \times XY \to XX) &= 0, \quad \xi(XX' \times XY \to XX') = 1, \quad \xi(XX' \times XY \to X'X') = 0, \\ \xi(XX' \times X'Y \to XX) &= 0, \quad \xi(X'X' \times X'Y \to XX') = 0, \quad \xi(X'X' \times X'Y \to X'X') = 1, \\ \xi(XX \times XY \to XY) &= 1, \quad \xi(XX \times XY \to X'Y) = 0, \\ \xi(XX \times X'Y \to XY) &= 1, \quad \xi(XX \times X'Y \to X'Y) = 0, \\ \xi(XX' \times XY \to XY) &= 0.5, \quad \xi(XX' \times XY \to X'Y) = 0.5, \\ \xi(XX' \times XY' \to XY) &= 0.5, \quad \xi(XX' \times X'Y \to X'Y) = 1, \\ \xi(X'X' \times XY \to XY) &= 0, \quad \xi(X'X' \times XY \to X'Y) = 1. \end{aligned}$$

In the United States, about 17,000 people have hemophilia. About one in 7,500 live male births has hemophilia and about one in 25,000,000 live female births has hemophilia. From the data, we find that almost all female babies with hemophilia are dead during the pregnancy. Thus, it is natural to assign the disadvantage factors as

$$\delta(XX) = \delta(XX') = \delta(XY) = 0,\\ \delta(X^\prime X') = 1,\\ \delta(X^\prime Y) = \delta.$$

Note that we take *δ* = 0 if male patients are completely cured. In the next subsections, we study the effect of male patient treatment by changing *δ*. The value of *δ* reflects the social development and standards of the general health care of particular geographical region where the hemophilia population is living.

#### **3.1 Random mating with** *δ* = 1

The most natural case of hemophilia would be random mating with fatal mortality. In this case, we let *δ*(X� Y) = *δ* = 1. We now focus on mutation rates *μ*(*X* → *X*� ) = *α* and *μ*(*X*� → *X*) = *β*.

Γ*ab* = 1.

*μ*(*X*� → *X*) = *β*, *μ*(*X*� → *X*�

) = *α*, *μ*(*X* → *X*) = 1 − *α*,

*Y* → *XX*�

*Y* → *XX*�

*X*� × *XY* → *XX*�

*Y* → *XX*�

*ξ*(*XX*� × *XY* → *XY*) = 0.5, *ξ*(*XX*� × *XY* → *X*�

*Y* → *XY*) = 0, *ξ*(*X*�

*X*� × *XY* → *XY*) = 0, *ξ*(*X*�

In the United States, about 17,000 people have hemophilia. About one in 7,500 live male births has hemophilia and about one in 25,000,000 live female births has hemophilia. From the data, we find that almost all female babies with hemophilia are dead during the pregnancy. Thus,

) = *δ*(*XY*) = 0, *δ*(*X*�

Note that we take *δ* = 0 if male patients are completely cured. In the next subsections, we study the effect of male patient treatment by changing *δ*. The value of *δ* reflects the social development and standards of the general health care of particular geographical region where

The most natural case of hemophilia would be random mating with fatal mortality. In this

Y) = *δ* = 1. We now focus on mutation rates *μ*(*X* → *X*�

*Y* → *XY*) = 0.5, *ξ*(*XX*� × *X*�

*ξ*(*XX* × *XY* → *XY*) = 1, *ξ*(*XX* × *XY* → *X*�

*Y* → *XY*) = 1, *ξ*(*XX* × *X*�

*X*�

*X*� × *X*�

*ξ*(*XX* × *X*�

*X*� × *X*�

*ξ*(*XX*� × *X*�

*ξ*(*X*�

*ξ*(*X*�

it is natural to assign the disadvantage factors as

the hemophilia population is living.

**3.1 Random mating with** *δ* = 1

case, we let *δ*(X�

*X*) = *β*.

*δ*(*XX*) = *δ*(*XX*�

) = 1 − *β*.

) = 0, *ξ*(*XX* × *XY* → *X*�

) = 0.5, *ξ*(*XX*� × *XY* → *X*�

*X*� × *X*�

*Y* → *X*�

*Y* → *X*�

*Y* → *X*�

*Y*) = *δ*.

*X*� × *XY* → *X*�

*X*� × *X*�

) = 1, *δ*(*X*�

) = 1, *ξ*(*XX* × *X*�

) = 0.5, *ξ*(*XX*� × *X*�

) = 1, *ξ*(*X*�

) = 0, *ξ*(*X*�

*X*� ) = 0,

*X*� ) = 0,

*X*� ) = 0,

*X*� ) = 0,

*X*� ) = 1,

) = 0.5,

*X*�

*Y* → *X*�

*Y* → *X*�

*Y* → *X*�

*Y*) = 0,

*Y*) = 0,

*Y*) = 0.5,

*Y*) = 0.5,

*Y*) = 1,

*Y*) = 1.

) = *α* and *μ*(*X*� →

*X*� × *XY* → *X*�

all, for the random mating, all of mating factors are given by 1 as

*μ*(*X* → *X*�

*ξ*(*XX* × *XY* → *XX*) = 1, *ξ*(*XX* × *XY* → *XX*�

*Y* → *XX*) = 0, *ξ*(*XX* × *X*�

*ξ*(*XX*� × *XY* → *XX*) = 0.5, *ξ*(*XX*� × *XY* → *XX*�

*Y* → *XX*) = 0, *ξ*(*XX*� × *X*�

*X*� × *XY* → *XX*) = 0, *ξ*(*X*�

*Y* → *XX*) = 0, *ξ*(*X*�

The mutation rates are given by

*ξ*(*XX* × *X*�

*X*� × *X*�

*ξ*(*XX*� × *X*�

*ξ*(*X*�

*ξ*(*X*�

The reproduction coefficients are found as

In fact, we determine the mutation rate *μ*(*X* → *X*� ) as about 3.3 <sup>×</sup> <sup>10</sup>−<sup>5</sup> using the demographic data of birth population, *XY* : *X*� *Y* = 0.9999 : 0.0001. The population equation at *α* = 3.3 × 10−<sup>5</sup> shows that the equilibrium adult population is calculated as

$$XX:XX':X'X' = 0.999868:0.000132:0,\\ XY:X'Y = 1:0,\tag{17}$$

where the value of 0.000132 was rather insensitive on the value of *β*. We found that this low sensitivity originates from the severe disadvantage factor in hemophilia. Although there is a report for existence of a gene self-repairing mechanism in the process of evolution (Avise, 1993), we assume that self-repairing is very rare. Throughout this section hereafter, we simply let *<sup>β</sup>* <sup>=</sup> 0.0 and *<sup>α</sup>* <sup>=</sup> 3.3 <sup>×</sup> <sup>10</sup>−<sup>5</sup> for hemophilia.

It is shown that these equilibrium population ratio values can be easily achieved not only analytically but also numerically (Lee et al., 2001). Numerical simulations show that these values become stable after about 100 generations, having no dependence on the initial state.

#### **3.2 Random mating with** *δ* = 0.1

We consider the case where male patients of hemophilia can be treated so that some of them could lead normal lives, although a genetic defect remains intact. However, since female baby patients are dead during the pregnancy, it is assumed that there is still no cure for female patients as it stands at present. In fact, when the disadvantage factors are given by *δ*(*XX*) = *δ*(*XX*� ) = *δ*(*XY*) = 0, *δ*(*X*� *Y*) = 0.1, and *δ*(*X*� *X*� ) = 1 with the same mutation rates as the above, we find the equilibrium population:

$$XX:XX':X'X' = 0.998121:0.001879:0,$$

$$XY:X'Y = 0.999125:0.000875,\tag{18}$$

in the random mating case where all Γ*ab* = 1. These numbers will be used as references to study the effect of the non-random mating below.

Recently an interesting article (Stonebraker et al., 2010) on hemophilia prevalence in different countries was published. The prevalence (per 100,000 males) for high income countries was 12.8 ± 6.0 (mean ± SD) whereas it was 6.6 ± 4.8 for the rest of the world. Within a country, there was a strong trend of increasing prevalence over time: the prevalence for Canada ranged from 10.2 in 1989 to 14.2 in 2008 and for the United Kingdom it ranged from 9.3 in 1974 to 21.6 in 2006. The data are consistent with the fact that the cure results in increasing hemophilia prevalence.

#### **3.3 Non-random mating with** *δ* = 0.1 **in minimizing P***n*(**XX**� ×**X**� **Y**)

Since rapid advances in molecular genetics have highlighted the potential use of genetic testing to screen for adult-onset chronic diseases (Burke et al., 2001), medical screening becomes more plausible.

We consider an artificial circumstance, where society-wide medical screening is available to control the mating factors. For example, we consider the case of hemophilia with a medical screening. The goal in this consideration is to find the equilibrium populations in that society. We assume that the medical screening helps male patients avoid marrying disease carrier

For the biological gene, we use the same mutation rates *μ*(*X* → *X*�

reproduction rates *ξ* in the population equations are written as *ξ* = *ξ*(*b*)*ξ*(*s*):

(*<sup>R</sup>* <sup>×</sup> *<sup>R</sup>* <sup>→</sup> *<sup>R</sup>*) = 1, *<sup>ξ</sup>*(*s*)

(*<sup>R</sup>* <sup>×</sup> *<sup>P</sup>* <sup>→</sup> *<sup>R</sup>*) = 1, *<sup>ξ</sup>*(*s*)

(*<sup>P</sup>* <sup>×</sup> *<sup>R</sup>* <sup>→</sup> *<sup>R</sup>*) = 1, *<sup>ξ</sup>*(*s*)

(*<sup>P</sup>* <sup>×</sup> *<sup>P</sup>* <sup>→</sup> *<sup>R</sup>*) = 0, *<sup>ξ</sup>*(*s*)

<sup>1</sup> ) = 0, *<sup>δ</sup>*(*T*(*F*)

<sup>3</sup> ) = 0, *<sup>δ</sup>*(*T*(*F*)

<sup>5</sup> ) = 1, *<sup>δ</sup>*(*T*(*F*)

<sup>1</sup> ) = 0, *<sup>δ</sup>*(*T*(*M*)

<sup>3</sup> ) = 0, *<sup>δ</sup>*(*T*(*M*)

These factors imply that hemophilia is still fatal for female patients, while the rich social gene

In order to analyze equilibrium population resulted from inbreeding between the normal and

With all the given parameters as the above with the inbreeding, we find the equilibrium

<sup>4</sup> : *<sup>T</sup>*(*F*)

We clearly observe that the existence of social gene drastically influences the inheritance of the biological genes. We believe that this influence is induced by the disadvantage factor of

In order to examine the role of the disadvantage factor in the evolution mechanism, we have

= 0.833195 : 0.154462 : 0.009700 : 0.002643 : 0 : 0,

<sup>1</sup> <sup>×</sup> *<sup>T</sup>*(*M*) <sup>1</sup> ).

> <sup>5</sup> : *<sup>T</sup>*(*F*) 6

= 0.837755 : 0.156138 : 0.005223 : 0.000884. (25)

*<sup>Z</sup>* <sup>=</sup> *Pn*(*T*(*F*)

<sup>3</sup> : *<sup>T</sup>*(*F*)

<sup>3</sup> : *<sup>T</sup>*(*M*) 4

*<sup>δ</sup>*(*T*(*F*)

*<sup>δ</sup>*(*T*(*F*)

*<sup>δ</sup>*(*T*(*F*)

*<sup>δ</sup>*(*T*(*M*)

*<sup>δ</sup>*(*T*(*M*)

<sup>2</sup> : *<sup>T</sup>*(*F*)

<sup>2</sup> : *<sup>T</sup>*(*M*)

reproduction rates *ξ*(*b*) as before.

For the social gene, we let the mutation rates be

*ξ*(*s*)

*ξ*(*s*)

*ξ*(*s*)

*ξ*(*s*)

We further assign the disadvantage factors as

can make male patients lead normal lives.

population ratios as

the genotype, *T*(*M*)

<sup>4</sup> .

rich genotypes, we set the objective function as

*T*(*F*) <sup>1</sup> : *<sup>T</sup>*(*F*)

*T*(*M*) <sup>1</sup> : *<sup>T</sup>*(*M*)

carried out a calculation by changing only *<sup>δ</sup>*(*T*(*M*)

), *μ*(*X*� → *X*) and the same

*μ*(*R* → *P*) = 0.2 and *μ*(*P* → *R*) = 0.1 (22)

(*R* × *R* → *P*) = 0,

(*R* × *P* → *P*) = 0,

(*P* × *R* → *P*) = 0,

<sup>2</sup> ) = 0,

<sup>4</sup> ) = 0,

<sup>6</sup> ) = 1,

<sup>2</sup> ) = 0,

(*P* × *P* → *P*) = 1. (23)

<sup>4</sup> ) = 0.1. (24)

<sup>4</sup> ) from 0.1 to 1. The resulting population

as a sample example. Also, we assign the reproduction rates *ξ*(*s*) with which the whole

Population Evolution in Hemophilia 63

women. This restriction lead us to choose the objective function as

$$Z = -P\_n(XX' \times X'Y)\_n$$

with which we expect a minimal value of *Pn*(*XX*� × *X*� *Y*) for maximizing *Z*. In consequence, with the previous disadvantage factors and mutation rates, the population equations give the equilibrium as

$$XX:XX':X'X' = 0.998089:0.001911:0,$$

$$XY:X'Y = 0.999110:0.000890.\tag{19}$$

The corresponding objective function *Z* is given by simply 0 as expected. We find the different numbers of the male patients and the female carriers from those of the random mating in Eq. (18).

#### **3.4 Non-random mating with** *δ* = 0.1 **in maximizing P***n*(**XX**� ×**X**� **Y**)

It is instructive to compare the above result with that of the opposite situation where the objective function is given by

$$Z = P\_n(XX' \times X'Y)$$

to maximize *Pn*(*XX*� × *X*� *Y*). For the same disadvantage factors and mutation rates as the above, the population equation gives the equilibrium as

$$XX:XX':X'X' = 0.999868:0.000132:0,$$

$$XY:X'Y = 0.999911:0.000089. \tag{20}$$

The corresponding objective function *<sup>Z</sup>* at equilibrium is given by 8.90937 <sup>×</sup> <sup>10</sup>−5, which means that all male patients marry with female carriers because *X*� *Y* is given by 0.000089 as shown in the above.

We find that the population ratios of Eq. (18) in random mating are between two extremal cases of Eqs. (19) and (20), which correspond to outbreeding and inbreeding of female carriers and male patients, respectively.

#### **3.5 Non-random mating in association of a social gene**

We now consider the effect of the social inheritance on the evolution of the biological genes in the case of hemophilia. We consider a two-gene system, where one of the relevant genes is hemophilia on X chromosome, and the other is a social gene. For simplicity, we assume that the social gene has two allele: Rich(*R*) and Poor(*P*). In this model, there are six female genotypes and four male genotypes:

$$\begin{aligned} T\_1^{(F)} &= XX : \mathbb{R}, \quad T\_2^{(F)} = XX : \mathbb{P}\_\prime \\ T\_3^{(F)} &= X^\prime X : \mathbb{R}, \quad T\_4^{(F)} = X^\prime X : \mathbb{P}\_\prime \\ T\_5^{(F)} &= X^\prime X^\prime : \mathbb{R}, \quad T\_6^{(F)} = X^\prime X^\prime : \mathbb{P}\_\prime \\ T\_1^{(M)} &= XY : \mathbb{R}, \quad T\_2^{(M)} = XY : \mathbb{P}\_\prime \\ T\_3^{(M)} &= X^\prime Y : \mathbb{R}, \quad T\_4^{(M)} = X^\prime Y : \mathbb{P}. \end{aligned}$$

*Z* = −*Pn*(*XX*� × *X*�

with the previous disadvantage factors and mutation rates, the population equations give the

The corresponding objective function *Z* is given by simply 0 as expected. We find the different numbers of the male patients and the female carriers from those of the random mating in Eq.

It is instructive to compare the above result with that of the opposite situation where the

The corresponding objective function *<sup>Z</sup>* at equilibrium is given by 8.90937 <sup>×</sup> <sup>10</sup>−5, which

We find that the population ratios of Eq. (18) in random mating are between two extremal cases of Eqs. (19) and (20), which correspond to outbreeding and inbreeding of female carriers

We now consider the effect of the social inheritance on the evolution of the biological genes in the case of hemophilia. We consider a two-gene system, where one of the relevant genes is hemophilia on X chromosome, and the other is a social gene. For simplicity, we assume that the social gene has two allele: Rich(*R*) and Poor(*P*). In this model, there are six female

*X* : *R*, *T*(*F*)

*X*� : *R*, *T*(*F*)

*Y* : *R*, *T*(*M*)

*Z* = *Pn*(*XX*� × *X*�

*Y*),

*X*� = 0.998089 : 0.001911 : 0,

*Y*) for maximizing *Z*. In consequence,

*Y* = 0.999110 : 0.000890. (19)

*Y* = 0.999911 : 0.000089. (20)

*Y* is given by 0.000089

×**X**� **Y**)

*Y*). For the same disadvantage factors and mutation rates as the

*Y*)

*X*� = 0.999868 : 0.000132 : 0,

<sup>2</sup> = *XX* : *P*,

*X* : *P*,

*X*� : *P*,

*Y* : *P*. (21)

<sup>4</sup> = *X*�

<sup>6</sup> = *X*�

<sup>4</sup> = *X*�

<sup>2</sup> = *XY* : *P*,

women. This restriction lead us to choose the objective function as

*XX* : *XX*� : *X*�

**3.4 Non-random mating with** *δ* = 0.1 **in maximizing P***n*(**XX**�

above, the population equation gives the equilibrium as

**3.5 Non-random mating in association of a social gene**

*T*(*F*)

*T*(*F*) <sup>3</sup> = *X*�

*T*(*F*) <sup>5</sup> = *X*�

*T*(*M*)

*T*(*M*) <sup>3</sup> = *X*�

<sup>1</sup> <sup>=</sup> *XX* : *<sup>R</sup>*, *<sup>T</sup>*(*F*)

<sup>1</sup> <sup>=</sup> *XY* : *<sup>R</sup>*, *<sup>T</sup>*(*M*)

*XX* : *XX*� : *X*�

*XY* : *X*�

means that all male patients marry with female carriers because *X*�

*XY* : *X*�

with which we expect a minimal value of *Pn*(*XX*� × *X*�

equilibrium as

objective function is given by

to maximize *Pn*(*XX*� × *X*�

as shown in the above.

and male patients, respectively.

genotypes and four male genotypes:

(18).

For the biological gene, we use the same mutation rates *μ*(*X* → *X*� ), *μ*(*X*� → *X*) and the same reproduction rates *ξ*(*b*) as before.

For the social gene, we let the mutation rates be

$$
\mu(R \to P) = 0.2 \quad \text{and} \quad \mu(P \to R) = 0.1 \tag{22}
$$

as a sample example. Also, we assign the reproduction rates *ξ*(*s*) with which the whole reproduction rates *ξ* in the population equations are written as *ξ* = *ξ*(*b*)*ξ*(*s*):

$$\begin{aligned} \mathfrak{F}^{(s)}(\mathbb{R}\times\mathbb{R}\to\mathbb{R}) &= 1, & \mathfrak{F}^{(s)}(\mathbb{R}\times\mathbb{R}\to P) &= 0, \\ \mathfrak{F}^{(s)}(\mathbb{R}\times P\to\mathbb{R}) &= 1, & \mathfrak{F}^{(s)}(\mathbb{R}\times P\to P) &= 0, \\ \mathfrak{F}^{(s)}(P\times\mathbb{R}\to\mathbb{R}) &= 1, & \mathfrak{F}^{(s)}(P\times\mathbb{R}\to P) &= 0, \\ \mathfrak{F}^{(s)}(P\times P\to\mathbb{R}) &= 0, & \mathfrak{F}^{(s)}(P\times P\to P) &= 1. \end{aligned} \tag{23}$$

We further assign the disadvantage factors as

$$
\begin{aligned}
\delta(T\_1^{(F)}) &= 0, & \delta(T\_2^{(F)}) &= 0, \\
\delta(T\_3^{(F)}) &= 0, & \delta(T\_4^{(F)}) &= 0, \\
\delta(T\_5^{(F)}) &= 1, & \delta(T\_6^{(F)}) &= 1, \\
\delta(T\_1^{(M)}) &= 0, & \delta(T\_2^{(M)}) &= 0, \\
\delta(T\_3^{(M)}) &= 0, & \delta(T\_4^{(M)}) &= 0.1.
\end{aligned}
\tag{24}
$$

These factors imply that hemophilia is still fatal for female patients, while the rich social gene can make male patients lead normal lives.

In order to analyze equilibrium population resulted from inbreeding between the normal and rich genotypes, we set the objective function as

$$Z = P\_{\mathfrak{n}}(T\_1^{(F)} \times T\_1^{(M)}).$$

With all the given parameters as the above with the inbreeding, we find the equilibrium population ratios as

$$\begin{aligned} &T\_1^{(F)}: T\_2^{(F)}: T\_3^{(F)}: T\_4^{(F)}: T\_5^{(F)}: T\_6^{(F)}\\ &= 0.833195: 0.154462: 0.009700: 0.002643: 0: 0, \\ &T\_1^{(M)}: T\_2^{(M)}: T\_3^{(M)}: T\_4^{(M)} \\ &= 0.837755: 0.156138: 0.005223: 0.000884. \end{aligned}$$

We clearly observe that the existence of social gene drastically influences the inheritance of the biological genes. We believe that this influence is induced by the disadvantage factor of the genotype, *T*(*M*) <sup>4</sup> .

In order to examine the role of the disadvantage factor in the evolution mechanism, we have carried out a calculation by changing only *<sup>δ</sup>*(*T*(*M*) <sup>4</sup> ) from 0.1 to 1. The resulting population

The current study does not include any dynamic change of mutation factors, which are expected especially in the social gene inheritance. The collective and the dynamic behavior

Population Evolution in Hemophilia 65

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**5. References**

ratios are given by

$$\begin{aligned} &T\_1^{(F)}: T\_2^{(F)}: T\_3^{(F)}: T\_4^{(F)}: T\_5^{(F)}: T\_6^{(F)}\\ &= 0.826752: 0.171908: 0.001127: 0.000213: 0: 0, \\ &T\_1^{(M)}: T\_2^{(M)}: T\_3^{(M)}: T\_4^{(M)} \\ &= 0.827356: 0.172040: 0.000604: 0. \end{aligned}$$

Comparing with Eq. (25), we clearly observe that the change of the disadvantage factor has a significant influence on the hemophiliac biological evolution. We also note that the ratio of rich to poor is higher in the diseased than in the healthy, clearly indicating the social gene effect of richness. Thus, we note again that existence of a particular social gene can have significant effect on the evolution process.

We note that a drawback of our approach is in the large number of tunable parameters. While mutation rates and recombination rates for biological genes may be adjusted by the coalescent method, it is hard to find an empirical basis for social genes. The social gene of last name probably gains some empirical support. However, if no support is provided, it is reasonable to perform sensitivity analysis by scanning around chosen parameters.

There are many genetic diseases other than hemophilia in the real world. As an example, the same approach may be applied to the case of congenital hypothyroidism (Calaciura et al., 2002). Expansion of the present scheme to other cases including more diverse social genes remains as a future study.

#### **4. Conclusion**

We have presented a theoretical scheme and a simulation program to study population genetics in a realistic complex situation. The result shows that the multiplicity of the gene loci greatly affect the demographic distribution of fractional population ratios. We suggest that more detailed demographic data including gene mutation (Porter, 1968; Strachan, 1996) and fitness is desirable to elaborate the theory further.

Treating social status as an inheritance trait, we introduce the concept of the social gene. Treating the social genes on equal footing with the biological genes, we build a unified framework to find the population evolution (Cavalli-Sforza & Bodmer, 1996) in the biological and the social inheritance. The nature of the social inheritance inevitably introduces the concept of non-random mating. This framework is used to investigate the detailed cases of hemophilia.

For hemophilia, we have presented a theoretical scheme to determine the mating factors uniquely in the non-random mating process by introducing the objective function concept. Finally, we have considered the effect of the social gene with two alleles, Rich and Poor on the biological gene of hemophilia. The result shows that the introduction of a social gene or social inheritances change the population evolution of the biological gene. We note that the social gene concept introduced here is different from polyphenism, for which multiple phenotypes can arise from a single genotype as a result of differing environment conditions. The reason for the difference is that the social gene is inherited with the biological gene, whereas the environment is not.

The current study does not include any dynamic change of mutation factors, which are expected especially in the social gene inheritance. The collective and the dynamic behavior are subjects of future studies.

#### **5. References**

14 Hemophilia

<sup>4</sup> : *<sup>T</sup>*(*F*)

Comparing with Eq. (25), we clearly observe that the change of the disadvantage factor has a significant influence on the hemophiliac biological evolution. We also note that the ratio of rich to poor is higher in the diseased than in the healthy, clearly indicating the social gene effect of richness. Thus, we note again that existence of a particular social gene can have

We note that a drawback of our approach is in the large number of tunable parameters. While mutation rates and recombination rates for biological genes may be adjusted by the coalescent method, it is hard to find an empirical basis for social genes. The social gene of last name probably gains some empirical support. However, if no support is provided, it is reasonable

There are many genetic diseases other than hemophilia in the real world. As an example, the same approach may be applied to the case of congenital hypothyroidism (Calaciura et al., 2002). Expansion of the present scheme to other cases including more diverse social genes

We have presented a theoretical scheme and a simulation program to study population genetics in a realistic complex situation. The result shows that the multiplicity of the gene loci greatly affect the demographic distribution of fractional population ratios. We suggest that more detailed demographic data including gene mutation (Porter, 1968; Strachan, 1996)

Treating social status as an inheritance trait, we introduce the concept of the social gene. Treating the social genes on equal footing with the biological genes, we build a unified framework to find the population evolution (Cavalli-Sforza & Bodmer, 1996) in the biological and the social inheritance. The nature of the social inheritance inevitably introduces the concept of non-random mating. This framework is used to investigate the detailed cases of

For hemophilia, we have presented a theoretical scheme to determine the mating factors uniquely in the non-random mating process by introducing the objective function concept. Finally, we have considered the effect of the social gene with two alleles, Rich and Poor on the biological gene of hemophilia. The result shows that the introduction of a social gene or social inheritances change the population evolution of the biological gene. We note that the social gene concept introduced here is different from polyphenism, for which multiple phenotypes can arise from a single genotype as a result of differing environment conditions. The reason for the difference is that the social gene is inherited with the biological gene, whereas the

= 0.826752 : 0.171908 : 0.001127 : 0.000213 : 0 : 0,

<sup>5</sup> : *<sup>T</sup>*(*F*) 6

= 0.827356 : 0.172040 : 0.000604 : 0. (26)

ratios are given by

*T*(*F*) <sup>1</sup> : *<sup>T</sup>*(*F*)

*T*(*M*) <sup>1</sup> : *<sup>T</sup>*(*M*)

significant effect on the evolution process.

remains as a future study.

**4. Conclusion**

hemophilia.

environment is not.

<sup>2</sup> : *<sup>T</sup>*(*F*)

<sup>2</sup> : *<sup>T</sup>*(*M*)

to perform sensitivity analysis by scanning around chosen parameters.

and fitness is desirable to elaborate the theory further.

<sup>3</sup> : *<sup>T</sup>*(*F*)

<sup>3</sup> : *<sup>T</sup>*(*M*) 4

**1. Introduction**

alternative hemostatic therapy with bypassing agents.

**2. Type of factor inhibitors**

**5**

**Hemophilia Inhibitors Prevalence,**

Hemophilia is a bleeding disorder that results from genetic alteration in production of coagulation factors that are important to maintain hemostasis. The commonest type is hemophilia A due to deficiency of factor VIII (FVIII), which is important zymogen co factor for clot formation. Hemophilia A is an X-linked disease that affects males at prevalence of 1:5000-10000. Hemophilia B is due to deficiency in factor (FIX) but less common with prevalence of 1:34,500 males. It is inherited also as X- linked. Although both disorders are rarely observed; they can be very serious (life threatening) and costly for families and countries. Treatment of hemophilia is based on replacement of the deficient factor. Two types of factor concentrates are available, plasma derived (pdFVIII/IX) and recombinant (rFVIII/IX) which are associated with variable incidence of inhibitor formation rates.The development of inhibitor is the most serious and challenging complication of hemophilia treatment with the enormous economic burden (1). FVIII inhibitors are immunoglobulin IgG (IgG1 and IgG4) antibodies that neutralize FVIII procoagulant activity in plasma. Inhibitors are usually classified according to their levels in plasma as a "high-titer" inhibitors, those with the highest activity >5 Bethesda Units (BU)/ml or a low-titer inhibitor type. In hemophilia A aproximately 60-70% of inhibitors are high titer inhibitors, and the remainder are low titer. Some patients develop transient inhibitors (usually low titer inhibitors that never exceed a titer of 5 BU/ml and disappear spontaneously with time (2). The development of inhibitors is associated with changes in the clinical picture with major effect on bleeding control, arthropathy status and overall quality of life. Patients with mild or moderate hemophilia may change to severe clinical behavior because of increase in factor clearance. Patients with inhibitors are resistant to the replacement therapy and thereby their bleeding symptoms become difficult to control and require either large doses of FVIII/IX or

During almost 50 years many studies have addressed different aspects of inhibitors issue from risk factors to diagnosis and management of patients who developed these antibodies.

Coagulation factor inhibitors can be divided to neutralizing antibodies that result in inactivation of the factor and non-neutralizing (i.e. non-inhibitory) antibodies that target non-functional epitopes on FVIII. The non-neutralizing antibodies become clinically relevant if they result in accelerated clearance of the transfused clotting factor (3) . Both types can by classified as:

**Causes and Diagnosis**

*King Faisal Specialist Hospital and RC*

Tarek M. Owaidah

*Saudi Arabia*

Li, C.C. (1976). *First Course in Population Genetics*, Boxwood, Pacific Grove.

### **Hemophilia Inhibitors Prevalence, Causes and Diagnosis**

Tarek M. Owaidah *King Faisal Specialist Hospital and RC Saudi Arabia*

#### **1. Introduction**

16 Hemophilia

66 Hemophilia

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*National Academy of Sciences*, 97 : 7360-7365.

Hemophilia is a bleeding disorder that results from genetic alteration in production of coagulation factors that are important to maintain hemostasis. The commonest type is hemophilia A due to deficiency of factor VIII (FVIII), which is important zymogen co factor for clot formation. Hemophilia A is an X-linked disease that affects males at prevalence of 1:5000-10000. Hemophilia B is due to deficiency in factor (FIX) but less common with prevalence of 1:34,500 males. It is inherited also as X- linked. Although both disorders are rarely observed; they can be very serious (life threatening) and costly for families and countries. Treatment of hemophilia is based on replacement of the deficient factor. Two types of factor concentrates are available, plasma derived (pdFVIII/IX) and recombinant (rFVIII/IX) which are associated with variable incidence of inhibitor formation rates.The development of inhibitor is the most serious and challenging complication of hemophilia treatment with the enormous economic burden (1). FVIII inhibitors are immunoglobulin IgG (IgG1 and IgG4) antibodies that neutralize FVIII procoagulant activity in plasma. Inhibitors are usually classified according to their levels in plasma as a "high-titer" inhibitors, those with the highest activity >5 Bethesda Units (BU)/ml or a low-titer inhibitor type. In hemophilia A aproximately 60-70% of inhibitors are high titer inhibitors, and the remainder are low titer. Some patients develop transient inhibitors (usually low titer inhibitors that never exceed a titer of 5 BU/ml and disappear spontaneously with time (2). The development of inhibitors is associated with changes in the clinical picture with major effect on bleeding control, arthropathy status and overall quality of life. Patients with mild or moderate hemophilia may change to severe clinical behavior because of increase in factor clearance. Patients with inhibitors are resistant to the replacement therapy and thereby their bleeding symptoms become difficult to control and require either large doses of FVIII/IX or alternative hemostatic therapy with bypassing agents.

During almost 50 years many studies have addressed different aspects of inhibitors issue from risk factors to diagnosis and management of patients who developed these antibodies.

#### **2. Type of factor inhibitors**

Coagulation factor inhibitors can be divided to neutralizing antibodies that result in inactivation of the factor and non-neutralizing (i.e. non-inhibitory) antibodies that target non-functional epitopes on FVIII. The non-neutralizing antibodies become clinically relevant if they result in accelerated clearance of the transfused clotting factor (3) . Both types can by classified as:

Hemophilia Inhibitors Prevalence, Causes and Diagnosis 69

Several risk factors for inhibitor formation have been hypothesized. Identification of these factors may help to predict inhibitor develoment and to choose the treatment approach minimising the potential risk in particular patient. The risk of developing inhibitors varies throughout the lifetime of a patient with haemophilia, with historical evidence suggesting that most of inhibitors develop during childhood before reaching the age of 12 years (13). The risk factors interact with each other and can be classified as "non-modifiable" and

These patient related factors that may enhance the risk of inhibitor development include a high-risk hemophilia genotype, co-stimulatory genotype–immunogenotype interactions,

Mutations in FVIII are major risk factors of inhibitor development predominantly in patients with severe form of disease (18). Several gene defects that increase the risk of factor VIII/IX inhibitors have been identified. Some mutations (so called null mutations) result in severe molecular defects with complete failure of FVIII or FIX proteins synthesis. High risk mutations include multi-domain mutations, large deletions/insertions and nonsense mutations which represent approximately 8% of all mutations in severe haemophilia A, as well as the intron-22 inversion with a prevalence of around 50%. The inhibitor formation in patients with high risk mutations ranges between 25% (intron 22 inv) and 60%-80%

Small deletions, missense and splice site mutations result in partial absence of FVIII protein and their prevalence in severe hemophilia is approximately 35% (19). Also in haemophilia B the genotype is a strong determinant of inhibitor risk; patients with gene deletions or rearrangements are at high risk of inhibitor formation. These mutations are present in approximately 50% of inhibitor patients, while the frameshift, premature stop, or splice-site mutations are present in approximately 20% of patients with inhibitor of FIX. The missense mutations, which constitute the majority of genotypes in haemophilia B are at very low risk of inhibitor formation (3,20). The prevalence of inhibitors in patients with haemophilia B

The discordance of inhibitor development and the type of mutation has been observed in patients with the same mutation of the F8 or F9 gene, including the siblings, suggesting the involvement of other genetic and environmental risk factors that may prevent or facilitate

Polymorphisms of the immune response genes, including the genes encoding the major histocompatibility complex (MHC) class II system, tumor necrosis factor-α(TNF-α), interleukin-10 (IL-10) and cytotoxic T-lymphocyte antigen-4 (CTLA-4), have been suggested to be the factors contributing to the risk of inhibitor (22,23,24,25). Some specific types of (HLA) genes may also be implicated in increasing the risk of inhibitor development (26). Ethnicity was also shown to play a role in development of inhibitors. African-Americans and Latinos with haemophilia A have higher inhibitor risk than Caucasians with prevalence of inhibitors in Black patients with hemophilia A twice of white patients (27,28). The estimated incidence of new inhibitors in Finnish patients was 10.3 per thousand patient years (29). In another study on inhibitors in Japanese populations the prevalence was as high as 29.7% (30). In a close population the prevalence of inhibitors in Chinese was as low

**4. Factors affecting development of inhibitors**

environmental, or so called "modifiable" risk factors.(14)

ethnicity and positive family history [15,16,17].

(multidomain mutations and large deletions) (17,18,).

and null mutations ranges from 6–60% (17).

inhibitor formation (21).

**a) Non modifiable risk factors.**

Occasionally, alloantibodies may be mistaken for autoantibodies. This occurs when an individual with a clinically silent mutation in FVIII (for example, a B-domain mutation) is exposed to wild-type FVIII. (4)

#### **3. Prevalence of inhibitor formation**

The overall prevalence of inhibitors is up to 30% in patients with hemophilia A and up to 5% in those with hemophilia B (9). Inhibitors are reported rarely in other coagulation factor deficiencies. Data on 294 individuals with deficiencies of FII, FV, FVII, FX, FXIII from North American Rare Bleeding Disorder Registry reported only 3% of patients with FV and FXIII deficiency who developed inhibitors following infusion of FFP and FXIII concentrate (10).

Risk factors for inhibitor development can be patient related (genetic, ethnicity or immune system), treatment related (type of product, exposure to FVIII/IX in terms of the age at the first treatment, treatment duration and intensity) or diagnostics related (type and sensitivity of test detecting the inhibitor, frequency of inhibitor testing). There are differences between the prevalence and incidence of factor VIII/IX inhibitors. Earlier studies reported consistently the incidence of inhibitor in the range of 25%–32%, although the prevalence eventually fell to approximately 12% as some antibodies disappeared over time (3). Some reports used both terminologies interchangeably which could be explained by difficulty to investigate inhibitor incidence due to the need for high patient number in a relatively uncommon disease. However, early studies were often undertaken on selected patients populations, using different assays for inhibitor detection and being mostly one-off studies on the proportion of inhibitors in particular patient population at a given time (11).

In several cohort studies an incidence rate of new inhibitors (number of new cases/ population at risk x the time at which new cases were ascertained) was determined in the absence of new product exposure with different incidence results. As an example, Kempton et al (2006) reported incidence rate of factor VIII inhibitors of 2.14 per 1000 person-years (12).

#### **4. Factors affecting development of inhibitors**

Several risk factors for inhibitor formation have been hypothesized. Identification of these factors may help to predict inhibitor develoment and to choose the treatment approach minimising the potential risk in particular patient. The risk of developing inhibitors varies throughout the lifetime of a patient with haemophilia, with historical evidence suggesting that most of inhibitors develop during childhood before reaching the age of 12 years (13). The risk factors interact with each other and can be classified as "non-modifiable" and environmental, or so called "modifiable" risk factors.(14)

#### **a) Non modifiable risk factors.**

68 Hemophilia

a. alloantibodies, those that develop in hemophiliacs exposed to exogenous FVIII or FIX. Most FVIII alloantibodies are directed against epitopes in the A2 and A3-C1 domains of FVIII. This binding interferes with the assembly of the FVIII-FIX complex. Antibodies directed against C2 domain affect the binding of FVIII to phospholipid and von Willebrand factor (vWF) and interfere with cleavage of FVIII by thrombin and FXa. In vitro the inactivation of factor VIII is time, temperature and pH dependent (4). Alloantibodies have type 1 reaction kinetics, which means that all FVIII added to

b. autoantibodies, those that suddenly appear in persons with normal F8 gene and previously normal plasma levels of FVIII, causing so called "acquired hemophilia". These inhibitors occur predominantly in the elderly patients, patients with autoimmune, inflammatory process and lymphoprolipherative disorders, and rarely in association with pregnancy (5,6 ) and result in serious bleeding manifestation with a high morbidity and mortality of 6%-20%. Currently 70%-80% of cases of acquired hemophilia are successfully treated with immunosuppressive therapy (7,8). In vitro FVIII autoantibodies present type 2 reactive kinetics with exponential decrease of FVIII, while even at a high titre of inhibitor some residual activity of factor may

Occasionally, alloantibodies may be mistaken for autoantibodies. This occurs when an individual with a clinically silent mutation in FVIII (for example, a B-domain mutation) is

The overall prevalence of inhibitors is up to 30% in patients with hemophilia A and up to 5% in those with hemophilia B (9). Inhibitors are reported rarely in other coagulation factor deficiencies. Data on 294 individuals with deficiencies of FII, FV, FVII, FX, FXIII from North American Rare Bleeding Disorder Registry reported only 3% of patients with FV and FXIII deficiency who developed inhibitors following infusion of FFP and FXIII

Risk factors for inhibitor development can be patient related (genetic, ethnicity or immune system), treatment related (type of product, exposure to FVIII/IX in terms of the age at the first treatment, treatment duration and intensity) or diagnostics related (type and sensitivity of test detecting the inhibitor, frequency of inhibitor testing). There are differences between the prevalence and incidence of factor VIII/IX inhibitors. Earlier studies reported consistently the incidence of inhibitor in the range of 25%–32%, although the prevalence eventually fell to approximately 12% as some antibodies disappeared over time (3). Some reports used both terminologies interchangeably which could be explained by difficulty to investigate inhibitor incidence due to the need for high patient number in a relatively uncommon disease. However, early studies were often undertaken on selected patients populations, using different assays for inhibitor detection and being mostly one-off studies on the proportion of inhibitors in particular patient population at a

In several cohort studies an incidence rate of new inhibitors (number of new cases/ population at risk x the time at which new cases were ascertained) was determined in the absence of new product exposure with different incidence results. As an example, Kempton et al (2006) reported incidence rate of factor VIII inhibitors of 2.14 per 1000 person-years (12).

haemophilia plasma is inhibited lineary.

be detectable. (4)

concentrate (10).

given time (11).

exposed to wild-type FVIII. (4)

**3. Prevalence of inhibitor formation**

These patient related factors that may enhance the risk of inhibitor development include a high-risk hemophilia genotype, co-stimulatory genotype–immunogenotype interactions, ethnicity and positive family history [15,16,17].

Mutations in FVIII are major risk factors of inhibitor development predominantly in patients with severe form of disease (18). Several gene defects that increase the risk of factor VIII/IX inhibitors have been identified. Some mutations (so called null mutations) result in severe molecular defects with complete failure of FVIII or FIX proteins synthesis. High risk mutations include multi-domain mutations, large deletions/insertions and nonsense mutations which represent approximately 8% of all mutations in severe haemophilia A, as well as the intron-22 inversion with a prevalence of around 50%. The inhibitor formation in patients with high risk mutations ranges between 25% (intron 22 inv) and 60%-80% (multidomain mutations and large deletions) (17,18,).

Small deletions, missense and splice site mutations result in partial absence of FVIII protein and their prevalence in severe hemophilia is approximately 35% (19). Also in haemophilia B the genotype is a strong determinant of inhibitor risk; patients with gene deletions or rearrangements are at high risk of inhibitor formation. These mutations are present in approximately 50% of inhibitor patients, while the frameshift, premature stop, or splice-site mutations are present in approximately 20% of patients with inhibitor of FIX. The missense mutations, which constitute the majority of genotypes in haemophilia B are at very low risk of inhibitor formation (3,20). The prevalence of inhibitors in patients with haemophilia B and null mutations ranges from 6–60% (17).

The discordance of inhibitor development and the type of mutation has been observed in patients with the same mutation of the F8 or F9 gene, including the siblings, suggesting the involvement of other genetic and environmental risk factors that may prevent or facilitate inhibitor formation (21).

Polymorphisms of the immune response genes, including the genes encoding the major histocompatibility complex (MHC) class II system, tumor necrosis factor-α(TNF-α), interleukin-10 (IL-10) and cytotoxic T-lymphocyte antigen-4 (CTLA-4), have been suggested to be the factors contributing to the risk of inhibitor (22,23,24,25). Some specific types of (HLA) genes may also be implicated in increasing the risk of inhibitor development (26).

Ethnicity was also shown to play a role in development of inhibitors. African-Americans and Latinos with haemophilia A have higher inhibitor risk than Caucasians with prevalence of inhibitors in Black patients with hemophilia A twice of white patients (27,28). The estimated incidence of new inhibitors in Finnish patients was 10.3 per thousand patient years (29). In another study on inhibitors in Japanese populations the prevalence was as high as 29.7% (30). In a close population the prevalence of inhibitors in Chinese was as low

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