STATISTICAL METHODS FOR STUDYING THE HEREDITY OF HAEMOPHILIA IN ANIMAL AND PLANT POPULATIONS

Transcription

1 STATISTICAL METHODS FOR STUDYI THE HEREDITY OF HAEMOHILIA I AIMAL AD LAT OULATIOS C. AHEL M. BOLDEA Banat University of Agricultural Sciences and Veterinary Medicine from Timişoara A. COJOCARIU Ofelia SOFRA Tibiscus University from Timişoara Abstract The present paper proposes a mathematical model for the evolution of haemophilia, which is a hereditary disease transmitted through the X chromosome (one of the two sex-determining chromosomes. The evolutions of the probabilities of the next generations are given by the formulas in Table 5 and Table 0. This evolution was computer-simulated in order to verify the mathematical model. Key words: Haemophilia, chromosome, simulation, algorithm, mathematical model Introduction Random mating (panmixia is an idealized model that best assesses the reality in animal and plant populations. The probabilistic study of mating is also necessary for establishing how certain traits, flaws or diseases are passed on from one generation to another. In haemophilia, the affected genes are the ones that carry the feminine sexual chromosomes, the X chromosomes. In the case of, the disease manifests and is transmitted to offspring if the X chromosome in the group is affected. In the case of fe, the disease appears only if both X chromosomes are affected. If only one of the two X chromosomes is affected, the disease does not become apparent but the respective female is a carrier of the disease and can pass it on to her offspring. In the first case, both X chromosomes are affected, the disease is lethal for fe in their first months of life or even before birth, therefore sick fe will not have offspring. 7 Romanian Statistical Review - Supplement nr. / 05

2 Theoretical background If we mark by X the female chromosome affected, then the following variants are possible for the offspring (Table. The offspring resulting from all possible pairing Table - healthy female - female carrier - sick female - healthy male X Y - sick male We consider a population of fe and, where α is the percentage of sick and β is the percentage of female carriers. (Table. The distribution of male and female population Table umber of fe ( Female population Male population X Y X Y umber of ( Revista Română de Statistică - Supliment nr. / 05 7

3 The pairing is considered to happen one time and at random. The probabilities for each type of offspring,,,, and and each type of pairing are given in The offspring probability for each type of pairing Table Type of pairing Two variants will be analysed. X X X X X Y Variant A. We will consider that all pairs will give birth to two offspring, including the cases where one or both are fe with haemophilia (. Counting the number of for each of the five types of individuals for each of the four types of pairs, we obtain Table. The number of resulting was multiplied by, admitting that for each pair there will be two so that the population remains constant in the second generation. The respective probabilities are obtained by dividing the number of favourable cases to the number of possible cases. 7 Romanian Statistical Review - Supplement nr. / 05

4 Type pairing of o. of favourable offspring ( ( umber of offspring after pairing o. of favourable offspring possible female offspring ( ( o. of favourable offspring ( ( o. of favourable offspring X Y Table possible male offspring ( ( ( ( ( ( ( ( ( ( ( ( 5 TOTAL ( ( [( ] ( ( The corresponding probabilities are calculated with the following relations: ( ( ( ( ( ( ( ( Relations (, (, (, ( represent the probabilities in the second generation (the following generation, actually. In the third generation, the probabilities will be calculated according to the same algorithm as in the previous generation. In practice, these probabilities become percentages of realization. Thus, Table 5 is obtained, which presents the evolution of generations for the respective probabilities in variant A. Revista Română de Statistică - Supliment nr. / 05 75

5 Evolution of probabilities for the next generation, variant A Table 5 eneration fe Table 6 shows that, if in the first generation all are healthy, starting with the second generation there will be some affected by haemophilia and female carriers, as well. Evolution of probabilities if all are healthy in the first generation Table 6 eneration X X fe X Y Table 7 makes it clear that if in the initial generation ( all fe are healthy, then in the second generation ( there will be as many healthy fe as there were healthy in the first generation. The in the second generation will all be healthy. In the third generation we will have female carriers and sick. Evolution of probabilities if all are healthy in the first generation Table 7 eneration X X fe Romanian Statistical Review - Supplement nr. / 05

6 Table 8 presents what happens if in the first generation ( all are sick and all fe are carriers: in the second generation ( all fe are carriers and half of the are sick. Evolution of probabilities if all are sick and all fe are carriers in Table 8 be eneration fe X Y Variant B. We will consider that all pairs have two viable offspring, meaning that if a sick female is born (, she will die soon after, and the pair will have another descendant to replace her. In this case, Table becomes Table 9. Type pairing of umber of favourable umber of offspring umber of favourable possible female umber of favourable umber of favourable X Y Table 9 possible male ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 5 TOTAL ( ( The corresponding probabilities will be the following: ( ( ( (5 Revista Română de Statistică - Supliment nr. / 05 77

7 (6 (7 (8 Thus, we obtain Table 0 of the evolution of generations for the respective probabilities in Variant B. Evolution of probabilities for the next generation, Variant B Tabel nr.0 eneration X X fe X Y ( ( Variants A and B are idealized and are taken as work hypotheses. In real life, not all pairs behave as in either of the two variants, but instead some behave as in A and others as in B. ractical application The software application that simulates this process functions as follows: a population is introduced,000,000 fe of the two types and as well as,000,000 of the two types and X Y, distributed in given percentage: α, β, - α, - β. A randomising algorithm is applied in both populations, to make sure that the pairing is at random. For variant A: Table is determined, with the respective probabilities. For example, for the first pair in the table - (a healthy female and a healthy male, two possibilities are introduced for the offspring: or. Of these two, one is randomly chosen, for instance. Then it is written in a file that contains the of the respective generation. The same is done 78 Romanian Statistical Review - Supplement nr. / 05

8 to all pairs. After that, each total result is divided to the number of fe and to the number of, respectively, in order to obtain the respective percentages from, which are compared to the theoretical probabilities. For each pair, two offspring are chosen, so that the population remains constant in the next generation. In addition, if results for an extraction from Table 5, then it is not counted anywhere. For variant B, the procedure is the same as for variant A, with the following difference: if a random extraction of a descendant results in an, then the extraction is annulled and a new extraction is made, in order to have two viable offspring for each pair. Conclusions Tables and give a few examples in order to show the degree of concordance between the theoretical results and the simulated results. Theoretical results and results of the simulation, variant A Table eneration % % % Fe % % % theoretical practical theoretical practical theoretical practical Revista Română de Statistică - Supliment nr. / 05 79

9 Theoretical results and results of the simulation, variant B Table eneration % % fe % % % % theoretical practical theoretical practical theoretical practical Table and Table show that the theoretical results and the results given by the computer are equal. In conclusion, for haemophilia, the probability evolutions for the next generations are given by the formulae in Table 5 and Table 0. Haemophilia is a hereditary disease, which is transmitted through the X chromosome. The probability evolutions for the next generations are given by the formulae in Table 5 and Table 0. Computer simulation confirms these relations (Table and Table. Bibliography. Anghel, C., M. Boldea A ew Distribution Law in Statistics, Stochastic Analysis and Applications vol. ( - 6, Ed. ova Science ublishers, Inc., ew York, 00. Anghel, C., M. Boldea Simularea pe calculator a unui proces binomial şi poissonian, Revista Română de Statistică, nr. ( - 5, Bucureşti, 000;. Anghel, C., M. Boldea Simularea pe calculator în transmiterea unor boli ereditare, Lucrări ştiinţifice, Facultatea de Agricultură, vol. XI (59-6, Ed. Agroprint Timişoara, 999;. C. anfil enetica sexelor, Editura Dacia Cluj-apoca 98; 5. Chiş, Codruţa, M. Chiş, M. Boldea Calcularea frecvenţelor de echilibru într-o algebră genetică, Cercetări Ştiinţifice, Facultatea de Hortcultură, Seria a VIII a, Biotehnologie şi Biodiversitate, (9-56, Ed. Agroprint, Timişoara, 00; 6. L. Bain, M. Engelhardt Introduction to robability and Mathematical Statistics, Boston 99; 7. T. Crăciun, M. ătraşcu Mecanismele eredităţii, Editura Albatros Bucureşti Romanian Statistical Review - Supplement nr. / 05