Linear and Nonlinear Models of Heredity for Blood Groups and Rhesus Factor

INTRODUCTION

Blood groups are distinguished by the blood’s antigenic properties. These properties are determined by the substances found on the surface of the red blood cells. There are approximately 200 blood group substances identified and categorized into 19 distinct systems. The most common system is the ABO system. The human ABO blood group was discovered by Karl Landsteiner in 1900 (Landsteiner, 1900) and its mode of inheritance through multiple alleles at a single generic locus was established by Felix Bernstein a quarter century later (Bernstein, 1925). The ABO blood group antigens appear to have been important throughout our evolution because the frequencies of different ABO blood types vary among different populations, suggesting that a particular blood type conferred a selection advantage.

There are three alleles or versions of the blood type gene: A, B and C. Since, humans are diploid organisms (meaning we carry a double set of chromosomes-one from each parent), blood types are determined by two alleles (Table 1). There are six possible combinations of such alleles: AA, BB, OO, AB, OA and OB. In genetic terms, these combinations are called genotypes and they describe the genes that an offspring inherited from his parents.

Table 1:	The ABO blood system

Table 2:	ABO phenotype in the offspring

The expression of the O allele is recessive to that of A and B, which are said to be co-dominant. Thus, the genotypes AO and AA express blood type A, BO and BB express blood type B, AB expresses blood type AB and OO expresses blood type O.

The ABO blood group antigens are encoded by one genetic locus, the ABO locus, which has three alternative (allelic) forms- A, B and O. It is well known that the heredity of blood groups behaves according to the Mendelian rules. A child receives one of the three alleles from each parent, giving rise to six possible genotypes and four possible blood types (phenotypes) (Table 2).

Offspring blood type is established by specific genes inherited from his parents. He receives one gene from his mother and one from his father; these two combine to establish his blood type.

Table 3:	Offspring’s blood type

Table 4:	Offspring’s Rhesus genes

As mentioned above, the heredity of blood groups behaves according to the Mendelian rules (Table 3).

According to Table 3, a group AB person cannot be the parent of a group O child, nor can a group O person be the parent of a group AB child. However, transmission of blood group AB in a family as if by a single chromosome or allele, instead of by two separate chromosomes or genes were reported and this led to the discovery of a very rare blood group called cis-AB. The ABO groups are inherited through multiple alleles at one locus, as seen by Bernstein nearly 80 yeas ago (Bernstein, 1925). Now concerning the inheritance of the ABO groups, the precise mechanism may change, although the triple allele theory of Bernstein is adequate for all practical purposes. The ABO blood group locus includes some sites where mutation can occur and between which, as an extreme rarity, crossing over can happen. The Rhesus system is the second most significant blood group system in human blood transfusion. Individuals either have, or do not have, the Rhesus factor (or Rh(D) antigen) on the surface of their red blood cells. This is usually indicated by Rh⁺ (does have the Rh(D) antigen) or Rh¯ (does not have the antigen) suffix to the ABO blood type.

A child inherits two Rhesus genes, one from each parent, where gene D corresponds to a positive Rhesus factor and gene d corresponds to a negative Rhesus factor (Table 4).

Offspring are Rhesus negative if they have inherited a d gene from each parent (d, d) and offspring are Rhesus positive if they inherited a D gene from both parents. If offspring have inherited a Rhesus positive gene D and a Rhesus negative gene d, they are most likely to be Rhesus positive as the D gene is more dominant as compared to the d gene. Hence, it is possible to have a Rhesus negative child and a Rhesus positive father. According Table 5, a husband and wife with negative Rhesus factor cannot be the parent of a positive Rhesus factor child. However, collected data shows the existence of families that contradict Mendel laws.

Table 5:	Offspring’s Rhesus factor

Thus, blood groups and Rhesus factor of parents do not determine unambiguously their offspring’s blood group and Rhesus factor (Table 3, 5). The transmission of blood group and its Rhesus factor from parents to their offspring are random and events that contradict Mendel laws increase this randomness. To study these transmissions we consider two types of stochastic modeling:

The theory of Markov chains is well known and describes a linear model.

QUADRATIC STOCHASTIC OPERATORS

Quadratic stochastic operators were first introduced by Bernstein (1924). Such operators frequently arise in models of mathematical genetics (Jenks, 1969; Kesten, 1970; Lyubich, 1971; Reed, 1997; Ganikhodjaev, 1999). Consider a biological population, such as a community of organisms that is closed with respect to reproduction. Assume that each individual in this population belongs to precisely one species i = 1,...,m. Below, we consider the scale of blood type with m = 4 and Rhesus factor with m = 2. The scale of species is such that the species of the parents i and j unambiguously determines the probability of every species k .for the first generation of direct descendants. Denote this probability, that is to be called the heredity coefficient, by p_ij,k. It is obvious that p_ij,k≥0 and

Assume that the population is so large that frequency fluctuations can be neglected. Then, the state of the population can be described by the m-tuple (x₁,x₂,...,x_m) of species probabilities, where, x_k is the fraction of the species k in the total population. In the case of panmixia (random interbreeding) the parent pairs i and j arise for a fixed state x =(x₁,x₂,...,x_m) with probability x_ix_j. Hence, the total probability of the species k in the first generation of direct descendants is defined by:

(1)

Let,

be the (m-1) dimensional canonical simplex in R^m. The transformation V : S^m-1⇒S^m-1 is called a quadratic stochastic operator if:

(2)

where, a: , b: and c:

for arbitrary i,j, k = 1,...,m.

Note that the condition (b) is not overloaded, since otherwise one can determine new heredity coefficients:

while preserving the operator V:

(3)

The set of transformations Eq. 2 or 3, which describes a model of heredity is called a quadratic stochastic operator. This model of heredity is uniquely determined by heredity coefficients:

Assume {x(k): k = 0,1,2,...} is the trajectory of the initial point x∈S^m-1, where, x(k+1) = V(x(k)) for all k = 0,1,2,... with x(0) = x.

To investigate limit behavior of trajectories and fixed points of the quadratic stochastic operator V, (2) plays an important role in many applied problems.

A fixed point of a quadratic stochastic operator V is a point x =a where V(a) = a.

A quadratic stochastic operator V is called regular if, for each initial point x∈S^m-1, the limit:

exists. Note that the limit point is a fixed point of a quadratic stochastic operator V.

Thus, the fixed points of a quadratic stochastic operator describe limit or long run behavior the trajectories of any initial points.

Let, us consider the linearization of the Eq. 3 near the fixed point a. Assume that a is an internal fixed point. We write y = x-a where, x∈S^m-1 implies:

Then, the evolution equation for y becomes:

where, J is a Jacobian that is a matrix whose entries are:

and the approximation is valid as long as y remains small.

Proposition 1: A Jacobian J, of quadratic stochastic operator V at a fixed point is a double- column stochastic matrix.

Proof: Direct calculus proves this statement.

Corollary: If J is a Jacobian of quadratic stochastic operator V at a fixed point, then:

All statements are simple reformulations corresponding to properties of stochastic matrices. J. The entries of the corresponding eigenvector b1 are all.

RHESUS FACTOR TRANSMISSION

Firstly, we consider a linear model of transmission, namely, the Markov chain. A Markov chain describes transmission, on some scale, from one of the parents to their offspring of the same gender. Note that only for such Markov chains, can one study their limiting distribution

or long run behavior. Below, we consider two Markov chains: first, a Markov chain that describes a transmission of Rhesus factor from fathers to their sons and second, one that describes transmission of Rhesus factor from mothers to their daughters. For the collected data, let Ns(Fx) be the number of sons of fathers Fx (fathers with Rhesus factor X) and N^Y_S (Fx) be the number of sons with Rhesus factor Y of fathers Fx where, X, Y∈{+,-}. Then:

To describe the transmission of Rhesus factor from fathers to their sons, we need to find the probability P_xy that a son will inherit a Rhesus factor Y from a from a father with Rhesus factor X, where, X,Y∈{+,-}. Thus, we denote:

Then, according to the collected data the transmission probability matrix of the first Markov chain has the form:

(4)

and the transmission probability matrix of the second Markov chain, which describes the transmission of Rhesus factor from a mother to her daughters, has the form:

(5)

Both Markov chains are regular with limiting distribution:

for the first Markov chain and

for the second one.

Now we consider nonlinear model of transmission, namely quadratic stochastic operators. Let a set of species be a set of Rhesus factor {1, 2}, where, 1 denotes positive Rhesus and 2 denotes negative Rhesus. To describe the transmission of Rhesus factor from parents to their offspring, we need to find the probability P_XY,Z that a child receives Rhesus factor Z from a father with Rhesus factor X and a mother with Rhesus factor Y, where, X, Y, Z∈{1,2}. Let, N(Fx,MY) be the number of offspring of fathers Fx and mothers My (fathers with Rhesus factor X and mothers with Rhesus factor Y) and N^Z(Fx,My) be the number of offspring with Rhesus factor Z of fathers Fx and mothers My. Then the transmission probability P_XY,Z is defined as:

(6)

Let,

be the heredity coefficients for i, j, k∈ {1,2}.

For the data collected and according to Eq. 6, we have the following:

(7)

where,

and the corresponding quadratic stochastic operator has the form:

(8)

where, x₁ is the fraction of the population with positive Rhesus factor and x₂ is the fraction of the population with negative Rhesus factor.

The Eq. 8 has single fixed point:

(9)

The Jacobian of the quadratic stochastic operator Eq. 8 at the fixed point has following form:

with eigenvalues λ₁ = 0.685 and λ₂ = 2. Then, the fixed point Eq. 9 is stable and any trajectory of the quadratic stochastic operator Eq. 8 converges to the fixed point Eq. 9. Thus, the quadratic stochastic operator Eq. 8 is a regular.

BLOOD TYPE TRANSMISSION

Now, let us consider transmission of blood type. As above, first we study the transmission of blood type from a father to his sons.

Let, N_S(F_x) be the number of sons of fathers F_x (fathers with blood group X) and N^Y_S(F_x) be the number of sons with blood group Y of fathers F_x where, X, Y∈{A, B, AB, 0}.Then .

The state of the population can be described by the quadruple (x_A, x_B, x_AB, x_O) where, x_A is the fraction of the population with blood group A. Similarly, fractions x_B, x_AB, x_O, represent blood groups B, AB and O, respectively.

The transmission probabilities matrix:

is defined as:

(10)

For the collected data of 3965 sons, using Eq. 10 we obtained the following transmission probability matrix:

(11)

Remark 1: According to Table 3, we should have P_AB,O = 0 and P_{O, AB} = 0. However, one can see that P_AB,O = 0.195 and P_{O, AB} = 0.040. The Markov chain with transmission probability matrix Eq. 11 has the following limiting distribution:

(12)

Similarly, in the case of transmission of blood types from a mother to her daughters, for the collected data of 5637 daughters, we have the following transmission probability matrix:

(13)

Remark 2: Note that according to Table 3, we should have P_AB,O = 0 and P_O,AB = 0. However, P_AB,O = 0173 and P_O,AB = 0.024. The Markov chain with transmission probability matrix Eq. 13 has the following limiting distribution:

(14)

Now, consider the transmission of blood groups from parents to their offspring. To describe this transmission we will apply quadratic stochastic operators.

Let a set of species be a set of blood types {A,B,AB,O}. To study the transmission of blood group from parents to their offspring we need to find a probability P_XY,Z that from a father with blood group X and a mother with blood group Z, their child receives the blood group Z, where X, Y, Z∈ {A,B,AB,O}. Let, N(F_X,M_Y) be the number of offspring of fathers F_X (with blood group X) and mothers M_Y (with blood group Y) and N^Z(F_X,M_Y) be the number of offspring with blood group Z of fathers F_X and mothers M_Y.

Then, the transmission probability P_XY,Z is defined as:

(15)

For brevity let’s replace symbols {A,B,AB,O} by {1,2,3,4}, respectively, so that blood group A corresponds to 1, blood group B corresponds to 2, blood group AB corresponds to 3 and blood group O corresponds to 4.

Then, the coefficients of heredity P_XY,Z for i,j,k∈{1,2,3,4} are defined in the following way:

where, X corresponds i, Y corresponds to j and Z corresponds to k.

Generally, the coefficients p_ij,k do not satisfy the condition p_ij,k = p_ji,k.

Let,

For the collected data we obtained:

Remark 3: One can see that, the values of some q_ij,k contradict those in Table 3. Clear examples of this can be seen in q_11,2, q_11,3 q_{44, 3} and so on.

The corresponding quadratic stochastic operator has the following form:

(16)

This quadratic stochastic operator has a single fixed point in simplex S³.

(17)

The Jacobian of the quadratic stochastic operator Eq. 16 at the fixed point has the following form:

with eigenvalues . Then the fixed point Eq. 17 is stable and any trajectory of quadratic stochastic operator Eq. 16 converges to fixed point Eq. 17. Thus, the quadratic stochastic operator Eq. 16 is a regular.