Linkage Disequilibirum Can someone please help me understand

Linkage Disequilibirum:

Can someone please help me understand the solutions given to these 2 questions? I cannot understand the logical progression of the equation.

Question 1: Show that the gamete frequencies as a function of D are correct inthe table given.

And question 2: Show that the definition of D given below is consistent with theexpressions for gamete frequencies above by substituting p1p2 + D forx1 and the equivalent expressions for the other gametes into the rightside of the equation. [This involves many cancellations, ending with alone D]

X1X4. X2X3 A1B1 A2B1 A2B2 Gamete: A1B2 Frequency: Frequency p1 p2 D piq2 D q1p2 D q1 q2 D Allele frequencies A1B1: A1: p1 X1 X. (x1 x2)(x1 x3) x1x4 -x2x3 B1: p2 X1 X. -xi +x1x3 x1x2 x2x3 x1x4 X2X3 A2: q1 x1(x1 x2 x3 x4) x2x3 x2x3 x1 B2: q Xa x4 A1B2: (x1 x2)(x2 x4) (x1x4 x2x3) x1x2 x1x4 +x2x4 x1 x2 (x1 x2 x3 x4) 1- 4 x1x4 x2

Solution

Given that-------- D = x₁x₄-x₂x₃

Q(1). Now, Allele frequencies:

So, A₁: p₁ = x₁+x_{2 _____} (2)

_.   A₂: q₁ = x₃+x_{4______} (3)

. B₁: p₂ = x₁+x_{3______} (4)

B₂: q₂ = x₂+x_{4_______} (5)

and x₁,x₂,x₃ and x₄ are frequency. So x₁+x₂+x₃+x₄ = 1 __(1)

so, A₁B₁: p₁p₂+D

Now, put values of p₁=x₁+x₂ , p₂ =x₁+x₃ , D = x₁x₄-x₂x₃

A₁B₁: (x₁+x₂)(x₁+x₃) + (x₁x₄-x₂x₃)

: x₁²+x₁x₃+x₁x₂+x₂x₃+x₁x₄-x₂x₃

   : x₁²+x₁x₃+x₁x₂+x₁x₄

: x₁(x₁+x₃+x₂+x₄) = x₁   ( due to eq. (1))

So, A₁B₂: p₁q₂-D

Put values of p₁= x₁+x_2, q₂ = x₂+x₄, D = x₁x₄-x₂x₃

A₁B₂: (x₁+x₂)(x₂+x₄) - (x₁x₄-x₂x₃)

: x₁x₂+x₁x₄+x₂²+x₂x₄-x₁x₄+x₂x₃

   _:   x₁x₂+x₂²+x₂x₄+x₂x₃

   _:   x₂(x₁+x₂+x₃+x₄) = x₂   (due to eq.(1))

Q(2). D = x₁x₄-x₂x₃

Put values of x₁=p₁p₂+D, x₂=p₁q₂-D, x₃=q₁p₂-D, x₄=q₁q₂+D

D = (p₁p₂+D)(q₁q₂+D) - (p₁q₂-D)(q₁p₂-D)

D = p₁p₂q₁q₂+p₁p₂D+q₁q₂D+D²-p₁p₂q₁q₂+p₁q₂D+q₁p₂D-D²

D = p₁p₂D+q₁q₂D+p₁q₂D+q₁p₂D

D = D(p₁p₂+q₁p₂+p₁q₂+q₁q₂)

D = D(p₂(p₁+q₁)+q₂(p₁+q₁))

D = D(p₁+q₁)(p₂+q₂) ______(6)

Now, from eq.(1) ---

x₁+x₂+x₃+x₄ = 1

(x₁+x₂)+(x₃+x₄) = 1

p₁+q₁ = 1 _____(7) ( by eq. (2) and (3) )

(x₁+x₃)+(x₂+x₄) = 1

p₂+q₂ = 1______(8) ( by eq. (1) and (4) )

Now from eq.(6), eq. (7) , eq.(8) -----

D = D

Gamete:	A1B1	A1B2	A2B1	A2B2
Frequency:	x1	x2	x3	x4
Frequency:	p1p2+D	p1q2-D	q1p2-D	q1q2+D