The Multiplication Method

 

Principle of Segregation

During the process of sex cell formation one, and only one, of a pair of alleles for a single gene go into each gamete.  These gametes are formed in equal frequencies. Subsequent fertilization is at random.

The Principal of segregation specifies that an individual who has the heterozygote genotype for the A gene of Aa makes two gametes A and a in equal frequencies ( A = 1/2 and a = 1/2).

Another way to write this is as a gametic array.

Gametic Array for the A gene  = 1/2A + 1/2a = 1 
(the sum of all the gametes adds to one).

Principle of Independent Assortment.

Alleles at one locus loci segregate into gametes independently of alleles at other loci. Such gametes are formed in equal frequencies: Subsequent fertilization is at random.

Now consider an individual with genotype AaBb.  This means that we are considering the genotype for two genes A and B. 

Each gene undergoes segregation.  Each gene has its own gametic array.  The A gene with genotype Aa makes two gametes A and a in equal frequencies and the B gene with genotype Bb makes two gametes B and b in equal frequencies. 

The gametic arrays for the A and B genes as a result of segregation are:

1/2A + 1/2a = 1

1/2B + 1/2b = 1

By the principle of independent assortment, the gametic array for the two genes considered at the same time can be calculated as the product of the gametic array for the A gene and the gametic array for the B gene.

(1/2A + 1/2a) X (1/2B + 1/2b) = 1 x 1

(1/2A*1/2B) + (1/2A*1/2b) + (1/2a*1/2B) +  (1/2a*1/2b) = 1

1/4 AB + 1/4Ab + 1/4aB + 1/4ab = 1

Thus, there are four gametes occurring in equal frequencies as predicted by the Principle of Independent Assortment.

Hint for Multiplying Gametic Arrays
  1. Multiply each term of the second array by each term of the first array.
  2. Add any similar terms.

 

Three or more genes:

The procedure is the same for 3,4 or more genes.  Each gene has a gametic array based on the principle of segregation.  The  gametic arrays for the single genes are multiplied together to get the gametic array for all the genes considered simultaneously. That is, the gametic array due to independent assortment.

Example:
Consider an individual with genotype AaBbCc. The separate gametic arrays would be:
1/2A + 1/2a = 1
1/2B + 1/2b = 1
1/2C + 1/2c = 1
Multiply all three arrays to get the final arrays.  This is most easily done sequentially.

The gametic array for AB is 1/4 AB + 1/4Ab + 1/4aB + 1/4 ab = 1 (calculated above).

  The gametic array for genes A, B and C is:

(1/4 AB + 1/4Ab + 1/4aB + 1/4 ab) *(1/2C + 1/2c) = 1*1

or

1/8ABC + 1/8AbC + 1/8aBC + 1/8abc + 1/8ABc + 1/8Abc + 1/8aBc + 1/8abc

Eight gametes formed in equal frequencies

 

Quick Links for Gametic, Genotypic and Phenotypic Arrays.
 
1 Representing Alleles, Genotype and Phenotypes
2 Multiplication Method; Gametic Arrays
3 Gametic Arrays make Genotypic Arrays
4 Phenotypic Arrays
5 Arrays and Independent Assortment
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