<biology knowledge background>:
two Cys(amino acid's name) can form a disulfide bond.
<the question>:
If the eight Cys residues in ribonuclease are allowed to combine at random forming four disulfide bonds, there are 105 different combinations. Show mathmetically this is the case.
To show mathematically that there are 105 different combinations of four disulfide bonds formed from eight cysteine (Cys) residues in ribonuclease, we can use the concept of combination.
First, let's start with the total number of ways to choose four disulfide bonds out of eight Cys residues. This can be calculated using the combination formula:
nCr = n! / (r!(n-r)!)
where n is the total number of items, r is the number of items being chosen, and the "!" denotes factorial.
In this case, we have eight Cys residues, and we want to choose four of them to form the disulfide bonds. So, we can calculate the combination as follows:
8C4 = 8! / (4!(8-4)!)
= 8! / (4!4!)
To simplify the calculation, let's expand the factorials:
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
4! = 4 x 3 x 2 x 1
Now, let's substitute these values into the combination formula:
8C4 = (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / [(4 x 3 x 2 x 1)(4 x 3 x 2 x 1)]
Now, simplify the numerator and denominator:
8C4 = (8 x 7 x 6 x 5) / (4 x 3 x 2 x 1)
= 1680 / 24
= 70
Therefore, there are 70 different combinations of choosing four disulfide bonds out of eight Cys residues. However, this calculation only considers the specific combination without considering the order of the disulfide bonds.
Since the question mentions combining at random, we need to consider the order of the disulfide bonds as well. To do this, we need to calculate the number of ways to arrange four disulfide bonds in a specific order.
The number of ways to arrange four disulfide bonds can be calculated using the permutation formula:
nPk = n! / (n-k)!
where n is the total number of items, and k is the number of items being arranged.
In this case, we have four disulfide bonds that need to be arranged.
4P4 = 4! / (4-4)!
= 4! / 0!
Any number divided by 0! is equal to 1. So, the number of ways to arrange four disulfide bonds is 1.
Now, to find the total number of different combinations with the specific order of disulfide bonds, we need to multiply the number of combinations (70) by the number of ways to arrange the disulfide bonds (1):
Total number of combinations = 70 x 1
= 70
Therefore, mathematically, there are 70 different combinations of four disulfide bonds formed from eight Cys residues in ribonuclease. This means that there are 70 different ways the eight Cys residues can combine at random to form these four disulfide bonds.