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ChE 359 Homework #3
Solutions prepared by Chris Singer
1.
(15 points) A protein may contain several cysteines, which may pair together to
form disulfide bonds.
If there is an even number
n
of cysteines,
n/2
disulfide
bonds can form.
Derive an expression for the number of different possible
disulfide pairing arrangements.
We first number the individual cysteines along
the chain.
The first cysteine along the chain (doesn’t matter if you start with
#1) can bond to any of the other
n
−
1
.
Two cysteines are then taken out of
consideration for future bonding.
The third cysteine can then bond to any of
the other
n
−
2
( )
−
1
.
The fifth cysteine can then bond to any of the other
n
−
4
( )
−
1
, and so on until all
bonds are formed.
Thus, the total possible
number
of
disulfide
pairing
arrangements
is
given
by:
W n
( ) =
n
−
1
( )
×
n
−
3
( )
×
n
−
5
( )
×
×
1
.
You can also derive this result by taking all the possible sequences of the
(distinguishable) cysteines. There are
n
!
, where each pair represents a bond.
To account for the fact that the order of the pair (12 vs 21) doesn't matter
and that it doesn't matter where in the sequence the pairs occur, you divide by
2
n
/ 2
and
n
2
Λ
Ν
Μ
Ξ
Π
Ο
!
respectively. This would give the expression:
W n
( ) =
n
!
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 Spring '10
 CynthiaLo
 Mole, Kinetics, the00

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