a
e
i
. Show that
h
R
i
= 0 and explain why that makes sense. Then, use the fact that we can
write
h
R
2
i
=
h
N
X
i
=1
N
X
j
=1
a
e
i
·
a
e
j
i
(13)
Explain both the merits and the problems with applying this kind of
analysis to pictures like that of the
E. coli
genome given in ﬁg. 8.6.
(b) As practice with the use of the binomial distribution and to make sure you
feel the
√
N
result given above “in your bones”, we will derive the endtoend
distance in one dimension by using the binomial distribution. Your goal is
to compute the average endtoend distance
h
x
i
and the standard deviation
q
h
x
2
i
. Use the fact that
x
= (
n
r

n
l
)
a
where
n
r
is the number of right
pointing monomers and
n
l
is the number of left pointing monomers. Make a
simple diagram of a onedimensional randomwalk conﬁguration that shows
the endtoend distance and how it depends upon the number of monomers
of each type. Explain your result for both of these averages and their signif
icance for the estimate of the genome size.
HINT: Eliminate
n
l
by using the relation
N
=
n
l
+
n
r
. To proceed, show
that the probability of having
n
r
steps pointing to the right out of a total of
N
steps is
p
(
n
r
,N
) =
N
!
n
r
!(
N

n
r
)!
p
n
r
r
p
n
l
l
,
(14)
where
p
r
is the probability of a right step and
p
l
is the probability of a left
step. Note that later we will invoke
p
r
=
p
l
= 1
/
2, but for now leave this as
is since it will help us with our analysis. Now, to obtain
h
x
i
and
q
h
x
2
i
all
5
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View Full Documentyou need to do is compute
h
n
r
i
and
h
n
2
r
i
. Show that these averages are of
the form
h
n
r
i
=
N
X
n
r
=0
n
r
N
!
n
r
!(
N

n
r
)!
p
n
r
r
p
N

n
r
l
,
(15)
and
h
n
2
r
i
=
N
X
n
r
=0
n
2
r
N
!
n
r
!(
N

n
r
)!
p
n
r
r
p
N

n
r
l
.
(16)
To evaluate these expressions, note that
h
n
r
i
=
p
r
∂
∂p
r
N
X
n
r
=0
N
!
n
r
!(
N

n
r
)!
p
n
r
r
p
N

n
r
l
.
(17)
But the binomial theorem tells us that
(
p
r
+
p
l
)
N
=
N
X
n
r
=0
N
!
n
r
!(
N

n
r
)!
p
n
r
r
p
N

n
r
l
.
(18)
With these tips, you have everything you need to evaluate the averages you
are asked to compute. When you get your result, make sure to connect it to
the result of part (a) and to explain how this can be used to ﬁnd the “size”
of a onedimensional DNA molecule.
(c) Using ﬁg. 8.6, make a best estimate at the number of basepairs in the
E. coli
genome, remembering that the Kuhn length
a
= 300 bp. Compare
your estimate to the observed genome length for
E. coli
and give a thoughtful
discussion of how the estimate might have gone wrong.
6
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 Winter '09
 Fourier Series, Dirac delta function, PBOC, pnr pl −nr

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