650
E DNA Splitting by Radiation
Eqs. (E.9) and (E.10) agree precisely with the results of Montroll and Simha 2 who
used a combinatorial argument.
E.2 Fragmentation and Dose
Here, a connection is drawn between the radiation dose given to a sample of
DNA and the cleavage probability. This will allow us to express the fragmentation
numbers in terms of the radiation suffered by the sample. At this stage, a simple but
reasonable target model will be taken in order to make the relationships clear.
Consider dividing the sample into M volumes (‘sites’), each of size v, small
enough so that, if radiation of sufficient energy is deposited into one site surrounding
a location on a DNA molecule, it will cause a cleavage of the DNA. Suppose the
radiation generates m localized deposition events randomly distributed throughout
the sample and of sufficient energy to cause a cleavage. The probability that a given
site is hit exactly k times by the first k events will be (1/M) k . The chance that
the remaining (m − k) radiation events hit the other sites is (1 − 1/M) m−k . But
the k events on a given site may have occurred interspersed in time among the m
depositions events. There are (m − k + 1)(m − k + 2) · · · m ways in which an
ordered set of the k events in the given site could have occurred among the remaining
(m − k) events which did not hit the given site. Any ordering of the k events is
equivalent, so that the number of ways that an unordered set of the k events can
occur is (m − k + 1)(m − k + 2) · · · m/k!. Therefore, the probability that a given
site experiences exactly k hits after m events among M loci is
p k =
m
k
1
M
k
1 −
1
M
m−k
,
(E.11)
where
m
k
is the binomial coefficient. The probability that a given locus is hit one
or more times will be
m
k=1
p k = 1 −
1 −
1
M
m
(E.12)
and the probability that it is hit two or more times will be
m
k=2
p k = 1 −
1 −
1
M
m−1
1 +
(m − 1)
M
.
(E.13)
2 E.W. Montroll, and R. Simha, Theory of depolymerization of long chain molecules, J. Chem Phys
8, 721 (1940).
E DNA Splitting by Radiation
Eqs. (E.9) and (E.10) agree precisely with the results of Montroll and Simha 2 who
used a combinatorial argument.
E.2 Fragmentation and Dose
Here, a connection is drawn between the radiation dose given to a sample of
DNA and the cleavage probability. This will allow us to express the fragmentation
numbers in terms of the radiation suffered by the sample. At this stage, a simple but
reasonable target model will be taken in order to make the relationships clear.
Consider dividing the sample into M volumes (‘sites’), each of size v, small
enough so that, if radiation of sufficient energy is deposited into one site surrounding
a location on a DNA molecule, it will cause a cleavage of the DNA. Suppose the
radiation generates m localized deposition events randomly distributed throughout
the sample and of sufficient energy to cause a cleavage. The probability that a given
site is hit exactly k times by the first k events will be (1/M) k . The chance that
the remaining (m − k) radiation events hit the other sites is (1 − 1/M) m−k . But
the k events on a given site may have occurred interspersed in time among the m
depositions events. There are (m − k + 1)(m − k + 2) · · · m ways in which an
ordered set of the k events in the given site could have occurred among the remaining
(m − k) events which did not hit the given site. Any ordering of the k events is
equivalent, so that the number of ways that an unordered set of the k events can
occur is (m − k + 1)(m − k + 2) · · · m/k!. Therefore, the probability that a given
site experiences exactly k hits after m events among M loci is
p k =
m
k
1
M
k
1 −
1
M
m−k
,
(E.11)
where
m
k
is the binomial coefficient. The probability that a given locus is hit one
or more times will be
m
k=1
p k = 1 −
1 −
1
M
m
(E.12)
and the probability that it is hit two or more times will be
m
k=2
p k = 1 −
1 −
1
M
m−1
1 +
(m − 1)
M
.
(E.13)
2 E.W. Montroll, and R. Simha, Theory of depolymerization of long chain molecules, J. Chem Phys
8, 721 (1940).