Stochastic Modeling of DNA Fragments Rejoining
Yongfeng Li , Hong Qian , Ya Wang and Francis A. Cucinotta
1
1
2
3
2
USRA, Division of Space Life Sciences, Houston TX, USA,
University of Washington, Seattle, WA, USA,
4
NASA, Lyndon B. Johnson Space Center, Houston TX, USA
Two Important Components for Stochastic Simulation
Abstract
When cells are exposed to ionizing radiation, DNA damage in the form of single strand
breaks (SSBs), double strand breaks (DSBs), base damage or their combinations are frequent events. It is known that the complexity and severity of DNA damage depends on
the quality of radiation, and the microscopic dose deposited in small segments of DNA,
which is often related to the linear transfer energy (LET) of the radiation. Experimental
studies have suggested that under the same dose, high LET radiation induces more small
DNA fragments than low-LET radiation, which affects Ku efficiently binding with DNA
end and might be a main reason for high-LET radiation induced RBE [1] since DNA DSB
is a major cause for radiation-induced cell death. In this work, we proposed a mathematical model of DNA fragment rejoining according to non-homologous end joining (NHEJ)
mechanism. By conducting Gillespie’s stochastic simulation, we found several factors
that impact the efficiency of DNA fragments rejoining. Our results demonstrated that
aberrant DNA damage repair can result predominantly from the occurrence of a spatial
distribution of DSBs leading to short DNA fragments. Because of the low efficiency
that short DNA fragments recruit repair protein and release the protein residue after
fragment joining, Ku-dependent NHEJ is significantly interfered with short fragments.
Overall, our work suggests that inhibiting the Ku-dependent NHEJ may significantly
contribute to the increased efficiency for cell death and mutation observed for high LET
radiation.
n
Reacting Species Distribution: S = Xn, XnE , XnEE , XnrE , Xnr , Xnrr , XnR
Propensity Functions:
F=
n
k1
∗ k2 E∗ E?
EX
n , V Xn Xm ,
V
o
o
k3Xnr∗, k3XnR .
where X also denotes the number of the copies of molecule X.
Gillespie’s Direct Method of Stochastic Simulation
4
3
Emory University, Atlanta GA, USA
Impact Factors of Mean Rejoining Time
The mean rejoining time can be affected by numerous factors. Figures (ac) exhibit a biphasic profile in the mean rejoining time, where jumps occurs
L∗
∗
at L̄ = L and L̄ = 2 . Moreover, (d) shows that the mean rejoining
time is an increasing function of the fraction of short DNA fragments.
(a) Effect of Volume
(b) Effect of Release Rate k3
(c) Effect of Initial Fragments MT
(d) Effect of Short Fragments
By Gillespie’s direct method [2], the stochastic simulation of system (1)
is performed as follows
0. Initialize DNA fragments distribution and propensity functions at t = 0.
k1
S0 = {Xn} ,
F0 =
EXn .
V
X
1. Calculate total propensity function atot =
ak .
ak ∈F
2. End simulation if atot = 0 or proceed otherwise.
3. Pick random variable (r1, r2) satisfying uniform distribution on [0, 1]2.
4. Calculate next reaction time τ and find firing reaction j in the following way
X
X
1
1
ln
τ=
,
ak > r2atot ≥
ak .
atot
r1
1≤k≤j
1≤k≤j−1
5. Update reacting species S and propensity functions F at t = t + τ and return to 1.
Hypothesis
There exist minimum length Lm and critical length L∗ such that a piece of
DNA fragment of length L can
• recruit no repair protein as L < Lm ;
• recruit repair proteins at only one end as Lm ≤ L < L∗ ;
• recruit repair proteins at both two ends simultaneously as L ≥ L∗.
Stochastic Simulation of Fragments Rejoining
Due to the randomness in the biochemical reactions, the kinetics and the
total rejoining time of fragments rejoining process vary from simulations
even if the same initial distribution is applied. Therefore the mean rejoining time is considered.
Mathematical Model
where k1 is the recruitment rate of repair protein, k2 the fragments rejoining rate and k3
the release rate of repair protein residue.
The kinetics of DSB repair is sometimes reflected by the loss of foci observed in the experiment (among many sample cells) at specific time
points. To compare with experimental data for the low and high LET
radiation induced DSB repair provided in [3], we have run 200 samples of
simulation and computed the averaged kinetics of fragments rejoining, as
shown below.
System of Biochemical Reactions

k1

E


X
+
E
−→
X
n

n,


k1

E
EE


X
+
E
−→
X
,

n
n



k2
 X E + X E −→
XR
n
m
n+m,

E
E∗ k2
r∗

X
+
X
−→
X

n
m
n+m,




E∗
E? k2
∗?

X
+
X
−→
X
,

n
m
n+m



k3

r∗ k3
 X R −→
Xn, Xn −→
n
n ≥ Lm,
n > L∗ ,
Lm ≤ n, m ≤ L∗
Lm ≤ n ≤ L∗ < m,
n, m > L∗,
Comparison with Experimental Data
(1)
Xn∗,
where Xn denotes the DNA fragment of length n (also the number of the molecules), E
the repair protein, r the protein residue that prevents recruitment of new repair protein
on one end of the fragment, R the protein residue that prevents recruitment of new
repair protein on both ends of the fragment; ∗ and ? may be either E, or r or none.
Conclusion
• The size of nucleus may affect the mean rejoining time, supporting the fact that cells
in the G2/M phase are more sensitive to radiation than cells in the G1 phase.
• The release rate of DNA repair protein residue may affect the mean rejoining time
when more short fragments are present. It is also one of main counts for the biphasic
profile of mean rejoining time.
• The presence of short DNA fragments may affect the mean rejoining time by
interfering the binding efficiency of Ku with DNA.
• Short DNA fragments that are not repairable may be related to the end points cell
death and mutation.
• The presence of short DNA fragments inhibits Ku-dependent NHEJ pathway due to
their low efficiency of binding Ku, that may contribute to the increased efficiency for
cell death and mutation by high LET radiation, to great extent.
• High LET radiation induces more short DNA fragments than low LET radiation of
the same dose, supporting that high LET induces more cell death or mutation than
low LET radiation.
• Our model revealed biphasic kinetics, where long DNA fragments are rejoined by fast
kinetics while short fragments by slow kinetics, consistent to the experimental
observation that DSBs are repaired in fast kinetics followed by slow kinetics.
References
[1] Wang H, Wang X, Zhang P, Wang Y (2008) The Ku-dependent non-homologous
end-joining but not other repair pathway is inhibited by high linear energy transfer
ionizing radiation. DNA Repair, 7(3): 725-733.
[2] Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annual Rev Phys
Chem, 58: 35-55.
[3] Asaithamby A, Hu B, Delgado O, Ding L-H, Story MD, Minna JD, Shay JW and
Chen DJ, Irreparable complex DNA double-strand breaks induce chromosome
breakage in organotypic three-dimensional human lung epithelial cell culture.
Nucleic Acids Res, 39 (13), (2011) 5474-5488.
The 23rd Annual NASA Space Radiation Investigators’ Workshop (2012)