A Stochastic Analysis of the Repair of Radiation-Induced Double-Strand Breaks
DNA
SHIOJENN TSENG Department of Mathematics, Tamkang University, Tamsui, Taiwan 25137, Republic of China
JYH-PING HSU Department of Chemical Engineering, National Taiwan Uniuersity, Taipei, Taiwan 10764, Republic of China Received 20 July 1989; revised 9 November I989
ABSTRACT A three-state stochastic model is described for the repair of radiation-induced double-strand breaks (DSBS) in DNA. If irradiated, a site or region in DNA is assumed to be in a potentially damaged state; this site may either become permanently damaged or be repaired after a certain period of time. The result of the analysis of the available experimental data reveals that the present two-parameter model is capable of interpreting the rapid decrease in the number of DSBs in the initial period, which cannot be predicted by previously proposed models. The stochastic analysis yields not only the temporal variation of the mean of the number of DSBs but also its variance, and therefore is a generalization of the conventional deterministic models.
INTRQDUCTION When exposed to an external radiation source, a DNA duplex often experiences lesions in which breaks of both of its strands occur. The so-called double-strand breaks (DSBS) in DNA have been found to be closely related to cell lethality [l-4]. Experimental evidence reveals that cells are capable of repairing this type of damage under appropriate conditions [l, 4-111. Since cell death correlates with the number of unrepaired DSBs, the temporal variation of the number of DSBs in a cell is of fundamental$mportance in evaluating its probability of survival after its exposure to radiation. Reported results concerning the repair of DNA DSBs after a radiation treatment are ample. Most previous studies, however, focused on the experimental aspect of the phenomenon. Relatively little attention was given to the theoretical analysis of the repair kinetics. Payne and Garrett M4THEMATICAL
BIOSCIENCES
OElsevier Science Publishing 655 Avenue of the Americas,
100:21-31
(1990)
Co., Inc., 1990 New York, NY 10010
21 00255564/90/$03.50
22
SHIOJENN
TSENG
AND JYH-PING
HSU
[12] derived a three-state stochastic representation describing the evolution of the joint probability distribution of the number of sites in each state. In their model, a site in a DNA duplex can exist either in an undamaged state or in one of the two damaged states. In special cases, this model leads to various models proposed by previous investigators [13, 141. Payne and Garrett [12] correctly pointed out the necessity of including the statistical fluctuations associated with both direct and indirect effects of the radiation field for studying the effect of radiation on cell survival. However, since the second damaged state is an absorbing state, to which all sites eventually go, care must be taken in employing their model. Based on the step-by-step removal of individual DSBs inside an autonomous repair unit, Pridal and Iokajicek [15] formulated a kinetic model. This model, although giving a different view of DSB repair kinetics, fails to interpret the rapid drop in the number of DSBs in the initial period of a typical radiation-induced DSB experiment. Thames [16] discusses the repair kinetics for both fractionated and continuous irradiations. Tobias [171 assumes that the repair rate of lesions contains a linear self-repair term and a quadratic cooperative term. This model is reformulated by Albright [18] in a stochastic manner. The lethal and potentially lethal model suggested by Curtis [19] hypothesizes that there exist two different types of lesions, irreparable lesions and repairable lesions (see also [14]). The correct repair mechanism is assumed to be first-order, and the misrepair mechanism is assumed to be secondorder. Goodhead [201 points out that the repair efficiency can be dose-dependent owing to inactivation of enzymes. The repair rate of lesions is assumed to be linearly dependent on the number of lesions in the so-called saturable repair. By defining five possible states for a DNA molecule, Ostashevsky [21] was able to obtain expressions for the dose and time dependences of the surviving fraction, the number of unrepaired DSBs, and the number of prematurely condensed chromosome fragments. Two types of repair mechanisms are considered, noncooperative and cooperative; the rate expressions are essentially linear. The aim of the present study is to modify the stochastic model of Payne and Garrett [12] in such a way that the status of a site in DNA is defined more realistically, and to provide a way of describing the rapid drop in the number of DSBs in the initial period of a radiation-induced DSB experiment.
MODELING Let us consider the DSB repair mechanism shown in Figure 1. When irradiated, a site or region in DNA is assumed to be in a potentially damaged state. This site can either become permanently damaged or be repaired after a certain period of time. Denote the random variables Nd
REPAIR OF RADIATION-INDUCED
23
DNA DSB
and N, as the number of permanently damaged sites and the number of repaired sites, respectively; specific values of Nd and N, will be represented by nd, nd = 0,1,2,. .., N,, and n,, n, = 0,1,2,. .., Nc, where N,, is the number of potentially damaged sites initially. Note that if the values of nd and n, are given, the number of potentially damaged sites, npr can be evaluated by np = N,, - nd - n,. Let P(nd,nr, t) be the probability that there are nd permanently damaged sites and n, repaired sites at time t. The probability that given nh permanently damaged sites and n; repaired sites at time s there will be nd damaged sites and n, repaired sites at time t (t > s> is denoted as p(nd, n,, 1; nh,n;, s). It is assumed that (a) p(nd + 1, n,, t + At; nd, n,, t) = Ad(nd, n,) At + o(At) (b) p(n&n,.+l,t + At;nd,nr,t)= hr(nd,n,)At + o(At) (c) p(n& n,, t + At&, n,, t) = 1 -[h&z& n,.)+ hr(n& n,)] At + o(At> where 0, t + At) denotes an infinitesimally small but finite time interval, hd(nd,nl) and Ar(nd,n,) are transition intensities, and o(At) satisfies the condition lim ~~,a[o(At)/Atl= 0. Assumption (a) states that the number of permanently damaged sites increases by one in the time interval (t, t + At) with probability A&d, n,) At + o(At). Similarly, (b) assumes that the probability that the number of repaired sites increases by one in the time interval (t, t + At) is Ar(nd, n,) At + o(At). All other possible transitions are assumed to have an infinitesimal probability of order o(At), as stated in (c). The above assumptions lead to the following equation: dP(ndrnr,t)
dt
=Ad(nd-l,nr)P(nd-l,nr,t) +A,Cnd,n,-l)p(nd,n,-l,t) -
[‘bhf,‘b)
FIG. 1. Schematic representation present study.
+‘bbdt%)]
P(n&nr,t)
of the DSB repair mechanism considered
(1)
in the
24
SHIOJENN TSENG AND JYH-PING HSU
or
dp(npf)=(D~l-l)hd(nd,n,)p(nd,n,,t) +(DnS1-l)Ar(nd,nr)P(nd,nr,f),
(2)
where p(nd,n,,t) is zero if either nd or n, is negative. The step operator D, when operating on a function of n, f(n), possesses the following properties [22]:
Qdf(n>l =f(n +I)
(3)
D;‘[f(n>l= f(n -1).
(4)
and
Equations (3) and (4) suggest that, for arbitrary have (see Appendix)
functions
f(n) and g(n), we
where ni = nd or n, = nr, and f - (N-?)
(14) + kz(No(Nd) - (N,N,> -(N:)), (15)
d(N,2>= k2[ -2(N,2)-2 dt
(N,N,)+(2No-l)(W)-(N,)+No].
(16)
26
SHIOJENN TSENG AND JYH-PING HSU
The solution to these equations, (N,N,) = (N,2) = 0, is
subject to the initial
conditions
(Ni)
=
(k,+k,N,-(k2-k,+2kINo)exp[-(k,+kz)t] (N,2) =(k, klNo + k,)*
+(klNo-kl)exp[-2(kl+k2)~]}, (17)
(iv%)
=
k,k,No(No-1){1-2exp[-(k,+k2)t]+exp[-2(kI+k2)t]], Ck, + k,)2
k2No {k,+k2No-(k,-k2+2k2No)exp[-(k,+k,)t] (k,+ k,)*
(N,2)=
+(k2No-k,)exp[-2(kI+k2)t]j.
The variance of Nd, (( Nd)), and the variance by, respectively,
of N,, ((N,.)),
(19)
are evaluated
((Nd)) = (N,z)- C&j2
(20)
C(N)> = V,2) - (W2.
(21)
and
Since Ndsb = No - N,, we have
((Ndsd) = C(W)>-
(22)
DISCUSSION The applicability of the present stochastic model is examined through analysis of the available experimental data. Frankenberg-Schwager et al. [7] performed the DSB repair experiment using yeast cells Succharomyces cerevisiae 211*B in stationary phase. Four levels of radiation dose were used; the temporal variation of the number of DSBs per cell was recorded. The averaged numbers of DSBs per cell, along with the values predicted by Equation (13), are illustrated in Figures 2 and 3. Also shown in these figures is an approximate 95% confidence band estimated by (Ndsb) f ‘I2 . The estimation of the value of the adjustable parameters, 1.96((&,)) k, and k,, is based on a least squares deviation criterion. As seen from
REPAIR OF RADIATION-INDUCED
27
DNA DSB (0) 2400 Gy
-----------40 -
01
----__ I
I
I
0
-_-__---__-
I
I
I
I
I
(b)
1500 Gy
~--___--_______---____~_ CI --------A_--_--_-----___
0
0
I
I
I
10
20
30
_
I
I
I
40
50
60
I
70
Time (hr) FIG. 2. The experimental data of Frankenberg-Schwager et al. [7] and the results predicted by the present model. (a) k, = 1.6131 X lo-‘, k, =5.7433X lo-*; (b) k, = 2.6081 x 10-2, k, = 1.1098X 10-l.
Figures 2 and 3, the present stochastic model successfully predicts the transient behavior of the DSB repair phenomenon. Intuitively, the transition intensity of a site from the potentially damaged state to either the damaged state or the repaired state depends on the radiation dose applied. This is because the degree of damage of a site varies with the level of radiation dose, as justified by the variation in the values of the adjustable parameters k, and k, at different levels of the radiation dose. Although both are three-state models, the present stochastic representation differs from the one proposed by Payne and Garrett [12] in that a site will be in a potentially damaged state if it is exposed to a radiation source; this site may become either permanently damaged or repaired afterwards. Since all sites will eventually go to the second damaged state (which corresponds to the permanently damaged state of the present study), Payne and Garrett’s model is appropriate if cells are continuously exposed to a
28
SHIOJENN TSENG AND JYH-PING HSU
900
Gy
(b) 300 Gy
-_--______-_-__-__----------CI b
‘L-1
0 0
10
I
20
I
I
30
40 Time
0 I
50
I
60
0 I 70
(hr)
FIG. 3. The experimental data of Frankenberg-Schwager et al. 171 and the results predicted by the present model. (a) k, = 3.9140X lo-‘, k, = 2.3837~ 10-l; (b) k, = 5.1691x10-*, k, = 3.0499x10-‘.
radiation source. Clearly, it should not be used to interpret the experimental data reported by Frankenberg-Schwager et al. [7], in which the exposure of cells to radiation occurs only in the initial period of the experiment. Figure 4 shows the variation in the variance of the number of DSBs as a function of time for each dose. A comparison of Figures 2 and 3 with Figure 4 reveals that the mean and the variance of Ndsb are of the same order of magnitude. Since the number of DSBs per cell is typically on the order of 100 [7], the fluctuation characteristic of the phenomenon can be significant. The present stochastic model contains two adjustable parameters, one in the permanently damaged rate expression and one in the repair rate expression. In contrast, three adjustable parameters are commonly used in conventional analyses [12, 151: the initial number of reparable DSBs, the number of irreparable DSBs, and the parameter describing the kinetics of the repair process. It should be pointed out that the first two of these parameters should not be treated as adjustable parameters. This is because their values are determined completely by the experimental conditions and
REPAIR OF RADIATION-INDUCED
29
DNA DSB
1500 Gy
900 Gy
300 Gy
Time
[hr)
FIG. 4. The variation in the variance of the number of DSBs as a function of time.
are experimentally measurable. Therefore the present more sound than those proposed previously.
model is physically
APPENDIX For convenience,
Equation
(3) is rewritten
mm&(n)
as
= CfeMn + 1).
(A-1)
n
n
Taking the transformation yields Cf(n)D,g(n) n
m = n + 1 on the right-hand
= Cf(m m
- l)g(m)
side of this equation
= &(m>f(m m
- 1).
(A.21
Since m is a dummy index, we have
Cf(n)kdn> = Cs(n)f(n - 1). n
Hence, we obtain,
by referring Cf(n)D,g(n) n
where fC - 1) = gW,
(A.31
n
to Equation
(4),
= Cg(n)K’f(n), n
+ 1) = 0. This is Equation
(5).
(A4
30
SHIOJENN TSENG AND JYH-PING HSU
Multiplying both sides of Equation (8) by the respective nd values, and summing all resultant equations over all possible values of nd and nr, yields
+
: : n,(D,;‘-l)k,(N,-n,-n,)p, n~=On,=o
(AS)
where p(n,, n,, t) is abbreviated as p for convenience. The left-hand side of this equation is the rate of variation of ( Nd), and its right-hand side can be simplified by resorting to Equation (5). Thus, we have
(A.6) Thus, Equation (9) is justified. Multiplying both sides of Equation (8) by the respective n, values and summing all resultant equations over all possible values of nd and n,, and following the same procedure as employed in the derivation of Equation (A.61, Equation (10) can be recovered. The governing equations for the second moments (N:), (N,N,), and (N,*) can be obtained in the same manner. REFERENCES M. A. Resnick and P. Martin, Repair of double-strand breaks in nuclear DNA of cerevisiae and its genetic control, Mol. Gen. Genet. 143:119-129 (1976). M. A. Resnick, Similar responses to ionizing radiation of fungal and vertebrate cells and the importance of DNA double-strand breaks, J. Theor. Biol. 71:339-346 (1978). A. M. R. Taylor, Unrepaired DNA strand breaks in irradiated At&a telangiectusia [ymphocytes suggested from cytogenic observations, Mufar. Res. S&407-413 (19781. D. Frankenberg, M. Frankenberg-Schwager, D. Blocher, and R. Harbich, Evidence for DNA double-strand breaks as the critical lesions in yeast cells irradiated with sparsely or densely ionizing radiation under oxic or anoxic conditions, Rudiat. Res. Saccharomyces
88:524-532
(1981).
K. S. Y. Ho, Induction of DNA double-strand breaks by X-rays in a radiosensitive strain of the yeast Saccharomyces cerevisiae, Mutat. Res. 30:327-334 (1975). M. A. Resnick, The repair of double-strand breaks in DNA: a model involving recombination, J. Theor. Biol. 59~97-106 (1976). M. Frankenberg-Schwager, M. Frankenberg, D. Blocher, and C. Adamcxyk, Repair of DNA double-strand breaks in irradiated yeast cells under nongrowth conditions, Radiat. Res. 82~498-510 (1980). M. Frankenberg-Schwager, M. Frankenberg, D. Blocher, and C. Adamcayk, The linear relationship between DNA double-strand breaks and the radiation dose (30
REPAIR
9
10
11 12 13 14 15 16 17 18 19 20 21 22
OF RADIATION-INDUCED
31
DNA DSB
MeV electrons) is converted into a quadratic function by cellular repair, Int. J. Radiat. Biol. 37:207-212 (19801. E. P. Clark, W. C. Dewey, and J. T. Lett, Recovery of CH cells from hyperthermic potentiation to X rays: repair of DNA and chromatin, Radiat. Res. 85:302-313 (1981). N. J. Sargentini
and K. C. Smith,
Characterization
and quantitation
of DNA strand
breaks requiring recA-dependent repair in X-irradiated Escherichia coli, Radial. Res. 105:180-186 (1986). T. M. Koval and E. R. Kazmar, DNA double-strand break repair in eukaryotic cell lines having radically different sensitivities, Radiat. Res. 113:268-277 (1988). M. G. Payne and W. R. Garrett, Models for cell survival with low LET radiation, Radiat. Res. 62:169-179 (1975). G. J. Dienes, A kinetic model of biological radiation response, Radiat. Res. 28:183-202 (1966). A. Kappos and W. Pohlit, A cybernetic model for radiation reactions in living cells I. Sparsely ionizing radiations; stationary cells, ht. J. Radiat. Biol. 22:51-65 (1972). I. Pridal and M. V. Lokajicek, A model of DSB-repair kinetics, J. Theor. Biol. 111:81-90 (1984). H. D. Thames, An incomplete repair model for fractionated and continuous irradiations, ht. J. Radiat. Biol. 47:319-339 (1985). C. A. Tobias, The repair-misrepair model in radiobiology. Comparison to other models, Radiat. Res. 104:S77-S95 (1985). N. Albright, A. Markov formulation of the repair-misrepair model of cell survival, Radiat. Res. 118:1-20 (1989). S. Curtis, Lethal and potentially lethal lesions induced by radiation-a unified model, Radiat. Res. 106:252-270 (1986). D. T. Goodhead, Saturable models of radiation action in mammalian cells, Radiat. Res. 104:S58-S67 (1986). J. Y. Ostashevsky, A model relating cell survival to DNA fragment loss and unrepaired double-strand breaks, Radiat. Res. 118:437-466 (1989). N. G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier/NorthHolland, New York, 1981, p. 145.
DNA
SHIOJENN TSENG Department of Mathematics, Tamkang University, Tamsui, Taiwan 25137, Republic of China
JYH-PING HSU Department of Chemical Engineering, National Taiwan Uniuersity, Taipei, Taiwan 10764, Republic of China Received 20 July 1989; revised 9 November I989
ABSTRACT A three-state stochastic model is described for the repair of radiation-induced double-strand breaks (DSBS) in DNA. If irradiated, a site or region in DNA is assumed to be in a potentially damaged state; this site may either become permanently damaged or be repaired after a certain period of time. The result of the analysis of the available experimental data reveals that the present two-parameter model is capable of interpreting the rapid decrease in the number of DSBs in the initial period, which cannot be predicted by previously proposed models. The stochastic analysis yields not only the temporal variation of the mean of the number of DSBs but also its variance, and therefore is a generalization of the conventional deterministic models.
INTRQDUCTION When exposed to an external radiation source, a DNA duplex often experiences lesions in which breaks of both of its strands occur. The so-called double-strand breaks (DSBS) in DNA have been found to be closely related to cell lethality [l-4]. Experimental evidence reveals that cells are capable of repairing this type of damage under appropriate conditions [l, 4-111. Since cell death correlates with the number of unrepaired DSBs, the temporal variation of the number of DSBs in a cell is of fundamental$mportance in evaluating its probability of survival after its exposure to radiation. Reported results concerning the repair of DNA DSBs after a radiation treatment are ample. Most previous studies, however, focused on the experimental aspect of the phenomenon. Relatively little attention was given to the theoretical analysis of the repair kinetics. Payne and Garrett M4THEMATICAL
BIOSCIENCES
OElsevier Science Publishing 655 Avenue of the Americas,
100:21-31
(1990)
Co., Inc., 1990 New York, NY 10010
21 00255564/90/$03.50
22
SHIOJENN
TSENG
AND JYH-PING
HSU
[12] derived a three-state stochastic representation describing the evolution of the joint probability distribution of the number of sites in each state. In their model, a site in a DNA duplex can exist either in an undamaged state or in one of the two damaged states. In special cases, this model leads to various models proposed by previous investigators [13, 141. Payne and Garrett [12] correctly pointed out the necessity of including the statistical fluctuations associated with both direct and indirect effects of the radiation field for studying the effect of radiation on cell survival. However, since the second damaged state is an absorbing state, to which all sites eventually go, care must be taken in employing their model. Based on the step-by-step removal of individual DSBs inside an autonomous repair unit, Pridal and Iokajicek [15] formulated a kinetic model. This model, although giving a different view of DSB repair kinetics, fails to interpret the rapid drop in the number of DSBs in the initial period of a typical radiation-induced DSB experiment. Thames [16] discusses the repair kinetics for both fractionated and continuous irradiations. Tobias [171 assumes that the repair rate of lesions contains a linear self-repair term and a quadratic cooperative term. This model is reformulated by Albright [18] in a stochastic manner. The lethal and potentially lethal model suggested by Curtis [19] hypothesizes that there exist two different types of lesions, irreparable lesions and repairable lesions (see also [14]). The correct repair mechanism is assumed to be first-order, and the misrepair mechanism is assumed to be secondorder. Goodhead [201 points out that the repair efficiency can be dose-dependent owing to inactivation of enzymes. The repair rate of lesions is assumed to be linearly dependent on the number of lesions in the so-called saturable repair. By defining five possible states for a DNA molecule, Ostashevsky [21] was able to obtain expressions for the dose and time dependences of the surviving fraction, the number of unrepaired DSBs, and the number of prematurely condensed chromosome fragments. Two types of repair mechanisms are considered, noncooperative and cooperative; the rate expressions are essentially linear. The aim of the present study is to modify the stochastic model of Payne and Garrett [12] in such a way that the status of a site in DNA is defined more realistically, and to provide a way of describing the rapid drop in the number of DSBs in the initial period of a radiation-induced DSB experiment.
MODELING Let us consider the DSB repair mechanism shown in Figure 1. When irradiated, a site or region in DNA is assumed to be in a potentially damaged state. This site can either become permanently damaged or be repaired after a certain period of time. Denote the random variables Nd
REPAIR OF RADIATION-INDUCED
23
DNA DSB
and N, as the number of permanently damaged sites and the number of repaired sites, respectively; specific values of Nd and N, will be represented by nd, nd = 0,1,2,. .., N,, and n,, n, = 0,1,2,. .., Nc, where N,, is the number of potentially damaged sites initially. Note that if the values of nd and n, are given, the number of potentially damaged sites, npr can be evaluated by np = N,, - nd - n,. Let P(nd,nr, t) be the probability that there are nd permanently damaged sites and n, repaired sites at time t. The probability that given nh permanently damaged sites and n; repaired sites at time s there will be nd damaged sites and n, repaired sites at time t (t > s> is denoted as p(nd, n,, 1; nh,n;, s). It is assumed that (a) p(nd + 1, n,, t + At; nd, n,, t) = Ad(nd, n,) At + o(At) (b) p(n&n,.+l,t + At;nd,nr,t)= hr(nd,n,)At + o(At) (c) p(n& n,, t + At&, n,, t) = 1 -[h&z& n,.)+ hr(n& n,)] At + o(At> where 0, t + At) denotes an infinitesimally small but finite time interval, hd(nd,nl) and Ar(nd,n,) are transition intensities, and o(At) satisfies the condition lim ~~,a[o(At)/Atl= 0. Assumption (a) states that the number of permanently damaged sites increases by one in the time interval (t, t + At) with probability A&d, n,) At + o(At). Similarly, (b) assumes that the probability that the number of repaired sites increases by one in the time interval (t, t + At) is Ar(nd, n,) At + o(At). All other possible transitions are assumed to have an infinitesimal probability of order o(At), as stated in (c). The above assumptions lead to the following equation: dP(ndrnr,t)
dt
=Ad(nd-l,nr)P(nd-l,nr,t) +A,Cnd,n,-l)p(nd,n,-l,t) -
[‘bhf,‘b)
FIG. 1. Schematic representation present study.
+‘bbdt%)]
P(n&nr,t)
of the DSB repair mechanism considered
(1)
in the
24
SHIOJENN TSENG AND JYH-PING HSU
or
dp(npf)=(D~l-l)hd(nd,n,)p(nd,n,,t) +(DnS1-l)Ar(nd,nr)P(nd,nr,f),
(2)
where p(nd,n,,t) is zero if either nd or n, is negative. The step operator D, when operating on a function of n, f(n), possesses the following properties [22]:
Qdf(n>l =f(n +I)
(3)
D;‘[f(n>l= f(n -1).
(4)
and
Equations (3) and (4) suggest that, for arbitrary have (see Appendix)
functions
f(n) and g(n), we
where ni = nd or n, = nr, and f - (N-?)
(14) + kz(No(Nd) - (N,N,> -(N:)), (15)
d(N,2>= k2[ -2(N,2)-2 dt
(N,N,)+(2No-l)(W)-(N,)+No].
(16)
26
SHIOJENN TSENG AND JYH-PING HSU
The solution to these equations, (N,N,) = (N,2) = 0, is
subject to the initial
conditions
(Ni)
=
(k,+k,N,-(k2-k,+2kINo)exp[-(k,+kz)t] (N,2) =(k, klNo + k,)*
+(klNo-kl)exp[-2(kl+k2)~]}, (17)
(iv%)
=
k,k,No(No-1){1-2exp[-(k,+k2)t]+exp[-2(kI+k2)t]], Ck, + k,)2
k2No {k,+k2No-(k,-k2+2k2No)exp[-(k,+k,)t] (k,+ k,)*
(N,2)=
+(k2No-k,)exp[-2(kI+k2)t]j.
The variance of Nd, (( Nd)), and the variance by, respectively,
of N,, ((N,.)),
(19)
are evaluated
((Nd)) = (N,z)- C&j2
(20)
C(N)> = V,2) - (W2.
(21)
and
Since Ndsb = No - N,, we have
((Ndsd) = C(W)>-
(22)
DISCUSSION The applicability of the present stochastic model is examined through analysis of the available experimental data. Frankenberg-Schwager et al. [7] performed the DSB repair experiment using yeast cells Succharomyces cerevisiae 211*B in stationary phase. Four levels of radiation dose were used; the temporal variation of the number of DSBs per cell was recorded. The averaged numbers of DSBs per cell, along with the values predicted by Equation (13), are illustrated in Figures 2 and 3. Also shown in these figures is an approximate 95% confidence band estimated by (Ndsb) f ‘I2 . The estimation of the value of the adjustable parameters, 1.96((&,)) k, and k,, is based on a least squares deviation criterion. As seen from
REPAIR OF RADIATION-INDUCED
27
DNA DSB (0) 2400 Gy
-----------40 -
01
----__ I
I
I
0
-_-__---__-
I
I
I
I
I
(b)
1500 Gy
~--___--_______---____~_ CI --------A_--_--_-----___
0
0
I
I
I
10
20
30
_
I
I
I
40
50
60
I
70
Time (hr) FIG. 2. The experimental data of Frankenberg-Schwager et al. [7] and the results predicted by the present model. (a) k, = 1.6131 X lo-‘, k, =5.7433X lo-*; (b) k, = 2.6081 x 10-2, k, = 1.1098X 10-l.
Figures 2 and 3, the present stochastic model successfully predicts the transient behavior of the DSB repair phenomenon. Intuitively, the transition intensity of a site from the potentially damaged state to either the damaged state or the repaired state depends on the radiation dose applied. This is because the degree of damage of a site varies with the level of radiation dose, as justified by the variation in the values of the adjustable parameters k, and k, at different levels of the radiation dose. Although both are three-state models, the present stochastic representation differs from the one proposed by Payne and Garrett [12] in that a site will be in a potentially damaged state if it is exposed to a radiation source; this site may become either permanently damaged or repaired afterwards. Since all sites will eventually go to the second damaged state (which corresponds to the permanently damaged state of the present study), Payne and Garrett’s model is appropriate if cells are continuously exposed to a
28
SHIOJENN TSENG AND JYH-PING HSU
900
Gy
(b) 300 Gy
-_--______-_-__-__----------CI b
‘L-1
0 0
10
I
20
I
I
30
40 Time
0 I
50
I
60
0 I 70
(hr)
FIG. 3. The experimental data of Frankenberg-Schwager et al. 171 and the results predicted by the present model. (a) k, = 3.9140X lo-‘, k, = 2.3837~ 10-l; (b) k, = 5.1691x10-*, k, = 3.0499x10-‘.
radiation source. Clearly, it should not be used to interpret the experimental data reported by Frankenberg-Schwager et al. [7], in which the exposure of cells to radiation occurs only in the initial period of the experiment. Figure 4 shows the variation in the variance of the number of DSBs as a function of time for each dose. A comparison of Figures 2 and 3 with Figure 4 reveals that the mean and the variance of Ndsb are of the same order of magnitude. Since the number of DSBs per cell is typically on the order of 100 [7], the fluctuation characteristic of the phenomenon can be significant. The present stochastic model contains two adjustable parameters, one in the permanently damaged rate expression and one in the repair rate expression. In contrast, three adjustable parameters are commonly used in conventional analyses [12, 151: the initial number of reparable DSBs, the number of irreparable DSBs, and the parameter describing the kinetics of the repair process. It should be pointed out that the first two of these parameters should not be treated as adjustable parameters. This is because their values are determined completely by the experimental conditions and
REPAIR OF RADIATION-INDUCED
29
DNA DSB
1500 Gy
900 Gy
300 Gy
Time
[hr)
FIG. 4. The variation in the variance of the number of DSBs as a function of time.
are experimentally measurable. Therefore the present more sound than those proposed previously.
model is physically
APPENDIX For convenience,
Equation
(3) is rewritten
mm&(n)
as
= CfeMn + 1).
(A-1)
n
n
Taking the transformation yields Cf(n)D,g(n) n
m = n + 1 on the right-hand
= Cf(m m
- l)g(m)
side of this equation
= &(m>f(m m
- 1).
(A.21
Since m is a dummy index, we have
Cf(n)kdn> = Cs(n)f(n - 1). n
Hence, we obtain,
by referring Cf(n)D,g(n) n
where fC - 1) = gW,
(A.31
n
to Equation
(4),
= Cg(n)K’f(n), n
+ 1) = 0. This is Equation
(5).
(A4
30
SHIOJENN TSENG AND JYH-PING HSU
Multiplying both sides of Equation (8) by the respective nd values, and summing all resultant equations over all possible values of nd and nr, yields
+
: : n,(D,;‘-l)k,(N,-n,-n,)p, n~=On,=o
(AS)
where p(n,, n,, t) is abbreviated as p for convenience. The left-hand side of this equation is the rate of variation of ( Nd), and its right-hand side can be simplified by resorting to Equation (5). Thus, we have
(A.6) Thus, Equation (9) is justified. Multiplying both sides of Equation (8) by the respective n, values and summing all resultant equations over all possible values of nd and n,, and following the same procedure as employed in the derivation of Equation (A.61, Equation (10) can be recovered. The governing equations for the second moments (N:), (N,N,), and (N,*) can be obtained in the same manner. REFERENCES M. A. Resnick and P. Martin, Repair of double-strand breaks in nuclear DNA of cerevisiae and its genetic control, Mol. Gen. Genet. 143:119-129 (1976). M. A. Resnick, Similar responses to ionizing radiation of fungal and vertebrate cells and the importance of DNA double-strand breaks, J. Theor. Biol. 71:339-346 (1978). A. M. R. Taylor, Unrepaired DNA strand breaks in irradiated At&a telangiectusia [ymphocytes suggested from cytogenic observations, Mufar. Res. S&407-413 (19781. D. Frankenberg, M. Frankenberg-Schwager, D. Blocher, and R. Harbich, Evidence for DNA double-strand breaks as the critical lesions in yeast cells irradiated with sparsely or densely ionizing radiation under oxic or anoxic conditions, Rudiat. Res. Saccharomyces
88:524-532
(1981).
K. S. Y. Ho, Induction of DNA double-strand breaks by X-rays in a radiosensitive strain of the yeast Saccharomyces cerevisiae, Mutat. Res. 30:327-334 (1975). M. A. Resnick, The repair of double-strand breaks in DNA: a model involving recombination, J. Theor. Biol. 59~97-106 (1976). M. Frankenberg-Schwager, M. Frankenberg, D. Blocher, and C. Adamcxyk, Repair of DNA double-strand breaks in irradiated yeast cells under nongrowth conditions, Radiat. Res. 82~498-510 (1980). M. Frankenberg-Schwager, M. Frankenberg, D. Blocher, and C. Adamcayk, The linear relationship between DNA double-strand breaks and the radiation dose (30
REPAIR
9
10
11 12 13 14 15 16 17 18 19 20 21 22
OF RADIATION-INDUCED
31
DNA DSB
MeV electrons) is converted into a quadratic function by cellular repair, Int. J. Radiat. Biol. 37:207-212 (19801. E. P. Clark, W. C. Dewey, and J. T. Lett, Recovery of CH cells from hyperthermic potentiation to X rays: repair of DNA and chromatin, Radiat. Res. 85:302-313 (1981). N. J. Sargentini
and K. C. Smith,
Characterization
and quantitation
of DNA strand
breaks requiring recA-dependent repair in X-irradiated Escherichia coli, Radial. Res. 105:180-186 (1986). T. M. Koval and E. R. Kazmar, DNA double-strand break repair in eukaryotic cell lines having radically different sensitivities, Radiat. Res. 113:268-277 (1988). M. G. Payne and W. R. Garrett, Models for cell survival with low LET radiation, Radiat. Res. 62:169-179 (1975). G. J. Dienes, A kinetic model of biological radiation response, Radiat. Res. 28:183-202 (1966). A. Kappos and W. Pohlit, A cybernetic model for radiation reactions in living cells I. Sparsely ionizing radiations; stationary cells, ht. J. Radiat. Biol. 22:51-65 (1972). I. Pridal and M. V. Lokajicek, A model of DSB-repair kinetics, J. Theor. Biol. 111:81-90 (1984). H. D. Thames, An incomplete repair model for fractionated and continuous irradiations, ht. J. Radiat. Biol. 47:319-339 (1985). C. A. Tobias, The repair-misrepair model in radiobiology. Comparison to other models, Radiat. Res. 104:S77-S95 (1985). N. Albright, A. Markov formulation of the repair-misrepair model of cell survival, Radiat. Res. 118:1-20 (1989). S. Curtis, Lethal and potentially lethal lesions induced by radiation-a unified model, Radiat. Res. 106:252-270 (1986). D. T. Goodhead, Saturable models of radiation action in mammalian cells, Radiat. Res. 104:S58-S67 (1986). J. Y. Ostashevsky, A model relating cell survival to DNA fragment loss and unrepaired double-strand breaks, Radiat. Res. 118:437-466 (1989). N. G. van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier/NorthHolland, New York, 1981, p. 145.
