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Analysis of mutant frequency curves and survival curves applied to the AL hybrid cell system
J. theor. Biol. (1988) 132, 113-117
Analysis of Mutant Frequency Curves and Survival Curves Applied to the AL Hybrid Cell System RODGER PARKER,* CHARLES WALDREN,t TOM K. HEI,:[: D. F. WONG* AND Z. L. PUCKT
*Johns Hopkins Medical Institutions, Baltimore, M D 21205; ¢ Eleanor Roosevelt Institute for Cancer Research, Denver, CO 80206; $ College of Physicians and Surgeons of Columbia University, New York, N Y 10032, U.S.A. (Received 16 July 1987, and in revised form 8 February 1988) A model is presented for the statistical analysis of survival curves and mutant frequency curves for a hybrid cell system. The derivation of the model is given in the Appendix, and depends on simple assumptions about the distribution of insults, their repair, and the loss of a marker that is not rescued. A single formula (5) is found which relates a survival curve to the mutant frequency curve, i.e., the response curve for production of mutants per 105 survivors induced by a mutagen. The analysis is applied to loss of the a~ gene in At-J1 hybrid cells submitted to Cesium 3,-rays. Previous experimental data using X-rays was reported by Waldren et al. (1986: Proc. Natl. Acad. Sci, USA 83, 4839.) Also, a derived formula (10), which predicts the probability that in a surviving cell a marker is lost and not rescued, will form the basis for testing the validity of the model in the future using new experimental data. Recently, W a i d r e n et al. (1986) have s h o w n for the A , hybrid cell system, that the response curve for p r o d u c t i o n o f mutants is a c o n c a v e function o f dose, x (see their fig. 2 which exhibits the m u t a n t yield o b t a i n e d for loss o f the h u m a n a, gene in x-irradiated AL-J1 hybrid cells). This result runs c o u n t e r to the expectation that p r o d u c t i o n o f mutants at low doses can be extrapolated linearly from values at higher doses. Using a mathematical m o d e l described in the A p p e n d i x , the following formulae were obtained. The survivor curve, P~(x) (see fig. 1 in W a l d r e n et al., 1986) is given by
p,(x) = e-t'~~)('- R(X))
( 1)
where
p(x) = expected n u m b e r o f insults R ( x ) = probability an insult is repaired. We assume that p(x)(1 - R ( x ) ) can be a p p r o x i m a t e d by
is the expected n u m b e r o f unrepaired insults and
p(x)(1 - R ( x ) ) = ax v.
(2)
If 3,> 1, then P~(x) has a shoulder; if 7 -< 1, P~(x) has no shoulder. W h e n 3' = 1, the log survivor curve, In P~(x), is linear. In experimental practice, 3 ,>- 1. If 3,> l, 113 0022-5193/88/090113+05 $03.00/0
© 1988 Academic Press Limited
114
R.
PARKER
ET
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then the s l o p e o f In P~(x) is zero at x = 0 ; o t h e r w i s e , w h e n 7 = 1, the s l o p e is - a for all x. Since 1 - R ( x ) is b e t w e e n 0 a n d 1 for all x, (2) i m p l i e s p ( x ) goes to co at least as fast as x r does. F i g u r e 1 shows the raw s u r v i v o r curve f o u n d r e c e n t l y by Hei for C e s i u m 3,-ray i r r a d i a t i o n , a l o n g with the fitted curve, using e x p r e s s i o n s (1) a n d (2). T a b l e 1 gives the a n a l y s i s o f v a r i a n c e table for the r e g r e s s i o n used in the curve-fitting. The overall F is significant at the 0.001 level. Using the m o d e l (see the A p p e n d i x ) , the f o r m u l a for P,,,j~(x), the p r o b a b i l i t y o f a m u t a t i o n given cell survival, is given by P,,,t,(x) = 1 - e -'c'~t R~c'~/mx~l
(3)
where R,..(x) = p r o b a b i l i t y an insult causes m a r k e r loss a n d the m a r k e r is not r e s c u e d , given that the cell survives the insult. TABLE 1
Analysis of variance table .for the regression in curve fitting the survival curve (see Fig. 1 ) Source Total Regression Residual
d.f.
Sum of squares
Mean square
Overall F
7 2 5
2-815 2-810 0-0046
-1-41 0.00093
-1500 --
i.CP
0.8~ll I~ \
o > > u~
0.6
0-4 O .o O
0-2
0
200 Cesium
400
600
800
gommo-roy dose ( r o d )
FIG. 1. Survival curve for AL~J~ cells. I , Experimental; • , curve fitted.
MUTANT
FREQUENCY
CURVES
AND
SURVIVAL
CURVES
115
Formula (3), when multiplied by 10 5, gives the model estimate for the response curve of production of mutants per 10 5 survivors versus dose. It is assumed that the detection of a marker loss mutation is contingent on the cell surviving and the marker not being rescued. In the case of the At-J1 hybrid cell, the marker being considered is the regionally m a p p e d genes for cell surface antigen a~ on h u m a n c h r o m o s o m e 11. Since many of the genetic lesions produced by irradiation are large c h r o m o s o m a l lesions and presumably deletions, the rescue processes involved are presumably translocations in which the marker liberated from its position on the h u m a n c h r o m o s o m e is reintegrated elsewhere in the genome. We assume that p(x)[R,..(x)/R(x)] can be a p p r o x i m a t e d by
p(x)[ R,..(x)/ R(x)] = fix ~.
(4)
Expression (4) is the formula for expected n u m b e r of incorrectly repaired insults given that a cell survives. In this context "incorrectly repaired" means the marker is lost and not rescued. If r < 1, then P,,,i.~(x) is concave; otherwise it is convex. In experimental practice, r -< 1. Since p(x) goes to 0o at least as fast as x, it follows that R * ( x ) / R ( x ) in practice goes to zero as x goes to 0o. Figure 2 gives the curve-fitted approximation for values of 105P,,,i.,(x) using Hei's data on Cesium ),-ray. The overall F in the analysis of variance for the regression was 340 which is significant at the 0.001 level. Equations (1) and (3) may be combined, by eliminating p(x), to yield
P,,i.,(x)= l - P,(x) k'''
(5)
:500
o>
I
200
0 e
o
IO0
ml 0
n
l
200
t
I
400
I 600
Cesium q a m m o - r o y (:lose ( r o d s )
FiG.
2. Response curve for production of a t mutants, m, Experimental; O, curve fitted.
116
r.
PARKER
ET
AL.
where
k ( x ) = R c , ( x ) / [ R ( x ) ( 1 - R(x))].
(6)
The parameter k ( x ) is equal to the left side of (4) divided by the left side of (2) and is interpretable as the expected number of insults which are incorrectly repaired divided by the expected n u m b e r of unrepaired insults. Equation (5) implies that k ( x ) is given by
k ( x ) = In (1 - P.,l~(x))/ln P~(x).
(7)
Applying expressions (1), (2), (3) and (4) to (7) yields the formula
k ( x ) = f i x ~'-v.
(8)
Ot
For the Cesium T-ray data which we have been analyzing: /3 -- 0.00011 a = 0-00026 z=0.51 y=l-5 so that for this data, by (8) we have that
k ( x ) ~- a / x
(9)
where a = 11/26. In the Appendix, it is shown that the conditional probability, Pm,.i,~t~.~(x), that in a surviving cell submitted to dose x, a marker will be lost and not rescued is given by:
P,,,r.lrat~,(x) = P,nl~(X)/(1 - e - p ' x l ) .
(10)
1 - P,,,,.i,,,~(x) is the probability of marker rescue. Since most rescues are presumably translocation events, the formulation offers a means, however rough, for estimating the incidence of translocations. Experiments are in progress to test the validity of (10).
REFERENCES WALDREN, C., CORRELL, L., SOGNIER, M. 8¢ PUCK, T. (1986). Measurement of Low Levels of x-ray
Mutagenesis in Relation to Human Disease. Proc. NatL Acad. Sci. USA 83, 4839.
MUTANT FREQUENCY CURVES AND SURVIVAL CURVES
117
APPENDIX Derivation of the Formulas
Let x = doses in rads Pj(x) = probability a cell incurs j insults each of which if unrepaired would kill the cell
R(x) = probability an insult is repaired. Then, Ps(x)= probability the cell survives, is given by Ps(x)= ~ P~(x)R(x) j.
(A1)
j=o
Assume that the number o f insults j is Poisson-distributed, i.e.,
P~(x) = p(x) i e-P°"/j!
(A2)
Substituting (A2) into (A1) and evaluating the sum gives
Ps(x) = e -"lx'(~-Rc~'.
(A3)
To derive the formula for P,,Is(X)= probability the cell survives but undergoes a mutation which is counted, we assume that the conditional probability that a cell which receives j insults will survive, but with a marker loss that is not rescued, is given by ( R ( x ) j - gc(x)J)/g(x) j
(A4)
where
R,.(x) = probability that a marker loss does not occur or if it occurs it is rescued. Then, P,,l~(X) is given by
Pmls(X) = ~ Pj(x)(R(x) j - R~'(x)i)/g(x) j.
(A5)
Substituting (A2) into (AS) and letting R,..(x)= R ( x ) - R , . ( x ) that the lost marker is not rescued gives
be the probability
j=l
P.,I~(x) = 1 - e -p~x~tR,'c~/mx~]
(A6)
The conditional probability, P,.r*I.,,~,(X), that in a surviving cell submitted to dose x, a marker will be lost and not rescued is given by
- R(x) j - R~.(x) j / ,~ Pr,,.i,,,,~,(x)-- =, t'j R-(-x~ / L=, Pj. p
(A7)
oc,
The term J ~ i = , PJ in (A7) is the probability that there a r e j insults which could cause marker loss, given that marker loss has occurred. The term (R(x) i R~(x)i)/R(x) j in (A7) is the probability that at least one of the j insults will result in marker loss such that the marker is not rescued. Applying expression (A5) in (A7), and using the fact that Po = 1 - Y . j : , Pj = e -~c'~, yields the formula
P,,,*rml~(x) = Pml.,(x)/(1 - e-Otx)).
(A8)
Analysis of Mutant Frequency Curves and Survival Curves Applied to the AL Hybrid Cell System RODGER PARKER,* CHARLES WALDREN,t TOM K. HEI,:[: D. F. WONG* AND Z. L. PUCKT
*Johns Hopkins Medical Institutions, Baltimore, M D 21205; ¢ Eleanor Roosevelt Institute for Cancer Research, Denver, CO 80206; $ College of Physicians and Surgeons of Columbia University, New York, N Y 10032, U.S.A. (Received 16 July 1987, and in revised form 8 February 1988) A model is presented for the statistical analysis of survival curves and mutant frequency curves for a hybrid cell system. The derivation of the model is given in the Appendix, and depends on simple assumptions about the distribution of insults, their repair, and the loss of a marker that is not rescued. A single formula (5) is found which relates a survival curve to the mutant frequency curve, i.e., the response curve for production of mutants per 105 survivors induced by a mutagen. The analysis is applied to loss of the a~ gene in At-J1 hybrid cells submitted to Cesium 3,-rays. Previous experimental data using X-rays was reported by Waldren et al. (1986: Proc. Natl. Acad. Sci, USA 83, 4839.) Also, a derived formula (10), which predicts the probability that in a surviving cell a marker is lost and not rescued, will form the basis for testing the validity of the model in the future using new experimental data. Recently, W a i d r e n et al. (1986) have s h o w n for the A , hybrid cell system, that the response curve for p r o d u c t i o n o f mutants is a c o n c a v e function o f dose, x (see their fig. 2 which exhibits the m u t a n t yield o b t a i n e d for loss o f the h u m a n a, gene in x-irradiated AL-J1 hybrid cells). This result runs c o u n t e r to the expectation that p r o d u c t i o n o f mutants at low doses can be extrapolated linearly from values at higher doses. Using a mathematical m o d e l described in the A p p e n d i x , the following formulae were obtained. The survivor curve, P~(x) (see fig. 1 in W a l d r e n et al., 1986) is given by
p,(x) = e-t'~~)('- R(X))
( 1)
where
p(x) = expected n u m b e r o f insults R ( x ) = probability an insult is repaired. We assume that p(x)(1 - R ( x ) ) can be a p p r o x i m a t e d by
is the expected n u m b e r o f unrepaired insults and
p(x)(1 - R ( x ) ) = ax v.
(2)
If 3,> 1, then P~(x) has a shoulder; if 7 -< 1, P~(x) has no shoulder. W h e n 3' = 1, the log survivor curve, In P~(x), is linear. In experimental practice, 3 ,>- 1. If 3,> l, 113 0022-5193/88/090113+05 $03.00/0
© 1988 Academic Press Limited
114
R.
PARKER
ET
AL.
then the s l o p e o f In P~(x) is zero at x = 0 ; o t h e r w i s e , w h e n 7 = 1, the s l o p e is - a for all x. Since 1 - R ( x ) is b e t w e e n 0 a n d 1 for all x, (2) i m p l i e s p ( x ) goes to co at least as fast as x r does. F i g u r e 1 shows the raw s u r v i v o r curve f o u n d r e c e n t l y by Hei for C e s i u m 3,-ray i r r a d i a t i o n , a l o n g with the fitted curve, using e x p r e s s i o n s (1) a n d (2). T a b l e 1 gives the a n a l y s i s o f v a r i a n c e table for the r e g r e s s i o n used in the curve-fitting. The overall F is significant at the 0.001 level. Using the m o d e l (see the A p p e n d i x ) , the f o r m u l a for P,,,j~(x), the p r o b a b i l i t y o f a m u t a t i o n given cell survival, is given by P,,,t,(x) = 1 - e -'c'~t R~c'~/mx~l
(3)
where R,..(x) = p r o b a b i l i t y an insult causes m a r k e r loss a n d the m a r k e r is not r e s c u e d , given that the cell survives the insult. TABLE 1
Analysis of variance table .for the regression in curve fitting the survival curve (see Fig. 1 ) Source Total Regression Residual
d.f.
Sum of squares
Mean square
Overall F
7 2 5
2-815 2-810 0-0046
-1-41 0.00093
-1500 --
i.CP
0.8~ll I~ \
o > > u~
0.6
0-4 O .o O
0-2
0
200 Cesium
400
600
800
gommo-roy dose ( r o d )
FIG. 1. Survival curve for AL~J~ cells. I , Experimental; • , curve fitted.
MUTANT
FREQUENCY
CURVES
AND
SURVIVAL
CURVES
115
Formula (3), when multiplied by 10 5, gives the model estimate for the response curve of production of mutants per 10 5 survivors versus dose. It is assumed that the detection of a marker loss mutation is contingent on the cell surviving and the marker not being rescued. In the case of the At-J1 hybrid cell, the marker being considered is the regionally m a p p e d genes for cell surface antigen a~ on h u m a n c h r o m o s o m e 11. Since many of the genetic lesions produced by irradiation are large c h r o m o s o m a l lesions and presumably deletions, the rescue processes involved are presumably translocations in which the marker liberated from its position on the h u m a n c h r o m o s o m e is reintegrated elsewhere in the genome. We assume that p(x)[R,..(x)/R(x)] can be a p p r o x i m a t e d by
p(x)[ R,..(x)/ R(x)] = fix ~.
(4)
Expression (4) is the formula for expected n u m b e r of incorrectly repaired insults given that a cell survives. In this context "incorrectly repaired" means the marker is lost and not rescued. If r < 1, then P,,,i.~(x) is concave; otherwise it is convex. In experimental practice, r -< 1. Since p(x) goes to 0o at least as fast as x, it follows that R * ( x ) / R ( x ) in practice goes to zero as x goes to 0o. Figure 2 gives the curve-fitted approximation for values of 105P,,,i.,(x) using Hei's data on Cesium ),-ray. The overall F in the analysis of variance for the regression was 340 which is significant at the 0.001 level. Equations (1) and (3) may be combined, by eliminating p(x), to yield
P,,i.,(x)= l - P,(x) k'''
(5)
:500
o>
I
200
0 e
o
IO0
ml 0
n
l
200
t
I
400
I 600
Cesium q a m m o - r o y (:lose ( r o d s )
FiG.
2. Response curve for production of a t mutants, m, Experimental; O, curve fitted.
116
r.
PARKER
ET
AL.
where
k ( x ) = R c , ( x ) / [ R ( x ) ( 1 - R(x))].
(6)
The parameter k ( x ) is equal to the left side of (4) divided by the left side of (2) and is interpretable as the expected number of insults which are incorrectly repaired divided by the expected n u m b e r of unrepaired insults. Equation (5) implies that k ( x ) is given by
k ( x ) = In (1 - P.,l~(x))/ln P~(x).
(7)
Applying expressions (1), (2), (3) and (4) to (7) yields the formula
k ( x ) = f i x ~'-v.
(8)
Ot
For the Cesium T-ray data which we have been analyzing: /3 -- 0.00011 a = 0-00026 z=0.51 y=l-5 so that for this data, by (8) we have that
k ( x ) ~- a / x
(9)
where a = 11/26. In the Appendix, it is shown that the conditional probability, Pm,.i,~t~.~(x), that in a surviving cell submitted to dose x, a marker will be lost and not rescued is given by:
P,,,r.lrat~,(x) = P,nl~(X)/(1 - e - p ' x l ) .
(10)
1 - P,,,,.i,,,~(x) is the probability of marker rescue. Since most rescues are presumably translocation events, the formulation offers a means, however rough, for estimating the incidence of translocations. Experiments are in progress to test the validity of (10).
REFERENCES WALDREN, C., CORRELL, L., SOGNIER, M. 8¢ PUCK, T. (1986). Measurement of Low Levels of x-ray
Mutagenesis in Relation to Human Disease. Proc. NatL Acad. Sci. USA 83, 4839.
MUTANT FREQUENCY CURVES AND SURVIVAL CURVES
117
APPENDIX Derivation of the Formulas
Let x = doses in rads Pj(x) = probability a cell incurs j insults each of which if unrepaired would kill the cell
R(x) = probability an insult is repaired. Then, Ps(x)= probability the cell survives, is given by Ps(x)= ~ P~(x)R(x) j.
(A1)
j=o
Assume that the number o f insults j is Poisson-distributed, i.e.,
P~(x) = p(x) i e-P°"/j!
(A2)
Substituting (A2) into (A1) and evaluating the sum gives
Ps(x) = e -"lx'(~-Rc~'.
(A3)
To derive the formula for P,,Is(X)= probability the cell survives but undergoes a mutation which is counted, we assume that the conditional probability that a cell which receives j insults will survive, but with a marker loss that is not rescued, is given by ( R ( x ) j - gc(x)J)/g(x) j
(A4)
where
R,.(x) = probability that a marker loss does not occur or if it occurs it is rescued. Then, P,,l~(X) is given by
Pmls(X) = ~ Pj(x)(R(x) j - R~'(x)i)/g(x) j.
(A5)
Substituting (A2) into (AS) and letting R,..(x)= R ( x ) - R , . ( x ) that the lost marker is not rescued gives
be the probability
j=l
P.,I~(x) = 1 - e -p~x~tR,'c~/mx~]
(A6)
The conditional probability, P,.r*I.,,~,(X), that in a surviving cell submitted to dose x, a marker will be lost and not rescued is given by
- R(x) j - R~.(x) j / ,~ Pr,,.i,,,,~,(x)-- =, t'j R-(-x~ / L=, Pj. p
(A7)
oc,
The term J ~ i = , PJ in (A7) is the probability that there a r e j insults which could cause marker loss, given that marker loss has occurred. The term (R(x) i R~(x)i)/R(x) j in (A7) is the probability that at least one of the j insults will result in marker loss such that the marker is not rescued. Applying expression (A5) in (A7), and using the fact that Po = 1 - Y . j : , Pj = e -~c'~, yields the formula
P,,,*rml~(x) = Pml.,(x)/(1 - e-Otx)).
(A8)