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A comment on the quantitative relationship between micronuclei and chromosomal aberrations
Mutation Research, 207 (1988) 33-36 Elsevier
33
MTRL 075
A comment on the quantitative relationship between micronuclei and chromosomal aberrations J o h n R.K. S a v a g e Medical Research Council, Radiobiology Unit, Chilton, Didcot, OXI 10RD (Great Britain)
(Accepted 9 July 1987)
Ke)'words" Quantitative relationship; Micronuclei; Chromosomal aberrations.
During a recent survey of micronucleus literature, I have been impressed by the number of authors who seem surprised that the observed frequencies of micronuclei (MN) are often so much lower than the frequencies of acentric fragments (AF) seen at first post-treatment metaphase. Without invoking the incredible cell-kinetic problems which plague the majority of MN systems, and which undoubtedly modify observed yields, it is easy to show that large disparities in the ratio, R = M N per daughter c e l l / A F per parent ceil, are to be expected on simple theoretical considerations, even in ideal populations. Imagine a cell with 1, and only 1 AF. (If it is a chromosome-type AF we will assume that the chromatids do not separate and the pair is transmitted as a unit. This is the basic assumption made in most theoretical calculations (Carrano and Heddle, 1973; Braselmann et al., 1986).) When this cell divides, the AF will pass to one daughter cell where it will either be included within the nucleus or excluded from it and form an MN in the cytoplasm. There are 4 exhaustive and Correspondence: Dr. John R.K. Savage, Medical Research Council, Radiobiology Unit, Chilton, Didcot, OXll 0RD (Great Britain).
mutually exclusive possibilities leading to 8 daughter cells, 2 of which will have MN. The relative frequency of the 8 daughters, and thus the observed frequency M N / c e l l will be a function of the probability with which the fragment is included in a nucleus, PI. Exclusion = ( 1 - P O . The m a x i m u m frequency of M N / c e l l for such cells is 0.5 when P~ = 0. In all published critical investigations, P~ is > 0, so that M N / A F of 1 / 3 - 1 / 4 or less are to be expected. Now consider a cell with 2 and only 2 AF. There are 16 possible segregations, 32 daughter ceils, 14 of which contain excluded fragments (12 with 1, 2 with 2). Again the relative frequencies depend on P~ but to calculate them, further assumptions have to be made. We will make 3: (a) That PI is a constant for all fragments irrespective of size or origin; (b) that fragments behave completely independently with respect to segregation; and (c) that each fragment leads to a single, visible MN; there is no degeneration and no fusion to influence frequency or distribution. Given these assumptions, we can readily show from binomial considerations that the relative frequencies of daughter cells with O: 1:2 MN a r e (PI) 2 + P I ( I - P I ) + 1/4(1-PI) 2 : PI(1-PI) + I/2(1-PI) 2 :
1/4(1
-
PI) 2.
0165-7992/88/$ 03.50 © 1988 Elsevier Science Publishers B.V. (Biomedical Division)
34 Similarly for cells with 3 and only 3 AF, the 128 daughter cells having 0:1:2:3 MN have relative frequencies: PI 3 + 1.5pi2(1-PI) + 0 . 7 5 P f f l - P 1 ) 2 + 0.125(1-PI)3 : 1.5PI2(1-Pt) + 1 . 5 P f f l - P I ) 2 + 0 . 1 2 5 ( 1 - P I ) 3 : 0 . 7 5 P f f I - P I ) 2 + 0.125(1-P1) 2 : 0 . 1 2 5 ( 1 - P I ) 3 and so on. At first sight, these distributions appear unusual, but they are, in fact, simple binomials. We can therefore generalise: Let F = 0.5 - 0.5P~. Then the MN distribution amongst all possible daughters of a cell having N fragments is the expansion of ( F + (1 - F ) ) 'v. In the case of mononucleate second-division cells Fcorresponds to the relative reduction, R, i.e. the ratio M N per daughter c e l l / A F per parent cell, given the simple assumptions used here. It is now a very easy matter to derive, for this ideal population, a mean frequency of M N / c e l l and its distribution, from any mean frequency and distribution of AF/cell. We choose a mean AF/cell and partition it to ceils with 0, 1, 2, . . . AF using a Poisson or some other pre-defined distribution. Each class frequency is doubled (2 daughters from 1 parent) and in turn partitioned into MN class frequencies using a chosen value of P~ to derive F. Summation then provides MN/cell, cells with MN and between-cell MN distribution - - expected in a truly random and unbiased sample of pure seconddivision cells derived from the original (and homogeneous) first-division AF bearing population. However, if only the mean MN frequency is required, then the pain of partitioning is unnecessary in such simple, hypothetical populations. Since Pj is a constant and directly related to F and R, M N / c e l l must be AF/ce[1 x F. If AF are distributed as a Poisson in parent cells, then, since we are summing a family of binomials with constant F, the resulting distribution of MN will also be Poisson. If the initial AF distribution is non-Poisson, which is often the case (Couzin and Papworth, 1979; Braselmann et al., 1986) the MN distribution will be non-Poisson, and partitioning will be necessary to obtain its form. Dose-response curves for M N / c e l l can be derived from those of AF/cell. Note that since each yield
YID
1.6 D. 1.2 Z
b S
1.0
2.0
3.0
4.0
/'
/
D J
R 0.3
z
0,8
LI,<
o.4 ~
/
O/
P, 0.4
/
F 0.3
~
(IF BN ceHs Pi 0.7 F 0.7)
1.0
2.0
3.0
4.0
D O S E (Gy)
Fig. 1. A hypothetical quadratic dose-response curve for radiation-induced acentric fragments (AF) and the derived micronucleus (MN) per mononucleatesecond-divisioncell when P~ = 0.4. See text for derivation methods. Inset: The plot Y/D with D giving the linear and quadratic coefficients for these two curves. If the MN curve was for binucleate (BN) cells than a Pz of 0.07 would be required.
of A F is reduced by the constant factor, R, the M N / c e l l curve has a lower slope. In particular, if AF/cell follows the familiar quadratic form, this reduced slope will be reflected in lower quadratic components, c~, /3. This is illustrated for a hypothetical case in Fig. 1. One of the sampling problems inherent in most populations is to obtain a pure sample of cells that have divided once, and only once, since clastogen treatment. Recently, several attempts have been made to overcome this difficulty (Joshi et al., 1982; Pincu et al., 1984; Fenech and Morley, 1985; Wakata and Sasaki, 1987). A relatively simple technique, gaining in popularity, is to use the agent cytochalasin B, which blocks cytokinesis without apparently affecting nuclear division, and results in binucleate (BN) cells. The scoring of MN is confined to BN ceils, since these must have divided once, and only once. Using the basic assumptions used for mononucelate cells and given a random and unbiassed sample of BN cells, we can apply the principles outlined to obtain estimates of M N / B N cell and
35 the b e t w e e n BN cell d i s t r i b u t i o n s . In this s i t u a t i o n we f i n d that F = PI so t h a t the p a r t i t i o n coefficients for a cell with N A F can be o b t a i n e d f r o m the e x p a n s i o n (Pl + (1 - P I ) ) n. F r o m this it follows t h a t , starting with the s a m e value o f AF/cell, the yield o f M N / B N cell will be 2 × MN/monon u c l e a t e cell, or l o o k e d at in a n o t h e r way, a c o m m o n r e d u c t i o n f a c t o r for m o n o n u c l e a t e a n d b i n u c l e a t e scores w o u l d indicate a c h a n g e in P~ (P~ = 1 - R for BN cells, 1 - 2 R for m o n o n u c l e a t e cells) (Fig. 1). U n f o r t u n a t e l y , we d o n o t have simple ideal p o p u l a t i o n s to deal with in the real w o r l d a n d so these c a l c u l a t i o n s c a n n o t be t r a n s l a t e d a n d a p p l i e d d i r e c t l y to a given e x p e r i m e n t a l set-up. I have p r e s e n t e d t h e m simply to show that o b s e r v a t i o n s o f low R, increasing d i s p a r i t y with dose a n d different shapes for AF and related MN d o s e - r e s p o n s e curves are n o t u n e x p e c t e d . S o m e o f the a s s u m p t i o n s I have m a d e for the c a l c u l a t i o n s are q u e s t i o n a b l e , a n d w o r t h y o f furt h e r investigation. P~ has a positive value in every system t h a t has b e e n used. In l y m p h o c y t e s it tends to be high, o f the o r d e r o f 0.8 (Pincu et al., 1984; C o u n t r y m a n a n d H e d d l e , 1976; F e n e c h a n d M o r l e y , 1985; Das a n d S h a r m a , 1987; c o n t r a S a s a k i a n d N o r m a n , 1967; B a u c h i n g e r et al,, 1986) a n d g e n e r a l l y lower (0.4-0.5) in f i b r o b l a s t s (Joshi et al., 1982; R o b e r t s et al., 1986; W a k a t a a n d S a s a k i , 1987). It is u n l i k e l y t h a t P~ is a c o n s t a n t irrespective o f size o r origin o f the f r a g m e n t a n d there are indications t h a t it m a y v a r y with dose ( C o n g e r , 1965). Ind e p e n d e n c y o f f r a g m e n t s is a n o t h e r p r o b l e m a r e a b o t h as r e g a r d s their s e g r e g a t i o n a n d their fusion in M N f o r m a t i o n . T h e r e is increasing evidence t h a t c h r o m o s o m e - t y p e A F d o s e p a r a t e a n d can be t r a n s m i t t e d singly (e.g. Das a n d S h a r m a , 1987). Several a u t h o r s have c o m m e n t e d on the p a u c i t y o f cells with m u l t i p l e M N (e,g. W a k a t a a n d Sasaki, 1987). F u r t h e r m o r e , n o t all f r a g m e n t s necessarily f o r m visible M N . H o m o g e n e i t y a n d u n i f o r m sensitivity o f the initial t a r g e t p o p u l a t i o n is a rare p h e n o m e n o n , a n d o b s e r v e d M N yields m a y v a r y with s a m p l e time even w h e n c o m p l i c a t i n g d i l u t i o n factors are
m i n i m a l (Brock a n d W i l l i a m s , 1985; R o b e r t s et al., 1986). C o n s e q u e n t l y , the m i t o t i c p e r t u r b a t i o n int r o d u c e d by a clastogen will p l a y h a v o c k b o t h with o b s e r v e d yields a n d with d i s t r i b u t i o n s (Savage a n d P a p w o r t h , 1973; C o u z i n a n d P a p w o r t h , 1979) so t h a t in the end, i n t e r p r e t a b l e a n d a n a l y s a b l e d o s e - r e s p o n s e curves m a y not be o b t a i n a b l e .
Acknowledgement I w o u l d like to t h a n k D a v i d P a p w o r t h for some h e l p f u l discussions on this topic.
References Bauchinger, M., E. Schmid and H. Braselmann (1986) Cell survival and radiation induced chromosome aberrations, II. Experimental findings in human lymphocytes analysed in first and second post-irradiation metaphases, Radiat. Environ. Biophys., 25, 253-260. Braselmann, H., M. Bauchinger and E. Schmid (1986) Cell survival and radiation induced chromosome aberrations, 1. Derivation of formulae for the determination of transmission and survival parameters of aberrations, Radiat. Environ. Biophys., 25, 243-251. Brock, W.A., and M. Williams (1985) Kinetics of micronucleus expression in synchronized irradiated Chinese hamster ovary cells, Cell Tissue Kinet., 18, 247-254. Carrano, A.V., and J.A. Heddle (1973) The fate of chromosome aberrations, J. Theoret. Biol., 38, 289-304. Conger, A.D. (1965) The fate of metaphase aberrations, Radiat. Botany, 4, 81-96. Countryman, P.I., and J.A. Heddle (1976) The production of micronuclei from the chromosome aberrations in irradiated cultures of human lymphocytes, Mutation Res., 41,321-332. Couzin, D., and D.G. Papworth (1979) The overdispersion between cells of chromosomal aberrations, J. Theoret. Biol., 80, 249-258. Das, B.C., and T. Sharma (1987) The fate of X-ray-induced chromosome aberrations in blood lymphocyte culture, Mutation Res., 176, 93-104. Fenech, M., and A.A. Morley (1985) Measurement of micronuclei in lymphocytes, Mutation Res., 147, 29-36. Joshi, G.P., W.J. Nelson, S.H. Revell and C.A. Shaw (1982) X-Ray induced chromosome damage in live mammalian cells, and improved measurements of its effects on their colony-forming ability, Int. J. Radiat. Biol., 41, 161-181. Pincu, M., D. Bass and A. Norman (1984) An improved micronuclear assay in lymphocytes, Mutation Res., 139, 61-65.
36 Roberts, C.J., G.R. Morgan and P.D. Holt (1986) A critical comparison of the micronucleus yield from high and low LET irradiation of plateau-phase cell populations, Mutation Res., 160, 237-242. Sasaki, M.S., and A. Norman (1967) Selection against chromosome aberrations in human lymphocytes, Nature (London), 214, 502-504. Savage, J.R.K., and D.G. Papworth (1973) The effect of variable G2 duration upon the interpretation of yield-time
curves of radiation induced chromatid aberrations, J. Theoret. Biol., 38, 17-38. Wakata, A., and M.S. Sasaki (1987) Measurement of micronuclei by cytokinesis-block method in cultured Chinese hamster cells: Comparison with types and rates of chromosome aberrations, Mutation Res., 190, 51-57. Communicated by R.J. Preston
33
MTRL 075
A comment on the quantitative relationship between micronuclei and chromosomal aberrations J o h n R.K. S a v a g e Medical Research Council, Radiobiology Unit, Chilton, Didcot, OXI 10RD (Great Britain)
(Accepted 9 July 1987)
Ke)'words" Quantitative relationship; Micronuclei; Chromosomal aberrations.
During a recent survey of micronucleus literature, I have been impressed by the number of authors who seem surprised that the observed frequencies of micronuclei (MN) are often so much lower than the frequencies of acentric fragments (AF) seen at first post-treatment metaphase. Without invoking the incredible cell-kinetic problems which plague the majority of MN systems, and which undoubtedly modify observed yields, it is easy to show that large disparities in the ratio, R = M N per daughter c e l l / A F per parent ceil, are to be expected on simple theoretical considerations, even in ideal populations. Imagine a cell with 1, and only 1 AF. (If it is a chromosome-type AF we will assume that the chromatids do not separate and the pair is transmitted as a unit. This is the basic assumption made in most theoretical calculations (Carrano and Heddle, 1973; Braselmann et al., 1986).) When this cell divides, the AF will pass to one daughter cell where it will either be included within the nucleus or excluded from it and form an MN in the cytoplasm. There are 4 exhaustive and Correspondence: Dr. John R.K. Savage, Medical Research Council, Radiobiology Unit, Chilton, Didcot, OXll 0RD (Great Britain).
mutually exclusive possibilities leading to 8 daughter cells, 2 of which will have MN. The relative frequency of the 8 daughters, and thus the observed frequency M N / c e l l will be a function of the probability with which the fragment is included in a nucleus, PI. Exclusion = ( 1 - P O . The m a x i m u m frequency of M N / c e l l for such cells is 0.5 when P~ = 0. In all published critical investigations, P~ is > 0, so that M N / A F of 1 / 3 - 1 / 4 or less are to be expected. Now consider a cell with 2 and only 2 AF. There are 16 possible segregations, 32 daughter ceils, 14 of which contain excluded fragments (12 with 1, 2 with 2). Again the relative frequencies depend on P~ but to calculate them, further assumptions have to be made. We will make 3: (a) That PI is a constant for all fragments irrespective of size or origin; (b) that fragments behave completely independently with respect to segregation; and (c) that each fragment leads to a single, visible MN; there is no degeneration and no fusion to influence frequency or distribution. Given these assumptions, we can readily show from binomial considerations that the relative frequencies of daughter cells with O: 1:2 MN a r e (PI) 2 + P I ( I - P I ) + 1/4(1-PI) 2 : PI(1-PI) + I/2(1-PI) 2 :
1/4(1
-
PI) 2.
0165-7992/88/$ 03.50 © 1988 Elsevier Science Publishers B.V. (Biomedical Division)
34 Similarly for cells with 3 and only 3 AF, the 128 daughter cells having 0:1:2:3 MN have relative frequencies: PI 3 + 1.5pi2(1-PI) + 0 . 7 5 P f f l - P 1 ) 2 + 0.125(1-PI)3 : 1.5PI2(1-Pt) + 1 . 5 P f f l - P I ) 2 + 0 . 1 2 5 ( 1 - P I ) 3 : 0 . 7 5 P f f I - P I ) 2 + 0.125(1-P1) 2 : 0 . 1 2 5 ( 1 - P I ) 3 and so on. At first sight, these distributions appear unusual, but they are, in fact, simple binomials. We can therefore generalise: Let F = 0.5 - 0.5P~. Then the MN distribution amongst all possible daughters of a cell having N fragments is the expansion of ( F + (1 - F ) ) 'v. In the case of mononucleate second-division cells Fcorresponds to the relative reduction, R, i.e. the ratio M N per daughter c e l l / A F per parent cell, given the simple assumptions used here. It is now a very easy matter to derive, for this ideal population, a mean frequency of M N / c e l l and its distribution, from any mean frequency and distribution of AF/cell. We choose a mean AF/cell and partition it to ceils with 0, 1, 2, . . . AF using a Poisson or some other pre-defined distribution. Each class frequency is doubled (2 daughters from 1 parent) and in turn partitioned into MN class frequencies using a chosen value of P~ to derive F. Summation then provides MN/cell, cells with MN and between-cell MN distribution - - expected in a truly random and unbiased sample of pure seconddivision cells derived from the original (and homogeneous) first-division AF bearing population. However, if only the mean MN frequency is required, then the pain of partitioning is unnecessary in such simple, hypothetical populations. Since Pj is a constant and directly related to F and R, M N / c e l l must be AF/ce[1 x F. If AF are distributed as a Poisson in parent cells, then, since we are summing a family of binomials with constant F, the resulting distribution of MN will also be Poisson. If the initial AF distribution is non-Poisson, which is often the case (Couzin and Papworth, 1979; Braselmann et al., 1986) the MN distribution will be non-Poisson, and partitioning will be necessary to obtain its form. Dose-response curves for M N / c e l l can be derived from those of AF/cell. Note that since each yield
YID
1.6 D. 1.2 Z
b S
1.0
2.0
3.0
4.0
/'
/
D J
R 0.3
z
0,8
LI,<
o.4 ~
/
O/
P, 0.4
/
F 0.3
~
(IF BN ceHs Pi 0.7 F 0.7)
1.0
2.0
3.0
4.0
D O S E (Gy)
Fig. 1. A hypothetical quadratic dose-response curve for radiation-induced acentric fragments (AF) and the derived micronucleus (MN) per mononucleatesecond-divisioncell when P~ = 0.4. See text for derivation methods. Inset: The plot Y/D with D giving the linear and quadratic coefficients for these two curves. If the MN curve was for binucleate (BN) cells than a Pz of 0.07 would be required.
of A F is reduced by the constant factor, R, the M N / c e l l curve has a lower slope. In particular, if AF/cell follows the familiar quadratic form, this reduced slope will be reflected in lower quadratic components, c~, /3. This is illustrated for a hypothetical case in Fig. 1. One of the sampling problems inherent in most populations is to obtain a pure sample of cells that have divided once, and only once, since clastogen treatment. Recently, several attempts have been made to overcome this difficulty (Joshi et al., 1982; Pincu et al., 1984; Fenech and Morley, 1985; Wakata and Sasaki, 1987). A relatively simple technique, gaining in popularity, is to use the agent cytochalasin B, which blocks cytokinesis without apparently affecting nuclear division, and results in binucleate (BN) cells. The scoring of MN is confined to BN ceils, since these must have divided once, and only once. Using the basic assumptions used for mononucelate cells and given a random and unbiassed sample of BN cells, we can apply the principles outlined to obtain estimates of M N / B N cell and
35 the b e t w e e n BN cell d i s t r i b u t i o n s . In this s i t u a t i o n we f i n d that F = PI so t h a t the p a r t i t i o n coefficients for a cell with N A F can be o b t a i n e d f r o m the e x p a n s i o n (Pl + (1 - P I ) ) n. F r o m this it follows t h a t , starting with the s a m e value o f AF/cell, the yield o f M N / B N cell will be 2 × MN/monon u c l e a t e cell, or l o o k e d at in a n o t h e r way, a c o m m o n r e d u c t i o n f a c t o r for m o n o n u c l e a t e a n d b i n u c l e a t e scores w o u l d indicate a c h a n g e in P~ (P~ = 1 - R for BN cells, 1 - 2 R for m o n o n u c l e a t e cells) (Fig. 1). U n f o r t u n a t e l y , we d o n o t have simple ideal p o p u l a t i o n s to deal with in the real w o r l d a n d so these c a l c u l a t i o n s c a n n o t be t r a n s l a t e d a n d a p p l i e d d i r e c t l y to a given e x p e r i m e n t a l set-up. I have p r e s e n t e d t h e m simply to show that o b s e r v a t i o n s o f low R, increasing d i s p a r i t y with dose a n d different shapes for AF and related MN d o s e - r e s p o n s e curves are n o t u n e x p e c t e d . S o m e o f the a s s u m p t i o n s I have m a d e for the c a l c u l a t i o n s are q u e s t i o n a b l e , a n d w o r t h y o f furt h e r investigation. P~ has a positive value in every system t h a t has b e e n used. In l y m p h o c y t e s it tends to be high, o f the o r d e r o f 0.8 (Pincu et al., 1984; C o u n t r y m a n a n d H e d d l e , 1976; F e n e c h a n d M o r l e y , 1985; Das a n d S h a r m a , 1987; c o n t r a S a s a k i a n d N o r m a n , 1967; B a u c h i n g e r et al,, 1986) a n d g e n e r a l l y lower (0.4-0.5) in f i b r o b l a s t s (Joshi et al., 1982; R o b e r t s et al., 1986; W a k a t a a n d S a s a k i , 1987). It is u n l i k e l y t h a t P~ is a c o n s t a n t irrespective o f size o r origin o f the f r a g m e n t a n d there are indications t h a t it m a y v a r y with dose ( C o n g e r , 1965). Ind e p e n d e n c y o f f r a g m e n t s is a n o t h e r p r o b l e m a r e a b o t h as r e g a r d s their s e g r e g a t i o n a n d their fusion in M N f o r m a t i o n . T h e r e is increasing evidence t h a t c h r o m o s o m e - t y p e A F d o s e p a r a t e a n d can be t r a n s m i t t e d singly (e.g. Das a n d S h a r m a , 1987). Several a u t h o r s have c o m m e n t e d on the p a u c i t y o f cells with m u l t i p l e M N (e,g. W a k a t a a n d Sasaki, 1987). F u r t h e r m o r e , n o t all f r a g m e n t s necessarily f o r m visible M N . H o m o g e n e i t y a n d u n i f o r m sensitivity o f the initial t a r g e t p o p u l a t i o n is a rare p h e n o m e n o n , a n d o b s e r v e d M N yields m a y v a r y with s a m p l e time even w h e n c o m p l i c a t i n g d i l u t i o n factors are
m i n i m a l (Brock a n d W i l l i a m s , 1985; R o b e r t s et al., 1986). C o n s e q u e n t l y , the m i t o t i c p e r t u r b a t i o n int r o d u c e d by a clastogen will p l a y h a v o c k b o t h with o b s e r v e d yields a n d with d i s t r i b u t i o n s (Savage a n d P a p w o r t h , 1973; C o u z i n a n d P a p w o r t h , 1979) so t h a t in the end, i n t e r p r e t a b l e a n d a n a l y s a b l e d o s e - r e s p o n s e curves m a y not be o b t a i n a b l e .
Acknowledgement I w o u l d like to t h a n k D a v i d P a p w o r t h for some h e l p f u l discussions on this topic.
References Bauchinger, M., E. Schmid and H. Braselmann (1986) Cell survival and radiation induced chromosome aberrations, II. Experimental findings in human lymphocytes analysed in first and second post-irradiation metaphases, Radiat. Environ. Biophys., 25, 253-260. Braselmann, H., M. Bauchinger and E. Schmid (1986) Cell survival and radiation induced chromosome aberrations, 1. Derivation of formulae for the determination of transmission and survival parameters of aberrations, Radiat. Environ. Biophys., 25, 243-251. Brock, W.A., and M. Williams (1985) Kinetics of micronucleus expression in synchronized irradiated Chinese hamster ovary cells, Cell Tissue Kinet., 18, 247-254. Carrano, A.V., and J.A. Heddle (1973) The fate of chromosome aberrations, J. Theoret. Biol., 38, 289-304. Conger, A.D. (1965) The fate of metaphase aberrations, Radiat. Botany, 4, 81-96. Countryman, P.I., and J.A. Heddle (1976) The production of micronuclei from the chromosome aberrations in irradiated cultures of human lymphocytes, Mutation Res., 41,321-332. Couzin, D., and D.G. Papworth (1979) The overdispersion between cells of chromosomal aberrations, J. Theoret. Biol., 80, 249-258. Das, B.C., and T. Sharma (1987) The fate of X-ray-induced chromosome aberrations in blood lymphocyte culture, Mutation Res., 176, 93-104. Fenech, M., and A.A. Morley (1985) Measurement of micronuclei in lymphocytes, Mutation Res., 147, 29-36. Joshi, G.P., W.J. Nelson, S.H. Revell and C.A. Shaw (1982) X-Ray induced chromosome damage in live mammalian cells, and improved measurements of its effects on their colony-forming ability, Int. J. Radiat. Biol., 41, 161-181. Pincu, M., D. Bass and A. Norman (1984) An improved micronuclear assay in lymphocytes, Mutation Res., 139, 61-65.
36 Roberts, C.J., G.R. Morgan and P.D. Holt (1986) A critical comparison of the micronucleus yield from high and low LET irradiation of plateau-phase cell populations, Mutation Res., 160, 237-242. Sasaki, M.S., and A. Norman (1967) Selection against chromosome aberrations in human lymphocytes, Nature (London), 214, 502-504. Savage, J.R.K., and D.G. Papworth (1973) The effect of variable G2 duration upon the interpretation of yield-time
curves of radiation induced chromatid aberrations, J. Theoret. Biol., 38, 17-38. Wakata, A., and M.S. Sasaki (1987) Measurement of micronuclei by cytokinesis-block method in cultured Chinese hamster cells: Comparison with types and rates of chromosome aberrations, Mutation Res., 190, 51-57. Communicated by R.J. Preston