The fate of chromosome aberrations

J. theor. Biol. (1973) 38, 289-304

The Fate of Chromosome Aberrations ANTHONY V. CARRANot

AND JOHN A. HEDDLE$

Laboratory of Radiobiology, University of California, San Francisco, California 94122, U.S.A. (Received 20 March

1972)

Chromosomal aberrations rapidly disappear from populations of dividing cells, but little is known about the details of the process. One may ask, for example, whether a cell with an acentric fragment is virtually certain to die after the tirst mitosis or whether it has a high probability of surviving to the second. Some recent data on aberrations in cultured human lymphocytes lead to estimates that fragments (and presumably the cells containing them) survive to the next mitosis about 30% of the time and dicentric chromosomes about 50%. These estimates were made without regard for the proliferation of normal cells, however, and so must be somewhat in error. In fact, when cell proliferation is taken into account, the most likely value of survival of the fragment itself is about 80 % (when both daughter cells are considered). Probable ranges of this value and of the other parameters considered are presented. It is hoped that this explicit formulation of a mathematical model will encourage further experimental examination of the effect of various aberrations upon cell populations.

1. Introduction It has long been known that chromosomal aberrations induced either by radiation or chemicals are selected against in proliferating systems (Sax, 1941; Yu & Sinclair, 1967; Dewey, Furman & Miller, 1970; Wolff, 1972). The details of this process are, however, unknown. Neither the killing efficiency of a specilic aberration nor the rate at which a given aberration will effect cell death is known with certainty. In some cases it has been found that induced chromosomal aberrations persist for long times in proliferating systems both in vivo and in vitro (Puck, 1958; Greenblatt, 1961; Bender & Gooch, 1963 ; Buckton, Dolphin & McLean, 1967; Bloom, Neriishi, Kamada & Iseki, 1967). t Work performed under the auspices of the U.S. Atomic Energy Commission. Supported in part by NIH Biophysics Training Grant No. S-T’Ol-GMO0829. $ Present address: Atkinson College, Biology Department, York University, Toronto, Canada. 289

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In studies with human lymphocytes, Norman, Sasaki, Ottoman & Fingerhut (1966) as well as Buckton, Smith & Court Brown (1967) have utilized the rate of disappearance of induced chromosomal aberrations to determine the life span of these blood cells in uiuo. In connection with these aberrations, Sasaki & Norman (1967) estimated the probability that a given aberration will survive the fist or second division in cultured human lymphocytes by Rejolnq

G, Break Acentric

Met-93

fragment x

Ring with acentric fraqment

n

---

l

I

u

Illcentric with acentrlc f ragm=ent

LJ irt--

--x_x, ‘/

A

1 w-.+

Tricentric with two acentric froynents 1’ ,-T--TFIG.

N

1. Formation

I W-L+

t

n

//

of radiation

induced G1 aberrations.

//

Parent hetophase)

Daughters ( telophase)

1

II

III

2. Possible fates of acentric fragment at cell division. I. Loss of fragment to both daughter nuclei. II. Splitting of fragment and incorporation into one or both daughter nuclei. III. Incorporation of fragment into one daughter nucleus. FIG.

FATE

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ABERRATIONS

measuring the aberration frequencies at the first, second, and third postirradiation mitoses. They did not, however, take into account the fact that the observed frequency of aberrations at subsequent mitoses is affected not only by the probability of survival of the aberration, but also by the death of some cells with aberrations. The frequency of unaffected cells in the population would thus be enhanced. Although the measurements themselves were vital in determining the longevity of lymphocytes, the estimated aberration survival probabilities must have been somewhat in error. The aberrations that Sasalci & Norman (1967) studied were induced in the G, phase of the cell cycle (after mitosis, but before DNA synthesis had begun) and so were all of the chromosome type. Chromosome aberrations comprise dicentrics, rings, tricentrics, and acentric fragments (Fig. 1). For a more detailed discussion see Wolff (1961). The possible fates of acentric fragments are shown in Fig. 2. In the first case the fragment, lacking a centromere, can be lost to both daughter nuclei. In the second case, the fragment may split such that both daughter nuclei receive a fragment or that only a single daughter receives one with the corresponding loss of the other piece. The third case occurs when the fragment maintains its paired state and is transmitted to one or the other of the daughter nuclei. The data obtained by Sasakl8c Norman (1967) indicate that the second case occurs less than 3 % of the time, i.e., in second division cells, fragments were primarily present in a paired configuration. Thus fragment transmission appears to be an all or none process: either both pieces are transmitted to a single daughter or the fragment is lost. Fragments which are not incorporated into

Parent (metophase)

63

Daughters ( telophase)

I

II

III

FIG. 3. Possible fates of a dicentric chromosome at cell division (fragment not shown). I. Fall free. II. Interlocked bridge. III. Criss-crossed bridge.

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the daughter nuclei become micronuclei. These micronuclei are incapable of chromosomal replication or condensation, except for those few which contain a nucleolar organizer (McLeish, 1954; Das, 1962), so that the fragments are not observed at the next mitosis. The fate of dicentrics poses a different problem. Figure 3 illustrates the three possible telophase configurations of a dicentric, If the two centromeres in a chromatid move to the same pole at mitosis a fall-free dicentric will ensue (unless the two chromatids are interlocked). Thus each daughter cell will contain a dicentric chromosome. On the other hand if the two centromeres in a chromatid move to opposite poles, or the chromatids interlock, a bridge will form at anaphase. In this latter case, the bridge may rupture (with unknown effect on progeny survival) or cytokinesis may fail to occur leading to the death of the cell or to polyploidy. The consequences of these events will be discussed later. The probability that dicentrics fall free or bridge is approximately 0.5 for each as shown by Conger (1965) in Tradescuntiu and would be expected on the basis of independent centromere behavior (Heddle, Wolff, Whissell & Cleaver, 1967). Conger (1965) also found that ring chromosomes fall free about 50 % of the time. Tricentrics should be transmitted 25 % of the time since a tricentric is the equivalent of two dicentrics and will be so treated in our analysis. For the remainder of this paper the number of “dicentrics” includes also rings and tricentrics, the latter being counted as two dicentrics. Since fragments are usually transmitted to only one of the two daughter nuclei, for every cell observed at the second mitosis with a fragment, one cell missing a fragment must have been formed. If that cell survived to the second mitosis it would be scored as normal insofar as that fragment is concerned, i.e., the loss of a fragment would not be detected by the usual scoring methods. There may thus be a contribution to the “normal” cells at the second mitosis from cells having fragments at the first. The size of the contribution depends upon the probability of survival of cells that have lost fragments. If the fragment is not transmitted to either daughter, both could contribute to the normal class at the second division. In their analysis of three successive divisions in human lymphocytes Sasaki & Norman (1967) have estimated that roughly 70 % of the acentric fragments are lost at the tist cell division following X-irradiation in G, and the remaining 30% are distributed wholly to a single daughter cell. By taking into account the influence of cell death and enhanced normal cells, we have derived equations to describe expected frequencies of dicentrics and fragments at the second mitosis given their frequencies at the tist. We have then used the data of Sasaki & Norman (1967) (Table 1) to determine what values of our parameters would satisfy these equations.

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2. Derivation of Formulae In deriving the formulae to predict the frequencies of fragments or dicentrics at the second mitosis from the values obtained at the fist, three parameters have been introduced : (i) T, the probability of transmission of an acentric fragment as whole (III, Fig. 2) to one daughter nucleus (transmission to either one or the other daughter nucleus being 2T); (ii) P, the probability that a cell, having lost an acentric fragment, will survive to a subsequent mitosis; and (iii) W, the probability that each daughter cell will receive a dicentric chromosome (I, I Fig. 3) and survive to a subsequent mitosis. In addition,

the following assumptions have been made.

(1) Dicentrics at the first post-irradiation mitosis have a single fragment associated with them (Sax, 1940). (2) Dicentrics are distributed among cells randomly, i.e., the distribution is Poisson (Norman, 1967). (3) The distribution of independent fragments is also Poisson, independent of the dicentrics (Wolff, 1959). (4) A fragment passes as a whole to one or the other daughter nucleus or is lost, i.e., only one of the daughters may receive a particular fragment, and if it does there will be two identical fragments (paired fragment) at the next mitosis (Sasaki & Norman, 1967). The anaphase behavior of each fragment and dicentric is independent of any other fragment or d¢ric. (5) Cells with and without aberrations are equally likely to pass from the first mitosis to the second except for the effect of the anaphase behavior of the dicentrics or fragments. (6) The fragments associated with dicentrics are equivalent to the independently induced fragments, i.e., have the same probability of transmission (T) and the same lethal effect (1 -P). (7) The probability that a cell will survive the loss off fragments is Pf. (8) If a cell forms a bridge at the first mitosis, it dies before reaching the second-otherwise it survives to the second with a probability of 1, except for the possible effect of a fragment. (9) The probability that a cell will survive the effect of d dicentrics is Wd. There are experimental data to support assumptions (1) through (4) and somewhat less directly assumption (5) (Savage, 1967). Assumptions (6)

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through (9) are neither supported nor contradicted by much experimental evidence (to our knowledge). They are discussed later with respect to their validity and to their influence upon our conclusions. At the first mitosis all possible classes of cells may be described by two parameters, the number of dicentrics, L!, and the number of fragments, J The number of fragments is equal to the number of independently induced fragments, x, plus the number associated with dicentrics, d (assumption (l)), i.e., f= d+x. (1) If these are independent random events, as is widely accepted, then the probability of a cell with d dicentrics andffragments, C(d,f), is given by the product of the two Poisson distributions, one of dicentrics with a mean of B and the other of independent fragments with a mean of 1 (assumptions (2) and (3)). Thus, the probability that a cell has exactly d dicentrics and f fragments is e-DiJd W,f)=dr

(A)

GENERAL

FORMULA

FOR

e-Xx/-d --

THE SURVIVAL

(f-d)!

(2)

’

OF CELLS

TO THE SECOND

MITOSIS

Since the probability of surviving the lethal effects of a dicentric and fragment are independent (assumption (8)), we may consider them separately. Consider the first daughter cell. If there are f fragments, each of whose probability of transmission into the first daughter nucleus is T, the probability of obtaining r fragments (r = 0, 1,. . . , f) in the daughter nucleus is the rth term of the binomial [T+ (1 - T)lf :

f! (f-r)!r!

- T’(l-

T)/-’

if the fragments behave independently (assumption (4)). If a cell losing a fragment has a probability P of surviving and this probability is independent of the other fragments (assumption (7)), a daughter cell receiving r fragments will be missing f-r so that its probability of surviving isPf-’ . The general term for the surviving cells (i.e., the probability of survival of the first daughter nucleus when all possible numbers of lost fragments are considered) thus becomes

J! (f-r)!r!

. T’(l-

j’-)‘-‘Pf-‘,

which is the rth term in the expansion of [T+ P( 1 - T)]/. If we now consider the other daughter cell, the number of fragments it can receive depends upon the number, r, that have passed to the first daughter.

FATE

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Thus, the probability of transmission, y, to the second daughter is a conditional probability. The probability of non-transmission is 1 -y. The total probability that the second daughter will get the fragment is, thus, the product of the probability that the first cell did not get it times the conditional probability of transmission, i.e., (1 -T)y. The two daughter cells are equivalent, however, so (1 - T)J’ = T. (3) Therefore

T

(4)

Y=l-T and l-y=

l-2T E.

(5)

The conditional probabilities of obtaining second daughter is given by the binomial --

0, 1, 2,. . . (f-r)

fragments in the

T

l-T+ whose ith term is

A daughter cell which receives i fragments has lost f- i and so has a probability of survival of Pf - i. The general term for survival is thus

which may be rewritten as (f(f-r-i)!

r> !

i! (&)‘(syPf-r-iPT

which is the ith term for the expansion of P(l-27) +-l-T

J-rp.

1

This is the probability of the second cell surviving given that the first received r fragments. The probability that the latter occurred is

f! (f-r)!r!

T’(l-

T)f-'

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so that the total probability

AND

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of the second cell surviving is

(6)

which is the expansion of [TP+T+P(~

-2~)lf

= [T+P(~

- T)ls,

(7)

i.e., the second daughter is equivalent to the first daughter as it should be. The total production of cells is thus 2(T+P-TP)f from a cell having f fragments, independent of the effect of dicentrics. This term is dependent only on T, P, andfand is the same for all cells havingffragments independent of the number of dicentrics they contain. Of course, the dicentrics influence both the survival and the number of fragments that a cell contains. From equation (2), the probability of obtaining C(d, f) equals e-bjjd -.-=

e-XX/-d

d!

e-6e-x __--

(f-d)!

(8)

f!

Since the probability of a dicentric failing to produce a bridge is W, the probability that the cell will survive d dicentrics is Wd (assumption (9)) and the total survival becomes

Since

is the general term for the expansion of ( WD + I)f, the survivors of all cells (considering loss due to both dicentrics and fragments) is f 2(T+P/=o (B)

GENERAL

FORMULA

FOR

TP)f EF THE

SURVIVAL

(Wa+x)/. OF

DICENTRICS

TO

THE

SECOND

MITOSIS

To determine the number of dicentrics that would survive, each surviving cell must be weighted by the number of dicentrics it contains. Since the effect

FATE

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297

ABERRATIONS

of fragments is independent of the effect of dicentrics, the number of surviving dicentrics is

fzo

wf d$o Wf)Wdd

W+P-

= i 2(T+P-TP)f f=O

d& Gg;

Wdd

=f~02(T+P-TP)fd~o~~~~~i(W~)dXf-d~.

(II)

But, --d;;;+

-dd = WD

( W4)dxf

.

13

(f-l>! (w@d- ‘Jff -d (d - 1)!(f- d) !

which is the general term for the expansion of

(12)

wa(wB+x)f-l

except for the d = 0 term which, however, equals 0. Thus, the formula which is valid for d 2 1 and, thus, forf2 1 gives the correct value because the term in f = 0 (i.e., d = 0) is 0. Th e t o t a1 number of surviving dicentrics is, therefore, ftl

2(T+P-

TP)”

e31! (f-l)!

WD(WD+X)f

-1.

0 3)

Thus, the predicted frequency of dicentrics at the second mitosis is the ratio equation (13) /(IO) or f J:l

2(T+P-TP)fWDe~(wD+X)” -__~-.. f 2(T+P-TP)f .r = 0

$2

2e-De-XW&T+p-TP)

2e-De-x

(WD+X)f y+p-*P) f-‘(W~+v=.’ /=I (.f-. ..-.~ I>! f (T+P- TywD+Af

-~~~. (14)

f=O

Inspection

of the terms reveals that

~~~(T+p--Tp)f-l(w~+~)f-l=

(f-l)!

fgo(T+P-

Tp)fLwD+x)"

j-!

(15)

'

so that the predicted frequency of dicentrics, E(D), is E(D) = Wi& T + P - TP).

T.B.

(16)

21

298 (C)

A. GENERAL

V.

FORMULA

CARRANO FOR

AND

THE

J.

SURVIVAL

A. OF

HEDDLE FRAGMENTS

TO

THE

SECOND

MITOSIS

In order to predict the number of fragments surviving, each surviving cell must be weighted by the number of fragments it contains. The number of fragments surviving with the first daughter cell becomes

f! (f-r)!r!

fT(f-l)! =(f-r)!(r-l)j

T’(l-T)‘-‘I’“-‘r

T’-‘(1

-T)‘-‘PS-‘,

(17)

which is the general term for the expansion of fTIT+P(l-T)]l-l

=fT(T+P-TP)‘-’

(18)

except for the r = 0 term. When r = 0 f! (f-r)!r!

T’(1 - T)f -‘pf -‘r = 0

for all ft T, and P so that the sum is the same. For second daughter cell, the number of fragments surviving becomes i(f - r)! (f-r-i)!i!

= (f - r)P’

(&y

& (

(ggy-‘-iPf-‘-ip

~~~~(~'!,,!

(T$~e1

(~l~m'wip~-'-i

>

=(fvr)P'

SF (

+T+p(;Iy) >c

1

I-'-l.

(20)

This is the conditional probability that the second cell survived with i fragments given that the first received r fragments. The combined probability is

which is the general term for the expansion of f-1 jfr+P(;Iy)]

(1-T))

=fT(T+P-TPIS-l

(22)

and which is identical to the result from the first daughter cell, as it should be. For the two daughter cells combined, the contribution to surviving fragments is 2fT(T+P-TP)/‘-‘.

FATE

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299

ABERRATIONS

This depends only on T, P, and f, and is, therefore, a constant for all cells having f fragments. The contribution to surviving fragments of all cells is

fzo YW+PTP)I-’

i

C(d,f)Wd

d=O

= -f 2fT(T+p-Tp)f-‘L f!

f=O

=

i?

f=l

2Te-“e-x

u---l)!

i

f!-

d=,,

[ d!(f-d)!

WQdXf-d

e-‘Je-X I

(T+P-TP)f-l(WD+X)f

(23)

because the value is zero when f = 0. The predicted frequency of fragments at the second mitosis is equation (23)/(10) or m 2Te-“emx (T+P-TP)f-1(WB+X)f-‘(WD+8) c /=I u--l>! m ze-ae-x (T+P-TP)/(Wb+X)’ c ___ f=O f! (WD+X)T 2 (T+P-TP)f-‘(WB+X)f-’

(f-l)!

f=l

f$o (T+P-Inspection

Tq)I(wD+~)’

-’

(24)

of the terms reveals that

m (T+P-TP)f-‘(WD+X)f-’ cf=1 (f-l>!

= fEo (T+P--

TJ’j;W’~+~lf,

(25)

so that the predicted fragment frequency becomes (WD+W)T.

Each fragment transmitted has two chromatids so that at the second mitosis each gives two fragments (one paired fragment). The frequency of fragments expected at the second division, E(F), then becomes E(F) = 2( WB + x) T.

(26)

3. Application of Formula Both E(F) and E(D) are linear functions in T, P, and W so their equations can be readily solved for the different parameters. In order to accomplish this, reasonable values of T, P, and W must be considered. Because P is a

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probability of survival, 0 I P _< I. I+‘, the probability of having a dicentric fall free and surviving to the next mitosis, can also vary in the range of 0 to I. Since T is the probability of transmission of a fragment to one daughter cell. it can assumevalues in the range 0 to 0.5. (A fragment can survive by being transmitted to either daughter so that the probability of transmission, 2T, varies from 0 to 1.) We have computed the most probable values of T, P, and W using the observed frequencies. The probable ranges have been computed by using the observed values plus or minus one standard error in the combinations giving the most extreme values. Since at least two values enter into each computation, the probable ranges should approximate the 95 ‘I;, confidence intervals. TABLE

1

Aberration frequencies in cultured human peripheral lymphocytes at the first and secondmitosis after 500 rads of X rays (Data of Sasaki and Norman, 1967) Aberration

type

Tricentrics Dicentrics centric Rings [ acentric

Post-irradiation First

mitosis Second

10 240 26 22

1 100 18 2

0 440

4 3

0 200

93 1.57

Acentric fragments Total no. of cells Frequency of dicentrics’ Frequency of independent fragments2 Frequency of fragment? (Independent-k those from dicentrics)

1.43io.09t 0.88zkO.O7t

0~76~.0~07t I .271:0.09-t

-

f Standard errors from the Poisson formula. 1 Dicentrics+centric rings+Z(tricentrics) total no. cells 2 Acentric unpaired rings+acentric unpaired fragments -frequency of dicentrics. total no. cells 3 Acentric unpaired rings+acentric unpaired fragments+2(acentric paired rings+ acentric paired fragments). total no. cells

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301

Initially,

we consider the limitations imposed on the ranges of T, P, and by graphical analysis utilizing the mean values from the data (Table 1) and their standard errors. The results of such an analysis are presented in Table 2. The standard errors limit T and W to approximately one-half of their theoretical range and P to about threequarters of its range. Since there are three parameters and only two equations, it is not possible to obtain a unique value for any of these. An example of the interaction of these variables can be obtained by utilizing the most probable values (observed) of the data and assuming various values of W. For example, rearranging equation (26) we obtain W. This is easily obtained

E(F)

(27)

T = 2(FvD+x)’

Rearrangement

of equation (16) yields p = E(D) - T WD Ws(l-T) ’

(28)

By substituting the mean value of the empirical data and varying W in the above two equations we obtain the results in Table 3. If approximately onehalf of the dicentrics bridge and these cells are lost, i.e., W= 0.5, then P N 1.0 and 2T z O-78. The implications are discussed below. TABLE 2

Limitations on range ofparameters T, P and W Parameter T

W P

Limited range 0.24 5 TI

0.44

Graph utilized W vs. T from equations (16) and (26)

0.48 I20~$0.88 0.45 5 w 2 1

0.281P11

W W

vs. T from equation (16) vs. T from equation (16)

TABLE 3

Most probable values of T and P Assumed value of II’ 0.53 0.60 0.70 1 .o

T 0.39 0.37 0.34 0.28

P 1.0 0.82 0.64 0.35

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4. Discussion In deriving the formulae several assumptions were made that must be considered. First, there exists the possibility that aberrations may kill a cell even if the cell does not lose a fragment or have a bridge. Second, the size of the fragment may differ for dicentric associated fragments and independent fragments and, consequently, the value of P may differ for each type. Third, the value of P may not be independent of the number of fragments lost. Examination of these situations indicates that if any of the original assumptions were invalid, the result would be to produce a deflated set of values for the parameters T, P, and W. Thus the true values would be higher than those calculated. Since those acentric fragments associated with dicentrics are formed by the union of two separate fragments (Fig. I), it is reasonable to expect that the dicentric associated fragments would, on the average, be twice the length of the independent fragments. Then, if P is the probability of a cell surviving after losing a fragment, dicentric fragments might be potentially more lethal to the cells than independent fragments. Again, this would eliminate cells containing dicentrics preferentially from the population. One could predict from this that in the second or subsequent mitosis following irradiation, the mean distance between centromeres in surviving dicentrics would be increased if larger fragments do, in fact, possess a greater killing potential. This is currently under investigation. The assumption that cell survival followingf fragment losses is the probability of occurrence offindependent events may not be correct. For example, deletions of similar regions of homologous chromosomes may be more lethal than non-homologous regions because a cell may be able to survive with only a haploid amount of part of its genetic material but unable to survive without any copies of those genes. The probability of survival would not be P2 for two such fragments. Of course, the likelihood of this particular event is very small compared to the total number of possible events, but other synergistic interactions may occur. The parameter, W, is in fact the probability that a dicentric will fall free and that the daughter cells, each containing a dicentric, will not die from this aberration. From Conger’s data (1965), it is evident that a dicentric does fall free with a probability of about O-5. Assuming this holds for human lymphocytes as well as Tradescantia, from the range of W calculated here, the dicentric transmitted into the daughter cells has no effect on their survival to the next mitosis. However, Saccardo (1971) has obtained approximately an 85 % transmissibility of a persistent dicentric in Pisum, and Sears & Camara (1952) have only occasionally observed bridge formation from a transmissible dicentric in the root tip mitoses of common wheat. Presumably such dicentrics

FATE OF CHROMOSOME ABERRATIONS

303

are unusual and thus do not much influence the data during the first few divisions nor the overall rate of cell survival. In deriving the formulae it was assumed that cell survival and dicentric survival are one and the same, i.e., in the calculation of the most probable values it was assumed that fall free dicentrics are the only ones to survive and that these constitute half the dicentrics. It may be, however, that cells can survive without the dicentric if it ruptures or even is dragged into a daughter cell. Heddle & Scott (1970) have shown that the probability of bridge rupture is related primarily to the exchange frequency. Another complication may be the occurrence of a breakage-fusion-bridge cycle in which the ends produced by bridge rupture recombine to form a new dicentric. In the latter case, W would be greater than 0.5 even with random centromere movement. If nonrandom movement of centromeres occurs in short dicentrics, a possibility mentioned by Bender (1969), then again W might be greater than O-5. The probable range we have computed is compatible with these possibilities. Evidently, the initial assumptions minimize the values for T, P, and Win all cases. Equation (16) demonstrates a dependence on these three parameters for the calculation of E(D). It is noteworthy, however, that E(F) (equation (26)) is independent of P. The implication of these results to biological systems indicate that, at least from the first to the second division following G, irradiation, cells do not inevitably die from fragment loss, i.e., P does not approach 0, and, in fact, there may be very little death. Death from chromosomal aberrations may therefore be a result of the induced asymmetrical exchange (dicentric, ring or tricentric) with failure of cytokinesis due to interlocked or crisscrossed bridges as a possible mechanism of lethality. Death and cellular disintegration would then be observed to occur while the cell is in mitosis or in the subsequent interphase. Death due to loss of genetic material may not be evident until later generations when the cell no longer possesses essential gene products, having depleted its intracellular pool. This agrees to some extent with the results observed by time lapse cinemicrography in the first few post-irradiation divisions in tissue culture systems (Hurwitz & Tolmach, 1969; Marin & Bender, 1966). In spite of the strong reciprocal relationship between P and W, T demonstrates only a small variation in the range of the likely values. Further, the minimum probable value of 2T (0.56), the probability that a fragment will be transmitted, is almost double the original estimate. 5. Conclusions

The derived formulae for the frequencies of dicentrics and fragments at the second post-irradiation division indicate that cell death between the first

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HEDDLE

and second division must be principally due to dicentric bridges. If it is assumed that one-half of the dicentrics form a bridge then it is necessary to conclude that about 80 % of the acentric fragments are transmitted paired to either daughter cell and that the daughter cell not receiving the fragment will survive to the subsequent mitosis. Fragment loss may, of course, produce an effect in later generations. Thus, in subsequeut generations, as the dicentric frequency decreases, the effect of fragment loss may become more pro-

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