Appendix: Randomness

Overview. The basic theoretical results on randomness in chromosome aberration production can be obtained by using contingency tables based on chromosome arm lengths (Savage and Papworth 1982). For the special, comparatively simple, case of two breaks, dominant at moderately low doses of sparsely ionizing radiations, the contingency table approach leads to various explicit equations, discussed below. In applications to high doses and/or complex aberrations, computer generalizations (Chen et al. 1995, 1996, 1997) of the explicit equations are needed, and the implications of randomness become model-dependent, e.g. are different for the breakage-and-reunion compared to the recombinational-repair model. The rest of this Appendix gives some details on the assumptions, consequences and empirical status of randomness, attempting inter alia to clear up some confusions which have arisen.

Randomness assumptions. It is sometimes useful to distinguish between randomness of DSB induction and randomness of misrejoining, related to the distinction between DSB allocation to regions in the genome and chromosome participation in aberrations (Savage 1991). Randomness of DSB induction would mean that just after an acute dose the average number of DSBs in any portion of the genome is proportional to the size (i.e. the genomic content, measured in Mbp). Averages over large DNA stretches, corresponding to the limit of resolution of the aberration assay, are meant; for example, whether randomness of DSB induction holds at the nucleosome level, for stretches as short as 200 bp, is not directly relevant in the usual aberration assays; even non-randomness in a substantial portion of a chromosome arm compared to other portions of that same arm, as suggested by some aberration experiments (Johnson et al. 1999, Xiao and Natarjan 1999a) and mutation experiments (e.g. Zhu et al. 1996), would usually not influence the frequencies scored in a FISH assay provided the average over the whole arm is representative of the genome as a whole.

Randomness of misrejoining would mean that the probability of DSB free ends interacting is the same regardless of which free ends are involved. Deviations from randomness of misrejoining occur due to proximity effects which bias for misrejoining of DSBs initially formed close together. Proximity effects bias for rings relative to simple interchanges (Savage and Papworth 1973), for insertions relative to S&S types 2F and 2G, and against complex interchanges relative to simple interchanges (Sachs et al. 1997a). By using interaction sites (Kellerer 1985, Savage 1996) these deviations from randomness can be systematically (though only approximately) incorporated into the calculations (Chen et al. 1996, 1997). Complete randomness of reactive DSB induction and misrejoining will now temporarily be assumed; later we outline changes in the predictions caused by taking proximity effects for misrejoining into account. Completeness will also be assumed.

Theoretical results for the case of one reactive DSB. The simplest situation is when aberration assays are used to estimate the number of breakpoints (i.e. DSBs which misrejoin rather than restituting) in a particular region of the genome. Randomness would mean that the average number of breakpoints is directly proportional to the size of the region observed (review in Johnson et al. 1999). Specifically, suppose in one experiment a portion of the genome is painted with a FISH colour, the size relative to the size of the whole genome being $f_a$. Suppose in an otherwise identical experiment a different fraction, $f_b$, of the genome is painted. Then randomness predicts the following simple proportionalities for the observed breakpoint frequencies $F_a$, $F_b$, and for the frequency $F_G$ of breakpoints in the whole genome:

$$\frac{F_a }{F_b} =\frac{f_a }{f_b}, \qquad \frac{F_G}{F_a} =\frac{1}{f_a} .$$ (1)

Theoretical results for the case of two reactive DSBs. The most frequent applications of randomness in the literature are to exchanges involving two and only two DSBs in situations where more complicated interactions are negligible. The basics are the following.

Symmetric and Asymmetric Aberrations. In this special case of exactly two DSBs, randomness of misrejoining implies equality for the frequencies of `symmetric' and `asymmetric' aberrations (Savage and Papworth 1982), specifically

dicentrics=translocations,   centric rings=pericentric inversions, (2)

$$\mbox {acentric rings=paracentric inversions}.$$ (3)

Translocations and Dicentrics. Further implications of randomness in the case of just two DSBs can be obtained by using proportionality to size for each of the two chromosome arms involved (Savage and Papworth 1982). We illustrate the results by considering several special cases, starting with translocations.

For translocations observed in one-colour painting, equations corresponding to Eq. 1 are (Lucas et al. 1992)

$$\frac{F_a}{ F_b} =\frac{f_a (1-f_a )}{ f_b (1- f_b )},\quad \frac... ...}^{46} f_i (1- f_i )} {2 f_a (1 - f_a )} \approx \frac{ 1}{2.05 f_a (1 - f_a )}$$ (4)

Here $f_a$ and $f_b$ are again the fractions of the genome painted in two experiments; $F_a , \; F_b ,$ and $F_G$ are again frequencies; and $f_i$ is the fractional size of the $i^{th}$ chromosome. The factor $\sum f_i (1- f_i )$ in the equation can be interpreted in terms of intrachanges (rings and inversions), as discussed below. The equality

$$\frac{ \sum_{i=1}^{46} f_i(1-f_i)}{2} \approx \frac{1}{2.05 },$$ (5)

used in Eq. 4 is approximate; it is based on human chromosome multiplicity and sizes. For most other organisms the number replacing 2.05 is also slightly larger than 2.

Eq. 4 also applies to dicentrics in the case where one homologue pair is painted, provided the convention is made that a dicentric involving both homologues is counted as contributing once to $F_G$ but not contributing to $F_a$ at all.

The reasoning behind Eq. 4 is the following (Savage and Papworth 1982, Lucas et al. 1992). With a fraction $f_a$ of the genome painted, a fraction $1-f_a$ is counterstained. For two DSBs, an observed interchange (i.e. an interchange involving a junction between FISH paint and counterstain) requires that one of the two DSBs is on the painted portion (probability $f_a$) and the other DSB is on the counterstained portion (probability $1-f_a$). This argument gives the equation for $F_a / F_b$. To get the formula for $F_G$, similar reasoning is applied to the chromosomes one at a time (Lucas et al. 1992), taking into account that each interchange involves two different chromosomes.

An Example. For example, each homologue of human chromosome 2 is about 255 Mbp in size, while the total for all 46 chromosomes is about 6370 Mbp. Thus if one homologue were painted, $f_a \approx 255/6370 \approx 0.04$. In the more realistic case of both homologues painted, $f_a \approx 0.08$. If the homologue were painted the frequency of observed translocations involving this homologue would, according to Eq. 4, be approximately 7.9% of the frequency for translocations in the entire genome, i.e.

$$F_a \approx 2.05 \times 0.04 \times 0.96 \times F_G \approx 0.079 F _ G$$ (6)

If both homologues are painted the equation gives approximately $2.05 \times 0.08 \times 0.92 \times F_G$ $\approx 0.151 F_G$. This latter number can also be obtained from Eq. 6 by adding the translocations for both homologues, as follows

$$2 \times 2.05 \times 0.04 \times 0.96 \times F_G - 2 \times 2.05 \times (0.04)^2 \times F_G = 2.05 \times 0.08 \times 0.92 \times F_G .$$ (7)

Note here that the there is an extra term, $2 \times 2.05 \times (0.04)^2 \times F_G$, which corrects for the fact that a translocation involving both of the homologues is not scored at all, rather than being scored twice.

Intrachanges. Formulae similar to Eq. 4, obtained by similar reasoning, hold for intrachanges (rings and inversions) (Sachs et al. 1993, Wu et al. 1997b, Lucas et al. 1999). We next state the results for rings but by using Eq. 2 analogous equations are obtained for inversions, or for the sum of rings and inversions. Let $p_i$ and $q_i$ denote the lengths of the short and long arms of the $i^{th}$ chromosome. Thus $f_i = p_i + q_i$ and for humans $i = 1, ... , 46$. Then for centric rings, with $F_j$ the frequency for the $j^{th}$ chromosome and $F_G$ that for the whole genome, one has

$$\frac{F_i }{ F_j} =\frac {p_i q_i }{p_j q _ j}, \qquad \frac{F_G }{ F_j} = \frac {\sum _ {i=1} ^ {46}p_i q_i} {p_j q_j }.$$ (8)

For acentric rings the same expressions hold provided $pq$ is everywhere replaced by $(p^2 + q^2 )$ (Wu et al. 1997b, Sachs et al. 1997b). When considering the sum of acentric and centric rings, the corresponding factor is $(p + q )^2$. The factor $(p + q )^2$ can be used to interpret the sum in Eq. 4 in terms of a correction factor for intrachanges. The correction factor accounts for the fact that if two breaks are on one chromosome the result of misrejoining is a ring or inversion rather than a dicentric or translocation. In equations, the sum obeys the following equality:

$$\sum f_i (1 -f_i ) = \sum f_i - \sum f_i^2 = 1 - \sum (p_i + q_i )^2 .$$ (9)

In the case under consideration, where there are just two DSBs, the arguments sketched above can be extended to a randomness prediction of the frequency for centric rings compared to dicentrics (or, equivalently by Eq. 2, pericentric inversions compared to translocations); for example for the whole genome randomness predicts the frequency ratio (Savage and Papworth 1973; Hlatky et al. 1992)

$$= \frac{2 \sum p_i q_i }{ \sum f_i (1- f_i)} \approx 1/86 .$$ (10)

If just one homologue pair is painted, the prediction for (painted centric rings)/(dicentrics showing a paint signal) is, for an average-sized chromosome, about half of that given in Eq. 10. In the equation the approximate factor 86 applies to normal human cells; for other cells it can be worked out from the equation provided the genome parameters $p_i$, $q_i$, and $f_i = p_i + q_i$ are given.

It is well known, however, that there are actually far more centric rings relative to dicentrics than Eq. 10 predicts (Savage and Papworth 1973, Hlatky et al. 1992), due to proximity effects.

Proximity effects and interaction sites. Proximity effects can typically be incorporated into a model by using `sites' or by using `interaction distances' (Kellerer 1985). Site models are presumably less realistic than interaction distance models but typically involve fewer adjustable parameters. Conceptually, sites are subregions of the cell nucleus within which interactions can take place (survey in Savage 1996). For example, CAS incorporates proximity effects quantitatively by dividing the nucleus into some number $S$ of sites, with interactions possible only between reactive DSBs in the same site (see the Methods section above). For whole-genome scoring, Eq. 10 is replaced in a site model, approximately, by

$$\approx 1/2N \approx S/86.$$ (11)

where $N$ is the average number of chromosomes contributing to one site and the second approximate equality holds for human cells provided, as in CAS, no chromosome contributes to more than one site. The remaining equations above are not altered.

Theoretical results for more than two DSBs: computer extensions of the equations. At doses of several Gy or higher, exchanges involving three or more breaks occur with significant frequencies. Then the above equations require some corrections, and in addition results for visibly complex aberrations are needed. Even for three breaks explicit equations similar to equation 4, or contingency tables, though putatively applicable, become enmeshed in a very complicated welter of contingencies. To make matters worse, the number of reactive breaks in an interaction site is not fixed at two or three or any given number. For low LET it is presumably governed by a Poisson distribution whose average is proportional to the genomic content of the site, and for high LET more complicated statistical distributions are needed (Kellerer 1985, Chen et al. 1997). In addition, the implications of randomness are in general model-dependent. Assuming doses so low that the frequency of complex aberrations is negligible, the results above (e.g. Eqs. 2 - 4, 8, 10, and 11) are valid no matter whether we assume the random-breakage-and-reunion model, the random recombinational repair model, or a random Revell-type exchange model. However, at higher doses, the predictions of the different models diverge.

Because of all these complications, explicit equations are problematical at higher doses. The only explicit equation currently known is that under the breakage-and-reunion model one has the following generalization of part of Eq. 2 (Lucas et al. 1996):

$$.$$ (12)

However, Monte Carlo computer simulation is a fully adequate substitute for explicit equations. One lets the computer make aberrations, cell by cell, using the probabilities implied by randomness and an aberration model. When enough cells have been simulated, numerical substitutes correcting Eqs. 2, 4, 8, 10, 11, and similar equations are obtained; one also obtains numerical values for additional results, e.g. on complex aberrations. Tables 1 and 5 of the text are examples.

To indicate the magnitude of the corrections to explicit equations involved at higher doses, Figure 5 shows the fraction of apparently simple interchanges predicted for each chromosome by Lucas' chromosome participation formula 4 and by the random breakage-and-reunion model, at low, intermediate and high doses. It is seen that for low doses, the formula is accurate, but at 9 Gy the actual model gives significant deviations from the formula approximating the model. Even at 3 Gy there are perceptible deviations; these deviations are in the same direction as, but much smaller than, the experimental deviations seen by Barquinero et al., i.e. the actual model result calculated by CAS is slightly but not very significantly closer to experiment than the Lucas approximation.

[Insert Fig. 5 + captions about here]

Fig. 3B of the main text is an example of a CAS prediction, on visibly complex aberrations, too complicated to be obtained even approximately with any known explicit formula, though the contingency table approach (Savage and Papworth 1982) would presumably lead to the same result if it could be carried through. More generally, randomness is used, directly or indirectly, throughout tables 1-5.

Experimental results on randomness. There is strong evidence that various predictions of randomness modulated by proximity hold, at least within 50% or better (e.g. Savage 1976, Chadwick and Leenhouts 1981, Savage and Papworth 1982, Savage 1991, Kovacs et al. 1994, Finnon et al. 1995, Muhlmann-Diaz and Bedford 1995, Lucas et al. 1996, Chen et al. 1996, 1997, Sachs et al. 1997a, Boei et al. 1998a, Johnson et al. 1998, 1999). The predictions which have been found to hold approximately include Eqs. 1, 2, 4, 8, 11, their corrections by computer simulations at moderately large doses, Eq. 12, and various additional results, also obtained by computer simulations, for visibly complex aberrations (e.g. Table 3 and Fig. 3B).

However, there is also considerable evidence that some deviations from the randomness predictions may occur. As regards Eq. 12, which is the most clear-cut case because equality is expected even at doses so high that a significant fraction of aberrations are complex (i.e. even when distortion effects are significant), there have been repeated reports that an excess of apparently simple translocations is observed, with ratios of apparently simple translocations to apparently simple dicentrics ranging from about 1 to about 2. For example, the two recent, very large, data sets used in the main text above (Barquinero et al. 1998, Simpson and Savage 1996), both do show some excess (Tables 1 and 5), though by much less than the factor of about 2 found in some earlier, smaller data sets.

In addition the Lucas chromosome participation formula, Eq. 4, underpredicts, somewhat, observed apparently simple aberrations involving most small chromosomes, and correspondingly overpredicts apparently simple aberrations involving most large chromosomes, for some large data sets on human lymphocytes (Knehr et al. 1996, Barquinero et al. 1998, Cigarran et al. 1998). More precise CAS calculations give at 3 Gy results basically similar to the Lucas formula (Fig. 5), though marginally closer to the lymphocyte data (Table 5). The present analysis confirms a substantial tendency for Chr. 3 to underparticipate, as compared to expectations based on randomness, not only in apparently simple aberrations, as pointed out by Barquinero et al., but also in visibly complex ones (Fig. 3, Table 5). For the fibroblast data no clear trend for large chromosomes to underparticipate in aberrations is present (Table 1). Many other possible deviations from randomness have been discussed (e.g. Xiao and Natarjan 1999b), but at present (apart perhaps from hot spots near centromeres, telomeres, and intrachromosomal telomeric sequences) do not appear to constitute systematic patterns applicable to different data sets.

Intra-arm intrachanges. Comparatively little is known experimentally about intra-arm intrachanges (i.e. paracentric inversions and acentric rings). Much as in Eq. 10 above, randomness also predicts various intrachange frequency ratios, e.g.

$$= \frac{ 2 \sum p_i q_i }{\sum (p_i^2 + q_i^2 )} \approx 0.79.$$ (13)

Here the relation involving sums is the general result for any genome and the numerical value 0.79 again refers specifically to the human genome; an exactly metacentric genome would have ratio 1.0. The data sets considered here did not attempt to measure acentric rings or inversions systematically. Other data sets suggest that in humans the observed ratio in Eq. 13 is typically smaller than the predicted value, 0.79 (Wu et al. 1997b, Sachs et al. 1997b, Bauchinger and Schmid 1998); a similar discrepancy exists for the plant Tradescantia (Savage and Papworth 1998). The reason is believed to be additional proximity effects, which bias for smaller intrachanges relative to larger intrachanges, i.e. are superimposed on the proximity effects already incorporated into CAS as described above.

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Figure 1: Two models for a visibly complex interchange. Panel A shows three chromosomes; one chromosome is painted with a FISH colour (shown black); the other two are counterstained (shown white). Centromeres, here shown as circles, are assumed recognizable (e.g. with a pancentromeric stain) throughout the analysis. Three DSBs are shown; for example the first DSB has free ends $a$ and $a '$. All DSBs shown are assumed to be reactive, i.e. capable of misrejoining. According to the breakage-and-reunion model, one possible exchange is for the DSB free ends to misrejoin in the reaction [ $a'c, \; ab',\; bc'$]; this is a cycle of order 3 (Sachs et al. 1999). This reaction produces the final pattern shown in panel B. In a FISH experiment, only the two components showing a FISH signal are scored. Even so, the interchange recognizably involves at least 3 DSBs, i.e. the pattern is visibly complex, because the number of centromeres on the two scored components is wrong for a complete reciprocal exchange involving just two DSBs. The particular pattern in panel B, with two bicoloured final components having three centromeres, is designated 2G in the S&S aberration classification system (Savage and Simpson 1994).

The same 2G observed pattern can be made according to the recombinational-repair mechanism, but more breaks are involved. The next two panels give an example. Panel C shows two radiation-induced DSBs, $aa'$ and $bb'$. Suppose the first DSB, $aa'$, `attacks' a third, counterstained, chromosome at the otherwise undamaged location $cc'$ and a complete reciprocal exchange [ $a'c, \; a c'$], occurs; suppose in addition $bb'$ attacks on the same arm, further from the centromere, and the complete reciprocal exchange [ $bd', \; b'd$] occurs. The outcome of these exchanges is shown in panel D. The sizes of some of the final components are different than in panel B, but the observed aberration pattern is again 2G.

Figure 2: An incomplete pattern. The conventions are the same as in Figure 1. Panel A shows final aberration pattern 2F in the S&S classification. Because neither of the counterstained segments contains a centromere, the pattern is a visibly complex interchange - at least three DSBs must have been involved, and this fact is evident even if only final components which show a FISH signal are examined.

Panel B shows a pattern sometimes observed, designated incomplete type I (Simpson and Savage 1996) or pattern 2BI (Barquinero et al. 1998). The optimal reclassification of the pattern as a complete pattern is somewhat problematic. One plausible explanation is that the acentric painted fragment is actually attached to an acentric counterstained fragment, just as in panel A, but the counterstained fragment is too small to be seen. Then the S&S pattern would be 2F, as in panel A. However the observed pattern could possibly also indicate an incomplete version of S&S pattern 2B (Simpson and Savage 1996).

Figure 3: Scatter plots for predicted and observed chromosome participation. Observed values of the per-cell frequency of certain aberrations for particular chromosomes painted, compared to the values calculated with CAS using the breakage-and-reunion model and the parameters of Table 2. Panel A. Values for apparently simple interchanges in the SS4 data. The SS4 values are obtained from Table 1, dividing the `2A+2B' row by the `Cells' row. For apparently simple interchanges the CAS values are approximately equal to those calculated with the Lucas formula (Appendix). The line is the best-fitting straight line. From the values shown, a correlation coefficient of 0.92 (p=0.01) was calculated. This correlation coefficient, and other correlation coefficients for chromosome participation, are shown in Table 3. Panels B-D respectively show plots similar to that in panel A, but for TVC in Table 1, 2A+2B in Table 5, and TVC in Table 5 respectively.

Figure 4. Dose-response relations. Fibroblast data (Simpson and Savage 1996) for apparently simple interchanges (ASI=2A+2B, squares) and total visibly complex patterns (TVC, diamonds) are shown; aberration frequencies are summed for the 6 homologue pairs painted one pair at a time. In panel A the parameters of Table 2 are used for the random breakage-and-reunion model (BR, solid lines) and random recombinational-repair model (RR, dotted lines). The latter is quite innacurate for the visibly complex aberrations. If only apparently simple interchanges are considered, the constraints on the model parameters are relaxed (panel B); then the optimal BR curve, obtained by replacing $\delta$=2.3 with $\delta$=2.15, is quite close to the data.

For apparently simple aberrations the BR model curves are quadratic only at low doses, and the RR curves are linear only at low doses. The downward curvatures seen at higher doses are mainly due to distortion, i.e. to the indirect effects of complex aberration formation (Savage and Papworth 1969); in the BR case saturation also plays a role (Chen et al. 1997). For aberrations which are simple rather than just apparently simple, distortion effects are even stronger (curves not shown).

Figure 5: Theoretical corrections to the Lucas chromosome participation formula at higher doses. Squares show the results of Eq. 4 for the relative frequency of observed apparently simple translocations that involve a particular homologue pair in a human female genome. 23 experiments with equal cell numbers and with one homologue pair painted in each experiment are assumed. The total is 200% because two differently numbered chromosomes are involved in each observed translocation. Corrections are needed for Eq. 4 at doses higher than a few Gy. CAS calculations, assuming the random breakage-and-reunion model with the B(ii) parameters of Table 2, are given for 1.5 Gy (light solid line), for 3 Gy (heavy solid line), and for 9 Gy (dotted line). The lines connect discrete points to guide the eye. The differences between the actual model predictions and the approximation given by Eq. 4 are due to distortion and saturation; they increase as the dose increases, and are more pronounced for truly simple translocations (curves not shown) than for the case shown here of apparently simple translocations.