Introduction

Subjecting cells to ionizing radiation during the G0/G1 phase of the cell cycle causes chromosome aberrations, through DNA breakage and large-scale rearrangement of the pieces. Such aberrations are informative about initial radiation damage, about DNA repair/misrepair pathways, and about cell nucleus ultrastructure. In addition, they are important to the main areas of applied radiobiology -- biodosimetry, cell killing during radiotherapy, and carcinogenesis risk estimation.

Models of aberration production. Biophysical models of aberration production are useful: they supply coherence to complex data sets; suggest mechanistically based extrapolation of information to different doses and/or different aberration types; and facilitate interrelation of experimentally distinct endpoints such as aberrations and mutations. For example, given a biophysical model, detailed information on simple and complex aberrations at one dose has implications for the frequency of simple aberrations at other doses.

Exchange-type chromosome aberrations involve two or more chromosome breaks (Savage 1976). The breaks will here be equated with DSBs (Cornforth 1998).¹Conventional wisdom in radiobiology, embodied both in the classic breakage-and-reunion model and the Revell model (Revell 1974, Savage 1998), is that almost all of the breaks involved are radiation-induced (Cornforth and Bedford 1993).

However, for several decades a different mechanism, recombinational repair (Chadwick and Leenhouts 1981, Resnick 1995), has also been considered. According to a current version of this third model (Chadwick and Leenhouts 1998), simple exchange-type aberrations, i.e. aberrations that involve two and only two breaks, are produced as follows. Radiation induces one DSB, which can next lead, in an enzymatic process, to a second break in an otherwise undamaged region of the genome having some DNA sequence homology with the region near the original DSB; a complete reciprocal exchange, i.e. a misrejoining with no free ends left over, can then occur. Thus only one of the two breaks involved is directly induced by radiation, and only one radiation track is involved in producing a simple exchange-type aberration. The model is therefore sometimes referred to as a `one-hit' aberration mechanism. In such a model, dose-rate effects and fractionation (i.e. split-dose) effects are usually attributed to saturable repair.

Significance and status of the recombinational-repair model. Recombinational repair is a linear no-threshold pathway. If this mechanism contributes substantially to aberration formation at laboratory doses, there are major implications. For relevant low-LET doses, the model predicts an approximately linear dose-response relation for simple exchange-type aberrations, rather than the approximately quadratic dose-response associated with two-track action (Kellerer 1985, Goodhead 1987, Sachs et al. 1997c). Having a linear dose response would affect the key question of how to extrapolate damage mechanisms quantitatively from laboratory doses to the much lower doses relevant for most risk estimation (Tucker et al. 1997, Brenner et al. 1998). A rationale given for the model has been the approximate linearity observed for dicentric dose-response curves when aberrations are induced by soft x-rays (overviews in Goodhead et al. 1993, Griffin et al. 1998).

However, the model is often regarded skeptically, for a number of reasons. With $\gamma$- and hard x-rays, typical dose-response curves for dicentrics are linear-quadratic rather than linear (Bauchinger 1995, Edwards 1997); this behavior holds even for photon energies as low as 5.4 keV (Roos and Schmid 1998); the deviations from quadratic behavior are at least partially due to the low-dose linearity which corresponds to some tracks making more than one DSB (Kellerer 1985, Goodhead 1987), and to distortion and saturation at higher doses (Norman and Sasaki 1966, Savage and Papworth 1969, Chen et al. 1997, Radivoyevitch et al. 1998). There is also direct evidence against one-hit models (Cornforth and Bedford 1993, Cornforth 1998). As far as molecular mechanisms are concerned, DNA sequence homology, associated with recombinational repair, is not required to produce chromosome aberrations in mammalian cells (Cornforth 1998, Jeggo 1998), although homology of short sequences may play an important role (Phillips and Morgan 1994); homologous repair, when it does occur, seems to be ineffective in producing translocations during G0/G1 (Richardson et al. 1998). Finally, the competing breakage-and-reunion model adequately approximates a considerable body of experimental results on aberrations (Sachs et al. 1997a).

Recently, some groups have nonetheless returned to the recombinational-repair model (Chadwick and Leenhouts 1998, Savage 1998). Among the evidence discussed, apart from the soft x-ray data mentioned above, has been the analysis of hard x-ray dose-response curves for FISH aberration patterns that are apparently simple, i.e. seem to be two-break exchange-type aberrations as far as analyzing colours and centromeres in a FISH experiment is concerned. In some cases apparently simple aberrations have increased less rapidly with dose than previously expected (Simpson and Savage 1996, Boei et al. 1998a), possibly suggesting a recombinational-repair type dose-response relation.

Visibly complex chromosome aberrations. A contemporary model must account for more than simple aberrations. Visibly complex interchanges (Savage and Simpson 1994), i.e. aberrations visibly involving at least three breaks on at least two chromosomes, are now copiously observed with FISH, in experiments with hard x-rays, soft x-rays, or other radiations (e.g. Griffin et al. 1995, 1998, Simpson and Savage 1996, Chen et al. 1995, 1996, 1997, Knehr et al. 1996, 1998, 1999, Wu et al. 1997a, Lindholm et al. 1998, Barquinero et al. 1998, Simpson et al. 1999, Ballarini et al. 1999). The recombinational-repair model does predict some visibly complex interchanges, resulting from superimposing two or more of its basic interactions described above (Savage 1998). In fact, one can prove that for one-colour (though not in general for multi-colour) painting², any observable aberration pattern allowed by the breakage-and-reunion model is also at least possible (though perhaps unlikely) according to the recombinational-repair model (Sachs et al. 1999).

However, the latter model's pathways for producing the most commonly observed visibly complex interchanges have very special requirements. The number of breaks, radiation-induced plus enzymatically-induced, required to produce a given visibly complex one-colour pattern is often larger than the minimum number required under the breakage-and-reunion model, and in addition the breaks must be situated in quite specific ways (Figure 1 gives an example). We reasoned that these stringent constraints might lead to severe underpredictions of the frequencies for visibly complex aberration patterns. To settle this numerical question requires quantitative modeling.

[Insert Figure 1 and its caption about here]

Testing models; randomness. It is possible to estimate the frequency of all kinds of apparently simple or visibly complex aberrations that a model predicts by assuming randomness, an assumption analyzed critically in the Appendix. Most papers on aberrations do use some randomness assumptions, explicitly or implicitly. Comparatively familiar consequences of randomness are those which hold, as approximations, for simple aberrations at low or moderate doses of sparsely ionizing radiations, up to several Gy. Examples include equality of `symmetric' and `asymmetric' aberrations (Savage and Papworth 1982, Lucas et al. 1996), Lucas' chromosome participation formula for whole-genome extrapolations of translocation frequencies measured for particular chromosomes with FISH (Lucas et al. 1992), a corresponding chromosome participation formula for centric rings (Appendix), etc. These standard consequences of randomness are model-independent in the sense that all three models mentioned above (breakage-and-reunion, Revell, and recombinational-repair) predict them as low/moderate-dose approximations.

In general, however, calculating the implications of randomness for aberration frequencies requires choosing one specific model. Then, approximating known deviations from randomness due to proximity effects in an oversimplified but consistent way (see Methods section), one finds modifications of the standard results at higher doses. For example there are systematic corrections to the Lucas chromosome participation formula (Appendix). One also obtains far-reaching interconnections and many different tests of the models. The standard results on chromosome participation in apparently simple aberrations can be extended to results on chromosome participation in the many kinds of visibly complex aberrations; frequencies of different kinds of aberrations in a particular experiment can be interrelated; dose-response relations can be predicted; data obtained with one particular FISH painting protocol can be compared in detail to data involving other FISH painting protocols or solid staining; etc. Monte Carlo computer simulations (Chen et al. 1995, 1996, 1997, 1998, Edwards et al. 1996, Ballarini et al. 1999) are needed for such calculations, which would otherwise be inordinately cumbersome.

In the case of the Revell model, calculations based on randomness have shown that the allowed aberration formation pathways, being highly constrained compared to the pathways of the breakage-and-reunion model (Edwards and Savage 1999), cannot produce nearly enough visibly complex aberrations of certain kinds, unless the constraints are in effect relaxed by clustering of DSBs along chromosomes, at Mbp or sub-Mbp scales (Sachs et al. 1999). For hard x-rays, more clustering is required than observations of DNA fragment sizes (e.g. in pulsed field gel electrophoresis experiments) seem to indicate.

For higher doses, calculations based on randomness also indicate that there are deviations from the conventional dose-response curves which have been used to argue for the recombinational-repair model. Pairwise misrejoining models like the random breakage-and-reunion model predict that even for low LET simple aberrations increase less rapidly than quadratically at doses higher than a few Gy (Savage and Papworth 1969, Sachs et al. 1997a, Boei et al. 1998a, Ballarini et al. 1999); at high enough doses simple aberrations are expected to increase only linearly, then sublinearly, and finally actually decrease as dose increases still further. This sub-quadratic behavior is related to distortion and to saturation. Distortion basically means that adding extra dose may transform simple aberrations into complex aberrations rather than adding more simple aberrations; the result is to slow and ultimately reverse the increase of truly simple aberrations, and even of apparently simple aberrations, as dose increases (Norman and Sasaki 1966, Savage and Papworth 1969); for example, if only one homologue-pair is painted no cell can have more than two apparently simple interchanges. In addition, saturation is expected, i.e. at sufficiently high doses the total number of misrejoined DSBs is expected to grow less rapidly than quadratically with increasing dose (reviews in Sachs et al. 1997c, Chen et al. 1997, Radivoyevitch et al. 1998).

Preview. These theoretical distortion and saturation effects will be quantified here. Monte Carlo simulations will be used to show that a random recombinational-repair dose-response curve for apparently simple aberrations is sub-linear at doses above a few Gy, due to distortion, not super-linear as one might have anticipated from the fact that apparently simple aberrations include some aberrations that are actually complex (Simpson and Savage 1996, Boei et al. 1998a). For apparently simple interchanges in fibroblasts (Simpson and Savage 1996) we will show that the breakage-and-reunion model gives a slightly more accurate dose-response relation than does the recombinational-repair model.

However, the most pronounced differences between the models concern visibly complex aberrations. We will show, assuming no DSB clustering along chromosomes, that the random recombinational-repair model, much like the Revell model, severely underpredicts the frequencies of certain visibly complex interchanges commonly observed with one-colour painting. In addition, for multi-colour painting, the recombinational-repair model fails to account for all the types of visibly complex patterns observed.