Robbing Peter to Pay Paul: Competition for Radiogenic Breaks During Rejoining Diminishes Curvature in the Dose Response for Simple Chromosome Exchanges

Abstract

The large majority of chromosome damage produced by ionizing radiations takes the form of exchange aberrations. For simple exchanges between two chromosomes, multi-fluor fluorescence in situ hybridization (mFISH) studies confirm that the dose response to X rays or gamma rays is quasilinear with dose. This result is in seeming conflict with generalized theories of radiation action that depend on the interaction of lesions as the source of curvature in dose-response relationships. A qualitative explanation for such “linearization” had been previously proposed but lacked quantitative support. The essence of this explanation is that during the rejoining of radiogenic chromosome breaks, competition for breaks (CFB) between different aberration types often results in formation of complex exchange aberrations at the expense of simple reciprocal exchange events. This process becomes more likely at high radiation doses, where the number of contemporaneous breaks is high and complex exchanges involving multiple breaks become possible. Here we provide mathematical support for this CFB concept under the assumption that the mean and variance for exchange complexity increase with radiation dose.

INTRODUCTION

Upward curvature in the dose response to sparsely ionizing radiations is a fundamental property of many biological end points. It is inextricably linked to radiobiological phenomena pertaining to the basic shape of dose-response relationships, and the biological response to changes in dose rate and dose fractionation. The most prominent biophysical explanation for this curvature is exemplified by the venerable theory of dual radiation action (TDRA) (1). Together with earlier-established cytogenetic theories (2, 3), it provides formal mathematical underpinnings for the ubiquitous linear-quadratic (alpha-beta) model Y=αD + βD², wherein the yield (Y) of lethal lesions is the sum of both a linear (αD) and quadratic (βD²) terms. The latter describes the characteristic upward curvature in the dose response to sparsely ionizing radiations, which is the focus of the current work. TDRA rests upon the principle of sublesion interaction in the context of cell killing. However, there is little doubt that its conceptual origins evolved much earlier from a consideration of chromosome aberration formation (4), whose dose-response relationships are most commonly described by the same equation.

This article represents the third of a three-part series that discusses various aspects of the βD² component as a source of curvature. The first part of this series dealt with the ability to discern marginal levels of curvature for exchange aberrations in response to gamma rays (5). In a second article, we scrutinized the usual assumption that lesions responsible for chromosome aberration and cell killing are necessarily distributed randomly among cellular target constituents according to Poisson statistics. By considering alternative distributions, we showed that the otherwise continuously-bending shape of the dose response to gamma rays could be made to approach a terminal exponential shape even though the mean was still described by the linear-quadratic function (6).

Here we seek to explain an unresolved issue in cytogenetics that has arisen since the implementation of combinatorial whole chromosome painting techniques like multi-fluor fluorescence in situ hybridization (mFISH) (7). Before such techniques came into widespread use, aberrations were visualized by staining mitotic figures with simple histological dyes such as Giemsa. The data from such experiments were consistent with the commonly held belief that virtually all exchanges between two chromosomes were simple reciprocal events involving the rejoining of precisely two radiogenic breaks. Evidence to the contrary (8) was largely overlooked until, using in situ hybridization techniques, it became possible to “paint” one, two or three whole chromosomes with colors that uniquely distinguish them from the remaining (unpainted) chromosomes in the genome. It soon became abundantly clear that many aberrations previously thought to be simple exchanges were not truly simple at all (9). Instead, they were complex, involving three (or more) chromosomes. This meant that many of the exchange aberrations previously classified as being simple were actually “pseudo-simple” complex rearrangements.

The biophysical interpretation of the shape of the dose-response curve for “bona fide” simple exchange aberrations became a topic of debate. Controversy reached a peak after Simpson and Savage (10, 11) made the bold prediction that the majority of curvature observed in the dose response (i.e., the βD² term) was due to complex exchanges. Because it simultaneously suggested that the dose response for truly simple exchanges was essentially linear, this claim was initially met with considerable skepticism (12). For one thing, it flew in the face of everything previously assumed about the formation of chromosome aberrations, particularly with respect to the lesion interaction models and the biophysical foundation that cytogenetics had always shared with TDRA. Nevertheless, when more sophisticated methods of analysis, such as 24-color mFISH, were brought to bear on the question, it was confirmed that Simpson and Savage had been essentially correct: Most all the curvature did come from complex exchanges. That ionizing radiation was later found capable of producing complex exchanges involving many chromosomes served to bolster the concept (13, 14). Since then, sophisticated modeling applied to mFISH data has shown that, while the dose response for true simple exchanges does contain scant curvature, it is, in fact, roughly linear (5).

It is important to recognize the implications of these collective results in the context of more generalized models of radiation action that have been proposed through the years, most notably those in fundamental conflict with TDRA. The apparent linearity harkened back to earlier suggestions that a single radiogenic break on only one chromosome was sufficient to induce an exchange (15), an idea that later gained momentum following the analysis of data from ultra-soft carbon K-shell X-ray experiments (16, 17). By extension, this line of reasoning led to the conclusion that curvature need not have anything to do with time-dependent pair-wise interaction of lesions (18), which is the central tenet of TDRA. Modifications made later to TDRA (19) helped to remove this source of conflict (20), though not entirely (21).

Here, we examine the plausibility of an alternative explanation for the quasi-linearity in question that is able to maintain harmony with the core concepts of TDRA. We suggest that the phenomenon results from a competition for rejoining among contemporary radiogenic breaks during exchange formation. A qualitative understanding of this explanation begins by imagining a constellation of three contemporary chromosome breaks, each on different chromosomes, whose six broken ends are all close enough in both time and space that they are capable of freely rejoining with one another. One possible rejoining outcome is that two of three chromosomes rejoin with one another, on the way towards forming a simple reciprocal exchange (Fig. 1A). This would leave the third chromosome with its two unrejoined broken ends, which can either join back together (restitute), remain unrejoined to form a terminal deletion or seek an illicit rejoining with distant open breaks elsewhere in the cell nucleus. If, on the other hand, a broken end from this third chromosome were to rejoin with breaks in either of the other two affected chromosomes, then, depending on the particular combination of pair-wise rejoinings that the cell chooses to make, the ability to form a simple exchange can be lost in favor of forming a complex 3-way exchange aberration (Fig. 1B). In that sense, one could say that the formation of the 3-way complex has “stolen away” breaks that might otherwise be destined to form a simple 2-way exchange, thus the title of this work, “Robbing Peter to Pay Paul”. The same would apply to more complicated complex exchanges, such as those involving the mis-rejoining of five breakpoints. One can imagine such a 5-way complex to form by “stealing away” breaks whose rejoining, if left undisturbed, would have formed a 4-way exchange. Similarly, the 4-ways form by sequestering breaks from would-be 3-ways, and so on. Since all these processes indirectly impinge on the formation of simple reciprocal exchanges, and since the relative abundance and complexity of complex aberrations is highly dose dependent (13), the net effect would be to reduce curvature in the dose response. We call this proposed mechanism the “Competition-for-Breaks” (CFB) hypothesis. The same concept was used to explain discrepancies between aberration data collected via 24-color mFISH, in comparison with data obtained by painting only a few chromosomes when the latter are extrapolated to full genome equivalency (22).

FIG. 1.

Alternative rejoining outcomes for a contemporary cluster of three initial radiogenic chromosome breaks. Leftmost elements in panels A and B represent the situation in G₁ phase of the cell cycle, which is where the exchanges actually form. Rightmost elements show the configuration at metaphase, where the exchanges are scored. Panel A: Constellation of three breaks in which rejoining is destined to produce a simple dicentric and associated compound acentric fragment. The dashed line connecting the two green chromosomes indicates that the fate of open breaks here is not yet committed. They could either remain open to yield a terminal deletion at metaphase, or (more likely) rejoin back together with one another (restitute). Panel B: Alternative scenario whereby breaks in the green chromosome have become involved in a three-way rejoining process that will produce a visibly complex aberration at mitosis. The instant that a broken end from the green chromosome commits to this complex rejoining, all possibility of producing a simple two-way exchange is eliminated.

In the current work, to test the plausibility of the CFB hypothesis, we quantitatively modeled exchange aberration formation as a probability distribution with a dose-dependent mean. This approach assumes that mean exchange complexity increases with radiation dose, and that the shape of this dose response, as well as the shape of the error distribution around the mean, affect the dose-response shapes for various aberration complexity classes such as simple or complex exchanges. For example, at low doses, where the mean aberration complexity is low, simple exchanges dominate. In contrast, at high doses, where the mean becomes high, complex exchanges with complexity index at or higher than the mean start to dominate. Simultaneously, the contribution of simple exchanges with complexity index below the mean declines because the sum of contributions of all aberration classes is defined to be 1.

We tested linear, quadratic and linear-quadratic dose dependences for the mean, and different error distributions (Poisson, negative binomial, and weighted negative binomial), on our data for exchange aberrations in irradiated human lymphocytes. The findings, described below, suggest this distribution-based approach is consistent with the data, providing intuitive support for the CFB concept.

MATERIALS AND METHODS

Cell Culture Conditions and Irradiations

Venous blood was obtained from a consenting healthy volunteer following procedures approved by the Institutional Review Board of the University of Texas Medical Branch. Blood was collected in heparinized BD Vacutainer tubes (Beckton-Dickinson, Franklin Lakes, NJ) and cultured in 5 ml of RMPI-1640 media in T25 tissue culture flasks. Cultures were irradiated with graded doses of ¹³⁷Cs gamma rays using a JL Shepherd & Associates Mark I-68A Irradiator (San Fernando, CA) at a dose rate of 85.6 cGy/min. Immediately after irradiation, cells were stimulated with phytohemagglutinin (Remel, Lenexa, KS) and mitotic cells were collected, fixed and spread onto slides 48 h later.

mFISH Analysis

Hybridization and karyotyping were conducted as described elsewhere (13). Briefly, fixed cells spread onto glass microscope slides were treated with acetone, RNase A and proteinase K before fixation in 3.7% formaldehyde. Slides were dehydrated through an ethanol series (70, 85 and 100%) and air dried before incubation in 72°C formamide (70%) in 2× SSC (0.3 M NaCl, 0.03 M sodium citrate) for 2 min to denature the chromosomal DNA. After dehydration through another ethanol series, 10 μl of denatured (10 min at 72°C) 24XCyte Human Multicolor FISH Probe (MetaSystems Group Inc., Boston, MA) was applied to each slide. Slides were covered with a 22 × 22-mm glass coverslip and sealed into position with rubber cement. Samples were allowed to hybridize for 48 h in a 37°C incubator. After hybridization, coverslips were removed, and the slides were washed for 2 min in 0.4× SSC containing IGEPAL (0.3%) nonionic detergent at 72°C. This was followed by a 30-s wash in 2× SSC (0.1% IGEPAL) at room temperature. Finally, 15 μl DAPI (MetaSystems) was applied to each slide and covered with a 24 × 40-mm coverslip.

Images of chromosome spreads were captured using a Zeiss Axio Imager.M1 epifluorescence microscope equipped with a CoolCube 1 digital high-resolution CCD camera (MetaSystems). Karyotypes were constructed from approximately 100 chromosome spreads using ISIS FISH Imagining System software (MetaSystems). Metaphase cells were analyzed by following previously established procedures (13). First, mPAINT descriptors were assigned to chromosomes involved in aberrations (23). Next, each rearrangement was brought to “pattern closure” by grouping elements in the most conservative way possible, minimizing the number of exchange breakpoints required. Reciprocal pair-wise rejoinings between one (as in the case of ring formation) or two chromosomes (i.e., dicentrics and translocations) were scored as simple exchanges. Exchanges involving three or more breakpoints were regarded as complex. This classification was also applied to incomplete exchanges where one or more elements failed to rejoin, as well as to so called “one-way” exchanges where one or more translocated segments of chromosome appeared to be missing, presumably because they were too small to be resolved by mFISH (24).

Aberration Complexity Metric (k)

For quantitative analysis, we tallied the number of breakpoints (BP) involved in each chromosome aberration on a cell-by-cell basis. We focused on exchanges, so terminal deletions were excluded. The resulting dataset consisted of exchanges with ≥2 BP per aberration. To simplify quantitative modeling of these data, we defined an exchange complexity index k = BP-2. Consequently, simple exchanges with 2 BP correspond to k = 0, 3-way exchanges correspond to k = 1, and so on. The distribution of k values at each radiation dose is provided in Table 1, and the fractional contributions of different k values to the total are provided in Table 2. Thus, the model developed below does not explicitly consider breaks from true terminal deletions as part of the competitive interaction process.

TABLE 1.

Summary of Aberration Complexities in Human Lymphocytes Irradiated at Different Doses

Dose (Gy)	Number of cells scored	Aberration complexity (k)
		0	1	2	3	4	5	6	7	8	9	10	15	Sum
0	49	1	0	0	0	0	0	0	0	0	0	0	0	1
0.25	99	9	0	0	0	1	0	0	0	0	0	0	0	10
0.5	100	8	1	0	0	0	0	0	0	0	0	0	0	9
1	103	28	1	2	0	0	0	0	0	0	0	0	0	31
2	100	58	12	3	1	0	0	0	0	0	0	0	0	74
4	100	162	43	18	12	4	1	1	2	0	0	0	0	243
6	143	336	85	62	21	18	11	10	6	1	1	1	2	554

Notes. The aberration complexity index k = BP-2, where BP is the number of breakpoints in a given exchange aberration. There were zero observations for k values from 11 to 14, so they are not shown in the table.

TABLE 2.

Fractional Contributions of Different Aberration Complexity (k) Values

Dose (Gy)	Aberration complexity (k)
	0	1	2	3 and 4	≥5
0	1.000	0.000	0.000	0.000	0.000
0.25	0.900	0.000	0.000	0.100	0.000
0.5	0.889	0.111	0.000	0.000	0.000
1	0.903	0.032	0.065	0.000	0.000
2	0.784	0.162	0.041	0.014	0.000
4	0.667	0.177	0.074	0.066	0.016
6	0.606	0.153	0.112	0.070	0.058

Note. The total of all fractional contributions at each radiation dose is 1.

Probability Distributions Used to Model Aberration Complexity

We used three different probability distributions, Poisson, negative binomial (NB) and weighted negative binomial (WNB), to model our data on aberration complexity (k), and compared the best fits of these distributions by information theoretic methods. According to the Poisson distribution, the probability P_Pois (k) of observing aberrations with complexity k is described by the following equation, where M(D) is the mean aberration complexity at radiation dose D:

$$P_{\text{Pois}}(k)=M{(D)}^k\times \exp [-M(D)]/k!].$$ (1)

We used the classic linear-quadratic (Table 3) function with a linear parameter a and a quadratic parameter b used to describe M(D). The notation using a and b was selected here for describing the dose response for aberration complexity to avoid confusion with the α and β notation used to describe the dose response for aberration frequency. To our knowledge, the issue of potential mechanistic justification for the dose-response shape for mean aberration complexity has not been explored previously in the literature. Due to lack of specific justifications for aberration complexity dose-response shapes, we considered the linear-quadratic (LQ) dependence, which has a mechanistic basis and long history in radiation biology. Specifically, the LQ formalism has a convenient mathematical form which approximates many more complex mechanistically-motivated radiobiological models (25, 26). The LQ function for aberration complexity dose response is parametrized as follows, where a and b are the linear and quadratic dose-response terms, respectively:

$$M(D)=a\times D+b\times D^2.$$ (2)

TABLE 3.

Best-fit Parameter Values for the Poisson, NB and WNB Distributions

Dose response for mean aberration complexity, for each distribution	Parameter	Meaning	Value	95% CIs		ΔAICc
Poisson
Linear	a	Linear dose-dependent coefficient for mean aberration complexity (Gy−1)	0.166	0.157	0.177	507.545
Quadratic	b	Quadratic dose-dependent coefficient for mean aberration complexity (Gy−2)	0.031	0.029	0.033	568.995
LQ	a	Linear and quadratic dose-dependent coefficients for mean aberration complexity	0.162	0.110	0.239	509.535
	b		0.001	0.000	0.010
NB
Linear	a	Linear dose-dependent coefficient for mean aberration complexity (Gy−1)	0.167	0.148	0.188	0.000
	r	Variance parameter	2.410	1.953	2.928
Quadratic	b	Quadratic dose-dependent coefficient for mean aberration complexity (Gy−2)	0.034	0.030	0.038	45.130
	r	Variance parameter	2.518	2.072	3.060
LQ	a	Linear and quadratic dose-dependent coefficients for mean aberration complexity	0.167	0.105	0.188	1.763
	b		0.000	0.000	0.002
	r	Variance parameter	2.409	1.982	2.928
WNB
Linear	a	Linear dose-dependent coefficient for mean aberration complexity (Gy1)	2.29 × 10−4	1.33 × 10−7	0.166	2.898
	r	Variance parameters	1.11 × 103	1.93	2.01 × 106
	q		0.448	0.000	2.73 × 105
Quadratic	b	Quadratic dose-dependent coefficient for mean aberration complexity (Gy−2)	4.43 × 10−5	0.000	2.56 × 10−2	64.726
	r	Variance parameters	1.08 × 103	7.68	1.53 × 105
	q		0.470	0.000	6.14 × 105
LQ	a	Linear and quadratic dose-dependent coefficients for mean aberration complexity	2.19 × 10−4	1.55 × 10−4	2.43 × 10−4	4.916
	b		0.000	0.000	9.93 × 10−4
	r	Variance parameters	1.17 × 103	8.52	1.60 × 105
	q		0.448	0.000	5.21 × 105

Note. As described in the main text ΔAICc values ≤6 represent similar degree of support from the data, whereas values >6 represent poor support.

The Poisson distribution is commonly used in radiological modeling, which assumes equality between variance and mean. We considered more complicated distributions with extra variance parameter(s). The NB distribution is frequently used for this purpose, and here we implemented a customized NB distribution parametrization. According to this distribution, P_NB(k) is the probability of observing aberrations with complexity k, which is described as follows, where Γ is the Gamma function, r is the “overdispersion” parameter and X = M(D) + 1/r:

$$P_{NB}(k)={(1/[r\times X])}^{1/r}\times {(M(D)/X)}^k\times \Gamma (k+1/r)/[\Gamma (1/r)\times k!].$$ (3)

If r ~ 0, there is no overdispersion and the variance and mean are equal, as in the Poisson distribution. On the other hand, if r > 0, the variance becomes greater than the mean and the ratio of variance to mean increases as the mean increases. M(D) in Eq. (3) is still described by Eq. (2).

We also considered the following WNB distribution with two variance parameters, q and r, where Y = k + 1/r:

$$P_{WNB}(k)=\frac{\left[r^k\times M{(D)}^k\times {[1+r\times M(D)]}^{-Y}\times (1+k\times q\times r)\times \Gamma (Y)\right]}{[(1+M(D)\times q\times r)\times \Gamma (1+k)\times \Gamma (1/r)]}.$$ (4)

Notably, the radiation-response function M(D) in Eq. (4) is still taken from Eq. (2), but it no longer represents the mean aberration complexity because the mean of the WNB distribution is also influenced by parameters r and q. The solution for the WNB mean is as follows:

$$WN{B}_{mean}=\frac{[M(D)\times (1+q\times r+M(D)\times q\times r\times (1+r))]}{[1+M(D)\times q\times r]}.$$ (5)

Model Fitting Procedure

We fitted each distribution (Poisson, NB and WNB) to the data from Table 1 by maximizing the log likelihood, using the sequential quadratic programming algorithm implemented in Maple 2020^® software. For the Poisson distribution, there were only two adjustable parameters, a and b, which determine the radiation dose dependence of mean aberration complexity [Eq. (2)]. For the NB distribution, there were three parameters: a, b, and the “overdispersion” parameter r. For the WNB distribution, there were four parameters: a, b, r and q. For each distribution, we also considered purely linear variants of the M(D) function [Eq. (2)] without the b × D² term, as well as purely quadratic variants without the a × D term.

All model parameters were restricted to positive values. This restriction was motivated by two reasons: 1. To maintain biological plausibility by preventing meaningless negative aberration complexity predictions at high doses; and 2. To enable the log likelihood maximization procedure to converge without errors by keeping all terms within the log function positive.

To maximize the probability of finding a global (rather than a local) optimum, each distribution-based model was refitted 1,000 times using random initial parameter values, and the best fit solution among all these attempts was selected as the optimum. Uncertainties (95% confidence intervals, CIs) were estimated for each adjustable parameter by profile likelihood.

The performances of each distribution-based model were compared using Akaike information criterion with sample size correction (AICc). AICc for the Mth modeled distribution (AICc_M) is given below, where Q_M is the number of adjustable parameters, LL_M is the maximized log-likelihood value for the Mth distribution, and N_tot is the total number of analyzed aberrations:

$$\begin{gathered} AIC{c}_M=-2\times L{L}_M+2\times Q_M \\ +2\times Q_M\times \left(Q_M+1\right)/\left(N_{tot}-Q_M-1\right). \end{gathered}$$ (6)

The model with the lowest AICc value (AICc_min) was considered to be best supported among the evaluated formalisms. The relative likelihood of the Mth model, called the evidence ratio (ER_M), can be expressed as:

$$E{R}_M=\exp \left[-\frac{1}{2}\times \Delta AIC{c}_M\right],\Delta AIC{c}_M=AIC{c}_M-AIC{c}_{min}.$$ (7)

ΔAICc_M values > 6 correspond to ER_M values < 0.05. For this reason, models with ΔAICc_M values > 6 are often considered to have significantly worse support than the best evaluated model.

RESULTS

Figure 2A shows, as a function of dose, the absolute frequencies of aberrations with different degrees of complexity, k, defined as breakpoints required for the formation of the exchange (BP) minus two. The k = 0 class consists of exchanges formed from the rejoining of two breakpoints (simple exchanges), while k = 1 includes three break complex exchanges and so on. Figure 2B shows the combined dose response for all exchange aberration types irrespective of complexity.

FIG. 2.

Absolute frequencies for exchange aberrations of varying complexity (see text). Panel A: Overall dose response for all categories of exchange. Panel B: Dose response for all exchange aberration types irrespective of complexity. Error bars are standard errors of the mean.

Visual inspection of these data suggests that the yield of simple exchanges (k = 0) initially increases with dose with upward curvature (i.e., positive second derivative) up to approximately 2 Gy. However, this curvature becomes scant, possibly even negative, at higher doses (4–6 Gy). In contrast, very complex exchanges (k ≥ 5) exhibit upward curvature throughout the tested dose range. Intermediate complexity classes expectedly show intermediate behaviors. As might also be expected, total aberration yields appear to increase with upward curvature at lower doses, with curvature diminishing at higher doses.

Parameter values and information theoretic support for the Poisson, NB and WNB fits to the data are shown in Table 3. Comparison of these results revealed the following findings. The Poisson error distribution was dramatically worse at describing these data (by >500 AICc units) than the NB and WNB distributions. The overall best-supported model variant (with ΔAICc = 0) was the linear NB fit. However, linear-quadratic NB, as well as linear or linear-quadratic WNB variants, could not be excluded with confidence because AICc score differences between them were small (<6 AICc units). Visual comparisons of the linear NB and Poisson distribution-based models are shown in Figs. 3 and 4.

FIG. 3.

Comparison of best-fit Poisson and NB distribution-based model predictions (curves) with observed data (symbols) for different aberration complexity classes from k=0 to k=4. In this and the subsequent figures, model parameters are listed in Table 3, and error bars on the data points represent 95% CIs, estimated using the score confidence interval method for binomial proportions (34).

FIG. 4.

Comparison of best-fit Poisson and NB distribution-based model predictions (curves) with observed data (symbols) for different aberration complexity classes with k ≥ 5.

In these model fits, we assumed no complex aberrations to occur in nonirradiated cells. To quantify the uncertainties associated with this assumption, we also fitted model variants where an extra adjustable parameter (intercept, c) was allowed to represent the mean aberration complexity in nonirradiated cells. In these model comparisons, the best-supported model with ΔAICc = 0 was the quadratic NB variant with c = 0.202 (95% CI: 0.111, 0.352), b = 0.023 (0.017, 0.029) Gy⁻² and r = 2.363 (1.974, 2.915). The second-best model with ΔAICc = 1.78 was the linear-quadratic NB variant with c = 0.168 (0.026, 0.345), a = 0.040 (0.000, 0.165) Gy⁻¹, b = 0.017 (0.000, 0.029) Gy⁻² and r = 2.397 (1.972, 2.913). Consequently, the uncertainty about the intercept value was considerable, but introducing it did not alter the main findings that the NB distribution described the data much better than the Poisson distribution.

Taken together, these results provide strong evidence for overdispersion of the aberration complexity distribution relative to Poisson. Although the details of the dose response for the mean aberration complexity could not be clarified using the data and methods described here, it appears that this dose response could be consistent with linearity.

Visualization of NB model behaviors over a wide dose range, from very low to very high doses (Fig. 5), illustrates dose-dependent shifts in the contributions of different aberration complexity classes. At low doses (<1 Gy), the best-supported model variant predicts that most exchange aberrations are simple (k = 0). At intermediate doses, the contribution of simple exchanges decreases, and the contributions of 3-way exchanges (k = 1) peak. At increasingly higher doses, the contribution of complex aberrations progressively increases.

FIG. 5.

Best-fit NB distribution-based model predictions for fractional contributions of different aberration complexity classes as function of radiation dose.

A dose-dependent increase in mean aberration complexity (Fig. 6) suggests that as dose increases, an increasing number of exchanges become complex (those with k values close to and above the mean), whereas the relative contribution of simple events (those with k values below the mean) gradually decreases with dose. Overall, our approach of modeling aberration complexity as a probability distribution with a dose-dependent mean supports the CFB concept. At each radiation dose, different aberration complexity classes “compete” for a finite number of breaks because the sum of fractional contributions of all aberration classes is by definition restricted to unity.

FIG. 6.

Comparison of best-fit NB distribution-based model predictions for mean aberration complexity as function of radiation dose (curve) with observed data (symbols). Error bars represent 95% CIs, estimated based on the normal distribution.

DISCUSSION

It is our position that the results of this investigation provide proof-of-principle support for the basic concepts embodied in the CFB hypothesis. We are well aware that accepting this premise forces a more nuanced view of the β × D² term as the source of curvature in the linear-quadratic model. Rather than representing lesion interaction between just two radiogenic lesions (breaks), we are instead forced to view it as an exceedingly complex interactive term that encompasses competitive rejoinings among multiple breakpoints on multiple chromosomes. It follows that the biophysical meaning of the β × D² term, as commonly envisioned, is a gross oversimplification that (perhaps coincidentally) approximates processes leading to curvature in the acute dose response for low-linear energy transfer (LET) exposures. In the context of CFB, the αD term continues to retain a general sense of harmony in relationship to the linear-quadratic model, which is to say that it represents the formation of exchange aberrations produced by two breaks caused by the passage of a single charged-particle track (in this case, electrons resulting from photo-absorptive events). However, it is clear that when viewed in the context of CFB, the classical biophysical interpretation of the β × D² term no longer retains its previous tangible meaning of representing the coming together of two breaks produced by two independent tracks.

Having said this, we hasten to add that the aforementioned argument does not negate the proposition that dose-rate and dose-fractionation effects involve fundamental time-dependent factors based on the pair-wise rejoining among radiogenic breaks. For independently-produced exchange aberrations, a minimum of two breaks are still necessary. While the formation of complex exchanges requires additional breaks, their rejoining (or mis-rejoining) still occurs in binary fashion. And since most initial radiogenic breaks disappear with time, either through restitution or mis-rejoining with other breaks, there will be diminishingly fewer potentially interacting pairs (or clusters) after radiation exposure. Thus, fewer exchanges (regardless of complexity) will be produced per unit total dose as the dose is protracted. Finally, this more “nuanced view” of β × D² term that would be forced by adopting the principles of CFB in no way detracts from its proven usefulness of the linear-quadratic model in predicting biological response to radiation, as numerous examples attest (27–31).

As for the shape of the dose-response relationship for total exchanges (Fig. 2), it should be noted that earlier researchers have offered a variety of explanations for the fact that upward curvature diminishes with increasing dose (32, 33). It should be pointed out, however, that these explanations were offered in the context of cytogenetic techniques that were largely incapable of detecting complex aberrations.

Our distribution-based analysis could not fully discern the details of the shape of the dose response for the mean aberration complexity (Table 3). For example, linear or linear-quadratic dependences with NB or WNB distributions achieved similar AICc support. The fact that chromosome aberrations figure prominently in cell killing, mutagenesis and carcinogenesis, suggests that the implications of the dose-response shape for mean aberration complexity could be far-reaching and warrant further investigation.