A polymer, random walk model for the size-distribution of large DNA fragments after high linear energy transfer radiation.

DNA double-strand breaks (DSBs) produced by densely ionizing radiation are not located randomly in the genome: recent data indicate DSB clustering along chromosomes. Stochastic DSB clustering at large scales, from > 100 Mbp down to < 0.01 Mbp, is modeled using computer simulations and analytic equations. A random-walk, coarse-grained polymer model for chromatin is combined with a simple track structure model in Monte Carlo software called DNAbreak and is applied to data on alpha-particle irradiation of V-79 cells. The chromatin model neglects molecular details but systematically incorporates an increase in average spatial separation between two DNA loci as the number of base-pairs between the loci increases. Fragment-size distributions obtained using DNAbreak match data on large fragments about as well as distributions previously obtained with a less mechanistic approach. Dose-response relations, linear at small doses of high linear energy transfer (LET) radiation, are obtained. They are found to be non-linear when the dose becomes so large that there is a significant probability of overlapping or close juxtaposition, along one chromosome, for different DSB clusters from different tracks. The non-linearity is more evident for large fragments than for small. The DNAbreak results furnish an example of the RLC (randomly located clusters) analytic formalism, which generalizes the broken-stick fragment-size distribution of the random-breakage model that is often applied to low-LET data.

Radiation-produced chromosome aberrations: colourful clues.

Ionizing radiation produces many chromosome aberrations. A rich variety of aberration types can now be seen with the technique of chromosome painting. Apart from being important in medicine and public health, radiation-produced aberrations act as colorful molecular clues to damage-processing mechanisms and, because juxtaposition of different parts of the genome is involved, to interphase nuclear organization. Recent studies using chromosome painting have helped to identify DNA double-strand-break repair and misrepair pathways, to determine the extent of chromosome territories and motions, and to characterize different aberration patterns left behind by different kinds of radiation.

Underprediction of visibly complex chromosome aberrations by a recombinational-repair ('one-hit') model.

PURPOSE: Published low-LET FISH data were used to test two models of chromosome aberration production based on breakage-and-reunion or recombinational repair. MATERIALS AND METHODS: Randomness of DNA double strand break induction and misrejoining is analyzed comprehensively and adopted as a working hypothesis. Proximity effects are approximated by using interaction sites. Model results are calculated using CAS (chromosome aberration simulator) Monte Carlo computer software with two adjustable parameters. CAS can emulate the specifics of any experimental painting protocol, allowing very detailed tests of the models. RESULTS: To reasonable approximation, breakage-and-reunion model predictions are consistent with low-LET FISH results, including two large, elaborate, one-paint data sets. An explicitly specified version of the recombinational-repair model severely underpredicts the frequency of the visibly complex aberration patterns most commonly observed with one-paint FISH, and is inconsistent with some observed multi-paint patterns. When high-dose effects (distortion and saturation) are taken into account quantitatively, a dose-response relation for apparently simple interchanges slightly favours the breakage-and-reunion model over the recombinational-repair model, despite being approximately linear over the dose range 2-6 Gy. CONCLUSIONS: The random breakage-and-reunion model gives comprehensive baseline predictions that are sufficiently accurate for the organization of experimental results. The data speak against complex aberrations being formed by the random recombinational repair pathway discussed here.

Random breakage and reunion chromosome aberration formation model; an interaction-distance version based on chromatin geometry.

PURPOSE: Using published FISH data for chromosome aberration production in human fibroblasts by hard X-rays to test a breakage-and-reunion model. METHODS: The model assumed pairwise misrejoining, random apart from proximity effects, of DNA double-strand break (DSB) free ends. CAS (chromosome aberration simulator) Monte Carlo computer software implementing the model was modified to use a distance algorithm for misrejoining instead of using DSB interaction sites. The modification (called CAS2) allowed a somewhat more realistic approach to large-scale chromatin geometry, chromosome territories and proximity effects. It required adding a third adjustable parameter, the chromosome territory intersection factor, quantifying the amount of intertwining among different chromosomes. RESULTS: CAS2 gave somewhat better results than CAS. A reasonable fit with a few discrepancies was obtained for the frequencies at three different radiation doses of many different aberration types and of aberrations involving various specific chromosomes in a large data set using one-paint FISH scoring. The optimal average chromosome territory intersection factor was approximately 1.1, indicating that, for an arbitrarily chosen location in the nucleus, on average slightly more than two chromosomes have very nearby loci. Without changing the three parameter values, a fit was also obtained for a corresponding, smaller, two-paint data set. CONCLUSIONS: A random breakage-and-reunion model incorporating proximity effects by using a distance algorithm gave acceptable approximations for many details of hard X-ray aberration patterns. However, enough discrepancies were found that the possibility of an additional or alternate formation mechanism remains.

Locations of radiation-produced DNA double strand breaks along chromosomes: a stochastic cluster process formalism.

Ionizing radiation produces DNA double strand breaks (DSBs) in chromosomes. For densely ionizing radiation, the DSBs are not spaced randomly along a chromosome: recent data for size distributions of DNA fragments indicate break clustering on kbp-Mbp scales. Different DSB clusters on a chromosome are typically made by different, statistically independent, stochastically structured radiation tracks, and the average number of tracks involved can be small. We therefore model DSB positions along a chromosome as a stationary Poisson cluster process, i.e. a stochastic process consisting of secondary point processes whose locations are determined by a primary point process that is Poisson. Each secondary process represents a break cluster, typically consisting of 1-10 DSBs in a comparatively localized stochastic pattern determined by chromatin geometry and radiation track structure. Using this Poisson cluster process model, which we call the randomly located clusters (RLC) formalism, theorems are derived for how the DNA fragment-size distribution depends on radiation dose. The RLC dose-response relations become non-linear when the dose becomes so high that DSB clusters from different tracks overlap or adjoin closely. The RLC formalism generalizes previous models, fits current data adequately and facilitates mechanistically based extrapolations from high-dose experiments to the much lower doses of interest for most applications.

Clustering of radiation-produced breaks along chromosomes: modelling the effects on chromosome aberrations.

PURPOSE: For high-LET radiations, and perhaps even for hard X-rays, DNA double-strand breaks (dsb) are clustered nonrandomly along chromosomes; disproportionately, many inter-dsb segments are less than a few Mbp (10(6) base pairs). The implications of such dsb clustering for chromosome aberrations are analysed. METHODS: Chromosome segments between different dsb within one dsb cluster are assumed too small to detect in the aberration assay. Enumeration or Monte-Carlo computer simulations are used to compute the relative frequencies of many observable aberration patterns: apparently simple or visibly complex. The theoretical predictions are compared with X-ray data for human fibroblasts, involving painted chromosomes 1, 2, 4, 5, 7 or 13. RESULTS AND CONCLUSIONS: Surprisingly, cryptic dsb multiplicity does not affect the frequency ratios predicted for aberration patterns by a random breakage-and-rejoining model. The model is generally consistent with current data on many different types of aberrations, whether or not dsb usually occur in cryptic clusters. For a Revell-type exchange model, however, the predictions do depend on clustering configurations; they gradually approach the predictions of the breakage-and-rejoining model as average cluster multiplicity increases. The model is consistent with the data, for example with the ratio of visibly complex to apparently simple aberrations, only if there is considerable dsb clustering even at low-LET, with approximately 1.5 or more reactive dsb per cluster on average.

The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships.

One of the fundamental tools in radiation biology is a formalism describing time-dose relationships. For example, there is a need for reliable predictions of radiotherapeutic isoeffect doses when the temporal exposure pattern is changed. The most commonly used tool is now the linear-quadratic (LQ) formalism, which describes fractionation and dose-protraction effects through a particular functional form, the generalized Lea-Catcheside time factor, G. We investigate the relationship of the LQ formalism to those describing other commonly discussed radiobiological models in terms of their predicted time-dose relationships. We show that a broad range of radiobiological models are described by formalisms in which a perturbation calculation produces the standard LQ relationship for dose fractionation/protraction, including the same generalized time factor, G. This approximate equivalence holds not only for the formalisms describing binary misrepair models, which are conceptually similar to LQ, but also for formalisms describing models embodying a very different explanation for time-dose effects, namely saturation of repair capacity. In terms of applications to radiotherapy, we show that a typical saturable repair formalism predicts practically the same dependences for protraction effects as does the LQ formalism, at clinically relevant doses per fraction. For low-dose-rate exposure, the same equivalence between predictions holds for early-responding end points such as tumor control, but less so for late-responding end points. Overall, use of the LQ formalism to predict dose-time relationships is a notably robust procedure, depending less than previously thought on knowledge of detailed biophysical mechanisms, since various conceptually different biophysical models lead, in a reasonable approximation, to the LQ relationship including the standard form of the generalized time factor, G.

Computer simulation of data on chromosome aberrations produced by X rays or alpha particles and detected by fluorescence in situ hybridization.

With fluorescence in situ hybridization (FISH), many different categories of chromosome aberrations can be recognized-dicentrics, translocations, rings and various complex aberrations such as insertions or three-way interchanges. Relative frequencies for the various aberration categories indicate mechanisms of radiation-induced damage and reflect radiation quality. Data obtained with FISH support a proximity version of the classic random breakage-and-reunion model for the formation of aberrations. A Monte Carlo computer implementation of the model, called the CAS (chromosome aberration simulator), is generalized here to high linear energy transfer (LET) and compared to published data for human cells irradiated with X rays or 238Pu alpha particles. For each kind of radiation, the CAS has two adjustable parameters: the number of interaction sites per cell nucleus and the number of reactive double-strand breaks (DSBs) per gray. Aberration frequencies for various painted chromosomes, of varying lengths, and for 11 different categories of simple or complex aberrations were simulated and compared to the data. The optimal number of interaction sites was found to be approximately 13 for X irradiation and approximately 25 for alpha-particle irradiation. The relative biological effectiveness (RBE) of alpha particles for the induction of reactive DSBs (which are a minority of all DSBs) was found to be approximately 4. The two-parameter CAS model adequately matches data for many different categories of aberrations. It can use data obtained with FISH for any one painting pattern to predict results for any other kind of painting pattern or whole-genome staining, and to estimate a suggested overall numerical damage indicator for chromosome aberration studies, the total misrejoining number.

Intra-arm and interarm chromosome intrachanges: tools for probing the geometry and dynamics of chromatin.

Many chromosome-type, exchange-type chromosomal aberrations produced by radiation are intrachanges, i.e. involve only one chromosome. It is assumed such intrachanges are formed by illegitimate reunion of two double-strand breaks (DSBs) on the chromosome. The yield of intra-arm intrachanges (acentric rings or paracentric inversions) relative to that of interarm intrachanges (centric rings or pericentric inversions) is larger than would occur if production and illegitimate reunion of DSBs were spatially random. The excess of intra-arm intrachanges is presumably due to proximity effects for illegitimate reunions, i.e. enhancement of the intrachange probability when two DSBs are formed close to one another. Radiation track structure may also play a role. Using a polymer description for "large-scale" chromatin geometry (>2 Mb), and using two alternate (rapid or slow motion) models for the way that DSBs move after they are produced, theoretical estimates are given for size distributions of intrachanges at low or high linear energy transfer (LET). The ratio of intra-arm to interarm intrachanges is derived from the size distribution and compared with data from the literature on centric rings, inversions, interstitial deletions and excess acentric fragments. Proximity effects enhance yields of intra-arm relative to interarm intrachanges at least severalfold and perhaps as much as 10-fold compared to expectations based on spatial randomness. We argue that further measurements of intra-arm and interarm intrachanges would be informative about large-scale chromatin structure and chromosome motion. Because inversions are more frequent than estimates of randomness would indicate, and are transmissible to daughter cells, their size distribution could also help characterize past exposure to high-LET radiation.

A convenient extension of the linear-quadratic model to include redistribution and reoxygenation.

PURPOSE: At present, the linear-quadratic model for cellular response to radiation can incorporate sublethal damage repair and repopulation. We suggest an extension, termed LQR, to include also the other two "Rs" of radiobiology, cell cycle redistribution, and reoxygenation. METHODS AND MATERIALS: In this approach, redistribution and reoxygenation are both regarded as aspects of a single phenomenon, which we term resensitization. After the first portion of a radiation exposure has decreased the average radiosensitivity of a diverse cell population by preferentially sparing less sensitive cells, resensitization gradually restores the average sensitivity of the population towards its previous value. The proposed LQR formula is of the same form as the original LQ formula, but with two extra parameters, an overall resensitization magnitude and a characteristic resensitization time. The LQR model assumes that resensitization is monotonic rather than oscillatory in time, i.e., always tends to increase average cellular sensitivity as overall time increases. We argue that this monotonicity assumption is likely to hold in clinical situations, though a possible extension is discussed to account for oscillatory decay of resensitization effects. RESULTS: The LQR model gives reasonable fits to relevant experimental data in the literature, reproducing an initial rise in cell survival, due to repair, as the treatment time is increased, followed by a resensitization-related decrease in survival due to redistribution and/or reoxygenation for treatment times of the order of the cell cycle time, and a final survival increase due to repopulation as the treatment time is increased still further. CONCLUSION: The LQR model is a simple and potentially useful extension of the LQ model for computing more realistic isoeffect relations for early responding tissues, including tumors, when comparing different radiotherapeutic protocols.

Chromosome aberrations produced by radiation: the relationship between excess acentric fragments and dicentrics.

Most chromosome aberrations produced by ionizing radiation develop from DNA double-strand breaks (DSBs). Published data on the yield and variance of excess acentric fragments after in vitro irradiation of human lymphocytes were compared with corresponding data on dicentrics. At low LET the number of excess acentric fragments is about 60% of the number of dicentrics, independent of dose and perhaps of dose rate, suggesting that dicentrics and excess acentric fragments arise from similar kinetics rather than from fundamentally different reactions. Only a weak dependence of the ratio on LET is observed. These results are quantified using generalizations of models for pairwise DSB interactions suggested by Brewen and Brock based on data for marsupial cells. By allowing singly incomplete and some "doubly incomplete" exchanges, the models can also account for the experimental observation that the dispersion for excess acentric fragments, a measure of cell-to-cell variance, is systematically larger than the dispersion for dicentrics. Numerical estimates of an incompleteness parameter are derived.

Influence of time-dependent stochastic heterogeneity on the radiation response of a cell population.

A solid tumor is a cell population with extensive cellular heterogeneity, which severely complicates tumor treatment by therapeutic agents such as ionizing radiation. We model the response to ionizing radiation of a multicellular population whose cells have time-dependent stochastic radiosensitivity. A reaction-diffusion equation, obtained by assuming a random process with the radiation response of a cell partly determined by competition between repair and binary misrepair of DNA double-strand breaks, is used. By a suitable transformation, the equation is reduced to that of an Ornstein-Uhlenbeck process so explicit analytic solutions are available. Three consequences of the model's assumptions are that (1) response diversity within a population increases resistance to radiation, that is, the population surviving is greater than that anticipated from considering an average cell; (2) resistant cell subpopulations preferentially spared by the first part of a prolonged radiation protocol are driven biologically into more radiosensitive states as time increases, that is, resensitization occurs; (3) an inverse dose-rate effect, that is, an increase in cell killing as overall irradiation time is increased, occurs in those situations where resensitization dominates effects due to binary misrepair of repairable damage. The results are consistent with the classic results of Elkind and coworkers on extra cell killing attributed to cell-cycle redistribution and are in agreement with some recent results on in vitro and in vivo population radiosensitivity. They also generalize the therapeutic paradigm that low dose rate or fractionated radiation can help overcome hypoxic radioresistance in tumors.

A Monte Carlo/Markov chain model for the association of data for chromosome aberrations and formation of micronuclei.

The micronucleus assay is a convenient, in situ method for observing cell damage resulting from exposure to clastogenic agents and has been widely used as a dosimeter of human exposure to radiation or chemicals. It also is a complement to the classic clonogenic cell survival assay in that it can be used to examine radiation damage vs dose as a function of cell type or radiation quality. Digitized imaging densitometry was conducted on CHO cells that have undergone one division and in which further cytokinesis was blocked to collect data on the distributions of percentage total cellular DNA per micronucleus and frequency of micronuclei per cell after gamma irradiation in G1 phase. Theoretical counterparts to both classes of distributions were generated by a Monte Carlo double-strand breakage (DSB) simulation to the CHO genome, followed by simulated repair of this initial damage using a Markov chain algorithm that assumes linear restitutions of single DSBs complete with quadratic, incomplete exchanges among pairs of DSBs. Micronuclei were presumed to consist of single acentric fragments (including fused acentric pairs). The empirical distributions, when compared to their fitted theoretical counterparts, suggest, inter alia, that: (1) a slight dependence of micronucleus size on dose exists, with a trend toward higher density in the 2-4% genome range, at the expense of the 0-2% range, with increasing dose; (2) the probability of exchange incompleteness is at least 20%; and (3) the dispersions of the micronucleus frequency distributions are progressively lower than their (essentially constant) counterparts with increasing dose. Suggested is a cooperative increase in the number of fragments per micronucleus with increasing dose. Beyond these specific results, however, it is clear that furthering the understanding of the connection between DNA aberrations and formation of micronuclei would further link these two large bodies of data.

Polymer models for interphase chromosomes.

The overall geometry of chromosomes in mammalian cells during interphase is analyzed. On scales larger than approximately 10(5) bp, a chromosome is modeled as a Gaussian polymer subjected to additional forces that confine it to a subvolume of the cell nucleus. An appropriate partial differential equation for the polymer Green's function leads to predictions for the average geometric length between two points on the chromosome. The model reproduces several of the experimental observations: (i) a square root dependence of average geometric distance between two marked chromosome locations on their genomic separation over genomic length scales from approximately 10(5) to approximately 10(6) bp; (ii) an approach of the geometric distance to a maximum value for still larger genomic separations of the two points; (iii) overall chromosome localization in subdomains of the cell nucleus, as suggested by fluorescent labeling of whole chromosomes and by radiobiological evidence. The model is also consistent with known properties of the 30-nm chromatin fiber. It makes a testable prediction: that for two markers a given number of base pairs apart on a given chromosome, the average geometric separation is larger if the configuration is near one end of the chromosome than if it is near the center of the chromosome.

DNA damage in non-proliferating cells subjected to ionizing irradiation at high or low dose rates.

Ionizing radiation damage to the genome of a non-cycling mammalian cell is analyzed using continuous time Markov chains. Immediate damage induced by the radiation is modeled as a batch Poisson arrival process of DNA double strand breaks (DSBs). Different kinds of radiation, for example gamma rays or alpha particles, have different batch probabilities. Enzymatic modulation of the immediate damage is modeled as a Markov process similar to the processes described by the master equation of stochastic chemical kinetics. An illustrative example is the restitution/complete exchange model, which postulates that radiation induced DSBs can subsequently either undergo enzymatically mediated repair (restitution) or can participate pairwise in chromosome exchanges, some of which make irremediable lesions such as dicentric chromosome aberrations. One may have rapid irradiation followed by enzymatic DSB processing or have prolonged irradiation with both DSB arrival and enzymatic DSB processing continuing throughout the irradiation period. A complete solution of the Markov chain is known for the case that the exchange rate constant is negligible so that no irremediable chromosome lesions are produced and DSBs are the only damage to the genome. Using PDEs for generating functions, a perturbation calculation is made assuming the exchange rate constant is small compared to the repair rate constant. Some non-perturbative results applicable to very prolonged irradiation are also obtained using matrix methods: Perron-Frobenius theory, variational methods and numerical approximations of eigenvalues. Applications to experimental results on expected values, variances and statistical distributions of DNA lesions are briefly outlined. Continuous time Markov chain models are the most systematic of those current radiation damage models which treat DSB-DSB interactions within the cell nucleus as homogeneous (e.g. ignore diffusion limitations). They contain most other homogeneous models as special cases, limiting cases or approximations. However, applying the continuous time Markov chain models to studying spatial dependence of DSB interactions, which is generally believed to be very important in some situations, presents difficulties.

DNA damage caused by ionizing radiation.

A survey is given of continuous-time Markov chain models for ionizing radiation damage to the genome of mammalian cells. In such models, immediate damage induced by the radiation is regarded as a batch-Poisson arrival process of DNA double-strand breaks (DSBs). Enzymatic modification of the immediate damage is modeled as a Markov process similar to those described by the master equation of stochastic chemical kinetics. An illustrative example is the restitution/complete-exchange model. The model postulates that, after being induced by radiation, DSBs subsequently either undergo enzymatically mediated restitution (repair) or participate pairwise in chromosome exchanges. Some of the exchanges make irremediable lesions such as dicentric chromosome aberrations. One may have rapid irradiation followed by enzymatic DSB processing or have prolonged irradiation with both DSB arrival and enzymatic DSB processing continuing throughout the irradiation period. Methods for analyzing the Markov chains include using an approximate model for expected values, the discrete-time Markov chain embedded at transitions, partial differential equations for generating functions, normal perturbation theory, singular perturbation theory with scaling, numerical computations, and certain matrix methods that combine Perron-Frobenius theory with variational estimates. Applications to experimental results on expected values, variances, and statistical distributions of DNA lesions are briefly outlined. Continuous-time Markov chains are the most systematic of those radiation damage models that treat DSB-DSB interactions within the cell nucleus as homogeneous (e.g., ignore diffusion limitations). They contain virtually all other relevant homogeneous models and semiempirical summaries as special cases, limiting cases, or approximations. However, the Markov models do not seem to be well suited for studying spatial dependence of DSB interactions, which is known to be important in some situations.

The ratio of dicentrics to centric rings produced in human lymphocytes by acute low-LET radiation.

Chromosome aberrations produced by ionizing radiation are assumed to develop from DNA double-strand breaks (DSBs) which interact pairwise, in an exchange event. Dicentrics and centric rings are aberrations that exemplify inter- and intrachromosomal exchanges, respectively. We show from a survey of published data that for acute low-LET irradiation of resting human lymphocytes the observed ratio of dicentrics to centric rings is approximately five times smaller than predicted by a pairwise interaction model which assumes complete randomness. Such a low ratio can be interpreted as evidence for a proximity effect, favoring exchanges of an intrachromosomal type. That is, since DSBs induced close together have an above-average chance of pairwise interaction, the observed excess of centric rings indicates that at the time of irradiation there is some degree of spatial confinement for the two arms of a single chromosome. Assuming the excess of centric rings is indeed due to proximity effects, the data are used to estimate that the volume of a domain, within which any one lymphocyte chromosome is localized at one instant during the G0/G1 phase, is at most approximately 20% of the nuclear volume.

Evolution of DNA damage in irradiated cells.

Ionizing radiation damage to a mammalian genome is modeled using continuous time Markov chains. Models are given for the initial infliction of DNA double strand breaks by radiation and for the enzymatic processing of this initial damage. Damage processing pathways include DNA double strand break repair and chromosome exchanges. Linear, saturable, or inducible repair is considered, competing kinetically with pairwise interactions of the DNA double strand breaks. As endpoints, both chromosome aberrations and the inability of cells to form clones are analyzed. For the post-irradiation behavior, using the discrete time Markov chain embedded at transitions gives the ultimate distribution of damage more simply than does integrating the Kolmogorov forward equations. In a representative special case explicit expressions for the probability distribution of damage at large times are given in the form used for numerical computations and comparisons with experiments on human lymphocytes. A principle of branching ratios, that late assays can only measure appropriate ratios of repair and interaction functions, not the functions themselves, is derived and discussed.

Dose-rate dependent stochastic effects in radiation cell-survival models.

When cells are subjected to ionizing radiation the specific energy rate (microscopic analog of dose-rate) varies from cell to cell. Within one cell, this rate fluctuates during the course of time; a crossing of a sensitive cellular site by a high energy charged particle produces many ionizations almost simultaneously, but during the interval between events no ionizations occur. In any cell-survival model one can incorporate the effect of such fluctuations without changing the basic biological assumptions. Using stochastic differential equations and Monte Carlo methods to take into account stochastic effects we calculated the dose-survival relationships in a number of current cell survival models. Some of the models assume quadratic misrepair; others assume saturable repair enzyme systems. It was found that a significant effect of random fluctuations is to decrease the theoretically predicted amount of dose-rate sparing. In the limit of low dose-rates neglecting the stochastic nature of specific energy rates often leads to qualitatively misleading results by overestimating the surviving fraction drastically. In the opposite limit of acute irradiation, analyzing the fluctuations in rates merely amounts to analyzing fluctuations in total specific energy via the usual microdosimetric specific energy distribution function, and neglecting fluctuations usually underestimates the surviving fraction. The MOnte Carlo methods interpolate systematically between the low dose-rate and high dose-rate limits. As in other approaches, the slope of the survival curve at low dose-rates is virtually independent of dose and equals the initial slope of the survival curve for acute radiation.