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J Theor Biol. Author manuscript; available in PMC 2010 November 21.

Published in final edited form as:

Published online 2009 August 11. doi: 10.1016/j.jtbi.2009.08.003

PMCID: PMC2774503

NIHMSID: NIHMS141326

The publisher's final edited version of this article is available at J Theor Biol

See other articles in PMC that cite the published article.

Ionizing radiation triggers oxidative stress, which can have a variety of subtle and profound biological effects. Here we focus on mathematical modeling of potential synergistic interactions between radiation damage to DNA and oxidative stress-induced damage to proteins involved in DNA repair/replication. When sensitive sites on these proteins are attacked by radiation-induced radicals, correct repair of dangerous DNA lesions such as double strand breaks (DSBs) can be compromised. In contrast, if oxidation of important proteins is prevented by strong antioxidant defenses, DNA repair may function more efficiently. These processes probably occur to some extent even at low doses of radiation/oxidative stress, but they are easiest to investigate at high doses, where both DNA and protein damage are extensive. As an example, we use data on survival of *Deinococcus radiodurans* after high doses (thousands of Gy) of acute and chronic irradiation. Our model of radiogenic oxidative stress is consistent with these data and can potentially be generalized to other organisms and lower radiation doses.

Ionizing radiation can damage all important cellular components, including DNA and proteins, both through direct ionization and through induction of oxidative stress. Radiogenic damage to DNA, such as double strand breaks (DSBs), which are typically difficult to repair and contribute greatly to clonogenic cell death, has been extensively studied (Barendsen, 1994; Iliakis et al., 2004; Kasten-Pisula et al., 2005; Saleh and El-Awady, 2005; Ward, 1990). Radiation-induced oxidative stress, which results in oxidation of proteins, lipids, and nucleotides, can have a variety of subtle and profound biological consequences, which are drawing increasing attention. For example, oxidative stress triggered by even quite low doses of radiation can produce an alteration of the cellular redox balance, which lasts for substantial time after exposure and may contribute to bystander effects, genomic instability, modified gene expression, elevated mutagenesis rates, changes in cell survival, proliferation, and differentiation (Azzam et al., 2002; Forman et al., 2002; Haddad, 2004; Hei, 2006; Mikkelsen, 2004; Rugo et al., 2002; Sangsuwan and Haghdoost, 2008; Schimmel and Bauer, 2002; Spitz et al., 2004; Tominaga et al., 2004).

Within this vast and complicated array of effects of radiogenic oxidative stress, in this article we focus on one aspect – potential interactions between oxidative damage to proteins and DNA damage repair. When sensitive sites on proteins involved in DNA repair and replication are oxidized by radiation-induced reactive oxygen and nitrogen species (ROS and RNS), the activity and fidelity of these proteins are altered, which may impede correct repair of DNA damage such as DSBs, enhancing cell death and mutagenesis (Adams et al., 1979; Bisby et al., 1982; Culard et al., 2003; Daly, 2009; Daly et al., 2007; Eon et al., 2001; Ghosal et al., 2005; Goodhead and Nikjoo, 1987; Kowalczyk et al., 2008; Saha et al., 1992). Such phenomena probably occur to some extent even at relatively low doses of radiation/oxidative stress (e.g. (Montaner et al., 2007)). However, they are easiest to investigate at high doses, where both DNA and protein damage are extensive (Adams et al., 1979; Bisby et al., 1982; Culard et al., 2003; Eon et al., 2001; Gerard et al., 2001; Goodhead and Nikjoo, 1987; Jolivet et al., 2006; Kowalczyk et al., 2008; Liu et al., 2003; Zahradka et al., 2006; Zimmermann et al., 1994), and interactions between them are probably most pronounced.

A good opportunity to study this aspect of radiation-induced oxidative stress is provided by certain prokaryotes, which have been evolutionarily optimized for coping with genotoxic agents such as desiccation, oxidative stress and UV radiation, and are, therefore, highly resistant to ionizing radiation (Blasius et al., 2008; Shukla et al., 2007). The best studied organism in this category is the bacterium *Deinococcus radiodurans*, which can survive acute exposure to several kGy of γ– or high-LET radiation without loss of viability and can proliferate at a normal rate under chronic γ-radiation at 50 or 60 Gy/h (Brim et al., 2006; Daly et al., 2004; Dewey, 1969; Lange et al., 1998; Zimmermann et al., 1994).

Here we propose a simple mathematical model, which is intended to investigate the potential synergistic relationship between oxidative stress, protein and DNA damage, using data on *D. radiodurans* as an example. The model is consistent with the observed patterns of cell survival for this organism under chronic irradiation and after acute exposures (e.g., (Battista et al., 1999; Blasius et al., 2008; Brim et al., 2006; Daly, 2006; Daly, 2009; Daly et al., 2007; Daly et al., 2004; Dewey, 1969; Ghosal et al., 2005; Hess, 2003; Jolivet et al., 2006; Lange et al., 1998; Liu et al., 2003; Makarova et al., 2001; Makarova et al., 2007; Mennecier et al., 2006; Shukla et al., 2007; White et al., 1999; Zahradka et al., 2006; Zhang et al., 2005; Zimmermann et al., 1994), reviewed by (Blasius et al., 2008; Daly, 2009)) and can assist in the interpretation of these patterns. Potentially, models such as the one presented here can enhance the understanding of radiation-induced oxidative stress at lower radiation doses and in other organisms, because the main model concepts are probably generalizable.

The main model assumptions are shown schematically in Fig. 1. More detailed discussion of these assumptions and their mathematical implementation is provided below.

Schematic representation of model assumptions: Radiation (lightning symbols) produces reactive oxygen species (ROS) and DNA double strand breaks (DSB). ROS can react with antioxidants (A) to form a complex (ROSC), which then decays, resulting in elimination **...**

During irradiation, reactive oxygen species and other radicals and oxidants (generically called *ROS* here) are generated, and can damage proteins (called *PR* here) which are needed for correct repair of DNA damage. Scavenging of radicals is accomplished by enzymatic and non-enzymatic antioxidants (generically called *A* here). Some radicals are also assumed to be inactivated by reacting with molecules in the cell which are not critical for survival; this mechanism is approximated by a first-order process. To reduce the number of adjustable parameters, we neglect several potentially substantial phenomena such as non-reversible ROS scavengers, a second-order process whereby ROS are inactivated by reacting with each other, ROS production under background conditions, multiple types of antioxidants and DNA repair proteins, etc. This set of assumptions is represented by the following system of differential equations:

$$\begin{array}{l}\text{dROS}(t)/\text{dt}={c}_{1}\phantom{\rule{0.2em}{0ex}}R-{c}_{2}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)\phantom{\rule{0.2em}{0ex}}A(t)-{c}_{3}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)\\ \text{dA}(t)/\text{dt}=-{c}_{2}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)\phantom{\rule{0.2em}{0ex}}A(t)+{c}_{4}\phantom{\rule{0.2em}{0ex}}\text{ROS}C(t)\\ \text{dROSC}(t)/\text{dt}={c}_{2}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)\phantom{\rule{0.2em}{0ex}}A(t)-{c}_{4}\phantom{\rule{0.2em}{0ex}}\text{ROSC}(t)\\ \text{dPR}(t)/\text{dt}={c}_{5}-[{c}_{6}+{c}_{7}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)]\phantom{\rule{0.2em}{0ex}}\text{PR}(t)\end{array}$$

(1)

Here *A* is the active form of the antioxidant, *ROSC* is the ROS-antioxidant complex (the temporarily inactive form of the antioxidant, which can be regenerated back to the active form *A*), and *R* is the radiation dose rate. The parameter interpretations, also presented in Table 1, are: *c _{1}* =

Default model parameter values and interpretations. Parameters *c*_{8} and *c*_{9} were estimated from the literature (Daly et al., 2007; Daly et al., 2004; Lange et al., 1998; Zahradka et al., 2006), and for the remaining ones arbitrary values were used. These **...**

The system in Eq. (1) can be simplified by applying an equilibrium assumption, i.e. that the active and inactive forms of the antioxidant (*A* and *ROSC*, respectively) always exist in equilibrium and the sum of their concentrations is equal to *A _{tot}*, where

$$\begin{array}{l}\text{dROS}(t)/\text{dt}={c}_{1}\phantom{\rule{0.2em}{0ex}}R-{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)/[{c}_{4}+{c}_{2}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)]-{c}_{3}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)\\ \text{dPR}(t)/\text{dt}={c}_{5}-[{c}_{6}+{c}_{7}\phantom{\rule{0.2em}{0ex}}\text{ROS}(t)]\phantom{\rule{0.2em}{0ex}}\text{PR}(t)\end{array}$$

(2)

Assuming that the kinetics of *ROS* production and removal are faster than those of protein turnover, Eq. (2) can also be simplified by assuming that *ROS* always exist at an equilibrium concentration *ROS _{eq}*, which is given by the following expression:

$$\begin{array}{l}{\text{ROS}}_{\text{eq}}=({c}_{2}\phantom{\rule{0.2em}{0ex}}{X}_{1}-{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}+{{X}_{3}}^{\xbd})/(2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}),\\ \text{where}\phantom{\rule{0.2em}{0ex}}{X}_{1}={c}_{1}\phantom{\rule{0.2em}{0ex}}R-{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}},\phantom{\rule{1.4em}{0ex}}{X}_{2}={c}_{1}\phantom{\rule{0.2em}{0ex}}R+{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}},\\ {X}_{3}={{c}_{2}}^{2}\phantom{\rule{0.2em}{0ex}}{{X}_{1}}^{2}+2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}}{X}_{2}+{({c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4})}^{2}\end{array}$$

(3)

It is expected that at low radiation dose rates the antioxidant concentration is sufficient to counteract *ROS* production, thereby maintaining *ROS _{eq}* at low values. At higher dose rates, the antioxidant becomes saturated and can no longer counteract accumulation of

The predicted equilibrium concentration of reactive radicals (*ROS*_{eq}, in arbitrary units) during irradiation at a constant dose rate (*R*).

The protein inactivation kinetics by *ROS* can then be estimated by substituting Eq. (3) into Eq. (2), using *ROS _{eq}* in place of

$$\begin{array}{l}{\text{PR}}_{\text{eq}}=2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{5}/({c}_{2}\phantom{\rule{0.2em}{0ex}}[{c}_{7}\phantom{\rule{0.2em}{0ex}}{X}_{1}+2\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{6}]-{c}_{7}\phantom{\rule{0.2em}{0ex}}[{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}-{{X}_{3}}^{\xbd}]),\\ \text{where}\phantom{\rule{0.2em}{0ex}}{X}_{1}={c}_{1}\phantom{\rule{0.2em}{0ex}}R-{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot},}\phantom{\rule{1.6em}{0ex}}{X}_{2}={c}_{1}\phantom{\rule{0.2em}{0ex}}R+{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot},}\\ {X}_{3}={{c}_{2}}^{2}\phantom{\rule{0.2em}{0ex}}{{X}_{1}}^{2}+2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}}{X}_{2}+{({c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4})}^{2}\end{array}$$

(4)

As intuitively expected, *PR _{eq}* as function of radiation dose rate (Eq. (4)) behaves in an inverse manner to

Radiation also generates multiple types of DNA damage, among which the most critical for cell survival are double strand breaks (*DSB*). In bacteria, antioxidants which protect proteins do not appear to protect DNA very well because different types of ROS may preferentially attack proteins vs. DNA (Daly et al., 2007). Consequently, the yield of DSBs per unit dose per base pair of DNA is similar in most bacteria under similar conditions (Gerard et al., 2001). Correct repair of DSBs (i.e. repair that is sufficient for cell survival, not necessarily for lack of mutation), however, is assumed to be dependent on the concentration of functional repair proteins (here generalized as *PR*). Of course, this set of assumptions, which is dictated by the need for reducing the number of model parameters, is highly simplistic and ignores multiple potentially important phenomena such as direct induction of DSBs by ROS, the existence of multiple types of DNA damage and damage repair proteins, etc. However, we believe that our assumptions capture some crucial aspects of the interactions between oxidative stress and DNA damage. They are modeled by the following differential equation, where *c _{8}* is the constant for DSB production by radiation and

$$\text{dDSB}(t)/\text{dt}={c}_{8}\phantom{\rule{0.2em}{0ex}}R-{c}_{9}\phantom{\rule{0.2em}{0ex}}\text{PR}(t)\phantom{\rule{0.2em}{0ex}}\text{DSB}(t)$$

(5)

At a constant dose rate, the equilibrium number of DSBs per cell (*DSB _{eq}*) can be calculated by substituting

$$\begin{array}{l}{\text{DSB}}_{\text{eq}}={c}_{8}\phantom{\rule{0.2em}{0ex}}R\phantom{\rule{0.2em}{0ex}}[{c}_{2}\phantom{\rule{0.2em}{0ex}}(2\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{6}+{c}_{7}\phantom{\rule{0.2em}{0ex}}{X}_{1})-{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}}{c}_{7}\phantom{\rule{0.2em}{0ex}}+{c}_{7}\phantom{\rule{0.2em}{0ex}}{{X}_{3}}^{\xbd}]/(2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{5}\phantom{\rule{0.2em}{0ex}}{c}_{9}),\\ \text{where}\phantom{\rule{0.2em}{0ex}}{X}_{1}={c}_{1}\phantom{\rule{0.2em}{0ex}}R-{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}},\phantom{\rule{1.4em}{0ex}}{X}_{2}={c}_{1}\phantom{\rule{0.2em}{0ex}}R+{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}},\\ {X}_{3}={{c}_{2}}^{2}\phantom{\rule{0.2em}{0ex}}{{X}_{1}}^{2}+2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}}{X}_{2}+{({c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4})}^{2}\end{array}$$

(6)

The behavior of *DSB _{eq}* as function of dose rate is shown in Fig. 4, and is consistent with the behavior of

By the time radiation exposure is over, i.e. when *t* = *Dose/R*, where *Dose* is the total radiation dose, the concentration of active protein (*PR _{d}*) can be calculated by using Eqs. (2) and (3) and assuming that the protein concentration before exposure was in equilibrium (i.e.

$$\begin{array}{l}{\text{PR}}_{d}={c}_{5}\phantom{\rule{0.2em}{0ex}}[2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{6}/{Y}_{2}+{c}_{7}\phantom{\rule{0.2em}{0ex}}({{Y}_{1}}^{\xbd}+{c}_{2}\phantom{\rule{0.2em}{0ex}}{X}_{1}-{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4})]{Y}_{2}/[{c}_{6}\phantom{\rule{0.2em}{0ex}}({c}_{7}\phantom{\rule{0.2em}{0ex}}{{Y}_{1}}^{\xbd}\\ +\phantom{\rule{0.2em}{0ex}}{c}_{7}\phantom{\rule{0.2em}{0ex}}({c}_{2}\phantom{\rule{0.2em}{0ex}}{X}_{1}-{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}})+2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{6})],\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}{X}_{1}={c}_{1}\phantom{\rule{0.2em}{0ex}}R-{c}_{4}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}},\\ {Y}_{1}={{c}_{4}}^{2}\phantom{\rule{0.2em}{0ex}}{({c}_{2}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}}+{c}_{3})}^{2}+2\phantom{\rule{0.2em}{0ex}}{c}_{1}\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}}R\phantom{\rule{0.2em}{0ex}}({c}_{3}-{c}_{2}\phantom{\rule{0.2em}{0ex}}{A}_{\text{tot}})+{({c}_{1}\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}R)}^{2},\\ {Y}_{2}=\text{exp}\phantom{\rule{0.1em}{0ex}}[-\text{Dose}\phantom{\rule{0.2em}{0ex}}({c}_{7}\phantom{\rule{0.2em}{0ex}}{{Y}_{1}}^{\xbd}+{c}_{2}\phantom{\rule{0.2em}{0ex}}(2\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{6}+{c}_{7}\phantom{\rule{0.2em}{0ex}}{X}_{1})-{c}_{3}\phantom{\rule{0.2em}{0ex}}{c}_{4}\phantom{\rule{0.2em}{0ex}}{c}_{7})/(2\phantom{\rule{0.2em}{0ex}}{c}_{2}\phantom{\rule{0.2em}{0ex}}{c}_{3}\phantom{\rule{0.2em}{0ex}}R)]\end{array}$$

(7)

The behavior of *PR _{d}* as function of dose and dose rate is shown in Fig. 5. At very low dose rates, where damage to protein by ROS is limited by antioxidant defenses, most of the protein remains active regardless of the cumulative radiation dose. At very high dose rates, the protein becomes inactivated as function of dose in an approximately first-order manner.

The predicted fraction of active protein, normalized relative to the background equilibrium value (i.e. *PR*_{dF} = *PR*_{d}/[*c*_{5}/c_{6}]) just after radiation exposure at various doses (in kGy) and dose rates (*R*, in kGy/h).

Assuming the irradiation is acute for the purposes of DNA repair (i.e. the total dose is delivered in such a short time that no DSBs can be repaired during exposure), the number of DSBs just after exposure is: *DSB _{d}* =

$$\begin{array}{l}\text{dDSB}(t)/\text{dt}=-{c}_{9}\phantom{\rule{0.2em}{0ex}}\text{PR}(t)\phantom{\rule{0.2em}{0ex}}\text{DSB}(t)\\ \text{dPR}(t)/\text{dt}={c}_{5}-{c}_{6}\phantom{\rule{0.2em}{0ex}}\text{PR}(t)\end{array}$$

(8)

Eq. (8) can be solved analytically to yield the following expressions (Eq. (9) below), where *PR _{d}* is given by Eq. (7):

$$\begin{array}{l}\text{DSB}(t)={c}_{8}\phantom{\rule{0.2em}{0ex}}\text{Dose}\phantom{\rule{0.2em}{0ex}}\text{exp}\phantom{\rule{0.1em}{0ex}}[-{c}_{9}\phantom{\rule{0.2em}{0ex}}({c}_{6}\phantom{\rule{0.2em}{0ex}}({c}_{5}\phantom{\rule{0.2em}{0ex}}t+{\text{PR}}_{\text{d}})+({c}_{5}\phantom{\rule{0.2em}{0ex}}-{c}_{6}\phantom{\rule{0.2em}{0ex}}{\text{PR}}_{d})\phantom{\rule{0.2em}{0ex}}\text{exp}\phantom{\rule{0.1em}{0ex}}[-{c}_{6}\phantom{\rule{0.2em}{0ex}}t]-{c}_{5})/{{c}_{6}}^{2}]\\ \text{PR}(t)=[({c}_{6}\phantom{\rule{0.2em}{0ex}}{\text{PR}}_{d}-{c}_{5})\phantom{\rule{0.2em}{0ex}}\text{exp}\phantom{\rule{0.1em}{0ex}}[-{c}_{6}\phantom{\rule{0.2em}{0ex}}t]+{c}_{5}]/{c}_{6}\end{array}$$

(9)

The cell survival predicted for some finite time available for repair (*T _{rep}*) is defined, according to standard assumptions that a single incorrectly repaired DSB is lethal to the cell, as

The predicted colony-forming unit survival (*S*_{cfu}) for *D. radiodurans* exposed to acute γ-radiation (on ice) in complete growth medium (curve), compared with observed data points from (Daly et al., 2004).

The model contains 11 parameters (*c _{1}* −

More insight into model behavior can be gained by measuring the sensitivity of model predictions to changes in each parameter. We performed both local and global sensitivity analyses. Local sensitivity was assessed by varying a given parameter by a selected factor (e.g. 1.5) above and below the default value, keeping all other parameters constant at their default values. Global sensitivity was estimated by calculating partial rank correlation coefficient (PRCC) for each parameter (described in the Appendix), according to the method reviewed by (Marino et al., 2008).

The mathematical model presented here can qualitatively and quantitatively describe two processes thought to be important for survival of bacteria at high doses of ionizing radiation: DNA double strand break (DSB) repair and protein oxidation. The interactions of these processes under conditions of severe radiation-induced oxidative stress are analyzed. Model predictions using some parameter values estimated from the literature and using freely-adjusted values for the remaining parameters (Table 1) were consistent with the observed survival curve of *D. radiodurans* exposed to acute γ-radiation (Daly et al., 2004) (Fig. 6), and with the ability of *D. radiodurans* to grow under constant dose rates of 0.05 or 0.06 kGy/h (Brim et al., 2006; Daly et al., 2004; Lange et al., 1998) by preventing excessive accumulation of DNA and protein damage (Figs. 3, ,4).4). As more information becomes available to estimate model parameters, the formalism can be tested more rigorously.

Local model sensitivity to varying the value of each parameter one at a time, keeping all other parameters at default values, was performed for colony-forming unit survival (*S _{cfu}*) after acute irradiation (Fig. 7), for equilibrium number of DSBs per cell (

For *S _{cfu}*, local sensitivity was also assessed numerically (Table 2) by estimating the effects of varying each parameter on the radiation dose required to reduce

Effects of varying parameter values on the shape of the survival curve for colony-forming units (*S*_{cfu}). Dose_{90} refers to the acute radiation dose (in kGy) required to reduce *S*_{cfu} to 90%; it is an estimate of the length of the “shoulder” **...**

As expected, sensitivity to a given parameter can be modulated by what outcome variable is tested (e.g. *DSB _{eq}* vs.

The local sensitivity analysis (Table 2 and Figs. 7--9)9) also largely confirmed the intuitive role of each parameter in the model. For example, it showed that the constants for ROS production by radiation (*c _{1}*), ROS removal by first-order kinetics (

Almost by definition, mathematical models are greatly simplified representations of complex biological processes. The model presented here focuses on the interactions between protein and DNA damage in the context of radiogenic oxidative stress, which have been suggested to be important for clonogenic survival of irradiated *D. radiodurans* and some other prokaryotes. Many aspects of this phenomenon, as well as multiple other factors known to be relevant for cell survival, have not been included in the model to improve its tractability and decrease the number of adjustable parameters. For example, for these reasons we neglected the following: metabolism-induced ROS, which can be important during and after irradiation (Daly et al., 2007; Ghosal et al., 2005); acceleration of protein turnover (e.g. degradation and excretion of damaged proteins and synthesis of their replacements) during and after irradiation (Blasius et al., 2008; Liu et al., 2003); and causes of cell death other than unrepaired/misrepaired DNA DSBs, e.g. severe global protein damage and activation of latent bacteriophages during DNA repair (Mennecier et al., 2006; Qiu et al., 2006). Also, it is important to note that parameter combinations other than the one we chose as the default (Table 1) may certainly be able to fit our selected data set just as well, particularly if the ratios between certain parameter values (e.g. between the ROS-related constants *c _{1}-c_{4}*) are kept constant. Additional data from future experimental studies will be needed to unambiguously determine these parameter values.

Despite its limitations, we believe that the model captures some crucial aspects of radiation-induced oxidative stress and its potentially synergistic relationship with DNA damage. In addition to being consistent with the selected experimental data set (clonogenic survival of *D. radiodurans* at different doses and dose rates), the model suggests some potentially useful insight and generalizations:

- For chronic irradiation, the model predicts that oxidative stress (ROS accumulation) and its consequences such as DNA repair protein oxidation can be largely suppressed by protective antioxidants at sufficiently low dose rates, but exhibit a dramatic release from suppression beyond a certain “threshold” dose rate, where the antioxidant capacity is saturated and overwhelmed. Above this threshold, protein damage will accumulate rapidly, thereby compromising DNA repair and making cell survival and proliferation impossible. Using the semi-arbitrary parameters chosen here (Table 1), the threshold dose rate should lie in the range of 100 - 1000 Gy/h (Figs. 2--4).4). This prediction needs to be tested by additional experimental data. Currently, the growth of
*D. radiodurans*under chronic exposure was assessed only for dose rates of 50 or 60 Gy/h (Brim et al., 2006; Daly et al., 2004; Lange et al., 1998), showing that under such conditions proliferation of this organism in complete growth medium is essentially unaffected. Assessing the proliferation capacity (or lack of it) of*D. radiodurans*under higher chronic dose rates can test whether or not a threshold dose rate exists and/or determine its value. - Because some parameters affect model predictions to different extents depending on dose rate, i.e. some are much more important at high dose rates and relatively unimportant at low dose rates or vice versa (Fig. 8), the ability of a given organism to counteract radiation effects at low dose rates and at high dose rates may not necessarily be correlated. In other words, if an organism is highly resistant to acute exposures, it may be quite sensitive to chronic irradiation, or the other way around. This is supported by mutants of
*D. radiodurans*which exhibit wild-type survival after acute exposures to several kGy, but cannot grow under chronic irradiation of 50-60 Gy/h, and vice versa (Hess, 2003). Also, other bacteria, such as*Enterococcus facium*, can grow well under 50 Gy/h, but are much more sensitive to acute exposures than*D. radiodurans*(Daly et al., 2004). - Similarly, the length of the shoulder and the steepness of the high-dose slope of the survival curve after acute irradiation may not necessarily be correlated, because they are determined to different extents by certain parameters (Table 2, Fig. 7). For example, it is possible for the model to generate a curve with a small shoulder and a shallow slope, or a large shoulder and a steep slope. This is qualitatively consistent with survival curve shape variability in
*D. radiodurans*as function of radiation LET (Dewey, 1969) and composition of the growth medium (Daly et al., 2007; Daly et al., 2004; Zhang et al., 2005), because these factors can modulate ROS production by radiation, the number and complexity of DSBs induced per unit dose, cellular antioxidant concentrations (e.g. of manganese ions), the ability to repair DSBs, protein turnover rates, and other relevant parameters.

Of course, the current formalism is only a preliminary attempt to model the interactions between oxidative stress and DNA damage repair. However, we believe that the basic approach presented here may potentially be applied to other organisms and lower radiation doses, because the main concepts and assumptions (Fig. 1) were intended to be quite general. For example, it has been shown experimentally that in mammalian cells ROS are removed by a combination of saturable and first-order kinetics (Makino et al., 2008; Sasaki et al., 1998), as assumed in the current model. Potential interference of ROS with DNA repair by oxidation of sensitive sites on DNA repair proteins may not occur to the same extent at lower radiation doses as at high doses, but may be important in some systems even on a subtle level – e.g. if the endpoint of interest is cell mutagenesis (and potential consequent carcinogenesis), rather than cell survival, then even small defects in DSB repair may become substantial.

Certainly, details of the model equations may need to be modified for particular organisms and situations. It seems likely that to apply this approach to mammalian cells, the main assumptions outlined here can still be used, but additional aspects may need to be considered. For example, it may be necessary to model some of the eukaryote-specific complexities of ROS production and removal (e.g. the role of radiation-damaged mitochondria in generating ROS even after irradiation has ended, the role of non-reversible antioxidants such as histones, etc.) and DSB repair (e.g. several competing non-homologous end joining and homologous recombination pathways). Also, DNA damage types other than DSBs may need to be considered for studying cell mutagenesis and carcinogenesis.

Research supported by National Cancer Institute grant 5T32-CA009529 (IS), National Institutes of Health grants P41 EB002033-09 and P01 CA-49062 (DJB).

Because the number of adjustable model parameters is large, estimates of global parameter sensitivity using the partial rank correlation coefficient (PRCC) were performed for the equilibrium number of DSBs/cell (*DSB _{eq}*) as function of radiation dose rate (

This method measures global model sensitivity to each parameter. A large positive PRCC (i.e. approaching +1) indicates that increasing the value of the given parameter substantially increases the model prediction. The converse is true for a large negative PRCC (i.e. approaching -1). The results are shown in Table A1. Some of the main patterns suggested by these PRCC values are discussed in the main text. This information can supplement the local parameter sensitivity calculations described in the main text, in Table 2 and in Figs. 7--99.

Parameter | DSB_{eq} | S_{cfu} | ||||
---|---|---|---|---|---|---|

R = 0.01 | R = 0.05 | R = 0.1 | R = 5.0 | D = 20 | D = 25 | |

c_{1} | 0.130 | 0.249 | 0.305 | 0.576 | -0.197 | -0.199 |

c_{2} | -0.019 | -0.017 | -0.017 | -0.006 | 0.010 | 0.007 |

c_{3} | -0.080 | -0.166 | -0.208 | -0.494 | 0.202 | 0.201 |

c_{4} | -0.071 | -0.106 | -0.120 | -0.147 | 0.034 | 0.035 |

c_{5} | -0.849 | -0.817 | -0.802 | -0.745 | 0.639 | 0.635 |

c_{6} | 0.834 | 0.781 | 0.748 | 0.478 | -0.495 | -0.483 |

c_{7} | 0.087 | 0.166 | 0.211 | 0.499 | -0.181 | -0.178 |

c_{8} | 0.844 | 0.813 | 0.799 | 0.740 | -0.312 | -0.297 |

c_{9} | -0.846 | -0.815 | -0.800 | -0.744 | 0.615 | 0.611 |

A_{tot} | -0.070 | -0.109 | -0.127 | -0.146 | -0.010 | -0.008 |

T_{rep} | NA | NA | NA | NA | 0.484 | 0.483 |

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