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Modeling cell response to low doses of photon irradiation—Part 1: on the origin of fluctuations

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Abstract

Intra- and inter-individual variability is a well-known aspect of biological responses of cells observed at low doses of radiation, whichever the phenomenon considered (adaptive response, bystander effects, genomic instability, etc.). There is growing evidence that low-dose phenomena are related to cell mechanisms other than DNA damage and misrepair, meaning that other cellular structures may play a crucial role. Therefore, in this study, a series of calculations at low doses was carried out to study the distribution of specific energies from different irradiation doses (3, 10 and 30 cGy) in targets of different sizes (0.1, 1 and \(10\, \upmu \hbox {m}\)) corresponding to the dimensions of different cell structures. The results obtained show a strong dependence of the probability distributions of specific energies on the target size: targets with dimensions comparable to those of the cell show a Gaussian-like distribution, whereas very small targets are very likely to not be hit. A statistical analysis showed that the level of fluctuations in the fraction of aberrant cells is only related to the fraction of aberrant cells and the number of irradiated cells, regardless of, for instance, the heterogeneity in cell response.

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Acknowledgments

This work was performed within the framework of the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program “Investissements d’ Avenir”(ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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Correspondence to Michaël Beuve.

Appendices

Appendix 1: LQD—LiQuiD water radiolysis

The LQD simulation code is quite standard in the literature, except for differences regarding some physical and computational details. It is one of three simulation codes dedicated to the simulation of water radiolysis by swift ions, electrons or photons. The determination of some simulation parameters, not available by calculation or from experimental data, resulted from the description of a water vapor system with empirical adjustments. These adjustments consisted in constraining the simulation results for 150 keV electrons to reproduce as best as possible the existing experimental radical yields of water radiolysis by high-energy photons or fast electrons. Four different kinds of collisional processes are considered and described in more detail in Gervais et al. (2006).

The one most important for energy deposition consists in the inelastic collisions associated with electronic transition in molecules, either ionization or excitation. The ionization mean free path dominates the high-energy regime above 100 eV. In this work, any possibility of collective excitation was neglected. For the water molecule, this makes little difference since collective excitations will decay mainly by ionization of the water molecule. The ionization mean free path is deduced from the empirical formulation proposed by Kim (2001), Kim and Rudd (1994). It has proved to be fairly accurate to describe the molecule ionization in gas phase, and it is not limited to the high-energy domain like the first Born approximation. It provides both total ionization cross section and energy differential cross section for each energy level of the water molecule. The angular distribution of electrons following an ionizing collision is simply modeled by using the following scheme, proposed by Grosswendt and Waibel (1978). If the kinetic energy K of the emitted electron is lower than 50 eV, the emission is isotropic. If K is between 50 and 200 eV, 90 % is uniformly sampled between \(\pi /2\), \(\pi /4\) and then the remaining 10 % is isotropic. Finally, if K exceeds 200 eV, the binary encounter approximation is used. In the same way, if the kinetic energy T of the incoming electron is lower than 100 eV, the angle of diffusion is uniformly sampled between 0 and \(\pi /4\). When T is larger than 100 eV, the binary encounter approximation is used. The ionization of the 1a1 level is followed by Auger electron emission. An electron is emitted isotropically with an energy of 480 eV. While its contribution to the total ionization is low (less than a few percent of single ionization), direct double ionization by electrons was included in a similar way as for single ionization but more roughly. The transferred kinetic energy is sampled in the same way as for the single ionization, but it is uniformly distributed between the two ejected electrons. The angles of emission are assumed to be uncorrelated and randomly chosen like for single ionization. Higher-order multiple ionization cross sections are typically one order of magnitude lower (Shevelko and Tawara 1998). They are neglected in the present work for electron transport.

The excitations of the water molecule into the a1b1 or b1a1 electronic states were described according to the work by Cobut et al. (1998). However, the cross-sectional formula was scaled in the low-energy range as suggested by Kim (2001) to avoid overestimation coming from the first Born approximation for energy close to the threshold. Any possibility for angular deflection of the incident electron associated with excitation was neglected since it is far lower than the deflections induced by elastic collision.

The second kind of process is the elastic collision leading to angular deflection of an electron. This process leads only to a tiny energy release in the medium, but it controls the diffusion of the electron in particular on the low-energy side. The elastic cross section is very difficult to obtain accurately from computation alone. Indeed, in the case of amorphous ice at 18 K, the structure factor and polarization of the medium have to be considered carefully in order to reproduce the experimentally observed structures (Michaud and Sanche 1987), and extrapolation from gas-phase results is off by a factor 50 below 100 eV. The present work was therefore based on the cross section deduced from experiments on amorphous ice at 18 K (Michaud et al. 2003). The measured elastic cross section includes the very low-energy acoustic-phonon modes. At low temperature, however, most of the phonons are in their ground state and only electron scattering by phonon excitation is allowed, while scattering by phonon deexcitation is forbidden. In contrast, at room temperature, a large number of phononic excited states are populated, and scattering by phonon deexcitation becomes possible. Since both processes are nearly equal in magnitude, the elastic cross section at room temperature is approximately twice the cross section at 18 K, as suggested by Michaud et al. (2003). The same holds true for all phononic modes that can be thermally activated. Such an enhancement by a factor 2 has been proposed by other authors for electron diffusion in liquid water (Cobut et al. 1998; Goulet et al. 1998; Muroya et al. 2002). For energies above 100 eV, the elastic cross section was extrapolated here by means of the following formula (Gervais et al. 2006):

$$\begin{aligned} \sigma = \dfrac{\sigma _0}{1+\gamma T} \end{aligned}$$
(20)

where \(\sigma _0 = 1.6\,\hbox {au}\) (\(1\,\hbox {au} = 2.798 \times 10^{-17}\,\hbox {cm}^{2}\)), \(\gamma = 0.75\,\hbox {au}\) and the kinetic energy of the electron T is expressed in atomic units (1 au = 27.2 eV).

The third kind of process concerns inelastic collisions associated with vibrational excitation of the medium. This process controls the slowing down of low-energy electrons when the electron kinetic energy becomes comparable to or lower than the lowest ionization threshold and then the thermalization distances. For this process, the cross sections measured by Michaud and Sanche (1987) for amorphous ice at low temperature (Cobut et al. 1998) were used. To improve the comparison of the simulation with experimental decay of solvated electrons (Bartels et al. 2000; Muroya et al. 2002), the values of the experimental cross sections were all enhanced by 15 %, with the exception of the two lowest phononic modes at 10 and 25 meV. To take account for the temperature effect on the electron–phonon interaction, the cross sections for both modes were enhanced by a factor 2. These values were extrapolated above 100 eV and below 3 eV with a formula proposed by Neff et al. (1980) to match the results of Michaud and Sanche (1987) on both energy sides.

The last kind of process, electron attachment to water molecule, was considered in the electron kinetic energy range 6.25–12.40 eV. The attachment cross section was adjusted to reproduce the yield of molecular hydrogen observed at short time (Pastina et al. 1999) while keeping a microsecond yield consistent with low LET experiments.

The diffusion of a subexcitation electron is in competition with the recombination with its parent cation. The probability of recombination is taken from the works of Cobut et al. (1998), Goulet and Jay-Gérin (1988) and Goulet et al. (1990) and enhanced by a factor 1.4 in order to improve the comparison with experimental results of Bartels et al. (2000) and Muroya et al. (2002). When the electron kinetic energy becomes lower than 0.025 eV, it is stopped.

Appendix 2: Fluctuations in a biological endpoint

Taking as example of a biological endpoint cell chromosomal aberrations, a cell may thus be classified in two states: aberrant (hereafter referred to as state 1) or normal (state 0). An average probability of aberrant cells is derived. Let \(\xi _j\) be the actual state (0 or 1) of the cell j. The number A of aberrant cells among the \(N_\mathrm {C}\) irradiated cells is calculated as:

$$\begin{aligned} A = \sum _{j=1}^{N_\mathrm {C}} \xi _j \end{aligned}$$
(21)

The mean value of A over all possible irradiation configurations reads:

$$\begin{aligned} \langle A\rangle = \sum _{j=1}^{N_\mathrm {C}} \langle \xi _i\rangle \end{aligned}$$
(22)

Ignoring the correlation between the responses of several cells, the mean value of \(A^2\) gives:

$$\begin{aligned} \langle A^2\rangle = \sum _{j=1}^{N_\mathrm {C}} \langle \xi _j^2\rangle + \sum _{(j,o)=1,j\ne o}^{N_\mathrm {C}} \langle \xi _j\rangle \langle \xi _o\rangle \end{aligned}$$
(23)

Assuming that the cell sampling process does not introduce any correlation in cell selection, i.e., that the number of sampled cells is low compared with the total number of available cells, one can deduce, for any cell j, the average over configurations:

$$\begin{aligned} \langle \xi _j\rangle&= P(0)\times 0 + P(1)\times 1 = p \end{aligned}$$
(24)
$$\begin{aligned} \langle \xi _j^2\rangle&= P(0) \times 0^2+ P(1)\times 1^2 = p \end{aligned}$$
(25)

where P(1) = p corresponds to the probability that the cell becomes aberrant and P(0) to the probability that the cell state remains normal.

The standard deviation qualifying the width of the statistical distribution of A verifies:

$$\begin{aligned} \begin{aligned} \sigma _A^2&= \langle (A-\langle A\rangle ^2)\rangle \\&= \langle A^2\rangle - \langle A\rangle ^2 \end{aligned} \end{aligned}$$
(26)

Other scenarios will be considered in the following sections.

Considering a single cell response

Here, it is considered that all cells have the same probability p of becoming aberrant. Consequently:

$$\begin{aligned} \langle A\rangle&= p{N_\mathrm {C}} \end{aligned}$$
(27)
$$\begin{aligned} \langle A^2\rangle&= p{N_\mathrm {C}} + {N_\mathrm {C}}({N_\mathrm {C}}-1)p^2 \end{aligned}$$
(28)
$$\begin{aligned} \sigma _A^2&= p{N_\mathrm {C}}(1-p) \end{aligned}$$
(29)

which are well-known results. One can define an estimator P of p by:

$$\begin{aligned} A = P N_\mathrm {C}\end{aligned}$$
(30)

with \(\langle P\rangle = p\). P is then the estimator of the probability of aberrant cells. An estimator of the relative standard error in the experimental estimation of p is, for large values of \({N_\mathrm {C}}\):

$$\begin{aligned} \dfrac{\sigma _P}{p} = \dfrac{\sigma _{A}}{\langle A\rangle } = \dfrac{\sqrt{1-p}}{\sqrt{p{N_\mathrm {C}}}} \end{aligned}$$
(31)

Therefore, the fluctuations simply depend on the probability of aberrant cells and the number of cells in the sample. For a low probability of aberrant cells (\(p \ll 1\)), it is approximated by:

$$\begin{aligned} \dfrac{\sigma _P}{p} = \dfrac{1}{\sqrt{\langle A\rangle }} \end{aligned}$$
(32)

All these are well-known results, but they constitute an introduction to the next section.

Considering a diversity of responses over a cell population

In this scenario, it is assumed that the cell response is heterogeneous over the sampled cell population. In other words, each cell j is characterized by an individual probability \(p_j\) to be aberrant. The process of selecting the \(N_\mathrm {C}\) cells to be irradiated is a random sampling over a large cell population characterized by a distribution of p values. Let thus f(p) be the distribution probability of the cell response for a individual. f(p) characterizes the level of diversity in cell responses for the considered endpoint. The average of A and \(A^2\) in Eqs. 27 and 28 has to be performed over both the irradiation configurations and the cell sampling configurations. The mean probability over the entire cell population of finding a cell in the state 0 (resp. 1) is:

$$\begin{aligned} \bar{P}(0) = \int f(p)\cdot P(0|p) \mathop {}\mathopen {}\mathrm {d}p \end{aligned}$$
(33)

and

$$\begin{aligned} \bar{P}(1) = \int f(p)\cdot P(1|p) \mathop {}\mathopen {}\mathrm {d}p \end{aligned}$$
(34)

where P(0|p) (resp. P(1|p)) refers to the probability that a cell gets the state 0 (resp. 1) when characterized by the response probability p. Since \(P(0|p)=1-p\) and \(P(1|p)=p\), and introducing the mean value of p over the cell population, one obtains:

$$\begin{aligned} \bar{p} = \int f(p)\cdot p \,\mathop {}\mathopen {}\mathrm {d}p \end{aligned}$$
(35)

It follows that:

$$\begin{aligned} \bar{P}(0)= & {} 1-\bar{p} \end{aligned}$$
(36)
$$\begin{aligned} \bar{P}(1)= & {} \bar{p} \end{aligned}$$
(37)

It is assumed that the cell sampling process does not introduce any correlation in cell selection, i.e., the number of sampled cells is low compared with the total number of available cells (f(p) is not affected by the sampling process). Hence, Eqs. 22 and 26 lead to:

$$\begin{aligned} \langle A\rangle&= \bar{p}{N_\mathrm {C}} \end{aligned}$$
(38)
$$\begin{aligned} \langle A^2\rangle&= \bar{p}{N_\mathrm {C}} + {N_\mathrm {C}}({N_\mathrm {C}}-1)\bar{p}^2 \end{aligned}$$
(39)
$$\begin{aligned} \langle \sigma _A^2\rangle&= \bar{p}{N_\mathrm {C}}(1-\bar{p}) \end{aligned}$$
(40)
$$\begin{aligned} \dfrac{\sigma _{\bar{P}}}{\bar{p}}&= \dfrac{\sqrt{1-\bar{p}}}{\sqrt{\bar{p}{N_\mathrm {C}}}} \end{aligned}$$
(41)

Equation 41 is equivalent to Eq. 31, which means that the combined random processes (cell sampling in an heterogeneous population, random irradiation processes) lead to the same relation as for a single random process (cf. previous section), for which \(p=\bar{p}\).

In other words, the whole random process can be viewed as a black box that randomly generates states 0 and 1 for the \({N_\mathrm {C}}\) cells with an average probability \(\bar{p}\). The details of the random generation do play a role in the probability \(\bar{p}\) that a cell is affected, but for a fixed value \(\bar{p}\), the stochastic fluctuations in the endpoint do not depend on the content of the black box.

As a conclusion, and somewhat a bit surprisingly, the diversity of responses within a cell population does not induce any increase in the stochastic fluctuations in the fraction of aberrant cells. This result is valid for any endpoint for which the individual response of a cell can be classified into two exclusive states (0 and 1).

Moreover, one can also draw another conclusion by applying this result to the irradiation mechanism itself. Indeed, the mechanism of irradiation that leads to chromosomal aberrations is likely made of various numerous random processes, including dose deposition at micro and nanoscale, number and position of the target structures, reactions of the damaged or implicated cell entities, etc. From this analysis, one can conclude that obviously all the details of these mechanisms control the average number of aberrant cells. However, except for this implication, these details do not affect the level of fluctuations. A high level of fluctuations is only due to the low fraction of aberrant cells and to the small number of cells under study.

Considering a time dependence of cell responses

One can try to apply the same approach by defining a more general distribution of cell responses f(pt) including the evolution of the cell response over the time t at which the cell sampling is performed. One would therefore define the average probability of aberrant cells by:

$$\begin{aligned} \bar{p} = \int f(p,t) \cdot p\mathop {}\mathopen {}\mathrm {d}p\mathop {}\mathopen {}\mathrm {d}t \end{aligned}$$
(42)

However, for such a random process, the protocol would be, for each of the \({N_\mathrm {C}}\) cells, to: (1) randomly generate a great number of values of time t at which this cell is selected, (2) randomly select this cell and (3) irradiate this cell.

Generally, measurements are performed with a large number of cells, which allows to cover the diversity in p over the population at the time t. However, the number of t values is limited, and for this reason, it is not enough to cover the domain of f(pt) (Eq. 42). Instead, a time-dependent mean value of p can be defined by:

$$\begin{aligned} \bar{p}(t) = \int f(p,t) \cdot p\mathop {}\mathopen {}\mathrm {d}p \end{aligned}$$
(43)

f(pt) may depend significantly on time t; thus, the mean response of the cell population \(\bar{p}(t)\) may evolve considerably with time. This evolution is not to be considered as fluctuations but instead as an evolution of the cell population of the individual over time. The measurements can then appear as non-reproducible. Deriving an average value of p over time would require measurements performed at many different times. This could lead to less fluctuations, but this protocol could be cumbersome to put in practice.

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Cunha, M., Testa, E., Komova, O.V. et al. Modeling cell response to low doses of photon irradiation—Part 1: on the origin of fluctuations. Radiat Environ Biophys 55, 19–30 (2016). https://doi.org/10.1007/s00411-015-0621-6

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