of counted colonies

growing probability under the control condition (dose=0, in this case)

xPE

plating efficiency (PE) 90%

Generally, cell survival curves are used for the quantitative representation of biological cellular responses. Cell survival curves, or in short, survival curves, are defined by the proportion of surviving cells (*S*) as a function of radiation dose (*D*), as in Fig. 5, and known to be different, depending on corresponding biological systems (mammalian cells, virus, yeast, bacteria, etc.), both in terms of shape and absolute value of *S* at a given dose. In this sense, "slope" of each survival curve implies degree of radiation sensitivity of corresponding biological systems.

Until now, many experiments to measure radiation survival curves have been performed for various species, including human cells. As a consequence of these experiments, many new findings have been revealed. For example, radiosensitivities for the corresponding species A, mammalian cells; B, *E. coli*; C, *E.coli* B/r; D, yeast; E, phage staph E; F, *Bacillus megaterium*; G, potato virus; and H, *M. radiodurans*, are comparable in the slope of the cell survival curves, and

Thus, the mammalian cells are most radiosensitive than other species. The differences of the radiosensitivity of various species are said to be correlated with its DNA content and efficiency

A > B > C > D > E > F > G > H. (2)

ered to mark the beginning of quantitative cellular radiation biology [10].

Interestingly, its slope is known to vary with DNA content [11].

they have the relation in radiosensitivity:

of the DNA repair system [12].


An affecting factor regarding the difference of species can be interpreted as the dependence of DNA content, and that of the radiation type and oxygen concentration can be interpreted as the dependence of the linear energy transfer (LET) values, i.e., average energy deposition per unit length. Here, the definition of the LET is described later. Clearly, these factors (2–4) are the external factors not relating to the biological activities inside the cell, and the following factors (5–8) seem to be the intracellular factors depending on the biological activities.

The extent of apoptosis seems to have been associated with the abrogation of *p53* function. It is well known that the gene *p53* mediates cellular responses to DNA damage, and results show that radiosensitivity is inversely correlated with the amount of DNA double-strand breaks (DSBs). In this case, the amount of DSBs should be the important feature to elucidate the affecting factor on cellular radiosensitivity. Moreover, cell lines derived from subjects with *ataxia telangiectasia* (AT) are three or four times sensitive to radiation than the normal cell lines, and no or very small shoulders appeared in survival curves for AT cell lines [17]. The genetic disorder responsible for the AT subject is the mutation of *ATM* gene, which is involved in the cellular response to DNA damage, especially in the DSB damage. Results of the *E. coli* mutant experiment clearly show that the *rec-* mutants have a higher radiosensitivity than that of *rec+*. The *rec* gene is involved in genetic recombination and DNA repair. These experimental facts also imply that the ability or capacity of DNA repair in specific cellular system plays an important role in radiosensitivity of cell.

The most complicated factor is the cell cycle. Generally, radiosensitivity is cyclically changed during cell cycle (Fig. 6), depending on the position of cell-cycle stages. To put it more precisely, radiation sensitivity is minimal when cells are irradiated in the early postmitotic (G1) and the premitotic (G2) phases of the cell-cycle and maximal in the mitotic (M) phase and late G1 or early synthesis (S) phases [19]. However, the pattern of the response cycle may vary depending on cell lines, especially on the length of G1 period [20]; typically, it is known to indicate a bimodal pattern. Generally, the origin of cell-cycle-dependent radiation sensitivity is supposed to be a consequence of some intracellular dynamics, e.g., the regulation mechanism of cellcycle checkpoint, repair ability of DNA damage, and higher-order structure of chromosomes; no explicit theoretical explanation exists presently for this cyclic response. <sup>8</sup> On the Dynamical Approach of Quantitative Radiation Biology

"cell time" shows irradiated time of cell cycle. Time zero represents mitosis (M in figure), and G1, S, and G2 represent corresponding cell-cycle stages. Adapted from [19] Radiosensitivity **Figure 6.** Plot of the surviving fraction of synchronized HeLa cells irradiated by X-ray. The horizontal axis "cell time" shows irradiated time of cell cycle. Time zero represents mitosis (M in figure), and G1, S, and G2 represent correspond‐ ing cell-cycle stages. Adapted from [19]

DNA content *↗* LET *↗* Dose rate *↗* Repair ability *↘* Cell cycle (cell time) *↘↗↘↗*

affecting quantities, except for the cell-cycle dependence, which has a cyclic relation

Fig. 6. Plot of the surviving fraction of synchronized HeLa cells irradiated by X-ray. The horizontal axis

Factor Small Large

Table 2. Summary of the factors that affect the radiosensitivity. Arrows inside the table show variation of the radiosensitivity. Radiosensitivity dependence has a linear relation with respect to the corresponding

and late G1 or early synthesis (S) phases [19]. However, the pattern of the response cycle may vary depending on cell lines, especially on the length of G1 period [20]; typically, it is known to indicate a bimodal pattern. Generally, the origin of cell-cycle-dependent radiation sensitivity is supposed to be a consequence of some intracellular dynamics, e.g., the regulation mechanism of cell-cycle checkpoint, repair ability of DNA damage, and higher-order structure of chromosomes; no explicit theoretical explanation exists presently for this cyclic response. Finally, the affecting factors will be summarized using quantitative indices (Table 2). It should be noted that the term "repair ability" in Table 2 seems not to be a quantitative index, but rather a qualitative index, it is thought that it can be measured, although not directly, but through other quantities, e.g., the DSB yield. Mathematical methods to analyze these experimental data and many attempts to understand the biological mechanisms involved in the variation of radiosensitivity under a variety of conditions are introduced in the next Finally, the affecting factors will be summarized using quantitative indices (Table 2). It should be noted that the term "repair ability" in Table 2 seems not to be a quantitative index, but rather a qualitative index, it is thought that it can be measured, although not directly, but through other quantities, e.g., the DSB yield. Mathematical methods to analyze these experimental data and many attempts to understand the biological mechanisms involved in the variation of radiosensitivity under a variety of conditions are introduced in the next section.


**Table 2.** Summary of the factors that affect the radiosensitivity. Arrows inside the table show variation of the radiosensitivity. Radiosensitivity dependence has a linear relation with respect to the corresponding affecting quantities, except for the cell-cycle dependence, which has a cyclic relation

#### **3. Mathematical representations**

The extent of apoptosis seems to have been associated with the abrogation of *p53* function. It is well known that the gene *p53* mediates cellular responses to DNA damage, and results show that radiosensitivity is inversely correlated with the amount of DNA double-strand breaks (DSBs). In this case, the amount of DSBs should be the important feature to elucidate the affecting factor on cellular radiosensitivity. Moreover, cell lines derived from subjects with *ataxia telangiectasia* (AT) are three or four times sensitive to radiation than the normal cell lines, and no or very small shoulders appeared in survival curves for AT cell lines [17]. The genetic disorder responsible for the AT subject is the mutation of *ATM* gene, which is involved in the cellular response to DNA damage, especially in the DSB damage. Results of the *E. coli* mutant experiment clearly show that the *rec-* mutants have a higher radiosensitivity than that of *rec+*. The *rec* gene is involved in genetic recombination and DNA repair. These experimental facts also imply that the ability or capacity of DNA repair in specific cellular system plays an

The most complicated factor is the cell cycle. Generally, radiosensitivity is cyclically changed during cell cycle (Fig. 6), depending on the position of cell-cycle stages. To put it more precisely, radiation sensitivity is minimal when cells are irradiated in the early postmitotic (G1) and the premitotic (G2) phases of the cell-cycle and maximal in the mitotic (M) phase and late G1 or early synthesis (S) phases [19]. However, the pattern of the response cycle may vary depending on cell lines, especially on the length of G1 period [20]; typically, it is known to indicate a bimodal pattern. Generally, the origin of cell-cycle-dependent radiation sensitivity is supposed to be a consequence of some intracellular dynamics, e.g., the regulation mechanism of cellcycle checkpoint, repair ability of DNA damage, and higher-order structure of chromosomes;

no explicit theoretical explanation exists presently for this cyclic response. <sup>8</sup> On the Dynamical Approach of Quantitative Radiation Biology

0 5 10 15 20

Fig. 6. Plot of the surviving fraction of synchronized HeLa cells irradiated by X-ray. The horizontal axis "cell time" shows irradiated time of cell cycle. Time zero represents mitosis (M in figure), and G1, S, and

**Figure 6.** Plot of the surviving fraction of synchronized HeLa cells irradiated by X-ray. The horizontal axis "cell time" shows irradiated time of cell cycle. Time zero represents mitosis (M in figure), and G1, S, and G2 represent correspond‐

Factor Small Large

Table 2. Summary of the factors that affect the radiosensitivity. Arrows inside the table show variation of the radiosensitivity. Radiosensitivity dependence has a linear relation with respect to the corresponding

and late G1 or early synthesis (S) phases [19]. However, the pattern of the response cycle may vary depending on cell lines, especially on the length of G1 period [20]; typically, it is known to indicate a bimodal pattern. Generally, the origin of cell-cycle-dependent radiation sensitivity is supposed to be a consequence of some intracellular dynamics, e.g., the regulation mechanism of cell-cycle checkpoint, repair ability of DNA damage, and higher-order structure of chromosomes; no explicit theoretical explanation exists presently for this cyclic response. Finally, the affecting factors will be summarized using quantitative indices (Table 2). It should be noted that the term "repair ability" in Table 2 seems not to be a quantitative index, but rather a qualitative index, it is thought that it can be measured, although not directly, but through other quantities, e.g., the DSB yield. Mathematical methods to analyze these experimental data and many attempts to understand the biological mechanisms involved in the variation of radiosensitivity under a variety of conditions are introduced in the next

DNA content *↗* LET *↗* Dose rate *↗* Repair ability *↘* Cell cycle (cell time) *↘↗↘↗*

affecting quantities, except for the cell-cycle dependence, which has a cyclic relation

**M G1 S G2**

cell time

Radiosensitivity

important role in radiosensitivity of cell.

48 Evolution of Ionizing Radiation Research

0

G2 represent corresponding cell-cycle stages. Adapted from [19]

0.2

cell survival

ing cell-cycle stages. Adapted from [19]

0.4

Many efforts have been done to understand a priori the actions of radiations on the living organisms from the early 1920s, and most of these types of studies have been done by physicists. The logical structure of this theory is basically based on the simple idea, that is, studying the behavior of the living organisms as the "response" to the radiation as the "action"; actually, it is a so-called dose-response relationship in today's radiation biology. Most of experiments in such studies have mainly used mortality or survival as the corresponding biological "effects." The surviving fraction of the various living organisms, including bacteria, virus, and drosophila egg, have been measured by irradiation experiments. It should be noted that the inactivation is used for the survival endpoint of virus or drosophila egg.

In a radiation biology, a plot of the surviving fraction of cells vs radiation dose is called survival curve. Generally, radiation dose is plotted along the horizontal axis (logarithm in some cases), and surviving fraction is plotted along the vertical axis. This kind of dose-response relationship obtained by measuring cell surviving fraction is a momentous indicator of the radiosensitivity of corresponding living organisms, up to now.

As functional forms of survival curves, two general types of functions are well known to represent various survival curves, exponential and sigmoid (Fig. 7). However, as described later, there is no theory that can clearly explain the difference of the functional form of these survival curves from biological mechanisms.

In such a situation, the most famous mathematical model to derive the function that has a good fit with the experimentally obtained survival data is the target theory that has been summar‐

section.

egg.

**3. Mathematical Representations**

8).

; therefore,

Will-be-set-by-IN-TECH 9

Many efforts have been done to understand a priori the actions of radiations on the living organisms from the early 1920s, and most of these types of studies have been done by physicists. The logical structure of this theory is basically based on the simple idea, that is, studying the behavior of the living organisms as the "response" to the radiation as the "action"; actually, it is a so-called dose-response relationship in today's radiation biology. Most of experiments in such studies have mainly used mortality or survival as the corresponding biological "effects." The surviving fraction of the various living organisms, including bacteria, virus, and drosophila egg, have been measured by irradiation experiments. It should be noted that the inactivation is used for the survival endpoint of virus or drosophila

In a radiation biology, a plot of the surviving fraction of cells vs radiation dose is called survival curve. Generally, radiation dose is plotted along the horizontal axis (logarithm in some cases), and surviving fraction is plotted along the vertical axis. This kind of dose-response relationship obtained by measuring cell surviving fraction is a momentous

As functional forms of survival curves, two general types of functions are well known to represent various survival curves, exponential and sigmoid (Fig. 7). However, as described later, there is no theory that can clearly explain the difference of the functional form of these

10 On the Dynamical Approach of Quantitative Radiation Biology Fig. 7. Two examples of survival curves. "E" means exponential and "S" means sigmoid In such a situation, the most famous mathematical model to derive the function that has a **Figure 7.** Two examples of survival curves. "E" means exponential and "S" means sigmoid

ized by Lea [21]. Target theory assumes that the "hit" of the discrete radiation to the radiationsensitive site called "target" may induce cell death in a broad sense; moreover, the radiation hit is assumed a random process which has a Poisson probability distribution. (Fig. 8). summarized by Lea [21]. Target theory assumes that the "hit" of the discrete radiation to the radiation-sensitive site called "target" may induce cell death in a broad sense; moreover, the

good fit with the experimentally obtained survival data is the target theory that has been

radiation hit is assumed a random process which has a Poisson probability distribution. (Fig.

Fig. 8. Schematic explanation of the target theory. Target theory assumes that each cell has some targets and radiation hits of all target may induce cell death. Left: Single radiation hit to the single target kills cell. Right: Single radiation hit to all of three targets kills cell **Figure 8.** Schematic explanation of the target theory. Target theory assumes that each cell has some targets and radia‐ tion hits of all target may induce cell death. Left: Single radiation hit to the single target kills cell. Right: Single radia‐ tion hit to all of three targets kills cell

*P*(*n*) = *e*

Let *P*(*n*) be the probability of *n* hits with expected value of hits *m*; then *<sup>−</sup><sup>m</sup> <sup>m</sup><sup>n</sup>* Let *P*(*n*) be the probability of *n* hits with expected value of hits *m*; then

$$P(n) = e^{-m} m^n n! \tag{3}$$

radiation dose *D*; thus, *p* = *αD*, where *α* is a proportional coefficient. Therefore, the expected hit number is written as *m* = *VαD* (4) There are several model types in target theory corresponding to the combination of required Let *V* be a target volume, and suppose that *p* denotes hit number per unit volume; then the expected value of hit number is *m*=*Vp*. Then, the hit number *p* may be proportional to the radiation dose *D*; thus, *p* =*αD*, where *α* is a proportional coefficient. Therefore, the expected hit number is written as

target and hit number for cell death; for simplicity, single-hit-single-target model and

At first, suppose each cell contains only one single target and single radiation hit kills (inactivates) the cell, then the cell survival probability *S*, i.e., non-hit (*n* = 0) probability, is

*<sup>−</sup><sup>m</sup>* = *e*

*S* = *P*(0) = *e*

Let *D*<sup>0</sup> be a radiation dose that has an expected hit number *m* = *VαD*<sup>0</sup> = 1; thus,

"E") in Fig. 7. Consequently, the probability of cell death for single-target cell is

*S* = *e*

1 *− S* = 1 *− e*

This survival curve equation (7) has a good fit with exponential-shaped curve (denoted as

expected value of hit number is *m* = *V p*. Then, the hit number *p* may be proportional to the

$$m = VaD\tag{4}$$

*Vα* = 1/*D*<sup>0</sup> (6)

*<sup>−</sup>D*/*D*<sup>0</sup> (7)

*<sup>−</sup>D*/*D*<sup>0</sup> . (8)

*<sup>−</sup>Vα<sup>D</sup>* (5)

There are several model types in target theory corresponding to the combination of required target and hit number for cell death; for simplicity, single-hit-single-target model and singlehit-multi-target model are described.

At first, suppose each cell contains only one single target and single radiation hit kills (inacti‐ vates) the cell, then the cell survival probability *S*, i.e., non-hit (*n* =0) probability, is

$$S = P\left(0\right) = e^{-m} = e^{-VaD} \tag{5}$$

Let *D*0 be a radiation dose that has an expected hit number *m*=*VαD*<sup>0</sup> =1 ; thus,

$$V\alpha = 1/D\_0\tag{6}$$

therefore,

ized by Lea [21]. Target theory assumes that the "hit" of the discrete radiation to the radiationsensitive site called "target" may induce cell death in a broad sense; moreover, the radiation hit is assumed a random process which has a Poisson probability distribution. (Fig. 8).

summarized by Lea [21]. Target theory assumes that the "hit" of the discrete radiation to the radiation-sensitive site called "target" may induce cell death in a broad sense; moreover, the radiation hit is assumed a random process which has a Poisson probability distribution. (Fig.

Radiation

hit

*<sup>n</sup>*! (3)

*<sup>−</sup>Vα<sup>D</sup>* (5)

*m* = *VαD* (4)

(4)

( )= ! -*m n Pn e mn* (3)

*Vα* = 1/*D*<sup>0</sup> (6)

*<sup>−</sup>D*/*D*<sup>0</sup> (7)

*<sup>−</sup>D*/*D*<sup>0</sup> . (8)

Radiation

hit hit

single hit-single target model single hit-multi target model

*P*(*n*) = *e*

Fig. 8. Schematic explanation of the target theory. Target theory assumes that each cell has some targets and radiation hits of all target may induce cell death. Left: Single radiation hit to the single target kills

**Figure 8.** Schematic explanation of the target theory. Target theory assumes that each cell has some targets and radia‐ tion hits of all target may induce cell death. Left: Single radiation hit to the single target kills cell. Right: Single radia‐

Let *V* be a target volume, and suppose that *p* denotes hit number per unit volume; then the expected value of hit number is *m* = *V p*. Then, the hit number *p* may be proportional to the radiation dose *D*; thus, *p* = *αD*, where *α* is a proportional coefficient. Therefore, the expected

Let *V* be a target volume, and suppose that *p* denotes hit number per unit volume; then the expected value of hit number is *m*=*Vp*. Then, the hit number *p* may be proportional to the radiation dose *D*; thus, *p* =*αD*, where *α* is a proportional coefficient. Therefore, the expected

There are several model types in target theory corresponding to the combination of required target and hit number for cell death; for simplicity, single-hit-single-target model and

At first, suppose each cell contains only one single target and single radiation hit kills (inactivates) the cell, then the cell survival probability *S*, i.e., non-hit (*n* = 0) probability, is

*m VD* = a

*<sup>−</sup><sup>m</sup>* = *e*

*S* = *P*(0) = *e*

Let *D*<sup>0</sup> be a radiation dose that has an expected hit number *m* = *VαD*<sup>0</sup> = 1; thus,

"E") in Fig. 7. Consequently, the probability of cell death for single-target cell is

*S* = *e*

1 *− S* = 1 *− e*

This survival curve equation (7) has a good fit with exponential-shaped curve (denoted as

*<sup>−</sup><sup>m</sup> <sup>m</sup><sup>n</sup>*

8).

Radiation

hit number is written as

hit number is written as

tion hit to all of three targets kills cell

; therefore,

miss

section.

egg.

50 Evolution of Ionizing Radiation Research

**3. Mathematical Representations**

survival curves from biological mechanisms.

Survival

E

**Figure 7.** Two examples of survival curves. "E" means exponential and "S" means sigmoid

single-hit-multi-target model are described.

cell. Right: Single radiation hit to all of three targets kills cell

Let *P*(*n*) be the probability of *n* hits with expected value of hits *m*; then

Let *P*(*n*) be the probability of *n* hits with expected value of hits *m*; then

10 On the Dynamical Approach of Quantitative Radiation Biology

In such a situation, the most famous mathematical model to derive the function that has a good fit with the experimentally obtained survival data is the target theory that has been

Fig. 7. Two examples of survival curves. "E" means exponential and "S" means sigmoid

Dose

Will-be-set-by-IN-TECH 9

Many efforts have been done to understand a priori the actions of radiations on the living organisms from the early 1920s, and most of these types of studies have been done by physicists. The logical structure of this theory is basically based on the simple idea, that is, studying the behavior of the living organisms as the "response" to the radiation as the "action"; actually, it is a so-called dose-response relationship in today's radiation biology. Most of experiments in such studies have mainly used mortality or survival as the corresponding biological "effects." The surviving fraction of the various living organisms, including bacteria, virus, and drosophila egg, have been measured by irradiation experiments. It should be noted that the inactivation is used for the survival endpoint of virus or drosophila

In a radiation biology, a plot of the surviving fraction of cells vs radiation dose is called survival curve. Generally, radiation dose is plotted along the horizontal axis (logarithm in some cases), and surviving fraction is plotted along the vertical axis. This kind of dose-response relationship obtained by measuring cell surviving fraction is a momentous

As functional forms of survival curves, two general types of functions are well known to represent various survival curves, exponential and sigmoid (Fig. 7). However, as described later, there is no theory that can clearly explain the difference of the functional form of these

S

indicator of the radiosensitivity of corresponding living organisms, up to now.

$$\mathcal{S} = e^{-D/D\_0} \tag{7}$$

This survival curve equation (7) has a good fit with exponential-shaped curve (denoted as "E") in Fig. 7. Consequently, the probability of cell death for single-target cell is

$$\mathbf{1} - \mathbf{S} = \mathbf{1} - e^{-D/D\_0}.\tag{8}$$

Next, suppose each cell contains two targets, and single radiation hit of both targets kills (inactivates) the cell. Moreover, suppose these two hit events are stochastically independent, then the cell death probability of two targets case *D*<sup>2</sup> equals the product of cell death probability of single target case, equation (8), therefore

$$^1D\_2 = \left(1 - S\right)\left(1 - S\right) = \left(1 - e^{-D/D\_0}\right)^2\tag{9}$$

Thus, the cell survival probability of two targets case, *S*2, is

$$S\_2 = 1 - \left(1 - e^{-D/D\_0}\right)^2;\tag{10}$$

consequently, cell survival probability of *k* targets case, *Sk* , is

$$S\_k = 1 - \left(1 - e^{-D/D\_0}\right)^k \tag{11}$$

Alternatively, this survival curve, equation (11), has a good fit with sigmoid-shaped survival curve (denoted as "S") in Fig. 7.

Studying the validity of target theory, various inactivation experiments of viruses by radiation have been done and examined [21]. Various measurement results have shown that there is a strong correlation between virus size (diameter) and inactivation dose (of X- or γ-rays); moreover, in some cases, target volume *V* equals the size of the virus itself. It should be noted that the word "target" used in the theory does not give a specific object; it is just a concept. Today, many experimental evidences suggest the DNA as the primary target for radiation.

Although the target theory can derive the functions to fit well with the survival curves of various experimental data, it is known that target theory gives higher surviving probability in very low-dose region than the real experimental data [22]. That is to say, actually, many cell deaths have occurred than expected from target theory in the very low-dose region. Moreover, it has been found that survival function obtained by target theory do not fit well with the experimental data not only in the low-dose region but in low-dose-rate irradiation.

In target theory, *D*<sup>0</sup> is generally called "mean lethal dose"; in a sense, one hit will kill the living organisms; however, it is just a parameter that can be estimated from fitting of experimental data to the target theory. On the contrary, by tuning this *D*0 parameter, target theory can derive any functions to fit well with almost all experimental data. It is clear that this kind of high possibility of application to fit experimental data represents the arbitrariness of the model; thus, the model is not supposed to capture the essential dynamics of phenomenon.

Here, another mathematical model which will explain the shape of cell survival curves is introduced. The so-called linear-quadratic model (L-Q model) has been proposed for the candidate expression of the cell survival curves,

$$S = e^{-\alpha D - \beta D^2} \tag{12}$$

where *S* is the surviving fraction for radiation dose *D* and α and *β* are proportionality constants. This kind of mathematical expression of the dose-response relationship can be found in many articles, e.g., [23, 24, 25, 26, 27].

L-Q model is often used to explain dose-response relationship of unstable chromosomal aberrations, e.g., dicentrics and rings ((b) and (d) in Fig. 9), which is thought to be the major cause of cell death. Rings and dicentrics can be seen and counted under the light microscope, and about 3 years of the half-life of lymphocytes carrying these aberrations have made it possible to estimate the accumulated radiation dose during a long period. Generally, this

consequently, cell survival probability of *k* targets case, *Sk* , is

curve (denoted as "S") in Fig. 7.

52 Evolution of Ionizing Radiation Research

( ) / <sup>0</sup> =1 1 - - - *<sup>k</sup> D D*

Alternatively, this survival curve, equation (11), has a good fit with sigmoid-shaped survival

Studying the validity of target theory, various inactivation experiments of viruses by radiation have been done and examined [21]. Various measurement results have shown that there is a strong correlation between virus size (diameter) and inactivation dose (of X- or γ-rays); moreover, in some cases, target volume *V* equals the size of the virus itself. It should be noted that the word "target" used in the theory does not give a specific object; it is just a concept. Today, many experimental evidences suggest the DNA as the primary target for radiation. Although the target theory can derive the functions to fit well with the survival curves of various experimental data, it is known that target theory gives higher surviving probability in very low-dose region than the real experimental data [22]. That is to say, actually, many cell deaths have occurred than expected from target theory in the very low-dose region. Moreover, it has been found that survival function obtained by target theory do not fit well with the

experimental data not only in the low-dose region but in low-dose-rate irradiation.

thus, the model is not supposed to capture the essential dynamics of phenomenon.

= - a b

candidate expression of the cell survival curves,

articles, e.g., [23, 24, 25, 26, 27].

In target theory, *D*<sup>0</sup> is generally called "mean lethal dose"; in a sense, one hit will kill the living organisms; however, it is just a parameter that can be estimated from fitting of experimental data to the target theory. On the contrary, by tuning this *D*0 parameter, target theory can derive any functions to fit well with almost all experimental data. It is clear that this kind of high possibility of application to fit experimental data represents the arbitrariness of the model;

Here, another mathematical model which will explain the shape of cell survival curves is introduced. The so-called linear-quadratic model (L-Q model) has been proposed for the

2

where *S* is the surviving fraction for radiation dose *D* and α and *β* are proportionality constants. This kind of mathematical expression of the dose-response relationship can be found in many

L-Q model is often used to explain dose-response relationship of unstable chromosomal aberrations, e.g., dicentrics and rings ((b) and (d) in Fig. 9), which is thought to be the major cause of cell death. Rings and dicentrics can be seen and counted under the light microscope, and about 3 years of the half-life of lymphocytes carrying these aberrations have made it possible to estimate the accumulated radiation dose during a long period. Generally, this

*D D S e* (12)

*<sup>k</sup> S e* (11)

Fig. 9. Schematic illustrations of the induction mechanism of chromosome aberrations. The four typical types of radiation-induced chromosome aberrations are presented. (a) One brake or cut is produced in one arm. "Gap" is defined as an achromatic lesion which has a smaller width than that of one chromatid and not separating. (b) Two breaks are produced in each arm of one chromatid. Then the broken ends may rejoin and make one ring and one fragment. (c)–(d) Two breaks are produced, one break in each arm of two chromosomes. Translocation (c) is made by rejoining of the proximal part, which has one centromere, and acentric fragment. Dicentric (d) is made by rejoining of the two proximal parts. Chromosome aberrations (c) and (d) are classified as the "exchange" **Figure 9.** Schematic illustrations of the induction mechanism of chromosome aberrations. The four typical types of ra‐ diation-induced chromosome aberrations are presented. (a) One brake or cut is produced in one arm. "Gap" is defined as an achromatic lesion which has a smaller width than that of one chromatid and not separating. (b) Two breaks are produced in each arm of one chromatid. Then the broken ends may rejoin and make one ring and one fragment. (c)–(d) Two breaks are produced, one break in each arm of two chromosomes. Translocation (c) is made by rejoining of the proximal part, which has one centromere, and acentric fragment. Dicentric (d) is made by rejoining of the two proxi‐ mal parts. Chromosome aberrations (c) and (d) are classified as the "exchange"

method is so-called biological dosimetry. Further examples of the chromosome aberration structure can be seen in the NIH Web site [28]. cause of cell death. Rings and dicentrics can be seen and counted under the light microscope, and about 3 years of the half-life of lymphocytes carrying these aberrations have made it possible to estimate the accumulated radiation dose during a long period. Generally, this method is so-called biological dosimetry. Further examples of the chromosome aberration

What is the phenomenological meaning of this function? It has been shown that the frequency of the chromosomal aberrations caused only by one break or damage has a linear dose dependence, *αD* ; however, that of induced by two chromosome breaks has a quadratic dose dependence, *βD* <sup>2</sup> (Fig. 10). In other words, frequency of the chromosome aberrations made from one break (one hit event) has a linear dose-response relationship, and then, frequency of the two-break (requires two hit)-induced aberrations has a quadratic dose-response relation‐ ship. Consequently, frequency of the chromosome aberrations per cell *X* may have the doseresponse relationship structure can be seen in the NIH Web site [28]. What is the phenomenological meaning of this function? It has been shown that the frequency of the chromosomal aberrations caused only by one break or damage has a linear dose dependence, *αD*; however, that of induced by two chromosome breaks has a quadratic dose dependence, *βD*<sup>2</sup> (Fig. 10). In other words, frequency of the chromosome aberrations made from one break (one hit event) has a linear dose-response relationship, and then, frequency of the two-break (requires two hit)-induced aberrations has a quadratic dose-response relationship. Consequently, frequency of the chromosome aberrations per cell *X* may have the dose-response relationship

$$X = a^\dagger D + \oint \vec{D}^2 \tag{13}$$

*D*<sup>2</sup> (13)

where *D* is the radiation dose and *α* ' and *β* ' being parameters. Moreover, it has been found that the relationship between chromosome aberrations per cell *X* and surviving fraction *S* has a relation [30]:

that the relationship between chromosome aberrations per cell *X* and surviving fraction *S* has

*D* + *β′*

*X* = *α′*

$$S = e^{-\gamma X} \tag{14}$$

therefore,

$$\mathbf{S} = \mathbf{e}^{-\rho \dot{\alpha}^\prime D - \eta \dot{\mathcal{P}}^2 D^2} = \mathbf{e}^{-aD - \beta D^2} \tag{15}$$

where *α* =*γα* ' , *β* =*γβ* ' . Will-be-set-by-IN-TECH <sup>13</sup>

Fig. 10. Dose effect curves for the different types of chromosome aberrations (gap and exchange) induced by X-ray in the *Vicia faba*. Clearly, gap shows a linear dose dependence while the exchange shows quadratic. Adapted from[29] **Figure 10.** Dose effect curves for the different types of chromosome aberrations (gap and exchange) induced by X-ray in the *Vicia faba*. Clearly, gap shows a linear dose dependence while the exchange shows quadratic. Adapted from[29]

a relation [30]: *S* = *e <sup>−</sup>γ<sup>X</sup>* (14) ; therefore, *S* = *e −γα′ D−γβ′ D*2 = *e <sup>−</sup>αD−βD*<sup>2</sup> (15) where *α* = *γα′* , *β* = *γβ′* . As stated above, the dose-response relationship between chromosome aberrations and radiation dose can be well explained from the relationship between the number of chromosome breaks and radiation dose using L-Q model. However, it may be said that L-Q model is rather descriptive than the dynamical.

As stated above, the dose-response relationship between chromosome aberrations and radiation dose can be well explained from the relationship between the number of chromosome breaks and radiation dose using L-Q model. However, it may be said that L-Q model is rather descriptive than the dynamical. Finally, one more interesting experimental data corresponding to the relation between Finally, one more interesting experimental data corresponding to the relation between radiation quality and dose is shown in Fig. 11. Here, each type of radiation has a different radiation quantity, linear energy transfer (LET) which is defined as the linear density of imparted energy. Therefore, LET is written mathematically as

radiation quality and dose is shown in Fig. 11. Here, each type of radiation has a different

$$\text{LET} = dEdl\tag{16}$$

LET <sup>=</sup> *dE dl* (16) where *dE* is the averaged energy deposition to the medium by charged particles in traversing a total length *dl*. In short, it is a quantity that represents averaged energy deposition around the tracks of charged particles (Fig. 12). Some examples of the LET values for different types of radiation are shown in Table 3. Moreover, Fig. 11 clearly shows that quality of the radiation where *dE* is the averaged energy deposition to the medium by charged particles in traversing a total length *dl*. In short, it is a quantity that represents averaged energy deposition around the tracks of charged particles (Fig. 12). Some examples of the LET values for different types of radiation are shown in Table 3. Moreover, Fig. 11 clearly shows that quality of the radiation determines the number of aberrations. It should be noted that the LET is not an actual, experimentally obtained value but just a practical, stochastically estimated value.

Fig. 11. Dose-response curves for dicentric chromosome aberration induction in human lymphocytes with various LET radiations. Adapted from [31] determines the number of aberrations. It should be noted that the LET is not an actual, **Figure 11.** Dose-response curves for dicentric chromosome aberration induction in human lymphocytes with various LET radiations. Adapted from [31]

The effects of ionizing radiation on the living organisms have been studied in many years to understand three principal physical aspects: 1) **quantity of radiation**, 2) **temporal distribution of radiation**, and 3) **radiation quality** [32]. Studies on the dose-response relationship including cell survival curves are considered to belong to the research field of quantity of radiation, and studies on the dose rate and fractionation [33] are considered to belong to the research field of temporal distribution. experimentally obtained value but just a practical, stochastically estimated value. The effects of ionizing radiation on the living organisms have been studied in many years to understand three principal physical aspects: 1) **quantity of radiation**, 2) **temporal distribution of radiation**, and 3) **radiation quality** [32] . Studies on the dose-response relationship including cell survival curves are considered to belong to the research field of quantity of radiation, and studies on the dose rate and fractionation [33] are considered to belong to the research field of temporal distribution. Historically, the concept of radiation quality has been established to study the biological

Historically, the concept of radiation quality has been established to study the biological effectiveness of ionizing radiation depending not only on the amount of absorbed radiation dose but also on the spatial distribution of energy deposition [34]. It is thought that the beginning of such study may have been closely related to the discovery of scientific evidence; i.e., DNA is the principal target for the biological radiation responses. For the sake of the presence of a small target in the cell, scientific knowledge of ionizing radiation, not only the macroscopic property (amount of absorbed radiation dose) but also detailed track structure (spatial distribution of energy deposition), has been required to study biological radiation responses. effectiveness of ionizing radiation depending not only on the amount of absorbed radiation dose but also on the spatial distribution of energy deposition [34]. It is thought that the beginning of such study may have been closely related to the discovery of scientific evidence; i.e., DNA is the principal target for the biological radiation responses. For the sake of the presence of a small target in the cell, scientific knowledge of ionizing radiation, not only the macroscopic property (amount of absorbed radiation dose) but also detailed track structure (spatial distribution of energy deposition), has been required to study biological radiation responses. In other words, A detailed information on how much damage on the DNA are produced by the different types of radiations with the same amount of dose has been necessary to estimate the radiation effects. Estimation or calculation of LET is one of such studies. As mentioned above, requirement of the detailed information concerning radiation quality


leads to plenty of studies that correspond to radiation track structure [35]; moreover, many

**Table 3.** Typical values of LET. Values taken from [12]

therefore,

where *α* =*γα* '

, *β* =*γβ* '

54 Evolution of Ionizing Radiation Research

Aberrations per Cell

quadratic. Adapted from[29]

, *β* = *γβ′* .

model is rather descriptive than the dynamical.

imparted energy. Therefore, LET is written mathematically as

a relation [30]:

where *α* = *γα′*

descriptive than the dynamical.

; therefore,

2 2

. Will-be-set-by-IN-TECH <sup>13</sup>

a b

DOSE

*<sup>−</sup>αD−βD*<sup>2</sup>

Fig. 10. Dose effect curves for the different types of chromosome aberrations (gap and exchange) induced by X-ray in the *Vicia faba*. Clearly, gap shows a linear dose dependence while the exchange shows

**Figure 10.** Dose effect curves for the different types of chromosome aberrations (gap and exchange) induced by X-ray in the *Vicia faba*. Clearly, gap shows a linear dose dependence while the exchange shows quadratic. Adapted from[29]

*S* = *e*

As stated above, the dose-response relationship between chromosome aberrations and radiation dose can be well explained from the relationship between the number of chromosome breaks and radiation dose using L-Q model. However, it may be said that L-Q model is rather

As stated above, the dose-response relationship between chromosome aberrations and radiation dose can be well explained from the relationship between the number of chromosome breaks and radiation dose using L-Q model. However, it may be said that L-Q

Finally, one more interesting experimental data corresponding to the relation between radiation quality and dose is shown in Fig. 11. Here, each type of radiation has a different radiation quantity, linear energy transfer (LET) which is defined as the linear density of

Finally, one more interesting experimental data corresponding to the relation between radiation quality and dose is shown in Fig. 11. Here, each type of radiation has a different radiation quantity, linear energy transfer (LET) which is defined as the linear density of

LET <sup>=</sup> *dE*

where *dE* is the averaged energy deposition to the medium by charged particles in traversing a total length *dl*. In short, it is a quantity that represents averaged energy deposition around the tracks of charged particles (Fig. 12). Some examples of the LET values for different types of radiation are shown in Table 3. Moreover, Fig. 11 clearly shows that quality of the radiation determines the number of aberrations. It should be noted that the LET is not an actual,

where *dE* is the averaged energy deposition to the medium by charged particles in traversing a total length *dl*. In short, it is a quantity that represents averaged energy deposition around the tracks of charged particles (Fig. 12). Some examples of the LET values for different types of radiation are shown in Table 3. Moreover, Fig. 11 clearly shows that quality of the radiation

experimentally obtained value but just a practical, stochastically estimated value.

*S* = *e −γα′ D−γβ′ D*2 = *e*

imparted energy. Therefore, LET is written mathematically as

*D D DD Se e* (15)

*<sup>−</sup>γ<sup>X</sup>* (14)

*dl* (16)

LET = *dEdl* (16)

(15)

= = - - --

*' '*

ga gb

Gap Exchange In other words, A detailed information on how much damage on the DNA are produced by the different types of radiations with the same amount of dose has been necessary to estimate the radiation effects. Estimation or calculation of LET is one of such studies. Radiation LET keV/*µ*m Cobalt-60 *γ*-rays 0.2 250-kV X-rays 2.0 10-MeV protons 4.7

Will-be-set-by-IN-TECH 15

As mentioned above, requirement of the detailed information concerning radiation quality leads to plenty of studies that correspond to radiation track structure [35]; moreover, many studies regarding radiation action that has a more broad timescale including very early physical (10-18-10-12 sec) and chemical (10-12-10-0 sec) process have been performed [36]. 150-MeV protons 0.5 2.5 MeV *α*-particles 166 2-GeV Fe ions 1000 Table 3. Typical values of LET. Values taken from [12]

> studies regarding radiation action that has a more broad timescale including very early physical (10*−*<sup>18</sup> *<sup>−</sup>* <sup>10</sup>*−*<sup>12</sup> sec) and chemical (10*−*<sup>12</sup> *<sup>−</sup>* 100 sec) process have been performed [36].

Fig. 12. Schematic explanation of the variation of the ionization density. LET is increased in the order from A to C (top to bottom). Black circles show positions of ionization **4. Analyzing the Various Survival Curves Based on the Knowledge of Figure 12.** Schematic explanation of the variation of the ionization density. LET is increased in the order from A to C (top to bottom). Black circles show positions of ionization

#### Survival curves of cells obtained under various conditions and mathematical expressions describing the survival curves have been introduced until now. It seems that the radiosensitivity and most of affecting factors have a simple proportional relationship; however, the interpretation of the cell-cycle-dependent nature of the radiosensitivity seems to be the most difficult. Can such a radiosensitivity really be explained by the mathematical **4. Analyzing the various survival curves based on the knowledge of modern molecular biology**

**Modern Molecular Biology**

discussed from the viewpoint of mathematical model and expected intracellular affecting factors. At first, factors that can affect the shape of cell survival curve are discussed from the target theory, equations (7) and (11). Clearly, these equations show that the radiation dose *D*, required target number to cell death *k*, and target volume *V* are all the affecting factors; moreover, radiosensitivity is inversely proportional to a required target number *k* and in proportion to the target volume *V* and radiation dose *D*. In order to determine a target volume and number, the information concerning the substance of the target should be needed. What is the actual target of radiation? Generally, it is believed that DNA is the principal target of Survival curves of cells obtained under various conditions and mathematical expressions describing the survival curves have been introduced until now. It seems that the radiosensi‐ tivity and most of affecting factors have a simple proportional relationship; however, the interpretation of the cell-cycle-dependent nature of the radiosensitivity seems to be the most difficult. Can such a radiosensitivity really be explained by the mathematical model mentioned above? Here, the cell-cycle-dependent cellular response has been briefly discussed from the viewpoint of mathematical model and expected intracellular affecting factors.

model mentioned above? Here, the cell-cycle-dependent cellular response has been briefly

radiation. At first, factors that can affect the shape of cell survival curve are discussed from the target theory, equations (7) and (11). Clearly, these equations show that the radiation dose *D*, required target number to cell death *k*, and target volume *V* are all the affecting factors; moreover, radiosensitivity is inversely proportional to a required target number *k* and in proportion to the target volume *V* and radiation dose *D*. In order to determine a target volume and number, the information concerning the substance of the target should be needed. What is the actual target of radiation? Generally, it is believed that DNA is the principal target of radiation.

In case of cell-cycle-dependent radiosensitivity, the radiation dose *D* is not the control parameter, and the required target number *k* for cell death (reproductive cell death) seems to be unchanged, because one chromosome aberration may induce reproductive cell death. Thus, the required target number to cell death *k* and target volume *V* are the factors that need to be considered now. The volume of the target seems to have almost the same value from the Mphase of the cell cycle to the early S-phase; occupying the volume of all chromosomes for the most condensed stage (M-phase) is ∼91.6*μm*<sup>3</sup> [37], whereas that of the interphase is estimated as the volume of cell nucleus; in case of 5.4*μm* diameter, interphase volume is almost the same as that of metaphase. On the contrary, target volume (DNA content) is doubled at the late S phase. By the interpretation from a target theory, increase in target volume causes increase in cell death. From Fig. 6, cell death seems to decrease during S-phase; this is not consistent with the analysis from the target theory.

In other words, A detailed information on how much damage on the DNA are produced by the different types of radiations with the same amount of dose has been necessary to estimate

Radiation LET keV/*µ*m Cobalt-60 *γ*-rays 0.2 250-kV X-rays 2.0 10-MeV protons 4.7 150-MeV protons 0.5 2.5 MeV *α*-particles 166 2-GeV Fe ions 1000

Will-be-set-by-IN-TECH 15

As mentioned above, requirement of the detailed information concerning radiation quality leads to plenty of studies that correspond to radiation track structure [35]; moreover, many studies regarding radiation action that has a more broad timescale including very early physical (10-18-10-12 sec) and chemical (10-12-10-0 sec) process have been performed [36].

> studies regarding radiation action that has a more broad timescale including very early physical (10*−*<sup>18</sup> *<sup>−</sup>* <sup>10</sup>*−*<sup>12</sup> sec) and chemical (10*−*<sup>12</sup> *<sup>−</sup>* 100 sec) process have been performed [36].

> Fig. 12. Schematic explanation of the variation of the ionization density. LET is increased in the order

**Figure 12.** Schematic explanation of the variation of the ionization density. LET is increased in the order from A to C

Survival curves of cells obtained under various conditions and mathematical expressions describing the survival curves have been introduced until now. It seems that the radiosensitivity and most of affecting factors have a simple proportional relationship; however, the interpretation of the cell-cycle-dependent nature of the radiosensitivity seems to be the most difficult. Can such a radiosensitivity really be explained by the mathematical model mentioned above? Here, the cell-cycle-dependent cellular response has been briefly discussed from the viewpoint of mathematical model and expected intracellular affecting

At first, factors that can affect the shape of cell survival curve are discussed from the target theory, equations (7) and (11). Clearly, these equations show that the radiation dose *D*, required target number to cell death *k*, and target volume *V* are all the affecting factors; moreover, radiosensitivity is inversely proportional to a required target number *k* and in proportion to the target volume *V* and radiation dose *D*. In order to determine a target volume and number, the information concerning the substance of the target should be needed. What is the actual target of radiation? Generally, it is believed that DNA is the principal target of

At first, factors that can affect the shape of cell survival curve are discussed from the target theory, equations (7) and (11). Clearly, these equations show that the radiation dose *D*, required target number to cell death *k*, and target volume *V* are all the affecting factors; moreover, radiosensitivity is inversely proportional to a required target number *k* and in proportion to the target volume *V* and radiation dose *D*. In order to determine a target volume and number, the information concerning the substance of the target should be needed. What is the actual target of radiation? Generally, it is believed that DNA is the principal target of radiation.

In case of cell-cycle-dependent radiosensitivity, the radiation dose *D* is not the control parameter, and the required target number *k* for cell death (reproductive cell death) seems to

Survival curves of cells obtained under various conditions and mathematical expressions describing the survival curves have been introduced until now. It seems that the radiosensi‐ tivity and most of affecting factors have a simple proportional relationship; however, the interpretation of the cell-cycle-dependent nature of the radiosensitivity seems to be the most difficult. Can such a radiosensitivity really be explained by the mathematical model mentioned above? Here, the cell-cycle-dependent cellular response has been briefly discussed from the

**4. Analyzing the Various Survival Curves Based on the Knowledge of**

**4. Analyzing the various survival curves based on the knowledge of**

viewpoint of mathematical model and expected intracellular affecting factors.

from A to C (top to bottom). Black circles show positions of ionization

Ionization

the radiation effects. Estimation or calculation of LET is one of such studies.

Table 3. Typical values of LET. Values taken from [12]

A

56 Evolution of Ionizing Radiation Research

B

C

factors.

**modern molecular biology**

radiation.

**Modern Molecular Biology**

(top to bottom). Black circles show positions of ionization

As seen in the above discussion, time variation in the target number and volume is not recognized to have a bimodal change in the cell cycle. In other words, it is difficult to discuss a cell-cycle-dependent cyclic change of radiosensitivity by using a target theory. Then, does any kind of intracellular biochemical molecule which is involved in the DNA repair vary periodically during cell cycle?

There are not so many quantitative time-series data about the intracellular density change of proteins involved in DNA repair; however, the experimental data for cyclin B1 [38] shows that the variations in radiosensitivity (several tens fold in range) during the cell cycle cannot be satisfactorily explained by the temporal variation in cellular molecules; the functional form is completely different from a radiosensitivity curve during cell cycle (Fig. 6). Then, what kind of dynamics will be related with this characteristic bimodal periodic variation? One of the possible reasons for the radiosensitivity variation during cell cycle is that some kind of biological changes as a consequence of the structural change of the chromosome may affect the radiosensitivity variation.

Chromosomes are known to change their structure and volume corresponding to cell cycle stages. Generally, higher-order chromosome structure is important for many intracellular dynamics including DNA repair [39, 40]. Regarding spatial structure, a part of the aspects are not yet understood, but it is thought to have roughly four hierarchical levels, i.e., nucleosome, 30 nm chromatin fiber, lampbrush structure, and chromosomal structure. These structures are also known to exhibit cell-cycle-dependent structural changes. The shortest human autosome, chromosome 21, contains 48×10<sup>6</sup> base pairs (bps) of DNA, and its extended length is about 16*μm*; however, its length is about 1*μm* during the metaphase, the most condensed state. Comparing the length of metaphase chromosomes to linear DNA, the packing ratio of DNA in metaphase chromosome is about 10,000.

As the first level of higher order compaction, 147 bps of DNA wraps 1.67 times around the histone octamer and makes unique unit structure called nucleosome core particle, and these units are connected by small amount of DNA, linker DNA. This basic repeating structure consists of nucleosome core particle and linker DNA is called "nucleosome," and it looks like a "beads-on-a-string" structure that corresponds to the state of interphase chromosome and the most basic unit structure of DNA packaging. Moreover, nucleosome is coiled into a 30 nm diameter helical structure known as the 30 nm chromatin fiber; however, their detailed packing manner is still unknown, and this is one of the expected basic structures in interphase. The most condensed metaphase chromosome is expected to have further two-step compaction from this 30 nm chromatin fiber, i.e., 300 nm "loop-like" (lampbrush) structure and 700 nm coiled structure (chromatid). Interphase chromosomes are known to occupy cell nucleus in spatially organized manner, i.e., chromosome territories (CTs), and on the other hand, their internal organized structures are poorly understood [41]. CTs have irregular shapes and occupy discrete compartments with little overlap; so interphase chromosomes are clearly separated inside a cell nucleus, respectively. Through the process of condensation, chromatin fiber increases its number of linkages and makes a condensed state, i.e., M-phase chromosome [42]. This process is very similar to the kind of phase transition from liquid to solid, freezing. Will-be-set-by-IN-TECH 17 The most condensed metaphase chromosome is expected to have further two-step compaction

In addition to its unique structural properties, chromosome is known to have extraordinary physical properties. Observations of stretching in the region of DSB using partially broken Xirradiated chromosomes led to the proposal of a governing equation for chromosome move‐ ment by Nicklas [43] as follows: from this 30 nm chromatin fiber, i.e., 300 nm "loop-like" (lampbrush) structure and 700 nm coiled structure (chromatid). Interphase chromosomes are known to occupy cell nucleus in spatially organized manner, i.e., chromosome territories (CTs), and on the other hand, their internal organized structures are poorly understood [41]. CTs have irregular shapes and occupy discrete compartments with little overlap; so interphase chromosomes are clearly separated inside a cell nucleus, respectively. Through the process of condensation, chromatin

fiber increases its number of linkages and makes a condensed state, i.e., M-phase chromosome

physical properties. Observations of stretching in the region of DSB using partially broken X-irradiated chromosomes led to the proposal of a governing equation for chromosome

$$F = Kl + \Gamma \frac{dl}{dt} \tag{17}$$

where *F* is the force; *K*, the elastic proportionality constant between the force and stretch *l*; and *Γ*, the proportionality constant between force and velocity. Figure 13 is a schematic description of the structural changes of chromosome during cell cycle. From this figure, structural parameters of chromosome, *L* and *R* are found to change periodically. Similarly, elastic coefficient of chromosome *K* in eq. (17) is also expected to change during cell cycle. movement by Nicklas [43] as follows: *<sup>F</sup>* <sup>=</sup> *Kl* <sup>+</sup> <sup>Γ</sup> *dl dt* (17) where *F* is the force; *K*, the elastic proportionality constant between the force and stretch *l*; and Γ, the proportionality constant between force and velocity. Figure 13 is a schematic description of the structural changes of chromosome during cell cycle. From this figure,

> structural parameters of chromosome, *L* and *R* are found to change periodically. Similarly, elastic coefficient of chromosome *K* in eq. (17) is also expected to change during cell cycle.

and NEB show phases of cell cycle. "NEB" is an abbreviation for "nuclear envelope breakdown." L and R are the "length" and "radius (thickness)" of chromosome, respectively. The subscripts M and I for the L and R show length and radius for mitosis and interphase, respectively, and L30nm shows a length of 30 nm chromatin fiber and R15nm shows that of radius In the recent years, many simulations using molecular dynamics (MD) on the DNA damage sites induced by ionizing radiation have been performed to study the detailed mechanics of **Figure 13.** Schematic explanation of the structural change of chromosome during cell cycle. M, G1, S, G2, and NEB show phases of cell cycle. "NEB" is an abbreviation for "nuclear envelope breakdown." L and R are the "length" and "radius (thickness)" of chromosome, respectively. The subscripts M and I for the L and R show length and radius for mitosis and interphase, respectively, and L30nm shows a length of 30 nm chromatin fiber and R15nm shows that of radius

Fig. 13. Schematic explanation of the structural change of chromosome during cell cycle. M, G1, S, G2,

the process of DNA damage and repair. These MD simulations have found some detailed states of the damage sites, though it is a limited condition, and one of the impressive findings is the very fast separating movement of the atoms that constitute damaged sites [44, 45]. Generally, the structure of the DNA damage sites is expected to be important for the repair In the recent years, many simulations using molecular dynamics (MD) on the DNA damage sites induced by ionizing radiation have been performed to study the detailed mechanics of the process of DNA damage and repair. These MD simulations have found some detailed states of the damage sites, though it is a limited condition, and one of the impressive findings is the very fast separating movement of the atoms that constitute damaged sites [44, 45]. Generally, the structure of the DNA damage sites is expected to be important for the repair probability which determines its ability. Thus, the dynamical aspects of the damaged sites accompanied by the structural transition of chromosome during cell cycle may affect the cell survival through a process of repair.

#### **5. Conclusion**

most condensed metaphase chromosome is expected to have further two-step compaction from this 30 nm chromatin fiber, i.e., 300 nm "loop-like" (lampbrush) structure and 700 nm coiled structure (chromatid). Interphase chromosomes are known to occupy cell nucleus in spatially organized manner, i.e., chromosome territories (CTs), and on the other hand, their internal organized structures are poorly understood [41]. CTs have irregular shapes and occupy discrete compartments with little overlap; so interphase chromosomes are clearly separated inside a cell nucleus, respectively. Through the process of condensation, chromatin fiber increases its number of linkages and makes a condensed state, i.e., M-phase chromosome [42]. This process is very similar to the kind of phase transition from liquid to solid, freezing. In addition to its unique structural properties, chromosome is known to have extraordinary physical properties. Observations of stretching in the region of DSB using partially broken Xirradiated chromosomes led to the proposal of a governing equation for chromosome move‐

Will-be-set-by-IN-TECH 17

The most condensed metaphase chromosome is expected to have further two-step compaction from this 30 nm chromatin fiber, i.e., 300 nm "loop-like" (lampbrush) structure and 700 nm coiled structure (chromatid). Interphase chromosomes are known to occupy cell nucleus in spatially organized manner, i.e., chromosome territories (CTs), and on the other hand, their internal organized structures are poorly understood [41]. CTs have irregular shapes and occupy discrete compartments with little overlap; so interphase chromosomes are clearly separated inside a cell nucleus, respectively. Through the process of condensation, chromatin fiber increases its number of linkages and makes a condensed state, i.e., M-phase chromosome [42]. This process is very similar to the kind of phase transition from liquid to solid, freezing. In addition to its unique structural properties, chromosome is known to have extraordinary physical properties. Observations of stretching in the region of DSB using partially broken X-irradiated chromosomes led to the proposal of a governing equation for chromosome

<sup>=</sup> + G *dl F Kl*

coefficient of chromosome *K* in eq. (17) is also expected to change during cell cycle.

where *F* is the force; *K*, the elastic proportionality constant between the force and stretch *l*; and *Γ*, the proportionality constant between force and velocity. Figure 13 is a schematic description of the structural changes of chromosome during cell cycle. From this figure, structural parameters of chromosome, *L* and *R* are found to change periodically. Similarly, elastic

*<sup>F</sup>* <sup>=</sup> *Kl* <sup>+</sup> <sup>Γ</sup> *dl*

where *F* is the force; *K*, the elastic proportionality constant between the force and stretch *l*; and Γ, the proportionality constant between force and velocity. Figure 13 is a schematic description of the structural changes of chromosome during cell cycle. From this figure, structural parameters of chromosome, *L* and *R* are found to change periodically. Similarly, elastic coefficient of chromosome *K* in eq. (17) is also expected to change during cell cycle.

M G1 S G2 NEB

Fig. 13. Schematic explanation of the structural change of chromosome during cell cycle. M, G1, S, G2, and NEB show phases of cell cycle. "NEB" is an abbreviation for "nuclear envelope breakdown." L and R are the "length" and "radius (thickness)" of chromosome, respectively. The subscripts M and I for the L and R show length and radius for mitosis and interphase, respectively, and L30nm shows a length of 30

**Figure 13.** Schematic explanation of the structural change of chromosome during cell cycle. M, G1, S, G2, and NEB show phases of cell cycle. "NEB" is an abbreviation for "nuclear envelope breakdown." L and R are the "length" and "radius (thickness)" of chromosome, respectively. The subscripts M and I for the L and R show length and radius for mitosis and interphase, respectively, and L30nm shows a length of 30 nm chromatin fiber and R15nm shows that of

In the recent years, many simulations using molecular dynamics (MD) on the DNA damage sites induced by ionizing radiation have been performed to study the detailed mechanics of the process of DNA damage and repair. These MD simulations have found some detailed states of the damage sites, though it is a limited condition, and one of the impressive findings is the very fast separating movement of the atoms that constitute damaged sites [44, 45]. Generally, the structure of the DNA damage sites is expected to be important for the repair

In the recent years, many simulations using molecular dynamics (MD) on the DNA damage sites induced by ionizing radiation have been performed to study the detailed mechanics of the process of DNA damage and repair. These MD simulations have found some detailed states

L<sup>I</sup> L30nm 2L<sup>I</sup> LM R<sup>I</sup> R15nm R<sup>I</sup> 2RM

*dt* (17)

*dt* (17)

ment by Nicklas [43] as follows:

58 Evolution of Ionizing Radiation Research

L R

radius

LM RM

nm chromatin fiber and R15nm shows that of radius

movement by Nicklas [43] as follows:

Some quantitative approaches in radiation biology based on the clonogenic method and obtained cell survival curves as a dose-response relationship are introduced. Cell survival curves seem to have a **universality** of its function, i.e., functional form of survival curves seems to be unchanged under various conditions including different species. Generally, the func‐ tional form of cell survival curve, especially by its slope, represents a quantity of radiosensi‐ tivity. Various factors affecting the radiosensitivity have been investigated to find macroscopic nature of living organisms, not to divide the system but to investigate the whole of it. Many mathematical models that describe cell survival curves have been presented; however, functional form of cell survival curves derived from, based on biological mechanism, does not yet exist. Mathematical or theoretical derivation of the functional form of cell survival may lead to understand the general theory of cellular responses to ionizing radiation, especially for the low-dose region.

Finally, the possibility that the structural change of chromosome affects the repair process is discussed. The survival curves that have a universality with respect to the dose-response relationship of cell should be studied further, to derive more analytically with the biological mechanism, also in order to progress the research on the secret of life.

#### **Author details**

Noriyuki B. Ouchi

Address all correspondence to: ouchi.noriyuki@jaea.go.jp

Research Group for Radiation and Biomolecular Science, Radiation Biology Research Divi‐ sion, Quantum Beam Science Center, JAEA, Japan

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