Radiation Protection Dosimetry (2015), Vol. 166, No. 1–4, pp. 142 –147 Advance Access publication 16 April 2015
doi:10.1093/rpd/ncv149
A MATHEMATICAL MODEL OF RADIATION-INDUCED RESPONSES IN A CELLULAR POPULATION INCLUDING CELL-TO-CELL COMMUNICATIONS Yuya Hattori1,*, Michiyo Suzuki2, Tomoo Funayama2, Yasuhiko Kobayashi2, Akinari Yokoya1,3 and Ritsuko Watanabe1 1 Research Group for Radiation Effect Analysis, Japan Atomic Energy Agency, Ibaraki, Japan 2 Microbeam Radiation Biology Group, Japan Atomic Energy Agency, Gunma, Japan 3 Research Group for Radiation and Biomolecular Science, Japan Atomic Energy Agency, Ibaraki, Japan
Cell-to-cell communication is an important factor for understanding the mechanisms of radiation-induced responses such as bystander effects. In this study, a new mathematical model of intercellular signalling between individual cells in a cellular population is proposed. The authors considered two types of transmission of signals: via culture medium and via gap junction. They focus on the effects that radiation and intercellular signalling have on cell-cycle modification. The cell cycle is represented as a virtual clock that includes several checkpoint pathways within a cyclic process. They also develop a grid model and set up diffusion equations to model the propagation of signals to and from spatially located cells. The authors have also considered the role that DNA damage plays in the cycle of cells which can progress through the cell cycle or stop at the G1, S, G2 or M-phase checkpoints. Results of testing show that the proposed model can simulate intercellular signalling and cell-cycle progression in individual cells during and after irradiation.
INTRODUCTION Non-targeted effects of radiation on living cells have been recognised as playing potentially important roles in multiple cellular systems exposed to ionising radiation. Bystander cells surrounding cells targeted for irradiation receive chemical substances from the exposed cells, and these substances can induce biological effects such as chromosome aberrations(1) or micronucleated or apoptotic cell formations(2). Experimental evidence of various kinds of cytokines as a bystander mediator and the role of mediators in the induction of genomic instability has also been recorded(3). From the theoretical side, several studies have tried to model and simulate intercellular signalling of bystander effects on important biological endpoints such as cell death, with taking into account diffusion of bystander mediators in a culture medium(4 – 6). However, another well-recognised pathway of signal transfer, through gap junctions(7), has not been considered in these previous models. To understand and evaluate the effect of radiation on cellular populations, the contributions of both pathways should be clarified. In this article, a prototype mathematical model of two types of radiation-induced intercellular signalling is presented in a cellular population. In this model, the fate of each cell in the population can be followed. Each cell is modelled as it progresses through successive cell-cycle phases. The effect of intercellular signalling on cell-cycle alternation was simulated with this model. The cell-cycle effect is thought to be one of the primary processes leading to biological consequences for the surviving fraction of the cellular
population, as suggested by the effect being greatly involved in genomic instability even after several iterations of mitoses. Modelling the bystander cell-cycle effect is the first step towards elucidating the mechanism of transgenerational effects, which can ultimately trigger radiation-induced carcinogenesis in low-dose regions. MODELING METHOD The cellular population is modelled by grids as shown in Figure 1. Each grid is a square with side length d. The grids can be categorized into three types: cells, medium and walls. Each cell grid consists of one cell and the culture solution. Each medium grid consists entirely of culture solution. The wall grids form a cell culture container. Although Figure 1 shows each grid as a cube, at present calculation is performed in only two dimensions. The simulation algorithm used in the study consists of four steps: (i) irradiation of the cell, (ii) generation and diffusion of radiation-induced intercellular signals, (iii) induction of damage in the cell by irradiation and radiation-induced intercellular signals and (iv) cellular response to the cell cycle after damage. The spatial and temporal evolution of the cellular population can be simulated according to the temporal progress of the steps. Step 1: Irradiation of the cell Assume that the total absorbed dose in each cell is given by accumulation of a single radiation track passing through the cell, which is defined as a radiation
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*Corresponding author: [email protected]
MATHEMATICAL MODEL OF RADIATION-INDUCED RESPONSES
hit. The expected value of the absorbed dose given by one radiation hit D1hit corresponds to an elementary dose(8). The probability of a radiation hit in a given cell over time interval Dt is calculated according to a Poisson distribution. When the number of radiation hits is given by Ki,j (t) as a function of time t, the total absorbed dose Ri,j (t), in cell grid i; j is Ri;j ðtÞ ¼ D1hit Ki;j ðtÞ
ð1Þ
The radiation quality and the dose rate can be considered by the values D1hit and Ki;j ðtÞ, respectively. Step 2: Generation and diffusion of radiation-induced intercellular signals
kinds of radiation-induced signals are considered in this study: one which is transferred through the culture medium and another that is transferred through gap junctions. The concentrations of the signals in grid position i; j at time t is represented by Mi;j ðtÞ and Gi;j ðtÞ for transfer by the culture medium and gap junction, respectively. In the case of medium transfer signals Mi;j ðtÞ, these are assumed to diffuse unconditionally into adjacent grids. In contrast, gap junction transfer signals Gi;j ðtÞ can diffuse into only adjacent cells. The time rates of Mi;j ðtÞ and Gi;j ðtÞ are also represented based on diffusion equations dMi;j ðtÞ wM X ¼ 2 fMk;l ðtÞ Mi;j ðtÞg pM Mi;j ðtÞ dt d k;l
Intercellular signalling is described by the diffusion of a transmitter substance into surrounding grids. Two 143
þ qM Ri;j ðtÞ ð2Þ
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Figure 1. Structure of the present model for calculating intercellular signalling and cell cycles.
Y. HATTORI ET AL.
dGi;j ðtÞ wG X ¼ 2 fGk;l ðtÞ Gi;j ðtÞg dt d k;l
ð3Þ
pG Gi;j ðtÞ þ qG Ri;j ðtÞ
Step 3: Damage to the cell Assume that radiation-induced damage that causes a response in the cell cycle is DNA damage. DNA damage is assumed to be able to be induced by three pathways: a radiation hit and the two kinds of intercellular signals. The initial DNA damage is simply assumed to be repaired over time with a fixed probability r. Namely, the amount of DNA damage Zi;j ðtÞ is defined as dZi;j ðtÞ ¼ aRi;j ðtÞ þ bMi;j ðtÞ þ cGi;j ðtÞ dt rZi;j ðtÞ
ð4Þ
where a, b and c are the damage-induction rates of Ri;j ðtÞ, medium transfer signal, Mi;j ðtÞ and gap-junction transfer, Gi;j ðtÞ, respectively. The values of the damageinduction rates, a, b and c, characterise differences in the effect of DNA damage according to source (radiation and intercellular signaling). The value of the repair rate, r, characterises the DNA repair efficiency. Step 4: The cell-cycle response The phase of the cell cycle for the cell in grid i; j at time t is represented by Si;j ðtÞ. Progress or arrest of the cell cycle is determined by a virtual clock (Figure 2) divided into six phases, ‘G0’, ‘G1’, ‘S’, ‘G2’, ‘M1’ or ‘M2’. The M-phase of the cell cycle is divided into ‘M1’ and ‘M2’ at the checkpoint. Si;j ðtÞ shows these phases, checkpoints (‘G1/S’, ‘S/G2’, ‘G2/M’ and ‘M1/M2’), and cell division (‘Division’). Si;j ðtÞ is defined by the position of the virtual clock hand,
Figure 2. Virtual clock to show cell-cycle phases.
represented by Ci;j ðtÞ. When the clock hand is progressing, dCi;j ðtÞ=dt is set to 1. When the clock hand stops, dCi;j ðtÞ=dt is 0. In this study, the clock hand is controlled by a two-threshold system using the amount of DNA damage Zi;j ðtÞ. The threshold value for cell-cycle arrest and the threshold value of cell death are represented by H Arrest and H Death, respectively. When Zi;j ðtÞ is larger than H Arrest, the clock hand stops at the time of the checkpoint. When Zi;j ðtÞ is larger than H Death, the location becomes a medium grid instei; jad of a cell grid, indicating cell death. Note that H Arrest , H Death. When Si;j ðtÞ coincides with ‘Division’, one medium grid at k; lis selected randomly from among the neighbourhood medium locations and becomes a cell location. If there is no medium location in the neighbourhood, then Si;j ðtÞ shows ‘G0’. After cell division, Ci;j ðtÞ and Ck;l ðtÞ are set to 0. RESULTS AND DISCUSSION A simulation system is prepared to confirm that the model system used in this study can simulate the radiation-induced response in a cellular population spatially and temporally for intercellular signalling and cell-cycle modification. Figure 3 shows an example of the simulation system. The system is composed of 10mm2 grids. Three cells A, B and C in different conditions were placed in the system. Cell B contacts cell A, but cell C is separate from the other cells. Only cell A was assumed to be irradiated. At the start, the cellcycle phases in these cell grids were set to ‘G2’. The values of parameters used in the simulation are given
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where wM and wG are the diffusion coefficients for medium and gap junction transfer, respectively, and d is the width of each grid. These signals are assumed to decay with time at constant rates pM and pG for medium and gap junction transfer, respectively. The amounts of intercellular signals are assumed to be generated in proportion to the absorbed radiation dose. The constants qM and qG are defined as the rate constants used to generate the medium transfer and gapjunction transfer signals, respectively, after irradiation with the absorbed dose Ri;j ðtÞ. The indices k; l show the position in the neighbourhood grid ðk [ fi 1; i; i þ 1g; l [ fj 1; j; j þ 1g; fk; l = i; jgÞ. Note that all grids can be targets for diffusion of medium transfer signals, although the target locations for diffusion of gap-junction transfer are limited to the adjacent cells.
MATHEMATICAL MODEL OF RADIATION-INDUCED RESPONSES
below. The aim of this study at this stage was confirmation of the performance of the model system. Therefore, some parameter values are not based on biophysical data. The irradiation condition was set to describe low linear energy transfer (LET) radiation with a lower dose rate. The absorbed dose given by one radiation hit ðD1hit Þ was set to be 0.001 Gy, which approximately corresponds to the elementary dose estimated for lowLET radiation such as 60Co g-rays(8). The expected value of the radiation hits in a cell location was set to be 100 hits per hour (0.1 Gy h21). Irradiation time was 30 min. The parameters for signal transfer were as follows: the diffusion coefficients of intercellular signalling for the medium transfer were set to wM ¼ 108 nm2 s21 by reference to the level of cytokine diffusion(9). The diffusion coefficients of intercellular signalling for the gap junction transfer were assumed to be much smaller than that for medium transfer. The parameter value was tentatively set to wG ¼ 109 nm2 s21. For other parameters, the following values were used: the rate constants of signal decay for medium transfer and gap junctions transfer were set to pM ¼ 0:019 s21 and s21, respectively. The signal production constants for medium transfer anpG ¼ 0:019d gap junction transfer were set to qM ¼ 1 hit21 and qG ¼ 1 hit21, respectively. The time step for the diffusion calculation was set to Dt ¼ 1 s to obtain sufficient accuracy. Regarding the DNA damage, the damage induction probability was tentatively set to be the same for all three pathways, with a ¼ b ¼ c ¼ 1. The probability for repair kinetics was set as r ¼ 0:00002s21. The cellular condition was modelled as follows: the times of the check points in the virtual clock were set to TG1=S ¼ 11 h, TS=G2 ¼ 19 h, TG2=M1 ¼ 23, TM1=M2 ¼ 23:5 h, and TDivision ¼ 24 h. The thresholds
Figure 4. Concentrations of the intercellular signals in cell grids A, B, and C. (a) Time course of the absorbed irradiation dose Ri;j ðtÞ. (b) Time course of Mi;j ðtÞ. (c) Time course of Gi;j ðtÞ.
for the DNA damage to affect the cell-cycle progression were set to H Arrest ¼ 0:0001 at TG2=M1 and H Death ¼ 100. Using the above model, the time course of the signal concentrations for two kinds of pathways were calculated, with the results shown in Figure 4. The time 0 is the start time of the irradiation. Figure 4a shows the time course of the absorbed dose Ri;j ðtÞ in cell A. Since the dose given by a single radiation hit was set to be constant for simplicity, only the time interval for each radiation hit is distributed; this follows a Poisson distribution. Figure 4b and c is, respectively, the time course of the signal concentrations Mi;j ðtÞ and Gi;j ðtÞ in cells A, B and C. The time course of Mi;j ðtÞ in cell A was revealed to coincide well with Ri;j ðtÞ. The signal concentration Mi;j ðtÞ in the cells B and C were smaller than that in cell A. The signal concentration Mi;j ðtÞ in cell C is the smallest among the cells. Decay of the signal with increasing distance from the exposed cell A is expected by the property of random diffusion of the signal
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Figure 3. Condition of each grid for the irradiation simulation.
Y. HATTORI ET AL.
CONCLUSIONS
Figure 5. Amount of DNA damage and the cell-cycle phase in cell grids A, B, and C. The top panel shows the time course of Zi;j ðtÞ in cell grids A, B, and C over 20 h. The bottom panel shows the time course of Si;j ðtÞ in each cell grid.
substance. The signal concentration Gi;j ðtÞ in cell A showed a similar time course to Mi;j ðtÞ. Gi;j ðtÞ in cell B was smaller than that in cell A, and no signal was observed in cell C. This is expected because there were no gap junctions connecting cells A and C. The time course of the amount of DNA damage and the cell-cycle phase in each cell were also evaluated. The top panel in Figure 5 shows that the amount of DNA damage increases during irradiation, and starts to decay after the irradiation. The DNA damage is induced in proportion to the absorbed dose and the concentration of the signals in each cell (Figure 4). Therefore, Zi;j ðtÞ of cell A is the largest and that in cell C is the smallest. From the bottom panel in Figure 5, cell-cycle arrest showed the longest period for cell A. When Zi;j ðtÞ in each cell grid is smaller than the threshold value for cell-cycle arrest (i.e. less than H Arrest ), Si;j ðtÞ progressed to the ‘M1’ phase. Similarly, other cells also showed cell-cycle arrest depending on Zi;j ðtÞ. As mentioned above, at the present stage the model assumptions are rough and should be improved by comparison with specific experimental data for each simulation step. Also, the parameters used for the presented calculation were chosen somewhat arbitrarily
A prototype of a new mathematical model of radiation-induced cellular response is presented. Test calculations showed that the model works as expected, following the fate of individual cells in a cellular population. However, the validity of the assumptions for each simulation steps should be examined in future research. The assumptions and parameter values should furthermore be examined by comparison with applicable experimental data. One of the features of the present model is that it can follow the fate of individual cells in each cell-cycle phase. Recent progress on techniques for following the time course of DNA damage and the cell cycles of individual cells is expected to provide adequate experimental information in near the future.
ACKNOWLEDGEMENTS The authors thank members of the Research Group for Radiation Effect Analysis, Research Group for Radiation and Biomolecular Science, and the Microbeam Radiation Biology Group at the Japan Atomic Energy Agency (JAEA) for their valuable suggestions and discussion. REFERENCES
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1. Nagasawa, H. and Little, J. B. Induction of sister chromatid exchanges by extremely low doses of alpha particles. Cancer Res. 52, 6394– 6396 (1992). 2. Prise, K. M., Belyakov, O. V., Folkard, M. and Michael, B. D. Studies of bystander effects in human fibroblasts using a charged particle microbeam. Int. J. Radiat. Biol. 74(6), 793–798 (1998). 3. Hei, T. K., Zhou, H., Chai, Y., Ponnaiya, B. and Ivanov, V. N. Radiation induced non-targeted response: mechanism and potential clinical implications. Curr. Mol. Pharmacol. 4(2), 96– 105 (2011). 4. Ballarini, F., Alloni, D., Facoetti, A., Mairani, A., Nano, R. and Ottolenghi, A. Modelling radiation-
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to examine the qualitative performance of the present model system. However, the results showed that the new model can potentially simulate intercellular signalling that causes DNA damage and the kinetics of cell-cycle arrest, as shown for cells B and C, which depend on the position in the cellular population. Although there have been few reports on bystander cell-cycle effects due to the technical difficulty of tracking the cell cycle of individual cells, recently, a live-cell imaging technique using the fluorescent ubiquitination-based cell-cycle indicator technique has helped to visualise the cell cycles of individual cells(10, 11). Such an approach is expected to soon provide applicable data that can be used to refine and improve the present model.
MATHEMATICAL MODEL OF RADIATION-INDUCED RESPONSES induced bystander effect and cellular communication. Radiat. Prot. Dosim. 122(1– 4), 244– 251 (2006). 5. Richard, M., Kirkby, K. J., Webb, R. P. and Kirkby, N. F. Cellular automaton model of cell response to targeted radiation. Appl. Radiat. Isot. 67, 443– 446 (2009). 6. Sasaki, K., Wakui, K., Tsutsumi, K., Itoh, A. and Date, H. A simulation study of the radiation-induced bystander effect: Modeling with stochastically defined signal reemission. Comput. Math. Method Med. 389095 (2012). 7. Azzam, E. I., de Toledo, S. M. and Little, J. B. Direct evidence for the participation of gap junction-mediated intercellular communication in the transmission of damage signals from alpha -particle irradiated to nonirradiated cells. Proc. Natl Acad. Sci. USA. 98(2), 473–478 (2001).
8. Booze, J. and Feinendgen, L. E. A microdosimetric understanding of low-dose radiation effects. Int. J. Radiati. Biol., 53, 13–21 (1988). 9. Jacobson, K. and Wojcieszyn, J. The translational mobility of substances within the cytoplasmic matrix. Proc. Natl Acad. Sci. USA, 81(21), 6747– 6751, (1984). 10. Sakaue-Sawano, A., et al. Visualizing spatiotemporal dynamics of multicellular cell-cycle progression. Cell. 132(3), 487– 498 (2008). 11. Kaminaga, K., Noguchi, M., Narita, A., Sakamoto, Y., Kanari, Y. and Yokoya, A. Visualization of cell cycle modifications by X-irradiation of single HeLa cells using fluorescent ubiquitination-based cell cycle indicators. Radiat. Prot. Dosim., 166, 91–94 (2015).
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doi:10.1093/rpd/ncv149
A MATHEMATICAL MODEL OF RADIATION-INDUCED RESPONSES IN A CELLULAR POPULATION INCLUDING CELL-TO-CELL COMMUNICATIONS Yuya Hattori1,*, Michiyo Suzuki2, Tomoo Funayama2, Yasuhiko Kobayashi2, Akinari Yokoya1,3 and Ritsuko Watanabe1 1 Research Group for Radiation Effect Analysis, Japan Atomic Energy Agency, Ibaraki, Japan 2 Microbeam Radiation Biology Group, Japan Atomic Energy Agency, Gunma, Japan 3 Research Group for Radiation and Biomolecular Science, Japan Atomic Energy Agency, Ibaraki, Japan
Cell-to-cell communication is an important factor for understanding the mechanisms of radiation-induced responses such as bystander effects. In this study, a new mathematical model of intercellular signalling between individual cells in a cellular population is proposed. The authors considered two types of transmission of signals: via culture medium and via gap junction. They focus on the effects that radiation and intercellular signalling have on cell-cycle modification. The cell cycle is represented as a virtual clock that includes several checkpoint pathways within a cyclic process. They also develop a grid model and set up diffusion equations to model the propagation of signals to and from spatially located cells. The authors have also considered the role that DNA damage plays in the cycle of cells which can progress through the cell cycle or stop at the G1, S, G2 or M-phase checkpoints. Results of testing show that the proposed model can simulate intercellular signalling and cell-cycle progression in individual cells during and after irradiation.
INTRODUCTION Non-targeted effects of radiation on living cells have been recognised as playing potentially important roles in multiple cellular systems exposed to ionising radiation. Bystander cells surrounding cells targeted for irradiation receive chemical substances from the exposed cells, and these substances can induce biological effects such as chromosome aberrations(1) or micronucleated or apoptotic cell formations(2). Experimental evidence of various kinds of cytokines as a bystander mediator and the role of mediators in the induction of genomic instability has also been recorded(3). From the theoretical side, several studies have tried to model and simulate intercellular signalling of bystander effects on important biological endpoints such as cell death, with taking into account diffusion of bystander mediators in a culture medium(4 – 6). However, another well-recognised pathway of signal transfer, through gap junctions(7), has not been considered in these previous models. To understand and evaluate the effect of radiation on cellular populations, the contributions of both pathways should be clarified. In this article, a prototype mathematical model of two types of radiation-induced intercellular signalling is presented in a cellular population. In this model, the fate of each cell in the population can be followed. Each cell is modelled as it progresses through successive cell-cycle phases. The effect of intercellular signalling on cell-cycle alternation was simulated with this model. The cell-cycle effect is thought to be one of the primary processes leading to biological consequences for the surviving fraction of the cellular
population, as suggested by the effect being greatly involved in genomic instability even after several iterations of mitoses. Modelling the bystander cell-cycle effect is the first step towards elucidating the mechanism of transgenerational effects, which can ultimately trigger radiation-induced carcinogenesis in low-dose regions. MODELING METHOD The cellular population is modelled by grids as shown in Figure 1. Each grid is a square with side length d. The grids can be categorized into three types: cells, medium and walls. Each cell grid consists of one cell and the culture solution. Each medium grid consists entirely of culture solution. The wall grids form a cell culture container. Although Figure 1 shows each grid as a cube, at present calculation is performed in only two dimensions. The simulation algorithm used in the study consists of four steps: (i) irradiation of the cell, (ii) generation and diffusion of radiation-induced intercellular signals, (iii) induction of damage in the cell by irradiation and radiation-induced intercellular signals and (iv) cellular response to the cell cycle after damage. The spatial and temporal evolution of the cellular population can be simulated according to the temporal progress of the steps. Step 1: Irradiation of the cell Assume that the total absorbed dose in each cell is given by accumulation of a single radiation track passing through the cell, which is defined as a radiation
# The Author 2015. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
Downloaded from http://rpd.oxfordjournals.org/ at University of Toronto on November 17, 2015
*Corresponding author: [email protected]
MATHEMATICAL MODEL OF RADIATION-INDUCED RESPONSES
hit. The expected value of the absorbed dose given by one radiation hit D1hit corresponds to an elementary dose(8). The probability of a radiation hit in a given cell over time interval Dt is calculated according to a Poisson distribution. When the number of radiation hits is given by Ki,j (t) as a function of time t, the total absorbed dose Ri,j (t), in cell grid i; j is Ri;j ðtÞ ¼ D1hit Ki;j ðtÞ
ð1Þ
The radiation quality and the dose rate can be considered by the values D1hit and Ki;j ðtÞ, respectively. Step 2: Generation and diffusion of radiation-induced intercellular signals
kinds of radiation-induced signals are considered in this study: one which is transferred through the culture medium and another that is transferred through gap junctions. The concentrations of the signals in grid position i; j at time t is represented by Mi;j ðtÞ and Gi;j ðtÞ for transfer by the culture medium and gap junction, respectively. In the case of medium transfer signals Mi;j ðtÞ, these are assumed to diffuse unconditionally into adjacent grids. In contrast, gap junction transfer signals Gi;j ðtÞ can diffuse into only adjacent cells. The time rates of Mi;j ðtÞ and Gi;j ðtÞ are also represented based on diffusion equations dMi;j ðtÞ wM X ¼ 2 fMk;l ðtÞ Mi;j ðtÞg pM Mi;j ðtÞ dt d k;l
Intercellular signalling is described by the diffusion of a transmitter substance into surrounding grids. Two 143
þ qM Ri;j ðtÞ ð2Þ
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Figure 1. Structure of the present model for calculating intercellular signalling and cell cycles.
Y. HATTORI ET AL.
dGi;j ðtÞ wG X ¼ 2 fGk;l ðtÞ Gi;j ðtÞg dt d k;l
ð3Þ
pG Gi;j ðtÞ þ qG Ri;j ðtÞ
Step 3: Damage to the cell Assume that radiation-induced damage that causes a response in the cell cycle is DNA damage. DNA damage is assumed to be able to be induced by three pathways: a radiation hit and the two kinds of intercellular signals. The initial DNA damage is simply assumed to be repaired over time with a fixed probability r. Namely, the amount of DNA damage Zi;j ðtÞ is defined as dZi;j ðtÞ ¼ aRi;j ðtÞ þ bMi;j ðtÞ þ cGi;j ðtÞ dt rZi;j ðtÞ
ð4Þ
where a, b and c are the damage-induction rates of Ri;j ðtÞ, medium transfer signal, Mi;j ðtÞ and gap-junction transfer, Gi;j ðtÞ, respectively. The values of the damageinduction rates, a, b and c, characterise differences in the effect of DNA damage according to source (radiation and intercellular signaling). The value of the repair rate, r, characterises the DNA repair efficiency. Step 4: The cell-cycle response The phase of the cell cycle for the cell in grid i; j at time t is represented by Si;j ðtÞ. Progress or arrest of the cell cycle is determined by a virtual clock (Figure 2) divided into six phases, ‘G0’, ‘G1’, ‘S’, ‘G2’, ‘M1’ or ‘M2’. The M-phase of the cell cycle is divided into ‘M1’ and ‘M2’ at the checkpoint. Si;j ðtÞ shows these phases, checkpoints (‘G1/S’, ‘S/G2’, ‘G2/M’ and ‘M1/M2’), and cell division (‘Division’). Si;j ðtÞ is defined by the position of the virtual clock hand,
Figure 2. Virtual clock to show cell-cycle phases.
represented by Ci;j ðtÞ. When the clock hand is progressing, dCi;j ðtÞ=dt is set to 1. When the clock hand stops, dCi;j ðtÞ=dt is 0. In this study, the clock hand is controlled by a two-threshold system using the amount of DNA damage Zi;j ðtÞ. The threshold value for cell-cycle arrest and the threshold value of cell death are represented by H Arrest and H Death, respectively. When Zi;j ðtÞ is larger than H Arrest, the clock hand stops at the time of the checkpoint. When Zi;j ðtÞ is larger than H Death, the location becomes a medium grid instei; jad of a cell grid, indicating cell death. Note that H Arrest , H Death. When Si;j ðtÞ coincides with ‘Division’, one medium grid at k; lis selected randomly from among the neighbourhood medium locations and becomes a cell location. If there is no medium location in the neighbourhood, then Si;j ðtÞ shows ‘G0’. After cell division, Ci;j ðtÞ and Ck;l ðtÞ are set to 0. RESULTS AND DISCUSSION A simulation system is prepared to confirm that the model system used in this study can simulate the radiation-induced response in a cellular population spatially and temporally for intercellular signalling and cell-cycle modification. Figure 3 shows an example of the simulation system. The system is composed of 10mm2 grids. Three cells A, B and C in different conditions were placed in the system. Cell B contacts cell A, but cell C is separate from the other cells. Only cell A was assumed to be irradiated. At the start, the cellcycle phases in these cell grids were set to ‘G2’. The values of parameters used in the simulation are given
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where wM and wG are the diffusion coefficients for medium and gap junction transfer, respectively, and d is the width of each grid. These signals are assumed to decay with time at constant rates pM and pG for medium and gap junction transfer, respectively. The amounts of intercellular signals are assumed to be generated in proportion to the absorbed radiation dose. The constants qM and qG are defined as the rate constants used to generate the medium transfer and gapjunction transfer signals, respectively, after irradiation with the absorbed dose Ri;j ðtÞ. The indices k; l show the position in the neighbourhood grid ðk [ fi 1; i; i þ 1g; l [ fj 1; j; j þ 1g; fk; l = i; jgÞ. Note that all grids can be targets for diffusion of medium transfer signals, although the target locations for diffusion of gap-junction transfer are limited to the adjacent cells.
MATHEMATICAL MODEL OF RADIATION-INDUCED RESPONSES
below. The aim of this study at this stage was confirmation of the performance of the model system. Therefore, some parameter values are not based on biophysical data. The irradiation condition was set to describe low linear energy transfer (LET) radiation with a lower dose rate. The absorbed dose given by one radiation hit ðD1hit Þ was set to be 0.001 Gy, which approximately corresponds to the elementary dose estimated for lowLET radiation such as 60Co g-rays(8). The expected value of the radiation hits in a cell location was set to be 100 hits per hour (0.1 Gy h21). Irradiation time was 30 min. The parameters for signal transfer were as follows: the diffusion coefficients of intercellular signalling for the medium transfer were set to wM ¼ 108 nm2 s21 by reference to the level of cytokine diffusion(9). The diffusion coefficients of intercellular signalling for the gap junction transfer were assumed to be much smaller than that for medium transfer. The parameter value was tentatively set to wG ¼ 109 nm2 s21. For other parameters, the following values were used: the rate constants of signal decay for medium transfer and gap junctions transfer were set to pM ¼ 0:019 s21 and s21, respectively. The signal production constants for medium transfer anpG ¼ 0:019d gap junction transfer were set to qM ¼ 1 hit21 and qG ¼ 1 hit21, respectively. The time step for the diffusion calculation was set to Dt ¼ 1 s to obtain sufficient accuracy. Regarding the DNA damage, the damage induction probability was tentatively set to be the same for all three pathways, with a ¼ b ¼ c ¼ 1. The probability for repair kinetics was set as r ¼ 0:00002s21. The cellular condition was modelled as follows: the times of the check points in the virtual clock were set to TG1=S ¼ 11 h, TS=G2 ¼ 19 h, TG2=M1 ¼ 23, TM1=M2 ¼ 23:5 h, and TDivision ¼ 24 h. The thresholds
Figure 4. Concentrations of the intercellular signals in cell grids A, B, and C. (a) Time course of the absorbed irradiation dose Ri;j ðtÞ. (b) Time course of Mi;j ðtÞ. (c) Time course of Gi;j ðtÞ.
for the DNA damage to affect the cell-cycle progression were set to H Arrest ¼ 0:0001 at TG2=M1 and H Death ¼ 100. Using the above model, the time course of the signal concentrations for two kinds of pathways were calculated, with the results shown in Figure 4. The time 0 is the start time of the irradiation. Figure 4a shows the time course of the absorbed dose Ri;j ðtÞ in cell A. Since the dose given by a single radiation hit was set to be constant for simplicity, only the time interval for each radiation hit is distributed; this follows a Poisson distribution. Figure 4b and c is, respectively, the time course of the signal concentrations Mi;j ðtÞ and Gi;j ðtÞ in cells A, B and C. The time course of Mi;j ðtÞ in cell A was revealed to coincide well with Ri;j ðtÞ. The signal concentration Mi;j ðtÞ in the cells B and C were smaller than that in cell A. The signal concentration Mi;j ðtÞ in cell C is the smallest among the cells. Decay of the signal with increasing distance from the exposed cell A is expected by the property of random diffusion of the signal
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Figure 3. Condition of each grid for the irradiation simulation.
Y. HATTORI ET AL.
CONCLUSIONS
Figure 5. Amount of DNA damage and the cell-cycle phase in cell grids A, B, and C. The top panel shows the time course of Zi;j ðtÞ in cell grids A, B, and C over 20 h. The bottom panel shows the time course of Si;j ðtÞ in each cell grid.
substance. The signal concentration Gi;j ðtÞ in cell A showed a similar time course to Mi;j ðtÞ. Gi;j ðtÞ in cell B was smaller than that in cell A, and no signal was observed in cell C. This is expected because there were no gap junctions connecting cells A and C. The time course of the amount of DNA damage and the cell-cycle phase in each cell were also evaluated. The top panel in Figure 5 shows that the amount of DNA damage increases during irradiation, and starts to decay after the irradiation. The DNA damage is induced in proportion to the absorbed dose and the concentration of the signals in each cell (Figure 4). Therefore, Zi;j ðtÞ of cell A is the largest and that in cell C is the smallest. From the bottom panel in Figure 5, cell-cycle arrest showed the longest period for cell A. When Zi;j ðtÞ in each cell grid is smaller than the threshold value for cell-cycle arrest (i.e. less than H Arrest ), Si;j ðtÞ progressed to the ‘M1’ phase. Similarly, other cells also showed cell-cycle arrest depending on Zi;j ðtÞ. As mentioned above, at the present stage the model assumptions are rough and should be improved by comparison with specific experimental data for each simulation step. Also, the parameters used for the presented calculation were chosen somewhat arbitrarily
A prototype of a new mathematical model of radiation-induced cellular response is presented. Test calculations showed that the model works as expected, following the fate of individual cells in a cellular population. However, the validity of the assumptions for each simulation steps should be examined in future research. The assumptions and parameter values should furthermore be examined by comparison with applicable experimental data. One of the features of the present model is that it can follow the fate of individual cells in each cell-cycle phase. Recent progress on techniques for following the time course of DNA damage and the cell cycles of individual cells is expected to provide adequate experimental information in near the future.
ACKNOWLEDGEMENTS The authors thank members of the Research Group for Radiation Effect Analysis, Research Group for Radiation and Biomolecular Science, and the Microbeam Radiation Biology Group at the Japan Atomic Energy Agency (JAEA) for their valuable suggestions and discussion. REFERENCES
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