2.2. Evaluating the percent contribution of HR to DSB repair
In order to evaluate the percent contribution of HR to the repair process, the quantitative model of radiation-induced DSB repair was used [
22]. The model consists of five main parts. The first one evaluates the initial yield of radiation-induced DSBs. The other parts are the quantitative models of NHEJ, HR, SSA, and Alt-EJ repair systems. To simulate the processing of DNA lesions by repair enzymes the mass-action chemical kinetics approach is used.
The kinetics of DSB induction and repair is simulates as
where
N0 =
NncDSB +
NcDSB;
VNHEJ,
VHR,
VSSA,
Vmicro-SSA, and
VAlt-EJ are the terms characterizing elimination of DSBs by the NHEJ, HR, SSA, micro-SSA, and Alt-EJ repair pathways respectively. Detalization of these terms is given in Eqs. A1 of
Appendix A. In Eq. (1),
Nir is the fraction of irreparable DSBs which corresponds to the level of γ-H2AX foci remained in the cell 24 h postirradiation. The rate of initial DSB induction is calculated as
, similarly as it is done in [23-25]. Here,
D is the dose of ionizing radiation (Gy) and
α(
L) is the slop coefficient of linear dose dependence which describes DSB induction per unit of dose (Gy
–1 per cell).
Enzymatic interactions occurred in the course of repair are simulated as follows. Within the NHEJ pathway, the stage of Ku binding to a DSB is represented by the following kinetic equation
where quantities in brackets denote time-dependent intracellular concentrations of repair complexes; K values with an appropriate subscript used to represent the dimensional reaction rate-constants. Here [DSB] is the number of DSBs that undergo binding by Ku,
is the level of resulting intermediate complex.
The stage of recruitment of the DNA-dependent protein kinase catalytic subunit (DNA-PKcs) and Artemis to a DSB site is described as
In this kinetic equation [DNA-PK] denotes a complex of Ku and DNA-PKcs. Art indicates that the mentioned intermediate complex is formed in the presence of Artemis.
The autophosphorylation of DNA-PKcs is represented by
where the superscript P defines the phosphorylated DNA-PK product.
The subsequent end-bridging process is described as a junction of two
constructs formed at the previous repair stage:
Here the [Bridge] intermediate complex characterizes the final product of the reaction.
The further assembling of the NHEJ repair complex promotes the recruitment of LigIV with its associated factors XRCC4 and XLF, and subsequent involvement of the polynucleotide kinase phosphatase (PNKP) with a break site:
The final step of the NHEJ pathway implying recruitment of a polymerase can be denoted as
In this consideration, after gap filling and ligation are finalized, it is accepted that the repair complexes dissociate, leaving the recovered double-strand DNA (dsDNA).
The first stages of HR associated with action of MRN, its co-factors (CtlP, ExoI, Dna2) and ATM are described as follows in the model:
where MRN (Mre11-Rad50-Nbs1) complex interacting with other repair factors is considered as a single complex and superscript P defines the phosphorylated species.
An involvement of the replication protein A (RPA) eliminating secondary structure of DNA and protecting single-stranded regions from other enzymatic activities is denoted by
The next HR step associated with formation of Rad51 filament is described with the following kinetic equation:
where Rad51par abbreviation denotes five biologically important Rad51 paralogs (Rad51B, Rad51C, Rad51D, XRCC2, and XRCC3), and [Rad51 filament] defines the
complex.
Formation of a displacement loop (D-loop) and two ways of its subsequent resolution are represented as follows:
where [DNA
inc] defines the incoming DNA duplex,
complex is assumed to contain Rad54 protein, and dHJ denotes the double Holiday junction.
The SSA pathway is given as the following set of kinetics equations. The first step assuming interaction with Rad52 is denoted by
where the
complex is the same as in Eq. (12).
The junction between Rad52 heptamer rings and each ssDNA termini allowing formation of a flapped structure is represented as
The cutting of the flapped ends by the ERCC1/XPF endonuclease and final ligation of a damaged site with LigIII complex are simulated by
Here, Rad52 and ERCC1/XPF assumed to dissociate from the processing site.
The simulation of the alternative end-joining pathways is based on the hypothesis suggesting two different Ku-independent repair mechanisms [
26]. First of them is represented by MMEJ which is sometimes considered as an independent end-joining mechanism distinct from the other pathways. Meanwhile, experimental evidence suggests that this type of repair represents a specific subclass of the SSA pathway called as micro-SSA [
27,
28]. On the basis of these considerations, in our model, rejoining via MMEJ is simulated as the additional part of DSBs which follow the SSA pathway described above.
To simulate the Alt-EJ pathway, an additional mechanistic explanation is proposed. According to the recent hypothesis, Alt-EJ requires activity of MRX complex and possibly exhibits the same initiation steps as HR [
26]. Therefore, the initial stages of Alt-EJ can be described by Eqs. (9)-(11). After the production of ssDNA, activity of PARP1 recruited to the DSB site is characterized by Eq. (21).
The kinetics of microhomology production via a polymerase activity is simulated with
In Eq. (23), [MicroHomol] denotes the yield of microhomology produced.
The final step of Alt-EJ is believed to rely on LigI activity [
26]. In the model, this stage is represented as follows
Kinetics of induction of γ-H2AX foci is simulated by summing up all active forms of DNA-PKcs and ATM, which are considered in the model, similarly as it was done in [
23]
where [H2AX] is the level of histone variant H2AX and
The mechanism of γ-H2AX foci dephosphorylation is assumed to be proportional to the amount of repaired DNA and its spontaneous decay with the corresponding rate constants
K11 and
K12
To describe the interactions of repair enzymes with their substrates, the mass-action chemical kinetics approach was used. A dynamic change of intracellular concentrations of main repair complexes was expressed as the following differential equations
where
is the intracellular level of the i-th repair complex,
is time, the functions
and
describe the complex accumulation and degradation, respectively. The dimensionless form of the system of ordinary differential equations (ODE) referred to simulation of NHEJ, HR, SSA and two alternative repair pathways as well as its parameters and initial conditions are presented in
Appendix A. In the present study, the solution of the ODE system is performed using the fourth-order Runge–Kutta method. The integration time-step is chosen as 10
–10 s in order to satisfy the fastest reactions being simulated in the model.
The initial yield (
N0) of DSBs was calculated as
where
D is the radiation dose (Gy) and
is the slop coefficient of linear dose dependence which describes DSB induction per unit of dose (Gy
–1 per cell). Parameters
a and
b used in this equation are also presented in
Appendix A. The particular details of parameter evaluation as well as simulation of each repair pathway can be found in [
22].
In order to simulate DSB rejoining in the asynchronous cell culture, we set a model cell cycle distribution similar to the distribution of HF19 cells observed in the work by [
29] where 40% of cells were in G
0/G
1, <10% in S phase and 50% in G
2/M. Assuming that S-phase cells include two equal sub-fractions of early-S and late-S cells, the final distribution was set as following: 45% of cells is in G
0/G
1 and early S phases, and 55% is in late S and G
2/M. Based on the data discussed in [
25,
27,
30,
31], the model provides the following pathways for the repair of non-clustered and clustered DSBs. In G
0/G
1 and early S phases non-clustered DSBs are likely to follow rejoining via NHEJ, while the limited number of complex DSBs may undergo PARP1-dependent Alt-EJ. The other pathways are assumed to be unavailable or masked by NHEJ in these phases. In the late S and G
2/M, non-clustered DSBs may follow NHEJ and may be a substrate for SSA and micro-SSA pathways. The clustered DSBs are suggested to proceed through HR, SSA, and Alt-EJ. Finally, in our calculations the models of corresponding repair mechanisms are processed according the following scheme
where numbers 0.45 and 0.55 represent the fractions of cells in G
0/G
1 and early S, and in late S and G
2/M phases, respectively,
NncDSB and
NcDSB denote amounts of non-clustered and clustered DSBs, and
T is the total size of the exponentially growing cell culture set to be 1×10
5 cells, as mentioned in [
29]. The model accounts for the very cell cycle-dependence of the repair pathways through the initiating the computation procedure with the levels of
NncDSB and
NcDSB parameters according to the cell cycle distribution of the cell culture used in the experiment.
The evaluation of the percent contribution of HR to DSB repair was done through the obtaining the calculated time-courses of γ-H2AX and Rad51 foci for the particular dose of X-rays. The HR contribution to PHR was evaluated as the following ratio
where
and
are the mean numbers of Rad51 and γ-H2AX foci respectively, counted per 24-hour simulation period.
To obtain a dependence of PHR on X-ray dose, the ODE system was run with a sufficiently small dose step equalling to 0.1 mGy within the range of 0–1000 mGy. This simulation procedure yielded a curve of PHR (
D) dependence that is expressed by the formula