Development and Implementation of a Metaphase DNA Model for Ionizing Radiation Induced DNA Damage Calculation

Abstract

Objective:

To develop a metaphase chromosome model representing the complete genome of a human lymphocyte cell to support microscopic Monte Carlo (MMC) simulation-based radiation-induced DNA damage studies.

Approach:

We first employed coarse-grained polymer physics simulation to obtain a rod-shaped chromatid segment of 730 nm in diameter and 460 nm in height to match Hi-C data. We then voxelized the segment with a voxel size of 11 nm per side and connected the chromatid with 30 types of pre-constructed nucleosomes and 6 types of linker DNAs in base-pair (bp) resolutions. Afterward, we piled different numbers of voxelized chromatid segments to create 23 pairs of chromosomes of 1~5 μm long. Finally, we arranged the chromosomes at the cell metaphase plate of 5.5 μm in radius to create the complete set of metaphase chromosomes. We implemented the model in gMicroMC simulation by denoting the DNA structure in a four-level hierarchical tree: nucleotide pairs, nucleosomes and linker DNAs, chromatid segments, and chromosomes. We applied the model to compute DNA damage under different radiation conditions and compared the results to those obtained with G0/G1 model and experimental measurements. We also performed uncertainty analysis for relevant simulation parameters.

Main Results:

The chromatid segment was successfully voxelized and connected in base-pair (bps) resolution, containing 26.8 mega bps (Mbps) of DNA. With 466 segments, we obtained the metaphase chromosome containing 12.5 Gbps of DNA. Applying it to compute the radiation-induced DNA damage, the obtained results were self-consistent and agreed with experimental measurements. Through the parameter uncertainty study, we found that the DNA damage ratio between metaphase and G0/G1 phase models was not sensitive to the chemical simulation time. The damage was also not sensitive to the specific parameter settings in the polymer physics simulation, as long as the produced metaphase model followed a similar contact map distribution.

Significance:

Experimental data reveal that ionizing radiation induced DNA damage is cell-cycle dependent. Yet, DNA chromosome models, except for the G0/G1 phase, are not available in the state-of-the-art MMC simulation. For the first time, we successfully built a metaphase chromosome model and implemented it into MMC simulation for radiation-induced DNA damage computation.

1. Introduction

Microscopic Monte-Carlo (MMC) simulation serves as a critical tool in quantifying ionizing radiation induced deoxyribonucleic acid (DNA) damage (Friedland et al., 2011; Incerti et al., 2016; Nikjoo et al., 2016; Schuemann et al., 2019; Tsai et al., 2020). DNA carries the basic genetic information that settles cell functions, and radiation-induced DNA damage plays a leading role in determining the final radiobiological effects. In MMC simulation, radiation-induced DNA damage is generally modeled through four steps: 1) the physical stage, which describes the step-by-step transport of the ionizing particles and their electromagnetic interactions with target molecules (mainly water molecules) at the femtosecond level; 2) the physicochemical stage that models the spontaneous association and dissociation of excited or ionized water molecules, from which free chemical radicals are formed; 3) the chemical stage, which governs the diffusion and chemical reactions of the induced radicals inside the water solvent. Depending on the lifetime of the chemicals of interest, this process could take place over nanoseconds or microseconds; and 4) the DNA damage quantification stage, which computes the physical and chemical DNA damages in the form of strand breaks and base damages upon a pre-constructed DNA structure.

Over the past few decades, with an increasing interest in MMC-based complex radiobiological phenomena study, there have been significant efforts to strengthen the reliability and versatility of the simulation frame (Nikjoo et al., 2016; Kyriakou et al., 2021; Incerti et al., 2010; Lai et al., 2021a; Boscolo et al., 2020; Sakata et al., 2019; Lai et al., 2021b; Abolfath et al., 2022). On the other hand, the geometry modeling of the biological target, the nucleus DNA, have also been found to impact the direct and indirect DNA damage computation (Lavelle and Foray, 2014). To make the simulation realistically applicable, it is essential to model the DNA geometry accurately.

Yet, performing detailed DNA modeling in MMC simulation has been challenging. One challenge comes from the DNA complexity itself. It is known that the human nucleus contains more than 6 Giga nucleotide pairs of DNA, with each nucleotide about 0.34 nanometers long, making the entire DNA about 2 meters long. The complete DNA set is folded into the cell nucleus at the micrometer level, thus making its topological structure extremely complex. Meanwhile, in the MMC simulation, we needed to overlap positions of physical and chemical events, which can be millions in one execution, with the DNA structure containing billions of nucleotides to obtain the damage sites. Unless organizing the DNA structure properly, it can be computationally expensive to perform DNA damage computation induced by ionizing radiation.

The modeling and implementation of DNA structures representing a complete set of human nucleus genomes in MMC simulation were not available until recently (Friedland et al., 2003; Nikjoo and Girard, 2012; Bernal et al., 2013; Friedland et al., 2011; Schuemann et al., 2019; Tsai et al., 2020). Based on the finest structure it contained, these models were built either at the base-pair (Friedland et al., 2003; Nikjoo and Girard, 2012; Tsai et al., 2020) or at the atomic (Bernal et al., 2013; Friedland et al., 2011) resolutions. Both kinds of models used a hierarchical approach to ease the computational challenge mentioned above. In the former, a human nucleus in G0/G1 phase was represented in a six-level hierarchical structure (Tsai et al., 2020). In the latter, except for the structures used in the base-pair-resolution model, atoms forming one base pair were also included. The hierarchical approach was inherited from previous development efforts for simple DNA structures (Charlton et al., 1989; Langowski and Heermann, 2007; Cremer et al., 2000; Kreth et al., 2004),which was also supported by the chromosome conformation capture (3C) data (Dekker et al., 2002), where the interphase chromosome was found to be multiscale and have topologically associated domains (TADs) (Nora et al., 2012; Sexton et al., 2012; Dixon et al., 2012) and compartments (Lieberman-Aiden et al., 2009).

However, as far as the authors know, there are no DNA models covering phases other than G0/G1 in the state-of-the-art MMC simulation, although multiple studies pointed out that radiation-induced DNA damage can be cell-cycle dependent (Dewey et al., 1970; Olive and Banath, 1993; Iliakis et al., 1991; Okayasu et al., 1988; Oleinick et al., 1984). For instance, in the x-ray irradiation experiment on synchronized Chinese Hamster cells (Dewey et al., 1970), the authors found that to kill a certain fraction of cells or to induce a certain number of aberrations, cells in the G1 phase required doses of 1.5-1.8 times greater than cells irradiated in metaphase, revealing the importance of considering cell cycle in the study of ionizing radiation induced DNA damage.

Compared to the G0/G1 phase, the DNA structures in other phases can be more complex as they associate with the disappearance of TADs and compartments, and the much denser compactions (Naumova et al., 2013). These geometrical changes made the DNA modeling for MMC simulation further challenging. In this work, we overcame this challenge by proposing a novel method to construct and implement a metaphase DNA model for the human lymphocyte cell nucleus in MMC simulation. The central idea is as follows. First, to make the model match the chromosome internal structure revealed by the Hi-C measurement (Lieberman-Aiden et al., 2009; Naumova et al., 2013), we applied the coarse-grained polymer physics simulation (Gibcus et al., 2018) via a proper geometry and interaction configuration and obtained a cylindrical chromatid segment as the building block. Second, to simultaneously reduce the computational burden and retain the geometry accuracy in MMC simulation, we constructed the segment by limiting its height to be the shortest length that can reproduce the Hi-C data, rebuilt it in a voxelized geometry and connected the polymer chain with pre-constructed base-pair-resolution structures of known orientations. After that, we piled up the building block along the axial direction to form the sister chromatids for each chromosome and placed the constructed chromosomes at the cell central plate. In this way, we obtained a complete DNA structure in metaphase. We implemented the developed chromosome model in MMC simulation and tested its performance in DNA damage computations. The model development, implementation, and test details are listed below.

2. Material and Methods

The internal configuration of the metaphase chromosome has not been significantly understood until recent technical development of chromosome conformation capture carbon copy (5C) (Dostie et al., 2006) and Hi-C (Lieberman-Aiden et al., 2009). Both 5C and Hi-C data revealed that the metaphase chromosome had no compartmentalization or TADs (Naumova et al., 2013). Deriving contact probability (P(s)) distribution from Hi-C data, it showed that pairs of genomic loci at genomic distances between 100 kilo-base (kb) and 10 mega-base (Mb) had a relatively high and stable contact frequency distribution while it dropped quickly at ~ 10 Mb (Figure 1(b)). Based on these findings, Naumova et al. (2013) proposed a metaphase chromosome model featuring consecutive loops for a chromatid containing 77 Mb. The model achieved a good agreement with Hi-C and microscopy measurement data. Recently, Gibcus et al. (2018) integrated genetic, genomic, and computational approaches to characterize the key steps in mitotic chromosome formation from the G2 nucleus to metaphase. Their experimental and modeling findings are consistent with the proposal of ‘loop extrusion’ by condensin protein complexes (Alipour and Marko, 2012; Goloborodko et al., 2016b), which has also been observed in vitro (Ganji et al., 2018; Banigan and Mirny, 2020). As a result of the loop extrusion, consecutive nested loops, including long outer and short inner loops, were formed as critical structures of the metaphase DNA (Figure 1(a)).

Figure 1.

(a) A polymer model of metaphase chromatid (Gibcus et al., 2018). The chromosomes were modeled as arrays of consecutive nested loops, including long outer loops formed by pairs of loop anchors in red and short inner loops created by pairs of anchors in blue. The red anchors were distributed along a central spiral scaffold and the blue ones were distributed radially around them. The diagram at the bottom right also illustrated the distribution of the outer and inner loops. (b) The contact distribution for K562 cells in metaphase, which was derived from (Naumova et al., 2013).

In this work, we started our modeling process by inheriting the key geometry features from the above findings. After that, we voxelized the geometry. The purpose of performing such a two-step construction was to ensure the finally obtained model was both profound in structure and efficient in damage computation. The overall modeling process is illustrated in Figure 2. We first performed a coarse-grained polymer physics simulation to obtain a short rod-shaped chromatid segment that matched Hi-C data for a human lymphocyte cell nucleus in metaphase. Based on the DNA density, we had each monomer represent one nucleosome. After that, we voxelized the segment by allowing at most one monomer in one voxel. If more than two monomers were found occupying the same voxel, the extra ones would be pushed to nearby empty voxels. These two sub-step simulations were performed iteratively until the voxelization could satisfy the predefined criteria. Third, we constructed 30 types of nucleosomes and 6 types of linker DNAs in base-pair resolution and known orientations. We then filled the monomer-occupied voxels with the nucleosomes and connected them through linker DNAs crossing voxel faces. In this way, we generated a voxelized chromatid segment. Finally, we axially piled several voxelized chromatid segments to form a sister chromatid by determining the specific number based on the chromosome length. We repeated the process to form chromatids for all 23 pairs of chromosomes and arranged them on the metaphase plate to create the complete genome model. The specific modeling process is then given in the following sub-sections.

Figure 2.

The flowchart of the metaphase DNA modeling process.

2.1. Metaphase DNA modeling

2.1.1. Coarse-Grained Polymer Physics Simulation

Coarse-grained polymer physics simulation is a powerful tool in simulating the behavior of complex systems with a simplified representation. In this work, we first employed this simulation method to build a metaphase chromatid segment in a low resolution but able to reflect the key chromosome geometry features. With the chromosome length measured in μm from literature (Claussen et al., 2002) and the size in bps, we estimated the density of lymphocyte cells in metaphase as ~ 50 Mbps/μm. It was comparable to the density of the chicken DT40 cells at late prometaphase (Gibcus et al., 2018). Meanwhile, noticing different human (human Hela S3, K562, and HFF1) cell lines shared similar contact distributions (that measured the physical contact probabilities of pairs of genomic loci at different genomic distances) in metaphase (Naumova et al., 2013), we reasonably assumed that the human lymphocyte cells also shared the similar contact map distribution. Furthermore, we found these contact maps were similar to that for the chicken DT40 cell lines at late prometaphase (Gibcus et al., 2018) between 100 kb and 10 Mb. We then reasonably started the construction of the metaphase chromatid segment for the lymphocyte cells upon the chromatid structure of DT40 cell lines at later prometaphase in (Gibcus et al., 2018). Specifically, noticing there is no existence of 30 nm chromatin fiber in the human metaphase DNA supported by accumulated experiments (Maeshima et al., 2010; Nishino et al., 2012; Eltsov et al., 2008), we represented the chromosome as a chain of 11 nm beads-on-string, with each bead representing one nucleosome of 200 base pairs (bps) in DNA. We initialized the chromosome distribution as shown in Figure 1(a), which included a spiral scaffold and consecutive nested outer and inner loops inside a cylinder, by setting parameters like radii R and r of the cylinder and the spiral scaffold, pitch (height per turn) size of the scaffold l_z, total turns N_l and average loop sizes l_out and l_in of outer and inner loops following that for late prometaphase in (Gibcus et al., 2018). Afterwards, we randomly selected a subset of monomers to be bases of the outer and inner loops to make them follow an exponential distribution with an average of l_out and l_in respectively, tethered the loop bases along the central spiral scaffold, attached the first and last monomers of the chromosome to the end centers of the cylinder and sampled the monomer positions of each loop radially via a 3D random walk simulation. Lastly, we applied the Langevin molecular dynamics simulation with OpenMM, a GPU-based high-performance molecular dynamics simulation platform (Eastman et al., 2017; Goloborodko et al., 2016a; Fudenberg et al., 2016), to equilibrate the chromosome scaffold.

We considered multiple interactions in the OpenMM-based simulation. First, a harmonic bond U_adjac = k_ad(r_ad − r)² was applied to adjacent two monomers with distance r along the polymer chain, where k_ad and r_ad are the stiffness constant and balanced distance respectively. Second, all adjacent three monomers are subject to a harmonic angular interaction in the form of U_angle = k_an(1 − cos(θ − π)) to make the monomers in the same loop connect smoothly without forming sharp angles. Here, θ is the angle formed by the two polymer strings connecting the three adjacent monomers. Third, a repulsive potential was applied to all monomers to avoid spatial overlapping. We modeled this potential with a three-stage function. When the distance r between any two monomers is greater than 11.55 nm, there is no repulsive force. If it is greater or equal to r₁ and smaller than 11.55 nm, $U_{\text{repulse},1}=5(1+(\frac{\sqrt{6∕7}\ast r}{11.55}{)}^{12}\ast \left((\frac{\sqrt{6∕7}\ast r}{11.55}{)}^2-1\right)\ast \frac{823543}{46656})k_{B}T$ with k_B the Boltzmann constant and T the temperature, which forms a soft and square-like potential. Otherwise U_repulse = (k_repulser)⁻¹ + U_repulse,1, which increases dramatically when two monomers are too close to each other. Fourth, we used a cylindrical boundary constraint of a harmonically-increasing potential U_boundary with a stiffness constant k_bound to keep all monomers confined in the chromatid rod. Fifth, we enforced a harmonic bond U_base with a stiffness constant k_ba between monomers forming a loop base. Sixth, we employed a tethering potential U_scaffold with a stiffness constant k_sca to the loop bases to keep their positions close to their initialized positions along the spiral scaffold. Finally, we imposed an additional harmonic potential U_ends with a stiffness constant k_end to the first and last monomers to the ends of the cylinder. In summary, we applied a total potential U(r) = U_adjac + U_angle + U_repulse + U_boundary + U_base + U_scaffold + U_ends in the simulation. The Langevin motion has the form of

$$m\ddot{\mathit{r}}=-\nabla \mathrm{U}(\mathbf{r})-\gamma \mathrm{m}\stackrel{.}{\mathit{r}}+\sqrt{2m\gamma k_{B}T}R(t),$$

where m is the particle’s mass, r is the position vector, γ is the damping coefficient, and R(t) is a delta-correlated stationary gaussian process.

After the simulation, we computed the contact probability distribution and compared it with that derived from Hi-C data for metaphase DNA geometry of K562 cell lines (Naumova et al., 2013). We repeated the simulation by adjusting the input parameters until obtaining a contact map closely following that from the Hi-C measurement. To make the obtained segment as short as possible to save memories in MMC simulation, we have constrained the total turns N_l to be as small as possible.

2.1.2. Voxelization of the Chromatid Segment

Post the polymer physics simulation, we voxelized the chromatid segment with a voxel size of 11 nm per side. The goal of the voxelization is to obtain a chromatid structure with at most one monomer in each voxel, considering that each monomer represents a nucleosome of about 11 nm in diameter and nucleosomes cannot overlap with each other. In the polymer physics simulation, due to the complex interactions among the monomers in a dense distribution, a portion of them would stay close to each other at the equilibrium state. Consequently, they could be grouped into the same voxel. We developed a strategy to push the redundant monomers to nearby empty voxels. However, sometimes it can be challenging to find enough nearby empty voxels if too many monomers occupied the same voxel. To solve the problem, we revisited the polymer physics simulation with a slight tuning of the interaction parameters until we could push all redundant monomers to a nearby empty voxel. We validated the chromatid segments pre- and post-voxelization via computing the angular distributions and the contact maps of the monomers and comparing with that derived from Hi-C data.

After voxelizing the chromatid segment at the nucleosome resolution, the next step was to smoothly connect them at the base-pair resolution. To achieve the goal, we constructed six types of linker DNAs and thirty types of nucleosomes as building blocks. Specific to the linker DNA, we first modeled an ‘extended’ nucleotide pair with three separate parts as shown in Figure 3(a): a central base pair in a cylindrical shape of 1 nm in diameter and 0.55 nm in height, and the left and right sugar phosphate groups of 0.9 nm diameter. The centers of the three parts were at the same vertical plane and the vectors connecting the centers formed an angle of 135° (Figure 3(a)). We helically piled seventeen of them with a pitch size of 3.4 nm to create a linker DNA as shown in Figure 3(b) to connect an edge center to a voxel face center. We rotated it to obtain six different types, which axes were along ±x, ±y and ±z directions, respectively. These linker DNAs were used to conduct all possible face-edge connections for all six faces of a voxel.

Figure 3.

(a) The nucleotide pair including a base pair (yellow) and two sugar-phosphate groups (red and blue). (b) A 5.5 nm long linker DNA containing 17 bps with a pitch size of 3.4 nm. (c) A straight nucleosome type having 200 bps in two orientations (correcting top and bottom faces at the xy plane). (d) A bent nucleosome type having 200 bps, with its two ends connecting to the center of the bottom xy and the right xz faces of the 11 nm³ cube. (e) Connecting the nucleosomes inside the voxelized geometry via direct nucleosome-nucleosome connection (left) and nucleosome-linker DNAs-nucleosome connection (right). The linking path is shown with the arrows, which composes four linker DNAs in three orientations.

As for the modeling of the nucleosome, we started from the nucleosome structure developed in our previous work (Tsai et al., 2020). It was built in the base-pair resolution, with each nucleotide pair of 0.34 nm in length. The nucleotide pairs were helically piled up to form a DNA double helix of 3.4 nm per turn and the DNA double helix was wound 1.75 times with a 2.7 nm pitch around a cylindrical histone octamer of 3.13 nm in radius. The formed nucleosome was 11 nm in diameter and 6 nm in height, containing 147 bps. To make the nucleosomes smoothly connected across the voxels, we extended the nucleosome bidirectionally along the axis of the histone octamer to make it 11 nm high and contain 200 bps. The specific extension was performed in two ways. Firstly, we placed the nucleosome 6 nm in height at the voxel’s center with the histone cylinder parallel to the cube edges. We used 53 bps to continually wind the DNA double helix along a parabola curve towards the centers of the two opposite faces as shown in Figure 3(c) (top and bottom faces). We called this a straight nucleosome type. Secondly, we placed the nucleosome at the voxel center along the line connecting centers of furthest parallel edges. We then continually wound the DNA double helix towards the centers of two adjacent faces with 53 bps to make the entire structure compose 200 bps, as shown in Figure 3(d). We named it a bent nucleosome. Depending on the starting and ending faces, we rotated the two nucleosome types to obtain 6 straight and 24 bent nucleosomes in total.

At this stage, we were ready to build a smoothly-connected chromatid segment inside the voxelized geometry. For a monomer i of interest, we labeled it with three sets of indices: (I_i, J_i, K_i) for its residual voxel v_i, (I_i−1, J_i−1, K_i−1) for the voxel v_i−1 containing the former genome-adjacent monomer and (I_i+1, J_i+1, K_i+1) for the voxel v_i+1 containing the latter genome-adjacent monomer. If ∣A_i−1 − A_i∣ == 1 and ∣B_i+1 − B_i∣ == 1 with A, B = I, J, K, voxel v_i shared faces with voxels v_i−1 and v_i+1. Based on the relative distribution of the two shared faces, we could easily determine the nucleosome type for voxel v_i. Otherwise, if ∣A_i±1 − A_i∣ ≠ 1, it meant voxel v_i was not adjacent to voxel v_i±1. In this situation, we developed a path to connect v_i to v_i±1 through linker DNAs crossing multiple voxel faces. After fixing the connection path, the nucleosome type for voxel v_i and the required linker DNAs for the connection were naturally determined. We illustrated both situations in Figure 3(e). We repeated the above process for all monomers except for the beginning and ending ones. To avoid duplicate usages of a specific voxel face for different nucleosome connections, we developed an algorithm to systematically distribute the connection paths for all genome-connected physical-nonadjacent monomers. As for the starting and ending monomers, they were bonded to the centers of the cylinder ends in the polymer physics simulation. In the voxelization process, we fixed them in the central voxels of the two end layers and attached one end of each nucleosome to the most-outside face of the corresponding voxel. We then figured out the voxel connections between them and the voxels containing the second or second-last monomers. In this way, we uniquely determined the nucleosome types for the starting and ending monomers.

2.1.3. Chromosome Construction and Arrangement

After obtaining the voxelized chromatid segment, we used it to construct the complete human genome in metaphase containing about 12.3 Gbps of DNA. Each human cell contains 22 paired non-sex chromosomes (autosomes) and a 23^rd paired sex chromosome. In metaphase, each chromosome replicated and remained joint at a midway point, forming sister chromatids. We extracted the chromatid length in units of bps for each chromosome type from the Genome Reference Consortium Human Build 37 (GRCh37) public data library. By comparing it with the size of the chromatid segment we built, we estimated the number of segments required to construct each chromatid and piled corresponding number of segments along the chromatid axial direction. We placed two same sets of chromatids next to each other at 100 nm away to form one chromosome. To mimic the chromosome “arm” structure formed by the centromere, we purposely shifted a pair of segments from left and right chromatids towards the middle line such that the entire chromosome was divided into short “p” and long “q” arm sections (Figure 4 (d)). As for the sex chromosome, we used the XY type. In this way, we constructed the entire human genome in metaphase.

Figure 4.

(a1) Loop distribution of the constructed chromatid segment before and after voxelization. The consecutive loops gradually expanded along the z direction, with the color slowly changing from blue to red. (a2) Monomer distributions before and after the voxelization for a representative loop. (b) Radial angle distribution of the consecutive outer loops. (c) Contact probabilities for the chromatid segment before (blue) and after (red) voxelization. The Hi-C data were for K562 cells in metaphase (Naumova et al., 2013). (d) A chromosome model with “p” and “q” arms. It contained 10 chromatid segments in each chromatid. (e) The chromosome distribution at the metaphase plate in two views. The sphere (left) and the circle (right) represent a lymphocyte cell of 5.5 μm in radius.

In our modeling of the cell nucleus DNA in G0/G1 phase, we arranged the entire genome structure in a sphere of 5.5 μm in radius to represent a lymphocyte cell nucleus (Tsai et al., 2020). In this work, we assumed the entire genome distributed on the central plate of a lymphocyte cell with 5.5 μm in radius. We semi-randomly packed all chromosomes at the metaphase plate by forcing long chromosomes to distribute in the cell’s central region. The finally obtained metaphase DNA model is as shown in Figure 4(e).

2.2. Implementation of the metaphase DNA model to gMicroMC simulation

We implemented the developed metaphase DNA model to gMicroMC simulation for DNA damage computation via a multi-level hierarchy approach. Specifically, at the chromosome level, we labeled each chromatid with its length, the chromosome index, and its center coordinate and orientation in the global coordinates system. At the chromatid segment level, we labeled each voxel with the nucleosome type and the index for the starting base pair. We also tagged the voxel faces with the linker DNA types and their starting base pairs. At the nucleosome level, we only recorded the nucleotide pair coordinates for one straight and one bent nucleosome type in the voxel local coordinates system and loaded them to gMicroMC simulation. We obtained the nucleotide pair coordinates for other nucleosome types via a simple rotation operation during the computation. Similarly, for the linker DNA, we also only recorded and loaded the nucleotide pair coordinates in the face local coordinates system for one linker DNA type and obtained the coordinates for other kinds via rotations.

During the gMicroMC simulation, for a given physical or chemical event, we could easily check if it was within the region of a certain chromatid based on the chromosome center coordinate and its length. If it was within the region, we computed the index of the chromatid segment with the known chromatid segment length. We then transferred the event coordinate from the global coordinates system to the local coordinates system of the chromatid segment and computed the voxel index in which the event was located. Afterward, we transferred the event from the chromatid segment local coordinates system to the voxel local coordinates system. We loaded in the corresponding nucleosome type and the linker DNA types for that voxel and calculated the distances between the event and all sugar-phosphate groups inside the voxel and over the voxel faces. We used the smallest distance to judge if there was damage based on the damage criteria stated in our previous work (Tsai et al., 2020). If yes, we recorded the event with the chromosome index, nucleotide pair index, and the notation of left or right strand.

2.3. Test Studies

2.3.1. DNA damage study

We performed DNA damage studies to test the practicability of applying the developed metaphase DNA model for radiation-induced DNA damage calculation. We selected monoenergetic 4.5 keV electrons and 0.5 MeV protons as primary particles to represent a low and a high linear energy transfer (LET) irradiation condition (~3 keV/μm and 50 keV/μm). To compare our simulation results with experimental measurements (Oleinick et al., 1984; Iliakis et al., 1991; Dewey et al., 1970), we also sampled primary electrons following a narrow energy spectrum and a broad energy spectrum to mimic x-ray and Co-60 irradiations by referring to the works of (Oleinick et al., 1984; Iliakis et al., 1991; Dewey et al., 1970; Hsiao and Stewart, 2007).

Under each irradiation condition, we randomly and uniformly sampled the primary particle positions and directions inside a cellular sphere of 5.5 um in radius. We then performed the physical transport and the chemical diffusion simulation until 1 ns with gMicroMC simulation. After that, we overlapped the physical energy deposition event locations and the · OH radical positions with the metaphase DNA geometry and the G0/G1 geometry (Tsai et al., 2020) to compute DNA damages in the form of double-strand breaks (DSBs) and single-strand breaks (SSBs) for both DNA models. We repeated the simulations fifteen times at different dose levels between 0.2 to 6 Gy.

With the computed DNA damage in metaphase and G0/G1 phase, we performed self-consistent studies among results yielded at different simulation conditions. We compared the damage with those measured in experiments under Co-60 and x-ray irradiations (Oleinick et al., 1984; Iliakis et al., 1991). We also studied the induced cell survival fractions (SFs) and compared them with experimental measurements (Dewey et al., 1970). Dewey et al. (1970) synchronized the Chinese Hamster Cells at mitotic phase and irradiated the cells at different time moments post-synchronization with x-rays. Based on the measured cell cycle for the cell line (Dewey et al., 1970), we interpreted the cells 0.8-hour post synchronization as in early G1 phase. We then correlated the SF measured at early G1 phase from (Dewey et al., 1970) to the DNA damages computed in G0/G1 phase under x-ray irradiation. After that, we predicted SF for metaphase with DNA damages computed at that phase and compared it with the corresponding experimental measurement (Dewey et al., 1970). The results are shown in Figures 5 to 7.

Figure 5.

Energy deposition events (blue) and · OH radical positions (red) at 1 ns for (a) electron irradiation and (b) proton irradiation under around 0.2 Gy. In the zoom-in image of (b), we show the · OH radicals distributed around the physical events.

Figure 7.

(a) Number of DNA DSBs as a function of dose under x-ray irradiation with G0/G1 phase (green dash line) and metaphase DNA models (green solid line). (b) The experimentally measured cell survival fractions (SFs) for synchronized Chinese hamster cells by X-rays (Dewey et al., 1970) in metaphase (black triangle), 0.8 hour after metaphase synchronization (early G1 phase, blue circle) and 1.5 hour after metaphase synchronization (late G1 phase, grey square). The fitted SF based on the alpha-beta model (black lined) for the early G1 phase and the predicted SF (red lined) for the metaphase cells using DNA damage data shown in (a). The yellow region represented the 15% uncertainty between the measured SF and the chromosome aberration results in (Dewey et al., 1970).

2.3.2. Parameter uncertainty study

In the above gMicroMC simulation process, we limited the chemical stage simulation time to 1 ns to effectively include the radical scavenging effect inside the cell nucleus (Lampe et al., 2018; Zhu et al., 2020). Yet, the specific value of this parameter may introduce uncertainty in the DNA damage computation results (Lai et al., 2020; Zhu et al., 2020). To quantify its impact on the computation of DNA damage in metaphase, we performed an uncertainty study by setting the chemical stage simulation time t_c to be 1 ns, 2.5 ns, 10 ns and 1 μs. With each t_c value, we computed the total SSBs and DSBs under 4.5 keV electron, 0.5 MeV proton and Co-60 irradiations for both metaphase and G0/G1 phase models. We then calculated the damage ratios between the metaphase and the G0/G1 phase models and plotted the results in Figure 8.

Figure 8.

DNA SSBs per Gy (left subfigure) and DSBs per Gy (right subfigure) as a function of chemical stage duration for both G0/G1 (cross marked) and metaphase phase (triangle marked) DNA models under 4.5 keV electron irradiation (red colored), 0.5 MeV proton irradiation (blue colored) and Co-60 irradiation (grey colored). At each time point, the damage ratio between metaphase and G0/G1 phase under each irradiation is also illustrated in the figure bottoms.

In the chromatid segment construction process, considering the complexity of the coarse-grained polymer physics simulation, different parameter settings may result in similar contact maps yet different monomer distributions inside the chromatid segment. This may bring a concern that the chromatid segment we built may not be reliable in the application of radiation-induced DNA damage computation. To mitigate this concern, we performed a DNA damage study under varied metaphase geometries. Specifically, we adjusted the polymer physics simulation parameters to rerun the simulation and obtained two new sets of chromatid segments. We voxelized them and used the voxelized segments to build two new metaphase chromosome models. We applied the new models to compute DNA damages under 4.5 keV electron irradiation and compared the results to that computed with the “standard” metaphase model we built above. The results are shown in Figure 9.

Figure 9.

(a) The contact maps for metaphase models obtained under different simulation conditions. (b) DNA SSBs per Gy (left subfigure) and DSBs per Gy (right subfigure) as a function of the corresponding metaphase DNA geometries at different chemical stage durations under 4.5 keV electron irradiation.

We performed all the above simulations with gMicroMC on a NVidia TITAN Xp GPU card with 3840 cores (at 1.58 GHz) and 10790.4 GFLOPS of single-precision processing power. We recorded the time cost for the DNA damage computations upon both metaphase and G0/G1 phase DNA models and compared their efficiency performances.

3. Results

3.1. Metaphase DNA Modeling

We list the tuned parameter values for the polymer physics simulation in Table 1. With these parameters, we obtained a chromatid segment containing 24 Mbps of DNA, of 730 nm in diameter and 460 nm in height. The longitudinal density of the chromatid segment is around 50 Mbp/μm. Its geometrical illustration is given in Figure 4(a1), which clearly shows the consecutive nested loops extend axially (from blue to red colors in Figure 4(a1)) and mix radially. The radial-angle distribution of the consecutive outer loops in Figure 4(b) exhibits a noisy periodic pattern, indicating the loops expand spirally along the z axis. The contact probability distribution with a cutoff distance of 51 nm (Gibcus et al., 2018) is given in Figure 4(c)), which matches well with the Hi-C data of the metaphase K562 cell lines in a broad range between 100 kb to 10 Mb. A slight dip around 4 Mb in the simulated model may be resulted from the trade-off between the geometry accuracy and the practicability of the voxelization.

Table 1.

Parameters and their values used in the polymer physics simulation.

Parameter	Value	Description	Stiffness Parameter ((kBT/nm2)	Value	Description
R (nm)	365	chromatid segment radius	k repulse	30	repulsive potential
r (nm)	25	spiral scaffold radius	k ad	661	adjacent monomer harmonics
lz (nm)	230	scaffold pitch size	k an	5	angular force
N z	2	number of pitches	k ba	102	base monomers
lout (kbp)	120	outer loop average size	k sca	0.075	scaffold potential
lin (kbp)	60	inner loop average size	k bound	5	boundary potential
rad (nm)	11	monomer balanced distance	k end	5	tethering potential for end monomers

We show the chromatid segment geometry after voxelization also in Figure 4(a1) and the monomer positions for a representative outer loop before and after voxelization in Figure 4(a2). From Figure 4(a1), the overall loop structures are similar while the local texture changes slightly pre- and post- voxelization. The regional differences can be viewed more clearly from the monomer distributions in Figure 4(a2). After voxelization (blue), the monomers are slightly stretched from their original positions (red) to fit the voxelized geometry. The contact probability distribution P(s) for the voxelized chromatid segment is shown in Figure 4(c). Compared to P(s) before voxelization, P(s) distribution only changes minorly after voxelization, consistent with the geometry visualization in Figures 4(a1) and (a2).

After connecting the chromatid in the voxelized geometry with nucleosomes and linker DNAs, the finally-obtained chromatid segment contains about 26.8 Mbps of DNA. With this information and the GRCh37 data, we estimate the required numbers of chromatid segments to construct each chromatid of the 22 autosomes as 10, 9, 8, 7, 7, 6, 6, 6, 6, 6, 5, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, and 2. It is 7 for the X chromosome and 2 for the Y chromosome. Figure 4(d) shows a representative chromosome created with the voxelized chromatid segments, featuring the “p” and “q” arms. The finalized chromosomes are ~1 to 5 μm in length and contain ~12.5 Gbps of DNA in total. Figure 4(e) shows the specific chromosomes distributions around the metaphase plate.

3.2. Test Studies

3.2.1. DNA damage study

We implemented the developed model in gMicroMC simulation and applied it to compute DNA damage under different irradiations at different dose levels. Figure 5 shows the energy deposition events (blue) and the · OH radical distributions (red) at 1 ns from gMicroMC simulation for 4.5 keV electron and 0.5 MeV proton irradiations. In both plots, the deposited energy is around 0.2 Gy. Compared to the low LET electron irradiation, the physical and chemical events from the high LET proton irradiation are much more condensed with a sparse distribution in the cell volume, which is within expectation.

In Figures 6 (a) and (b), we show the mean DNA damage in the forms of SSBs and DSBs from the Co-60 (grey solid line), electron (red solid line) and proton (blue solid line) irradiations on the metaphase DNA geometry. At each dose level, we also show the standard deviation of the damage as the error bar. From the plotting, along with the increase of the radiation dose, the numbers of SSBs and DSBs increase for all irradiations, partially revealing that the damage could be adequately computed with the newly-implemented DNA model. Except for that, we also observe two interesting phenomena. First, the number of SSBs from the proton irradiation is lower than that from the electron, while the number of DSBs is higher. It is consistent with the damage behavior as a function of LET obtained with the G0/G1 DNA model by others (Tang et al., 2020). Second, both SSBs and DSBs from protons are associated with much larger error bars than from electron and Co-60 irradiations. The uncertainty is typically below 3% for Co-60 irradiation and below 5% for electron irradiation, while it can reach as high as 10% for proton irradiation. We interpret the large fluctuation of SSBs and DSBs induced by proton irradiation as the consequence of the sparse proton track distributions over the cellular volume, which has largely-varied chances to meet with a chromosome structure in different simulation runs.

Figure 6.

(a) Number of DNA single strand breaks (SSBs) and (b) number of DNA double strand breaks (DSBs) as a function of dose under Co-60 irradiation (grey colored), 4.5 keV electron irradiation (red colored) and 0.5 MeV proton irradiation (blue colored) with the metaphase DNA model (solid line) and G0/G1 model (dashed line).

In Figures 6(a) and (b), we also show the mean and standard deviation of DNA damage from the Co-60 (grey dashed line), electron (red dashed line) and proton (blue dashed line) irradiations on the G0/G1 DNA geometry. Like the metaphase DNA situation, SSBs decrease and DSBs increase along with the increase of the LET (Co-60 and electron to proton), consistent with the previous publications (Tang et al., 2020). Meanwhile, a larger fluctuation is also observed for the proton than for the Co-60 and electron irradiations. Comparing the DNA damage between G0/G1 phase and metaphase under the same irradiation condition, the fluctuation for the latter is larger than for the former. The fluctuation for the proton irradiation on the metaphase DNA structure is the largest. We understand it as a synergy effect between the heterogenous proton tracks and the heterogenous metaphase DNA geometry on the DNA damage quantification.

Averaging the obtained SSBs over different runs and different dose levels, we obtain the average number of SSBs as 1464, 1268 and 972 per Gy for Co-60, electron and proton irradiations in G0/G1 phase. They are 2190, 1998, and 1594 per Gy in metaphase. The corresponding DSBs are 53, 70 and 169 per Gy for G0/G1 phase and 80, 118 and 286 per Gy for metaphase. Considering the G0/G1 phase model contains 6.2 Gbp and metaphase model contains 12.5 Gbp, we find the ratio of SSBs per Gy per Gbp between the G0/G1 phase and the metaphase under the Co-60 irradiation is ~1.35 (1464/6.2 vs. 2190/12.5). It is slightly higher than the average ratio of percent SSBs between asynchronous and mitotic V79 cells under Co-60 irradiation, which is calculated as ~ 1.2 from Figure 4 of (Oleinick et al., 1984). Yet, considering a portion of the asynchronous V79 cells are at the mitotic phase, SSBs per Gy per Gbp for asynchronous cells should be lower than for that for cells in G0/G1 phase. Hence, it is reasonable to get a slightly higher SSB ratio between G0/G1 and metaphase cells than between asynchronous and mitotic cells.

We show DSBs as a function of dose for x-ray irradiation in Figure 7 (a). The average DSB per Gy is 104 for metaphase and 69 for G0/G1 phase. The ratio between the two is ~1.5. It is comparable to the ratio measured between G2+M phase and G1 phase CHO cells exposing to x-ray irradiation (Iliakis et al., 1991) within the uncertainty range, which is 2.1 ± 0.7 (derived from Table 3 of (Iliakis et al., 1991)).

As for the SF study, noticing that DSBs are intrinsically 300 times more cytotoxic than SSBs (Tounekti et al., 2001), we reasonably assumed SF was primarily determined by the number of DSBs (N_DSB) for the x-ray irradiation. Based on the number of DSBs, we fit the experimental SF (blue circle in Figure 7(b)) for early G1 phase with the alpha/beta model as SF(D) = exp(−0.98 * D − 0.00 * D²). From Figure 7(a), we have N_DSB = 68.6 *D (D measured in Gy) for DSBs computed under x-ray irradiation on the G0/G1 phase DNA model. Replacing D in SF(D) by D = N_DSB/68.6, we obtain SF as a function of N_DSB as SF(N_DSB) = exp(−0.98 * N_DSB/68.6) for early G1 (G0/G1) phase. From this function, SF is uniquely determined by N_DSB. We assume the relationship also holds for the metaphase DNA. From Figure 7(a), we have N_DSB = 103.7 * D for metaphase DNA model under x-ray irradiation. Replacing N_DSB with 103.7 * D , we predict SF as a function of D for the metaphase as SF(D) = exp(−0.98 * (103.7 * D)/68.6), which is shown as the red line in Figure 7(b). Comparing to the SF for the mitotic cells extracted from the experimental measurement (triangle-marked in Figure 7(b)) (Dewey et al., 1970), the precited SF is only slightly higher than the measurement (yellow region in the plot represents the 15% uncertainty range, which was observed between the lethality and the chromosome aberration quantifications), indicating the accuracy of our model in quantifying radiation-induced DNA damages.

3.2.2. Uncertainty study

In Figure 8, we show the mean and standard deviation of SSBs and DSBs per Gy obtained at different chemical simulation time t_c under electron (red colored), proton (blue colored) and Co-60 (grey colored) irradiations for G0/G1 phase (cross marked) and metaphase (triangle marked) DNA models. Generally, both SSBs and DSBs per Gy decrease when t_c increases for all irradiation conditions. Yet, the ratios of SSBs or DSBs between metaphase and G0/G1 phase are not changing significantly. The ratio of SSBs for electron irradiation remains at the level of 1.57 ± 0.07 at different t_c values. It is 1.50 ± 0.03 for Co-60. As for proton irradiation, it increases slightly from 1.64 ± 0.14 at 1 ns to 1.79 ± 0.15 at 1 us. The ratio of DSBs slightly increases from 1.69 ± 0.18, 1.50 ± 0.12, and 1.70 ± 0.16 at 1 ns to 1.80 ± 0.24, 1.60 ± 0.13, and 1.77 ± 0.16 at 1 us for electron, Co-60 and proton irradiations, respectively. Overall, all ratios remain in the uncertainty range, indicating that it is reliable to compare DNA damage between metaphase and G0/G1 phase models at t_c = 1 ns.

Figure 9 depicts the impacts of different metaphase DNA configurations on the DNA damage computation. As shown in Figure 9(a), we plot contact maps for two new metaphase geometries (“a”, dashed dot line and “b”. dashed line) that are obtained under different simulation conditions from that listed in Table 1. For instance, to get geometry “a”, the average outer and inner loop sizes l_out and l_in is increased to 600 and 200 kbp, the loop size follows a normal distribution, the stiffness of the scaffold potential k_sca is reduced to 0.038 k_BT/nm², etc. As for geometry “b”. the bond balance distance r_ad is slightly decreased to 10.9 nm, the stiffness of the repulsive potential k_repulse is decreased to 15 k_BT/nm², etc. Comparing the contact distribution, geometry “a” has a much larger drop-off of contact probability at around 5 Mb while geometry “b” follows a similar contact distribution with our standard metaphase geometry used in all the above simulations. Applying them to simulate DNA damage under 4.5 keV electron irradiation, the results are shown in Figure 9(b). From the figure, SSBs and DSBs generated from all metaphase geometries share a similar distribution, which is distinct from that produced from the G0/G1 phase model. SSBs and DSBs from geometry “b” are within 5% of that from the standard geometry, while that from geometry “a” are within 10%. These results show that metaphase geometries sharing a similar contact probability distribution also generate a similar DNA damage result, indicating the reliability of using the metaphase model we built to compute radiation-induced DNA damage.

Lastly, it is found the execution time of the DNA damage calculation with the metaphase model in gMicroMC simulation is comparable to that with the G0/G1 phase model. Both are at the few-seconds level. It shows the effectiveness of using the multi-level hierarchical approach to present the metaphase chromosome geometry in MMC simulation.

4. Discussion

In this work, for the first time, we develop a DNA model to represent the complete genome for a human lymphocyte cell in metaphase to support radiation-induced DNA damage study. We employ the coarse-grained polymer physics simulation to obtain a rod-shaped chromatid segment that features the spiral scaffold and consecutive nested loop structures. We voxelize the segment and connect the chromatid with 30 types of nucleosomes and 6 types of linker DNAs in known orientations. We pile different numbers of the voxelized chromatid segments to create the chromosomes of different lengths and arrange the chromosomes at the cell metaphase plate of 5.5 μm in radius to create the complete set of the metaphase DNA model. We implement the model into gMicroMC simulation with denoting the structure in a four-level hierarchical tree: the nucleotide pairs, the nucleosomes and linker DNAs, the chromatid segments and the chromosomes. We apply the model to study DNA damage under different radiation conditions. The obtained DSBs and SSBs are self-consistent and generally agree with the experimental measurements.

With the DNA modeling method developed in this work, we open a way to model highly-condensed DNA geometries to be implemented in MMC simulation. It enables the study of ionizing radiation induced DNA damage as a function of cell cycle with MMC simulation. This not only extends the application scope of MMC simulation, but also provides a starting point to model the different radiobiological responses at different cell phases. Although in this work we focus on modeling the metaphase DNA geometry, there is no foreseen barrier to extend the developed method to model other cell phases. It is our next step work to develop DNA models to cover the entire cell cycle and implement them into MMC simulation.

From our simulation results, when the ionizing radiation tracks and the DNA geometry are both spatially heterogeneous, a high fluctuation on the yields of SSBs and DSBs can be observed. Extending it to the tumor level, this means heterogenous DNA damage among the cell population could be induced by the heterogeneous radiation tracks and the heterogeneous DNA geometries even under a homogeneous dose painting. This could trigger a heterogenous cell responses to ionizing radiation. It indicates the importance on considering the target chromosome geometry and the ionizing radiation tracks in the microscopic DNA damage quantification and the macroscopic radiation-based tumor treatment.

In Figure 7(b), the predicted SF for metaphase cells from our computation is slightly higher than that from experimental measurements. The difference between the two may originate from different levels of repair capacities between metaphase and G0/G1 phase cells (Kato et al., 2008; Kinner et al., 2008), which was not considered in our SF prediction process. For instance, Kato et al. (2008) reported that the dephosphorylation process of γH2AX, a sensitive molecular marker of DNA damage and repair, is slower in irradiated metaphase cells than in G0/G1 phase cells, indicating the limited accessibility of dephosphorylation enzyme in metaphase cells. If including these findings as a reduced DNA damage repair capability in metaphase into SF computation, the obtained curve can be further suppressed from the predicted line, moving closer to the measured SF in metaphase.

In our DNA damage calculation process, we first transported the ionizing particles and the induced chemical radicals in water solvent and then overlapped the induced event positions with the DNA geometry to obtain the damage sites. This could bring uncertainty to the computed results from two aspects. First, the interaction cross sections between the ionizing particle and the water molecule could be different from that between the particle and the DNA molecule, especially when the DNA is in a highly-condensed format. Second, with the highly-condensed DNA structure in metaphase, the chemical species could be more easily scavenged by the DNA molecules and the histone proteins once it falls into the chromosome region, preventing its further diffusion inside the geometry. Under these two effects, the ionizing events and the chemical radicals may have a further-localized distribution, impacting the quantification of the damage pattern. As for the latter, the concurrent transport framework developed in our previous work (Lai et al., 2021b) may help elucidate its impact. It is our next step work to examine these two potential issues.

5. Conclusion

For the first time, we develop a metaphase DNA model to represent the complete genome of a human lymphocyte cell to support the study of DNA damage induced by ionizing radiation. In the model development process, we apply the coarse-grained polymer physics simulation and the voxelization technique to make the obtained model not only profound to match Hi-C data but also efficient in DNA damage computation. We implement the model to gMicroMC to compute DNA damage for both low-LET electron and high-LET proton irradiations. The obtained results are self-consistent and consistent with experimental measurements. It indicates the successful development and implementation of the metaphase DNA model in microscopic Monte Carlo simulation-based radiation-induced DNA damage computation.