For simplicity, we did not include rotational effects in our simulations. For many of the configurations we simulated, rotation would result in a more even spread of dose across the crystal volume (particularly at the edges) but the key effect being explored (the impact of PE escape on damage reduction) would not be significantly affected by rotation of the crystal.
3.1. Prediction of Extended Lifetimes for Small Crystals and Beams, and Higher Energies
Our findings suggest that in some cases the lack of inclusion of PE escape leads to deviations from the behaviour predicted by RADDOSE-3D [
9] (see, for example, [
14,
16,
17]). This conclusion is supported by both the similarity in output to RADDOSE-3D under conditions where PE escape is unlikely to affect dose (or when it has been ignored in our models), and the divergence in output under conditions where PE escape is expected to be significant.
A comparison of
Figure 1 and
Figure 2 shows that results that account for PE escape and those that do not exhibit a similar convergence as the beam width is increased, whether or not the crystal size is also increased so that it matches the beam width. The reason for this is that PEs are produced everywhere within the beam footprint, in both crystal and solvent. If the crystal is large, the majority of the PEs produced within it will not escape it. By contrast, if the crystal is small, many PEs will escape but if the beam footprint includes a significant portion of the surrounding solvent, a roughly equal number of PEs will be scattered into the crystal, offsetting the effect of those that escape it. For the sample that we modelled, the chemical compositions and densities of the crystal and surrounding solvent are not significantly different in terms of the production and propagation of PEs. This is a consequence of the fact that the cross-sections and average binding energies between solvent and crystals are quite similar, leading to only a small difference in energy-deposition profile. Our choice of sample, lysozyme, contains no atoms heavier than sulphur and the only heavy atoms in the solvent we modelled are sodium and chlorine. The presence of more heavy atoms in either the crystal (e.g., metalloproteins) or the solvent (e.g., cacodylate) could lead to a more significant difference between the crystal and the solvent in the production of PEs.
The most striking discrepancy between results that included PE escape and those that did not occurred when simulating a 1 μm crystal in a 1 μm beam (FWHM) at high energies (⩾15 keV) (
Figure 3). Here, our simulations predict increased diffraction efficiency compared to RADDOSE-3D by a factor of close to 170% at 15 keV, 300% at 20 keV, 380% at 25 keV and 400% at 30 keV. This discrepancy occurs because the IMFP of the PEs increases with energy and the resulting energy deposition is therefore spread over a larger area. The implication is that for micron-sized crystals illuminated by similarly sized beams, simulations that neglect PE escape over estimate the dose received, becoming increasingly less accurate as the incident X-ray photon energy increases. It has been previously observed that conducting MX experiments with smaller crystals at higher energies may allow experimenters to maximise the reduced radiation damage afforded by PE escape [
17,
20,
21]. However, a systematic experimental study, isolating the effect of beam-energy in radiation damage has not, to our knowledge, been carried out and will be the focus of further work.
3.2. Simulations Enabling Qualitative and Semi-Quantitative Interpretation of Published Data
In early simulation work carried out by Nave and Hill [
21], they used simple CASINO models to estimate that an 8 μm crystal illuminated with 30 keV X-rays would benefit from a three-fold increase in diffraction efficiency due to PE escape. Although not included in their model, they noted the need to minimise the material surrounding the crystal to prevent PEs being scattered into it from the solvent, and the possibility of leveraging the anisotropic emission of PEs due to the beam polarisation to minimise damage. The largest crystal we simulated was 5 μm, which saw an approximate four-fold improvement in diffraction efficiency at 30 keV when PE escape was accounted for; which is consistent with Nave and Hill’s result. By including the solvent in our models, we have been able to gauge the effect of illuminating the volume around the crystal and, as discussed in the previous section, our results support the conclusion that PE production outside the crystal must be minimised if appreciable damage-reduction is to be achieved. Although Nave and Hill assumed larger beams and suggested physically limiting the volume of the solution surrounding the crystal, the increasing availability of microfocus X-ray beams may permit the same ends to be achieved through focusing, provided the crystal can be positioned within the beam with suitable precision to limit PE production outside the crystal.
Following, and extending upon, the work of Nave and Hill, Cowan and Nave [
20] used coarse Monte Carlo models to study the impact of PE escape on diffraction efficiency. When neglecting PE escape, but accounting for Compton scattering, they observed a variation in diffraction efficiency of up to 40% between 1 keV and 40 keV, with a plateau of maximum efficiency between 20 keV and 30 keV. Under the same conditions, our model shows a similar trend, though the calculated values vary over a slightly wider range. Incorporating a model for PE escape, Cowan and Nave observed an improvement in diffraction efficiency at higher energies over a wide range of crystal sizes, though the effect was stronger for smaller crystals. In their model, a 1 μm crystal saw a roughly five-fold increase in diffraction efficiency at 12 keV when accounting for PE escape, with the improvement above 20 keV being around 15-fold. This is consistent with our results (
Figure 4) for the 1 μm crystal in a 1 μm beam, where the effect of PEs being generated in the solvent (which has been ignored in their model) is minimal. As in the model that excluded PE escape, Cowan and Nave’s results show a broad plateau in diffraction efficiency for the 1 μm crystal between 20 keV and 30 keV, before it drops off slightly at higher energies. Our models also show a similar trend, though the diffraction efficiency (across all beam and crystal sizes) continues to improve as the energy is increased up to 30 keV, albeit at a reduced rate. This suggests that the increased cross-section for Compton scattering at higher energies offsets the reduction in dose due to PE escape. However, as shown in
Figure 3, the discrepancy between our control simulation (that did included Compton scattering) and the RADDOSE-3D data (that did not include Compton scattering) is small, even at higher energies, and the discrepancy between them and the simulations that did include PE escape grows much more slowly above 20 keV than it did between 9 keV and 20 keV. Since this study was undertaken, RADDOSE-3D has been updated to include incoherent scatter and Bury et al. [
19] observed that this inclusion only begins to have a significant effect on dose prediction above 40 keV. These results suggest that the increase in Compton scattering alone does not account for the plateau in diffraction efficiency at higher energies. A complementary interpretation is that much of the PE energy is already escaping the crystal at 20 keV and the increase in electron IMFP above this energy simply serves to scatter the PEs further into the solvent. There is some support for this idea in
Figure 4, as the gradient of the diffraction efficiency of the 1 μm crystal in a 1 μm beam (the configuration compared in
Figure 3) drops off slightly at higher energies compared to the gradients for the larger crystals. Cowan and Nave [
20], who simulated crystals up to 20 μm, observed a similar effect, with larger crystals continuing to benefit from increased diffraction efficiency as the beam energy was increased above the 20 keV to 30 keV where it plateaued for the 1 μm crystal.
In Holton and Frankel’s [
27] paper on the minimum crystal size required to collect a complete data set, they estimated the effect of PE escape by first calculating the dose in the absence of PE escape and then scaling this dose by a so-called
Nave-Hill capture factor,
, which they approximated as
where
R was the radius of the spherical crystals they modelled,
was the range of a PE with energy
E, and
and
were the dose values with and without accounting for PE escape, respectively. This capture factor was designed to equate to unity for “large” crystals at 12.4 keV and was intended to provide only a coarse indication of the benefit of PE escape. We used global dose values from our simulations to calculate
as the ratio
. We also calculated
using the RHS of Equation (
2), taking
as the maximum penetration depth of electrons in our CASINO simulations and substituting half the crystal edge-length for
R. The two methods of calculation produced values for
within a factor of 2 of each other. Our simulations did not have the spherical-symmetry of those performed by Holton and Frankel, but, given the coarse nature of their estimate, this substitution is expected to provide a reasonable point of comparison. The similarity in values of
indicate that our simulation results are comparable to the Holton and Frankel model.
In addition to qualitative comparisons to previous modelling studies, a semi-quantitative assessment of our model can be conducted by comparison to recent results obtained by Finfrock et al. [
16,
18] and Sanishvili et al. [
17], who performed experimental studies of radiation damage in large crystals by exposing sub-volumes using micro-focus and line-focus beams. They observed that the radiation damage footprint extended between 1.5 μm and 5 μm beyond the actual beam footprint, showing the spread of PEs beyond the diffraction volume. Both experimenters used beam energies close to 18 keV. In the Monte Carlo stage of our models (implemented using CASINO), energy-deposition extended approximately 5 μm beyond the interaction point at 20 keV, which is in-line with these previous observations.
Since this study was conducted, a new version of RADDOSE-3D [
19] has been released that includes Compton scattering and PE escape, neither of which was included in the version we used. As noted above, Bury et al. [
19] found that the inclusion of Compton scattering made negligible difference below 20 keV, and only minimal difference between 20 keV and 40 keV. Conversely, Bury et al. found the inclusion of PE escape to have a very significant impact on dose predictions for microcrystals. They simulated crystal sizes from 1 μm to 100 μm for a single beam energy of 12.4 keV and observed a non-negligible reduction in dose for crystals as large as 20 μm under these conditions. For a 10 μm crystal, they observed a reduction in dose of approximately 20% when accounting for PE escape, and reduction of over 95% for a 1 μm crystal. In our simulations, we observed a reduction of 72% for the 1 μm crystal in a 12 keV beam, when the beam FWHM was matched to the crystal (and a much lower reduction when the beam was larger). This discrepancy can be accounted for by the fact that we simulated the solvent around the crystal, thereby accounting for PEs scattered into the crystal as well as out of it, while the new version of RADDOSE-3D assumes an isolated crystal, neglecting the effect of the surrounding solvent. In the future, a comprehensive comparison of our simulations and this new version of RADDOSE-3D could provide useful insights.
3.3. Permitting a Real Space Interpretation of Crystal Damage
The idea that diffracted intensity is approximately proportional to crystal volume is well-established [
7] and the implication is that damage leading to a reduced intensity of diffracted radiation can be interpreted in real-space as an apparent “shrinking” of the diffraction volume of the crystal. Note, this does not mean that any actual protein material is being lost from the volume of the initial crystal, but that it becomes so disordered that it no longer contributes meaningfully to the diffraction signal. Recently, Coughlan et al. [
14,
15] reported such shrinking of the diffraction volume when using Bragg coherent diffractive imaging (BCDI) to produce 2D and 3D real-space images of lysozyme crystals that had been subjected to high dose burns (several hundred MGy).
Warkentin et al. [
26] also recently modelled the change in the intensity of diffracted radiation per unit area for a crystal in a Gaussian beam, as it undergoes radiation damage. They observed that, due to the flux being highest in the beam centre, the portion of the crystal nearest the beam centre produced the largest number of diffracted photons, but also underwent radiation damage the fastest. Crucially, however, their model did not account for PE escape. To estimate which regions of the crystal contributed the most diffracted photons at the detector, they weighted each contribution by both the flux density in that part of the beam and the degree of damage already suffered. They observed a broadening of this contribution over time; initially, the central portion of the crystal contributed most strongly to the diffracted signal but by the time the spot intensity had dropped to half its initial value, their models predicted that the contribution to diffracted intensity was more or less equal across the whole FWHM of the beam. After sufficient exposure, radiation damage progressed to the point that only the fringes of the crystal (those in the lower flux edge of the Gaussian beam) contributed coherent scattered photons.
The snapshots of the diffraction volume in
Figure 5 show a reduction in the total contribution to diffracted signal and shrinking and flattening of the diffraction contribution—meaning photons contributing to Bragg peaks are coming from different regions of the crystal. The flattening and shrinking of the diffraction volume is anisotropic as PEs are preferentially scattered at lower angles to the polarisation axis of the X-ray beam (here the
x-axis), resulting in slower damage rates along the axis perpendicular to polarisation. As highlighted in the summary graph in
Figure 5, our simulations show the same trend as Warkentin’s models [
26], with an initial peak in the centre of the crystal and a flattening of the diffraction contribution across the whole crystal with increased dose. The overall shrinking of the diffraction volume is also in-line with the observations of Coughlan et al. [
14,
15] and may provide an initial model for future experimental examination of this phenomenon.
3.4. Micro-Crystallography Could Benefit Significantly from Higher X-Ray Energies
Predictions produced by our model suggest significant benefits could be gained by optimising MX experiments along multiple parameter-axes, particularly when using smaller crystals and higher beam energies. The increase in electron IMFP at higher energies, coupled with a decrease in damage-causing interactions relative to elastic scatter, suggests that, all other things being equal, the greatest increase in diffraction efficiency could be gained from using higher energy beams. As highlighted in
Table 1, although the coherent scattering cross-section falls with increasing beam energy, it does so more slowly than the combined cross-sections for incoherent scattering and photoelectric absorption. Although experiments at higher energy take longer (roughly twice as long at 20 keV and four times as long at 30 keV) due to the reduced cross-section for coherent scattering, the ratio of diffracted intensity to dose-accumulation (damage) would be roughly doubled for experiments above 20 keV, relative to those at 12 keV. This is even before accounting for the effect of PE escape; as shown in
Figure 4, our simulations predict that PE escape could be responsible for increasing diffraction efficiency by up to 30 times, relative to the diffraction efficiency at 12 keV in cases where PE escape is not considered. This is assuming the use of micro-crystals and micro-focus beams so that the effect of PE escape is large, but even for a 5 μm crystal/beam, our calculations show PE escape contributing to an increase in diffraction efficiency. Given the relative rate at which the cross-sections change and the longer IMFP for higher-energy PEs, conducting experiments at higher energies may be advantageous even with larger crystals in more conventional MX experiments.
Several experiments [
16,
17,
18] have shown reduced damage occurring due to PE escape, but have only explored a narrow parameter space thus far; a more thorough examination, particularly focussed on the benefit of using higher energy beams, could be guided by our results and data obtained could subsequently be used to validate and improve our model. Should experiments show quantitative agreement with our predictions, our model could be used to provide more accurate predictions of dose rate, and therefore crystal lifetime in micro-crystallography experiments, compared to currently available radiation damage software packages.