Kinetic Modeling of the X-ray-induced Damage to a Metalloprotein

Abstract

It is well known that biological samples undergo x-ray-induced degradation. One of the fastest occurring x-ray-induced processes involves redox modifications (reduction or oxidation) of redox-active cofactors in proteins. Here we analyze room temperature data on the photoreduction of Mn ions in the oxygen evolving complex (OEC) of photosystem II, one of the most radiation damage sensitive proteins and a key constituent of natural photosynthesis in plants, green algae and cyanobacteria. Time-resolved x-ray emission spectroscopy with wavelength-dispersive detection was used to collect data on the progression of x-ray-induced damage. A kinetic model was developed to fit experimental results, and the rate constant for the reduction of OEC MnIII/IV ions by solvated electrons was determined. From this model, the possible kinetics of x-ray-induced damage at variety of experimental conditions, such as different rates of dose deposition as well as different excitation wavelengths, can be inferred. We observed a trend of increasing dosage threshold prior to the onset of x-ray-induced damage with increasing rates of damage deposition. This trend suggests that experimentation with higher rates of dose deposition is beneficial for measurements of biological samples sensitive to radiation damage, particularly at pink beam and x-ray FEL sources.

Introduction

The high brilliance of third generation x-ray synchrotron sources and newly constructed x-ray free electron lasers allows for new types of measurements to analyze the structure and function of biological molecules.^1–4 However, the associated increase in x-ray flux and rate of x-ray dose deposition exacerbates the problem of x-ray-induced damage to biological molecules and its effect on electronic and geometrical structures.

Here, we present an experimental and theoretical analysis of the x-ray-induced damage to the photosystem II (PSII) metalloprotein complex during exposure to the full flux achievable at a 3^rd generation synchrotron source. PSII is an integral component of natural photosynthesis. The oxygen-evolving complex (OEC) of PS II contains a Mn₄Ca core. In active protein, the Mn centers are present in both Mn^IV and Mn^III oxidation states. Earlier studies show that x-ray-induced damage to PS II manifests in the reduction of Mn centers to Mn^{II 5–8} and breakage of Mn di-µ-oxo units.^7,8 To understand the origins of this effect, we review the stages of radiation damage known for biological specimens in the hard x-ray (5–15 ke V) energy range, typical for protein crystallography and x-ray spectroscopy.

Primary damage is considered to arise from electrons generated in a cascade (timescale ~tens of fs) after the interaction of x-rays with matter via absorption and inelastic scattering. The relaxation cascade eventually results in a large number of photoelectrons with energies of a few to several tens of eV. The number and energy distribution of these electrons can be theoretically estimated.⁹ Subsequent radiolytic reactions caused by products of primary damage are classified as secondary damage. The consequences of primary and secondary radiation damage reactions include the breakage of chemical bonds, generation of free radicals and changes in the redox state of cofactors; these are commonly referred to as specific damage. In crystallography experiments, the alteration of individual biological molecules due to specific damage eventually causes long-range rearrangements of molecules in the crystal and a loss of crystalline order. This process is referred to as global damage. A number of experimental and modeling studies aimed at gaining a better understanding of the temporal and spatial progression of x-ray-induced damage in biological samples have been reported.^10–18

Temperature has a major effect on the rate of x-ray-induced damage. Experimentally, low temperatures (LT’s) of 10–20K drastically decrease the rate of x-ray-induced damage.^5,6,12 Until recently^{3,5,6,19–25} for PSII and other biological samples, cryogenic measurements have provided the only feasible method of data collection.^8,26–28 Unfortunately, LT data collection is not applicable to dynamic studies of biological samples. Some transient intermediates can be trapped by freeze quench and analyzed at LT, however, others, such as the proposed S₄ state in PS II, escape cryogenic trapping due to extremely short lifetimes. In such cases, time-resolved x-ray techniques (x-ray absorption spectroscopy, XAS, and x-ray emission spectroscopy, XES) remain the methods of choice as they can characterize changes in the electronic structure at particularly short times.

Recently, we demonstrated a methodology allowing for room temperature data collection capable of monitoring OEC state transitions with high time resolution.⁵ The high fluxes required to obtain experimental spectra make characterization and understanding of x-ray-induced damage to the OEC very important. Here, we use kinetic modeling to extract the rate of reaction of the OEC with free radicals generated in solution. The developed model can be further verified by analyzing damage kinetics observed with higher/lower rates of dose deposition, as well as the use of different excitation wavelengths.

Materials and Methods

Experimental Methods

PSII-enriched thylakoid-membrane particles were prepared from supermarket spinach.^29,30 PSII was stored in a buffer using sucrose as a cryo-protectant: 0.4M sucrose, 5mM CaCl₂, 5mM MgCl₂, and 15mM NaCl, 50mM MES, pH 6.0. The oxygen evolution activity of PSII was measured by a Clark-type electrode in a Hansatech oxygraph. The activity of the preparation was 300 µmol O₂/(mg Chl · hr) or greater under constant saturating illumination at 25°C utilizing 0.3mM 2,6-dichloro-1,4-benzoquinone (DCBQ) as an artificial electron acceptor. The Chl a:b ratio was derived from the optical absorbance of chlorophyll extracted with 80% acetone:20% water solution. This was measured with a Cary300 Bio UV-visible spectrophotometer. For all samples, this ratio was ~2.5:1 which indicates high enrichment of membrane particles with PSII. To ensure the high quality of the samples, LT X-band EPR spectra were recorded for the S₁ and S₂ states. S₂ state samples were obtained by illuminating S₁ state samples with 120W Halogen lamp for 30 minutes while in a cold bath of ethanol and dry ice to maintain a sample temperature of 195K, after which they were immediately frozen in liquid nitrogen. All sample preparation, handling and storage environments were completely dark, save for dim green light when unavoidable, to prevent state transitions. In addition, samples were maintained at a constant temperature of approximately 4°C during preparation. To simulate fully damaged PSII with 100% of Mn ions in the Mn^II oxidation state, 100mM MnCl₂ tetrahydrate stock solution was added to the PSII pellet samples. At the synchrotron radiation facility, the PSII samples for XES were prepared immediately prior to the measurements as described in the following:

1. After slow thawing of PSII stock preparations (~30 mg Chl/mL) stored in liquid nitrogen, the undiluted PSII pellet was spread onto polycarbonate holders with 4µm polypropylene tape backing stretched flat.

2. The samples were then left to partially dry for ~1.5 hrs on ice under the flow of pure N₂. This ensured the highest PSII concentration possible in the beam.

X-ray emission spectra were collected at the Advanced Photon Source (APS) at Argonne National Laboratory on insertion device beamlines 20-ID and 14-ID (BioCARS). The radiation was monochromatized by a Si(111) double crystal monochromator. Focusing was done using Rh coated KB mirrors operated at 4 mrad glancing incidence. A He-filled chamber (I₀) with a beam clean-up pinhole was placed before the sample to monitor the intensity of the x-rays. The monochromator was calibrated via the KMnO₄ pre-edge located at 6543.3eV. In addition, MnO emission spectra and Fe foil XANES were taken periodically to monitor and correct for any shift in energy calibration. To reduce the dose rate at 20-ID, a defocused mode was used with a ~105 × 85 µm² projection of the beam onto the sample surface, 45° to the beam (angled to acquire the highest percent of analyzed fluorescence). A scan program synchronized with the shutter also provided sample protection from the beam during motor movements.

The BioCARS beamline provided a unique pulsed pink-beam capability for XES experiments. The undulator gap and white beam slits¹ were set to give an undulator spectrum peaking at ~7.85 keV with a width of ~500eV FWHM. To prevent unwanted fluorescence originating from the GaP spectrometer crystals, high energy x-rays were suppressed by using the Si stripes of the KB mirrors with mirror angles of 3.8 mrad. The incident flux was monitored downstream of these mirrors via a photodiode. The high-heat-load chopper¹ was set to produce 44 µs pulses and scans were synchronized with a pulse repetition rate of 41.1 Hz. In between pulses, all shutters were closed to save the sample from unnecessary degradation. The vertical mirror was defocused to give a final projection size of ~50 × 120 µm² (V × H) onto the sample surface at 45° to the beam. Details concerning spectrometer calibration are described in Davis et al.⁵

Short working distance (SWD), miniature x-ray emission spectrometers (miniXES) were used to record Mn Kβ emission. These spectrometers use multiple flat Bragg analyzers to reflect x-ray fluorescence onto a Pilatus100k (Dectris) 2D-PSD^31–34. The miniXES spectrometer design is based on the observation that a micro-focused incident beam permits one to obtain good energy resolution from a flat analyzer crystal that is only a few centimeters from the sample.^31–34 We utilized two different miniXES configurations suitable for the analysis of Mn Kβ emission. The first instrument uses the Ge 440 reflection in a Johansson arrangement,³⁵ and allows for a 100eV collection range containing the Kβ’, Kβ_1,3 (emission from the 3p level) and Kβ“ peaks and demonstrates an instrumental energy resolution of ~1.5eV. To improve the energy resolution, the second design instead uses the GaP 440 reflection in a von Hamos configuration.³⁶ This von Hamos design has a decreased collection range of 50eV, including only the Kβ main lines. It demonstrates a higher energy resolution of ~0.3eV which is comparable to that of spectrometers utilizing SBCA’s.²⁶ More details can be found in Davis et al.⁵

Measurements on the dark-adapted S₁ state of PSII were done at room temperature (RT). Undamaged PSII Mn Kβ_1,3 spectra were obtained and damage studies were performed. Fresh PSII (protected by synchronized shutter) was exposed to the full intensity micro-focused x-ray beam (e.g. 10¹² photons/s at 20-ID). PSD exposures were collected in defined time intervals throughout the irradiation – e.g. the shortest for 20-ID at ~7 ms. In order to obtain the desired signal-to-noise, multiple images were summed together for different points on the sample corresponding to the same total irradiation period. These divisions allowed us to monitor the x-ray-induced damage in time.

Mathematical Methods

The mathematical model we use to fit the experimental data⁵ consists of a system of ordinary differential equations (ODEs) with respect to time. To account for the considerable thickness of studied PS II samples (~ 1mm) and the effect of attenuation on both the incident x-ray beam and generated fluorescence, we introduce steps of 10µm, over which we assume constant r_c (equal to the number of radicals created by the photons absorbed in this sample thickness), and solve the system for each 10µm depth interval. This gives us a set of rates for each step in the ‘z’ direction (into the sample); see equation (1).

$$r_c=(\Phi )(\zeta )(e^{-\alpha z_i})e^{-\alpha z_i}-e^{-\alpha z_{i+1}}$$ (1)

where the first term, Φ, represents the flux per unit volume (e.g.1.1 × 10⁷ photons s⁻¹µm⁻³); the second term, ζ, the number of radicals created per absorbed photon (e.g. ~190); the first exponential converts the flux to the transmitted flux at any given depth, z_i, given by Beer-Lambert’s law, I/I₀= e^−αz, where α = 1.46 × 10⁻³ µm⁻¹ in the case of bulk sample approximated as water; and, the final term provides the number of photons absorbed in each 10µm slice respectively. Note that z₁ represents position ‘zero’, the surface of the sample; it follows that z₁₀₀ is 990µm into the sample.

We then take a weighted average over those rate values whose individual importance is again determined by Beer-Lambert’s Law. This method provides reaction rates (k₁ and k₂) for every 10µm step into the depth of the sample (fitting by assuming all data points originated in that layer), see Table S1, as well as ‘weighted average’ values of k₁ and k₂ to simulate the experimental fluorescence data. We apply the same weighting to the calculated Mn²⁺ content shown in Figure 3 as a solid line.

Figure 3.

The data presented in Figure 2 is fit by the proposed model (green – ode45; blue – ode15s). All beamtimes were considered a unified data set. Note that the percent error was not taken into account during the fit. Error bars are present to provide a measure of uncertainty in the data, as before.

We employ either MATLAB’s ode45 or ode15s algorithm iteratively in conjunction with the MATLAB function lsqcurvefit to fit the experimental data with the proposed model and to find the optimal values for k₁ and k₂ parameters. Ode45 is a one-step ODE solver utilizing the Dormand-Prince method.³⁷ In order to use this solver, it is necessary to remove the stiffness of the problem by dividing out a factor of 10⁶. Ode15s, while often less accurate, relies on numerical differentiation formulas and is efficient at solving stiff problems. Our results indicate an equivalency between these two solvers, Figure 3. To fit these data, we run lsqcurvefit with appropriate starting values and conditions (R(t=0)=0 and [rad(t=0)] = 0) on the numerical solution to equation 5 below. To allow for these initial values of [rad(t=0)], we predominantly use ode15s, unless specified.

Results and Discussion

Experimental analysis of x-ray-induced damage to PS II

In this work, we monitor the progression of x-ray-induced damage to PS II via Mn Kβ XES. The dispersive nature of the miniXES spectrometers allows us to collect data with high time resolution, Figure 1A. XES involves the excitation of an inner shell electron into the continuum and subsequent repopulation of the newly created hole, Figure 1B, insert. Kβ emission lines correspond to 3p → 1s transitions. While the Kβ spectrum is ~8 times less intense than Kα emission (2p → 1s transition), it has higher sensitivity to the electronic structure of the Mn center due to the exchange interaction between the 3p and 3d orbitals. This interaction creates a multiplet spread of 15eV from which the doublet (Kβ′ and Kβ_1,3) arises. The position of the more intense Kβ_1,3 peak is often used to determine changes in oxidation state.^26,38 As the oxidation state decreases, more electrons are present in the 3d level increasing the valence spin and leading to a greater exchange interaction (increased splitting) between 3d and 3p. Thus, if only considering the Kβ_1,3 peak, it appears to shift to lower energies with the reduction of Mn ions. We use this shift to monitor the extent of x-ray-induced photoreduction to the Mn cluster, Figure 1B. Undamaged room temperature S₁ state XES data were reported previously.⁵ We analyzed the x-ray-induced damage progression by ‘overexposing’ the sample, Figure 1B. Currently, it is accepted that OEC in the S₁ state contains two Mn^III and two Mn^IV that are all susceptible to reduction to Mn^II. This reduction can be monitored by XES and any spectral shift to higher energies. X-ray emission spectra of Mn^II solution added into PSII samples were taken to mimic fully damaged S₁ state samples.

Figure 1.

In XES experiments (photon-in-photon-out spectroscopy) the sample is excited with an x-ray photon of sufficient energy to create a 1s hole - above the K-edge absorption energy for Mn in this work. Repopulation of this 1s hole from other energy levels (3p in the case of Kβ emission) results in the emitted x-ray fluorescence which is then analyzed by Bragg reflection from a suitable crystal plane. (A) The x-ray fluorescence is reflected via Bragg scattering from a flat analyzer crystal onto the position sensitive detector. Pixel-to-energy calibration allows us to reconstruct the XES spectrum. Detector read out is possible for different time intervals indicated as Δt. (B) Effect of the progression of x-ray-induced damage on the Mn Kβ PSII XES spectrum. Data are shown for consecutive 100ms exposures. The shift of the XES spectrum to higher energy is due to the photoreduction of the OEC containing Mn^III and Mn^IV ions to Mn^II.

Utilizing previously collected undamaged S₁ state XES, we created calibration spectra composed of a chosen ratio of these data and XES of Mn^II in solution. These spectra then served as a comparison to the collected data for determining the percent reduction to Mn^II, see Figure 2. Comparisons of experimental and calibration spectra were done visually. The difference in the spectrometers (damage studies were done with two different spectrometers, see Figure 2 caption) created spectral shape variations due to differences in energy resolution. To account for uncertainties in the visual determinations of the peak center, we conservatively estimated the error to be ±10% for each point.

Figure 2.

X-ray-induced damage to the OEC expressed as a percentage of Mn^II plotted versus photon dose for RT. RT data collected from the Johansson (red, excitation energy 7.5 keV), and von Hamos miniXES (light blue and orange – corresponding to different beamtimes, excitation energy 7.09 keV) spectrometers, respectively. The dark blue points were calculated from the data of Grabolle et al. (2006) reporting exponential decay with a rate of damage k=0.9 min⁻¹ for RT measurements taken with a flux ~10¹² photons s⁻¹ mm⁻²(i.e. 10⁶ photons s⁻¹ µm⁻²). This is 100 times less than the flux presented in this paper. Note that they also reported an exposure time of ~200s resulting in 90% Mn^II (the green point). See Table 1 for more details regarding the fill patterns during the respective data sets.

For increased signal-to-noise, we sacrificed time resolution by not only summing all data corresponding to the same irradiation period (usually 20 ms – monochromatic beam, or each ~40µs pulse – pink beam), but also sets of three (e.g. total of 60 ms) or five (e.g. total of 100 ms) sequential exposure periods. Any spectral shifts occurring in these intervals would be averaged together and could potentially skew the data. In order to correct for this, we approximated the shifts in Kβ_1,3 as linear for small timescales. It follows that the damage observed in an averaged set of exposures actually reflects the damage caused by half the dose deposited during that time frame plus the dose deposited in the time interval/s preceding. This assumption is only a reasonable first approximation at very short timescales, comparable with minimum data collection interval.

We compared our results with the only available PSII room temperature damage study reported by Grabolle et al.,⁶ Figure 2. Both data sets fall within the same order of magnitude, Figure 2, in spite of 100 times higher rate of dose deposition used in our study. However, we did not observe the reported lag phase prior to the onset of damage, which could be attributed to differences in the rate of dose deposition.

Kinetic Modeling of x-ray-induced damage at room temperature

Earlier RT damage analysis by Grabolle et al. describes the kinetic of damage by a single exponential model with k_main= 0.9 min⁻¹.⁶ Such a model sheds little light on the mechanism of damage. It is critical to develop a more detailed model that can be used to predict damage under the condition of increased rate of dose deposition, available with pink beam mode at 3^rd generation synchrotrons and new ultrafast high brilliance sources such as the LCLS. Depending on the effect of dose deposition rate on the damage profile, it may be possible to collect data prior to the onset of damage even with very intense sources.^4,19,24

Unlike in x-ray crystallographic studies detecting x-ray scattering, the generation of x-ray emission requires absorption of the x-rays by samples (the process producing the damage). Due to the low dilution of our target metal ions, it is highly unlikely that direct absorption by Mn and the creation of Auger electrons (less prevalent for heavier elements such as Mn) is the main source of damage. Instead, we hypothesize that absorption into the bulk sample and subsequent radiolysis of water creates radicals whose interaction with the probed species dominate room temperature x-ray damage in biological samples. We approximate PSII-enriched thylakoid membrane fragments as water for the purposes of x-ray absorption.

In its most general form, the rate equations for a reaction of the OEC with radicals can be written as

$$\frac{\mathrm{d}[{\mathrm{S}}_1(t)]}{\mathrm{d}t}=k_2[{\mathrm{S}}_1(t)][\text{rad}(t)]$$ (2)

and

$$\frac{\mathrm{d}[\text{rad}(t)]}{\mathrm{d}t}=r_c-r_d-r_r,$$ (3)

where [S₁(t)] represents the concentration of undamaged Mn (Mn^IV and Mn^III ions) in the S₁ state of PSII, [rad(t)] represents the concentration of radicals, k₂ is the rate constant associated with the radical-Mn ion interaction, and r_c, r_d, and r_r represent the rates of radical creation, decay and interaction with S₁ respectively. Aqueous electrons, e⁻_aq, and hydroxyl radicals, OH· are the most prevalent and reactive species created through water radiolysis with a G value (the number of radicals of a particular species created after the deposition of 100eV of energy into water) of approximately 2.6–2.7.^39–42 This implies that for every photon with energy 7090eV, about 190 radicals (e⁻_aq as well as OH·) are created. However, OH· is a highly oxidizing species. As the damage in PSII is manifest through the reduction of the Mn₄Ca cluster, we propose that the effects of the hydroxyl radical are either secondary processes, or with regards to the Mn ions, non-dominant. Aqueous electrons, in contrast, are a highly reducing species, with a high G value similar to that of the OH· radical.^39–42

With a sample thickness of 1mm, spot size of ~105×85µm², and an average flux of 10¹² photons/s, the absorption by the sample is 8.5×10⁴ photons/sµm³. Assuming that, as a first approximation, only e⁻_aq contribute to the damage, we obtain a rate of radical production, r_c= 1.57×10⁷ radicals/s·µm³ or 26mM/s. While this is a reasonable approximation, it averages the effect of attenuation by the sample and does not take into account the uneven distribution of radicals through the sample volume due to the exponential absorption term. Given the short-lived nature of e⁻_aq ’s and consequent limited effect of diffusion, it is more accurate to consider r_c as a function of depth to account for attenuation occurring through the sample. See Materials and Methods for additional details. The rate of radical production (r_c) is independent of the radical concentration and can, thus, be considered zeroth order. However, according to previous studies, the ‘decay’ of e⁻_aq is a first order reaction.⁴³

Considering r_r to be the rate of reaction with PSII, we can now rewrite equation (3) as the following:

$$\frac{\mathrm{d}[\text{rad}(t)]}{\mathrm{d}t}=r_c-k_1[\text{rad}(t)]-k_2[{\mathrm{S}}_1(t)][\text{rad}(t)]$$ (4)

where r_c is determined iteratively through the sample thickness and k₁ is the rate constant of e⁻_aq decay. The solution to this equation must be determined numerically. However, we first combine equations (2) and (4) analytically to yield

$$\frac{{\mathrm{d}}^2\mathrm{R}(\mathrm{t})}{\mathrm{d}{\mathrm{t}}^2}+\frac{\mathrm{d}\mathrm{R}(\mathrm{t})}{\mathrm{d}\mathrm{t}}({\mathrm{k}}_1+\mathrm{A}{\mathrm{k}}_2{\mathrm{e}}^{-{\mathrm{k}}_2\mathrm{R}(\mathrm{t})})-{\mathrm{r}}_{\mathrm{c}}=0$$ (5)

where R(t) = ∫[rad(t)]dt and A = [S₁(0)] = (1.5)*3×10⁵ PSII/µm³ or 0.75mM. Note that the S₁ state has Mn ions in both the III and IV oxidation states. To account for the presence of the Mn^IV ions, requiring two electrons to reduce to Mn^II, we multiply the initial concentration of undamaged Mn by 1.5. This is a simplification that assumes Mn^IV and Mn^III ions have similar reduction kinetics. So far, no bi-exponential kinetics has been reported for the rate of photoreduction for Mn^III versus Mn^IV ions. This simplification will slightly lower the rate constant (k₂) of Mn reduction by substituting the sequential reduction of Mn^IV to Mn^III and then to Mn^II by introducing instead two Mn ions capable of single electron reduction.

As shown in Figures 2 and 3, the [S₁(t)] data is most easily displayed as a percentage of Mn^II accumulated in the sample. To fit the experimental data, we therefore use the equation

$$\%\mathrm{M}{\mathrm{n}}^{II}=100(1-e^{-k_{2}R(t)}).$$ (6)

These fits (Figure 3 and 4, Table 2) yield the weighted mean rate constants k₁ = 4.0 × 10⁷ s⁻¹ and k₂ = 17.1 µm³/(#radicals·s) at a χ² value of ~3.2. Note that this value indicates an overestimation of error (±10%) in visually assigning percent damage. Converting to more common units for k₂, we obtain k₂ = 1.03×10¹⁰ M⁻¹ s₋₁. k₁ = 4.0 × 10⁷ s⁻¹ gives a lifetime for aqueous electrons of τ ~2.5×10⁻⁸ s in the PSII sample. According to Cercek & Cercek,⁴³ the aqueous electron has a lifetime τ ~ 4.5×10⁻⁸ s in protein-enriched environments at room temperature, lending credibility to model values for k₁ and k₂.

Figure 4.

(a) Model prediction for the number of undamaged PSII S₁ state centers per unit volume as a function of time, given monochromatic beam conditions and ignoring the bunch structure (i.e. ‘continuous’ ~1.1 × 10⁸ photons/s·µm²). (b) Assuming a sample thickness of 10µm, a comparison between ‘continuous’ (blue) and ‘pulsed’ – i.e. accounting for bunch structure – (red) models for monochromatic beam in 24-bunch mode, see Table 1 for additional details regarding this mode.

Table 2.

The values of the least squares fit using ode15s are displayed here.

Point	Time (ms)*	Percentage Mn Experiment	Percentage Mn Theory	Weighted Mean Residual
t1	24.7	0	0	0
t2	26.4	5	0.400	−4.59
t3	30.0	3	1.20	−1.79
t4	50.0	5	5.62	0.594
t5	74.1	7	10.7	3.65
t6	79.3	12	11.8	−0.272
t7	90.0	10	13.9	3.86
t8	150	22	25.2	3.09
t9	150	23	25.2	2.10
t10	162	23	27.3	4.20
t11	172	30	29.1	−0.970
t12	185	35	31.1	−3.95
t13	210	32	35.0	3.00
t14	250	46	40.9	−5.03
t15	270	40	43.7	3.70
t16	271	41	43.9	2.93
t17	284	53	45.5	−7.40
t18	330	47	51.2	4.34
t19	350	50	53.5	3.67
t20	396	68	58.43	−9.26

^*Note that the values of time were calculated by taking the dosage accumulated and assuming an incident flux of 10¹² photons/s.

Up to this point in the modeling, we ignored the time structure of the x-ray beam obtained from ~100 ps electron bunches with variable spacing (depending on the fill mode of APS, see Table 1). It happens that the data presented in Fig. 2 were collected in different fill modes, see Table 1. If we compare the lifetime of aqueous electrons (~21 ns) with the spacing between electron bunches we see that this lifetime is longer than the x-ray pulse spacing in 324 bunch mode (11.4 ns), however, the ~153 ns spacing of the 24 bunch model is longer than the lifetime of solvated electrons. Hybrid fill has a mixed pattern with more complicated dependencies, Table 1. Experimentally we were insensitive to these differences, Fig 2. To verify the simplification of substitution of the pulsed x-ray beam by continuous, one we compared the predicted damage using values obtained for the rate constants, k₁ and k₂, assuming an incident flux of 10¹² photons/s and a beam projection ~105×85µm², Figure 4b. Computationally this was done by solving the differential equations for each electron bunch iteratively and compounding the radicals produced. Note that to perform our pulsed beam equivalency tests, we approximate ~ps x-ray bunches as non-zero initial radical concentrations ([rad(t=0)]≠0 instant radical injection) and set a rate of radical production, r_c=0.

Table 1.

Comprehensive description of basic beam parameters including fill patterns, and bunch spacing as well as the corresponding spectrometer type and beamline for all data presented in Figure 2 and referenced in Figure 5.

#	Data	Fill Pattern	Spacing	Spectrometer	Beamline
1		324 singlets (0.31 mA)	11.37 ns (uniform)	Johansson	20-ID
2		24 singlets (4.25mA)	153 ns (uniform)	von Hamos	20-ID
3		Hybrid (single 16 mA isolated bunch, 8 groups of 7 11mA bunches)	(non-uniform) symmetrical 1.594 µs surround 16mA bunch →51 ns between consecutive 11mA bunches	von Hamos	20-ID
4		324 singlets (0.31 mA)	11.37 ns (uniform)	von Hamos	14-ID

Prediction of the progression of x-ray-induced damage for variable rate of dose deposition

Assuming the rate constants (k₁ and k₂) are independent of flux, we can use the model to predict damage progression with different rates of dose deposition. While some previous studies indicate a damage progression independent of dose rate, we propose this is affected by the magnitudes of deposition rates considered. Note the rates we compare range from 126 kGy/s (monochromatic) to 4,500 kGy/s (pink beam, during the pulse), much higher than those considered earlier.¹⁷ We consider three typical rates: monochromatic beam, characteristic of ID beamlines at a 3^rd generation synchrotron source; pink beam, produced at ID beamlines without the use of a monochromator; and an x-ray free electron laser beam, such as the one generated at LCLS. Experimentally, we observed greater than ~6.7×10⁶ photons/µm² to ~1.1×10⁷ photons/µm² (corresponding to 7.6×10³ Gy to 1.3×10⁴ Gy, note that this was calculated using the full sample thickness at an energy of 7.1keV) as the dosage limit for 5% damage from ~1mm of sample (monochromatic beam),⁵ while the model calculations for such a thickness indicate a dosage threshold at ~2.5×10⁶ photons/µm² (corresponding to 2.8×10³ Gy, given the same parameters). We consider these two numbers to be in good agreement. To simplify the modeling for different rates of dose deposition, we compare calculations for a single 10µm slice with 1×1µm² cross-section, Table 3. According to the model, the dosages required for 5% damage during both pink beam (~3×10¹⁵photons/s – during the pulses) and monochromatic beam (~10¹² photons/s) are similar: 5.3 × 10⁵ photons/µm² (corresponding to 6.7×10⁴ Gy assuming an incident energy of 7.85keV and ignoring undulator spread) in pink beam mode, and 5.0 × 10⁵ photons/µm2 (or 5.7×10⁴ Gy) in monochromatic mode – assuming 10µm sample thickness. Thus, in spite of a 3000 times increase in the rate of dose deposition, the effective damage remains overwhelmingly dependent on the total number of deposited photons.

Table 3.

Comparison of predictions for damage progression.

Facility	Rate of dose deposition, photons per sec*	Number of photons to 5% damage	Dosage in Gy to 5% damage*	Time to 5% damage
APS monochromator	1 × 1012	5.0 × 105	5.7 × 104	~4.5ms
APS pink beam	3 × 1015	5.3 × 105	6.7 × 104	~1µs
LCLS	1 × 1025	109	1.3 × 108	~0.55ps

^*Assuming 7.1keV for monochromatic beam and 7.85keV (ignoring undulator spread) for pink and LCLS beams on 10µm thick PSII sample of area 1×1µm² assuming continuous deposition. Note that for beam delivered as pulses, the values for dose deposition rate are those calculated for the dosing time only.

For heat load management, the pink beam is delivered in the form of µs pulses using a rotating high heat load shutter. Figure 5 shows the progression of damage as a function of number of x-ray pulses (~40µs/pulse), each of which deposits ~3×10¹¹ photons. We average the pulses in sets of two in order to increase statistics. This is possible, as we do not observe any obvious damage within the first two pulses. Experimentally, we observed ~1.4×10⁸ photons/µm² (corresponding to 1.8×10⁵ Gy at 7.85keV peak undulator energy, and full sample depth) as the upper dose limit for a ~1mm thick sample. This corresponds to a ~20 fold increase in the dosage threshold between monochromatic (threshold minimally ~6.7×10⁶ photons/µm², or 7.6×10³ Gy) and pink (~1.4×10⁸ photons/µm², or 1.8×10⁵ Gy) beams.

Figure 5.

Effect of the progression of x-ray-induced damage on the Mn Kβ PSII XES spectrum given pulsed (~40us pulse duration) pink beam exposure. Data are shown in sets of two sequential pulses (~3 × 10¹¹ photons/pulse) averaged for improved statistics. The shift of the XES spectrum to higher energy is due to the photoreduction of the OEC containing Mn^III and Mn^IV ions to Mn^II. We do not see a shift within the first 2–3 pulses. Indicating a higher dosage threshold than that of monochromatic beam, experimentally determined to be minimally 6×10¹⁰ photons.

The effect appears to be stronger than the one predicted by model. We propose that the simplicity of our model is responsible for the discrepancies between the pink beam damage threshold experimentally versus the model prediction. Other radicals and/or processes such as those described above likely affect the concentration of solvated electrons responsible for reduction of the OEC. For instance, recombination between the solvated electrons and OH· radicals can become significant at higher rates of dose deposition (recombination rate constant, 3.02 × 10¹⁰ M⁻¹s⁻¹ according to Ryuji et al.⁴⁴). It has been proposed previously that short, high intensity x-ray pulses could be used to acquire data prior to the onset of damage processes such as those initiated by photoelectrons, Auger electrons and radiolysis.

Before concluding, we discuss the application of our model to the vastly higher dose rates achievable at new x-ray FEL sources such as the LCLS. According to our model, the exposure time to reach 5% damage in the Mn oxidation state is ~1ps for a single pulse at LCLS (taken as 10²⁵ photons/s – during the pulses ~100fs duration). This indicates a dose limit of 10¹⁰ photons/µm² for a 10µm sample thickness, Figure 6. The one picosecond damage time is longer than the 100 fs data collection time, implying that even higher number of photons can be used within a single LCLS pulse. However, caution should be exercised in case other physical/chemical processes become significant with such ultra-high rates of dose deposition.^2,4,45

Figure 6.

Considering a 10µm slice of sample and spot size of 10 × 10 µm², we apply our model of pulsed beam structure for a single pulse at LCLS (10¹² photons). Note that, as done with all our pulsed model runs, we assume this dosage is deposited instantaneously at t=0. Inset: Time to 5% damage given 1 pulse is ~1ps, well after the experimental pulse duration of ~100fs.

Timescales for the LCLS approach these required for the generation of primary electrons ~10 fs and time for the generation of solvated electrons about 10⁻¹³ s.⁴⁴ Thus, at such a source, data might be acquired with the least damage, as damaging species are created on the timescale of data collection, and thus do not have time to chemically interact.²⁵ The factor of ~20,000 increase in the deposited dose at LCLS resulting in the same extent of damage is an exciting outcome of the presented model. It suggests that experimentation with higher rates of dose deposition is beneficial for measurements of biological samples sensitive to radiation damage. A few comments must be made here. As pink beam and x-ray free electron laser sources are typically delivered in the form of pulses (composed of sets of ‘bunches’ inherent to the ring), it is beneficial in the experimental design to create a single pulse delivering the threshold dose (the dose just below the onset of x-ray-induced damage). However, the application of high rates of dose deposition might result in the creation of other radicals or initiation of processes, for which, the presented model does not account.