Probabilistic Estimate of |F_oa| from FEL Data

Abstract

The method of the joint probability distribution function was applied in order to estimate the normal structure factor amplitudes of the anomalous scatterer substructure in a FEL experiment. The two-wavelength case was examined. In this, the prior knowledge of the moduli $\left|F_1^{+}\right|,\left|F_1^{-}\right|,\left|F_2^{+}\right|,\left|F_2^{-}\right|$ was used to predict the value of $\left|F_{0a}^{}\right|$, which is the structure factor amplitude arising from the normal scattering of the heavy atom anomalous scatterers. The mathematical treatment provides a solid theoretical basis for the RIP (Radiation-damage Induced Phasing) method, which was originally proposed in order to take the radiation damage induced by synchrotron radiation sources into account. This was further adapted to exploit FEL data, where the crystal damage is usually more massive.

1. Introduction

The properties of large tunability, high monochromaticity, small beam divergence and large power of the modern synchrotron beamlines have made MIR, MAD and MIRAS popular tools for determining the crystal structure solution of proteins. Such phasing procedures are usually organized into two steps. In the first step, the amplitudes of the structure factors corresponding to the anomalous scatterers or to the heavy atom substructure are estimated. After this, they are used to solve the substructure via the Direct or Patterson methods [1,2,3,4,5,6]. In the second step, the full protein structure is identified using the prior information of the substructure [7,8,9,10,11,12,13].

Despite the increased efficiency of the modern phasing methods, a bottleneck is still present, which is related to the capacity of the method in growing sufficiently large and well-ordered crystals for a large class of biologically important proteins (e.g., membrane proteins). We can use X-ray free-electron lasers (FEL) to overcome the difficulty. Furthermore, due to their unprecedented high X-ray fluence [14,15], the diffraction effects may be measured even in nanocrystals [15,16,17,18,19].

However, the extreme high fluence [20] causes severe radiation damage [21] although diffraction before destruction is allowed as an effect of the ultrashort X-ray pulses. During femtosecond pulses, an ionization process [15] occurs, with this effect usually described as diffraction during ionization [22].

Therefore, radiation damage is of major concern if FEL data are collected for phasing purposes. According to Blake and Phillips [23], the induced damage quickly reduces the diffractive power of the cryocooled crystals, increases the vibrational atomic parameters and consequently, increases the cell volume. Local variations of the electron density are also generated. Heavy-atom substructures seem to be particularly sensitive to damage and as a consequence, dispersive signals can be obscured in SAD-MAD experiments.

More recently, the radiation damage has been used to phase macromolecular structures [24,25]. This method is mainly focused on synchrotron radiation sources and has been named RIP (Radiation-damage Induced Phasing). The data collected before the damage and data collected from a damaged crystal are used as the isomorphous data.

The RIP approach has also been applied to FEL data, where the damage is usually more massive and of a different nature. Indeed, the damage generated by a synchrotron source is mainly constituted of the breaking of bonds and implies atomic movements. Due to the very short pulse-length, the FEL damage takes the ionization of the atoms into account and does not imply atomic movement.

A detailed description of the physical phenomena occurring during the ultrashort X-ray pulses has been described in recent papers [22,26,27,28,29,30]. These papers studied the evolution of different electron populations for some selected state charges during an X-ray pulse and demonstrated how the neutrality of heavy atoms may be depleted into high-charge states. The practical physical effects may be condensed in a short statement. In a FEL experiment, nanocrystals may be used to provide different sets of diffraction data, with one corresponding to the undamaged crystal (obtained via low fluence pulses) and one or more with high fluence pulses.

The interaction X-ray pulse-crystal affects both the coherent atomic scattering factor and the dispersion effects. Son et al. [22] suggested that a basic assumption may be made in order to ensure that the FEL anomalous dispersion experiment is highly informative. This assumption is that the contribution of the light atoms to the anomalous scattering may be neglected when the photon energy is near to the inner-shell ionization threshold of the heavy atoms. They also showed that the K-shell edge is shifted from the standard position by a certain quantity, which depends on the specific charge state of the atom (accordingly, different configurations of the same heavy atom species may be simultaneously present in the crystal). Furthermore, $\Delta f_j$ and $f_j^{″}$ amplitudes also change according to the configuration selected for the different charge states.

In this paper, we will consider a simplified model for both the damaged and undamaged crystals, in which diffraction data with anomalous scattering effects are available. In the state 1, where radiation damage is assumed to be absent, the jth scattering factor may be represented by:

$${[f_j^{}]}_1=f_j^0+{[\Delta f_j^{\prime }]}_1+i{[f_j^{″}]}_1={[f_j^{\prime }]}_1+i{[f_j^{″}]}_1$$

In the state 2, where radiation damage occurs, we obtain:

$${[f_j^{}]}_2=f_j^0+{[\Delta f_j^0]}_2+{[\Delta f_j^{\prime }]}_2+i{[f_j^{″}]}_2=f_j^0+{[\Delta f_j]}_2+i{[f_j^{″}]}_2$$

In both cases, the scattering factor may be represented (see the definition at Section 1) by:

$$f_j^{}=f_j^0+\Delta f_j+i{f}_j^{″},$$

where $\Delta f_j$ includes the variations of both the normal scattering factor and the real component of the anomalous scatterer. Thus, we can conclude that:

• for the state 1 $\Delta f_j={[\Delta f_j^{\prime }]}_1$ and $f_j^{″}={[f_j^{″}]}_1$;
• for the state 2 $\Delta f_j={[\Delta f_j^0]}_2+{[\Delta f_j^{\prime }]}_2$ and $f_j^{″}={[f_j^{″}]}_2$.

The above two states are usually generated in a FEL experiment by two different fluences, although the same wavelength is used. This situation has no counterpart in a SAD experiment and is closer to a two-wavelength MAD experiment, where the two different states correspond to two different pairs $(\Delta f_{}^{\prime },f_{}^{″})$. The substantial difference between FEL and the two-wavelength MAD case is not due to the use of a different number of wavelengths (the final MAD formulas do not depend on the number of wavelengths but on the states corresponding to them), but this is attributed to the fact that the crystal damage at high fluence involves the use of $f_j^0+{[\Delta f_j^0]}_2$ rather than $f_j^0$ in a FEL experiment.

The continuous ionization of the heavy atoms during FEL experiments makes it difficult to directly apply MAD techniques to FEL data [22]. Indeed, the present theory cannot take into account the large number of electronic states accessible to heavy atoms during a high-fluence FEL experiment and the dynamics of these states during measurements. Our model is static. In other words, we assume that the structure factors do not change during measurements taken from the damaged crystal. The model would be certainly more representative of the dynamic nature of the damage if it might include a time-dependent term, but this is outside of our present capacity.

However, the fact that many protein structures have been solved using FEL data obtained by Molecular Replacement techniques suggests that data collected from an undamaged crystal (low fluence) and data collected from a damaged crystal (high fluence) have sufficient internal consistency to be useful in attempts to determine the protein crystal structure. This was the main reason that encouraged us to describe a probabilistic theory integrating FEL damage into the framework of the anomalous dispersion techniques [12].

This work focuses on the first step of a phasing procedure, which involves finding the heavy atom positions according to the diffraction data.

2. The Mathematical Model

A necessary condition for finding the heavy atom substructure from the diffraction amplitudes of undamaged and damaged crystals involves having sufficiently accurate estimates of the $R_{0H}$ amplitudes. Once available, the Direct or Patterson methods can use these estimates to locate the heavy atom positions. The most general approach for estimating $R_{0H}$ involves the calculation of the joint probability distribution function:

$$P\left(E_{0H},E_{}^{+},E_d^{+},E_{}^{-},E_d^{-}\right).$$

(1)

From this (see below), we can derive the conditional estimate as follows:

$$<R_{0H}|R_{}^{+},R_d^{+},R_{}^{-},R_d^{-}>.$$

where the superscripts + and − indicate that the structure factors are calculated for the reflections h and −h, respectively.

The use of Equation (1) requires the preliminary modelling of the structure factors. Their algebraic form should reflect the physical phenomena occurring in a FEL experiment. According to Section 1, the isomorphous derivatives created in such experiments are similar, although they are not identical to those available in traditional MAD or MIRAS cases. Thus, it may be worthwhile to compare these derivatives in order to discover similarity and differences in addition to profiting from the mathematical theories already developed for treating the usual MAD and MIRAS cases [5,13,31].

Our FEL model involves the following assumptions:

(1)

For light atoms, $\Delta f_j$ is assumed to be negligible both for the damaged and for the undamaged crystals. In our model, we set $\Delta f_j=0$ for j = 1, …, L. Furthermore, $f_j^0$ is assumed to be the same for the jth atom independently of whether this refers to the undamaged or damaged crystal.

(2)

For heavy atoms, the scattering factor of the jth atom $f_j^0$ is described by two different functions according to whether the atom is considered in the undamaged or in the damaged crystal. As a result, $\Delta f_j$ will assume different values.

(3)

The atomic positions in the damaged and in the undamaged crystals coincide [26,32]. Indeed, there is no evidence so far that the pulse duration (about a few femtoseconds) produces detectable changes in the heavy atom positions in femtosecond X-ray nanocrystallography even if the simulations of radiation dynamics in proteins suggests some correlated movement of the heavy atoms [33].

(4)

The observed amplitudes $F_{}^{+},F_d^{+},F_{}^{-},F_d^{-}$ are affected by errors, which are calculated as:

$${\mu }^{+}=\left|{\mu }^{+}\right|exp\left(i{\vartheta }^{+}\right) \mathrm{and} {\mu }^{-}=\left|{\mu }^{-}\right|exp\left(i{\vartheta }^{-}\right),$$

$${\mu }_d^{+}=\left|{\mu }_d^{+}\right|exp\left(i{\vartheta }_d^{+}\right) \mathrm{and} {\mu }_d^{-}=\left|{\mu }_d^{-}\right|exp\left(i{\vartheta }_d^{-}\right)$$

These represent the measurement errors relative to the undamaged and the damaged crystal, respectively. The errors are assumed to be complex quantities because they influence both the real and the imaginary components of the structure factors. Since $R_{0H}$ is not measured and is instead estimated using the probabilistic approach, no error will be associated with it. Furthermore, the errors will be assumed to be uncorrelated and thus, we can obtain:

$$<{\mu }_{}^{+}>=<{\mu }_d^{+}>=<{\mu }_{}^{-}>=<{\mu }_d^{-}>=0$$

and

$$<{\mu }_{}^{+}{\mu }_d^{+}>=<{\mu }_{}^{+}{\mu }_{}^{-}>=<{\mu }_{}^{-}{\mu }_d^{-}>=<{\mu }_d^{+}{\mu }_d^{-}>=0$$

According to Equations (1)–(4), the following mathematical model will be adopted:

$$A_{}^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_j^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_j^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_{}^{+}\right|cos{\vartheta }_{}^{+}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2},$$

$$B_{}^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^o\sin \left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_j^{\prime }\sin \left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+}{\displaystyle \sum _{j=1}^H{f}_j^{″}\cos \left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_{}^{+}\right|\sin {\vartheta }_{}^{+}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2},$$

$$A_{}^{-}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_j^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+}{\displaystyle \sum _{j=1}^H{f}_j^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_{}^{-}\right|cos{\vartheta }_{}^{-}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2}$$

$$B_{}^{-}=\left\{-{\displaystyle \sum _{j=1}^L{f}_j^{o}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}-{\displaystyle \sum _{j=1}^H{f}_j^{\prime }sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+}{\displaystyle \sum _{j=1}^H{f}_j^{″}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_{}^{-}\right|sin{\vartheta }_{}^{-}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2},$$

$$A_d^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{+}\right|cos{\vartheta }_d^{+}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2},$$

$$B_d^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^o\sin \left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }\sin \left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}\cos \left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{+}\right|\sin {\vartheta }_d^{+}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2},$$

$$A_d^{-}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{-}\right|cos{\vartheta }_d^{-}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2},$$

$$B_d^{-}=\left\{-{\displaystyle \sum _{j=1}^L{f}_j^{o}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}-{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{-}\right|sin{\vartheta }_d^{-}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2}.$$

It is important to notice that for heavy atoms:

$$f_j^{\prime }=f_j^0+\Delta f_j^{\prime },$$

(2a)

$$f_{dj}^{\prime }=f_{dj}^0+\Delta f_{dj}^{\prime },$$

(2b)

where

$$\Delta f_{dj}^{\prime }=\Delta f_{0j}^{}+\Delta f_j^{\prime }.$$

(2c)

${[\Delta f_j^{\prime }]}_1$ and ${[\Delta f_j^{\prime }]}_2$ are the real components of the anomalous scattering of the jth heavy atom for the undamaged and damaged crystals, respectively.

We then considered the mathematical model adopted for the usual two-wavelength MAD experiment by Giacovazzo and Siliqi [12]. In order to facilitate an easier comparison with the above FEL model, we slightly changed the original Giacovazzo and Siliqi notation. We denoted $E^{+}\equiv (A_{}^{+}+i{B}^{+}), E^{-}\equiv (A_{}^{-}+i{B}^{-})$ as the normalized structure factors corresponding to the first wavelength. $E_d^{+}\equiv (A_d^{+}+i{B}_d^{+}),$ $E_d^{-}\equiv (A_d^{-}+i{B}_d^{-})$ is denoted as the normalized structure factor corresponding to the second MAD wavelength, while $E_{0H}^{}$ is denoted as the structure factor corresponding to the anomalous scatterer substructure (anomalous scattering is not included). L and H is the number of non-hydrogen light atoms and the number of the anomalous scatterers in the unit cell, respectively. Thus, the Giacovazzo and Siliqi model may be rewritten as follows:

$$A^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_j^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_j^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }^{+}\right|cos{\vartheta }^{+}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2},$$

$$B^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_j^{\prime }sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_j^{″}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }^{+}\right|sin{\vartheta }^{+}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2},$$

$$A^{-}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_j^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_j^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }^{-}\right|cos{\vartheta }^{-}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2},$$

$$B^{-}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_j^{\prime }sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_j^{″}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }^{-}\right|sin{\vartheta }^{-}}\right\}/{\left(\epsilon {\sum }_N\right)}^{1/2},$$

$$A_d^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{+}\right|cos{\vartheta }_d^{+}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2},$$

$$B_d^{+}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{+}\right|sin{\vartheta }_d^{+}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2},$$

$$A_d^{-}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{-}\right|cos{\vartheta }_d^{-}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2},$$

$$B_d^{-}=\left\{{\displaystyle \sum _{j=1}^L{f}_j^{o}sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)}+{\displaystyle \sum _{j=1}^H{f}_{dj}^{\prime }sin\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)-}{\displaystyle \sum _{j=1}^H{f}_{dj}^{″}cos\left(2\pi \mathbf{h}{\mathbf{r}}_j\right)+\left|{\mu }_d^{-}\right|sin{\vartheta }_d^{-}}\right\}/{\left(\epsilon {\sum }_{Nd}\right)}^{1/2}$$

where

$$f_j^{\prime }=f_j^0+\Delta f_j^{\prime }$$

(3a)

and

$$f_{dj}^{\prime }=f_j^0+\Delta f_{dj}^{\prime }.$$

(3b)

Now $f_j^0$ does not vary when considered for the first or for the second wavelength independently of whether Equation (3) refers to heavy or to light atoms.

Comparing Equation (3) with Equation (2) reveals the main difference between the FEL and the usual two-wavelength MAD mathematical model. The FEL data imply a supplementary decay of the scattering factor of the heavy atoms, which is generated by the crystal damage when passing from the first to the second wavelength. However, there is significantly similarity between the two models so thus, most of the mathematics developed for MAD by Giacovazzo and Siliqi [12] may also be used for FEL with small modifications, which is shown in Section 3.

3. The Joint Probability Distribution $\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{0}\mathit{H}}\mathbf{,}{\mathbf{E}}_{}^{\mathbf{+}}\mathbf{,}{\mathbf{E}}_{\mathit{d}}^{\mathbf{+}}\mathbf{,}{\mathbf{E}}_{}^{\mathbf{-}}\mathbf{,}{\mathbf{E}}_{\mathit{d}}^{\mathbf{-}}\mathbf{\right)}$

The characteristic function of

$$P\left(A_{0H},A_{}^{+},A_d^{+},A_{}^{-},A_d^{-},B_{oH},B_{}^{+},B_d^{+},B_{}^{-},B_d^{-}\right)$$

may be written down as:

$$\begin{array}{l}C\left(u_{0H},u_{}^{+},u_d^{+},u_{}^{-},u_d^{-},v_{0H},v_{}^{+},v_d^{+},v_{}^{-},v_d^{-}\right)\approx \\ exp\{-\frac{1}{4}[k_{11}\left(u_{0H}^2+v_{0H}^2\right)+k_{22}\left(u_{}^{+2}+v_{}^{+2}\right)+k_{33}\left(u_d^{+2}+v_d^{+2}\right)+k_{44}\left(u_{}^{-2}+v_{}^{-2}\right)\\ +k_{55}\left(u_d^{-2}+v_d^{-2}\right)+2k_{24}\left(u_{}^{+}{u}_{}^{-}-v_{}^{+}{v}_{}^{-}\right)+2k_{35}\left(u_d^{+}{u}_d^{-}-v_d^{+}{v}_d^{-}\right)\\ +2k_{29}\left(u_{}^{+}{v}_{}^{-}+v_{}^{+}{u}_{}^{-}\right)+2k_{3,10}\left(u_d^{+}{v}_d^{-}{+v}_d^{+}{u}_d^{-}\right)+2k_{34}\left(u_d^{+}{u}_{}^{-}-v_d^{+}{v}_{}^{-}\right)\\ +2k_{23}\left(u_{}^{+}{u}_d^{+}+v_{}^{+}{v}_d^{+}\right)+2k_{25}\left(u_{}^{+}{u}_d^{-}-v_{}^{+}{v}_d^{-}\right)+2k_{2,10}\left(u_{}^{+}{v}_d^{-}+v_{}^{+}{u}_d^{-}\right)\\ +2k_{28}\left(u_{}^{+}{v}_d^{+}-v_{}^{+}{u}_d^{+}\right)+2k_{12}\left(u_{0H}^{}{u}_{}^{+}+v_{0H}^{}{v}_{}^{+}\right)+2k_{14}\left(u_{0H}^{}{u}_{}^{-}-v_{0H}^{}{v}_{}^{-}\right)\\ +2k_{17}\left(u_{0H}^{}{v}_{}^{+}-v_{0H}^{}{u}_{}^{+}\right)+2k_{19}\left(u_{0H}^{}{v}_{}^{-}+v_{0H}^{}{u}_{}^{-}\right)+2k_{13}\left(u_{0H}^{}{u}_d^{+}+v_{0H}^{}{v}_d^{+}\right)\\ +2k_{15}\left(u_{0H}^{}{u}_d^{-}-v_{0H}^{}{v}_d^{-}\right)+2k_{18}\left(u_{0H}^{}{v}_d^{+}-v_{0H}^{}{u}_d^{+}\right)+2k_{1,10}\left(u_{0H}^{}{v}_d^{-}+v_{0H}^{}{u}_d^{-}\right)\end{array}$$

(4)

where $u_{0H},u_{}^{+},u_d^{+},u_{}^{-},u_d^{-},v_{0H},v_{}^{+},v_d^{+},v_{}^{-},v_d^{-}$ are the carrying variables associated with

$A_{0H},A_{}^{+},A_d^{+},A_{}^{-},A_d^{-},B_{oH},B_{}^{+},B_d^{+},B_{}^{-},B_d^{-}$, respectively.

Equation (4) can also be written as:

$$C=exp\left\{-\frac{1}{4}\left(\overline{\mathbf{U}}\mathbf{K}\mathbf{U}\right)\right\}$$

(5)

where

$$\overline{U}=\left[u_{0H},u_{}^{+},u_d^{+},u_{}^{-},u_d^{-},v_{0H}{,v}_{}^{+},v_d^{+},v_{}^{-},v_d^{-}\right]$$

and K is the symmetric square matrix. The elements of K are specified as follows:

$$k_{11}=k_{66}=1$$

$$k_{22}=k_{77}=e_{}^{+}=1+{\sigma }_{}^{+2} \mathrm{with} {\sigma }_{}^{+2}=<|{\mu }_{}^{+}{|}^2>/\left(\epsilon {\sum }_N\right)$$

$$k_{33}=k_{88}=e_d^{+}=1+{\sigma }_d^{+2} \mathrm{with} {\sigma }_d^{+2}=<|{\mu }_d^{+}{|}^2>/\left(\epsilon {\sum }_{Nd}\right)$$

$$k_{44}=k_{99}=e_{}^{-}=1+{\sigma }_{}^{-2} \mathrm{with} {\sigma }_{}^{-2}=<|{\mu }_{}^{-}{|}^2>/\left(\epsilon {\sum }_N\right)$$

$$k_{55}=k_{10,10}=e_d^{-}=1+{\sigma }_d^{-2} \mathrm{with} {\sigma }_d^{-2}=<|{\mu }_d^{-}{|}^2>/\left(\epsilon {\sum }_{Nd}\right)$$

$$k_{12}=k_{14}=k_{67}=-k_{69}=S_9/{\left({\sum }_H^0{\sum }_N\right)}^{1/2}$$

$$k_{13}=k_{15}=k_{68}=-k_{6,10}=S_{11}/{\left({\sum }_H^0{\sum }_{Nd}\right)}^{1/2}$$

$$k_{16}=k_{27}=k_{38}=k_{49}=k_{5,10}=0$$

$$k_{17}=k_{19}=k_{46}=-k_{26}=S_{10}/{\left({\sum }_H^0{\sum }_N\right)}^{1/2}$$

$$k_{18}=k_{1,10}=-k_{36}=k_{56}=S_{12}/{\left({\sum }_H^0{\sum }_{Nd}\right)}^{1/2}$$

$$k_{23}=k_{45}=k_{9,10}=k_{78}={\left({\sum }_L^0+S_5+S_6\right)/\left({\sum }_N{\sum }_{Nd}\right)}^{1/2}$$

$$k_{24}=-k_{79}=\left({\sum }_L^0+S_1\right)/{\sum }_N$$

$$k_{25}=k_{34}=-k_{7,10}=-k_{89}={\left({\sum }_L^0+S_5-S_6\right)/\left({\sum }_N{\sum }_{Nd}\right)}^{1/2}$$

$$k_{28}=-k_{37}=+k_{4,10}=-k_{59}={\left(S_7-S_8\right)/\left({\sum }_N{\sum }_{Nd}\right)}^{1/2}$$

$$k_{29}=k_{47}=S_3/{\sum }_N$$

$$k_{2,10}=k_{39}=k_{48}=k_{57}=\left(S_7+S_8\right)/{\left({\sum }_N{\sum }_{Nd}\right)}^{1/2}$$

$$k_{35}=-k_{8,10}=\left({\sum }_L^0+S_2\right)/{\sum }_{Nd}$$

$$k_{3,10}=k_{58}=S_4/{\sum }_{Nd},$$

where

$$S_1=\sum _H\left(f_j^{\prime 2}-f_j^{″2}\right), S_2=\sum _H\left(f_{dj}^{\prime 2}-f_{dj}^{″2}\right), S_3=2\sum _H{f}_j^{\prime }{f}_j^{″}$$

$$S_4=2\sum _H{f}_{dj}^{\prime }{f}_{dj}^{″}, S_5=\sum _H{f}_j^{\prime }{f}_{dj}^{\prime }, S_6=\sum _H{f}_j^{″}{f}_{dj}^{″}$$

$$S_7=\sum _H{f}_j^{\prime }{f}_{dj}^{″}, S_8=\sum _H{f}_j^{″}{f}_{dj}^{\prime }, S_9=\sum _H{f}_j^o{f}_j^{\prime }$$

$$S_{10}=\sum _H{f}_j^o{f}_j^{″}, S_{11}=\sum _H{f}_j^o{f}_{dj}^{\prime }, S_{12}=\sum _H{f}_j^o{f}_{dj}^{″}$$

The formal expressions of the $k$ and S parameters coincide with those obtained by Giacovazzo and Siliqi for the usual two-wavelength MAD experiments. However, the $f_{dj}^{\prime }$, according to Equation (2b), implies a change in the normal jth heavy atoms scattering factor, which will influence the values of all the S parameters that involves $f_{dj}^{\prime }$.

In our mathematical treatment, $\Delta f_{0j}^{}$ is implicitly assumed to be negative (it describes the passage from a neutral to an ionic state). As $\Delta f_j^{\prime }$ is also usually negative, the value of $\Delta f_{dj}^{\prime }=\Delta f_{0j}^{}+\Delta f_j^{\prime }$ is expected to be largely negative, particularly for high ionization states. This effect may strongly reduce the $f_{dj}^{\prime }$ values. As a result, the S parameters $f_{dj}^{\prime }$ are expected to be smaller for FEL data than for standard MAD data.

In accordance with the above considerations, the joint probability distribution function $P\left(A_{oH},A_{}^{+},A_d^{+},A_{}^{-},A_d^{-},B_{oH},B_{}^{+},B_d^{+},B_{}^{-},B_d^{-}\right)$ may be obtained by the Fourier inversion of the characteristic function of Equation (5). Through the change of variables, we can obtain:

$$A_{0H}=R_{0H}cos{\phi }_{0H}, B_{0H}=R_{0H}sin{\overleftrightarrow{\phi }}_{0H}$$

$$A_{}^{+}=R^{+}cos{\phi }_{}^{+}, B_{}^{+}=R^{+}sin{\phi }_{}^{+}$$

$$A_{}^{-}=R^{-}cos{\phi }_{}^{-}, B_{}^{-}=R^{-}sin{\phi }_{}^{-}$$

$$A_d^{+}=R_d^{+}cos{\phi }_d^{+}, B_d^{+}=R_d^{+}sin{\phi }_d^{+}$$

$$A_d^{-}=R_d^{-}cos{\phi }_d^{-}, B_d^{-}=R_d^{-}sin{\phi }_d^{-}.$$

The following joint probability distribution function is obtained:

$$\begin{array}{l}P\left(R_{0H},R^{+},R^{-},R_d^{+},R_d^{-},{\phi }_a,{\phi }_{}^{+},\dots ,{\phi }_d^{-}\right)\approx {\pi }^{-5}{R}_{0H}{R}^{+}{R}^{-}{R}_d^{+}{R}_d^{-}{\left[\det \mathbf{K}\right]}^{-1/2}\\ exp\{-[{\lambda }_{11}{(R_{0H}^{})}^2+{\lambda }_{22}{(R_{}^{+})}^2+{\lambda }_{33}{(R_d^{+})}^2+{\lambda }_{44}{(R^{-})}^2+{\lambda }_{55}{(R_d^{-})}^2\\ +2{\lambda }_{23}{R}^{+}{R}_d^{+}cos\left({\phi }_{}^{+}-{\phi }_d^{+}\right)-2{\lambda }_{28}{R}^{+}{R}_d^{+}\sin \left({\phi }_{}^{+}-{\phi }_d^{+}\right)\\ +2{\lambda }_{24}{R}^{+}{R}^{-}cos\left({\phi }_{}^{+}+{\phi }_{}^{-}\right)+2{\lambda }_{29}{R}^{+}{R}^{-}\sin \left({\phi }_{}^{+}+{\phi }_{}^{-}\right)\\ +2{\lambda }_{25}{R}^{+}{R}_d^{-}cos\left({\phi }_{}^{+}+{\phi }_d^{-}\right)+2{\lambda }_{2,10}{R}^{+}{R}_d^{-}\sin \left({\phi }_{}^{+}+{\phi }_d^{-}\right)\\ +2{\lambda }_{34}{R}^{-}{R}_d^{+}cos\left({\phi }_{}^{-}+{\phi }_d^{+}\right)+2{\lambda }_{39}{R}^{-}{R}_d^{+}\sin \left({\phi }_{}^{-}+{\phi }_d^{+}\right)\\ +2{\lambda }_{35}{R}_d^{+}{R}_d^{-}cos\left({\phi }_d^{+}+{\phi }_d^{-}\right)+2{\lambda }_{3,10}{R}_d^{+}{R}_d^{-}\sin \left({\phi }_d^{+}+{\phi }_d^{-}\right)\\ +2{\lambda }_{45}{R}^{-}{R}_d^{-}cos\left({\phi }_{}^{-}-{\phi }_d^{-}\right)-2{\lambda }_{4,10}{R}^{-}{R}_d^{-}\sin \left({\phi }_{}^{-}-{\phi }_d^{-}\right)\\ +2{\lambda }_{12}{R}_{0H}{R}^{+}cos\left({\phi }_{0H}^{}-{\phi }_{}^{+}\right)-2{\lambda }_{17}{R}_{0H}{R}^{+}\sin \left({\phi }_{0H}^{}-{\phi }_{}^{+}\right)\\ +2{\lambda }_{13}{R}_{0H}{R}_d^{+}cos\left({\phi }_{0H}^{}-{\phi }_d^{+}\right)-2{\lambda }_{18}{R}_{0H}{R}_d^{+}\sin \left({\phi }_{0H}^{}-{\phi }_d^{+}\right)\\ +2{\lambda }_{14}{R}_{0H}{R}^{-}cos\left({\phi }_{0H}^{}+{\phi }_{}^{-}\right)-2{\lambda }_{19}{R}_{0H}{R}^{-}\sin \left({\phi }_{0H}^{}+{\phi }_{}^{-}\right)\\ +2{\lambda }_{15}{R}_{0H}{R}_d^{-}cos\left({\phi }_{0H}^{}+{\phi }_d^{-}\right)+2{\lambda }_{1,10}{R}_{0H}{R}_d^{-}\sin \left({\phi }_{0H}^{}+{\phi }_d^{-}\right)]\},\end{array}$$

(6)

where ${\lambda }_{pq}$ are the elements of the matrix K⁻¹.

The distribution found in Equation (6) is the main result of this paper.

In accordance with Giacovazzo and Siliqi and based on Equation (6), the conditional probability may be obtained:

$$P\left(R_{0H}|R_{}^{+},R_{}^{-},R_d^{+},R_d^{-}\right)$$

(7)

From this, the conditional mean value is:

$$<\left(R_{0H}|R_{}^{+},R_{}^{-},R_d^{+},R_d^{-}\right)>\approx \frac{1}{2}{\left(\pi /{\lambda }_{11}\right)}^{1/2}{\left(1+\frac{4}{\pi }\frac{X^2}{{\lambda }_{11}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$

(8)

where:

X² = Q₁²+ Q₂²

$$Q_1={\lambda }_{12}{R}_1+{\lambda }_{13}{R}_2+{\lambda }_{14}{G}_1+{\lambda }_{15}{G}_2$$

$$Q_2={\lambda }_{17}{R}_1+{\lambda }_{18}{R}_2-{\lambda }_{19}{G}_1-{\lambda }_{1,10}{G}_2.$$

The details of the mathematical derivation may be found in the referenced paper. In this paper, the standard deviation associated with the estimate of Equation (8) was calculated to be:

$${\sigma }_{R_{0a}}={\left[<R_{oa}^2|..>-<R_{oa}^{}|..{>}^2\right]}^{1/2}={\left[\left(1-\frac{\pi }{4}\right){\lambda }_{11}^{-1}\right]}^{1/2}$$

so that

$$\frac{<R_{oa}^{}|..>}{{\sigma }_{R_{oa}^{}}}={\left[\frac{\pi /4+X^2/{\lambda }_{11}}{1-\pi /4}\right]}^{1/2}.$$

(9)

Since λ₁₁ is always expected to be positive, the expected values of $<R_{oa}^{}>$ and ${\sigma }_{R_{oa}^{}}$ are always positive. The reflections with the largest values of the ratio found in Equation (9) are likely to be the most useful ones.

4. Conclusions

We have described a probabilistic approach for modelling the crystal structure factors when the crystal is damaged due to the strong impact of FEL radiation on the crystal sample. This involves using corresponding data to solve the crystal structure. The calculations show that the mathematical formulation described by Giacovazzo and Siliqi [12] may also be applied to FEL data, provided that suitable modifications are introduced in the parameters depending on the scattering factors.