Determination of the number of water molecules in the proton pathway of bacteriorhodopsin using neutron diffraction data

Abstract

It has been shown that water molecules participate in the proton pathway of bacteriorhodopsin. Large efforts have been made to determine with various biophysical methods the number of water molecules involved. Neutron diffraction H₂O/D₂O exchange experiments have been often used to reveal the position of water even with low-resolution diffraction data. With this technique, care must be taken with the limitations of the difference Fourier method which are commonly applied to analyze the data. In this paper we compare the results of the difference Fourier method applied to measured diffraction data (not presented here) and models with those from alternative methods introduced here: (1) a computer model calculation procedure to determine a label's scattering length density based on a comparison of intensity differences derived from models and intensity differences from our measurements; (2) a method based on the Parseval formula. Both alternative methods have been evaluated and tested using results of neutron diffraction experiments on purple membranes (Hauss et al. 1994). Our findings indicate that the difference Fourier method applied to low-resolution diffraction data can successfully determine the position of localized water molecules but underestimates their integrated scattering length density in the presence of labels in other positions. Furthermore, we present the results of neutron diffraction experiments on purple membranes performed to determine the number of water molecules in the projected area of the Schiff base at 86%, 75% and 57% relative humidity (r.h.). We found 19±2 exchangeable protons at 75% r.h., which means at least 8–9 water molecules are indispensable for normal pump function.

Appendix

Appendix 1: the simulation method

The procedure to determine the position and scattering length density of a LL was evaluated using the known structure of BR (Edholm et al. 1995). The neutron scattering length density of a unit cell was projected onto the plane of the membrane to derive a model of the natural dry membrane (NatMod). At the in-plane position of the Schiff base the scattering length of 10 H₂O or 10 D₂O molecules was added on a circular area of ~9 Å² per monomer. The additional scattering length simulated the presence of a file of 10 H₂O or 10 D₂O molecules perpendicular to the plane of PM. The structures created in this way simulated experimental samples equilibrated in a D₂O or H₂O atmosphere, respectively. Fourier transformation of these structures provided intensity pairs and their differences. These differences were then compared with intensity differences derived from trial structures, which were produced as described below. As similarity parameters between the sets of the intensity differences, we used the correlation coefficient, the R factor and χ². Systematic trials with up to three labels per monomer, equivalent to 10, 6 and 3 water molecules, at the same time gave the following results concerning the method.

The CC is a very effective criterion to determine the position of the label but less effective for the determination of its strength. The R and χ² are less sensitive in position and more sensitive in strength. The R factor gave more reliable results than χ² and therefore in the following only CC and R are used as similarity parameters. If a label has strength comparable to the Fourier termination error, it is impossible to determine its position working in real space. For a resolution of 7 Å (S=0.1429) the termination error is about 10% of the strength of the strongest label. The localization of the two larger labels was sufficiently good. It was also possible to determine their strength. Unfortunately, this was not possible in the case of the weakest label.

The procedure for the case of both a LL and an EL together was evaluated using as a model for an EL a H₂O or D₂O monolayer defined by an area covering the lipid area of the PM elementary cell. This area was derived from density maps representing H/D exchange at high r.h. The LL was equivalent to 10 water molecules at (x=18.4, y=0.5) with a radius of 1.0 Å. All coordinates in this article are given in Å. These labels (both D₂O and H₂O) were added to NatMod to simulate hydrated PM at high r.h. The Fourier transform of the derived structures provided a set of pairs of intensities for PM-D₂O and PM-H₂O, respectively. Intensity differences were calculated without scaling of the data to each other. For the sake of the evaluation of the simulation method, these intensity differences represent the measured differences. During the simulation procedure we have tried to match them and the intensity differences from trial modifications of NatMod.

During trials we realized that it was more convenient to start the calculations with the determination of the EL and then proceed to determine the LL. The region of the EL was varied stepwise and for every step also the scattering length that we add on NatMod was varied. This was done, first in a coarse way by varying the level that determines the area in steps of 5% and then repeating in steps of 1% around the best area found. For every level the scattering length was varied until no further improvement of the similarity parameters was obtained.

After the determination of the EL followed the determination of the LL. This was done by adding the EL on NatMod and started the calculations as for LL. We repeated then the determination of the EL starting with NatMod modified by the LL previously found. The cyclic procedure was terminated when no further improvement of the comparison parameters was obtained.

In the case of the model, our simulation method revealed perfectly the location and the strength of the added labels. Also in the case of scaled intensities the results accurately represented the model, even if the converging was slower.

After successful evaluation of the method, we applied it to data from neutron diffraction experiments.

Coarse determination of the position of a label

The steps used were:

• 1. To determine a first estimate of the label position, a test label scanned the asymmetric part (one third) of the elementary cell in steps of 4 Å in both directions.

• 2. For every label position the CC and R-factor were calculated between experimental (ΔI _e) and calculated intensity (ΔI _m) differences. A candidate for the coarse label position is the one with the highest CC.

These steps were repeated with labels of different radius and strength. The position corresponding to the best CC and R was chosen.

Fine determination of a label position

The above procedure was repeated by scanning the region around the position that had been defined by the coarse determination. The step width was 0.5 Å. The strength and radius of the test label was that defined at the end of the above section. The final position of the label was the one with the highest CC.

Determination of a label's strength and size

The first procedure was repeated, holding the test label at the position that had been defined by the second procedure and varying its strength and size. The best strength/radius combination was the one corresponding to the minimal R.

The above three steps defined accurately the position, the strength and the size of a localized area label. In the case where more than one LL was present, the described cyclic procedure determined the strongest of them. In order to determine a possible second label, we started the first procedure with the NatMod, modified by adding the first label. At the end of the above procedure the second label was determined.

To refine the strength and the position of the first label, the first procedure had to be started again with the NatMod modified by adding the previously determined second label. After fixing the parameters for the first label, the second one could be refined.

The cyclic procedure had to be terminated if the similarity parameters did not improve any more. Trials with scaled intensities showed no effect on the final result, but the procedure converged more slowly.

We tested the above simulation method by applying it to the scaled neutron diffraction data of BR, with specifically deuterated retinals (Hauss et al. 1994). In the sample called D11, eleven protons of the retinal have been exchanged with deuterons, while in the sample called D5, only five have been exchanged near the Schiff base linkage.

Using the proposed simulation method, the label D11 has been localized at (x=20.3, y=12.1), close to the position at (x=20.8, y=12.6) given by Hauss et al. (1994). The simulation method overestimated the number of deuterons to 12.3 with an error of +11%. The radius was found to be 1.5 Å.

In a next step the determination of the D11 label was tested by calculating CC and R with the intensity differences weighted by (1−ε), where ε is the relative error in intensity. The result was again a position at (x=20.3, y=12.1), the number of deuterons 9.9 and the radius=0.5 Å.

Finally, the method was tested by reducing the diffraction resolution, taking into account 25 (S=0.1183) intensities instead of the full set of 35 intensities (S=0.1429). No remarkable differences in the results were observed.

Considering D5, its position has been found at (x=18.9, y=4.4), with the scattering length corresponding to 7 deuterons and a radius of 4.4 Å. The simulation method overestimated both the strength and size of the label. The reason for this deviation is that D5 represents only 0.5% of the maximum contrast in the elementary cell.

Considering that the simulation method leads to very good results concerning the position and the strength of D11 and that 11 deuterons are equivalent to 5.5 D₂O molecules, it is reasonable that the application of this method to our data for the H₂O/D₂O exchange experiments at different r.h. will reveal the correct number of water molecules at the projected area of the Schiff base.

We used the following definitions for the similarity parameters:

$$ {\rm CC} = {{N\sum {\left( {\Delta I_{\rm m} \Delta I_{\rm e} } \right)} - \sum {\Delta I_{\rm m} \sum {\Delta I_{\rm e} } } } \over {\sqrt {N\sum {\left( {\Delta I_{\rm m} } \right)^2 - \left( {\sum {\Delta I_{\rm m} } } \right)^2 } } \sqrt {N\sum {\left( {\Delta I_{\rm e} } \right)^2 - \left( {\sum {\Delta I_{\rm e} } } \right)^2 } } }}^{} $$

(3)

$$ R = 100{{\sum {\left| {\left| {\Delta I_{\rm e} } \right| - \left| {\Delta I_{\rm m} } \right|} \right|} } \over {\sum {\left| {\Delta I_{\rm e} } \right|} }} $$

(4)

$$ \chi ^2 = {{\sum {\left( {\Delta I_{\rm m} - \Delta I_{\rm e} } \right)} ^2 } \over {\Delta I_{\rm e} }} $$

(5)

where N is the number of intensities.

Appendix 2: calculation of the total scattering length of a label according to the Parseval formula

The scattering density distribution is reconstructed from the structure factors according to:

$$ \rho (x) = {1 \over L}\sum\limits_{h = - \infty }^\infty {F(h)\exp ( - 2\pi ihx)} $$

(6)

In the above equation, "x" and "h" stand for "x,y" and "h,k", respectively.

In the following we drop the factor 1/L, since in the final formula we introduce a factor adapting numerical results to theory. Integrating ρ(x) over x we obtain:

$$ \int\limits_{ - \infty }^\infty {\rho (x){\rm d}x} = \int\limits_{ - \infty }^\infty {\sum\limits_{h = - \infty }^\infty {F(h)} \exp ( - 2\pi ihx)\,{\rm d}} x = \sum\limits_{h = - \infty }^\infty {F(h)} \int\limits_{ - \infty }^\infty {\exp ( - 2\pi ihx)\,} {\rm d}x $$

(7)

According to a property of the delta function, the last integral of the above equation is δ(h). This means that the total scattering density over the entire elementary cell originates from F(0), while $\sum\limits_{h \ne 0} {F(h)} \exp ( - 2\pi ihx)$ gives positive and negative contributions, which cancel out in the sum over the entire elementary cell. After this introduction it is obvious that we can rewrite ρ(x) as follows:

$$ \rho (x) = \rho _0 + \rho _x \,\,\,{\rm with}\,\,\,\rho _0 = F(0)\,\,\,{\rm and}\,\,\,\rho _x = \sum\limits_{h \ne 0} {F(h)} \exp ( - 2\pi ihx) $$

(8)

So at every one position "x" the scattering density ρ(x) is ρ _x (with the property $\sum\limits_x {\rho _x } = 0$) over a homogeneous background of ρ₀. For the numerical calculations we replace ρ _x with ρ _i . The above considerations are valid also for the DF maps.

According to the Parseval formula (Killingbeck and Cole 1971), if F(s) is the Fourier transform of ρ(x), then:

$$ \int\limits_{ - \infty }^\infty {\left| {\rho (x)} \right|^2 {\rm d}x = } \int\limits_{ - \infty }^\infty {\left| {F(s)} \right|^2 {\rm d}s} \,\,\,{\rm or}\,\,{\rm numerically}\,\,\,\sum\limits_{\rm i} {\left( {\rho _{\rm 0} + \rho _i } \right)^2 = \sum\limits_j {I_j } } $$

(9)

where l _j =|Fj|² and "j" indexes the reciprocal space.

From Eq. (9):

$$ \sum\limits_i {\rho _i^2 + 2\rho _0 } \sum\limits_i {\rho _i } + \sum\limits_i {\rho _0^2 = \sum\limits_j {I_j } } $$

(10)

Because $\sum\limits_i {\rho _i } = 0$, Eq. (10) gives:

$$ \sum\limits_i {\rho _i^2 + } \sum\limits_i {\rho _0^2 = \sum\limits_{j \ne 0} {I_j } } + I_0 $$

(11)

In our experiments:

$$ I_0 = 0 \Rightarrow \rho _0 = 0 \Rightarrow \sum\limits_i {\rho _0^2 } = 0 $$

(12)

Finally:

$$ \sum\limits_i {\rho _i^2 } = \sum\limits_{j \ne 0} {I_j } $$

(13)

In a difference map, the only source of $\sum\limits_i {\rho _i^2 }$ is the label and $I_j = \left| {\Delta F_j } \right|^2$. So the sum provides the sum of the square of the local scattering length density over the area of the label. So far we have not used any phases at all but we still do not know ΔF.

According to the cosine rule:

$$ \left| {\Delta F} \right|^2 = \left| {F^{{\rm nl}} } \right|^2 + \left| {F^{\rm n} } \right|^2 - 2\left| {F^{{\rm nl}} } \right|\left| {F^{\rm n} } \right|\cos (\phi ^{{\rm nl}} - \phi ^{\rm n} ) $$

(14)

As is made clear from Fig. 2, cos(φ^nl−φⁿ)≈1, at least for labels similar to a file of 10 D₂O molecules perpendicular to the membrane plane, so we can rewrite Eq. (14) as:

$$ \left| {\Delta F} \right|^2 = \left( {\left| {F^{{\rm nl}} } \right| - \left| {F^{\rm n} } \right|} \right)^2 \,\,\,\,{\rm and}\,\,\,\,I = \left| {\Delta F} \right|^2 = \left( {\left| {F^{{\rm nl}} } \right| - \left| {F^{\rm n} } \right|} \right)^2 = \left( {\Delta \left| F \right|} \right)^2 $$

(15)

Fig. 2.

The value of cos(φ^nl−φⁿ) from Eq. (14) versus resolution for data from model calculations. The label is a circular distribution of scattering length density at the in-plane position of a Schiff base equivalent to 10 D₂O molecules

The method according to Parseval avoids the use of phases but it cannot avoid the more or less arbitrary splitting of the overlapping reflections to calculate Δ|F|. The diffraction data at 15% r.h. (in D₂O and H₂O atmospheres) show significant intensity differences in eight reflections with Miller indices (1,0), (1,1), (1,2), (3,0), (4,0), (4,1), (1,4), (4,3). Four of them, (1,2), (4,1), (1,4), (4,3), are overlapping. Significant differences between data measured at 15% r.h. in D₂O and data at 0% r.h. show 10 reflections: (1,0), (1,1), (2,0), (1,2), (3,0), (2,2), (4,0), (3,2), (4,1), (1,4). Four of them, (1,2), (3,2), (4,1), (1,4), are overlapping.

Since ρ(x,y) is unknown, in order to apply Eq. (13) we must assume that the label is a homogeneous distribution of a mean scattering length density d per matrix point over an area of a total of n matrix points. This is justified for low resolution. So Eq. (13) is rewritten as:

$$ nd^2 = \sum\limits_{j \ne 0} {I_j } $$

(16)

If n is known, then:

$$ d = \sqrt {{{\sum\limits_j {I_j } } \over n}} $$

(17)

or the total scattering length of the label is:

$$ D = nd = \sqrt {n\sum\limits_j {I_j } } $$

(18)

Model calculations using a label each time spread on a different number of matrix points n showed that if ρ _i is the calculated scattering density of the label, then Eq. (13) holds for any resolution by introducing a constant proportionality factor before ΣI _j . This factor accounts for any constant coefficients needed for the correct application of the Fourier transform, which we omitted so far. The same calculations showed that for low resolution or for narrow distribution of the label the calculated $\sum\limits_i {\rho _i^2 }$ is much smaller than that from the model. In order to apply Eq. (18) to low-resolution data, we determined a correction dependent on n as ~1/n. Considering the above, Eq. (18) takes the form:

$$ D = nd = 9175\sqrt {\sum\limits_j {I_j } } $$

(19)

In order to test Eq. (19), we applied it to experimental data of known total scattering length (Hauss et al. 1994). Sample D11 (known to have 11 protons/monomer replaced by deuterons at the retinal ring) gave 10.5 D/H or only 5% less. D5 (known to have 5 protons/monomer replaced by deuterons at the retinal end) gave 7.3 D/H or 45% more. This is a satisfactory result, given that D5 is a weak label. With the Parseval formula we were able to determine the D11 label with an accuracy of 5% and the weak D5 label with 45% accuracy.

Before applying Eq. (19), the data must have been scaled according to NatMod.

Appendix 3: integration of a label's scattering length density

In order to sum up the scattering length density over the region of the label, we have firstly to define this region and secondly to adjust the zero level of the difference density map. The latter is necessary because of the absence of ΔF(0,0), which has the effect of suppressing the label by raising the zero level. In the presence of only one label (three in a threefold symmetry) it is possible to restore the zero level to its original height by demanding that the area of the density map outside the label must fulfill the condition $\sum\limits_i {\rho _i } = 0$. So the first step is to define the area of the label. This is a somewhat complicated task, given that we deal with 2D maps and that the position as well as the area of the label can vary with the resolution and other parameters. We applied two methods to define the region of the label. According to the first, starting from the position of a maximum in the scattering length density we radially scan this region and define all matrix points as belonging to the label until we have found a minimum. According to the second method, matrix points around a maximum belong to the label if their scattering length density exceeds a predefined level compared to the total contrast in the elementary cell. The second step is to exclude the area of the label from the 128×128 matrix and calculate a suitable offset to ensure that $\sum\limits_i {\rho _i } = 0$ in the non-label regions. In a third step we add this offset to the whole matrix and sum up over the already defined region of the label.

Appendix 4: the peak heights in difference Fourier

A label is reconstructed by applying:

$$ \Delta \rho (x,y) = {1 \over A}\sum\limits_{h \ne 0} {\sum\limits_{k \ne 0} {\left| {F_{hk}^{\rm l} } \right|\exp (i\phi _{hk}^{\rm l} )\exp ( - 2\pi i(hx + ky))} } $$

(20)

In difference Fourier, $\left| {F_{hk}^{\rm l} } \right|\exp (i\phi _{hk}^{\rm l} )$ is approximately replaced by $(\left| {F_{hk}^{{\rm nl}} } \right| - \left| {F_{hk}^{\rm n} } \right|)\exp (i\phi _{hk}^{\rm n} )$:

$$ \Delta \rho (x,y) = {1 \over A}\sum\limits_{h \ne 0} {\sum\limits_{k \ne 0} {(\left| {F_{hk}^{{\rm nl}} } \right| - \left| {F_{hk}^{\rm n} } \right|)\exp (i\phi _{hk}^{\rm n} )\exp ( - 2\pi i(hx + ky))} } $$

(21)

The features of the DF map are determined by the following contributions (Blundell and Johnson 1976):

$$ \eqalign{ & (\left| {F_{hk}^{{\rm nl}} } \right| - \left| {F_{hk}^{\rm n} } \right|)\exp (i\phi _{hk}^{\rm n} ) = {1 \over {\left| {F_{hk}^{{\rm nl}} } \right| + \left| {F_{hk}^{\rm n} } \right|}}\left| {F_{hk}^{\rm l} } \right|^2 \exp (i\phi _{hk}^{\rm n} ) \cr & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {1 \over {\left| {F_{hk}^{{\rm nl}} } \right| + \left| {F_{hk}^{\rm n} } \right|}}\left| {F_{hk}^{\rm n} } \right|\left| {F_{hk}^{\rm l} } \right|\exp (i\phi _{hk}^{\rm l} ) \cr & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + {1 \over {\left| {F_{hk}^{{\rm nl}} } \right| + \left| {F_{hk}^{\rm n} } \right|}}\left| {F_{hk}^{\rm n} } \right|\left| {F_{hk}^{\rm l} } \right|\exp (i( - \phi _{hk}^{\rm l} + 2\phi _{hk}^{\rm n} )) \cr} $$

(22)

In our model calculations, $\left\langle {\left| {F^{\rm l} } \right|} \right\rangle \approx 0.1\left\langle {\left| {F^{\rm n} } \right|} \right\rangle$ and $\left\langle {\left| {F^{{\rm nl}} } \right|} \right\rangle \approx \left\langle {\left| {F^{\rm n} } \right|} \right\rangle$, where the symbol 〈〉 denotes the average.

The first right-hand term of Eq. (22) is a weak reconstruction of the native structure of the order 0.05|F ^l|. The second right-hand term of Eq. (22) contributes with half of |F ^l|. An inspection of the behavior of the factor |F ⁿ|/(|F ⁿ|+|F ^nl|) versus S (Fig. 3) shows that this factor fluctuates around 0.5 and that for resolution S>0.4 the amplitudes of the fluctuations clearly become very small. This is the reason why the upper curve (squares) of Fig. 1 reaches asymptotically the value predicted by the DF method.

Fig. 3.

The factor |F ⁿ|/(|F ⁿ|+|F ^nl|) of Eq. (22) versus resolution using data from our model calculations

Finally, the third term in Eq. (22) contributes with noise because φ^l and 2φⁿ are not correlated. The manifestation of the absence of correlation between φ^l and φⁿ in our model calculations is shown in Fig. 4 as a graph of their correlation coefficient versus resolution. In the same figure the correlation coefficient between two random variables in the range (−180, 180) is also presented. The similarity of these curves convinces us that φ^l and 2φⁿ are not correlated.

Fig. 4.

Curve A: the correlation coefficient between φ^l and φⁿ (full line) versus resolution. Curve B: the correlation coefficient between two random variables in the range −180 to 180 (dashed line) versus size of the statistical samples mapped to resolution