Development

There have been several attempts to enhance the quality of the electron density maps obtained by MEM, since the electron densities obtained by MEM may suffer from noise and artifacts (Jauch and Palmer, 1993; Jauch, 1994; de Vries et al., 1994;

Roversi et al., 1998; Palatinus and van Smaalen, 2002). Some of the main reasons for these noise and artifacts are

• inappropriate weighting scheme wi

• use of uninformational PRIOR densities (e.g. uniform PRIOR)

• series termination effects

• inaccuracy of data and their quality

• wrong choice of the value of χ²aim

Jauch and Palmer (1993) were the first to show that the distribution of normal-ized residuals for the MEM electron density is non-Gaussian. But according to the criterion of MEM (Eq. 2.6), the residual distribution

FM EM(Hi)

σ(Hi) = Fobs(Hi)−FM EM(Hi)

σ(Hi) (2.11)

should be a Gaussian distribution. This problem has been observed in the standard version of MEM which uses the value wi = 1. This lead to large normalized residual values for very few low-order reflections and the remaining reflections however pos-sessing very small residuals. To overcome this problem an ad hocweighting scheme is applied by Hofmann et al. (2007a) in the F-constraints (Eq. 2.6) as suggested by de Vries et al. (1994): where |H_i| is the length of the scattering vector of Bragg reflection i and n are small positive integers. This weighting scheme leads to reduced residuals of low-order reflections by giving larger weight to those reflections (with short scattering vectors) and in turn giving rise to a Gaussian distribution of normalized residuals.

Several tests have confirmed this and suggested the optimum choice of n = 4 for obtaining best electron density map (de Vries et al., 1994; Hofmann et al., 2007a;

2.2. MAXIMUM ENTROPY METHOD 15

Netzel et al., 2008). We have performed the MEM calculation in the Chapter 5 by using weights according to Eq. (2.12) withn= 4.

Initially a flat prior density (total number of electrons in the unit cell, which are uniformly distributed over the volume of the unit cell) has been used for the electron density calculation using the MEM (Sakata and Sato, 1990). This approach has resulted in the existence of noise and artifacts (non-nuclear maxima) in the electron density map, whose magnitudes are larger than the effects of chemical bonding (Sakata and Sato, 1990; Iversen et al., 1995; Palatinus and van Smaalen, 2002). To overcome this problem de Vries, Briels and Feil (1996) first proposed the idea of using a non-uniform prior density and established the absence of non-nuclear maxima in Si-Si bonds in crystalline silicon, which was present before in the electron density map analyzed by Sakata and Sato (1990). Palatinus and van Smaalen (2002) also confirm a reduction in noise and artifacts in the MEM density by employing a non-unform prior, which is generated by using the coordinates and ADPs from the IAM.

From this one can understand that the magnitude of noise and artifacts depends on the type of the prior density used (van Smaalen and Netzel, 2009), since the maximum value of the entropy is obtained for ρk = ρ^prior_k (Eq. 2.4). Deviation of the ρk from ρ^prior_k always leads to a lowering of the entropy, but it is allowed to do so if it is required to fit the data (Eq. 2.7). Noise and artifacts increase with increase in magnitude of this difference. Therefore it has been recommended to use the IAM as prior for electron-density analysis using the MEM (Palatinus and van Smaalen, 2002; Hofmann et al., 2007a;b; Netzel et al., 2008; van Smaalen and Netzel, 2009). However van Smaalen and Netzel (2009) suggested the idea of using multipole model as an alternative choice for prior density in the MEM, since it is more informative than IAM and probably closer to the true densities. With this idea, in Chapter 5 we report MEM electron densities calculated with either a MP model, an invariom model (multipoles transferred from the database and not varied in the structure refinement) and an IAM model created by high-order refinement (IAM-HO) as prior, with the purpose to investigate the effect of different prior densities on the MEM.

Although the series termination effects in the MEM are by far not as big a problem as in conventional Fourier synthesis of electron densities, still it can be one of the reasons for artifacts (de Vries et al., 1994; Gilmore, 1996). This might arise due to limited number of reflections available from the data set (Jauch, 1994), which can be suppressed by employing the sufficiently informative prior i.e. non-uniform prior (Palatinus and van Smaalen, 2005). MEM generally de-emphasises the series

termination but does not remove them (Gilmore, 1996).

The other main source of errors which produces artifacts in MEM is the inac-curacy of the data specially at higher scattering angles. This happens due to de-creasing scattered intensities with inde-creasing scattering angle and the corresponding structure factors may be measured as weak or unobserved. To overcome this prob-lem, Palatinus and van Smaalen (2005) have suggested the method of prior-derived F-constraints (PDC) (Palatinus and van Smaalen, 2005) with

C_F^{P DC}2 =−χ²_aim+ ¹ exper-iment. The standard uncertainties σ(H_j) are chosen to be equal to the smallest standard uncertainty amongst the experimental data. The iterations are performed with the summation of Eq. (2.13). The calculated structure factors by the method of PDC gives a good estimate for structure factors at high-angle reflections and PDC enhances the quality of electron density map obtained by MEM (Palatinus and van Smaalen, 2005; Hofmann et al., 2007a;b; Netzel et al., 2008; van Smaalen and Netzel, 2009). However to consider employing the method of PDC in MEM, the minimum resolution of the experimental data should be available up to sin(θ)/λ = 0.9 ˚A⁻¹.

The choice of optimalχ²_aim is very important to get a good-quality MEM electron density map, otherwise MEM electron density map will have under-fitted data or noise (Netzel et al., 2008; van Smaalen and Netzel, 2009), since the value of χ²_aim determines the point of convergence through the criterion CF²=0 (Eq. 2.6). If the PDC (C_F^{P DC}2 ) is included, the MEM still checks the convergence through theCF²=0 on the experimental data only Eq. (2.6).

The standard version of MEM employs χ²aim = 1 (Skilling and Bryan, 1984;

Sakata and Sato, 1990; Tanaka et al., 2002). However it is recommended to de-termine the value of χ²aim for each individual MEM calculation (Hofmann et al., 2007b; Netzel et al., 2008; van Smaalen and Netzel, 2009). One way to evaluate the optimum χ²aim values is by comparing the difference Fourier map and dynamic deformation density map generated at different sections of the studied molecule for different χ²_aim values. For the optimum value of χ²_aim, the difference Fourier map

2.2. MAXIMUM ENTROPY METHOD 17

Figure 2.1: C1-C2-N plane of density maps ofD, L-serine for the IAM model (compare to Chapter 5). (a, b, c) difference Fourier map with contours at 0.05 e/˚A³; (d, e, f) dynamic deformation density with contours at 0.05 e/˚A³ ; and (g, h, i) MEM density with contours at 0.2 e/˚A³ up to 2.5 e/˚A³. For (a, d, g) χ²_aim = 0.2; (b, e, h)χ²_aim = 0.55; and (c, f, i) χ²_aim = 0.9. Solid lines denote positive values, dotted values denote negative values and dashed lines are zero contour.

needs to be featureless [see for example Fig. 2.1(b)] and the dynamic deformation density map should exhibit smooth features [Fig. 2.1(e)] (Hofmann et al., 2007b;

Netzel et al., 2008; van Smaalen and Netzel, 2009). Too large values of χ²_aim lead to under-fitted data and it will possess larger residual densities in the difference Fourier map [Fig. 2.1(c)]. Too small values will lead to the over-fitting of the data. Noise will be added to the electron density [Fig. 2.1(g)], such that the difference Fourier map will be flat [Fig. 2.1(a)]. Therefore the optimum χ²aim value can easily be determined by examining these maps. The corresponding electron density obtained will be free of noise and artifacts.

By employing all the above extensions in the computer program BayMEM (van Smaalen et al., 2003), we have obtained the electron densities by MEM for three amino acids and a tripeptide and described in the Chapter 5.