Section 2.4.4.4. Treatment of errors in phase evaluation: Blow and Crick formulation

As shown in Section 2.4.2.3, ideally, protein phase angles can be evaluated if two isomorphous heavy-atom derivatives are available. However, in practice, conditions are far from ideal on account of several factors such as imperfect isomorphism, errors in the estimation of heavy-atom parameters, and the experimental errors in the measurement of intensity from the native and the derivative crystals. It is therefore desirable to use as many derivatives as are available for phase determination. When isomorphism is imperfect and errors exist in data and heavy-atom parameters, all the circles in a Harker diagram would not intersect at a single point; instead, there would be a distribution of intersections, such as that illustrated in Fig. 2.4.4.1. Consequently, a unique solution for the phase angle cannot be deduced.

Figure 2.4.4.1 | top | pdf | Distribution of intersections in the Harker construction under non-ideal conditions.

The statistical procedure for computing protein phase angles using multiple isomorphous replacement (MIR) was derived by Blow & Crick (1959). In their treatment, Blow and Crick assume, for mathematical convenience, that all errors, including those arising from imperfect isomorphism, could be considered as residing in the magnitudes of the derivative structure factors only. They further assume that these errors could be described by a Gaussian distribution. With these simplifying assumptions, the statistical procedure for phase determination could be derived in the following manner.

Consider the vector diagram, shown in Fig. 2.4.4.2, for a reflection from the ith derivative for an arbitrary value for the protein phase angle. Then, If corresponds to the true protein phase angle , then coincides with . The amount by which differs from , namely, is a measure of the departure of from . is called the lack of closure. The probability for being the correct protein phase angle could now be defined as where is the normalization constant and is the estimated r.m.s. error. The methods for estimating will be outlined later. When several derivatives are used for phase determination, the total probability of the phase angle being the protein phase angle would be where the summation is over all the derivatives. A typical distribution of plotted around a circle of unit radius is shown in Fig. 2.4.4.3. The phase angle corresponding to the highest value of would obviously be the most probable protein phase, , of the given reflection. The most probable electron-density distribution is obtained if each is associated with the corresponding in a Fourier synthesis.

Figure 2.4.4.2 | top | pdf | Vector diagram indicating the calculated structure factor, , of the ith heavy-atom derivative for an arbitrary value for the phase angle of the structure factor of the native protein.

Figure 2.4.4.3 | top | pdf | The probability distribution of the protein phase angle. The point P is the centroid of the distribution.

Blow and Crick suggested a different way of using the probability distribution. In Fig. 2.4.4.3, the centroid of the probability distribution is denoted by P. The polar coordinates of P are m and , where m, a fractional positive number with a maximum value of unity, and are referred to as the `figure of merit' and the `best phase', respectively. One can then compute a `best Fourier' with coefficients The best Fourier is expected to provide an electron-density distribution with the lowest r.m.s. error. The figure of merit and the best phase are usually calculated using the equations where are calculated, say, at intervals (Dickerson et al., 1961). The figure of merit is statistically interpreted as the cosine of the expected error in the calculated phase angle and it is obviously a measure of the precision of phase determination. In general, m is high when and are close to each other and low when they are far apart.