The process of histogram matching

Zhang, K. Y. J.; Cowtan, K. D.; Main, P.

doi:10.1107/97809553602060000687

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 15.1, pp. 315-316 | 1 | 2 |

Section 15.1.2.2.3. The process of histogram matching

K. Y. J. Zhang,^a K. D. Cowtan^b ^* and P. Main^c

^a Division of Basic Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N., Seattle, WA 90109, USA,^bDepartment of Chemistry, University of York, York YO1 5DD, England, and ^cDepartment of Physics, University of York, York YO1 5DD, England
Correspondence e-mail: cowtan+email@ysbl.york.ac.uk

15.1.2.2.3. The process of histogram matching

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Zhang & Main (1990a) demonstrated that, at better than 4 Å resolution, the histogram for an MIR map is generally significantly different from the ideal distribution calculated from atomic coordinates. The obvious course is therefore to alter the map in such a way as to make its density histogram equal to the ideal distribution. Unfortunately, there are an infinite number of maps corresponding to any chosen density distribution, so we must choose a systematic method of altering the map.

The conventional method of performing such a modification is to retain the ordering of the density values in the map. The highest point in the original map will be the highest point in the modified map, the second highest points will correspond in the same way, and so on.

Mathematically, this transformation is represented as follows. Let $[P(\rho)]$ be the current density histogram and $[P'(\rho)]$ be the desired distribution, normalized such that their sums are equal to 1. The cumulative distribution functions, $[N(\rho)]$ and $[N'(\rho)]$ , may then be calculated: $[\eqalign{N(\rho) &= {\textstyle\int\limits_{\rho_{\min}}^{\rho}} P(\rho)\ \hbox{d} \rho,\cr N'(\rho') &= {\textstyle\int\limits_{\rho_{\min}}^{\rho'}} P'(\rho)\ \hbox{d} \rho.} \eqno(15.1.2.12)]$ The cumulative distribution function of a variable transforms a value chosen from the distribution into a number between 0 and 1, representing the position of that value in an ordered list of values chosen from the distribution.

The transformation may, therefore, be performed in two stages. A density value is taken from the initial distribution and the cumulative distribution function of the initial distribution is applied to obtain the position of that value in the distribution. The inverse of the cumulative distribution function for the desired distribution is applied to this value to obtain the density value for the corresponding point in the desired distribution. Thus, given a density value, ρ, from the initial distribution, the modified value, ρ′, is obtained by $[\rho' = N'^{-1} \left[{N(\rho)}\right]. \eqno(15.1.2.13)]$ The distribution of ρ′ will then match the desired distribution after the above transformation. The transformation of an electron-density value by this method is illustrated in Fig. 15.1.2.3. The transformation in equation (15.1.2.13) can be achieved through a linear transform represented by $[\rho'_{i} = a_{i} \rho_{i} + b_{i}, \eqno(15.1.2.14)]$ where $[i = \left\{1, \ldots, n\right\}]$ and n is the number of density bins. The above linear transform is sufficient if the number of density bins is large enough. An n value of about 200 is usually quite satisfactory.

Figure 15.1.2.3| top | pdf |

Transformation of density ρ to $[\rho'_{\rm mod}]$ by histogram matching.

Various properties of the electron density are specified in the density histogram, such as the minimum, maximum and mean density, the density variance, and the entropy of the map. The mean density of the ideal map can be obtained by $[\overline{\rho} = {\textstyle\int\limits_{\rho_{\min}}^{\rho_{\max}}} {\rho P(\rho)\ \hbox{d}\rho}. \eqno(15.1.2.15)]$ The variance of the density in the ideal map can be obtained by $[\sigma (\rho) = \left({\overline {\rho^{2}} - \overline{\rho}^{2}}\right)^{1/2}, \eqno(15.1.2.16)]$ where $[\overline{\rho^{2}} = {\textstyle\int\limits_{\rho_{\min}}^{\rho_{\max}}} {\rho^{2} P(\rho)\ \hbox{d}\rho}. \eqno(15.1.2.17)]$ The entropy of the ideal map can be calculated by $[S = - {\textstyle\int\limits_{\rho_{\min}}^{\rho_{\max}}} {P(\rho)} \rho \ln (\rho)\ \hbox{d}\rho. \eqno(15.1.2.18)]$

Therefore, the process of histogram matching applies a minimum and a maximum value to the electron density, imposes the correct mean and variance, and defines the entropy of the new map. The order of electron-density values remains unchanged after histogram matching.

Histogram matching is complementary to solvent flattening since it is applied to the protein region of a map, whereas solvent flattening only operates on the solvent region of the map. The same envelope that was used for isolating the solvent region can be used to determine the protein region of the cell. An alternative approach is to define separate solvent and protein masks, with uncertain regions excluded from either mask and allowed to keep their unmodified values.

References

Zhang, K. Y. J. & Main, P. (1990a). Histogram matching as a new density modification technique for phase refinement and extension of protein molecules. Acta Cryst. A46, 41–46.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.1, pp. 315-316