Molecular replacement is a method of solving the phase problem

Phase problem

In physics the phase problem is the name given to the problem of loss of information concerning the phase that can occur when making a physical measurement. The name itself comes from the field of x-ray crystallography, where the phase problem has to be solved for the determination of a structure...

 in X-ray crystallography

X-ray crystallography

X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...

. MR relies upon the existence of a previously solved protein structure which is homologous (similar) to our unknown structure from which the diffraction data is derived.

The first goal of the crystallographer is to obtain an electron density map, density being related with diffracted wave as follows:


With usual detectors the intensity is being measured, so all the information about phase () is lost.
Then, in the absence of phases (Φ), we are unable to complete the shown Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 relating the experimental data from X-ray crystallography

X-ray crystallography

X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...

 (in reciprocal space) to real-space electron density, into which the atomic model is built. MR tries to find the model which fits best experimental intensities among known structures.

Principles of Patterson-based molecular replacement

However, we can derive a Patterson map

Patterson function

The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT....

, which is an interatomic vector map created by squaring the structure factor amplitudes and setting all phases to zero. This vector map contains a peak for each atom related to every other atom, with a large peak at 0,0,0, where vectors relating atoms to themselves "pile up". Such a map is far too noisy to derive any high resolution structural information—however if we generate Patterson maps for the data derived from our unknown structure, and from the structure of a previously solved homologue, in the correct orientation and position within the unit cell, the two Patterson maps should be closely correlated. This principle lies at the heart of MR, and can allow us to infer information about the orientation and location of an unknown molecule with its unit cell.

Due to historic limitations in computing power, an MR search is typically divided into two steps: rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

 and translation

Translation

Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...

.

Rotation function

In the rotation function, our unknown Patterson map is compared to Patterson maps derived from our known homologue structure in different orientations. Historically r-factors and/or correlation coefficients

Pearson product-moment correlation coefficient

In statistics, the Pearson product-moment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...

 were used to score the rotation function, however, modern programs use maximum likelihood

Maximum likelihood

In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....

-based algorthims. The highest correlation (and therefore scores) are obtained when the two structures (known and unknown) are in similar orientation(s)—these can then be output in Euler angles

Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...

 or spherical polar angles.

Translation function

In the translation function, the now correctly oriented known model can be correctly positioned by translating it to the correct co-ordinates within the asymmetric unit. This is accomplished by moving the model, calculating a new Patterson map, and comparing it to the unknown-derived patterson map. This brute-force search is computationally expensive and fast translation functions are now more commonly used. Positions with high correlations are output in Cartesian coordinates.

The next step

Following this, we should have correctly oriented and translated phasing models, from which we can derive phases which are (hopefully) accurate enough to derive electron density maps. These can be used to build and refine an atomic model of our unknown structure.