  GNU-Darwin Web CCP4i: Graphical User Interface MOLREP Theory

The following comes straight from Appendix A in the README from MOLREP, and can now also be found in the CCP4 program documentation for MOLREP (which may be more up-to-date).

### Molecular replacement method

There are two major steps in the Molecular replacement method: orientation and translation search. They are performed by Rotation and Translation function. Both of them are correlation functions (or overlapping functions) between observed and calculated model Pattersons.

#### Rotation function

```     ROT(R) = Irad Pobs(r) * Pcalc(R,r) dr
```

where

 R operator of rotation Irad integral inside a sphere in the centre of Patterson with radius=rad Pobs observed Patterson Pcalc calculated Patterson for rotated (R) model

#### Translation function

```     TR(s)  = Icell Pobs(r) * Pcalc(s,r) dr  =

= Sum ( I Pobs(r) * Pcalc_ij_(s,r) dr) =
i#j

= Sum TRij(s)
i#j
```

where

 s vector of translation I integral i,j cryst. symmetry operator numbers Pcalc_ij_(s,r) calculated Patterson for model corresponding to ith operator and model corresponding to jth operator TRij(s) translation function of Pattersons Pobs(r) and Pcalc_ij_(s,r)

Translation function is the sum of translation functions for every pair of different cryst. symmetry operators.

The best rotation function algorithm is the Crowther Fast Rotation Function which is used in the program MOLREP. It utilizes FFT. The program can compute the Rotation Function for three different orientations of model and average them. That reduces the noise of Rotation function.

The translation function algorithm was developed by the author and performs calculations in reciprocal space using FFT. There are two major differences from other translation function:

1. Instead of summation of translation function for two operators TRij we use their multiplication which results in a map with far more contrast.
2. Finally we can multiply translation function by Packing function to remove peaks corresponding to incorrect solution with bad packing.

#### Packing function

Packing function is overlapping function:

```
P(s) = Sum ( I  Roi(r) * Roj(r) dr )
i#j  cell
```

where Roi(r) is the electron density of model which corresponds to ith cryst. symmetry operator. The algorithm of calculation of Packing function is similar to the one for translation function and performed by the same program.

```     TR(s)  = [  M  TRij(s) ] * P(s)
i#j
```

where M means multiplication of different TRij.

#### Secrets of success

Success of molecular replacement method depends on:

1. quality of the model, homology, conformation
2. quality of experimental data
3. scaling |F|_obs and |F|_calc
4. low resolution limit
5. high resolution limit

#### Patterson scaling

For scaling we use a completely new strategy based on scaling of two Patterson maps. This method makes it possible to have a very good approximation for scaling problem even if only low resolution data is available where other methods do not work. Scaling by Patterson also is useful for Cross rotation function where the cell is different for the model and the unknown structure.

#### Low resolution cut-off (Boff)

Low resolution cut-off introduces systematical errors in the electron density especially near the surface of the model. It is the series termination effect. Instead of using the usual low resolution cut-off, the program multiplies the modules of structure factors by a special coefficient:

```     Fnew = Fold (1-exp(-Boff*s^2))
```

where Boff = 4resmin^2. Boff is called "soft low resolution cut-off", which allows the removal of structure factors in this resolution range without introducing the series termination effect.

#### The use of a priori knowledge of similarity and completeness of the model

For low similarity the high resolution reflections are weighted down. Program uses for this additional overall factor Badd:

```     Fnew = Fold exp(-Badd*s^2)
```

Value of similarity 'SIM' can be : from 0.1 to 1.0. It directly corresponds to Badd ranging from (B_limit-Boverall) to (-Boverall), where B_limit refers to the so-called 'Optical Resolution'. SIM="1" means normalized F will be used.

For low completeness, e.g. when there are several molecules in the asymmetric unit, the contribution of low resolution reflections is weighted down. To manage the completeness of model, the program uses low resolution cut-off (Boff). Completeness of model 'COMPL' can be : from 0.2 to 1.0. It corresponds to Boff: from 400 to 1600.

#### Optical Resolution

Optical resolution is defined as an expected minimum distance between two resolved peaks in the electron density map. It is computed from the width of the Patterson origin peak. For reference, see MOLREP program documentation.

### Functions of real space searching

#### Spherically averaged phased translation function (SAPTF)

SAPTF gives expected position of model in Electron density map by the comparison of spherically averaged density of model with locally spherically averaged observed density.

```     SAPTF(s) = Irad(s) Robs(r,s) * Rcalc(r) dr
```

where

 Irad(s) integral inside a sphere centred in point s of electron density with radius=rad Robs spherically averaged around point s observed electron density Rcalc spherically averaged around origin of coordinate system calculated electron density for model

#### Phased Rotation function (PRF)

PRF gives the orientation of model placed in some point of Electron density.

```     PROT(O) = Irad(s) Robs(r) * Rcalc(O,r) dr
```

where

 O operator of rotation Irad(s) integral inside a sphere centred in point s of electron density with radius=rad Robs observed electron density Rcalc calculated Electron density for rotated (O) model

#### Phased Translation function (PTF)

Translation search in Elecron density map.

```     PTR(s)  = Icell Robs(r) * Rcalc(s,r) dr
```

where

 s vector of translation I integral Robs observed electron density Rcalc(s,r) calculated electron density for model placed in the vector s

#### Fitting two models

Fitting through Electron density. Second model (see MODEL_2) is the target model which is converted to electron density. To search the best overlapping of electron densities of models there are two algorithms:

1. Rotation function (Patterson space) and Phased Translation function (Real space)
2. All functions in real space. Spherically averaged phased translation function gives expected position for model. Phased Rotation function for expected position gives orientation. Phased Translation function checks and refines the translation vector