# Grids and maps¶

## Volumetric grid¶

### Introduction¶

Macromolecular models are often accompanied by 3D data on an evenly spaced, rectangular grid (but note that the spacing in different directions may differ). The data may represent electron density, a mask of the protein area, or any other scalar data.

In Gemmi such a data is stored in a class called Grid. Actually, it is a set of classes for storing different types of data: floating point numbers, integers or boolean masks.

Grid dimension are given in variables nu, nv, nw. The data layout is Fortran-style contiguous, i.e. point (1,1,1) is followed by (2,1,1).

Grid classes also store:

• unit cell dimensions (to know Cartesian coordinates of grid nodes),
• and crystallographic symmetry (to know symmetry-equivalent grid nodes).

If the symmetry is not set (or is set to P1) we effectively have a box with periodic boundary conditions (PBC).

#### C++¶

The `gemmi/grid.hpp` header defines:

```template<typename T=float> struct Grid;
```

which stores dimensions and data:

```int nu, nv, nw;
std::vector<T> data;
```

The data point can be accessed with:

```T Grid<T>::get_value(int u, int v, int w) const
void Grid<T>::set_value(int u, int v, int w, T x)
```

#### Python¶

In Python we have classes FloatGrid (for maps), Int8Grid (for masks). (We will add other classes such and ComplexGrid when we see a use for it.)

The constructor may take grid dimensions or a NumPy array as an argument:

```>>> import gemmi
>>> grid = gemmi.FloatGrid(12, 12, 12)
>>> grid.nu, grid.nv, grid.nw
(12, 12, 12)
```
```>>> grid2 = gemmi.FloatGrid(numpy.zeros((30, 31, 32), dtype=numpy.float32))
>>> grid2.nu, grid2.nv, grid2.nw
(30, 31, 32)
```

Values are accessed with functions get_value() and set_value():

```>>> grid.set_value(1, 1, 1, 7.0)
>>> grid.get_value(1, 1, 1)
7.0
>>> # we can test wrapping of indices (a.k.a. periodic boundary conditions)
>>> grid.get_value(-11, 13, 25)
7.0
```

The data can be also accessed through the buffer protocol. It means that you can use it as a NumPy array (Fortran-style contiguous) without copying the data:

```>>> import numpy
>>> array = numpy.array(grid, copy=False)
>>> array.dtype
dtype('float32')
>>> array.shape
(12, 12, 12)
>>> numpy.argwhere(array == 7.0)
array([[1, 1, 1]])
```

(It does not make gemmi dependent on NumPy – gemmi talks with NumPy through the buffer protocol, and it can talk with any other Python library that supports this protocol.)

### Symmetry¶

The main advantage that Grid has over a generic 3D array is that it understands crystallographic symmetry. After setting the symmetry we can use a family of symmetrize functions that performs operations on symmetry-equivalent grid points. For example, we can set all equivalent points to the value calculated as a minimum, maximum or a sum of values of the equivalent points.

In C++ we directly set the spacegroup property:

```const SpaceGroup* spacegroup;
```

Similarly in Python:

```>>> grid.spacegroup = gemmi.find_spacegroup_by_name('P2')
```

Now let us use one of the symmetrizing functions:

```>>> # the point (1, 1, 1) was already set to 7.0
>>> grid.set_value(0, 0, 0, 0.125)  # a special position
>>> grid.sum()  # for now only two points: 7.0 + 0.125
7.125
>>> grid.symmetrize_max()  # applying symmetry
>>> grid.sum()  # one point got duplicated, the other is on rotation axis
14.125
```

In C++ we have a templated function that can perform any operation on symmetry-equivalent points:

```template<typename Func> void Grid::symmetrize(Func func)
```

Python bindings provide the following specializations:

```>>> grid.symmetrize_min()      # minimum of equivalent values
>>> grid.symmetrize_max()      # maximum
>>> grid.symmetrize_abs_max()  # value corresponding to max(|x|)
>>> #grid.symmetrize_sum()     # sum symmetry-equivalent nodes
```

### Unit cell¶

The unit cell parameters (in a member variable `unit_cell`: UnitCell) enable conversion between coordinates and grid points.

The unit cell should be set using `Grid<T>::set_unit_cell()`, which in addition to setting `unit_cell` sets also `spacing`, the spacing between grid points that is precalculated for efficiency.

```>>> grid.set_unit_cell(gemmi.UnitCell(45, 45, 45, 90, 82.5, 90))
>>> grid.unit_cell
<gemmi.UnitCell(45, 45, 45, 90, 82.5, 90)>
>>> grid.spacing
(3.7179..., 3.75..., 3.7179...)
```

Each grid point (u, v, w) can now be expressed in fractional or Cartesian coordinates:

```>>> grid.get_fractional(6, 6, 6)
<gemmi.Fractional(0.5, 0.5, 0.5)>
>>> grid.get_position(6, 6, 6)
<gemmi.Position(25.4368, 22.5, 22.3075)>
```

### Grid point¶

Grid contains a little helper class (GridBase<T>::Point in C++) that bundles grid point coordinates (u, v, w: int) and a pointer to the value in grid (value). This bundle is obtained with getter:

```>>> grid.get_point(0, 0, 0)
<gemmi.FloatGridBase.Point (0, 0, 0) -> 0.125>
>>> _.u, _.v, _.w, _.value
(0, 0, 0, 0.125)
```

or when iterating the grid:

```>>> for point in grid:
...   if point.value != 0.: print(point)
<gemmi.FloatGridBase.Point (0, 0, 0) -> 0.125>
<gemmi.FloatGridBase.Point (1, 1, 1) -> 7>
<gemmi.FloatGridBase.Point (11, 1, 11) -> 7>
```

The point can be converted to its index (position in the array):

```>>> point = grid.get_point(6, 6, 6)
>>> grid.point_to_index(point)
942
```

to fractional coordinates:

```>>> grid.point_to_fractional(point)
<gemmi.Fractional(0.5, 0.5, 0.5)>
```

and to orthogonal (Cartesian) coordinates in Angstroms:

```>>> grid.point_to_position(point)
<gemmi.Position(25.4368, 22.5, 22.3075)>
```

The other way around, we can find the grid point nearest to a position:

```>>> grid.get_nearest_point(_)
<gemmi.FloatGridBase.Point (6, 6, 6) -> 0>
```

### Interpolation¶

To get a value corresponding to an arbitrary position, you may use trilinear interpolation of the 8 nearest nodes, or tricubic interpolation that uses 64 nodes.

C++

```T Grid<T>::interpolate_value(const Fractional& fctr) const
T Grid<T>::interpolate_value(const Position& ctr) const

double Grid<T>::tricubic_interpolation(const Fractional& fctr) const
double Grid<T>::tricubic_interpolation(const Position& ctr) const

// calculates also derivatives
std::array<double,4> Grid<T>::tricubic_interpolation_der(double x, double y, double z) const
```

Python

```>>> grid.interpolate_value(gemmi.Fractional(1/24, 1/24, 1/24))
0.890625
>>> grid.interpolate_value(gemmi.Position(2, 3, 4))
2.0333263874053955
>>> grid.tricubic_interpolation(gemmi.Fractional(1/24, 1/24, 1/24))
1.283477783203125
>>> grid.tricubic_interpolation(gemmi.Position(2, 3, 4))
2.6075661737715046
>>> # calculate also derivatives in directions of unit cell axes
>>> grid.tricubic_interpolation_der(gemmi.Fractional(1/24, 1/24, 1/24))
[1.283477783203125, 35.523193359375, 36.343505859375, 35.523193359375]
```

The cubic interpolation is smoother than linear, but may increase the noise. This is illustrated on the plots below, which shows density along two lines in a grid that was filled with random numbers from [0, 1). Trilinear interpolation is blue, tricubic – red. The left plot shows density along a line in a random direction, the right plot – along a line parallel to one of the axes. Implementation notes

Tricubic interpolation, as described on Wikipedia page and in Appendix B of a PHENIX paper, can be implemented either as 21 cubic interpolations, or using method introduced by Lekien & Marsen in 2005, which involves 64x64 matrix of integral coefficients (see also this blog post). The latter method should be more efficient, but gemmi uses the former, which takes ~100 ns. If you’d like to speed it up or to get derivatives, contact developers.

Optimization for Python

If you have a large number of points, making a Python function call each time would be slow. If these points are on a regular 3D grid (which may not be aligned with our grid) call `interpolate_values()` (with s at the end) with two arguments: a 3D NumPy array (for storing the results) and a Transform that relates indices of the array to positions in the grid:

```>>> # first we create a numpy array of the same type as the grid
>>> arr = numpy.zeros([32, 32, 32], dtype=numpy.float32)
>>> # then we setup a transformation (array indices) -> (position [A]).
>>> tr = gemmi.Transform()
>>> tr.mat.fromlist([[0.1, 0, 0], [0, 0.1, 0], [0, 0, 0.1]])
>>> tr.vec.fromlist([1, 2, 3])
>>> # finally we calculate interpolated values
>>> grid.interpolate_values(arr, tr)
>>> arr[10, 10, 10]  # -> corresponds to Position(2, 3, 4)
2.0333264
```

(If your points are not on a regular grid – get in touch – there might be another way.)

### ASU and MaskedGrid¶

Sometimes we want to focus on a part of the grid only. For this, we have class MaskedGrid that combines two Grid objects, using one of them as a mask for the other.

When an element of the mask is 0 (false), the corresponding element of the other grid is unmasked and is to be used. This is the same convention as in NumPy MaskedArray.

The primary use for MaskedGrid is working with asymmetric unit (asu) only:

```>>> asu = grid.asu()
>>> asu
<gemmi.MaskedFloatGrid object at 0x...>
>>> asu.grid is grid
True
>>> asu.mask
<gemmi.Int8Grid(12, 12, 12)>
>>> sum(point.value for point in asu)
7.125
>>> for point in asu:
...   if point.value != 0: print(point)
<gemmi.FloatGridBase.Point (0, 0, 0) -> 0.125>
<gemmi.FloatGridBase.Point (1, 1, 1) -> 7>
```

### Setting value in areas¶

First, to set the whole grid to the same value use:

```>>> grid.fill(0)
```

To set the grid points in a certain radius from a specified position use:

```void Grid<T>::set_points_around(const Position& ctr, double radius, T value)
```
```>>> grid.set_points_around(gemmi.Position(25, 25, 25), radius=3, value=10)
>>> numpy.argwhere(array == 10)
array([[6, 6, 7],
[6, 7, 7]])
```

This function, to be efficient, ignores symmetry. At the end we should call one of the symmetrizing functions:

```>>> grid.symmetrize_max()
```

While we could use the above functions for masking the molecule (or bulk solvent) area, we have specialized functions to create a bulk solvent mask.

### Solvent mask¶

Gemmi implements a variant of the most popular method for calculating the bulk solvent area. This method was introduced in CNS. It uses van der Waals (or similar) atomic radii r and two parameters: rprobe and rshrink.

• We mark the area in radius r + rprobe of each atom as non-solvent (0). r usually depends on the element, but some programs use the same radius for all atoms. The extra margin rprobe is largely cancelled in the next step.
• We shrink the non-solvent area by rshrink. All the 0’s in a distance rshrink from 1’s are changed to 1, shrinking the solvent volume. Both rprobe and rshrink have the same order of magnitude. Jiang & Brünger (1994) proposed rprobe = 1.0 Å and rshrink = 1.1 Å.
• The above procedure eliminates small solvent islands. If it is not sufficient, we can explicitly remove islands (contiguous areas of 1’s) up to a certain volume. This step was added for compatibility with Refmac.

Here is how to create a mask identical as `phenix.mask`:

```>>> masker = gemmi.SolventMasker(gemmi.AtomicRadiiSet.Cctbx)
>>> st = gemmi.read_structure('../tests/1orc.pdb')
>>> grid = gemmi.Int8Grid()
>>> # take space group and unit cell from Structure,
>>> # and set size based on the specified minimal spacing
>>> grid.setup_from(st, spacing=1.0)
>>> masker.put_mask_on_int8_grid(grid, st)
```

The parameters of SolventMasker can be inspected and changed:

```>>> masker.atomic_radii_set
<AtomicRadiiSet.Cctbx: 1>
>>> masker.rprobe
1.11
>>> masker.rshrink
0.9
>>> masker.island_min_volume  # 0 = unused
0.0
>>> masker.constant_r  # 0 = unused
0.0
```

The example above uses a parameter set based on cctbx. We also have a few others sets. You can create mask similar to Refmac (but due to unintended features of solvent masking in Refmac, the results are not identical):

```>>> masker = gemmi.SolventMasker(gemmi.AtomicRadiiSet.Refmac)
```

or a mask with Van der Waals radii from Wikipedia and rprobe = 1.0 Å and rshrink = 1.1 Å, as in the original Jiang & Brünger paper:

```>>> masker = gemmi.SolventMasker(gemmi.AtomicRadiiSet.VanDerWaals)
```

or with constant radius, similarly to the NCSMASK program from CCP4:

```>>> masker = gemmi.SolventMasker(gemmi.AtomicRadiiSet.Constant, 3.0)
```

If the mask is to be FFT-ed to structure factors, store it on FloatGrid (function `put_mask_on_float_grid` in Python and `put_mask_on_grid` in C++). See the section about bulk solvent coorection for details and examples.

## MRC/CCP4 maps¶

We support one file format for storing the grid data on disk: MRC/CCP4 map. The content of the map file is stored in a class that contains both the Grid class and all the meta-data from the CCP4 file header.

The CCP4 format has a few different modes that correspond to different data types. Gemmi supports:

• mode 0 – which correspond to the C++ type int8_t,
• mode 1 – corresponds to int16_t,
• mode 2 – float,
• and mode 6 – uint16_t.

CCP4 programs use mode 2 (float) for the electron density, and mode 0 (int8_t) for masks. Mask is 0/1 data that marks part of the volume (e.g. the solvent region). Other modes are not used in crystallography, but may be used for CryoEM data.

The CCP4 format is quite flexible. The data is stored as sections, rows and columns that correspond to a permutation of the X, Y and Z axes as defined in the file header. The file can contain only a part of the asymmetric unit, or more than an asymmetric unit (i.e. redundant data). There are two typical approaches to generate a crystallographic map:

• old-school way: a map covering a molecule with some margin around it is produced using CCP4 utilities such as `fft` and `mapmask`,
• or a map is made for the asymmetric unit (asu), and the program that reads the map is supposed to expand the symmetry. This approach is used by the CCP4 clipper library and by programs that use this library, such as `cfft` and Coot.

The latter approach generates map for exactly one asu, if possible. It is not possible if the shape of the asu in fractional coordinates is not rectangular, and in such case we must have some redundancy. On average, the maps generated for asu are significantly smaller, as compared in the UglyMol wiki.

Nowadays, the CCP4 format is rarely used in crystallography. Almost all programs read the reflection data and calculate maps on the fly.

### C++¶

#### Reading¶

To read and write CCP4 maps you need:

```#include <gemmi/ccp4.hpp>
```

We normally use float type when reading a map file:

```gemmi::Ccp4<float> map;
map.read_ccp4_map("my_map.ccp4");
```

and int8_t when reading a mask (mask typically has only values 0 and 1, but in principle it can have values from -127 to 128):

```gemmi::Ccp4<int8_t> mask;
mask.read_ccp4_map("my_mask.ccp4");
```

If the grid data type does not match the file data type, the library will attempt to convert the data when reading.

#### setup()¶

`read_ccp4_map()` reads the data from file into a Grid class, keeping the same axis order and the same dimensions as in the file. But the functions that operate on the grid data (such as `get_position()`, `interpolate_value()`, `symmetrize()`) expect that the grid covers the whole unit cell and that the axes are in the X,Y,Z order. So before calling a function that uses either the symmetry or the unit cell parameters we need to setup the grid as required:

```map.setup(GridSetup::Full, NAN);
```

The second argument in this call is a value to be used for unknown values, i.e. for values absent in the input file (if the input file does not cover the whole asymmetric unit).

#### Writing¶

To write a map to a file:

```// the file header needs to be prepared/updated with an explicit call
int mode = 2; // ccp4 file mode: 2 for floating-point data, 0 for masks
bool update_stats = true; // update min/max/mean/rms values in the header
map.update_ccp4_header(mode, update_stats);

map.write_ccp4_map(filename);
```

By default, the map written to a file covers the whole unit cell. To cover only a given box, call `set_extent()` before writing the map. Traditionally, CCP4 program MAPMASK was used for this. To cover a molecule with 5Å margin do:

```map.set_extent(calculate_fractional_box(structure, 5));
```

After calling `set_extent()` we have the same situation as before calling `setup()` – some grid functions may not work correctly.

### Python¶

The Python API is similar.

```>>> m = gemmi.read_ccp4_map('../tests/5i55_tiny.ccp4')
>>> m
<gemmi.Ccp4Map with grid 8x6x10 in SG #4>
>>> m.grid  # tiny grid as it is a toy example
<gemmi.FloatGrid(8, 6, 10)>
>>> m.grid.spacegroup
<gemmi.SpaceGroup("P 1 21 1")>
>>> m.grid.unit_cell
<gemmi.UnitCell(29.45, 10.5, 29.7, 90, 111.975, 90)>
>>> m.setup()
>>> m.grid
<gemmi.FloatGrid(60, 24, 60)>
>>> m.write_ccp4_map('out.ccp4')
```

For the low-level access to header one can use the same getters and setters as in the C++ version.

```>>> m.header_float(20), m.header_float(21)  # dmin, dmax
(-0.5310382843017578, 2.3988280296325684)
>>> m.header_i32(28)
0
>>> m.set_header_i32(28, 20140)
>>> m.header_str(57, 80).strip()
'Created by MAPMAN V. 080625/7.8.5 at Wed Jan 3 12:57:38 2018 for A. Nonymous'
```

To write map covering the model with 5Å margin (equivalent of running MAPMASK with XYZIN and BORDER 5) do:

```>>> st = gemmi.read_structure('../tests/5i55.cif')
>>> m.set_extent(st.calculate_fractional_box(margin=5))
>>> m.write_ccp4_map('out.ccp4')
```

Let us end with two examples.

Example 1. A short code that draws a contour plot similar to mapslicer plots (see Fig. 3 in this CCP4 paper if you wonder what is mapslicer). To keep the example short we assume that the lattice vectors are orthogonal.

```import numpy
from matplotlib import pyplot
import gemmi

# toxd_aupatt.map is generated by \$CCP4/examples/unix/runnable/patterson
ccp4 = gemmi.read_ccp4_map('/tmp/wojdyr/toxd_aupatt.map')
ccp4.setup()
arr = numpy.array(ccp4.grid, copy=False)
x = numpy.linspace(0, ccp4.grid.unit_cell.a, num=arr.shape, endpoint=False)
y = numpy.linspace(0, ccp4.grid.unit_cell.b, num=arr.shape, endpoint=False)
X, Y = numpy.meshgrid(x, y, indexing='ij')
pyplot.contour(X, Y, arr[:,:,40])
pyplot.gca().set_aspect('equal', adjustable='box')
pyplot.show()
``` Example 2. A tiny utility that compares two masks (maps with 0/1 values) of the same size, printing a summary of matches and mismatches:

```\$ python examples/maskdiff.py old_mask.ccp4 new_mask.ccp4
Size: 240 x 300 x 270  and  240 x 300 x 270
0-0      5006818  25.76%
1-1     13496058  69.42%
0-1       937124   4.82%
1-0            0   0.00%
total   19440000
```

Here is the script:

```import sys
import numpy
import gemmi

def maskdiff(path1, path2):
mask1 = gemmi.read_ccp4_mask(path1)
mask1.setup()
arr1 = numpy.array(mask1.grid, copy=False)
mask2 = gemmi.read_ccp4_mask(path2)
mask2.setup()
arr2 = numpy.array(mask2.grid, copy=False)
print("Size: %d x %d x %d  and  %d x %d x %d" % (arr1.shape + arr2.shape))
if arr1.shape != arr2.shape:
sys.exit("Different sizes. Exiting.")
t = 2 * (arr1 != 0) + (arr2 != 0)
for (a, b) in [(0, 0), (1, 1), (0, 1), (1, 0)]:
n = numpy.count_nonzero(t == 2*a+b)
print('%d-%d %12d %6.2f%%' % (a, b, n, 100.*n/arr1.size))
print('total %10d' % arr1.size)

if __name__ == '__main__':
if len(sys.argv) != 3:
sys.exit("Usage: maskdiff.py map1 map2")
maskdiff(sys.argv, sys.argv)
```