## Calculating a 3D map from topographic surface images

The atomic force microscope produces a topographic map of the surface of a specimen convolved with the shape of the tip.

A 2D image is interpreted as height data defining the edge of an envelope in 3D space. To ensure a smooth appearance when isosurfaced, this edge is generated using a function which approximates the integral of a Gaussian function:

where σ is very similar to the Gaussian width parameter sigma, a = 1.618 and h(x,y) is the height value at pixel (x,y) in the topographic image. The expansion of a 2D image to a 3D edge or surface requires at a minimum the specification of the new z dimension (in pixels), the range of surface convolution (in Å), and the sampling (in Å/pixel):

bsurf -v 7 -expand 20 -pos 13,17 -sam 1.5 -resol 3.5 -inv brcy1.tif brcy1.mrc

This command line will generate a volume, 20 voxels thick and with the minimum and maximum of the surface placed at 13 and 17 angstrom, respectively. The resolution option smooths the edge.

## Bacteriorhodopsin example

The required starting files for this example:

bacteriorhodopsin_example.tgz

These include the cytoplasmic and extracellular AFM topographs of bacteriorhododopsin crystals:

These topographs were used to generate the two surfaces:

bsurf -v 7 -expand 80 -pos 67,75 -sam 1 -ori 62,62,40 -resol 3.5 wt_cyt.tiff wt_cyt.mrc

bsurf -v 7 -expand 80 -pos 56,61 -sam 1 -ori 62,62,40 -resol 3.5 wt_ext.tiff wt_ext.mrc

The extracellular surface was then rotated to bring it into the same frame as the cytoplasmic surface:

bint -verb 7 -rot 0,1,0,180 -fill 0 wt_ext.mrc wt_ext_rot.mrc

These surfaces were then combined into one volume:

bop -verb 7 -small 1,0 wt_ext_rot.mrc wt_cyt.mrc wt_2surf.mrc

The result is a volume bounded by the two surfaces. The atomic structure of the bacteriorhododopsin trimer (1QHJ) was then position between the surfaces, with the top surface facing the cytoplasm: