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The "jump" between direct and reciprocal spaces, mediated by a Fourier transform represented by the electron density function
|Readers who have arrived at this chapter
in a sequential manner will notice that, apart from the phase
the relationship between the diffraction pattern (reciprocal space) and
the crystal structure (direct space) is mediated by a Fourier
transform represented by the electron density function: ρ(xyz), (see the drawing on the left).
Readers will also know that the relationship between these two spaces is "holistic", meaning that the value of this function, at each point in the unit cell of coordinates (xyz), is the result of "adding" the contribution of "all" structure factors (diffracted waves: amplitudes and phases) contained in the diffraction pattern.
They will also remember that the diffraction pattern contains many structural factors (several thousand for a simple structure, and hundreds of thousands for a protein structure).
|Moreover, the number of points in the
unit cell, where the ρ function has to be calculated, is very
high. In a cell of about 100 x 100 x 100 Angstrom3,
would be necessary to calculate at least 1000 points in every unit cell
direction to obtain a resolution of 100/1000, which equals 0.1
Angstrom in each
direction. This means calculating at least 1000 x 1000 x 1000 =
1,000,000,000 points (one billion points) and at each point to
structure factors F(hkl).
It should therefore be clear that, regardless of the difficulties of the phase problem, solving a crystal structure implies the use of computers.
Finally, the analysis of a crystal or molecular structure also implies calculating many geometric parameters that define interatomic distances, bond angles, torsional angles, molecular surfaces, etc., using the atomic coordinates (xyz).
the reasons described above, since the beginning of the use of
Crystallography as a discipline to determine molecular
and crystal structures, crystallographers have devoted special
attention to the development of calculation tools to facilitate
crystallographic work. With this aim, and even before the
computers appeared, the crystallographers introduced the so-called
strips," which were widely used in all Crystallography laboratories.
The Beevers-Lipson strips
The Beevers-Lipson strips
The Beevers-Lipson strips (which were strips of paper containing the values for some trigonometric functions) were used in laboratories to speed up the calculations (by hand) of the Fourier transforms (see above: the electron density function, for example).
These strips were introduced in 1936 by A.H. Beevers and H. Lipson. In the 1960s, more than 300 boxes were distributed to nearly all the laboratories in the world. The nightmare was maintaining this box, which had a narrow base, upright, and mainting the strips in order!
ENIAC (Electronic Numerical Integrator and Computer, 1945) -- the very first electronic computer. Some pictures of the rooms where it was installed.
ENIAC, short for Electronic Numerical Integrator And Computer, was the first general-purpose electronic computer, whose design and construction were financed by the United States Army during the Second World War. It was the first digital computer capable of being reprogrammed to solve a full range of computing problems, especially calculating artillery firing tables for the U.S. Army's Ballistic Research Laboratory.
The ENIAC had immediate importance. When it was announced in 1946, it was heralded in the press as a "Giant Brain". It boasted speeds one thousand times faster than electro-mechanical machines, a leap in computing power that no single machine has matched. This mathematical power, coupled with general-purpose programmability, excited scientists and industrialists.
Besides its speed, the most remarkable thing about ENIAC was its size and complexity. ENIAC had 17,468 vacuum tubes, 7,200 crystal diodes, 1,500 relays, 70,000 resistors, 10,000 capacitors and around 5 million hand-soldered joints. It weighed 27 tons, was roughly 2.6 m by 0.9 m by 26 m, took up 63 m², and consumed 150 kW of power.
Later, with the development of Electronics and Microelectronics, which introduced integrated circuits, computers became accessible to crystallographers, who flocked to these facilities with large boxes of "punched cards" (the only means for data storage at that time), containing the diffraction intensities and their own computer programs.
Around the early 1970s, and for over a decade, crystallographers became a nightmare for the managers and operators of the so-called "computing centers,'' running in some universities and research centers.
A punch card or punched card (or punchcard or Hollerith card or IBM card), is a piece of stiff paper which contains digital information represented by the presence or absence of holes in predefined positions. It was used by crystallographers until the end ot the 1970s.
|Punched paper tape (shown in yellow) and
different magnetic tapes used for data storage during the
1970s and 1980s.
In the 1980s the laboratories of Crystallography became "flooded" with computers, which for the first time gave crystallographers independence from the large computing centers. The VAX series of computers (sold by the company Digital Equipment Corporation) marked a splendid era for crystallographic calculations. They allowed the use of magnetic tapes and the first hard disks, with limited capacity (only a few hundred MB) -- very big and heavy, but they eliminated the need for the tedious punched cards. Nostalgics should have a look into this link.!!!
Over the years, crystallographic computing has become easy and affordable thanks to personal computers (PC), which meet nearly all the needs of most conventional crystallographic calculations, at least concerning crystals of low and medium complexity (up to hundreds of atoms). Their relative low price and their ability to be assembled into "farms" (for distributed calculation) provide crystallographers the best solution for almost any type of calculation.
However, the crystallography applied to macromolecules not only needs what we could call "hard" computing. The management of large electron density maps, which are used to build the molecular structure of proteins, as well as the subsequent structural analysis, requires more sophisticated computers with powerful graphic processors and, if possible, with the capability of displaying 3-dimensional images using specialized glasses...
A typical graphic computer used to visualize 3-dimensional electron density maps and structures. The processor and the screen are complemented by an infrared transmitter (black box on the screen) and the glasses used by the crystallographer.
The current computing facilities represent a big jump respect to the capabilities available during the mid-twentieth century, as it is shown in the representation of the structural model used for the structural description of penicillin, based on three 2-dimensional electron density maps...
Three-dimensional model of the structure of penicillin, based on the use of three 2-dimensional electron density maps, as used by Dorothy C. Hodgkin, Nobel laureate in 1964
And even 3d maps where also used!...
Representation of 3d electron density maps used until the middle of the 1970's. The contours are lines of electron density and show the positions of individual atoms in the structure
A typical computer (of the VAX series) used in many Crystallography laboratories during the 1980s.
A typical personal computer (PC) used in the 2000s
A typical PC-farm used in the 2000s
A typical personal computer commonly used since 2010 for crystallographic calculations
|Crystallographic computer programs|
|Macromolecules:||The Web-Book of the Department of Crystallography & Structural Biology (CSIC)|
|Of general interest:||The crystallographic software list maintained by the International Union of Crystallography - (IUCr)|
|Structural databases and databanks||P=public; L=license needed|
|Metals and intermetallic compounds||CRYSTMET||L|
|Organic and organometallic compounds||CSD||L|
|Proteines, Nucleic acids and large complexes||PDB||P|