- Email: [email protected]
The Patterson function
TIBS - J u ~ 1984
328 compare to the two M*ADP and M*AMP.PNP states? (2) Are there two M*ATP states? (3) Are there two states for the actomyosin complex? (4) How does the binding of actin to myosin affect the structure of myosin? (5) Where are the actin binding and ATPase catalytic sites located on the globular head, and how do they interact? (6) What is the three-dimensional molecular structure of the myosin globular head? Finally, techniques are needed which allow the observation of myosin structural transitions in a working fiber. Two techniques which have great potential, and are presently being used, are timeresolved X-ray diffraction of muscle fibers using synchrotron radiation28 and fluctuation analysis of chromophores attached to myosin crossbridges in fibers 29.
Acknowledgements The author is supported by the National Institutes of Health (New Investigator Research Award, AM3071201), the Muscular Dystrophy Association, the American Heart Association (Illinois
Affiliate), the Research Corporation, and the Apple Educational Foundation.
References A more complete appreciation of the work of the many contributing laboratories may be obtained from the references (in particular Refs 5, 8, 17, 23, 24 and 27). 1 Hill, T. L. (1977) Free Energy Transduetion in Biology, Academic Press 2 Huxley, H. (1969) Science 114, 1356: Huxley, A. F. and Simmons, R. M. Nature 233, 533-538 3 Highsmith, S. and Cooke, R. (1983) in Cell and Muscle Motility, Vol. 4 (Dowben, R. M. and Shay, J. W., eds), Plenum Press 4 Harrington, W. F. (1979) Proc. NatlAcad. Sci. USA 76, 5066-5070 5 Sleep, J. A. and Smith, S. J. (1981) Current Topics in Bioenergetics 11,239-286 6 Weeds, A. G. and Pope, B. (1977) J. Mol. Biol. 111,129-157 7 White, H. D. and Taylor, E. W. (1976) Biochemistry 15, 5818-5826 8 Eisenberg, E. and Greene, L. E. (1980) Annu. Rev. Physiol. 42, 293-309 9 Morita, F. (1977) J. Biochemistry 81,313--320 10 Bechet, J.-J., Breda, C., Guinand, S., Hill, M. and d'Albis, A. (1979) Biochemistry 18, 4080-4089 11 Shriver, J. W. and Sykes, B. D. (1981) Biochemistry 20, 2004-2012 12 Shriver, J. W. and Sykes, B. D. (1981) Biochemistry 20, 6357-6362 13 Shriver, J. W. and Sykes, B. D. (1982) B/o-
chemistry 21, 3022-3028 14 Shrivel J. W., Kay, L. E. and Sykes, B. D. (1982) Biophysical J. 37, 54a 15 Trybus, K. M. and Taylor E. W. (1982) Biochemistry 21, 1284-1294 16 Chock, S. P. (1981) J. Biol. Chem. 256, 1096110966 17 Kourad, M. and Goody, R. S. (1982) Eur. J. Biochem. 128, 547-555 18 Swenson, C. and Ritchie, P. A. (1979) Biochemistry 18, 3654-3658 19 White, H. D. (1977) Biophysical J. 17, 40a 20 Burke, M., Reisler, E. and Harrk, gton, W. F. (1976) Biochemistry 15, 1923-1927 21 Chalovich, J. M., Greene, L. E. and Eisenberg, E. (1983) Proc. Natl Acad. Sci. USA 80, 4909--4913 22 Wells, J. A. and Yount, R. G. (1982) Methods in Enzymology, Vol. 85, pp. 93-116, Academic Press 23 Kuhn, H. J. (1981) J. Muscle Res. Cell Motility 2, 7-44 24 Tregear, R. T. and Marston, S. B. (1979) Annu. Rev. Physiol. 41,723-736 25 Stein, R. B., Gordon, T. and Shriver, J. (1982) Biophysical J. 40, 97-107. (See also Bressler, B. H. (1981) Can. J. Physiol. and Pharmacology 59, 548-554) 26 Hammes, G. (1983) Proc. Natl Acad. Sci. USA 79, 6881-6884 27 Morales, M. F. and Botts, J. (1979) Proc. Natl Acad. Sci. USA 76, 3857-3859 28 Huxley, H. E., Simmons, R. M., Faruqi, A. R., Kress, M., Bordas, J. and Koch, M. H. J. (1983) J. Mol. Biol. 169, 469 29 Borejdo, J., Putman, S. and Morales, M. F. (1979) Proc. Natl Acad. Sci. USA 76, 6346
50 years ago The Patterson function Jenny P. Glusker distribution in the diffraction pattern. When he inquired of chemists which were the interesting structures to look at, he was sent to see Pope and Barlow. They had predicted that alkali halides consisted of ions arranged as closely packed spheres, and Pope urged W. L. Bragg to investigate their structures. This prediction was thereby experimentally verified. Several other interesting small structures, such as diamond with tetrahedrally coordinated carbon atoms and C-C distances of 1.54 A, were determined. Bragg's method depended on high symmetry in the structure so that only a few parameters had to be derived from the diffraction pattern. For example, hexamethylenetetramine, C6N4Hx2, crystallizes in a cubic unit cell and, using Jenny P. Glusker is at the Fox Chase Cancer the symmetry of the space group of the Center, Institute for Cancer Research, 7701 crystal, it was shown that the structure Burholme Avenue, Philadelphia, PA 19111, USA. (excluding hydrogen atoms) could be
It is now 72 years since the first X-ray diffraction experiment was carried out and 50 years since the first diffraction patterns of proteins were obtained, and it was not until that year, 1934, that it became possible for X-ray diffraction data from crystals of most compounds to be interpreted in terms of atomic arrangements within the crystals. The Patterson function made this possible. In 1912 Max von Laue realized that if Xrays were wave-like (rather than particlelike) they should be diffracted by a grating with a very small repeat spacing, such as that provided by crystals. This idea was confirmed experimentally by Friedrich and Knipping. W. L. Bragg then showed how structural information could be extracted from the intensity
~) 1984,ElsevierSciencePublishersB,V, Amsterdam 1137li- 5067/84/$02.(1I
defined by two unknown parameters; these were determined by an examination of the diffraction intensities. However, it was very frustrating to the chemist that only the simplest of compounds, only those that crystallized with high
A. L. Patterson
TIBS
- July
1984
symmetry, could have their structures determined. Diffraction patterns of more complex compounds could readily be obtained, but they were generally uninterpretable in terms of structure. In some cases a possible model of the structure could be built, its diffraction pattern calculated, and compared with that observed; this 'trial-and-error' method of solving structures was timeconsuming and not very satisfactory in practice. So there is a period from 1914 to 1934 during which very few structure determinations by X-ray diffraction from crystals were reported. The only gleam of hope occurred when Cork showed that when crystals are isomorphous (with almost identical atomic arrangements but with one atom replaced by a different atom), a comparison of their diffraction patterns would give structural information. This gave rise to the method of isomorphous replacement. Unfortunately not many small structures are truly isomorphous and therefore this method was rarely applicable (although it is now used in all macromolecular structure determinations). Lindo Patterson realized that this was an important problem to solve. Born in New Zealand in 1902, he was educated in Canada and England and studied physics at McGill University, where he obtained a bachelor's degree in 1923 and a master's degree in 1924. He obtained a fellowship to study with W. H. Bragg at the Davy-Faraday Laboratory of the Royal Institution in London for two years. Bragg was, at that time, interested in the structures of naphthalene and anthracene, while Mfiller and Shearer were studying longchain compounds. Therefore Patterson studied some compounds with characteristics intermediate between them, the phenylaliphatic acids. Since structure determinations could not be contemplated at that time, only unit cell dimensions were determined. To do this Patterson had to build his own X-ray equ~ment. He then went to work with Hermann Mark at the Kaiser-WilhelmInstitute in Dahlem to study the theory of particle-size line broadening, and finally returned to McGill where he obtained his Ph.D. in 1928. He moved to the Rockefeller Institute in New York to work from 1929-1931 with R. W. G. Wyckoff on cyclohexane derivatives. As he later wrote, he had an 'obsession with the notion that something was to be learned about structural analysis from Fourier theory'1; in other words he knew that something had to be done about the problem of determining crystal
(
,o
329
!I
( ( a
)
(
a/2
J,
Fig. 1. Potassium dihydrogen phosphate. Electron density on the left and hkO Patterson map (origin at the center) on the right. The structure is projected down the c axis. The space group is I42d with the positions of potassium and phosphorus fixed by symmetry positions. However, the oxygen atom positions could be found from the Patterson map as shown. In this view P lies directly over K +. P . . . . . 0 vector~ are drawn on each map. Note that the alternate orientation of each phosphate group (as required by the space group) is also indicated in the Patterson map (left).
structures from diffraction patterns and had an intuitive feeling that this could be done mathematically. Fourier had derived mathematical expressions that could be used to describe a periodic (regularly repeating) function (as found in a crystal). This was done by forming the sum of a set of cosine waves (also regularly repeating functions) of appropriately differing amplitudes, frequencies, and relative phases. Such expressions were first used in X-ray crystallography by W. H. Bragg. Two-dimensional Fourier summations were first introduced by W. L. Bragg (following a suggestion from his father, W. H. Bragg) in his study of diopside and are still used today in the calculation of electron-density maps. By 1930 Lindo was spending hour after hour looking through the tables of contents of the mathematical journals in the New York Public Library, searching for clues on how to proceed. He wrote 'Whenever the title was promising I looked through the paper and whenever the paper was promising I read it carefully'. After two more years, spent at the Johnson Foundation in Philadelphia studying biological materials, he had saved up enough money to support himself for a year so that he could work exclusively on this 'obsession'. He went as an unpaid guest scientist to the laboratory of Bert Warren at MIT. The person who knew most at that time about Fourier integrals was Norbert Wiener. Fortunately both Wiener and Patterson were fans of Gilbert and Sullivan operas and together they sang songs from them. During the singing of the songs Lindo would interpose a question such as 'What do you know about a function representable by a Fourier
series when you know only the amplitudes of the Fourier coefficients?' Weiner replied 'You know the Faltung'. The Faltung is the integral involving the electron density at a point x, and at another point x + t; if both points have high electron density then there will be a peak at a point t in the Faltung and this then represents the vector between two points of high electron density. Lindo wrote 'The understanding of the Faltung came, of course, from the work on fiquids and their radial distributions. Warren with Gingrich and others had perfected the techniques used by Debye and Menke 2 in the study of the X-ray scattering from liquids. These were based on the original suggestions of Zernike and Prins 3. Warren and Gingrich 4 had already had the idea that these methods applied to powders would give the radial distribution in a crystal. While trying to learn about their work, I noticed that the mathematical form of the theory given by Debye and Menke would be identical with that of the Faltung if the integrations over random orientation were left out and the randomness of choice of origin was left in. What was immediately apparent was that the crystal contained atoms and that the Faltung of a set of atoms was very special in that it would consist of a set of atom-like peaks whose centers were specified by the distances between the atoms in the crystal.' Thus the Patterson or ]_FI2 series was proposed. Lindo had only three days from the realization that the I_FI2 series would give information on atomic arrangements to the deadline for the 1934 spring meeting of the American Physical Society. Papers were submitted by Warren on the radial distribution in carbon black and by
T1BS - July 1984
330 Gingrich and Warren on the radial distribution in powders. Patterson only had time, in view of the primitive computing equipment available, to determine the 'Patterson map' for one projection of potassium dihydrogen phosphate (Fig. 1). He called this a n '[El 2 series', but it is generally known as a Patterson function 5'6. It is computed using the squares of the structure amplitudes of the diffracted X-ray beams (directly related to the intensities) and the indexing (order) of the diffracted beam. Cosine waves, all with identical phases (0°) are summed. The coefficients are obtained directly from the measured intensities and no phase information is required. The Patterson function contains peaks at the positions (relative to the origin) of all possible interatomic vectors between the atoms of the crystal under study. Thus the distances between any two atoms are specified, as for liquids and powdered solids; but, in addition, the direction of the vector between them is also specified. The height of each peak in the Patterson map is proportional to the product of the atomic numbers of the two atoms at the ends of the vector. In addition, the symmetry of the Patterson map depends on the space group of the crystal, differing from it only by the additional presence of a center of symmetry. H. F. Judson 7, in The Eighth Day of Creation compares the Patterson function to a cocktail party to which one hundred strangers have been invited. If each person's shoes are nailed to the floor, he wrote, then handshakes must involve different directions and lengths of arms. In all nearly five thousand such handshakes must take place, but they give information on the relative arrangement of guests. Just as a very important step in structure determination was taken, when it was shown that the structure amplitudes of the diffracted beams are the coefficients of the Fourier series that must be used to represent electron density, another important step was taken when Patterson realized that the squares of the structure amplitudes of the diffracted beams could be used as coefficients in a Fourier series that would then contain all interatomic vectors. The important feature of the Patterson series is that it does not depend on any knowledge of the relative phases of the diffracted beams, which is information that is lost in the diffraction experiment. Only the intensities of the diffracted beams are required, and these are experimentally measurable. In addition, the Patterson
map depends only on the crystal class, knowledge of which may narrow the choice of possible space groups so that information on the space group may often be obtained. Harker 8 extended the interpretation of Patterson maps and showed that certain portions of the Patterson map, depending on the space group of the crystal under study, contain a large proportion of the readily interpretable structural information. This was a particularly useful discovery in assisting in the analysis in the days when computing was very tedious and expensive because then only a small portion of the Patterson map need be calculated. It is not a simple task to obtain the actual atomic arrangement in a crystal from a Patterson map, especially if some atoms have similar atomic numbers and many coincident interatomic vectors. There are n 2 interatomic vectors (peaks) in the Patterson map of a unit cell containing n atoms (n at the origin because n atoms are zero distance from themselves). As n increases to the value found for a protein, say 103, the number of vectors increases to 106 in the same volume. Thus the number of vectors in a given volume is sufficiently large that the resolution of individual peaks representing individual interatomic vectors may be difficult or impossible. However, if the unit cell contains no more than about 100 light atoms together with one or more heavy atoms (large atomic number), then the vectors between the heavy atoms will dominate the Patterson map, and the structure solution may be much simplified. There are several methods for solving a Patterson map that involve transcribing this map upon itself with different relative origins (minimum function vector superposition map) 9, by rotating the map if it is suspected that there are two or more identical groups in the crystal (rotation function) l°, or by comparing the map with a vector map calculated for known molecular fragments of the molecule in the crystal under study n. In a minimum-function map 19, the origin of the Patterson map is put in turn on each of the symmetry-related positions of a known atom (such as a heavy atom). The smaller value at each point in the superposed functions is then recorded; this minimum-function map will contain only peaks obtained from all the maps used in the superpositions and will not be affected by a high peak that is found on only one map. This superposition process is repeated until the structure is revealed. The rotation function, described by Rossmann and Blow 1°,
measures the degree of coincidence over a given volume when one Patterson function is rotated on another identical one. In the case of proteins such a rotation of a Patterson map upon itself may indicate some symmetry in the macromolecular structure under study. In the third method, when part of the structure of a molecule is known, one can try to find the entire arrangement in the crystal by seeking a fit between the experimental Patterson function and the computed vector map of the known fragment u of the molecule. A match between the two maps will indicate the orientation of the known fragment of the structure in the unit cell, from which approximate phases and an approximate electron-density map can be calculated. Recently the fiftieth anniversary of the first X-ray photographs of proteins was celebrated. This work showed that crystalline proteins had sufficient order for the determination of their atomic structure to high resolution. Today the structures of many such macromolecules proteins, nucleic acids and viruses are known in detail as a result of the use of the Patterson function. While different (although related) methods are used for small structures today the Patterson map and isomorphous replacement methods are the principal tools used to interpret diffraction patterns of macromolecules. Now that tuned synchrotron radiation can be used in diffraction experiments, and hence anomalous scattering effects can be readily studied, Patterson methods 12a3 may be developed further as aids in analyses of complicated and asymmetrical structures. -
References 1 Patterson, A. L. (1962) in Fifty Years of X-ray Diffraction (Ewald, P. P., ed.), A. Oosthoek 2 Debye, P. and Menke, H. (1930) Physik. Z. 31,797-798 3 Zernike, F. and Prins, J. A. (1927) Z. far Physik 41,184-194 4 Warren, B. E. and Gingrich, N. S. (1934) Phys. Rev. 46, 368-372 5 Patterson, A. L. (1934) Phys. Rev. 46, 372-376 6 Patterson, A. L. (1935) Z. far Krist. A90, 517-542 7 Judson, H. F. (1979) The Eighth Day of Creation: Makers of the Revolution in Biology, Simon and Schuster 8 Harker, D. (1936) J. Chem. Phys. 4, 381 9 Beevers, C. A. and Robertson, J. H. (1950) Acta Cryst. 3, 164 10 Rossmann, M. G. and Blow, D. M. (1962) Acta Cryst. 15, 24-31 11 Nordman, C. E. and Nakatsu, K. (1963) J. Amer. Chem. Soc. 85, 353-354 12 Patterson, A. L. (1963) Acta Cryst. 16, 1255-1256 13 Okaya, Y., Saito, Y. and Pepinsky, R. (1955) Phys. Rev. 98, 1857
328 compare to the two M*ADP and M*AMP.PNP states? (2) Are there two M*ATP states? (3) Are there two states for the actomyosin complex? (4) How does the binding of actin to myosin affect the structure of myosin? (5) Where are the actin binding and ATPase catalytic sites located on the globular head, and how do they interact? (6) What is the three-dimensional molecular structure of the myosin globular head? Finally, techniques are needed which allow the observation of myosin structural transitions in a working fiber. Two techniques which have great potential, and are presently being used, are timeresolved X-ray diffraction of muscle fibers using synchrotron radiation28 and fluctuation analysis of chromophores attached to myosin crossbridges in fibers 29.
Acknowledgements The author is supported by the National Institutes of Health (New Investigator Research Award, AM3071201), the Muscular Dystrophy Association, the American Heart Association (Illinois
Affiliate), the Research Corporation, and the Apple Educational Foundation.
References A more complete appreciation of the work of the many contributing laboratories may be obtained from the references (in particular Refs 5, 8, 17, 23, 24 and 27). 1 Hill, T. L. (1977) Free Energy Transduetion in Biology, Academic Press 2 Huxley, H. (1969) Science 114, 1356: Huxley, A. F. and Simmons, R. M. Nature 233, 533-538 3 Highsmith, S. and Cooke, R. (1983) in Cell and Muscle Motility, Vol. 4 (Dowben, R. M. and Shay, J. W., eds), Plenum Press 4 Harrington, W. F. (1979) Proc. NatlAcad. Sci. USA 76, 5066-5070 5 Sleep, J. A. and Smith, S. J. (1981) Current Topics in Bioenergetics 11,239-286 6 Weeds, A. G. and Pope, B. (1977) J. Mol. Biol. 111,129-157 7 White, H. D. and Taylor, E. W. (1976) Biochemistry 15, 5818-5826 8 Eisenberg, E. and Greene, L. E. (1980) Annu. Rev. Physiol. 42, 293-309 9 Morita, F. (1977) J. Biochemistry 81,313--320 10 Bechet, J.-J., Breda, C., Guinand, S., Hill, M. and d'Albis, A. (1979) Biochemistry 18, 4080-4089 11 Shriver, J. W. and Sykes, B. D. (1981) Biochemistry 20, 2004-2012 12 Shriver, J. W. and Sykes, B. D. (1981) Biochemistry 20, 6357-6362 13 Shriver, J. W. and Sykes, B. D. (1982) B/o-
chemistry 21, 3022-3028 14 Shrivel J. W., Kay, L. E. and Sykes, B. D. (1982) Biophysical J. 37, 54a 15 Trybus, K. M. and Taylor E. W. (1982) Biochemistry 21, 1284-1294 16 Chock, S. P. (1981) J. Biol. Chem. 256, 1096110966 17 Kourad, M. and Goody, R. S. (1982) Eur. J. Biochem. 128, 547-555 18 Swenson, C. and Ritchie, P. A. (1979) Biochemistry 18, 3654-3658 19 White, H. D. (1977) Biophysical J. 17, 40a 20 Burke, M., Reisler, E. and Harrk, gton, W. F. (1976) Biochemistry 15, 1923-1927 21 Chalovich, J. M., Greene, L. E. and Eisenberg, E. (1983) Proc. Natl Acad. Sci. USA 80, 4909--4913 22 Wells, J. A. and Yount, R. G. (1982) Methods in Enzymology, Vol. 85, pp. 93-116, Academic Press 23 Kuhn, H. J. (1981) J. Muscle Res. Cell Motility 2, 7-44 24 Tregear, R. T. and Marston, S. B. (1979) Annu. Rev. Physiol. 41,723-736 25 Stein, R. B., Gordon, T. and Shriver, J. (1982) Biophysical J. 40, 97-107. (See also Bressler, B. H. (1981) Can. J. Physiol. and Pharmacology 59, 548-554) 26 Hammes, G. (1983) Proc. Natl Acad. Sci. USA 79, 6881-6884 27 Morales, M. F. and Botts, J. (1979) Proc. Natl Acad. Sci. USA 76, 3857-3859 28 Huxley, H. E., Simmons, R. M., Faruqi, A. R., Kress, M., Bordas, J. and Koch, M. H. J. (1983) J. Mol. Biol. 169, 469 29 Borejdo, J., Putman, S. and Morales, M. F. (1979) Proc. Natl Acad. Sci. USA 76, 6346
50 years ago The Patterson function Jenny P. Glusker distribution in the diffraction pattern. When he inquired of chemists which were the interesting structures to look at, he was sent to see Pope and Barlow. They had predicted that alkali halides consisted of ions arranged as closely packed spheres, and Pope urged W. L. Bragg to investigate their structures. This prediction was thereby experimentally verified. Several other interesting small structures, such as diamond with tetrahedrally coordinated carbon atoms and C-C distances of 1.54 A, were determined. Bragg's method depended on high symmetry in the structure so that only a few parameters had to be derived from the diffraction pattern. For example, hexamethylenetetramine, C6N4Hx2, crystallizes in a cubic unit cell and, using Jenny P. Glusker is at the Fox Chase Cancer the symmetry of the space group of the Center, Institute for Cancer Research, 7701 crystal, it was shown that the structure Burholme Avenue, Philadelphia, PA 19111, USA. (excluding hydrogen atoms) could be
It is now 72 years since the first X-ray diffraction experiment was carried out and 50 years since the first diffraction patterns of proteins were obtained, and it was not until that year, 1934, that it became possible for X-ray diffraction data from crystals of most compounds to be interpreted in terms of atomic arrangements within the crystals. The Patterson function made this possible. In 1912 Max von Laue realized that if Xrays were wave-like (rather than particlelike) they should be diffracted by a grating with a very small repeat spacing, such as that provided by crystals. This idea was confirmed experimentally by Friedrich and Knipping. W. L. Bragg then showed how structural information could be extracted from the intensity
~) 1984,ElsevierSciencePublishersB,V, Amsterdam 1137li- 5067/84/$02.(1I
defined by two unknown parameters; these were determined by an examination of the diffraction intensities. However, it was very frustrating to the chemist that only the simplest of compounds, only those that crystallized with high
A. L. Patterson
TIBS
- July
1984
symmetry, could have their structures determined. Diffraction patterns of more complex compounds could readily be obtained, but they were generally uninterpretable in terms of structure. In some cases a possible model of the structure could be built, its diffraction pattern calculated, and compared with that observed; this 'trial-and-error' method of solving structures was timeconsuming and not very satisfactory in practice. So there is a period from 1914 to 1934 during which very few structure determinations by X-ray diffraction from crystals were reported. The only gleam of hope occurred when Cork showed that when crystals are isomorphous (with almost identical atomic arrangements but with one atom replaced by a different atom), a comparison of their diffraction patterns would give structural information. This gave rise to the method of isomorphous replacement. Unfortunately not many small structures are truly isomorphous and therefore this method was rarely applicable (although it is now used in all macromolecular structure determinations). Lindo Patterson realized that this was an important problem to solve. Born in New Zealand in 1902, he was educated in Canada and England and studied physics at McGill University, where he obtained a bachelor's degree in 1923 and a master's degree in 1924. He obtained a fellowship to study with W. H. Bragg at the Davy-Faraday Laboratory of the Royal Institution in London for two years. Bragg was, at that time, interested in the structures of naphthalene and anthracene, while Mfiller and Shearer were studying longchain compounds. Therefore Patterson studied some compounds with characteristics intermediate between them, the phenylaliphatic acids. Since structure determinations could not be contemplated at that time, only unit cell dimensions were determined. To do this Patterson had to build his own X-ray equ~ment. He then went to work with Hermann Mark at the Kaiser-WilhelmInstitute in Dahlem to study the theory of particle-size line broadening, and finally returned to McGill where he obtained his Ph.D. in 1928. He moved to the Rockefeller Institute in New York to work from 1929-1931 with R. W. G. Wyckoff on cyclohexane derivatives. As he later wrote, he had an 'obsession with the notion that something was to be learned about structural analysis from Fourier theory'1; in other words he knew that something had to be done about the problem of determining crystal
(
,o
329
!I
( ( a
)
(
a/2
J,
Fig. 1. Potassium dihydrogen phosphate. Electron density on the left and hkO Patterson map (origin at the center) on the right. The structure is projected down the c axis. The space group is I42d with the positions of potassium and phosphorus fixed by symmetry positions. However, the oxygen atom positions could be found from the Patterson map as shown. In this view P lies directly over K +. P . . . . . 0 vector~ are drawn on each map. Note that the alternate orientation of each phosphate group (as required by the space group) is also indicated in the Patterson map (left).
structures from diffraction patterns and had an intuitive feeling that this could be done mathematically. Fourier had derived mathematical expressions that could be used to describe a periodic (regularly repeating) function (as found in a crystal). This was done by forming the sum of a set of cosine waves (also regularly repeating functions) of appropriately differing amplitudes, frequencies, and relative phases. Such expressions were first used in X-ray crystallography by W. H. Bragg. Two-dimensional Fourier summations were first introduced by W. L. Bragg (following a suggestion from his father, W. H. Bragg) in his study of diopside and are still used today in the calculation of electron-density maps. By 1930 Lindo was spending hour after hour looking through the tables of contents of the mathematical journals in the New York Public Library, searching for clues on how to proceed. He wrote 'Whenever the title was promising I looked through the paper and whenever the paper was promising I read it carefully'. After two more years, spent at the Johnson Foundation in Philadelphia studying biological materials, he had saved up enough money to support himself for a year so that he could work exclusively on this 'obsession'. He went as an unpaid guest scientist to the laboratory of Bert Warren at MIT. The person who knew most at that time about Fourier integrals was Norbert Wiener. Fortunately both Wiener and Patterson were fans of Gilbert and Sullivan operas and together they sang songs from them. During the singing of the songs Lindo would interpose a question such as 'What do you know about a function representable by a Fourier
series when you know only the amplitudes of the Fourier coefficients?' Weiner replied 'You know the Faltung'. The Faltung is the integral involving the electron density at a point x, and at another point x + t; if both points have high electron density then there will be a peak at a point t in the Faltung and this then represents the vector between two points of high electron density. Lindo wrote 'The understanding of the Faltung came, of course, from the work on fiquids and their radial distributions. Warren with Gingrich and others had perfected the techniques used by Debye and Menke 2 in the study of the X-ray scattering from liquids. These were based on the original suggestions of Zernike and Prins 3. Warren and Gingrich 4 had already had the idea that these methods applied to powders would give the radial distribution in a crystal. While trying to learn about their work, I noticed that the mathematical form of the theory given by Debye and Menke would be identical with that of the Faltung if the integrations over random orientation were left out and the randomness of choice of origin was left in. What was immediately apparent was that the crystal contained atoms and that the Faltung of a set of atoms was very special in that it would consist of a set of atom-like peaks whose centers were specified by the distances between the atoms in the crystal.' Thus the Patterson or ]_FI2 series was proposed. Lindo had only three days from the realization that the I_FI2 series would give information on atomic arrangements to the deadline for the 1934 spring meeting of the American Physical Society. Papers were submitted by Warren on the radial distribution in carbon black and by
T1BS - July 1984
330 Gingrich and Warren on the radial distribution in powders. Patterson only had time, in view of the primitive computing equipment available, to determine the 'Patterson map' for one projection of potassium dihydrogen phosphate (Fig. 1). He called this a n '[El 2 series', but it is generally known as a Patterson function 5'6. It is computed using the squares of the structure amplitudes of the diffracted X-ray beams (directly related to the intensities) and the indexing (order) of the diffracted beam. Cosine waves, all with identical phases (0°) are summed. The coefficients are obtained directly from the measured intensities and no phase information is required. The Patterson function contains peaks at the positions (relative to the origin) of all possible interatomic vectors between the atoms of the crystal under study. Thus the distances between any two atoms are specified, as for liquids and powdered solids; but, in addition, the direction of the vector between them is also specified. The height of each peak in the Patterson map is proportional to the product of the atomic numbers of the two atoms at the ends of the vector. In addition, the symmetry of the Patterson map depends on the space group of the crystal, differing from it only by the additional presence of a center of symmetry. H. F. Judson 7, in The Eighth Day of Creation compares the Patterson function to a cocktail party to which one hundred strangers have been invited. If each person's shoes are nailed to the floor, he wrote, then handshakes must involve different directions and lengths of arms. In all nearly five thousand such handshakes must take place, but they give information on the relative arrangement of guests. Just as a very important step in structure determination was taken, when it was shown that the structure amplitudes of the diffracted beams are the coefficients of the Fourier series that must be used to represent electron density, another important step was taken when Patterson realized that the squares of the structure amplitudes of the diffracted beams could be used as coefficients in a Fourier series that would then contain all interatomic vectors. The important feature of the Patterson series is that it does not depend on any knowledge of the relative phases of the diffracted beams, which is information that is lost in the diffraction experiment. Only the intensities of the diffracted beams are required, and these are experimentally measurable. In addition, the Patterson
map depends only on the crystal class, knowledge of which may narrow the choice of possible space groups so that information on the space group may often be obtained. Harker 8 extended the interpretation of Patterson maps and showed that certain portions of the Patterson map, depending on the space group of the crystal under study, contain a large proportion of the readily interpretable structural information. This was a particularly useful discovery in assisting in the analysis in the days when computing was very tedious and expensive because then only a small portion of the Patterson map need be calculated. It is not a simple task to obtain the actual atomic arrangement in a crystal from a Patterson map, especially if some atoms have similar atomic numbers and many coincident interatomic vectors. There are n 2 interatomic vectors (peaks) in the Patterson map of a unit cell containing n atoms (n at the origin because n atoms are zero distance from themselves). As n increases to the value found for a protein, say 103, the number of vectors increases to 106 in the same volume. Thus the number of vectors in a given volume is sufficiently large that the resolution of individual peaks representing individual interatomic vectors may be difficult or impossible. However, if the unit cell contains no more than about 100 light atoms together with one or more heavy atoms (large atomic number), then the vectors between the heavy atoms will dominate the Patterson map, and the structure solution may be much simplified. There are several methods for solving a Patterson map that involve transcribing this map upon itself with different relative origins (minimum function vector superposition map) 9, by rotating the map if it is suspected that there are two or more identical groups in the crystal (rotation function) l°, or by comparing the map with a vector map calculated for known molecular fragments of the molecule in the crystal under study n. In a minimum-function map 19, the origin of the Patterson map is put in turn on each of the symmetry-related positions of a known atom (such as a heavy atom). The smaller value at each point in the superposed functions is then recorded; this minimum-function map will contain only peaks obtained from all the maps used in the superpositions and will not be affected by a high peak that is found on only one map. This superposition process is repeated until the structure is revealed. The rotation function, described by Rossmann and Blow 1°,
measures the degree of coincidence over a given volume when one Patterson function is rotated on another identical one. In the case of proteins such a rotation of a Patterson map upon itself may indicate some symmetry in the macromolecular structure under study. In the third method, when part of the structure of a molecule is known, one can try to find the entire arrangement in the crystal by seeking a fit between the experimental Patterson function and the computed vector map of the known fragment u of the molecule. A match between the two maps will indicate the orientation of the known fragment of the structure in the unit cell, from which approximate phases and an approximate electron-density map can be calculated. Recently the fiftieth anniversary of the first X-ray photographs of proteins was celebrated. This work showed that crystalline proteins had sufficient order for the determination of their atomic structure to high resolution. Today the structures of many such macromolecules proteins, nucleic acids and viruses are known in detail as a result of the use of the Patterson function. While different (although related) methods are used for small structures today the Patterson map and isomorphous replacement methods are the principal tools used to interpret diffraction patterns of macromolecules. Now that tuned synchrotron radiation can be used in diffraction experiments, and hence anomalous scattering effects can be readily studied, Patterson methods 12a3 may be developed further as aids in analyses of complicated and asymmetrical structures. -
References 1 Patterson, A. L. (1962) in Fifty Years of X-ray Diffraction (Ewald, P. P., ed.), A. Oosthoek 2 Debye, P. and Menke, H. (1930) Physik. Z. 31,797-798 3 Zernike, F. and Prins, J. A. (1927) Z. far Physik 41,184-194 4 Warren, B. E. and Gingrich, N. S. (1934) Phys. Rev. 46, 368-372 5 Patterson, A. L. (1934) Phys. Rev. 46, 372-376 6 Patterson, A. L. (1935) Z. far Krist. A90, 517-542 7 Judson, H. F. (1979) The Eighth Day of Creation: Makers of the Revolution in Biology, Simon and Schuster 8 Harker, D. (1936) J. Chem. Phys. 4, 381 9 Beevers, C. A. and Robertson, J. H. (1950) Acta Cryst. 3, 164 10 Rossmann, M. G. and Blow, D. M. (1962) Acta Cryst. 15, 24-31 11 Nordman, C. E. and Nakatsu, K. (1963) J. Amer. Chem. Soc. 85, 353-354 12 Patterson, A. L. (1963) Acta Cryst. 16, 1255-1256 13 Okaya, Y., Saito, Y. and Pepinsky, R. (1955) Phys. Rev. 98, 1857