J. Mol. Biol. (1975) 98, 161-177
Structure of Deoxyhemoglobin A Crystals Grown from Polyethylene Glycol Solutions K. B. WXRDt, B. C. WISH~.R, E. E. LATTMAN$AND W. E. Love.
Thomas C. Jenldna Department of Biophysics John~ Hopldns University Baltimore, Md 21218, U.S.A. (Received 17 Mar~h 1975) The structure of a new crystal form of deoxyhemoglobin A grown from polyethylene glycol solutions has been determined at 3.5 A resolution. The molecular orientations and positions were found by means of rotation and translation functions using the squared molecular transform of horse deoxyhemoglobin. Phases were calculated using atomic co-ordinates previously determined for deoxyhemoglobin A grown in another crystalline form. A difference Fourier synthesis showed minor structural differences near intermolecular contacts, the heine groups, and the 81 carboxyl terminus. Some of these differences may be caused by the different crystalline environment; others may be due to errors in the analytical method. These apparent structural differences will be useful for interpreting results of a similar analysis of deoxyhemoglobin S crystals grown from the same solvent. 1. I n t r o d u c t i o n Many factors influence the growth of protein crystals. Variables such as temperature, pH, ionic strength, protein concentration and solvent polarity affect crystal structure presumably because they alter the relatively weak forces which determine the secondary, tertiary, and quarternary structure of proteins. Thus, changes in the nature of ion-pair bonds, hydrogen bonds, and hydrophobic interactions produce slight changes in protein conformation and lattice-packing arrangements which result in different crystal structures. Conversely, the intermolecular contacts can influence the local conformation of the polypeptide chain. Protein crystals, however, invariably have a large solvent content, ~with typical values ranging from 30 to 78~/0 solvent (Matthews, 1968). Thus there are rather large interstitial spaces between molecules, and only a few areas of contact between them. I t is therefore argued that crystalpacking forces should have a negligible effect on protein conformation. However, few investigations of the influences of lattice forces and solvent properties on protein conformation have been based on an actual comparison of protein structures derived from crystal-structure analyses of different crystals. Drenth et al. (1971) compared electron density maps and atomic models of t Present address: Laboratory for the Structure of Matter, Naval Research Laboratory, Washington, D.C. 20375, U.S.A. Present address : Structural Biology Laboratory, Rosenstiel Basic Medical Sciences Research Center, Brandeis University, Waltham, Mass. U.S.A. 11
161
162
K. B. W A R D
ET
AL.
subtilisin Nero and subtilisin BPN'. These two enzymes have identical sequences but their tertiary structures were determined independently from crystals grown under quite dissimilar conditions. Differences in conformation between the two crystallographically independent molecules in tosyl-~-chymotrypsin crystals were studied by Birktoft et al. (1969). Tulinsky and co-workers (Vandien & Tulinsky, 1973; Tulinsky et al., 1973) have described differences between the two independent molecules in ~-ehymotrypsin crystals and, in addition, have investigated changes in tertiary structure caused by variation in~pH. Although these studies all indicate that crystallization conditions have no major effect on protein tertiary structure, they do show that lattice forces and solvent properties sometimes cause small changes in local conformation. Since mechanistic explanations of protein function often rely heavily upon atomic co-ordinates obtained from crystal-structure analysis (Lipscomb, 1971; Dickerson, 1972), the small effects of molecular packing and solvent properties may be important. Changes in conformation caused by crystallization depend upon the mechanism by which the protein is precipitated. An organic solvent increases the activity coefficient of a protein because its low dielectric constant increases the electrostatic free energy of the protein's charged ionic groups (Edsall, 1947). On the other hand, proteins crystallize from concentrated salt solutions probably because the salt ions compete with protein molecules for the water necessary for solvation (Cohn & Ferry, 1943). The non-ionic polymer, polyethylene glycol, is routinely used for fractional precipitation of protein solutions (Kaufman, 1969; Polson et al., 1964) and has occasionally been used to obtain protein crystals (Janssen & Ruelius, 1968; Swaney & Klotz, 1971 ; Epp et al., 1971). Evidence has been presented that suggests that PEGt precipitates proteins by a mechanism different from those mentioned above (Laurent & Killander, 1964; Iverius & Laurent, 1967; Zeppezauer & Brishammer, 1965; Laurent, 1963; Edmond & Ogston, 1968; Kaufman, 1969). It was found, for example, that less concentrated solutions of PEG (expressed as g/ml) are necessary to cause a fixed decrease in protein solubihty when PEG of greater average molecular weight is used. In addition, larger proteins are precipitated by PEG solutions of lower concentration than is needed to precipitate proteins with smaller volumes. These two observations are consistent with the steric exclusion mechanism suggested by Laurent (1963). He proposed that PEG solutions contained regions into which proteins could not penetrate because of the irregular network formed by the long polymeric chains. Higher molecular weight PEG would form these networks at lower concentrations, and larger proteins would be more readily excluded from these regions. If this idea is correct, then PEG acts not only by competing with the protein for solvent molecules but also by sterically excluding the protein from part of the volume of the solution and thus increasing its chemical activity until its solubility limit is reached. Crystals of both deoxy Hb A and deoxy Hb S that grow from solutions of PEG are different from any crystals of hemoglobin previously reported (Wishner & Love, 1973; iYIuirhead et al., 1967). A structure analysis of these Hb S crystals is currently underway using the molecular replacement technique, and the results at low resolution are presented in the following paper (Wishner et al., 1975). One goal of this investigation is th~ comparison of the deoxy Hb S molecular structure with the "t"Abbreviation used: PEG, polyethylene glyool; Hb, hemoglobin.
DEOXYHEMOGLOBIN
A CRYSTALS
168
s t r u c t u r e of d e o x y H b A d e t e r m i n e d b y M u i r h e a d & Greer (1970), who s t u d i e d c r y s t a l s g r o w n f r o m a m m o n i u m s u l f a t e solutions. H o w e v e r , t h e p r i m a r y s t r u c t u r e s o f H b S a n d H b A differ a t o n l y one p o s i t i o n , a n d c r y s t a l s o f h g a n d e d H b S a n d H b A give d i f f r a c t i o n p a t t e r n s t h a t a r e indist i n g u i s h a b l e ( P e r u t z e$ al., 1951). H e n c e i t is l i k e l y t h a t d e o x y H b S a n d d e o x y H b A differ o n l y s l i g h t l y i n s t r u c t u r e . A p p a r e n t s t r u c t u r a l differences d u e t o errors i n h e r e n t i n t h e m o l e c u l a r r e p l a c e m e n t technique, or m i n o r differences in c o n f o r m a t i o n c a u s e d b y c r y s t a l l i z a t i o n c o n d i t i o n s m i g h t be difficult t o d i s t i n g u i s h f r o m m o r e i m p o r t a n t differences r e s u l t i n g f r o m t h e single a m i n o a c i d r e p l a c e m e n t . Therefore, t h e s t r u c t u r e o f t h e d e o x y H b A c r y s t a l s g r o w n f r o m P E G solutions h a s b e e n d e t e r m i n e d using t h e s a m e t e c h n i q u e s e m p l o y e d in t h e H b S analysis.
2. Materials and Methods (a) GryaaUiza~n A stock solution containing 50 g of P E G a n d 100 ml of HaO was m a d e from P E G having an average molecular weight of 6000 (J. T. B a k e r Chemical Co., Phillipsburg, N.J. 08865). Small glass tubes conta~nlng mixtures of solutions of 0-1 g deoxy H b A / m l and stock P E G were filled with nitrogen, sealed and stored a t 4~ Crystals grew in those tubes in which the P E G composition was 15 to 20~o (v/v) of the stock solution a n d which h a d been a d j u s t e d to a p H between 6.4 a n d 7.5 with 0-01 ~ p h o s p h a t e buffer. The crystals used in ~his s t u d y grew a t p H 7.1 in 17% P E G stock solution. The crystals exhibit the s y m m e t r y of space group P21212 a n d h a v e lattice constants, a = 97.07 A, b = 99.42 A, and c = 65.98 A. There is one t e t r a m e r p e r asymmetric unit. (b) Data collection I n t e n s i t y d a t a wore collected from a single crystal on a Syntox P21 four-circle diffractometer using Cu K ~ radiation reflected from a graphite monochromater. A n ~-scan technique was used a n d b a c k g r o u n d intensities were measured on b o t h sides of each reflection. F i v e reference reflections were regularly monitored during the d a t a collection, a n d during the t o t a l collection time of 87 h the average variation in intensity for these reflections was less t h a n 7 ~/o. No correction was applied for deterioration due to irradiation. A t o t a l of 8549 reflections were collected within a unique o c t a n t of d a t a between 1]20 A a n d 1/3.5 A. A n W-scan (Arndt & Willis, 1966) was m a d e in 5 ~ stops a b o u t the [007] direction a n d the variation in intensity of the 007 reflection was t a k e n as a measure of t h e relative transmission of X - r a y s through the crystal as a function of W. B y meaxm of this empirical absorption curve, a correction was then applied to each reflection in a m a n n e r similar to t h a t described b y N o r t h e$ al. (1968). (c) Molecular trans]orms R o t a t i o n a n d translation function searches were performed using molecular transforms derived from the atomic co-ordinates of the ~1~1 dimer of horse deoxyhemoglobin k i n d l y supplied b y McLachlan & F e r m i (personal communication). The source of t h e co-ordinates was a 2.8 A Fourier m a p obtained b y means of multiple h e a v y - a t o m isomorphous replacem e n t (Bolton & Perutz, 1970). The co-ordinates h a d been refined b o t h b y the model. building a n d real-space refinement programs of D i a m o n d (1966,1971). The local origin a n d co-ordinate axes were redefined as follows: the x axis was parallel to the b axis of the horse hemoglobin crystal; t h e z axis was parallel to the a axis of the horse hemoglobin crystal, a n d therefore to the molecular d y a d ; the origin of the local co-ordinate system was on the a axis of the horse hemoglobin crystal a t --0.254 of t h e u n i t cell edge. The local origin defined here is a p p r o x i m a t e l y a t t h e center of the hemoglobin tetramer. The transform of t h e horse hemoglobin t e t r a m e r was evaluated, to a resolution of 6 A, on a n orthogonal grid with spacings of 11200 A, using a n overall t e m p e r a t u r e factor B = 20 A 2. The
164
K. B. W A R D E T A L .
transform of the "z,Sz dimer was calculated to 5.5 ~. resolution on a n orthogonal grid with spacings 1/128 .~. The use of large lattice constants ensured fine sampling for the molecular transforms. (d) Calcu/a~ion of the rotation function The orientation of the hemoglobin molecule within the u n i t cell of the crystal was determined b y using the modified rotation function described b y L a t t m a n & Love (1970), R(C) = Y Fm2(Crh)I(h), h
where I(h) is a sot of intensities, _~ma is the squared transform of a n isolated molecule, C is a matrix which rotates _~m2, a n d C T is C transposed. Largo values of R(C) occur when C rotates the isolated test molecule into the orientation that corresponds to an orientation of a molecule within the crystal. Because the transform of a n isolated molecule is used, R(C) can be computed more quickly b y means of this algorithm t h a n b y the one originally described b y Rossma~n & Blow (1962). Fm 2 was calculated (see above) on a grid sufficiently fine to allow values at the non-integer lattice points, Crh, to be found b y interpolation. ,Only reflections with Bragg spacings d < 10 ~ were used since low-order data include appreciable scattering from the solvent. The modified intensities, I(h) = Io(h) -- Io were employed, where io was the average value of the observed intensities, ]o(h), within the annulus of data used (6 A < d < 10 A). This effectively removed the origin of the Patterson function of the crystal when a l l of the data within this range were used (Lipson & Cochran, 1966). When, because of the expense of the calculation, searches were made with only a fraction of the data, the origin removal was m a i n t a i n e d by choosing this fraction from among the most positive and most negative values of I(h) such t h a t I(h') = 0 , h
where the index h' was taken only over those reflections actually used in the calculation. The matrix C (Sz, 82, 8a) was computed using the convention of Rossmann & Blow (1962) for Eulerian angles. The use of constant sampling intervals in 8z, 8u, a n d 8a is inefficient and produces maps containing distorted peaks because the volume of angle space associated with each sample point varies drastically with 8~. A n y two orientations of a molecule can be related b y a rotation, Xa, which can be taken as a definition of the angular distance between the two orientations. L a t t m a n (1972) has suggested t h a t levels of constant 82 be sampled along 8+ = 81 + 83 and 8_ ---- 81 - - 8 3 in increments given b y 82
=
Za
8. = xdcos(82/2) and
8_ = Xd/sin(82/2)z,
where Xd is a constant. Sampling along the locally orthogonal axes, 8+, 82 and 8_, ensures t h a t the angular distance, xa, between sample points remains constant, and requires only 2/~r times as m a n y points as conventional samphng. Steps of xa = 10~ were used in preliminary searches, while 2 ~ and 1~ steps were used for subsequent fine sampling of selected regions. The ~=2 for the al/]z dimer has s y m m e t r y T a n d the s y n n n e t r y of the crystal's intensity distribution is m m m . Thus the s y m m e t r y of their rotation function is P2zab (Tollin et aL, 1966). The angular ranges explored were 0
Structure of Deoxyhemoglobin A Crystals Grown from Polyethylene Glycol Solutions K. B. WXRDt, B. C. WISH~.R, E. E. LATTMAN$AND W. E. Love.
Thomas C. Jenldna Department of Biophysics John~ Hopldns University Baltimore, Md 21218, U.S.A. (Received 17 Mar~h 1975) The structure of a new crystal form of deoxyhemoglobin A grown from polyethylene glycol solutions has been determined at 3.5 A resolution. The molecular orientations and positions were found by means of rotation and translation functions using the squared molecular transform of horse deoxyhemoglobin. Phases were calculated using atomic co-ordinates previously determined for deoxyhemoglobin A grown in another crystalline form. A difference Fourier synthesis showed minor structural differences near intermolecular contacts, the heine groups, and the 81 carboxyl terminus. Some of these differences may be caused by the different crystalline environment; others may be due to errors in the analytical method. These apparent structural differences will be useful for interpreting results of a similar analysis of deoxyhemoglobin S crystals grown from the same solvent. 1. I n t r o d u c t i o n Many factors influence the growth of protein crystals. Variables such as temperature, pH, ionic strength, protein concentration and solvent polarity affect crystal structure presumably because they alter the relatively weak forces which determine the secondary, tertiary, and quarternary structure of proteins. Thus, changes in the nature of ion-pair bonds, hydrogen bonds, and hydrophobic interactions produce slight changes in protein conformation and lattice-packing arrangements which result in different crystal structures. Conversely, the intermolecular contacts can influence the local conformation of the polypeptide chain. Protein crystals, however, invariably have a large solvent content, ~with typical values ranging from 30 to 78~/0 solvent (Matthews, 1968). Thus there are rather large interstitial spaces between molecules, and only a few areas of contact between them. I t is therefore argued that crystalpacking forces should have a negligible effect on protein conformation. However, few investigations of the influences of lattice forces and solvent properties on protein conformation have been based on an actual comparison of protein structures derived from crystal-structure analyses of different crystals. Drenth et al. (1971) compared electron density maps and atomic models of t Present address: Laboratory for the Structure of Matter, Naval Research Laboratory, Washington, D.C. 20375, U.S.A. Present address : Structural Biology Laboratory, Rosenstiel Basic Medical Sciences Research Center, Brandeis University, Waltham, Mass. U.S.A. 11
161
162
K. B. W A R D
ET
AL.
subtilisin Nero and subtilisin BPN'. These two enzymes have identical sequences but their tertiary structures were determined independently from crystals grown under quite dissimilar conditions. Differences in conformation between the two crystallographically independent molecules in tosyl-~-chymotrypsin crystals were studied by Birktoft et al. (1969). Tulinsky and co-workers (Vandien & Tulinsky, 1973; Tulinsky et al., 1973) have described differences between the two independent molecules in ~-ehymotrypsin crystals and, in addition, have investigated changes in tertiary structure caused by variation in~pH. Although these studies all indicate that crystallization conditions have no major effect on protein tertiary structure, they do show that lattice forces and solvent properties sometimes cause small changes in local conformation. Since mechanistic explanations of protein function often rely heavily upon atomic co-ordinates obtained from crystal-structure analysis (Lipscomb, 1971; Dickerson, 1972), the small effects of molecular packing and solvent properties may be important. Changes in conformation caused by crystallization depend upon the mechanism by which the protein is precipitated. An organic solvent increases the activity coefficient of a protein because its low dielectric constant increases the electrostatic free energy of the protein's charged ionic groups (Edsall, 1947). On the other hand, proteins crystallize from concentrated salt solutions probably because the salt ions compete with protein molecules for the water necessary for solvation (Cohn & Ferry, 1943). The non-ionic polymer, polyethylene glycol, is routinely used for fractional precipitation of protein solutions (Kaufman, 1969; Polson et al., 1964) and has occasionally been used to obtain protein crystals (Janssen & Ruelius, 1968; Swaney & Klotz, 1971 ; Epp et al., 1971). Evidence has been presented that suggests that PEGt precipitates proteins by a mechanism different from those mentioned above (Laurent & Killander, 1964; Iverius & Laurent, 1967; Zeppezauer & Brishammer, 1965; Laurent, 1963; Edmond & Ogston, 1968; Kaufman, 1969). It was found, for example, that less concentrated solutions of PEG (expressed as g/ml) are necessary to cause a fixed decrease in protein solubihty when PEG of greater average molecular weight is used. In addition, larger proteins are precipitated by PEG solutions of lower concentration than is needed to precipitate proteins with smaller volumes. These two observations are consistent with the steric exclusion mechanism suggested by Laurent (1963). He proposed that PEG solutions contained regions into which proteins could not penetrate because of the irregular network formed by the long polymeric chains. Higher molecular weight PEG would form these networks at lower concentrations, and larger proteins would be more readily excluded from these regions. If this idea is correct, then PEG acts not only by competing with the protein for solvent molecules but also by sterically excluding the protein from part of the volume of the solution and thus increasing its chemical activity until its solubility limit is reached. Crystals of both deoxy Hb A and deoxy Hb S that grow from solutions of PEG are different from any crystals of hemoglobin previously reported (Wishner & Love, 1973; iYIuirhead et al., 1967). A structure analysis of these Hb S crystals is currently underway using the molecular replacement technique, and the results at low resolution are presented in the following paper (Wishner et al., 1975). One goal of this investigation is th~ comparison of the deoxy Hb S molecular structure with the "t"Abbreviation used: PEG, polyethylene glyool; Hb, hemoglobin.
DEOXYHEMOGLOBIN
A CRYSTALS
168
s t r u c t u r e of d e o x y H b A d e t e r m i n e d b y M u i r h e a d & Greer (1970), who s t u d i e d c r y s t a l s g r o w n f r o m a m m o n i u m s u l f a t e solutions. H o w e v e r , t h e p r i m a r y s t r u c t u r e s o f H b S a n d H b A differ a t o n l y one p o s i t i o n , a n d c r y s t a l s o f h g a n d e d H b S a n d H b A give d i f f r a c t i o n p a t t e r n s t h a t a r e indist i n g u i s h a b l e ( P e r u t z e$ al., 1951). H e n c e i t is l i k e l y t h a t d e o x y H b S a n d d e o x y H b A differ o n l y s l i g h t l y i n s t r u c t u r e . A p p a r e n t s t r u c t u r a l differences d u e t o errors i n h e r e n t i n t h e m o l e c u l a r r e p l a c e m e n t technique, or m i n o r differences in c o n f o r m a t i o n c a u s e d b y c r y s t a l l i z a t i o n c o n d i t i o n s m i g h t be difficult t o d i s t i n g u i s h f r o m m o r e i m p o r t a n t differences r e s u l t i n g f r o m t h e single a m i n o a c i d r e p l a c e m e n t . Therefore, t h e s t r u c t u r e o f t h e d e o x y H b A c r y s t a l s g r o w n f r o m P E G solutions h a s b e e n d e t e r m i n e d using t h e s a m e t e c h n i q u e s e m p l o y e d in t h e H b S analysis.
2. Materials and Methods (a) GryaaUiza~n A stock solution containing 50 g of P E G a n d 100 ml of HaO was m a d e from P E G having an average molecular weight of 6000 (J. T. B a k e r Chemical Co., Phillipsburg, N.J. 08865). Small glass tubes conta~nlng mixtures of solutions of 0-1 g deoxy H b A / m l and stock P E G were filled with nitrogen, sealed and stored a t 4~ Crystals grew in those tubes in which the P E G composition was 15 to 20~o (v/v) of the stock solution a n d which h a d been a d j u s t e d to a p H between 6.4 a n d 7.5 with 0-01 ~ p h o s p h a t e buffer. The crystals used in ~his s t u d y grew a t p H 7.1 in 17% P E G stock solution. The crystals exhibit the s y m m e t r y of space group P21212 a n d h a v e lattice constants, a = 97.07 A, b = 99.42 A, and c = 65.98 A. There is one t e t r a m e r p e r asymmetric unit. (b) Data collection I n t e n s i t y d a t a wore collected from a single crystal on a Syntox P21 four-circle diffractometer using Cu K ~ radiation reflected from a graphite monochromater. A n ~-scan technique was used a n d b a c k g r o u n d intensities were measured on b o t h sides of each reflection. F i v e reference reflections were regularly monitored during the d a t a collection, a n d during the t o t a l collection time of 87 h the average variation in intensity for these reflections was less t h a n 7 ~/o. No correction was applied for deterioration due to irradiation. A t o t a l of 8549 reflections were collected within a unique o c t a n t of d a t a between 1]20 A a n d 1/3.5 A. A n W-scan (Arndt & Willis, 1966) was m a d e in 5 ~ stops a b o u t the [007] direction a n d the variation in intensity of the 007 reflection was t a k e n as a measure of t h e relative transmission of X - r a y s through the crystal as a function of W. B y meaxm of this empirical absorption curve, a correction was then applied to each reflection in a m a n n e r similar to t h a t described b y N o r t h e$ al. (1968). (c) Molecular trans]orms R o t a t i o n a n d translation function searches were performed using molecular transforms derived from the atomic co-ordinates of the ~1~1 dimer of horse deoxyhemoglobin k i n d l y supplied b y McLachlan & F e r m i (personal communication). The source of t h e co-ordinates was a 2.8 A Fourier m a p obtained b y means of multiple h e a v y - a t o m isomorphous replacem e n t (Bolton & Perutz, 1970). The co-ordinates h a d been refined b o t h b y the model. building a n d real-space refinement programs of D i a m o n d (1966,1971). The local origin a n d co-ordinate axes were redefined as follows: the x axis was parallel to the b axis of the horse hemoglobin crystal; t h e z axis was parallel to the a axis of the horse hemoglobin crystal, a n d therefore to the molecular d y a d ; the origin of the local co-ordinate system was on the a axis of the horse hemoglobin crystal a t --0.254 of t h e u n i t cell edge. The local origin defined here is a p p r o x i m a t e l y a t t h e center of the hemoglobin tetramer. The transform of t h e horse hemoglobin t e t r a m e r was evaluated, to a resolution of 6 A, on a n orthogonal grid with spacings of 11200 A, using a n overall t e m p e r a t u r e factor B = 20 A 2. The
164
K. B. W A R D E T A L .
transform of the "z,Sz dimer was calculated to 5.5 ~. resolution on a n orthogonal grid with spacings 1/128 .~. The use of large lattice constants ensured fine sampling for the molecular transforms. (d) Calcu/a~ion of the rotation function The orientation of the hemoglobin molecule within the u n i t cell of the crystal was determined b y using the modified rotation function described b y L a t t m a n & Love (1970), R(C) = Y Fm2(Crh)I(h), h
where I(h) is a sot of intensities, _~ma is the squared transform of a n isolated molecule, C is a matrix which rotates _~m2, a n d C T is C transposed. Largo values of R(C) occur when C rotates the isolated test molecule into the orientation that corresponds to an orientation of a molecule within the crystal. Because the transform of a n isolated molecule is used, R(C) can be computed more quickly b y means of this algorithm t h a n b y the one originally described b y Rossma~n & Blow (1962). Fm 2 was calculated (see above) on a grid sufficiently fine to allow values at the non-integer lattice points, Crh, to be found b y interpolation. ,Only reflections with Bragg spacings d < 10 ~ were used since low-order data include appreciable scattering from the solvent. The modified intensities, I(h) = Io(h) -- Io were employed, where io was the average value of the observed intensities, ]o(h), within the annulus of data used (6 A < d < 10 A). This effectively removed the origin of the Patterson function of the crystal when a l l of the data within this range were used (Lipson & Cochran, 1966). When, because of the expense of the calculation, searches were made with only a fraction of the data, the origin removal was m a i n t a i n e d by choosing this fraction from among the most positive and most negative values of I(h) such t h a t I(h') = 0 , h
where the index h' was taken only over those reflections actually used in the calculation. The matrix C (Sz, 82, 8a) was computed using the convention of Rossmann & Blow (1962) for Eulerian angles. The use of constant sampling intervals in 8z, 8u, a n d 8a is inefficient and produces maps containing distorted peaks because the volume of angle space associated with each sample point varies drastically with 8~. A n y two orientations of a molecule can be related b y a rotation, Xa, which can be taken as a definition of the angular distance between the two orientations. L a t t m a n (1972) has suggested t h a t levels of constant 82 be sampled along 8+ = 81 + 83 and 8_ ---- 81 - - 8 3 in increments given b y 82
=
Za
8. = xdcos(82/2) and
8_ = Xd/sin(82/2)z,
where Xd is a constant. Sampling along the locally orthogonal axes, 8+, 82 and 8_, ensures t h a t the angular distance, xa, between sample points remains constant, and requires only 2/~r times as m a n y points as conventional samphng. Steps of xa = 10~ were used in preliminary searches, while 2 ~ and 1~ steps were used for subsequent fine sampling of selected regions. The ~=2 for the al/]z dimer has s y m m e t r y T a n d the s y n n n e t r y of the crystal's intensity distribution is m m m . Thus the s y m m e t r y of their rotation function is P2zab (Tollin et aL, 1966). The angular ranges explored were 0
