THE JOURNAL OF BIOLOGICAL CHEMISTRY Vol. 248, No. 9, Issue of May 10, pp. 3327-3334, 1973

Pwded in U.S.A.

Carp Muscle Calcium-binding Protein

III. PHASE REFINE,1IEKT USING THE TANGENT FORMULA

(Received for publication, October 30, 1972)

WAYNE A. HENDRICKSONANDJEROMEI~ARLE

From the Laboratory for the Structure of Matter, Naval Research Laboratory, Washington, D. C. 20375

SUMMARY

The tangent formula for crystallographic phase determina- tion has been used to refine and extend the phase informa- tion for the crystal structure of carp muscle calcium-binding protein. Phases from the multiple isomorphous replace- ment analysis were used to initiate the tangent formula calculations. New phases were developed in a cautious, gradual procedure which continued to emphasize the iso- morphous replacement data. Probability measures were instrumental in combining the phase information from the tangent formula with that from isomorphous replacement. The new electron density map which finally resulted showed a more highly resolved and readily interpreted image of the molecule than was evident in earlier maps. Most impor- tantly, the new map permitted the interpretation of a second calcium-binding site.

Kretsinger and Nockolds (I) used the method of multiple iao- morphous replacement in their determination of the structure of the calcium-binding protein from carp muscle. Unfortunately, however, the heavy atoms were bound at one and the same major site in all three of the isomorphous derivatives which thy used. As a result, notwithstanding the inclusion of anom- alous scattering data in the analysis, the phase ambiguity char- acteristic of a single isomorphous replacement was not fully resolved for many of the reflections. Furthermore, phases were unavailable for a large number of high index (high resolution) structure amplitudes of the native crystals since the corresgond- ing diffraction data from derivative crystals had not been meas- ured. These shortcomings of the initial phase determination prompted a desire to improve the accuracy and extent of phasing for this protein structure and motivated the application of the tangent formula described in this report.

The tangent formula of Karle and Hauptman (2) has the po- tential to refine a set of approximate crystallographic phases and to evaluate additional phases as well. The possible effec- tiveness of this relationship has been amply shown in the crystal structure analyses of numerous smaller molecules. However, attempts at using the tangent formula to refine aud extend pro- tein phases (e.g. for cytochrome c (3), carbosypeptidase A (4), and sperm whale myoglobin (5)) have met Kith disappointingly

limited success. These results are not altogether unexpected since the restricted extent of data and molecular size of proteins make the protein application of the tangent formula a theoreti- cally marginal problem. In this application to carp muscle calcium-binding protein, special care has been taken to avoid pitfalls of the problem by using techniques learned in similar studies of lamprey hemoglobin and Glycera hemoglobin.

COMPUTATIONAl. PROCEDURE

Data-Data from crystals of Component B of the calcium- binding protein from carp muscle were provided us by Kretsinger and Nockolds (1). The space group for these crystals is C2 and the lattice constants are a = 28.2, b = 61.0, c = 54.3 A, and p = 95.0”. The data comprised 5055 reflections which had been phased by isomorphous replacement and anomalous scat- tering measurements from three derivatives, 1102 observations for which no phase information had been secured, and 2626 “unobserveds” (reflections showing less than the minimum observable intensity). Altogether the data extended to a reso- lution of roughly 1.8 A; phased reflections reached to spacings of about 2.0 A. Diffraction data were presented in the form of structure factor amplitudes. Information concerning phase values was presented in terms of Blow-Crick phase probability distributions, PO(~), computed at 5” intervals (1).

Phase Probability Disfributions-For convenience in handling the phase data in ensuing calculations, the phase probability distributions were cast in the simplified form,

PC(q) = exp (K + A cos ‘p + B sin (o + C cos 2~ + D sin 2~) (1)

by a fitting procedure described previously (6). Least squares weights of w(p) = P,(q) had performed unfailingly in previous tests, but in this application 57 of the 5055 fits using these weights left relative residuals,

s 2a / J=,(P) - P&P) 1 & 2a Po(ddv

0 /s 0

exceeding 1.0. In these cases, fittings a-ith weights of Pb1’2(p) and PO’/“(p) were tried as well as unimodal fits by exp(k cos(cp - cpc)) with cpc as the centroid phase and k derived from the figure- of-merit, m, according to m = Zi(k)/l&) where 1, is a Bessel function of imaginary argument. The one of thcsc with the lowest relative residual was in each case a satisfactory fit, i.e. less than 1.0. An example typical of the fit of Equation 1 to the Blow-Crick distributions, PO@), is shown in Fig. 1. Over all the fits, the mean relative residual was 0.026 and the mean

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FIG. 1. Typical phase probability distribution. The stepped CUTW is the phase probability distribution for reflection 5,5,% sampled at 5” intervals as provided by Kretsinger and Nockolds (1). The smooth c7Lrz’e is the fitting made to these data with Equa- tion 1. The relative residual (defined in the text) of this fit is 0.049 as compared to the average of 0.026 over-all fits. The phase ppl marks the absolute maximum, and ‘p2 the secondary maximum, of this bimodal distribution.

discrepancies, ns calculated from actual and fitted distributions,

\Tere 0.0008 for figures-of-merit and 0.08” for centroid phases. X major site common t’o all three derivatives dominated the

phase information and the minor sites were all within 1.1 A of this site in the y direction. This introduced a pseudo mirror of

symmetry into the isomorphous replacement phasing and nearly all reflections were left with a phase ambiguity centered at either

0 or P. The anomalous scattering was quite effective in re- solving these phase ambiguities, but not completely so (see Fig. 2) and bimodality persisted in many of the phase probability distributions. Since a procedure of choosing between most probable phase alternatives had proved effective in combining tangent formula information with single isomorphous replace- ment data (7-9), a generalization of this approach seemed promising for the asymmetrically bimodal distributions of this

problem. The first step in this analysis was to locate the ex- treme values of the distributions. At the estrema, dP(cp) /& =

0. Applying this condition to Equation 1 and remembering the identit,y co& + sin”0 = 1, one arrives at

and

2/ = (40~~ + Bx2 - 2D)/(4Cz + A) (2)

16(Cz + D2)x’ + S(AC f BD)x3 + (A2 + B2 - 16(Cz + D*))x* (3)

- (4BD + 8AC)x + (402 - AZ) = 0

where x = cos qextremum and y = sin ~extremur,i. For each re- fiection the quartic equation, 3, was solved by computer and its real roots (four if bimodal, two if unimodal) mere substituted into Equation 2 to determine completely the phases at extrema. Reflections with C=D=O or B=D=O were treated as special cases. The extrema were sorted into maxima and minima by direct inspection of Z’(q) at the extreme points. The discrimina- tire probability, PI, that the phase be at the absolute maximum,

PHASETANGLE

FIG. 2. Frequency distribution of centroid phase angles. The solid line shows the distribution of IS0 phase angles and the distri- bution of ISO+TAX phase angles is given by the dolted line. Only the 3410 general reflections (k # 0) for which both IS0 and TAN phases were calculated are included. The frequencies shown are sampled in 10” intervals so a unitorm distribution would have 94.7 reflections within each sampling interval. The peaks at 0 and ?r reflect the persistence of the pseudo mirror of symmetry which was introduced into the isomorphous replacement phase information by the occurrence of all heavy atom sites at approxi- mately the same levei in VJ.

1

6 .7 .a PROBABILITY

.9

FIG. 3. Frequency distribution of discriminative probabilities for bimodal phase probability distributions. The discriminative probability is PI = P(&/P(& + P&X) where (Do and (Ok are posi- tions of the absolute and secondary maxima in a phase probability distribution, P(p). It discriminates between phase alternatives and measures the extent to which the phase ambiguity has been re- solved. The concentration of these probabilities near 0.5 indi- cates that the phase ambiguity was essentially unbroken for most of the 1365 bimodal IS0 distributions.

-

cp~, rather than the secondary maximum, (~2, was ascertained for each of the 1365 bimodal distributions. A frequency histogram of these probabilities is shown in Fig. 3. In addition, intrinsic figures-of-merit were calculated for the phase alternatives, (ol

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and cp2, by a generalization of Equation 6 of Reference 7. Inte- grations for these were carried over the intervals between the absolute minimum and the respective adjacent maxima.

ATormaZized Struclure Facfors-Normalized structure factor amplitudes, 1 E 1, were derived from the structure amplitudes, j F 1, for use in the tangent formula calculations. Since quite a high proportion of the data were “unobserveds,” expected values for these? structure amplitudes were needed to properly perform the normalization. These expected values were estimated in shells of sine/X under the assumption the “unobserveds” are distinguished as falling below a particular threshold intensity. For noncentrosymmetrical crystals, the normalized intensities, - - z = E2 = I/I (I being the local average of intensities, I), are distributed according to P(z)& = exp(-z)dz. It follows that the fraction of reflections with z greater than a threshold z’, i.e. the fraction of “observed” reflections, is f = exp( -z’) and the normalized intensity value for an “unobserved” is (z), < p’ = (1 - f(l - hlf))/(l - f). The factor, q = f,,,r/(l - lnf) de- rived from the “observed” data to relate normalized intensities to intensities can then be applied to yield the expected values,

m unobserved = s, f or unobserved structure amplitudes. Having these values, the calculation of normalized structure factors proceeded straightforwardly as in the lamprey hemo- globin allnlysis. The K values were based on the atomic com- position established by the sequence analysis (10) and were fitted by a K curve of the analytic form K(sinB/X) = exp[a + b(sint9/X)C] (11) which was then used in computing 1 E 1 values. Least squares parameters of the K curve were a = 0.1258, b = 33.35, and c = 2.474. The absolute scale indicated by the K curve, K(0) = exp(u) = 1.13, differs little from the unit scale set by a standard Wilson plot (1).

Tangent Formula Ccblculations-The intention behind the phase computations employing the tangent formula was to refine the phases determined by isomorphous replacement and to extend the phasing to those structure factors lacking other phase infor- mation. The tangent formula (2),

CkEkEh-k sin (qk + vh-k) Bh tan lph) = -&Eke&k COS (Pk + vh-k) = Ah

(4)

is suited to this task since it affords an estimate of a given phase, cph, in terms of reflections of known phase. However, the tangent formula can lead to somewhat specious, although self- consistent, results, especially when applied with limited data such as are obtained from proteins, and care must be taken to avoid these t.endencies. The procedure followed here was pat- terned after the tangent formula applications to lamprey hemo- globin (9) and orthorhombic Glyceru hemoglobin crystals (8). Phases were developed gradually, emphasizing the more physi- cally based isomorphous replacement data over nascent tangent formula information and using only highly reliable phases as “knowns.” Figures-of-merit were used to judge the reliability of isomorphous replacement phases and reliability factors, a, were used as measures of reliability for tangent formula phases. The reliability factor,

(5d

is the parameter of dispersion for the probability distribution W),

f%‘) = N exP (0~ COS k’ - k’h))) (5b)

associated wit,h the tangent formula. (a, = Cfj” where fj is

an atomic scattering factor and the sum is over all atoms in the primitive unit cell.) It depends directly on the number, mag- nitude, and phasing consistency of the terms which contribute to the tangent formula and is related in an inverse manner to the variance of the tangent formula phase. Actually, the cy factors defined in Equation 5u were reduced by an appropriate amount by using Equations 18, 19, 20 of Reference 9 to account for errors propagated from inaccuracies in the “known” phases. The tangent-formula calculations were performed with a modi- fied version of the program by Brenner and Gum (9, 13).

The first cycle of calculat.ions involved only reflections with I E / values over 1.5. These numbered 790, of which 426 had initial figures-of-merit exceeding 0.8, the reliability level chosen as an acceptance cutoff for “known” phases. After this and each succeeding cycle, the newly calculated tangent formula phase information (TAN) was incorporated with the isomorphous replacement information (ISO) to derive a new set of “known” phases for the next cycle.

Reflections with bimodal and unimodal IS0 distributions were treated separately in this combining of ISOl and TAN information. In the bimodal case, a choice was made between the IS0 alternatives using an extension of the procedure de- scribed for single isomorphous replacement data (7). Accord- ingly, the normalized probability that the phase be the more probable of the two alternatives was computed from the products of TAN and IS0 probabilities (from Equations 1 and 5b) at each of the IS0 maxima. If this probability exceeded 0.7 and the intrinisic figure-of-merit, m’, for the chosen alternative was greater than 0.8, then this phase was given “known” status. In the unimodal case, which included cent.ric reflections and bi- modal reflections which were essentially unimodal (P1 > 0.9 or m > 0.8) as well as truly unimodal reflections, the TAN infor- mation was primarily used in a negative sense. Among uni- modal reflections with reliable IS0 phases (m > 0.8), if the TAN phase was also designated as reliable (al > 2.0) but none- theless agreed rather poorly with the IS0 phase ([ prso - qTAN j > 50”), then this reflection was rejected from the “known?‘; otherwise, it was retained as a “known” and kept at the IS0 phase. Only when the IS0 determination of a phase was rather poor (m < 0.8 if unimodal, m’ < 0.8 if bimodal) or lacking (no derivative) was the TAN information allowed its full weight, and then only if the indication was highly reliable (a > 6.0). The IS0 and TAN phase probability distributions for reflections meeting these conditions were multiplicatively combined and the centroid phase of the product distribution was accepted as a “known.”

The revised set of “known” phases thus determined were used in a second computation of TAN phases; and then the whole process was iterated twice more. Thereupon, reflections with 1.5 > j E j > 1.4 were added to those with j E I > 1.5 for phasing in the next cycle and to serve as potential “knowns” in following cycles. After three cycles including I E Is down to 1.4, additional reflections were introduced by again lowering the acceptance level on 1 E 1 values by 0.1. The refinement was continued in a succession of like steps until the minimal I E I value reached 0.8. In all, this took 26 cycles. However, the results from cycle 24 were taken as final since statistics discussed below indicated optimum accuracy at that point. The 4506 TAN phases determined in this effectively final cycle were based on 1974 “known” reflections (7664 when symmetry expanded).

1 The abbreviations used are: ISO, phase information from the multiple isomorphous replacement method with anomalous scat- tering data included; TAN, tangent formula phase information.

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FIG. 4. Phase discrepancies during the tangent formula phase development. The average discrepancy between IS0 and TAN phases, ( 1 FAN - ‘peso 1 ), is recorded for each 1 E ( category after each cycle of tangent formula calculations. Reflections are cate- gorized according to 1 E 1 magnitude; those reflections with 1 E 1 > 1.5 are identified by Emin = 1.5, those with 1.4 < 1 E 1 < 1.5 are designated as Emin = 1.4, and so on. In order of descending 1 E 1 the numbers of reflections in each category are 558, 257, 283, 339, 425, 520, 512, and 516. All general reflections (k # 0) except those without IS0 phase information are included.

1 I 1 I I

2 4

Reliability 6Factor, CY* ICI 12

FIG. 5. Relation between phase discrepancy and the TAN re- liability factor. Average phase discrepancies, (IVAN - qrsol), after tangent formula cycle 24 are plotted as a function of the re- liability factor 01. The reflections included in these averages are the same as those used in Fig. 4. Of the total 3410 data, 1770 fall into categories with (Y > 2.0 and 588 have (Y > 4.0.

The preponderance of these “know& retained their IS0 phase values, but the influence of TAN cycle 23 was felt through 273

choices between IS0 bimodal alternatives, 5 multiplicative

combinations and 538 rejections. Cycle 24 took 27.4 min of

FIG. 6. Distribution of phasing information as function of sin 0/x. The distribution of the native reflections measured as having “observable” intensity is given by the solid line. The dot-dash line shows the distribution of reflections with IS0 phase informa- tion, i.e. those for which derivative data were also measured, and the dashed line presents the distribution of reflections for which TAN phases were calculated. Those reflections given TAN phases but lacking IS0 information are distributed according to the dotted line. Bounds for the sin e/x intervals were chosen at points equidistant in (sin 0/W. Thus, the abscissa of the figure is linear in (sin e/x)3 and each sampling interval represents the same number of lattice points. This number is approximately 440 re- flections, the upper border line of the figure. The difference be- tween this constant and the native “observed” reflections, the solid line, consists of reflections measured as “unobserved” as well as unmeasured reflections. Note that the resolution limit, dmi,, is related to sin 0/X according to dmin = (2 sin S/W so, for ex- ample, sin ($0, = 0.25 corresponds to 2 A resolution.

, ’ t I I I 0 .I0 .I5 .20

Sin 6 25 29

-I-

FIG. 7. Dependence of the TAN reliability factor on sin B/X. Average values of the TAN reliability factor, LY, are plotted as a function of sin e/x. The categories of sin B/X are the same as those used in Fig. 6. The averages shown by the solid line are over-all TAN phased reflections, i.e. those indicated by the dashed line in Fig. 6; the dashed line represents averages over TAN phased reflections which lack IS0 information, i.e. those indicated by the dotted line in Fig. 6.

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PH IE

PHE 70

II1

PHE 29 I I I

FIG. 8. Comparisons of sections through the electron density distributions of phenylalanine side chains. All of the 10 phenyl- alanines in the structure are represented. The left-hand member of each pair of frames is from the IS0 map and the right-hand sec- tion is from the ISO+TAN map. Superimposed on each ISO+ TAN image is a skeleton of the atomic configuration fitted to the

density distribution. The plane of section for each group was de- termined by fitting a least squares plane, weighted by ISO+TAN electron density values, to the grid points within the phenyl ring density. A special program was used to compute the Fourier syntheses in these arbitrary sections.

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CDC 3800 computer time and the average time per cycle was 7.6 min.

RESULTS AND DISCUSSION

Statistical Assessment-Throughout the phase development, progress was monitored by statistics comparing TAN and IS0 phases among reflections having both kinds of information. Mean phase discrepancies, (Ap) where Acp = 1 cprso - (PT.&N 1 taken in the direction around the phase circle giving the smaller difference, were calculated as a function of sinB/X, 1 E /, a, and m after each cycle. In calculating discrepancies, cprso was set at the centroid phase except in bimodal cases, and there the more probable of the alternative phases was used. Phase dis- crepancies, as shown in Fig. 4, decreased rather steadily for each category of 1 E 1 until after cycle 24 when some increases began. Meanwhile, the root mean square relative phase discrepancy, (@c~)“l(V))~‘~ where the variance, V, estimated from reliability parameters is V = VIso + VTAN = 2(1 - m) + 2(1 - II(a)/

IO(a)) (9), remained relatively constant. This means that the improvement in accuracy predicted by the increase in mean LY from 0.76 initially to 2.27 after cycle 24 was indeed realized in the reduced discrepancies. Further verification of the merit of (Y as a relaibility estimator for TAN phases is given by the de- pendence of (Ap) on a: shown in Fig. 5.

There are several ways in which the final TAN phase informa- tion can be incorporated with pre-existing IS0 information to produce a new Fourier map. The most straightforward ap- proach is to multiply the independent phase probability distri- butions together and to then use the centroid phases and figures- of-merit from the composite distributions in computing the

PH

(b)

PHE

I -

(a) (b) FIG. 9. Comparisons of different phasing contributions to

phenylalanines-70 and -102. Sections were prepared as described in the legend for Fig. 8. The IS0 phases and figures-of-merit for the 5055 reflections with IS0 information were used in computing Sections a. Frames b and c included the same reflections as used

Fourier. A more conservative procedure, from the viewpoint of the weight given the TAN information, is to combine the data in a manner akin to the treatment used between cycles in the phase development above. Since the statistics showed the probability parameters, o(, to be valid indicators of reliability, the multiplicative approach was adopted. (Actually, the multi- plication was effected by simply adding exponents in probability functions having the form of Equation 1 (14).) The resulting ISO+TAN map was reasonable and readily interpreted, so no alternative maps were calculated.

In all, 6000 phased reflections contributed to the final ISO+ TAN map. These consisted of 1494 reflections with only IS0 information, 945 with only TAN information, and 3561 with both; 157 observations, all with j E 1 < 0.8, remain unphased. The distribution in sinB/X of the amplitude observations and the phase information is shown in Fig. 6. Inclusion of the TAN information served to increase the mean figure-of-merit from 0.776 to 0.840 for the 3561 reflections which also had IS0 phase information. These values coupled with mean figures of 0.732 for IS0 only phases and 0.522 for TAN only phases led to an over-all mean figure-of-merit of 0.763 for the ISO+TAN phases. In the main, reflections beyond sinB/X = 0.25 were phased solely by TAN information. Reliability factors, (Y, for TAN only re- flections compare favorably with those at lower sin%/1 (Fig. 7) and average 1.69 versus 2.27 over-all. Thus it appears that phasing for the new map has been effectively extended to 1.85 A resolution. In addition, there is every indication that the incorporation of tangent formula results has improved the phasing generally and particularly so at low resolution (Fig. 7).

(cl

(d)

(d) in a, but IS0 phases and ISO+TAN figures-of-merit were used for b and both ISO+TAN phases and ISO+TAN figures-of-merit were used for c. Sections d are based on the complete ISO+TAN phasing with all 6000 reflections.

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Comparison of Electron Density dlaps-The electron density map based on the combined ISO+TAN phasing was sectioned and contoured at the same intervals as those used for the IS0 map: a/40, b/80, c/80, and 0.215 e/AS. This map was then sent to R. H. Kretsinger at the rniversity of Virginia for interpreta- tion. There the sections were traced onto acrylic sheets and mounted, in a Richards box for optical superposition on the Kendrew skeletal model which had been fitted to the ISO-phased electron density distribution. A preliminary comparison of the matches of the IS0 and IS0 +ThN maps to the model indicated t.hat while the two maps were basically similar, the ISOfTAN map conformed more closely to t’he density distribut.ions ex- pected of the groups in select, unambiguously interpretable re- gions. This apparent improvement sufficed to warrant a read- justment of the model to a best fit with the new ISO+TAN elect,ron density distribution.

For the most part the readjustments to the model const,ituted small refinements which corroborated and strengthened the original interpretation. Thus the basic course of the polypep- t,ide chain remained as designated earlier (15). However, the

reinterpretation did lead to significant changes of local configura- tion in several regions. Most importantly, the ISO+TAN map brought the realization that the CD loop binds a calcium atom (1, 15). Several factors contributed to enhance the inter- pretability of the new map: the electron density protrusions of carbonyl groups were enhanced thereby making these landmarks more noticeable; the identifying characteristics of several side chains, notably the flattened appearance of aromatic rings, were accentuated; the density at sulfur and calcium positions became more distinct; the continuity of density was strengthened at some weak places along the chain; and spurious joins between groups in van der Waals contact were diminished. In sum, Kretsinger gained the subjective impression that the new map was more readily and surely interpretable than was its predeces- sor (1).

One’s impressions about the relative ease of interpretation of two maps do provide valuable evidence regarding the quality of the maps. But they are perforce subjective where ideally the comparison should be quantitative and objective. Neither is the statistical evaluation of the previous section adequate. The statistical measures do indicate an improvement in phases, but

Fro. 10. Stereoviews of the calcium-binding loops. The direc- tion of view looks out from the molecular interior along the local inbramolecular diad. Parts a and b compare the IS0 and ISOf TAN electron density maps of the calcium sites. Map a was com- puted with IS0 phasing whereas ISO+TAN phases were used in computing Mup b. The bounds of each section enclose an area 14.4 X 21.6 A centered at the local 2-fold axis and the sections are 0.9 A apart. The calcium positions in a and b are indicated by small nuts. Part c is the portion of the atomic model fitted to the

ISO+TAN map (1) which corresponds to the region shown in a and 6. This includes the calcium atoms, the CD loop peptide Asp 51-Gln-Asp-Lys-Ser-Gly-Phe 57-Ilu-Glu-Glu-Asp-Glu 62, and the homologous peptide Asp 90-Ser-Asp-Gly-Asp-Gly-Lys 96-Ilu- Gly-Val-Asp-Glu 101 of the EF loop. For clarity, side chains other than those involved in calcium coordination are truncated at Cg. The a-carbon atoms are shown as jilled circles and the cal- cium atoms are drawn stippled.

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the connection between these measures and the quality of Fourier maps has not been fully established. Unfortunately, a meaning- ful comparison of maps should involve the separate comparison of each map to an external standard such as the density distribu- tion for a refined atomic model. In the absence of accurate atomic parameters it is difficult to devise a measure for quantify- ing the interpretability and accuracy of electron density maps. The comparison of electron density sections shown below are offered as qualitative substitutes for the desired quantitative evaluation.

The complete set of phenylalaniue side chains was chosen as a representative cross-section of the structure for purposes of comparing the IS0 and ISO+TXN electron density maps. Comparative sections through the planes of the 10 phenyl rings arr shown in Fig. 8. At this resolution (-2A) a phenylalauyl side chain is cspectcd to appear as a handled disk of density with a central dimpling of the disk if the spatial and temporal ordering of this moiety permit. Most of the phenylalaninc density dis- tributions are in accord with these expectations, but phcnylala- nine-2 and phenylalanine~57 which occur on the molecular sur- face (see Table V of Reference 1) are hardly distinguishable as phenylalanincs. Generally the IS0 +TXN rcprcsentation gives an improved appearance for the ring although the ISO+TAN scrtion for l)hc~iylalnl~ine-47 is a distortion of a weak but recog- nizable ring in thr IS0 section. The improvement in the ap- prnrance of the IYO+TAX section as against the IS0 section is particularly striking for l.‘hen!-lalanille-70 and phenylalanine- 102. .\s shown in Fig. 9, this improvement is due to the changes in phase at&s and is not a simple enhancement of density due to increased figures-of-merit. Fig. 9 also reveals that the bulk of the iml)rovement came from the refincmcnt of IS0 phases rather than from the extension of phasing to previously unphased reflections.

The most siguificant achievement of this tangent-formula refinement of ljhases has been the reiuterprctation of the CD loop as a sccsond calcium-binding site which is related by a 2-fold axis to the EF calcium-binding loop (I). In a view down the local intramolecular diad, Fig. 10 compalcs the atomic model with thcx ISO and ISO+T*\N rlrctron density distributions of the calcium-hiiictitlg loops. Thr density distribution at the CD calcium position ill the IS0 +TAN map is more spherical, better resolved, alld morel intcnsc than is the comparable feature in the IS0 map. Tllis improvement, together with greater distinc- tiveness of the coordinating side chains facilitated the reinter- pretation. Whether the CD calcium might not also have been discovered on re-examination of the IS0 map is now a moot point, but certainly its identification could not have been made as confidently as I&W based on the IS0 +TAN map.

The tangent formula, due to its relation to the squared struc- ture, tends to develop phases which exaggerate the electron density at the positions of the heavier atoms. Although this tendency is diminished when T1N and IS0 information are combined, the effect is still evident as shown by accentuated densities at calcium and sulfur positions (Table I). This ex- aggerated enhancement of density is not, however, actually deleterious; rather it aids the localization of heavy atoms as noted before (9) and above and also helps in identification. In particular, the enhancement ratios for the calcium atoms help to identify them as heavier than sulfur and of the same species. (Incidentally, the relative peak heights at the calcium and sul- fur positions indicate that the calcium sites are equally and nearly fully occupied.) The tangent formula can also distort and exaggerate the electron density distributions of peaks not

TABLE I Cotiparison of electron den&y peak heigh/s

Peak values of several salient features in the electron densit) distribution, p(ISO), based solely on IS0 phase information are compared with the corresponding values from the distribution, p(ISO+TAN), calculated using combined IHO+TAN informa- tion. Electron density values are relative to the average electron density of the crystal.

Feature

-SH of Cys 18. Calcium of the CD loop. Calcium of the EF loop, Unexplained peak. Highest other IS0 peak. Highest other ISO+TAN pea Mean height of other 36 peaks

with p,,x(ISO) > 1.08.. c mmI-TAd’/C nm.4’.

~max USO+TAN)

e/A 3 e/n3

1.39 1.97 1.30 2.32 1.25 2.35 2.67 5.93 1.6 (2.1)

(1.4) 2.2

l.lG 1.42

i Ratio

1.42 1.78 1.89 2.23 1.3 1.6

1.22 1.13

associated with heavy atoms (3-5, 9). However, this ISO+ TAN map showed little distortion and the enhancement, ratio fol peaks, 1.22, was not much greater than the over-all amltlifi- cation of density, 1.13.

There is one disturbing feature of the map which remains ~11. esplained. It is an ellipsoidal mass which rises to thr highest electron den&- in the structure. .Is shown in Table I this unexplained pclak behaved as 1~.ould an atom or group heavier than calcium, Jet there is no chemical or sterc~ochemical evi- dence for such a constituent (1). Either it is an atom or group of surprising identity which behaved as expected in the tangent formula refiucmnlt or it is a spurious product of the IS0 phasiug which the tangent formula calculations perpetuated and iu- tensified. In any case, the evaluation of this work cannot be considered complete until this peak is esplained.

dclinowledgments~~~e thank Doctors Clivc Nockolds and Robert K&singer for proTiding the data for this atlalysi~. This work has also profited from discussions with Robert Iirctsillger and we are most grateful for his help.

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Wayne A. Hendrickson and Jerome KarleTANGENT FORMULA

Carp Muscle Calcium-binding Protein: III. PHASE REFINEMENT USING THE

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