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Quantification of phase improvement in the tangent formula application to carp muscle calcium-binding protein
APPENDIX
of Phase Improvement in the Tangent Formula Application to Carp Muscle Calcium-binding Protein
Quantification
W. HENDRICIESON
Naval Research l&natory Washington, D.C. 20375, U.S.A. A procedure for improving phases by use of the tangent formula produced a new electron density map for the carp muscle calcium-binding protein which seemed more readily and surely interpretable than was the initial map (Hendrickson & Karle, 1973). Indeed, the new map brought the first realization of a second calcium-binding site. However, while it is the interpretability of the map produced with a given set of phases which is of ultimate concern, appraisals of interpretability are largely subjective and other measures are needed for a quantitative evaluation of the phases. There were no means for such a quantification earlier. But now that the atomic model for the calcium-binding protein has been relined (Moews & Kretsinger, 1974), it is possible finally to quantify those tangent formula results. The phases derived from the refined model offer a solid and quite independent standard against which other sets of phases can be compared. Average phase discrepancies among various pertinent phase sets for the calciumbinding protein are given in Table Al. The basis of reflections for the overall comparisons is the subset common to all the phase sets, i.e. those reflections having phase information from the refined model (MODEL, F-6h from Moews & Kretsinger, 1974), from the isomorphous replacement method (ISO), from the tangent formula calculations (TAN) and from the probabilistically combined isomorphous replacement and tangent formula information (ISOTAN). In addition to the overall discrepancies, averages are also given for two subcategories of these reflections: those for which the IS0 information is particularly weak (figure of merit, m, less than 0.5) and those for which the TAN information is especially strong (reliability factor, u, greater than 4.0). A more complete picture of the dependence of phase discrepancies on the reliability factor is elaborated in Figure Al. These phase comparisons confirm the earlier subjective conclusion that the tangent formula procedure had brought about an improvement in phases. However, the actual extent of improvement is seen to be quite modest : ISOTAN phases average only about 2” more accurate than the initial IS0 phases. Indeed, except at the highest levels of TAN reliability as judged by CC,the TAN phases themselves are less accurate than the IS0 phases. It is only by virtue of the probability procedures used to combine the IS0 and TAN information that any improvement is realized. Of course, the extent of phase improvement is greater than average for certain subclasses of the data. In particular, improvement is most appreciable where it is most needed, namely among reflections with the weakest initial IS0 phasing. Also, the accuracy of TAN phases and the extent of phase improvement is strongly correlated with the reliability factor, u. 226
PHASE
IMPROVEMENT
IN
THE
TANGENT
227
FORMULA
FIQ. Al. Relation between phase discrepancies and the TAN reliability fa&or. Average phase discrepancies, < A$ >, for the comparisons of TAN, IS0 and ISOTAN phases against the MODEL phases are plotted as a function of the TAN reliability factor, a. TAN velues are shown by the dashed line, IS0 valuee are given by the dotted line, and ISOTAN values are represented by the solid line.
The data given in Figure Al and Table Al also lend credence to this tangent formula procedure as a whole and in particular to the reliability factor on which it is heavily baaed. The decrease of phase discrepancy with increasing a is as theoretically expected, thus confirming that a is a good measure of TAN phase accuracy. Further verification of the procedure is implicit in the improvement of ISOTAN over IS0 phases despite the inferior accuracy of the TAN phases. The effectiveness of this combining of information is attributable to the a values, which are the characteristic parameters of TAN phase probability distributions. The relative weighting of TAN to IS0 information given by these experimental a values proves to be imperceptibly different from optimal. Uniform modification of the a values by a factor of 05 to 15 leaves the overall MODEL verse ISOTAN phase discrepancy between 57-l and 57-Z”, while TABLE
Al
Average phase discrepancies among various &z-se sets Category
Overall a > 4.0 171< 0.6
No. of refleations 3103 624 460
MODEL VW8U-3
IS0 69.4 60.2 76.3
MODEL ver.m.9 ISOTAN 67.2 46-9 69.0
MODEL V@WU8
TAN 67.2 57-4 70.1
IS0 vt?mua
IS0 vereua
ISOTAN
TAN
28.1 21.0 66.7
68-S 56.2 77.2
ISOTAN Ver’Bus
TAN 50-6 39.9 29.2
The average phase disorepanoy, < AI$ >, between two phase sets, A and B, is defined as the mean of the smaller of #A - 4BI and 360” - ]+A - $BI, i.e. the mean of difference as taken in the dire&ion of the shorter path around the phase circle.
228
W.
HENDRICKSON
further decrease or increase of the factor quickly drives the discrepancy t,o the MODEL versus TAN ext,remes. Another aspect of this tangent formula application which can only now be evaluated is the accuracy of extended phases, that is, TAN phases for reflections beyond the resolution limit of IS0 phasing. There were 792 such extended reflections, nearly all of them in the region of 2.2 A to l-8 A spacings. The MODEL versus TAN phase discrepancies for these data average 67-2” overall and 59.8” for reflections with a > 4.0. These values compare favorably with those of 67.2 and 57.4”, respectively, reported in Table Al for the lower resolution, refinement data. Thus the phase extension appears to have been effective. Finally, it must be noted that the unexplained feature of highest electron density in the IS0 map proves to have been an artifact of the IS0 phasing. Unfortunately, in spite of the cautions, gradualness, and selectivity of this tangent formula procedure, the spurious peak was perpetuated and enhanced. This experience demonstrates an inherent shortcoming of tangent formula refinement on limited data, but it also offers hope that greater accuracy than that seen here might obtain from refinement’s on data which are free of such systematic errors. I wish to thank P. C. Moews and R. H. Kretsinger calculations.
for sending me the results of their
REFERENCES Hendrickson, W. A. & Karle, J. (1973). J. Biol. Chem. 248, 3327-3334. Moews, P. C. $ Kretsinger, R. H. (1974). J. Mol. Biol. 91, 229-232.
of Phase Improvement in the Tangent Formula Application to Carp Muscle Calcium-binding Protein
Quantification
W. HENDRICIESON
Naval Research l&natory Washington, D.C. 20375, U.S.A. A procedure for improving phases by use of the tangent formula produced a new electron density map for the carp muscle calcium-binding protein which seemed more readily and surely interpretable than was the initial map (Hendrickson & Karle, 1973). Indeed, the new map brought the first realization of a second calcium-binding site. However, while it is the interpretability of the map produced with a given set of phases which is of ultimate concern, appraisals of interpretability are largely subjective and other measures are needed for a quantitative evaluation of the phases. There were no means for such a quantification earlier. But now that the atomic model for the calcium-binding protein has been relined (Moews & Kretsinger, 1974), it is possible finally to quantify those tangent formula results. The phases derived from the refined model offer a solid and quite independent standard against which other sets of phases can be compared. Average phase discrepancies among various pertinent phase sets for the calciumbinding protein are given in Table Al. The basis of reflections for the overall comparisons is the subset common to all the phase sets, i.e. those reflections having phase information from the refined model (MODEL, F-6h from Moews & Kretsinger, 1974), from the isomorphous replacement method (ISO), from the tangent formula calculations (TAN) and from the probabilistically combined isomorphous replacement and tangent formula information (ISOTAN). In addition to the overall discrepancies, averages are also given for two subcategories of these reflections: those for which the IS0 information is particularly weak (figure of merit, m, less than 0.5) and those for which the TAN information is especially strong (reliability factor, u, greater than 4.0). A more complete picture of the dependence of phase discrepancies on the reliability factor is elaborated in Figure Al. These phase comparisons confirm the earlier subjective conclusion that the tangent formula procedure had brought about an improvement in phases. However, the actual extent of improvement is seen to be quite modest : ISOTAN phases average only about 2” more accurate than the initial IS0 phases. Indeed, except at the highest levels of TAN reliability as judged by CC,the TAN phases themselves are less accurate than the IS0 phases. It is only by virtue of the probability procedures used to combine the IS0 and TAN information that any improvement is realized. Of course, the extent of phase improvement is greater than average for certain subclasses of the data. In particular, improvement is most appreciable where it is most needed, namely among reflections with the weakest initial IS0 phasing. Also, the accuracy of TAN phases and the extent of phase improvement is strongly correlated with the reliability factor, u. 226
PHASE
IMPROVEMENT
IN
THE
TANGENT
227
FORMULA
FIQ. Al. Relation between phase discrepancies and the TAN reliability fa&or. Average phase discrepancies, < A$ >, for the comparisons of TAN, IS0 and ISOTAN phases against the MODEL phases are plotted as a function of the TAN reliability factor, a. TAN velues are shown by the dashed line, IS0 valuee are given by the dotted line, and ISOTAN values are represented by the solid line.
The data given in Figure Al and Table Al also lend credence to this tangent formula procedure as a whole and in particular to the reliability factor on which it is heavily baaed. The decrease of phase discrepancy with increasing a is as theoretically expected, thus confirming that a is a good measure of TAN phase accuracy. Further verification of the procedure is implicit in the improvement of ISOTAN over IS0 phases despite the inferior accuracy of the TAN phases. The effectiveness of this combining of information is attributable to the a values, which are the characteristic parameters of TAN phase probability distributions. The relative weighting of TAN to IS0 information given by these experimental a values proves to be imperceptibly different from optimal. Uniform modification of the a values by a factor of 05 to 15 leaves the overall MODEL verse ISOTAN phase discrepancy between 57-l and 57-Z”, while TABLE
Al
Average phase discrepancies among various &z-se sets
Overall a > 4.0 171< 0.6
No. of refleations 3103 624 460
MODEL VW8U-3
IS0 69.4 60.2 76.3
MODEL ver.m.9 ISOTAN 67.2 46-9 69.0
MODEL V@WU8
TAN 67.2 57-4 70.1
IS0 vt?mua
IS0 vereua
ISOTAN
TAN
28.1 21.0 66.7
68-S 56.2 77.2
ISOTAN Ver’Bus
TAN 50-6 39.9 29.2
The average phase disorepanoy, < AI$ >, between two phase sets, A and B, is defined as the mean of the smaller of #A - 4BI and 360” - ]+A - $BI, i.e. the mean of difference as taken in the dire&ion of the shorter path around the phase circle.
228
W.
HENDRICKSON
further decrease or increase of the factor quickly drives the discrepancy t,o the MODEL versus TAN ext,remes. Another aspect of this tangent formula application which can only now be evaluated is the accuracy of extended phases, that is, TAN phases for reflections beyond the resolution limit of IS0 phasing. There were 792 such extended reflections, nearly all of them in the region of 2.2 A to l-8 A spacings. The MODEL versus TAN phase discrepancies for these data average 67-2” overall and 59.8” for reflections with a > 4.0. These values compare favorably with those of 67.2 and 57.4”, respectively, reported in Table Al for the lower resolution, refinement data. Thus the phase extension appears to have been effective. Finally, it must be noted that the unexplained feature of highest electron density in the IS0 map proves to have been an artifact of the IS0 phasing. Unfortunately, in spite of the cautions, gradualness, and selectivity of this tangent formula procedure, the spurious peak was perpetuated and enhanced. This experience demonstrates an inherent shortcoming of tangent formula refinement on limited data, but it also offers hope that greater accuracy than that seen here might obtain from refinement’s on data which are free of such systematic errors. I wish to thank P. C. Moews and R. H. Kretsinger calculations.
for sending me the results of their
REFERENCES Hendrickson, W. A. & Karle, J. (1973). J. Biol. Chem. 248, 3327-3334. Moews, P. C. $ Kretsinger, R. H. (1974). J. Mol. Biol. 91, 229-232.