Ultramlcroscopy 47 (1992) 256-265 North-Holland
Assessment of resolution in biological electron crystallography Robert M Glaeser Molecular and Cell Btology Department, Stanley/Donner ASU, and Cell and Molecular Btology Dwtston, Lawrence Berkeley Laboratory, Umverstty of Cahfornta, Berkeley, CA 94720, USA
and Kenneth H Downing Cell and Molecular Btology Dwtston, Donner Laboratory, Lawrence Berkeley Laboratory, Unwerstty of Cahforma, Berkeley, CA 94720, USA Recewed 2 August 1991, at Edltorml Office 13 May 1992
The resolution of images or density maps produced by electron microscopyand electron crystallographycan be objectwely defined in terms of the spatml frequency of the highest resolution diffraction spot, or Fourier coeffloent, included m the data processing In practice, this objectwe definition of resolution is expected to be too optimistic ff the amphtudes of the highest resolution structure factors are too weak, ff the population of high resolution reflections is too sparse, or ff the signal-to-noise ratio of the high resolution data is too low Calculated examples are presented here which dlustrate how the apparent resolution m images of a membrane protein, bactenorhodopsln, can be reduced from a nominal value of 3 5 ,~ by weak amphtudes, sparse data or high noise levels These calculations provide concrete examples which can serve as a guide when estimating whether the objective defimtlon of image resolution Is hkely to correspond to a practical, structurally useful estimate of ~mage resolution
I. Introduction W h e n p e r f o r m i n g a crystallographic structure analysis it 1s n a t u r a l to characterize the r e s o l u t i o n o b t a i n e d by specifying the Bragg spacing, or period, c o r r e s p o n d i n g to the highest r e s o l u t i o n diffraction spot that is e n t e r e d into the structure analysis T h e same c h a r a c t e r i z a t i o n of resolution can also be used for structural studies of n o n crystalline objects I n the case of crystals, the obvious p r e s e n c e of a distinct diffraction spot at a specified reciprocal lattice position lends a sense of objectivity a n d q u a n t i t a t i v e vahdlty to this deftnltlOn of r e s o l u t i o n A l t h o u g h not as self-evident as the p r e s e n c e of a diffraction spot, equally objective criteria for d e f i n i n g the r e s o l u t i o n in terms of the signal-to-noise ratio or similar criteria are in use for n o n - c r y s t a l h n e objects [7] Thus, in all areas of electron crystallography of biologl-
cal m a c r o m o l e c u l e s [5], there are objective criteria to use in deciding w h e t h e r a given F o u r i e r c o m p o n e n t should be e n t e r e d into a structure analysis, the highest resolution F o u r i e r compon e n t so used provides an u n a m b i g u o u s , operatlonal definttlon of the r e s o l u t i o n of the structure analysis I n spite of the q u a n t i t a t i v e basis a n d the freed o m from ambiguity of the accepted definition of resolution described above, it is widely recognized that th~s o p e r a t i o n a l d e f i n i t i o n may be of little practical relevance in m a n y circumstances F o r example, is a projection m a p ( t w o - & m e n sional image) really a "3 5 A m a p " if the data are relatively c o m p l e t e only to a lower resolution, say 7 A, a n d there are only a h a n d f u l of additional F o u r i e r c o m p o n e n t s out to 3 5 .~9 O r is the map still a "3 5 A m a p " if virtually all diffraction spots are p r e s e n t to 3 5 .~ but with a strong G a u s s m n
0304-3991/92/$05 00 © 1992 - Elsevier Soence Pubhshers B V All rights reserved
R M Glaeser, K H Downmg / Assessment of resolutton m btologtcal electron crystallography
roll-off (l e "temperature factor") due to experimental factors or crystalline disorder9 Should a map really be called a "3 5 .~" map if most of the data beyond a certain lower resolution, say 7 A, has such a low signal-to-noise ratio that including the high resolution data adds as much noise as signal9 In all of these cases, the higher resolution features - or at least parts of them - are there in the map, but does it make any sense to say that the map is a high resolution representation of the structure if the features are too incomplete, too weak in mtenslty, or too deeply buried in the noise to be seen9 The definition of resolution which is stated in the first paragraph above IS probably not of as much practical use as ItS quantltatwe objectivity might suggest The problems that are likely to arise due to sparslty of data, down-weighting of high frequency terms and noise have been widely recognized, but they have been discussed mainly in informal conversation Relatwely little has been done to explore the point at which each such factor may become significant in reducing the practical resolution to some value lower than that of the highest resolution Fourier component It seemed worthwhile now to demonstrate the influence of such effects by computer simulations, starting with a complete, high quality data set The intention is to provide concrete examples of the influence of these effects and thereby to decrease the degree of uncertainty as to how important they may be in reducing the practical resolution to a value less than that usually cited as the "objectwe resolution" The test example chosen for this work IS the two-dimensional (2D) projection of bacterIorhodopsin, an integral membrane protein which forms well ordered 2D crystals within the natwe cell membrane of Halobactertum halobtum [1] A complete, 3 5 A resolution set of structure factor amplitudes in the hkO plane has been published for these crystals [6] Application of a Gausslan roll-off to this data results m a progresswe loss of resolution, as expected, as the magnitude of the "temperature factor" Increases Ehminatlon of the weakest, high resolution Fourier components has little effect on the map, as expected, but ehmlnatlon of all save a few of the strongest o
257
reflections above 7 ,~ effectively reduces the resolution to 7 .~, again as expected The addition of high levels of noise, roughly equal m amphtude to the strongest high resolution reflections, does reduce the resolution of the map, but the practical resolution of the strongest high resolution features is found to be remarkably robust in the presence of high levels of noise The conclusions drawn from these simulations may vary in detail in a structure-dependent way Even so, these simulations should serve as a useful guide for future work in which there is reason to think that the objectwe definition of resolution discussed in the first paragraph above may be too optimistic, and there is a desire to estimate - in practical terms - the working resolution of a gwen structure analysis
2. Computer simulations 2 1 Model data set The Initial 3 5 .~ resolution data set of structure factor amplitudes and phases for 2D crystals of bacterlorhodopsin are those published by Henderson et al [6] The amplitudes are derived from electron diffraction patterns, and they are therefore free of the experimental errors and reduction in amplitudes that would occur for values derived directly from images The phases are derived from images, and they clearly have an Increasing amount of error at high resolution For the purpose of these simulations, however, the published data now represent a model structure, and in that sense they are "perfect data", free from noise and distortions The advantage in using this experimental data set, rather than some other type of model data set, is that we know that the experimental data constitute a reahstlc model of the sort of data that occurs in the practice of electron crystallography 2 2 Gausstan reduction m amphtudes The structure factor amphtudes that are derwed from the Fourier transforms of electron
R M Glaeser, K H Downmg / Assessment of resolutton m btologtcal electron crystallography
258
microscope images are strongly reduced in magnitude at high resolution, Ill comparison to the electron dlffractton amplitudes [4,6] We have
chosen to model this experimental reduction in amplitudes by a Gausslan functton In this case the tmage amplitudes, [ F ( s ) i m a g e I, and the
=0
+'p R
R=60,,
l / s , might effectively delete those structure factors from the Fourier synthesis The extstance of a Gaussian parameter, B ~2, s h o u l d t h e r e f o r e r o u g h l y h m t t t h e r e s o l u t i o n to a v a l u e s i m i l a r to v/B A T h e a c t u a l results, o b t a i n e d in o u r s i m u l a t i o n s w i t h b a c t e r l o r h o d o p s m " m o d e l " s t r u c t u r e f a c t o r s , b e h a v e in a w a y t h a t is s o m e w h a t m o r e c o m p l i c a t e d t h a n e x p e c t e d A p p l i c a t i o n o f a G a u s s l a n filter w i t h = 4 5 A r e s u l t s in a m a p o f p o o r e r q u a l i t y t h a n is o b t a i n e d w i t h a s h a r p , 4 5 A c u t o f f in r e s o l u t i o n T h i s r e s u l t m a y b e d u e in p a r t to t h e fact that there are no Fourier coefficients, not e v e n o n e s w i t h a m p l i t u d e s r e d u c e d to less t h a n e - 1 , at r e s o l u t i o n s h i g h e r t h a n 3 5 .& I f t h e y w e r e p r e s e n t , l e if t h e initial d a t a set e x t e n d e d to h i g h e r r e s o l u t i o n , t h e n it is likely t h a t t h e m a p c o r r e s p o n d i n g to a 4 5 .& G a u s s l a n c u t o f f m i g h t
264
R M Glaeser, K H Downmg / Assessment of resolution m btologtcal electron crystallography
have a closer similarity to the map produced by a sharp 4 5 .~ cutoff An important point which is demonstrated in the high resolution simulations is the fact that significant distortions or loss of resolution can occur, due to Gausslan attenuation of amplitudes, if one must rely only on images On the other hand, if accurate phases can be obtained from images, even in the presence of a severe Gausslan filter, and if amplitudes can be obtained from electron diffraction patterns rather than from images, then the quality of the map is greatly improved At lower resolution, in the range of 10 A, the correspondence between a Gausslan cutoff and a sharp cutoff begins to break down While a progresslve loss of resolution still occurs for increasing values of B, other types of changes also appear in the low resolution map that are poorly represented by a sharp resolution cutoff At low resolution the number of coefficients entering the Fourier synthesis is greatly reduced The resultlng loss of redundancy in the data may be one factor that leads to an lnequlvalence between maps produced with a sharp cutoff and those produced with a Gaussmn cutoff Experimental hmltatlons normally are also responsible for the mablhty to fully recover phases for all of the available diffraction spots The weakest reflections are expected to be the first ones to be lost, and the simulations shown in fig 3 model this effect When on!y a few of the strongest reflections above 7 A resolution are retained, even with their proper, electron-diffraction derived amplitudes, there is little improvement in the map as compared to the 7 ,~ map In order for the "high resolution" map to really capture the important features of the correct map, the high resolution spots included have to represent something like half or two thirds of the intensity In the 7-3 5 A zone In the case of bacterlorhodopsln, this fraction of the total intensity is reached after only one-fifth to one-third of the theoretically avmlable reflections are included By the same token, the loss of 15% of the diffraction power, representing 47% of the available measurements between 7 and 3 5 A, has very little effect on the map o
The presence of increasing levels of noise is expected to lead to maps with progressively lower resolution, especially when spurious high resolution features that are due solely to noise are removed by suitable averaging of independent data sets Even without averaging, however, down-weighting of noisy data, i e reflections with uncertain phases, must lead to a resolution cutoff of a type that would contain similarities to both effects investigated above, l e the systematic "removal" of weaker reflections and an overall filterlng that is biased against higher resolution reflections In view of these prior expectations, it is surprising to see how robust the high resolution map is in the presence of noise Even with a phase residual of 70 ° (for all data between 7 and 3 5 A), the map continues to be a useful representation of the true (1 e model) 3 5 .~ map The validity of the map which is obtained w~th such a high phase residual can be judged by comparing the three independent realizations of the high resolution projection to each other, to their 3-fold average, and to the original, high resolution map At this noise level the average I Q is between 7 and 8, corresponding to an average figure of merit m the range 0 7 - 0 8 Although a reflection whose figure of merit is 0 7 contributes as much power to the noise in the map as ~t does to the signal [3] the signal itself is attenuated (by the weighting factor) by far less drastic a factor than e ~ Seen in this way, there is some rationale for understanding why a map of useful quality xs still produced, even when the phase residual is as high as 70 ° On the other hand, the quality of the high resolution map is badly affected when the noise level is further doubled, bringing the phase residual to 85 ° At this point all reflections have an I Q 8 (or worse), and the corresponding average weight (figure of merit) is well below 0 7 We wish to emphasize that the results of our simulations with the bacterxorhodopsln data set cannot give a guide which can be applied m a precise or universal way to all structures To some degree one should expect the effects of Gaussian filtration, incomplete data, and no~se to alter the effective resolution of projection maps in a structure-dependent way Even so, the simulations presented here should be of some general o
R M Glaeser, K H Downmg / Assessment of resolunon m btologtcal electron crystallography
i n t e r e s t b e c a u s e t h e y give c o n c r e t e e x a m p l e s o f effects t h a t p r e v i o u s l y o n e w o u l d have h a d to guess a b o u t , b a s e d u p o n intuition I n t u i t i o n o r " q u a l l t a t w e f e e l " for such effects has t h e shortc o m i n g t h a t o n e p e r s o n m a y differ quite s u b s t a n tially f r o m a n o t h e r in w h a t t h e y envision in t h e i r m i n d ' s eye T h e c o n c r e t e e x a m p l e s c a l c u l a t e d h e r e can serve the p u r p o s e o f f o u n d i n g o u r respective i n t u i t i o n on a c o m m o n , accessible b a s e of information
Acknowledgements W e t h a n k D r M e h m e t S a n k a y a for stimulating us to c a r r y o u t t h e w o r k r e p o r t e d h e r e by mvltlng o u r p a r t i c i p a t i o n in t h e 1991 E M S A Sympossum on R e s o l u t i o n m t h e M i c r o s c o p e W e also t h a n k m a n y c o l l e a g u e s over t h e y e a r s with w h o m we have informally d i s c u s s e d the issues now add r e s s e d in this p a p e r , b u t most especially D r
265
U e h A e b l , w h o s e p e r s u a s w e a n d forceful p e r spectlve has m a d e t h e s e questions really stick m o u r m i n d s This w o r k was s u p p o r t e d by t h e Office of Health and Environmental Research, United S t a t e s D e p a r t m e n t o f Energy, u n d e r c o n t r a c t no D E - A C 0 3 - 7 6 S F 0 0 0 9 8 a n d by t h e N a t i o n a l Instttutes o f H e a l t h P r o g r a m ProJect g r a n t G M 36884
References [1] A E Blaurock and W Stoeckemus, Nature (London) New Blol 233 (1971) 155 [2] D R Brdhnger, K H Downing and R M Glaeser, J Stat Planning Inference 25 (1990) 535 [3] R E Dlckerson, J C Kendrew and B E Strandberg, Acta Cryst 14 (1961) 1188 [4] K H Downing, Sctence 251 (1991) 53 [5] R M Glaeser, Annual Rev Phys Chem 36 (1985)543 [6] R Henderson, J M Baldwin, K H Downing, J Lepault and F Zemhn, Ultramlcroscopy 19 (1986) 147 [7] M Unser, B L Trus, J Frank and A C Steven, Ultramlcroscopy 30 (1989) 459
Assessment of resolution in biological electron crystallography Robert M Glaeser Molecular and Cell Btology Department, Stanley/Donner ASU, and Cell and Molecular Btology Dwtston, Lawrence Berkeley Laboratory, Umverstty of Cahfornta, Berkeley, CA 94720, USA
and Kenneth H Downing Cell and Molecular Btology Dwtston, Donner Laboratory, Lawrence Berkeley Laboratory, Unwerstty of Cahforma, Berkeley, CA 94720, USA Recewed 2 August 1991, at Edltorml Office 13 May 1992
The resolution of images or density maps produced by electron microscopyand electron crystallographycan be objectwely defined in terms of the spatml frequency of the highest resolution diffraction spot, or Fourier coeffloent, included m the data processing In practice, this objectwe definition of resolution is expected to be too optimistic ff the amphtudes of the highest resolution structure factors are too weak, ff the population of high resolution reflections is too sparse, or ff the signal-to-noise ratio of the high resolution data is too low Calculated examples are presented here which dlustrate how the apparent resolution m images of a membrane protein, bactenorhodopsln, can be reduced from a nominal value of 3 5 ,~ by weak amphtudes, sparse data or high noise levels These calculations provide concrete examples which can serve as a guide when estimating whether the objective defimtlon of image resolution Is hkely to correspond to a practical, structurally useful estimate of ~mage resolution
I. Introduction W h e n p e r f o r m i n g a crystallographic structure analysis it 1s n a t u r a l to characterize the r e s o l u t i o n o b t a i n e d by specifying the Bragg spacing, or period, c o r r e s p o n d i n g to the highest r e s o l u t i o n diffraction spot that is e n t e r e d into the structure analysis T h e same c h a r a c t e r i z a t i o n of resolution can also be used for structural studies of n o n crystalline objects I n the case of crystals, the obvious p r e s e n c e of a distinct diffraction spot at a specified reciprocal lattice position lends a sense of objectivity a n d q u a n t i t a t i v e vahdlty to this deftnltlOn of r e s o l u t i o n A l t h o u g h not as self-evident as the p r e s e n c e of a diffraction spot, equally objective criteria for d e f i n i n g the r e s o l u t i o n in terms of the signal-to-noise ratio or similar criteria are in use for n o n - c r y s t a l h n e objects [7] Thus, in all areas of electron crystallography of biologl-
cal m a c r o m o l e c u l e s [5], there are objective criteria to use in deciding w h e t h e r a given F o u r i e r c o m p o n e n t should be e n t e r e d into a structure analysis, the highest resolution F o u r i e r compon e n t so used provides an u n a m b i g u o u s , operatlonal definttlon of the r e s o l u t i o n of the structure analysis I n spite of the q u a n t i t a t i v e basis a n d the freed o m from ambiguity of the accepted definition of resolution described above, it is widely recognized that th~s o p e r a t i o n a l d e f i n i t i o n may be of little practical relevance in m a n y circumstances F o r example, is a projection m a p ( t w o - & m e n sional image) really a "3 5 A m a p " if the data are relatively c o m p l e t e only to a lower resolution, say 7 A, a n d there are only a h a n d f u l of additional F o u r i e r c o m p o n e n t s out to 3 5 .~9 O r is the map still a "3 5 A m a p " if virtually all diffraction spots are p r e s e n t to 3 5 .~ but with a strong G a u s s m n
0304-3991/92/$05 00 © 1992 - Elsevier Soence Pubhshers B V All rights reserved
R M Glaeser, K H Downmg / Assessment of resolutton m btologtcal electron crystallography
roll-off (l e "temperature factor") due to experimental factors or crystalline disorder9 Should a map really be called a "3 5 .~" map if most of the data beyond a certain lower resolution, say 7 A, has such a low signal-to-noise ratio that including the high resolution data adds as much noise as signal9 In all of these cases, the higher resolution features - or at least parts of them - are there in the map, but does it make any sense to say that the map is a high resolution representation of the structure if the features are too incomplete, too weak in mtenslty, or too deeply buried in the noise to be seen9 The definition of resolution which is stated in the first paragraph above IS probably not of as much practical use as ItS quantltatwe objectivity might suggest The problems that are likely to arise due to sparslty of data, down-weighting of high frequency terms and noise have been widely recognized, but they have been discussed mainly in informal conversation Relatwely little has been done to explore the point at which each such factor may become significant in reducing the practical resolution to some value lower than that of the highest resolution Fourier component It seemed worthwhile now to demonstrate the influence of such effects by computer simulations, starting with a complete, high quality data set The intention is to provide concrete examples of the influence of these effects and thereby to decrease the degree of uncertainty as to how important they may be in reducing the practical resolution to a value less than that usually cited as the "objectwe resolution" The test example chosen for this work IS the two-dimensional (2D) projection of bacterIorhodopsin, an integral membrane protein which forms well ordered 2D crystals within the natwe cell membrane of Halobactertum halobtum [1] A complete, 3 5 A resolution set of structure factor amplitudes in the hkO plane has been published for these crystals [6] Application of a Gausslan roll-off to this data results m a progresswe loss of resolution, as expected, as the magnitude of the "temperature factor" Increases Ehminatlon of the weakest, high resolution Fourier components has little effect on the map, as expected, but ehmlnatlon of all save a few of the strongest o
257
reflections above 7 ,~ effectively reduces the resolution to 7 .~, again as expected The addition of high levels of noise, roughly equal m amphtude to the strongest high resolution reflections, does reduce the resolution of the map, but the practical resolution of the strongest high resolution features is found to be remarkably robust in the presence of high levels of noise The conclusions drawn from these simulations may vary in detail in a structure-dependent way Even so, these simulations should serve as a useful guide for future work in which there is reason to think that the objectwe definition of resolution discussed in the first paragraph above may be too optimistic, and there is a desire to estimate - in practical terms - the working resolution of a gwen structure analysis
2. Computer simulations 2 1 Model data set The Initial 3 5 .~ resolution data set of structure factor amplitudes and phases for 2D crystals of bacterlorhodopsin are those published by Henderson et al [6] The amplitudes are derived from electron diffraction patterns, and they are therefore free of the experimental errors and reduction in amplitudes that would occur for values derived directly from images The phases are derived from images, and they clearly have an Increasing amount of error at high resolution For the purpose of these simulations, however, the published data now represent a model structure, and in that sense they are "perfect data", free from noise and distortions The advantage in using this experimental data set, rather than some other type of model data set, is that we know that the experimental data constitute a reahstlc model of the sort of data that occurs in the practice of electron crystallography 2 2 Gausstan reduction m amphtudes The structure factor amphtudes that are derwed from the Fourier transforms of electron
R M Glaeser, K H Downmg / Assessment of resolutton m btologtcal electron crystallography
258
microscope images are strongly reduced in magnitude at high resolution, Ill comparison to the electron dlffractton amplitudes [4,6] We have
chosen to model this experimental reduction in amplitudes by a Gausslan functton In this case the tmage amplitudes, [ F ( s ) i m a g e I, and the
=0
+'p R
R=60,,
l / s , might effectively delete those structure factors from the Fourier synthesis The extstance of a Gaussian parameter, B ~2, s h o u l d t h e r e f o r e r o u g h l y h m t t t h e r e s o l u t i o n to a v a l u e s i m i l a r to v/B A T h e a c t u a l results, o b t a i n e d in o u r s i m u l a t i o n s w i t h b a c t e r l o r h o d o p s m " m o d e l " s t r u c t u r e f a c t o r s , b e h a v e in a w a y t h a t is s o m e w h a t m o r e c o m p l i c a t e d t h a n e x p e c t e d A p p l i c a t i o n o f a G a u s s l a n filter w i t h = 4 5 A r e s u l t s in a m a p o f p o o r e r q u a l i t y t h a n is o b t a i n e d w i t h a s h a r p , 4 5 A c u t o f f in r e s o l u t i o n T h i s r e s u l t m a y b e d u e in p a r t to t h e fact that there are no Fourier coefficients, not e v e n o n e s w i t h a m p l i t u d e s r e d u c e d to less t h a n e - 1 , at r e s o l u t i o n s h i g h e r t h a n 3 5 .& I f t h e y w e r e p r e s e n t , l e if t h e initial d a t a set e x t e n d e d to h i g h e r r e s o l u t i o n , t h e n it is likely t h a t t h e m a p c o r r e s p o n d i n g to a 4 5 .& G a u s s l a n c u t o f f m i g h t
264
R M Glaeser, K H Downmg / Assessment of resolution m btologtcal electron crystallography
have a closer similarity to the map produced by a sharp 4 5 .~ cutoff An important point which is demonstrated in the high resolution simulations is the fact that significant distortions or loss of resolution can occur, due to Gausslan attenuation of amplitudes, if one must rely only on images On the other hand, if accurate phases can be obtained from images, even in the presence of a severe Gausslan filter, and if amplitudes can be obtained from electron diffraction patterns rather than from images, then the quality of the map is greatly improved At lower resolution, in the range of 10 A, the correspondence between a Gausslan cutoff and a sharp cutoff begins to break down While a progresslve loss of resolution still occurs for increasing values of B, other types of changes also appear in the low resolution map that are poorly represented by a sharp resolution cutoff At low resolution the number of coefficients entering the Fourier synthesis is greatly reduced The resultlng loss of redundancy in the data may be one factor that leads to an lnequlvalence between maps produced with a sharp cutoff and those produced with a Gaussmn cutoff Experimental hmltatlons normally are also responsible for the mablhty to fully recover phases for all of the available diffraction spots The weakest reflections are expected to be the first ones to be lost, and the simulations shown in fig 3 model this effect When on!y a few of the strongest reflections above 7 A resolution are retained, even with their proper, electron-diffraction derived amplitudes, there is little improvement in the map as compared to the 7 ,~ map In order for the "high resolution" map to really capture the important features of the correct map, the high resolution spots included have to represent something like half or two thirds of the intensity In the 7-3 5 A zone In the case of bacterlorhodopsln, this fraction of the total intensity is reached after only one-fifth to one-third of the theoretically avmlable reflections are included By the same token, the loss of 15% of the diffraction power, representing 47% of the available measurements between 7 and 3 5 A, has very little effect on the map o
The presence of increasing levels of noise is expected to lead to maps with progressively lower resolution, especially when spurious high resolution features that are due solely to noise are removed by suitable averaging of independent data sets Even without averaging, however, down-weighting of noisy data, i e reflections with uncertain phases, must lead to a resolution cutoff of a type that would contain similarities to both effects investigated above, l e the systematic "removal" of weaker reflections and an overall filterlng that is biased against higher resolution reflections In view of these prior expectations, it is surprising to see how robust the high resolution map is in the presence of noise Even with a phase residual of 70 ° (for all data between 7 and 3 5 A), the map continues to be a useful representation of the true (1 e model) 3 5 .~ map The validity of the map which is obtained w~th such a high phase residual can be judged by comparing the three independent realizations of the high resolution projection to each other, to their 3-fold average, and to the original, high resolution map At this noise level the average I Q is between 7 and 8, corresponding to an average figure of merit m the range 0 7 - 0 8 Although a reflection whose figure of merit is 0 7 contributes as much power to the noise in the map as ~t does to the signal [3] the signal itself is attenuated (by the weighting factor) by far less drastic a factor than e ~ Seen in this way, there is some rationale for understanding why a map of useful quality xs still produced, even when the phase residual is as high as 70 ° On the other hand, the quality of the high resolution map is badly affected when the noise level is further doubled, bringing the phase residual to 85 ° At this point all reflections have an I Q 8 (or worse), and the corresponding average weight (figure of merit) is well below 0 7 We wish to emphasize that the results of our simulations with the bacterxorhodopsln data set cannot give a guide which can be applied m a precise or universal way to all structures To some degree one should expect the effects of Gaussian filtration, incomplete data, and no~se to alter the effective resolution of projection maps in a structure-dependent way Even so, the simulations presented here should be of some general o
R M Glaeser, K H Downmg / Assessment of resolunon m btologtcal electron crystallography
i n t e r e s t b e c a u s e t h e y give c o n c r e t e e x a m p l e s o f effects t h a t p r e v i o u s l y o n e w o u l d have h a d to guess a b o u t , b a s e d u p o n intuition I n t u i t i o n o r " q u a l l t a t w e f e e l " for such effects has t h e shortc o m i n g t h a t o n e p e r s o n m a y differ quite s u b s t a n tially f r o m a n o t h e r in w h a t t h e y envision in t h e i r m i n d ' s eye T h e c o n c r e t e e x a m p l e s c a l c u l a t e d h e r e can serve the p u r p o s e o f f o u n d i n g o u r respective i n t u i t i o n on a c o m m o n , accessible b a s e of information
Acknowledgements W e t h a n k D r M e h m e t S a n k a y a for stimulating us to c a r r y o u t t h e w o r k r e p o r t e d h e r e by mvltlng o u r p a r t i c i p a t i o n in t h e 1991 E M S A Sympossum on R e s o l u t i o n m t h e M i c r o s c o p e W e also t h a n k m a n y c o l l e a g u e s over t h e y e a r s with w h o m we have informally d i s c u s s e d the issues now add r e s s e d in this p a p e r , b u t most especially D r
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U e h A e b l , w h o s e p e r s u a s w e a n d forceful p e r spectlve has m a d e t h e s e questions really stick m o u r m i n d s This w o r k was s u p p o r t e d by t h e Office of Health and Environmental Research, United S t a t e s D e p a r t m e n t o f Energy, u n d e r c o n t r a c t no D E - A C 0 3 - 7 6 S F 0 0 0 9 8 a n d by t h e N a t i o n a l Instttutes o f H e a l t h P r o g r a m ProJect g r a n t G M 36884
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