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An experimental study of baseline reproducibility and its effect on high-speed sedimentation equilibrium data
ANALYTICAL
An
45,
BIOCHEMISTRY
Experimental Effect
86-99
Study
on Hig’h-Speed THOMAS
(197%
of Baseline
Reproducibility
and
Sedimentation
Equilibrium
Data1
A. HORBETT2 of
University
Department
of
AND
Washington,
Biochemistry, Received
March
DAVID
Its
C. TELLER
School of Me&c&e, Seattle, Washington 98106 24, 1971
Since its introduction by Yphantis (1) in 1964, high-speed sedimentation equilibrium determination of protein molecular weights has become an important technique to the biochemist. However, as pointed out by Yphantis and in the experience of subsequent users of this technique, the high-speed (or meniscus depletion) method gives less reproducible results than the older low-speed technique (2,3). One of the most important types of error leading to this decreased reproducibility is baseline variability. It is the purpose of this report to present the results of a study undertaken to measure the magnitude of this variability, to determine some of its causes, and to develop an improved method for baseline determinations. Ansevin et al. (4) recently reported a new, externally loaded ultracentrifuge cell that can be cleaned and refilled without disassembly. This cell was designed to, and apparently does, obviate the difficulties associated with baseline determination in the Yphantis-style six-channel cell assemblies (1) used here. However, for economic reasons alone, the widespread use of the older cell will probably continue for some time. Second, the results in this report show that, for speeds up to 32,000 rpm, baseline determinations as accurate as those obtained with the external loading cell of Ansevin et al. (4) can be obtained with the older style cell provided the procedure recommended here is used. This procedure employs a low-speed picture of the resuspended sample taken after the high-speeed equilibrium experiment’. ‘Supported by a Public Health Service Research Grant, National Institute of General and Medical Science. ‘The material presented is taken, in part, from a thesis 9. Horbctt to the Graduate Faculty of the University of fulfillment of the requirements for the Ph.D. degree. 86 @ 1972 by Academic Press, Inc.
GM
13401,
submitted Washington
from by in
the
Thomas partial
BASELINE
87
REPRODUCIBILITY
METHODS
Ultracentrifugation was performed as previously described (5). The cell assembly consisted of a single-sector cell housing, a Yphantis-style (1) six-channel charcoal-filled Epon centerpiece, sapphire windows, and interference window holders. The screw ring gasket was greased with Spinkote prior to cell assembly. The white window gaskets were discarded after every run. The cell was filled with an excess of water so that the baseline pattern could be determined throughout each channel. Pictures were taken within about 10 min of reaching speed unless otherwise noted. To allow better control of the window and window liner positions relative to the window holder, a window alignment scribe mark (A, Fig. 1) was made from the center of the keyway slot across the edge of the window holder which faces the centerpiece in the assembled cell. Two window liner alignment marks were made on the same edge, but placed opposite the two ends of a carefully centered window liner (Marks B, Fig. 1).
Scribe
Scribe
Scribe
Mark
Mark
B
A
FIG. 1. Scribed window holder assembly. Scribe marks A and B are described in the text. This figure also illustrates the use of a Bakelite clip in conjunction with the Teflon window liner which allows reproducible positioning of the window in the holder.
Data Collection and Reduction
Plate reading was performed on the modified Nikon microcomparator previously described (5). Care was taken to make the x-direction of the microcomparator stage travel parallel to the radial axis of the fringe rn\-elol) since misorientation of tllc plate could result in an apparent rotation of the baseline. The plates were aligned by rotating the stage until the y-micrometer positions of the central white light fringes in the inner and outer reference hole patterns were judged to be the same within reading error (3 to 5 p). (A movable gate in the optical tube allowed a
88
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A-94~01
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AND
TELLER
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X~mm.) FIG. 2. An example of baseline variability. Data from four baseline pictures were individually smoothed (see Appendix). The two baselines shown (A and B) resulted from averaging the smoothed baselines in the pairs indicated in the figure. The pictures were taken at 3200 x-pm prior to four separate high-speed equilibrium experiments at 32,000 rpm. The runs were done with the same cell, by the same person, and within a few days of one another.
white light picture of the reference holes and a monochromatic picture of the cell to be taken separately without moving the plate.) A transparent bull&ye pattern taped to the view screen aided in finding the center of these white light fringes, but all other readings were made electronically with the modified microcomparator (5). Readings were taken every 200 p throughout each channel and all pictures were read at the same radial positions in order to facilitate comparisons between separate baselines. In comparing baselines, we feIt it important to assess the contribution of random error to baseline va~ability. First, comparisons between smoothed estimates of baselines were therefore also made. Second, we have found it useful to smooth baselines before using the data to correct the high-speed equilibrium fringe pattern for baseline deviations. Third, routine use of the suggested low speed end-of-run baseline procedure described below should be accompanied by occasiona checks on the baseline reproducibility of the cell between speeds and quantitative estimates of the variability can be made using the computational procedure we have developed. Finally, an average baseline based on several separate runs is sometimes the best estimate one can make. The computational procedures we used in smoothing and comparing baselines is: described in detai1 in the Appendix.
BASELINE
89
REPRODUCIBILITY
RESULTS Figure 2 shows a typical example of baseline variability. These data were obtained from runs performed using the same cell within a few days of one another. As can be seen, baseline A is very similar in shape but rotated relative to baseline B. The average absolute difference in baseline height between these two baseline patterns is 7 pi (0.025 fringe). Figures 3 and 4 show the effect on the number-average and weightaverage molecular weights, M(n) and M (w) , in applying these two baselines to one set of high-speed sedimentation equilibrium data. The data are from an experiment on a-ohymotrypsin (Worthington CDI-6LD) under conditions in which the protein reversibly dimerizes. The effect on M(n) at 2 fringes concentration is approximately 10%. This seems much larger than one might at first ascribe to the 7 p difference in baseline, which is only about twice the random error level. The explanation is that a systematic error is propagated throughout the dat.a in such a way that it leads to a rotation of the whole M(n) curve in much the same way as the baselines themselves are rotated. The effect on M(W) is seen to be much smaller, amount,ing to about 5% difference in the M(w) curves
*A l B
Concentration
(frlngesl
3. Effect of baseline on M(n). Baseline A or B of Fig. 2 was used to correct the observed fringe pattern of one eq~~b~~ run and the resultant two sets of data were used to compute the two data sets shown. FIG.
90
20.0 0I
HORBETT
AND
1
6
I
4
Concentration
TELLER
8
I
10
I
12
I
14
(fringes)
FIG. 4. Effect of baseline on M(W). Baseline A or B of Fig. 2 was used to correct the observed fringe pattern of one equilibrium run and the resultant two sets of data were used to compute the two data sets shown.
at 2 fringes concentration. This difference in the sensitivity of M(n) and M(w) to systematic errors is intrinsic in their calculation (6). The effects of baseline variability are thus seen to be both large and systematic in nature. These conclusions led us to a study of some of the causes of the variability, and to a series of measurements of the magnitude of the variability under several conditions. One cause of baseline rotation is illustrated in Fig. 5. It is a picture of the keyway slot area of two upper window holders-a new one and a damaged one. The upper window holder damage resulted from failure to use grease (Spinkote) on the screw ring gasket. During cell tightening, enough friction occurred to carry the upper window holder past the keyway in the cell housing. In these cells, the window holder is aluminum and the keyway is steel. Thus, when the tightened cell was used, the centrifugal force encountered was strong enough to press the previously misaligned window holder back into the keyway and the window holder became gouged as shown in Fig. 5. In subsequent usage of the cell, the gouged side of the damaged window holder allowed the window holder and its window to be rotated past the keyway as the screw ring was tightened. The resultant window misorientation was sufficient to give baseline shifts as large as those illustrated in Fig. 2.
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REPRODUCIBILITY
FIG. 5. Comparison of damaged (upper) and new (lower) The two holders were placeed on a microcomparator stage was then made by photographing the enlarged image appearing
window holder slots. and this photograph on the viewscreen.
Table 1 shows part of the results of a study of baseline reproducibility. The root-mean-square (RMS) difference figures were calculated using differences in fringe displacement at each of approximately 40 equally spaced z-positions (200 p apart) in each channel. The smoothed data were obtained using the least-squares procedures described in the Appendix. The first line shows that our plate reader’s reproducibility in
Baseline
TABLE 1 Reproducibilityn R.&IS.
Purpose
of experiment
Reading erroti Photographicc noise Run-to-rund variability Blakelite Teflon Teflon + Bakelite Changing Teflon
clip
No. of runs
No. of photos
No. of channels
No. of points
6 1
6 5
18 15
1440 600
5 5 5 Fj
5 5 5 5
12e 15 15 15
480 600 600 600
Raw
difference,
fi
Smoothed
1.1 2 3
0.7 1.1
2.4 3.4 4.0 3.7
1.9 3.0 3.2 3.3
n All pictures at 32,000 rpm. * Each of six separate phot,os was read t,wice and t’he t,wo sets of readings from each photo were compared pairwise; the tabulated R.M.S. difference vahles are the average of the values obtained from the pairwisr comparisons. r Five pictures of the same baseline rml were taken and compared. d The cell was disassembled, cleaned, and refilled bet,ween each run. e Third channel fringes unreadable in three runs.
92
HORBETT
AND
TELLER
rereading the same plate is 1.1 ,Ubetween untreated data and 0.7~ between smoothed data. The second line shows that photographic plate noise, as measured by reading the same baseline from five different pictures of one run, does contribute significantly to the data variability. However, most of this contribution can be smoothed out, as seen by the reduction of the R.M.S. difference to 1.1 p in the smoothed data. This is only 0.4 ,u greater than the differences in smoothed data observed for rereading a plate. The third line shows that the standard window liner material, Bakelite, allowed baseline reproducibility nearly equal to the untreated data R.M.S. difference to be expected between pictures. However, the variation in smoothed data in these runs is 0.8 ,X greater than the difference in smoothed data observed between pictures. This is a reflection of the fact that systematic rotation did occur and contributed significantly to the variability from run to run. In the course of this study of baseline reproducibility with Bakelite window liners, the fringes in the third channel were often so severely blurred that readings were not possible. The occurrence of such distortion was not correlated with significant baseline rotations, however, since it occurred in three of the five experiments performed with Bakelite liners in which excellent reproducibility was observed (Table 1, line 3). A general method for preventing window distortion is to replace the Bakelite liners with replicas made from 0.015” Teflon sheets (Universal Plastics Co., Seattle). We have never observed window distortion when Teflon liners were used. In Table 1, the fourth line represents the results of five runs done with one set of Teflon liners. In the smoothed data, the 3.0 p run-to-run variability observed with Teflon is 1.1 p worse than observed when Bakelite was used. Thus, whether the use of Teflon liners is advisable depends on the speed of the run. We have not had window distortion problems below about 28,000 rpm. The Teflon liners are much more flexible and slippery than Bakelite liners and therefore do not work as well in holding the window in place during manipulation. To overcome this problem and to determine whether this was the cause of the lowered baseline reproducibility, a small section of Bakelite was placed in the window holder between the ends of the Teflon liner (see Fig. 1). The combination liner was oriented with the Bakelite opposite the keyway slot. The Bakelite clip acts as a spring to hold the window in position. The fifth line of Table 1 shows, however, that this procedure does not improve reproducibility. The last line of Table 1 shows that replacement of the Teflon liners after each run with new Teflon liners does not change their performance with regard to baseline reproducibility.
BASELINE
93
REPRODUCIBILITY
Previous comparisons of low- and high-speed pictures of a single baseline run in which the low-speed picture was taken first have shown unacceptably large variation, apparently because the windows shift position with increasing speed. Figure 2 illustrates the type of rotation of the baselines often encountered in his type of experiment. Table 2 shows an attempt to determine whether taking the low-speed picture after first spinning the cell at high speed would improve the reliability of the lowspeed picture procedure. The first line shows that the run-to-run variability at 3,200 rpm was 2.6 p in the smoothed data, compared to the 3.6 p variability at 32,000 rpm shown on the second line. When the 32,000 rpm and 3,200 rpm pictures of the same run were compared, it was found that the “between-speed” variability, 2.2 ,u in the smoothed data (Table 2, line 3) was well below either the 32,000 or 3,200 rpm run-to-run variabilities. These data suggest that an improved way to perform baseline determinations is to use a low-speed picture of the still-assembled but vigorously shaken cell after the sedimentation equilibrium run is done. The speed effect experiments so far discussed were carried out without removing the cell from the rotor between speeds and required only brief exposure of the cell to the higher speeds. However, the suggested endof-run baseline procedure involves removing the cell from the rotor, shaking it vigorously, and then replacing it in the rotor and taking a picture at low speed. Furthermore, a typical sedimentation equilibrium experiment would have involved spinning the cell for at least 8 hr and often much longer at the higher speed. Therefore, a direct test of the proposed end-of-run baseline procedure was performed. A cell was assembled and filled exactly as for a sedimentation equilibrium experiment, except water was used in all channels. Four pictures were taken: (1) at 32,000 rpm as soon as possible after reaching that speed (about 0.5
Effect
of Speed
TABLE 2 on Baseline Reproducibilitya
No. of runs
No. of phot’os
5 5 5
5 r ;b
R.M.S. Purpose
of experiment
Run-to-run variability at 3,200 rpm at 32,000 rpm Variability between speeds, within a run
No. of channels
15 14 28
= The cell was disassembled, cleaned, and refilled tures were taken first, then the rotor was brought
difference,
No. of points
Raw
Smoothed
600 560 1120
3.7 4.5 3.2
2.6 3.6 2 2
w
between runs. The 32,000 rpm pio to 3200 rpm for the second picture.
94
HORBETT
40
Z36 I
-
24 -
18 Zero
Time
2. * 22.5 3 3200
Hours rpm.
4=
% 5
Shakm
1
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TELLER
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AND
3
3
4
9
3200
4 4 344
:I 44
rpm.
rpm. I
24:;3
1
4
4 4 34
3 3
431
:443 43
4
4
1 4 14 21323 31
2 :33 2
341
33
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11
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1
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10
11
12
13
1A
15
16
17
X MICROMETER FIG.
I
1
1 12 43
:32:
43411 12422 33 0.
rpm.
At 32000
Cdl,
4 34:4;3
12-
At 32000
6. Baseline
reproducibility
with
READINGS
(mm.)
speed and time, unsmoothed
data.
hr) ; (2) at 32,000 rpm after 22.5 hr; (3) at 3,200 rpm as soon as possible after decelerating the rotor directly after picture 2; (4) at 3,200 rpm after removing the cell from the rotor, shaking vigorously, and replacing it in the rotor. The unsmoothed and smoothed baseline readings on one of the three channels are shown in Figs. 6 and 7. The numbers used to represent the data in these figures correspond to readings from the four pictures described. The readings from the four pictures in the other two channels were similarly superimposeable. The average R.M.S. derivat,ion between the unsmoothed baseline data was 2.0~ for the centripetal channel data of Fig. 6, 1.9 for the middle channel, and 2.6 for the outermost channel. In the smoothed data, the R.M.S. deviations were 1.0, 0.9, and 2.0~~ respectively. It is clear from this experiment 48 c 2
1.
536
-
Zero
2 * 22.5 4. 3’
2 e
q.24
Tim
At 32000
Hours
At
3200 Shaken rpm. Cdl,
-
32000 3200
rpm. rpm. rpm. 4
Ill
ti 2
41
3333322223333.:;3333 114444444444444:::44
c3
>- 12
-
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14422
4
0 9
‘
10
11
12
13
X MICROMETER FIG. 7. Baseline reproducibility
14
READINGS
15
16
17
(mm.)
with speed and time, smoothed
data,
BASELINE
95
REPRODUCIBILITY
that the procedure employed in the proposed end-of-run baseline technique can give an excellent estimate of the high-speed baseline. Three warnings to the prospective user of the end-of-run baseline technique are in order at this point. It has been observed in two separate instances that, when the cell barrel becomes distorted and fits much more tightly into the rotor hole than a new barrel, the baselines shifted an unacceptably large amount between 32,900 and 3,200 rpm. That is, the 32,000 versus 3,200 variability became larger t,han the variability between separate 32,000 rpm baselines in three cases. It is not known at present whether cell distortion is an adequate measure of when the low-speed end-of-run baseline procedure becomes unreliable. Therefore, use of the end-of-run baseline procedure requires an occasional check on the performance of the cell with respect to between-speed variability. This is readily done with a water-filled cell by comparing its baseline at the low and high speed to be used in the experiments. Another pitfall of the end-of-run baseline technique is due to t’he material under investigation, which may not redistribute uniformly when the cell is shaken or may sediment significantly even at t’he low speed. This problem can be overcome by performing a separate baseline experiment as described by Yphantis (1). The third precaution to be taken in the use of the end-of-run baseline technique regards the maximum speed at which the technique will still work well. In high-speed equilibrium experiments, a speed of 32,000 rpm is appropriate for a protein of molecular weight of 25,000 gm/mole (6,7). This speed is somewhat lower than originally suggested by Yphantis (l), but is based on improved experimental criteria (6). All the experiments reported here refer to a maximum speed of 32,000 rpm. Ansevin et al. (4) report changes in baseline patterns wit’h time at speeds above 40,000 rpm with standard cell assemblies.Also, the maximum operating speed for the Yphantis-style (1) centerpiece used here is listed as 48,000 rpm by the manufacturer (8). Thus, it might well be impossible to use the end-of-run technique at speeds of 48,000 rpm or more, while only experimentation by the user with a water-filled cell as described above will determine this technique’s reliability between 32,000 and 48,000 rpm. DISCUSSION 1. The principal conclusion of this study is that ‘baseline variability
is of the high-speed
a major contributor to the decreased reproducibility sedimentation equilibrium method. 2. Our measurements show that baseline variability is not due to plate reading errors or photographic noise alone. These causes contribute only 1.1 p after smoothing. As much as 1.5 p
96
HORBETT
AND
TELLER
additional variability arises from systematic shifts, probably due to window rotation. Some cases of window rotation can be prevented by greasing the screw ring gasket prior to every run, but in general the only remedy appears to be the exercise of extreme care in collecting baseline data. At present, the best procedure for minimizing baseline errors is to use a baseline taken from a low-speed picture of the agitated cell after the high-speed run. SUMMARY
Baseline variability is an important source of error in molecular weight determination with the high-speed sedimentation equilibrium method when Rayleigh interference optics are employed. Using Yphantis-style six-channel cell assemblies (1) , we have measured this variability under several conditions at 32,000 rpm and below. These measurements have led to the conclusion that the baseline for speeds up to at least 32,000 rpm should be measured from a picture taken at low speed at the end of the high-speed experiment. ACKNOWLEDGMENT The authors gratefully acknowledge the assistance of Grace Tsien in this study. It was her skill and patience in reading the many pictures involved which made her study possible. REFERENCES 1. 2. 3. 4.
YPHANTIS, D. A., BiochemLtry 3, 297 (1964). RICHARDS, E. G., AND SCHACHMAN, H. K., J. Phys. VAN HOLDE, K. E., AND BALDWIN, R. L., J. Phys. ANSEVIN, A. T., RQARK, D. E., AND YPHANTIS,
Chem. Chem.
D. A.,
63, 1578 (1959). 62, 734 (1953). Anal. Biochem. 34,
237
(1970). 5. TELLER, 6. TELLER,
D. C., Anal. Biochem. 19, 256 (1967). D. C., HORBETT, T. A., RICHARDS, E. N. Y. Acad. Sci. 164, 66 (1969). 7. ROARK, D. E., AND YPHANTIS, D. A., Ann. N. 8. Spinco Division of Beckman Instruments, Rotors, Cells, Accessories,” Spinco SB-269A,
G., AND SCXACHMAN,
H. K., Ann.
Sci. 164, 245 (1969). Inc., “Analytical Ultracentrifuge p. 12 (1970). Y. Acad.
APPENDIX
A Fortran IV computer program was written to process the baseline data in the following way. The y-micromet,er readings from a single channel were “zeroed” by subtracting the minimum y-micrometer reading from all readings (Table A-l, steps l-2). This ‘Lzeroed7’ but otherwise “raw” data was used to estimate a “smooth” baseline by a repetitive smoothing and filtering procedure (as summarized in Table A-l, steps 3-5). A smoothed baseline was first obtained using least-squares smoothing formulas given by Hildebrand (1) for points equally spaced on the
BASELINE
Summarv
of Comput’ational
97
REPRODUCIBILITY
TABLE A-l Procedures Used
in Baseline
Comparisons
A. Perform
steps l-5 for each channel of each baseline run. Obtain data from a single channel. 2. “Zero” the data. 3. Calculate three smoothed baselines, using linear, quadratic, or cubic formula. In each case, use the repetitive smoothing and filtering procedure of steps a-f. a. Calculate smoothed baseline I from raw data. b. Use smoothed baseline I to filter raw data. c. Calculate smoothed baseline II from filtered raw data. d. Calculate smoothed baseline III from smoothed baseline II. e. Use smoothed baseline III to filter smoothed baseline II. f. Calculate smoothed baseline IV from filtered, smoothed baseline II. 4. Construct a composite smooth baseline by choosing at each point among the three estimates given in step 3, according to the minimum value of the efficiency function. 5. Smooth the result of step 4 with a three-point floating average. B. Perform steps 6-10 twice for all baseline runs to be compared, first with raw baselines, then with smoot,hed baselines. 6. Group data according to channel and perform steps 7-9 on each group separately. 7. Reposition all baselines in the group to make average displacement of each baseline the same as all others. 8. Calculate weighted average fringe displacement and weighted R.M.S. deviation at each radial position in the channel. 9. Calculate whole channel average R.M.S. deviation from data of step 8. 10. Calculate whole cell average R.M.S. deviation by averaging result of step 9 from each group; result is “R.M.S. difference-raw” or “R.M.S. differencesmooth.” 1.
z-axis (step 3a). The overall root-mean-square (R.M.S.) deviation between the raw and smoothed baselines was calculated. A “filtered” dat,a set was then constructed from the raw data by replacing a raw datum with the smoothed estimate if the absolute deviation between the two exceeded 1.5 times the overall rms deviation (step 3b). This filtered raw data was smoothed again (step 3~). The resultant smoothed data set was then filtered in the same way, namely, by comparing it to a smoothed version of itself (steps 3c, 3d, 3e) and the filtered, smoothed data were smoothed once more to give the final set of smoothed data (step 3f). Seven points were used in obtaining the smoothed estimate of the central point of the interval. The formulas used corresponded to fitting a linear, quadratic, and cubic polynomial over the interval. For every point, an estimate of the corresponding smoothed value was calculated with all three types of smoothing formula (step 3). For the points near
98
HORBETT
AND
TELLER
the ends of the baseline not at, the center of a seven point interval, special “end point” formulas were used (1). The choice for the smoothed estimate at the central point of each interval (step 4) was the one given by the polynomial with the minimum value of the following function: efliciency
= R.M.S.
$& II2 ( 1 where R.M.S. is the root-mean-square deviation between the raw and smoothed baseline estimates in the interval, N is the number of points in the interval, and n is the degree of the polynomial (1). This formula provides an estimate of the efficiency with which the polynomial fits the data since it takes into account the degree of t,he polynomial used. The resulting set of smoothed data was smoothed once more using a threepoint moving average (step 5) since it was observed that the juncture of two regions of point,s which had been smoothed by polynomials of different degree was often discontinuous in appearance relative to the smooth-appearing regions to either side. A difficulty in comparing differences in fringe positions between baselines arises from the fact that the whole of each baseline is subject to an error in placement governed by the random error in the value of the minimum y-micrometer reading, due to the “zeroing” procedure. Thus, for example, two readings of the same baseline might differ by 5 p in the value of the minimum y-micrometer reading. This difference is propagated into all data points by the ‘Lzeroing” procedure and would cause the average difference between the curves to be dominated by the error in the minimum y-micrometer reading. To avoid this problem, the average fringe position of each baseline was calculated, and the absolute value of the difference between these averages was added to all the fringe positions (both raw and smoothed estimates) of the baseline with the smaller average fringe position (steps 6 and 7, Table A-l). This procedure makes the average displacement of all baselines to be compared the same. This average position of all baselines thus serves as a new reference level for fringe displacements, replacing the zero position of individual baselines. In order to compare these repositioned baselines it was necessary to use weight,ed averaging procedures because the quality of the fringes varied between runs, especially before Teflon window liners were used. The weighting factor used was: w, = (a + RZY In this equation, D, is the deviation between raw and smoothed baseline estimates at radial position x and R, is the R.M.S. deviation between
BASELINE
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REPRODUCIBILITY
smoothed and raw baseline estimates for all seven points in the interval centered at Z. The smoothed baseline estimates used in calculation of R, were made using the same smoothing formula used for the smoot’hed estimate at Z. This weighting factor minimizes the contribut8ion of a “noisy” baseline to the average baseline. Comparison of repositioned baselines was performed by calculating the weighted average fringe displacement and weighted R.M.S. deviation at each radial position in the channel (step 8, Table A-l). Inspection of these data allows detection of any regions of the channel where fringe positions are especially variable. To characterize the performance of the entire channel, an unweighted average of these rms deviations was calculated (step 9, Table A-l). Finally, these whole channel averages from each of t’he three channels in the cell were averaged to provide an estimate of the performance of the whole cell in a given series of baseline runs that were to be compared (step 10, Table A-l). The “R.M.S. difference-raw” or “R.M.S. difference-smoothed” referred to in the tables and text are this whole cell average R.M.S. deviat,ion among unsmoothed or smoothed baselines, respectively. REFERENCE 1.
F. B., “Introduction Hill, New York, 1956.
HILDEBRAND,
to Numeriral
Analyses,”
pp.
258-311.
McGraw-
An
45,
BIOCHEMISTRY
Experimental Effect
86-99
Study
on Hig’h-Speed THOMAS
(197%
of Baseline
Reproducibility
and
Sedimentation
Equilibrium
Data1
A. HORBETT2 of
University
Department
of
AND
Washington,
Biochemistry, Received
March
DAVID
Its
C. TELLER
School of Me&c&e, Seattle, Washington 98106 24, 1971
Since its introduction by Yphantis (1) in 1964, high-speed sedimentation equilibrium determination of protein molecular weights has become an important technique to the biochemist. However, as pointed out by Yphantis and in the experience of subsequent users of this technique, the high-speed (or meniscus depletion) method gives less reproducible results than the older low-speed technique (2,3). One of the most important types of error leading to this decreased reproducibility is baseline variability. It is the purpose of this report to present the results of a study undertaken to measure the magnitude of this variability, to determine some of its causes, and to develop an improved method for baseline determinations. Ansevin et al. (4) recently reported a new, externally loaded ultracentrifuge cell that can be cleaned and refilled without disassembly. This cell was designed to, and apparently does, obviate the difficulties associated with baseline determination in the Yphantis-style six-channel cell assemblies (1) used here. However, for economic reasons alone, the widespread use of the older cell will probably continue for some time. Second, the results in this report show that, for speeds up to 32,000 rpm, baseline determinations as accurate as those obtained with the external loading cell of Ansevin et al. (4) can be obtained with the older style cell provided the procedure recommended here is used. This procedure employs a low-speed picture of the resuspended sample taken after the high-speeed equilibrium experiment’. ‘Supported by a Public Health Service Research Grant, National Institute of General and Medical Science. ‘The material presented is taken, in part, from a thesis 9. Horbctt to the Graduate Faculty of the University of fulfillment of the requirements for the Ph.D. degree. 86 @ 1972 by Academic Press, Inc.
GM
13401,
submitted Washington
from by in
the
Thomas partial
BASELINE
87
REPRODUCIBILITY
METHODS
Ultracentrifugation was performed as previously described (5). The cell assembly consisted of a single-sector cell housing, a Yphantis-style (1) six-channel charcoal-filled Epon centerpiece, sapphire windows, and interference window holders. The screw ring gasket was greased with Spinkote prior to cell assembly. The white window gaskets were discarded after every run. The cell was filled with an excess of water so that the baseline pattern could be determined throughout each channel. Pictures were taken within about 10 min of reaching speed unless otherwise noted. To allow better control of the window and window liner positions relative to the window holder, a window alignment scribe mark (A, Fig. 1) was made from the center of the keyway slot across the edge of the window holder which faces the centerpiece in the assembled cell. Two window liner alignment marks were made on the same edge, but placed opposite the two ends of a carefully centered window liner (Marks B, Fig. 1).
Scribe
Scribe
Scribe
Mark
Mark
B
A
FIG. 1. Scribed window holder assembly. Scribe marks A and B are described in the text. This figure also illustrates the use of a Bakelite clip in conjunction with the Teflon window liner which allows reproducible positioning of the window in the holder.
Data Collection and Reduction
Plate reading was performed on the modified Nikon microcomparator previously described (5). Care was taken to make the x-direction of the microcomparator stage travel parallel to the radial axis of the fringe rn\-elol) since misorientation of tllc plate could result in an apparent rotation of the baseline. The plates were aligned by rotating the stage until the y-micrometer positions of the central white light fringes in the inner and outer reference hole patterns were judged to be the same within reading error (3 to 5 p). (A movable gate in the optical tube allowed a
88
HORBETT
A-94~01
4.
B -941.01,
AND
TELLER
,wix cm.01
.
.. l ’
. . . .. . . . .*
B:
l . ..*g.**“-***. l
.
.
l l
0.
.
. . . .. . . .
.
/*
.
. ‘*A
l
* l
***
lB
*
X~mm.) FIG. 2. An example of baseline variability. Data from four baseline pictures were individually smoothed (see Appendix). The two baselines shown (A and B) resulted from averaging the smoothed baselines in the pairs indicated in the figure. The pictures were taken at 3200 x-pm prior to four separate high-speed equilibrium experiments at 32,000 rpm. The runs were done with the same cell, by the same person, and within a few days of one another.
white light picture of the reference holes and a monochromatic picture of the cell to be taken separately without moving the plate.) A transparent bull&ye pattern taped to the view screen aided in finding the center of these white light fringes, but all other readings were made electronically with the modified microcomparator (5). Readings were taken every 200 p throughout each channel and all pictures were read at the same radial positions in order to facilitate comparisons between separate baselines. In comparing baselines, we feIt it important to assess the contribution of random error to baseline va~ability. First, comparisons between smoothed estimates of baselines were therefore also made. Second, we have found it useful to smooth baselines before using the data to correct the high-speed equilibrium fringe pattern for baseline deviations. Third, routine use of the suggested low speed end-of-run baseline procedure described below should be accompanied by occasiona checks on the baseline reproducibility of the cell between speeds and quantitative estimates of the variability can be made using the computational procedure we have developed. Finally, an average baseline based on several separate runs is sometimes the best estimate one can make. The computational procedures we used in smoothing and comparing baselines is: described in detai1 in the Appendix.
BASELINE
89
REPRODUCIBILITY
RESULTS Figure 2 shows a typical example of baseline variability. These data were obtained from runs performed using the same cell within a few days of one another. As can be seen, baseline A is very similar in shape but rotated relative to baseline B. The average absolute difference in baseline height between these two baseline patterns is 7 pi (0.025 fringe). Figures 3 and 4 show the effect on the number-average and weightaverage molecular weights, M(n) and M (w) , in applying these two baselines to one set of high-speed sedimentation equilibrium data. The data are from an experiment on a-ohymotrypsin (Worthington CDI-6LD) under conditions in which the protein reversibly dimerizes. The effect on M(n) at 2 fringes concentration is approximately 10%. This seems much larger than one might at first ascribe to the 7 p difference in baseline, which is only about twice the random error level. The explanation is that a systematic error is propagated throughout the dat.a in such a way that it leads to a rotation of the whole M(n) curve in much the same way as the baselines themselves are rotated. The effect on M(W) is seen to be much smaller, amount,ing to about 5% difference in the M(w) curves
*A l B
Concentration
(frlngesl
3. Effect of baseline on M(n). Baseline A or B of Fig. 2 was used to correct the observed fringe pattern of one eq~~b~~ run and the resultant two sets of data were used to compute the two data sets shown. FIG.
90
20.0 0I
HORBETT
AND
1
6
I
4
Concentration
TELLER
8
I
10
I
12
I
14
(fringes)
FIG. 4. Effect of baseline on M(W). Baseline A or B of Fig. 2 was used to correct the observed fringe pattern of one equilibrium run and the resultant two sets of data were used to compute the two data sets shown.
at 2 fringes concentration. This difference in the sensitivity of M(n) and M(w) to systematic errors is intrinsic in their calculation (6). The effects of baseline variability are thus seen to be both large and systematic in nature. These conclusions led us to a study of some of the causes of the variability, and to a series of measurements of the magnitude of the variability under several conditions. One cause of baseline rotation is illustrated in Fig. 5. It is a picture of the keyway slot area of two upper window holders-a new one and a damaged one. The upper window holder damage resulted from failure to use grease (Spinkote) on the screw ring gasket. During cell tightening, enough friction occurred to carry the upper window holder past the keyway in the cell housing. In these cells, the window holder is aluminum and the keyway is steel. Thus, when the tightened cell was used, the centrifugal force encountered was strong enough to press the previously misaligned window holder back into the keyway and the window holder became gouged as shown in Fig. 5. In subsequent usage of the cell, the gouged side of the damaged window holder allowed the window holder and its window to be rotated past the keyway as the screw ring was tightened. The resultant window misorientation was sufficient to give baseline shifts as large as those illustrated in Fig. 2.
BASELINE
91
REPRODUCIBILITY
FIG. 5. Comparison of damaged (upper) and new (lower) The two holders were placeed on a microcomparator stage was then made by photographing the enlarged image appearing
window holder slots. and this photograph on the viewscreen.
Table 1 shows part of the results of a study of baseline reproducibility. The root-mean-square (RMS) difference figures were calculated using differences in fringe displacement at each of approximately 40 equally spaced z-positions (200 p apart) in each channel. The smoothed data were obtained using the least-squares procedures described in the Appendix. The first line shows that our plate reader’s reproducibility in
Baseline
TABLE 1 Reproducibilityn R.&IS.
Purpose
of experiment
Reading erroti Photographicc noise Run-to-rund variability Blakelite Teflon Teflon + Bakelite Changing Teflon
clip
No. of runs
No. of photos
No. of channels
No. of points
6 1
6 5
18 15
1440 600
5 5 5 Fj
5 5 5 5
12e 15 15 15
480 600 600 600
Raw
difference,
fi
Smoothed
1.1 2 3
0.7 1.1
2.4 3.4 4.0 3.7
1.9 3.0 3.2 3.3
n All pictures at 32,000 rpm. * Each of six separate phot,os was read t,wice and t’he t,wo sets of readings from each photo were compared pairwise; the tabulated R.M.S. difference vahles are the average of the values obtained from the pairwisr comparisons. r Five pictures of the same baseline rml were taken and compared. d The cell was disassembled, cleaned, and refilled bet,ween each run. e Third channel fringes unreadable in three runs.
92
HORBETT
AND
TELLER
rereading the same plate is 1.1 ,Ubetween untreated data and 0.7~ between smoothed data. The second line shows that photographic plate noise, as measured by reading the same baseline from five different pictures of one run, does contribute significantly to the data variability. However, most of this contribution can be smoothed out, as seen by the reduction of the R.M.S. difference to 1.1 p in the smoothed data. This is only 0.4 ,u greater than the differences in smoothed data observed for rereading a plate. The third line shows that the standard window liner material, Bakelite, allowed baseline reproducibility nearly equal to the untreated data R.M.S. difference to be expected between pictures. However, the variation in smoothed data in these runs is 0.8 ,X greater than the difference in smoothed data observed between pictures. This is a reflection of the fact that systematic rotation did occur and contributed significantly to the variability from run to run. In the course of this study of baseline reproducibility with Bakelite window liners, the fringes in the third channel were often so severely blurred that readings were not possible. The occurrence of such distortion was not correlated with significant baseline rotations, however, since it occurred in three of the five experiments performed with Bakelite liners in which excellent reproducibility was observed (Table 1, line 3). A general method for preventing window distortion is to replace the Bakelite liners with replicas made from 0.015” Teflon sheets (Universal Plastics Co., Seattle). We have never observed window distortion when Teflon liners were used. In Table 1, the fourth line represents the results of five runs done with one set of Teflon liners. In the smoothed data, the 3.0 p run-to-run variability observed with Teflon is 1.1 p worse than observed when Bakelite was used. Thus, whether the use of Teflon liners is advisable depends on the speed of the run. We have not had window distortion problems below about 28,000 rpm. The Teflon liners are much more flexible and slippery than Bakelite liners and therefore do not work as well in holding the window in place during manipulation. To overcome this problem and to determine whether this was the cause of the lowered baseline reproducibility, a small section of Bakelite was placed in the window holder between the ends of the Teflon liner (see Fig. 1). The combination liner was oriented with the Bakelite opposite the keyway slot. The Bakelite clip acts as a spring to hold the window in position. The fifth line of Table 1 shows, however, that this procedure does not improve reproducibility. The last line of Table 1 shows that replacement of the Teflon liners after each run with new Teflon liners does not change their performance with regard to baseline reproducibility.
BASELINE
93
REPRODUCIBILITY
Previous comparisons of low- and high-speed pictures of a single baseline run in which the low-speed picture was taken first have shown unacceptably large variation, apparently because the windows shift position with increasing speed. Figure 2 illustrates the type of rotation of the baselines often encountered in his type of experiment. Table 2 shows an attempt to determine whether taking the low-speed picture after first spinning the cell at high speed would improve the reliability of the lowspeed picture procedure. The first line shows that the run-to-run variability at 3,200 rpm was 2.6 p in the smoothed data, compared to the 3.6 p variability at 32,000 rpm shown on the second line. When the 32,000 rpm and 3,200 rpm pictures of the same run were compared, it was found that the “between-speed” variability, 2.2 ,u in the smoothed data (Table 2, line 3) was well below either the 32,000 or 3,200 rpm run-to-run variabilities. These data suggest that an improved way to perform baseline determinations is to use a low-speed picture of the still-assembled but vigorously shaken cell after the sedimentation equilibrium run is done. The speed effect experiments so far discussed were carried out without removing the cell from the rotor between speeds and required only brief exposure of the cell to the higher speeds. However, the suggested endof-run baseline procedure involves removing the cell from the rotor, shaking it vigorously, and then replacing it in the rotor and taking a picture at low speed. Furthermore, a typical sedimentation equilibrium experiment would have involved spinning the cell for at least 8 hr and often much longer at the higher speed. Therefore, a direct test of the proposed end-of-run baseline procedure was performed. A cell was assembled and filled exactly as for a sedimentation equilibrium experiment, except water was used in all channels. Four pictures were taken: (1) at 32,000 rpm as soon as possible after reaching that speed (about 0.5
Effect
of Speed
TABLE 2 on Baseline Reproducibilitya
No. of runs
No. of phot’os
5 5 5
5 r ;b
R.M.S. Purpose
of experiment
Run-to-run variability at 3,200 rpm at 32,000 rpm Variability between speeds, within a run
No. of channels
15 14 28
= The cell was disassembled, cleaned, and refilled tures were taken first, then the rotor was brought
difference,
No. of points
Raw
Smoothed
600 560 1120
3.7 4.5 3.2
2.6 3.6 2 2
w
between runs. The 32,000 rpm pio to 3200 rpm for the second picture.
94
HORBETT
40
Z36 I
-
24 -
18 Zero
Time
2. * 22.5 3 3200
Hours rpm.
4=
% 5
Shakm
1
>
TELLER
-
3
:2
AND
3
3
4
9
3200
4 4 344
:I 44
rpm.
rpm. I
24:;3
1
4
4 4 34
3 3
431
:443 43
4
4
1 4 14 21323 31
2 :33 2
341
33
I
11
'
1
I
I
I
I
I
I
10
11
12
13
1A
15
16
17
X MICROMETER FIG.
I
1
1 12 43
:32:
43411 12422 33 0.
rpm.
At 32000
Cdl,
4 34:4;3
12-
At 32000
6. Baseline
reproducibility
with
READINGS
(mm.)
speed and time, unsmoothed
data.
hr) ; (2) at 32,000 rpm after 22.5 hr; (3) at 3,200 rpm as soon as possible after decelerating the rotor directly after picture 2; (4) at 3,200 rpm after removing the cell from the rotor, shaking vigorously, and replacing it in the rotor. The unsmoothed and smoothed baseline readings on one of the three channels are shown in Figs. 6 and 7. The numbers used to represent the data in these figures correspond to readings from the four pictures described. The readings from the four pictures in the other two channels were similarly superimposeable. The average R.M.S. derivat,ion between the unsmoothed baseline data was 2.0~ for the centripetal channel data of Fig. 6, 1.9 for the middle channel, and 2.6 for the outermost channel. In the smoothed data, the R.M.S. deviations were 1.0, 0.9, and 2.0~~ respectively. It is clear from this experiment 48 c 2
1.
536
-
Zero
2 * 22.5 4. 3’
2 e
q.24
Tim
At 32000
Hours
At
3200 Shaken rpm. Cdl,
-
32000 3200
rpm. rpm. rpm. 4
Ill
ti 2
41
3333322223333.:;3333 114444444444444:::44
c3
>- 12
-
,:i:‘4;337:1 132 34
14422
4
0 9
‘
10
11
12
13
X MICROMETER FIG. 7. Baseline reproducibility
14
READINGS
15
16
17
(mm.)
with speed and time, smoothed
data,
BASELINE
95
REPRODUCIBILITY
that the procedure employed in the proposed end-of-run baseline technique can give an excellent estimate of the high-speed baseline. Three warnings to the prospective user of the end-of-run baseline technique are in order at this point. It has been observed in two separate instances that, when the cell barrel becomes distorted and fits much more tightly into the rotor hole than a new barrel, the baselines shifted an unacceptably large amount between 32,900 and 3,200 rpm. That is, the 32,000 versus 3,200 variability became larger t,han the variability between separate 32,000 rpm baselines in three cases. It is not known at present whether cell distortion is an adequate measure of when the low-speed end-of-run baseline procedure becomes unreliable. Therefore, use of the end-of-run baseline procedure requires an occasional check on the performance of the cell with respect to between-speed variability. This is readily done with a water-filled cell by comparing its baseline at the low and high speed to be used in the experiments. Another pitfall of the end-of-run baseline technique is due to t’he material under investigation, which may not redistribute uniformly when the cell is shaken or may sediment significantly even at t’he low speed. This problem can be overcome by performing a separate baseline experiment as described by Yphantis (1). The third precaution to be taken in the use of the end-of-run baseline technique regards the maximum speed at which the technique will still work well. In high-speed equilibrium experiments, a speed of 32,000 rpm is appropriate for a protein of molecular weight of 25,000 gm/mole (6,7). This speed is somewhat lower than originally suggested by Yphantis (l), but is based on improved experimental criteria (6). All the experiments reported here refer to a maximum speed of 32,000 rpm. Ansevin et al. (4) report changes in baseline patterns wit’h time at speeds above 40,000 rpm with standard cell assemblies.Also, the maximum operating speed for the Yphantis-style (1) centerpiece used here is listed as 48,000 rpm by the manufacturer (8). Thus, it might well be impossible to use the end-of-run technique at speeds of 48,000 rpm or more, while only experimentation by the user with a water-filled cell as described above will determine this technique’s reliability between 32,000 and 48,000 rpm. DISCUSSION 1. The principal conclusion of this study is that ‘baseline variability
is of the high-speed
a major contributor to the decreased reproducibility sedimentation equilibrium method. 2. Our measurements show that baseline variability is not due to plate reading errors or photographic noise alone. These causes contribute only 1.1 p after smoothing. As much as 1.5 p
96
HORBETT
AND
TELLER
additional variability arises from systematic shifts, probably due to window rotation. Some cases of window rotation can be prevented by greasing the screw ring gasket prior to every run, but in general the only remedy appears to be the exercise of extreme care in collecting baseline data. At present, the best procedure for minimizing baseline errors is to use a baseline taken from a low-speed picture of the agitated cell after the high-speed run. SUMMARY
Baseline variability is an important source of error in molecular weight determination with the high-speed sedimentation equilibrium method when Rayleigh interference optics are employed. Using Yphantis-style six-channel cell assemblies (1) , we have measured this variability under several conditions at 32,000 rpm and below. These measurements have led to the conclusion that the baseline for speeds up to at least 32,000 rpm should be measured from a picture taken at low speed at the end of the high-speed experiment. ACKNOWLEDGMENT The authors gratefully acknowledge the assistance of Grace Tsien in this study. It was her skill and patience in reading the many pictures involved which made her study possible. REFERENCES 1. 2. 3. 4.
YPHANTIS, D. A., BiochemLtry 3, 297 (1964). RICHARDS, E. G., AND SCHACHMAN, H. K., J. Phys. VAN HOLDE, K. E., AND BALDWIN, R. L., J. Phys. ANSEVIN, A. T., RQARK, D. E., AND YPHANTIS,
Chem. Chem.
D. A.,
63, 1578 (1959). 62, 734 (1953). Anal. Biochem. 34,
237
(1970). 5. TELLER, 6. TELLER,
D. C., Anal. Biochem. 19, 256 (1967). D. C., HORBETT, T. A., RICHARDS, E. N. Y. Acad. Sci. 164, 66 (1969). 7. ROARK, D. E., AND YPHANTIS, D. A., Ann. N. 8. Spinco Division of Beckman Instruments, Rotors, Cells, Accessories,” Spinco SB-269A,
G., AND SCXACHMAN,
H. K., Ann.
Sci. 164, 245 (1969). Inc., “Analytical Ultracentrifuge p. 12 (1970). Y. Acad.
APPENDIX
A Fortran IV computer program was written to process the baseline data in the following way. The y-micromet,er readings from a single channel were “zeroed” by subtracting the minimum y-micrometer reading from all readings (Table A-l, steps l-2). This ‘Lzeroed7’ but otherwise “raw” data was used to estimate a “smooth” baseline by a repetitive smoothing and filtering procedure (as summarized in Table A-l, steps 3-5). A smoothed baseline was first obtained using least-squares smoothing formulas given by Hildebrand (1) for points equally spaced on the
BASELINE
Summarv
of Comput’ational
97
REPRODUCIBILITY
TABLE A-l Procedures Used
in Baseline
Comparisons
A. Perform
steps l-5 for each channel of each baseline run. Obtain data from a single channel. 2. “Zero” the data. 3. Calculate three smoothed baselines, using linear, quadratic, or cubic formula. In each case, use the repetitive smoothing and filtering procedure of steps a-f. a. Calculate smoothed baseline I from raw data. b. Use smoothed baseline I to filter raw data. c. Calculate smoothed baseline II from filtered raw data. d. Calculate smoothed baseline III from smoothed baseline II. e. Use smoothed baseline III to filter smoothed baseline II. f. Calculate smoothed baseline IV from filtered, smoothed baseline II. 4. Construct a composite smooth baseline by choosing at each point among the three estimates given in step 3, according to the minimum value of the efficiency function. 5. Smooth the result of step 4 with a three-point floating average. B. Perform steps 6-10 twice for all baseline runs to be compared, first with raw baselines, then with smoot,hed baselines. 6. Group data according to channel and perform steps 7-9 on each group separately. 7. Reposition all baselines in the group to make average displacement of each baseline the same as all others. 8. Calculate weighted average fringe displacement and weighted R.M.S. deviation at each radial position in the channel. 9. Calculate whole channel average R.M.S. deviation from data of step 8. 10. Calculate whole cell average R.M.S. deviation by averaging result of step 9 from each group; result is “R.M.S. difference-raw” or “R.M.S. differencesmooth.” 1.
z-axis (step 3a). The overall root-mean-square (R.M.S.) deviation between the raw and smoothed baselines was calculated. A “filtered” dat,a set was then constructed from the raw data by replacing a raw datum with the smoothed estimate if the absolute deviation between the two exceeded 1.5 times the overall rms deviation (step 3b). This filtered raw data was smoothed again (step 3~). The resultant smoothed data set was then filtered in the same way, namely, by comparing it to a smoothed version of itself (steps 3c, 3d, 3e) and the filtered, smoothed data were smoothed once more to give the final set of smoothed data (step 3f). Seven points were used in obtaining the smoothed estimate of the central point of the interval. The formulas used corresponded to fitting a linear, quadratic, and cubic polynomial over the interval. For every point, an estimate of the corresponding smoothed value was calculated with all three types of smoothing formula (step 3). For the points near
98
HORBETT
AND
TELLER
the ends of the baseline not at, the center of a seven point interval, special “end point” formulas were used (1). The choice for the smoothed estimate at the central point of each interval (step 4) was the one given by the polynomial with the minimum value of the following function: efliciency
= R.M.S.
$& II2 ( 1 where R.M.S. is the root-mean-square deviation between the raw and smoothed baseline estimates in the interval, N is the number of points in the interval, and n is the degree of the polynomial (1). This formula provides an estimate of the efficiency with which the polynomial fits the data since it takes into account the degree of t,he polynomial used. The resulting set of smoothed data was smoothed once more using a threepoint moving average (step 5) since it was observed that the juncture of two regions of point,s which had been smoothed by polynomials of different degree was often discontinuous in appearance relative to the smooth-appearing regions to either side. A difficulty in comparing differences in fringe positions between baselines arises from the fact that the whole of each baseline is subject to an error in placement governed by the random error in the value of the minimum y-micrometer reading, due to the “zeroing” procedure. Thus, for example, two readings of the same baseline might differ by 5 p in the value of the minimum y-micrometer reading. This difference is propagated into all data points by the ‘Lzeroing” procedure and would cause the average difference between the curves to be dominated by the error in the minimum y-micrometer reading. To avoid this problem, the average fringe position of each baseline was calculated, and the absolute value of the difference between these averages was added to all the fringe positions (both raw and smoothed estimates) of the baseline with the smaller average fringe position (steps 6 and 7, Table A-l). This procedure makes the average displacement of all baselines to be compared the same. This average position of all baselines thus serves as a new reference level for fringe displacements, replacing the zero position of individual baselines. In order to compare these repositioned baselines it was necessary to use weight,ed averaging procedures because the quality of the fringes varied between runs, especially before Teflon window liners were used. The weighting factor used was: w, = (a + RZY In this equation, D, is the deviation between raw and smoothed baseline estimates at radial position x and R, is the R.M.S. deviation between
BASELINE
99
REPRODUCIBILITY
smoothed and raw baseline estimates for all seven points in the interval centered at Z. The smoothed baseline estimates used in calculation of R, were made using the same smoothing formula used for the smoot’hed estimate at Z. This weighting factor minimizes the contribut8ion of a “noisy” baseline to the average baseline. Comparison of repositioned baselines was performed by calculating the weighted average fringe displacement and weighted R.M.S. deviation at each radial position in the channel (step 8, Table A-l). Inspection of these data allows detection of any regions of the channel where fringe positions are especially variable. To characterize the performance of the entire channel, an unweighted average of these rms deviations was calculated (step 9, Table A-l). Finally, these whole channel averages from each of t’he three channels in the cell were averaged to provide an estimate of the performance of the whole cell in a given series of baseline runs that were to be compared (step 10, Table A-l). The “R.M.S. difference-raw” or “R.M.S. difference-smoothed” referred to in the tables and text are this whole cell average R.M.S. deviat,ion among unsmoothed or smoothed baselines, respectively. REFERENCE 1.
F. B., “Introduction Hill, New York, 1956.
HILDEBRAND,
to Numeriral
Analyses,”
pp.
258-311.
McGraw-