Distinguishing Small Molecular Mass Differences of Proteins by Mass Spectrometry

Distinguishing Small Molecular Mass Differences of Proteins by Mass Spectrometry

ANALYTICAL BIOCHEMISTRY ARTICLE NO. 260, 204 –211 (1998) AB982692 Distinguishing Small Molecular Mass Differences of Proteins by Mass Spectrometry ...

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ANALYTICAL BIOCHEMISTRY ARTICLE NO.

260, 204 –211 (1998)

AB982692

Distinguishing Small Molecular Mass Differences of Proteins by Mass Spectrometry M. Kirk Green,* Martha M. Vestling,*,1 Murray V. Johnston,*,2 and Barbara S. Larsen† *Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware 19711; and †Central Research & Development, DuPont Company, P.O. Box 80228, Wilmington, Delaware 19880-0228

Received October 27, 1997

Electrospray ionization–Fourier transform ion cyclotron resonance (ESI–FTICR) mass spectrometry allows for high-resolution, accurate mass analysis of multiply charged ions of proteins. In the work described here, the ability of ESI–FTICR to distinguish small differences in molecular mass is evaluated. Ubiquitin was used as an internal mass calibration standard to measure the molecular mass of cytochrome c, myoglobin, and several carbonic anhydrase isoforms. Mass calibration was based on the tallest isotopic peak of each ubiquitin charge state. Ubiquitin performed well as an internal standard because its charge states covered the appropriate mass range, interference was minimal, and the tallest peak was easily identified. The peak masses of cytochrome c (12.5 kDa) and myoglobin (17 kDa) were measured to an accuracy of about 0.02 Da (<2 ppm). However, errors of 1.0 Da were observed for some individual determinations because of the difficulty in identifying the tallest peak. When the technique was applied to bovine carbonic anhydrase II, even combining data from several charge states did not yield an unequivocal assignment of the tallest peak, resulting in a mass assignment of 29,023.7 or 29,024.7. Similarly, measurements of two isoforms with a mass difference of 1 Da, human carbonic anhydrase I, pI 6.0 and 6.6, yielded overlapping values for the mass of the tallest peak. However, these two isoforms were clearly distinguished by (a) identification of the tallest peak using a measurement of average mass as a guide and (b) comparison of the isotopic peak intensity patterns. © 1998 Academic Press

1 Current address: Department of Chemistry, University of Wisconsin–Madison, 1101 University Avenue, Madison, WI 53706. 2 To whom correspondence should be addressed. Fax: 302-8316335 E-mail: [email protected].

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With the introduction of ionization techniques such as matrix-assisted laser desorption ionization (MALDI)3 and electrospray ionization (ESI), mass spectrometry has become an increasingly powerful tool for biochemical studies. Mass spectrometry offers the ability to measure the molecular masses of whole proteins with far more accuracy than traditional techniques such as gel electrophoresis or size exclusion chromatography, opening new possibilities for identification, structure confirmation, and detection of modifications. While MALDI–time of flight (TOF) instrumentation has theoretically infinite mass range and recent developments have dramatically improved resolving power, accurate mass measurements (better than 10 ppm) are still limited to ,104 Da (1). ESI has the ability to impart multiple charges to the analyte molecules, lowering their m/z so that species with molecular weights in the 104–106 Da range are observed in the range of m/z 500 – 4000 (2, 3). In particular, the combination of ESI with Fourier transform ion cyclotron resonance (FTICR) mass spectrometry has enabled the observation of mass spectra of large molecules (10 –100 kDa) with high resolution (4 – 6) and accuracies within 1 Da (4, 5). Mass accuracy of this order is potentially useful for at least two applications: confirmation of structure or identity by mass measurement, and discerning small differences between proteins. For example, deamidation of a residue increases the molecular weight of a protein by 1 Da; formation of a disulfide bridge reduces the molecular weight by 2 Da; a number of residue substitutions result in molecular weight changes of less than 5 Da. However, with any mass spectrometric technique mass accuracy becomes poorer with increasing molecular weight. Thus, while it ap3

Abbreviations used: MALDI, matrix-assisted laser desorption; ESI, electrospray ionization; FTICR, Fourier transform ion cyclotron resonance; TOF, time of flight; BCA, bovine carbonic anhydrase. 0003-2697/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.

MASS DIFFERENCES OF PROTEINS BY MASS SPECTROMETRY

pears that 1 Da accuracy for species weighing ,20 kDa can be achieved by ESI in combination with quadrupole (2, 7), magnetic sector (7), or FTICR instrumentation, this type of accuracy becomes more and more difficult to achieve for larger species, both because the relative accuracy requirement increases and because of decreasing resolution at higher mass. A problem particular to FTICR is that while the position of individual peaks can be measured with high accuracy (for example, a 5 ppm measurement accuracy implies a mass accuracy of 1 Da for a 200-kDa ion), conversion of an observed m/z to a useful molecular weight is not always straightforward. The presence of isotopes, particularly 13C, results in wide isotopic distributions of peaks, rather than a single m/z, and the problem becomes one of either determining where the average of the distribution lies or determining the position of identifiable peaks within the distribution. Determining the weighted average of the observed isotopic distribution is conceptually simple and provides a straightforward measurement of average mass (8, 9); however, this may be subject to errors due to overlap with the isotopic envelopes of species of similar molecular weight, for example, sodium adducts, as well as distortion of envelope shapes (5, 10) and baseline drifts. Zubarev et al. (11) suggest minimizing interference by taking the centroid of the part of an isotopic distribution (whether or not resolved) above 50% relative intensity; that is, peaks smaller than 50% of the tallest peak of the cluster are ignored, and only the portions of the remaining peaks above 50% relative intensity are used in calculating an average mass. Furthermore, the authors show that the molecular weight determined in this fashion is consistently and predictably about 0.5 Da below the true average molecular weight, and therefore can be corrected to within 0.1 Da. The above-mentioned uncertainty in isotopic abundances means that even a very accurate average molecular weight does not necessarily translate into an unambiguous formula. The least ambiguous measurement is that of the monoisotopic mass (9, 12), that is, the mass of the species which contains all 12C, all 14N, etc. Unfortunately, while the monoisotopic peak of a protein can be directly observed up to about 10 kDa, much beyond this point it becomes too small to be reliably identified, and its position must be inferred from the observed isotopic envelope. This requires an unambiguous determination of the identity of the observed peaks, i.e., which peak corresponds to the nominal incorporation of a given number of 13C’s. This task becomes more difficult as the number of atoms in the analyte molecule increases and the signal for the ion is split into an increasing number of isotopic peaks, with differences in intensity between neighboring peaks becoming smaller and smaller. Another approach is to measure the position of the tallest peak of the distri-

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bution (the ‘‘peak top mass’’ (12)) and compare this to the position of the tallest peak in a calculated spectrum. This approach was used in the present study, and is more direct than inferring the position of the monoisotopic peak, but still depends on identifying which peak corresponds to the ‘‘tallest’’ peak. One approach which solves the problem of identifying the monoisotopic peak is to produce 13C, 15N-depleted proteins, which renders the monoisotopic peak visible up into the 100-kDa range (13). While it is certainly effective, this approach requires a significant investment of time and effort, and is practical only for proteins which can be expressed by bacteria. Alternatively, if sufficient sample is available, isotopic abundances for that particular sample could be measured in separate work, and these values used to generate valid calculated spectra for peak identification. However, this would significantly increase the total sample requirement. In the present study, we examine the ability of a commercially available ESI–FTICR system to achieve accurate mass measurement of proteins and determine at what point peak identification becomes a serious problem. Different approaches to correctly assigning isotope peaks are examined. Furthermore, the application of these approaches to the mass measurement of two proteins differing in mass by only one Da, the pI 6.0 and 6.6 isoforms of human carbonic anhydrase I, is demonstrated. EXPERIMENTAL

Spectra were obtained on a Bruker BioAPEX FTICR equipped with a 7 Tesla magnet and an Analytica electrospray source. Ions were accumulated in the hexapole for 200 –500 ms before being moved into the ICR cell. The analyzer pressure was 2–3 3 10210 torr. Proteins were obtained from Sigma and were electrosprayed as 5–7 mM solutions in methanol/water/acetic acid (50/49.5/0.5) at a flow rate of 0.5 ml/min using a Cole Parmer 74900 Series syringe pump. Countercurrent nitrogen drying gas flows were generally kept quite low (,1 liter/min). Whereas purifying the carbonic anhydrase samples by HPLC removed impurities and decreased adduct formation, it was found to be unnecessary for obtaining good quality spectra, and all of the spectra shown here were obtained with unpurified samples. Typically, spectra were obtained in broadband mode over the range m/z 800 –2500, and 500 K data points were collected. Resolving power of the spectra was 5– 8 3 105. No apodization was used, except as noted to improve signal-to-noise ratio and peak shape. Isotope distributions were calculated using an implementation of Yergey’s XMASS (14) program.

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FIG. 1.

ESI–FTICR spectrum of ubiquitin. Numbers indicate the charge states [M 1 Hn]n1.

RESULTS AND DISCUSSION

Whereas there are several programs available for calculating theoretical masses of compounds, there is some ambiguity in the mass obtained for a calibrant or a proposed unknown because of variations in the precision of the calculations and the differences in the values used for isotopic masses and abundances. In a comparison of five different programs (14 –17), while calculations of the monoisotopic mass for bovine carbonic anhydrase II (BCA II, mol wt 29 kDa) were spread over only 2 mDa (18), the average mass calculations covered a range of 0.2 Da because of the different isotopic abundances used by the different programs. Unfortunately, there is no ‘‘right’’ table of isotopic abundances because of natural variation in, for example, the 12C/13C ratio (19, 20). The ESI–FTICR spectrum of ubiquitin, the internal calibrant used for this work, is shown in Fig. 1. Ubiquitin was chosen because the prominent charge state peaks cover the m/z range of interest (700 –1200), and they are widely spaced, minimizing interference with analyte peaks, and the isotope envelope is easy to interpret. Theoretical and observed spectra of the 110 charge state of ubiquitin are shown in Fig. 2. Note that while the monoisotopic peak is discernible, the tallest isotopic peak (nominally six 13C’s) was used for calibration in this work. Figure 3 shows a mass spectrum of a mixture of ubiquitin (internal standard), cytochrome c, and myo-

globin. Note that in the acidic spraying solution used, myoglobin loses the noncovalently bound heme and is observed as apomyoglobin. In this case, the 112 to 16 charge states of ubiquitin were used to calibrate over the range m/z 712 to 1224, with an internal consistency of 1 ppm. Table 1 lists the m/z values for the tallest peak for each charge state of cytochrome c and apo-

FIG. 2. Calculated and experimental spectra of the 110 charge state of ubiquitin.

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MASS DIFFERENCES OF PROTEINS BY MASS SPECTROMETRY

FIG. 3. ESI-FTICR spectrum of a mixture of ubiquitin, cytochrome c and myoglobin. U, ubiquitin peaks; C, cytochrome c peaks; M, apomyoglobin peaks; *, noise spike. Additional peaks due to sodium and phosphate adducts are present. Inset: Expansion of the 115 charge state region of cytochrome c. I, monoisotopic peak.

myoglobin within the calibration range, together with the corresponding mass calculated for the zero charge state. Two observations stand out: (a) all calculated values for the tallest peak agree with the theoretical value in the first decimal place, with a deviation of at most 3 units in the second decimal place, suggesting the potential for accuracy significantly greater than the accuracy to which the average mass can be determined, given the uncertainty in isotopic abundances (vide infra); and (b) this excellent consistency is sometimes marred by the inconsistency in the whole number mass, that is, the inability to reliably identify the peak which corresponds to the calculated tallest peak. The average of the tallest peaks for apomyoglobin differs from the theoretical value by 0.01 Da, less than 1 ppm, showing excellent agreement. The value for the tallest peak of cytochrome c, however, is 12,358.35 Da, whereas the calculated mass for reduced cytochrome c is 12,359.34! Assuming that the tallest peak has been correctly assigned, the simplest explanation for this apparent discrepancy of 1 Da is that the heme is in the oxidized state; i.e., the singly charged species is M1, containing Fe31, rather than MH1, containing Fe21. In fact, cytochrome c as purchased is in the oxidized state. Thus, for example, the tallest peak of the 151 charge state, corresponding to nominal incorporation of seven 13 C’s, appears at (12,359.3431 1 15 3 1.0073)/15 5 824.8963 (from Table 1, 824.8975). Assignment of the tallest peak in this case can be confirmed by the mono-

TABLE 1

Molecular Masses Corresponding to the Tallest Isotopic Peak for Selected Charge States of Apomyoglobin and Cytochrome c Charge state

m/z

Mra

Apomyoglobin 20 19 18 17 16 15

848.5565 893.1120 942.6748 998.1257 1060.4425 1131.0737 Averageb: Theoretical valuec:

17 16 15 14 13 12

727.9681 773.4683 824.8975 883.7474 951.6476 1030.9540 Averageb: Theoretical valuec:

16950.98 16949.99b 16950.02b 16951.01 16950.96 16951.00 16,950.99 6 .02 16,951.00

Cytochrome c 12358.33 12359.38b 12358.35 12358.36 12358.32 12359.36b 12,358.34 6 .02 12,359.34d

a Calculated from molecular mass 5 (m/z 2 1.0073) 3 charge state. b Peaks differing by 1 mass unit from the majority were excluded from the average. c Calculated from the sequence. d Assuming the heme Fe is reduced, Fe12.

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FIG. 4. ESI–FTICR spectrum of bovine carbonic anhydrase II (BCA II) with ubiquitin as internal calibrant, and an expansion of the region around m/z 856. While the tallest peak of ubiquitin is readily apparent, selection of the tallest BCA II peak is less certain.

isotopic peak (only clearly visible in apodized spectra at high S/N ratio, Fig. 2). It appears at 824.4288 (observed 824.4268), the position expected for the oxidized species. It should be mentioned that the low-abundance 54 Fe and 55Fe isotopes have been neglected in the calculation of monoisotopic mass. It was recently reported (21) that Fe in tuna cytochrome c is in the reduced state in the gas phase, and this was attributed to reduction during the electrospray process. This was based on identification of the smallest observed peak of the 115 charge state isotopic cluster (m/z 802.39) as the monoisotopic peak (calculated values for the monoisotopic peak were 802.41 for the reduced species and 802.34 for the oxidized species). However, the tallest peak was at 802.79, and apparently corresponded to incorporation of only six 13 C’s, whereas in the calculated spectrum (not shown in Ref. 21), the tallest peak corresponds to incorporation of seven 13C’s. The inference is that the monoisotopic peak was in fact too small to be observed (theoretical intensity 1.3%), resulting in an incorrect assignment of the peak corresponding to nominal incorporation of one 13C as the monoisotopic peak, and an incorrect determination of the oxidation state of Fe in tuna cytochrome c ions. In a nozzle-skimmer dissociation/collision-induced dissociation study of some heme-containing biomol-

ecules, it was reported that while myoglobin and hemoglobin, which have noncovalently bound heme, release the heme as an oxidized group upon dissociation, cytochrome c, in which the heme is covalently bound, releases it as a reduced species (22). Our observation that the parent ion in the gas phase contains oxidized heme implies that the heme becomes reduced during the dissociation. The difficulty in picking the tallest peak can be attributed to a combination of noise and the fact that with increasing molecular weight, the differences in

TABLE 2

Molecular Masses of Tallest Peaks for the Carbonic Anhydrases Trial Compound

1

2

3

Theoretical value

Bovine carbonic anhydrase II BCA II, pI 5.9 BCA II, pI 5.4 HCA I, pI 6.0 HCA I, pI 6.6

29,024.72 29,023.65 28,996.64 28,781.40 28,780.40

29,023.72 29,023.77 28,996.55 28,781.38 28,781.41

29,023.70 29,024.70 28,995.52 28,780.40 28,779.33

29,023.74a 29,023.74a 28,996.70a 28,781.40 28,780.40

a

Assuming all four uncertain residues are in the acid form.

MASS DIFFERENCES OF PROTEINS BY MASS SPECTROMETRY

intensity between the tallest peaks in a cluster decrease, so that better signal quality is required. It should be noted that noise has two components: instrument noise and the statistical noise of the ion population selected for the measurement. The obvious remedy is simply to acquire more scans to improve the S/N ratio; however, this is not always practical, and certainly requires more time and sample. The simplest deconvolution approach to measuring the tallest peak is to sum the intensities of the peaks from each charge state which correspond to the same nominal mass (determined from mass 5 (m/z 2 1.0073) 3 charge state) and choose the most intense summed peak as the tallest peak. When we applied this approach to spectra of ubiquitin/cytochrome c/myoglobin mixtures (data not shown), the three determinations of the molecular mass of the cytochrome c ion were all correct within 0.01 Da. However, one of the determinations of the most intense summed peak of myoglobin was 1.02 Da low. This illustrates that while summing intensities improves the reliability of the choice of the tallest peak, there is still a very real possibility of misidentification. Several other approaches have been discussed in the literature. Mathematical deconvolution of the electrospray spectra (7, 23–25) to give a ‘‘zerocharge’’ spectrum effectively averages all the charge states together, and somewhat alleviates the problem of tallest peak identification. McLafferty and co-workers have suggested a least-squares fit of the theoretical intensity profile of the isotope peaks of a protein with average amino acid composition (‘‘averagine’’) of appropriate molecular weight to the observed intensity profile (10)]. Recently, Marshall (26) has reported the application of maximum entropy methods to deconvolute the monoisotopic mass from the observed isotopic envelope, an approach which offers both enhanced effective resolution and a lessened susceptibility to the problem of ambiguity in the identification of the tallest peak. However, it requires specialized mathematical tools. The work of Zubarev et al. (11) discussed in the introduction represents an alternative approach to peak identification, since determination of the centroid of the isotope distribution is not subject to the 61 Da uncertainty. Furthermore, since the true average is 0 –1 Da greater than the m/z value of the tallest peak (11), the 50% cutoff centroid, which is approximately 0.5 Da lower than the correct average mass, should be within 60.5 Da of the tallest peak; that is, the tallest peak in the zero charge spectrum is the one closest to the 50% cutoff centroid value. This method is appealing in its computational simplicity and is discussed in more detail below. Figure 4 shows the ESI spectrum of bovine carbonic anhydrase II with ubiquitin as the internal standard. The inset shows an expansion of the region around the

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134 state of carbonic anhydrase and the 110 state of ubiquitin. While the individual peaks of carbonic anhydrase are well resolved, visually picking out the tallest peak is clearly very difficult. Furthermore, the spectrum in Fig. 2 required 64 scans, and improving S/N by collecting additional scans would quickly become prohibitive in terms of instrument time required. In a case such as this, some form of averaging is clearly necessary. Table 2 summarizes the results obtained by summing corresponding peaks as described above for three different lots of bovine carbonic anhydrase II—Sigma’s ‘‘bovine carbonic anhydrase II,’’ ‘‘bovine carbonic anhydrase II, pI 5.9,’’ and ‘‘bovine carbonic anhydrase II, pI 5.5.’’ Note that for the BCAs, however, not all charge states were resolved with equal efficacy, and some displayed poor S/N. Hence, some were discarded for the calculations. It should be mentioned that there is some uncertainty in the literature value of the molecular weight for BCA II, due to uncertainty in the identity of three Asp/Asn and one Glu/Gln residues (27). The value given as the theoretical value in the table assumes all four residues are in the acid form. Given the difficulty in unambiguously identifying the tallest peak, the question then arises as to whether FTICR is capable of discerning small mass differences. To address this question, we chose two isoforms of human carbonic anhydrase I (HCA I), the pI 6.6 and 6.0 isoforms. The pI 6.0 isoform is 1 Da heavier than the pI 6.6 isoform, due to the hydrolysis of one Asn or Gln residue (28). The two isoforms were run in triplicate back to back. The results obtained by simple summing of corresponding peaks as described above (Table 2) are ambiguous: although the average of the values obtained for the pI 6.0 isoform is clearly higher, the overlap in values obtained for the tallest peak of the two species makes it difficult to state with much confidence that there is, in fact, a difference. As discussed above, using the 50% centroid should provide a reliable guide to picking the tallest peak. The values of the 50% cutoff centroid for the HCA I measurements are listed in Table 3, along with the corresponding closest isotope peaks and the theoretical values for the tallest peak from Table 2. In every case, the theoretically tallest peak has been identified, and the two isoforms are clearly differentiated. Thus, in cases where the tallest peak is not readily apparent by direct examination of an intensity pattern, the centroid offers a simple and reliable guide to picking the tallest peak. The least-squares fitting approach of McLafferty and co-workers (10) discussed above was also tried, with the isotopic pattern generated by the ‘‘averagine’’ formula C1279H1993N352O383S11 (average Mr 28,781.7). As discussed previously (10), only peaks above a threshold relative intensity (30% in this work) were used in the calculation. In all cases, the correct peak was identified as the tallest peak by this

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GREEN ET AL. TABLE 3

50% Centroid Masses and Closest Peaks Trial Compound HCA I, pI 6.0 HCA I, pI 6.6

50% centroid Closest peak 50% centroid Closest peak

1

2

3

28,781.42 28,781.40 28,780.26 28,780.40

28,781.40 28,781.38 28,780.34 28,780.41

28,781.33 28,781.40 28,780.26 28,780.33

method as well. Both methods are expected to have a similar performance, but the least-squares fitting requires somewhat more calculation. Figure 5 shows a comparison of the profiles for the two HCAs (average of three determinations), together with the calculated profile for the pI 6.6 isoform. It is readily apparent that the pI 6.0 envelope is shifted to high mass of the pI 6.6 envelope by approximately 1 Da, as expected. Surprisingly, the match of calculated and experimental profiles for the 6.6 isoform is not particularly good: the experimental profile is shifted 0.2– 0.7 Da to low mass of the calculated one. This reflects deviations of peak intensities from their expected values, rather than a shift in peak positions, since the positions of the peaks themselves are in very good agreement. One possibility is that the pI 6.6 sample contains a contaminant 1–2 Da lower in molecular weight; while this is certainly possible, the most likely candidate would be an HCA I with one less hydrolyzed

Theoretical value

28,781.40 28,780.40

Asn/Gln residue. Such are unknown, and presumably would have a different pI. A third possibility is that the isotopic abundances in the pI 6.6 sample are different enough from those assumed to shift the intensities; the discrepancy is about the maximum possible suggested by Beavis (19). Another possibility is that the apparent shift is an experimental artifact; as mentioned earlier, isotope distributions may be distorted in FTICR (5, 10). Thus it appears that, at least up to a molecular weight of 30 kDa, FTICR performed with a 7 T instrument is capable of discerning a mass difference of 1 Da in two large proteins. It should be stated that 30 kDa does not necessarily represent the limit of this capability. Carbonic anhydrase was chosen simply because it was the largest protein for which baseline isotopic resolution could be readily obtained on our instrument. However, baseline isotopic resolution is not essential for accurate mass determination, and the mass limit for 1 Da accuracy for this type of instrument is presumably higher. CONCLUSIONS

FIG. 5. Isotope pattern (isotope peak intensities) of HCA I, pI 6.0 and 6.6, compared with the calculated pattern for the pI 6.6 isoform. The shift in the envelope by 1 Da on going from pI 6.6 to 6.0 reflects deamidation of one pI 6.6 residue. Note that while peak positions agree very well, the observed pI 6.6 profile is shifted about 0.4 Da to low mass of the calculated profile. Error bars represent standard deviations in intensity from triplicate runs.

FTICR is capable of highly accurate mass measurement on proteins. However, the promised accuracy is compromised by a number of factors, the most important of which is the unambiguous identification of the isotopic peaks. Simply picking the peak, which appears to be the largest, can easily result in an error of 1.0 Da. Averaging the data for the different charge states decreases the chance for error, but does not eliminate it. Two approaches which minimize the chances of error are using the calculated centroid of the distribution to find the largest peak, and least-squares matching of experimental and calculated isotope peak patterns. However, all methods are vulnerable to instrumental distortion of the isotopic profile. Use of an internal standard is recommended for highest accuracy. In this work, ubiquitin proved satisfactory as an internal standard, and measurements for which this internal standard were self-consistent to within 1 ppm resulted in mass measurements with the same range of mass accuracy. The ability of FTICR to reliably distinguish a mass difference of 1 Da between two proteins of mass 30

MASS DIFFERENCES OF PROTEINS BY MASS SPECTROMETRY

kDa, human carbonic anhydrase I and II, has been demonstrated. However, more effort will be required to make such applications routine and extend the mass range for reliability and accuracy. ACKNOWLEDGMENTS This work was supported by a grant from the National Science Foundation (M.V.J., Grant 9629672), a Grant Opportunities for Academic Liaison with Industry (GOALI) supplement from the National Science Foundation (M.V.J. and B.S.L., Grant 9300644), the University of Delaware, and the DuPont Company. The authors thank G. Kruppa, D. P. Ridge, G. Nicol, and T. Jones for helpful suggestions.

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