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For puri ed samples of small molecules studied on low-resolution mass spectrometers, and for singly charged larger molecules, fragment ions are readily identi ed by straightforward calculations using average mass values The presence or absence of observable fragments (including the parent ion) are known to depend on the type.

has only one real root for s > 0, and that this root is positive for s > 0, equal to 1 for s 0, and bounded by 1. Introduce Y s : y s=b 1= n 1 . To investigate the singularities of this solution of Eq. (6.35), note that dY s Y n s : ds 1 nsY n 1 s 6:36

(15.4)

The only pair sn ; Y sn that satis es Eq. (6.35) and for which the denominator on the right-hand side of Eq. (6.36) becomes zero is the pair n 1 1 n 1 sn n n and Y sn n : n 1 6:37

TABLE 15.5. Nominal, Monoisotopic, and Average Mass for Some Elements and Moleculesa Element or Molecule Carbon Hydrogen Nitrogen Oxygen Sulfur Phosphorus Tripeptide C27H29N3O7 Porcine insulin C256H381N65O76S6

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To see this, write Y s 1 sY n s 1 1=n nsY n 1 s Y s , which equals 1 Y s =n for s sn. Hence, sn as speci ed in Eq. (6.37) (by its principal value) is the only singularity of Y s . ^ ^ Note that for a solution Y s of Eq. (6.35) with Y 0 T 1 holds that ^ limjsj30 jY s j I. Next, we derive a series expansion in powers of s for the solution Y s with Y 0 1. It is obtained by using Lagrange's theorem (cf. Whittaker and Watson [37, Chap. 7]). Put q s : Y s 1; so that q s s 1 q s n ; q 0 0: 6:39 6:38

respectively according to equation (15.2).

Let C be the circle with center at q 0 and radius q sn Y sn 1 1= n 1 . Choosing jsj < sn , it is seen that, for q P C, jqj > jsk1 qjn : 6:40

In daltons (Da).

After a new node is added, the system will be reorganized to that in Figure 15.12. Now the two new redundant data block, denoted by c0,1 and c1,1 , are computed from

Consider a circle CE that is concentric with C but has a radius that is E smaller 0 < E ( 1 . Since q and 1 q n are regular for q inside and on CE , while Eq. (6.40) holds on C, Eq. (6.39) seen as an equation in q has one root in the interior of C, and Lagrange's theorem gives a series development of the zero q s of Eq. (6.39) in powers of s. For jsj < sn, q s

c0,1 =

6:41

of ion source (e.g., hard vs. soft) as well as ionization source variables (such as ionization and acceleration voltages). Biomolecules such as proteins and oligonucleotides present mass spectra that are complicated by the presence of multiply charged families of ions. The parent species, for example, generated by a soft ionization method such as FAB of ESI, will yield several m/z peaks in which m is equal to a constant plus (protein) or minus (nucleic acid) a variable number of proton masses, while z is a variable. Egg white lysozyme, for example, yields ESI mass spectra with ne parent peaks between m/z values of 1194 and 1791; the corresponding z values are 12 8 [see Fig. 15.9(a)].

Observe that p r Y br n 1 1 q br n 1 , so that, for 0 < Re br n 1 < sn, p r 1 Hence

(15.5)

6:42

Figure 15.9. Typical protein ESI mass spectra for egg white lysozyme (a,b) and BSA (c,d), showing an algorithm-based MW calculation (HyperMass reconstruct).14 [Reprinted, with permission, from From Gary Siuzdak, Mass Spectrometry for Biotechnology , Academic Press, 1996. Copyright # 1996 by Academic Press, Inc.]

c1,1 =

6:43

6:44

(15.6)

(Continued )

By applying Theorem 2 of Doetsch [22, p. 159], an asymptotic series for Pn 1 t can be obtained from Eq. (6.43). We thus obtain the main result of this section. For t 3 I, and for every nite positive integer H, 1 Pn 1 t

Comparing equation (15.5) with equation (15.3) we can observe that they share four common terms in v j v0 , v1 , v2 , v3 . Hence we can rewrite equation (15.5) as follows:

1 n 1

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