# N-Protonated Isomers and Coulombic Barriers to Dissociation of Doubly Protonated Ala_{8}Arg

## Abstract

Collision-induced dissociation (or tandem mass spectrometry, MS/MS) of a protonated peptide results in a spectrum of fragment ions that is useful for inferring amino acid sequence. This is now commonplace and a foundation of proteomics. The underlying chemical and physical processes are believed to be those familiar from physical organic chemistry and chemical kinetics. However, first-principles predictions remain intractable because of the conflicting necessities for high accuracy (to achieve qualitatively correct kinetics) and computational speed (to compensate for the high cost of reliable calculations on such large molecules). To make progress, shortcuts are needed. Inspired by the popular mobile proton model, we have previously proposed a simplified theoretical model in which the gas-phase fragmentation pattern of protonated peptides reflects the relative stabilities of N-protonated isomers, thus avoiding the need for transition-state information. For singly protonated Ala_{ n } (*n* = 3–11), the resulting predictions were in qualitative agreement with the results from low-energy MS/MS experiments. Here, the comparison is extended to a model tryptic peptide, doubly protonated Ala_{8}Arg. This is of interest because doubly protonated tryptic peptides are the most important in proteomics. In comparison with experimental results, our model seriously overpredicts the degree of backbone fragmentation at N_{9}. We offer an improved model that corrects this deficiency. The principal change is to include Coulombic barriers, which hinder the separation of the product cations from each other. Coulombic barriers may be equally important in MS/MS of all multiply charged peptide ions.

## Keywords

Collision-induced dissociation Conformation Coulombic barrier Fragmentation Gas phase Helix Mass spectrometry Mobile proton model Peptide Polyalanine Proton affinity Quantum chemistry Tryptic## Introduction

Tandem mass spectrometry (MS/MS) of protein digests is invaluable in high-throughput proteomics studies. Thus, understanding and predicting gas-phase fragmentation of peptide ions is of both fundamental and practical interest. Fragmentation at peptide bonds yields “sequence ions,” which are useful for inferring the sequence of amino acid residues in the peptide. In the absence of specific chemical effects [1], fragmentation is usually explained with the “mobile proton model” [2, 3]. In this model, protons in an energized peptide ion may become “mobile” if their number exceeds the number of basic residues, so that at least one proton can be considered as not affixed to any particular location. Backbone fragmentation requires that this mobile proton migrate to the nitrogen atom of a backbone amide bond, weakening the peptide linkage. Thus, the N-protonated isomer is the gateway to fragmentation at that position along the backbone.

In an MS/MS experiment, the relative abundance of fragment ions mirrors the corresponding rate coefficients for dissociation of the parent peptide ion. Rate coefficients may be computed from first principles [4, 5, 6]. However, such computations are more difficult and expensive than for small molecules because polypeptides are so much larger and so conformationally flexible. In an attempt to simplify the computations, we proposed that the relative electronic energies of N-protonated isomers be used as surrogates for the energies of the corresponding transition states [7]. The computations are further simplified by ignoring vibrational zero-point energy (ZPE) and restricting interest to the low-energy limit of collisional energy (where enthalpy dominates). This is a strong approximation, but was supported by comparison of theory and experiment for small, singly protonated polyalanines. Although a promising result, it was only for singly charged peptides, which are of minor importance in proteomics. For greater relevance to proteomics, we have now extended our calculations to doubly protonated Ala_{8}Arg. Ala_{8}Arg is a tryptic peptide (i.e., C-terminal Arg or Lys), the most common type in bottom-up sequencing. It is doubly protonated, as in typical experiments, so there is a mobile proton on activation.

To predict unimolecular dissociation rates, the most popular theories are statistical, largely because of their conceptual and computational simplicity [6, 8]. For a canonical, equilibrated, thermal distribution of molecular internal energies, transition-state theory (TST) is the typical choice. For a microcanonical ensemble in which all molecules have the same internal energy, Rice–Ramsperger–Kassel–Marcus (RRKM) theory [11, 12] is typical. MS/MS, unfortunately, is a nonthermal process in which the ensemble is neither canonical nor microcanonical. Nevertheless, it is usually treated as if it were canonical and characterized by an “effective” temperature [13]. The most prominent example of this practice is probably the “kinetic method” of measuring relative proton affinities [14, 15, 16]. We accept this popular approximation when considering MS/MS of peptide ions. In particular, we start from ordinary TST.

*k*(

*T*), assuming unit transmission efficiency, is given by Equation 1 [17]:

where *k* _{B} is the Boltzmann constant, *h* is the Planck constant, *T* is the effective temperature, *R* is the gas constant, and ∆*G* ^{‡} = ∆*H* ^{‡}−*T*∆*S* ^{‡} are pseudo-thermodynamic differences of Gibbs energy, enthalpy, and entropy between the transition state and the reacting peptide ion.

The rate coefficient depends exponentially on the Gibbs energy of activation. This is one reason that it is so difficult to accurately predict rate coefficients ab initio. For example, if the uncertainty in ∆*G* ^{‡} is ±10 kJ/mol, which is good even for a thermochemical prediction for a large molecule, the corresponding uncertainty in the rate coefficient at *T* = 500 K is a *multiple* of 11 (or of 1/11). In the present application to peptide ions, it is hoped and expected that the chemical similarity among the competing transition states will cause many errors to cancel when relative rates are evaluated. However, it is unreasonable to expect the theoretical precision to match that of experimental measurements.

where ∆*E* ^{‡} is the classical barrier height. The classical barrier height is simply the energy difference between the global minimum (peptide ion) and the saddle point (transition state) on the Born–Oppenheimer potential energy surface.

*T*), whereas at high CID energy the entropy will dominate. Here we are addressing only the low-energy limit. Thus, we ignore the effect of entropy, which is equivalent to assuming that ∆

*S*

^{‡}is the same for all N-protonated isomers. We justify this approximation by noting the close chemical similarity among the N-protonated isomers and among their associated transition states, as is done when the “kinetic method” is applied to determining proton affinities [18]. For the same reason, we expect heat capacities and vibrational ZPEs [19] to be nearly equal. Assuming that these contributions are constant across all amide scissions leads to a simplified expression for the rate coefficient, shown in Equation 3:

*i*specifies an N-protonated isomer and the constant

*A*is the product of the quantities that have been assumed to be constant. Experimentally, branching fractions (i.e., relative intensities) are available, not rate coefficients. The branching fraction,

*f*

_{ i }, for fragmentation at position N

_{ i }is given by Equation 4:

*E*

^{‡}, in red) to break a particular amide bond determines the rate of fragmentation at that bond, as shown in Equation 3. When collisional or infrared activation raises the effective temperature of the ion ensemble, small populations of high-energy isomers are created. Some of these isomers are protonated on a backbone N atom, weakening the corresponding amide bond. We use the symbol

*E** to denote the additional energy required to break an amide bond that has already been N-protonated. Our major assumption is that

*E** is the same for all amide bonds. This approximation is shown in Equation 5 and Figure 2, where

*E*

_{ i }is the energy of an N-protonated isomer, relative to the most stable conformation of the most stable isomer of the peptide ion:

Thus, the amide bond with the lowest activation energy (lowest ∆*E* ^{‡}) is assumed to be the amide bond with the most stable N-protonated isomer (lowest *E* _{ i }).

*E** is assumed to be a constant:

According to our model, the likelihood of fragmentation at an amide bond can be inferred from the stability of the corresponding N-protonated isomer. It is not necessary to know the barrier heights or the value of *E**. Pechan and Gwaltney [20] constructed essentially the same hypothesis independently. An alternative thermodynamic hypothesis, that dissociation energies giving rise to products can be used to predict which fragments will not form, has been explored by Obolensky et al. [21, 22].

*κ*

_{ j }is the rate of fragmentation at amide position

*j*,

*r*

_{ j }(

*T*) is the temperature-dependent rate coefficient for the fragmentation reaction, and

*n*

_{ j }is the population of the isomer protonated on N

_{ j }. Note that

*r*

_{ j }(

*T*) is not the same as the corresponding

*k*

_{ j }(

*T*) in our model, because it refers to a different reactant. The branching (i.e., fragment intensity) fractions,

*f*

_{ j }, are then given by Equation 8:

*r*

_{ j }(

*T*) is given by Equation 9, which is analogous to Equation 3:

*E** is the same for each N-protonated isomer, the rate coefficients are the same for all N-protonated isomers. The equilibrium speciation is given by Equation 10, again using our other approximations:

*E** is the same for all

*j*.:

Clearly Equation 11 is the same as Equation 6, and the models are numerically equivalent.

Although we have made many assumptions to simplify the problem, it is still a heavy computational burden. The most demanding task is the global geometry optimization (i.e., obtaining a plausible candidate for the lowest-energy conformation of each structural isomer).

The simple model described above addresses only the relative rates of amide bond cleavage, not the ultimate product ions (a_{ n }, b_{ n }, c_{ n }, y_{ n }, etc.) that may be obtained in a real CID experiment. At high energies, the effects of entropy will dominate and would have to be included in a corresponding model. Moreover, the dissociation products may themselves have enough internal energy to dissociate, leading to secondary reactions. Similarly, in many experiments the ion acceleration is not mass selective, so the observed MS/MS spectrum is contaminated by MS/MS spectra of fragment ions that formed from the original peptide ions (i.e., by MS^{ n } spectra) [23].

Ala_{8}Arg has nine backbone nitrogen atoms, numbered from N_{1} at the N-terminus to N_{9} for the C-terminal residue. In this study, we considered nine structural isomers (tautomers) of doubly protonated H_{2}Ala_{8}Arg^{2+}. All are protonated on the guanidine group of the C-terminal Arg, with the other (“mobile”) proton on one of the backbone nitrogen atoms. For notational convenience, the isomer protonated on N_{ j } will be labeled N*j*. For example, the isomer protonated on N_{1} is labeled N1.

## Theoretical Methods

We used the same procedure as in our previous study [7]. Each of the N-protonated isomers of H_{2}Ala_{8}Arg^{2+} was subjected to a conformational search using the Monte Carlo multiple minima (MCMM) method [24] as implemented in the program MacroModel [25, 26, 27]. The OPLS2005 force field [28] was used for these calculations as implemented in MacroModel. Both *cis* and *trans* conformations of the peptide bonds were included, because N-protonation nearly eliminates the barrier for *cis*–*trans* isomerization [7, 29].

For each isomer, 50,000 local minima were obtained by MCMM/OPLS2005. All unique structures below 41.8 kJ/mol relative energy, plus 100 structures uniformly distributed between 41.8 and 200 kJ/mol, were subjected to single-point energy evaluations using the Hartree–Fock (HF) method and the 3-21G basis set (HF/3-21G). Relative energies were reevaluated at this level, and all structures above 50 kJ/mol were discarded. The remaining structures were refined by HF/3-21G geometry optimization followed by single-point energy evaluation using locally correlated, second-order perturbation theory accelerated by a resolution-of-the-identity approximation (RI-LMP2/cc-pVDZ//HF/3-21G). LMP2 was chosen instead of density functional theory (DFT) because it includes van der Waals interactions yet has little basis-set superposition error [30, 31]. The structure with the lowest RI-LMP2 energy was taken as the optimum geometry. Vibrational ZPEs were assumed to be similar for all conformations of all isomers [19] and were not evaluated. The Gaussian software package [32] was used for the HF calculations and the QChem package [33, 34] was used for the RI-LMP2 calculations.

_{9}instead of N

_{1}. Following the suggestion of an anonymous reviewer, the final structure for each of the nine isomers was re-refined with use of DFT and a larger basis set. We selected the M06-2X [35] and ωB97X-D [36] functionals [30, 37, 38, 39]. Geometries were reoptimized with use of the M06-2X functional and 6-311G(d,p) basis sets. ZPEs were added (harmonic, unscaled), also computed at the M06-2X/6-311G(d,p) level, and single-point energies were computed at the RI-LMP2/cc-pVDZ level as before. Compared with our earlier computation, the energies of N-protonated isomers (relative to the N1 isomer) decreased by 4–12 kJ/mol, not enough to change any conclusions. However, when the DFT calculations were repeated with the dispersion-corrected ωB97X-D functional, the conformation of the N1 isomer changed significantly during the local minimization, decreasing its energy by 41 kJ/mol. This new conformation makes N1 the most stable N-protonated isomer, as expected, at all self-consistent field levels of theory. The ZPE-corrected RI-LMP2 energies of the other amide-protonated isomers increased slightly by 2–6 kJ/mol (relative to the original N1 conformation). Table 1 shows the isomer energies relative to the improved N1 conformation, as computed with seven quantum chemistry methods. The close agreement between models A and B shows that the cc-pVDZ basis set is adequate for computing RI-LMP2 energies. Comparison of models C and D shows the minor effect of the method used for geometry refinement (M06-2X or ωB97X-D functional). Models E and F are DFT-only models, without RI-LMP2 energies, and disagree noticeably. The difference between models F and G represents vibrational ZPE. Our original model (model A) is the least expensive model listed. Compared with the average of the other six models, its root-mean-square difference is only 2.2 kJ/mol.

Relative Energies, *E* _{ i } (in kJ/mol), of Backbone N-protonated Isomers of H_{2}Ala_{8}Arg^{2+} (N_{1} is the Terminal Amine and N_{9} is Within the Arg^{+} Residue)

Isomer | Model A | Model B | Model C | Model D | Model E | Model F | Model G |
---|---|---|---|---|---|---|---|

N1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

N2 | 110 | 115 | 99 | 103 | 98 | 116 | 124 |

N3 | 124 | 125 | 111 | 117 | 109 | 134 | 141 |

N4 | 127 | 122 | 116 | 118 | 105 | 132 | 142 |

N5 | 110 | 107 | 101 | 103 | 87 | 118 | 126 |

N6 | 111 | 110 | 96 | 101 | 99 | 125 | 133 |

N7 | 62 | 62 | 50 | 53 | 43 | 71 | 79 |

N8 | 40 | 38 | 32 | 35 | 29 | 53 | 61 |

N9 | 30 | 29 | 22 | 27 | 19 | 39 | 44 |

Use of empirical force fields for conformational searching is standard practice [40], and is done because of its low computational cost. The substantial change in conformation for the N1 isomer, described above, is a cautionary example of failure of this widespread practice, at least when standard force fields are used. Unfortunately, conformational searching on a DFT or HF potential-energy surface is usually too costly to be feasible.

As an explicit test of our assumption that *E** is the same for all N-protonated isomers, we also computed transition states starting from the best conformation for each isomer (i.e., without conformational searching within the transition-state search). The transition-state calculations were done at the ωB97X-D/6-311G(d,p) level and were for the reaction of each N-protonated isomer to the corresponding oxazolone b_{ n } and y_{9-n } ions.

*r*) was gradually increased. All other degrees of freedom were energy-minimized during each dissociative scan. (Only local energy minimization was possible during the dissociative scan because of the tremendous cost of global optimizations. Consequently, the computed energies are only upper bounds.) These scans, which were difficult to converge geometrically, were done at the ωB97X-D/6-31G(d) level; the small basis set was selected to improve computational efficiency. (Minima and transition states were also re-refined with use of this basis set, to obtain compatible energies for comparison.) The C–N distance was increased until the energy showed a clear, negative distance dependence as expected from Coulombic repulsion between the b

_{ n }and y

_{9-n }ions. That repulsive portion of the energy curve was fitted to a Coulomb function:

*E*

_{∞}(the asymptotic, dissociated energy) and

*r*

_{0}(the effective distance between charge centers at zero C–N distance).

*C*= 1389.35 kJ mol

^{-1}Å is the Coulomb constant for two charges of +

*e*and was held fixed. The fitted function was extrapolated to smaller distances to find where it crosses the non-Coulombic part of the potential. The crossing point at the largest distance provides a crude estimate for the Coulombic dissociation barrier. To obtain a dissociation energy (vibrationless

*D*

_{e}; Equation 13) that is easier to compute than

*E*

_{∞}, (local) energy minimizations were done for the isolated b

_{ n }and y

_{9-n }ions:

## Results and Discussion

The purpose of the present study was to test our working hypothesis (Figure 2) on a doubly charged tryptic peptide. Part of our hypothesis is that vibrational ZPE can be ignored. Table 1 shows the relative energies of the nine N-protonated isomers of H_{2}Ala_{8}Arg^{2+}, as obtained by use of seven different computational protocols (see the footnotes to Table 1). The different quantum chemistry models all produce the same general trend, irrespective of whether ZPE is included. However, the quantitative reliability is modest, as shown by the scatter within each row of Table 1.

### Comparison with Experiments

There are nine backbone nitrogen atoms in Ala_{8}Arg. For the corresponding nine N-protonated isomers of H_{2}Ala_{8}Arg^{2+}, relative energies (*E* _{ i }) are listed in Table 1. By hypothesis, we expect the low-energy MS/MS fragmentation of H_{2}Ala_{8}Arg^{2+} to reflect the relative energies. That is, greater stability of the isomer protonated at N_{ k } should correspond with greater cleavage at that position (e.g., to *b* _{ k-1} and *y* _{10-k } fragment ions), as expected from Equation 6. However, at high collision energies, energy barriers are less important, being overwhelmed by entropic effects.

_{2}Ala

_{8}Arg

^{2+}under three different conditions: high-energy collisional dissociation (nominal 10 eV with N

_{2}collision gas), at a nominal collision energy of 20 eV (He collision gas), and at an unusually low nominal energy of 5 eV (He). Figure 3 shows the experimental cleavage fraction at each peptide bond, including all ion products (a

_{ n }, b

_{ n }, y

_{9-n }, etc.) that can plausibly be ascribed to a particular peptide linkage. (However, we cannot rule out secondary fragmentation of b ions in the experiment [43], which would skew the distribution toward the N-terminus.) Our model is applicable only at the low-energy limit, where entropic effects are minor. As shown in Figure 3, lower collision energy favors dissociation closer to the C-terminus, with fragmentation at N

_{8}dominating at the lowest energy studied. The effect of collision energy is weaker for H

_{2}Ala

_{7}Arg

^{2+}and stronger for H

_{2}Ala

_{9}Arg

^{2+}(see the electronic supplementary material for plots analogous to those in Fig. 3). We propose that this fragmentation pattern, termed “class III” by Yang et al., can be explained qualitatively by the results in Table 1, supporting the hypothesis illustrated in Fig. 2. However, fragmentation at N

_{9}was never dominant in the experiments, as we would predict from Table 1. (Similarly, fragmentation at N

_{8}appears suppressed for H

_{2}Ala

_{7}Arg

^{2+}, and fragmentation at N

_{10}appears suppressed for H

_{2}Ala

_{9}Arg

^{2+}.) This discrepancy for the N9 isomer indicates that an important effect is missing from our model.

### Transition-State Energies

_{9}, we performed explicit transition-state computations for the isomers from N3 to N9. The transition states were for oxazolone formation, leading to (b, y) ion pairs. This provides a value of

*E** for each isomer, to see how similar those values actually are, and also provides values for

*ΔE*

_{ i }

^{‡}. The results are shown in Table 2, as obtained with the ωB97X-D/6-311G(d,p) and ωB97X-D/6-31G(d) quantum chemistry models. Variation in the value of

*E** does not explain why fragmentation at N

_{9}is suppressed. On the contrary, we find the lowest activation energy (153 kJ/mol) for breaking the bond at N

_{9}. Overall, the mean value of

*E** is 97 kJ/mol (standard deviation 18 kJ/mol), but the values range from 72 to 121 kJ/mol; they are not constant as we have assumed. On the other hand, this is a narrower range than the range of the

*E*

_{ i }(Table 1). Stated as an approximate equation,

Values of *E** and of ∆*E* ^{‡} (in kJ/mol) Computed with the ωB97X-D Functional and Two Basis Sets

Isomer | 6-311G(d,p) | 6-31G(d) | ||
---|---|---|---|---|

| ∆ | | ∆ | |

N3 | 72 | 213 | 69 | 214 |

N4 | 100 | 242 | 96 | 238 |

N5 | 89 | 215 | 85 | 213 |

N6 | 77 | 210 | 77 | 207 |

N7 | 121 | 200 | 125 | 204 |

N8 | 111 | 172 | 109 | 170 |

N9 | 109 | 153 | 111 | 154 |

Of course, the numerical results will depend somewhat on the theoretical model used to generate them. For example, the smaller 6-31G(d) basis set, with values shown in Table 2, results in *E** = 96 ± 20 kJ/mol. The values shown in Equation 14 are the best estimates from this study.

### Coulombic Barriers

_{ n }ion plus the y

_{9-n }ion co-product). However, the reaction products are both cations, so there may be a Coulombic barrier between the transition state and the completely dissociated products. If that Coulombic barrier is high enough, it will be rate limiting. Starting near each transition state, we computed the energy while gradually increasing the length of the breaking C–N bond. This was continued until the energy showed simple Coulombic distance dependence. The resulting energies, along with a Coulomb function fitted to the long-distance points, are shown in Figure 4 for the N9 isomer. The energy discontinuities are caused by conformational stick–slip during the dissociative scan, and would presumably be reduced or eliminated by use of global geometry optimization at every C–N distance. However, that would require an impractical amount of computing time.

_{1}is the terminal, primary amine. Isomer N2 is ignored because it yields a b

_{1}ion, which cannot be stabilized by oxazolone or diketopiperazine formation.) The estimated values of \( \varDelta {E}_{\mathrm{Coul}}^{{}^{\ddagger }} \) exceed the corresponding chemical barriers (∆

*E*

^{‡}), except for N7 and N8. One may expect that the fragmentation rate coefficient (and therefore the fragmentation probability) at each amide position will reflect the greater of \( \left(\varDelta {E}^{{}^{\ddagger }},\varDelta {E}_{\mathrm{Coul}}^{{}^{\ddagger }}\right) \) for each N-protonated isomer. We denote these higher barriers as

Fitting Constants for Equation 12 and Coulombic Barriers Estimated from ωB97X-D/6-31G(d) Dissociative Scans

Isomer | | | ∆ |
---|---|---|---|

N3 | 233 | 11.9 | 287 |

N4 | 203 | 7.7 | 265 |

N5 | 186 | 10.8 | 236 |

N6 | 174 | 8.2 | 239 |

N7 | 125 | 8.6 | 177 |

N8 | 112 | 12.0 | 166 |

N9 | 172 | 5.8 | 222 |

*f*

_{1}) against \( \varDelta {E}_{\max}^{{}^{\ddagger }} \). The regression line is close to five of the seven points, but for N4 there is substantially less fragmentation than expected, and for N6 there is more. \( \varDelta {E}_{\max}^{{}^{\ddagger }} \) is the best predictor that we have found for the experimental fragmentation fractions. This is not surprising because it represents the most rigorous (and most expensive) procedure that we investigated.

We have no satisfactory explanation for the underprediction and overprediction of fragmentation at N_{6} and N_{4} respectively. There is probably something missing from our model, such as entropic effects, but there could also be a shortcoming in the calculations that supplied the numbers to the model. For example, we did not attempt global geometry optimizations of reaction products. If the conformation that we used for the b_{5} or y_{4} ion is especially bad, then the corresponding Coulombic barrier will be lower than we think and more fragmentation will be expected. For N4, there is less fragmentation than expected, so a different explanation would be needed. A possibility is that the b_{3} ion is especially fragile and prone to secondary fragmentation to yield b_{2}. If so, then apparent intensity would be shifted from N_{4} to N_{3} in the experimental data. Correcting for this shift would move the point for N_{4} higher and the point for N_{3} lower in Figure 5.

*E*

_{∞}: \( \varDelta {E}_{\mathrm{Coul}}^{{}^{\ddagger }}={E}_{\infty } \) + (55 ± 6) kJ/mol (mean ± standard deviation). This suggests that the value of \( \varDelta {E}_{\mathrm{Coul}}^{{}^{\ddagger }} \) may be estimated from the energy of the corresponding dissociated (b, y) ions:

*D*

_{e}(Table 4, model H) are very close to the values of

*E*

_{∞}in Table 3, as they should be.

Dissociation Energies, *D* _{e} (Equation 13) (kJ mol^{-1}), for Creation of Separated (b_{ n-1}, y_{10-n }) Ion Pairs from the N1 Isomer of H_{2}Ala_{8}Arg^{2+}

N | Model A | Model H |
---|---|---|

N | 192 | 235 |

N | 190 | 204 |

N | 177 | 187 |

N | 157 | 175 |

N | 101 | 125 |

N | 99 | 113 |

N | 153 | 173 |

Obolensky et al. [21, 22] suggested the use of product energies to predict peptide ion fragmentation, although they proposed it for the dissociation of singly charged peptide ions, which do not have Coulombic barriers. In particular, they suggested that fragmentation will not be observed when *D* _{e} > 50 kcal/mol (209 kJ/mol). None of the dissociation energies exceed 209 kJ/mol, although the energies for dissociation at N_{3} and N_{4} come close. Indeed, these are the sites with the least fragmentation (Figure 1), qualitatively supporting the suggestion of Obolensky et al.

### Improved Theoretical Model

In our original model (Figure 2), we assumed that *E** is the same for all amide bonds. Our direct computations of *E** for H_{2}Ala_{8}Arg^{2+} show that it takes a range of values (Table 2) around 97 kJ/mol. This is summarized by Equation 14. We also assumed that the transition states dictate the rate coefficients for dissociation. Although this may be true for singly charged ions, for this doubly charged peptide there are Coulombic barriers for the nonbonded, b ion–y ion complex to dissociate to separated b and y ions. The Coulombic barriers, \( \varDelta {E}_{\mathrm{Coul}}^{{}^{\ddagger }} \), are often higher than the chemical barriers, therefore controlling the kinetics. The values of \( \varDelta {E}_{\mathrm{Coul}}^{{}^{\ddagger }} \) cover a wide range (Table 3), but the *reverse* Coulombic barriers are fairly similar, as summarized by Equation 17.

- 1.
Compute the relative energies of the N-protonated isomers,

*E*_{ i }, after global geometry optimization. - 2.
Estimate the chemical barrier, \( \varDelta {E}_i^{{}^{\ddagger }} \), at each amide bond with Equation 14.

- 3.
Compute the energies of the separated b and y ions to obtain the dissociation energy,

*D*_{e}, giving rise to each (b_{ i-1}, y_{10-i }) pair of product ions, Equation 13. - 4.
Estimate the Coulombic barrier, \( \varDelta {E}_{\mathrm{Coul},\ i}^{{}^{\ddagger }} \), for separating each (b

_{ i-1}, y_{10-i }) ion pair with Equation 17. - 5.
The estimated barrier to fragmentation at the

*i*th amide bond, \( \varDelta {E}_{\max,\ i}^{{}^{\ddagger }} \), is the larger of the two barriers, Equation 15.

### Potential Energy Diagrams

_{9}, which is predicted by our model to dominate but is shown experimentally [42] to be of secondary intensity. While pursuing an explanation, we explored features of the potential-energy surface beyond Figure 2. As described earlier, we computed energies of the transition states for oxazolone (b, y) formation, of the b ion–y ion complexes, of the Coulombic barriers to dissociation, and of the (b, y) reaction products. These energies are listed in the various tables and assembled graphically in Figure 7. The Coulombic barriers are substantial and cannot be ignored. In particular, the low intensity of fragmentation at N

_{9}is explained by the high Coulombic barrier for the product ions to escape the initial b

_{8}–y

_{1}complex.

### Conformation

_{2}Ala

_{8}Arg

^{2+}. Quantitative details (i.e., atomic coordinates) are available in the electronic supplementary material. Rationalizing these structures is beyond the scope of the present report. The N9 structure (protonated on N

_{9}) is an alpha helix with the guanidinium group participating in the helix (Figure 8). The N8 structure is like N9 except that the helix ends at (protonated) N

_{8}. The N7 structure has the guanidinium group pointing away from the helix, which ends at N

_{7}. The N6 structure is analogous to N7. The first five residues of the N5 structure form a crude helix, with the remaining residues coiling to the left and the guanidinium group directed away. Thus, for structures N5 through N9, the mobile charge stabilizes an N-terminal helix but no C-terminal helix, as would be expected from electrostatic (helix dipole [44, 45, 46, 47]) considerations. The N4 structure is surprisingly extended. The N3 structure is folded in the middle, like an incipient beta sheet. The N2 structure is a better beta sheet, folded in the middle, with the guanidinium group excluded. The N1 structure (protonated on the terminal amine group) is most unusual: a loose, left-handed helix with the guanidinium group pointing away and off the helix axis. Except for N1, which is chemically different from the other isomers, the relative stabilities (Table 1) appear correlated with the amount of alpha helix in the structure. Greater distance between the two nominal charge centers does not correspond with increased stability.

## Conclusions

The purpose of this study was to test our earlier, approximate model on a doubly charged, tryptic peptide. Our principal conclusions were not anticipated, and remain important regardless of the fate of our model: (1) dissociation kinetics may be controlled by Coulombic barriers, which impede the separation of product ion pairs; (2) empirical force fields may not be adequate for conformational searching.

Regarding the intended test of our model, the numerical results are summarized by the potential-energy diagram shown in Figure 7. For this doubly charged peptide, there are significant Coulombic barriers hindering the separation of product (b, y) ions from each other. The rate-controlling barrier is taken to be the larger of the transition-state energy and the Coulombic barrier. The result is imperfect, but much of the experimental trend in low-energy fragmentation is modeled correctly (Figure 5). To construct an economical computational procedure, the transition-state energy may be estimated from the energy of its associated N-protonated isomer, and the Coulombic barrier may be estimated from the energy of its asymptotic product ions. Thus, expensive calculations of transition states and of Coulombic barriers are avoided. The resulting estimates model the data nearly as well as the rigorous calculations. However, it is unknown whether this approximate procedure is applicable to doubly charged tryptic ions in general or only to H_{2}Ala_{8}Arg^{2+}.

The new computational procedure can be considered as a descendant of our earlier model and of the model suggested by Obolensky et al. [21, 22]. Like the earlier procedures, it is applicable only in the low-energy limit of MS/MS. For modeling the higher energies that are more typical in analytical work, it will be necessary to include the effects of entropy. On the other hand, errors in barrier heights will be less important at higher collision energies.

## Electronic Supplementary Material

Plots of experimental fragmentation sites for H_{2}Ala_{7}Arg^{2+} and H_{2}Ala_{9}Arg^{2+}; atomic coordinates, energies and drawings for isomers N1 through N9; atomic coordinates and energies for transition states for isomers N3 through N9; plots of dissociative energy scans for isomers N1 through N9. (48 pages)

## Notes

### Acknowledgements

We thank anonymous reviewers for suggesting better theory and bigger basis sets, and for insisting on a detailed explanation of the N9 discrepancy.

### Compliance with Ethical Standards

### Conflict of Interest

The authors declare that they have no competing interests.

## Supplementary material

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