# Phase Correction for Absorption Mode Two-Dimensional Mass Spectrometry

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

_{min}. Scheme 1 shows the initial phase of each pulse, which leads to a corresponding phase offset for the signals of ions that are excited from the center of the ICR cell in that pulse.

_{0}. The precursor ions rotate during the encoding delay t

_{1}at their own cyclotron frequency. The second pulse has a starting phase of 2πf

_{min}(T

_{1}+ t

_{1}). The phase difference between the second pulse and the precursor ion motion is 2πt

_{1}(f

_{ICR}− f

_{min}) where f

_{ICR}is the cyclotron frequency of the ion, f

_{min}is the minimum frequency in the pulse, and t

_{1}is the encoding delay. Therefore, depending on the phase difference, precursors are excited to higher radii or de-excited towards the center of the ICR cell. The frequency of the precursor ion radius modulation is f

_{ICR}− f

_{min}. At the end of the second pulse, the radius of the precursor ions can be expressed as [2,11,40]:

_{m}with a method that has a maximum fragmentation efficiency at the center of the ICR cell (e.g., electron capture dissociation, or ECD, and infrared multiphoton dissociation, or IRMPD) is applied to the precursor ions [14,22]. Ions within the fragmentation zone are fragmented. The abundance of the fragment ions is modulated with the same frequency as the radius of the precursors, i.e., f

_{ICR}–f

_{min}. The third pulse has a starting phase of 2πf

_{min}(2T

_{1}+ t

_{1}+ τ

_{m}) and excites all ions to high radius before detection. Fragment ions start accruing phase during the third pulse [38].

_{p,h}accrued by precursor ions that are not fully de-excited to the center of the ICR cell at the end of the second pulse, can be expressed with a quadratic phase dependence as:

_{0}′, c

_{1}′, and c

_{2}′ are the parameters for the quadratic phase incurred during excitation, f

_{1}is the cyclotron frequency of the ion, t

_{1}is the encoding delay, T

_{1}is the duration of the second pulse, τ

_{m}is the fragmentation period, T

_{2}is the duration of the third pulse, and T

_{3}is the delay between excitation and detection [26,27,38].

_{f,h}accrued by fragment ions and unfragmented precursor ions that have been fully de-excited to the center of the ICR cell at the end of the second pulse, can be expressed with a quadratic phase dependence as:

_{0}″, c

_{1}″, and c

_{2}″ are the parameters for the quadratic phase incurred during excitation, f

_{2}is the cyclotron frequency of the ion, t

_{1}is the encoding delay, T

_{1}is the duration of the first and the second pulse, τ

_{m}is the fragmentation period. As a rule, in phase-sensitive Fourier transform mass spectrometry, a shift of origin induces a zero order phase shift (c

_{0}″ in Equation (3)), a delay induces a first order phase rotation (c

_{1}″ in Equation (3)), and a frequency-dependent excitation induces a second order phase rotation (c

_{2}″ in Equation (3)).

_{f,v}for precursor ion signal can be expressed with a linear phase dependence as:

_{1}is the cyclotron frequency of the ion, t

_{1}is the encoding delay, f

_{min}is the lowest frequency in the pulse, and T

_{1}is the duration of the second pulse.

_{f,v}of the fragment ion signal is shifted by π [23,24,38]. The phase of the fragment ion signal can therefore be expressed as:

_{1}is the cyclotron frequency of the precursor ion, t

_{1}is the encoding delay, f

_{min}is the lowest frequency in the pulse, and T

_{1}is the duration of the second pulse.

_{1}amplitude modulation, a data set recorded in 2D mass spectrometry experiments requires processing using hypercomplex Fourier transformation [39,40,41,42,43]. The resulting spectrum is a 4-quadrant data set:

_{1}and f

_{2}are the frequencies, and i, j, and k are constants following the rules:

_{1}, f

_{2}) along the f

_{1}and f

_{2}axes:

## 3. Experimental Methods

#### 3.1. Sample Preparation

_{biotin}.

#### 3.2. Instrument Parameters

_{pp}and the delay between excitation and detection was 3 ms.

_{pp}. The encoding delay incremented 1024 times by 50 μs, corresponding to a 10 kHz bandwidth, with 1 scan per increment. The signal was folded over 14 times, leading to a frequency range of 214,659.79–224,659.79 Hz (corresponding to m/z 482.177–504.602 for precursor ion modulation) [25]. The parameters of the excitation-detection sequence were identical to the parameters used for the standard tandem mass spectrum, with transients lasting 489.34 ms (512k datapoints).

#### 3.3. Data Processing

_{1}delay. The demodulation frequency was determined to be 74,659.79 Hz. The exact demodulation frequency was offset by 68 Hz from the lowest frequency entered in the Apex Control software (Bruker Daltonik GmbH, Bremen, Germany). The correct demodulation frequency was measured by processing the 2D mass spectrum with the nominal frequency entered in the software, which causes peak-splitting on the autocorrelation line, and measuring the frequency difference between the split peaks. Digital demodulation with the corrected frequency eliminated the peak-splitting [17]. The parameters determined for the one-dimensional tandem mass spectrum for quadratic phase correction were then applied to each row in the spectrum. The imaginary part of each row of the data set was then removed to reduce the computer memory burden. Then, each column of the data set was apodised with a slightly shifted sine-bell window (with a maximum of the bell at 15% of the transient), zero-filled twice and Fourier transformed. The theoretical linear phase correction determined above was applied on each column without modification. The imaginary part of the signal in each column was dropped and the dataset stored on disk.

## 4. Results and Discussion

_{24}

^{4+}fragment of the monomethylated histone peptide and compares the phase-corrected absorption mode mass spectrum to the magnitude mode mass spectrum. Both the resolving power and the signal-to-noise ratio are almost double for the phase corrected absorption mode spectrum than for the magnitude mode spectrum, as predicted by theory [26,27].

_{min}(2T

_{1}+ t

_{1}+ τ

_{m}), in which f

_{min}is the minimum frequency in the pulse, T

_{1}is the duration of the first pulse, τ

_{m}the irradiation period, and t

_{1}is the incremental delay. The term 2πf

_{min}(2T

_{1}+ τ

_{m}) is constant throughout the experiment (and can be corrected with a constant phase offset in the phase correction function), but 2πf

_{min}t

_{1}is different for each transient.

_{min}t

_{1}in each transient. In magnitude mode, digital demodulation is used to reduce the number of harmonics visible in the 2D mass spectrum [11,17]. Here, by applying digital demodulation before the quadratic phase correction in the horizontal dimension, we can eliminate the effect of the continuous phase pulse generator on the phase of fragment ion signals.

^{13}C isotope of [M+6H]

^{6+}ions of the monomethylated histone peptide (K7 1m). The fragment z

_{24}

^{4+}contains 24 out of the 26 residues of the peptide. Consequently, if the precursor ion contains exactly one

^{13}C, then the probability of z

_{24}

^{4+}retaining the

^{13}C is very high [11,25,49,50,51]. In Figure 2, the relative intensity of the 1 ×

^{13}C isotope of z

_{24}

^{4+}is much higher than the relative intensity of the

^{12}C isotope. We can also see a peak for the 2 ×

^{13}C isotope of z

_{24}

^{4+}, which is caused by scintillation noise from the fragment of the 2 ×

^{13}C isotope of the precursor ion. The isotopic distribution shown in Figure 2 is a consequence of the precursor ion signals being isotopically resolved in the vertical dimension. In the horizontal dimension, the phase-corrected absorption mode has doubled the resolving power and signal-to-noise ratio of the magnitude mode, as expected [26,27].

^{12}C isotope of the c

_{6}fragment of the histone peptides. Since the modifications are all on the 7th residue from the N-terminus, all histone peptides in the sample have identical c

_{6}fragments. This precursor ion scan is therefore an appropriate column of the data set to test the linear phase correction function. Equations (4) and (5) predict that the phase correction function has a slope of f

_{N}T

_{1}, in which f

_{N}is the Nyquist frequency in the vertical dimension (i.e., 10 kHz in the present experiment) and T

_{1}is the duration of the first pulse (i.e., 739 μs). Therefore, the theoretical first order phase correction should be of 7.39 turns over the whole spectrum. The resulting spectrum is well-phased, but we observed that 7.8 turns give slightly better results (see Appendix A). Figure 3 (left) shows that the linear phase correction function yields a well-corrected spectrum in the vertical dimension. A comparison with the extracted precursor ion scan from the magnitude mode spectrum (Figure 3 right) shows that the absorption mode spectrum has doubled the resolving power and signal-to-noise ratio compared to the magnitude mode spectrum, as is predicted [26,27]. However, as this experiment was acquired in narrowband mode vertically, the frequency bandwidth is about twenty times narrower than in broadband 2D mass spectra, which may lead to a different phase correction [25].

_{6}fragment of the dimethylated histone peptide (Figure 3 centre) shows that the peaks are baseline- resolved. The full-width at half-maximum (FWHM) of the peaks is approximately 35 mDa (R = 14,000 at m/z 490), which is sufficient to isotopically resolve the fragments of biomolecules up to 14 kDa.

_{14}

^{2+}fragment of the monomethylated histone peptide shows that clearly. Figure 4a shows mostly fragment ion peaks with very few artefacts. Figure 4b shows the negative values in the same region, which correspond to the 2ω harmonic of the ion signal in the vertical dimension [14]. The modulation frequency of the unmodified and the monomethylated peptides is between 146.34 and 147.34 kHz.

_{24}

^{4+}fragment of the unmodified histone peptide with frequency-to-mass conversion that has been adapted to the 2ω harmonics. An additional phase correction of −33° was applied to the extracted precursor ion scans to correctly phase the 2ω harmonic peaks. As has been discussed in a previous article, the harmonic signals in a 2D mass spectrum can be used to increase the resolving power, just like they can in one-dimensional mass spectra [25,52,53,54,55]. Here, we almost double the resolving power from R = 14,000 to R = 27,000.

^{12}C isotopes of all four histone peptides on the autocorrelation line of the 2D mass spectrum. A comparison between equations (2) and (4) shows that, in the horizontal fragment ion dimension, the phase of precursor ion signals behaves differently from the phase of fragment ion signals, and that the phase correction function for precursor ion signals depends on t

_{1}. Furthermore, as shown in equations (2) and (3), in the vertical precursor ion dimension, the phase of precursor ion signals is shifted by −180° compared to the phase of fragment ion signals [38]. Figure 5 confirms that, with phase correction functions that aim to correct the phases of fragment ion signals, precursor ion peaks on the autocorrelation line have negative intensities, and that their phase shifts with their cyclotron frequency (and their m/z ratio).

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Appendix A

- # Import necessary modules
- import spike
- from spike.File import Apex
- # Load 2D data set:
- d = Apex.Import_2D(“histonepeptide_2D_000002.d”)
- # Apodisation with a sine bell with a maximum at 15% of the transient, double zero-filling in #the horizontal dimension, Fourier transform in the horizontal dimension
- d.apod_sin(axis = 2, maxi = 0.15).chsize(d.size1,4*d.size2).rfft(axis = 2)
- # Digital demodulation by multiplying every row by${e}^{-2\pi j{f}_{min}{t}_{1}}$:
- d.flipphase(0.0, 180*d.axis1.htoi(74,659.79), axis = 1)
- # Quadratic phase correction in the horizontal dimension with -9/360+564(f
_{2}/f_{N})+4595.7(f_{2}/f_{N})^{2}:* - d.phase(-9, 564.0, 4595.7, 0.0, axis = 2)
- # Discard of the imaginary part of the data in the horizontal dimension
- d.real(axis = 2)
- # Apodisation with a sine bell with a maximum at 15% of the transient, double zero-filling in the vertical dimension, Fourier transform in the vertical dimension
- d.apod_sin(axis = 1, maxi = 0.15).chsize(4*d.size1,d.size2).rfft(axis = 1)
- # Linear phase correction in the vertical dimension with 59.9/360+7.8(f
_{1}/f_{N}):* - d.phase(59.9, 7.8, 0.0, 0.00, axis = 1)
- # Discard of the imaginary part of the data in the vertical dimension
- d.real(axis = 1)
- # Offset of the frequencies for narrowband 2D mass spectrometry
- offset = 74659.79
- d.axis1.offsetfreq = 14*d.axis1.specwidth + offset
- d.axis1.leftpoint = 0

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**Scheme 1.**Pulse sequence for two-dimensional mass spectrometry and evolution of phases for (

**a**) precursor ion radii, (

**b**) fragment ion motion, (

**c**) unfragmented precursor ion motion, (

**d**) excitation pulses. (

**e**) Duration of each segment of the pulse sequence. The factor α (resp. β) is related to the moment that the swept frequency of the chirp reaches the cyclotron frequency of the fragment (resp. unfragmented precursor).

**Figure 1.**Tandem mass spectrum of the [M + 6H]

^{6+}histone peptides with ECD fragmentation in absorption mode (

**a**) before phase correction and (

**b**) after phase correction. Insert: zoom-in on the z

_{24}

^{4+}fragment ion isotopic distribution of the monomethylated histone peptide (K7 1m), comparison between magnitude mode and phase-corrected absorption mode (FWHM: full-width at half-maximum).

**Figure 2.**Extracted fragment ion scan from the 2D mass spectrum for the 1 ×

^{13}C isotope of [M + 6H]

^{6+}ions of the monomethylated histone peptide (K7 1m) at m/z 489.453, zoom-in on fragment ion z

_{24}

^{4+}, comparison between absorption mode and magnitude mode.

**Figure 3.**Precursor ion scan extracted from the absorption mode 2D mass spectrum for the

^{12}C isotope of the c

_{6}fragment of the histone peptides (m/z 616.38892), zoom-in on the c

_{6}fragment ion isotopic distribution in the 2D mass spectrum for the dimethylated histone peptide (K7 2m), comparison between the absorption mode and magnitude mode of the precursor ion scan of the c

_{6}fragment.

**Figure 4.**Phase-corrected absorption mode 2D mass spectrum of histone peptides, zoomed in between m/z 600-700 horizontally and m/z 485-495 vertically, (

**a**) positive values plotted (insert: zoom-in on the isotopic distribution of the c

_{14}

^{2+}fragment of the methylated histone peptide) and (

**b**) negative values plotted (insert: vertical precursor ion scans of the isotopic distribution of z

_{24}

^{4+}of the unmodified histone peptide, with frequency-to-mass conversion for the 2ω harmonic).

**Figure 5.**Extracted precursor ion scans for the

^{12}C isotopes of the four histone peptides from the phase-corrected absorption mode 2D mass spectrum.

**Table 1.**Comparison between theoretically and empirically calculated coefficients for the phase correction functions in the vertical dimension (linear phase correction function) and in the horizontal dimension (quadratic phase correction function). The theoretical values are obtained from equation 28 of ref. [28], ignoring image charge effect, and ref [38]. Here, f

_{min}is the lowest frequency in the excitation pulse and ∆f the frequency range in the excitation pulse, T

_{1}is the duration of the first encoding pulse, T

_{2}the duration of the third excitation pulse and T

_{3}is the delay between excitation and detection, f

_{N1}is the Nyquist frequency in the vertical precursor ion dimension and f

_{N2}is the Nyquist frequency in the horizontal fragment ion dimension (see Scheme 1).Zero order coefficients are expressed in degrees, first and second order coefficients are expressed in turns over the whole spectrum.

Coefficient | Theory | Theoretical Value | Empirical Value |
---|---|---|---|

0 order (vertical) | ${f}_{\mathrm{min}}{T}_{1}$ | 88.2 | 59.9 |

1st order (vertical) | ${f}_{N1}{T}_{1}$ | 7.44 | 7.8 |

0 order (horizontal) | N/A | N/A | −9 |

1st order (horizontal) | $\frac{{T}_{2}{f}_{\mathrm{min}}+{T}_{3}\Delta f}{\Delta f}{f}_{N2}$ | 632 | 564 |

2nd order (horizontal) | $-\frac{{T}_{2}}{\Delta f}{f}_{N2}^{2}$ | 4607 | 4595 |

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**MDPI and ACS Style**

Delsuc, M.-A.; Breuker, K.; van Agthoven, M.A.
Phase Correction for Absorption Mode Two-Dimensional Mass Spectrometry. *Molecules* **2021**, *26*, 3388.
https://doi.org/10.3390/molecules26113388

**AMA Style**

Delsuc M-A, Breuker K, van Agthoven MA.
Phase Correction for Absorption Mode Two-Dimensional Mass Spectrometry. *Molecules*. 2021; 26(11):3388.
https://doi.org/10.3390/molecules26113388

**Chicago/Turabian Style**

Delsuc, Marc-André, Kathrin Breuker, and Maria A. van Agthoven.
2021. "Phase Correction for Absorption Mode Two-Dimensional Mass Spectrometry" *Molecules* 26, no. 11: 3388.
https://doi.org/10.3390/molecules26113388