High-Resolution Intact Protein Analysis via Phase-Modulated, Stepwise Frequency Scan Ion Trap Mass Spectrometry

Mass spectrometry (MS) using an electron multiplier for intact protein analysis remains limited. Because of the massive size and complex structure of proteins, the slow flight speed of their ions results in few secondary electrons and thus low detection sensitivity and poor spectral resolution. Thus, we present a compact ion trap-mass spectrometry approach to directly detect ion packets and obtain the high-resolution molecular signature of proteins. The disturbances causing deviations of ion motion and mass conversion have been clarified in advance. The radio frequency waveform used to manipulate ions is proposed to be a sequence of constant-frequency steps, interconnected by short time-outs, resulting in least dispersive distortion. Furthermore, more such constant-phase conjunctions are arranged in each step to compensate for fluctuations resulting from defects in the system and operation. In addition, two auxiliary pulses are generated in the right phase of each step to select ions of a specific secular state to detect one clean and sharp spectral line.This study demonstrates a top-down approach for the MS measurement of cytochrome C molecules, resulting in a spectral profile of the protein in its natural state at a resolution of 20 Da. Additionally, quick MS scans of other proteins were performed.


Table of contents
The dynamic equations governing both the in-trap axial and radial motions, driven by a simple sinusoidal wave, are described by Mathieu's differential equation in the time domain.
This in-trap motion maintains a constant angular momentum around the trap axis of symmetry.
In the radial equation, a "centrifugal force" term is neglected for ions with zero initial azimuthal speed.In reality, the ion trajectory circles around an off-radial-center axis and never passes through the trap axis of symmetry (Figure S3A).For cases in which the mass spectrometry (MS) process involves varying the frequency of the driving RF waveform, similar to a "frequency scan," the stability diagram and its secular coordinates are expected to deform, as the dynamics in the time domain no longer fit Mathieu's equation.Consequently, stable MS measurements must be externally calibrated using reference molecules, with the assumption that the dynamics are adiabatically disturbed and do not considerably deviate from the ideal described by Mathieu's equation as below (Eq.S1).
Ideal Mathieu's equation in perfect electric field: where u represents the displacement of in-trap motion, 2 is the phase angle of RF-waveform, and a, q are respectively the DC and AC parameters.
The dynamics of in-trap ions explicitly depends on the phase of the driving RF waveform.Given a fixed AC amplitude and DC offset, with the phase as the independent variable, both the axial and radial equations of motion remain similar to Mathieu's equation in the phase domain.However, both equations include an additional "driving" term owing to a dispersive coupling between the "velocity" of the in-trap ion and the "frequency variation" of the RF waveform.Accordingly, the differential equation of in-trap motion is as below: Non-ideal Mathieu's equation revised with real perturbations: wherein the homogeneous part is sinuously driven in the phase domain, similarly formulated to Mathieu's equation in the time domain.The inhomogeneous part ( ̅ ) contains all the abovementioned non-ideal terms, owing to field defects, buffer gas collisions, ion-ion interactions, and dispersive coupling.
In the phase domain, the "frequency variation" of the RF waveform can be understood and investigated as a definite source of deviations from the stability diagram derived by Mathieu's equation.Other non-ideal but real perturbations caused by high-order electric fields, buffer gas collisions, and ion-ion interactions contribute significant errors and uncertainty to the MS measurement of in-trap ions.

Dynamics of In-Trap Motion: Constant-Phase Conjunction to Bridge Constant-Frequency Steps.
As analyzed above, the "frequency scan" in the phase domain introduces dispersive coupling, which tends to distinctly distort the stability diagram.For small molecules, the same MS approach of a slow, continuous frequency scan can be used with the aid of external calibration.However, for much heavier and more complex molecules, a stepwise frequency-scan approach is devised.In this MS sequence, the waveform frequency is set to a constant value corresponding to one specific m/z for each step.During such a "constant-frequency" step, the in-trap motion aligns with the stability diagram derived from Mathieu's equations.
Consequently, all in-trap ions can be accurately assessed for instability after a reasonable frequency jump to the next step.
A crucial design element for a reasonable frequency jump is using the constant-phase Below is to evaluate the influence on the in-trap oscillations, whenever such a CPC is applied.Following the integration of the differential equation, Eq.S2, infinitesimally near the phase to apply CPC, we have two implications: (A) To the first order, the ion's velocity disturbance introduced by one CPC is linearly proportional to the ion's instantaneous displacement.The dispersive contribution of such a CPC of the lowest order, is to the second order (Eq.S3, as below).(B) To the second order, the ion's displacement disturbance introduced by one CPC is much smaller than the displacement itself (Eq.S4, as below).
Below is the influence on the in-trap oscillations whenever such a CPC is applied.
where  represents the displacement (either axial or radial); 2  is the phase at which one CPC is applied; and  and  are respectively the DC and AC parameters in Mathieu's equation, which determine the coordinates of the stability diagram.In particular, 2() denotes the phase of the simple sinusoidal wave, synthesized as a function of time by the embedded system of the in-trap MALDI mass spectrometer.
Thus, when one CPC is used to bridge the waveform between MS steps, the in-trap motion is perturbed to exhibit a slight velocity variation that is proportional the instantaneous displacement, while the displacement variation remains almost zero.That is, CPCs perturb intrap ions, and preserve them inside the trap.

Dynamics of In-Trap Motion: Using CPCs to Achieve Secular Modulation.
In a prolonged MS scan, dissipation and fluctuations hinder the preparation of a "statespecific" packet of ions from the in-trap ion cloud.Nonetheless, the proportional linearity in property (A) allows for the realization of secular-phase modulation through additional CPCs, provided that the modulation periodicity is longer, far different from the periodicity of microoscillation.
According to the formula Eq.S3, the velocity variation of in-trap ions is trivial, whenever their displacement is right at zero as the CPC is applied.Otherwise, the velocity of in-trap ions will change a bit, as the CPC is applied.Therefore, when the secular motion of intrap ions is nearly coincident to the alternating periodicity of applying CPCs, their motion will be "modulated".And, in the long run, their secular displacement "zeros" can be gradually synchronous to each other as if those CPCs are trivially applied.As for those in-trap ions with secular motion far different from the alternating periodicity, no modulation will happen since CPCs are effectively applied at random.
Specifically, for the detection of ejected ions from the end-cap electrodes, the instantaneous axial displacement of in-trap ions would be gradually reduced to near-zero, periodically aligning with certain effective phases whenever alternating CPCs are applied secularly.Consequently, in-trap ions of the same m/z will eventually have their motion modulated to the same secular phase, allowing them to be synchronously ejected in the axial direction.
Given that additional CPCs are arranged in one step of 12 RF cycles (Figure 4D or 4E), in-trap ions of the same m/z will have their motion effectively modulated while scanning around the constant-frequency steps corresponding to β = 1/6, 1/3, 1/2, 2/3, and 1. Near these steps, the period of the driving RF waveform is nearly matched in integer multiples to the secular periodicity of in-trap motion.In contrast, for steps that do not coincide with these "integer" values, their modulation becomes almost random and thus trivial.
Furthermore, whenever the in-trap motion is effectively modulated, its state transitions into one of the secular degeneracies over the long term.The integer periodicity ratio of secular oscillation to micro-oscillation defines secular degeneracy, which is 2/β (Table S2).For example, with modulation at β = 2/3 (Figure S3), there are three possibilities for aligning the primary zeros of in-trap motion concerning the phase of applying an effective "CPC."Notably, two of the three primary zeros always exhibit opposite polarity trends compared with the third zero.
In the MS scan, once the motion of all in-trap ions has been effectively modulated, the secularphase synchronization of ion-trap motion provides a phase-sensitive routine for high-resolution resonance detection.Additionally, as illustrated in Figure S3, two auxiliary pulses at specific phases are employed to quickly trigger resonance ejection.The correct phase for resonance is closely related to the "effectiveness" of secular modulation.Figure S4 illustrates the results of phase probing.In this case, a sinusoidal wave is not used for the auxiliary resonance, as it is too slow and thus disrupts the secular modulation.

Figure S2.
Stepwise frequency changes for two mass regions.
Given the same DMZ, the frequency jump is significantly larger for the small mass range.The frequency conversion is detailed in Table S1.(2) The counting of steps nearby integer degeneracy is to estimate the effectiveness of secular modulation.The smaller is the DMZ, the more effective is the modulation.

Figure S3 .
Figure S3.Schematic of the motion of an in-trap ion and three possible phase alignments for the effectively modulated motion of in-trap ions.The trajectory of an in-trap ion, illustrated in the axial degree of freedom.The ion motion results from the combination of micro-oscillation and secular oscillation.In the case of β = 2/3, the in-trap motion is characterized by three primary zeros, with significant zero-crossings.(B) The diagram illustrates the phase alignment for the axial motion of multiple in-trap ions after the ions of the same m/z have been modulated into the secular state of β = 2/3.The ions are aligned with respect to the effective CPCs, as indicated on the RF waveform.This alignment presents three possibilities for the primary zeros of axial oscillation.Two auxiliary pulses, with the appropriate magnitude, are positioned at the correct phase to excite the secular singlet, rather than at the other two phases for the secular doublet.

Figure S4 .
Figure S4.MS measurement used to probe resonance ejection.Only at the correct phases can the two auxiliary HV pulses quickly trigger resonant ejection.The results demonstrate the secular characteristic of β = 2/3, with the three resonance phases sharply separated by 120°.If the phase is incorrect, no resonance occurs.However, the poor spectral line due to residual unstable ejection indicates that the modulation is also affected by the pulses.Further adjustments to their strength and polarity are needed to preserve the effectiveness of secular modulation.Figure 9 illustrates the optimized results, where the spectral lines of the singlet and doublet in the case of β = 2/3 are identified.

Dynamics of In-Trap Motion: Mathieu's Scheme in the Phase Domain.
For the equations, see (2.60) and (2.61) of "Quadrupole Mass Spectrometry and Its Applications," P.H. Dawson (ed.), Elsevier 1976.The stability diagram derived from two Mathieu's equations requires the driving RF waveform to have a fixed amplitude, frequency, and DC offset.The diagram location

Table S2 .
Number of steps near an integer degeneracy under different DMZs.Relationship between secular degeneracy, period and frequency of the in-trap motion.