Mass Spectrometry in Biophysics

Derivation: Magnetic Sector Mass Analyzer

An ion of mass $m$, charge $q = ze$ (where $z$ is the charge state and $e$ the elementary charge) is first accelerated through a potential difference $V$, gaining kinetic energy:

$$zeV = \frac{1}{2}mv^2 \implies v = \sqrt{\frac{2zeV}{m}}$$

The ion then enters a magnetic field $B$ perpendicular to its velocity. The Lorentz force provides centripetal acceleration:

$$zevB = \frac{mv^2}{r}$$

Solving for $r$: $r = mv/(zeB)$. Substituting the expression for $v$:

$$r = \frac{m}{zeB}\sqrt{\frac{2zeV}{m}} = \frac{1}{B}\sqrt{\frac{2mV}{ze}}$$

Solving for $m/z$:

$$\boxed{\frac{m}{z} = \frac{B^2 r^2 e}{2V}}$$

By scanning $B$ or $V$ while keeping $r$ fixed (at the detector position), different $m/z$ values are selected sequentially.

Derivation: Time-of-Flight (TOF) Mass Analyzer

In a TOF analyzer, all ions are accelerated through the same potential $V$ and then drift through a field-free region of length $L$. The drift time depends on the ion velocity, which in turn depends on $m/z$.

From $zeV = \frac{1}{2}mv^2$, the velocity is $v = \sqrt{2zeV/m}$. The flight time through the drift tube is:

$$t = \frac{L}{v} = L\sqrt{\frac{m}{2zeV}}$$

Solving for $m/z$:

$$\boxed{\frac{m}{z} = \frac{2eVt^2}{L^2}}$$

Mass resolution: The resolving power is defined as$R = m/\Delta m$, where $\Delta m$ is the smallest mass difference that can be distinguished. For TOF:

$$R = \frac{m}{\Delta m} = \frac{t}{2\Delta t}$$

where $\Delta t$ is the temporal peak width. Modern reflectron TOF instruments achieve $R > 50{,}000$ by correcting for initial kinetic energy spread. Orbitrap and FT-ICR analyzers achieve $R > 10^6$.

Derivation: The Rayleigh Limit

A charged droplet of radius $R$ carrying charge $q$ experiences two competing forces: surface tension $\gamma$ tends to minimize the surface area (keep the droplet spherical), while Coulomb repulsion tends to tear it apart.

The surface energy of a sphere is:

$$E_{\text{surface}} = 4\pi R^2 \gamma$$

The electrostatic self-energy of a uniformly charged sphere is:

$$E_{\text{Coulomb}} = \frac{q^2}{8\pi\varepsilon_0 R}$$

Rayleigh's stability analysis considers the energy change when the sphere deforms into a prolate spheroid. The droplet becomes unstable when the Coulomb energy equals twice the surface energy (for the $l = 2$ deformation mode). Setting$\partial^2 E / \partial \epsilon^2 = 0$ for the deformation parameter$\epsilon$ gives the Rayleigh limit:

$$\boxed{q_R = 8\pi\sqrt{\varepsilon_0 \gamma R^3}}$$

When the droplet charge exceeds $q_R$, Coulomb fission occurs: the droplet ejects small, highly charged offspring droplets containing approximately 2% of the parent mass but 15% of the charge. This process repeats as solvent evaporates, ultimately yielding bare, multiply charged analyte ions.

Charge state distribution: For proteins under denaturing conditions, the average charge state follows an empirical relationship with molecular weight:

$$z_{\text{avg}} \approx 0.0778 \sqrt{MW}$$

A 50 kDa protein thus acquires approximately $0.0778 \sqrt{50000} \approx 17$charges. Under native conditions, proteins acquire fewer charges due to their compact structure, with the maximum charge limited by the solvent-accessible surface area.

Derivation: Collision-Induced Dissociation Energy

In collision-induced dissociation (CID), a precursor ion collides with a neutral gas (N$_2$, Ar, or He). The maximum energy available for internal excitation in a single collision is limited by the center-of-mass collision energy:

$$E_{\text{cm}} = E_{\text{lab}} \cdot \frac{m_{\text{gas}}}{m_{\text{ion}} + m_{\text{gas}}}$$

where $E_{\text{lab}}$ is the laboratory-frame kinetic energy of the ion. For a peptide of mass 1000 Da colliding with N$_2$ (28 Da) at$E_{\text{lab}} = 30$ eV:

$$E_{\text{cm}} = 30 \times \frac{28}{1000 + 28} = 0.82 \text{ eV}$$

Multiple collisions deposit enough internal energy to break covalent bonds. For peptides, the weakest bonds along the backbone are the amide bonds, leading to characteristic fragment ion series.

Peptide fragmentation nomenclature: Along the peptide backbone, cleavage of the amide bond C(O)–NH generates:

• b ions: N-terminal fragments retaining the charge on the N-terminal side. $b_n$ contains residues 1 through $n$.
• y ions: C-terminal fragments retaining the charge on the C-terminal side. $y_n$ contains the last $n$ residues.

For a peptide of $n$ residues, $b_i + y_{n-i} = M + H$ (where$M$ is the neutral peptide mass). The mass differences between consecutive$b$ or $y$ ions correspond to amino acid residue masses:

$$b_{i+1} - b_i = \text{residue mass of amino acid } (i+1)$$

$$y_{i+1} - y_i = \text{residue mass of amino acid from C-terminus}$$

By reading the mass ladder, the peptide sequence can be determined de novo or matched against a protein database. The mass accuracy of modern instruments ($< 5$ ppm) enables unambiguous identification of most amino acids (except the isobaric pair Leu/Ile at 113.084 Da).

Detailed Example: De Novo Sequencing from an MS/MS Spectrum

Consider a peptide with [M+2H]$^{2+}$ at $m/z = 530.28$, giving$M = 2 \times 530.28 - 2 \times 1.008 = 1058.54$ Da. The MS/MS spectrum shows the following dominant y-ion series (singly charged):

y$_1$ = 175.12, y$_2$ = 304.16, y$_3$ = 417.24, y$_4$ = 530.33, y$_5$ = 627.38, y$_6$ = 740.47, y$_7$ = 869.51, y$_8$ = 940.55

The mass differences between consecutive y ions reveal the sequence (reading from C-terminus):

• y$_2$ - y$_1$ = 129.04 $\to$ Glu (E, 129.043)
• y$_3$ - y$_2$ = 113.08 $\to$ Leu/Ile (L/I, 113.084)
• y$_4$ - y$_3$ = 113.09 $\to$ Leu/Ile (L/I, 113.084)
• y$_5$ - y$_4$ = 97.05 $\to$ Pro (P, 97.053)
• y$_6$ - y$_5$ = 113.09 $\to$ Leu/Ile (L/I, 113.084)
• y$_7$ - y$_6$ = 129.04 $\to$ Glu (E, 129.043)
• y$_8$ - y$_7$ = 71.04 $\to$ Ala (A, 71.037)

Reading from C-terminus: R-E-L-L-P-L-E-A-... The remaining mass at the N-terminus ($M + 1.008 - y_8 = 1059.55 - 940.55 = 119.00$) corresponds to Thr (T, 101.048) plus the N-terminal H (1.008) and additional mass. Database searching confirms the peptide.

Mass accuracy requirements: At 5 ppm mass accuracy on a 1000 Da peptide, the mass tolerance is 0.005 Da. Since the closest amino acid pair (Gln/Lys) differs by 0.036 Da, this accuracy is sufficient for unambiguous identification of all residues except Leu/Ile (identical mass to 4 decimal places, distinguishable only by specialized fragmentation methods such as ECD or ETD).

Mass Analyzer Comparison

Different mass analyzers offer distinct trade-offs between resolution, mass range, speed, and sensitivity:

• Quadrupole: Four parallel rods with oscillating RF/DC fields act as a mass filter. Resolution $R \sim 1{,}000\text{--}4{,}000$. Fast scanning, robust, widely used for quantitation (SRM/MRM mode).
• Time-of-flight (TOF): Resolution$R \sim 20{,}000\text{--}60{,}000$ with reflectron. Theoretically unlimited mass range. Very fast (all ions detected simultaneously). Ideal for MALDI and intact protein analysis.
• Orbitrap: Ions orbit around a central spindle electrode. Frequency of axial oscillation: $\omega_z = \sqrt{(z/m) \cdot k}$ where$k$ is a field constant. Resolution $R \sim 100{,}000\text{--}1{,}000{,}000$. Image current detection enables exquisite mass accuracy ($< 1$ ppm).
• FT-ICR: Ions orbit in a magnetic field at the cyclotron frequency $\omega_c = zeB/m$. Highest resolution ($R > 10^7$) and mass accuracy ($< 0.1$ ppm). Requires superconducting magnets (7–15 T).

Derivation: HDX Exchange Kinetics

The exchange of a backbone amide hydrogen is catalyzed by both acid and base:

$$k_{\text{int}} = k_A[\text{H}^+] + k_B[\text{OH}^-] + k_W$$

where $k_{\text{int}}$ is the intrinsic (unprotected) exchange rate. At physiological pH ($\sim 7$), base catalysis dominates, giving rates of$\sim 1\text{--}100$ s$^{-1}$ for unstructured peptides.

In a folded protein, an amide hydrogen must undergo a local unfolding event before exchange can occur. In the EX2 regime (dominant under most conditions):

$$\text{NH}_{\text{closed}} \underset{k_{cl}}{\overset{k_{op}}{\rightleftharpoons}} \text{NH}_{\text{open}} \xrightarrow{k_{\text{int}}} \text{ND}$$

When $k_{cl} \gg k_{\text{int}}$ (EX2 limit), the opening/closing equilibrium is established before exchange occurs, and the observed exchange rate is:

$$k_{\text{ex}} = k_{\text{int}} \cdot \frac{K_{\text{op}}}{K_{\text{op}} + 1} \approx k_{\text{int}} \cdot K_{\text{op}}$$

where $K_{\text{op}} = k_{\text{op}} / k_{\text{cl}} \ll 1$ for well-folded regions. The protection factor quantifies the slowing of exchange:

$$\boxed{P = \frac{k_{\text{int}}}{k_{\text{ex}}} = \frac{1}{K_{\text{op}}} = \frac{k_{\text{cl}}}{k_{\text{op}}}}$$

The protection factor is related to the free energy of local unfolding:

$$\Delta G_{\text{op}} = -RT\ln K_{\text{op}} = RT\ln P$$

Protection factors range from $\sim 1$ (fully exposed loops) to$> 10^8$ (buried core residues), corresponding to local stability differences of 0 to $> 11$ kcal/mol.

HDX-MS workflow: Protein in H$_2$O is diluted into D$_2$O, incubated for various times (10 s to hours), quenched at pH 2.5/0°C (slowing exchange $\sim 10^6$-fold), digested with pepsin, and peptides analyzed by LC-MS. The mass increase of each peptide ($+1.006$ Da per deuterium) reports on exchange kinetics in that region.

Derivation: Charge State and Compactness

Under native conditions (aqueous, neutral pH, ammonium acetate buffer), proteins retain their folded conformation during electrospray. The maximum charge a protein can acquire is limited by the solvent-accessible surface area (SASA). For a roughly spherical protein of radius $R$, the Rayleigh charge limit is:

$$z_{\max} \propto R^{3/2} \propto MW^{1/2}$$

Under denaturing conditions, the unfolded protein exposes more surface area, acquiring more charges. The charge state ratio between native and denatured forms reports on structural compactness:

$$\frac{z_{\text{native}}}{z_{\text{denatured}}} \approx \left(\frac{R_{\text{native}}}{R_{\text{denatured}}}\right)^{3/2} \approx \left(\frac{R_g^{\text{native}}}{R_g^{\text{denatured}}}\right)^{3/2}$$

Typically, $z_{\text{native}} / z_{\text{denatured}} \approx 0.5\text{--}0.7$.

Derivation: Collision Cross-Section from Ion Mobility

In ion mobility spectrometry, ions drift through a buffer gas (typically N$_2$ or He) under a weak electric field $E$. The drift velocity $v_d$ is proportional to $E$:

$$v_d = KE$$

where $K$ is the ion mobility. The Mason-Schamp equation relates $K$to the rotationally averaged collision cross-section $\Omega$:

$$\boxed{\Omega = \frac{3ze}{16N}\sqrt{\frac{2\pi}{\mu k_B T}} \cdot \frac{1}{K}}$$

where $N$ is the buffer gas number density, $\mu = m_{\text{ion}}m_{\text{gas}}/(m_{\text{ion}} + m_{\text{gas}})$is the reduced mass, and $k_B T$ is the thermal energy. The collision cross-section$\Omega$ provides a measure of the ion's projected area.

For globular proteins, $\Omega$ scales with molecular weight approximately as:

$$\Omega \propto MW^{2/3}$$

consistent with a sphere whose volume is proportional to mass. Deviations from this scaling indicate non-spherical shapes or conformational changes. Unfolded proteins show$\Omega$ values 50–100% larger than their native forms.

Derivation: Mason-Schamp Equation from Kinetic Theory

The Mason-Schamp equation can be derived from the kinetic theory of ion transport. An ion drifting through a buffer gas at low field experiences a balance between the electric force and the frictional drag from collisions:

$$zeE = \frac{m_{\text{ion}}v_d}{\tau}$$

where $\tau$ is the mean time between collisions. From the kinetic theory of gases, the collision frequency is $1/\tau = N\Omega\bar{v}$, where $\bar{v}$ is the mean relative velocity. For a Maxwell-Boltzmann distribution of relative velocities:

$$\bar{v} = \sqrt{\frac{8k_BT}{\pi\mu}}$$

Substituting and applying correction factors from rigorous kinetic theory (the first-order Chapman-Enskog approximation introduces a factor of $3\pi/16$):

$$K = \frac{v_d}{E} = \frac{3ze}{16N\Omega}\sqrt{\frac{2\pi}{\mu k_BT}}$$

Rearranging for $\Omega$ gives the Mason-Schamp equation. The key physical insight is that larger ions (larger $\Omega$) experience more drag and drift more slowly, arriving at the detector later. This separates ions by shape and size in addition to$m/z$, adding an orthogonal dimension of structural information.

Typical CCS values for proteins: Ubiquitin (8.6 kDa):$\Omega \approx 1{,}000$ Å$^2$; cytochrome c (12.4 kDa):$\sim 1{,}500$ Å$^2$; BSA (66.4 kDa):$\sim 5{,}000$ Å$^2$; GroEL (800 kDa):$\sim 25{,}000$ Å$^2$. These values agree well with CCS computed from crystal structures using trajectory or projection methods.

Native MS Applications in Structural Biology

Stoichiometry determination: Native MS directly measures the mass of intact complexes, revealing subunit stoichiometry. For example, native MS showed that the 20S proteasome is a 28-subunit complex ($\alpha_7\beta_7\beta_7\alpha_7$, ~700 kDa), and that the GroEL-GroES complex consists of GroEL$_{14}$-GroES$_7$ (~870 kDa).

Binding affinity measurement: By quantifying the relative intensities of free and bound species at different ligand concentrations, native MS can determine dissociation constants $K_d$ for protein-ligand interactions. The ratio of bound to free protein is:

$$\frac{[\text{PL}]}{[\text{P}]} = \frac{[\text{L}]}{K_d}$$

Native MS is particularly powerful for studying heterogeneous mixtures where multiple binding partners or states coexist, as each species appears at a distinct $m/z$. It has been used to study membrane protein complexes (solubilized in detergent micelles or lipid nanodiscs), viral capsids ($> 10$ MDa), and the assembly pathways of large macromolecular machines.