NMR Spectroscopy in Biophysics

Derivation: Larmor Frequency and Energy Splitting

The Zeeman Hamiltonian for a spin in a magnetic field is:

$$\hat{H} = -\boldsymbol{\mu} \cdot \mathbf{B}_0 = -\gamma \hbar \hat{I}_z B_0$$

For a spin-1/2 nucleus ($I = 1/2$), the allowed magnetic quantum numbers are$m_I = +1/2$ ($\alpha$ state) and $m_I = -1/2$($\beta$ state). The eigenvalues of $\hat{I}_z$ are$m_I \hbar$, so the energy levels are:

$$E_{m_I} = -\gamma \hbar m_I B_0$$

For the two states:

$$E_\alpha = -\frac{1}{2}\gamma \hbar B_0, \quad E_\beta = +\frac{1}{2}\gamma \hbar B_0$$

(assuming $\gamma > 0$, as for $^1$H). The energy splitting is:

$$\boxed{\Delta E = E_\beta - E_\alpha = \gamma \hbar B_0 = \hbar \omega_0}$$

where we identify the Larmor frequency:

$$\boxed{\omega_0 = \gamma B_0}$$

For $^1$H with $\gamma/2\pi = 42.576$ MHz/T and$B_0 = 14.1$ T (a 600 MHz spectrometer):

$$\nu_0 = \frac{\omega_0}{2\pi} = 42.576 \times 14.1 = 600.3 \text{ MHz}$$

The Boltzmann population difference at thermal equilibrium determines the net magnetization and thus the signal strength. At temperature $T$:

$$\frac{N_\alpha}{N_\beta} = \exp\left(\frac{\Delta E}{k_B T}\right) = \exp\left(\frac{\hbar \gamma B_0}{k_B T}\right)$$

Since $\hbar \gamma B_0 \ll k_B T$ at room temperature (for $^1$H at 14.1 T, $\Delta E / k_B T \approx 10^{-4}$), we expand the exponential:

$$\frac{N_\alpha - N_\beta}{N} \approx \frac{\hbar \gamma B_0}{2 k_B T}$$

This tiny population excess ($\sim 1$ in $10^5$) is what makes NMR inherently insensitive compared to optical spectroscopies, but also what makes it exquisitely sensitive to local electronic environments.

Classical Picture: Precession

Classically, the torque on the magnetic moment is $\boldsymbol{\tau} = \boldsymbol{\mu} \times \mathbf{B}_0$. Since $\boldsymbol{\tau} = d\mathbf{J}/dt = d(\hbar \mathbf{I})/dt$ and$\boldsymbol{\mu} = \gamma \hbar \mathbf{I}$:

$$\frac{d\boldsymbol{\mu}}{dt} = \gamma \boldsymbol{\mu} \times \mathbf{B}_0$$

This describes precession of $\boldsymbol{\mu}$ about $\mathbf{B}_0$at the Larmor frequency $\omega_0 = \gamma B_0$. The bulk magnetization$\mathbf{M} = \sum_i \boldsymbol{\mu}_i$ obeys the same equation, forming the basis for the Bloch equations.

Derivation: The Bloch Equations

Starting from the equation of motion for the magnetization in a field $\mathbf{B}$:

$$\frac{d\mathbf{M}}{dt} = \gamma \mathbf{M} \times \mathbf{B}$$

Bloch added relaxation terms to account for the return to equilibrium. The longitudinal magnetization $M_z$ recovers toward the equilibrium value $M_0$with time constant $T_1$ (spin-lattice relaxation), while the transverse components $M_x, M_y$ decay to zero with time constant $T_2$(spin-spin relaxation):

$$\frac{dM_x}{dt} = \gamma(\mathbf{M} \times \mathbf{B})_x - \frac{M_x}{T_2}$$

$$\frac{dM_y}{dt} = \gamma(\mathbf{M} \times \mathbf{B})_y - \frac{M_y}{T_2}$$

$$\frac{dM_z}{dt} = \gamma(\mathbf{M} \times \mathbf{B})_z - \frac{M_z - M_0}{T_1}$$

$T_1$ relaxation (spin-lattice): Energy exchange between nuclear spins and the surrounding "lattice" (molecular tumbling, vibrations). After a perturbation (e.g., 180° pulse), $M_z$ recovers exponentially:

$$M_z(t) = M_0\left(1 - 2e^{-t/T_1}\right)$$

$T_2$ relaxation (spin-spin): Loss of phase coherence among precessing spins due to local field inhomogeneities. After a 90° pulse creating transverse magnetization:

$$M_{xy}(t) = M_0 e^{-t/T_2}$$

The relationship $T_2 \leq T_1$ always holds. For small molecules in solution,$T_1 \approx T_2 \sim 1\text{--}10$ s, while for large proteins,$T_2$ can be much shorter ($\sim 10\text{--}100$ ms) than $T_1$.

Derivation: Free Induction Decay and Fourier Transform

After a 90° pulse tips $M_z$ into the transverse plane, the precessing magnetization induces a signal in the receiver coil. In the rotating frame at the carrier frequency $\omega_{rf}$, the complex transverse magnetization for a single resonance at offset $\Omega = \omega_0 - \omega_{rf}$ is:

$$M^+(t) = M_x(t) + iM_y(t) = M_0 e^{i\Omega t} e^{-t/T_2}, \quad t \geq 0$$

This is the free induction decay (FID). The NMR spectrum is obtained by Fourier transformation:

$$S(\omega) = \int_0^\infty M^+(t) e^{-i\omega t}\, dt = \int_0^\infty M_0 e^{i(\Omega - \omega)t} e^{-t/T_2}\, dt$$

Evaluating the integral:

$$S(\omega) = M_0 \left[\frac{-1}{i(\Omega - \omega) - 1/T_2}\right]_0^\infty = \frac{M_0}{1/T_2 - i(\Omega - \omega)}$$

Separating real and imaginary parts:

$$\text{Re}[S(\omega)] = \frac{M_0 / T_2}{(1/T_2)^2 + (\Omega - \omega)^2} \quad \text{(absorption Lorentzian)}$$

$$\text{Im}[S(\omega)] = \frac{M_0 (\Omega - \omega)}{(1/T_2)^2 + (\Omega - \omega)^2} \quad \text{(dispersion Lorentzian)}$$

The absorption lineshape is a Lorentzian centered at $\omega = \Omega$ with full width at half maximum $\Delta\omega_{1/2} = 2/T_2$, or in Hz,$\Delta\nu_{1/2} = 1/(\pi T_2)$. Longer $T_2$ gives narrower lines.

Derivation: Chemical Shift from Electron Shielding

The local field experienced by a nucleus is modified by the shielding tensor $\boldsymbol{\sigma}$:

$$\mathbf{B}_{\text{local}} = (1 - \sigma)\mathbf{B}_0$$

where $\sigma$ is the isotropic shielding constant (typically $10^{-6}$ to$10^{-4}$ for $^1$H). The resonance frequency becomes:

$$\boxed{\omega = \gamma B_0(1 - \sigma)}$$

Rather than reporting absolute frequencies, chemists use the chemical shift$\delta$ relative to a reference compound (TMS for $^1$H and $^{13}$C):

$$\delta = \frac{\omega - \omega_{\text{ref}}}{\omega_{\text{ref}}} \times 10^6 = \frac{\sigma_{\text{ref}} - \sigma}{1 - \sigma_{\text{ref}}} \times 10^6 \approx (\sigma_{\text{ref}} - \sigma) \times 10^6 \text{ (ppm)}$$

Sources of shielding in proteins:

• Diamagnetic shielding: Circulation of electrons in the ground state around the nucleus, proportional to electron density
• Paramagnetic shielding: Involves excited electronic states; dominant for $^{13}$C and $^{15}$N
• Ring current effects: The delocalized $\pi$electrons in aromatic rings (Phe, Tyr, Trp, His) create a ring current that generates an induced magnetic field. Nuclei above/below the ring are shielded (upfield shift), while those in the plane are deshielded (downfield shift). The ring current shift is:$$\Delta\delta \propto \frac{1 - 3\cos^2\theta}{r^3}$$where $r$ and $\theta$ are the distance and angle from the ring center
• Hydrogen bonding: Deshields the bonded proton; the amide$^1$H chemical shift correlates with H-bond length in proteins

Typical protein chemical shift ranges: Backbone amide $^1$H: 6–10 ppm; $^1$H$\alpha$: 3–5 ppm; methyl $^1$H: 0–1 ppm; $^{13}$C$\alpha$: 45–65 ppm;$^{15}$N amide: 100–135 ppm.

Multidimensional NMR Pulse Sequences

COSY (COrrelation SpectroscopY): Detects through-bond (J-coupling) correlations between nuclei. A simple two-pulse sequence (90°–$t_1$–90°–$t_2$) produces cross-peaks connecting J-coupled nuclei. The cross-peak intensity depends on$\sin(\pi J t_1)$, where $J$ is the coupling constant.

NOESY (Nuclear Overhauser Effect SpectroscopY): Detects through-space correlations via dipolar cross-relaxation. The mixing time $\tau_m$allows magnetization transfer between nuclei close in space ($< 5$ Å). Cross-peak intensity is proportional to $1/r^6$.

HSQC (Heteronuclear Single-Quantum Coherence): Correlates$^1$H with a directly bonded heteronucleus ($^{15}$N or $^{13}$C). The $^{15}$N-HSQC spectrum serves as the "fingerprint" of a protein, with one peak per backbone amide. For a well-folded protein of $N$ residues, expect approximately $N - 1$ peaks (minus prolines, which lack amide NH).

Triple-resonance experiments: For backbone assignment, experiments like HNCA, HN(CO)CA, HNCO, CBCA(CO)NH correlate amide $^1$H-$^{15}$N with $^{13}$C$\alpha$, $^{13}$C$\beta$, and$^{13}$CO of the same and preceding residues, enabling sequential assignment.

Derivation: NOE and the Solomon Equations

Consider a two-spin system (spins I and S) with dipolar coupling. The Solomon equations describe the time evolution of the longitudinal magnetizations:

$$\frac{dI_z}{dt} = -\rho_I (I_z - I_0) - \sigma_{IS}(S_z - S_0)$$

$$\frac{dS_z}{dt} = -\sigma_{IS}(I_z - I_0) - \rho_S (S_z - S_0)$$

where $\rho_I, \rho_S$ are the auto-relaxation rates and $\sigma_{IS}$ is the cross-relaxation rate. The dipolar relaxation rates depend on the internuclear distance$r$ and the rotational correlation time $\tau_c$:

$$\sigma_{IS} = \frac{1}{10}\left(\frac{\mu_0}{4\pi}\right)^2 \frac{\gamma_I^2 \gamma_S^2 \hbar^2}{r^6} \left[6J(2\omega_0) - J(0)\right]$$

where the spectral density function for isotropic tumbling is:

$$J(\omega) = \frac{2}{5} \frac{\tau_c}{1 + \omega^2 \tau_c^2}$$

The steady-state NOE enhancement when spin S is saturated is:

$$\eta = \frac{I_z - I_0}{I_0} = \frac{\sigma_{IS}}{\rho_I} \cdot \frac{\gamma_S}{\gamma_I}$$

The critical distance dependence:

$$\boxed{\sigma_{IS} \propto \frac{1}{r^6}}$$

This steep distance dependence means that NOE is essentially a short-range effect: observable only for proton pairs within $\sim 5$ Å. The NOE changes sign depending on molecular size: positive for small molecules ($\omega_0\tau_c \ll 1$), zero at$\omega_0\tau_c \approx 1.12$, and negative for large molecules ($\omega_0\tau_c \gg 1$).

Initial rate approximation: For short mixing times$\tau_m$, the NOE buildup is linear:

$$\text{NOE}(I \leftarrow S) \approx \sigma_{IS} \tau_m \propto \frac{\tau_m}{r_{IS}^6}$$

Structure determination: In a NOESY spectrum, cross-peak volumes are proportional to $1/r^6$. By calibrating against a known distance (e.g., geminal CH$_2$ protons at 1.78 Å), distances are extracted:$r_{ij} = r_{\text{ref}}(V_{\text{ref}}/V_{ij})^{1/6}$. These serve as upper bound distance restraints for structure calculation by simulated annealing or distance geometry.

Derivation: Lipari-Szabo Model-Free Analysis

The relaxation of a $^{15}$N nucleus is dominated by the dipolar interaction with the directly bonded $^1$H and the $^{15}$N chemical shift anisotropy (CSA). The relaxation rates are expressed in terms of spectral density functions:

$$\frac{1}{T_1} = \frac{d^2}{4}\left[J(\omega_H - \omega_N) + 3J(\omega_N) + 6J(\omega_H + \omega_N)\right] + c^2 J(\omega_N)$$

$$\frac{1}{T_2} = \frac{d^2}{8}\left[4J(0) + J(\omega_H - \omega_N) + 3J(\omega_N) + 6J(\omega_H) + 6J(\omega_H + \omega_N)\right] + \frac{c^2}{6}\left[4J(0) + 3J(\omega_N)\right]$$

$$\text{NOE} = 1 + \frac{d^2}{4T_1}\frac{\gamma_H}{\gamma_N}\left[6J(\omega_H + \omega_N) - J(\omega_H - \omega_N)\right]$$

where $d = (\mu_0/4\pi)\gamma_H\gamma_N\hbar/r_{NH}^3$ (dipolar coupling constant) and $c = \omega_N \Delta\sigma / \sqrt{3}$ (CSA contribution).

The Lipari-Szabo model-free spectral density: Rather than assuming a specific motional model, Lipari and Szabo (1982) showed that the spectral density can be approximated as:

$$\boxed{J(\omega) = \frac{2}{5}\left[\frac{S^2 \tau_m}{1 + \omega^2\tau_m^2} + \frac{(1-S^2)\tau}{1 + \omega^2\tau^2}\right]}$$

where $1/\tau = 1/\tau_m + 1/\tau_e$, and:

• $\tau_m$ is the overall rotational correlation time of the molecule
• $S^2$ is the generalized order parameter($0 \leq S^2 \leq 1$), measuring the spatial restriction of the internal motion.$S^2 = 1$ means completely rigid; $S^2 = 0$ means unrestricted motion
• $\tau_e$ is the effective internal correlation time for fast motions

Physical interpretation of $S^2$: For a bond vector confined to a cone of semi-angle $\theta$:

$$S^2 = \left[\frac{\cos\theta(1 + \cos\theta)}{2}\right]^2$$

Typical values: backbone amide N-H in structured regions: $S^2 \approx 0.85\text{--}0.90$; in flexible loops: $S^2 \approx 0.5\text{--}0.7$; in disordered termini:$S^2 < 0.4$.

Extracting $\tau_m$: The overall tumbling time can be estimated from the $T_1/T_2$ ratio. For backbone amides in structured regions (where $S^2$ is similar), $T_1/T_2$ is approximately constant and depends primarily on $\tau_m$:

$$\frac{T_1}{T_2} \approx \frac{6J(\omega_N)}{4J(0) + 3J(\omega_N)} \xrightarrow{\omega_N\tau_m \gg 1} \frac{3}{2}\omega_N^2\tau_m^2$$

The Stokes-Einstein relation predicts $\tau_m = 4\pi\eta r_H^3 / (3k_BT)$, giving approximately $\tau_m \approx 0.5$ ns per kDa for globular proteins at 25°C in water.