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Experimental Techniques

The experimental methods that reveal molecular structure at atomic resolution. From X-ray crystallography and cryo-EM to NMR spectroscopy and mass spectrometry — the tools that make structural biology possible.

1. X-ray Crystallography

X-ray crystallography remains the gold standard for atomic-resolution structure determination. When X-rays scatter from a crystal lattice, constructive interference produces a diffraction pattern governed by Bragg's law:

$$n\lambda = 2d\sin\theta$$

where $n$ is the diffraction order, $\lambda$ the X-ray wavelength,$d$ the inter-plane spacing, and $\theta$ the Bragg angle. The diffraction pattern encodes the structure through the structure factor:

$$F(\mathbf{h}) = \sum_{j=1}^{N} f_j \exp(2\pi i \, \mathbf{h} \cdot \mathbf{r}_j)$$

The electron density is recovered by inverse Fourier transform, but phases are lost in the measurement (the "phase problem"):

$$\rho(\mathbf{r}) = \frac{1}{V}\sum_{\mathbf{h}} |F(\mathbf{h})| \, e^{i\phi(\mathbf{h})} \, e^{-2\pi i \, \mathbf{h} \cdot \mathbf{r}}$$

Molecular Replacement

Uses known homologous structure as phasing model via Patterson function rotation/translation search

MAD/SAD Phasing

Anomalous scattering from heavy atoms (Se-Met) provides phase information via Bijvoet differences

Isomorphous Replacement

Heavy atom derivatives (MIR/SIR) introduce measurable intensity changes for phasing

2. Cryo-Electron Microscopy

The "resolution revolution" in cryo-EM now routinely achieves near-atomic resolution without crystallization. Samples are flash-frozen in vitreous ice and imaged by transmission electron microscopy. The image contrast is described by the Contrast Transfer Function (CTF):

$$\text{CTF}(s) = -\sin\left(\frac{\pi C_s \lambda^3 s^4}{2} + \pi \Delta f \lambda s^2\right)$$

where $C_s$ is the spherical aberration coefficient, $\lambda$ the electron wavelength, $s$ the spatial frequency, and $\Delta f$ the defocus. Single-particle reconstruction involves:

Image Processing Pipeline

1. Motion correction (beam-induced movement)
2. CTF estimation and correction
3. Particle picking (template/neural network)
4. 2D classification (remove junk)
5. Ab initio 3D reconstruction
6. 3D refinement and sharpening

Resolution Metrics

Resolution is assessed by the Fourier Shell Correlation (FSC) between two independent half-maps:

$$\text{FSC}(s) = \frac{\sum F_1(s) \cdot F_2^*(s)}{\sqrt{\sum|F_1(s)|^2 \cdot \sum|F_2(s)|^2}}$$

Resolution defined at FSC = 0.143 (gold-standard)

3. NMR Spectroscopy

Nuclear Magnetic Resonance provides atomic-level structural and dynamic information for proteins in solution. The resonance frequency of a nucleus depends on its chemical environment through the chemical shift:

$$\delta = \frac{\nu_{\text{sample}} - \nu_{\text{ref}}}{\nu_{\text{ref}}} \times 10^6 \text{ (ppm)}$$

Chemical Shift

Reports on local electronic environment. Secondary structure affects backbone shifts (CSI method).

NOE (Nuclear Overhauser Effect)

Distance restraints between protons within ~5 Angstroms. The NOE intensity scales as:

$$I_{\text{NOE}} \propto r^{-6}$$

Relaxation

T1 (spin-lattice) and T2 (spin-spin) relaxation report on molecular dynamics from ps to ms timescales.

$$M_z(t) = M_0(1 - e^{-t/T_1})$$

4. Mass Spectrometry

Mass spectrometry measures the mass-to-charge ratio (m/z) of ionized molecules with extraordinary precision. In proteomics, it identifies proteins, maps post-translational modifications, and quantifies expression levels.

$$\frac{m}{z} = \frac{M + nH^+}{n}$$

where M is the molecular mass, n is the number of charges, and H+ is the proton mass. Time-of-flight (TOF) analyzers separate ions by their flight time:

$$t = L\sqrt{\frac{m}{2zV}}$$

where L is the flight path length and V is the accelerating voltage. Native mass spectrometry preserves non-covalent interactions, enabling analysis of intact protein complexes.

5. Fluorescence Spectroscopy

Fluorescence techniques probe molecular structure, dynamics, and interactions with extreme sensitivity — down to single molecules. The Stokes shift between absorption and emission reflects vibrational relaxation in the excited state.

FRET (Single-Molecule)

smFRET measures donor-acceptor distances in individual molecules, revealing conformational heterogeneity hidden in ensemble measurements:

$$E = \frac{1}{1 + (R/R_0)^6} = \frac{I_A}{I_A + \gamma I_D}$$

Fluorescence Lifetime

Time-resolved fluorescence decays report on the local environment through the fluorescence lifetime:

$$I(t) = I_0 \exp(-t/\tau)$$

Typical lifetimes: 1-10 ns for organic fluorophores

Python Simulation: X-ray Diffraction Pattern

This simulation calculates the X-ray diffraction pattern from a 2D crystal lattice with a two-atom basis. It computes structure factors, d-spacings, Bragg angles, and reconstructs the electron density via Fourier synthesis.

X-ray Diffraction Pattern Simulation

Python

Structure factor calculation, Bragg angles, and electron density reconstruction for a 2D crystal

script.py118 lines

import numpy as np

# X-ray Diffraction Pattern Simulation for a 2D Crystal Lattice
# ==============================================================

print("=" * 65)
print("X-RAY DIFFRACTION PATTERN FROM A 2D CRYSTAL LATTICE")
print("=" * 65)

# Lattice parameters
a = 5.0   # lattice constant a (Angstroms)
b = 5.0   # lattice constant b (Angstroms)
gamma = 90.0  # angle between a and b (degrees)
gamma_rad = np.radians(gamma)

print(f"\nLattice: a = {a} A, b = {b} A, gamma = {gamma} degrees")

# Basis atoms (fractional coordinates)
# Simple two-atom basis (like NaCl in 2D)
basis = [
    (0.0, 0.0, 1.0),   # atom 1 at origin, scattering factor f1
    (0.5, 0.5, 0.8),   # atom 2 at center, scattering factor f2
]

print(f"Basis: {len(basis)} atoms per unit cell")
for i, (fx, fy, f) in enumerate(basis):
    print(f"  Atom {i+1}: ({fx}, {fy}), f = {f}")

# Calculate structure factor F(hk)
# F(hk) = sum_j f_j * exp(2*pi*i*(h*x_j + k*y_j))

def structure_factor(h, k, basis_atoms):
    """Calculate structure factor for Miller indices (h, k)."""
    F = 0.0 + 0.0j
    for (xj, yj, fj) in basis_atoms:
        phase = 2.0 * np.pi * (h * xj + k * yj)
        F += fj * np.exp(1j * phase)
    return F

# Calculate d-spacing and Bragg angles
wavelength = 1.5406  # Cu K-alpha (Angstroms)
print(f"\nX-ray wavelength: {wavelength} A (Cu K-alpha)")

print("\n--- Structure Factors and Diffraction Data ---")
print(f"{'(h,k)':>8} {'|F(hk)|^2':>12} {'d-spacing(A)':>14} {'2theta(deg)':>14} {'Intensity':>10}")
print("-" * 65)

reflections = []
for h in range(-4, 5):
    for k in range(-4, 5):
        if h == 0 and k == 0:
            continue

F = structure_factor(h, k, basis)
        intensity = abs(F)**2

if intensity < 0.01:
            continue

# d-spacing for 2D rectangular lattice
        d_inv_sq = (h/a)**2 + (k/b)**2
        if d_inv_sq == 0:
            continue
        d = 1.0 / np.sqrt(d_inv_sq)

# Bragg's law: n*lambda = 2*d*sin(theta)
        sin_theta = wavelength / (2.0 * d)
        if abs(sin_theta) > 1.0:
            continue

two_theta = 2.0 * np.degrees(np.arcsin(sin_theta))

if h >= 0 and (h > 0 or k > 0):  # avoid duplicates
            reflections.append((h, k, intensity, d, two_theta))

# Sort by 2theta
reflections.sort(key=lambda x: x[4])

max_I = max(r[2] for r in reflections) if reflections else 1
for h, k, I, d, tt in reflections:
    rel_I = 100 * I / max_I
    bar = '#' * int(rel_I / 4)
    print(f"  ({h},{k}){' ' * (5 - len(f'({h},{k})'))} {I:12.4f} {d:14.4f} {tt:14.4f} {rel_I:8.1f}% {bar}")

# Electron density calculation (1D projection)
print("\n--- Electron Density (1D projection along x) ---")
n_points = 40
x_vals = np.linspace(0, a, n_points)
rho = np.zeros(n_points)

for i, x in enumerate(x_vals):
    for h in range(-6, 7):
        for k in range(-6, 7):
            F = structure_factor(h, k, basis)
            phase = 2.0 * np.pi * h * x / a
            rho[i] += np.real(F * np.exp(-1j * phase))

rho_min = np.min(rho)
rho_max = np.max(rho)
rho_norm = (rho - rho_min) / (rho_max - rho_min)

print(f"{'x (A)':>8} {'rho (a.u.)':>12} {'Density Map':>20}")
for i in range(0, n_points, 2):
    bar = '#' * int(rho_norm[i] * 30)
    print(f"{x_vals[i]:8.3f} {rho[i]:12.4f}  |{bar}")

# Bragg's law verification
print("\n--- Bragg's Law Verification ---")
print(f"n * lambda = 2 * d * sin(theta)")
print(f"lambda = {wavelength} A")
for h, k, I, d, tt in reflections[:6]:
    theta = tt / 2.0
    lhs = wavelength
    rhs = 2 * d * np.sin(np.radians(theta))
    print(f"  ({h},{k}): 2d*sin(theta) = 2*{d:.4f}*sin({theta:.2f}) = {rhs:.4f} A (lambda = {lhs:.4f} A)")

print("\n[Simulation complete]")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Computation: Bragg Angle Calculator

This Fortran program calculates Bragg diffraction angles for common crystal structures including NaCl and protein crystals. It applies selection rules for face-centered cubic lattices and computes resolution as a function of scattering angle.

Bragg Angle Calculator

Fortran

Diffraction angle calculations for NaCl and protein crystal structures with resolution analysis

bragg_calculator.f90126 lines

program bragg_calculator
  ! Bragg Angle Calculator for Common Crystal Planes
  ! Calculates diffraction angles for various X-ray wavelengths
  ! and crystal structures
  implicit none

integer, parameter :: dp = selected_real_kind(15, 307)
  real(dp) :: a, d, sin_theta, theta, two_theta, wavelength
  real(dp) :: pi
  integer :: h, k, l, n_refl
  character(len=20) :: crystal_name

pi = 4.0_dp * atan(1.0_dp)

write(*,'(A)') '======================================================'
  write(*,'(A)') ' BRAGG ANGLE CALCULATOR FOR COMMON CRYSTAL STRUCTURES'
  write(*,'(A)') '======================================================'
  write(*,'(A)') ''
  write(*,'(A)') ' Bragg''s Law: n*lambda = 2*d*sin(theta)'
  write(*,'(A)') ''

! X-ray wavelengths
  write(*,'(A)') '--- Common X-ray Sources ---'
  write(*,'(A,F8.5,A)') ' Cu K-alpha:  ', 1.5406_dp, ' Angstroms'
  write(*,'(A,F8.5,A)') ' Mo K-alpha:  ', 0.7107_dp, ' Angstroms'
  write(*,'(A,F8.5,A)') ' Co K-alpha:  ', 1.7902_dp, ' Angstroms'
  write(*,'(A,F8.5,A)') ' Cr K-alpha:  ', 2.2909_dp, ' Angstroms'
  write(*,'(A)') ''

! Crystal 1: NaCl (FCC, a = 5.640 A)
  crystal_name = 'NaCl (FCC)'
  a = 5.640_dp
  wavelength = 1.5406_dp  ! Cu K-alpha

write(*,'(A,A,A,F8.4,A)') ' Crystal: ', trim(crystal_name), ', a = ', a, ' A'
  write(*,'(A,F8.5,A)') ' Wavelength: ', wavelength, ' A (Cu K-alpha)'
  write(*,'(A)') ''
  write(*,'(A6,A6,A6,A12,A12,A12)') &
    'h', 'k', 'l', 'd (A)', 'theta(deg)', '2theta(deg)'
  write(*,'(A)') '------------------------------------------------------'

n_refl = 0
  do h = 0, 5
    do k = 0, h
      do l = 0, k
        if (h == 0 .and. k == 0 .and. l == 0) cycle

! FCC selection rule: h,k,l all even or all odd
        if (.not. (mod(h+k, 2) == 0 .and. mod(k+l, 2) == 0 &
            .and. mod(h+l, 2) == 0)) cycle

! d-spacing for cubic
        d = a / sqrt(real(h*h + k*k + l*l, dp))

! Bragg's law
        sin_theta = wavelength / (2.0_dp * d)
        if (abs(sin_theta) > 1.0_dp) cycle

theta = asin(sin_theta) * 180.0_dp / pi
        two_theta = 2.0_dp * theta

write(*,'(I6,I6,I6,F12.4,F12.4,F12.4)') h, k, l, d, theta, two_theta
        n_refl = n_refl + 1
      end do
    end do
  end do
  write(*,'(A,I3,A)') ' Total reflections: ', n_refl, ''
  write(*,'(A)') ''

! Crystal 2: Protein crystal (typical)
  crystal_name = 'Lysozyme (Tet.)'
  a = 79.1_dp  ! Angstroms (tetragonal, a=b)
  wavelength = 1.5406_dp

write(*,'(A,A,A,F8.2,A)') ' Crystal: ', trim(crystal_name), ', a = ', a, ' A'
  write(*,'(A)') ' (Large unit cell = many closely spaced reflections)'
  write(*,'(A)') ''
  write(*,'(A6,A6,A6,A12,A12,A12)') &
    'h', 'k', 'l', 'd (A)', 'theta(deg)', '2theta(deg)'
  write(*,'(A)') '------------------------------------------------------'

n_refl = 0
  do h = 0, 6
    do k = 0, 6
      l = 0  ! Only h,k,0 plane for simplicity
      if (h == 0 .and. k == 0) cycle

d = a / sqrt(real(h*h + k*k, dp))
      sin_theta = wavelength / (2.0_dp * d)
      if (abs(sin_theta) > 1.0_dp) cycle

theta = asin(sin_theta) * 180.0_dp / pi
      two_theta = 2.0_dp * theta

if (k <= h) then
        write(*,'(I6,I6,I6,F12.4,F12.4,F12.4)') h, k, l, d, theta, two_theta
        n_refl = n_refl + 1
      end if
    end do
  end do
  write(*,'(A,I3,A)') ' Total reflections (h,k,0): ', n_refl, ''

! Resolution calculation
  write(*,'(A)') ''
  write(*,'(A)') '--- Resolution vs 2theta ---'
  write(*,'(A12,A14)') '2theta(deg)', 'Resolution(A)'
  write(*,'(A)') '----------------------------'
  do h = 5, 80, 5
    sin_theta = sin(real(h, dp) / 2.0_dp * pi / 180.0_dp)
    d = wavelength / (2.0_dp * sin_theta)
    write(*,'(F12.1,F14.3)') real(h, dp), d
  end do

write(*,'(A)') ''
  write(*,'(A)') '--- Contrast Transfer Function (Cryo-EM) ---'
  write(*,'(A)') ' CTF(s) = -sin(pi*Cs*lambda^3*s^4/2 + pi*Df*lambda*s^2)'
  write(*,'(A)') ''
  write(*,'(A)') ' Typical parameters:'
  write(*,'(A)') '   Cs (spherical aberration) = 2.7 mm'
  write(*,'(A)') '   lambda (300 kV) = 0.0197 A'
  write(*,'(A)') '   Defocus range: 0.5 - 3.0 um'
  write(*,'(A)') ''

write(*,'(A)') '[Computation complete]'

end program bragg_calculator

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

MIT: X-ray Crystallography

MIT lecture on X-ray crystallography fundamentals, Bragg's law derivation, structure factor calculation, and the phase problem.

Introduction to Cryo-EM

Overview of cryo-electron microscopy including sample preparation, image processing, single-particle reconstruction, and recent resolution achievements.

Key Concepts Summary

X-ray Crystallography

Bragg diffraction from crystal lattices provides atomic-resolution structures via Fourier synthesis

Cryo-EM

Flash-frozen samples imaged without crystals; single-particle reconstruction now achieves sub-2 Angstrom resolution

NMR Spectroscopy

Solution-state structure and dynamics from chemical shifts, NOEs, and relaxation measurements

Mass Spectrometry

Precise m/z measurements identify proteins, map modifications, and analyze intact complexes

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