Spin-Label Electron Paramagnetic Resonance Spectroscopy

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Author
1 Introduction
	1.1 Introduction
	1.2 Electron Spin, Magnetic Moment and Resonance Absorption
	1.3 Angular Anisotropy of the Nitroxide Spectrum
	1.4 Rotational Dynamics and Spin-Label Lineshapes
	1.5 Transition Probabilities and Selection Rules
	1.6 Spin–Lattice Relaxation and Saturation
	1.7 Spin–Spin Relaxation and Linewidths
	Appendix A1: Spin Labels
	Recommended Reading
2 The Nitroxide EPR Spectrum
	2.1 Introduction
	2.2 Nitroxide Spin Hamiltonian
	2.3 Angular Variation of Nitroxide EPR Spectra
	2.4 EPR Powder Patterns
	2.5 Inhomogeneous Broadening
	2.6 Voigt Lineshapes and Spectral Convolution
	2.7 Concluding Summary
	Appendix A2: Spin-Hamiltonian Tensors
	References
3 Hyperfine Interactions and g-Values
	3.1 Introduction
	3.2 Hückel Molecular Orbitals for a Nitroxide
	3.3 Isotropic Nitrogen Hyperfine Couplings
	3.4 Anisotropic Nitrogen Hyperfine Couplings
	3.5 Paramagnetic Shifts in Nitroxide NMR
	3.6 (sup[13])C-Hyperfine Couplings
	3.7 (sup[13])C-Hyperfine Couplings of β-C Atoms
	3.8 Proton Hyperfine Couplings
	3.9 (sup[17])O-Hyperfine Couplings
	3.10 (sup[14])N-Quadrupole Couplings
	3.11 Electric Field Gradients and p-Orbital Occupancies
	3.12 g-Tensor of Nitroxides
	3.13 Concluding Summary
	Appendix A3.1: Electron Spin Densities of Nitroxides
	Appendix A3.2: Proton Hyperfine Couplings of Nitroxides
	References
4 Polarity Dependence
	4.1 Introduction
	4.2 Aprotic Environments
		4.2.1 Reaction Fields
		4.2.2 Onsager and Block–Walker Models for the Reaction Field
		4.2.3 Influence of Local Electric Fields on Nitroxides
		4.2.4 Isotropic Hyperfine Couplings in Aprotic Media
	4.3 Local Fields from Fixed Electric Charges or Dipoles
		4.3.1 Effect of Fixed Electric Charges on Hyperfine Couplings
		4.3.2 Effect of Fixed Electric Dipoles on Hyperfine Couplings
		4.3.3 Effect of Polarity and Fixed Charges on Isotropic g-Values
	4.4 Protic Environments
		4.4.1 Isotropic Hyperfine Couplings in Protic Media
		4.4.2 Isotropic (sup[17])O-Hyperfine Couplings in Protic Media
		4.4.3 Isotropic g-Values in Protic Media
		4.4.4 g-Tensor Anisotropy in Protic Media
	4.5 Quadrupole Interactions and Polarity: (sup[14])N-ESEEM of Nitroxides
	4.6 Membrane Polarity Profiles
	4.7 Concluding Summary
	References
5 Spin Relaxation Theory
	5.1 Introduction
	5.2 Time-Dependent Perturbation Theory for Random Fluctuations: Correlation Functions
	5.3 Spin Hamiltonian and Angular Averages
	5.4 Transition Probabilities for a Nitroxide
		5.4.1 Electron Spin Relaxation
		5.4.2 Nuclear Spin Relaxation
		5.4.3 Cross Relaxation
	5.5 Spectral Densities for a Nitroxide
		5.5.1 Hyperfine Anisotropy (END)
		5.5.2 g-Value Anisotropy (Zeeman Interaction)
		5.5.3 END-Zeeman Cross Term
		5.5.4 Spin-Rotation Interaction
	5.6 Spin–Lattice Relaxation Rates
		5.6.1 Electron Spin–Lattice Relaxation
		5.6.2 Nuclear Spin–Lattice Relaxation
		5.6.3 Cross Relaxation
	5.7 Secular, Pseudosecular and Non-secular Transverse Relaxation
	5.8 Nitroxide Transverse (Spin–Spin) Relaxation Rates
	5.9 Concluding Summary
	Appendix A5: Transition Probabilities and Transverse Relaxation Rates for Fast Anisotropic Rotational Diffusion
		A5.1 Electron Spin–Lattice Relaxation
		A5.2 Nuclear Spin–Lattice Relaxation
		A5.3 Cross Relaxation
		A5.4 Electron Transverse Relaxation
	References
6 EPR Lineshape Theory
	6.1 Introduction
	6.2 Bloch Equations and Lorentzian Lineshape
	6.3 Exchange-Coupled Bloch Equations
	6.4 Bloch Equations with Slow Sudden Jumps
	6.5 Bloch Equations Coupled by Slow Rotational Diffusion
	6.6 Density-Matrix Methods
	6.7 Slow-Motion Simulations: Stochastic Liouville Equation
	6.8 A Simple Example: Axial g-Tensor
	6.9 Example Including Axial (sup[15])N-Hyperfine Structure
	6.10 Slow Motion for the Full (sup[14])N-Nitroxide
	6.11 Composite Motions on Different Timescales
	6.12 Concluding Summary
	Appendix A6.1: Clebsch–Gordan Coefficients Needed for Slow-Motion Calculations
	Appendix A6.2: Solution of Stochastic Liouville Equation for Axially Anisotropic Rotation
	References
7 Dynamics and Rotational Diffusion
	7.1 Introduction
	7.2 Rotational Diffusion Equation
	7.3 Rotational Friction Coefficients
	7.4 Slip Boundary Conditions
	7.5 Isotropic Rotational Diffusion
	7.6 Anisotropic Rotational Diffusion
	7.7 Linewidths and Fast Rotational Motion
	7.8 Correlation Times for Fast Isotropic Rotational Diffusion
	7.9 Correlation Times for Fast Anisotropic Rotational Diffusion
	7.10 Measurements at High Field/Frequency
	7.11 Fast Non-axial Rotation and HF-EPR
	7.12 Free Rotation and Jump Diffusion
	7.13 Slow Rotational Diffusion and Anisotropy
	7.14 Slow-Motion Calibrations for Outer Splittings and Linewidths
	7.15 Concluding Summary
	Appendix A7.1: Shape Factors for Rotational Dynamics of a General Ellipsoid
	Appendix A7.2: Spin-Hamiltonian Tensor Anisotropies
	References
8 Dynamics and Orientational Ordering (Liquid Crystals and Membranes)
	8.1 Introduction
	8.2 Restricted Torsional Libration
	8.3 Orientational Order Parameters
	8.4 Motional Models for Uniaxial Order Parameters
	8.5 Independent Ordering Components, Segmental Motion
	8.6 Restricted Off-Axis Amplitudes and Lateral Ordering (HF-EPR)
	8.7 Motional-Narrowing Theory: Linewidths and Order
	8.8 Spin–Lattice Relaxation and Cross Relaxation with Orientational Ordering
	8.9 Linewidths and Lateral Ordering
	8.10 Angular-Dependent Linewidths and Ordering: Strong Jump Model
	8.11 Linewidths and Ordering: Brownian Diffusion
	8.12 Ordering and Slow Motion
	8.13 Order-Parameter Calculations from Powder Samples
	8.14 Dynamics of Combined Slow and Segmental Motions: Lipid Chains
	8.15 Timescale Separation and Multifrequency EPR
	8.16 Concluding Summary
	Appendix A8.1: Clebsch–Gordon Coefficients
	Appendix A8.2: Allowance for Nitrogen Nuclear Quantization Axis
	Appendix A8.3: Slow-Motion Order Parameter Calibrations for Different(sub[R])
	References
9 Spin–Spin Interactions
	9.1 Introduction
	9.2 Magnetic Dipole–Dipole Interactions – Like Spins and Strong Coupling
	9.3 Dipolar Powder Spectra: The Pake Doublet
	9.4 Dipolar Coupling of Unlike Spins
	9.5 Relaxation by Magnetic Dipole–Dipole Interaction
		9.5.1 Dipolar Relaxation by Rotational Diffusion
		9.5.2 Dipolar Relaxation by Translational Diffusion
		9.5.3 Translational Diffusion Coefficients and Dipolar Relaxation
	9.6 Exchange Interaction and Exchange Integral
	9.7 Heisenberg Spin Exchange
	9.8 Nitroxide Biradicals
	9.9 Exchange Probability and Exchange Dynamics
	9.10 Spin-Exchange Frequency and EPR Lineshapes
	9.11 Exchange-Induced Dispersion
	9.12 Line Shifts and Re-encounters in Spin-Exchange Spectra
	9.13 Translational Diffusion and Friction Coefficients
	9.14 Bimolecular Collision Rates and Translational Diffusion
	9.15 Spin Exchange Constant and Bimolecular Collisions
	9.16 Translational Diffusion in Membranes
	9.17 Dipolar-Induced Magnetization Transfer and Dispersion Lineshapes
	9.18 Concluding Summary
	Appendix A9.1: First-Derivative Absorption Lineshapes with Heisenberg Spin Exchange
	Appendix A9.2: Broadening, Line Shifts and Dispersion Admixture by Heisenberg Spin Exchange
	References
10 Spin–Lattice Relaxation
	10.1 Introduction
	10.2 Effective Spin–Lattice Relaxation Times
	10.3 Reduction Factors for CW Electron–Electron Double Resonance (ELDOR)
	10.4 Dependence of Spin–Lattice Relaxation on Rotational Dynamics
	10.5 Paramagnetic Spin–Lattice Relaxation Enhancement
	10.6 Paramagnetic T(sub[1]) -Relaxation Enhancements by Heisenberg Exchange
	1 10.7 Paramagnetic Enhancement by Weak/Intermediate Heisenberg Exchange
	10.8 Paramagnetic Enhancement by Heisenberg Exchange between Charged Species
	10.9 Paramagnetic Enhancement by Static Magnetic Dipole–Dipole Interaction
	10.10 Paramagnetic Enhancement by Dynamic Magnetic Dipolar Interaction
	10.11 Spin–Lattice Relaxation Enhancement by Exchange Processes
	10.12 Relaxation by Slow Two-Site Exchange
	10.13 Concentration Dependence of Heisenberg Exchange Between Spin Labels
	10.14 Concluding Summary
	Appendix A10.1: Effective Spin–Lattice Relaxation Times and ELDOR Reduction Factors for (sup[15])N-Nitroxides
	Appendix A10.2: Further CW-ELDOR Reduction Factors for (sup[14])N-Nitroxides
	Appendix A10.3: Cross Relaxation and m(sub[I])-Dependent Intrinsic Spin–Lattice Relaxation Rates for (sup[14])N-Nitroxides
	Appendix A10.4: Heisenberg-Exchange Rate Constants for Paramagnetic Relaxants
	References
11 Nonlinear and Saturation-Transfer EPR
	11.1 Introduction
	11.2 Progressive Saturation: Bloch Equations
	11.3 Progressive Saturation: Inhomogeneous Broadening
	11.4 Field Modulation: Modulation-Coupled Bloch Equations
	11.5 Progressive Saturation: Sudden-Jump Rotational Mobility
	11.6 Progressive Saturation: Brownian Rotational Diffusion
	11.7 Nonlinear Displays: Out-of-Phase Spectra (ST-EPR)
	11.8 T(sub[1])-Sensitive Nonlinear EPR Displays
	11.9 First-Harmonic, Out-of-Phase Absorption V(sub[1])'-EPR
	11.10 Second-Harmonic, Out-of-Phase Absorption V(sub[2])'-EPR Intensities
	11.11 Example of Very Slow Two-Site Exchange in Lipid–Protein Interactions
	11.12 Saturation Transfer EPR: Ultraslow Rotational Motion
	11.13 Saturation-Transfer EPR: V(sub[2])'-Lineshapes
	11.14 Rotational Diffusion of Membrane Proteins
	11.15 ST-EPR Simulations with the Bloch Equations
	11.16 Stochastic-Liouville ST-EPR Simulations
	11.17 Concluding Summary
	Appendices: Calibrations of Nonlinear EPR Spectra for T(sub[1]), and for τ(sub[R])
	Appendix A11.1: Calibrations for T(sub[1])-Measurements in Progressive-Saturation CW-EPR with Molecular Motion
	Appendix A11.2: Calibrations for T -Measurements in First-Harmonic Out-of-Phase V(sub[1])'- EPR with Molecular Motion
	Appendix A11.3: Calibrations τ(sub[R]) for in Second-Harmonic Absorption, Out-of-Phase V(sub[2])'-EPR
	References
12 Saturation-Recovery EPR and ELDOR
	12.1 Introduction
	12.2 Saturation-Recovery Detection
	12.3 Nuclear Relaxation in (sup[15])N- and (sup[14])N-Nitroxides
	12.4 Rate Equations for SR-EPR and SR-ELDOR of (sup[15])N-Nitroxides
	12.5 Rotational Dynamics and SR-EPR, SR-ELDOR of (sup[15])N-Nitroxides
	12.6 Rate Equations for SR-EPR and SR-ELDOR of (sup[14])N-Nitroxides
	12.7 Rotational Dynamics and SR-EPR, SR-ELDOR of (sup[14])N-Nitroxides
	12.8 Solid-State (Vibrational) Contributions and Glassy Solvents
	12.9 Spin–Lattice Relaxation Enhancements in SR-EPR
	12.10 Slow Two-Site Exchange and SR-EPR
	12.11 Heisenberg Spin Exchange in SR-EPR
	12.12 (sup[14])N-(sup[15])N Nitroxide Pairs and Heisenberg Exchange
	12.13 Slow Rotational Diffusion for SR-ELDOR and SR-EPR
	12.14 Saturation-Recovery EPR and Molecular Ordering
	12.15 Concluding Summary
	References
13 Spin-Echo EPR
	13.1 Introduction
	13.2 Microwave Pulses in the Vector Model
	13.3 Finite Pulse Power and Width
	13.4 Vector Model for Primary Spin Echo
	13.5 Product Spin-Operator Method for Primary Spin Echo
	13.6 Density Matrix, Pulses and Transverse Relaxation
	13.7 Primary-Echo Decay and Rotational Dynamics (Phase-Memory Time)
	13.8 Rotational Diffusion and Primary-Echo Decay
	13.9 Stochastic-Liouville Simulations of Primary-Echo Decay
	13.10 Slow-Motion Simulations of Primary-Echo Decay
	13.11 Primary-Echo Decay in Sudden-Jump Model
	13.12 Experimental Phase-Memory Times
	13.13 Primary Echo-Detected Spectra and Librational Dynamics
	13.14 Spin–Spin Interactions and Spectral Diffusion
		13.14.1 Jump Model for Short τ
		13.14.2 Sudden Jump Model: General Case
		13.14.3 Gauss–Markov Model: Long τ
	13.15 Spin–Spin Interactions and Instantaneous Diffusion
	13.16 Vector Model for Stimulated Echoes
	13.17 Unwanted Echoes and Phase Cycling
	13.18 Stimulated Echo and Rotational Dynamics
	13.19 Stimulated-Echo Decay in Sudden-Jump Model
	13.20 Stimulated Echo and Spin–Spin Interaction
	13.21 Experimental Stimulated-Echo Decays
	13.22 Concluding Summary
	Appendix A13: Lorentzian Spectral Diffusion
		A13.1 Conditional Probabilities and Spectral Diffusion
		A13.2 Echo Decays
		A13.3 Stationary Probabilities and Spectral Diffusion
	References
14 ESEEM and ENDOR: Hyperfine Spectroscopy
	14.1 Introduction
	14.2 Echo Envelope Modulation
	14.3 Two-Pulse ESEEM for I = 1/2 Nuclei
	14.4 Two-Pulse ESEEM for I = 1 Nuclei
	14.5 Three-Pulse ESEEM
	14.6 Standard Intensities for 3-Pulse ESEEM
	14.7 Water-Penetration Profiles and H-Bonding in (sup[2])H-ESEEM
	14.8 ESEEM in Disordered Samples: Powder Spectra
	14.9 ESEEM Powder Lineshapes
	14.10 Electron–Nuclear Double Resonance (ENDOR)
	14.11 Continuous-Wave Proton ENDOR
	14.12 Orientation Selection in Powder ENDOR
	14.13 Distances from ENDOR Frequencies
	14.14 Pulse ENDOR
		14.14.1 Davies ENDOR
		14.14.2 Example of Davies ENDOR
		14.14.3 Mims ENDOR
		14.14.4 Example of Mims ENDOR
	14.15 ELDOR-Detected NMR
		14.15.1 ELDOR-Detected NMR of (sup[15])N-Nitroxides
		14.15.2 ELDOR-Detected NMR of (sup[14])N-Nitroxides
	14.16 Concluding Summary
	References
15 Distance Measurements
	15.1 Introduction
	15.2 Dipolar Pair Spectra
	15.3 Dipolar Convolution and Deconvolution
	15.4 Dipolar Deconvolution
	15.5 Random Distribution of Spins
	15.6 Absolute-Value First Moment and Mean Dipolar Splitting
	15.7 Second Moment of Dipolar Lineshape
	15.8 Gaussian Distance Distributions
	15.9 Statistical Theory of Dipolar Broadening for Dilute Spins
	15.10 Echo-Detected ELDOR (DEER)
		15.10.1 Three-Pulse DEER
		15.10.2 Four-Pulse DEER
	15.11 Processing and Analysing 4-Pulse DEER Signals
		15.11.1 Background DEER Signal
		15.11.2 Spin-Pair Distributions and Spin Counting
		15.11.3 Tikhonov Regularization and Gaussian Fitting
	15.12 Explicit Distance Distributions in Shells of Uniform Density
		15.12.1 Spherical Surface and Volume Distance-Distributions
		15.12.2 Sphere-within-Shell Distances
		15.12.3 Distance Distribution within Spherical Shell
	15.13 Excluded Volume in Background DEER Signals
	15.14 Orientation Selection in DEER Signals
	15.15 Concluding Summary
	Appendix A15.1: General Expressions for Distance Distribution in Spherical Shells
	Appendix A15.2: Decay Function α for Excluded Volume
	References
16 Site-Directed Spin Labelling (SDSL)
	16.1 Introduction
	16.2 Side-Chain Accessibilities and Mobilities
	16.3 Nitroxide Scanning: Secondary Structure
		16.3.1 Soluble Proteins
		16.3.2 Membrane Proteins
	16.4 β-Sheets and β-Barrel Proteins
	16.5 Mobility Mapping
		16.5.1 Angular Amplitudes
		16.5.2 Rotational Rates
		16.5.3 Order Parameters
	16.6 MTSSL (R1) Side-Chain Rotamers
	16.7 χ4/χ5 Lineshape Simulations for R1-Side Chains
		16.7.1 Water-Exposed Non-interacting Sites on α-Helicves
		16.7.2 Lipid-Exposed Non-interacting Sites in Transmembrane Helices
		16.7.3 Lipid-Exposed Sites on a Transmembrane β-Barrel
		16.7.4 Water-Exposed Residues in β-Sheets
	16.8 Lineshape Simulations from Molecular-Dynamics Trajectories
		16.8.1 Order Parameters from MD
		16.8.2 Orientation Potentials and SLE Simulations
		16.8.3 Brownian Dynamics Trajectories
		16.8.4 Spin Hamiltonian and Trajectory of Transverse Magnetization
		16.8.5 Fourier Transformation
		16.8.6 Trajectories
		16.8.7 Discrete Markov States
	16.9 SR-EPR: Spin–Lattice Relaxation and Conformational Exchange
	16.10 TOAC Spin-Label Residue in Peptides: Helix Orientations
	16.11 Distances Between Site-Directed Spin Labels: α-Helices, β-Strands
		16.11.1 α-helices
		16.11.2 β-sheets
		16.11.3 Comparison with Crystal Structures
	16.12 Rotamer Libraries
	16.13 Helix Assembly and Interhelical Distances: Coiled Coils and α-Bundles
		16.13.1 Coiled Coils
		16.13.2 Example of 4-Stranded Coiled Coil: SNARE Complex
		16.13.3 α-Helical Bundles
	16.14 Beta-Sheet Topology: Interstrand Separations
		16.14.1 α-Crystallin Chaperone Fold
		16.14.2 Amyloid Fibril Cross-β Structure
	16.15 DEER and SDSL: Tertiary Fold, Subunits and Docking
		16.15.1 Homo-oligomers
		16.15.2 Triangulation and Distance Geometry
		16.15.3 Distance Restraints in Modelling
		16.15.4 Docking Subunits
		16.15.5 Conformational Changes
	16.16 Concluding Summary
	Appendix A16: Additional Conformational and Distance Data
	References
Fundamental Physical Constants
	Symbols
Appendices A–M: Fundamentals
	Appendix A: Units and Conversions
	Appendix B: Vectors, Matrices and Tensors
	Appendix C: Quantum Mechanical Basics
	Appendix D: Schrödinger Equation and Heisenberg Equation of Motion
	Appendix E: Angular-Momentum/Spin Operators and their Matrix Elements
	Appendix F: Magnetic Field in Quantum Mechanics
	Appendix G: Quantum Mechanical Perturbation Theory
	Appendix H: Spin-Hamiltonian Diagonalization
	Appendix I: Rotation of Axes
	Appendix J: Second-Order Hyperfine Shifts
	Appendix K: Atomic Structure and Molecular Bonding
	Appendix L: g-Values for a p-Electron
	Appendix M: Time-Dependent Perturbation Theory and Selection Rules
	References
Appendices N–V: Specialist Topics
	Appendix N: Spin Density-Matrix
	Appendix O: Relaxation Theory with Density Matrices
	Appendix P: Product-Operator Formalism for Pulse EPR
	Appendix Q: Addition of Angular Momenta and Wigner 3j-symbols
	Appendix R: Rotation Operators, Euler Angles and Wigner Rotation Matrices
	Appendix S: Irreducible Spherical Tensors
	Appendix T: Fourier Transforms, Convolutions and Correlation Functions
	Appendix U: Moments of EPR Lineshapes, and Traces of Spin Operators
	Appendix V: Spherical Harmonics and Legendre Polynomials
	References
Index