Elements of continuous wave EPR theory

EPR spectroscopy is based on the interaction between an external magnetic field (B0)

with a dipole (e) associated to the spin angular momentum of the electron. The dipole can orient itself parallel or antiparallel to the magnetic field B0, and an oscillating

magnetic field (B1) with the proper energy can promote the transition of the electron

from one orientation to another. The interaction of the dipole e with the external field B0 is called Zeeman interaction and its energy is given by:

𝐸 = − 𝜇⃗⃗⃗⃗ 𝐵𝑒⃗⃗⃗⃗ 0 Equation 1

where e is the magnetic moment associated to the electron spin angular momentum, 𝑆 :

𝜇_𝑒 = −𝑔𝛽_𝑒𝑆 Equation 2

In equation 2,  is the electron Bohr magneton. The g factor is a proportionality factor that relates the observed magnetic moment μ of a particle to its angular momentum quantum number and for the free electron its value is ge = 2.0023. When the electron is

located in a molecular system, the g value differs from ge in dependence of the elctron

distribution in the molecule. 𝑆 is the spin angular momentum and for an electron, that has an electron spin quantum number of s=½, it can only assume the value |𝑆| = √𝑠(𝑠 + 1)ℏ = 3 2⁄ ℏ. Its projection on the axis parallel to the external magnetic field B0,

conventionally the z axis, can only assume two values, depending on the spin magnetic quantum number mS: since Sz=mSħ, then Sz can assume the values of +1/2 or -1/2 (in

ħ units). Therefore two energy levels can be derived from Equation 1: E = ±(1/2)geB0.

The energy difference, in ħ units, between these two levels is therefore dependent to the external field B0:

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E = h = geB0 Equation 3

The dependence of E on B0 is schematized in Figure 1.12. The simplest EPR

experiment is the cw EPR experiment. It consists in irradiating the sample with an oscillating magnetic field with a frequency  that equals the energy gap (Equation 3); the strength of the magnetic field is swept to induce a transition of the unpaired electron between the two energy levels. The typical EPR signal, shown in Figure 1.12, is collected as the first derivative of the actual signal; this allows to improve the signal-to-noise ratio and to better observe scarcely resolved signals. In EPR spectroscopy at X band, the static field B0 is about 0.3 - 0.8 mT, while the oscillating magnetic field lies in the

microwave range. The intensity of the EPR signal is proportional to the microwave absorption that is dependent on the population difference between the two energy levels. The population ratio is given in equation 4, and the population difference, in the case of E<<kT, is given in equation 5.

(N+1/2)/(N-1/2) = exp (-E / kT) Equation 4

(N-1/2 - N+1/2)/(N-1/2) = E/kT Equation 5

To increase the difference of population, and therefore to improve the EPR signal, it is possible to increase the magnetic field B0 or to lower the temperature T. To restore the

Boltzmann equilibrium of the populations after the energy absorption, relaxation pathways such as the spin-spin relaxation and the spin-lattice relaxation take place. If the relaxation pathways are not efficient, the system can reach the saturation (i.e., the two energy levels are equally populated and therefore no signal can be measured). Moreover, electrons can experience a perturbation of their local environment due to the influence of magnetic nuclei. This interaction is called hyperfine interaction and can result in a local Figure 1.12 Schematic representation of the Zeeman interaction. At the proper resonance frequency h, there is the EPR transition (one line represented as the first derivative of the signal).

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enhancement or a local reduction of the external B0. The resonance conditions in

equation 3 are modified to include the hyperfine interaction A:

E = h = geB0+ mIA Equation 6

For a nucleus with nuclear quantum number I (characteristic of a given nuclide), mI can

assume 2I+1 values. It is then clear that the hyperfine interaction of an electron with a nucleus with nuclear spin I leads to the splitting of the electron transition into 2I+1 transitions, as represented in Figure 1.13 for the case of an hyperfine interaction of an electron (S=1/2) with a nucleus with I=1. The hyperfine interaction between an electron and a nucleus has two components, an isotropic component and a dipolar one. The isotropic splitting, defined as a0, is proportional to the electron spin density on the

nucleus, and is determined by the electronic wavefunction. The dipolar component is dependent on the spatial orientation, with respect to B0, of the vector between the two

magnetic moments. Being anisotropic, the hyperfine interactions can be described with a tensor, A, that in its principal axes system assumes a diagonal form:

𝑨 = |

𝐴𝑥𝑥 0 0

0 𝐴𝑦𝑦 0

0 0 𝐴𝑧𝑧

|

Moreover, also the Zeeman interaction of the electron spin with the external field H is anisotropic, therefore the g factor, introduced in equation 2, is a tensor that can be represented, in its principal axes system, in diagonal form as well:

𝒈 = |

𝑔_𝑥𝑥 0 0

0 𝑔_𝑦𝑦 0

0 0 𝑔_𝑧𝑧|

Figure 1.13 Schematic representation of the Zeeman and hyperfine interaction of a spin (S=1/2) with a nucleus with I=1. The three EPR transitions are represented as the first derivative of the signal.

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1.3.2 Nitroxides

Rarely biological samples are measurable with EPR spectroscopy at physiological temperature, since they do not have paramagnetic centers in their structure. With the exception of metalloproteins, biological samples as proteins and peptides must be modified with paramagnetic probes. The most commonly used probes are nitroxides, that contain an unpaired electron on the p orbital of the N―O bond of a nitroxide derivative; they are usually sterically hindered 5- or 6-member rings, and example is given in Figure 1.14. Nitroxides have been extensively used to label proteins at a specific site (Site Directed Spin Labeling, SDSL). The probe MTSSL (1-oxyl-2,2,5,5- tetramethylpyrroline-3-methyl) methanethiosulfonate) is covalently linked to a Cys residue of the protein. It is also possible to mutate the protein to have Cys residues available for the labeling in the desired positions. The MTSSL probe is used to study protein partition between membrane and solution, orientation and dynamics of the protein in membrane environment and solvent accessibility in different domains of the protein. To study peptides, the amino acid TOAC (2,2,6,6-tetramethylpiperidine-1oxyl- 4-amino-4-carboxylic acid) has been often employed, that, differently from MTSSL, must be incorporated during the peptide synthesis and represents a direct modification on the primary sequence. TOAC is a C_{-tetrasubstituted, sterically hindered, amino-acid} that promotes helical formation: thus, its presence could alter the secondary structure of the peptide, so its position must be chosen carefully. It is also possible to prepare - peptides modified with a -amino-acid, like POAC ((3R-4R)-4-amino-1-oxyl-2,2,5,5- tetramethylpirrolidine-3-carboxylic acid).

All the nitroxides used as EPR probes, some of which are described above, are based on the same principle: the unpaired electron (S=1/2) interacts with the 14_{N nucleus}

(I=1). The hyperfine interaction between them generates a splitting of the electron signal into three transitions (2I+1=3, see equation 6 and Figure 1.13). The magnetically active site of these nitroxides is the N—O fragment, the structure of which is reported in Figure 1.15. The unpaired electron is localized in a 2p orbital on the N—O group. Typical principal values of the hyperfine tensor A for nitroxide spin labels are Axx ≈ Ayy

≈ 0.7 mT and Azz ≈ 3.5 mT. Since the difference between the x and y component is

Figure 1.14 Three common EPR probes: from left, MTSSL, TOAC, POAC.

S S O O N O H₃N O O N O H3N COO N O

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usually very small, the tensor is assumed to have an axial symmetry. Instead, the g tensor has a rhombic symmetry, and typical values of the g tensor for a nitroxide spin label are gxx ≈ 2.0085, gyy ≈ 2.0065, gzz ≈ 2.0027. The fact that A and g tensors are anisotropic

means that EPR spectroscopy is sensitive to changes in orientation of the nitroxide and to its rotational motion.