EPR spectroscopy

C H A P T E R

4 EPR spectroscopy W.R. Hagen Department of Biotechnology, Delft University of Technology, Delft, The Netherlands O U T L I N E Why electron paramagnetic resonance spectroscopy?

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What is electron paramagnetic resonance spectroscopy? 122 Anisotropy

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A comparison of electron paramagnetic resonance versus NMR 126 Electron paramagnetic resonance spectrometer

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What (bio)molecules give electron paramagnetic resonance?

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Basic theory and simulation of electron paramagnetic resonance 133 Saturation

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Concentration determination

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Hyperfine interactions

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High-spin systems

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Applications overview

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Test questions

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Answers to test questions

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References

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Why electron paramagnetic resonance spectroscopy? Electron paramagnetic resonance (EPR) spectroscopy in biology is applicable to paramagnetic molecules with one (low spin) or more (high spin) unpaired electrons, that is, radicals and transition metal ion complexes. This chapter explains the basic phenomena whose understanding is required for a meaningful analysis and (bio)chemical interpretation of spectroscopy, namely electronic Zeeman interaction, resonance, anisotropy, saturation, hyperfine interaction, and zero-field interactions. The concept of the spectral powder pattern from randomly oriented samples is treated, and its computed simulation by

Practical Approaches to Biological Inorganic Chemistry DOI: https://doi.org/10.1016/B978-0-444-64225-7.00004-3

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© 2020 Elsevier B.V. All rights reserved.

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unit-sphere walking is introduced. The biologically key application of spin counting or quantitative EPR is addressed also in relation to the notion of effective spins. In biochemistry spectrometers are used for two reasons: to determine a concentration or to determine a structure. Measured as a function of time, a change in concentration gives a reaction rate, and a change in structure affords information on a reaction mechanism. The reactions that we are interested in are typically conversions of a substrate into a product under the influence of a biocatalyst. By far the easiest way to measure a concentration is by UV
vis spectroscopy. If a substrate and its product do not have a measurable absorption in the spectral range from near-UV to near-IR, we turn to other detection methods. EPR (electron paramagnetic resonance) spectroscopy can only detect systems with unpaired electrons. Most metabolites in living cells are relatively stable organic molecules; these are so-called closed-shell systems, that is, they have an even number of electrons arranged in a pairwise manner (according to the Pauli exclusion principle) such that no electron remains unpaired, and no EPR is possible. Only in relatively rare cases does a substrate or product have an odd number of electrons with one electron unpaired. In the language of quantum mechanics (QM) these molecules are called spin-one-half-systems, or S 5 1/2. Some of these radicals (or paramagnets) may have a color and others don’t, but they all have an EPR spectrum. A more consequential application area of EPR spectroscopy pertains not to the substrate but to the biocatalyst, which is typically an enzyme with an active site that may encompass a radical, but much more frequently a transition metal ion, or a cluster of metal ions. By definition transition ions are open-shell systems, that is, they have partially filled d- or f-shells, and they have one (S 5 1/2), or more (S . 1/2) unpaired electrons (i.e., they carry paramagnetism) in at least one of their common oxidation states. Thus biological EPR spectroscopy is predominantly a means to study the structure and functioning of active sites of enzymes. This usually includes the quantitative measurement of concentration of spin systems for the determination of the stoichiometry of paramagnetic prosthetic groups per enzyme molecule. It sometimes also encompasses figuring out the mutual magnetic interaction of spin systems within an enzyme molecule to pin down geometrical constraints. This chapter is an introduction to the subject of continuous-wave EPR of biomolecules and their models. EPR is a quantum-mechanical phenomenon, and the theory of EPR likewise makes ample use of QM methods. Knowledge of QM is not required to read this chapter, nor to do a range of useful EPR experiments on biological systems. Key equations are given here without derivation. Those interested in a more extensive treatment of the subject, including derivation of equations, are referred to the book Biomolecular EPR spectroscopy (Hagen, 2009).

What is electron paramagnetic resonance spectroscopy? EPR spectroscopy is the absorption of microwave radiation between energy levels of molecules. In EPR one does not vary the frequency ν of the radiation (or the wavelength λ 5 c/ν in which c 5 299,792,458 m/s is the speed of light), but one uses a monochromatic source at a single fixed frequency. The most commonly used frequency is in the range

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9
10 GHz (part of the X-band). The unit of energy used in EPR is the “wavenumber” or “reciprocal cm.” A frequency of ν 5 9.5 GHz means an energy hνD0.3 cm21 (so λD3 cm). This is a very small energy quantum. Compare an optical transition at 500 nm, which represents an energy quantum of (1/500)nm21 or 20,000 cm21. The small molecular energy splittings required for EPR are due to quantization of the electron spin S and require an external dipole magnet: when the paramagnetic molecule is placed in an axial magnetic field B, which is a field with a north and a south pole (i.e., a vector B⃑ ), the unpaired electron behaves like a little bar magnet but with the quantum-mechanical property that its orientation in the field can assume only two values “parallel” or “antiparallel” to the external field. These orientations correspond to a higher and a lower energy state of the molecule between which microwave energy can be absorbed. The spin quantum numbers are mS 5 11/2 and mS 5
1/2 and in the common so-called “bra-ket” notation of QM the states are denoted as |mS 5 11/2i and |mS 5
1/2i. The two states are “degenerate” (i.e., they have identical energies) when there is no external field (when the magnet is switched off). Since the frequency ν is fixed we have to vary the field B to create a spectrum. So in contrast to optical spectroscopy where we submit an invariant molecule to radiation of continuously varying energy, in EPR we throw radiation of invariant energy on a molecule whose paramagnetism is continuously varied by means of a scanning magnetic field. Absorption occurs when the splitting of the two spin energy levels, caused by the magnetic field, happens to be exactly equal to the energy of the microwave. This is called the resonance condition, hν 5 gβB, in which h 5 6.62607015 3 10234 Js is Planck’s constant and β 5 9.27400999 3 10224 J/T (T is tesla; 1 T10,000 gauss) is the Bohr magneton. The two electron spin energy levels are E 5 6gβB/2. The proportionality constant g is what is determined in an EPR experiment: the g value is specific for the molecule under study; it contains electronic information of (bio)chemical relevance. Therefore we rewrite the resonance condition in the practical form g 5 0:714477

ν ðMHzÞ B ðgaussÞ

ð4:1Þ

As an estimate of practical magnetic field values we can use the (theoretical) g value for a free electron in vacuo, ge 5 2.00232, and a microwave frequency of ν 5 9500 MHz. This requires a field of BD3390 gauss, or 0.339 T, a field that is readily produced with an electromagnet. Since radicals are molecules with a reactive, delocalized, loosely bound unpaired electron, they typically exhibit g values close to ge, say 2.00260.005, for carbon, nitrogen, and/or oxygen-based radicals (and with somewhat greater deviations from ge for heavier atoms like sulfur). This does not hold for transition metals which exhibit more pronounced deviations from ge (e.g., for Cu(II) complexes, typically, 2.0 # g # 2.5) because the unpaired electron’s movement (in QM language, its orbital angular momentum) is more strongly influenced by the metal nucleus. Furthermore, in high-spin systems (more than one unpaired electron) the measured g value, or effective g value (see below), can essentially take any value between zero and infinity as a consequence of magnetic interaction between the different unpaired electrons. The interaction between the electron spin S and a magnetic field B, which causes the splitting of otherwise degenerate spin states, is called the electronic Zeeman interaction.

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If Eq. (4.1) were a sufficient description of EPR, then all spectra would consist of a single line, and the frequency of the microwave source would be an irrelevant choice. In practice EPR spectra can be fairly complex with many details, and they usually change with the frequency (note a change in microwave frequency typically means the use of two or more spectrometers that operate with different source frequencies). There are four main reasons why this complexity (therefore increased chemical information content) occurs. Firstly, the electronic Zeeman interaction between unpaired electrons and the magnet (S2B interaction) is essential for EPR to occur, but it is not the only magnetic interaction to determine the spectrum. Also nuclei can have a spin, I, and this nuclear magnet can interact not only with the external magnet (I2B or nuclear Zeeman interaction, which is the basis for NMR spectroscopy) but also with the electron spin (S2I or hyperfine interaction). If a paramagnet has more than one unpaired electron (i.e., a high-spin system), these electron spins can mutually interact (S2S or zero-field interaction). An equivalent effect occurs for nuclear high-spin systems (I2I or quadrupole interaction). When a system has more than one paramagnet, either within one molecule or between molecules, this can lead to dipolar interaction between spins (another form of S2S interaction). Finally, many metalloproteins contain clusters of metals (i.e., metal ions at a mutual distance of one or two chemical bonds, for example, in iron
sulfur clusters) leading to the coupling of electron spins associated with individual metal ions into a new system spin, or cluster spin (in QM language this is called exchange interaction). Since only the Zeeman interactions are linear in the field (therefore linear in the frequency) and all other interactions are independent of the field, changing the microwave frequency (and therefore the field) changes the relative weight of different interactions and thus changes the EPR spectrum. Note that the quadrupole interaction and, especially, the nuclear Zeeman interaction are very often too weak to be resolved in regular EPR, however, they are observed in more elaborate double-resonance experiments like electron-nuclear double resonance (ENDOR) or electron spin echo envelope modulation (ESEEM) spectroscopy. Secondly, as a consequence of the nonspherical structure of molecules the abovementioned interactions are all dependent on the orientation of the dipole magnet (the vector ⃑B ) with respect to the molecule (or with respect to a Cartesian coordinate system defined by the molecular structure of the paramagnet). In other words, the EPR spectrum depends on the direction of orientation of the molecule in the magnetic field, and even if the electronic Zeeman interaction is the only relevant spectral determinant, the spectrum is almost never a single line. Thirdly, if we use a microwave to excite an S 5 1/2 molecule from its lower spin energy level, or ground state, to its higher level, or excited state, then the molecule subsequently has to return to its ground state again by some mechanism if only to be ready to absorb a next microwave quantum and thus to maintain the spectrum in time. This so-called “spin relaxation” influences the shape and, particularly, the width of EPR spectra, and it makes the spectral shape dependent on the temperature T and on the intensity P (for power) of the microwaves. Fourthly, all molecules are subject to conformational distribution: the exact relative atomic coordinates slightly (or not so slightly) vary in a real sample from one molecule to the other. This means that each molecule has a slightly different g value (and values for the other interactions), which in turn leads to EPR broadening and spectral changes. This phenomenon is called “g strain.”

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Anisotropy

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Understanding EPR means to understand the nature and the practical consequences of these phenomena, and this is the subject of the remainder of this chapter.

Anisotropy The unpaired electron of a paramagnetic molecule does not only experience the external magnetic field, but it also “sees” the molecular structure as it is part of the molecule’s electronic structure. Electrons as moving charges represent a magnetic field, and this internal field adds to, or subtracts from, the external field in the Zeeman interaction: hν 5 geβ(Bext 1 Bint). By convention, this is usually written as an observed shift in the g value of the free electron: hν 5 (ge 1 Δg)βB, or simply hν 5 gβB, where g, or rather the deviation from ge, now contains electronic (and therefore structural) information on the molecule. Axial magnetic fields are vectors in 3D space, for example, in the space defined by the coordinates of the paramagnetic molecule. The external field is the dominant one, so it determines the direction along which the EPR spectroscopist “looks at” the molecule. In other words, the orientation of the molecular structure in the external field is of the essence: rotation of the molecule in the field (or, alternatively, rotation of the magnet around the molecule) will result in a change of the EPR spectrum. Each orientation has its own spectrum, and so a single molecule can give rise to an infinite number of different spectra. If the spectra are determined by the electronic Zeeman interaction only, then they all consist of a single absorption line. Single-molecule EPR spectroscopy does not exist (yet?), because the signal of a single molecule is too weak to be detected. However, a single crystal consists of many identical molecules all with the same orientation in space. A single crystal of identical S 5 1/2 molecules gives a single-line Zeeman EPR spectrum; its g value changes when the crystal is rotated. Our biomolecular or model-compound samples will usually not be single crystals; they are homogeneous solutions, or frozen solutions, or powder samples. Each molecule has a different orientation and the EPR of such a sample is the sum of many different single-line spectra. All these spectra have different g values, because the internal field seen by the unpaired electron depends on the orientation of the molecule in the external monitoring field. The result, illustrated in Fig. 4.1, is a specific spectral shape called the “powder pattern”; it covers a defined field range (or g value range) between two extreme values. Most EPR spectrometers produce the first derivative of this powder absorption spectrum, and since the slope of the powder pattern rapidly changes around the three g values gx, gy, gz (together also called the g tensor or—by mathematics purists—the g matrix), corresponding to the molecular x, y, z-axes, one gets the impression that the EPR derivative spectrum consists of three “peaks.” This three-featured form (a peak, a derivative, and a negative peak), called the “rhombic powder pattern,” is the general fingerprint of an S 5 1/2 system without any specific symmetry properties (gx6¼gy6¼gz). Note that the labeling with (x, y, z) is arbitrary; one might just as well use (z, y, x) or (x, z, y) or (a, b, c) or (1, 2, 3), etc. Symmetric properties of the coordination complex may simplify the EPR pattern. A metal ion at the center of a perfect octahedron with six identical ligands with identical metal-to-ligand bond lengths will give

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FIGURE 4.1 The construction of an X-band (ν 5 9.5 GHz) EPR powder pattern (the black trace) by summation of single-orientation spectra from individual molecules in the XYZ molecular axis system with respect to the magnetic field vector B. Red traces are for B along one of the molecular axes, and blue traces are examples from a large number of spectra from intermediate orientations. The label ZY, for example, means that B is oriented halfway between the molecular Z- and X-axis. The magenta trace is the observable first derivative of the EPR powder absorption spectrum with g values: gx 5 1.62, gy 5 2.09, gz 5 2.95 (which could stem from a low-spin heme Fe(III) complex). Note that the spectra from individual molecules at different orientations have been given different widths to illustrate linewidth variation common in experimental spectra.

a single (derivative) line spectrum, called an isotropic pattern, with gx 5 gy 5 gz. Such a highly symmetrical structure is not likely to occur in biology. However, for metalloproteins we do frequently find (near) “axial” spectra with gz6¼gyDgx. An axial pattern can occur when a perfect octahedron is elongated (or compressed) along one of the axes (which is then defined as the z-axis). More generally, if one, or two of the ligands along the z-axis are different form the others (as in many tetrapyrrole complexes, for example in hemoproteins) the local structure and the EPR spectrum can be nearly axial. Strictly speaking axiality of the EPR spectrum reflects axiality in the electronic structure, that is, in the wave function(s) occupied by the unpaired electron, and may not necessarily be retraceable to an exact axial geometry. Several so-called blue copper proteins, such as plastocyanin with His, His, Met, and Cys ligands, exhibit axial EPR, although the copper coordination is a very strongly deformed NNSS (nitrogen-nitrogen-sulfur-sulfur) tetrahedron. In summary the basic EPR spectrum is a one-, two-, or three-featured pattern, but it is not always easy to link this to structural symmetry.

A comparison of electron paramagnetic resonance versus NMR The EPR and the NMR effect were both discovered in the mid-1940s, and their subsequent technical and theoretical developments initially took a parallel course. It is instructive to compare the two spectroscopies in terms of present-day similarities and differences. The simplest system giving EPR is one with a single unpaired electron, S 5 1/2. Most nuclear magnetic resonance spectroscopy, for example, 1H-NMR, is done on

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A comparison of electron paramagnetic resonance versus NMR

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systems with a nuclear spin I 5 1/2. There is a strong analogy between S 5 1/2 and I 5 1/2 in that they both give rise to two quantized energy states in an external magnetic field by means of a Zeeman interaction, although in EPR, due to the negative sign of the electron charge, the orientation antiparallel to the field is the lowest in energy, while in NMR the parallel orientation is the ground state. Both the nuclear Zeeman interaction and the electron Zeeman interaction are measured in the form of a spectroscopic shift. The electron shielding-induced chemical shift away from the resonance frequency of a standard compound such as tetramethylsilane is the NMR equivalent of the internal fieldinduced shift Δg from the free-electron value ge in EPR. Of course for the same external field strength, nuclear spin energy level differences are much smaller than the separations of electron spin energy levels, and the radiation to induce nuclear resonance is in the radio frequency range (MHz instead of GHz), which is one of the reasons why NMR spectroscopy typically has a lower concentration sensitivity than EPR spectroscopy. Arguably the most important historical divergence between NMR and EPR was the emergence of commercial pulsed-NMR spectrometers starting in the late 1960s. From then on it became routine to apply a broad spectrum of frequencies (i.e., a short singlefrequency pulse turning into a frequency range by virtue of Heisenberg’s uncertainty principle) to a sample in a single-valued, invariant external magnetic field. In this approach the recording of a single NMR spectrum takes a short time, and thus extensive averaging becomes practical. Even today such an experiment is technically impossible in EPR spectroscopy because one cannot make a sufficiently short GHz pulse of sufficient intensity and homogeneity to cover all the frequencies of an EPR spectrum at constant field. Therefore EPR still uses monochromatic continuous wave (CW) radiation in combination with a varying magnetic field, and the recording of a single EPR spectrum typically takes a few minutes, mainly to allow for noise reduction by means of a low-pass filter. Pulsed versions of EPR have also been developed based on nanosecond electronics, but there is a paradigmatic difference in the application of pulsed EPR versus that of pulsed NMR. EPR studies always start with the CW experiment; pulsed EPR is an optional “next step” or “advanced” follow-up experiment providing information additional to the CW data and obtained at additional cost and effort. Because the EPR pulse cannot cover the whole spectrum, it is used to measure a small range of the spectrum at increased resolution. Technically this is typically set up in the form of a double-resonance experiment. With the CW spectrum known, the magnetic field is fixed at a value corresponding to a single point of the powder spectrum, and then a pulse of radiation is applied to probe a small frequency spectral range around the fixed powder point. Today a variety of pulsed-EPR methods are at one’s disposition such as pulsed ENDOR, ESEEM, pulsed electron
electron double resonance (ELDOR) also called PELDOR or DEER (double electron
electron resonance). These techniques are usually only available in specialized laboratories, and each one has its own possibilities and limitations. Their details are beyond the scope of this chapter. From the perspective of the biochemist there is also a major conceptual difference in the application of NMR versus EPR spectroscopy. A high-resolution solution NMR experiment affords a global picture of all protons (and/or 13C, 15N, etc.), that is, of all parts of a biomacromolecule. On the contrary, EPR only looks at paramagnets and thus typically provides a picture of a local spot of the macromolecule, for example, a metal coordination complex

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in a metalloprotein. In other words, the EPR result is spatially limited and spectroscopically more simple. On the other hand, the focus point is usually on the most important part of the molecule, the active site. And here the two methods show considerable complementarity because the NMR resonances are extensively broadened (frequently beyond detection) for the nuclei near the paramagnetic center. Both S 5 1/2 EPR and I 5 1/2 NMR exhibit anisotropy, however, in liquid-state NMR tumbling of molecules affords complete averaging of this pattern to single-line spectra for each nucleus even in large biomacromolecules, hence its “high resolution.” In EPR this only works for relatively small molecules; proteins are so big that their tumbling on the EPR timescale is too slow to average anisotropy away, and therefore, for example, metalloproteins always give anisotropic, powder pattern spectra even when in solution. In solidstate NMR one does observe powder patterns, very similar to the EPR ones, as spectra for individual nuclei. Finally, “increasing the frequency” has a rather different connotation in NMR as compared to EPR. Over the years the proton NMR frequency has been steadily increased from tens of megahertz to around 1 GHz at present, concomitant with improving technology for the generation of stable, homogeneous static magnetic fields first with electromagnets and subsequently with superconducting solenoids. Each frequency field increase has led to an increase in sensitivity and an increase in resolution in these nuclear Zeeman interaction dominated spectra. On the contrary, concentration sensitivity of the EPR detection system typically decreases with increasing frequency above X-band (9
10 GHz) due to several reasons such as increasingly noisy detection diodes or increased power losses in highfrequency microwave components. Sensitivity also decreases with decreasing frequency below X-band mainly due to a less favorable Boltzmann distribution of molecules over the spin energy levels (see below). The combined effect of these trends is that starting more than six decades ago (Bagguley and Griffiths, 1947) until this day EPR at X-band has generally been found to be clearly the optimal choice in terms of sensitivity (although not necessarily in terms of spectral resolution).

Electron paramagnetic resonance spectrometer The vast majority of EPR spectrometers operate at a frequency in the range 9
10 GHz, which is part of the X-band of c. 8
12 GHz. The most efficient, or least lossy, way to transfer microwaves at X-band frequencies is the waveguide, typically a rectangular tube made of brass filled with air. The waves move through the inner “skin” of the waveguide, that is, through a layer a few micrometers thick on the inside of the guide. The second-best choice for transport in X-band is the coaxial cable, an inner and outer conductor separated by a dielectric such as Teflon. Thus the microwave part of the spectrometer is a spaghetti of components connected by waveguide and coaxial cable, most of which is, however, invisible to the operator, because it is built in a box called “the bridge.” This design gives the spectrometer a somewhat austere look, in which only a single piece of waveguide sticks out of the bridge to terminate in the heart of the machine called the cavity, centered in between the poles of the magnet (Fig. 4.2).

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FIGURE 4.2 Drawing of a conventional continuous-wave EPR spectrometer consisting of a dipolar electromagnet, a microwave bridge (with waveguide schematic) with a reflection resonator or cavity, and a computer console whose screens exhibit an EPR spectrum and a tuning mode pattern.

The cavity is also called the resonator, but note that this name has nothing to do with the resonance (the R in EPR) of the sample. The bridge-coupled cavity can be described as an electronic resonator circuit (also tank circuit or tuned circuit) with a quality factor Q. In brief this means that the inner dimensions of the cavity and the materials properties of its inner walls are such that a unique frequency in X-band affords the sustaining of a standing-wave pattern with an energy density that is Q-times greater than in a setup in which the cavity would be absent. Typical Q factors for X-band cavities are roughly of the order of 5000, and the sensitivity of the spectrometer is approximately increased by this factor compared to that of a simple transmission or reflection instrument without a cavity.

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The “mode pattern,” or the microwave electromagnetic field lines in the cavity, are such that the magnetic component of the microwave (i.e., the field required to make EPR transitions) is maximal along a vertical line through the middle, where the electric component of the microwave (i.e., the field that only disturbs the EPR measurement by nonresonant absorption) is minimal. The shape of an EPR sample in an X-band cavity is a vertical cylinder with a length of c. 15 mm and a diameter of c. 1 mm for aqueous samples and a diameter of c. 3.5 mm for all other samples including frozen aqueous solutions. This gives approximate volumes of 15 and 175 μL held in quartz tubes; the reduced diameter for watery samples is to partially compensate for the increased nonresonant absorption of microwaves due to the high dielectric constant of water, which is some 25 times higher than that of ice. Microwave intensities are expressed in decibel attenuation with respect to a reference value in watts. The decibel scale is a logarithmic one: n ðdBÞ2100:n

ð4:2Þ

so for a source of 200 mW maximal power (which is a common value for commercial spectrometers) the attenuated power in dB versus watt is as in Table 4.1 (in which mW 5 milliwatt, μW 5 microwatt, nW 5 nanowatt). The source of X-band microwaves is either a klystron (vacuum tube technology) or a Gunn diode with a typical initial output power of c. 400 mW slowly decreasing over the source’s lifetime ( . 10,000 hours of operation). This power is “leveled” (i.e., reduced) to a fixed value of 200 mW to assure constant output over time. The “road map” of the microwaves is then as follows (cf. Fig. 4.2): a wave of 200 mW leaves the source. A small amount of this intensity (c. 1% or
20 dB) is “coupled out” to the reference arm by a device called a directional coupler. The remaining 99% intensity in the main arm can be reduced to the required level (see “Saturation” section) by means of an attenuator, with the maximal attenuation for a good spectrometer being typically 60 dB (i.e., 1,000,000 times attenuation to 200 nW). After the attenuator the main wave enters a circulator, a device that can be thought of as a right-hand roundabout, and is forced to go into the waveguide that ends with the cavity. Waves reflected back from the cavity pass the circulator to go to the third TABLE 4.1 Conversion of decibel attenuation to power in watt for a source leveled at 200 mW. Decibel

0

22

24

26

28

0

200 mW

126 mW

79.6 mW

50.2 mW

31.7 mW

210

20.0 mW

12.6 mW

7.96 mW

5.02 mW

3.17 mW

220

2.00 mW

1.26 mW

796 μW

502 μW

317 μW

230

200 μW

126 μW

79.6 μW

50.2 μW

31.7 μW

240

20.0 μW

12.6 μW

7.96 μW

5.02 μW

3.17 μW

250

2.00 μW

1.26 μW

796 nW

502 nW

317 nW

260

200 nW

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arm with the detector diode, and any wave reflected from there is almost completely caught in the dead end of the fourth arm, called “load” to be converted into heat waste. As an essential prelude to actual measurement the spectrometer has to be “tuned,” which means that we have to (1) adjust the frequency, (2) make the spectrometer reflectionless, and (3) adjust the reference arm. The microwave frequency of the source in the bridge is tunable over a small range (9
10 GHz in X-band) to make it correspond to the unique lowest “eigenfrequency” (or the frequency of the fundamental mode) of the loaded cavity with a sample tube and (if necessary) a cooling system in place. This is accomplished by inspection of an oscilloscope tracing of the power reflected from the cavity as a function of a small frequency scan, say 10 MHz, around the set microwave frequency (cf. the right-hand screen on the computer in Fig. 4.2) and tuning this mode pattern to be symmetric with the “dip” centered. Then the spectrometer is made “reflectionless,” which is brought about in practice by adjusting a Teflon screw, located at the back of the cavity, with a metal end plate in front of a little aperture called the iris between the waveguide and the cavity until the dip in the tuning pattern is maximally deepened, and no current is measured at the detection diode. Reflectionless means that a standing wave is set up in the cavity whose energy dissipates only through the cavity’s side walls with no loss by means of back-reflection into the waveguide. In the language of electronics the impedance of the cavity is now “matched” to that of the rest of the system. This delicately balanced system will be detuned when, in an actual experiment, some of the radiation is absorbed by the sample, and as a result some radiation will be reflected out to cause a voltage change over the detection diode. However, for proper operation of the diode the voltage change should not be around zero but around a finite value called the voltage bias. One can create this bias by slightly detuning the cavity with the iris screw, but the disadvantage of this approach is that the spectrometer has to be readjusted every time the power, that is, the intensity of the microwaves, is changed by adjustment of the attenuator. To avoid this complication we use the reference arm through which a constant fraction of the unattenuated microwave is directly passed to the detection diode. The reference arm has its own attenuator, usually called “bias,” to optimize its output power to the characteristics of the diode (i.e., typically a diode current output of 200 μA). The reference arm also has a device called “phase shifter” to make the phase of the microwave equal to that through the main arm, which is accomplished by adjusting the phase shifter until the tuning mode pattern is perfectly symmetric. The apparatus is now ready to run a spectrum, which means that we scan the magnetic field (the x-axis of the spectrum) over the required range over a period of typically a few minutes. The field is produced by a water-cooled electromagnet, or in high-frequency EPR (c. ν $ 90 GHz) by a superconducting solenoid. To further increase spectrometer sensitivity we use a technique called phase-sensitive detection: the very slowly varying magnetic field is modulated with a very rapidly varying (100 kHz) small sinusoidal field of the order of 61 gauss, and the EPR signal from the detection diode is measured with a lock-in amplifier, that is, a device that takes an in-phase 100 kHz “look” at the signal. The result of this technique is twofold: (1) the EPR signal-to-noise ratio increases because electronic system noise at frequencies other than 100 kHz is not amplified, and (2) we obtain, by necessity, the first derivative of the EPR absorption spectrum. Note that the choice for 100 kHz is again one of optimization: the noise in an EPR spectrometer is found to be

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mainly of the “pink” or “one-over-f” (1/f ) type, which means that low-frequency noise dominates and noise amplitude decreases with increasing noise frequency, so the fieldmodulation frequency should be as high as possible. On the other hand, modulation fields of frequencies significantly above 100 kHz have increasing difficulty in penetrating the thin side walls of the cavity and reaching the sample.

What (bio)molecules give electron paramagnetic resonance? All compounds with one or more unpaired electrons give an EPR spectrum. The very vast majority of biologically relevant radicals have one single unpaired electron: a doublet with S 5 1/2 (e.g., flavin radicals; amino acid-based radicals; nitric oxide, NO; superoxide, O2•—). In rare cases one finds two unpaired electrons: a triplet with S 5 1/2 1 1/2 5 1, the most notorious example being molecular oxygen, O2. Radicals with more than two unpaired electrons exist, but none has yet been reported for a biological system. Occasionally, it may be possible to excite a diamagnetic system (no unpaired electrons) by continuous illumination with UV
visible light into the cavity to an excited triplet state with two unpaired electrons. If the lifetime of the triplet is sufficiently long it can be detected with cw-EPR, for example, photosynthetic reaction centers (Dutton et al., 1972). Transition ions can have up to five unpaired d-electrons—a quintet state with S 5 5/2 (e.g., high-spin Fe31)—or up to seven unpaired f-electrons—a heptet state with S 5 7/2 (Gd31)—and exchange-coupled clusters of transition ions can have many unpaired electrons, for example, the P-cluster with 8Fe in the nitrogenase enzyme, although it contains only d-ions, has seven unpaired electrons and a system spin of S 5 7/2 (Hagen et al., 1987). From an EPR spectroscopist’s point of view it is practical to divide all molecular systems into four groups: diamagnets (S 5 0), doublet systems (S 5 1/2), half-integer highspin systems (S 5 n/2), and integer high-spin systems (S 5 n). Diamagnets have a ground state that is not split by a magnetic field, that is, a singlet system, and they cannot have an EPR spectrum. Doublet systems are “easy” not only because they have only a single electron Zeeman transition, but also because electron spin relaxation (see “Saturation” section) is typically slow, which means that their EPR can be measured at relatively high temperatures (i.e., ambient temperature or nitrogen-flow temperatures). However, note that relaxation is usually not slow for metal clusters with a system spin of S 5 1/2. Half-integer high-spin systems are “not so easy” because they have more than one electron Zeeman transition, and because their relaxation is usually so fast that cooling with cryogenic helium gas is required to obtain sharp lines. Integer high-spin systems are “difficult” because usually they have very broad, asymmetric, and weak EPR features from (almost) forbidden transitions. Some S 5 n/2 systems behave as “effective” S 5 1/2 systems, which means that only one of the several Zeeman transitions is detectable, but the relaxation can still be fast. Some S 5 n systems behave as effective S 5 0 systems, which means that none of the several possible Zeeman transitions is detectable, and so these systems are called “EPR silent.” An overview of these classes each with an iron
protein example is given in Table 4.2.

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TABLE 4.2 Overview of classes of spin systems. A

S50

Diamagnets

Fe(II) in oxy-myoglobin

B

Low spin

S 5 1/2

Fe(III) in ferrimyoglobin-sulfide

C

High spin

S 5 n/2

Fe(III) in rubredoxin

C’

Single-transition high spin

Effective S 5 1/2

Fe(III) in ferrimyoglobin

D

Integer spin

S5n

Fe(II) in deoxy-myoglobin

D’

EPR silent integer spin

Effective S 5 0

Fe(II) in rubredoxin

TABLE 4.3 Examples of biologically relevant spin systems. S50

Most organic molecules; Complexes of all main group elements; Low-spin Fe(II), Co(III), square planar Ni(II), Cu(I), Zn(II), Mo(VI), W(VI); Clusters, for example, [2Fe
2S]21, [4Fe
4S]21

S 5 1/2

Most organic radicals, for example, flavin radicals, quinone radicals, amino acid radicals; Most inorganic radicals, for example, nitric oxide NO, superoxide O2•2; Low-spin Fe(III), low-spin Co(II), Ni(III), Ni(I), Cu(II), Mo(V), W(V); Clusters, for example, [2Fe
2S]11, [3Fe
4S]11, [4Fe
4S]31, [4Fe
4S]11, [Fe(II)
O
Fe(III)]

S 5 n/2

Mn(II), Mn(IV), high-spin Fe(III), high-spin Co(II); Clusters, for example, linear [3Fe
4S]11, some [4Fe
4S]11

S5n

Biradicals (triplets), for example, light-excited reaction centers, molecular oxygen; Mn(III), high-spin Fe(II), Fe(IV), high-spin Ni(II); Clusters, for example, [3Fe
4S]0, [Cu(II)
heme Fe(III)] in cytochrome oxidase

Table 4.3 gives a more extensive list of biologically relevant systems and the spins of their electronic ground states.

Basic theory and simulation of electron paramagnetic resonance EPR spectroscopy of single crystals from biomolecules is rare because, for example, protein crystals are much smaller than the X-band sample size of 175 μL, and they do not give sufficient signal intensity. EPR samples are almost always (frozen) solutions of biomolecules, which means that they contain very many molecules (c. 1017 for a millimolar solution), each one with a different orientation with respect to the external axial magnetic field. To quantitatively understand EPR spectra we must carry out what is called a “walk over the unit sphere,” which means that we conceptually place the molecule of interest at the origin of an x, y, z Cartesian axes system (i.e., the molecular axes system), and then let a vector from the origin, of unit length, parallel to the magnetic field B, sample “all” possible orientations with respect to the molecule. This is done by defining a sphere around the

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FIGURE 4.3 Sketch of a walk over the unit sphere. The solid blue arrow is a unit vector along the magnetic field B with polar angles θ and φ in the Cartesian XYZ molecular axis system. The broken blue arrow is the projection of the unit vector onto the XY plane. The paramagnet is at the origin; the surface of a surrounding unit sphere is divided into fragments of equal area defined in terms of the polar angles as d cos θ 3 dφ. The unit vector along B samples each of these fragments once.

origin with a radius of unity, dividing up the surface of this sphere in a large number of little areas of equal size and shape, and letting the unit vector subsequently point toward each one of these little areas. The “walking” of the unit vector is conveniently defined in polar coordinates (1, θ, ϕ) as shown in Fig. 4.3, and we must make steps in cos θ (i.e., not in θ), and in ϕ to keep the little areas constant in size. For a rhombic spectrum as in Fig. 4.1, this means that we must solve the equation for the field position of the EPR absorption, B 5 0.714477  ν/g with g(θ, ϕ) in terms of sines and cosines: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðθ; ϕÞ 5 g2x l2x 1 g2y l2y 1 g2z l2z ð4:3Þ in which we used the definition of the so-called direction cosines lx 5 sin θ cos ϕ; ly 5 sin θ sin ϕ; lz 5 cos θ and the equation for the intensity (or transition probability) X ðg2i 2 g22 l2i g4i Þ I ðθ; ϕÞ 5 g21

ð4:4Þ

ð4:5Þ

i 5 x;y;z

by making equidistant steps in cos θ and in ϕ. For example, to generate a simulation such as the spectrum in Fig. 4.1, we write a computer program that has three nested FOR-loops (or DO-loops), two for the walk over the unit sphere, and one for a scan through a line shape function, F, for example, a Gaussian distribution (Eq. 4.6):   2 ðln 2ÞðBr 2BÞ2 F 5 Ir exp ð4:6Þ W2 in which the subscript r stands for “resonance,” so Ir is the transition probability at resonance field Br, B is a scan through the field, and W is the half width at half height of the Gaussian line. The program is as follows:

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Saturation

Pseudo-code: a basic program to generate EPR spectra of S 5 1/2 systems. FOR

cosθ 5 1 to 0

! step from z to xy plane

ϕ 5 0 to π/2 gr 5 g(θ,ϕ) Ir 5 I(θ,ϕ)

! step from x to y ! compute g value

Br 5 0.714484 * ν / gr FOR b 5 Br- 3.16 W to Br 1 3.16 W IF Bstart , b , BendTHEN F 5 F 1 Irexp[-ln2(Br-b)2/W2]

! compute resonance field ! step in Gaussian line shape ! check spectrum limits ! compute amplitude

FOR

! compute intensity

The scan in field goes out to 3.16 times the linewidth W (half width at half height) to where the Gaussian has 0.1% of its maximal intensity. This example program is written in “pseudo code” and should be rewritten in your favorite programming language in order to work. Alternatively, you can of course use existing programs, some of which can be downloaded for free from the internet. Note also that the walk over the unit sphere in the example is actually over only one octant, because the other seven octants give identical g (θ,ϕ) and I(θ,ϕ) values. Also fortunately we do not really have to calculate 1017 different molecular orientations; typically some 100 steps in cos θ and 100 in ϕ, that is, c. 104 orientations, are enough, which means that an increase in the number of steps will not change the shape of the final simulation. For a digital spectrum of say 1024 points this means calculating c. 107 amplitudes, which your garden-variety laptop will accomplish in less than a second.

Saturation How much microwave power should we use? How far should we open the attenuator of the main arm in the bridge of Fig. 4.2? There is no danger of destroying the sample: the maximum output power is typically 200 mW, which is some 5000 times less than what one uses in a household microwave oven. In X-band an aqueous sample at room temperature subjected to full power may warm up by a few degrees, which is why most work on aqueous samples is done at reduced power of
10 dB (i.e., 20 mW) or less. The temperature of a frozen sample in a cryogenic flow of nitrogen or helium will not even noticeably change at full power. From a statistical viewpoint higher power means more transitions per unit time and therefore higher EPR amplitude. However, the phenomenon of saturation limits the maximum power that we may use. This is readily illustrated on the energy level scheme of a simple S 5 1/2 system. At resonance the two levels are separated by an energy difference ΔE 5 hν, and this means a Boltzmann population distribution n1 5 n0e2ΔE/kT, in which k 5 0.69503476 cm21/K is the Boltzmann constant. In other words, for a total number of n0 1 n1 molecules we will have n0 in the ground state and n1 in the excited state. For a sample in X-band (ν 5 9.5 GHz; λ 5 3.156 cm) at room temperature (T 5 295 K) we find n1/n0 5 0.9985, that is, a very small difference in population indeed. Inducing transitions by microwave absorption will

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further reduce this small difference, the more so at higher power levels. This reduction is counteracted by relaxation, that is, the falling back to the ground state of excited molecules by dissipation of the energy difference ΔE to the surroundings (in EPR literature these surroundings are also called “lattice” or “bath”). If the relaxation cannot keep up with the microwave power input, then there will be a net decrease in population difference. At high microwave power the difference will eventually become zero and the possibility for net microwave absorption will be abolished: the spin system is completely saturated and the EPR signal has disappeared. In summary with increasing microwave power the EPR amplitude increases linearly with low power, then levels off at higher power, then decreases at even higher power, and eventually disappears at very high power. This amplitude versus power relation is usually measured and plotted for normalized amplitudes, that is, corrected for the power. Experimentally, this is done as follows. The power expressed in decibels is a logarithmic scale; the gain, or electronic amplification, on EPR spectrometers is also expressed on a logarithmic scale either in hardware on older spectrometers or in software on newer spectrometers. In other words, one can only choose from a limited number of gain values, and these are 1.25, 1.6, 2.0, 2.5, 3.2, 4.0, 5.0, 6.3, 8.0, and 10 times a power of 10. These numbers are equal to 100.n times a power of 10, where n 5 1 through 10, that is, a logarithmic scale. In practice this means that when a signal is not saturated, increasing the power by m 3 2 dB and at the same time decreasing the gain by m steps should afford the same EPR amplitude. So we can make a saturation graph (or “power plot”) by running a spectrum at a power of, say,
30 dB and a gain of, say, 4.0 3 103; then we take the next spectrum at higher power 5
28 dB and lower gain 5 3.2 3 103, and so on until we reach a power/gain combination at which the EPR amplitude starts to decrease. We have then reached the onset of power saturation, and we should go back one step (2 dB) to the optimal measuring condition in terms of signal-tonoise ratio for the given temperature. This, then, is the setting to obtain “publicationquality” data. The experiment is schematically illustrated by the green trace in Fig. 4.4: increasing the power from the very low value of
60 dB initially leaves the normalized amplitude unaffected. When we reach a power of c.
30 dB the signal starts to decrease, and at the highest power of 0 dB we face some 90% saturation. The optimal power value for this case would be approximately
32 dB, that is, the spectrum has maximal signal-to-noise ratio without being significantly saturated. There are two good reasons why saturation is to be avoided. Firstly, under (partial) saturation the signal amplitude is no longer linear in the applied power (expressed in dB) and so determination of the spin concentration (cf. next section) versus an external standard is no longer possible. Secondly, relaxation rate, and therefore saturation, is anisotropic: its extent with increasing power is different for different parts of the spectrum. Therefore under partial saturation a spectrum will change shape in a complex manner for whose analysis no theory is available to date. Fig. 4.4 gives the power plots for one single EPR signal taken at four different sample temperatures with T1 . T2 . T3 . T4. The values of the Ts are not specified because the quantitative temperature dependence of relaxation rates in EPR is usually rather complex.

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FIGURE

4.4 Theoretical power plots showing saturation as a collapse of normalized EPR amplitude with increasing microwave power. Traces are shown for four different temperatures with T1 . T2 . T3 . T4. At lower temperature the onset of saturation occurs at lower power. At T1 the signal is only slightly saturable; at T4 the signal cannot be measured under nonsaturating conditions.

Qualitatively, however, we can take the relaxation rate to always decrease with decreasing temperature: it is easier to saturate a signal at low T than at high T. This has the very important practical implication that for a given EPR signal we have to experimentally determine the optimal microwave power for each temperature at which we want to measure. In literature one frequently finds reports of the temperature-dependence of an EPR spectrum taken at a single, intermediate power level. It should be obvious from Fig. 4.4 that such an approach is far from optimal. Suppose, for example, that we take spectra at power 5
30 dB for all four Ts. At T1 the signal is unsaturated, but the signal-to-noise ratio is suboptimal; at T2 we have a near-optimal situation with high signal-to-noise and hardly any saturation; at T3 however the signal is seriously saturated (and therefore deformed) at this power; and at T4 there is hardly any intensity left. In fact for the example of Fig. 4.4 it is impossible to measure an EPR spectrum at T4 with any available power level without serious saturation problems. T4 is simply too cold for this sample. This situation is commonly encountered, for example, for S 5 1/2 systems, such as [2Fe
2S]11 proteins, recorded in X-band at temperatures close to the boiling point of liquid helium, that is, T  4.2 K. As a final practical observation on the subject of saturation note that modern computerized EPR spectrometers may offer the possibility to produce 2D power saturation data which are stacks of spectra taken automatically at fixed intervals of power values. Unfortunately these plots usually do not conform to the format in Fig. 4.4. In contrast the spectral amplitudes are corrected for the used power, so that the spectra taken at low power have correspondingly very small amplitudes, and it is not easily possible to make a reliable estimate of the onset of saturation. Correcting the amplitudes of the individual spectra for the used power would solve the problem, and when the commercial software that comes with the spectrometer does not provide for this option, writing a little routine that just does this job may significantly save time and avoid frustration.

Concentration determination Arguably one of the most useful applications of EPR in biochemistry is “spin counting,” that is, the determination of the concentration of spin systems. Picture a complex protein

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containing several paramagnets, for example the enzyme succinate dehydrogenase that contains a [2Fe
2S] cluster, a [3Fe
4S] cluster, a [4Fe
4S] cluster, a heme, and an FAD (flavin adenine dinucleotide) radical. Determination of their stoichiometry (at a given redox potential) would be a very difficult task indeed if one did not have quantitative EPR spectroscopy available. Since the intensity of a spectrum is defined by Eq. (4.5), one can simply take the area under the EPR absorption envelope (cf. the black trace in Fig. 4.1), corrected for the intensity expression, as a measure for the concentration of a paramagnet. An absolute concentration is obtained by comparison with the area under the EPR absorption envelope from a standard compound of known concentration. A commonly used standard is a frozen solution of the hydrated Cu21 ion such as 10 mM CuSO4 1 10 mM HCl 1 2 M NaClO4 with g values 2.404, 2.076, and 2.076 (Hagen, 2006). Note that addition of the “nonligand” perchlorate is to avoid di- and polymerization of the copper ions which would otherwise lead to severely broadened spectra through dipolar interaction. And adding 10 mM HCl is not the same as titrating to pH 2. In these analytical experiments the anisotropic intensity expression in Eq. (4.5) is usually replaced with an approximating average scalar according to Aasa and Va¨nnga˚rd (1975): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ðg2x 1 g2y 1 g2z Þ ðgx 1 gx 1 gx Þ 2 ð4:7Þ 1 I5 3 9 3 In practice the procedure is as follows: run a spectrum of the paramagnet with unknown concentration C Ð ÐU; read the approximate g values from the spectrum (cf. Fig. 4.1); take the second integral U (i.e., the area under the EPR absorption spectrum); and calculate the scalar intensity factor I in Eq. (4.7) for normalization. Do the same with Ð Ð the standard compound of known concentration CK, and then correct the final number K for any difference in experimental measuring conditions between the unknown and the standard. When the two spectra are taken under identical experimental conditions, we have RR IK CU 5 RR U ð4:8Þ CK I K U Most of the experimental differences between unknown and standard require linear corrections, for example, the modulation amplitude (M), the electronic amplification (G), the sample tube diameter (d), and/or the sample temperature (T). However, if the two spectra have different x-axis dimensions, that is, different magnetic field scan widths (W), then this correction should be taken squared, because the EPR derivative spectrum is integrated once to obtain the absorption and then once more to obtain the area under this absorption. Note also that a difference in used microwave power (P) in dB requires a logarithmic correction. Altogether, the concentration CU is now obtained as RR   IK MK GK dK TU WK 2 ðPU 2PK Þ CU 5 RR U 10 20 ð4:9Þ CK K I U M U G U d U TK W U Sometimes it may be preferable to replace the experimental spectrum with its simulation, for example, when integration of the experimental spectrum is unreliable because the

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Hyperfine interactions

139

FIGURE 4.5 Spin counting in a complex EPR spectrum. The experimental X-band (ν 5 9.25 GHz) spectrum of oxidized beef heart cytochrome c oxidase is simulated as a sum of four signals with intensity ratios 1:1:0.04:0.0005. The relative intensities of the simulated spectral components show that the dinuclear mixedvalence copper-A center and the cytochrome a are stoichiometric, that the EPR-silent dimer of copper-B and cytochrome a3 is slightly uncoupled affording a small amount of high-spin heme Fe(III), and that the preparation is very slightly contaminated with “dirty” iron.

baseline is of poor quality, or because the spectrum overlaps with other spectra, or because the spectrum extends over a field range that is not covered by the used electromagnet. And internal stoichiometries (e.g., of the different paramagnets in a multicenter protein) follow directly from the weighing factors used in the simulation. An example is given in Fig. 4.5: the spectrum of oxidized bovine cytochrome oxidase (EXP) is simulated (SIM) with four different spectral components, CuA, cyt a, cyt a3, dirty Fe, with relative stoichiometries 1:1:0.04:0.0005. These ratios make biochemical sense: CuA is the mixed-valence [Cu(II)
Cu(I)] cluster with S 5 1/2, cyt a is the low-spin heme of cytochrome a with S 5 1/2, and they occur once per cytochrome oxidase molecule. The other two metal centers, CuB (S 5 1/2) and cytochrome a3 (S 5 5/2), are antiferromagnetically exchange-coupled to a cluster spin S 5 5/2
1/2 5 2, which is not detectable under the used conditions. A small fraction of the Cu
Fe cluster is uncoupled (presumably by reduction of CuB to Cu(I)), and we observe a high-spin heme a3 spectrum of low intensity. The fourth component is a contamination with non-specifically bound high-spin Fe(III) of very low intensity. Note that this analysis could not have been based on inspection of the amplitudes of the four spectral components.

Hyperfine interactions Many nuclei have a nuclear spin I6¼0 and they are experienced by unpaired electrons in EPR as extra magnets, affording S2I or hyperfine interactions. Nuclei studied with NMR are mainly limited to those with I 5 1/2 (because the complexity of NMR spectroscopy with I $ 1/2 rapidly increases with I value), with two possibilities for the nuclear spin

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quantum number mI 5 1/2 and mI 5
1/2, corresponding to two energy states in an external field written in QM language with the “bra-ket” labeling as |mI 5 1/2i and |mI 5
1/2i. This limitation to I 5 1/2 does not usually hold for EPR because the S2I hyperfine interaction is a perturbation to the electronic Zeeman interaction (S2I is much weaker than S2B), and therefore allowed transitions are limited by the selection rule ΔmI 5 0. For example, for the copper ion Cu21, with S 5 1/2 and I 5 3/2, one finds only four EPR transitions, in combining bra-ket notation: |mS(i); mI(k)i2|mS(j); mI(k)i (i.e., the mI value does not change). The copper EPR line is split by a magnitude A (the hyperfine splitting expressed in gauss) into four lines centered around the field corresponding to the g value. So for an S 5 1/2 system with a single nuclear spin the number of EPR transitions is equal to 2I 1 1. This description holds under the condition that the hyperfine interaction is indeed significantly less than the Zeeman interaction, which is a reasonable assumption at X-band and higher frequencies. At low frequencies (c. 1 GHz or less for copper) the assumption no longer holds, and the spectra become more complex and their analysis more complicated. Just like the Zeeman interaction, the hyperfine interaction also is generally anisotropic, that is, different for different molecular orientations in the external magnetic field. The magnitude of the splitting in terms of the polar angles θ and ϕ of Fig. 4.3, used to define the direction cosines of Eq. (4.4), is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   A li gi 5 g22 l2x g4x A2x 1 l2y g4y A2y 1 l2z g4z A2z ð4:10Þ in which g is defined by Eq. (4.3), and from which it can be seen that the three features in a powder pattern (cf. the black trace in Fig. 4.1), namely the low-field absorption-like peak around gz, the intermediate-field derivative-like line around gy, and the high-field negative absorption-like peak around gx, will each be split into 2I 1 1 features, but with different splitting magnitudes Ai (together also called the A-tensor or the hyperfine tensor) for the different g values. Spectra with hyperfine structure can now be analyzed by simulation using Eq. (4.3) for the anisotropic g value and Eq. (4.10) for the anisotropic hyperfine splitting with the resonance field defined as Bres 5

X hν 2 AmI gβ mI

ð4:11Þ

A simple, but practically important, example is given in Fig. 4.6. Spin traps are diamagnetic compounds that readily react with unstable radicals to from meta-stable paramagnetic adducts. The spin trap 5,5-dimethyl-pyrroline N-oxide (DMPO) reacted with superoxide radical, O2•2 (added as KO2), affords the spectrum given in the red trace (exp). The unpaired electron of the superoxide delocalizes over the DMPO molecule, and consequently has hyperfine interaction with a nitrogen (14N with I 5 1) and a hydrogen (1H with I 5 1/2) nucleus. Because the adduct is a small molecule in water, all anisotropy is averaged away by tumbling, and one observes a single g value and single hyperfine splitting A values according to the isotropic resonance field expression Bres 5

hν N H H 2 AN iso mI 2 Aiso mI giso β

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ð4:12Þ

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Hyperfine interactions

FIGURE 4.6 Analysis of isotropic hyperfine interaction. The spin trap 5,5-dimethyl-pyrroline N-oxide (DMPO) has been mixed with superoxide and the room temperature aqueous solution EPR spectrum of the adduct has been simulated as a single peak split by hyperfine interaction with one nitrogen and one proton with AN  AH.

which has been used to simulate the spectrum in the black trace (sim). To understand how such a spectrum comes about the figure also gives simulations for the theoretical cases of interaction only with a single nitrogen (blue) or only with a single proton (green). The experimental spectrum can be thought of as a single line split into three by the nitrogen, with subsequent splitting of each of these three into two by the hydrogen. In general this should result in a total of six lines, however, since the splittings from the nitrogen and from the hydrogen by chance happen to be identical (AN 5 AH 5 15 gauss) some lines overlap and the result is a four-line spectrum with 1:2:2:1 intensity pattern. A more complicated example, now involving anisotropy, of the hydrated vanadyl ion (VOSO4 dissolved in acidified water) is analyzed in Fig. 4.7. Vanadium proteins with the vanadium reduced to V(IV) exhibit similar spectra (but with more noise due to the lower concentration). The spin system is S 5 1/2 and I 5 7/2, so we expect a powder pattern whose features are split into 2 3 7/2 1 1 5 8 lines. The structure of the hydrated VO21 ion (five H2O ligands) is quasi octahedral with the molecular z-axis defined along the V 5 O bond. The x- and y-axes are equivalent, so the symmetry is axial, and this implies that gx 5 gy and also Ax 5 Ay (but this is not generally the case in vanadium proteins, since their metal binding sites are of lower symmetry). The red trace in Fig. 4.7 is the experimental X-band spectrum taken with the sample immersed in liquid nitrogen (i.e., T 5 77 K). The black trace is an axial simulation. The blue trace is a simulation of what the spectrum would look like if there were no hyperfine interaction. The green trace is a simulation of the EPR absorption spectrum for a single molecular orientation (the “parallel” orientation), namely, for the external magnetic field B along the molecular z-axis; similarly, the magenta trace is for a single molecular orientation (“perpendicular”), namely, for B along the molecular x-axis (or, for that matter, anywhere else in the xy plane). Note that the hyperfine splitting in the z-direction, Az, affords positive peaks on the low-field side but negative peaks of the high-field side of the powder pattern. A good estimate of the value Az is obtained by taking the difference in field position between the highest-field negative peak and the lowest-field positive peak divided by seven (i.e., 2 3 I). Also from the average field

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FIGURE 4.7 Analysis of anisotropic hyperfine interaction. The 9.42 GHz frozen solution spectrum (T 5 77 K) of the S 5 1/2 system V41 in vanadyl sulfate in acidic water is split by hyperfine interaction with the 51V nucleus (I 5 7/2). The hydrated vanadyl ion VO(H2O)521 has axial symmetry with g|| 5 1.927, g\ 5 1.972, A|| 5 201 gauss, A\ 5 75 gauss. The blue trace “nohyp” is a simulation with I 5 0; the green trace “parl” simulates the absorption EPR for θ 5 0 (i.e., along the z-axis), and the magenta trace “perp” is for θ 5 90 degrees (i.e., in the xy plane).

position of these eight peaks (or of the first and the last one) one can make a good estimate of the gz value. It is more difficult to estimate the gx and Ax values from the “messy” middle part of the spectrum. One of the reasons can be appreciated by inspection of the single-orientation spectrum of the magenta trace: the distance between the eight hyperfine peaks is not constant, and the width of the peaks is also not constant. The first phenomenon is called a “second-order effect,” which means that Eq. (4.9) (derived using perturbation theory) is not exactly correct for these large hyperfine interactions and requires a significant correction in X-band. The details are beyond the scope of this chapter but the required equations can be found in Hagen (2009). The variable line width is a reflection of conformational distribution of the VO(H2O)521 structure, which leads to an mI-dependence of the width of the form W ðmI Þ 5 W0 1 c1 mI 1 c2 m2I

ð4:13Þ

The bottom line is that approximate g and A values may be estimated directly from the spectrum, however, accurate analysis of spectra of the type shown in Fig. 4.7 requires computer simulation affording high-quality fits (Hagen, 2009). Even without this sophisticated analysis, inspection of hyperfine structure can be very useful because simple “line counting” provides direct information on the chemical elements involved in the spin system. For example, in the spectrum of Fig. 4.7 with two Az peaks on the low-field side resolved and three Az (negative) peaks on the high-field side,

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TABLE 4.4 Biological transition metal ions and their nuclear spin. Metal

Isotope mass number

Spin (abundance)

EPR hyperfine lines (intensity)

V

51

7/2

8

Mn

55

5/2

6

Fe

54, 56, 57, 58

0, 1/2 (2%)

1, 2(1%)

Co

59

7/2

8

Ni

58, 60, 61, 62, 64

0, 3/2 (1%)

1, 4 (0.25%)

Cu

63, 65

3/2

4

Mo

92, 94, 95, 96, 97, 98, 100

0, 5/2 (25%)

1, 6(4%)

W

180, 182, 183, 184, 186

0, 1/2 (14%)

1, 2(7%)

it is straightforwardly established that there are eight Az peaks in total, and therefore that I 5 7/2. In a biochemical setting this would define the spectrum to be either from a lowspin Co(II) complex or from a V(IV) complex, since V and Co are the only biometals with I 5 7/2. Since crystal-field theory dictates that Co(II) has g . ge and V(IV) has g , ge (ge 5 2.0023) the distinction is readily made. There are no other splittings in addition to those due to the vanadium nucleus, which is consistent with all ligands being oxygen (16O has I 5 0). The protons (1H has I 5 1/2) of the water ligands are apparently too far away from the unpaired electron to afford resolved splittings, but they could be studied with special techniques, for example, ENDOR or ESEEM. Table 4.4 lists metal isotopes that are relevant in the frame of hyperfine structure in biological EPR; the mass numbers of isotopes with a nuclear spin are underlined. Note that when the percentage natural abundance of an isotope with nuclear spin is less than 100, the remainder is made up of isotopes with I 5 0, and the EPR spectrum is the sum of hyperfine split and unsplit spectra. For example, natural molybdenum consists of the isotopes with mass number 92, 94, 95, 96, 97, 98, and 100, and the EPR spectrum is for 25.5% split into six lines and for 74.5% unsplit. Since the splitting into six reduces the amplitude by a factor of six, the amplitudes of the split spectrum and the unsplit spectrum relate as 25.5/6 versus 74.5 or in percentages 5.4% versus 94.6%. Sometimes the symmetry of (biological) complexes is so low that it is not clear how to define a Cartesian axis system, with related direction cosines (cf. Eq. 4.4), in terms of the molecular structure. Analysis of the EPR may then reveal that the axis system for the g values defined by Eq. (4.3) differs from the axis system for the hyperfine A values defined by Eq. (4.10). Mathematically this means that Eq. (4.10) has to be rotated in 3D space with respect to Eq. (4.3) (in formal wording the g tensor and the A tensor diagonalize in different axes systems), and the EPR analysis becomes very involved. In practice this so-called “axes noncolinearity” from low molecular symmetry can be qualitatively recognized in the EPR spectrum as small extra peaks and shoulders in between the hyperfine lines (Hagen, 2009).

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TABLE 4.5 Biologically relevant ligand atoms and their nuclear spin. Ligand atom (intensity)

Isotope mass number

Spin (abundance)

Superhyperfine lines (intensity)

H

1, 2

1/2, 1(0.01%)

2, 3

C

12, 13

0, 1/2 (1.1%)

1, 2 (0.5%)

N

14, 15

1, 1/2 (0.4%)

3, 2

O

16, 17

0, 5/2 (0.04%)

1, 6 (0.07%)

F

19

1/2

2

P

31

1/2

2

S

32, 33, 34

0, 3/2 (0.8%)

1, 4 (0.2%)

Cl

35, 37

3/2

4

As

75

3/2

4

Se

76, 77, 78, 80, 82

0, 1/2 (7.6%)

1, 2 (3.8%)

Br

79, 81

3/2

4

I

127

5/2

6

The donor atoms of ligands that are coordinated by a metal can also have a nuclear spin. In EPR this nuclear spin may be detected through superhyperfine interaction, that is, the interaction of the electron spin S with the ligand atom nuclear spin I. Its observation implies that the unpaired electron(s) of the metal has spin density at the ligand atom, which is a direct manifestation of the covalent character of the coordination bond. In biological EPR spectroscopy it is used as an identifier of structure (and change of structure following a reaction) of prosthetic groups. An especially relevant case for enzymology is the observation of superhyperfine splittings from atoms (possibly enriched in certain isotopes) of a substrate that binds directly to the metal ion in an active site of an enzyme. Some biologically relevant ligand atom isotopes and their nuclear spins are given in Table 4.5; again the mass numbers of nuclei with I6¼0 are underlined. Possibly the most common superhyperfine pattern found in the EPR of biological systems is the three-line pattern of the metal-coordinated nitrogen in histidine ligand. A notorious example is that of NO-reacted ferrous heme in enzymes as exemplified in Fig. 4.8 (Fraaije et al., 1996). In particular the gy derivative-shaped feature is split into three by the nitrogen from the axial NO ligand, and each of these three lines is again split (by a smaller amount) by the nitrogen from the other axial ligand which is a histidine nitrogen. The experiment does not only unequivocally establish histidine as an axial ligand, it also shows that, after binding, the unpaired electron of the NO radical has delocalized over the iron onto the histidine nitrogen where it now has some finite spin density.

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High-spin systems

FIGURE 4.8 EPR spectrum at 9.18 GHz (top) of a heme-containing enzyme (catalase-peroxidase) reacted with NO to form an Fe(II)
NO paramagnet with S 5 1/2. The rhombic spectrum is split by a strong superhyperfine interaction with the NO nitrogen and a weaker superhyperfine interaction with a histidine nitrogen. The bottom trace is a simulation with Azyx 5 20.5, 12, 10 gauss for the NO nitrogen and Azyx 5 6.5, 6.5, 6.5 gauss for the histidine nitrogen. Reproduced with permission from Fraaije, M.W., Roubroeks, H.P., Hagen, W.R., Van Berkel, W.J.H., 1996. Purification and characterization of an intracellular catalaseperoxidase from Penicillium simplicissimum. Eur. J. Biochem. 235, 192
198.

High-spin systems High-spin systems (S . 1/2) are intrinsically more complex than S 5 1/2 systems because they have more than two (namely 2S 1 1) magnetic sublevels, and this has two important consequences: (1) the number of possible EPR transitions is greater than one, and (2) the observed spectral features define “effective” g values, that is, the real g values from the Zeeman interaction are dramatically shifted by the S2S interaction between unpaired electrons (also called zero-field interaction because it is always present even when the magnet is turned off). We use the example of S 5 5/2 (e.g., high-spin Fe31) to illustrate these matters. There are six sublevels labeled with the magnetic quantum numbers mS 5 15/2, 13/2, 11/2,
1/2,
3/2,
5/2, which could in principle afford up to 15 EPR transitions. In practice, however, this number is usually seriously restricted by the selection rule ΔmS 5 1 and by the fact that some zero-field interlevel splittings are greater than the microwave quantum hν (which makes a transition impossible). The ΔmS 5 1 selection rule is a quantum mechanical shorthand notation to state that some of the possible transitions have low probability, and are practically undetectable. In many high-spin biological systems and model compounds at X-band frequencies the zero-field S2S interaction is much stronger than the Zeeman S2B interaction, and this results in a grouping of sublevels in so-called Kramers pairs (or doublets) with mS 5 6n/2 separated by an energy Δ . hν. For S 5 5/2

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FIGURE 4.9 S 5 5/2 spin manifold for axial symmetry (E 5 0) and for maximal rhombicity (E/D 5 1/3). Effective g values are given for all three intradoublet spectra assuming a real isotropic g 5 2.00.

we find the pairs mS 5 61/2, mS 5 63/2, and mS 5 65/2, which in hemoproteins (e.g., ferrimyoglobin) are separated by Δ1 5 2D (see below) and Δ2 5 4D with typically D  10 cm21, that is, energy differences much greater than the X-band quantum of c. 0.3 cm21, cf. the left-hand panel of Fig. 4.9. Together with the ΔmS 5 1 selection rule only a single transition is possible, namely, the intradoublet transition within the mS 5 61/2 pair. This would make the system very similar to a low-spin S 5 1/2 one; indeed it is therefore called an “effective” spin 1/2 system. There are two important differences with real S 5 1/2 systems. Firstly, the real g values for a system with a half-filled outer electron shell (Fe31 is d5) are predicted to be very close to ge, however, for this effective S 5 1/2 system we find two of the effective g values close to g 5 6. Secondly, although the mS 5 63/2 and mS 5 65/2 doublets may not contribute to the EPR spectrum, they are populated by molecules, and so the temperature dependence of the EPR intensity is more complex than that of a real S 5 1/2 system. At high temperature only one-third of all molecules is in the mS 5 61/2 doublet, so when counting spins we have to multiply the double integral value by three. At very low temperature, for example, 4.2 K, only the mS 5 61/2 doublet may be populated, and so no correction is required. At intermediate temperatures a correction has to be made according to the Boltzmann distribution over the three doublets (see below). In 3D space the anisotropic Zeeman interaction S2B and the anisotropic hyperfine interaction S2I are characterized by sets of three parameters: gx, gy, gz (the g tensor), and Ax, Ay, Az (the A tensor). Similarly, the anisotropic zero-field interaction S2S between electrons is characterized by the set of parameters Dx, Dy, Dz (the D tensor). However, in this case it can be shown that the three parameters are not independent, Dx2 1 Dy2 1 Dz2 5 0, and so the set can be reduced to two independent parameters, D 

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147

3Dz/2 and E  (Dx 2 Dy)/2. In the case of axial symmetry we have gx 5 gy and Ax 5 Ay, but Dx 5 Dy implies that E 5 0, so an axial zero-field interaction is described by a single parameter D. In X-band the hyperfine interaction is a perturbation to the Zeeman interaction (S2I ,, S2B) leading to characteristic splittings of the EPR spectrum as described in the previous section. Similarly, if the zero-field interaction is a perturbation to the Zeeman interaction (S2S ,, S2B), it leads to splittings of the EPR spectrum. For example the S 5 1 EPR of organic biradicals is split by D/geβ and (D63E)/geβ gauss (cf. Wasserman et al., 1964). However, for metal complexes we noted above that in X-band the zero-field interaction is usually dominant (S2S .. S2B) and, therefore the spectrum of a high-spin transition ion complex does not show spectral splittings from S2S interactions. And while we have generally valid simple analytical expressions for the resonance field as a function of g and A values (cf. Eqs. 4.3 and 4.10) the situation is somewhat different for systems with dominant zero-field interaction. In the EPR literature this is also called the “weak-field” case, since the Zeeman interaction, which depends on the magnetic field, is weak compared to the field-independent zero-field interaction. For the lowest doublet (mS 5 61/2) of an S 5 5/2 system (e.g., high-spin ferric heme proteins) the following equations hold in the weak-field limit (Hagen, 1981): " #  2 gx βBx 4E eff gx 5 3gx 1 2 2 ð4:14Þ D 2D2 " #  2 gy βBy 4E eff gy 5 3gy 1 2 1 ð4:15Þ D 2D2 geff z 5 gz

ð4:16Þ

in which the superscript “eff” of the left-hand symbols stands for effective g values, or real g values shifted by the zero-field interaction. The effective g values are the observables of an effective S 5 1/2 system and their angular dependence is as in Eq. (4.3). Note that the right-hand sides of Eqs. (4.14)
(4.16) contain a total of five unknowns, so with three observables the system is underdetermined when the EPR spectrum is measured at one frequency only. For the 3d5 high-spin system Fe31 the real g values happen to be very close to ge but even this simplification does not help much since, due to the large value of D eff eff compared to the X-band microwave quantum, the effective values gx and gy are very insensitive to the value of D unless one would collect multifrequency data including spectra at high frequencies well above 100 GHz (Van Kan et al., 1998). Thus it turns out that we can only determine the quantity E/D with some accuracy, because the splitting eff eff between gx and gy is approximately equal to 48E/D (assuming gx 5 gy 5 2). This is an example of a very general observation that the effective g values of half integer high-spin (S 5 n/2) systems, in particular metalloproteins, are dominated by a single parameter η 5 E/D also known as the rhombicity parameter. We can estimate the value of D from a fit of the temperature dependence of the EPR intensity to a Boltzmann distribution over all the sublevels of the spin manifold. For S 5 5/2 the relation is

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FIGURE 4.10 Rhombogram for S 5 5/2. Effective g values as a function of rhombicity are given for all three intradoublet spectra: the red traces are for the 61/2 doublet, the black traces are for the 63/2 doublet, and the blue traces are for the 65/2 doublet. The real g tensor is assumed to be isotropic, g 5 2.00. The arrows indicate rhombicities for the real spectra in Fig. 4.11.



21 I ~ 11e22D=kT 1e26D=kT T21

ð4:17Þ

in which T is the absolute temperature and it is assumed that D .. E. An example of such a determination for an S 5 7/2 system can be found in Hagen et al. (1987). To understand high-spin EPR spectra generally requires making simulations based on energy matrix diagonalization techniques. This complex subject is beyond the scope of this chapter (it is addressed in detail, e.g., in Hagen, 2009). However, its application results in effective g values as a function of rhombicity η 5 E/D only. Combined with the theoretical result that rhombicity is limited to 0 # η # 1/3 (Troup and Hutton, 1964) this allows for the practical procedure of getting effective g values from tables or figures or fast computer programs (Hagen, 2009). An example is given in Fig. 4.10. These plots are called “rhombograms”; the rhombicity E/D is on the x-axis and it runs from zero to its theoretical maximum 1/3. On the y-axis are effective g values, and the eff eff eff graph defines gx , gy , and gz for the intradoublet spectra from the three doublets mS 5 61/2 (red), mS 5 63/2 (black), and mS 5 65/2 (blue). So a given S 5 5/2 compound will have its unique rhombicity value η 5 E/D, related to the molecular symmetry at the paramagnetic center, which defines, according to Fig. 4.10, three sets of g values for three overlapping spectra. The graph also gives an indication of the relative intensity of these three spectra: the closer the three g values are to each other the narrower the spectrum and the higher its amplitude will be. Note, however, that spectra are taken on a linear

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High-spin systems

FIGURE 4.11 Examples of S 5 5/2 spectra (red traces) and their simulations (black traces) based on effective g values from rhombograms with rhombicity η 5 E/D. The top spectrum (ν 5 9.18 GHz; T 5 15 K) is from high-spin heme Fe(III) in PAC (Penicillium simplicissimum atypical catalase), and the bottom spectrum (ν 5 9.30 GHz; T 5 90 K) is from high-spin Fe(III) in Escherichia coli iron-superoxide dismutase.

magnetic field scale, that is, on a reciprocal g value scale. So moving g values toward zero will rapidly increase the field range of the spectrum and thus rapidly decrease its amplitude. In fact, geff 5 0 means a resonance at infinite field and thus an infinitely wide scan range and an infinitely small amplitude. In other words, the transition is forbidden. This is what we noted above to be the case for ferrimyoglobin, for which the energy level of the left-hand panel of Fig. 4.9 holds: two of the three possible spectra have geff 5 0, and therefore are totally unobservable. On the other hand, the Fe(III) in the protein rubredoxin, coordinated by four cysteine sulfurs in a strongly deformed tetrahedron, has η 5 E/D  1/3, for which the right-hand panel of Fig. 4.9 holds: all three spectra are allowed, but the one from the middle doublet will have the highest amplitude by far, because it incidentally has an isoeff eff eff tropic geff value (gx 5 gy 5 gz 5 4:29). Fig. 4.11 gives two experimental example spectra for cases of intermediate rhombicity: the slightly rhombic (η 5 0.023) heme iron site in a catalase, and the rather rhombic (η 5 0.238) iron site in a superoxide dismutase. Their effective g values are indicated by arrows on the top of the rhombogram of Fig. 4.10. Hagen (2009) gives rhombograms and example spectra for S 5 3/2
9/2. All metalloproteins and models are subject to conformational distributions, which in turn lead to distributions in EPR parameters, that is, g strain. Where a distribution in g values leads to broadening and skewing of the main features in the powder pattern, and a distribution in A values leads to nuclear-orientation dependent line width (cf. Eq. 4.13), a distribution in D values leads to broadening of all features in high-spin spectra,

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however, with the broadening increasing over the field scan. In other words, the low-field features of spectra from S 5 n/2 systems with large zero-field splittings always exhibit the best resolution (Hagen, 2007). Thus far, we have limited the discussion to half-integer spin systems. Integer spin systems with S 5 1, 2, 3, etc., form a special class from the viewpoint of EPR spectroscopy. Many of these systems are “EPR silent,” which means that no signal is found in X-band EPR. This situation is very different from that of the half-integer spins, which should always afford at least one signal at any frequency. The reason is that the equivalents of Kramers pair doublets in half-integer spins are non-Kramers pair doublets in integer spin. According to QM Kramers pairs are always degenerate (i.e., they have the same energy) in zero field; non-Kramers pairs do not have this restriction, and, in fact they are always split in zero field (with the exception of axial S 5 1 systems) (Hagen, 2009). If this splitting happens to be greater than the microwave quantum hν, then no transitions are possible at frequency ν, and one has to turn to a spectrometer with (much) higher frequency to detect EPR. When the zero-field intradoublet splitting is small (Δ , hν) then EPR is possible, but the transition probability is low for normal spectrometers in which the external field B is perpendicular to the magnetic component B’ of the microwave (B\B’). One frequently has to use a special cavity, the parallel-mode resonator, in which the external field is parallel to the magnetic component of the microwave, B||B’, to get reasonably sharp spectra of reasonable intensity (Hagen, 1982). See Hagen (2006, 2009) for further theoretical and experimental details. A brief, practical summary of integer-spin X-band EPR of metalloproteins and models is as follows. S 5 2 is the most commonly observed system. S 5 1 systems are rarely detectable. S 5 2 systems are either EPR silent or give a single, broad line at low field with geff $ 8 (e.g., the [3Fe
4S]0 cluster). S . 2 systems in biology are restricted to metal clusters (e.g., the P-cluster in nitrogenase enzymes), and they may exhibit a single, relatively sharp line in parallel-mode EPR with geff  4S (Hagen, 2009).

Applications overview EPR is a “something for everyone” spectroscopy: practical and useful EPR applications on biomolecules and models can range from very simple to very involved experiments and analyses. “Wow, this protein contains a metal cofactor ‘X’!” could well be the verbal synopsis of a breakthrough result from a 5-min, first-trial EPR measurement. Such an identification does not necessarily require any knowledge of EPR theory, as it can be based on fingerprint correlation, that is, comparing the EPR spectrum (e.g., shape, peak positions, broadening with increasing temperature) with literature data on characterized systems. Rather more often, however, some qualitative or (semi-)quantitative understanding of EPR may support assignments. For example, “This signal is split into four lines, so it must be from a copper complex, and since the Az value is in the range of 30
100 gauss we must have a type-I, or ‘blue’ copper protein.” Or, “The gz . g\ so EPR combined with ligandfield theory stipulated that this Cu21 probably has an axially elongated octahedral coordination.” In other words, the element identification usually also affords a qualitative

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conclusion about the coordination environment. And, of course, observing a copper EPR signal from a protein also immediately defines the oxidation state of the metal to be 2 1 . Input of slightly more effort can make the experiment quantitative, and thus considerably increase its biological relevance. Establishing nonsaturating measurement conditions and then counting spins from the doubly-integrated EPR with respect to an external standard, provides a concentration of the metal, which, when compared to protein concentration, allows for important interpretational discriminations such as “The Cu/protein ratio is approximately unity, so this must be a copper protein” versus “The Cu/protein ratio is c. 0.05, so this is either a heavily demetallated copper protein, or, perhaps more likely, it is a noncopper protein that in the course of the purification procedure became contaminated with extraneous copper.” Observation and qualitative understanding of superhyperfine splittings (the “super” means from ligands, not from the central metal) can identify ligands such as nitrogens in metal coordination. For example, observation of a line that is split into five lines with an intensity ratio of 1:2:3:2:1, would be consistent with coordination by two equivalent 14Ns (i.e., with the same A value). Alternatively, a line that is split into nine lines of equal intensity, would be consistent with coordination by two nonequivalent Ns (i.e., with significantly different A values). A chemically more sophisticated form of this type of experiment is to isotopically label an enzyme’s substrate (or better a nonconvertible substrate analogue), and then, after reaction of substrate and enzyme, to look for hyperfine splittings, for example, in the ENDOR of the metal in the catalytic center of the enzyme. Identification of such splittings from, for example, 2H, 13C, or 15N, not only proves that the substrate directly binds to the metal (which is also shown by a change in the EPR upon incubation with substrate), but it may also reveal which part of the substrate is coordinated by the metal. By the way, observation of superhyperfine interaction in EPR is the most direct way a (bio)chemist has available to show that a coordination bond possesses covalent character: it proves that the unpaired electron of the metal must be delocalized and spend some of its time on the ligand. Clustering of metal ions, especially in iron
sulfur clusters, is another phenomenon that is readily identified in EPR experiments. Since iron
sulfur clusters in proteins are relatively labile entities (they contain acid-labile sulfur, so the cluster disintegrates upon acidification), iron is easily liberated from the protein and iron content is readily determined colorimetrically. Suppose a protein is found to contain c. 4 irons. If the EPR of the protein only shows a single spin system, then the straightforward conclusion is that all irons are electronically coupled by exchange interaction (more precisely superexchange, because the coupling is not directly between irons, but rather over two chemical bonds via the sulfurs). Interaction of electrons by exchange is significant only over a distance of one or two chemical bonds, but through-space interaction between electrons (i.e., dipolar interaction) is significant over longer distances. This particular form of S2S interaction gives small splittings and shoulders in X-band EPR for spin systems at mutual distances of the order ˚ . Its observation is a qualitative indicator for the fact that two centers “see each of 5
15 A ˚ , and therefore are located in the same protein other,” that is, are separated by some 10 A complex. Quantitative interpretation in terms of accurate distance and mutual orientation is possible but requires involved analyses usually based on data taken at several microwave frequencies.

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If the identification of a signal with a particular structure is unequivocal, then one can also forget about EPR theory and use the spectrometer as a black box. In these types of experiments the amplitude of a signal is usually measured as a function of some external parameter, for example, redox potential (to determine reduction potentials), concentration (to determine binding affinities), reaction time (to determine rate constants), acidity (to determine pKs), etc. The experimental challenges are now not so much in the EPR spectroscopy, but rather in the setups to chemically prepare the EPR samples, for example, anaerobicity, rapid mixing, rapid freezing, etc. On a final note, let us consider the particular situation of EPR spectroscopy on complex biological systems such as multicenter enzymes, complete respiratory chains, cell organelles such as mitochondria, or even whole prokaryotic or eukaryotic cells. We are now facing a situation of several-to-many overlapping spectra while at the same time the concentration of the individual paramagnets drops down to the micromolar range or less. Nevertheless, it is particularly on this battlefield that biological EPR has savored some of its most consequential victories such as the initiation of the ever-expanding field of iron
sulfur biochemistry that started off with the discovery of an “unusual” EPR signal with g 5 1.94 in mammalian heart mitochondria some six decades ago (Beinert and Sands, 1960). A range of special measures has been developed to support this type of experimentation: (1) procedures to maximize concentration, for example, by centrifugation of samples in EPR tubes; (2) statistics on EPR from multiple, similar preparations; (3) extensive deconvolution of spectra by thermodynamic (redox equilibrium titrations) and/or kinetic (rapid redox reactions) resolution; (4) deliberate partial power saturation and/or magnetic field overmodulation of EPR; and (5) molecular biological constructs that overexpress paramagnetic metalloproteins of interest. Some of these approaches (and their putative pitfalls) were discussed in more detail in a recent review (Hagen, 2018).

Test questions [q1] (A) The g value of a radical measured at ν 5 9 GHz is g 5 2.006; what is its g value at ν 5 250 GHz? (B) Why does an S 5 1/2 EPR spectrum generally have three “peaks”? [q2] (A) Why do EPR spectrometers have a resonator/cavity? (B) Why does the microwave bridge of an EPR spectrometer have a reference arm? [q3] (A) What is the EPR equivalent of the extinction coefficient in optical spectroscopy? (B) The detection limit for X-band EPR of a reduced [2Fe
2S] cluster is c. 5 μM. How many mg of an 11 kDa ferredoxin do you have to prepare at minimum to measure a spectrum? [q4] In a small coordination complex in water an S 5 1/2 metal ion is coordinated by four nitrogens. (A) How many EPR lines do we observe when the hyperfine interaction with all 14Ns is of the same strength? (B) And how many lines do we get if each N has a very different A value? [q5] In Fig. 4.7, why does the vanadium hyperfine pattern in the z-direction give positive peaks at low field but negative peaks at high field?

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[q6] With reference to the rhombogram in Fig. 4.10, what approximately would be the relative intensities of the three subspectra of an S 5 5/2 system with rhombicity E/D 5 0.11? [q7] Why are some integer-spin systems EPR silent?

Answers to test questions [a1] (A) A real g value is frequency-invariant: if ν goes up, then Bres goes up too, and their ratio is constant, therefore also g 5 hν/βB is constant. (B) The derivative of the EPR absorption spectrum emphasizes the turning points of the powder pattern. [a2] (A) Creating a standing microwave increases the spectrometer’s sensitivity by Q  5000. (B) Diverging a fixed, small amount of power to the detection diode makes it possible to change the power to the sample without need to retune the spectrometer. [a3] (A) The EPR extinction coefficient is unity, and the EPR intensity is given by Eq. (4.5) or is approximated by the average scalar in Eq. (4.10). (B) 1 mM is 11 mg/mL, so 5 μM in 0.2 mL is 0.011 mg. [a4] (A) 9. (B) 81. [a5] Experimental EPR spectra are the first derivative of EPR absorption spectra. The first derivative gives the slope of the absorption (the latter is positive by definition); approaching the end of the spectrum the slope must be negative in order to return to the zero level of the baseline. [a6] The anisotropy in the effective g tensor of the 61/2 and the 63/2 doublet are similar, and, therefore the two subspectra will have comparable intensity. Anisotropy for the 65/2 intradoublet transition is very extensive, so the intensity of this subspectrum will be extremely low. [a7] For integer-spin systems (also non-Kramers systems) it is possible that all splittings between the energy levels of the spin manifold are greater than the microwave quantum.

References Aasa, R., Va¨nnga˚rd, T., 1975. EPR signal intensity and powder shapes: a reexamination. J. Magn. Reson. 19, 308
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533. Beinert, H., Sands, R.H., 1960. Studies on succinic and DPNH dehydrogenase preparations by paramagnetic resonance (EPR) spectroscopy. Biochim. Biophys. Res. Commun. 3, 41
46. Dutton, P.L., Leigh, J.S., Seibert, M., 1972. Primary processes in photosynthesis: in situ ESR studies on the light induced oxidized and triplet state of reaction center bacteriochlorophyll. Biochem. Biophys. Res. Commun. 46, 406
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198. Hagen, W.R., 1981. Dislocation strain broadening as a source of anisotropic line width and asymmetrical line shape in the electron paramagnetic resonance spectrum of metalloproteins and related systems. J. Magn. Reson. 44, 447
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Hagen, W.R., 1982. EPR of non-Kramers doublets in biological systems: characterization of an S 5 2 system in oxidized cytochrome c oxidase. Biochim. Biophys. Acta 708, 82
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634. Hagen, W.R., Wassink, H., Eady, R.R., Smith, B.E., Haaker, H., 1987. Quantitative EPR of an S 5 7/2 system in thionine-oxidized MoFe proteins of nitrogenase; a redefinition of the P-cluster concept. Eur. J. Biochem. 169, 457
465. Troup, G.J., Hutton, D.R., 1964. Paramagnetic resonance of Fe31 in kyanite. Br. J. Appl. Phys. 15, 1493
1499. Van Kan, P.J.M., Van der Horst, E., Reijerse, E.J., Van Bentum, J.M., Hagen, W.R., 1998. Multi-frequency EPR spectroscopy of myoglobin; spectral effects for high-spin Fe(III) ion at high magnetic fields. J. Chem. Soc. Faraday Trans. 94, 2975
2978. Wasserman, E., Snyder, I.C., Yager, W.A., 1964. ESR of the triplet states of randomly oriented molecules. J. Chem. Phys. 41, 1763
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