Pulse Dipolar Electron Spin Resonance: Distance Measurements

Abstract

In recent years electron spin resonance (ESR) has provided the means to obtain structural constraints in the field of structural biology on the nanoscale by measuring distances between paramagnetic species, which usually have been nitroxide spin-labels. These ESR methods enable the measurement of distances over the wide range from ca. 6–10 Å to nearly 90 Å. While cw methods may be used for the shortest distances, it is the pulse methods that enable this wide range, as well as determination of the distributions in distance. In this chapter we first describe the underlying theoretical concepts for understanding the principal pulse methods of double quantum coherence (DQC)-ESR and double-electron–electron-resonance (DEER), which we collectively refer to as Pulse-Dipolar ESR Spectroscopies (PDS). We then provide technical aspects of pulse ESR spectrometers required for high quality PDS studies. This is followed by an extensive description of sensitivity considerations in PDS, based largely upon our highly sensitive 17.3 GHz pulse spectrometer at ACERT. This description also includes a comparison of the effectiveness of the respective PDS pulse methods. In addition, the newer methods of 5-pulse DEER, which enables longer distances to be measured than by standard DEER, and 2D-DQC, which provides a convenient mapping for studying orientational coherence between spin labels and their interspin vector, are described.

Appendix

1.1 Signals in 3,4,5-Pulse DEER Sequences

Here, we derive the expression given by Eq. (40) for PELDOR/DEER (cf. Fig. 5a), using the spin Hamiltonian given by Eqs. (13) and (14) (which neglects pseudosecular terms), in the absence of pulses. We express H ₀ in the frame of reference doubly rotating with frequencies ω ₁ and ω ₂ of mw pulses applied, respectively, to spins A and B, having their Larmor frequencies at Ω _a and Ω _b (cf. Slichter, p. 279, and assumptions therein [95]). Note that Eqs. (13) and (14) use spins 1 and 2, but for DEER pulse sequences we number spins by the subscripts a and b. In this frame of reference H ₀ becomes

$$ {H_0}={\omega_a}{S_{az }}+{\omega_b}{S_{bz }}+a{S_{az }}{S_{bz }} $$

(63)

In Eq. (63) a is as in Eq. (14), ω _a and ω _b are the Larmor frequency offsets from ω ₁ and ω ₂ respectively. We further assume the following set of inequalities a ≪ γ _e B _1a(b) ≪ |ω ₁ − ω ₂|. The first inequality allows us to neglect the dipolar coupling during the pulse, the second ensures that there may be only a small overlap of pulse excitations at the two frequencies, but we will retain related terms that may be produced in the course of calculations of the signal. (The first inequality, as related to A-spins, makes it easier for one to consider pseudosecular terms in conducting a more detailed analysis). Note that for a pair of spins, depending on angle θ, one of or both spins may contribute to the echo. We can assume that the first spin is always an A spin, but the second spin can be either A or B spin. To simplify this matter, we use when needed the subscripts numbering spins as 1 or 2.

The amplitude V(t) of the echo signal that we are interested in computing is given by the trace, Tr(S _a+ ρ(t))/Tr(S _a+ S _a−), where ρ(t) is the density matrix measured at time t after the first pulse in the sequence. Therefore in the end we retain in ρ only the terms in S _a−. We will follow the evolution of single-quantum in-phase coherences of spin 1, S _1a± created by the first π/2 pulse. They evolve due to the dipolar coupling aS _1z S _2z into anti-phase coherences S _1a± S _2z and vice versa; the process thus interconverts these coherences leading to their modulation by the dipolar frequency a/2 as described in Sect. 2.3.1. These coherences under the action of pulses and free evolution periods will turn out as detectable S _a− carrying this dipolar modulation.

The pulse sequences for 3-pulse PELDOR and 4-pulse DEER can be expressed in arrow representation respectively as:

$$ \begin{array}{lll} {P_{1a }}(\pi /2){\mathop{\longrightarrow}\limits^{H}_{0}({t}_{1}-t)}\;\;{P_{2b }}(\pi ){\mathop{\longrightarrow}\limits^{H_0(t) }}\;\;{P_{3a }}(\pi ){\mathop{\longrightarrow}\limits^{H}_{0}({t}_{1}+{t_{\mathrm{ e}}})}\;\;echo \hfill \cr {P_{1a }}(\pi /2)\mathop{\longrightarrow}\limits^{H}_{0}({t}_{1})\;\;{P_{2a }}(\pi ){\mathop{\longrightarrow}\limits^{H_0(t) }}\;\;{P_{3b }}(\pi ){\mathop{\longrightarrow}\limits^{H}_{0}({t}_{2}-t)}\;\;{P_{4a }}(\pi )\mathop{\longrightarrow}\limits^{H}_{0}({t}_{2}-{t_1}+{t_{\mathrm{ e}}})\;\;echo, \end{array} $$

(64)

where H ₀(t) denotes free evolution for the duration of t due to H ₀ and P _ka(b) is the pulse propagator for kth pulse applied nominally at the frequency ω _a or ω _b . The primary echo produced by pulses 1 and 3 in 3-pulse PELDOR corresponds to a coherence pathway, p = (+1, −1). In 4-pulse DEER based on the refocused echo created by pulses 1, 2, and 4 coherences pass through a p = (−1, +1, −1) pathway. We describe the action of π-pulses by introducing probability p _ka or p _kb for the spin at ω _a or ω _b , respectively, to be flipped by the kth pulse. (We may drop the subscript a (or b), when unimportant.) The probability not to be flipped, q _kc , is then 1 − p _kc (where c is a or b). We denote the amount of S _1a± produced by the first π/2 pulse as h _1a . Note that q, p, and h correspond to standard amplitude factors for the action of selective pulses, for example, as defined in the literature [35]. For a spin at a resonant frequency offset ω from the frequency of the RF pulse, the probabilities p and q to be flipped or not flipped by the pulse with nominal flip angle β is given by

$$ p={\sin^2}(\beta u/2)/{u^2},\quad q=1-p $$

(65)

Here, ${u^2}=1+{\omega^2}/\omega_{\mathrm{ mw}}^2$ and ω _mw = γ _e B ₁.

To manage free evolution, we introduce operators H _z  ≡ ω _a S _1z and Ω ₁₂ ≡ aS _1z S _2z . Then the free evolution propagator is exp[−i(H _z  + Ω ₁₂)]. Note that H _z and Ω ₁₂ commute and we can consider them separately and write for the free evolution of shift operators S _1a± due to H _z or Ω ₁₂ the following:

$$ \begin{array}{lll} {S_{{1a\pm }}}{\mathop{\longrightarrow}\limits^{{{H_{z }}}t}}\;\;\;{S_{{1a\pm }}}{{\mathrm{ e}}^{{\mp \mathrm{ i}{\omega_a}t}}} \hfill \cr {S_{{1a\pm }}}{\mathop{\longrightarrow}\limits^{{{\varOmega_{12 }}}t}}\;\;{S_{{1a\pm }}}{(\rm cos}(at/2)\mp \mathrm{ 2i}{S_{2z }}{\rm sin}(at/2))\equiv {S_{{1a\pm }}}{D_{{\pm t}}} \end{array} $$

(66)

We numbered the spins in Eq. (66). Note that S _z may correspond to spin 2 at ω _a or ω _b , since pseudosecular terms are neglected and the evolution due to weak dipolar coupling is then given by Eq. (15). Since first-order coherences of A-spins pass through the prescribed pathway and all pulses applied during the evolution are nominally π-pulses, we need to consider only the following actions of the pulses:

$$ \begin{array}{lll} {S_{{1a\pm }}}{\mathop{\longrightarrow}\limits^{{{P_{kb }}}}}\;\;\;{q_{kb }}{S_{{1a\pm }}},\quad \quad {S_{{1a\pm }}}{\mathop{\longrightarrow}\limits^{{{P_{ka }}}}}\;\;\;{p_{ka }}{S_{{1a\mp }}} \hfill \cr {S_{2z }}{\mathop{\longrightarrow}\limits^{{P_{kc}}}}\;\;\;\;({q_{kc}}-{p_{kc}}){S_{2z }}\end{array} $$

(67)

Here, P _k represents the action of pulse k and subscript a or b is added to indicate at what frequency the pulse is applied. Other spin manipulations lead to pathways that do not contribute to the echo of interest. In the following, we drop the subscripts numbering spins. Since pulse excitations at the two frequencies have only small overlap, Eq. (67) is good approximation. We will disregard unessential phase shifts [98] introduced into S _a± by the pulses applied at ω _b . From Eqs. (66) and (67) we find that D _t has the following properties:

$$ {D_t}{\mathop{\longrightarrow}\limits^{{{H_{kc }}}}}\;\;{q_{kc }}{D_t}+{p_{kc }}D_t^{*},\quad \quad D_t^{*}={D_{-t }},\quad \quad {D_{{{t_1}+{t_2}}}}={D_{{{t_1}}}}{D_{{{t_2}}}} $$

(68)

We first compute the final density operator ρ _f for 3-pulse PELDOR by tracking the coherence pathway that lead to S _a−. We thus start from S _a+ produced by the first π/2 pulse. Equations 67 and 68 reduce our task to merely writing all ensuing “dipolar trajectories”. By repeatedly applying Eqs. (67) and (68) to S _a+, the following sequence of transformations defines the detectable density matrix element in PELDOR:

$$ \begin{array}{lll} {\rho_0}={S_{az }}{\mathop{\ \longrightarrow}\limits^{{{P_{1a }}}}}\;\;\;{h_{1a }}{S_{a+ }}{\mathop{\ \longrightarrow}\limits^{{{H_0}}(t)}}\;\;\;\;{h_{1a }}{S_{a+ }}{D_t}{{\mathrm{ e}}^{{-\mathrm{ i}{\omega_a}t}}} \cr {\mathop{\longrightarrow}\limits^{{{P_{2b }}}}}\;\;\;\;{h_{1a }}{q_{2a }}{S_{a+ }}({q_2}{D_t}+{p_2}{D_{-t }}){{\mathrm{ e}}^{{-\mathrm{ i}{\omega_a}t}}}{\mathop{\longrightarrow}\limits^{{{H_0}}(\tau -t)}}\;\;\;{h_{1a }}{q_{2a }}{S_{a+ }}({q_2}{D_{\tau }}+{p_2}{D_{{\tau -2t}}}){{\mathrm{ e}}^{{-\mathrm{ i}{\omega_a}\tau }}} \cr {\mathop{\longrightarrow}\limits^{{{P_{3a }}}}}\;\;{\ S_{a- }}{h_{1a }}{q_{2a }}{p_{3a }}({q_2}{q_3}{D_{\tau }}+{q_2}{p_3}{D_{{-\tau }}}+{p_2}{q_3}{D_{{\tau -2t}}}+{p_2}{p_3}{D_{{2t-\tau }}}){{\mathrm{ e}}^{{-\mathrm{ i}{\omega_a}\tau }}} \cr {\mathop{\longrightarrow}\limits^{{{H_0}}(\tau +\delta {t_{\mathrm{ e}}})}}\;\;\;{\ S_{a- }}{h_{1a }}{q_{2a }}{p_{3a }}({q_2}{q_3}{D_0}+{q_2}{p_3}{D_{{-2\tau }}}+{p_2}{q_3}{D_{-2t }}+{p_2}{p_3}{D_{{2t-2\tau }}}){{\mathrm{ e}}^{{\mathrm{ i}{\omega_a}\delta {t_{\rm e}}}}}\end{array} $$

(69)

Coefficients p _k and q _k inside the brackets refer to spin 2, which may be at ω _b or ω _a . The spin echo amplitude, V at time 2τ + δt _e is then taken as the trace: Tr(S _a+ S _a−(2τ + δt _e))/Tr(S _a+ S _a−). For simplicity, we neglect dipolar evolution during δt _e and after retaining detectable in-phase coherences by substituting D _2t with their real parts, cos(at), we arrive at the expression for the echo signal

$$ \begin{array}{lll} V(\tau, t,\delta {t_{\mathrm{ e}}})=\left\langle {{h_{1a }}{q_{2a }}{p_{3a }}[{q_2}{q_3}+{q_2}{p_3}\cos (a\tau )+{p_2}{q_3}\cos (at)} \right. \cr {{\left. {\quad +{p_2}{p_3}\cos (a(t-\tau ))]{{\mathrm{ e}}^{{\mathrm{ i}{\omega_a}\delta {t_{\rm e}}}}}} \right\rangle}_{a,b }}\end{array} $$

(70)

The term in exp(iω _a δτ _e) together with all other frequency-dependent factors (p, q, h) after averaging over ω _a,b produces the spin echo shape, V(δt _e) so that Eq. (70) becomes equivalent to Eq. (40). The dipolar modulation in Eq. (70) is represented by the two terms: ~q ₃[1 − p ₂(1 − cos(at))] and ~ p ₃ p ₂ cos(a(t − τ)). The first term is the well-known formula for the PELDOR/DEER signal [30, 73]. The second “back-in-time” signal is relatively small if 〈p ₃ p ₂〉 _a,b  ≪ 〈p ₂〉 _a,b . Usually, this is the case for DEER (but in the single-frequency DEER analog, “2 + 1,” both signals are comparable [38]). In Eq. (70), there are two more terms that are constant in t: one, which is time independent, corresponds to unaffected spin B; whereas another term in cos(aτ) corresponds to the dipolar signal between A spins in the limit of very small a (when pseudosecular term can be neglected). To fully account for their effects more detailed calculations have to be carried out, for example ones based on a modified product operator method as described by Borbat and Freed [35]. Then Eq. (70) becomes at first somewhat unwieldy (e.g., such as an approximate expression given by Raitsimring [106]), but it will simplify practically to Eq. (70) when the “+1” pumping pulse has only a small overlap with the rest of the pulses.

Derivation of the expression analogues to Eq. (70), but for 4-pulse DEER adds one more step to Eq. (69) doubling the number of terms in dipolar signals to a total of eight,

$$ V(t)\propto {B_a}\sum\limits_{k=1}^8 {{B_{bk }}\cos (a{t_k})} $$

(71)

where B _a  = h _1a p _2a q _3a p _4a . Only four terms in Eq. (71) have a dependence on the position t of the pump pulse. Table 1 compiles B _bk and respective time variables, t _k defined in different ways for these terms.

Table 1 t-Dependent dipolar pathways in four-pulse sequence

The dipolar pathways in the 5-pulse DEER sequence were studied in [171] by employing a similar approach. We can describe them qualitatively using the data from Table 1. Signal (1) is the standard 4-pulse DEER signal whereas signals 2–4 are relatively weak. In the 5-pulse DEER sequence, (2) or (4) are no longer weak, since the extra pulse 5 following pulse 4 makes p ₄ greater than p ₃, thereby suppressing (1) and developing the 5-pulse dipolar signal (2). Alternatively, the extra pulse may have position 5′ right before pulse 2 and develops (4) due to increased p ₂ and suppresses (1).