Estimation of distance-distribution probabilities from pulsed electron paramagnetic resonance (EPR) data of two dipolar interaction coupled nitroxide spin labels using doubly rotating frames and least-squares fitting

Abstract

A method, based on the doubly rotating frame (DRF) technique to calculate the basis DEER (Double Electron–Electron Resonance) signals [Physica B: Condensed Matter, 625, 413,511 (2022)] accurately by numerical techniques over a range of $r$ values, where $r$ is the distance between the two nitroxides in a biradical in a biological system, has been exploited to calculate the probabilities of distance distribution, $P\left( r \right),$ by the use of Tikhonov regularization. It is demonstrated here by applying it to the data reported by Lovett et al. [J. Magn. Reson., 223, 98–106 (2012)] on a sample of bis-nitroxide nanowire, P1, in deuterated ortho-terphenyl solvent with 5% BnPy (d14-oTP/BnPy) in semi-rigid state. An improvement in the agreement of the calculated signal with respect to the experimental signal and thus in the probabilities of the distance distribution, $P\left( r \right)$, so obtained, is found, as compared to that obtained using the kernel signals based on analytical expressions.

Graphical abstract

Appendices

Appendix A: Doubly rotating frame (DRF) technique to calculate DEER signal

In the DRF technique, there are used two rotating frames, one for all times at resonance with the observer spins characterized by the g-value, g_obs, except for the duration of the pump pulse, which resonates with the pump spins at a different g-value, g_pump, from that of the observer spins. At any instant, both, the observer and pump spins, are treated in the same rotating frame, implying that when the observer spins are precisely in their rotating frame, the pump spins see a rotating frame which is not coincident with their rotating frame and vice versa. In a theoretical treatment, the rotating frame for the pump spins can be treated by changing the intensity of the external magnetic field during the action of the pump pulse by an amount equivalent to the difference in the frequency of the pump pulse from that of the observer pulse. Specifically, a difference of 65 $MHz$, in a typical experiment [2], is equivalent to a difference in the intensity of the external field by -$65.0\left( {MHz} \right)/2.8\left( {MHz/G} \right){ } \approx$ -23.2 G.

In order to calculate the signal for a polycrystalline sample, one uses a $\left( {\theta ,\phi } \right)$ grid over the unit sphere, keeping the rotating frame fixed for the observer spins for all orientations, except for those during the application of the pump pulse at a fixed value of g = g_obs. It can be chosen for a fixed orientation, say for $\theta = \phi = 0^{ \circ }$, for the choice of the Euler angles (${\upalpha }_{1}$ = $0^{ \circ }$, ${\upbeta }_{1}$, ${\upgamma }_{1}$) for the observer spins appropriately. Then, the resulting coefficient $C_{1}$ of the $S_{{z_{1} }}$ term as calculated for the observer spins in the static spin Hamiltonian, Eq. (2.1), is, in general, not zero, implying that it is not coincident with its rotating frame at the applied field, $B_{0}$_, which requires that the Zeeman term is zero. However, by changing the reference of energy by subtracting both $C_{2}$ and $C_{1}$ by $C_{1} ,$ the coefficient of the $S_{{z_{1} }}$ term is rendered zero. Then, the observer spins are precisely in their rotating frame. The pump spins with the orientations $\left( {\theta ,\phi } \right)$ of their dipolar axes with respect to the external magnetic field distributed over the unit sphere will have their coefficients of the $S_{{z_{2} }}$ terms in the static spin Hamiltonian, Eq. (2.1), changed to $C_{2} {-}C_{1} { } \ne$ 0, so that these spins are not in their rotating frame, as expected. On the other hand, during the application of the pump pulse, the external magnetic field is changed so it is at resonance for the pump spins at g = g_pump, or equivalently at the pump frequency, as listed in Table 1 One then keeps the same value of the coefficients for the observer spin,$C_{1P}$ (= $C_{{1P_{0} }} )$, i.e., that calculated for $\theta = \phi = 0,$ for the external magnetic field intensity $B_{0} - ~\Delta B$, for all $\left( {\theta ,\phi } \right) {\text{values}}$ over the unit sphere. As for the pump spins, the coefficient of the $S_{{z_{2} }}$ term, $C_{2P}$, as calculated for the external magnetic field $B_{0}$—${\Delta }$ B, is, in general, not zero, so that they are not in their rotating frames. In order to change the reference frame to the rotating frame of the pump spins for the various $(\theta ,\phi$) values over the unit sphere, the reference of energy is now changed by $C_{2P}$, so that the coefficient of $S_{{z_{2} }}$ is changed rom $C_{2P}$ to $C_{2P} - C_{2P} = 0$_, so that the pump spins are precisely in their rotating frame, whereas the coefficient of $S_{{z_{1} }}$ for the observer spins is changed from $C_{1P} \left( { = C_{{1P_{0} }} } \right)$ to $C_{1P} - C_{2P} \left( { = C_{{1P_{0} }} - C_{2P} } \right)$, so that the observer spins are not precisely in their rotating frame.

Table 1 The values of the parameters used in the simulations of the DEER signals of the coupled nitroxide biradical, for the data of Lovett et al. [2]

Appendix B: Flowchart for the calculation of three-pulse DEER signal.