Estimation of Distance Distributions Between Gd³⁺ Ion Pairs with a Significant Zero-Field Splitting from Pulsed EPR DEER Data

Abstract

DEER (Double Electron–Electron Resonance) kernel signals are calculated, using the double rotating-frames (DRF) technique, for the various distances, r, between two coupled Gd³⁺ ions, distributed randomly in a biological system. This is done numerically, using full diagonalization of the spin-Hamiltonian matrix, taking into account the zero-field splitting of the Gd³⁺ion with spin $S=7/2$. These kernel signals are then used to estimate the probabilities of the distance distribution, P(r), between the pairs of various Gd³⁺ ions. This is accomplished using Tikhonov regularization, as implemented in the software DeerAnalysis [Jeschke et al. Appl Magn Reson 30(3): 473–498, (2006)], with the kernel signals calculated here by the DRF technique for different r values. This procedure is successfully illustrated by applying it to calculate the probabilities, P(r) from the reported experimental four-pulse DEER data for a sample of Gd ruler 1₅ in D₂O/glycerol-d₈ (i) at Q-band [Doll et al. J Magn Reson 259: 153–162, 2015] and (ii) at W-band [Dalaloyan et al. Phys Chem Chem Phys 17(28): 18464–18476, (2015)]. These results are found to be about the same, within the RMSD maximum and minimum bounds, from those obtained using analytical kernel signals, valid for hard (infinite) pulses for spin ½, so that the ZFS is not included, at the lower frequency (~ 35 GHz) of Q-band. However, at the higher frequency of W-band (~ 95 GHz), there are found slight, but distinct, differences beyond the RMSD maximum/minimum bounds, in the P(r) values calculated by the two approaches. In application scenarios to the commonly investigated proteins, where the errors are large, the results obtained by the two approaches would be about the same, at least for the data obtained at Q- and W-bands, if not at higher frequencies.

Appendices

Appendix A.1

1.1 Flowchart for the Calculation of Four-Pulse DEER Signal

Appendix A.2

2.1 Expressions Referred to in the Flowchart A.1

The evolution of the density matrix under the application of a pulse is described by

$$\rho \left({t}_{0}+{t}_{p}\right)={e}^{-i\left({H}_{0}+{H}_{p}\right){t}_{p}}\rho \left({t}_{0}\right) {e}^{i\left({H}_{0}+{H}_{p}\right){t}_{p}},$$

(A.1)

where ${H}_{p};\, \mathrm{with/} p = \mathrm{obs},\mathrm{ pump},$ denote the Hamiltonians for the observer and pump pulses, respectively [27], given as

$${H}_{obs}=\frac{{\upomega }_{1obs}}{2}\left({e}^{-i\upphi }{\left({S}_{1+}\right)}_{(\mathrm{5,6}) }+{e}^{i\upphi }{\left({S}_{1-}\right)}_{(\mathrm{6,5})}\right),$$

(A.2a)

$${H}_{pump}=\frac{{\omega }_{1pump}}{2}\left({e}^{-i\upphi }{\left({S}_{2+}\right)}_{(\mathrm{4,5}) }+{e}^{i\upphi }{\left({S}_{2-}\right)}_{(\mathrm{5,4})}\right).$$

(A.2b)

In Eqs. (A.2a) and (A.2b) ${\omega }_{1p};\mathrm{with} p=\mathrm{obs},\mathrm{ pump}$ are the amplitudes of the observer and pump pulses, respectively, in frequency units $(={\upgamma }_{e}{B}_{1p});{B}_{1p}=$ amplitudes of the pulses in Gauss (Table 1) and ${\upgamma }_{e}$ is the gyromagnetic ratio of the electron $;\upphi$ is the phase of the pulse and${S}_{k\pm }$; $k=\mathrm{1,2}$ are the raising/lowering operators of the $k$ th electronic spin of the coupled $G{d}^{3+}$ ions system in the $64 \times 64$ direct product Hilbert space. The indices in (A.2a) and (A.2b) indicate that the pulse Hamiltonians are adjusted to the specific transitions, $-3/2 \leftrightarrow -1/2$ and$-1/2 \leftrightarrow 1/2$, which define the observer and pump pulses, respectively.

The density matrix for the system of coupled $G{d}^{3+}$ ions evolves during free evolution, i.e., in the absence of any pulse, governed by the LVN (Liouville von Neumann) equation, as follows:

$$\rho \left({t}_{0}+t\right)={e}^{-{iH}_{0}t}\rho \left({t}_{0}\right){e}^{i{H}_{0}t} ,$$

(A.3)

where $\rho \left({t}_{0}\right)$ is the density matrix just after the application of a pulse at time ${t}_{0}$ and $\rho \left({t}_{0}+t\right)$ is the density matrix after free evolution over the time interval $t$, subsequent to the application of a pulse.

The complex time-domain DEER signal for this case is expressed as

$${S}^{(DEER)}\left(t, \theta , \phi ,{\mathrm{\alpha }}_{1}=0,{\upbeta }_{1},{\upgamma }_{1},{\mathrm{\alpha }}_{2},{\upbeta }_{2},{\upgamma }_{2}\right)=Tr\left({S}_{1+}{\rho }_{f}(t)\right)=Tr\left({S}_{2+}{\rho }_{f}(t)\right)$$

(A.4)

for the orientation, ($\uptheta ,\upphi$), of the static magnetic field with respect to the dipolar axis of the two $G{d}^{3+}$ ions, whose dipole moments are oriented with respect to the respective molecular frames, denoted by five independent Euler angles $\{{\upbeta }_{1},{\upgamma }_{1},{\mathrm{\alpha }}_{2},{\upbeta }_{2},{\upgamma }_{2}\}$, choosing the arbitrary angle ${\mathrm{\alpha }}_{1}=0,$ as shown in Fig. 2, exhibiting the configuration of the two coupled $G{d}^{3+}$ ions [27]. In Eq. (A.4), ${\rho }_{f}\left(t\right)$ is the final density matrix $.$

Appendix B

3.1 Estimation of the Probability Distribution, ${\varvec{P}}\left({\varvec{r}}\right),$ from the Experimental Signal Using DeerAnalysis, Modified for Double Rotating-Frames (DRF)-Calculated Kernel Signals

• 1. Determine the range of the distance, $r$, between the two $G{d}^{3+}$ ions, over which the probability distribution is non-zero, as calculated by the application of DeerAnalysis software.

• 2. Simulate the four-pulse DEER signals using the DRF technique for a good number of evenly distributed points, ${r}_{n}$, over this range.

• 3. Use the Spline function in Matlab to derive 512 time-domain signals by the cubic-spline technique, each calculated as a function of time in 512 even steps over the range of time from 0 to ${{\varvec{\uptau}}}_{2}$ (listed inTable 1) from the m DRF-calculated signals in the previous step.

• 4. Normalize the time-domain signals, as simulated by the DRF technique, for the various ${r}_{n}$ values, by dividing each by its maximum value.

• 5. Stack these signals in a 512 $\times$ 512 matrix, $K(t,r)$, where $t,r$ denote the rows and columns, respectively.

• 6. In the DeerAnalysis Matlab file, “make_bas_Tikh.m”, make the following replacements: (i) $rmin$ and rmax by the minimum and maximum of range of distance $r$, respectively, determined in step 1; (ii) $dt$ by ${\uptau }_{2}/512$; (iii) the already present analytically calculated kernel signals,$K(t,r)$,by the DRF-calculated kernel signals, $K(t,r)$, as described in step 5

• 7. 7. In the Matlab file “get_Tikhonov_new.m” in DeerAnalysis folder, replace the conditional statement “if length(tdip) > 2048”, which allows one to use already calculated and stored analytically calculated kernel signals,$K(t,r)$, by the statement “if length(tdip) > ${n}_{RF}$, where the integer ${n}_{RF}$ < 512”, i.e., ${n}_{RF}=200$ in the present case, for one to use the DRF-calculated kernel signals of size (512, 512) as in step 5.

• 8. Delete “pake_base_tikh_512.mat” file from the DeerAnalysis folder, so that the already stored analytically calculated kernel signals, $K(t,r)$, are removed, allowing the use of the DRF-calculated kernel signals, $K(t,r)$. Then open the “DeerAnalysis.m” file, run it, and wait until the DeerAnalysis workspace loads up (about 5 min).

• 9. When the workspace opens, click on the button “load”, then from the “Data sets” box load the experimental data file (with the ending.DTA).

• 10. In the “Distance Analysis” box, select “Tikhonov” and “L curve” to obtain plots of L-shaped curve and the probability distribution as a function of distance.