Optimising broadband pulses for DEER depends on concentration and distance range of interest

Abstract EPR distance determination in the nanometre region has become an important tool for studying the structure and interaction of macromolecules. Arbitrary waveform generators (AWGs), which have recently become commercially available for EPR spectrometers, have the potential to increase the sensitivity of the most common technique, double electron–electron resonance (DEER, also called PELDOR), as they allow the generation of broadband pulses. There are several families of broadband pulses, which are different in general pulse shape and the parameters that define them. Here, we compare the most common broadband pulses. When broadband pulses lead to a larger modulation depth, they also increase the background decay of the DEER trace. Depending on the dipolar evolution time, this can significantly increase the noise level towards the end of the form factor and limit the potential increase in the modulation-to-noise ratio (MNR). We found asymmetric hyperbolic secant (HS {1,6} ) pulses to perform best for short DEER traces, leading to a MNR improvement of up to 86 % compared to rectangular pulses. For longer traces we found symmetric hyperbolic secant (HS {1,1} ) pulses to perform best; however, the increase compared to rectangular pulses goes down to 43 %.


S1.3 Resonator profile
The resonator profile was measured by a series of nutation experiments at different frequencies as described in the literature (Doll and Jeschke, 2014). It was measured over 300 MHz with a step size of 10 MHz. The magnetic field was co-stepped.
The nutation frequencies were calculated by a Fourier transformation of the nutation traces.

S1.4 DEER
All DEER experiments were measured with the standard four pulse DEER sequence: 2 − 1 − obs − − pump − 1 + 2 − − obs − 2 − echo The delay between the /2 and the pulse in the observer channel 1 was 400 ns. The dipolar evolution time 2 was 8 µs.
For all DEER experiments with rectangular and Gaussian pump pulses the pump frequency was set to 34.00 GHz. The magnetic field was chosen such that the pump lies on the maximum of the nitroxide spectrum. We used the phase cycling ((x) [x] xp x) as suggested by (Tait and Stoll, 2016) and nuclear modulation averaging as suggested by (Jeschke, 2012).

S1.5 DEER optimisation
For the optimisation measurements we used a python script that can automatically perform several DEER experiments after another. We shifted the magnetic field from 1.2090 T to 1.2113 T for an observer pulse of 33.91 GHz and from 1.2097 T to 1.2119 T for an observer position of 33.93 GHz to ensure that the pump pulse is on the maximum of the nitroxide spectrum. Figure S1 illustrates the idea with a fixed observer frequency of 33.93 GHz. The shift of the pump spin is indicated by the red line.

S1.6 Pulse calculations
For rectangular and Gaussian pulses, we used the pulses that are generated by Bruker Xepr software. For Gaussian pulses the FWHM is defined by FWHM = p 2√2 ln (2) . All other pulses were calculated with the pulse function from the easyspin (Version 5.2.21) package for MATLAB R2018b (Stoll and Schweiger, 2006). The resulting pulse shapes were normalised to amplitude values between -1 and 1 and loaded into Xepr.

S1.7 Integration window
The integration window was determined by recording a series of 300 Hahn echoes in transient mode. We evaluated the signal-to-noise (SNR) with different integration windows and determined the integration window with the maximum SNR.

S1.8 Inversion profiles
Inversion profiles for broadband shaped pulses were measured with the pulse sequence.
broadband shaped pulse − 1 − 2 − 2 − obs − 2 − echo The inversion profiles were measured as the echo intensity as function of the frequency offset of the initial broadband shaped pulses. The /2 and pulses were rectangular pulses with a fixed frequency of 34 GHz.

S2 The MNR as the function of merit
Here, we want to discuss whether the MNR is a suitable function of merit for the determination of distance distributions and up to which time point in the DEER trace, the MNR needs to be evaluated to serve this purpose. Therefore, we performed simulations with a model distance distribution 0 that is based on the narrow distance distribution of the model system used in this study. We approximated the experimentally obtained distance distribution with a Gaussian with a mean at 5.08 nm and a standard deviation of 0.08 nm. We varied the background density in ten steps from = 0.01 1/µs to = 0.3 1/µs in combination with a low, medium and high noise level (noise 0 = 0.02, 0.05 and 0.1) that was added to the DEER trace. The background dimension was set to = 3 and a modulation depth of 0.5 was used. The DEER traces were simulated in the time domain up to 8 µs. For each parameter set we generated ten different traces. To compare the background correction by division (Jeschke et al., 2006) with the kernel inclusion approach as described in (Fábregas Ibáñez and Jeschke, 2020) we analysed all simulated DEER traces with both methods. We did not fit the background but used the true background function. The regularisation parameter was chosen according to the generalised cross-validation method. The quality of the resulting distance distributions was estimated by the Euclidian distance from the true distance 0 : The MNR of the form factor was calculated as described in the main text up to a limit of 7 µs according to equation (13) of the main text. Figure S2: The Euclidian distance of the real and calculated distance distribution as defined in equation (1) is plotted as a function of the MNR. Each dot represents a simulated DEER trace with either low ( 0 = 0.02, green), medium ( 0 = 0.05, blue) and high ( 0 = 0.1, red) noise. The background correction was performed by (a) dividing the DEER trace by the background and (b) including the background in the kernel.
In Fig. S2, the quality of the determined distance distribution was plotted as a function of the determined MNR for both a background correction by division (Fig. S2a) and a kernel inclusion approach (Fig. S2b). For each noise level the MNR only depends on the density of the background as all other parameters are kept constant and only the background density is varied. So a lower MNR corresponds to a higher background density rate and vice versa. For the low noise level ( 0 = 0.02), the quality of the determined distance distributions only varies a little for different background density rates. For medium ( 0 = 0.05) and high ( 0 = 0.1) noise levels, however, the dependency of the quality of the determined distance distribution decreases significantly with a decreasing MNR. If the MNR is only evaluated up to an early point of the form factor, the information of the background decay rate is lost in this case and is not properly included in the MNR as the MNR would then depend nearly exclusively on the given noise level. . The grey area shows the area that is covered by the calculated distance distribution for ten exemplary DEER traces. The mean of the shaded area is drawn in blue and the true distance is drawn in green.
A closer inspection reveals that whereas the obtained distance distributions for high background densities reproduce the mean of the distance distribution correctly, they overestimate the width of the distribution and the distance appears to be broader as it is (see Fig. S3 for an exemplary data set). Depending on the information that shall be obtained by the DEER measurements, the mean of the distance distribution might suffice. However, if high resolution distance distributions shall be obtained, it seems to be important to optimise the MNR up to the limit which is given by equation (13) of the main text. The comparison of both background correction methods shows that the kernel inclusion gives better results particularly for a high noise and a high background decay. It should therefore be considered as the superior method. However, the correlation between the quality of the determined distance distribution and the MNR is still valid. This is why, we consider the MNR as a proper function of merit, even if the kernel inclusion approach is used.
For a more comprehensive study, the effect of the MNR on the quality of the obtained distance distribution could also be tested for distance distributions with different distance ranges and widths. Such a detailed study was, however, beyond the scope of the this manuscript.

S3 Determination of the integration window
To determine the ideal integration window we recorded a series of Hahn echoes and calculated the SNR ratio for different integration window lengths. The results show that for rectangular pulses the ideal integration window is typically longer than the -pulse length (Fig. S4). An improvement of up to 14 % for a -pulse length of 28 ns and an ideal integration window of 44 ns was achieved. 8 For Gaussian pulses the ideal integration window is typically smaller than the -pulse length (Fig. S5). S4 Parameters for the observer pulse Table S1: Parameters for the rectangular observer pulses. The pulse length is referring to the -pulse.

Amp. [%]
[ns] Length of integration window  10 S5 The MNR for rectangular and Gaussian pump pulses evaluated up to 7 µs  Figure S6: The excitation profiles of the observer (blue) and pump pulses (red) of (a) rectangular and (b) Gaussian pulses.

S6 Inversion profiles for rectangular and Gaussian pulses
The light blue profiles are for the /2 observer pulses and the dark blue profiles for the observer pulse. The rectangular observer pulses have an amplitude of 60 %, a pulse length of 32 ns ( on observer) and 16 ns (pump), and the Gaussian have an amplitude of 100 %, a length of 56 ns ( on observer) and 34 ns (pump). It can be seen that the spectral overlap can be reduced with Gaussian pulses. The pulse amplitudes of the pump pulse are always 100 %.

S7 Simulations of spin inversion trajectories
We simulated the effect of an HS{1,1} pulse with a pulse length of 100 ns, a truncation parameter of = 8/ , a frequency sweep range from -55 MHz and 55 MHz. We performed the numerical simulation in the density operator framework with MATLAB R2018b. The maximum of the 1 -field was set to 30 MHz, which corresponds to the maximum of the resonator profile. This pulse shows a good inversion between a frequency range of approximately -40 MHz and 40 MHz (Fig. S7a). In Fig. S7b, the inversion of a spin packet with an offset of -40 MHz and 40 MHz are shown. It can be seen that the spins are inverted in a time window between roughly 20 ns and 80 ns, making an effective pulse length of 60 ns. This would correspond to a minimum distance of 2.32 nm. Note that these numbers were only obtained by visual inspection, so this should only be considered as a qualitative discussion. The spin flip behaviours is also different for different pulses. 13 S8 The MNR for broadband pump pulses evaluated up to 7 µs   We recorded the inversion profiles of the best performing pulses and compared them with the simulations in order to detect potential deviations. The results in Fit. S10 shows that the experimentally recorded inversion profiles reproduce the general trends of the simulations. Nonetheless, there are some deviations that are probably caused by the spectromeer and that shall be discussed here.
It can be noticed that for HS{1,6} and HS{1,1} pulses the measured inversion profiles do not reach the inversion profile of the simulation. For HS{1,6} the overall inversion efficiency is a bit lower than expected and for HS{1,1} pulses a bump in the centre of the frequency sweep was found. The simulations can, however, predict the fact that HS{1,6} have a higher inversion efficiency than HS{1,1} pulses. For WURST and chirp pulses the experimentally recorded inversion profiles reach the inversion efficiency of the simulation.
The experimental inversion profiles of HS{1,6}, HS{1,1} and chirp pulses have a larger inversion range than what is predicted by the simlations. This can increase the overlap with the observer pulse and therefore reduce the echo intensity.
But as the inversion range is only a little bit larger, we consider this not to be particularly worrysome. For HS{1,6} and HS{1,1} pulses, the larger inversion range could compensate the reduced inversion efficiency For the chirp pulses the frequency range as well as the inversion efficiency of the experimental and simulated inversion profiles agree. There are some minor deviations in the pattern of the oscillations that are present in the inversion profile, which we do not expect to have a large effect on the performance of the pulse.
18 S12 Full DEER traces Figure S11: Comparison of the performance of DEER with rectangular pulses (green) and with Gaussian observer pulses and the HS{1,1} pulse from table 1 that yielded the best MNR (blue). The form factors are shown in (a) and the corresponding distance distributions in (b). One 10 minute scan was recorded for both experiments. The corresponding DEER traces are depicted in Fig. S17.

S13 The influence of the length of broadband shaped pump pulses
Tests with broadband shaped pump pulses with pulse lengths of 200 ns and 400 ns showed that they do not lead to an overall performance increase. This is shown here exemplary by comparing the performance of HS{1,1} pump pulses and Gaussian observer pulses (Fig. S12). There are indeed some pump pulses (for example a HS{1,1} pulse with = 10/ and Δ = 110 MHz) that show an improvement with a longer pulse length, however there is no overall gain by using a pump pulse length of 200 ns. We noticed that a major problem with longer broadband shaped pump pulses is that the intensity of the echo can be reduced

20
A comparison of the calculated inversion profiles of the respective pulses (Fig. S14a-c) shows that, whereas the 100 ns pulse should lead to an incomplete inversion, a nearly complete inversion can be expected for the longer pulses. Furthermore, the longer pulses should have slightly steeper excitation flanks. Those trends can indeed be found for the measured inversion profiles. There are some deviations of the measured and calculated inversion profiles. The measured inversion profile of the 100 ns pulse shows an increased frequency width compared to the calculated profile. Furthermore, there is bump in the centre of the frequency sweep. The measured inversion profiles of the longer pulses show the expected steep frequency flanks that can also be seen in the simulation. The inversion profiles of the 200 ns and the 400 ns pulses show a small asymmetry around the centre of the frequency sweep. We assign these deviations to instrumental pulse distortions caused by the spectrometer. It is expected that steeper excitation flanks lead to a smaller overlap with the observer pulses and therefore a smaller effect on the echo intensity. Despite this is the case here as well (Fig. 14d), the overlap is not reduced completely and 400 ns pulse still has some remaining spectral overlap with the observer pulses. We assume that the contradictive findings concerning the echo intensity here are caused by this remaining small overlap. It could become more perturbing for longer pulses as the overall energy of the pulses increases with the pulse length and therefore potential disturbances might be enhanced. A measurement with an larger offset between the pulses at 130 MHz shows that the echo decrease is indeed reduced (Fig. S13b) when the overlap gets smaller. Despite leading to a higher echo intensity, such a high offset is not favourable for nitroxode-nitroxide DEER, because of the limited width of the nitroxide spectrum.

S15 Comparison of bandwidth compensated and non-bandwidth compensated pulses
We tested the performance of a bandwidth compensation for HS{1,6}, WURST, chirp and HS{1,1} pump pulses. The observer pulses were rectangular with an offset of 90 MHz and an observer π pulse length of 28 ns. We estimated the effect of bandwidth compensation with the help of the 2p parameter. For WURST and chirp pulses, a bandwidth compensation lead to an improvement of 3.0 % and 3.2 %. However, for HS{1,6} and HS{1,1} pulses, we observed a decrease of 10.5% and 2.6% (data not shown). As a bandwidth compensation requires a measurement of the resonator profile before each DEER measurement and did not always result in an increase in performance, we decided to stick to pulses without bandwidth compensation.
22 S16 The MNR for rectangular and Gaussian pump pulses evaluated up to 2 µs 23 S17 The MNR for broadband pump pulses evaluated up to 2 µs Table S9: MNR for the different broadband shaped pulses with a rectangular observer pulse. The MNR has been evaluated up to 2 µs. [GHz] Obs.

Amp. [%]
Pump  Table S10: MNR for the different broadband shaped pulses with a Gaussian observer pulse. The MNR has been evaluated up to 2 µs. The chirp pulse where no r time is specified is a pulse without the quartersine smoothing. [GHz] Obs.

Amp. [%]
Pump  S11: MNR for the different broadband shaped pulses with a Gaussian observer pulse. The MNR has been evaluated up to 2 µs. The chirp pulse where no r time is specified is a pulse without the quartersine smoothing. [GHz] Obs.

Amp. [%]
Pump  The corresponding form factors are depicted in Fig. S11.

S21 Calculation of the background-dependent performance of broadband shaped pulses
To estimate the influence of broadband shaped pulses for different maximum distances and concentrations, we performed some analytical calculations. The background decay reduces the echo-intensity and therefore decreases the signal-to-noise ratio towards the end of the DEER trace. Whereas the measured trace ( ) has a constant noise level 0 , the background corrected form factor has an increasing noise level towards the end: where ( ) is the noise of the form factor and is the background density. Here, we assumed a 3D background. As discussed in the main text, the form factor is truncated at a time truncation to exclude the later part. An integration from = 0 to = truncation , with truncation as the dipolar evolution time, yields the average noise in the form factor √< 2 >= 0 √ 1 2 truncation (exp(2 truncation ) − 1).
The modulation-to-noise (MNR) as the ratio of the modulation depth and the average noise is then described by: As both the modulation depth and the background density k directly depend on the inversion efficiency, a linear dependence can be expected between them. Indeed, we experimentally found an approximately linear correlation between them (Fig. S16). Whereas the 2 value captures a decrease in echo intensity it will miss the effect of a larger background decay. We chose exemplary parameters that resembled our experimental findings. For the modulation depth, we assumed an increase from 30 % to 50 %, which corresponds to the modulation depths that we found for rectangular and the best HS{1,1} pump pulse. For the density of the background we assumed an increase about the same factor: S = 5 3 R with S as the background density for the broadband shaped pulse and R as the background density for the rectangular pulses. As the sample with a concentration of 80 µM of doubly-labelled ligand had a background density R of approximately 0.1 with rectangular pulses, we tested R -values from 0 to 0.15 1/µs to keep it in a realistic range. According to equation (5)  .
(5) 28 Figure S18: The MNR-ratio of adiabatic and rectangular pulses as a function of the background density (with rectangular pulses) and the truncation -time. The corresponding maximum distance according to equation (13) of the main text is also depicted. As the background density reflects the concentration the x-axis is a measure for the concentration of the spin centres. In our sample with 80 µM, we had a background density R of 0.1 with rectangular pulses. Figure S18 shows that the performance of shaped pulses can heavily depend on the circumstances of the measurement. For a maximum distance below 4 nm ( truncation ≈ 5 µs), a MNR increase can be expected for all realistic concentration ranges.
This is not the case if a longer distance shall be detected. For maximum distances around 5 nm, the MNR increase goes to 1 for high background densities of R = 0.15 1/µs, which corresponds to very high concentrations > 100 µM. Typical concentrations for DEER measurements are around 50 µM, which here corresponds to a R ≈ 0.06 1/µs. For this concentration, a significant increase in the MNR can only be expected up to a truncation time of truncation = 10 µs, which is equal to a maximum distance of approximately 6 nm.
As broadband shaped pulses are particularly interesting for long distances, the calculations were performed up to a rather long truncation time of 25 µs (maximum distance of approximately 7.5 nm). For distances in the range > 6 nm, only with concentrations in the range of 10-30 µM ( R ≈ 0.01-0.04 1/µs) a significant increase in the MNR due to broadband shaped pump pulses can be expected. The MNR increase drops quickly when higher concentrations are used. For a maximum distance of 7.5 nm and for concentrations over approximately 40 µM no increase can be expected any more due to broadband shaped pulses. If a concentration of 80 µM is used, the MNR is about to decrease to roughly 40 % when switching to broadband shaped pulses. It is known that diluting the sample is favourable if long distances shall be detected because it increases the phase memory time of the echo (Schmidt et al., 2016). When broadband shaped pump pulses, the higher background decay adds an additional point for carefully choosing the concentration of the sample and it seems to be advisable to avoid high concentrations.