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 %.
Introduction
In the last few years DEER (double electron–electron resonance) has developed
into an important technique for the determination of distances in the
nanometre range (Jeschke, 2012; Milov et al., 1981, 1984) and in particular into a suitable tool
for studying biological macromolecules (e.g. proteins (Jeschke, 2012;
Robotta et al., 2014) or RNA/DNA (Grytz et al., 2017; Kuzhelev et al.,
2018)). As many bio-macromolecules do not contain paramagnetic centres, for
many DEER experiments spin labels are introduced with the help of
site-directed spin labelling (Hubbell et al., 1998). Although many different
types of spin labels have been introduced in the last years, ranging from
trityl (Abdullin et al., 2015; Jassoy et al., 2018), Gd(III) (Collauto et
al., 2016; Dalaloyan et al., 2015; Mahawaththa et al., 2018), copper(II)
(Wort et al., 2019) to photoexcitable spin labels (Di Valentin et al., 2014;
Hintze et al., 2016), just to mention a few examples, nitroxide labels are
still amongst the most widely used tags.
Increasing the sensitivity of DEER spectroscopy is an active field of
research (Borbat et al., 2013; Breitgoff et al., 2017; Doll et al., 2015;
Jeschke et al., 2004; Lovett et al., 2012; Milikisiyants et al., 2019;
Polyhach et al., 2012; Tait and Stoll, 2016; Teucher and Bordignon, 2018). A
very elegant approach to increasing DEER sensitivity has been made possible
by the availability of arbitrary waveform generators with time resolution in
the nanosecond region as they allow the generation of broadband microwave
pulses (Doll et al., 2013; Doll and Jeschke, 2017; Spindler et al., 2017).
Here, we compare nitroxide–nitroxide DEER performance for different types of
pulses and different sample conditions. The paper is organised as
follows. In Sect. 1 we will give a brief overview of the pulse shapes that
are compared in this paper. In Sect. 2, we will describe the
experimental details and the compounds that have been used in this study. In
Sect. 3, we will present and discuss the experimental results. We will
compare rectangular, Gaussian and different types of broadband-shaped pulses
on a commercial spectrometer. In order to give them a fair comparison, the
parameters for each pulse family will be optimised. In doing so, we will
provide a step-by-step guide to how an experimental optimisation for DEER
can be performed. The larger inversion efficiency of broadband-shaped pump
pulses that leads to a higher modulation depth will also lead to a higher
background decay and therefore potentially limit the signal gain that is
promised by broadband-shaped pump pulses. We set out to examine this effect
for the presented pulse families in detail and show that different types of
broadband-shaped pulses are ideal for different spin concentrations and
distance ranges.
In magnetic resonance experiments, a pulse is generated by a time-dependent
field B1 that is applied perpendicularly to the
B0 field which defines the z direction. All pulses in this
paper can be described in terms of an amplitude function A(t) and a
frequency function ω(t).
The resulting B1 field in the rotating frame is
1B1,x(t)=A(t)cos(ρ(t)),2B1,y(t)=A(t)sin(ρ(t)),
where the phase ρ(t) is defined as ρ(t)=∫0tω(t′)dt′.
Rectangular pulses are described by ω(t)=0 and
A(t)=|B1| during the pulse, i.e. by a B1
field with a constant phase and intensity. The sidebands of the sinc-shaped
excitation profile of rectangular pulses increase the overlap of the
observer and pump pulses in DEER, resulting in so-called “2+1” artefacts at
the end of the DEER trace. It has recently been shown that those artefacts
can be reduced by replacing the rectangular pulses with Gaussian pulses
(Teucher and Bordignon, 2018). Gaussian pulses also have a frequency
function of ω(t)=0 but an amplitude function
A(t)=exp-4ln(2)t2FWHM2.
FWHM describes the full width at half maximum of the pulse in the time
domain (Teucher and Bordignon, 2018). Here and in the following equations
the time axis is defined such that t=0 lies in the centre of the pulses.
During a rectangular or Gaussian pulse the magnetisation vector is rotated
around a fixed axis with an angle that is independent of the
initial orientation of the magnetisation vector. Such pulses are therefore
called uniform rotation pulses (Kobzar et al., 2012). As rectangular and
Gaussian pulses have a fixed frequency, they are also referred to as
monochromatic pulses.
One of the most significant challenges in EPR spectroscopy is the limited
excitation bandwidth of rectangular and also Gaussian pulses compared to the
width of many EPR spectra. In the case of nitroxide–nitroxide DEER, a
significant part of the EPR spectrum contributes neither to observing
nor to pumping when using rectangular pulses.
Using broadband-shaped pulses, the excitation bandwidth can be increased
(Doll et al., 2013). Broadband-shaped pulses differ from rectangular
and Gaussian pulses mainly in that they do not have a constant frequency,
but the frequency is swept over a given range during the pulse, which allows
the excitation bandwidth to be increased. In an accelerated frame, which rotates
with the instantaneous excitation frequency of the pulse, the effective
field rotates from the +z to -z directions (Baum et al., 1985;
Deschamps et al., 2008; Garwood and DelaBarre, 2001; Kupče and Freeman,
1996). Under adiabatic conditions the magnetisation follows the effective
field on its way from +z to -z (Baum et al., 1985; Doll et al., 2013).
Pulses that induce this kind of spin flip behaviour are called
point-to-point rotation pulses. This approach allows the generation of
pulses that have a large excitation bandwidth and that are, above a certain
threshold, more insensitive to the resonator profile than rectangular pulses
(Baum et al., 1985). Their ability to flip spins from the +z to the
-z axis makes such broadband-shaped pulses perfect candidates for the pump
pulse in the DEER pulse sequence. Their larger excitation profile has the
potential to result in a larger modulation depth and therefore a larger
sensitivity (Bahrenberg et al., 2017; Doll et al., 2015; Spindler
et al., 2013; Tait and Stoll, 2016).
Intuitively, a high adiabaticity means that the effective magnetic field
moves more slowly from +z to -z, making it easier for the spins to
follow, thus resulting in a higher inversion efficiency.
The adiabaticity Q is formally defined as (Kupče and Freeman, 1996)
Q=2πνeff|dθ/dt|.
Here, νeff is the strength of the effective magnetic
field and θ is its polar angle in the accelerated frame. The pulses
have a good inversion efficiency, if Q≫1 (Deschamps et al., 2008). In
general, the adiabaticity changes during the duration of the pulse and is
different for spins with different frequency offsets. Adiabatic pulses are
typically quantified by their minimum adiabaticity Qmin.
Chirp pulses have a constant amplitude function and a linear frequency
function ω(t)=fstart+pt, where p=Δf/tp is a sweep constant, tp is the pulse
length and Δf=fend-fstart.
fstart and fend are the start and end
frequencies of the frequency sweep. The minimum adiabaticity
Qmin is reached when a spin is on resonance with the pulse
frequency (Doll et al., 2013):
Qmin=2πν12tpΔf,
where ν1 is the nutation frequency which increases linearly with |B1|. Qmin increases with the pulse length but decreases with the
sweep width. The frequency width for a pump pulse should be chosen such that
a large part of the spectrum is excited without having significant spectral
overlap with the pulses at the observer frequency. The steep flanks at the
beginning and the end of the rectangular amplitude profile lead to
distortions in the excitation profiles of chirp pulses, because the initial
effective magnetic field is not aligned with the z axis. Smoothing both ends
of the pulses with a quarter sine wave can reduce theses distortions (Böhlen
and Bodenhausen, 1993). The smoothing can be adapted by changing the rising
time trise. Following the logic so far, the pulse length
should be chosen as long as possible to enable a very high adiabaticity.
However, a broadband-shaped pulse flips spins with different offsets at
different times. When used as a pump pulse in DEER, this results in a shift
of the dipolar oscillations and an artificial broadening of smaller
distances in the distance distribution. Therefore, the pulse length should
be chosen such that (Breitgoff et al., 2019)
tp<Tdd4,
with the dipolar evolution time Tdd of the shortest expected
distance.
In addition to chirp pulses, there are more elaborate pulses employing more
elaborate frequency and amplitude functions. The most common ones are WURST
(wideband, uniform, smooth truncation) and HS (hyperbolic secant) pulses.
The trends discussed so far are valid for them as well. However, they
feature additional parameters that can be used to tune the steepness of the
corresponding excitation profiles.
WURST pulses have a linear frequency sweep as well but a different amplitude
function than chirp pulses (Kupče and Freeman, 1995b; Spindler et al.,
2017):
A(t)=Amax1-sinπttpn.
The effect of the parameter n determining the steepness of the amplitude
function will be discussed below.
HS pulses have non-linear frequency sweeps and are described by the
following amplitude and frequency functions:
8A(t)=sechβ2h-1ttph,9ω(t)=Δf2tanhβ2-1tanhβttp,
with order parameter h and truncation parameter β. The effects of
β will be discussed below. A common choice for h is to set h=1.
These pulses have an offset-independent adiabaticity and a rather
rectangular excitation profile (Baum et al., 1985; Tannús and Garwood,
1996). Increasing the order h of an HS pulse will lead to a higher
adiabaticity at the maximum of the excitation profile but less steep
flanks (Breitgoff et al., 2019). A compromise can be found by using an
asymmetric HS pulse where the flank close to the observer is made steep by
an order of 1 and where the other flank has a higher order for a higher
adiabaticity (Doll et al., 2016). Symmetric pulses with an order parameter
of h=1 will be referred to as HS{1,1};
asymmetric pulses where the first part of the pulse has an order parameter
of h=1 and the second half has h=6, as suggested by Doll et al. (2016),
are referred to as HS{1,6} (Doll et al., 2016).
The measured DEER trace V(t) is the product of the form factor F(t) that
contains the required intramolecular distance information and a
background function B(t) (Jeschke, 2012):
V(t)=F(t)⋅B(t).
The background decay is caused by the intermolecular interactions of the
observer spin with pump spins of surrounding molecules. Assuming that the
spins are homogenously distributed, the background decay can be described by
an exponential decay (Jeschke, 2016):
B(t)=exp(-(k|t|)d/3),
where d is a dimensionality constant and the decay constant k is described
by the following equation (Hu and Hartmann, 1974; Pannier et al., 2000):
k=2Naπμ093hg2μe2fc.
Here, c is the spin concentration, f the inversion efficiency of the pump
pulse, μe the Bohr magneton, μ0
the magnetic field constant, Na the Avogadro number and g
the isotropic g factor of the nitroxide.
Materials and methodsSample preparation
Wheat germ agglutinin (WGA) was purchased from Sigma-Aldrich (article-no.:
L9640) as lyophilised powder and used without further purification. The
doubly spin-labelled tetravalent ligand (1) was synthesised in the lab of
Valentin Wittmann. Details of synthesis and characterisation can be found in
Rohse et al. (2020)). For the WGA-ligand samples investigated in this study
solutions of WGA and the tetravalent ligand were prepared separately in
deionised water. The protein concentration of the WGA solution was
determined spectrophotometrically.
WGA-ligand samples were prepared by mixing WGA and ligand solutions
resulting in a 2:1 molar excess of WGA compared to the ligand referring to
the final sample volume. The 2-fold excess on protein was chosen to prevent a
free, unbound ligand in solution. The sample solution was lyophilised and
the resulting powder was dissolved in D2O (Magnisolv, Cas-no.: 7789200,
article: S571556621) and 20 % (v/v) deuterated glycerin (Sigma-Aldrich,
lot-no. MBBB5255, article: 447498-1G) as cryoprotectant. Unless stated
otherwise we used a sample concentration of 160 µM WGA and 80 µM
ligand; 60 µL of solution was filled into 3 mm outer diameter quartz
sample tubes (ER 221 TUB/2, part no. E221003), shock-frozen in liquid
nitrogen before measurement and placed in the probe head precooled to 50 K.
Samples were stored at -80∘C with unfreezing avoided.
EPR experiments
All experiments have been performed on a Bruker Elexsys E580 spectrometer at
Q band (34 GHz). The spectrometer is equipped with a SpinJet-AWG unit
(Bruker) and a 150 W pulsed travelling-wave tube (TWT). All samples were
measured in 3 mm outer diameter sample tubes in an overcoupled ER5106QT-2
resonator (Bruker). The quality factor Q of the overcoupled resonator is
approximately 200.
The samples were cooled to 50 K with a Flexline helium recirculation system
(CE-FLEX-4K-0110, Bruker Biospin, ColdEdge Technologies) comprising a cold
head (expander, SRDK-408D2) and an F-70H compressor (both SHI cryogenics,
Tokyo, Japan), controlled by an Oxford Instruments Mercury ITC.
DEER measurements were recorded with the standard four-pulse DEER sequence
(Pannier et al., 2000), an eight-step phase cycle (Tait and Stoll, 2016) and
nuclear modulation averaging (Jeschke, 2012). The dipolar evolution time was
set to 8 µs and the time step to 8 ns.
We analysed the DEER traces with DeerAnalysis2019 (Jeschke et al., 2006). We
performed a background correction resulting in a background function with a
dimension of d=3.5. The form factor was analysed with Tikhonov
regularisation and a regularisation parameter chosen by the generalised
cross-validation criterion (Edwards and Stoll, 2018).
A crucial parameter for pulsed dipolar spectroscopy is the
modulation-to-noise parameter MNR=λn, with the modulation depth
λ and the noise level n. We calculated the noise similarly to
published procedures by the standard error from a fit with a smoothing
spline (Bahrenberg et al., 2017; Breitgoff et al., 2019; Mentink-Vigier et
al., 2013). We excluded the first 10 data points from the form factor because
the spline typically showed some deviations at the start of the trace.
Unless stated otherwise, the upper limit for the noise calculation was 7 µs.
We used the η2P parameter which has been suggested by
Doll et al. (2015) and already been used by other authors (Doll et al.,
2015; Spindler et al., 2013; Tait and Stoll, 2016). The η2P value is defined as the difference between two distinct
time points in the DEER trace and therefore does not require the
measurement of full DEER traces. We recorded short DEER traces with eight data
points only and calculated η2P as the difference of
the phase-corrected DEER trace at the zero time V(0) minus the first minimum
of the DEER trace V(tmin).
For a more detailed description of materials and methods, see Sect. S1 in the Supplement.
Results and discussion
In order to study the performance of DEER using different pulses, we used
the doubly nitroxide-labelled tetravalent ligand bound to wheat germ
agglutinin dimer (WGA) as a model system (Fig. 1). The ligand binds with a
very high affinity to WGA and features a narrow distance distribution (FWHM=0.2nm) at 5.1 nm (Rohse et al., 2020). We performed DEER experiments
with different combinations of pulses. In the following, we will refer to a
combination of rectangular observer and pump pulses as RR, to a combination
of Gaussian observer and pump pulses as GG, to a combination of
rectangular observer and broadband-shaped pump pulses as RS and to a
combination of Gaussian observer and broadband-shaped pump pulses as GS.
The structure of the tetravalent ligand with its two
spin-(2,2,5,5-tetramethyl-3-pyrrolin-1-yloxycarbonyl) labels and the ligand
bound to WGA. The visualisation of the dimeric WGA with the subunits
coloured in grey and blue is based on the crystal structure (PDB entry 2X52; Schwefel et al., 2010). In red, a schematic representation of the ligand
is overlaid to the crystal structure. The ligand is suggested to bind with
its four GlcNAc moieties to the primary binding sites of WGA. Blue balls and
arrows indicate nitroxide spin labels.
As stated by Eq. (10), the measured raw data not only consist of
the desired form factor, but also include a background contribution emerging from
intermolecular interactions. A common way to deal with this is to fit the
background according to Eq. (11) and divide the raw data by the fit to
obtain the form factor that can then be transformed into a distance
distribution (Jeschke, 2012; Jeschke et al., 2006). When measuring DEER
traces, a precise distance determination is desired. Since for an
experimental parameter optimisation the true underlying distance
distribution is unknown, a metric is needed that is based on the recorded
data. The MNR of the form factor is suitable for this case as it increases
with an increasing modulation depth and an increasing echo intensity. As the
noise of the form factor increases towards its end due to the division by
the background, the MNR goes down with a stronger background decay. It can
therefore capture the fact that a larger background decay leads to less
reliable distance distributions, as has recently been investigated by
Fábregas Ibáñez and Jeschke (2020) in a detailed study. In
their paper they also suggest a different method for background correction
that treats the background by directly including it in the kernel that is
needed to calculate the distance distribution from the DEER trace. As this
methods renders the calculation of a form factor redundant, a MNR cannot be
directly obtained by it. Even though this new method has shown itself to
give more reliable distance distributions in the case of large background
decays, its performance still drops with an increasing background. Therefore,
we still consider the MNR that is obtained by the background correction by
division to be the best measure to optimise settings for DEER
measurements experimentally.
The evaluation of the noise of the entire DEER trace is not always feasible.
It depends on the maximum distance rmax that is to be
detected up to which part the form factor is of interest. Here, we truncated
the form factor for the calculation of the MNR at 3 times the
oscillation period of the maximum distance that is of interest (Edwards and
Stoll, 2018):
τtruncation=3rmaxnm352MHz.
This corresponds to roughly three dipolar oscillations in the form factor.
In this case of a distance at 5.1 nm this is equivalent to a truncation time
τtruncation≈7µs. A simulation with a
model distance reveals that in order to obtain the correct width of the
distance distribution, a good MNR up to this time point can be necessary and
a good MNR of only the first part of the form factor is not as reliable when
the credibility of the obtained distance distribution is to be estimated.
The details of this study can be found in Sect. S2.
Performance comparison for rectangular and Gaussian pulses
The pump pulse frequency was set to 34.00 GHz, which is the maximum of the
resonator profile (Fig. 2a). The magnetic field was set such that we pumped
on the maximum of the nitroxide spectrum (Fig. 2b). To optimise the settings
for RR and GG we tested observer pulses with a frequency offset of 90
and 70 MHz between the pump and observer pulses, respectively. To check for
different excitation profile widths of the observer pulse, we tested
settings with observer pulse amplitudes of 100 % and 60 %. The pulse
length was always adjusted to get π/2 and π pulses. The observer
pulse lengths for both tested frequency offsets were identical in all
experiments owing to the similar values of the resonator profile at both
observer frequencies (33.91 and 33.93 GHz, Fig. 2a). For rectangular
observer pulses the pulse lengths were 28 and 32 ns; for Gaussian
observer pulses, they were 56 and 74 ns. For the pump pulse we kept the
amplitude fixed at 100 %, which resulted in pulse lengths of 16 ns for
rectangular and 34 ns for Gaussian pulses. As we used Gaussian pulses that
were generated by Xepr, the FWHM of the Gaussian pulses was automatically
defined by the software as FWHM=tp/22ln(2),
and we did not optimise this parameter. An overview of all observer pulse
settings can be found in Tables S1 and S2 in the Supplement.
(a) Resonator profile with both tested observer frequencies (blue)
and the pump (red) frequencies for the rectangular pulses and (b) the
nitroxide spectrum with the positions of the two tested observer frequencies
and the pump frequency.
For optimum observation of the spin echo modulation in DEER traces, it has
been suggested to record the echo in transient mode and then perform a
digital integration over a product of the recorded echo with a Gaussian
filter (Pribitzer et al., 2017). This procedure is not ideal for commercial
spectrometers as the transient recording of the echo drastically increases
the spectrometer overhead time. Therefore, we performed a direct integration
of the spin echo. We optimised the integration window for each parameter set
for a maximum MNR by recording a series of Hahn echoes. Compared with
commonly used integration lengths equal to the π-pulse length for
rectangular pulses (Jeschke, 2007), we find settings where a 14 %
increase in the SNR can be achieved by choosing a larger integration window.
For Gaussian pulses, we find that it is typically preferable to choose
integration windows that are shorter than the π-pulse length. More
details can be found in Sect. S2.
The rectangular and Gaussian pulses with the best performance.
The best MNR for the setting RR was found to be 35 (Table 1). It was
achieved for an offset of 70 MHz and 60 % intensity. The best MNR for GG
was 41 at an offset of 70 MHz as well and a pulse amplitude of 100 %
(Table 1). This corresponds to a 17 % increase in the MNR of Gaussian
pulses compared to rectangular pulses. This is in contrast to the findings
of Teucher and Bordignon (2018), who found that Gaussian observer pulses
have a slightly lower MNR that rectangular pulses. The exact reason for the
deviating results is not entirely clear to us: we assume that this is due to
their different setup with a homebuilt resonator that has slightly
different properties than our commercial one.
As expected, the missing sidebands of the Gaussian pulses allow the usage of
higher pulse amplitudes. This hints that for the chosen parameters the pulse
overlap is indeed a limiting factor for rectangular pulses. The modulation
depth for RR and GG is approximately 30 % in both cases, but Gaussian
pulses seem to have the advantage of a higher echo intensity, probably due
to a lower pulse overlap. For RR and GG, the small offset of 70 MHz
performed better than a larger offset of 90 MHz, most likely due to the
different echo intensities at the corresponding positions in the EPR
spectrum (Fig. 2b). The results for all RR and GG settings can be found in
Tables S3 and S4.
Broadband-shaped pulses
We set out to investigate several broadband-shaped pulses, i.e. chirp,
WURST, HS{1,1} and HS{1,6} pulses for the settings RS and GS. Unless specified
otherwise, we used pump pulse lengths of 100 ns. According to Eq. (6), the
pulse length of 100 ns corresponds to a minimum accessible distance of
rmin=2.75nm. For the determination of shorter distances
we also tested chirp pulses with a length of 36 ns (referred to as short
chirp pulses below), which corresponds to a distance of rmin=1.96nm. Such a distance limit should be suitable for most practical
applications. Despite the fact that longer broadband-shaped pump pulses
should give higher inversion efficiencies, we found that they do not result
in a better performance for DEER. As the minimum accessible distance also
increases when longer pump pulses are used, we did not test pump pulses
longer than 100 ns. This is discussed in more detail in Sect. S13.
Spins are not flipped within the whole pulse duration, but only a smaller
fraction of it (Spindler et al., 2013). Simulations with an
HS{1,1} pulse with a length of tp=100ns, a frequency sweep width of Δf=110MHz and β=8/tp show distances up to rmin=2.32nm
could be detectable (see Sect. S7). It is, however, hard to generalise this effect
as the spin trajectories for different broadband pulses are not necessarily
the same.
Calculated inversion profiles of broadband pulses normalised to
ν1=30MHz, which corresponds to the maximum of the
measured resonator profile. (a) Chirp pulses with a frequency width of 120 MHz, a length of 100 ns and a rising time of 10 ns (green), a rising time of
30 ns (blue), a length of 36 ns and no quarter sine smoothing (red). (b) WURST pulses with a frequency width of 120 MHz, a pulse length of 100 ns and
a value for n of 6 (green) and 24 (blue). (c) HS{1,1} pulses with a frequency width of 90 MHz and truncation
parameters of 4 (green) and 10 (blue). (d) An HS{1,6} (green) and HS{1,1} (blue)
pulse with a width of 90 MHz and a pulse length of 100 ns. The truncation
parameter was 10 in both cases.
Figure 3 shows the calculated excitation profiles of some of the tested
pulses. The calculated excitation profiles are normalised to a ν1 field strength of 30 MHz, which we achieved with our
setup at the maximum of the resonator profile. Under such conditions, the
long chirp pulses have an adiabaticity of around 5, i.e. a chirp pulse with
a length of 100 ns, a sweep width of 120 MHz and a ν1
strength of 30 MHz with a calculated adiabaticity of 4.7. A short chirp pulse
with a length of 36 ns (and otherwise unchanged parameters) has an
adiabaticity of 1.7 due to the higher-frequency sweep rate. Although this
value is rather low, the calculations show that short chirp pulses achieve a
nearly complete inversion efficiency around the maximum of the excitation
profile (Fig. 3a). On the other hand, the excitation profile is rather broad,
with many sidebands. The finite length of the pulses creates an additional
distortion. By smoothing the edge with a quarter sine, this disturbance can
be reduced (Fig. 3a). A higher rising time will lead to a more properly
defined excitation profile with fewer sidebands, but the overall width of the
excitation profile is reduced (see Fig. 3a).
WURST pulses (Fig. 3b) are characterised by an additional parameter n. A
high value of n results in a more rectangular shape of the pulse and leads
to distortions in the excitation profile around the maximum (Kupče and
Freeman, 1995a, b; O'Dell, 2013). Small values of n lead to excitation
profiles with very steep and well-defined side flanks. However, for small n
very long pulse durations are needed to achieve a high inversion efficiency.
As long pulses are not feasible, because they limit the minimum distances
that can be resolved, we chose to stick to 100 ns pulses and test the values
for n of 6, 12 and 24, for which a reasonable excitation profile can be
expected (Fig. 3b).
In Fig. 3c we show the comparison of the excitation profile of
HS{1,1} pulses for truncation parameters of
β=4/tp and β=10/tp. For
β=10/tp the inversion efficiency is smaller than
for β=4/tp; however, the excitation profile is well
defined and does not show the sideband oscillations that can be seen for the
latter.
Owing to their higher adiabaticity, HS{1,6}
pulses feature higher inversion efficiency than HS{1,1} pulses with otherwise equal parameters (Fig. 3d) while
maintaining the steep frequency flank towards the observer profile at the
lower-frequency end.
Calculated inversion profiles of a WURST (n=12, green) and a
HS{1,6} (truncation parameter of
6/tp, blue) pulse with a pulse length of 100 ns and a sweep
width of 100 MHz normalised to ν1=30MHz, which
corresponds to the maximum of the measured resonator profile.
For HS{1,1} and HS{1,6} pulses a frequency sweep width of 50 to 110 MHz was
tested. WURST and chirp pulses tend to have narrower excitation profiles
for a given frequency sweep width at the tested parameters (see Fig. 4). We
therefore chose to use higher-frequency sweep widths for WURST than for HS
pulses to achieve a similar excitation bandwidth.
As the bandwidth of the resonator and the width of the spectrum are limited,
there is an optimum offset between the two pulses that minimises the overlap
but is not too large for the resonator bandwidth. We tested offsets from a
range of 70 to 130 MHz. The offset is defined as the difference between
the observer frequency and the centre of the frequency sweep of the
broadband-shaped pulses. For the optimisation measurements, the frequency of
the observer channel was fixed and the frequency of the pump pulse was
changed stepwise. We shifted the magnetic field with the pump pulse so that
we always pumped on the maximum of the spectrum (see Sect. S1). During the
increase in the offset, the position of the observer pulses in the spectrum
will change as the spectrum is shifted, with the pump pulse resulting in a
decrease in the echo for higher offsets. Table 2 shows an overview of all
tested pump pulse parameter sets.
The parameters for the broadband-shaped pump pulses.
We used the same parameters for the observer pulses as before, meaning that
we tried rectangular and Gaussian observer pulses at microwave frequencies
of 33.91 and 33.93 GHz at 100 % and 60 % amplitude, respectively
(see Tables S1 and S2), and combined them with all the broadband-shaped
pulses from Table 2. This results in 504 different settings (Table 2) for
the pump pulse and 8 different settings for the observer pulses, which gives
a total of 4032 different DEER settings. As the measurement of full DEER
traces and subsequent determination of the MNR would be very time-consuming,
we used the η2P parameter as an estimation for the
MNR. This was suggested by Doll et al. (2015) and already used by other
authors (Spindler et al., 2013; Tait and Stoll, 2016). As it
requires only two points of the DEER trace, the measurement time can be
drastically reduced. However, it has the disadvantage that artefacts, e.g. echo-crossing artefacts or nuclear modulation, might remain undetected.
Therefore, we decided to additionally perform phase cycling and nuclear
modulation averaging. For different observer pulse settings, the η2P parameters are not necessarily comparable, because η2P assumes a constant absolute noise level. However, this
noise level could change with different integration windows. Hence, we
identified the best chirp, WURST, HS{1,1} and
HS{1,6} pulse for each observer pulse setting
and recorded full DEER traces of them, giving a total number of 16 traces of
types RS and GS each.
Heat maps with the η2P values for 4p-DEER
measurements with an observer pulse length of 56 ns, Gaussian pulses and an
integration window of 56 ns. The observer frequency is at 33.93 GHz. The
pump pulse length is 100 ns. Each heat map shows a different pump pulse
type: (a) HS{1,6}, (b) WURST, (c) chirp with
quarter sine, and (d) HS{1,1}.
Exemplary heat maps showing the η2P for Gaussian
observer pulses and 100 ns pump pulses can be found in Fig. 5. We could
identify several trends that were true for all observer pulse settings.
HS{1,1} and HS{1,6}
pulses have higher maximum η2P values than chirp and
WURST pulses. HS{1,1} and HS{1,6} have their highest η2P values for
smaller offsets than chirp and WURST pulses. This fits to the steeper flanks
in their excitation profiles and a resulting smaller overlap with the
observer frequency. Nonetheless, the overall range of reasonable offsets for
all pulses is rather small and within a range of 80 and 100 MHz, meaning
that for nitroxide–nitroxide DEER and our setup the width of the spectrum
and the resonator profile has a more crucial influence in choosing the right
offset than the excitation profiles of the different pump pulses.
HS{1,1} and HS{1,6}
pulses have smaller ideal frequency widths of 90 and 110 MHz, whereas
for chirp and WURST pulses the frequency widths seem to be ideal at 120
and 160 MHz. This fits to the already mentioned observation that WURST
pulses have smaller excitation profiles for a given sweep width with the
used parameters than HS{1,1} and
HS{1,6} pulses (see Fig. 4). Interestingly,
despite their lower adiabaticity, the short chirp pulses with a length of 36 ns had a larger η2P value than the chirps with a
length of 100 ns for all observer pulse settings. Quarter sine smoothing
does not necessarily lead to a better performance of the short chirp pulses.
For the WURST pulses, a value of n=6 gives the best performance with all
observer pulses. For different observer pulses, we find that the best
performance of HS{1,1} and HS{1,6} pulses can be achieved with β parameters ranging
from 6/tp to 10/tp.
For all observer pulse settings, we identified the best parameter set for
each pulse family, resulting in a maximised η2P. We
then recorded a full DEER trace for each family and compared them by their
MNRs. All results for the full DEER traces can be found in Tables S5 and S6.
Table 2 shows the parameters and observer pulses that resulted in the best
performing chirp, WURST, HS{1,1} and
HS{1,6} pulses for the full DEER traces. We
found that also for broadband-shaped pump pulses Gaussian observer pulses
outperform rectangular ones. Again, this hints that Gaussian observer pulses
can successfully reduce the frequency overlap with the pump pulse due to
their missing sidebands. In all scenarios we found that an observer pulse
that is positioned with a 70 MHz offset to the maximum of the resonator
profile performs better than an observer pulse position with a 90 MHz offset
to the maximum of the resonator profile. The offset to the broadband-shaped
pump pulse, however, does not change on average, which means that in the
former case the observer and pump pulses have a more symmetric positioning
around the maximum.
The parameters of the observer and pump pulses that gave the best MNR for each pump pulse type. All observer
pulses are Gaussian pulses with a pulse length of 74 ns for a 60 % intensity and 56 ns for a 100 % intensity. The observer
frequency was 33.93 GHz in all cases. The MNR was evaluated up to 7 µs.
Pump pulseObs.tπΔfOffsetMNRMod depth λAmp. (%)(ns)(MHz)(MHz)HS{1,6} (β=10/tp)6010011090450.61WURST (n=6)6010016090400.63Chirp (no smoothing)1003612080450.49HS{1,1} (β=8/tp)10010011090500.52
Broadband-shaped pump pulses lead to a larger modulation depth than
rectangular and Gaussian pulses. Whereas for non-broadband pulses the
modulation depth is limited to around 30 % with our setup, we achieved an
increase of up to 63 % with WURST pulses. HS{1,6} pulses also lead to high modulation depths of 61 %.
For chirp and HS{1,1} pulses smaller modulation
depths of approx. 50 % were observed. However, the highest modulation
depth will not necessarily lead to the highest MNR, as can be seen in Table 3. This is due to a larger background decay of pulses with a higher
inversion efficiency and will be analysed in the next section. Due to a
higher bandwidth overlap, broadband-shaped pulses will also reduce the echo
intensity more strongly than rectangular or Gaussian pulses. HS{1,1} pulses seem to be a good compromise between a high
modulation depth, a high echo intensity and a background decay that is not
too steep. They resulted in the highest MNR of 50 with a pulse length of 100 ns, an offset of 90 MHz, a frequency bandwidth of 110 MHz and β=8/tp with the observer pulses being Gaussian pulses with
an amplitude of 100 % and a frequency of 33.93 GHz. Interestingly, this
performance is achieved although the broadband pulse does not achieve a
complete inversion (Fig. S8d in the Supplement). The modulation depth in that case increased
to 52 % (Fig. S11). This corresponds to an MNR increase of 43 %
compared to RR and 22 % compared to GG. To estimate the lower limit of
distances that can be determined with such a 100 ns pulse, we performed a
simulation to see when the spins are actually flipped during the experiment
(see Sect. S7). A visual inspection reveals that most spins are flipped between 20 and 80 ns within the pulse duration, making it an effective length of 60 ns where the spin flips occur, which would correspond to a minimum
detectable distance limit of 2.3 nm instead of 2.8 nm for a 100 ns spin-flip
period.
Depending on the resonator and the microwave amplifier, different
B1 field strengths are available on different spectrometers.
However, as the inversion efficiency of broadband-shaped pulses is less
dependent on the B1 field strength, as is the case for
rectangular and Gaussian pulses, which always require a proper adjustment of
the pulse length, we assume the findings here to be rather generalisable. In
order to discuss this more quantitatively, we simulated inversion profiles of
the best performing pulses from Table 3 for different B1 field
strengths.
The inversion profiles of the best performing (a) HS{1,6}, (b) WURST, (c) chirp and (d) HS{1,1} pulses with the parameters from Table 3. They were
simulated with a B1 field strength of ν1=20MHz (green), 30 MHz
(blue) and 40 MHz (red). These field strengths correspond to π-pulse
lengths of 25.0, 16.7 (which approximately corresponds to our setup) and
12.5 ns. The B1 field here is depicted as the Rabi
frequency.
We compare the pulse profiles with ν1=30MHz, which
corresponds to our setup with the cases where lower (ν1=20MHz) or higher (ν1=40MHz) B1 field
strengths are reached. Figure 6 shows how the different pulses behave when
different B1 field strengths are used. The WURST pulse (Fig. 6b) shows the least variation for different B1 field
strengths. As expected, the inversion efficiency drops a little bit for
ν1=20MHz. But this drop seems to be rather
insignificant, and good modulation depths can still be expected. The decrease
in inversion efficiency is a bit more significant for the HS{1,6} pulse, so that a small reduction in the modulation depth
is possible here. Both pulse profiles do not show significant changes when a
higher B1 field strength is used. The HS{1,1} pulse has a massive drop in inversion efficiency when
going to lower B1 field strengths. This does not come as a
surprise as the inversion efficiency is already incomplete at
ν1=30MHz. Here, it might be advantageous to reduce the
β parameter of the HS{1,1} pulse. As
has been stated earlier, this will increase the inversion efficiency. For a
higher B1 field strength of ν1=40MHz
the inversion efficiency of this HS{1,1} will
increase. Therefore, a higher modulation depth comparable to the
HS{1,1} pulse is expected. As this will also
increase the background decay, a higher MNR is not guaranteed. The chirp
pulse also shows a rather strong decrease in the inversion efficiency for a
ν1=20MHz. However, the inversion efficiency also
decreased for a higher B1 field strength of ν1=40MHz. This rather unexpected behaviour is probably caused by an
insufficient smoothing of the edges of the chirp pulse. With higher
B1 field strength the initial effective magnetic field
vector in the accelerated frame becomes less aligned with the z axis.
Therefore, smoothing becomes more important. In Fig. S10, we compared the
inversion profiles of 36 and 100 ns chirp pulses with and without quarter
sine smoothing. When quarter sine smoothing is applied, chirp pulses can
with a length of 36 ns indeed reach a high inversion efficiency with
ν1=40MHz. (Fig. S10b). As the width of the inversion
profile of this chirp pulse drops significantly for smaller B1 field strengths, it is only advisable to implement a quarter sine
smoothing with chirp pulses of a length of 36 ns when enough microwave power
is available. The situation looks different for chirp pulses with a pulse
length of 100 ns. Here, the inversion profile looks very similar for all
tested B1 field strengths. Particularly for smaller
B1 field strengths we expect 100 ns chirp pulses to
outperform chirp pulses with a length of 36 ns.
Another crucial parameter for DEER measurements that can vary from setup to
setup is the width of the resonator profile. Here, we have a FWHM of
approximately 200 MHz. Larger widths do not seem to be necessary because
they would exceed the width of the spectrum of the nitroxide. If only a
smaller width is available, the offset between pump and observer pulses
might need to be reduced. This would increase the overlap between the
observer and pump pulses. This problem could be overcome by either using
longer pump pulses or reducing the frequency width of the broadband-shaped
pulses. As a narrower resonator profile is also necessarily steeper, it
might also be necessary to perform a resonator bandwidth compensation as
suggested by Doll et al. (2013). Performing a resonator bandwidth
compensation with our setup does not give a significant advantage in the
η2P value (see Sect. S15). This is probably due to the
rather flat resonator profile in the region with maximum sensitivity where
the pump pulse is applied.
Background behaviour
The broader excitation profile of broadband-shaped pulses will increase the
background decay, which results in a higher noise level of broadband-shaped
pump pulses compared to rectangular or Gaussian pump pulses. We find an
approximately linear relation between the modulation depth and the
background decay (see Sect. S19). To investigate this effect more deeply, we
evaluated the MNR of the experimental DEER form factors, excluding the later
part of the form factor and only taking into account the first part up to a
truncation time τtruncation. Truncation of the DEER trace
will not change the modulation depth, but due to the background decay, the
noise level will be different. Figure 7 shows the MNR of broadband pulses as
a function of the truncation time τtruncation. As
expected, the MNR decreases with increasing τtruncation for all pulses, because of the increase in the noise. However,
the rate of the decrease in MNR is different for different pulse types,
which means that the relative performance of the pulses also depends on the
length of the DEER trace and therefore on the distance between the spin
centres that is supposed to be measured.
The MNR value as a function of the dipolar evolution time up to
which the noise has been evaluated, i.e. the truncation time τtruncation. The sample has a concentration of 80 µM
of the spin-labelled ligand. The maximum distance according to Eq. (13) is
depicted in the upper x axis. The line between the points is only a guide
for the eyes.
It turns out that HS{1,6} and WURST pulses with
their higher modulation depths have the highest MNR for short DEER traces,
whereas for longer traces HS{1,1} and chirp
pulses are better. The background decay seems to play a decisive role for
the MNR and the pulses resulting in a high modulation depth also have a
larger background decay. As the background decay causes the noise level to
increase with increasing dipolar evolution time, its influence is less
pronounced for short DEER traces, where the high modulation depth seems to
be leading to a high MNR. For longer traces, a high modulation depth is
linked to a strong background decay and a high noise level towards the end
of the trace. Therefore, the MNR of pump pulses generating a high modulation
depth decreases more strongly than for pulses effecting a smaller modulation
depth. This means that HS{1,1} and chirp pulses
perform better for longer traces.
As rectangular and Gaussian pump pulses have rather small modulation depths,
the corresponding decrease in the MNR due to the background decay is also
rather small, which means that the improvement achievable with broadband-shaped pulses is greater for shorter DEER traces. For short truncation times
τtruncation of 2 µs we observe an increase in
MNR from 44 for rectangular pulses (RR) to 82 for the best broadband-shaped
pulse (RS), which was an HS{1,6} pulse in this
case. This corresponds to an increase of 86 %. For long truncation times
τtruncation of 7 µs, this increase goes down to
43 %. This means that the MNR improvement that can be achieved by
broadband-shaped pulses can be drastically dependent on the length of the
measured DEER trace and therefore on the distance range to be covered by the
measurement. For a concentration of 80 µM, a high MNR improvement can
be achieved if the maximum distance of interest is below 4 nm with a pulse
that achieves a high modulation depth. This would correspond to the
HS{1,6} and WURST pulse in this case. If longer
distances up to 5 nm are to be detected, it seems to be advantageous to use
pulses that might not give the highest modulation depth in order to reduce
the background decay. An extrapolation for higher truncation times shows
that if even longer distances are of interest, broadband-shaped pulses will
not give a better MNR compared to rectangular pulses. Here, it is necessary
to reduce the background decay by using lower concentrations.
The performance of all the pulses at τtruncation=2µs can be found in Tables S7–S10. The chirp, WURST, HS{1,1} and HS{1,6} pulse resulting
in the best MNR are summarised in Table 4. For the broadband-shaped pulses
there were some minor changes in the parameters that gave the MNR when the
truncation time was set to a shorter value of ttruncation=2µs. For RR and GG there were changes in the best
parameter settings.
The parameters of the observer and pump pulses resulting in the best
MNR for each pulse type when the MNR was evaluated up to τtruncation=2µs. All observer pulses are
Gaussian pulses with a pulse length of 74 ns for 60 % intensity and 56 ns
for 100 % intensity. The observer frequency was 33.93 GHz for all pulses.
To check for a concentration-dependent performance of broadband-shaped
pulses, we also prepared a sample with a lower concentration of 30 µM
ligand and 60 µM WGA and performed DEER measurements with the optimised
parameter settings for the short chirp, WURST, HS{1,1} and HS{1,6} pulses. We did,
however, not check observer frequencies of 33.91 GHz, since they always
performed worse than an observer position of 33.93 GHz. For RR we tested an
offset of 70 MHz and 60 % intensity; for GG we tested an offset of 70 MHz
as well, but an intensity of 100 %, as these settings performed best
before.
This sample showed almost no background for all used pump pulses (see Fig. S17b). As the influence of the background is minimised due to the low
concentration, we expected to find the trends in the MNR as for the case of
the high concentrated samples and short truncation times. Figure 8 shows the
MNR as a function of the truncation time point τtruncation up to which the noise has been evaluated. As expected, no
significant decrease in the MNR with higher truncation times τtruncation was found. Without a significant background the
noise towards the end of the background-corrected form factor does not
increase significantly. The decrease in the MNR found for the high
concentration sample was therefore not observed here. For some pulses there
is a slight increase in the MNR with the truncation time; however, we
attributed this behaviour to a numerical uncertainty in the analysis.
The MNR as a function of the dipolar evolution time up to which
the noise has been evaluated. The sample has a concentration of 30 µM
of the spin-labelled ligand. The maximum distance according to Eq. (13) is
depicted in the upper x axis. The line between the points is only a guide
for the eyes.
With the low concentration sample, we found an MNR of 40 and a modulation
depth of 31 % for rectangular pulses; for Gaussian pulses we found an MNR
of 47 and a modulation depth of 30 %. Thus, also at lower spin
concentrations the Gaussian pulses lead to a similar modulation depth to
rectangular pulses, but again to an overall higher MNR. Table 5 shows the
results for the different broadband-shaped pump pulses in combination with
the observer pulse with which they performed best.
The parameters of the observer and pump pulses resulting in the best MNR for each pulse type. All observer pulses are Gaussian pulses with pulse lengths of 74 ns for 60 % intensity and 56 ns for 100 % intensity. The observer frequency was 33.93 GHz for all pulses. The MNR was evaluated up to 7 µs.
Pump pulseObs.tπΔfOffsetMNRMod depth λAmp. (%)(ns)(MHz)(MHz)HS{1,6} (β=10/tp)10010011090610.55WURST (n=6)60100160100540.59Chirp (no smoothing)1003612080580.46HS{1,1} (β=8/tp)10010011090650.47
Table 5 shows the optimised parameters for the different pump pulses. All
results can be found in Table S11. The parameters found for the observer and
pump pulses differ slightly from the parameters identified for the high
concentration sample, but lie in a similar range.
The broadband-shaped pump pulses resulted in a modulation depth that is a
bit lower than for the sample with the high concentration. The MNR was lower
as well. Furthermore, the order of performance of the different pulse types
changed. While we expected HS{1,6} and WURST
pulses with their high modulation depths to perform better than
HS{1,1} and chirp pulses for a sample less
susceptible to background influence, HS{1,1}
pulses actually performed best and WURST pulses were the worst
broadband-shaped pulses. HS{1,1} pulses lead to
an increase in the MNR of 60 % compared to rectangular pulses. This is
also lower than the 86 % increase that was obtained for the 80 µM
ligand concentration. The reason for the change in this behaviour is
probably a difference in the resonator profile that we noticed compared to
the other sample with the higher concentration (see Sect. S22). The achieved
B1 field was a bit lower for this sample, which changes the
performance of the pulses. However, HS{1,1} and
HS{1,6} pulses both give a good MNR with a high
concentration as well as with a low concentration.
When the MNR is to be increased by using broadband-shaped pulses to detect
long distances >5nm, lower concentrations are preferable as
they reduce the enhancement of the background decay. Here, switching to a
concentration of 30 µM of the doubly labelled ligand was enough to
significantly reduce the influence of the background. In Sect. S19 we performed
analytical calculations to estimate the potential MNR increase that can be
achieved by switching to broadband-shaped pulses for different
concentrations and distance ranges. For maximum distances below 4 nm an
increase in the MNR can be expected for all concentrations up to
approximately 100 µM. The situation is different if distances over 6 nm are to be detected. A significant gain can only be expected for smaller
concentrations in the range between 10 and 30 µM. For higher
concentrations the MNR gain drops quickly. For higher concentrations in the
range of 80 µM a MNR decrease has to be expected in this distance
regime. This is discussed in more detail in Sect. S21.
Conclusion and outlook
We have compared various broadband-shaped pulses as pump pulses for DEER
spectroscopy in Q band performed on samples with nitroxide spin labels and
investigated under which circumstances they perform best. By increasing the
inversion profile, broadband-shaped pulses can increase the modulation depth
from 30 % with rectangular pulses up to
60 %. However, with a larger inversion profile of broadband-shaped pulses
the overlap with the observer pulse and the background decay will also
increase. Both of those effects will tend to reduce the MNR. The overall MNR
increase will therefore be a compromise between the increase in the
modulation depth and the smaller echo and larger background contribution.
Systematic analysis of a trial-and-error optimisation has shown that the
performance of broadband-shaped pulses depends on the dipolar evolution time
and the concentration of spin centres. Larger dipolar evolution times mean
that the background has decayed more strongly by the end of the form factor.
Pulses with a higher inversion efficiency will produce a larger background
decay and their performance decreases more strongly for longer traces than for
pulses with a smaller inversion efficiency. We found HS{1,1} and HS{1,6} in combination
with Gaussian observer pulses to give a good MNR for high as well as low
spin concentrations. HS{1,1} have a lower
inversion efficiency and therefore a lower modulation depth, but they perform
better with longer traces needed for longer distances. The exact parameters
depend on the setup, but with values of β=8/tp or
β=10/tp, tp=100ns, and Δf=110MHz and an offset of 80 or 90 MHz we typically achieved good
results. If a high modulation depth which is particularly suitable for short
distances should be achieved, HS{1,6} and WURST
pulses are the best pulses. Good parameters are β=10/tp, tp=100ns, Δf=110MHz
and an offset of 90 MHz for HS{1,6} and n=6,
tp=100ns, Δf=160MHz and an offset of 90
or 100 MHz for WURST pulses.
Data availability
The raw data can be downloaded at 10.5281/zenodo.3726735
(Scherer et al., 2020).
The supplement related to this article is available online at: https://doi.org/10.5194/mr-1-59-2020-supplement.
Author contributions
AS, ST, SW and MD conceived the research idea and designed the conducted
experiments. AS conducted the EPR experiments and analysed the results with
ST. The spin-labelled ligand was synthesised in the lab of VW. AS prepared
all the figures and wrote the draft manuscript. All the authors discussed the
results and revised the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We thank Philipp Rohse for the synthesis of the spin-labelled ligand.
Jörg Fischer is thanked for sample preparation. This project
received funding from the European Research Council (ERC) under the European
Union's Horizon 2020 research and innovation programme (grant agreement
number: 772027 – SPICE – ERC-2017-COG). Andreas Scherer is grateful for
financial support from the Zukunftskolleg of the University of Konstanz.
Andreas Scherer, Sonja Tischlik and Sabrina Weickert are grateful for financial support from the
Konstanz Research School Chemical Biology (KoRS-CB).
Financial support
This work was supported financially by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant no. 772027‐SPICE‐ERC‐2017‐COG).
Review statement
This paper was edited by Daniella Goldfarb and reviewed by two anonymous referees.
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