ABSTRACT The exchange of molecules between different physical or chemical environments due to diffusion or chemical transformations has a crucial role in a plethora of fundamental processes

such as breathing, protein folding, chemical reactions and catalysis. Here, we introduce a method for a single-scan, ultrafast NMR analysis of molecular exchange based on the diffusion

coefficient contrast. The method shortens the experiment time by one to four orders of magnitude. Consequently, it opens the way for high sensitivity quantification of important transient

physical and chemical exchange processes such as in cellular metabolism. As a proof of principle, we demonstrate that the method reveals the structure of aggregates formed by surfactants

ELECTROLYTE SOLUTIONS Article Open access 05 January 2023 INTRODUCTION The direct observation of molecular exchange is challenging, because typically the processes occur in a liquid or gas

within a solid matrix. Nuclear magnetic resonance (NMR) spectroscopy provides the means to study dynamics of molecules non-invasively, without tracers, and even inside optically opaque

matter1,2. Two-dimensional (2D) exchange spectroscopy (EXSY) enables comprehensive analysis of complex exchange processes3. However, the 2D measurements are time consuming, as the experiment

has to be repeated from tens to hundreds of times with an incremented evolution time. Ultrafast (UF) NMR spectroscopy relies on spatial encoding of incremented evolution times into the

layers of a sample by exploiting the principles of magnetic resonance imaging (MRI)4,5. In theory, the UF approach is capable of delivering any kind of 2D NMR spectra in a single scan6, and

spatial encoding in discrete layers by using frequency selective pulses along with gradients have been exploited in the measurement of single-scan EXSY spectrum7. Furthermore, under-sampled,

small flip-angle single-scan EXSY was demonstrated based on the phase accrual during echo times8. In those experiments, the indirect dimension was heavily under-sampled (only three points

collected) and only a small part of the full magnetization was used in the detection due to the small flip angle excitation. On the other hand, the method could probe 5–20 mixing times in a

single scan, leading to a comprehensive characterization of the exchange process. Recently, we have demonstrated that the principles of spatial encoding can be extended to multidimensional

relaxation and diffusion correlation experiments under the concept of ultrafast Laplace NMR (UF LNMR)9,10. Here, we show that UF LNMR allows also single-scan exchange measurements via a

Laplace NMR contrast (in this case molecular diffusion coefficient _D_). The LNMR contrast is especially useful in the case of physical exchange, such as diffusion-driven pore-to-pore

exchange of fluid molecules in porous materials11 or intracellular to extracellular exchange of metabolites in cancer cell suspension12, when the exchanging sites are not resolved in

spectrum. On the other hand, the LNMR contrast can be exploited in the investigation of chemical exchange as well, if the chemical interaction changes the diffusion coefficient (or

relaxation time) of the spins. Contrary to the single-scan EXSY methods described above, the method introduced here is based on continuous spatial encoding and non-under-sampled data.

RESULTS ULTRAFAST DIFFUSION EXCHANGE EXPERIMENT The pulse sequence for the conventional diffusion exchange spectroscopy (DEXSY) experiment (Fig. 1a)13 includes two diffusion-encoding blocks

separated by the mixing time _τ_M. The experiment correlates the initial and final diffusion coefficients. The change in _D_ indicates that the molecules have moved to a different physical

or chemical environment during the mixing period. The conventional DEXSY experiment is extremely slow, because each point corresponding to both first and second diffusion encoding has to be

collected in a separate experiment. As illustrated in Fig. 1c, if the numbers of points collected in the indirect and direct dimensions are _N_ and _M_, the experiment requires _N_ × _M_

repetitions. For example, in order to collect 50 data points in each direction, the experiment has to be repeated at least 502 = 2500 times (multiplied by potential additional scans for

signal averaging and phase cycling). For typical samples, the repetition time (including the relaxation delay) of the experiment varies from 5 to 60 s, leading to a one scan experiment time

of 3.5–42 h. Our UF counterpart of the DEXSY experiment is shown in Fig. 1b. The first diffusion-encoding block before _τ_M includes simultaneous frequency-swept and gradient pulses. The

frequency of the former pulses is linearly increasing with time, while the gradient pulses make the Larmor frequency of nuclei linearly dependent on position. Therefore, the initial π/2

pulse excites the spins in the bottom of the sample tube first and those in the top last. The first π pulse is switched on at the midpoint of the π/2 pulse, and its length is a half of the

π/2 pulse. The spins in different layers form an echo at the midpoint of the first diffusion-encoding block, and the echo time is linearly dependent on position (zero in the top and equal to

the length of the π/2 pulse in the bottom). However, no collective echo is observed due to the different phases of the frequency-swept pulses experienced by the spins in different layers14.

The quadratic spatial dependence of the phases is compensated out by the second set of identical frequency-swept pulses arranged symmetrically around the midpoint of the encoding block,

along with another gradient pulse with an opposite polarity. Overall, the first encoding block consists of a double spin echo with the total echo time linearly dependent on the position. If

the gradients are strong enough, the decay of the signal is dominated by diffusion instead of _T_2 relaxation. Therefore, the longitudinal magnetization profile represents the diffusion

decay curve (see “Methods” section). After the mixing period, the magnetization profile is read by applying the principles of MRI, i.e., using read gradients within a CPMG15 loop. Due to the

presence of the strong gradients, the signal decay in the CPMG loop is also dominated by diffusion instead of _T_2 relaxation (see “Methods” section). Therefore, the data equivalent to

conventional DEXSY experiment is measured by a single-scan, whereas, as explained above, the conventional experiment requires hundreds to thousands of repetitions. The UF approach introduced

here accelerates the DEXSY experiment in two groundbreaking ways: first, the data corresponding to the indirect dimension is probed in a single scan based on spatial encoding; second,

contrary to the conventional DEXSY13, the direct dimension is detected in a single scan based on CPMG accompanied with gradients. In some studies, CPMG sequence has been used to detect

diffusion in constant field gradient experiments16, to accelerate pulsed-field-gradient diffusion experiments17 and to remove effects of field-inhomogeneity and exchange18, but it has not

been used in DEXSY experiments or LNMR experiments aiming at determination of distributions of diffusion coefficients by the Laplace inversion. The drawback of the UF DEXSY experiment is

that the spectral resolution is lost due to the spatial encoding. A partial spectral resolution can be achieved by exciting one peak from the spectrum by using frequency selective pulses in

the CPMG loop, as was done in this work. On the other hand, quite often in the case of physical exchange the resonances of exchanging sites are overlapping, and spectral resolution does not

provide any additional information. EXCHANGE THROUGH A LIPID BILAYER To demonstrate the feasibility and utility of the UF DEXSY based molecular exchange experiment, we studied the exchange

of water in an aqueous sodium decanoate surfactant system. Surfactants are a class of chemicals that lowers the surface tension of liquids. They are widely exploited in industrial

applications as detergents, wetting agents, foaming agents, emulsifiers etc19. They have also significant effect on the properties of aerosols, which are key components of the climate

system20,21. Decanoic acid is one of the amphiphiles found from carbonaceous meteorites22. Our sample included 717 mM of sodium decanoate in water (10% H2O and 90% D2O). 1H NMR experiments

were performed using a 400 MHz spectrometer at room temperature. Conventional diffusion-ordered spectroscopy (DOSY)23 analysis revealed a bimodal diffusion coefficient distribution of water

molecules with significantly different _D_ values, 6.0 × 10−11 and 1.9 × 10−9 m2 s−1 (Fig. 2b). The higher value corresponds to the _D_ of free water, whereas the lower _D_ value is equal to

the _D_ of decanoate (Fig. 2b). The results indicate that decanoate molecules form vesicles (Fig. 2a)24 and some water molecules are encapsulated inside the vesicles. The encapsulated water

molecules diffuse at the same rate as the vesicles. According to the Stokes-Einstein equation25 and the _D_ of decanoate, the diameter of the spherical vesicles is about 6.4 nm. Because the

length of the decanoate molecule is about 1.2 nm, the diameter of the water capsule inside the vesicle is about 1.6 nm. The data from a UF DEXSY measurement after the Fourier transform of

the spatial encoding dimension is shown in Fig. 3a. The signal of water was exclusively selected by using frequency selective pulses in the CPMG loop (Fig. 1b). A row and a column of the 2D

data shown on the top and left of the figure shows clearly the existence of two diffusion components decaying with different rates; the rapidly decaying part corresponds to the fast

diffusing free water and the slowly decaying part represents the encapsulated water. The data processing is explained in detail in “Methods” section. The 2D diffusion exchange maps resulting

from 2D Laplace inversions26,27 are shown in Fig. 3b, c (see also Supplementary Fig. 1). The maps include two diagonal peaks corresponding to water molecules located in the same site before

and after the mixing time, as well as two off-diagonal cross peaks with their coordinates corresponding to _D_ of free water in the first dimension and that of encapsulated water in the

second dimension or vice versa. The cross peaks reveal unambiguously the exchange of water molecules between the two sites due to diffusion through the decanoate bilayer28. The intensity of

the cross-peaks increases first with mixing time due to increased number of water molecules passing through the bilayer and later it is decreasing due to effect of _T_1 relaxation (Fig. 3d).

To quantify the exchange rates, the integrals of the diagonal and cross peaks were fit with a two-site exchange model3 (Fig. 3d). The exchange rates (_k_ = 28 ± 6 s−1, _k_FE = 21 ± 5 s−1,

_k_EF = 7.0 ± 1.1 s−1) are in good agreement with the conventional DEXSY analysis (see “Methods” section and Supplementary Fig. 2). However, the experiment time of a single conventional

DEXSY experiment with 8 scans and 16 diffusion encoding gradient steps in each dimension was 34 h, while the UF DEXSY measurement with 64 scans took only 1 h (the repetition time in both

experiments was long, 60 s, because _T_1 of H2O was 11 s). The overall duration of the exchange experiments comprising 6 different mixing times was decreased from 9 days to 7 h by switching

from the conventional to UF approach. This is a very significant reduction of the instrument time and it reduces substantially the possibilities for sample degradation during the

experiments. Furthermore, much higher number of points were collected in the UF (71 and 32 points in the indirect and direct dimensions) than in the conventional experiments (16 points in

the both dimensions), which led significantly better resolution in the UF DEXSY maps (cf. Fig. 3 and Supplementary Figs. 1 and 2). DISCUSSION A corresponding single-scan UF DEXSY measurement

(including the relaxation delay) takes only 1 min, while a one-scan conventional experiment with the same resolution (71 × 32 data points) is 2272-times longer (38 h). We estimated that the

spatial encoding lowers the sensitivity of the single-scan UF experiment by the factor of 9 as compared to the one-scan conventional experiment. However, the sensitivity per unit time

(i.e., SNR in the experiment with the experiment time equal to that of the conventional experiment) is increased by the factor of 6, because the UF DEXSY experiment can be repeated 2272

times during a single conventional DEXSY experiment. We note that replacing the PGSE or PGSTE type second diffusion encoding block in the conventional DEXSY experiment by the CPMG block in

the UF DEXSY experiment does not lower the sensitivity of the experiment, as full magnetization is detected in the CPMG loop. However, it has some consequences on the resolution, as the

maximum resolutions in the indirect and direct dimensions are coupled, i.e., one cannot be increased without decreasing another (see “Methods” section, subsection Resolution in UF DEXSY

experiments). Anyway, sufficient resolution can be achieved in both dimensions in typical liquid state studies, but the resolution may be a limiting factor in gas phase studies. The spatial

encoding block used in the UF DEXSY experiment shown in Fig. 1b differs from the spatial encoding blocks used in earlier diffusion studies10,29,30,31. In the earlier studies, the spatial

encoding was based on the modified PGSE and PGSTE sequences. Here, it is based on the double spin-echo sequence, in which the gradient is on for the whole sequence. Consequently, the spatial

encoding of diffusion requires less time than in the case of PGSE and PGSTE based sequences, in which the pulsed gradients are switched on only for a short time. Another benefit of the

current UF DEXSY sequence is that it can serve also as UF _T_2–_T_2 exchange sequence, if the gradients are small enough so that _T_2 decay dominates over _D_ decay (see “Methods” section).

On the other hand, the spatial encoding block used in this work can be replaced by any other spatial diffusion encoding block supplemented with a π/2 pulse for storing magnetization along

longitudinal direction for the mixing time. The PGSE and PGSTE based spatial diffusion encoding blocks do not require frequency-swept π/2 pulses, which are challenging to calibrate, and they

do not include two simultaneous frequency-swept pulses. Therefore, they may be more robust alternatives in many applications. For the completeness, we note also that, in addition to the

spatial encoding, there are many other approaches for accelerating NMR diffusion measurements, which are based on small flip angle pulses and trains of diffusion gradients, as well as

non-uniform sampling32,33,34,35,36,37. The UF approach has the potential to exploit modern nuclear spin hyperpolarization techniques38,39 to boost the sensitivity of the DEXSY experiment by

several orders of magnitude, because the whole 2D data can be measured in a single scan after the hyperpolarization process10. This makes much smaller amounts of substances (even

physiological concentrations)40 observable. In the conventional DEXSY experiment, the hyperpolarization process should be repeated before each repetition of the experiment, altogether from

hundreds to thousands of times. In practise, this is not feasible because, for example, the build-up of dynamic nuclear polarization (DNP) takes from tens of minutes to hours12. Therefore,

the UF approach has a potential to improve simultaneously both the time efficiency and sensitivity of molecular exchange analysis by several orders of magnitude. Combining UF DEXSY with

hyperpolarization should be feasible, as hyperpolarization has been successfully exploited in other kinds of UF experiments8,10,12,41, even with gases42. The high efficiency and sensitivity

of the UF DEXSY can be utilized in a variety of important molecular exchange processes in chemistry, biochemistry and medicine, including those in cancer cells12, exosomes, aerosols,

catalysts, porous media etc. The method is non-invasive and it does not require the use of tracers. It provides information, which is challenging to reach with any other methods. For

example, the conventional EXSY cannot probe the water exchange phenomena described in this communication, because the exchange sites are not resolved in an NMR spectrum, and the standard

optical methods, such as nanoparticle tracking analysis43, cannot detect such small (6.4 nm) vesicles, which were observed in the UF DEXSY analysis. It is possible to switch from diffusion

to _T_2 relaxation contrast in the exchange analysis, if small gradients are used in the UF measurements (see “Methods” section), in which case the contrast reflects predominantly rotational

motion of the molecules instead of translational44. The time window of molecular processes observable by the UF DEXSY can be extended at least by an order of magnitude by exploiting

long-lived states45,46, enabling the investigation of slow exchange processes. Because the UF LNMR experiments are based on spin echoes, they are applicable even with low-field, portable and

affordable NMR devices47 with inhomogeneous magnetic fields48,49. Therefore, hyperpolarized, mobile UF LNMR has the potential to bring advanced, high sensitivity and cost-efficient analysis

of molecular exchange processes outside laboratories. METHODS EXPERIMENTAL The sample included 717 mM sodium decanoate (Sigma-Aldrich, purity ≥98%) in water (90% D2O, 10% H2O) in a sealed 5

 mm NMR tube. The pH of the sample was about 8. NMR experiments were carried out on a Bruker Avance III 400 MHz spectrometer equipped with 5 mm BBO probe at room temperature. Conventional

DEXSY experiments: The number of diffusion gradient steps was 16 in both dimensions and the amplitudes of the gradients were linearly increased from 26.7 to 508 mT m−1. The length of the

diffusion gradients (_δ_) was 2.6 ms and the diffusion delay (∆) was 100 ms. The number of accumulated scans was 8, the relaxation delay was 60 s and the experiment time was 34 h. Altogether

6 experiments were performed with the mixing time varying from 5 ms to 5 s. Ultrafast DEXSY experiments: The lengths of 90° and 180° frequency-swept pulses were 15 and 7.5 ms, respectively,

and the sweeping width Δ_ν_ was 144 kHz. The amplitude of the spatial/diffusion encoding gradient _G_SD was 508 mT m−1. The number of echoes in the CPMG loop was 32, and the echo time was 8

 ms. The number of complex data points collected for each echo was 128, dwell time was 5 μs and the acquisition time of an echo was 1.28 ms. The amplitude of the trapezoidal read gradient

_G_RD was 268 mT m−1, the gradient ramp time was 0.5 ms and the overall length of the gradient was 3.94 ms (including the ramping times). The length of Gaussian frequency selective 90° and

180° in the CPMG loop was 1 ms. The number of accumulated scans was 64, the relaxation delay was 60 s and the experiment time was 1 h 5 min. Altogether 6 experiments were performed with the

mixing time _τ_M varying from 30 ms to 5 s. There are small artefacts at the edges and in the center of the spatial encoding profile of the UF DEXSY experiment due to imperfect initial/final

performance of the frequency-swept pulses (see Supplementary Fig. 3). Before the 2D Laplace inversion, the data points suffering from the artefacts, as well as the data outside the region

affected by the frequency-swept inversion pulse were removed, and the _z_ axis was converted into the _t_1 axis using Eq. (7) shown below. 2D Laplace inversions were performed using the

Iterative Thresholding Algorithm for Multi-exponential Decay (ITAMeD)27. THEORETICAL BACKGROUND OF THE UF DEXSY EXPERIMENT The frequency _ν_F of the frequency-swept pulse is linearly

increasing with time _t_: $$\nu _{\mathrm{F}} = \frac{{\Delta \nu }}{{t_{\mathrm{F}}}}t,\,{\mathrm{when}}\, - \frac{{t_{\mathrm{F}}}}{2} \le t \le \frac{{t_{\mathrm{F}}}}{2}.$$ (1) Here,

Δ_ν_ is the sweep width and _t_F is the length of the pulse. The phase of the frequency-swept pulse, _ϕ_F, is quadratically dependent on time: $$\phi _{\mathrm{F}} = \phi _{\mathrm{F}}^0 +

{\int} 2 \pi \nu _{\mathrm{F}}{\mathrm{d}}t = \phi _{\mathrm{F}}^0 + \frac{{\pi \Delta \nu }}{{t_{\mathrm{F}}}}t^2.$$ (2) Here, \(\phi _{\mathrm{F}}^0\) is the phase at _t_ = 0. In the

presence of the spatial/diffusion encoding gradient _G_SD, the Larmor frequency of the nuclei, _ν_L, is linearly dependent on position, _z_: $$\nu _{\mathrm{L}} = \frac{{\gamma

G_{{\mathrm{SD}}}}}{{2\pi }}z.$$ (3) Here, _γ_ is the gyromagnetic ratio of the nuclei. The spins at position _z_ are excited/inverted when \(\nu _{\mathrm{F}} = \nu _{\mathrm{L}}\) (the

validity of the instantaneous excitation/inversion approximation is discussed in ref. 50), and, according to Eqs. (1) and (3), the excitation/inversion time instant of the spins is linearly

dependent on position: $$t\left( z \right) = \frac{{\gamma G_{{\mathrm{SD}}}t_{\mathrm{F}}}}{{2\pi \Delta \nu }}z.$$ (4) Because the minimum and maximum frequencies of the frequency-swept

pulse are \(- \frac{{\Delta \nu }}{2}\) and \(+ \frac{{\Delta \nu }}{2}\), the maximum and minimum positions affected by the pulse are $$z_{{\mathrm{max}}/{\mathrm{min}}} = \pm \frac{{\pi

{\mathrm{\Delta }}\nu }}{{\gamma G_{{\mathrm{SD}}}}}.$$ (5) According to Eqs. (2) and (4), the phase of the frequency-swept pulse experienced by spins at _z_, \(\phi _{\mathrm{F}}\left( z

\right)\), is quadratically dependent on position: $$\phi _{\mathrm{F}}\left( z \right) = \phi _{\mathrm{F}}^0 + \frac{{\gamma ^2G_{{\mathrm{SD}}}^2t_{\mathrm{F}}}}{{4\pi \Delta \nu }}z^2.$$

(6) In the UF DEXSY experiment, the initial π/2 frequency-swept pulse excites the spins in the bottom of the spatial encoding region right in the beginning of the pulse and those in the top

at the end of the pulse. In between these extremes, the excitation time instant is linearly dependent on position (see Eq. 4). The first frequency-swept π pulse is switched on in the

midpoint of the initial π/2 pulse, and its length, \(t_{\mathrm{F}}^\pi\), is half of the length of the π/2 pulse, \(t_{\mathrm{F}}^{\pi /2}\). As it is sweeping the same frequency range

Δ_ν_, the sweep rate of the π pulse is double as compared to the π/2 pulse. Therefore, the π pulse affects the spins always in the halfway between the excitation and the end of the initial

π/2 pulse. The second frequency-swept π/2 and π pulses are switched on simultaneously, and they are identical to the first pulses. However, the gradient has an opposite polarity during those

pulses. Consequently, the pulses affect the spins in the top first and those in the bottom last (see Eq. 4). Overall, the first diffusion encoding period comprises a double spin echo with

the total echo time linearly varying with position (zero in the top, \(2t_{\mathrm{F}}^{\pi /2}\) in the bottom), and the second π/2 rotates magnetization along the longitudinal direction

for the mixing time _τ_M. As the phase of the first π/2 pulse, \(\phi _{\mathrm{F}}^{\pi /2}(z)\), is quadratically dependent on position (see Eq. 6), it is inducing a quadratic dependence

of the phase of the spins on the position _z_. However, this is compensated out by the second π/2 pulse with the same phase dependence (the phase is dependent on \(G_{{\mathrm{SD}}}^2\), and

therefore the change in polarity of the gradient does not change the phase). Similarly, the first π pulse is inducing its own spatial phase dependence (note that \(\phi _{\mathrm{F}}^{\pi

/2}(z) \ne \phi _{\mathrm{F}}^\pi (z)\), because \(t_{\mathrm{F}}^{\pi /2} = 2t_{\mathrm{F}}^\pi\)), which is compensated out by the second π pulse. The overall double spin echo time, _t_1,

as a function of the position _z_ is $$t_1\left( z \right) = 2\left( {1 - \frac{{\gamma G_{{\mathrm{SD}}}}}{{\pi {\mathrm{\Delta }}\nu }}z} \right)t_{\mathrm{F}}^{\pi /2},{\mathrm{when}} -

\frac{{\pi \Delta \nu }}{{\gamma G_{{\mathrm{SD}}}}} \le z \le + \frac{{\pi \Delta \nu }}{{\gamma G_{{\mathrm{SD}}}}}.$$ (7) The amplitude of the echo is50,51 $$E_1\left( z \right) =

E_1^0\exp \left[ { - \left( {\frac{1}{{T_2}} + \frac{{\gamma ^2G_{{\mathrm{SD}}}^2\left[ {t_1(z)} \right]^2}}{{48}}D_1} \right)t_1\left( z \right)} \right],$$ (8) where \(E_1^0\) is the

initial signal amplitude and _D_1 is the diffusion coefficient during the first diffusion encoding block. The first term in the parentheses accounts for the signal decay due to _T_2

relaxation while the second term represents decay due to molecular diffusion in the presence of gradient _G_SD. The echo amplitude in the CPMG loop in the second diffusion encoding block

is50,51 $$E_2\left( {t_2} \right) = E_0^2\exp \left[ { - \left( {\frac{1}{{T_2}} + \frac{{\gamma ^2G_{{\mathrm{Reff}}}^2\tau ^2}}{3}D_2} \right)t_2} \right],$$ (9) where _τ_ is the time

between the π/2 and π pulses and _t_2 is the time variable of the second dimension. _G_Reff is the amplitude of a gradient pulse with the length of _τ_ and its area equal to the that of the

read gradient (_G_R) pulse. Strictly speaking, Eq. (9) is valid for a constant gradient, but, here, the read gradient was switched off for the RF pulses. As a good approximation, _G_Reff was

used in Eq. (9) instead of _G__R_. In the UF DEXSY experiments, the diffusion decay dominated over the _T_2 decay due to the use of strong gradients. Therefore, the _T_2 terms in Eqs. (8)

and (9) can be neglected. Furthermore, there was a distribution of diffusion coefficients in the sample instead of a single _D_. Consequently, the overall signal amplitude observed in the

DEXSY experiment is $$E\left[ {t_1\left( z \right),t_2} \right] = \, E_0{\int} P \left( {D_1,D_2} \right)\exp \left[ { - \frac{{\gamma ^2G_{{\mathrm{SD}}}^2\left[ {t_1\left( z \right)}

\right]^2}}{{48}}D_1t_1\left( z \right)} \right]\\ \exp \left[ { - \frac{{\gamma ^2G_{{\mathrm{Reff}}}^2\tau ^2}}{3}D_2t} \right]{\mathrm{d}}D_1{\mathrm{d}}D_2.$$ (10) The 2D distribution of

diffusion coefficients, _P_(_D_1, _D_2), was solved by a Laplace inversion based on non-negativity constraint and Tikhonov regularization with the _l_1-norm penalty function27. EXCHANGE

RATES The exchange rates were obtained by fitting a two-site exchange model into amplitudes of DEXSY peaks (see Fig. 3d and Supplementary Fig. 1)3. In the fits, _T_1 of free water was fixed

to be 10.86 s, which is the value determined by an inversion recovery experiment. In the case of UF DEXSY, the observed exchange rate was _k_ = 28 ±  6 s−1 (_k_EF = 21 ± 5 s−1, _k_FE = 7.0 ±

 1.1 s−1), the relative populations of the free and encapsulated water pools were _x__F_ = 0.76 ± 0.02 and _x__E_ = 0.24 ± 0.02, and the relaxation rate in the encapsulated pool _R_1E = 

1/_T_1E = 8.5 ± 1.1 s−1. In the case of conventional DEXSY, corresponding values were _k_ = 32 ± 6 s−1 (_k_EF = 22 ± 4 s−1, _k_FE = 10 ± 2 s−1), _x_F = 0.70 ± 0.04, _x_E = 0.30 ± 0.03, and

_R_1E = 1/_T_1E = 7.1 ± 0.8 s−1. The exchange rates are in good agreement within the experimental error. There are slight deviations in the populations and relaxation rates in the

encapsulated pool, most probably because of much smaller amount of points collected in the conventional experiment and, consequently, different kind of probing of especially the initial part

of the signal decay. ALTERNATIVE ANALYSIS OF THE CONVENTIONAL DEXSY DATA In the analysis of the conventional and UF DEXSY data described above, we did not take into account the exchange

during the diffusion encoding periods. To check that this does not lead to significant errors in the resulting exchange rate values, we performed an alternative analysis of the DEXSY data

measured with the mixing time of 1 s, using a two-site exchange model taking into account the exchange during the diffusion period52,53,54,55. According to the model, the observed signal is

$$S\left( {b_1,b_2,\Delta ,\tau _{\mathrm{M}}} \right) = S_0\left[ {\left. {\exp \left( { - b_1{\boldsymbol{D}}} \right)\exp \left( { - {\boldsymbol{K}}} \right)\exp \left( { - \tau

_{\mathrm{M}}[{\boldsymbol{R}}_1 + {\boldsymbol{K}}]} \right)\exp \left( { - b_2{\boldsymbol{D}}} \right)\exp \left( { - {\boldsymbol{K}}} \right)} \right){\boldsymbol{M}}} \right]$$ (11)

Here, \(b_i = \Delta \gamma ^2G_{\mathrm{D}}^2\delta _i^2\), where _δ__i_ is the length of the gradient pulse, _i_ = 1, 2 and Δ is the diffusion delay, _D_ is the diffusion matrix

$${\boldsymbol{D}} = \left[ {\begin{array}{*{20}{c}} {D_{\mathrm{A}}} & 0 \\ 0 & {D_{\mathrm{B}}} \end{array}} \right],$$ (12) _K_ is the exchange matrix $${\boldsymbol{K}} = \left[

{\begin{array}{*{20}{c}} {k_{\mathrm{A}}} & { - k_{\mathrm{B}}} \\ { - k_{\mathrm{A}}} & {k_{\mathrm{B}}} \end{array}} \right],$$ (13) _R_1 is the longitudinal relaxation matrix

$${\boldsymbol{R}}_1 = \left[ {\begin{array}{*{20}{c}} {R_{1,{\mathrm{A}}}} & 0 \\ 0 & {R_{1,{\mathrm{B}}}} \end{array}} \right],$$ (14) and _M_ is the magnetization vector

magnetization vector (_M__A_ + _M_B = 1) $${\boldsymbol{M}} = [M_{\mathrm{A}}M_{\mathrm{B}}]^{\mathrm{T}}.$$ (15) Using the conservation of mass, _k_A = _k_B_M_B/_M_A. The fit of Eq. (11)

with the DEXSY data resulted in the following parameter values: _k_EF = 19.5 ± 0.3 s−1, _k_FE = 8.4 ± 0.2 s−1, _x_F = 0.70 ± 0.01, _x_E = 0.30 ± 0.01, _D_E = (5.3 ± 0.2) × 10−11 m2 s−1, _D_F

 = (1.49 ± 0.03) × 10−9 m2 s−1, _T_1E = 9.98 ± 0.03 s and _T_1F = 9.96 ± 0.03 s. The exchange rates are in agreement with the previous analysis within the error limits, confirming that it is

reasonably good approximation to neglect exchange during the diffusion encoding. Other parameters are also in good agreement with the previous parameters. RESOLUTION IN UF DEXSY EXPERIMENTS

The strength (or length) of the spatial and diffusion encoding gradient _G_SD has to be adjusted so that the decay of the transverse magnetization is sufficient, i.e., the smallest value of

the transverse magnetization is about 5–10% of the highest value (see Supplementary Fig. 3). The smaller _D_, the higher _G_SD. After setting an appropriate _G_SD, the sweeping width of the

frequency-swept pulses has to be adjusted so that the spatial encoding region matches with the sensitive region of the coil. The higher _G_SD, the wider sweep width. Wide sweep width means

high number of points in the pulse shape, but this is not a limiting factor in modern spectrometers. Therefore, there are no significant limitations related to the spatial encoding of

indirect dimension. Naturally, the maximum strength of the gradient sets the limit to the smallest observable diffusion coefficient. The strength and length of the gradient pulses in the

CPMG loop, _G_DD and _G_RD, as well as the echo time and number of echoes are selected so that the decay of signal during the CPMG loop is sufficient and a sufficient number of echoes is

observed. The stronger gradients, the faster decay. Once the _G_DD and _G_RD values have been set, one has to calculate a sufficient time step for collecting data points of echoes, so that

the field-of-view is larger than the spatial encoding region, as well as the number of collected points, which is high enough for a reasonable spatial resolution. The length of _G_RD pulse

determines the maximum number of collected points. In this work, 128 complex data points were collected per each echo, which resulted in a high spatial resolution (78 μm) and 71 points in

the indirect dimension of the UF DEXSY data after the removal of the extra point outside the spatial encoding region. Altogether 32 echoes were collected, which resulted in a sufficient

resolution in the direct dimension. The resolution in the direct dimension can be increased by shortening the echo time and the length of _G_RD pulse. This will lead to lowered maximum

spatial resolution. Therefore, the maximum resolutions in the indirect and direct dimensions are coupled; if one is increased, another is decreased. The smaller _D_, the higher _G_RD is

required to get sufficient diffusion decay in the CPMG loop. The higher _G_RD, the higher maximum spatial resolution. Therefore, the smaller _D_, the higher overall resolution in the UF

DEXSY experiments, i.e., the overall maximum resolution is improved when studying more viscous systems or larger molecules or aggregates. Naturally, this requires that shortened _T_2 will

not become a limiting factor. On the other hand, higher _D_ will lead to decreased maximum resolution in UF DEXSY experiments. _D_ of gases is typically about three orders of magnitude

higher than that of liquids, and therefore resolution in gas phase studies may be low. In summary, although the maximum resolutions in the indirect and direct dimensions are coupled, i.e.,

one cannot be increased without decreasing another, sufficient resolution can be achieved in both dimensions in typical liquid state studies. However, the resolution may be a limiting factor

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acknowledge funding from the European Research Council under Horizon 2020, projects Ultrafast Laplace NMR (grant agreement no. 772110) and SURFACE (grant agreement no. 717022), Academy of

Finland (grant nos. 289649, 294027, 308238, 314175, 319216, 321701 and 323480), KAUTE foundation, Finnish Cultural Foundation–Fanny and Yrjö Similä fund, Tauno Tönning Foundation, Orion

research foundation, Fortum Foundation and The University of Oulu Scholarship Foundation–Science Fund. The financial support from the Kvantum institute (University of Oulu) is also

acknowledged. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * NMR Research Unit, University of Oulu, P.O.Box 3000, FIN-90014, Oulu, Finland Otto Mankinen, Vladimir V. Zhivonitko, Anne Selent, 

Sarah Mailhiot, Sanna Komulainen, Susanna Ahola & Ville-Veikko Telkki * NANOMO Research Unit, University of Oulu, P.O. Box 3000, FIN-90014, Oulu, Finland Nønne L. Prisle Authors * Otto

Mankinen View author publications You can also search for this author inPubMed Google Scholar * Vladimir V. Zhivonitko View author publications You can also search for this author inPubMed 

Google Scholar * Anne Selent View author publications You can also search for this author inPubMed Google Scholar * Sarah Mailhiot View author publications You can also search for this

author inPubMed Google Scholar * Sanna Komulainen View author publications You can also search for this author inPubMed Google Scholar * Nønne L. Prisle View author publications You can also

search for this author inPubMed Google Scholar * Susanna Ahola View author publications You can also search for this author inPubMed Google Scholar * Ville-Veikko Telkki View author

publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS O.M.: implementation of the UF DEXSY experiment, carrying out the NMR experiments, analysis of the

results, writing the first version of the manuscript. V.V.Z.: planning and implementation of the UF DEXSY experiment. A.S.: planning and preparation of the surfactant samples. S. M.: time

domain analysis of the DEXSY data. S.K.: planning and preparation of the surfactant samples. N.L.P.: ideation and planning of surfactant vesicle samples and Stokes-Einstein analysis. S.A.

implementation of the UF DEXSY experiment. V.-V.T.: corresponding author, planning the study and UF DEXSY experiment, writing the polished version of the manuscript. All the authors

contributed on the final version of the manuscript. CORRESPONDING AUTHORS Correspondence to Otto Mankinen or Ville-Veikko Telkki. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare

no competing interests. ADDITIONAL INFORMATION PEER REVIEW INFORMATION _Nature Communications_ thanks Bernhard Blümich, and the other, anonymous, reviewer(s) for their contribution to the

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CITE THIS ARTICLE Mankinen, O., Zhivonitko, V.V., Selent, A. _et al._ Ultrafast diffusion exchange nuclear magnetic resonance. _Nat Commun_ 11, 3251 (2020).

https://doi.org/10.1038/s41467-020-17079-7 Download citation * Received: 10 March 2020 * Accepted: 11 June 2020 * Published: 26 June 2020 * DOI: https://doi.org/10.1038/s41467-020-17079-7

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