Zero-field nuclear magnetic resonance of chemically exchanging systems

Zero- to ultralow-field (ZULF) nuclear magnetic resonance (NMR) is an emerging tool for precision chemical analysis. In this work, we study dynamic processes and investigate the influence of chemical exchange on ZULF NMR J-spectra. We develop a computational approach that allows quantitative calculation of J-spectra in the presence of chemical exchange and apply it to study aqueous solutions of [15N]ammonium (15N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{H}}_4^ +$$\end{document}H4+) as a model system. We show that pH-dependent chemical exchange substantially affects the J-spectra and, in some cases, can lead to degradation and complete disappearance of the spectral features. To demonstrate potential applications of ZULF NMR for chemistry and biomedicine, we show a ZULF NMR spectrum of [2-13C]pyruvic acid hyperpolarized via dissolution dynamic nuclear polarization (dDNP). We foresee applications of affordable and scalable ZULF NMR coupled with hyperpolarization to study chemical exchange phenomena in vivo and in situations where high-field NMR detection is not possible to implement.


Table of Contents
Supplementary Figure 4. Left -kd values extracted from the experimental data using two simulation models described above ("all" vs. "random" proton dissociation) and different intrinsic linewidth values ( Figure 7. Diagram of the chemical exchange process leading to the loss of nuclear "spin memory" in ammonia. Nuclear spin states are denoted in the computation basis, i.e., | ⟩ = |0⟩, | ⟩ = |1⟩. Initially, all four 1 H atoms are polarized (state | ⟩, or |0000⟩). One exchange event returns hydrogen spin with equal probability of states |0⟩ and |1⟩. Figure 8. Normalized entropy in the spin system as a function of the number of proton exchange events (in each exchange event, upcoming proton is considered to be unpolarized). Blue -exchange in hypothetical A2X system, green -exchange in hypothetical A3X system, Red -exchange in A4X system.  ⇌ A3X + 2B, where (A3X)B2 is a spin system of 2,2-dihydroxypropionic acid, A3X refers to a spin system of pyruvic acid and B refers to unpolarized 1 H atoms of water. Association exchange rates were calculated from the corresponding equilibrium constant K * + = -. ⁄ .

Supplementary Tables
Supplementary Table 1. Integrals of 1 H NMR peaks corresponding to CH3 groups of pyruvic acid (S1) and its hydrated form (S2) as a function of pH. Fraction of hydration is calculated as S2/(S1+S2).

Supplementary Notes
Values for the integrals of 1 H NMR peaks corresponding to CH3 groups of pyruvic acid (S1) and its hydrated form (S2) as a function of pH are given in the Supplementary Table 1. Hydration fraction was calculated as S2/(S1+S2) and plotted in Figure 3b of the main text.
where K L and K MNL are density matrices corresponding to a nitrogen atom and ammonium ion, respectively. (A and X represent 1 H and 15 N nuclei, respectively). Liouvillian superoperators Q Q L = − U U L + Q Q L and Q Q MNL = − U U MNL + Q Q MNL describe coherent evolution (defined by Hamiltonian superoperators, U U _ K = [ U _ , K]). For calculating Redfield relaxation superoperators, we used two relaxation models: (i) local fluctuating magnetic fields (step-by-step computational approach is given in Ref. 1) and (ii) intramolecular dipole-dipole interaction (the calculational approach is described in Refs. 2 and 3). The results of the calculations comparing the two relaxation models for The second (and more realistic) way of simulating proton dissociation considers the fact that each of the protons has equal probability for dissociation, therefore where we used a notation similar to that above. In this case, partial trace operator was taken as follows High-field 15 N NMR spectra were calculated for the species AnX by taking the Fourier transform of their time-dependent NMR signal S(t), which was calculated as follows: ( ) = Trlm n o + n p q K MrL ( )s.
Here n o and n p are nuclear spin operators corresponding to the spin X (written in the product operator basis of the spin system AnX), and K MrL is a density matrix of the system AnX.
Calculation of proton exchange in ammonium considers equilibrium constant determining the ratio of dissociation and association rates: From Supplementary Equations 6, one finds the association rateas and the molar fractions of conjugate forms w* { y = 1 1 + K -10 E€* , w* x = Exchange rates were extracted from the simulation and fitting the experimental data (Supplementary Table 2-3). Both simulation approaches (simultaneous dissociation from ammonium of all four hydrogen atoms and dissociation of a random hydrogen atom) give similar results, i.e., an initial broadening of the resonances with increased kd until they merge and subsequent narrowing of the single NMR resonance (Supplementary Figure 3).
Supplementary Figure 4 shows that two simulation models ("all" vs. "random" proton dissociation) do not differ significantly in the regime of slow exchange (pH < 3), and give similar values for kd (Supplementary Table 3). In the fast exchange regime (pH > 4) the value of kd is four times larger for the "random" dissociation model, as expected. Importantly, for the slow and fast exchange regimes, the linewidth is significantly affected by the intrinsic linewidth (FWHH0). For the pH values below 3, dissociation is linearly proportional to the concentration of hydroxyl anions (Supplementary Figure 5) demonstrating the fact that the rate-determining step is transition of a hydrogen ion from ammonium ion to [OH -]. 5 Zero-field NMR spectra are calculated in a way similar to high-field (Supplementary Equation 5) but considering the effect of all spins in the system and considering different NMR detection geometry: here n ‰ M Š and n ‰ L are nuclear spin operators corresponding to the spins A and X, respectively (written in the product operator basis of the spin system AnX), and K MrL is the density matrix of the system AnX. Figure 6 demonstrates the decline of the amplitude of the 15 N-ammonium as a function of the pH of the solution.

Supplementary
Since ZULF NMR signal of 15 NH4 is mainly determined by polarization of protons (due to much larger gyromagnetic ratio of 1 H compared to 15 N spins), one can determine the loss of proton polarization as a function of number of exchange events (Supplementary Figure 7). Let us compute how many proton exchange events are required for the system to go from 100% polarized state to the totally non-polarized state.
We start from composing a vector of state populations ƒ for the system with four identical spins: Here the states are , , , , and spins are indistinguishable. Therefore, the thermal equilibrium spin state is Let us introduce the "spin off" operator, Pr N→d , that describes dissociation of a random spin in the system The "spin on" operator describes association of the random unpolarized spin: After N complete exchanges, the state • will be given by In order to compare the final state with the thermal-equilibrium state, we define the normalized entropy ( ) of the system as where _ • is the -th element of the vector • , n is the number of spins (four, in the case of ammonium), _ is the partition function of the state corresponding to . Supplementary Figure 8 shows the entropy as a function of number of exchange events in a four-spin system (corresponding to ammonium) as well as two hypothetical 3-and 2-spin systems. One can see that the entropy quickly grows as the number of exchange events increases.
One can see that 10 exchange events bring the system close to thermal equilibrium; this fact can be used to explain why the ZULF NMR signals of ammonium disappear under fast exchange rate (see text).
Proton dissociation from pyruvic acid (PA) results in formation of pyruvate (P) ion. The equilibrium constant for hydrogen dissociation for this process is known, pK -e = 2.2. 6 Hydrated form of pyruvic acid (PAH) is a weaker acid and upon proton dissociation forms pyruvate hydrate (PH) as characterized by the corresponding equilibrium constant, K -g = 3.6 . 6 Equations describing equilibrium concentrations of the corresponding compounds are the following: One can easily derive from Supplementary Equation 16-17 that hydration fraction ( ) measured in high-field NMR is the following function of pH: = K * (α + 10 €*E€• •© ) K * (α + 10 €*E€• •© ) + 10 €*E€• •© + 1 .
As discussed in the main text, the fact that the NMR line at 2JCH is broader than the NMR line at JCH have three possible explanations (see Supplementary Figures 10-12).