INDIANA: An in-cell diffusion method to characterize the size, abundance and permeability of cells

Abstract

NMR and MRI diffusion experiments contain information describing the shape, size, abundance, and membrane permeability of cells although extracting this information can be challenging. Here we present the INDIANA (IN-cell Diffusion ANAlysis) to simultaneously and non-invasively measure cell abundance, effective radius, permeability and intrinsic relaxation rates and diffusion coefficients within the inter- and intra- cellular populations. The method couples an experimental dataset comprising stimulated-echo diffusion measurements, varying both the gradient strength and the diffusion delay, together with software to fit a model based on the Kärger equations to robustly extract the relevant parameters. A detailed error analysis is presented by comparing the results from fitting simulated data from Monte Carlo simulations establishing its effectiveness. We note that for parameter typical of mammalian cells the approach is particularly effective, and the shape of the underlying cells does not unduly affect the results. Finally, we demonstrate the performance of the experiment on systems of suspended yeast and mammalian cells. The extracted parameters describing cell abundance, size, permeability and relaxation are independently validated.

Introduction

Magnetic resonance experiments provide a powerful method to non-invasively analyse molecules within cells and tissues (CITE), and diffusion methods are frequently employed^1–3. Applications range from the medical, where diffusion based MRI measurements can be used diagnostically^4,5, through to characterising protein folding^6,7 and localising biomolecules in cellular mixtures⁸. There continues to be significant development of diffusion methodology including the FEXSY method⁹, triple quantum experiments¹⁰ and improved diffusion tensor imaging methods¹¹. For biological systems, the high abundance and slow relaxtion of water make it an excellent probe. The restricted motion of water restrained within internal cavities of cells and tissues, the transport of water across permeable membranes via channels such as aquaporin^12–14 and variations in intrinsic relaxation can all complicate analysis of data (Fig. 1A).

Figure 1.

A Factors that need to be considered when analysing water diffusion experiments in cellular suspensions. The cell abundance, radius and permeability, as well as intra- and extra-cellular diffusion coefficients and relaxation rates will be significant. B STE pulse sequence used to measure diffusion. As magnetisation is stored longitudinally in this experiment, longer times can be experimentally accessed, that allows the cellular spaces to be more fully explored. C(i) Simulated signal intensities from diffusion experiments when the signal comprises two independent, non interacting compoents as a function of q². The slow/fast diffusion coefficents were set to 200/2000 gm² s^-1 in a 1:9 population ratio. C(ii) the same data plotted as a function of b together with the cases where there is no exchange. In the cases where there is no exchange and constant diffusion coefficients, all curves overlay. D(i) demonstrates the failure of a simple biexponential model to explain the diffusion data observed for a real system of mammalian 3T3 cells. The curves do not overlay, and there are significant systematic deviations between the data and the best fit. D(ii) Inclusion of exchange and restriction effects using the model set out in this paper (equations (8) and (10)) results in good fits to the data, and physically reasonable fitted parameters (Fig 5).

In both NMR and MRI diffusion experiments, magnetisation is typically first encoded a position dependent phase by application of a pulsed field gradient of strength G is applied for duration δ After a delay Δ, where magnetisation is held either transverse in a spin-echo (SE) or longitudinal in stimulated echo (STE) experiment, a decoding pulsed field gradient is applied. Signals originating from molecules that have moved away from their original position are attenuated. Where the particle of interest undergoes translational self-diffusion, signal will decay according to¹⁵:

$$S(b)=S(0)e^{-bD}$$ (1)

where in the limit Δ>δ, b = (Δ – <δ/3)q^{2 16}, q = Gγδ, and γ is the gyromagnetic ratio of the nucleus under study.

The restrictive effects of a cavity on translational diffusion has been treated by Neuman¹⁷ under the assumption that the displacement probability distribution will remain Gaussian. This assumption breaks down at high gradients and in the long-time limit^18,19, where distinctive ‘diffusion-diffraction’ patterns are expected whose profile is dominated by shape of the cavity, not the local diffusion coefficient. The size of mono-disperse polystyrene²⁰ has been measured using this method. When the sample under study is heterogeneous, as is typically the case for biological samples¹⁸, this effect is rarely observed experimentally.

Reflecting the need to combine both restricted diffusion and permeability, Kärger developed a model^21,22 where water can exchange between an interior, and an exterior pool. The displacement probability distributions are implicitly assumed to be Gaussian and so Kärger equations are compatible with the results of Neuman.

The heterogeneity of human tissue means that the assumption of two pools can be inappropriate. Monte Carlo simulations have been used to examine the effects of water exchanging between 2 and 3 pool models with differing underlying geometries^23–25 tailored to specific applications. These have been followed by applications to study the microstructure of perfused glial cells²⁶, rat brain tissue²⁷, blood²⁸ and bovine optic nerve²⁹. More recently, a method has been proposed for analysing cancer tissue diffusion data that includes restricted diffusion but neglects exchange³⁰.

For in-cell magnetic resonance experiments, the geometry is less complex than inside living tissue. In such applications, it is desirable to have a single model that can be applied to give useful information that is insensitive to shape of the cavity. Moreover, when applying such a model, it is important to have a reasonable expectation for the accuracy of fitted parameters obtain, and where the likely systematic errors are likely to be.

To address these challenges, here we present the INDIANA (IN-cell Diffusion ANAlysis) experimental protocol and software package. Water diffusion data from an STE diffusion experiment (Fig 1B) is acquired, varying both G and A. The data is analysed by a model that treats the cells as spheres following the approach of Neuman¹⁷, and allows water to exchange with a freely moving exterior population using Karger^21,22 equations, that have been modified to include the effects of intrinsic relaxation. The outputs from the model are cell count (interior population), cell radius, permeability (cell interior to exterior exchange rate), as well as the intrinsic relaxation rates and diffusion coefficients from the interior and exterior populations.

Regions where we expect acceptable parameters are identified by fitting data simulated using Monte-Carlo models to our model (Fig 5). We demonstrate that accounting for a physically reasonable distribution of cavity sizes and the expected variation in gradient strengths in typical NMR spectrometers³¹ in simulations is necessary to resemble real data, and effectively extend the region of validity of the approach. The accuracy of the fitting parameters will be dependent on the choice of G and Δ, but for regions of parameter space typical for mammalian cells, the accuracy is remarkably high. Moreover, the results are relatively insensitive to the specific geometry of the underlying cell.

Figure 5.

Error plots made by comparing fitted parameters using our model (equations (8) and (10)) to values set in the simulations. Error bars in model parameters are determined by a bootstrapping analysis, where a subset of curves is selected whose total number is equal to the number simulated but selecting from values of Δ at random with replacement, a process repeated 1000 times. Where a parameter’s value is not varied within a particular plot its value is fixed as follows: D_in = 500 μm² s^-1, D_out = 2000 μm² s^-1, k_ex = 3.55s^-1, R = 10 μm and P_in = 0.15. In Figures A-D the values of Rⁱⁿ₁ = R^ex₁ = 0 s^-1. In Figures E and F the values of R₁ = 0.25 s^-1 when they are not being varied.

Experimental data can be acquired in as little as 4 minutes and we show in applications to budding yeast (Saccharomyces cerevisiae), 3T3 and HeLa mammalian cells the parameters we extract are consistent with those published previously (Fig 6). Overall, the experimental and analysis tools are well suited for in-cell NMR studies to quickly and efficiently determine properties of the underlying cells in a manner that that is not overly dependent on the underlying geometry. Software for the INDIANA analysis is written in Python and can be freely downloaded.

Figure 6.

NMR data collected on real systems of (A) yeast, (B) 3T3 and (C,D) HeLa cells with varying cell densities fitted to the model as described in section 4.1. The data is well explained by the model, and the extracted parameters are in good agreement with those obtained independently (Tables 1,2).

Results

We first introduce an analysis that demonstrates the need to use a Karger-type model when analysing water diffusion data. Temporarily neglecting relaxation, in the case where two freely diffusing, but non-interacting populations are present, A and B, signal decay will follow a bi-exponential form:

$$S(b)=S_A(0)e^{-b{D}_A}+S_B(0)e^{-b{D}_B}.$$ (2)

Datasets acquired with varying G and Δ will overlay when plotted against b (Fig 1 Ci,ii). Data acquired on a sample of suspended 3T3 cells (Fig 1D) do not overlay, demonstrating the need for more sophisticated analysis. Individual datasets at constant Δ with varying G decay typically take a bi-exponential form although at smaller values of b, signal decay can appear to be mono-exponential (Fig 1Ci). Restricted diffusion effects, cell permeability and differential relaxation are all expected to significantly affect the measured signal (Fig 2,3). Following the approach of Kärger, for a stimulated echo diffusion experiment, the differential equation for the attenuation of signal S from the interior (in) and external (out) populations due to diffusion, relaxation and exchange will be:

$$\frac{d}{dt}\left(\begin{array}{c}{S}_{in}\\ S_{out}\end{array}\right)=\left(\begin{array}{cc}-q^2{D}_{in}-k_{10}-R_1^{in}& k_{\text{OI}}\\ k_{\text{Io}}& -q^2{D}_{out}-k_{OI}-R_1^{out}\end{array}\right)\left(\begin{array}{c}{S}_{in}\\ S_{out}\end{array}\right)$$ (3)

Figure 2.

A Visual representation of a single Monte-Carlo single particle trajectory (Methods, A). The simulations account for distinct intra- and extra-cellular environments characterised by separate diffusion coefficients as well as exchange between the two environments and differential relaxation in the two phases. B A probability distribution of displacements for different diffusion times can be determined. C The trajectories can be combined to simulate experimental data (equation (21)).

Figure 3.

Simulated data neglecting relaxation was fitted to equation (8) showing the effect of exchange between freely diffusing populations on the diffusion attenuation. The slow and fast diffusion coefficient were 200 and 2000 μm² s^-1’, weighted with a 1:9 ratio. In general, data acquired at variable Δ do not overlay. The simulations are perfectly accounted for by fitting to the Kärger equations^21,22 (solid lines). B(i) Simulated data showing the effect of restricted diffusion. In each case, particles are diffusing within an impermeable sphere with the diffusion coefficient set to 500 μm² s^-1. In each case a distribution of spheres is imposed of stated average radius and standard deviation set to 20% of the average value, reflecting size distributions observed in cellular systems. Data simulated for variable values of Δ do not overlay. Data are fitted to the Neuman equation¹⁷ (solid lines equation (10)). B(ii) Schematic showing single particle trajectories for case when average radius is set to 10 μm. The number of collisions with the membrane increases with the diffusion time and the diffusion attenuation increasingly becomes a property of the restricting geometry rather than the viscosity.

Where D_in and D_out are the intra- and extra-cellular diffusion coefficients, k_IO/k_OI are the in-to-out/out-to-in exchange rates, and $R_1^{in}/R_1^{out}$ are the intra-/extra-cellular longitudinal relaxation rates. For a spin-echo diffusion experiment, the relevant relaxation rates will be transverse (R ₂) rather than longitudinal (R ₁). This equation is of the form $\frac{dS}{dt}=\rho S$, whose general solution is given by S(q,Δ)=e^p(q)ΔS(0,0). The matrix exponential can be evaluated by constructing a matrix with Eigenvalues along the diagonal (D) together with a matrix of Eigenvalues (P). In the case where the chemical shift in the two compartments is identical such as for water in typical cellular samples, a detection operation that combines the signals is required:

$$S(q,\Delta )=P{e}^{D\Delta }{P}^{-1}\left(\begin{array}{l}{p}_{in}\hfill \\ p_{out}\hfill \end{array}\right)\left(\begin{array}{c}11\end{array}\right)S(0,0)$$ (4)

where p_in and p_out and the equilibrium populations. The Eigenvalues of -p are

$${\lambda }^{\pm }=\frac{1}{2}({\xi }^{+}\pm \varphi ),$$ (5)

expressed in terms of the following substitutions:

$$\begin{array}{c}{\xi }^{\pm }=q^2{D}^{\pm }+R_1^{\pm }+k_{ex}(p_{out}\pm p_{in}),\\ \varphi =\sqrt{{({\xi }^{-})}^2+4k_{ex}^2{p}_{in}{p}_{out}}\text{and}\\ D^{\pm }=D_{in}\pm D_{out},R_1^{\pm }=R_1^{in}\pm R_1^{out}\end{array}$$ (6)

and the overall exchange rate is given by k_ex = k_OI + k_IO, the reciprocal of the lifetime where k_OI = p_ink_ex and k_IO = p_outk_ex. In the case where the resonances of the interior and exterior populations can be spectroscopically resolved then:

$$\begin{array}{c}\frac{S_{in}(q,\Delta )}{S_{in}(0,0)}=p_{in}(e^{-\lambda +\Delta }+(\varphi +k_{ex}-q^2{D}^{-}-R^{-})\frac{e^{-{\lambda }^{-}\Delta }-e^{-{\lambda }^{+\Delta }}}{2\varphi })\\ \frac{S_{out}(q,\Delta )}{S_{out}(0,0)}=p_{out}(e^{-\lambda -\Delta }+(\varphi -k_{ex}-q^2{D}^{-}-R^{-}(\frac{e^{-{\lambda }^{+}\Delta }-e^{-\lambda -\Delta }}{2\varphi })\end{array}$$ (7)

In the case where the chemical shift is identical in the interior and exterior partitions, as is typically the case for water in cellular systems, then the combined signal will be:

$$\frac{S(q,\Delta )}{S(0,0)}=e^{-{\lambda }^{-\Delta }}+(\varphi -k_{ex}+(p_{in}-p_{out})(q^2{D}^{-}+R^{-}))\frac{e^{-{\lambda }^{+}\Delta }-e^{-{\lambda }^{-\Delta }}}{2\varphi }$$ (8)

When the exchange rate is slow when compared to the difference in diffusion coefficients $|q^2{D}^{-}+R_1^{-}|\gg k_{ex}$ (Fig 3A, equation (27)), the diffusion curves retain a strong bi-exponential character and the data for different delays overlay. As the diffusion time increases to an intermediate regime (Fig 3A), the curves no longer overlay, and the specific value of q at which the two phases apparently intersect vary with Δ. Physically, this corresponds to slower diffusing population having an increased probability to move into the fast diffusing phase at longer times, where it experiences greater signal attenuation. As the exchange rate continues to increase into the fast regime, $|q^2{D}^{-}+R_1^{-}|\ll k_{ex}$ the curves once again overlay and fit well to a single exponential whose apparent diffusion coefficient (ADC) is equal to the population-weighted average diffusion coefficient (Fig 3A, equation (28)). The Kärger equations are recovered by setting $R_1^{in}=R_1^{out}=0$ ^21,22.

Following the results of Murday, Cotts³² and Neuman¹⁷, if the intra-cellular displacement probability distribution for particles within an impermeable spherical cavity of radius r is approximately Gaussian, then:

$$S(b)=S(0)e^{-b{D}_{in}^{app}}$$ (9)

Where the apparent internal diffusion coefficient is

$$D_{in}^{app}=\frac{2\text{Φ}(D_{in},\delta ,\Delta ,R)}{D_{in}^2{\delta }^2(\Delta -\delta /3)}$$ (10)

where

$$\text{Φ}(D,\delta ,\Delta ,r)=\sum _{n=1}^{\infty }\frac{1}{{\alpha }_n^6(r^2{\alpha }_n^2-2)}(2{\alpha }_n^{2}D\delta -2+2e^{-{\alpha }_n^{2}D\delta }+2e^{-{\alpha }_n^{2}D\Delta }-e^{-{\alpha }_n^{2}D(\Delta -\delta )}-e^{-{\alpha }_n^{2}D(\Delta +\delta )})$$ (11)

is a complicated sum, requiring α_n, the n^th root of:

$$({\alpha }_{n}r)J_{3/2}^{\prime }({\alpha }_{n}r)-\frac{1}{2}{J}_{3/2}({\alpha }_{n}r)=0$$ (12)

where J is a Bessel function of the first kind and J’ is its first derivative. The roots of the Bessel function equation are readily obtained numerically. Typically, the first twenty roots are sufficient for the sum to get an error of approximately 0.1% in the applications described here.

With $\zeta =\frac{D\Delta }{R}$, then in the limit Δ ≫δ and ζ ≪1, it follows that $\sum _{n=1}^{\infty }\frac{1}{{\alpha }_n^2(r^2{\alpha }_n^2-2)}=\frac{r^2}{10}$, and we recover equation (1), reflecting the free diffusion limit where particles rarely hit the cavity walls:

$$D_{app}=\frac{2D_{in}^3{\delta }^2}{D_{in}^2{\delta }^2}\sum _{n=1}^{\infty }\frac{1}{(r^2{\alpha }_n^2-2)}=D_{in}$$ (13)

In the limit Δ ≫ δ and ζ ≫ 1, it follows that $\sum _{n=1}^{\infty }\frac{1}{r^2{\alpha }_n^2-2}=\frac{1}{2}$:

$$D_{app}=\frac{2}{\Delta }\sum _{n=1}^{\infty }\frac{1}{{\alpha }_n^2(r^2{\alpha }_n^2-2)}=\frac{r^2}{5\Delta }$$ (14)

In this limit, the decay of signal intensity will be independent of Δ reflecting the limit where the results are independent of the internal diffusion coefficient.

To determine the practical validity of the formula, Monte-Carlo simulations were performed on particles held within a spherical cavity (Fig 3B, C). Data was simulated with variable values of G and Δ (Fig 3B, points) and fitted to equation (10). At relatively short times, where a molecule has relatively few collisions with the cavity walls, and the Neuman approximation is excellent (Fig 4A). At longer times where molecules are colliding with the surface of the cavity many times, the expected distribution is no longer Gaussian (Fig 4A inset, equation (29)) and the theory breaks down, evidenced by negative curvature in the diffusion plots (Fig. 3). When qr = 2π, the gradient pitch is the size of the radius of the sphere, dS/dq reaches a minimum and the signal takes a negative sign, leading to the characteristic ‘diffusion diffraction’ pattern^3,20 (Fig. 4). A series expansion of the exact result reveals that the Neuman formula is exact to 2^nd order in qr, and will deviate substantially from the expected result when $qr\gg \sqrt{10}$, (equation (31)). Empirically we find the Neuman formula to be numerically accurate to within a few percent in the range qr ≤ 2 (Fig 4C).

Figure 4.

A Simulated displacement distributions, P(z), for different diffusion times calculated from 500,000 simulated single particle trajectories where each particle is started in a random position in an impermeable spherical cavity with a 10 μm radius and internal diffusion coefficient 500 gm² s^-1. The coloured dashed lines indicate Gaussian fits to the simulated distributions. The distributions are Gaussian at short times, but at long times, there are significant deviations (inset), indicating a breakdown in the model proposed by Neuman. The simulations tend towards the exact expected probability distribution expected at long times (equation(29)). B Simulated displacement distributions as for A, for a Gaussian distribution of impermeable spherical particles with average radii 10 μm and standard deviation 2 μm, as expected for a preparation of HeLa cells³⁴. The distributions are Gaussian (inset). C-F Calculated NMR signal at Δ = 0.5 s with average gradient strengths (G) varying from 4 to 80 G cm^-1 and 3 = 3 ms. For C and D the NMR signal is calculated using the displacement probability distribution in A, assuming a uniform radius whereas E and F are calculated using the distribution in B that includes a range of cell sizes. For C and E the signal iscalculated assuming perfectly uniform gradients whereas in D and F the effects of non-uniform gradients are explicitly included (Fig S1). The solid lines in C-F show the signal attenuation predicted by the Neuman equation. The combined effect of including a distribution, and non-uniform gradients is to extend the limit of the theory from qr ≤ 2 to qr ≤ 4. G-H Simulated NMR decay curves for typical experimental parameters: varying both G from 4 to 60 Gcm^-1 and Δ = 0.01 to 0.5 s with 8 = 2 ms. The data is fitted to the Neuman result (equation (10)) assuming perfect gradients and the displacement distribution in A for G and non-uniform gradients and the distribution in B for H. The fitted diffusion coefficients are 483±7 (G) and 480±10 μm² s^-1 (H) and radii are 10.82±0.03 (G) and 10.15±0.05 μm (H). Error bars in model parameters are determined by a bootstrapping analysis, where a subset of curves is selected whose total number is equal to the number simulated but selecting from values of Δ at random with replacement, a process repeated 1000 times. Including the effects of non-uniform gradients and accounting for an expected range of sphere sizes effectively extends the region of validity of the Neuman result (equation (10)).

Diffusion diffraction effects are not typically observed in experimental data analysing tissue and cells, though they are expected from simulations (Fig 4). Biological samples contain a heterogeneous distribution of particle sizes. HeLa cells, for example, are expected to have a variation of ±20% in a typical population^33,34. While the overall displacement probability distributions in simulations from uniform spheres are not Gaussian when ζ≫ 1, including a distribution of sphere sizes in simulations results in distributions that are close to Gaussian (Fig 4B). Moreover, the gradients executed in typical NMR spectrometers are not expected to be uniform. When both of these physically reasonable effects are taken into account in Monte-Carlo simulations, diffusion diffraction effects are largely averaged away. Comparing simulations including these effects to the Neuman result reveal that the region of validity is extended qr ≤ 4, and the accuracy of radii obtained from fitting the data is improved (Fig 4H). For an experiment looking at water restrained within cells of radius 10 μm, typical experimental values with a maximum gradient G = 60 G cm^-1 and δ=2 ms, the product qr is 3.3, which is outside the useful limit of the Neuman result. By including the distribution of sphere sizes in simulations, the Neuman formula nevertheless continues to provide an excellent description of restriction effects.

Overall, our model is expressed by equation (8), where D_in is replaced by an apparent diffusion coefficient $D_{in}^{app}$ in equation (10). The fitting parameters can be rearranged to give the cell density in cells per unit volume,

$$\frac{N_{cell}}{V_{tot}}=\frac{3p_{in}}{4\pi r^3}$$ (15)

and the permeability, a measure of net water flux, in m s^-1

$$\kappa =\frac{k_{IO}r}{3}=\frac{p_{out}{k}_{ex}r}{3}$$ (16)

Determining the accuracy of fitted parameters

To quantitatively determine the regions of parameter space where we expect to obtain reliable fitting parameters, an extensive set of Monte-Carlo simulations were performed systematically varying all parameters, which were then fitted globally to the model (equations (8) and (10)). Experimentally accessible values of G and A were used, remaining in the limit qr < 4. The returned parameters were compared to those employed in the simulation to determine accuracy. The uncertainties in the numerical fitting process were established by a boot-strapping procedure³⁵. Overall, this analysis reveals regions of parameter space where the theory is expected to return reliable parameters and where to expect systematic errors.

The following baseline values were used in the simulation, chosen to reflect typical values expected for an NMR/MRI sample containing mammalian cells: D_in = 500 μm² s^-1, D_out = 2000 μm² s^-1, k_ex = 3.55 s^-1, R = 10 μm and P_in = 0.15. Individual parameters were then systematically varied over a range with the others fixed (r Fig 5A, p_in 5B kex Fig 5C, aspect ratio of the cavity Fig 5D Rⁱⁿ₁ Fig 5E, R₀^out Fig 5F).

Overall, fitted parameters fall within 50% of the true value (Fig. 5, green) indicating that the model can be used with confidence when analysing water diffusion data from eukaryotic cells (Fig. 5). The analysis identifies regions of parameter space where the error from bootstrapping is smaller than the actual error. This indicates regions of parameter space where systematic errors are expected. Relatively large discrepancies in parameters are observed when the exchange rate tends towards the fast regime (equation (28)), where k_ex > 40 s^-1 (Fig 5C). Previously determined exchange rates from cells tend to be < 10 s^{-1 33,36,39}, an area where the model is accurate. Notably, the bulk diffusion coefficient in the interior of the cell D_in is often poorly determined (Fig 5A-E, orange), a factor likely due to our specific choice of G and Δ values.

By contrast, the effective cell radius is typically well determined (Fig 5A), and the fitted parameters are tolerant to both large changes in cell abundance (Fig 5B), and substantial changes in the geometry of the cell (Fig 5D). The radius was underestimated when cells exceed 30 pm, as the 0.5 s maximum value of Δ used here is insufficient to lead to significant restriction of particles. The fitted radius corresponds to the effective radius of the equivalent sphere whose volume matches that of the cells. Thus, while we expect a large variation in cell shape, the fitted parameters nevertheless provide a robust measurement of the overall cell volume (Fig 5D).

The effects of differential longitudinal relaxation can also be significant (Fig 5E,F), as has been noted previously⁴⁰. The R₁ of bulk water is small (~0.25 s^-1) and will not have a major impact on the on the data. Within biological tissue or in the presence of paramagnetic reagents the longitudinal relaxation rate can be significantly higher and failure to account for this can result in significant errors in the fitted values for other parameters. R^out₁ is typically well determined (Fig 5G,H), but Rⁱⁿ₁ is poorly determined unless its value becomes larger than ~0.75 s-¹. Below this limit, the effects of Rⁱⁿ₁ are imperceptible in the data. For example in Fig 5F Rⁱⁿ₁ shows consistent systematic errors, and the fitted value tends to zero. In these simulations, its maximum contribution to the data is ca. 1% (Rⁱⁿ₁ 0.25 s^-1). When Rⁱⁿ₁ is greater than 0.75 s^-1, the contribution to the raw data is substantially larger, and the parameter can be reliably measured free of systematic errors (Fig. 5E).

Experimental application to cell samples

To obtain experimental validation of our analysis, suspensions of yeast cells and mammalian 3T3 and HeLa cells were prepared (Methods, H). STE diffusion data with variable G and Δ were acquired (Methods, G). The data were analysed by the INDIANA software, and the reported parameters from fitting were physically reasonable (Table 1). The fitted radii, permeabilities and cell counts were consistent with the expected values (Table 2). In the case of the smaller yeast cells, the radius will be systematically slightly overestimated (Fig. 5A), which leads to a slight overestimate of the cell count. For the mammalian cells, the agreement is improved. Similarly, the intra-cellular diffusion coefficients and the small longitudinal relaxation values are likely to be systematically inaccurate (Fig. 5).

Table 1.

Derived parameters from NMR analysis. Error bars are taken from 1000 bootstrap runs.

Sample	Din / μm2 s-1	Dout / μm2 s-1	kex /s-1	Radius/ μm	pin	R1,in /s-1	R1,out /s-1	Cell density / ml-1	κ/ μm s-1
Yeast (S cerevisiae)	20±4	1250±15	1.23± 0.15	3.57± 0.18	0.092± 0.03	0.86± 0.13	0.81± 0.07	5.3×108± 8×107	1.33± 0.09
3T3	826±65	1980± 44	4.45± 0.24	12.42± 0.11	0.086± 0.03	0.00± 0.00	0.29± 0.03	1.1×107± 4×105	16.84±0.79
HeLa Sample 1	291±62	1310± 69	7.39± 0.66	10.38± 0.18	0.169± 0.012	0.00± 0.09	0.51± 0.07	3.95×107± 6.6±106	21.25±1.78
HeLa Sample 2	344±86	1750± 61	6.43± 0.49	10.69± 0.22	0.046± 0.003	0.00± 0.02	0.28± 0.06	8.99×106± 1.25×106	21.85±1.10

Table 2. Radius and exchange parameters determined in this study compared to literature measurements of these values.

Sample	Radius (here) / μm	Radius (literature) / μm	kIO(here)/ s-1	kIO(literature) / s-1	Cell density (NMR) / cells/ml	Cell density (independent) ^/ cells/ml
Yeast(S. cerevisiae)	3.57	3.9741, 2.8842 (average sizes)	1.16	1.4943 ‡,~1.8036	5.3×108	~2.1×109
3T3	12.42	5.45 to 14.3344	4.07	---	1.1×107	—
HeLa (sample 1)	10.38	10.5±2.233,34	6.14	8.433 §	3.95×107	~3.5×107
HeLa (sample 2)	10.69	10.5±2.233,34	6.13	8.433 §	8.99×106	~1.1×107

^{^}cell density estimated from OD600 measurement for yeast, and in a coulter counter (HeLa).

^‡Temperature of measurement is not stated.

^§Measurement is made at 310 K rather than 298 K so exchange rate/permeability is expected to be higher³⁶.

Discussion

INDIANA is a method to characterise the cell density, size, permeability, as well as intrinsic relaxation and diffusion coefficients of cell suspensions. Experimentally, STE water diffusion spectra can be obtained for a range of G and Δ values on either an MRI or an NMR spectrometer. The data are then globally analysed using equations (8) and (10). A freely downloadable software package is provided to facilitate this analysis.

The advantage of the STE over the SE experiment is that as R₁ ≪ R₂, longer values of b can be accessed, a regime where restricted diffusion effects are more prominent and slower exchange rates can be reliably fitted. The need to analyse the data using a model that contains restricted diffusion and exchange can be identified if analysis of the data are not well explained using equation (2). More generally, as permeability can significantly impact data, a multi-exponential model that neglects exchange is unlikely to be suitable⁴⁵.

Following the approaches of Kärger and Neuman, our model is valid when the displacement distributions of water are approximately Gaussian. Where a sample comprises uniform spheres, simulations reveal the model is reasonable in the regime qr < 2. In the case where there is a distribution of particle sizes, as expected in biological samples, we establish that the region of validity is effectively extended, to qr < 4. By comparing the parameters obtained from fitting our model to an extensive set of simulations, we determine the region of parameter space where we expect the regions of parameter space effectively encompassed by our theory. This region encompasses the regions likely to be encountered when studying eukaryotic cells by NMR.

Our approach assumes that cells are spherical which in principle presents a significant limitation. By simulating data with non-spherical geometries, and fitting our spherical model, we nevertheless obtain an ‘effective’ radius (Fig 5D). The volume associated with this effective sphere is in excellent agreement with the volume of simulated non-spherical cavity suggesting that the effective radius returned from our analysis provides generally useful information even in the case where cells are non-spherical, a finding that establishes the generality of the method.

While we have focused here on the water where both interior and exterior portions are detected at the same chemical shift value, this analysis could in principle be carried out on a resonance from any molecule that can be spectroscopically resolved. Expressions are produced for the case where the interior and exterior chemical shifts differ (equation (7)). This approach will be particularly powerful where a resonance has been isotopically enriched, allowing background-free measurements. For MRI applications where there are limitations on total experiment time, it will be possible to apply compressed sensing algorithms to reduce the number of q and Δ values required.

With the tremendous growth of in-cell NMR and MRI applications, our method provides a straightforward route to determine the abundance, permeability, size and relaxation characteristics of cellular preparations, in a non-invasive fashion. There is also clear potential for using these experiments in the context of diffusion-MRI to obtain information on cell suspensions.

Methods

A. Monte Carlo simulations

Monte-Carlo simulations were performed similarly to those described previously^23,46. Space is discretised into cubes each with either a semi-permeable sphere or a cylinder at its centre, with identical cubes extending in all directions (Fig 2). A particle is initialised at a random point in space such that the average initial positions of the particles reflects the intra- and extra-cellular mole fractions. The extra- and intra-cellular spaces are each characterised by a single diffusion coefficient, D_out and D_in respectively. The current displacement, and an intensity factor are then updated for each step of duration δt:

$$\begin{array}{c}x(t+\delta t)=x(t)+\delta x_{step}\\ y(t+\delta t)=y(t)+\delta y_{step}\\ z(t+\delta t)=z(t)+\delta z_{step}\\ I(t+\delta t)=\{\begin{array}{c}\text{inside:}I(t)e^{R_1^{in}\delta t}\\ \text{outside:}I(t)e^{R_1^{ou}\delta t}\end{array}\end{array}$$ (17)

The displacement step sizes δ_xstep, δ_ystep, and δ_zstep are determined by independently drawing a random number from a Gaussian distribution of standard deviation $\sqrt{2D\delta t}$ where the value of D selected depends on whether the particle is inside, or outside the cell. Total simulation times were as specified (typically 0.5 s) and δt was typically 10 μs, a value that provides a compromise between overall simulation time and accuracy, values that necessitate 50,000 individual steps. The final signal intensity is obtained by summing the final values over a set of independent particle trajectories:

$$\frac{S(q,\Delta )}{S(0,0)}=\text{Re}(\sum _{Particles}I(\Delta )e^{-iqz})=\sum _{Particles}I(\Delta )\cos (qz)$$ (18)

Typically, δ was set to 2 ms and G was varied quadratically in the range 4 to 60 G cm^-1. mimicking commonly used experimental values. Under conditions where attenuation occurs more quickly these values are adjusted such that at the highest applied gradient field strength the signal is attenuated by approximately two orders of magnitude.

After each step, the new position of the particle is checked. If particles have just crossed the cell membrane, we determine if we will accept this jump. The in-to-out or out-to-in probabilities are calculated^23,47:

$$P_{IO}=2s\frac{\kappa }{2D_{in}}{P}_{OI}=2s\frac{\kappa }{2D_{out}}$$ (19)

where s is the distance between the particle and cell surface prior to the jump, and k, is the cell’s permeability, which is in turn is related to the in-to-out rate constant k_IO ³⁶:

$$\kappa =k_{IO}\frac{V}{A}$$ (20)

where V is the cell volume and A is the cell surface area. In the case of spherical cells $\kappa =k_{IO}\frac{r}{3}$ where r is the radius. A uniform random number between 0 and 1 is drawn. If the relevant probability is less than this number, the crossing is accepted and if not, the jump is rejected, and the particle is translated back into the compartment from which it came. Treating the collisions as specular reflections gives no discernible advantage in terms of accuracy, as reported previously^48,49.

The Monte-Carlo simulations were implemented in Python, and the source code is available (LINK). Typically 500,000 trajectories were assembled. To account for variable sizes of the cells, the radius of a cell within a newly encountered cube was drawn from a Gaussian distribution whose average radius was set within the simulation and the standard deviation set to twenty percent of the average value to account for variations in size observed in cellular systems (Fig 4). As the single particle trajectories are all independent, the calculation can be easily run in parallel.

The Monte Carlo simulations were performed over a wide range of conditions parametrised by D_in, D_out, k_ex, r, p_in (in all cases) and Rⁱⁿ₁ and R^out₁ (in Figures 5E and 5F only), to validate the robustness of the theoretical model (Fig 5). The time required to simulate a single set of parameters, parallelised over 20 2.4 GHz Intel Xeon CPUs was approximately 50 minutes.

To model the effects of non-uniform gradient profiles expected in typical instruments³¹, we evaluate the following (Fig S1):

$$\frac{S(q,\Delta )}{S(0,0)}=\sum _{G=G_{min}}\sum _{Particles}I(\Delta )P(G)e^{iq(G)z(\Delta )}$$ (21)

where P(G) is the probability of a given gradient strength.

B. Fitting theory to simulations and data

Simulated NMR data and simulations were fit to the model (equations (8) and (10)) using a python implementation of Lmfit⁵⁰ (Fig 3C). Uncertainties in individual signal intensities were estimated from repeated measurements, and found to be approximately a constant percentage over several orders of magnitude of signal (Fig S3). To account for this weighting, fitting was performed on the logarithm of the signal intensity. No prior information is fed into the fitting procedure and the only restraint placed on the free parameters is that their values must be ≥ 0. The fitted values were compared to those specified in the simulations (Fig 5) to determine their accuracy. The values obtained on ‘real’ experimental systems (Fig 6) were in excellent agreement with independently validated values.

C. Detailed derivation of equation (8)

To calculate the detected signal intensity, we need to evaluate the following

$$S(q,\Delta )=e^{\rho \Delta }S(0,0)O$$

Where ρ is the evolution matrix from equation (3), S(0) contains the equilibrium signal intensity, and O is an operator that combines the populations to give a single observed signal, appropriate for when the chemical shift of the interior and exterior populations cannot be distinguished. The matrix exponential can be evaluated by taking a square matrix with the eigenvalues on the diagonal (D), the matrix of eigenvectors P and its inverse:

$$S(q,\Delta )=P{e}^{-D\Delta }{P}^{-1}\left[\begin{array}{l}{p}_{in}\hfill \\ p_{out}\hfill \end{array}\right]\left[\begin{array}{c}11\end{array}\right]S(0,0)$$ (22)

With the eigenvalues λ^± are defined in terms of the substitutions ξ^± and ϕ, defined earlier (equation (6)):

$$P=\left(\begin{array}{cc}1& 1\\ \frac{-2k_{ex}{p}_{out}}{{\xi }^{-}+\varphi }& \frac{-2k_{ex}{p}_{out}}{{\xi }^{-}-\varphi }\end{array}\right)$$ (23)

The matrix exponential is given by:

$$P{\text{e}}^{-D\Delta }{P}^{-1}=\frac{1}{2\varphi }\left(\begin{array}{cc}-({\xi }^{-}-\varphi )e^{-{\lambda }^{-\Delta }}+({\xi }^{-}+\varphi )e^{-{\lambda }^{+}\Delta }& 2k_{ex}{p}_{in}(e^{-{\lambda }^{-\Delta }}-e^{-{\lambda }^{+}\Delta })\\ 2k_{ex}{p}_{out}(e^{-{\lambda }^{-\Delta }}-e^{-{\lambda }^{+}\Delta })& ({\xi }^{-}+\varphi )(e^{-{\lambda }^{-}\Delta }-e^{-{\lambda }^{+}\Delta })\end{array}\right)$$

Multiplying by the equilibrium populations is required for the final result (equation (8)). An alternative form is:

$$\frac{S(q,\Delta )}{S(0,0)}=\frac{1}{4k_{ex}\varphi }(({(q^2{D}^{-}+R^{-})}^2-{(k_{ex}-\varphi )}^2)e^{-{\lambda }^{+}\Delta }-({(q^2{D}^{-}+R^{-})}^2-{(k_{ex}+\varphi )}^2)e^{-{\lambda }^{-}\Delta })$$

D. Fast and slow exchange limits

The expression reduces to the expected form in the fast and slow exchange limits.

Rearranging ϕ allows for fast and slow exchange definitions:

$$\varphi =\sqrt{{(q^2{D}^{-}+R_1^{-})}^2+2(q^2{D}^{-}+R_1^{-})k_{ex}(p_{out}-p_{in})+k_{ex}^2}$$

Slow exchange is where $q^2{D}^{-}+R_1^{-}\mid >>k_{ex}$, and fast exchange is where $|q^2{D}^{-}+R_1^{-}|<<k_{ex}$. In these two limits:

$$\begin{array}{l}{\varphi }_{fast}=k_{ex}+(p_{out}-p_{in})(q^2{D}^{-}+R_1^{-})\\ {\varphi }_{slow}=k_{ex}(p_{out}-p_{in})+(q^2{D}^{-}+R_1^{-})\end{array}$$ (24)

which leads to the following Eigenvalues:

$$\begin{array}{l}{\lambda }_{slow}^{-}=q^2{D}_{out}+R_1^{out}+k_{\text{OI}}\\ {\lambda }_{slow}^{+}=q^2{D}_{in}+R_1^{in}+k_{IO}\\ {\lambda }_{fast}^{-}=p_{in}(q^2{D}_{in}+R_1^{in})+p_{out}(q^2{D}_{out}+R_1^{out})=q^2\overline{D}+{\overline{R}}_1\\ {\lambda }_{fast}^{+}=k_{ex}\end{array}$$ (25)

Notably, λ^-_fast tends to the combined population averaged value of D and R₁, and the slow exchange values tend towards the values expected for the isolated species. In the slow exchange limit, signal from the interior, exterior and total become:

$$\begin{array}{l}\frac{S_{in}(q,\Delta )}{S_{in}(0,0)}=p_{in}{e}^{-{\lambda }_{\text{sow}}^{+}\Delta }+\frac{k_{ex}{p}_{in}{p}_{out}}{k_{ex}(p_{out}-p_{in})+(q^2{D}^{-}+R_1^{-})}(e^{-{\lambda }_{\text{dow}}^{-}\Delta }-e^{-{\lambda }_{\text{dow}}^{+}\Delta })\\ \frac{S_{out}(q,\Delta )}{S_{out}(0,0)}=p_{out}{e}^{-{\lambda }_{\text{ilow}}^{-}\Delta }+\frac{k_{ex}{p}_{in}{p}_{out}}{k_{ex}(p_{out}-p_{in})+(q^2{D}^{-}+R_1^{-})}(e^{-{\lambda }_{slow}^{-}\Delta }-e^{-{\lambda }_{\text{stow}}^{+}\Delta })\\ \frac{S_{slow}(q,\Delta )}{S_{slow}(0,0)}=p_{in}{e}^{-{\lambda }_{slow}^{+}\Delta }+p_{out}{e}^{-{\lambda }_{slow}^{-}\Delta }+\frac{2k_{ex}{p}_{in}{p}_{out}}{k_{ex}(p_{out}-p_{in})+(q^2{D}^{-}+R_1^{-})}(e^{-{\lambda }_{slow}^{-}\Delta }-e^{-{\lambda }_{\text{sow}}^{+}\Delta })\end{array}$$ (26)

In the limit where k_ex=0, we recover the expected population weighted bi-exponential decay for the total signal

$$\begin{array}{l}\begin{array}{c}\frac{S_{in,k_{ex}=0}(q,\Delta )}{S_{in}(0,0)}=p_{in}{e}^{-(q^2{D}_{in}+R_1^{in})\Delta }\\ \frac{S_{out,k_{ex}=0}(q,\Delta )}{S_{out}(0,0)}=p_{out}{e}^{-(q^2{D}_{out}+R_1^{out})\Delta }\end{array}\\ \frac{S_{k_{ex}=0}(q,\Delta )}{S(0,0)}=p_{in}{e}^{-(q^2{D}_{in}+R_1^{in})\Delta }+p_{out}{e}^{-(q^2{D}_{out}+R_1^{out})\Delta }\end{array}$$ (27)

Finally, in the fast exchanging limit we obtain the following expressions:

$$\begin{array}{l}\frac{S_{f\text{fat},in}(q,\Delta )}{S_{in}(0,0)}=p_{in}(e^{-{\lambda }_{幻x}^{\Delta }}+\frac{p_{out}(q^2{D}^{-}+R_1^{-})(e^{-k_{\alpha }\Delta }-e^{-{\lambda }_{jad}^{-}\Delta })}{k_{ex}+(p_{out}-p_{in})(q^2{D}^{-}+R_1^{-})})\\ \frac{S_{fast,out}(q,\Delta )}{S_{out}(0,0)}=p_{out}(e^{-{\lambda }_{{\text{fun}}^{-}\Delta }}-\frac{p_{in}(q^2{D}^{-}+R_1^{-})(e^{-k_{\alpha }\Delta }-e^{-{\lambda }_{fad}^{-}\Delta })}{k_{ex}+(p_{out}-p_{in})(q^2{D}^{-}+R_1^{-})})\\ \frac{S_{fast}(q,\Delta )}{S(0,0)}=e^{-{\lambda }_{\text{fau}}^{-}\Delta }=e^{-(q^2\overline{D}+\overline{R})\Delta }\end{array}$$ (28)

Where the total signal decays with a single exponential whose decay constant is the population average of the diffusion and relaxation rates.

E. Long time limit for restricted diffusion within a sphere

In the case of restricted diffusion, we can calculate the probability distribution in the long time limit. The displacement within a sphere in spherical polar co-ordinates will be dz = r(cosθ₁ — cosθ₂). The probability of a given displacement will depend on the area (rsinθ)². The un-normalised displacement distribution will be:

$$P(z)={\int }_0^{\pi }{\int }_0^{\pi }{\sin }^3{\theta }_1{\sin }^3{\theta }_2\delta (r(\cos {\theta }_1-\cos {\theta }_2)-z)d{\theta }_{1}d{\theta }_2$$

Changing variables such that u = cosθ₁ and v = cosθ₂ :

$$P(z)={\int }_{-1}^1{\int }_{-1}^1(1-u^2)(1-v^2)\delta (r(u-v)-z)dudv$$

Following a final change of variables $\kappa =\frac{\left|z\right|}{r}$and we arrive at the following symmetric normalised distribution in the range 2 > k > 2, and zero otherwise:

$$P(\kappa )=\frac{3}{160}(2-\kappa {)}^3(4+\kappa (6+\kappa ))$$ (29)

This can be converted to expected NMR signal intensity^51,52:

$$S(q)={\int }_{-\infty }^{\infty }P(z)e^{iqz}dz=\frac{9{(qr\cos (qr)-\sin (qr))}^2}{{(qr)}^6}$$ (30)

We note that the Neuman approximation for restricted diffusion does not converge on this limit. For shorter times and specific gradient strengths encountered on typical NMR and MRI spectrometers, the Neuman approximation is sufficiently accurate. This can be proven from taking the expansion of equation (30) to 6^th order:

$$\frac{S_i}{S_0}=1-\frac{{(qr)}^2}{5}+\frac{3{(qr)}^4}{175}-\frac{4{(qr)}^6}{4725}\dots$$ (31)

reveals that the distribution will be effectively Gaussian providing the quartic term is smaller than the quadratic term. The Neuman formula is exact in this limit to 4^th order in qr (equation (14)). The formal requirement for this limit will be 10>>(qr)². In more practical terms, the Neuman result is numerically accurate to within a per cent providing qr ≤ 2. This is a challenge for experiments, as 10 μm spheres, analysed with a gradient of 60 G cm^-1 applied for 2 ms, looking at water where y is 26,700 G^-1 rads s^-1 exceed this limit. The limit of applicability is extended to qr ≤ 4 when we account for a distribution of sphere sizes, and non-uniform gradients inherent in all spectrometers is taken into account (Fig. 4).

F. Continuous diffusion theory

The results from Monte-Carlo simulations were cross-validated with calculations conducted with a macroscopic perspective. In one dimension, the concentration in a region of space will be updated according Fick’s second law of diffusion:

$$\frac{dc}{dt}=D\frac{d^{2}c}{d{x}^2}$$

Which can be numerically integrated using the following scheme:

$$c(i,t+\delta t)=c(i,t)+D\frac{\delta t}{\delta x^2}(c(i+1,t)+c(i-1,t)-2c(i,t))$$ (32)

which is readily expanded to 2 and 3 dimensions. A spherical cavity was maintained in the calculation by establishing boundaries where $x_i^2+y_j^2+z_k^2<r^2$. A desired total concentration was placed into a single square at a specific location f along the z-axis, which was then allowed to evolve for the desired amount of time according to equation (32). The concentration in all cubes inside the sphere, c_ijk(f) were saved at desired times Δ. Starting locations f were varied uniformly from 0 to r, and the spherical average was calculated by determining the displacement probabilities along the z axis by rotating the initial position co-ordinates positions. For a given rotation of θ about the x axis, the probability distribution will be:

$$P(\delta z,f,\theta )=\sum _{ijk}{c}_{ijk}(f)\delta (y_j\sin \theta +z_k\cos \theta -\delta z)$$

Which can be spherically averaged to yield the final distribution:

$$P(\delta z)={\int }_0^r{\int }_0^{\pi }P(\delta z,f,\theta )f^2\sin \theta d\theta df$$

Which is converted to NMR signal intensity using

$$\frac{S(q,\Delta )}{S(0,0)}={\int }_{-2R}^{2R}P(z)\cos (qz)dz$$ (33)

The ratio Dδt/δx² was set to 0.15, and space was split into cubic boxes of side length δx 1 μm, the sphere’s radius was set to 20 μm, δ= 2 ms, γ= 26,700 G rads s^-1 and 100 gradient strengths in the range 0-60 G cm^-1, varied quadratically. The interior diffusion coefficient was set to 10^-9 m² s^-1, a value expected for water free in solution. With this value, the time step δt becomes 1.5 ms and for 10,000 steps, the total simulation time was 1.5 s which takes approximately 5 minutes on a single Intel i7 processor. The distributions from this method were in excellent agreement with those obtained from Monte-Carlo simulations (Fig S2) demonstrating the equivalence of the two approaches.

G. NMR experimental methods

All NMR experiments were carried out at 298 K on a 14.1 T (600 MHz) spectrometer equipped with a 5 mm z-axis gradient triple resonance room temperature probe room temperature probe. A STE pulse sequence with variable diffusion delays were applied. In all experiments δ = 2 ms. For yeast sample, 11 quadratically spaced gradient strengths from 8-60 G cm^-1 were used and Δ = 20, 40, 60, 80, 100, 200, 300, 400 and 500 ms. For 3T3 sample,11 quadratically spaced gradient strengths were used from 1-60 G cm^-1 and Δ = 50, 100, 200, 300, 400 and 500 ms. For HeLa cell sample, 11 quadratically spaced gradient strengths from 8-60 G cm^-1 were used and Δ = 50, 100, 200, 300, 400 and 500 ms. All spectra were Fourier transformed and phased using NMRPipe⁵³. In each spectrum the water peak is integrated using Python scripts, making use of the nmrglue module⁵⁴, and water peak intensities were carried forward for analysis as described for the simulated datasets in the previous section.

H. Cell preparation methods

Yeast cell samples were prepared according to previously published protocols^36,55. Briefly, dried yeast was mixed with tap water in a ratio of approximately 1:3 and the mixture was left for three days during which it was shaken periodically to allow for the release of carbon dioxide bubbles. The mixture is subsequently transferred to an NMR tube and 5% D2O added.

In the case of 3T3 cells, cells were grown in T-175 flasks and detached from their substrate by incubation with trypsin-EDTA for 5 min. Cells were then re-suspended in 250 μL Dulbecco’s phosphate buffered saline (DPBS) with 5% D2O and transferred to a Shigemi tube for NMR experiments.

For HeLa cells, cells were grown in DMEM with 10% FBS in petri dishes and detached from plates using mechanical scraping (using petri dishes gives access to maximal area for scraping cells). Cells were subsequently re-suspended in 250 μL foetal bovine serum (FBS), with 5% D₂O and transferred to a Shigemi tube for NMR experiments.