Acquiring and processing ultrafast biomolecular 2D NMR experiments using a referenced-based correction

Abstract

Thanks to their special spatiotemporal encoding/decoding scheme, ultrafast (UF) NMR sequences can deliver arbitrary 2D spectra following a single excitation. Regardless of their nature, these sequences have in common their tracing of a path in the ${\hbox {F}_{1}}$–$t_{2}$ plane, that will deliver the spectrum being sought after a 1D Fourier transformation versus $t_{2}$. This need to simultaneously digitize two domains, tends to impose bandwidth limitations along all spectral axes. Along the $t_{2}$/${\hbox {F}_{2}}$ dimension this problem is exacerbated by the fact that odd and even time points are not equispaced, and by additional artifacts such as time shifts between time points sampled while under the action of positive and negative decoding gradients. As a result, odd and even $t_{2}$ points are typically Fourier transformed separately, halving the potential spectral width along this dimension. While this halving of the ${\hbox {F}_{2}}$ span can be overcome by an interlaced Fourier transform, this post-processing is seldom used because of its sensitivity to hardware inaccuracies requiring even finer corrections of the even/odd $t_{2}$ data points. These corrections have so far been done manually, but are challenging to implement when dealing with low signal-to-noise ratio signals like those associated with biomolecular NMR experiments. This study introduces an algorithm for an automatic correction of all even/odd ultrafast NMR inconsistencies, based on the acquisition of a reference scan on the solvent. This algorithm was verified experimentally using an ${}^{1}\hbox {H}$-${}^{13}\hbox {C}$ UF-HSQC variant on ubiquitin at 600 MHz. Features of this method as well as of the interlaced Fourier transformation in general, are discussed.

Appendix: The interlaced Fourier transform

As Eq. (8) used a version of the interlaced FT that has been modified from what has been reported in the literature, we devote this Appendix to justify this form as well as to introduce the behavior resulting from a peak folding into the interlaced FT-augmented spectral window from outside. We start with two signals: $S^{(1)}$ of length M / 2 made up of the odd samples only, without extra zeros in between, and $S^{(2)}$ also of length M / 2 made up of the even samples only. We assume the sample points of each set are separated by a duration $\tau$ and are shifted from zero by $\Delta _{1}$ and $\Delta _{2}$, respectively. Thus $S^{(1)}$ is measured at times $\Delta _{1}+m\tau$, and similarly $S^{(2)}$ is measured at times $\Delta _{2}+m\tau$, where $m=0,\ldots ,(M/2-1)$ and the time delays $\Delta _{1}$, $\Delta _{2}$ and $\tau$ are precisely known. ($\tau$, $\Delta _{1}$ and $\Delta _{2}$ are the generic versions of $T_{c}$, $\Delta t_{n}^{\mathrm {odd}}$and $\Delta t_{n}^{\mathrm {even}}$ in the main text.) Applying a standard FT to signals $S^{(1)}$ and $S^{(2)}$ results in two spectra $\hat{S}^{(1)}$ and $\hat{S}^{(2)}$, each with a spectral resolution of $\delta \omega =2\pi /(\tau \cdot M/2)=4\pi /(M\tau )$ and a bandwidth of $\delta \omega \cdot M/2=2\pi /\tau$. We assume now that the actual peaks could resonate over a range of frequencies that doubles the latter badwidth to $4\pi /\tau$, so that the spectrum is defined by M amplitudes $a_{m}$ with $m=-M/2,\ldots ,(M/2-1)$. As the Fourier transforms of $S^{(1)}$ and $S^{(2)}$ cannot distinguish between frequencies that are $\delta \omega \cdot M/2$ away, the transformed signals $\hat{S}^{(1)}$ and $\hat{S}^{(2)}$ are actually given by

$$ \hat{S}_{m}^{(1)}= \frac{1}{2}a_{m}e^{i\delta \omega m\Delta _{1}}+\frac{1}{2}a_{m-M/2}e^{i\delta \omega (m-M/2)\Delta _{1}},$$

(13a)

$$ \hat{S}_{m}^{(2)}= \frac{1}{2}a_{m}e^{i\delta \omega m\Delta _{2}}+\frac{1}{2}a_{m-M/2}e^{i\delta \omega (m-M/2)\Delta _{2}},$$

(13b)

where the different phase coefficients reflect the delays $\Delta _{1}$ and $\Delta _{2}$ defining the beginning of each set’s acquisition, and where the 1 / 2 factor is a normalization needed when compared to a FT of a full signal of length M. Solving the above two sets of equations for $a_{m-M/2}$ and $a_{m}$ with $m=0,\ldots ,(M/2-1)$, we get

$$ a_{m-M/2}=2\frac{\hat{S}_{m}^{(2)}e^{-i\delta \omega m\Delta _{2}}-\hat{S}_{m}^{(1)}e^{-i\delta \omega m\Delta _{1}}}{e^{-i\delta \omega M\Delta _{2}/2}-e^{-i\delta \omega M\Delta _{1}/2}},$$

(14a)

and

$$ a_{m} =2\frac{\hat{S}_{m}^{(2)}e^{-i\delta \omega (m-M/2)\Delta _{2}}-\hat{S}_{m}^{(1)}e^{-i\delta \omega (m-M/2)\Delta _{1}}}{e^{i\delta \omega M\Delta _{2}/2}-e^{i\delta \omega M\Delta _{1}/2}}.$$

(14b)

Using

$$ \delta \omega =2\pi /(\tau \cdot M/2)=4\pi /(M\tau ) $$

(15)

the above results can also be rewritten as:

$$\begin{aligned} a_{m-M/2}=&\frac{ie^{i\pi (\Delta _{1}+\Delta _{2})/\tau }}{\sin \left[ \pi (\Delta _{2}-\Delta _{1})/\tau \right] }\nonumber \\&\times \left[ \hat{S}_{m}^{(2)}e^{-i2\pi (2m/M)\Delta _{2}/\tau }-\hat{S}_{m}^{(1)}e^{-i2\pi (2m/M)\Delta _{1}/\tau }\right] , \end{aligned}$$

(16a)

$$\begin{aligned} a_{m}=&\frac{-ie^{-i\pi (\Delta _{1}+\Delta _{2})/\tau }}{\sin \left[ \pi (\Delta _{2}-\Delta _{1})/\tau \right] }\nonumber \\&\times \left[ \hat{S}_{m}^{(2)}e^{-i2\pi (2m/M-1)\Delta _{2}/\tau }-\hat{S}_{m}^{(1)}e^{-i2\pi (2m/M-1)\Delta _{1}/\tau }\right] , \end{aligned}$$

(16b)

where $m=0,\ldots ,(M/2-1)$.

Applying the 1D interlaced FT above to each of the N columns of $S_{m,n}$ in a 2D UF experiment and making the changes

$$ \tau\rightarrow T_{c}$$

(17a)

$$ \Delta _{1},\Delta _{2}\rightarrow \Delta t_{n}^{\mathrm {odd}},\Delta t_{n}^{\mathrm {even}}$$

(17b)

$$ S_{m}^{(1)},S_{m}^{(2)}\rightarrow S_{m,n}^{\mathrm {odd}},S_{m,n}^{\mathrm {even}}$$

(17c)

$$ a_{m},a_{m-M/2}\rightarrow \hat{S}_{m,n}^{\mathrm {interlaced}},\hat{S}_{m-M/2,n}^{\mathrm {interlaced}},$$

(17d)

gives the expressions of Eq. (8) in the main text:

$$\begin{aligned} \hat{S}_{m-M/2,n}^{\mathrm {interlaced}}&= \frac{ie^{i\pi (\Delta t_{n}^{\mathrm {odd}}+\Delta t_{n}^{\mathrm {even}})/T_{c}}}{\sin \left[ \pi (\Delta t_{n}^{\mathrm {even}}-\Delta t_{n}^{\mathrm {odd}})/T_{c}\right] }\nonumber \\&\times \left[ \hat{S}_{m,n}^{\mathrm {even}}e^{-i2\pi (2m/M)(\Delta t_{n}^{\mathrm {even}}/T_{c})}\right. \nonumber \\&\left. \;\;-\hat{S}_{m,n}^{\mathrm {odd}}e^{-i2\pi (2m/M)(\Delta t_{n}^{\mathrm {odd}}/T_{c})}\right] , \end{aligned}$$

(18a)

$$\begin{aligned} \hat{S}_{m,n}^{\mathrm {interlaced}}=& \frac{-ie^{-i\pi (\Delta t_{n}^{\mathrm {odd}}+\Delta t_{n}^{\mathrm {even}})/T_{c}}}{\sin \left[ \pi (\Delta t_{n}^{\mathrm {even}}-\Delta t_{n}^{\mathrm {odd}})/T_{c}\right] }\nonumber \\&\times \left[ \hat{S}_{m,n}^{\mathrm {even}}e^{-i2\pi (2m/M-1)\Delta t_{n}^{\mathrm {even}}/T_{c}}\right. \nonumber \\&\left. \;\;-\hat{S}_{m,n}^{\mathrm {odd}}e^{-i2\pi (2m/M-1)(\Delta t_{n}^{\mathrm {odd}}/T_{c})}\right] , \end{aligned}$$

(18b)

Having defined the interlaced FT, we now examine the fate of a signal which comes from outside the transform’s augmented target spectral width. Assume for this a peak with angular frequency $\delta \omega \cdot (m+pM/2)$, where $p=0,\pm 1,\pm 2,\ldots$. (Note that the cases $p=0,-1$ actually cover the “allowed” spectral width already treated above.) After Fourier transforming the even and odd signals separately we get

$$ S_{m}^{(1)}=\frac{1}{2}a_{m+pM/2}e^{i\delta \omega (m+pM/2)\Delta _{1}},$$

(19a)

$$ S_{m}^{(2)}=\frac{1}{2}a_{m+pM/2}e^{i\delta \omega (m+pM/2)\Delta _{2}}.$$

(19b)

Plugging these relations into the solution above for $a_{m}$ and “not knowing” that the signal originates from outside the expected SW, results in

$$\begin{aligned} a_{m-M/2}^{\mathrm {eff.}}=&-e^{i\pi (p+1)(\Delta _{2}+\Delta _{1})/\tau }\nonumber \\&\times a_{m+pM/2}\frac{\sin \left[ \pi p(\Delta _{2}-\Delta _{1})/\tau \right] }{\sin \left[ \pi (\Delta _{2}-\Delta _{1})/\tau \right] }, \end{aligned}$$

(20a)

$$\begin{aligned} a_{m}^{\mathrm {eff.}}=&e^{-i\pi p(\Delta _{1}+\Delta _{2})/\tau }\nonumber \\&\times a_{m+pM/2}\frac{\sin \left[ \pi (p+1)(\Delta _{2}-\Delta _{1})/\tau \right] }{\sin \left[ \pi (\Delta _{2}-\Delta _{1})/\tau \right] }, \end{aligned}$$

(20b)

where $a_{m+pM/2}$ is the true amplitude of the given peak. It follows from these expressions that (1) each peak outside the expected spectral width (i.e., $p\ne 0,-1$) splits into two folded peaks at indices m and $m-M/2$, and (2) the magnitude of the peaks is given by a ratio of two sines, with a maximum amplification of the true peak amplitude $a_{m+pM/2}$ by a factor of p or $p+1$ arising when $\Delta _{2}-\Delta _{1}=0 {\text{ or }} \tau$, but also with the possibility that this scaling value may reach zero (with the implicit assumption that $0\le \Delta_{2}-\Delta_{1} \le \tau$).