Recent advances in theory (Laage et al. 2008; Bertini et al. 2011b; Hu et al. 2011; Loening et al. 2012; Bjerring et al. 2012; Nielsen et al. 2012; Westfeld et al. 2012; Lamley and Lewandowski, 2012; Giffard et al. 2012) and in sample preparation techniques (Lewandowski et al. 2011a; Akbey et al. 2012) have pushed the limits of solid state NMR (SSNMR) close to that of state-of-the-art solution NMR for the determination of structure (Bertini et al. 2010a; Knight et al. 2011; Luchinat et al. 2012; Huber et al. 2012; Knight et al. 2013; Bhaumik et al. 2013) and dynamics (Lewandowski et al. 2010; Lewandowski et al. 2011b; Knight et al. 2012; Lewandowski 2013; Haller and Schanda, 2013; Schanda et al. 2010; Zinkevich et al. 2013) for micro- to nanocrystalline systems, and even in the case of systems lacking long-range order like fibrils (Petkova et al. 2002; Tycko and Ishii 2003; Debelouchina et al. 2010; Bertini et al. 2011c; Lewandowski et al. 2011a; Bayro et al. 2011; Parthasarathy et al. 2011; Habenstein et al. 2011; Habenstein et al. 2012; Lopez del Amo et al. 2012; Qiang et al. 2012; Lv et al. 2012), large aggregates (Sun et al. 2009; Loquet et al. 2010; Loquet et al. 2012; Byeon et al. 2012; Yan et al. 2013; Loquet et al. 2013), membrane proteins(Ader et al. 2009; Lange et al. 2010; Weingarth and Baldus, 2013; Cady et al. 2010; Hong et al. 2011; Hong 2006; Hong and Schmidt-Rohr 2013; Petkova et al. 2003; Harbison et al. 1985; Marassi and Opella 2000; Opella and Marassi 2004; Marassi et al. 2011; Ding et al. 2013; Opella 2013; Murray et al. 2013; Yang et al. 2011; Hefke et al. 2011; Ullrich and Glaubitz 2013; Ketchem et al. 1993; Sharma et al. 2010) and protein-mineral hybrids (Long et al. 2001; Goobes et al. 2006; Goobes et al. 2007a; Goobes et al. 2007b; Roerich and Drobny 2013; Fragai et al. 2013a). Nevertheless, a wealth of systems still results inaccessible to both solution and solid state NMR.

Very large biomolecular assemblies have solution NMR lines that are broad beyond detection (Wider 2005; Fernández and Wider 2003; Riek et al. 1999; Fiaux et al. 2002; Guo et al. 2008; Tugarinov et al. 2006; Tugarinov et al. 2005a; Tugarinov et al. 2005b; Bermel et al. 2007; Bermel et al. 2008; Bermel et al. 2010). To solve this problem, specific sample preparation and labelling schemes were proposed (Tugarinov et al. 2006; Matzapetakis et al. 2007). Since the lines in SSNMR are independent of the molecular weight of the system, (Marassi et al. 1997) due to the absence of CSA relaxation (Haeberlen and Waugh 1969) [and of Curie-spin relaxation in the case of paramagnetic solids (Kervern et al. 2007)], SSNMR may be considered the technique of choice for atomic-level structural characterization of large biomolecular systems.

In many cases, SSNMR requires the sample to be crystalline and hydrated for obtaining high resolution spectra (Martin and Zilm 2003), but crystal contacts are known to affect in some cases the native structure of the protein (Barbato et al. 1992; Fischer et al. 1999; Skrynnikov et al. 2000; Chou et al. 2001; Poon et al. 2007; Bertini et al. 2008; Bertini et al. 2009). Furthermore, when crystals are obtained, X-ray diffraction gives much simpler access to structural information. Manipulation of the sample (i.e.: precipitation, freezing or lyophilisation) can deteriorate the quality of the spectra (Martin and Zilm 2003; Linden et al. 2011; Pauli et al. 2000; Fragai et al. 2013b).

Mainz et al. proposed that suppressing the rotational diffusion in protein solution could result in a sample that could circumvent the limitations of solution and solid state NMR. This was afforded by “adding glycerol and employing low temperature and high protein concentrations” (Mainz et al. 2009). Indeed, these conditions were shown not to be sufficient to suppress rotational diffusion (Ravera et al. 2013b). Conversely, it has been demonstrated that large macromolecules in solution, sealed in a MAS rotor and spun at the usual MAS NMR frequencies, undergo sedimentation as in a real ultracentrifuge (Bertini et al. 2011d). This can explain the results by (Mainz et al. 2009) and opens the more general opportunity of studying large soluble molecules by SSNMR.

Sedimented solute NMR (SedNMR) (Bertini et al. 2011d; Bertini et al. 2012b; Polenova 2011; Mainz et al. 2012; Gardiennet et al. 2012; Baldwin et al. 2012; Ravera et al. 2013a; Bertini et al. 2013; Luchinat et al. 2013; Mainz et al. 2013) is an ideal tool to address all those soluble biomolecular systems that are too large for solution NMR and that do not crystallize: the sample remains in the buffer used for the solution studies, interactions can be followed like in a usual NMR titration, and the system is always hydrated so as to give highly resolved spectra. The larger the molecules, the easier to obtain them in a good sedimented form.

To foresee the outcome of the SedNMR experiment, we have adapted the usual equation of sedimentation equilibrium (Van Holde and Baldwin 1958) so as to parametrically take into account the pelleting of the macromolecule (Bertini et al. 2011d; Bertini et al. 2012b; Bertini et al. 2012a; Bertini et al. 2013). The macromolecular concentration c(r) at any distance r from the rotation axis of the rotor or of the ultracentrifuge is given by:

$$ c(r) = \frac{{c_{l} }}{{Ae^{{ - kr^{2} }} + 1}}, $$
(1)

where c l is the limiting concentration (see Table 1), k is given by:

$$ k = \frac{{M(1 - \rho_{solvent} /\rho_{biomolecule} )\omega^{2} }}{2RT}, $$
(2)

and A is an integration constant that needs to be determined according to the law of conservation of mass. For the geometry of the SSNMR rotor A evaluates analytically to:

Table 1 Parameters for Eqs. 1–3, as given in the web interface
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$$ A = \frac{{\exp \left[ {\frac{{M(1 - \rho_{solvent} /\rho_{biomolecule} )\omega^{2} b^{2} }}{2RT}\left( {1 - \frac{{c_{0} }}{{c_{l} }}} \right)} \right] - 1}}{{1 - \exp \left[ { - \frac{{M(1 - \rho_{solvent} /\rho_{biomolecule} )\omega^{2} b^{2} }}{2RT}\frac{{c_{0} }}{{c_{l} }}} \right]}} $$
(3)

As noted elsewhere, the method finds an intrinsic sensitivity limitation in the fact that the starting concentration can optimistically reach 60 % of the corresponding crystal for highly soluble proteins such as BSA (Venturi et al. 2008; Lundh 1980; Lundh 1985; Andersson and Hovmoller 2000). Furthermore, the rotors designed for ultra-fast MAS, which can be used to achieve high resolution spectra (Bertini et al. 2010b; Knight et al. 2011; Webber et al. 2012; Asami et al. 2012) and site-specific dynamics information (Lewandowski et al. 2011b; Lewandowski et al. 2010; Schanda et al. 2010; Haller and Schanda 2013), suffer from their small volume and are also penalized by their smaller internal radius that requires higher molecular weights for sedimentation as compared to the other rotors (Bertini et al. 2012b; Bertini et al. 2013).

We thus proposed (Bertini et al. 2012b) that ultracentrifugal devices, like the one described by Böckmann et al. (Böckmann et al. 2009), could be used to form and funnel the sediment directly in the NMR rotor, thus increasing the amount of sample in the rotor. This approach was successfully applied by Gardiennet et al. (Gardiennet et al. 2012) for sedimenting a 59 kDa dodecameric helicase (total molecular weight 708 kDa) and, successively, by Gelis et al. (Gelis et al. 2013) to sediment the ribosome.

In this case, Eqs. 1 and 2 retain their validity, but (Bertini et al. 2012b) the integration constant A needs now to be determined according to the following equation:

$$ \pi r_{1}^{2} \int\limits_{{b_{0} }}^{{b_{1} }} {c(h)dh + \pi } \int\limits_{{b_{1} }}^{{b_{2} }} {\left( {\frac{{h_{p} - h + b_{1} }}{{h_{p} }}r_{1} } \right)^{2} c(h)dh + \pi r_{2}^{2} } \int\limits_{{b_{2} }}^{{b_{3} }} {c(h)dh + \pi r_{3}^{2} } \int\limits_{{b_{3} }}^{{b_{4} }} {c(h)dh = c_{0} V_{device} } $$
(4)

where the meaning of the parameters r 1, r 2, r 3 and h p and of the integration limits are shown in Fig. 1. These parameters reflect the geometry of the device described in (Bertini et al. 2012a).

Fig. 1
figure 1

Schematics of the relevant parameters for the device geometry

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The integrals in Eq. 4 are evaluated with the Romberg method as implemented in SciPy (Oliphant 2007).

We have thus developed a user-friendly web interface, based on a framework (Bertini et al. 2011a) that we developed within the WeNMR project (Wassenaar et al. 2012), that allows for the calculation of the concentration profile as a function of distance from the rotation axis. Its appearance is shown in Fig. 2.

Fig. 2
figure 2

Appearence of the SedNMR web interface available in WeNMR (http://py-enmr.cerm.unifi.it/access/index/sednmr). The user can tune the parameters using the sliders, according to the properties of the sample under investigation. The graphical output displays the concentration profile as a function of the distance from the rotation axis that would be obtained at a given MAS frequency (upper panel) and the fraction of immobilized protein as a function of the MAS frequency for the selected rotors (lower panel). The relevant parameters are listed in Table 1, and given with their physically significant range. Left “Rotor” option, where calculations are performed for the commercially available Bruker and Agilent rotors. Right “Device” option, where the calculations are performed for the device geometry

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The web tool allows for downloading the plot datapoints in a two-column format (which can be imported into a spreadsheet) for offline visualization of the result. All the corresponding parameters values are provided in the RTF format.

The parameter “Threshold for immobilization” that is used to calculate the fraction of immobilized protein in the “ROTOR” option is set to 86 % to match the experimentally determined profile for sedimented ferritin as a function of the rotation rate.(Bertini et al. 2011d; Bertini et al. 2012b) This value represents the value of concentration at which the self-crowding exerted by the protein molecules in solution on each other is enough to abolish the rotational motions to such an extent that a solid-like signal is observed (i.e. the concentration at which the solute becomes a sediment). This is qualitatively expected to occur if the rotational correlation time is slower than the MAS period, so that the nuclear interactions become mechanically averaged.(Haeberlen and Waugh 1969; Matti Mariq and Waugh 1979) There are several models through which a theoretical or empirical calculation of this value could be performed.(Koenig and Brown 1990; Balbo et al. 2013)

Among the parameters listed in Table 1, the most notable is the limiting concentration. This value ranges from the lowest observed value (200 mg/ml for DNAb Helicase) to the theoretical limit of close packing of cubes. A number of experimental observations are given in Table 2. With the data taken from Table 2 we have used the functionality of the sedNMR web tool (http://py-enmr.cerm.unifi.it/access/index/sednmr) to calculate some example results, as summarized in Tables 3 and 4. In Tables 3 and 4, the fraction of immobilized protein with respect to the total protein in the rotor is presented for commonly used rotors. The protein is supposed to have a concentration of 100 mg/ml, and the rotor is supposed to spin at its maximum allowed spinning rate

Table 2 Limiting concentrations observed for some biomolecular sediments
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Table 3 Fraction of immobilized protein, starting from a 150 mg/ml solution at the maximum spinning rate for some commercially available rotors
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Table 4 Fraction of immobilized protein, starting from a 150 mg/ml solution at the maximum spinning rate for some commercially available rotors
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