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J Biomol NMR. Author manuscript; available in PMC 2016 July 1.

Published in final edited form as:

Published online 2015 June 5. doi: 10.1007/s10858-015-9951-6

PMCID: PMC4782772

NIHMSID: NIHMS757743

Witold Andrałojć,^{a} Konstantin Berlin,^{b} David Fushman,^{b,}^{*} Claudio Luchinat,^{a,}^{c,}^{*} Giacomo Parigi,^{a,}^{c} Enrico Ravera,^{a,}^{c} and Luca Sgheri^{d}

Corresponding authors: David Fushman, Email: ude.dmu@namhsuf, Claudio Luchinat, Email: ti.ifinu.mrec@tanihcul

The publisher's final edited version of this article is available at J Biomol NMR

See other articles in PMC that cite the published article.

Long-range NMR data, namely residual dipolar couplings (RDCs) from external alignment and paramagnetic data, are becoming increasingly popular for the characterization of conformational heterogeneity of multidomain biomacromolecules and protein complexes. The question addressed here is how much information is contained in these averaged data. We have analyzed and compared the information content of conformationally averaged RDCs caused by steric alignment and of both RDCs and pseudocontact shifts caused by paramagnetic alignment, and found that, despite the substantial differences, they contain a similar amount of information. Furthermore, using several synthetic tests we find that both sets of data are equally good towards recovering the major state(s) in conformational distributions.

Biological macromolecules are inherently flexible objects and often accomplish
their task through extensive conformational rearrangement (Sicheri and Kuriyan, 1997; Pickford and Campbell,
2004; Zhang and Zuiderweg, 2004; Tonks, 2006; Chuang et
al., 2010). Characterization of such rearrangements and the relevant conformational
states can provide important clues about the mechanisms underlying biological function. This
however is a challenging task because the system is underdetermined, implying a large
degeneracy in the reconstructed solutions, and requires extensive experimental work often
involving multiple techniques (Bonvin and Brunger,
1996; Choy and Forman-Kay, 2001; Svergun et al., 2001; Burgi et al., 2001; Clore and Schwieters,
2004; Schroeder et al., 2004; Iwahara et al., 2004; Bertini et al., 2004a; Blackledge, 2005;
Lindorff-Larsen et al., 2005; Fragai et al., 2006; Tolman and Ruan,
2006; Boehr et al., 2006; Ryabov and Fushman, 2006; Chen et al., 2007; Bernadò et al.,
2007; Bertini et al., 2007; Ryabov and Fushman, 2007; Lange et al., 2008; Hulsker et al., 2008;
Korzhnev and Kay, 2008; Nodet et al., 2009; Boehr et al.,
2009; Stelzer et al., 2009; Huang and Grzesiek, 2010; Fisher et al., 2010; Bashir et al., 2010;
Rinnenthal et al., 2011; Bothe et al., 2011; Fisher and Stultz,
2011; Berlin et al., 2013; Russo et al., 2013; Guerry et al., 2013; Kukic et al., 2014;
Ravera et al., 2014; Torchia, 2015). Therefore, it is important to know the information
content provided by various experimental methods in order to decide on an optimal set of
experiments *a priori*.

Residual dipolar couplings (RDC) (Lohman and Maclean, 1978) are widely used as a source of information on biomolecular structure and dynamics (Tolman, 2001; Tolman and Ruan, 2006; Berlin et al., 2013; Ravera et al., 2014). They arise in the presence of partial molecular orientation, which can be achieved by interactions with alignment media surrounding the molecule (Tolman et al., 1995; Tjandra and Bax, 1997; Hansen et al., 1998; Losonczi and Prestegard, 1998; Ramirez and Bax, 1998; Wang et al., 1998; Al-Hashimi et al., 2000; Prestegard et al., 2000; Zweckstetter and Bax, 2001; Lakomek et al., 2008) and/or by the preferential orientation of the molecule itself in a magnetic field due to its magnetic susceptibility anisotropy (Lohman and Maclean, 1978; Tolman et al., 1995; Zhang et al., 2003; Latham et al., 2008; Ravera et al., 2014; Musiani et al., 2014). RDCs obtained by alignment induced by an external orienting medium, herein referred to as diamagnetic RDCs (dRDC), depend on the nature of the interactions of the biomolecule with the medium. These interactions can be steric and/or electrostatic and, because of this, dRDC are reporters also on the overall shape of the macromolecule and/or its charge distribution (Zweckstetter and Bax, 2000; Zweckstetter, 2008; Berlin et al., 2009; Maltsev et al., 2014). On the other hand, RDCs caused by molecular self-alignment, often induced by the presence of a paramagnetic center with an anisotropic magnetic susceptibility, herein termed paramagnetic RDCs (pRDC), only depend on the orientation of the internuclear vectors in the reference frame of the magnetic susceptibility tensor and are generally independent of the shape of the molecule. However, the presence of an anisotropic magnetic susceptibility also gives rise to pseudocontact shifts (PCS) (Kurland and McGarvey, 1970), which are reporters on the positions of the nuclei in the principal axis frame of the magnetic susceptibility tensor centered on the paramagnetic site, and therefore contain information about the structure/shape of a molecule. The use of paramagnetism-induced restraints (Gochin and Roder, 1995a; Gochin and Roder, 1995b; Banci et al., 1996; Banci et al., 1998; Bertini et al., 2001a; Gaponenko et al., 2004; Bertini et al., 2005; Diaz-Moreno et al., 2005; Jensen et al., 2006; Bertini et al., 2008; Schmitz et al., 2012; Yagi et al., 2013b) is becoming increasingly popular because of the introduction of lanthanide binding tags (Barthelmes et al., 2011; Wöhnert et al., 2003; Rodriguez-Castañeda et al., 2006; Su et al., 2006; John and Otting, 2007; Pintacuda et al., 2007; Zhuang et al., 2008; Su et al., 2008b; Su et al., 2008a; Keizers et al., 2008; Häussinger et al., 2009; Su and Otting, 2010; Hass et al., 2010; Man et al., 2010; Das Gupta et al., 2011; Saio et al., 2011; Swarbrick et al., 2011b; Swarbrick et al., 2011a; Bertini et al., 2012a; Liu et al., 2012; Kobashigawa et al., 2012; Cerofolini et al., 2013; Yagi et al., 2013a; Gempf et al., 2013; Loh et al., 2013), that extend the range of applications from paramagnetic metalloproteins (Banci et al., 1996; Banci et al., 1997) (or proteins in which the naturally occurring metal can be replaced by a paramagnetic one (Allegrozzi et al., 2000; Bertini et al., 2001a; Bertini et al., 2001c; Bertini et al., 2001b; Bertini et al., 2003; Bertini et al., 2004b; Balayssac et al., 2008; Bertini et al., 2010a; Luchinat et al., 2012b)) to, in principle, any protein.

Given the various possibilities and limited resources, choosing the optimal set of observables for the characterization of protein conformational heterogeneity is important. In this work we analyze the information content associated with the two commonly used types of experimental data (dRDC and paramagnetic data) and discuss their features and advantages and pitfalls. Specifically, we want to understand what information can be recovered and to what extent. Importantly, the methodology that we develop below is not limited to dRDC or paramagnetic data, and can be applied to any set of experimental observables.

We focus on analyzing the ensemble information content of three specific types of NMR restraints, dRDC, pRDC, and PCS, in the case of proteins composed of two domains connected by a flexible linker. We have used the two-domain protein calmodulin (CaM) as a test case. As done previously (Bertini et al., 2007; Berlin et al., 2013), we assume that all three types of NMR restraints considered here represent a population-weighted average of the corresponding values for the individual conformers, and therefore have a linear dependence on the ensemble populations, such that

$$\mathbf{y}={\mathbf{a}}_{1}{x}_{1}+\dots +{\mathbf{a}}_{N}{x}_{N}+\epsilon =\mathbf{\text{Ax}}+\epsilon $$

(1)

where **y** is a length-*L* column vector
representing the experimental data (dRDC, pRDC, PCS, or some combination thereof),
**A** is an *L*x*N* prediction matrix consisting
of *N* column-vectors **a**_{j}
(*j*=1…*N*) representing the predicted data for
each of the *N* conformers, *x _{j}* is the
population weight for the

Since in general recovering **x** from equation 1 is an ill-posed problem, having an infinite number of
solutions, we seek to recover the minimum ensemble (sparsest solution) satisfying the
experimental observables, which we express as a constrained linear least-squares problem
(Berlin et al., 2013),

$${\mathbf{x}}^{*}=\underset{\mathbf{x}}{\text{argmin}}{\Vert \mathbf{W}(\mathbf{\text{Ax}}-\mathbf{y})\Vert}_{2}\text{s.t.}\mathbf{x}\ge 0,{\Vert \mathbf{x}\Vert}_{0}=M$$

(2)

where **W** is the weight matrix that non-uniformly weighs the
residuals between **y** and **Ax**, *M* is the desired
ensemble size, ‖…‖_{2} is the Euclidian norm, and
‖**x**‖_{0} is the _{0} quasinorm of
**x**, i.e. the number of nonzero elements in **x**. Typically the
experimental errors are assumed to be uncorrelated, in which case **W** is simply
a diagonal matrix with **W**_{ii} = 1/σ_{i}, where
σ_{i} is the estimated experimental error of the *i*th
observation *y _{i}*. For simplicity, for the rest of the manuscript
we will drop

For steric dRDC data, we generate the prediction matrix **A** using
program PATI (Berlin et al., 2009; Berlin et al., 2013), which assumes the presence of a
steric planar alignment medium (Fig. 1A).
Electrostatically induced RDCs were similarly simulated using PALES (Zweckstetter, 2008). The absolute scaling in the predicted dRDC values
is regulated by changing the value of the parameter “liquid crystal
concentration” (Zweckstetter and Bax,
2000) that controls the distance between the planar steric barriers. In the SES
model the absolute scaling of the predicted dRDC is treated as an implicit parameter since
the sum of all weights $(\sum _{j}{x}_{j})$ is not constrained (Berlin et al.,
2013).

Schematic illustration of the relationship between the conformation of a
multidomain protein and the alignment tensor for the two experimental methods considered
her (or: alignment tensor caused by external and internal alignment). In the case of
partial **...**

For pRDC and PCS, without loss of generality, we can assume that a metal ion tag
is located on the first (rigid) domain of the protein (Bertini et al., 2003). Therefore, the position of the metal ion relative to that
domain is the same for all conformers. So, instead of performing the prediction of pRDC
and PCS values for both domains, we obtain the prediction matrix **A** for a
two-domain rigid system by first deriving the magnetic susceptibility anisotropy tensor
(and metal ion’s position) from the experimental data for the first domain, and
then use these tensors to predict the matrix **A** values for the second domain
based on its position relative to the first domain (Fig.
1B). This formulation assumes that the distribution of the relative positions of
the two domains is independent of the orientation of the magnetic susceptibility
anisotropy tensor in the magnetic field (Bertini et al.,
2002a).

Given a specific conformer, the pRDC values in the **A** matrix are thus
predicted by first deriving the vector containing the 5 independent components of the
alignment tensor, **S***, directly from the experimental data for the first
domain:

$${\mathbf{S}}^{*}=\underset{\mathrm{s}}{\text{argmin}}{\Vert {\mathbf{V}}_{1}\mathbf{S}-{\mathbf{y}}_{1}\Vert}_{2}$$

(3)

where **V**_{1} is a 5-column matrix, the elements of
which depend on the orientations of the normalized bond vectors in the fixed frame (Losonczi et al., 1999; Valafar and Prestegard, 2004; Berlin et al.,
2009; Simin et al., 2014) and
**y**_{1} are the observed experimental pRDC values for the first
domain. Then, using the derived **S***, we predict the pRDC for the second domain
of the *j*th conformer (**A*** _{pRDC,j}*) as

$${\mathbf{A}}_{\mathit{\text{pRDC,j}}}={\mathbf{V}}_{2j}{\mathbf{S}}^{*}$$

(4)

where **V**_{2j} is the 5-column matrix
of the bond vectors for the second domain in the *j*th conformer.

Similarly, the PCS values for the first domain can be used to derive the
magnetic susceptibility anisotropy tensor **T***, represented by a 3×3
traceless symmetric matrix, and the metal ion’s position **p*** (computed
by alternating between solving a non-linear least-squares problem for **p***, and
a linear problem for **T***). These values are then used to predict the PCS for
the second domain of the *j*th conformer
(**A**_{PCS,j}). The elements of the
**A**_{PCS,j} vector are the PCSs predicted for each
nucleus *i* of the second domain, according to the relationship

$${\mathbf{A}}_{\mathit{\text{PCS,j,i}}}\approx \frac{1}{12\pi {\Vert {\mathbf{r}}_{\mathit{\text{ij}}}\Vert}_{2}^{5}}\text{tr}\left(\right[\begin{array}{ccc}\hfill 3{r}_{\mathit{\text{ij}},1}^{2}-{\Vert {\mathbf{r}}_{\mathit{\text{ij}}}\Vert}_{2}^{2}\hfill & \hfill 3{r}_{\mathit{\text{ij}},1}{r}_{\mathit{\text{ij}},2}\hfill & \hfill 3{r}_{\mathit{\text{ij}},1}{r}_{\mathit{\text{ij}},3}\hfill \\ \hfill 3{r}_{\mathit{\text{ij}},1}{r}_{\mathit{\text{ij}},2}\hfill & \hfill 3{r}_{\mathit{\text{ij}},2}^{2}-{\Vert {\mathbf{r}}_{\mathit{\text{ij}}}\Vert}_{2}^{2}\hfill & \hfill 3{r}_{\mathit{\text{ij}},2}{r}_{\mathit{\text{ij}},3}\hfill \\ \hfill 3{r}_{\mathit{\text{ij}},1}{r}_{\mathit{\text{ij}},3}\hfill & \hfill 3{r}_{\mathit{\text{ij}},2}{r}_{\mathit{\text{ij}},3}\hfill & \hfill 3{r}_{\mathit{\text{ij}},3}^{2}-{\Vert {\mathbf{r}}_{\mathit{\text{ij}}}\Vert}_{2}^{2}\hfill \end{array}\left]{\mathbf{T}}^{*}\right)$$

(5)

where **r**_{ij} =
[*r*_{ij,1},
*r*_{ij,2},
*r*_{ij,3}] is the vector connecting the metal
ion (located at **p***) and the *i*th atom in the
*j*th conformer, and tr(…) designates the trace of a matrix. The
elements of the tensor **T*** and the components of the alignment tensor
**S*** are related to one another by a proportionality constant (Bertini et al., 2002b), so that each of the two can be
easily calculated from the other.

Similarly to dRDC for multiple alignment media, pRDC and PCS from multiple metal
ion derivatives (determined from the **S*** and **T*** tensors,
respectively, of the corresponding metals) can be combined together in a single
**A** matrix of predicted data.

Since the scaling of the predicted dRDC values has an uncertainty (Berlin et al., 2013), when recovering SES ensembles
using dRDC, we allow the total sum of **x**, $\sum _{j}{x}_{j}$, to float, and only use the restraint **x** ≥ 0 (see
Eq. 2).

By contrast, the values of pRDC and PCS are determined without any adjustable scaling factor, and thus the two datasets can be directly combined into a single population-constrained pRDC+PCS SES problem,

$${\mathbf{x}}^{*}=\underset{\mathbf{x}}{\text{argmin}}{\Vert \left[\begin{array}{c}\hfill {\mathbf{y}}_{\mathit{\text{pRDC}}}\hfill \\ \hfill {\mathbf{y}}_{\mathit{\text{PCS}}}\hfill \end{array}\right]-\left[\begin{array}{c}\hfill {\mathbf{A}}_{\mathit{\text{pRDC}}}\hfill \\ \hfill {\mathbf{A}}_{\mathit{\text{PCS}}}\hfill \end{array}\right]\mathbf{x}\Vert}_{2}\text{s.t.}\mathbf{x}\ge 0,\sum _{j}{x}_{j}\le c,{\Vert \mathbf{x}\Vert}_{0}=M$$

(6)

where *c* is the upper bound on the total population
weight. Since $\sum _{j}{x}_{j}$ represents the total population weight $\sum _{j}{x}_{j}$ should be 1. However, we allow for the sum of the weights to be less
than 1, since we aim at recovering the sparsest ensemble representing the major states
(potentially there could be a very large set of transient minor states). The validity of
the recovered solution can be evaluated from the geometrical interpretation of pRDC: a
solution is a convex combination of a set of conformers such that the averaged pRDC belong
to the polyhedron with vertices in the conformers (see Figure S5) (Gardner et al., 2005; Longinetti et
al., 2006). Since the problem is underdetermined, there will be many solutions,
and the SES method chooses to limit the number of vertices to *M*. In order
to find a solution with this constraint, we need to use a *c*<1 in
Eq. (6). This is equivalent to shrinking
the vertices of the polyhedron towards the origin by a factor *c* and
renormalizing the weighting factors to 1. However, since the origin is an acceptable point
(Sgheri, 2010a) and the set is convex, the
shrunk vertices will be anyway acceptable points. In other words, if *c* is
relatively close to 1, the conformers representing the vertices are anyway good
representatives of the conformational freedom of the system. Finally, the $\sum _{j}{x}_{j}\le 1$ restraint prevents from finding unphysical solutions.

SES ensemble recovery was implemented using the multi-orthogonal matching
pursuit (MOMP) algorithm (Berlin et al., 2013). We
modified the MOMP method to handle the $\sum _{j}{x}_{j}\le c$ requirement using the active set method (O’Leary, 2009) to restrain our solution for each iteration of MOMP.
Given that there are two restraints on **x: x** ≥ 0 and $\sum _{j}{x}_{j}\le c$ during each iteration of the MOMP algorithm there are four possible sets
of active restraints: (i) no restraints are active; $\text{(ii)}\sum _{j}{x}_{j}\le c$ restraint is active; (iii) the **x** ≥ 0 restraint is
active; or (iv) both **x** ≥ 0 and $\sum _{j}{x}_{j}\le c$ are active. To summarize, the constrained least-squares problem is
solved as follows: update the solution using conjugate gradient (CG) method; if the
solution violates **x** ≥ 0 or $\sum _{j}{x}_{j}\le c$, solve the linearly constrained linear least-squares problem by using a
“feasible direction” method (O’Leary, 2009); if the solution still violates **x** ≥
0, drop this solution from a list of possible solutions stored in a priority queue. This
procedure is repeated for all propagated solutions from the previous iteration.

The time vs. accuracy tradeoff in the MOMP algorithm is controlled by how many
top solutions, *K*, from the current iteration are propagated to the next
iteration of MOMP (Berlin et al., 2013). In order
to improve the memory requirement for running SES using very large *K*
values (>10^{6}), we modified the algorithm used to solve the
overdetermined linear least-squares problem for each iteration of SES, when a new solution
must be computed right after one new column is added to the list of active columns (see
Supporting Information in
(Berlin et al., 2013)). In the previous
implementation (Berlin et al., 2013), the
least-squares solution was efficiently updated by doing a rank-1 update of the QR
decomposition. However, this approach requires us to store *K* QR
decompositions during each iteration. In our current updated version, we switched to an
iterative CG least-squares solver, which requires that we only store the
previous-iteration solution, rather than the QR decomposition. This significantly reduced
the SES memory footprint for large *K*. The full **A**^{T}
**A** matrix required for the CG algorithm is never explicitly formed, and
instead the multiplication step in the CG algorithm is computed as
**A**^{T}(**Ax**). With the CG implementation we are able to
run SES on a 10 GB RAM desktop for *K*=10^{6}, without any
sacrifice in computational time or accuracy, as compared to the previous
implementation.

The Maximum Occurrence (MaxOcc) of each and every conformer is defined as the maximum weight that it can obtain when part of a conformational ensemble without violating the constraints of the experimental data. No restriction is posed on the number of conformations to be included in the ensemble. Maximum occurrence (MaxOcc) can be interpreted as the maximum fraction of time that a conformation can exist, when taken together with any ensemble of conformations with optimized weights (Longinetti et al., 2006; Bertini et al., 2007; Sgheri, 2010b; Bertini et al., 2010b; Das Gupta et al., 2011; Luchinat et al., 2012a; Bertini et al., 2012b; Bertini et al., 2012c; Cerofolini et al., 2013).

We formulate MaxOcc as a convex regularization problem, where for each conformer
*j* we find the weight vector **x** which minimizes

$$\underset{\mathbf{x}}{\text{argmin}}\{{\Vert \mathbf{A}\mathbf{x}-\mathbf{y}\Vert}_{2}^{2}+\lambda {({x}_{j}-{x}_{\mathit{\text{MO}}})}^{2}+\lambda (1-{x}_{\mathit{\text{MO}}}-\sum _{i=1,i\ne j}^{N}{x}_{i}{)}^{2}\}\text{s.t.}\mathbf{x}\ge 0$$

(7)

where *x _{MO}* is the desired weight of the
conformation

Calculations are also performed to determine the maximum occurrence of a region (MaxOR) defined in the conformational space of the protein (Andralojc et al., 2014). The MaxOR, similar to MaxOcc, is defined as the maximum weight that a region in conformational space (composed of multiple structures) can have in an ensemble without causing a violation of the experimental restraints. First, the highest-MaxOcc structures are clustered according to their positions using a k-means algorithm as implemented in the Python library SciPy (Jones et al., 2001). The number of clusters is set to the highest value yielding reproducible clustering by the algorithm. Once the clusters are built, small regions are defined around the centers of the clusters, which include all conformations within a given distance Δ from the center of the cluster. The MaxORs of these regions are determined by solving

$$\underset{\mathbf{x}}{\text{argmin}}\{{\Vert \mathbf{A}\mathbf{x}-\mathbf{y}\Vert}_{2}^{2}+\lambda [{({x}_{\mathit{\text{MO}}}-\sum _{i\in C}{x}_{i})}^{2}-{(1-{x}_{\mathit{\text{MO}}}-\sum _{i\in D}{x}_{i})}^{2}\left]\right\}\text{s.t.}\mathbf{x}\ge 0$$

(8)

where *x _{MO}* is the fixed value that must
correspond to the sum of the weights of all conformations within the region, and C and D
indicate the structures within and outside that region, respectively. Again, the largest

An important theoretical question that we would like to answer *a
priori*, before performing any time-consuming simulation or experiment, is how
much information for ensemble recovery is contained in dRDC vs. pRDC vs. PCS and in dRDC vs.
pRDC+PCS combined. For example, intuitively, dRDC should contain more information than pRDC,
since dRDC contain shape/size-related information, while the relative informational content
of PCS is harder to intuitively quantify. To what extent combining pRDC with PCS yields
better results than each of these data separately? Is the information provided by pRDC+PCS
similar to that provided by dRDC? Would using several different metal ions be needed to
obtain results comparable to those obtained with multiple sets of dRDC, or do they produce a
better set of experimental data for the characterization of the conformational
heterogeneity?

In order to answer these questions, we analyzed several algebraic properties of eight experimentally feasible datasets: (i) single-alignment medium dRDC; (ii) single-metal ion pRDC; (iii) single-metal ion PCS; (iv) single-metal ion pRDC+PCS combined; and (v-viii) datasets analogous to (i-iv) but with three alignment media or thee metal ions. We will refer to the one and three media/metal ions datasets as the one- and three-restraint datasets, respectively.

The datasets were generated for a pool of 32723 conformers of calmodulin (CaM), a protein composed of two rigid domains connected by a 4-residue flexible linker (Barbato et al., 1992; Tjandra et al., 1995; Chou et al., 2001; Kukic et al., 2014). This large pool of sterically allowed conformations of the protein was taken from reference (Bertini et al., 2010b), where it was generated using the program RanCh (Bernadò et al., 2007), For each conformer and for each aligning medium or metal ion, a set of dRDCs, pRDCs, and PCSs was generated, as described in the Theory section.

The paramagnetic restraints consisted of PCS of the amide H atoms and pRDC of amide N-H pairs of the C-terminal domain of CaM induced by the presence of a paramagnetic center in its N-terminal domain. Three metals with non-coinciding magnetic susceptibility tensors (corresponding to the experimental ones obtained for Tb(III), Tm(III), and Dy(III) CaM) were used to generate three sets of PCSs (132 observations in total) and pRDCs (112 observations in total). The magnetic susceptibility anisotropy tensors were taken from reference (Bertini et al., 2009).

The simulated diamagnetic restraints were amide ^{15}N-^{1}H
dRDCs (219 in total) induced in both CaM domains by 3 independent external alignment
media: flat uncharged discs and either positively or negatively charged rods. In the first
case, dRDCs were generated using PATI (Berlin et al.,
2009), in the other cases using PALES (Zweckstetter and Bax, 2000; Zweckstetter,
2008). In both cases, the calculation of the alignment tensors, and of the
corresponding dRDC, are performed under the assumption that the protein’s
conformations are rigid during the time course of its interaction with the alignment
medium. As a word of caution we note that every interaction of a protein with the
alignment medium might actually perturb its conformation, and these interactions can occur
on a timescale that is slower than the conformational averaging itself. The assumption
that the averaged dRDCs correspond to a weighted average of the RDCs calculated for the
individual conformations, although universally used, might fall short in representing the
real physical picture.

The first and simplest analysis we performed was aimed at evaluating the
theoretical information content of the eight different datasets described above. This was
done through the spectral analysis of the prediction matrix **A** for each
dataset. The spectral analysis measures the number of significant linearly independent
components present in the data, by counting the eigenvalues corresponding to linearly
independent eigenvectors. This directly provides an upper bound on the number of
independent conformers we can hope to extract. Trying to recover a larger number of
independent conformers would result in overfitting. The results are shown in Fig. 2A,D.

SVD decomposition (left panels), histogram of column correlations (center
panels), and condition number of randomly subselected set of columns (right panels), for
the eight described datasets. The results for a single medium/metal ion are shown on the
top, **...**

As shown in Eq. 3, any vector of
RDC values (either pRDC or dRDC) from a rigid domain can be expressed as a matrix
**V**, which can be determined from the orientations of the bond vectors of
that domain, multiplied by the 5 independent components of the alignment tensor matrix.
Since there is a linear dependence of the observed data on the 5 components of the
alignment tensor, we expect the **A** matrix for dRDC to have rank 10 (5
independent parameters for each of the two domains), and for pRDC to have rank 5, since
only the second domain data are used for ensemble recovery. The number of unknowns in the
paramagnetic case is also smaller because the alignment tensor for the first domain (5
parameters) can be easily determined from PCS and pRDC measured for this domain, as they
are not averaged by conformational variability.

Numerical spectral analyses of the generated prediction matrices for dRDC and
pRDC (Fig. 2A,D) support our theoretical analysis,
and show that the number of singular values of matrix **A** for one-restraint
dRDC and pRDC data is 10 and 5, respectively. Going from 1 to 3 alignment tensors triples
the number of non-zero singular values for dRDC and pRDC, as would be expected for
linearly independent alignments. The large decrease in the magnitude of singular values
for the last 10 dRDC and 5 pRDC non-zero singular values in the three-restraint datasets
likely reflects the difficulty in experimentally obtaining three fully independent
alignment tensors. The larger magnitudes of dRDC singular values compared to the singular
values for pRDC are not related to their information content, but merely reflect the
relative strength of diamagnetic versus paramagnetic alignment in the simulated data. On
the contrary, it is the decrease in the relative magnitude of the singular values with
respect to the largest value, calculated from a set of data, that reflects the difficulty
in exploiting the associated restraints, and is hence ultimately related to the
information content.

Similarly, the observed PCS data for a rigid domain which is not containing the
paramagnetic ion (i.e. for the second domain) can be expressed using 8 parameters: the 5
independent components/parameters defining the **T** tensor, and the 3 parameters
describing the metal-ion’s position **p** with respect to this domain.
However, since the observed PCS vector **y** is not linearly related to
**p**, the rank of **A**_{PCS} (calculated from the PCSs in
the second domain) is much higher than 8, and greater than that for dRDC or pRDC datasets.
The rank of **A**_{PCS} is actually close to (up to) the number of
observations; however as Figure 2A,D show, the
magnitude of the singular values decreases very rapidly. This decrease reflects the strong
difference in the PCS values between conformers where the C-terminal (second) domain is
close to the metal ion (paramagnetic center) and those where it is far away. After the
first ≈15 entries, in the one-restraint case the singular values are very small
because similar PCS values are calculated for conformers not very far from one another and
for nuclei which are spatially close to several other nuclei. When using three sets of
metal ions, the number of conformers with large and different PCS values increases. Thus,
the decrease in the magnitude of the singular values is significantly slower than in the
case of a single metal ion (Fig. 2A–D).

One major advantage of using metal ions instead of steric alignment is that both
pRDC and PCS are collected from the same biochemical construct. Thus, two independent
datasets can be directly combined, as described in Eq. 6. When combining these datasets, a significantly slower decay in singular
values of **A** is obtained compared to the pRDC and PCS datasets analyzed
independently. This supports the accepted intuition that pRDC and PCS provide orthogonal
structural restraints (pRDCs are very sensitive to orientation, PCSs mostly provide
distance restraints).

The spectral analysis of the **A** matrices suggests that pRDC+PCS and
even PCS alone provide better restraints for ensemble selection than dRDC. However,
singular values are not an exhaustive description of the overall vector distribution.
Therefore, we directly analyzed the distribution of correlations between all columns of
the matrix **A** calculated for dRDC, pRDC, and PCS. The uncentered correlation
distributions between all pairs of columns are shown in Fig.
2B,E. The more uncorrelated the columns of each specific **A**
(**A**_{dRDC}, **A**_{pRDC},
**A**_{PCS}) the smaller the chance that an alternative conformer can
explain the same subset of experimental data, thus decreasing the number of viable
alternative ensembles. In the optimal case, all columns would have zero correlation, and
the ensemble solution would be unique.

Figure 2B,E clearly demonstrate that even though the number of singular values of PCS is larger than that of dRDC and pRDC, the correlation distribution is actually significantly worse than for any other dataset, so that their information content could not be larger. The higher correlation for large fraction of the conformers reflects a distribution of PCS where very large changes occur in proximity of the metal ion only, whereas almost no change occurs far away from the metal ion. Additional metal ions can significantly improve the distribution of correlations, although it remains poor with respect to that of the other restraints.

Since pRDCs are distance-independent, they provide a more uniform distribution of values, so that their correlation distribution is much better than for PCS. The pRDC distribution is anyway worse than that of dRDC in the one-restraint case; it significantly improves, essentially to the level of dRDC, in the three-restraint case. Interestingly, the dRDC distribution changes only slightly between one and three restraints, which suggests that the information contained in the additional dRDC datasets is more redundant than in the pRDC case.

Combining pRDC with PCS results in a better correlation distribution than for pRDC and PCS individually. In turn, the correlation distribution of pRDC+PCS is very similar to that of dRDC in the one-restraint case and actually somewhat better in the three-restraint case.

While the correlation plots in Fig. 2B,E
provide an estimate of the **A** matrix column vector distribution, they do not
directly tell how well ensembles greater than two can be recovered, nor do they take
signal-to-noise ratio into account. To assess how well larger ensembles can be recovered,
we computed the mean and standard deviation of the relative error from a synthetically
generated **y** data (with added Gaussian error) for
*M*=1,…,10 columns. The mean and standard deviation were computed
by randomly sampling, for each *M* value, *M* columns and
uniformly at random generating the associated population weights **x**. The
synthetic **y** was generated as y=**Ax**+*N*(0,1), where
*N*(0,1) is the zero-mean Gaussian distribution with σ=1. The
vector **x*** and the associated relative error,
||**x-x***||_{2}/||**x**||_{2},
were recovered by solving Eqs. 2 and 6. In order to guarantee a less than
0.1% relative error with greater than 99.999% confidence using Chernoff
bound, the process was repeated 40000 times for each *M*. The results for
all datasets are shown in Fig. 2C,F.

For the one-restraint datasets, dRDC has lower relative error than pRDC, PCS, or pRDC+PCS. As expected, there is a rapid growth in pRDC errors due to the low matrix rank, and high errors overall in PCS due to the high correlation between columns. In the case of the three-restraint datasets, dRDC has significantly lower relative error than pRDC, even though on the correlation plot the two distributions are very similar. Interestingly, combining pRDC+PCS yields only slightly higher error rate than for dRDC.

In the previous sections we theoretically analyzed the information content of 8 datasets of synthetic dRDC, pRDC and/or PCS data. Here we perform a direct comparison of the performance of the different restraints in recovering information on the structural variability of the system. To achieve this, we determined i) the minimum-size sparsest ensemble solution using the Sparse Ensemble Selection (SES) method (Berlin et al., 2013) and ii) the conformations (as well as the regions in the conformational space) with the highest MaxOcc values. In this way it becomes possible to analyze the accuracy of the recovered solutions from the different sets of synthetic averaged data.

For this purpose, we devised three simulations modeling i) extensive mobility around a single conformation, ii) two-site exchange with limited mobility around each center, and iii) two-site exchange with a reduced difference in the orientations of the two centers. In each of the simulations, the two-domain protein CaM was allowed to sample different, well defined, parts of its sterically allowed conformational space. Synthetic restraints were calculated as weighted averages over the values of dRDC, pRDC, and PCS of the individual conformations belonging to the sampled regions. These average data were perturbed with a Gaussian error with a standard deviation of 1, 2, or 3 Hz for pRDC and dRDC and of 0.01, 0.02, or 0.03 ppm for PCS.

In the following descriptions of the simulated conformational ensembles, the N-terminal domain of CaM is taken as the frame of reference, and each conformation is described by the different position and orientation of the C-terminal domain with respect to the N-terminal domain. The exact details of each simulation, although described accurately for completeness, are not crucial for the success of the ensemble recovery attempts.

In this first simulation we consider the case of conformational variability centered at a single extended conformation of CaM. The sampled ensemble consists of all the conformers, present in the pool of the 32723 sterically allowed conformers, within a distance Δ (measured as a combination of translation and rotation) from the central extended structure (Fig. 3A) (Bertini et al., 2012b). Specifically, this distance is defined as:

$$\Delta =d+f(1-\text{cos}\alpha )$$

(9)

where *d* is the translation of the center of mass of the
C-terminal domain from the central structure, and α is the angle of rotation
from the central structure, calculated as α =
acrccos(|*q _{c}* ·

This simulation models the case of a two-site exchange, with limited mobility
allowed around each of the two main conformers (Fig. 3B). The two centers were separated by approximately 30 Å and
their C-terminal domains were rotated by ca. 140° with respect to each other.
The mobility around each center was simulated as in the previous case with the threshold
on Δ set to 10 Å and *f* equal to 42.7 Å, which
corresponds to a maximum allowed angular displacement with respect to the central
conformer of 80° (α=40°).

This simulation is similar to Simulation 2, with the difference that the
angular distance between the two sites was decreased almost twofold (Fig. 3C). Sites with more similar orientations are
likely to present a bigger challenge in ensemble recovery using restraints which depend
on the domain orientations. The distance between the centers (both distinct from those
used in Simulation 2) is 30 Å while the difference in orientation of the
C-terminal domains is now 80°. The threshold of Δ and the value of
*f* used to simulate the residual mobility around each center were the
same as in Simulation 2, hence the same upper limit on the angle α.

We applied the SES method to these simulated datasets and analyzed how the various restraints affect the recovery of the main conformations contained in the synthetic ensembles used to generate the data. The recovered ensembles were evaluated in terms of their sizes (number of major states) and of the proximity of recovered structures to the centers of the synthetic ensembles (in terms of spatial and angular displacement). As already mentioned in the Theory section, the ensemble size was chosen using the L-curve method (Berlin et al., 2013) (see Figure S6).

The results are presented in Tables 1, ,22 and and3. In3. In general, dRDCs allowed a reasonably accurate recovery of the major states that were used to generate the synthetic datasets (see, for instance, Fig. 4A). However, in all three simulations, in some solutions one additional conformer was recovered, albeit with a relatively low weight. This additional conformer either belongs to the distribution of conformers around one of the main centers (as in Simulation 1 with error of 1 Hz and 2 Hz, and in Simulation 2 with error of 2 Hz, Fig. 4B) or is positioned in-between the two major states (as in Simulation 2 with error of 1 Hz, Fig. 4C). In the first case its presence may reflect conformational heterogeneity; in the second case it is likely related to artifacts. The latter may arise because, ‘average conformers’ can be more compatible with the averaged experimental observables than any of the actually sampled conformations taken individually.

SES recovery using dRDC data. Color code for the C-terminal domain: green -
simulated conformers in the centers of the regions, red, blue, and yellow - reconstructed
conformers with highest, intermediate, and lowest weight, respectively. (A) An ensemble
with **...**

Results of Simulation 1. The table reports the sizes of the recovered ensembles,
the specific weights ascribed to their constituent conformers, and for each of these
conformers their spatial and angular displacement from the center of the original
ensemble. **...**

Results of Simulation 2. The table reports the sizes of the recovered ensembles,
the specific weights ascribed to their constituent conformers and for each of these
conformers indicates the closest of the two original sites together their spatial and
angular **...**

Results of Simulation 3. The table reports the sizes of the recovered ensembles,
the specific weights ascribed to their constituent conformers and for each of these
conformers indicates the closest of the two original sites together their spatial and
angular **...**

In the case of pRDCs, the right number of major states was always recovered (Fig. S1), and in the corresponding conformers the domains were oriented with an accuracy comparable to that achieved with dRDC. It should be recalled that pRDCs contain no information whatsoever on the relative positions of the domains, which therefore results in inaccuracy of their positioning.

PCS data alone in two out of three simulations were sufficient to recover the correct solutions (Fig. S2) in terms of ensemble sizes and locations of the major states (with the accuracy similar to dRDC). However in Simulation 3, where the two states are more alike to one another, the calculations provide only a single state (Fig. S2B) situated in-between the two actual centers (in terms of both translation and orientation). The recovery of such an incorrect state is most likely, as already mentioned for dRDC, the outcome of the averaging of the experimental observables. Using all the paramagnetic data together (i.e., pRDC and PCS) improved the robustness of the recovery: both translations and orientations were satisfactory accurate in all cases (Fig. 5). The translation and rotation with respect to the conformers at the center of the distributions were within 4 Å and 16° for Simulation 1, 8 Å and 34° for Simulation 2, and 3 Å and 18° or 7 Å and 9° for Simulation 3 (1 Hz and 0.01 ppm error case). The ensemble recovery is robust, as increased errors did not noticeably affect the accuracy of solutions.

SES recovery with all the paramagnetic data. Color code for the C-terminal
domain: green - simulated conformers in the centers of the regions, red, blue -
reconstructed conformers with higher and lower weight, respectively. (A) Simulation 1, (B)
Simulation **...**

In conclusion, diamagnetic RDC, as well as the combination of paramagnetic RDC and PCS, are both equally suitable restraints for the recovery of the major states present in conformational ensembles. Special attention should be paid to the fact that, occasionally, ‘average conformers’ may be recovered.

Similar to the SES analysis, we performed MaxOcc analysis on the same datasets.
From the MaxOcc values, it is possible to determine which conformers can be sampled with
the largest weights. In order to speed the computational analysis up, we used random
sampling to detect regions of with potentially high MaxOcc conformers, and then expanded
those regions, to find the globally best solution. To do this, we first computed MaxOcc
for 400 conformers, randomly chosen from the generated pool (Bertini et al., 2010b; Bertini et al.,
2012b; Cerofolini et al., 2013). Then the
conformers with the highest MaxOcc (up to 0.8 of the MaxOcc of the highest scoring
conformer) were selected and the MaxOcc of their neighboring conformers (in the
conformational space) were calculated. The procedure was repeated until no more neighbors
with high MaxOcc were found. The neighboring conformers scored at each iteration were
chosen using Eq. 9 with the threshold on
Δ of 5 Å and *f*=40 Å. If the final distribution of
the highest MaxOcc conformers was broad, the analysis was supplemented by the Maximum
Occurrence of Regions (MaxOR) approach, which permitted to discriminate between the cases
of high MaxOcc conformers corresponding to conformers actually sampled by the protein and
the cases of high MaxOcc conformers corresponding to conformers arising from data
averaging (Andralojc et al., 2014).

The results of the MaxOcc analysis are reported in Table 4 for all three simulations. In Simulation 1, for both the paramagnetic and diamagnetic data, the analysis revealed that all the conformers with the highest MaxOcc (from 0.8 to 1 of the highest MaxOcc, corresponding to 0.58–0.73 for the paramagnetic data and 0.57–0.71 for the dRDC) form a single, relatively compact, region in the conformational space (Fig. 6A,C). In order to quantify its agreement with the original distribution, the center of the region was calculated by averaging the translational and orientation parameters of the highest MaxOcc conformers. The conformation so obtained was then compared with the conformation at the center of the original distribution. As shown in Table 4 and Fig. 6B,D, the agreement was very good in terms of spatial and angular displacement for both the diamagnetic and the paramagnetic data, either for 1 Hz/0.01 ppm or for 3 Hz/0.03 ppm errors.

MaxOcc results for Simulation 1. Each conformation is represented by a triad of
Cartesian axes, centered at the center of mass of the C-terminal domain. Color code for
A,C – according to the MaxOcc value (0.0-blue, 0.8-red), (A) The conformers **...**

The MaxOcc/MaxOR analysis. For each simulation the spatial and angular
displacement of the center of the ensemble of the best scoring conformers from the center
of the actually sampled distribution is reported, together with the indication of the
closest **...**

In simulation 2, i.e. the case of two well separated conformational regions,
when dRDC are used, the highest MaxOcc conformers are positioned in two distinct, clearly
separated regions (Fig. 7A), the centers of which are
positioned very close to the centers of the actually sampled distribution (Table 4, Fig. 7B).
When paramagnetic data (PCS+pRDC) are used, the highest MaxOcc (0.41–0.51)
conformers are positioned in one elongated, banana-shape region in the conformational
space (Fig. 8A), which includes the two actually
sampled centers, but also many conformers situated between them (their high score is an
outcome of conformational averaging as described in the SES results paragraph). From these
results, one cannot conclude whether the studied conformational ensemble mainly reflects a
two-site exchange case or the sampling of all the conformations within the determined
region. In order to distinguish between these two cases, MaxOR calculations were
performed. The highest MaxOcc conformers were clustered in 5 regions, shown in Fig. 8B, which include all conformations with distance
Δ≤ 5Å from the central conformation (calculated using eq. 9, with *f* =147 Å).
The MaxOR values for these regions are reported in Table S1 (diagonal entries). All regions have similar MaxOR values (up
to 0.60), not much higher than the largest MaxOcc values for the individual conformations.
If however MaxOR values are calculated for pairs of regions (off-diagonal entries of Table S1), strong differences arise.
All pairs yielding the highest MaxOR (0.90–1.00) are composed of regions at the
opposite sides of the distribution of the highest MaxOcc conformers, whereas all pairs
composed of the regions located on the same side of the distribution or more importantly
containing a region in the middle, have significantly lower MaxOR (up to 0.63 and 0.78,
respectively). This strongly suggests the occurrence of a two-site exchange model. The
pair of regions with the highest MaxOR has their central conformations in nice agreement
with the conformations in the center of the distributions in the synthetic ensemble, with
an accuracy comparable to that obtained by SES (Table
4, Fig. 8D).

MaxOcc results for Simulation 2 with dRDC with error of 1 Hz. Each conformation
is represented by a triad of Cartesian axes, centered at the center of mass of the
C-terminal domain. (A) The conformers with the highest MaxOcc, color code: according to
the **...**

MaxOcc/MaxOR results for Simulation 2 with paramagnetic data (1 Hz error for
pRDC and 0.01 ppm error for PCS). Each conformation is represented by a triad of Cartesian
axes, centered at the center of mass of the C-terminal domain. (A) The conformers with **...**

In simulation 3, for both the paramagnetic and diamagnetic data, the conformers recovered by MaxOcc form elongated regions comprising both the two centers and conformers situated between them (Figs. S3A and S4A). MaxOR was thus applied in both cases. As in the previous simulation, no single region has MaxOR significantly higher than the others, but the analysis of pairs of regions indicated again the occurrence of a two-site exchange (Tables S2 and S3). The two central conformations of the synthetic ensemble were identified with good accuracy (Table 4, Figs. S3D and S4D) using both kinds of experimental restraints. Again, the results are robust, as increased errors did not largely affect the accuracy of the solutions.

The performed MaxOcc/MaxOR analysis, as it appears from Table 4 as a whole, confirms the conclusion from the SES results that paramagnetic and diamagnetic restraints are equally useful for the recovery of conformational ensembles.

In many experimental studies RDCs have been shown to be precious restraints for analyzing molecular conformational freedom (Montalvao et al., 2014; Ravera et al., 2014; Camilloni and Vendruscolo, 2015; Torchia, 2015). Here we compared paramagnetic and diamagnetic RDCs and found substantial differences in their information content in the case of multidomain proteins. We found that the information content of dRDC is larger than that of pRDC in terms of number of singular values, and this reflects the shape dependence of dRDC. However, since the internal alignment due to paramagnetism also gives rise to PCSs, the total informational content recovered in a paramagnetic experiment is at least on par with dRDCs.

We have performed several simulations to evaluate the capability of recovering the conformational variability of two-domain proteins by the use of two different approaches, SES and MaxOcc/MaxOR. The main states of the protein were recovered reasonably well for both paramagnetic and diamagnetic datasets, with both approaches (see Tables 1–4 and also Table S4). Even for rather large experimental errors, we have found that both datasets still retain the ability of recovering the main conformational states, thus resulting appealing for the analysis of averaged experimental data possibly also in the case of large systems, where RDCs are affected by large errors. Of course, since the problem is underdetermined, a correct reconstruction of the main states may be unsuccessful for different rather unpredictable conformational distributions.

Such analysis suggests that pRDC+PCS provide a very promising alternative to dRDC data. It is important to note that this analysis does not include modeling error, which is harder to quantify. Therefore, our analysis does not capture the principal advantages of pRDC+PCS over dRDC, in that it does not require assumption of a barrier model in order to predict the alignment. In addition, one has to consider that the interactions of the protein with the alignment medium might actually perturb the system, and that these interactions can occur on a timescale that is slower than the conformational averaging itself, so that the assumption that the measured dRDCs can be represented as a population-weighted average of the RDCs for the individual (rigid) conformers may fall short in representing the real physical picture.

Finally, the availability of a number of rigid lanthanide-binding tags nowadays may make the acquisition of three independent metal ion datasets more practical and safer than the acquisition and prediction of three independent alignment media. One current limitation of using metal ions is the low signal-to-noise ratio in pRDC and PCS data, which could potentially be improved with better technology and methodology.

This work has been supported by Ente Cassa di Risparmio di Firenze, MIUR PRIN 2012SK7ASN, NIH grant GM065334, European Commission projects BioMedBridges No. 284209, pNMR No. 317127, and Instruct, part of the European Strategy Forum on Research Infrastructures (ESFRI) and supported by national member subscriptions. Specifically, we thank the EU ESFRI Instruct Core Centre CERM, Italy.

**Compliance with Ethical Standards**

Conflict of Interest: The authors declare that they have no conflict of interest.

This article does not contain any studies with human participants or animals performed by any of the authors.

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