Integrating NOE and RDC using sum-of-squares relaxation for protein structure determination

Abstract

We revisit the problem of protein structure determination from geometrical restraints from NMR, using convex optimization. It is well-known that the NP-hard distance geometry problem of determining atomic positions from pairwise distance restraints can be relaxed into a convex semidefinite program (SDP). However, often the NOE distance restraints are too imprecise and sparse for accurate structure determination. Residual dipolar coupling (RDC) measurements provide additional geometric information on the angles between atom-pair directions and axes of the principal-axis-frame. The optimization problem involving RDC is highly non-convex and requires a good initialization even within the simulated annealing framework. In this paper, we model the protein backbone as an articulated structure composed of rigid units. Determining the rotation of each rigid unit gives the full protein structure. We propose solving the non-convex optimization problems using the sum-of-squares (SOS) hierarchy, a hierarchy of convex relaxations with increasing complexity and approximation power. Unlike classical global optimization approaches, SOS optimization returns a certificate of optimality if the global optimum is found. Based on the SOS method, we proposed two algorithms—RDC-SOS and RDC–NOE-SOS, that have polynomial time complexity in the number of amino-acid residues and run efficiently on a standard desktop. In many instances, the proposed methods exactly recover the solution to the original non-convex optimization problem. To the best of our knowledge this is the first time SOS relaxation is introduced to solve non-convex optimization problems in structural biology. We further introduce a statistical tool, the Cramér–Rao bound (CRB), to provide an information theoretic bound on the highest resolution one can hope to achieve when determining protein structure from noisy measurements using any unbiased estimator. Our simulation results show that when the RDC measurements are corrupted by Gaussian noise of realistic variance, both SOS based algorithms attain the CRB. We successfully apply our method in a divide-and-conquer fashion to determine the structure of ubiquitin from experimental NOE and RDC measurements obtained in two alignment media, achieving more accurate and faster reconstructions compared to the current state of the art.

Appendix

The residual dipolar coupling and Saupe tensor

We give here a brief introduction to RDC and the Saupe tensor, while a detailed exposition can be found in (Tolman and Ruan 2006) for example. Let $\varvec{v}_{nm}$ be the unit vector denoting the direction of the bond between nuclei n and m. Let b be the unit vector denoting the direction of the magnetic field. The RDC $D_{nm}$ due to the interaction between nuclei n and m is

$$\begin{aligned} D_{nm} = D_{nm}^\text {max} \left\langle \frac{3 (\varvec{b}^T \varvec{v}_{nm})^2 -1}{2} \right\rangle _{t,e}. \end{aligned}$$

(67)

$D_{nm}^\text {max}$ is a constant depending on the gyromagnetic ratios $\gamma _n,\gamma _m$ of the two nuclei, the bond length $r_{nm}$, and the Planck’s constant h as

$$\begin{aligned} D_{nm}^\text {max} = -\frac{\gamma _n \gamma _m h}{2 \pi ^2 r_{nm}^3}, \end{aligned}$$

(68)

and $\langle \ \cdot \rangle _{t,e}$ denotes the ensemble and time averaging operator. As presented, RDC depends on the relative angle between the magnetic field and the bond. By extension of terminology, we refer to the normalized RDC

$$\begin{aligned} r_{nm} =D_{nm}/D_{nm}^\text {max} \end{aligned}$$

(69)

as simply the RDC.

It is conventional to interpret the RDC measurement in the molecular frame. More precisely, we treat the molecule as being static in some coordinate system, and the magnetic field direction being a time and sample varying vector. In this case the RDC becomes

$$\begin{aligned} D_{nm} = D_{nm}^\text {max} \varvec{v}_{nm}^T \varvec{S} \varvec{v}_{nm}, \end{aligned}$$

(70)

where the Saupe tensor S is defined as

$$\begin{aligned} \varvec{S} = \frac{1}{2}(3 \varvec{B} - \varvec{I}_3),\quad \varvec{B} = \left\langle \varvec{bb}^T \right\rangle _{t,e}. \end{aligned}$$

(71)

We note that $\varvec{S}$ is symmetric and $\text {Tr}(\varvec{S}) = 0$. In order to use RDC for structural refinement of a protein, $\varvec{S}$ is usually first determined from a known structure (known $\varvec{v}_{nm}$) that is similar to the protein.

We now detail a classical way of obtaining the Saupe tensor from a known template structure (Losonczi et al. 1999). Using the fact that $\varvec{S}$ is symmetric and $\text {Tr}(\varvec{S}) = 0$, Eq. (70) can be rewritten as

$$r_{nm} = ({\varvec{v}_{nm}}_2^2-{\varvec{v}_{nm}}_1^2) \varvec{S}(2,2) + ({\varvec{v}_{nm}}_3^2-{\varvec{v}_{nm}}_1^2) \varvec{S}(3,3) + 2 {\varvec{v}_{nm}}_1 {\varvec{v}_{nm}}_2 \varvec{S}(1,2) + 2 {\varvec{v}_{nm}}_1 {\varvec{v}_{nm}}_3 \varvec{S}(1,3) + 2 {\varvec{v}_{nm}}_2 {\varvec{v}_{nm}}_3 \varvec{S}(2,3)$$

(72)

where ${\varvec{v}_{nm}}_i$, $i=x,y,z$ are the different components of $\varvec{v}_{nm}$ in the molecular frame. When there are L RDC measurements, Eq. (72) results in L linear equations in five unknowns ($\varvec{S}(2,2),\varvec{S}(3,3),\varvec{S}(1,2),\varvec{S}(1,3)$ and $\varvec{S}(2,3)$), that can be written in matrix form as

$$\begin{aligned} \varvec{A} s = \varvec{r},\quad \varvec{s} = \begin{bmatrix} \varvec{S}(2,2)\\ \varvec{S}(3,3)\\ \varvec{S}(1,2)\\ \varvec{S}(1,3)\\ \varvec{S}(2,3) \end{bmatrix} \in {\mathbb {R}}^5, \quad \varvec{r} =\begin{bmatrix} r_{n_1m_1}\\ \vdots \\r_{n_L m_L} \end{bmatrix} \in {\mathbb {R}}^M \end{aligned}$$

(73)

and $\varvec{A}\in {\mathbb {R}}^{L\times 5}$. An ordinary least squares procedure can be used to estimate s if $\varvec{A}$ has full rank. This is also referred to as the SVD procedure in (Losonczi et al. 1999).

Sum-of-squares relaxation

In this section, we explain why the convex relaxation presented in “Convex relaxation with only RDC constraints” section is coined SOS. The polynomial optimization problem

$$\begin{aligned} p_1 = \min _{\varvec{x}\in {\mathbb {R}}^n} f(\varvec{x})\quad \text {s.t.}\ h(\varvec{x})=0, \end{aligned}$$

(74)

where $f(\varvec{x}),h(\varvec{x})$ are polynomial functions, can be expressed equivalently as

$$\begin{aligned} \max _{d} d\quad \text {s.t.}\ f(\varvec{x})-d\ge 0\ \ \ \text {on}\ h(\varvec{x})=0. \end{aligned}$$

(75)

This is equivalent to

$$\begin{aligned} d_1 = \max _{d,t_{\varvec{\alpha }}} d\quad \text {s.t.}\ f(\varvec{x})-d+\left(\sum _{\varvec{\alpha }}t_{\varvec{\alpha }}\varvec{x}^{\varvec{\alpha }}\right) h(\varvec{x})\ge 0\ \ \ \forall \ \varvec{x} \end{aligned}$$

(76)

[(Blekherman et al. 2011), Chapter 3], which is the dual problem to (74). However, due to the NP-hardness in testing the non-negativity of a polynomial (Blekherman et al. 2011), we further restrict the search space from the set of non-negative polynomials to the set of SOS polynomials:

$$\begin{aligned} d_2 = \max _{d,t_{\varvec{\alpha }}} d\quad \text {s.t.}\ f(\varvec{x})-d+\left(\sum _{\varvec{\alpha }}t_{\varvec{\alpha }}\varvec{x}^{\varvec{\alpha }}\right) h(\varvec{x})\ \text {is SOS}. \end{aligned}$$

(77)

This results in a standard SDP

$$\begin{aligned} \max _{d,\varvec{P}\succeq 0,t_{\varvec{\alpha }}} d\quad \text {s.t.}\ f(\varvec{x})-d+\left(\sum _{\varvec{\alpha }}t_{\varvec{\alpha }}\varvec{x}^{\varvec{\alpha }}\right) h(\varvec{x}) = [\varvec{x}]^T_{t} \varvec{P} [\varvec{x}]_{t} \end{aligned}$$

(78)

for some specific choices of t. Since $p_1 = d_1\ge d_2$, solving (78) provides a lower bound to (74). Indeed, the dual of (78) is exactly the type of convex relaxations presented in “Convex relaxation with only RDC constraints” section for optimization problems of the form (74).

Cramér–Rao lower bound

In this section, we introduce a classical tool from statistics, the Cramér–Rao bound (CRB) (Casella and Berger 2002), to give perspective on the lowest possible error any unbiased estimator can achieve when estimating coordinates from noisy RDC measurements. We first describe the CRB for general point estimators. Let $\varvec{\theta }\in {\mathbb {R}}^n$ be a multidimensional parameter which is to be estimated from measurements $\varvec{x} \in {\mathbb {R}}^{m}$. Suppose $\varvec{x}$ is generated from the distribution $p(\varvec{x}\vert \varvec{\theta }).$ The Fisher information matrix (FIM) is defined as the $n\times n$ matrix

$$\begin{aligned} \varvec{I}(\varvec{\theta }) = \mathbb {E}[(\nabla _{\varvec{\theta }} \ln p(\varvec{x} \vert \varvec{\theta })) (\nabla _{\varvec{\theta }} \ln p( \varvec{x} \vert \varvec{\theta }))^T] \end{aligned}$$

(79)

where expectation is taken with respect to the distribution $p(\varvec{x}\vert \varvec{\theta })$ and the gradient $\nabla _{\varvec{\theta }}$ is taken with respect to $\varvec{\theta }$. For any unbiased estimator $\hat{\varvec{\theta }}$ of $\varvec{\theta }$, that is $\mathbb {E}(\hat{\varvec{\theta }}) = \varvec{\theta }$, the following relationship holds:

$$\begin{aligned} \mathbb {E}[(\hat{\varvec{\theta }} - \varvec{\theta })(\hat{\varvec{\theta }} - \varvec{\theta })^T] \succeq \varvec{I}(\varvec{\theta })^{-1} \end{aligned}$$

(80)

if $\varvec{I}(\varvec{\theta })$ is invertible. Therefore the total variance of the estimator $\hat{\varvec{\theta }}$ is lower bounded by $\text {Tr}(\varvec{I}(\varvec{\theta })^{-1})$. We remark that for an unbiased estimator, its variance and the mean-squared error are the same, therefore we often use these terms interchangeably.

We also introduce the CRB in the case when $\varvec{\theta }$ and $\hat{\varvec{\theta }}$ are constrained to be in the set $\{\varvec{\theta }\vert \ f(\varvec{\theta })= 0\}$ where $f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^k$ (Stoica and Ng 1998). Let $\varvec{Df}(\varvec{\theta })\in {\mathbb {R}}^{k\times n}$ be the gradient matrix of f at $\varvec{\theta }$ with full row rank, and $\varvec{Q}\in {\mathbb {R}}^{n\times (n-k)}$ be a set of orthonormal vectors satisfying

$$\begin{aligned} \varvec{Df}(\varvec{\theta })\varvec{Q} = 0 \end{aligned}$$

(81)

i.e. $\varvec{Q}$ is an orthonormal basis of the null space of $\varvec{Df}(\varvec{\theta })$. In this case, for any unbiased estimator $\hat{\varvec{\theta }}$ satisfying $f(\hat{\varvec{\theta }}) = 0$, the CRB is then

$$\begin{aligned} \mathbb {E}[(\hat{\varvec{\theta }} - \varvec{\theta })(\hat{\varvec{\theta }} - \varvec{\theta })^T] \succeq \varvec{Q}(\varvec{Q}^T \varvec{I}(\varvec{\theta }) \varvec{Q})^{-1} \varvec{Q}^T \end{aligned}$$

(82)

if $\varvec{Q}^T \varvec{I}(\varvec{\theta }) \varvec{Q}$ is invertible.

CRB for the variance of coordinate estimator

We are now ready to investigate the CRB for estimating atomic positions from RDC data. Let $\varvec{\zeta }= [\varvec{\zeta }_1,\ldots ,\varvec{\zeta }_K]\in {\mathbb {R}}^{3\times K}$ be the coordinates of the atoms we want to estimate. We aim to derive a lower bound $\text {Tr}(\varvec{Q}(\varvec{Q}^T \varvec{I}(\varvec{\zeta }) \varvec{Q})^{-1} \varvec{Q}^T)$ of $\mathbb {E}[\text {Tr}((\hat{\varvec{\zeta }}-\varvec{\zeta })^T(\hat{\varvec{\zeta }}-\varvec{\zeta }))]$ for any unbiased estimator $\hat{\varvec{\zeta }}$ of $\varvec{\zeta }$, and compare

$$\begin{aligned} \sqrt{\frac{\text {Tr}(\varvec{Q}(\varvec{Q}^T \varvec{I}(\varvec{\zeta }) \varvec{Q})^{-1} \varvec{Q}^T)}{K}} \end{aligned}$$

(83)

with the RMSD of the solutions from RDC-SOS and RDC–NOE-SOS in Fig. 3.

We assume that the RDC measurements are generated through the noise model in (64). This noise model is used to get an expression for $\varvec{I}(\varvec{\theta })$. There are several sets of equality constraints that need to be considered when deriving $\varvec{Q}$. We assume that within each rigid unit, the distance between any pair of atoms is fixed. We therefore have a set of equality constraints

$$\begin{aligned} d_{nm}^2 = \Vert \varvec{\zeta }_n - \varvec{\zeta }_m\Vert _2^2,\quad (n,m)\in E_\text {fixed} \end{aligned}$$

(84)

where $E_\text {fixed}$ consists of all atom pairs within each and every rigid unit. Without loss of generality, we also consider the constraint

$$\begin{aligned} \varvec{\zeta }\varvec{1} = 0 \end{aligned}$$

(85)

which implies the points $\varvec{\zeta }_1,\ldots ,\varvec{\zeta }_K$ are centered at zero. This is due to the fact that

$$\begin{aligned} \text {Tr}((\hat{\varvec{\zeta }}& - \varvec{\zeta })^T(\hat{\varvec{\zeta }} - \varvec{\zeta })) \nonumber \\= & {} \text {Tr}((\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c - t \varvec{1}^T)^T(\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c - t \varvec{1}^T))\nonumber \\= & {} \text {Tr}((\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c)^T(\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c)) + (1/K)\Vert t\Vert _2^2\nonumber \\& -2 \text {Tr}((\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c)^T t \varvec{1}^T )\nonumber \\= & {} \text {Tr}((\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c)^T(\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c)) + (1/K)\Vert t\Vert _2^2\nonumber \\\ge & {} \text {Tr}((\hat{\varvec{\zeta }}_c -\varvec{\zeta }_c)^T(\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c)) \end{aligned}$$

(86)

where $\varvec{\zeta }_c$ and $\hat{\varvec{\zeta }}_c$ denote the zero centered coordinates and coordinate estimators, and t is the relative translation between $\varvec{\zeta }$ and $\hat{\varvec{\zeta }}$. Equation (86) implies that deriving a lower bound for $\mathbb {E}[\text {Tr}((\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c)^T(\hat{\varvec{\zeta }}_c - \varvec{\zeta }_c))]$ is sufficient for obtaining a lower bound for $\mathbb {E}[\text {Tr}((\hat{\varvec{\zeta }}-\varvec{\zeta })^T(\hat{\varvec{\zeta }}-\varvec{\zeta }))]$. When there are atoms that are constrained to lie on the same plane, we need to add the constraint that any three vectors in the plane span a space with zero volume, i.e.

$$\begin{aligned} \det ([\varvec{\zeta }_i-\varvec{\zeta }_j,\varvec{\zeta }_k-\varvec{\zeta }_l,\varvec{\zeta }_m-\varvec{\zeta }_n]) = 0 \end{aligned}$$

(87)

for atoms i, j, k, l, m, n in the same plane.

We first start with deriving an expression for the Fisher information matrix when RDC data are generated through (64). From (64) and (65), the likelihood function for the coordinates is

$$\begin{aligned}&p(\{r_{nm}\}_{(n,m)\in E_\text {RDC}} \vert \varvec{\zeta }_1,\ldots ,\varvec{\zeta }_K) =\nonumber \\&\underset{\begin{array}{c} (n,m)\in E_\text {RDC}\\ j=1,2 \end{array}}{{{{\Pi }}}} \frac{1}{\sqrt{2\pi \sigma ^2}}\nonumber \\&\exp \bigg (-\frac{\left( (\varvec{\zeta }_n-\varvec{\zeta }_m)^T \varvec{S}^{(j)}(\varvec{\zeta }_n-\varvec{\zeta }_m)-r_{nm}^{(j)} d_{nm}^2\right) ^2}{2 d_{nm}^4 \sigma ^2}\bigg ) \end{aligned}$$

(88)

and the log-likelihood is (up to an additive constant)

$$\begin{aligned}&l(\{r_{nm}\}_{(n,m)\in E_\text {RDC}} \vert \varvec{\zeta }_1,\ldots ,\varvec{\zeta }_K)\nonumber \\&:=\ln p(\{r_{nm}\}_{(n,m)\in E_\text {RDC}} \vert \varvec{\zeta }_1,\ldots ,\varvec{\zeta }_K) \nonumber \\&= \sum _{\begin{array}{c} (n,m)\in E_\text {RDC}\\ j=1,2 \end{array}} \frac{-((\varvec{\zeta }_n-\varvec{\zeta }_m)^T \varvec{S}^{(j)}(\varvec{\zeta }_n-\varvec{\zeta }_m) - r_{nm}^{(j)} d_{nm}^2)^2}{2 d_{nm}^4 \sigma ^2}\nonumber \\&= -\sum _{\begin{array}{c} (n,m)\in E_\text {RDC}\\ j=1,2 \end{array}} \frac{(\varvec{e}_{nm}^T\varvec{\zeta }^T \varvec{S}^{(j)}\varvec{\zeta }\varvec{e}_{nm} - r_{nm}^{(j)} d_{nm}^2)^2}{2 d_{nm}^4 \sigma ^2} \end{aligned}$$

(89)

where $\varvec{e}_{nm} = \varvec{e}_n - \varvec{e}_m$. The derivative of l with respect to $\text {vec}(\varvec{\zeta })$ is then

$$\begin{aligned}&\nabla _{\text {vec}(\varvec{\zeta })} l=\nonumber \\&-\sum _{\begin{array}{c} (n,m)\in E_\text {RDC}\\ j=1,2 \end{array}} \frac{2(\varvec{e}_{nm}^T\varvec{\zeta }^T \varvec{S}^{(j)}\varvec{\zeta }\varvec{e}_{nm} - r_{nm}^{(j)} d_{nm}^2)}{d_{nm}^4 \sigma ^2}\nonumber \\&\qquad \qquad \qquad \times(\varvec{e}_{nm}\varvec{e}_{nm}^T\otimes \varvec{S}^{(j)})\text {vec}(\varvec{\zeta }). \end{aligned}$$

(90)

It follows from the noise model (64) and the independence of $\varvec{\epsilon }_{nm}^{(j)}$’s that the Fisher information matrix

$$\begin{aligned}&\varvec{I}(\varvec{\zeta })=\mathbb {E}((\nabla _{\text {vec}(\varvec{\zeta })} l) (\nabla _{\text {vec}(\varvec{\zeta })} l)^T) = \nonumber \\&4\sum _{\begin{array}{c} (n,m)\in E_\text {RDC}\\ j=1,2 \end{array}}\nonumber \\&\qquad \qquad \qquad \frac{(\varvec{e}_{nm}\varvec{e}_{nm}^T\otimes \varvec{S}^{(j)})\text {vec}(\varvec{\zeta })\text {vec}(\varvec{\zeta })^T (\varvec{e}_{nm}\varvec{e}_{nm}^T\otimes \varvec{S}^{(j)})}{\sigma ^2 d^4_{nm}} \end{aligned}$$

(91)

Having the Fisher information matrix, we now incorporate the constraints in (84) and (85) in order to obtain a bound as in (82). Stacking the equality constraints (84) into a $\vert E_\text {fixed}\vert \times 1$ matrix, we get

$$\begin{aligned} f(\text {vec}(\varvec{\zeta })) := \begin{bmatrix} \varvec{e}_{n m}^T\varvec{\zeta }^T \varvec{\zeta }\varvec{e}_{nm} - d_{n m}^2 \end{bmatrix}_{(n ,m )\in E_\text {fixed}} = 0 \end{aligned}$$

(92)

The gradient matrix is thus

$$\begin{aligned} \varvec{Df}(\text {vec}(\varvec{\zeta })) = \text {vec}(\varvec{\zeta })^T \begin{bmatrix}(\varvec{e}_{nm} \varvec{e}_{nm}^T \otimes \varvec{I}_3)\end{bmatrix}_ {(n ,m )\in E_\text {fixed}} \end{aligned}$$

(93)

where $\varvec{Df}(\text {vec}(\varvec{\zeta })) \in {\mathbb {R}}^{\vert E_\text {fixed} \vert \times 3K}$. We note that $\varvec{Df}(\text {vec}(\varvec{\zeta }))$ is known as the rigidity matrix (Jackson 2007), and the vectors in its null space indicate the direction of infinitesimal motion the atoms can take without violating (84). Even in the case when all pairwise distances between the atoms are known, there is still a 6-dimensional null space for $\varvec{Df}(\text {vec}(\varvec{\zeta }))$, corresponding to an infinitesimal global rotation and translation to the coordinates $\varvec{\zeta }$ that preserves all pairwise distances. We now augment $f(\text {vec}(\varvec{\zeta }))=0$ with the centering constraint $\varvec{\zeta }\varvec{1} = 0$, and this augments $\varvec{Df}(\text {vec}(\varvec{\zeta }))$ with three rows $\varvec{1}^T \otimes \varvec{I}_3$, i.e.

$$\begin{aligned} \varvec{Df}(\text {vec}(\varvec{\zeta })) = \begin{bmatrix}\text {vec}(\varvec{\zeta })^T [(\varvec{e}_{nm} \varvec{e}_{nm}^T \otimes \varvec{I}_3)]_ {(n ,m )\in E_\text {fixed}}\\ \varvec{1}^T \otimes \varvec{I}_3\end{bmatrix} \end{aligned}$$

(94)

The inclusion of such centering constraint eliminates the three dimensional subspace in the kernel of the rigidity matrix that corresponds to the translational degree of freedom. Let $\varvec{Q}$ be an orthonormal basis that spans the null space of $\varvec{Df}(\text {vec}(\varvec{\zeta }))$. Together with (91) and (82) we obtain the desired CRB. We omit detailing the derivative for constraint (87) but simply note that the inclusion of such constraints eliminates the out of plane infinitesimal motion for atoms lying on rigid planar unit.

Inclusion of NOE constraints

We have so far neglected the use of NOE measurements when deriving the CRB. Unlike RDC, the NOE restraints remain more qualitative, with imprecise upper and lower bound (Bonvin et al. 1996) due to the $r^{-6}$ scaling of the interaction. In protein structural calculation, it is customary to include a flat potential well-like penalty [(e.g. (12)] in addition to the RDC log-likelihood function derived from RDC, or treat the backbone NOE as inequality constraints on the distances. In any of these cases, when the coordinates $\varvec{\zeta }$ strictly satisfy both upper and lower bounds on the distances, the CRB is exactly the same as the CRB derived in “CRB for the variance of coordinate estimator” section (Gorman and Hero 1990) since the CRB only depends on the local curvature of the log-likelihood function around $\varvec{\zeta }.$ Therefore when the noise on RDC is large and the NOE restraints are active in determining a coordinate estimator $\varvec{\zeta }$, the CRB may no longer serve as a lower bound for the mean squared error of $\hat{\varvec{\zeta }}$. In particular, it is possible for $\hat{\varvec{\zeta }}$ to have a mean squared error lower than the CRB due to the bias introduced by the NOE (by favoring solutions that satisfies the distance bounds), as observed in Fig. 3. A fundamental results in statistical estimation theory-the bias-variance trade-off (Wasserman 2013), states that the mean squared error of an estimator can be obtained from the summation of the variance and squared bias of the estimator. It is possible that with the expense of having some bias, the variance of an estimator can be greatly reduced, resulting a mean squared error that is lower than the CRB (Wasserman 2013, Chapter 7).

Observed Fisher information matrix and protein variability

We remark that since the LHS of (80) (or (82)) is the covariance matrix for the estimator $\hat{\varvec{\theta }}$, the leading eigenvectors of ${\varvec{I}(\varvec{\theta })}^{-1}$ give the direction of the greatest variations of the protein based on the observed data, whereas the corresponding eigenvalues give the variance (amplitude) of variations. When deriving the CRB, we use the FIM (91) which is obtained from averaging over the distribution of the data. An estimator $\widehat{\varvec{I}(\varvec{\theta })}$ of the FIM can be obtained from the observed data, by replacing $\varvec{\theta }$ in FIM by its maximum-likelihood estimator $\hat{\varvec{\theta }}$ and plugging in the observed data (in our case the observed RDC $r_{nm}$) instead of taking expectation over the distribution of the data. The direction for which an estimator $\hat{\varvec{\theta }}$ has the greatest variance can be estimated by the top eigenvector of $\widehat{\varvec{I}(\varvec{\theta })}^{-1}$. In the constrained case, we compute the top eigenvector of $\hat{\varvec{Q}} (\hat{\varvec{Q}}^T \widehat{\varvec{I}(\varvec{\theta })}^{-1} \hat{\varvec{Q}}) \hat{\varvec{Q}}^T$ where $\hat{\varvec{Q}}$ are computed based on $\hat{\varvec{\theta }}$ instead of $\varvec{\theta }$. In Fig. 5a, we demonstrate the variation of the ubiquitin fragment for residue 1–7 (with 159 atoms) using the eigenvector of estimated FIM. In Fig. 5b we show the largest 10 eigenvalues of the inverse of the estimated FIM. As we see, there is one prominent mode of variation for this protein fragment. We note that this procedure of determining the modes of protein variation bear resemblance to normal mode analysis (Case 1994). In such analysis, the Hessian for the pseudo energy function of a protein near a local minimum is first determined. Then the normal modes are determined by the eigenvectors of the Hessian matrix. If we treat the log-likehood function as some pseudo energy function, our FIM-based analysis of the modes of atomic displacement corresponds to the classical normal modes analysis.

Fig. 5

a Variation of the structure of a ubiquitin fragment (residue 1–7). We compare the RDC–NOE-SOS solution $\hat{\varvec{\zeta }}$ (solid) with two other structures (dashed) obtained from adding the top eigenvector of $\hat{\varvec{Q}} (\hat{\varvec{Q}}^T \widehat{\varvec{I}(\varvec{\theta })}^{-1} \hat{\varvec{Q}}) \hat{\varvec{Q}}^T$ multiplied by small scalars to $\hat{\varvec{\zeta }}$. b The largest 10 eigenvalues of the inverse of FIM.

Infinitesimal rigidity and invertibility of the Fisher information matrix

In this subsection, we study the infinitesimal rigidity (Liberti et al. 2014) of the protein structure given RDC and distance measurements and how it guarantees invertibility of the Fisher information matrix. Let a framework with coordinates $\varvec{\zeta }\in {\mathbb {R}}^{3\times K}$ be constrained by

$$\begin{aligned} (\varvec{\zeta }_n - \varvec{\zeta }_m)^T (\varvec{\zeta }_n - \varvec{\zeta }_m) = d_{nm}^2,\quad (n,m)\in E_\text {fixed}, \end{aligned}$$

(95)

and

$$\begin{aligned}&(\varvec{\zeta }_n - \varvec{\zeta }_m)^T \varvec{S}^{(j)}(\varvec{\zeta }_n - \varvec{\zeta }_m) = r^{(j)}_{nm},\nonumber \\&j=1,\ldots ,N,\ (n,m)\in E_\text {RDC}. \end{aligned}$$

(96)

In order to derive a condition for infinitesimal rigidity, we first let $\text {vec}(\varvec{\zeta }(s))$ be a curve in dimension ${\mathbb {R}}^{3K}$ parameterized by s, where $\varvec{\zeta }(0)$ satisfies (95) and (96). Taking derivative of the constraints in (95) and (96) with respect to s at $s=0$, we have

$$\begin{aligned}&\left[ \begin{array}{ll} &{} \text {vec}{{\left( \varvec{\zeta } \left( 0 \right) \right) }^{T}}{{[{\varvec{e}_{nm}}{\varvec{e}_{nm}}^{T}\otimes {\varvec{I}_{3}}]}_{(n,m)}}\in {{E}_\text{fixed}} \\ &{} \text {vec}{{\left( \varvec{\zeta }\left( 0 \right) \right) }^{T}}{{[{\varvec{e}_{nm}}{\varvec{e}_{nm}}^{T}\otimes {\varvec{S}^{(j)}}]}_{(n,m)}}\in {{E}_\text{RDC}},j\in [1,N] \\ \end{array} \right] \frac{d}{ds}\text {vec}(\varvec{\zeta } (0)) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\varvec{R}(\varvec{\zeta }(0))\frac{d}{ds}\text {vec}(\varvec{\zeta }(0))=0. \end{aligned}$$

(97)

The null space of the generalized rigidity matrix $\varvec{R}(\varvec{\zeta }(0))$ with dimension $(\vert E_\text {fixed} \vert + \vert E_\text {RDC} \vert )\times 3K$ represents the direction of infinitesimal motion such that $\varvec{\zeta }(s)$ satisfies the constraints (95), (96) for infinitesimally small s. If $\varvec{R}(\varvec{\zeta }(0))$ only has a three dimensional nullspace, i.e. the global translations in x, y, z-directions, we say the framework $\varvec{\zeta }(0)$ along with the constraints (95) and (96) is infinitesimally rigid.

Now we verify that the constrained Fisher information matrix is invertible if $\varvec{R}(\varvec{\zeta }(0))$ has a three dimensional null space corresponds to global translation of the points. We define $\text {ker}(\varvec{A})$ to be the kernel of a matrix $\varvec{A}$ and $\text {range}(\varvec{A})$ to be the column space of $\varvec{A}$. Let $\varvec{Q}$ again be the basis of the nullspace of $\varvec{Df}(\text {vec}(\varvec{\zeta }))$ defined in (94) such that $\varvec{Df}(\text {vec}(\varvec{\zeta }))\varvec{Q} = 0$. Let $\varvec{v}$ satisfies $\begin{aligned} \varvec{Q}^T\varvec{I}(\varvec{\zeta }) \varvec{Q} \varvec{v} =0. \end{aligned}$ $\varvec{Q}^T \varvec{I}(\varvec{\zeta }) \varvec{Q} \varvec{v} = 0$ if and only if $\varvec{v}\in \text {ker}(\varvec{Q})$ or $\varvec{Q}\varvec{v}\in \text {ker}(\varvec{I})$. Since the columns of $\varvec{Q}$ are linearly independent, $\varvec{Q}\varvec{v}\ne 0$ unless $\varvec{v}=0$. This means $\varvec{Q}^T \varvec{I}(\varvec{\zeta }) \varvec{Q} \varvec{v} = 0$ if and only if $\varvec{v} = 0$ or $\varvec{Q}\varvec{v} \in \text {ker}(\varvec{I})\cap \text {range}(\varvec{Q}) = \text {ker}(\varvec{I})\cap \text {range}(\varvec{Q}) = \text {ker}(\varvec{I})\cap \text {ker}(\varvec{Df}(\text {vec}(\varvec{\zeta }))).$ Therefore if

$$\begin{aligned} \text {ker}(\varvec{I})\cap \text {ker}(\varvec{Df}(\text {vec}(\varvec{\zeta }))) = \emptyset , \end{aligned}$$

or in other words

$$\begin{aligned} \text {range}(\varvec{I}) \cup \text {range}(\varvec{Df}(\text {vec}(\varvec{\zeta })))={\mathbb {R}}^{3K} \end{aligned}$$

(98)

then $\varvec{Q}^T \varvec{I}(\varvec{\zeta }) \varvec{Q}$ is invertible. From the form of the (91), it is easy to show that the range condition (98) is satisfied if and only if the range of

$$\begin{aligned} \left[ \begin{array}{ll} &{} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\varvec{1}^{T}}\otimes {{\varvec{I}}_{3}} \\ &{} \,\,\,\,\,\,\,\,\,\text {vec}{{\left( \zeta \left( 0 \right) \right) }^{T}}{{[{{e}_{nm}}{{e}_{nm}}^{T}\otimes {\varvec{I}_{3}}]}_{(n,m)}}\in {{E}_\text{fixed}} \\ &{} \text {vec}{{\left( \varvec{\zeta } \left( 0 \right) \right) }^{T}}{{[{\varvec{e}_{nm}}{\varvec{e}_{nm}}^{T}\otimes {\varvec{S}^{(j)}}]}_{(n,m)}}\in {{E}_\text{RDC}},j\in [1,N] \\ \end{array} \right] =\left[ \begin{array}{ll} &{} {\varvec{1}^{T}}\otimes {\varvec{I}_{3}} \\ &{} \varvec{R}\left( \varvec{\zeta } \left( 0 \right) \right) \\ \end{array} \right] \end{aligned}$$

(99)

is ${\mathbb {R}}^{3K}$. Then we arrive at the conclusion that if the framework $\varvec{\varvec{\zeta }}$ is infinitesimally rigid with the null space of $\varvec{R}(\varvec{\zeta })$ being the global translations, the constrained Fisher information matrix defined as $\varvec{Q}^T \varvec{I}(\varvec{\zeta }) \varvec{Q}$ is invertible.

In Yershova et al. (2011), it is shown that if there exists RDC measurements for a bond in the peptide plane and a bond in the CA-body in a single alignment media, the solutions of the protein structure form a discrete set. Therefore under this condition, there is no infinitesimal motion other than global translation such that the protein framework satisfies the RDC and NOE constraints. We can thus compute the CRB safely under such condition.