De novo determination of internuclear vector orientations from residual dipolar couplings measured in three independent alignment media

Abstract

The straightforward interpretation of solution state residual dipolar couplings (RDCs) in terms of internuclear vector orientations generally requires prior knowledge of the alignment tensor, which in turn is normally estimated using a structural model. We have developed a protocol which allows the requirement for prior structural knowledge to be dispensed with as long as RDC measurements can be made in three independent alignment media. This approach, called Rigid Structure from Dipolar Couplings (RSDC), allows vector orientations and alignment tensors to be determined de novo from just three independent sets of RDCs. It is shown that complications arising from the existence of multiple solutions can be overcome by careful consideration of alignment tensor magnitudes in addition to the agreement between measured and calculated RDCs. Extensive simulations as well applications to the proteins ubiquitin and Staphylococcal protein GB1 demonstrate that this method can provide robust determinations of alignment tensors and amide N–H bond orientations often with better than 10° accuracy, even in the presence of modest levels of internal dynamics.

Appendix

Separation of the Q value into components arising from structural quality and noise

The Q value is used to assess the level of agreement between a structural model and a single RDC dataset. It can be written as follows:

$$ Q= \frac{\left\| {\vec{{\mathbf{d}}}-{\bf BB}^{+}\;\vec{{\mathbf{d}}}} \right\|}{\left\| {\vec{{\mathbf{d}}}} \right\|} =\sqrt{\frac{\sum\limits_i {\left( {d_{i,meas} -d_{i,calc} } \right)^{2}} }{\sum\limits_i {d_{i,meas}^2 } }} $$

(A1)

where ${\vert}{\vert}\quad {\vert}{\vert}$ denotes the norm and $\vec{{\mathbf{d}}}$ is a column vector consisting of the RDC measurements. The matrix B is of dimension N  ×  5 where N is the number of dipolar interactions for which RDC measurements have been made and the matrix B ⁺ is its Moore-Penrose pseudoinverse. Each row of the matrix B contains the irreducible tensorial description of the specific dipolar interaction tensor. Contributions to a computed Q value can arise from errors in the measured RDCs themselves or structural and dynamic deviations from the coordinates embodied in B. To distinguish, we write the set of measured couplings $\vec{{\mathbf{d}}}=\vec{{\mathbf{d}}}^{\prime}+\varvec{\varepsilon},$ in which $\vec{{\mathbf{d}}}^{{\prime}}$ is the set of true couplings and $\varvec{\varepsilon}$ is a vector containing the experimental errors. Substitution into Eq. A1 leads to,

$$ Q=\frac{\left\|\left({\mathbf{I}}-{\mathbf{BB}}^{+} \right)\;\vec{{\mathbf{d}}}+\left({\mathbf{I}}-{\mathbf{BB}}^{+} \right)\;\varvec{\varepsilon}\right\|}{\left\| {\vec{{\mathbf{d}}}} \right\|} $$

(A2)

in which I is the identity matrix. Note that if there are no experimental errors, $\varvec{\varepsilon}={\mathbf{0}},$ then the Q value depends only on the first term in the numerator and is solely an assessment of structural quality. On the other hand if the structural model B is perfect then only the second term will be non-zero and it will be solely related to the magnitude of experimental errors. From Eq. A2, one can arrive at the following relationship under the assumption that experimental errors are uncorrelated with the structural model B,

$$ Q^{2}=Q_{struct}^2 +Q_{noise}^2 $$

(A3)

with

$$ Q_{struct}=\frac{\left\| {\left({\mathbf{I}}-{\mathbf{BB}}^{+}\right)\;\vec{{\mathbf{d}}}} \right\|}{\left\|{\vec{{\mathbf{d}}}}\right\|},\quad \quad Q_{noise} =\frac{\left\|\left({\mathbf{I}}-{\mathbf{BB}}^{+} \right)\;\varvec{\varepsilon}\right\|}{\left\|\vec{{\mathbf{d}}} \right\|} $$

(A4)

It is the value of Q_struct that is normally desired and thus it would be useful if Q_noise could be estimated. We start by writing the error vector $\varvec{\varepsilon}$ in terms of a normalized vector $\varvec{\varepsilon}_{0}$ and the estimated random error specified by σ _D . Given a normalized N-dimensional vector, its elements form a distribution with $\sigma=1/\hbox{sqrt}(\hbox{N}).$ This leads to the following expression for $\varvec{\varepsilon}.$

$$ \varvec{\varepsilon}=\sqrt{N} \sigma _D \varvec{\varepsilon}_{0} $$

(A5)

Considering that B is rank 5 and that ${\mathbf{BB}}^{+}$ represents an orthogonal projector (Albert 1972) which projects an N dimensional vector onto a 5 dimensional subspace, the following relationships can be derived,

$$ {\mathbf{BB}}^{+}\varvec{\varepsilon}_{0}\;=\;\sqrt{\frac{5}{N}} \varvec{\varepsilon}_{0}^{\prime },\quad \quad \left({{\mathbf{I}}-{\mathbf{BB}}^{+}}\right)\varvec{\varepsilon}_{0} \;=\;\sqrt{\frac{N-5}{N}}\varvec{\varepsilon}_{0}^{\prime \prime} $$

(A6)

given that $\varvec{\varepsilon}_{0}^\prime$ and $\varvec{\varepsilon}_{0}^{\prime\prime}$ are both normalized N-dimensional vectors. This leads to the desired expression for Q _noise .

$$ Q_{noise}=\frac{\left\|{\left({{\mathbf{I}}-{\mathbf{BB}}^{+}} \right)\;\varvec{\varepsilon}} \right\|}{\left\| {\vec{{\mathbf{d}}}} \right\|}\;=\frac{\sqrt{N-5}\sigma _D \left\| {\varvec{\varepsilon}_{0}^{\prime \prime }} \right\|}{\left\| {\vec{{\mathbf{d}}}} \right\|} =\sqrt{\frac{N-5}{N}}\frac{\sigma _D }{rms\left( {\vec{{\mathbf{d}}}} \right)} $$

(A7)

Errors in estimation of alignment tensor magnitudes based on observed d_min and d_max

Recalling the expression for the estimated generalized degree of order (GDO) from the observed values of d_min and d_max,

$$ \varphi_{est}=\frac{1}{\kappa}\sqrt{\frac{4}{3}\left({d_{\rm max}^2 +d_{\rm min}^2 +d_{\rm min} d_{\rm max} } \right)} $$

(A8)

we note that in the absence of experimental errors, φ _est represents an absolute lower bound for the actual value of φ. In the presence of experimental errors, the lower bound, φ _lower , will be reduced below that of φ _est according to the propagated uncertainty in φ from the measurements d_min and d_max. The expression for σ_φ is obtained by evaluation of,

$$ \sigma_\varphi^2 =\left({\frac{\partial \varphi }{\partial d_{\rm max} }\sigma _D}\right)^{2}+\left({\frac{\partial\varphi}{\partial d_{\rm min}}\sigma_D}\right)^{2} $$

(A9)

under the assumption of axial symmetry (η = 0), which produces the maximum propagation of error into φ. Finally, one obtains the desired expression for σ_φ,

$$ \sigma_\varphi=\frac{1}{\kappa}\sigma _D $$

(A10)

Recalling the expression for φ _est in Eq. A8, this allows a lower bound for φ to be established as follows,

$$ \varphi_{lower}=\varphi_{est}-2\sigma_\varphi=\frac{1}{\kappa }\left[{\sqrt{\frac{4}{3}\left({d_{\rm max}^2+d_{\rm min}^2+d_{\rm min}d_{\rm max}}\right)}-2\sigma_D}\right] $$

(A11)

Establishing an upper bound requires an additional piece of information. Namely, the upper limit on the extent to which φ _est underestimates the actual value of φ due to noncoincidence of internuclear vectors with the Z and Y principal axes of alignment corresponding to A_zz and A_yy. To do this a uniform distribution of internuclear vector orientations will be assumed. Under this assumption, the extent of solid angle on the unit sphere occupied by one of a set of N internuclear vectors is equal to 4π/N and the semiangle for a cone spanning that solid angle can be described by the angle λ, which satisfies the following equation,

$$ \frac{1}{4\pi}\int\limits_0^{2\pi}{d\phi \int\limits_0^\lambda{\sin \theta d\theta }}=\frac{1}{N} $$

(A12)

This leads to the following result for λ,

$$ \lambda=\arccos\left({1-\frac{2}{N}}\right) $$

(A13)

Thus one can say that each internuclear vector inhabits its own cone on the surface of the unit sphere with a semi-angle given by λ. While it is not geometrically possible to cut a sphere up into perfect cones, the deviation from this simplified picture is expected to be very small. For a uniform distribution of vectors, each vector can thus be considered to lie at the center of its respective cone and choice of a random vector on the sphere cannot deviate from one of the preexisting N vectors by more than the angle λ. Within this framework, the maximum possible underestimation of A_zz and A_yy occurs for vectors which have spherical coordinates (λ, 90) and (90-λ, 90), respectively, relative to the true principal axes,

$$ A_{zz,est} \left( {\max } \right)=A_{zz} \left[ {1-\frac{6+2\eta }{N}+\frac{6+2\eta }{N^{2}}} \right];\quad \quad A_{yy,est} \left( {\max } \right)=A_{zz} \left[ {-\frac{1}{2}\left( {1+\eta } \right)+\frac{6+2\eta }{N}-\frac{6+2\eta }{N^{2}}} \right] $$

(A14)

From the above expressions, it is apparent that the largest possible underestimation occurs for cases of highest asymmetry (η = 1). As estimation of the asymmetry is subject to greater uncertainty than for A_zz, we derive an expression for the maximum possible underestimation in the GDO for the case of η = 1,

$$ \begin{aligned} \varphi_{est}\left(\hbox{max}\right)&=\sqrt{\frac{4}{3}\left( {A_{zz,est}^2 \left( {\max } \right)+A_{yy,est}^2 \left( {\max } \right)+A_{zz,est} \left( {\max } \right)A_{yy,est} \left( {\max } \right)} \right)} \\ &=\sqrt{\frac{4}{3}}A_{zz} \left( {1-\frac{8}{N}+\frac{8}{N^{2}}} \right)=\varphi \left( {1-\frac{8}{N}+\frac{8}{N^{2}}} \right)\\ \end{aligned} $$

(A15)

This leads to the following expression for the maximum difference between the estimated and true values of the GDO assuming a uniform distribution of internuclear vectors and the absence of dynamic averaging,

$$ \delta\varphi_{\rm max}=\varphi-\varphi_{est}(\hbox{max})=\varphi \left({\frac{8}{N}-\frac{8}{N^{2}}} \right) $$

(A16)

An estimate for the upper bound in the magnitude of alignment can then be obtained after some algebraic simplification utilizing results shown in Eqs. A10, A15, and A16,

$$ \varphi_{upper}=\varphi_{est}+\delta\varphi_{\rm max}+2\sigma_\varphi=\frac{1}{\kappa }\left[{\left( {\frac{1}{1-\frac{8}{N}+\frac{8}{N^{2}}}} \right)\sqrt{\frac{4}{3}\left( {d_{\rm max}^2+d_{\rm min}^2+d_{\rm min}d_{\rm max}}\right)}+2\sigma_D}\right] $$

(A17)