Automated error-tolerant macromolecular structure determination from multidimensional nuclear Overhauser enhancement spectra and chemical shift assignments: improved robustness and performance of the PASD algorithm

Abstract

We report substantial improvements to the previously introduced automated NOE assignment and structure determination protocol known as PASD (Kuszewski et al. (2004) J Am Chem Soc 26:6258–6273). The improved protocol includes extensive analysis of input spectral data to create a low-resolution contact map of residues expected to be close in space. This map is used to obtain reasonable initial guesses of NOE assignment likelihoods which are refined during subsequent structure calculations. Information in the contact map about which residues are predicted to not be close in space is applied via conservative repulsive distance restraints which are used in early phases of the structure calculations. In comparison with the previous protocol, the new protocol requires significantly less computation time. We show results of running the new PASD protocol on six proteins and demonstrate that useful assignment and structural information is extracted on proteins of more than 220 residues. We show that useful assignment information can be obtained even in the case in which a unique structure cannot be determined.

Appendix: Domain determination using a maximum likelihood fitting procedure

In order to fit subregions of structures which do not have high overall similarity, we have implemented a version of the maximum likelihood (ML) algorithm developed by Theobald and Wuttke (2006a) with a minor simplifying alteration which yields slightly improved results. In short, we have implemented the algorithm outlined in the supplementary material of Theobald and Wuttke (2006b) in which the following quantity is maximized:

$$ -\frac{1}{2}\sum_i^N || ({\mathbf{X}}_i+{\mathbf{1}}_K{\mathbf{t}}_i^T){\mathbf{R}}_i - {\mathbf{M}}||_{{\varvec{\Upsigma}}^{-1}}^2 - \frac{3N}{2}\ln |{\varvec{\Upsigma}}|, $$

(17)

where $|{\mathbf{U}}|$ denotes the determinant of matrix U and $||{\mathbf{A}}||_{\mathbf B} = \hbox{Tr } {\mathbf{A}}^T{\mathbf{BA}}.$ ${\mathbf{X}}_i$ is a K × 3 matrix of coordinates of the input structures, ${\mathbf{1}}_K$ is a K-dimensional vector with all elements set to one, ${\mathbf{R}}_i$ and ${\mathbf{t}}_i$ are, respectively, the rotation matrix and translation vector determined in the fitting process, while M corresponds to the average coordinates

$$ {\mathbf M} = \frac{1}{N} \sum_i^N {\mathbf X}_i{\mathbf R}_i. $$

(18)

${\varvec{\Upsigma}}$ is the K × K coordinate covariance matrix whose inverse weights the fit of the coordinates such that coordinates with larger variances do not contribute as much to the fit. If ${\varvec{\Upsigma}}$ is set to the identity matrix, Eq. 17 reduces to standard least squares coordinate fitting. Coordinate precision can be expressed in terms of ${\varvec{\Upsigma}}$ in a form analogous to the standard least squares RMSD:

$$ \hbox{RMSD}_{ML} = \sqrt{\frac{K}{Tr {\varvec{\Upsigma}}^{-1}}}. $$

(19)

Maximizing Eq. 17 balances two objectives: making structures as similar as possible to the mean, while minimizing the structure spread. The maximum likelihood estimate for the covariance matrix is

$$ {\varvec{\Upsigma}} = \frac{1}{3N} \sum_i^N [({\mathbf{X}}_i+{\mathbf{1}}_K{\mathbf{t}}_i^T){\mathbf{R}}_i-{\mathbf{M}}][ ({\mathbf{X}}_i+{\mathbf{1}}_K{\mathbf{t}}_i^T){\mathbf{R}}_i-{\mathbf{M}}]^ T $$

(20)

where the sum is over all structures to fit. Expressions for the ML estimates of ${\mathbf{R}}_i$ and the associated coordinate translation ${\mathbf{t}}_i$ can be found in Theobald and Wuttke (2006b). ML coordinate fitting is an iterative process since each structure’s translation and rotation depend on ${\varvec{\Upsigma}}$, which in turn depends on the translation and rotation. However, convergence typically occurs fairly rapidly (in fewer than 30 iterations).

Now, strictly speaking, ${\varvec{\Upsigma}}$ cannot be inverted because it always has zero eigenvalues due in part to invariance of overall translation and rotation. In Theobald and Wuttke (2006b) trial values of ${\varvec{\Upsigma}}$ are perturbed such that the eigenvalues obey an inverse gamma distribution. However, as the off-diagonal covariances are fairly meaningless (and hence not considered in their default algorithm), they resort to approximating the eigenvalues as the diagonal atomic variances. We find the whole procedure cumbersome and unwarranted, since the diagonal elements are poor estimates of the true eigenvalues. Instead we simply perturb ${\varvec{\Upsigma}}$ with a small value:

$$ {\varvec{\Upsigma}} \rightarrow {\varvec{\Upsigma}} + \varepsilon {\mathbb 1} $$

(21)

where ${\mathbb 1}$ is a K × K unit matrix and ɛ is a small value (typically 10⁻⁴). For multiple systems we find that this procedure works slightly better (converges in fewer iterations) and gives nearly identical fits to the method of Theobald and Wuttke (2006b).

In our iterative domain determination method we take the 50 structures of the second PASD structure calculation, fit them using this modified fitting procedure, and collect those atoms with a fit positional RMSD threshold less than ρ_thresh. We consider residues to be contiguous if their primary sequence difference is less than D _min, the number of residues in the smallest domain considered. If RMSD _ML  < 1.5 Å we consider the selected atoms to be in a single domain. Otherwise, we repeat the procedure, considering only this subset of atoms, and we decrement ρ_thresh by Δρ_thresh. This process is repeated until the first domain is determined. Successive domains are determined by repeating the procedure, excluding the atoms in the previously determined domains. We use the parameters ρ_thresh = 3.5 Å, Δρ_thresh = 0.5 Å (decremented every other iteration), and D _min = 20 residues. For the ThTP domain determination it should be noted that the domain identification was found to be fairly insensitive to the RMSD threshold value. A script implementing this domain determination algorithm is now distributed with the Xplor-NIH package.

Glossary of terms and symbols

Active assignment

An NOE assignment which contributes to the linear (Pass 1) or quadratic (Pass 2) restraint terms. Whether an assignment is active or inactive is determined from its assignment likelihoods via the procedure described in Section “Determination of active peak assignments”.

Active peak

An NOE peak with one or more active assignments.

Assignment likelihood λ(i,j)

The probability of the correctness of assignment j of peak i. λ _p is the previous likelihood of an assignment based on previously obtained information; in Pass 1 λ _p is denoted $\lambda_p^n$ and is based on the network contact map, while in Pass 2 previous likelihoods $\lambda_p^v$ are based on distance violations of the structures calculated in Pass 1. The violation likelihood λ _v is the probability of correctness of an assignment based on distance violations in the current structure. The overall peak assignment likelihood λ _o is a weighted average of previous and violation likelihoods. The assignment likelihood λ _a is used to determine which single assignment to use for a given peak during Pass 2.

Broad tolerance Δ _B

The size of chemical shift bins used in the initial assignment procedure. [Section “Shift assignment stripe correction”]

Calibration peak

NOE peaks corresponding to intraresidue or backbone sequential connectivities, used for stripe correction and network analysis. [Section “Shift assignment stripe correction”]

Characteristic violation distance Δr _c

Distance used in determining assignment likelihood λ _v . Smaller values reduce the likelihood of assignments with large violations. [Eq. 13]

Linear NOE potential E _lin

Energy term used in Pass 1 which is linear in NOE violation. [Eq. 6]

Network score R(a,b)

The residue pair score between residues a and b, based on connectivities deduced from the initial collection of possible NOE assignments. R′(a,b) is the normalized score used for assigning initial likelihoods; associated assignments are specified as active for R′ > R _c . Larger R′ corresponds to a larger number of connections. [Eqs. 1 and 2]

Peak assignment

A specific NOE peak assignment relating a single peak to a pair of assigned chemical shifts.

Previous likelihood weight w _p

Weight determining the contribution of λ _p and λ _v to λ _o . [Eq. 14]

Quadratic NOE potential E _quad

Energy term used in Pass 2 which is quadratic in NOE violation. [Eq. 10]

Repulsive distance potential E _repul

Energy term used in Pass 1 which repels atoms associated with shift assignments which are inactive. [Eq. 11]

Stripe coverage C

The fraction of calibration peaks consistent with a particular chemical shift assignment. [Section “Shift assignment stripe correction”]

Symmetry partners

Two NOE peaks with from- and to- assignments reversed.

Tight tolerance Δ _T

The size of chemical shift bins used during peak assignment after the stripe correction procedure. [Section “Shift assignment stripe correction”]