Simultaneous single-structure and bundle representation of protein NMR structures in torsion angle space

Abstract

A method is introduced to represent an ensemble of conformers of a protein by a single structure in torsion angle space that lies closest to the averaged Cartesian coordinates while maintaining perfect covalent geometry and on average equal steric quality and an equally good fit to the experimental (e.g. NMR) data as the individual conformers of the ensemble. The single representative ‘regmean structure’ is obtained by simulated annealing in torsion angle space with the program CYANA using as input data the experimental restraints, restraints for the atom positions relative to the average Cartesian coordinates, and restraints for the torsion angles relative to the corresponding principal cluster average values of the ensemble. The method was applied to 11 proteins for which NMR structure ensembles are available, and compared to alternative, commonly used simple approaches for selecting a single representative structure, e.g. the structure from the ensemble that best fulfills the experimental and steric restraints, or the structure from the ensemble that has the lowest RMSD value to the average Cartesian coordinates. In all cases our method found a structure in torsion angle space that is significantly closer to the mean coordinates than the alternatives while maintaining the same quality as individual conformers. The method is thus suitable to generate representative single structure representations of protein structure ensembles in torsion angle space. Since in the case of NMR structure calculations with CYANA the single structure is calculated in the same way as the individual conformers except that weak positional and torsion angle restraints are added, we propose to represent new NMR structures by a ‘regmean bundle’ consisting of the single representative structure as the first conformer and all but one original individual conformers (the original conformer with the highest target function value is discarded in order to keep the number of conformers in the bundle constant). In this way, analyses that require a single structure can be carried out in the most meaningful way using the first model, while at the same time the additional information contained in the ensemble remains available.

Appendix

Bimodal averages and standard deviation of torsion angles

If the values $S = \{ \phi_{1} , \ldots ,\phi_{n} \}$ of a torsion angle ϕ are clustered in two separate regions it makes little sense to determine an average value. Instead it is meaningful to split the set S into two disjoint subsets S ₁ and S ₂ for the purpose of computing two bimodal average values ${{\overline{\phi}}_{1}} = \arg \sum\nolimits_{{k \in s_{1} }} {e^{{i\phi_{k} }} }$ and ${{\overline{\phi}}_{1}} = \arg \sum\nolimits_{{k \in s_{2} }} {e^{{i\phi_{k} }} }$ of the torsion angle values in S ₁ and S ₂, respectively. The choice of S ₁ and S ₂ is optimal if it minimizes the bimodal standard deviation

$$ \sigma_{\phi }^{(2)} = \sqrt {\frac{1}{n}\sum\limits_{k =1}^{n} {\min \left( {\left| {\phi_{k} - {{\overline{\phi}}_{1}}} \right|,2\pi - \left| {\phi_{k} - {{\overline{\phi}}_{1}} }\right|,\left| {\phi_{k} - {{\overline{\phi}}_{2}} }\right|,2\pi - \left| {\phi_{k} - {{\overline{\phi}}_{2}} }\right|} \right)^2} } $$

that results from summing for each torsion angle value ϕ _k the squared deviation from the closer of the two bimodal average values ${{\overline{\phi}}_{1}}$ and ${{\overline{\phi}}_{2}}$, taking into account the periodicity.

It would be computationally inefficient to evaluate $\sigma_{\phi }^{(2)}$ for each of the 2ⁿ possible choices of the subsets S ₁ and S ₂. To determine a good approximation of the optimal bimodal average values in polynomial time, we first calculate the n × n matrix of torsion angle differences $\Updelta \phi_{ij} = \min \left( {\left| {\phi_{i} - \phi_{j} } \right|,2\pi - \left| {\phi_{i} - \phi_{j} } \right|} \right)$. For all pairs (i, j) with $\Updelta \phi_{ij} > \pi /4$ (to avoid splitting into two hardly separated clusters), we compute $\widetilde{{\phi_{1} }} = \arg \sum\nolimits_{{k:\Updelta \phi_{ki} \le \Updelta \phi_{kj} }} {e^{{i\phi_{k} }} }$ and $\widetilde{{\phi_{2} }} = \arg \sum\nolimits_{{k:\Updelta \phi_{ki} > \Updelta \phi_{kj} }} {e^{{i\phi_{k} }} }$. (In the exponential functions i denotes the imaginary unit $\sqrt {-1,}$ otherwise the index i.) The deviations of the individual torsion angle values $\phi_{k}$ from $\widetilde{{\phi_{1} }}$ and $\widetilde{{\phi_{2} }}$ are given by $\delta_{1k} = \min \left( {\left| {\phi_{k} - \widetilde{{\phi_{1} }}} \right|,2\pi - \left| {\phi_{k} - \widetilde{{\phi_{1} }}} \right|} \right)$ and $\delta_{2k} = \min \left( {\left| {\phi_{k} - \widetilde{{\phi_{2} }}} \right|,2\pi - \left| {\phi_{k} - \widetilde{{\phi_{2} }}} \right|} \right)$ for k = 1, …, n. The corresponding subsets are $S_{1} = \{ k\left| {\delta_{1k} \le \delta_{2k} } \right.\}$ and $S_{2} = \{ k\left| {\delta_{1k} > \delta_{2k} } \right.\}$. We choose the optimal subsets S ₁ and S ₂ from the pair (i, j) that yields the largest value of $\left| {\sum\nolimits_{{k \in S_{1} }} {e^{{i\phi_{k} }} } } \right| + \left| {\sum\nolimits_{{k \in S_{2} }} {e^{{i\phi_{k} }} } } \right|$ to obtain the bimodal average values ${{\overline{\phi}}_{1}}$ and ${{\overline{\phi}}_{2}}$. If S ₂ contains more elements than S ₁, we exchange the values of ${{\overline{\phi}}_{1}}$ and ${{\overline{\phi}}_{2}}$ such that ${{\overline{\phi}}_{1}}$ always corresponds to the cluster with the larger number of elements.