Practical Manual
 

Summary

Introduction
 
 

Summary.

One of the most obvious applications of the previously inaccessible long-range angular information afforded by RDC is the accurate definition of domain orientation in multi-module macromolecules or complexes.

MODULE is a novel program developed to allow the determination of alignment tensor parameters for individual or multiple domains in macromolecules from residual dipolar couplings and to facilitate their manipulation to construct low-resolution models of macromolecular structure.

For multi-domain systems the program determines the relative orientation of individual structured domains, and provides graphical user-driven rigid-body modeling of the different modules relative to the common tensorial frame.

Translational freedom in the common frame, and equivalent rotations about the diagonalized (x,y,z) axes, are used to position the different modules in the common frame to find a model in best agreement with experimentally measured couplings alone or in combination with additional experimental or covalent information.

Introduction.

While NMR spectroscopy is now successfully established as the most important technique for the high resolution structure determination of small to medium sized, compact macromolecules in the solution state (Wüthrich 1986, Clore & Gronenborn 1998), the method has some notable limitations for more complex molecular systems.

The basic experimental parameter used for the determination of molecular structure (nuclear Overhauser effect-nOe) becomes difficult to measure in large protonated molecules due to prohibitive relaxation effects, making the determination of structure beyond 30kD unrealistic using classical techniques.

Moreover, modular or elongated proteins, and large RNA superstructures, encounter the serious problem of ill-defined relative orientation of different domains, due to inadequate local structural information at interfacial or hinge regions.
 
 

The relative orientation of different domains is however known to be closely correlated to physiological function while the characterization of the exact nature of molecular interaction in reaction complexes clearly holds the key to understanding macromolecular function.

The last five years has seen a rapid acceleration in the search for viable, alternative sources of structural information for the resolution of long-range orientation in systems of more complex geometry (Tjandra 1999).

In particular weak alignment of proteins prevents complete averaging of the dipolar interaction, while retaining the solution properties necessary for high resolution NMR. This alignment can exist naturally, due to the paramagnetic properties of the molecule (Tolman et al 1995), or can, more generally, be induced by solvation in liquid crystal media (Tjandra & Bax 1997).

The residual dipolar coupling (RDC) measured under these conditions provides geometric information relative to the common alignment frame of the form

A_a and A_r are the axial 1/3(A_zz-(A_xx+A_yy)/2) and rhombic 1/3(A_xx-A_yy) components of the alignment tensor, and {q,j} the vector orientation relative to this tensor, r_ij is the internuclear distance and S the local order parameter.

In the presence of a known alignment tensor, the geometric information available from a single measured coupling for an individual internuclear vector is highly degenerate. As shown in the example below, even for a significantly rhombic tensor, a continuum of orientations exist which can give rise to the same measured residual dipolar coupling (assuming this coupling has neither the maximum or minimum values).

This degeneracy can be raised if we can measure more couplings in a domain of known structure, and whose relative orientation in the domain is known. In the right hand side of the figure shown above, we have sketched the equivalent orientations for an imaginary sub-structure conissiting of differently oriented vectors. There are now four equivalent orientations of the differently valued couplings (colour coded in the figure) which are in agreement with measure values.

This four-fold degeneracy is inherent to the orientation of any three-dimensional structure relative to a molecular alignment tensor, and derives from simple symmetry operations (180° rotations around A_xx, A_yy and A_zz). Despite this inherent four-fold orientational degeneracy, the ability to determine domain orientation is a very powerful complement to classical structure determination and forms the basis of many recent studies of the molecular architecture of multidomain systems (Cai et al 1998, Skrynnikov et al 1999, Mollova et al 2000), and protein-ligand complexes (Weaver and Prestegard 1998, Olejniczak et al 1999 et al).

These data are complementary to many readily available sources of structural constraint currently used to build models of molecular assemblies, whether these are experimental, such as intermolecular nOe measured between interacting surfaces (Clore 2000) or predicted from existing structural information, for example electrostatic or hydrophobic surface calculations.
 
 
 
 

Interpretation of residual dipolar couplings for the determination of domain orientation requires tools specifically developed for the manipulation of sub-structures within a reference calculation frame.

As currently available molecular modeling packages are not yet adapted to handling this kind of specific analysis, we have developed Module to facilitate alignment tensor degterination and incorporated graphical display of the tensor relative to the three-dimensional atomic coordinates, as well as correlation plots of the measured and calculated couplings for the selected datasets.

Module also provides graphical user-driven, rigid-body modeling of the individual modules of multi-domain assemblies by simple cursor-driven manipulation, for the determination of relative position of structural motifs with respect to the common alignment tensor A.

How does MODULE work ?.

The program allows the user to define regions to be taken into consideration as separate structural entities using the grahical display of the primary sequence.

The alignment tensor will be calculated for each unit, and the unit considered structurally intact throughout the procedure. This region is not necessarily contiguous in primary sequence, for example in domain-swapped assemblies, nucleic acid structures (where paired strands may be taken as structurally inseparable) or multi-partner molecular complexes. This choice is performed using a simple cursor selection in the graphical interface.

Tensor eigenvalues and eigenvectors are then extracted using least-squares minimization of the target function over all couplings associated with a given domain:

(2)

where s_ijis the uncertainty in the experimentally measured coupling. The minimization algorithm searches the {A_a,A_r,a,b,g} parametric space by random variation of these parameters, using a combination of simulated annealing (Metropolis et al. 1953), temperature regulation using fuzzy logic (Leondes 1997), and Levenberg-Marquardt minimization (Press et al. 1988) which we have previously developed for the determination of the rotational diffusion tensor from heteronuclear relaxation measurements (Dosset et al 2000). The couplings are calculated with the appropriate pre-factors in equation (1), including the gyromagnetic ratio and the inter-nuclear distance, which can be chosen to be a either a standard fixed distance from an interactive table, or the actual distance (Å) present in the coordinate file.

The traceless molecular alignment tensor has an inherent degeneracy if A_a and A_r are allowed to take any values — to avoid confusion Module applies relevant transformations to place the minimum within the reference frame (|A_xx| < |A_yy|< |A_zz|, -p<a,g<+p, 0 < b < p). The three axes of these tensors are then superimposed graphically on the structural motifs and correlation plots presented for each different coupling type, as well as c² value for the fit of the RDC data for each module.

If we then assume that the different domains present in the molecule or complex experience negligible mobility relative to the each other, they will experience the same interaction with the liquid crystal, and consequently the same aligning forces, and will therefore be governed by the same alignment tensor A. If the eigenvalues of the tensors determined for the separate domains are significantly different, the amplitude of the relative domain motion can no doubt be estimated, although an appropriate analysis is beyond the scope of this paper (Fischer et al 1999).

Assuming similar eigenvalues, the relative orientation of the different sub-structures can be determined by aligning the domains such that all tensors are collinear (it should be remembered that we cannot exclude interdomain motion even if the eigenvalues are similar, and that in all cases inter-domain orientation representing the averaged couplings will be determined).

The program Module simply reorients each domain, and associated tensors, into a common graphical display frame (this can be considered to be the frame in which all tensors are diagonal). There is an inherent degeneracy of relative orientation present, due to the equivalence of any combination of vectors with respect to 180° rotations about any of the alignment tensor axes (A_xx, A_yy and A_zz) (Al-Hashimi et al 2000).

These equivalent orientations can be viewed by the user, who can then position the different modules using the graphical interface (cursor-controlled) with respect to each other using only these equivalent orientations and three-dimensional translational freedom with respect to the diagonalized frame.

The entire coordinate space available with these degrees of freedom is equivalent with respect to the sum of the target functions (equation 2) for the different modules. In the case of an axially symmetric alignment tensor (i.e. negligible rhombicity), it is possible to select a specific mode allowing rotation of the molecule about the unique axis A^_zz, as all of these positions are equivalent in this case.

In the case of a covalently bonded multimer, the program highlights the bonded partners at the junction between the selected modules and indicates the distance between the bonded atoms, so that the user can gauge the most likely relative positioning of the different domains. Automatic domain positioning is performed by the program to provide an initial model, by minimizing the function

with respect to the relative positions of the different oriented modules. d_^ij are the distances between the covalently bound atoms at each module junction. The positions can also be manually adapted to find a more intelligent solution. Once the preferred orientation has been found, the model can be fixed, and the coordinates written to a standard format coordinate file, or transferred to a standard molecular dynamics package for further refinement under RDC constraint forces (Clore et al 1998, Tsui et al 2000, Hus et al 2000).

Acknowledgements.

This work was supported by the Commisariat à l’Energie Atomique and the Centre National de la Recherche Scientifique.

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