nucleus 2 sequential (n+1) intraresidual

H1’ H6/H8 w w

H2’ H6/H8 s w

H3’ H6/H8 m m

H2’ H1’ w s

H6 H5 w s

Orientational Restraints

Since distance restraints derived from NOE data only report on short- to medium-range distances, restraints lack for long stretched RNA sequences as in double helices. Without these long-range restraints the structure calculations of RNA biomacromolecules can hardly converge. Stems may therefore bend during the structural calculation, although they are linear in solution. In addition, regions with intrinsic conformational flexibility may prevent a restriction of the local conformation, due to the lack of distance or dihedral restraints which cannot be determined experimentally.

In this case, residual dipolar couplings (RDC) can improve the structure, since they provide angular information of individual I-S vectors in respect to the main magnetic field across the whole molecule.^[159-162] In solution the dipolar interactions between the dipoles I and S normally average to zero due to the rapid molecular tumbling. By introduction of an aligning medium, e.g. phages, this averaging is hindered and a residual alignment (~0.1%) under steric and/or electrostatic influences is maintained which leads to residual dipolar couplings up to a few tens of Hertz. In RNA, RDCs between carbons or nitrogens and their attached protons are usually measured. Hereby the protonated carbons are more advantageous, due to their higher abundance and their wider angular distribution in respect to the global alignment tensor.

In contrast, N-H vectors only exist in the nucleobase and mostly lie parallel to each other in a plane perpendicular to the main component of an axial symmetric alignment tensor which is often found for RNA hairpin structures.

These conditions decrease the orientational information that can be derived from N-H residual dipolar couplings.

RDCs can be determined in a HSQC where the decoupling between the two spins is switched off. During the chemical shift evolution period (direct or indirect dimension) scalar and dipolar couplings then evolve leading to a signal splitting. The difference between the splittings in a solution containing an alignment medium and an isotropic solution represents the RDC. The RDC can be converted into an angular correlation of the measured vector to the molecular alignment tensor, orienting the vectors in respect to each other and irrespective of their distances:

( )

{

}

π γ γ π

μ 3 1 2

4 2

RDC_IS ⁰ _{2 3} S ¹₂ cos² ₂¹ sin² cos r

h zz IS S

I − +

⎟⎠

⎜ ⎞

⎝

−⎛

= (Equation 6),

where Szz (main component) and η (asymmetry) are parameters of the alignment tensor and θ and φ are the polar angles relating the vector to the molecule-fixed principal axis system (PAS) of the tensor. The PAS can be determined primarily with a starting structure and under the assumption of mainly steric interactions between aligning medium and the molecule, e.g.

with the program PALES.^[163] For RNA, negatively charged filamentous bacteriophages (Pf1) are often used, which align mainly by steric influences but also electrostatic repulsions between the negative phages and the negative phosphodiester backbone play a role.^[164-166] Due to the aromatic groups stacked in helical stems, RNA biomacromolecules possess a high diamagnetic susceptibility, which induces a field alignment of the molecule.^{[159, 160]} The degree of alignment scales with the squared value of the magnetic field, so that measurements of the splittings at two distinct fields enables to predict the splitting at a field strength of zero, which corresponds to the scalar coupling. The two magnetic field strengths should be reasonably separated to assure a precise analysis. However, the reduced resolution at lower fields needs to be accounted for. And since the overall alignment is low, the resulting RDCs are only about one tenth of RDCs determined with an alignment medium. The scalar coupling of RNA imino

groups, ¹JNH, can additionally be calculated with the empirically determined equation 7, since the value is proportional to the imino chemical shift δH.^[167]

Hz

To determine a structure of a single molecule or a complex a minimization of a target function is required, which calculates the agreement between a conformation of the investigated system and the set of con- and restraints defined for this system. This minimization is performed by either of two major techniques: (i) metric matrix distance geometry (DG) and (ii) Cartesian or torsion angle restrained molecular dynamics (rMD). The metric matrix distance geometry technique is e.g. implemented in the programs DIG-II^[168]

and DIANA^[169] and calculates the structures from a matrix of atomic distances. The second technique, rMD, is realized in the programs like AMBER^[170], CHARMM^[171] and CNX/CNS^[172]. In the case of rMD, structures are calculated from selected force field constraints and the experimental restraints with the aim to minimize the potential function, Vtot:

NMR ff

tot V V

V = + (Equation 8)

with the potential Vff including the force field parameters:

tics

and the potential VNMR containing the experimental restraints:

RDC

The terms ωi herein describe the force constants that weight the corresponding potential. These constants are predefined in the chosen force field (e.g. AMBER^[173], GROMACS^{[174, 175]}, OPLS^{[176, 177]}) which comprehends parameters for bond lengths, bond angles, torsion angles, impropers, van der Waals- and electrostatic interactions. The weighting factors for the

experimental potentials are defined empirically, as described above for the NOE distance restraints.

Different forms of the potential functions, Vi, can be selected. Mostly based on a biharmonic function, flat-bottom (or flat-well) or soft potentials are often used (Figure 19). The chosen boundaries of the respective parameter define the extent of the flat-bottom, while the upper limit may be restrained gently with a soft flat-bottom potential.

Figure 19: Graphical representation of possible potentials, Vi, used for con- and restraint parameters in structure calculations. The black line represents a biharmonic function, the red line a flat-bottom potential dependent on a lower and upper bound and the blue dashed line for soft flat-bottom potential.

In the following the programs ARIA and HADDOCK are shortly introduced which are recent developments based on the restrained molecular dynamics technique. The presented setup parameters are extracted from the TAR RNA/pyrimidinyl-ligand investigation.^[5]

ARIA

The program ARIA 1.2^[152] is based on the program CNS 1.1 (Crystallography and NMR Systems)^[172]. Its advantage is the use of ambiguous distance

preliminary structure the NOESY cross peaks are calibrated and ambiguous assignments are weighted to the possible contributing distances. With these distances and the other non-distance restraints, CNS calculates a number of structures following a simulated annealing (SA) protocol with torsion angle dynamics (TAD). Starting at a high initial temperature (e.g. 10000 K) SA is performed for a defined number of steps. The simulation temperature is then reduced stepwise in three cooling stages (e.g. to 2000 K and 1000 K) to almost 0 K. In the distinct stages and iterations the contributions of the different restraints are weighted individually. After each iteration the lowest energy structures are analyzed to optimize the calibration and weighting of the ambiguous cross peaks for the following structure calibration.

HADDOCK

If a structure of a biomacromolecule is known and the interactions with ligands do not change the structure globally, the program HADDOCK 2.1 (High Ambiguity Driven Docking)^{[178, 179]} can be used to model this complex.

It is a further development of ARIA and predicts possible interactions between the macromolecule and the ligand(s) based on the structures and ambiguous interaction restraints (AIRs). These AIRs are derived from biochemical and/or biophysical interaction data. Biochemical data can, for example, be extracted from mutagenesis experiments while NMR spectroscopy provides biophysical interaction data derived from chemical shift perturbations and/or from intermolecular NOEs. This interaction data is then introduced as ambiguous distance between defined sites on the interacting molecules to drive the docking process. For NOEs a manually classification as described above for the distance restraints is suitable. At first, arbitrary starting structures of the complex are generated from reliable structures of the macromolecule and the ligand(s) determined previously. A rigid docking is then performed at a high temperature (e.g. 2000 K) in iteration 1 followed by a first cooling stage to a temperature around the experimental setup (e.g.

300 K). Starting from this temperature a semi-flexible docking is allowed

during a cooling step to a lower temperature (e.g. 50 K) where the flexible regions on each molecule are user defined. In the final cooling stage down to a temperature of 0 K, more flexible residues are selected to allow local structural adjustments of the macromolecule and the ligand(s). The restraints introduced for the ARIA calculations before can also be implemented to maintain the overall structural integrity. This is especially important in the case of RNA complexes, since HADDOCK strongly uses electrostatic interactions for the complex assembly and repulsions along the phosphodiester backbone may lead to distortions within the RNA structure.