Calculation Of 3d Structure Using Nmr Restraints

The ultimate goal of most biological NMR studies is to obtain an accurate 3D structure (conformation) of the molecule. This cannot be done by human judgement and analysis of individual pieces of evidence; there is far too much data and we need an unbiased method for finding the best 3D structure that is most consistent with the NMR data. Structure calculations have been done since computers became available on a variety of organic compounds and biological molecules. The various forces exerted by covalent bonds (bond lengths, bond angles, planarity of double bonds, Van der Waals attraction, hard sphere repulsion, etc.) are summarized in a force field (e.g., Amber or cvff). The goal is to search the "conformational space" (the total of all possible conformations) to find the minimum of energy as defined by the force field. This is a big challenge for a linear polymer like a protein because the number of possible conformations is astronomical, as defined by the various dihedral angles ($, X1, X2, etc.) for each residue resulting from free rotation around single bonds. Even the most sophisticated structure calculations cannot define a protein's conformation without restraints defined by experimental observations. The "folding problem" in proteins is far from solved: no one can predict the 3D shape of a protein simply by knowing its amino acid sequence.

12.9.1 NMR Experimental Restraints

The NMR data enter into the structure calculation in the form of "restraints": limitations placed on H-H distances and H-N-C-H« dihedral angles based on the observation of NOE crosspeaks or J couplings. The specific form of the restraints is a "penalty function" that adds to the total energy of a conformation if the distance or dihedral angle is outside the limits defined by the NMR data. For example, a strong NOE between two protons indicates that the distance should be less than 2.8 A, so any time that distance is exceeded we "penalize" the structure by adding to its total energy. Exceeding the restraint distance is called a "violation," and the larger the violation the more energy is added. The energy gradient behaves like a force, pulling the two protons together. It's like tying a rubber band between the two protons: if they are within the restraint distance (in this case 2.8 A), the rubber band is slack and there is no force. If the restraint is violated (distance greater than 2.8 A) the rubber band is taut and exerts an attractive force between the two protons. As the violation increases the force also increases. A similar torsional force is introduced for dihedral angles by adding an energy penalty any time the angle gets outside the range of angles indicated by the measured J coupling. In this way the process of energy minimization is simultaneously maintaining the bond distances and angles defined by covalent geometry and trying to satisfy all of the NOE distances and dihedral angles defined by the NMR data.

NOE distance restraints are determined from the intensities of the NOESY crosspeaks. There is a theoretical relationship between the initial rate of NOE buildup (as mixing time is increased) and the inverse sixth power of the distance between two protons. In practice, it is very difficult to measure accurate distances in protein NMR, so NOESY crosspeaks are sorted into "bins" representing, for example, very strong (<2.9 A), strong (<3.3 A), medium (<4.0 A), weak (<5.0 A), and very weak (<6.5 A). The dividing points for these intensity categories are determined by measuring NOE intensities corresponding to well-known distances such as cross-strand Ha-Ha in a j-sheet or Ha(i) to HN(i + 3) in an a-helix, a process known as calibration of the NOE intensities. Once a few accurate distances are associated with specific NOE crosspeak volumes, we can use the 1/r6 rule to calculate the volume cutoffs for sorting NOEs into the distance bins. Notice that these categories are merely upper limits of distance between two protons; lower limits are not used. In the case of overlapped NOE crosspeaks it is best to reduce the restraint to the next bin (e.g., from strong to medium) or to the least restrictive bin (weak) for severe overlap. The temptation to define the distance restraints very tightly must be resisted; the molecule will tie up itself like a tangled ball of yarn if we force the NOE restraints too hard. Keep in mind that it is not the precision of any one NOE restraint that gives us an accurate structure, but rather the combination of a very large number (often in the thousands) of relatively imprecise and loosely enforced measurements that, taken together, can lead to a very well defined 3D structure. Figure 12.34 shows a typical penalty function for NOE distance restraints. The penalty is a quadratic function defined by the square of the violation distance:

E = k(r — ro)2 if r>ro; E = 0 if r < ro where r is the distance between the two protons and ro is the maximum distance set by the NOE restraint. The force constant k is the same for all distance restraints and determines the tightness of the "spring" or "rubber band" connecting the two protons. It is important to realize that the NOE is not an attractive force! It is an interaction observed by NMR that allows us to apply an artificial force in the structure calculation.

The nondegenerate geminal pairs are usually named according to their chemical shifts (e.g., j downfield of j) rather than their stereochemical relationships (pro-R and pro-S). In structure calculations, this usually is dealt with by creating a "pseudo-atom" right between the pro-R and pro-S positions in 3D space. The NOE restraints are applied to the pseudoatom and not to the real atoms, and the distance limit is increased a bit to account for the ambiguity (we do not really know which restraint applies to which of the two positions in space). Similarly, a pseudoatom is created at the center of the three equivalent protons of a CH3 group, and the distance restraint is applied to the pseudoatom.

NOE distance restraints

0123456789 Interproton distance r (A)

Figure 12.34

Similar restraints are generated for dihedral angles based on measured J coupling constants using the Karplus relation. The HN-N-Ca-Ha dihedral angle is determined by measuring the HN-Ha J value from the DQF-COSY crosspeak fine structure of each residue. The HN-Ha dihedral angle is related to the O angle of protein conformational analysis: for an ideal a-helix the J coupling should be small (3.9 Hz) and for an extended strand of a ^-sheet the J coupling should be large (8.9 Hz). Typically, if the coupling constant 3 JHN-Ha is greater than 8 Hz, we assume an extended conformation and the O dihedral angle is restricted to the range -150° to —90o (Fig. 12.35, left If the coupling constant is less than 6 Hz we assume a helical conformation and the O angle is restricted to the range —90° to —40° (Fig. 12.35, right If the coupling constant is between 6 and 8 Hz the conformation is probably changing rapidly between these two extremes so that the J value is averaged over all conformations to an intermediate value. In this case no restrictions are placed on the O angle. Figure 12.35 shows the energy penalty function for a "helical" dihedral restraint (□) and for an "extended" or ^-sheet dihedral restraint (■).

The mathematics of computational chemistry is very complicated and we will only attempt to describe the steps in the process in general terms. The first step is to generate a group of structures that satisfy the NMR restraints while providing the largest possible diversity of starting conformations. This process is called "embedding" and uses a technique called distance geometry. We might generate 50 different structures to ensure a wide variety of starting points. At this point, we have some very unhappy molecules with bonds stretched and twisted in bizarre ways. The next step is energy minimization, where we allow each structure to "relax" under the influence of the total energy function (covalent force field plus NMR restraint energy) and move down the energy hill (gradient) toward a minimum of energy. There are many pitfalls in this journey because the molecule might find a "local minimum" of energy, like a dry lake basin in the desert, such that all directions leading

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